Subgroups — #31736468

Order 17152.

1818 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22574010	C₂	1	2	yes	maximal, complemented
2	#22573169	C₂	1	4	yes	maximal, complemented
3	#22573110	C₂	1	4	yes	maximal, complemented
4	#22573077	C₂	1	4	yes	maximal, complemented
5	#22573077	C₂	1	2	yes	maximal, complemented
6	Q₈×D₅₃₆	C₂	1	2	yes	maximal, complemented
7	#22570748	C₂	1	2	yes	maximal, complemented
8	#22570723	C₂	1	1	yes	maximal, characteristic, complemented
9	#22570041	C₂	1	4	yes	maximal, complemented
10	#22570008	C₂	1	2	yes	maximal, complemented
11	#22569062	C₂	1	2	yes	maximal
12	C₂×C₄×D₅₃₆	C₂	1	1	yes	maximal, characteristic
13	C₆₇×#2960	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²×D₅₃₆	C₂²	1	1	yes	characteristic
15	#16864107	—	2	2	no	—
16	#16864106	C₂²	1	4	yes	complemented
17	#16864106	C₂²	1	8	yes	complemented
18	#16864102	—	2	4	no	—
19	Q₈×D₂₆₈	C₂²	1	4	yes	complemented
20	#16864061	—	2	2	no	—
21	#16864024	C₂²	1	4	yes	—
22	#16864021	—	2	4	no	—
23	#16863953	C₂²	1	2	yes	—
24	C₂²⋊Q₈×D₆₇	—	2	4	no	—
25	#16863870	C₂²	1	4	yes	—
26	C₂×C₄×D₂₆₈	C₂²	1	2	yes	—
27	#16863805	C₂²	1	8	yes	complemented
28	#16863801	C₂²	1	1	yes	characteristic, complemented
29	#16863801	C₂²	1	4	yes	complemented
30	#16863801	C₂²	1	1	yes	characteristic, complemented
31	#16863798	C₂²	1	2	yes	complemented
32	#16863798	C₂²	1	4	yes	complemented
33	#16863798	C₂²	1	2	yes	complemented
34	#16863794	C₂²	1	2	yes	complemented
35	#16863697	C₂²	1	8	yes	complemented
36	#16863697	C₂²	1	8	yes	complemented
37	#16863697	C₂²	1	8	yes	complemented
38	#16863687	C₂²	1	4	yes	complemented
39	#16863687	C₂²	1	8	yes	complemented
40	#16863677	C₂²	1	4	yes	complemented
41	#16863677	C₂²	1	8	yes	complemented
42	#16863509	C₂²	1	8	yes	—
43	C₄×D₅₃₆	C₂²	1	2	yes	—
44	C₄×D₅₃₆	C₂²	1	2	yes	—
45	C₄×D₅₃₆	C₂²	1	2	yes	—
46	C₄×D₅₃₆	C₂²	1	2	yes	—
47	C₄×D₅₃₆	C₂²	1	4	yes	—
48	C₄×D₅₃₆	C₂²	1	4	yes	—
49	#16863438	C₂²	1	4	yes	—
50	#16863434	C₂²	1	2	yes	—
51	#16863434	C₂²	1	2	yes	—
52	C₂×#14523977	C₂²	1	1	yes	characteristic
53	#16863402	C₂²	1	4	yes	—
54	C₂×#14523965	C₂²	1	2	yes	—
55	#533×C₆₇	C₂²	1	1	yes	characteristic
56	#446×C₆₇	C₂²	1	2	yes	complemented
57	#446×C₆₇	C₂²	1	2	yes	complemented
58	C₈×Q₈×C₆₇	C₂²	1	2	yes	complemented
59	C₈×Q₈×C₆₇	C₂²	1	2	yes	complemented
60	#424×C₆₇	C₂²	1	2	yes	—
61	#423×C₆₇	C₂²	1	1	yes	characteristic
62	#405×C₆₇	C₂²	1	2	yes	—
63	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
64	C₂×Q₈×D₆₇	—	2	4	no	—
65	C₂²×D₂₆₈	C₂³	1	2	yes	—
66	C₂²×C₄×D₆₇	—	2	2	no	—
67	C₂×D₅₃₆	Q₈	1	8	yes	complemented
68	C₂×D₅₃₆	C₂³	1	2	yes	—
69	C₂×D₅₃₆	C₂³	1	2	yes	—
70	#14524043	C₂³	1	8	yes	—
