Subgroups — #31732072

Order 17152.

2620 conjugacy classes of subgroups.

The subgroup lattice has 746 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22574983	C₂	1	2	yes	maximal, complemented
2	#22574978	C₂	1	2	yes	maximal, complemented
3	#22572014	C₂	1	2	yes	maximal, complemented
4	#22572009	C₂	1	2	yes	maximal, complemented
5	#22572003	C₂	1	4	yes	maximal, complemented
6	#22571997	C₂	1	4	yes	maximal, complemented
7	#22571996	C₂	1	4	yes	maximal, complemented
8	#22570637	C₂	1	1	yes	maximal, characteristic, complemented
9	#22569930	C₂	1	2	yes	maximal, complemented
10	#22569929	C₂	1	2	yes	maximal, complemented
11	#22569560	C₂	1	4	yes	maximal, complemented
12	C₂×#16863514	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#443×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
14	#16864232	C₂²	1	4	yes	complemented
15	#16864213	—	2	2	no	—
16	#16864213	C₂²	1	4	yes	complemented
17	#16864210	C₂²	1	4	yes	complemented
18	#16864209	—	2	2	no	—
19	#16864208	—	2	4	no	—
20	#16864208	C₂²	1	8	yes	complemented
21	#16864203	C₂²	1	8	yes	complemented
22	#16864202	—	2	4	no	—
23	#16864201	—	2	2	no	—
24	#16864201	C₂²	1	4	yes	complemented
25	#16864198	C₂²	1	4	yes	complemented
26	#16864197	—	2	2	no	—
27	#16864196	—	2	4	no	—
28	#16864196	C₂²	1	4	yes	complemented
29	#16864194	C₂²	1	4	yes	complemented
30	#16864193	—	2	4	no	—
31	C₂×#14524065	—	2	4	no	—
32	C₂×#14524065	C₂²	1	1	yes	characteristic
33	C₂×#14524065	C₂²	1	2	yes	complemented
34	C₂×#14524065	C₂²	1	2	yes	complemented
35	#135×D₆₇	—	2	4	no	—
36	C₂×#14524049	C₂²	1	2	yes	complemented
37	#16863946	C₂²	1	2	yes	complemented
38	#16863780	C₂²	1	2	yes	complemented
39	#16863780	C₂²	1	2	yes	complemented
40	#16863776	C₂²	1	1	yes	characteristic, complemented
41	#16863773	C₂²	1	1	yes	characteristic, complemented
42	#16863769	C₂²	1	2	yes	complemented
43	#16863672	C₂²	1	4	yes	complemented
44	#16863671	C₂²	1	4	yes	complemented
45	#16863667	C₂²	1	4	yes	complemented
46	#16863666	C₂²	1	4	yes	complemented
47	#16863660	C₂²	1	8	yes	complemented
48	#16863659	C₂²	1	8	yes	complemented
49	#16863580	C₂²	1	8	yes	complemented
50	#16863575	C₂²	1	8	yes	complemented
51	#16863570	C₂²	1	8	yes	complemented
52	#16863565	C₂²	1	8	yes	complemented
53	#16863514	C₂²	1	2	yes	complemented
54	#16863514	C₂²	1	2	yes	complemented
55	#16863514	C₂²	1	2	yes	complemented
56	#16863514	C₂²	1	2	yes	complemented
57	C₂×#14523979	C₂²	1	1	yes	characteristic, complemented
58	C₂×#14523971	C₂²	1	2	yes	complemented
59	#16863378	C₂²	1	4	yes	complemented
60	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
61	C₂×#131×C₆₇	C₂²	1	2	yes	complemented
62	#443×C₆₇	C₂²	1	2	yes	complemented
63	#443×C₆₇	C₂²	1	2	yes	complemented
64	#443×C₆₇	C₂²	1	2	yes	complemented
65	#443×C₆₇	C₂²	1	2	yes	complemented
66	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic, complemented
67	C₂×#104×C₆₇	C₂²	1	2	yes	complemented
68	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
69	#14524091	—	2	4	no	—
70	#14524090	—	2	4	no	—
71	#14524090	C₂³	1	8	yes	—
72	#14524089	—	2	4	no	—
73	#14524089	C₂³	1	8	yes	—
74	#14524088	—	2	4	no	—
75	#14524088	—	4	4	no	—
76	C₂×#11761628	—	2	8	no	—
