Subgroups — #30878337

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	#22569664	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863511	C₂	1	2	yes	maximal, complemented
3	#22568952	C₂	1	1	yes	maximal, characteristic, complemented
4	#22561036	C₂	1	1	yes	maximal, characteristic, complemented
5	#22560982	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560912	C₂	1	2	yes	maximal, complemented
7	#22560536	C₂	1	2	yes	maximal, complemented
8	#22560483	C₂	1	2	yes	maximal, complemented
9	C₄⋊C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#14523970	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2906	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524063	C₂²	1	2	yes	—
13	C₂×#14524049	C₂²	1	2	yes	—
14	#16863946	C₂²	1	1	yes	characteristic
15	#16863618	C₄	1	2	yes	complemented
16	#16863611	C₄	1	2	yes	complemented
17	#16863606	C₄	1	1	yes	characteristic, complemented
18	#16863601	C₄	1	1	yes	characteristic, complemented
19	#16863596	C₄	1	1	yes	characteristic, complemented
20	#16863591	C₄	1	1	yes	characteristic, complemented
21	#16863511	—	2	8	no	—
22	#16863473	—	2	1	no	—
23	#16863466	—	2	2	no	—
24	#16863461	—	2	1	no	—
25	C₂×#14523978	C₂²	1	2	yes	—
26	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
27	C₂×#14523970	C₂²	1	1	yes	characteristic
28	C₂×#14523970	C₂²	1	1	yes	characteristic
29	C₂×#14523969	C₂²	1	2	yes	—
30	C₂×#14523969	C₂²	1	2	yes	—
31	#16863390	C₂²	1	2	yes	—
32	#16863362	C₂²	1	2	yes	—
33	C₄×#11761612	C₂²	1	1	yes	characteristic
34	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
35	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
36	#16863160	C₂²	1	2	yes	—
37	#16863155	C₂²	1	1	yes	characteristic
38	#16863155	C₂²	1	1	yes	characteristic
39	#16863150	C₂²	1	2	yes	—
40	C₂×#131×C₆₇	C₂²	1	2	yes	—
41	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
42	#424×C₆₇	—	2	2	no	—
43	#424×C₆₇	C₂²	1	2	yes	—
44	#423×C₆₇	—	2	1	no	—
45	#423×C₆₇	C₂²	1	1	yes	characteristic
46	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
47	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
48	C₂×#11761628	C₂³	1	2	yes	—
49	#14524063	—	2	4	no	—
50	#14524063	—	2	4	no	—
51	C₂×#11761618	C₂×C₄	1	2	yes	—
52	C₂×#11761616	C₂×C₄	1	2	yes	—
53	C₂×#11761615	C₂×C₄	1	2	yes	—
54	C₂×#11761613	C₂×C₄	1	2	yes	—
55	#14524049	—	2	4	no	—
56	#14524049	—	4	2	no	—
57	C₂×#11761612	C₂³	1	1	yes	characteristic
58	C₂×#11761612	C₂³	1	1	yes	characteristic
59	C₂×#11761611	—	2	2	no	—
60	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
61	C₂×C₈×D₆₇	—	2	2	no	—
62	#14524043	C₂×C₄	1	1	yes	characteristic