71	#14524043	C₂³	1	2	yes	—
72	#14524043	C₂³	1	2	yes	—
73	#14524042	—	2	4	no	—
74	#14524035	C₂³	1	8	yes	—
75	#14524035	C₂³	1	4	yes	—
76	#14524034	—	2	4	no	—
77	#14524023	—	2	8	no	—
78	#14524023	—	2	4	no	—
79	#14524022	—	2	2	no	—
80	#14524022	—	2	2	no	—
81	#14524021	—	2	8	no	—
82	#14524021	—	2	8	no	—
83	#14524021	—	2	8	no	—
84	#14524020	—	2	8	no	—
85	#14524019	—	2	8	no	—
86	#14524019	—	2	4	no	—
87	C₄×Dic₁₃₄	—	2	2	no	—
88	C₄×Dic₁₃₄	—	2	2	no	—
89	C₄×D₂₆₈	—	2	2	no	—
90	C₄×D₂₆₈	—	2	2	no	—
91	C₄×D₂₆₈	C₂³	1	8	yes	—
92	C₄×D₂₆₈	C₂³	1	8	yes	—
93	C₄×D₂₆₈	C₂³	1	4	yes	—
94	C₄×D₂₆₈	C₂³	1	4	yes	—
95	#14524005	—	2	4	no	—
96	#14524005	—	2	4	no	—
97	#14524002	C₂³	1	4	yes	—
98	C₂×C₄⋊Dic₆₇	C₂³	1	2	yes	—
99	C₄⋊C₄×D₆₇	—	2	8	no	—
100	C₄⋊C₄×D₆₇	—	2	4	no	—
101	C₂×(C₂×D₆₇)⋊C₄	—	2	2	no	—
102	C₂²⋊C₄×D₆₇	—	2	8	no	—
103	#14523977	C₂³	1	2	yes	—
104	#14523977	C₂³	1	2	yes	—
105	#14523977	C₂³	1	2	yes	—
106	#14523977	C₂³	1	2	yes	—
107	#14523977	C₂³	1	2	yes	—
108	#14523977	C₂³	1	8	yes	—
109	#14523977	C₂³	1	2	yes	—
110	#14523975	C₂³	1	4	yes	—
111	#14523975	C₂³	1	4	yes	—
112	#14523975	C₂³	1	8	yes	—
113	#14523965	C₂³	1	8	yes	—
114	#14523965	C₂³	1	8	yes	—
115	#14523965	C₂³	1	4	yes	—
116	#14523965	C₂³	1	4	yes	—
117	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
118	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
119	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
120	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
121	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
122	#125×C₆₇	D₄	1	2	yes	—
123	C₂²⋊Q₈×C₆₇	D₄	1	2	yes	—
124	C₂²⋊Q₈×C₆₇	D₄	1	2	yes	—
125	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
126	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
127	#117×C₆₇	C₂³	1	2	yes	—
128	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
129	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
130	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
131	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
132	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
133	#97×C₆₇	C₂³	1	4	yes	—
134	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
135	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
136	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
137	C₄×C₈×C₆₇	C₂³	1	4	yes	—
138	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
139	C₂³×D₆₇	—	2	2	no	—
140	C₂×Dic₁₃₄	—	2	2	no	—
141	C₂×Dic₁₃₄	—	2	2	no	—
142	C₂×Dic₁₃₄	—	2	8	no	—
143	Q₈×D₆₇	—	8	8	no	—
144	C₂×D₂₆₈	—	2	2	no	—
145	C₂×D₂₆₈	—	2	4	no	—
146	C₂×D₂₆₈	C₂×Q₈	1	8	yes	complemented
147	C₂×D₂₆₈	C₂⁴	1	4	yes	—
148	C₂²×Dic₆₇	—	2	2	no	—
149	C₂×C₄×D₆₇	—	2	4	no	—