77	C₂×#11761628	C₂³	1	2	yes	—
78	C₂×#11761628	C₂³	1	4	yes	—
79	#41×D₆₇	—	2	8	no	—
80	#41×D₆₇	—	4	4	no	—
81	#14524076	—	2	4	no	—
82	#14524076	—	2	8	no	—
83	#14524076	—	2	8	no	—
84	#14524076	C₂³	1	8	yes	—
85	#14524075	—	2	8	no	—
86	#14524074	—	2	4	no	—
87	#14524073	C₂³	1	8	yes	—
88	#14524072	—	2	8	no	—
89	#14524071	—	2	4	no	—
90	#14524071	—	2	8	no	—
91	#14524071	—	2	8	no	—
92	#14524071	C₂³	1	8	yes	—
93	#14524070	—	2	8	no	—
94	#14524069	—	2	4	no	—
95	#14524068	C₂³	1	8	yes	—
96	#14524067	—	2	8	no	—
97	#14524066	—	2	4	no	—
98	#14524065	—	2	8	no	—
99	#14524065	—	2	4	no	—
100	#14524065	—	2	8	no	—
101	#14524065	—	2	4	no	—
102	#14524065	—	2	2	no	—
103	#14524065	—	2	2	no	—
104	#14524065	—	2	2	no	—
105	#14524065	—	2	2	no	—
106	#14524065	—	2	2	no	—
107	#14524065	—	2	2	no	—
108	#14524065	—	2	2	no	—
109	#14524065	—	2	2	no	—
110	#14524065	D₄	1	2	yes	complemented
111	#14524065	D₄	1	2	yes	complemented
112	#14524065	C₂³	1	2	yes	—
113	#14524065	C₂³	1	2	yes	—
114	#14524064	—	2	8	no	—
115	#14524063	—	2	8	no	—
116	#14524062	—	2	4	no	—
117	C₂×Dic₂₆₈	—	2	4	no	—
118	C₂×Dic₂₆₈	D₄	1	1	yes	characteristic, complemented
119	C₂×Dic₂₆₈	C₂³	1	2	yes	—
120	C₂×Dic₂₆₈	C₂³	1	2	yes	—
121	Q₁₆×D₆₇	—	2	4	no	—
122	C₂×#11761617	—	2	8	no	—
123	C₂×#11761617	D₄	1	2	yes	complemented
124	C₂×#11761617	C₂³	1	4	yes	—
125	C₂×#11761617	C₂³	1	4	yes	—
126	SD₁₆×D₆₇	—	2	8	no	—
127	C₂×D₅₃₆	—	2	4	no	—
128	C₂×D₅₃₆	D₄	1	1	yes	characteristic, complemented
129	C₂×D₅₃₆	C₂³	1	2	yes	—
130	C₂×D₅₃₆	C₂³	1	2	yes	—
131	D₈×D₆₇	—	2	4	no	—
132	#14524049	—	2	2	no	—
133	#14524049	—	2	2	no	—
134	#14524049	C₂³	1	4	yes	—
135	#14524049	C₂³	1	4	yes	—
136	C₂×#11761612	C₂³	1	2	yes	—
137	C₂×#11761611	—	2	2	no	—
138	C₂×C₈×D₆₇	—	2	2	no	—
139	#14524043	C₂³	1	2	yes	—
140	#14524040	C₂³	1	2	yes	—
141	#14524031	C₂³	1	4	yes	—
142	#14524017	—	4	2	no	—
143	C₄×D₂₆₈	—	4	2	no	—
144	#14523979	C₂³	1	2	yes	—
145	#14523979	C₂³	1	2	yes	—
146	#14523971	C₂³	1	4	yes	—
147	#14523971	C₂³	1	4	yes	—
148	#14523961	C₂³	1	8	yes	—
149	#14523959	C₂³	1	8	yes	—
150	C₂×#41×C₆₇	C₂³	1	2	yes	—
151	#135×C₆₇	D₄	1	2	yes	complemented
152	#135×C₆₇	D₄	1	2	yes	complemented
153	#135×C₆₇	C₂³	1	2	yes	—
154	#135×C₆₇	C₂³	1	2	yes	—
155	C₂×Q₁₆×C₆₇	D₄	1	1	yes	characteristic, complemented
156	C₂×SD₁₆×C₆₇	D₄	1	2	yes	complemented
157	C₂×D₈×C₆₇	D₄	1	1	yes	characteristic, complemented
158	#131×C₆₇	—	2	2	no	—
159	#131×C₆₇	—	2	2	no	—
160	#131×C₆₇	C₂³	1	4	yes	—
161	#131×C₆₇	C₂³	1	4	yes	—
162	C₂×M₁₆×C₆₇	—	2	2	no	—
163	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
164	C₂²×C₈×C₆₇	—	2	2	no	—
165	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
166	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
167	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
168	#108×C₆₇	C₂³	1	2	yes	—
169	#108×C₆₇	C₂³	1	2	yes	—
170	#104×C₆₇	C₂³	1	4	yes	—
171	#104×C₆₇	C₂³	1	4	yes	—
172	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
173	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
174	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
175	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
176	#11761629	—	2	8	no	—
177	#11761629	—	4	4	no	—
178	#11761628	—	2	2	no	—
179	#11761628	—	2	2	no	—
