Subgroups — #30878372

Order 17152.

892 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569790	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863514	C₂	1	1	yes	maximal, characteristic, complemented
3	#22568930	C₂	1	2	yes	maximal, complemented
4	C₂²×#14523979	C₂	1	1	yes	maximal, characteristic, complemented
5	#22561097	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560957	C₂	1	1	yes	maximal, characteristic, complemented
7	#22560858	C₂	1	2	yes	maximal, complemented
8	#22560509	C₂	1	2	yes	maximal, complemented
9	#22560487	C₂	1	2	yes	maximal, complemented
10	#22560088	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×C₂²⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524065	C₂²	1	1	yes	characteristic
13	C₂×#14524049	C₂²	1	2	yes	—
14	C₂²×#11761612	C₂²	1	2	yes	—
15	#16863907	C₂²	1	2	yes	—
16	#16863637	C₄	1	1	yes	characteristic, complemented
17	#16863637	C₄	1	1	yes	characteristic, complemented
18	#16863632	C₄	1	1	yes	characteristic, complemented
19	#16863632	C₄	1	1	yes	characteristic, complemented
20	#16863625	C₄	1	2	yes	complemented
21	#16863625	C₄	1	2	yes	complemented
22	#16863514	—	2	4	no	—
23	#16863471	—	2	2	no	—
24	#16863470	—	2	2	no	—
25	#16863464	—	2	2	no	—
26	(C₂²⋊D₆₇)⋊C₈	—	2	2	no	—
27	C₂×#14523979	—	2	4	no	—
28	C₂×#14523979	C₂²	1	1	yes	characteristic
29	C₂×#14523979	C₂²	1	2	yes	—
30	C₂×#14523979	C₂²	1	1	yes	characteristic
31	#16863434	C₂²	1	1	yes	characteristic
32	#16863422	C₂²	1	2	yes	—
33	C₂×#14523971	C₂²	1	2	yes	—
34	C₂×#14523971	C₂²	1	2	yes	—
35	#16863393	C₂²	1	2	yes	—
36	#16863362	C₂²	1	2	yes	—
37	#16863351	C₂²	1	2	yes	—
38	#16863156	C₂²	1	2	yes	—
39	#16863156	C₂²	1	2	yes	—
40	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
42	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
43	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
44	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
45	#336×C₆₇	C₂²	1	1	yes	characteristic
46	#336×C₆₇	C₂²	1	1	yes	characteristic
47	C₂×#11761628	C₂³	1	2	yes	—
48	#14524065	—	2	2	no	—
49	#14524065	—	2	2	no	—
50	C₂×Dic₂₆₈	C₂×C₄	1	1	yes	characteristic
51	C₂×#11761617	C₂×C₄	1	2	yes	—
52	C₂×D₅₃₆	C₂×C₄	1	1	yes	characteristic
53	#14524049	—	2	4	no	—
54	#14524049	—	4	2	no	—
55	C₂×#11761612	—	2	4	no	—
56	C₂×#11761612	—	2	2	no	—
57	C₂×#11761612	—	4	2	no	—
58	C₂×#11761612	C₂³	1	2	yes	—
59	C₂×#11761612	C₂³	1	2	yes	—
60	C₂×#11761611	—	2	2	no	—
61	C₂²×C₆₇⋊C₈	—	2	2	no	—
62	C₂×C₈×D₆₇	—	2	2	no	—
63	#14524026	—	2	2	no	—
64	#14524022	C₂×C₄	1	2	yes	—
65	#14524022	C₂×C₄	1	2	yes	—
66	#14524017	C₂×C₄	1	2	yes	—
67	#14524017	C₂×C₄	1	2	yes	—
68	#14524012	—	2	2	no	—
69	C₄×C₂²⋊D₆₇	—	2	2	no	—
70	C₄×C₂²⋊D₆₇	—	2	2	no	—
71	#14524002	C₂³	1	2	yes	—
72	#14523979	—	2	4	no	—
73	#14523979	—	2	2	no	—
74	#14523979	D₄	1	4	yes	complemented
75	#14523977	C₂×C₄	1	1	yes	characteristic
76	#14523977	C₂×C₄	1	1	yes	characteristic
77	#14523975	C₂×C₄	1	2	yes	—
78	Dic₆₇⋊C₈	—	2	2	no	—
79	Dic₆₇⋊C₈	—	2	2	no	—
80	#14523971	—	2	4	no	—
81	#14523971	—	4	2	no	—
82	#14523968	C₂×C₄	1	2	yes	—
83	#14523968	C₂×C₄	1	2	yes	—