150	C₂×C₄×D₆₇	—	2	8	no	—
151	C₂×C₄×D₆₇	—	2	8	no	—
152	C₂×C₄×D₆₇	—	2	4	no	—
153	C₂×C₄×D₆₇	—	8	4	no	—
154	D₅₃₆	C₂×Q₈	1	8	yes	complemented
155	D₅₃₆	C₂×Q₈	1	8	yes	complemented
156	C₄⋊Dic₆₇	—	2	2	no	—
157	C₄⋊Dic₆₇	—	2	8	no	—
158	C₄⋊Dic₆₇	—	2	2	no	—
159	C₄⋊Dic₆₇	—	2	2	no	—
160	C₄⋊Dic₆₇	C₂⁴	1	4	yes	—
161	C₄⋊Dic₆₇	C₂⁴	1	4	yes	—
162	C₄⋊Dic₆₇	C₂⁴	1	4	yes	—
163	Dic₆₇⋊C₄	—	2	4	no	—
164	Dic₆₇⋊C₄	—	2	8	no	—
165	Dic₆₇⋊C₄	—	2	8	no	—
166	Dic₆₇⋊C₄	—	2	4	no	—
167	C₂²⋊Dic₆₇	—	2	4	no	—
168	(C₂×D₆₇)⋊C₄	—	2	8	no	—
169	(C₂×D₆₇)⋊C₄	—	2	8	no	—
170	(C₂×D₆₇)⋊C₄	—	2	4	no	—
171	(C₂×D₆₇)⋊C₄	—	2	4	no	—
172	C₄×Dic₆₇	—	2	4	no	—
173	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
174	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
175	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
178	M₁₆×C₆₇	—	2	4	no	—
179	C₂×C₈×C₆₇	—	2	2	no	—
180	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	complemented
181	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
182	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
183	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
184	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
185	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
186	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
187	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
188	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
189	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
190	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
191	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
192	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
193	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
194	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
195	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
196	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
197	C₄²×C₆₇	C₂⁴	1	4	yes	—
198	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
199	C₂²×D₆₇	—	2	4	no	—
200	C₂²×D₆₇	—	2	8	no	—
201	C₂²×D₆₇	—	8	2	no	—
202	C₂²×D₆₇	—	8	2	no	—
203	Q₈×C₆₇	—	2	4	no	—
204	Q₈×C₆₇	—	4	2	no	—
205	C₂×C₄×C₆₇	—	2	2	no	—
206	C₂×C₄×C₆₇	—	4	1	no	—
207	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
208	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
209	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
210	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
211	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
212	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
213	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
215	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