63	#14524040	C₂×C₄	1	1	yes	characteristic
64	#14524031	C₂×C₄	1	2	yes	—
65	#14524017	—	2	1	no	—
66	#14524017	—	2	1	no	—
67	C₄×D₂₆₈	—	2	1	no	—
68	C₄×D₂₆₈	—	2	1	no	—
69	C₂×C₄×Dic₆₇	—	2	1	no	—
70	C₂×C₄×Dic₆₇	—	2	1	no	—
71	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
72	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
73	#14523978	—	2	4	no	—
74	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
75	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
76	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
77	#14523970	—	2	2	no	—
78	#14523970	D₄	1	4	yes	complemented
79	#14523969	—	2	4	no	—
80	#14523969	—	4	2	no	—
81	#14523966	C₂×C₄	1	2	yes	—
82	#14523966	C₂×C₄	1	2	yes	—
83	#14523963	C₂×C₄	1	2	yes	—
84	#14523963	C₂×C₄	1	2	yes	—
85	(C₂×D₆₇)⋊C₈	—	2	2	no	—
86	(C₂×D₆₇)⋊C₈	—	2	2	no	—
87	#14523954	—	2	1	no	—
88	C₄×C₆₇⋊C₈	—	2	1	no	—
89	C₈×Dic₆₇	—	2	4	no	—
90	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
91	#131×C₆₇	—	2	4	no	—
92	#131×C₆₇	—	4	2	no	—
93	C₂×M₁₆×C₆₇	—	2	2	no	—
94	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
95	C₂²×C₈×C₆₇	—	2	2	no	—
96	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
97	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
98	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
99	C₄×D₄×C₆₇	—	2	1	no	—
100	C₄×D₄×C₆₇	—	2	1	no	—
101	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
102	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
103	#117×C₆₇	—	2	1	no	—
104	#117×C₆₇	C₂³	1	1	yes	characteristic
105	C₂×C₄²×C₆₇	—	2	1	no	—
106	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
107	C₄⋊C₈×C₆₇	—	2	2	no	—
108	C₄⋊C₈×C₆₇	—	2	2	no	—
109	C₄⋊C₈×C₆₇	—	2	1	no	—
110	C₄⋊C₈×C₆₇	—	2	1	no	—
111	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
112	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
113	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
114	#11761628	—	2	2	no	—
115	#11761628	—	2	2	no	—
116	#11761628	—	2	2	no	—
117	#11761628	—	2	2	no	—
118	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
119	C₂×D₂₆₈	—	2	1	no	—
120	C₂×D₂₆₈	—	2	1	no	—
121	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
122	C₂×C₂²⋊D₆₇	—	2	2	no	—
123	C₂²×Dic₆₇	—	2	1	no	—
124	C₂²×Dic₆₇	—	2	1	no	—
125	C₂×C₄×D₆₇	—	2	2	no	—
126	#11761618	—	2	4	no	—
127	#11761616	—	2	4	no	—
128	#11761615	—	2	4	no	—
129	#11761613	—	2	4	no	—
130	#11761612	—	2	1	no	—
131	#11761612	—	2	1	no	—
132	#11761612	—	2	1	no	—
133	#11761612	C₂×D₄	1	2	yes	—
134	#11761611	—	4	4	no	—
135	C₂×C₆₇⋊C₈	—	2	2	no	—
136	C₂×C₆₇⋊C₈	—	2	1	no	—
137	C₂×C₆₇⋊C₈	—	2	2	no	—
138	C₂×C₆₇⋊C₈	—	2	1	no	—