84	#14523965	C₂×C₄	1	2	yes	—
85	#14523965	C₂×C₄	1	2	yes	—
86	(C₂×D₆₇)⋊C₈	—	2	2	no	—
87	(C₂×D₆₇)⋊C₈	—	2	2	no	—
88	#14523953	—	2	2	no	—
89	C₈×Dic₆₇	—	2	2	no	—
90	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
91	C₂²×C₈×C₆₇	—	2	2	no	—
92	C₂²×C₈×C₆₇	—	2	2	no	—
93	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
94	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
95	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
96	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
97	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
98	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
99	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
100	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
101	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
102	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
104	C₂²⋊C₈×C₆₇	D₄	1	1	yes	characteristic
105	C₂²⋊C₈×C₆₇	D₄	1	1	yes	characteristic
106	C₂²⋊C₈×C₆₇	D₄	1	1	yes	characteristic
107	C₂²⋊C₈×C₆₇	D₄	1	1	yes	characteristic
108	C₄×C₈×C₆₇	—	2	2	no	—
109	C₄×C₈×C₆₇	—	2	2	no	—
110	#95×C₆₇	—	2	1	no	—
111	#11761628	—	2	2	no	—
112	#11761628	—	2	2	no	—
113	#11761628	—	2	2	no	—
114	#11761628	—	2	2	no	—
115	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
116	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
117	C₂×C₂²⋊D₆₇	—	2	2	no	—
118	C₂×C₂²⋊D₆₇	—	2	2	no	—
119	C₂×C₂²⋊D₆₇	—	2	2	no	—
120	C₂×C₄×D₆₇	—	2	2	no	—
121	Dic₂₆₈	—	2	2	no	—
122	#11761617	—	2	4	no	—
123	D₅₃₆	—	2	2	no	—
124	#11761612	—	2	2	no	—
125	#11761612	—	2	2	no	—
126	#11761612	—	2	2	no	—
127	#11761612	—	4	4	no	—
128	#11761612	C₂×D₄	1	4	yes	—
129	#11761611	—	4	4	no	—
130	C₂×C₆₇⋊C₈	—	2	2	no	—
131	C₂×C₆₇⋊C₈	—	2	2	no	—
132	C₂×C₆₇⋊C₈	—	4	4	no	—
133	C₈×D₆₇	—	4	4	no	—
134	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
135	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
136	Dic₆₇⋊C₄	—	2	2	no	—
137	Dic₆₇⋊C₄	—	2	2	no	—
138	C₂²⋊Dic₆₇	—	2	2	no	—
139	C₂²⋊Dic₆₇	—	2	2	no	—
140	C₂²⋊Dic₆₇	—	2	2	no	—
141	(C₂×D₆₇)⋊C₄	—	2	2	no	—
142	(C₂×D₆₇)⋊C₄	—	2	2	no	—
143	C₄×Dic₆₇	—	2	2	no	—
144	C₂⁴×C₆₇	#41	1	1	yes	characteristic
145	C₂²×C₄×C₆₇	—	2	1	no	—
146	C₂²×C₄×C₆₇	—	2	1	no	—
147	C₂²×C₄×C₆₇	—	2	1	no	—
148	C₂²×C₄×C₆₇	—	2	1	no	—
149	C₂²×C₄×C₆₇	—	2	1	no	—
150	C₂²×C₄×C₆₇	—	2	1	no	—
151	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
152	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
153	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
156	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
157	C₂×C₈×C₆₇	—	2	1	no	—
158	C₂×C₈×C₆₇	—	2	1	no	—
159	C₂×C₈×C₆₇	—	2	1	no	—
160	C₂×C₈×C₆₇	—	2	1	no	—
161	C₂×C₈×C₆₇	—	2	1	no	—
162	C₂×C₈×C₆₇	—	2	1	no	—
163	C₂×C₈×C₆₇	—	2	2	no	—
164	C₂×C₈×C₆₇	—	2	2	no	—
165	C₂×C₈×C₆₇	—	2	2	no	—
166	C₂×C₈×C₆₇	—	2	2	no	—
167	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
168	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
169	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
170	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
171	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