216	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
217	C₂×Dic₆₇	—	2	4	no	—
218	C₂×Dic₆₇	—	2	4	no	—
219	C₂×Dic₆₇	—	2	4	no	—
220	C₂×Dic₆₇	—	8	2	no	—
221	D₂₆₈	—	2	8	no	—
222	D₂₆₈	—	2	4	no	—
223	D₂₆₈	C₂²×Q₈	1	8	yes	—
224	C₄×D₆₇	—	4	8	no	—
225	C₄×D₆₇	—	8	8	no	—
226	C₄×D₆₇	—	8	4	no	—
227	C₄×D₆₇	—	8	8	no	—
228	Dic₁₃₄	—	4	4	no	—
229	Dic₁₃₄	—	8	8	no	—
230	Dic₁₃₄	—	8	2	no	—
231	C₈×C₆₇	—	2	1	no	—
232	C₈×C₆₇	—	2	1	no	—
233	C₈×C₆₇	—	2	4	no	—
234	C₈×C₆₇	C₂²×Q₈	1	2	yes	—
235	C₈×C₆₇	C₂²×Q₈	1	2	yes	—
236	C₂²×C₆₇	—	2	1	no	—
237	C₂²×C₆₇	—	2	1	no	—
238	C₂²×C₆₇	D₄×Q₈	1	2	yes	—
239	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
240	C₂×D₆₇	—	2	8	no	—
241	C₂×D₆₇	—	4	4	no	—
242	C₂×D₆₇	—	4	4	no	—
243	C₂×D₆₇	—	8	8	no	—
244	C₄×C₆₇	—	2	1	no	—
245	C₄×C₆₇	—	2	4	no	—
246	C₄×C₆₇	—	2	1	no	—
247	C₄×C₆₇	—	2	4	no	—
248	C₄×C₆₇	—	2	1	no	—
249	C₄×C₆₇	D₄×Q₈	1	2	yes	—
250	C₄×C₆₇	#576	1	2	yes	—
251	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
252	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
253	Dic₆₇	—	4	4	no	—
254	Dic₆₇	—	4	4	no	—
255	Dic₆₇	—	4	4	no	—
256	#61370	—	67	1	no	maximal, Sylow
257	C₂×C₆₇	—	2	2	no	—
258	C₂×C₆₇	C₂×D₄×Q₈	1	1	yes	characteristic
259	C₂×C₆₇	C₂×#576	1	1	yes	characteristic
260	C₂×C₆₇	#3584	1	1	yes	characteristic
261	D₆₇	—	4	8	no	—
262	#3501	—	67	2	no	—
263	#3389	—	67	4	no	—
264	#3384	—	67	4	no	—
265	#3380	—	67	2	no	—
266	#3380	—	67	4	no	—
267	Q₈×D₈	—	67	2	no	—
268	#3164	—	67	2	no	—
269	#3160	—	67	1	no	—
270	#3081	—	67	4	no	—
271	#3077	—	67	2	no	—
272	#2960	D₆₇	1	1	yes	characteristic, complemented
273	#2948	—	67	2	no	—
274	C₂×C₄×D₈	—	67	1	no	—
275	C₆₇	#61370	1	1	yes	characteristic, complemented, Sylow
276	C₂²×D₈	—	67	1	no	—
277	#554	—	67	4	no	—
278	#554	—	67	8	no	—
279	#554	—	134	2	no	—
280	#554	—	134	4	no	—
281	D₄×Q₈	—	67	4	no	—
282	D₄×Q₈	—	134	2	no	—
283	#544	—	67	4	no	—
284	#543	—	134	4	no	—
285	#533	—	67	2	no	—
286	#533	C₂×D₆₇	1	1	yes	characteristic
287	C₂×C₂²⋊Q₈	—	134	4	no	—
288	#518	—	67	4	no	—
289	C₂×C₄×D₄	—	67	2	no	—
290	#501	—	67	8	no	—
291	#500	—	67	1	no	—
292	#500	—	67	4	no	—
293	#500	—	67	1	no	—
294	#499	—	67	2	no	—
295	#499	—	67	4	no	—
296	#499	—	67	2	no	—
297	#498	—	67	2	no	—
298	#478	—	67	8	no	—
299	#478	—	67	8	no	—
300	#478	—	67	8	no	—
301	#476	—	67	4	no	—
302	#476	—	67	8	no	—
303	#474	—	67	4	no	—
304	#474	—	67	8	no	—
305	#446	C₂×D₆₇	1	2	yes	complemented
306	#446	C₂×D₆₇	1	2	yes	complemented
307	C₈×Q₈	C₂×D₆₇	1	2	yes	complemented
308	C₈×Q₈	C₂×D₆₇	1	2	yes	complemented
309	#442	—	67	8	no	—
310	C₄×D₈	—	67	2	no	—
311	C₄×D₈	—	67	4	no	—
312	C₄×D₈	—	67	2	no	—
313	C₄×D₈	—	67	4	no	—
314	C₄×D₈	—	67	2	no	—