139	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
140	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
141	C₈×D₆₇	—	4	4	no	—
142	C₄⋊Dic₆₇	—	2	1	no	—
143	C₄⋊Dic₆₇	—	2	1	no	—
144	(C₂×D₆₇)⋊C₄	—	2	2	no	—
145	(C₂×D₆₇)⋊C₄	—	2	2	no	—
146	C₄×Dic₆₇	—	2	4	no	—
147	C₄×Dic₆₇	—	2	1	no	—
148	C₄×Dic₆₇	—	2	1	no	—
149	C₄×Dic₆₇	—	2	1	no	—
150	C₄×Dic₆₇	—	4	2	no	—
151	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
152	#41×C₆₇	—	2	1	no	—
153	#41×C₆₇	—	2	1	no	—
154	#41×C₆₇	C₂×D₄	1	2	yes	—
155	#41×C₆₇	#41	1	2	yes	—
156	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
157	C₂×D₄×C₆₇	—	2	1	no	—
158	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
159	C₂²×C₄×C₆₇	—	2	1	no	—
160	C₂²×C₄×C₆₇	—	2	1	no	—
161	C₂²×C₄×C₆₇	—	2	1	no	—
162	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
163	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
164	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
165	M₁₆×C₆₇	—	2	2	no	—
166	M₁₆×C₆₇	—	2	2	no	—
167	M₁₆×C₆₇	—	4	4	no	—
168	C₂×C₈×C₆₇	—	2	2	no	—
169	C₂×C₈×C₆₇	—	2	2	no	—
170	C₂×C₈×C₆₇	—	2	2	no	—
171	C₂×C₈×C₆₇	—	4	4	no	—
172	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
173	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
174	C₄⋊C₄×C₆₇	—	2	1	no	—
175	C₄⋊C₄×C₆₇	—	2	1	no	—
176	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
177	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
178	C₂²⋊C₄×C₆₇	—	2	1	no	—
179	C₂²⋊C₄×C₆₇	—	2	1	no	—
180	C₂²⋊C₄×C₆₇	—	2	1	no	—
181	C₄²×C₆₇	—	2	1	no	—
182	C₄²×C₆₇	—	2	1	no	—
183	C₄²×C₆₇	—	2	1	no	—
184	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
185	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
186	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
187	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
188	C₂³×C₆₇	—	2	1	no	—
189	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
190	C₂²×D₆₇	—	2	2	no	—
191	Q₈×C₆₇	—	2	1	no	—
192	Q₈×C₆₇	C₄×D₄	1	2	yes	—
193	D₄×C₆₇	—	2	2	no	—
194	D₄×C₆₇	—	2	2	no	—
195	D₄×C₆₇	—	2	1	no	—
196	D₄×C₆₇	C₄×D₄	1	2	yes	—
197	C₂×C₄×C₆₇	—	2	1	no	—
198	C₂×C₄×C₆₇	—	2	2	no	—
199	C₂×C₄×C₆₇	—	2	2	no	—
200	C₂×C₄×C₆₇	—	2	2	no	—
201	C₂×C₄×C₆₇	—	2	1	no	—
202	C₂×C₄×C₆₇	—	2	1	no	—
203	C₂×C₄×C₆₇	—	4	1	no	—
204	C₂×C₄×C₆₇	—	4	2	no	—
205	C₂×C₄×C₆₇	—	8	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	C₄×D₄	1	1	yes	characteristic
209	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
210	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
211	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
212	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