172	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
173	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
174	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
175	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
176	C₂×C₈×C₆₇	#41	1	2	yes	—
177	C₂²⋊C₄×C₆₇	—	2	4	no	—
178	C₄²×C₆₇	—	2	1	no	—
179	C₄²×C₆₇	—	2	1	no	—
180	C₄²×C₆₇	—	2	1	no	—
181	C₄²×C₆₇	—	2	1	no	—
182	C₂³×C₆₇	—	2	1	no	—
183	C₂³×C₆₇	—	2	1	no	—
184	C₂³×C₆₇	—	2	2	no	—
185	C₂³×C₆₇	—	4	1	no	—
186	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
187	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
188	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
189	C₂²×D₆₇	—	2	2	no	—
190	C₂×C₄×C₆₇	—	2	1	no	—
191	C₂×C₄×C₆₇	—	2	1	no	—
192	C₂×C₄×C₆₇	—	2	1	no	—
193	C₂×C₄×C₆₇	—	2	1	no	—
194	C₂×C₄×C₆₇	—	2	1	no	—
195	C₂×C₄×C₆₇	—	2	1	no	—
196	C₂×C₄×C₆₇	—	2	1	no	—
197	C₂×C₄×C₆₇	—	2	1	no	—
198	C₂×C₄×C₆₇	—	2	1	no	—
199	C₂×C₄×C₆₇	—	2	1	no	—
200	C₂×C₄×C₆₇	—	2	2	no	—
201	C₂×C₄×C₆₇	—	2	2	no	—
202	C₂×C₄×C₆₇	—	4	1	no	—
203	C₂×C₄×C₆₇	—	4	1	no	—
204	C₂×C₄×C₆₇	—	4	1	no	—
205	C₂×C₄×C₆₇	—	4	1	no	—
206	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
207	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
208	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
209	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
210	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
211	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
212	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
213	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
214	C₂²⋊D₆₇	—	4	2	no	—
215	C₂²⋊D₆₇	—	4	2	no	—
216	C₂²⋊D₆₇	—	4	4	no	—
217	C₂²⋊D₆₇	—	8	2	no	—
218	C₂×Dic₆₇	—	2	2	no	—
219	C₂×Dic₆₇	—	2	2	no	—
220	C₂×Dic₆₇	—	2	2	no	—
221	D₂₆₈	—	2	2	no	—
222	D₂₆₈	—	2	2	no	—
223	C₄×D₆₇	—	4	2	no	—
224	C₄×D₆₇	—	4	2	no	—
225	Dic₁₃₄	—	2	2	no	—
226	Dic₁₃₄	—	2	2	no	—
227	C₈×C₆₇	—	2	1	no	—
228	C₈×C₆₇	—	2	1	no	—
229	C₈×C₆₇	—	2	1	no	—
230	C₈×C₆₇	—	2	1	no	—
231	C₈×C₆₇	—	2	1	no	—
232	C₈×C₆₇	—	2	1	no	—
233	C₈×C₆₇	C₄×D₄	1	2	yes	—
234	C₈×C₆₇	C₄×D₄	1	2	yes	—
235	C₆₇⋊C₈	—	2	4	no	—
236	C₆₇⋊C₈	—	4	2	no	—
237	C₂²×C₆₇	—	2	1	no	—
238	C₂²×C₆₇	—	2	1	no	—
239	C₂²×C₆₇	—	2	1	no	—
240	C₂²×C₆₇	—	2	1	no	—
241	C₂²×C₆₇	—	2	1	no	—
242	C₂²×C₆₇	—	2	1	no	—
243	C₂²×C₆₇	—	2	1	no	—
244	C₂²×C₆₇	—	4	2	no	—
245	C₂²×C₆₇	—	4	2	no	—
246	C₂²×C₆₇	#443	1	2	yes	complemented
247	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
248	C₂²×C₆₇	#548	1	1	yes	characteristic
249	C₂²×C₆₇	#553	1	1	yes	characteristic
250	C₂×D₆₇	—	4	2	no	—
251	C₂×D₆₇	—	4	2	no	—
252	C₂×D₆₇	—	4	2	no	—
253	C₄×C₆₇	—	2	1	no	—
254	C₄×C₆₇	—	2	1	no	—
255	C₄×C₆₇	—	4	2	no	—
256	C₄×C₆₇	—	4	2	no	—
257	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
258	C₄×C₆₇	#520	1	1	yes	characteristic
259	C₄×C₆₇	D₄²	1	1	yes	characteristic
260	C₄×C₆₇	#547	1	1	yes	characteristic
261	Dic₆₇	—	4	2	no	—
262	Dic₆₇	—	4	2	no	—
263	Dic₆₇	—	4	2	no	—
264	#23018	—	67	1	no	maximal, Sylow
265	C₂×C₆₇	—	2	1	no	—
266	C₂×C₆₇	—	2	1	no	—
267	C₂×C₆₇	—	2	2	no	—
268	C₂×C₆₇	—	4	1	no	—
269	C₂×C₆₇	#2361	1	1	yes	characteristic