180	#11761628	—	2	4	no	—
181	#11761628	—	2	4	no	—
182	#11761628	—	2	4	no	—
183	#11761628	—	2	4	no	—
184	#11761628	—	2	8	no	—
185	#11761628	—	2	8	no	—
186	#11761628	—	4	8	no	—
187	#11761628	—	4	8	no	—
188	#11761628	C₂×D₄	1	4	yes	—
189	#11761628	C₂⁴	1	4	yes	—
190	#11761627	—	2	8	no	—
191	#11761627	—	4	4	no	—
192	#11761627	—	4	8	no	—
193	#11761627	—	4	8	no	—
194	C₂×Dic₁₃₄	—	2	8	no	—
195	C₂×Dic₁₃₄	C₂×D₄	1	2	yes	—
196	C₂×Dic₁₃₄	C₂⁴	1	4	yes	—
197	Q₈×D₆₇	—	2	8	no	—
198	Q₈×D₆₇	—	4	4	no	—
199	C₂×D₂₆₈	—	2	8	no	—
200	C₂×D₂₆₈	—	4	2	no	—
201	C₂×D₂₆₈	C₂×D₄	1	2	yes	—
202	C₂×D₂₆₈	C₂⁴	1	4	yes	—
203	C₂×C₂²⋊D₆₇	—	2	2	no	—
204	C₂×C₂²⋊D₆₇	—	2	4	no	—
205	C₂×C₂²⋊D₆₇	—	4	8	no	—
206	D₄×D₆₇	—	2	8	no	—
207	D₄×D₆₇	—	4	4	no	—
208	D₄×D₆₇	—	4	8	no	—
209	D₄×D₆₇	—	4	8	no	—
210	C₂×C₄×D₆₇	—	2	2	no	—
211	C₂×C₄×D₆₇	—	2	4	no	—
212	C₂×C₄×D₆₇	—	4	8	no	—
213	Dic₂₆₈	—	2	8	no	—
214	Dic₂₆₈	—	2	2	no	—
215	Dic₂₆₈	—	2	2	no	—
216	Dic₂₆₈	—	2	2	no	—
217	Dic₂₆₈	—	2	2	no	—
218	Dic₂₆₈	C₂×D₄	1	2	yes	—
219	Dic₂₆₈	C₂×D₄	1	2	yes	—
220	#11761618	—	2	8	no	—
221	#11761617	—	2	8	no	—
222	#11761617	—	2	8	no	—
223	#11761617	—	2	4	no	—
224	#11761617	—	2	4	no	—
225	#11761617	—	2	4	no	—
226	#11761617	—	2	4	no	—
227	#11761617	C₂×D₄	1	4	yes	—
228	#11761617	C₂×D₄	1	4	yes	—
229	#11761616	—	2	8	no	—
230	#11761615	—	2	8	no	—
231	D₅₃₆	—	2	8	no	—
232	D₅₃₆	—	2	2	no	—
233	D₅₃₆	—	2	2	no	—
234	D₅₃₆	—	2	2	no	—
235	D₅₃₆	—	2	2	no	—
236	D₅₃₆	C₂×D₄	1	2	yes	—
237	D₅₃₆	C₂×D₄	1	2	yes	—
238	#11761613	—	2	8	no	—
239	#11761612	—	2	2	no	—
240	#11761612	C₂⁴	1	4	yes	—
241	#11761611	—	2	4	no	—
242	#11761611	—	2	4	no	—
243	C₂×C₆₇⋊C₈	—	2	2	no	—
244	C₈×D₆₇	—	2	4	no	—
245	C₈×D₆₇	—	2	4	no	—
246	C₄⋊Dic₆₇	—	4	2	no	—
247	(C₂×D₆₇)⋊C₄	—	4	4	no	—
248	#41×C₆₇	—	2	2	no	—
249	#41×C₆₇	—	2	2	no	—
250	#41×C₆₇	C₂×D₄	1	4	yes	—
251	#41×C₆₇	C₂⁴	1	4	yes	—
252	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
253	C₂×D₄×C₆₇	—	2	2	no	—
254	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
255	C₂²×C₄×C₆₇	—	2	2	no	—
256	C₂²×C₄×C₆₇	—	2	1	no	—
257	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
258	Q₁₆×C₆₇	C₂×D₄	1	2	yes	—
259	Q₁₆×C₆₇	C₂×D₄	1	2	yes	—
260	SD₁₆×C₆₇	C₂×D₄	1	4	yes	—
261	SD₁₆×C₆₇	C₂×D₄	1	4	yes	—
262	D₈×C₆₇	C₂×D₄	1	2	yes	—
263	D₈×C₆₇	C₂×D₄	1	2	yes	—
264	M₁₆×C₆₇	—	2	2	no	—
265	M₁₆×C₆₇	—	2	4	no	—
266	M₁₆×C₆₇	—	2	4	no	—
267	M₁₆×C₆₇	C₂⁴	1	4	yes	—
268	C₂×C₈×C₆₇	—	2	2	no	—
269	C₂×C₈×C₆₇	—	2	4	no	—
270	C₂×C₈×C₆₇	—	2	4	no	—
271	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
272	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
273	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
274	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
275	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
276	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
277	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
278	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
279	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
280	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
281	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
282	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
283	C₄²×C₆₇	—	2	1	no	—