213	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
214	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
215	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
216	C₂²⋊D₆₇	—	4	2	no	—
217	C₂²⋊D₆₇	—	4	2	no	—
218	C₂×Dic₆₇	—	2	2	no	—
219	C₂×Dic₆₇	—	2	2	no	—
220	C₂×Dic₆₇	—	2	2	no	—
221	C₂×Dic₆₇	—	2	2	no	—
222	C₂×Dic₆₇	—	2	2	no	—
223	C₂×Dic₆₇	—	4	1	no	—
224	C₂×Dic₆₇	—	4	1	no	—
225	D₂₆₈	—	2	2	no	—
226	D₂₆₈	—	2	2	no	—
227	D₂₆₈	—	4	2	no	—
228	D₂₆₈	—	8	1	no	—
229	C₄×D₆₇	—	4	2	no	—
230	C₄×D₆₇	—	4	2	no	—
231	Dic₁₃₄	—	2	2	no	—
232	Dic₁₃₄	—	2	2	no	—
233	C₈×C₆₇	—	2	2	no	—
234	C₈×C₆₇	—	2	2	no	—
235	C₈×C₆₇	—	4	2	no	—
236	C₆₇⋊C₈	—	2	2	no	—
237	C₆₇⋊C₈	—	2	2	no	—
238	C₆₇⋊C₈	—	2	2	no	—
239	C₆₇⋊C₈	—	4	1	no	—
240	C₂²×C₆₇	—	2	1	no	—
241	C₂²×C₆₇	—	2	1	no	—
242	C₂²×C₆₇	—	2	2	no	—
243	C₂²×C₆₇	—	4	1	no	—
244	C₂²×C₆₇	—	4	1	no	—
245	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
246	C₂²×C₆₇	#547	1	1	yes	characteristic
247	C₂²×C₆₇	#553	1	1	yes	characteristic
248	C₂×D₆₇	—	4	2	no	—
249	C₂×D₆₇	—	4	2	no	—
250	C₂×D₆₇	—	4	2	no	—
251	C₄×C₆₇	—	2	2	no	—
252	C₄×C₆₇	—	2	1	no	—
253	C₄×C₆₇	—	2	1	no	—
254	C₄×C₆₇	—	2	1	no	—
255	C₄×C₆₇	—	4	1	no	—
256	C₄×C₆₇	—	4	1	no	—
257	C₄×C₆₇	#443	1	2	yes	complemented
258	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
259	C₄×C₆₇	#520	1	1	yes	characteristic
260	C₄×C₆₇	D₄²	1	1	yes	characteristic
261	C₄×C₆₇	#548	1	1	yes	characteristic
262	Dic₆₇	—	2	4	no	—
263	Dic₆₇	—	4	2	no	—
264	Dic₆₇	—	4	2	no	—
265	#23015	—	67	1	no	maximal, Sylow
266	C₂×C₆₇	—	2	1	no	—
267	C₂×C₆₇	—	2	1	no	—
268	C₂×C₆₇	—	4	2	no	—
269	C₂×C₆₇	#2361	1	1	yes	characteristic
270	C₂×C₆₇	C₂×#443	1	1	yes	characteristic
271	C₂×C₆₇	#2956	1	1	yes	characteristic
272	D₆₇	—	8	2	no	—
273	#3035	—	67	1	no	—
274	C₂×#443	—	67	2	no	—
275	#2935	—	67	1	no	—
276	#2906	D₆₇	1	1	yes	characteristic, complemented
277	#1962	—	67	1	no	—
278	#1950	—	67	1	no	—
279	#1937	—	67	2	no	—
280	#1876	—	67	2	no	—
281	#1866	—	67	2	no	—
282	C₄×C₄⋊C₈	—	67	1	no	—
283	C₄×#104	—	67	1	no	—
284	C₆₇	#23015	1	1	yes	characteristic, complemented, Sylow
285	C₂×#135	—	67	2	no	—
286	C₂×#131	—	67	2	no	—
287	C₂×#131	C₂×D₆₇	1	2	yes	—
288	#532	—	67	1	no	—
289	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
290	#464	—	67	2	no	—
291	#463	—	67	2	no	—
292	#462	—	67	1	no	—
293	#461	—	67	1	no	—
294	#460	—	67	1	no	—
295	#459	—	67	1	no	—
296	#443	—	134	8	no	—
297	#436	—	134	1	no	—
298	#435	—	134	2	no	—
299	C₈×D₄	—	134	1	no	—
300	C₂×#108	—	67	2	no	—
301	#424	—	2	2	no	—
302	#424	C₂×D₆₇	1	2	yes	—
303	#423	—	2	1	no	—
304	#423	C₂×D₆₇	1	1	yes	characteristic
305	C₂×C₄⋊C₈	—	2	1	no	—