270	C₂×C₆₇	C₂×#443	1	1	yes	characteristic
271	C₂×C₆₇	#2957	1	1	yes	characteristic
272	D₆₇	—	8	2	no	—
273	#3050	—	67	1	no	—
274	C₂×#443	—	67	1	no	—
275	#2933	—	67	2	no	—
276	C₂²×#108	—	67	1	no	—
277	#1974	—	67	1	no	—
278	#1945	—	67	1	no	—
279	#1929	—	67	2	no	—
280	#1871	—	67	2	no	—
281	#1866	—	67	2	no	—
282	#1772	—	67	1	no	—
283	C₈×C₂²⋊C₄	D₆₇	1	1	yes	characteristic, complemented
284	C₆₇	#23018	1	1	yes	characteristic, complemented, Sylow
285	C₂×#135	—	67	1	no	—
286	C₂×#131	—	67	2	no	—
287	C₂²×M₁₆	—	67	2	no	—
288	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
289	#525	—	67	2	no	—
290	#467	—	67	1	no	—
291	#467	—	67	1	no	—
292	#466	—	67	1	no	—
293	#466	—	67	1	no	—
294	#465	—	67	2	no	—
295	#465	—	67	2	no	—
296	#443	—	134	4	no	—
297	#436	—	134	2	no	—
298	#435	—	134	2	no	—
299	#435	—	134	2	no	—
300	C₈×D₄	—	134	2	no	—
301	C₂×#108	—	67	1	no	—
302	C₂×#108	—	67	2	no	—
303	C₂×#108	—	67	1	no	—
304	C₂×#108	—	134	4	no	—
305	#427	—	67	1	no	—
306	#424	—	67	2	no	—
307	C₂×#104	—	67	2	no	—
308	C₂×#104	—	67	2	no	—
309	#416	—	67	2	no	—
310	#408	—	67	2	no	—
311	C₂×C₂²⋊C₈	C₂×D₆₇	1	1	yes	characteristic
312	#405	—	67	2	no	—
313	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
314	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
315	C₄×C₂²⋊C₄	C₂×D₆₇	1	1	yes	characteristic
316	#339	—	67	2	no	—
317	#339	—	67	2	no	—
318	#336	C₂×D₆₇	1	1	yes	characteristic
319	#336	C₂×D₆₇	1	1	yes	characteristic
320	C₂×#41	—	67	2	no	—
321	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
322	#135	—	134	2	no	—
323	#135	—	134	2	no	—
324	C₂×Q₁₆	—	67	1	no	—
325	C₂×SD₁₆	—	67	2	no	—
326	C₂×D₈	—	67	1	no	—
327	#131	—	268	2	no	—
328	#131	—	134	4	no	—
329	C₂×M₁₆	—	268	2	no	—
330	C₂×M₁₆	—	67	2	no	—
331	C₂×M₁₆	—	67	2	no	—
332	C₂×M₁₆	—	134	4	no	—
333	C₂×M₁₆	—	134	2	no	—
334	C₂×M₁₆	—	134	2	no	—
335	C₂²×C₈	—	2	2	no	—
336	C₂²×C₈	—	2	2	no	—
337	C₂²×C₈	—	134	2	no	—
338	C₂²×C₈	—	134	2	no	—
339	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
340	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
341	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
342	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
343	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
344	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
345	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
346	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
347	#123	—	134	2	no	—
348	C₂²⋊Q₈	—	67	2	no	—
349	C₂²⋊Q₈	—	67	2	no	—
350	#121	—	67	2	no	—
351	#121	—	67	2	no	—
352	#120	—	134	2	no	—
353	C₄×D₄	—	134	2	no	—
354	C₄×D₄	—	134	2	no	—
355	#117	—	67	2	no	—
356	C₂×C₂²⋊C₄	—	2	1	no	—
357	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
358	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
359	#108	—	67	4	no	—
360	#108	—	134	4	no	—
361	#108	—	134	2	no	—
362	#107	—	67	1	no	—
363	#107	—	67	1	no	—
364	#106	—	67	2	no	—
365	C₄⋊C₈	—	134	2	no	—
366	C₄⋊C₈	—	134	2	no	—
367	#104	—	268	2	no	—
368	#104	—	134	4	no	—
369	#103	—	67	2	no	—
370	#103	—	67	2	no	—
371	#102	—	67	2	no	—
372	#102	—	67	2	no	—
373	C₂²⋊C₈	—	134	2	no	—