284	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
285	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
286	C₂³×C₆₇	—	2	2	no	—
287	C₂³×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
288	C₂²×D₆₇	—	2	2	no	—
289	C₂²×D₆₇	—	2	4	no	—
290	C₂²×D₆₇	—	4	8	no	—
291	Q₈×C₆₇	—	2	2	no	—
292	Q₈×C₆₇	C₂²×D₄	1	4	yes	—
293	D₄×C₆₇	—	2	4	no	—
294	D₄×C₆₇	—	2	4	no	—
295	D₄×C₆₇	—	2	2	no	—
296	D₄×C₆₇	C₂²×D₄	1	4	yes	—
297	C₂×C₄×C₆₇	—	2	2	no	—
298	C₂×C₄×C₆₇	—	2	1	no	—
299	C₂×C₄×C₆₇	—	2	4	no	—
300	C₂×C₄×C₆₇	—	2	4	no	—
301	C₂×C₄×C₆₇	—	2	1	no	—
302	C₂×C₄×C₆₇	—	4	2	no	—
303	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
304	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
305	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
306	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
307	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
308	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
309	C₂²⋊D₆₇	—	2	4	no	—
310	C₂²⋊D₆₇	—	2	4	no	—
311	C₂²⋊D₆₇	—	4	8	no	—
312	C₂²⋊D₆₇	—	4	8	no	—
313	C₂²⋊D₆₇	—	4	4	no	—
314	C₂²⋊D₆₇	—	4	4	no	—
315	C₂×Dic₆₇	—	2	2	no	—
316	C₂×Dic₆₇	—	2	4	no	—
317	C₂×Dic₆₇	—	4	8	no	—
318	D₂₆₈	—	2	2	no	—
319	D₂₆₈	—	2	4	no	—
320	D₂₆₈	—	2	4	no	—
321	D₂₆₈	—	4	8	no	—
322	D₂₆₈	—	8	4	no	—
323	D₂₆₈	C₂²×D₄	1	4	yes	—
324	C₄×D₆₇	—	2	4	no	—
325	C₄×D₆₇	—	2	4	no	—
326	C₄×D₆₇	—	4	8	no	—
327	C₄×D₆₇	—	4	8	no	—
328	C₄×D₆₇	—	4	4	no	—
329	C₄×D₆₇	—	4	4	no	—
330	Dic₁₃₄	—	2	2	no	—
331	Dic₁₃₄	—	2	4	no	—
332	Dic₁₃₄	—	2	4	no	—
333	Dic₁₃₄	—	4	8	no	—
334	Dic₁₃₄	C₂²×D₄	1	4	yes	—
335	C₈×C₆₇	—	2	4	no	—
336	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
337	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
338	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
339	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
340	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
341	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
342	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
343	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
344	C₆₇⋊C₈	—	2	4	no	—
345	C₂²×C₆₇	—	2	4	no	—
346	C₂²×C₆₇	—	4	1	no	—
347	C₂²×C₆₇	—	8	2	no	—
348	C₂²×C₆₇	D₄²	1	1	yes	characteristic
349	C₂²×C₆₇	D₄²	1	1	yes	characteristic
350	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
351	C₂×D₆₇	—	2	4	no	—
352	C₂×D₆₇	—	4	4	no	—
353	C₂×D₆₇	—	8	2	no	—
354	C₂×D₆₇	—	8	4	no	—
355	C₂×D₆₇	—	16	8	no	—
356	C₄×C₆₇	—	2	2	no	—
357	C₄×C₆₇	—	2	4	no	—
358	C₄×C₆₇	—	4	1	no	—
359	C₄×C₆₇	D₄²	1	1	yes	characteristic
360	C₄×C₆₇	D₄²	1	1	yes	characteristic
361	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
362	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
363	Dic₆₇	—	2	4	no	—
364	Dic₆₇	—	4	4	no	—
365	#61105	—	67	1	no	maximal, Sylow
366	C₂×C₆₇	—	2	1	no	—
367	C₂×C₆₇	—	2	1	no	—
368	C₂×C₆₇	—	2	1	no	—
369	C₂×C₆₇	—	4	4	no	—
370	C₂×C₆₇	C₂×D₄²	1	1	yes	characteristic
371	D₆₇	—	4	4	no	—
372	D₆₇	—	8	4	no	—
373	#3584	—	67	2	no	—
374	#3584	—	67	2	no	—
375	#3293	—	67	2	no	—
376	#3292	—	67	2	no	—
377	#3291	—	67	4	no	—
378	#3290	—	67	4	no	—
379	#3290	—	67	4	no	—
380	#3147	—	67	1	no	—
381	#3067	—	67	2	no	—
382	#3067	—	67	2	no	—
383	#3018	—	67	4	no	—
384	C₂×#443	—	67	1	no	—
385	C₂×#443	D₆₇	1	1	yes	characteristic, complemented