306	C₂×C₄⋊C₈	—	67	1	no	—
307	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
308	C₂×#104	—	67	2	no	—
309	C₂×#104	—	67	1	no	—
310	C₂×#104	—	67	1	no	—
311	C₂×#104	—	67	2	no	—
312	#416	—	67	2	no	—
313	#408	—	67	2	no	—
314	C₄×M₁₆	—	67	1	no	—
315	C₂×C₄×C₈	—	67	2	no	—
316	C₄³	—	67	1	no	—
317	#341	—	67	2	no	—
318	#339	—	67	1	no	—
319	#339	—	67	1	no	—
320	#336	—	67	2	no	—
321	C₂×#41	—	67	2	no	—
322	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
323	#135	—	134	4	no	—
324	#135	—	134	4	no	—
325	C₂×Q₁₆	—	67	2	no	—
326	C₂×SD₁₆	—	67	2	no	—
327	C₂×SD₁₆	—	67	2	no	—
328	C₂×D₈	—	67	2	no	—
329	#131	—	2	4	no	—
330	#131	—	4	2	no	—
331	#131	—	268	2	no	—
332	#131	—	134	4	no	—
333	C₂×M₁₆	—	2	2	no	—
334	C₂×M₁₆	—	67	1	no	—
335	C₂×M₁₆	—	67	1	no	—
336	C₂×M₁₆	—	134	2	no	—
337	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
338	C₂²×C₈	—	2	2	no	—
339	C₂²×C₈	—	67	2	no	—
340	C₂²×C₈	—	134	2	no	—
341	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
342	C₄⋊Q₈	—	67	1	no	—
343	#127	—	67	1	no	—
344	#124	—	67	2	no	—
345	#121	—	134	1	no	—
346	#121	—	134	1	no	—
347	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
348	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
349	C₄×D₄	—	2	1	no	—
350	C₄×D₄	—	2	1	no	—
351	C₄×D₄	—	134	1	no	—
352	C₄×D₄	—	134	1	no	—
353	C₄×D₄	C₄×D₆₇	1	1	yes	characteristic
354	C₄×D₄	C₄×D₆₇	1	1	yes	characteristic
355	#117	—	2	1	no	—
356	#117	C₂²×D₆₇	1	1	yes	characteristic
357	C₂×C₄²	—	2	1	no	—
358	C₂×C₄²	—	67	1	no	—
359	C₂×C₄²	—	67	1	no	—
360	C₂×C₄²	—	134	1	no	—
361	C₂×C₄²	—	134	1	no	—
362	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
363	#108	—	134	4	no	—
364	C₄⋊C₈	—	2	2	no	—
365	C₄⋊C₈	—	2	2	no	—
366	C₄⋊C₈	—	2	1	no	—
367	C₄⋊C₈	—	2	1	no	—
368	C₄⋊C₈	—	67	1	no	—
369	C₄⋊C₈	—	67	1	no	—
370	C₄⋊C₈	—	67	2	no	—
371	C₄⋊C₈	C₂²⋊D₆₇	1	1	yes	characteristic
372	C₄⋊C₈	C₂²⋊D₆₇	1	1	yes	characteristic
373	C₄⋊C₈	C₂²⋊D₆₇	1	2	yes	—
374	#104	—	268	2	no	—
375	#104	—	67	4	no	—
376	#104	—	134	4	no	—
377	#104	—	134	2	no	—
378	#103	—	67	2	no	—
379	#103	—	67	2	no	—
380	#102	—	67	2	no	—
381	#102	—	67	2	no	—
382	C₂²⋊C₈	—	134	2	no	—
383	C₂²⋊C₈	—	134	2	no	—
384	#97	—	134	1	no	—
385	C₄×C₈	—	134	4	no	—
386	C₄×C₈	—	134	1	no	—
387	#41	—	2	1	no	—
388	#41	—	2	1	no	—
389	#41	—	134	2	no	—
390	#41	—	134	2	no	—
391	#41	—	134	2	no	—
392	#41	—	134	2	no	—
393	#41	D₄×D₆₇	1	2	yes	—
394	#41	#11761629	1	2	yes	—
395	C₂×Q₈	—	67	2	no	—
396	C₂×Q₈	C₂×C₄×D₆₇	1	1	yes	characteristic
397	C₂×D₄	—	2	1	no	—
398	C₂×D₄	—	67	2	no	—
399	C₂×D₄	—	134	2	no	—
400	C₂×D₄	—	134	1	no	—
401	C₂×D₄	—	134	1	no	—
402	C₂×D₄	C₂×C₄×D₆₇	1	1	yes	characteristic