374	C₂²⋊C₈	—	134	2	no	—
375	C₂²⋊C₈	D₂₆₈	1	1	yes	characteristic
376	C₂²⋊C₈	D₂₆₈	1	1	yes	characteristic
377	C₂²⋊C₈	D₂₆₈	1	1	yes	characteristic
378	C₂²⋊C₈	D₂₆₈	1	1	yes	characteristic
379	#97	—	134	2	no	—
380	C₄×C₈	—	2	2	no	—
381	C₄×C₈	—	2	2	no	—
382	C₄×C₈	—	134	2	no	—
383	#95	—	2	1	no	—
384	C₂⁴	#11761628	1	1	yes	characteristic
385	#41	—	134	2	no	—
386	#41	—	134	2	no	—
387	#41	—	134	2	no	—
388	#41	—	134	2	no	—
389	C₂×Q₈	—	67	2	no	—
390	C₂×D₄	—	67	2	no	—
391	C₂×D₄	—	134	2	no	—
392	C₂×D₄	—	134	2	no	—
393	C₂×D₄	—	134	2	no	—
394	C₂²×C₄	—	2	1	no	—
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	2	1	no	—
397	C₂²×C₄	—	2	1	no	—
398	C₂²×C₄	—	2	1	no	—
399	C₂²×C₄	—	2	1	no	—
400	C₂²×C₄	—	134	2	no	—
401	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
402	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
403	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
404	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
405	C₂²×C₄	#11761628	1	1	yes	characteristic
406	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
407	Q₁₆	—	134	2	no	—
408	SD₁₆	—	134	4	no	—
409	D₈	—	134	2	no	—
410	M₁₆	—	268	4	no	—
411	M₁₆	—	268	4	no	—
412	M₁₆	—	67	4	no	—
413	M₁₆	—	134	2	no	—
414	M₁₆	—	134	2	no	—
415	M₁₆	—	134	2	no	—
416	C₂×C₈	—	2	1	no	—
417	C₂×C₈	—	2	1	no	—
418	C₂×C₈	—	2	1	no	—
419	C₂×C₈	—	2	1	no	—
420	C₂×C₈	—	2	1	no	—
421	C₂×C₈	—	2	1	no	—
422	C₂×C₈	—	2	2	no	—
423	C₂×C₈	—	2	2	no	—
424	C₂×C₈	—	2	2	no	—
425	C₂×C₈	—	2	2	no	—
426	C₂×C₈	—	268	4	no	—
427	C₂×C₈	—	268	4	no	—
428	C₂×C₈	—	134	2	no	—
429	C₂×C₈	—	134	2	no	—
430	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
431	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
432	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
433	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
434	C₂×C₈	D₄×D₆₇	1	2	yes	—
435	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
436	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
437	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
438	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
439	C₂×C₈	#11761627	1	2	yes	—
440	C₄⋊C₄	—	67	2	no	—
441	C₄⋊C₄	—	67	2	no	—
442	C₄⋊C₄	—	134	2	no	—
443	C₄⋊C₄	—	134	2	no	—
444	C₂²⋊C₄	—	2	4	no	—
445	C₂²⋊C₄	—	134	2	no	—
446	C₂²⋊C₄	—	134	2	no	—
447	C₂²⋊C₄	—	134	2	no	—
448	C₂²⋊C₄	—	134	2	no	—
449	C₂²⋊C₄	—	134	2	no	—
450	C₄²	—	2	1	no	—
451	C₄²	—	2	1	no	—
452	C₄²	—	2	1	no	—
453	C₄²	—	2	1	no	—
454	C₄²	—	134	2	no	—
455	C₂³	—	2	2	no	—
456	C₂³	—	2	1	no	—
457	C₂³	—	2	1	no	—
458	C₂³	—	4	1	no	—
459	C₂³	—	134	2	no	—
460	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
461	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
462	C₂³	C₂×#11761628	1	1	yes	characteristic
463	Q₈	—	134	2	no	—
464	Q₈	—	134	2	no	—
465	D₄	—	268	4	no	—
466	D₄	—	268	2	no	—
467	D₄	—	268	2	no	—
468	D₄	—	536	2	no	—
469	D₄	—	134	2	no	—
470	D₄	—	134	2	no	—
471	C₂×C₄	—	2	1	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	2	1	no	—
474	C₂×C₄	—	2	1	no	—
475	C₂×C₄	—	2	1	no	—
476	C₂×C₄	—	2	1	no	—
477	C₂×C₄	—	2	1	no	—
478	C₂×C₄	—	2	1	no	—