386	C₆₇	#61105	1	1	yes	characteristic, complemented, Sylow
387	#585	—	67	4	no	—
388	#578	—	67	4	no	—
389	#578	—	67	4	no	—
390	#578	—	134	2	no	—
391	#578	—	134	2	no	—
392	#577	—	67	8	no	—
393	#577	—	67	8	no	—
394	#577	—	134	4	no	—
395	#577	—	134	4	no	—
396	#576	—	67	4	no	—
397	#576	—	67	4	no	—
398	#576	—	134	2	no	—
399	#576	—	134	2	no	—
400	#575	—	67	4	no	—
401	#575	—	67	4	no	—
402	#575	—	134	4	no	—
403	#575	—	134	4	no	—
404	C₂×#135	—	67	1	no	—
405	C₂×#135	—	67	2	no	—
406	C₂×#135	—	67	2	no	—
407	C₂×#135	—	134	4	no	—
408	C₂×#135	—	134	4	no	—
409	C₂×#135	C₂×D₆₇	1	1	yes	characteristic
410	C₂×#131	—	67	2	no	—
411	C₂×#131	C₂×D₆₇	1	2	yes	complemented
412	#532	—	67	2	no	—
413	#495	—	67	2	no	—
414	#495	—	67	2	no	—
415	#494	—	67	1	no	—
416	#493	—	67	1	no	—
417	#492	—	67	2	no	—
418	#473	—	67	4	no	—
419	#473	—	67	4	no	—
420	#472	—	67	4	no	—
421	#472	—	67	4	no	—
422	#471	—	67	8	no	—
423	#471	—	67	8	no	—
424	#456	—	67	8	no	—
425	#455	—	67	8	no	—
426	#454	—	67	8	no	—
427	#453	—	67	8	no	—
428	#443	—	67	2	no	—
429	#443	—	67	2	no	—
430	#443	—	67	2	no	—
431	#443	—	67	2	no	—
432	#443	C₂×D₆₇	1	2	yes	complemented
433	#443	C₂×D₆₇	1	2	yes	complemented
434	#443	C₂×D₆₇	1	2	yes	complemented
435	#443	C₂×D₆₇	1	2	yes	complemented
436	C₂×#108	—	67	1	no	—
437	C₂×#108	C₂×D₆₇	1	1	yes	characteristic, complemented
438	C₂×#104	—	67	2	no	—
439	C₂×#104	C₂×D₆₇	1	2	yes	complemented
440	#413	—	67	4	no	—
441	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic, complemented
442	2₋¹⁺⁴	—	67	8	no	—
443	2₋¹⁺⁴	—	134	4	no	—
444	2₋¹⁺⁴	—	134	4	no	—
445	H(5,𝔽₂)	—	268	4	no	—
446	H(5,𝔽₂)	—	67	8	no	—
447	H(5,𝔽₂)	—	134	4	no	—
448	H(5,𝔽₂)	—	134	4	no	—
449	C₂×#41	—	268	4	no	—
450	C₂×#41	—	67	2	no	—
451	C₂×#41	—	67	4	no	—
452	C₂×#41	—	134	8	no	—
453	C₂×#41	—	134	8	no	—
454	C₂×#41	C₂²×D₆₇	1	2	yes	—
455	#137	—	67	8	no	—
456	#137	—	67	8	no	—
457	#137	—	134	8	no	—
458	#137	—	134	8	no	—
459	#137	—	134	4	no	—
460	#137	—	134	4	no	—
461	#137	—	134	8	no	—
462	#137	—	134	8	no	—
463	AGL(1,ℤ/8)	—	67	8	no	—
464	AGL(1,ℤ/8)	—	67	8	no	—
465	AGL(1,ℤ/8)	—	134	8	no	—
466	AGL(1,ℤ/8)	—	134	8	no	—
467	AGL(1,ℤ/8)	—	134	4	no	—
468	AGL(1,ℤ/8)	—	134	4	no	—
469	AGL(1,ℤ/8)	—	134	8	no	—
470	AGL(1,ℤ/8)	—	134	8	no	—
471	#135	—	67	2	no	—
472	#135	—	67	2	no	—
473	#135	—	67	2	no	—
474	#135	—	67	2	no	—
475	#135	—	134	4	no	—
476	#135	—	134	8	no	—
477	#135	—	134	4	no	—
478	#135	—	134	8	no	—
479	#135	—	134	8	no	—
480	#135	—	134	4	no	—
481	#135	—	134	8	no	—
482	#135	—	134	4	no	—
483	#135	—	134	2	no	—
484	#135	—	134	2	no	—
485	#135	—	134	2	no	—
486	#135	—	134	2	no	—
487	#135	—	134	2	no	—
488	#135	—	134	2	no	—
489	#135	—	134	2	no	—
490	#135	—	134	2	no	—
491	#135	D₂₆₈	1	2	yes	complemented
492	#135	D₂₆₈	1	2	yes	complemented
493	#135	C₂²×D₆₇	1	2	yes	—
494	#135	C₂²×D₆₇	1	2	yes	—
495	C₂×Q₁₆	—	67	1	no	—
496	C₂×Q₁₆	—	67	2	no	—
497	C₂×Q₁₆	—	67	2	no	—
498	C₂×Q₁₆	—	134	4	no	—
499	C₂×Q₁₆	—	134	4	no	—
500	C₂×Q₁₆	D₂₆₈	1	1	yes	characteristic, complemented
501	C₂×SD₁₆	—	67	2	no	—
502	C₂×SD₁₆	—	67	4	no	—
503	C₂×SD₁₆	—	67	4	no	—
504	C₂×SD₁₆	—	134	8	no	—
505	C₂×SD₁₆	—	134	8	no	—
506	C₂×SD₁₆	D₂₆₈	1	2	yes	complemented
507	C₂×D₈	—	67	1	no	—