403	C₂²×C₄	—	2	1	no	—
404	C₂²×C₄	—	2	1	no	—
405	C₂²×C₄	—	2	1	no	—
406	C₂²×C₄	—	134	1	no	—
407	C₂²×C₄	—	134	2	no	—
408	C₂²×C₄	—	134	1	no	—
409	C₂²×C₄	#11761628	1	1	yes	characteristic
410	C₂²×C₄	#11761628	1	1	yes	characteristic
411	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
412	Q₁₆	—	134	4	no	—
413	SD₁₆	—	134	4	no	—
414	SD₁₆	—	134	4	no	—
415	D₈	—	134	4	no	—
416	M₁₆	—	2	2	no	—
417	M₁₆	—	2	2	no	—
418	M₁₆	—	4	4	no	—
419	M₁₆	—	268	4	no	—
420	M₁₆	—	67	2	no	—
421	M₁₆	—	134	1	no	—
422	M₁₆	—	134	1	no	—
423	M₁₆	—	134	1	no	—
424	C₂×C₈	—	2	2	no	—
425	C₂×C₈	—	2	2	no	—
426	C₂×C₈	—	2	2	no	—
427	C₂×C₈	—	4	4	no	—
428	C₂×C₈	—	268	4	no	—
429	C₂×C₈	—	67	2	no	—
430	C₂×C₈	—	67	2	no	—
431	C₂×C₈	—	134	1	no	—
432	C₂×C₈	—	134	2	no	—
433	C₂×C₈	—	134	1	no	—
434	C₂×C₈	—	134	2	no	—
435	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
436	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
437	C₄⋊C₄	—	2	1	no	—
438	C₄⋊C₄	—	2	1	no	—
439	C₄⋊C₄	—	134	1	no	—
440	C₄⋊C₄	—	134	1	no	—
441	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
442	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
443	C₂²⋊C₄	—	2	1	no	—
444	C₂²⋊C₄	—	2	1	no	—
445	C₂²⋊C₄	—	2	1	no	—
446	C₂²⋊C₄	—	134	2	no	—
447	C₂²⋊C₄	—	134	2	no	—
448	C₄²	—	2	1	no	—
449	C₄²	—	2	1	no	—
450	C₄²	—	2	1	no	—
451	C₄²	—	268	2	no	—
452	C₄²	—	67	2	no	—
453	C₄²	—	134	4	no	—
454	C₄²	—	134	1	no	—
455	C₄²	—	134	1	no	—
456	C₄²	—	134	1	no	—
457	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
458	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
459	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
460	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
461	C₂³	—	2	1	no	—
462	C₂³	—	134	2	no	—
463	C₂³	C₂×#11761628	1	1	yes	characteristic
464	Q₈	—	2	1	no	—
465	Q₈	—	134	2	no	—
466	Q₈	—	134	2	no	—
467	Q₈	#14524005	1	2	yes	—
468	D₄	—	2	2	no	—
469	D₄	—	2	2	no	—
470	D₄	—	2	1	no	—
471	D₄	—	268	2	no	—
472	D₄	—	268	2	no	—
473	D₄	—	268	2	no	—
474	D₄	—	536	1	no	—
475	D₄	—	134	2	no	—
476	D₄	—	134	2	no	—
477	D₄	#14524005	1	2	yes	—
478	C₂×C₄	—	2	1	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	1	no	—
483	C₂×C₄	—	2	1	no	—
484	C₂×C₄	—	4	1	no	—
485	C₂×C₄	—	4	2	no	—
486	C₂×C₄	—	8	1	no	—
487	C₂×C₄	—	268	1	no	—
488	C₂×C₄	—	268	1	no	—
489	C₂×C₄	—	268	2	no	—
490	C₂×C₄	—	268	2	no	—
491	C₂×C₄	—	134	2	no	—
492	C₂×C₄	—	134	2	no	—
493	C₂×C₄	—	134	2	no	—
494	C₂×C₄	—	134	2	no	—
495	C₂×C₄	—	134	2	no	—
496	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
497	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
498	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