479	C₂×C₄	—	2	1	no	—
480	C₂×C₄	—	2	1	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	4	1	no	—
484	C₂×C₄	—	4	1	no	—
485	C₂×C₄	—	4	1	no	—
486	C₂×C₄	—	4	1	no	—
487	C₂×C₄	—	268	2	no	—
488	C₂×C₄	—	268	2	no	—
489	C₂×C₄	—	134	2	no	—
490	C₂×C₄	—	134	2	no	—
491	C₂×C₄	—	134	2	no	—
492	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
493	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
494	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
495	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
496	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
497	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
498	C₂×C₄	#14524089	1	1	yes	characteristic
499	C₂×C₄	#14524090	1	1	yes	characteristic
500	C₈	—	2	1	no	—
501	C₈	—	2	1	no	—
502	C₈	—	2	1	no	—
503	C₈	—	2	1	no	—
504	C₈	—	2	1	no	—
505	C₈	—	2	1	no	—
506	C₈	—	268	2	no	—
507	C₈	—	134	4	no	—
508	C₈	(C₂²⋊D₆₇)⋊C₄	1	2	yes	—
509	C₈	(C₂²⋊D₆₇)⋊C₄	1	2	yes	—
510	C₂²	—	2	1	no	—
511	C₂²	—	2	1	no	—
512	C₂²	—	2	1	no	—
513	C₂²	—	2	1	no	—
514	C₂²	—	2	1	no	—
515	C₂²	—	2	1	no	—
516	C₂²	—	2	1	no	—
517	C₂²	—	4	2	no	—
518	C₂²	—	4	2	no	—
519	C₂²	—	268	2	no	—
520	C₂²	—	268	2	no	—
521	C₂²	—	268	2	no	—
522	C₂²	#16863514	1	2	yes	complemented
523	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
524	C₂²	#16864055	1	1	yes	characteristic
525	C₂²	#16864092	1	1	yes	characteristic
526	C₄	—	2	1	no	—
527	C₄	—	2	1	no	—
528	C₄	—	4	2	no	—
529	C₄	—	4	2	no	—
530	C₄	—	268	2	no	—
531	C₄	—	268	2	no	—
532	C₄	—	268	2	no	—
533	C₄	C₂×(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
534	C₄	#16863876	1	1	yes	characteristic
535	C₄	D₄×D₂₆₈	1	1	yes	characteristic
536	C₄	#16864050	1	1	yes	characteristic
537	C₂	—	2	1	no	—
538	C₂	—	2	1	no	—
539	C₂	—	2	2	no	—
540	C₂	—	4	1	no	—
541	C₂	—	536	2	no	—
542	C₂	#22563223	1	1	yes	characteristic
543	C₂	C₂×#16863514	1	1	yes	characteristic
544	C₂	#22569127	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₈×C₂²⋊C₄×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#22569790	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₈×C₂²⋊C₄	D₆₇
hypocenter	C₆₇	#23018
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₂²⋊C₈ ◁ C₈×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×#16863514	C₂	semidirect
3. #22568930	C₂	semidirect
4. C₂²×#14523979	C₂	semidirect
5. #22561097	C₂	semidirect
6. #22560957	C₂	semidirect
7. #22560858	C₂	semidirect
8. #22560509	C₂	semidirect
9. #22560487	C₂	semidirect
10. #22560088	C₂	semidirect
11. C₈×C₂²⋊C₄×C₆₇	C₂	semidirect
16. #16863637	C₄	semidirect
17. #16863637	C₄	semidirect
18. #16863632	C₄	semidirect
19. #16863632	C₄	semidirect
20. #16863625	C₄	semidirect
21. #16863625	C₄	semidirect
74. #14523979	D₄	semidirect
246. C₂²×C₆₇	#443	semidirect
283. C₈×C₂²⋊C₄	D₆₇	semidirect
284. C₆₇	#23018	semidirect
522. C₂²	#16863514	semidirect

Maximal subgroups

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

subgroup	index
#22569790	2
C₂×#16863514	2
#22568930	2
C₂²×#14523979	2
#22561097	2
#22560957	2
#22560858	2
#22560509	2
#22560487	2
#22560088	2
C₈×C₂²⋊C₄×C₆₇	2
#23018	67