508	C₂×D₈	—	67	2	no	—
509	C₂×D₈	—	67	2	no	—
510	C₂×D₈	—	134	4	no	—
511	C₂×D₈	—	134	4	no	—
512	C₂×D₈	D₂₆₈	1	1	yes	characteristic, complemented
513	#131	—	2	2	no	—
514	#131	—	2	2	no	—
515	#131	—	67	4	no	—
516	#131	—	67	4	no	—
517	#131	—	134	2	no	—
518	#131	—	134	2	no	—
519	#131	C₂²×D₆₇	1	4	yes	—
520	#131	C₂²×D₆₇	1	4	yes	—
521	C₂×M₁₆	—	2	2	no	—
522	C₂×M₁₆	—	67	2	no	—
523	C₂×M₁₆	—	134	2	no	—
524	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
525	C₂²×C₈	—	2	2	no	—
526	C₂²×C₈	—	134	2	no	—
527	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
528	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
529	C₄⋊Q₈	—	67	2	no	—
530	#127	—	67	2	no	—
531	#124	—	67	4	no	—
532	#121	—	268	2	no	—
533	C₄×D₄	—	268	2	no	—
534	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
535	#108	—	67	2	no	—
536	#108	—	67	2	no	—
537	#108	C₂²×D₆₇	1	2	yes	—
538	#108	C₂²×D₆₇	1	2	yes	—
539	#104	—	67	4	no	—
540	#104	—	67	4	no	—
541	#104	C₂²×D₆₇	1	4	yes	—
542	#104	C₂²×D₆₇	1	4	yes	—
543	#101	—	67	8	no	—
544	#100	—	67	8	no	—
545	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
546	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
547	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
548	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
549	#41	—	2	2	no	—
550	#41	—	2	2	no	—
551	#41	—	268	8	no	—
552	#41	—	268	8	no	—
553	#41	—	268	4	no	—
554	#41	—	268	4	no	—
555	#41	—	268	8	no	—
556	#41	—	268	8	no	—
557	#41	—	67	4	no	—
558	#41	—	67	4	no	—
559	#41	—	134	8	no	—
560	#41	—	134	8	no	—
561	#41	—	134	8	no	—
562	#41	—	134	8	no	—
563	#41	—	134	4	no	—
564	#41	—	134	4	no	—
565	#41	—	134	4	no	—
566	#41	—	134	4	no	—
567	#41	—	134	2	no	—
568	#41	—	134	2	no	—
569	#41	C₂×D₂₆₈	1	4	yes	—
570	#41	C₂³×D₆₇	1	4	yes	—
571	C₂×Q₈	—	268	4	no	—
572	C₂×Q₈	—	67	2	no	—
573	C₂×Q₈	—	67	4	no	—
574	C₂×Q₈	—	134	8	no	—
575	C₂×Q₈	—	134	8	no	—
576	C₂×Q₈	C₂×D₂₆₈	1	2	yes	—
577	C₂×D₄	—	2	2	no	—
578	C₂×D₄	—	268	2	no	—
579	C₂×D₄	—	268	8	no	—
580	C₂×D₄	—	268	4	no	—
581	C₂×D₄	—	268	8	no	—
582	C₂×D₄	—	268	8	no	—
583	C₂×D₄	—	67	2	no	—
584	C₂×D₄	—	67	4	no	—
585	C₂×D₄	—	134	8	no	—
586	C₂×D₄	—	134	4	no	—
587	C₂×D₄	—	134	8	no	—
588	C₂×D₄	—	134	2	no	—
589	C₂×D₄	C₂×D₂₆₈	1	2	yes	—
590	C₂²×C₄	—	2	2	no	—
591	C₂²×C₄	—	2	1	no	—
592	C₂²×C₄	—	268	8	no	—
593	C₂²×C₄	—	134	4	no	—
594	C₂²×C₄	—	134	2	no	—
595	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
596	Q₁₆	—	67	2	no	—
597	Q₁₆	—	67	2	no	—
598	Q₁₆	—	134	8	no	—
599	Q₁₆	—	134	8	no	—
600	Q₁₆	—	134	2	no	—
601	Q₁₆	—	134	2	no	—
602	Q₁₆	—	134	2	no	—
603	Q₁₆	—	134	2	no	—
604	Q₁₆	C₂×D₂₆₈	1	2	yes	—
605	Q₁₆	C₂×D₂₆₈	1	2	yes	—
606	SD₁₆	—	67	4	no	—
607	SD₁₆	—	67	4	no	—
608	SD₁₆	—	134	8	no	—
609	SD₁₆	—	134	8	no	—
610	SD₁₆	—	134	8	no	—
611	SD₁₆	—	134	8	no	—
612	SD₁₆	—	134	4	no	—
613	SD₁₆	—	134	4	no	—
614	SD₁₆	—	134	4	no	—
615	SD₁₆	—	134	4	no	—
616	SD₁₆	C₂×D₂₆₈	1	4	yes	—
617	SD₁₆	C₂×D₂₆₈	1	4	yes	—
618	D₈	—	67	2	no	—
619	D₈	—	67	2	no	—
620	D₈	—	134	8	no	—
621	D₈	—	134	8	no	—
622	D₈	—	134	2	no	—
623	D₈	—	134	2	no	—
624	D₈	—	134	2	no	—
625	D₈	—	134	2	no	—
626	D₈	C₂×D₂₆₈	1	2	yes	—
627	D₈	C₂×D₂₆₈	1	2	yes	—
628	M₁₆	—	2	2	no	—
629	M₁₆	—	2	4	no	—
630	M₁₆	—	2	4	no	—
631	M₁₆	—	67	4	no	—