499	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
500	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
501	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
502	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
503	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
504	C₂×C₄	#14524088	1	1	yes	characteristic
505	C₂×C₄	#14524091	1	1	yes	characteristic
506	C₈	—	2	2	no	—
507	C₈	—	2	2	no	—
508	C₈	—	4	2	no	—
509	C₈	—	268	1	no	—
510	C₈	—	134	2	no	—
511	C₈	—	134	2	no	—
512	C₈	—	134	2	no	—
513	C₂²	—	2	1	no	—
514	C₂²	—	2	1	no	—
515	C₂²	—	2	2	no	—
516	C₂²	—	4	1	no	—
517	C₂²	—	4	1	no	—
518	C₂²	—	268	2	no	—
519	C₂²	—	268	2	no	—
520	C₂²	—	268	2	no	—
521	C₂²	C₂×C₄×C₂²⋊D₆₇	1	1	yes	characteristic
522	C₂²	#16864047	1	1	yes	characteristic
523	C₂²	#16864094	1	1	yes	characteristic
524	C₄	—	2	1	no	—
525	C₄	—	2	2	no	—
526	C₄	—	2	1	no	—
527	C₄	—	2	1	no	—
528	C₄	—	4	1	no	—
529	C₄	—	4	1	no	—
530	C₄	—	268	2	no	—
531	C₄	—	268	2	no	—
532	C₄	—	134	4	no	—
533	C₄	#16863511	1	2	yes	complemented
534	C₄	C₂×#14524005	1	1	yes	characteristic
535	C₄	#16863877	1	1	yes	characteristic
536	C₄	D₄×C₂²⋊D₆₇	1	1	yes	characteristic
537	C₄	#16864057	1	1	yes	characteristic
538	C₂	—	2	1	no	—
539	C₂	—	2	1	no	—
540	C₂	—	4	2	no	—
541	C₂	—	536	2	no	—
542	C₂	#22563225	1	1	yes	characteristic
543	C₂	C₂×#16863511	1	1	yes	characteristic
544	C₂	#22569116	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#2906	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#22569664	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#2906	D₆₇
hypocenter	C₆₇	#23015
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈ ◁ #2906

Complements

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

N	H ≅ G/N	type
2. C₂×#16863511	C₂	semidirect
3. #22568952	C₂	semidirect
4. #22561036	C₂	semidirect
5. #22560982	C₂	semidirect
6. #22560912	C₂	semidirect
7. #22560536	C₂	semidirect
8. #22560483	C₂	semidirect
9. C₄⋊C₈×Dic₆₇	C₂	semidirect
10. C₄×#14523970	C₂	semidirect
11. C₆₇×#2906	C₂	semidirect
15. #16863618	C₄	semidirect
16. #16863611	C₄	semidirect
17. #16863606	C₄	semidirect
18. #16863601	C₄	semidirect
19. #16863596	C₄	semidirect
20. #16863591	C₄	semidirect
78. #14523970	D₄	semidirect
257. C₄×C₆₇	#443	semidirect
276. #2906	D₆₇	semidirect
284. C₆₇	#23015	semidirect
533. C₄	#16863511	semidirect

Maximal subgroups

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

subgroup	index
#22569664	2
C₂×#16863511	2
#22568952	2
#22561036	2
#22560982	2
#22560912	2
#22560536	2
#22560483	2
C₄⋊C₈×Dic₆₇	2
C₄×#14523970	2
C₆₇×#2906	2
#23015	67