632	M₁₆	—	134	2	no	—
633	M₁₆	—	134	4	no	—
634	M₁₆	—	134	4	no	—
635	M₁₆	C₂³×D₆₇	1	4	yes	—
636	C₂×C₈	—	2	2	no	—
637	C₂×C₈	—	2	4	no	—
638	C₂×C₈	—	2	4	no	—
639	C₂×C₈	—	134	2	no	—
640	C₂×C₈	—	134	4	no	—
641	C₂×C₈	—	134	4	no	—
642	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
643	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
644	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
645	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
646	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
647	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
648	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
649	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
650	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
651	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
652	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
653	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
654	C₄⋊C₄	—	268	2	no	—
655	C₂²⋊C₄	—	268	4	no	—
656	C₄²	—	2	1	no	—
657	C₄²	C₂³×D₆₇	1	1	yes	characteristic
658	C₄²	C₂³×D₆₇	1	1	yes	characteristic
659	C₂³	—	2	2	no	—
660	C₂³	—	268	8	no	—
661	C₂³	—	134	4	no	—
662	C₂³	—	134	2	no	—
663	C₂³	#14524089	1	1	yes	characteristic
664	Q₈	—	2	2	no	—
665	Q₈	—	268	8	no	—
666	Q₈	—	67	4	no	—
667	Q₈	—	134	4	no	—
668	Q₈	—	134	4	no	—
669	Q₈	—	134	2	no	—
670	Q₈	C₂²×D₂₆₈	1	4	yes	—
671	D₄	—	2	4	no	—
672	D₄	—	2	4	no	—
673	D₄	—	2	2	no	—
674	D₄	—	268	8	no	—
675	D₄	—	268	8	no	—
676	D₄	—	268	4	no	—
677	D₄	—	268	4	no	—
678	D₄	—	268	8	no	—
679	D₄	—	536	4	no	—
680	D₄	—	67	4	no	—
681	D₄	—	134	4	no	—
682	D₄	—	134	4	no	—
683	D₄	—	134	2	no	—
684	D₄	—	134	4	no	—
685	D₄	—	134	4	no	—
686	D₄	C₂²×D₂₆₈	1	4	yes	—
687	C₂×C₄	—	2	2	no	—
688	C₂×C₄	—	2	1	no	—
689	C₂×C₄	—	2	4	no	—
690	C₂×C₄	—	2	4	no	—
691	C₂×C₄	—	2	1	no	—
692	C₂×C₄	—	4	2	no	—
693	C₂×C₄	—	268	8	no	—
694	C₂×C₄	—	268	8	no	—
695	C₂×C₄	—	268	8	no	—
696	C₂×C₄	—	268	4	no	—
697	C₂×C₄	—	268	4	no	—
698	C₂×C₄	—	134	4	no	—
699	C₂×C₄	—	134	2	no	—
700	C₂×C₄	—	134	4	no	—
701	C₂×C₄	—	134	4	no	—
702	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
703	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
704	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
705	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
706	C₂×C₄	#14524089	1	1	yes	characteristic
707	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
708	C₈	—	2	4	no	—
709	C₈	—	134	4	no	—
710	C₈	C₂×D₄×D₆₇	1	1	yes	characteristic
711	C₈	C₂×D₄×D₆₇	1	1	yes	characteristic
712	C₈	C₂×D₄×D₆₇	1	1	yes	characteristic
713	C₈	C₂×D₄×D₆₇	1	1	yes	characteristic
714	C₈	C₂²×D₂₆₈	1	1	yes	characteristic
715	C₈	C₂²×D₂₆₈	1	1	yes	characteristic
716	C₈	C₂²×D₂₆₈	1	1	yes	characteristic
717	C₈	C₂²×D₂₆₈	1	1	yes	characteristic
718	C₂²	—	2	4	no	—
719	C₂²	—	4	1	no	—
720	C₂²	—	8	2	no	—
721	C₂²	—	268	4	no	—
722	C₂²	—	536	2	no	—
723	C₂²	—	536	4	no	—
724	C₂²	—	1072	8	no	—
725	C₂²	—	134	4	no	—
726	C₂²	D₄×D₂₆₈	1	1	yes	characteristic
727	C₂²	D₄×D₂₆₈	1	1	yes	characteristic
728	C₂²	C₂×#14524089	1	1	yes	characteristic
729	C₄	—	2	2	no	—
730	C₄	—	2	4	no	—
731	C₄	—	4	1	no	—
732	C₄	—	268	4	no	—
733	C₄	—	134	4	no	—
734	C₄	D₄×D₂₆₈	1	1	yes	characteristic
735	C₄	D₄×D₂₆₈	1	1	yes	characteristic
736	C₄	C₂²×D₄×D₆₇	1	1	yes	characteristic
737	C₄	C₂³×D₂₆₈	1	1	yes	characteristic
738	C₂	—	2	1	no	—
739	C₂	—	2	1	no	—
740	C₂	—	2	1	no	—
741	C₂	—	4	4	no	—
742	C₂	—	268	4	no	—
743	C₂	—	536	4	no	—
744	C₂	C₂×D₄×D₂₆₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₂₆₈
commutator, cosocle	C₂×C₄×C₆₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂×#443×C₆₇	C₂
socle	C₂×C₆₇	C₂×D₄²
hypercenter	C₂×#443	D₆₇
hypocenter	C₆₇	#61105
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#443

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #22574983	C₂	semidirect
2. #22574978	C₂	semidirect
3. #22572014	C₂	semidirect
4. #22572009	C₂	semidirect
5. #22572003	C₂	semidirect
6. #22571997	C₂	semidirect
7. #22571996	C₂	semidirect
8. #22570637	C₂	semidirect
9. #22569930	C₂	semidirect
10. #22569929	C₂	semidirect
11. #22569560	C₂	semidirect
12. C₂×#16863514	C₂	semidirect
13. C₂×#443×C₆₇	C₂	semidirect
14. #16864232	C₂²	semidirect
16. #16864213	C₂²	semidirect
17. #16864210	C₂²	semidirect
20. #16864208	C₂²	semidirect
21. #16864203	C₂²	semidirect
24. #16864201	C₂²	semidirect
25. #16864198	C₂²	semidirect
28. #16864196	C₂²	semidirect
29. #16864194	C₂²	semidirect
33. C₂×#14524065	C₂²	semidirect
34. C₂×#14524065	C₂²	semidirect
36. C₂×#14524049	C₂²	semidirect
37. #16863946	C₂²	semidirect
38. #16863780	C₂²	semidirect
39. #16863780	C₂²	semidirect
40. #16863776	C₂²	semidirect
41. #16863773	C₂²	semidirect
42. #16863769	C₂²	semidirect
43. #16863672	C₂²	semidirect
44. #16863671	C₂²	semidirect
45. #16863667	C₂²	semidirect
46. #16863666	C₂²	semidirect
47. #16863660	C₂²	semidirect
48. #16863659	C₂²	semidirect
49. #16863580	C₂²	semidirect
50. #16863575	C₂²	semidirect
51. #16863570	C₂²	semidirect
52. #16863565	C₂²	semidirect
53. #16863514	C₂²	semidirect
54. #16863514	C₂²	semidirect
55. #16863514	C₂²	semidirect
56. #16863514	C₂²	semidirect
57. C₂×#14523979	C₂²	semidirect
58. C₂×#14523971	C₂²	semidirect
59. #16863378	C₂²	semidirect
61. C₂×#131×C₆₇	C₂²	semidirect
62. #443×C₆₇	C₂²	semidirect
63. #443×C₆₇	C₂²	semidirect
64. #443×C₆₇	C₂²	semidirect
65. #443×C₆₇	C₂²	semidirect
66. C₂×#108×C₆₇	C₂²	semidirect
67. C₂×#104×C₆₇	C₂²	semidirect
68. C₂×C₄×C₈×C₆₇	C₂²	semidirect
110. #14524065	D₄	semidirect
111. #14524065	D₄	semidirect
118. C₂×Dic₂₆₈	D₄	semidirect
123. C₂×#11761617	D₄	semidirect
128. C₂×D₅₃₆	D₄	semidirect
151. #135×C₆₇	D₄	semidirect
152. #135×C₆₇	D₄	semidirect
155. C₂×Q₁₆×C₆₇	D₄	semidirect
156. C₂×SD₁₆×C₆₇	D₄	semidirect
157. C₂×D₈×C₆₇	D₄	semidirect
385. C₂×#443	D₆₇	semidirect
386. C₆₇	#61105	semidirect
411. C₂×#131	C₂×D₆₇	semidirect
432. #443	C₂×D₆₇	semidirect
433. #443	C₂×D₆₇	semidirect
434. #443	C₂×D₆₇	semidirect
435. #443	C₂×D₆₇	semidirect
437. C₂×#108	C₂×D₆₇	semidirect
439. C₂×#104	C₂×D₆₇	semidirect
441. C₂×C₄×C₈	C₂×D₆₇	semidirect
491. #135	D₂₆₈	semidirect
492. #135	D₂₆₈	semidirect
500. C₂×Q₁₆	D₂₆₈	semidirect
506. C₂×SD₁₆	D₂₆₈	semidirect
512. C₂×D₈	D₂₆₈	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22574983	2
#22574978	2
#22572014	2
#22572009	2
#22572003	2
#22571997	2
#22571996	2
#22570637	2
#22569930	2
#22569929	2
#22569560	2
C₂×#16863514	2
C₂×#443×C₆₇	2
#61105	67