315	C₄×D₈	—	67	2	no	—
316	#428	—	67	4	no	—
317	#427	—	67	2	no	—
318	#427	—	67	2	no	—
319	C₂×#107	—	67	1	no	—
320	#424	C₂×D₆₇	1	2	yes	—
321	#423	C₂×D₆₇	1	1	yes	characteristic
322	#418	—	67	4	no	—
323	C₂×#102	—	67	2	no	—
324	#405	C₂×D₆₇	1	2	yes	—
325	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
326	C₂²×Q₈	—	134	4	no	—
327	C₂²×D₄	—	67	2	no	—
328	C₂³×C₄	—	134	2	no	—
329	C₂×D₈	—	67	8	no	—
330	C₂×D₈	—	67	2	no	—
331	C₂×D₈	—	67	2	no	—
332	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
333	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
334	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
335	C₄⋊Q₈	—	67	2	no	—
336	C₄⋊Q₈	—	67	2	no	—
337	C₄⋊Q₈	—	67	8	no	—
338	C₄⋊Q₈	—	134	4	no	—
339	C₄⋊Q₈	D₂₆₈	1	1	yes	characteristic
340	C₄⋊Q₈	D₂₆₈	1	1	yes	characteristic
341	#125	—	67	4	no	—
342	#125	—	67	8	no	—
343	#125	—	134	4	no	—
344	#125	D₂₆₈	1	2	yes	—
345	C₂²⋊Q₈	—	134	4	no	—
346	C₂²⋊Q₈	—	134	8	no	—
347	C₂²⋊Q₈	—	134	2	no	—
348	C₂²⋊Q₈	—	134	2	no	—
349	C₂²⋊Q₈	—	134	4	no	—
350	C₂²⋊Q₈	—	134	8	no	—
351	C₂²⋊Q₈	—	134	8	no	—
352	C₂²⋊Q₈	—	134	8	no	—
353	C₂²⋊Q₈	—	134	8	no	—
354	C₂²⋊Q₈	—	134	8	no	—
355	C₂²⋊Q₈	D₂₆₈	1	2	yes	—
356	C₂²⋊Q₈	D₂₆₈	1	2	yes	—
357	C₄×Q₈	—	134	2	no	—
358	C₄×Q₈	—	134	2	no	—
359	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
360	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
361	C₄×D₄	—	67	4	no	—
362	C₄×D₄	—	67	4	no	—
363	C₄×D₄	—	67	8	no	—
364	C₄×D₄	—	67	8	no	—
365	C₄×D₄	—	134	4	no	—
366	C₄×D₄	—	134	4	no	—
367	C₄×D₄	—	134	2	no	—
368	C₄×D₄	—	134	2	no	—
369	#117	—	67	4	no	—
370	#117	C₂²×D₆₇	1	2	yes	—
371	C₂×C₄⋊C₄	—	67	2	no	—
372	C₂×C₄⋊C₄	—	134	8	no	—
373	C₂×C₄⋊C₄	—	134	4	no	—
374	C₂×C₂²⋊C₄	—	134	2	no	—
375	C₂×C₂²⋊C₄	—	134	8	no	—
376	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
377	#107	—	67	2	no	—
378	#107	—	67	2	no	—
379	#107	—	67	2	no	—
380	#107	—	67	2	no	—
381	#107	—	67	2	no	—
382	#107	—	67	2	no	—
383	#107	—	67	8	no	—
384	#106	—	67	4	no	—
385	#106	—	67	4	no	—
386	#106	—	67	8	no	—
387	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
388	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
389	C₄⋊C₈	C₂²×D₆₇	1	4	yes	—
390	C₄⋊C₈	C₂²×D₆₇	1	4	yes	—
391	#102	—	67	4	no	—
392	#102	—	67	4	no	—
393	#102	—	67	8	no	—
394	#102	—	67	8	no	—
395	#97	C₂²×D₆₇	1	4	yes	—
396	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
397	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
398	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
399	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
400	C₄×C₈	C₂²×D₆₇	1	4	yes	—
401	C₂⁴	—	134	2	no	—
402	C₂×Q₈	—	536	8	no	—
403	C₂×Q₈	—	134	2	no	—
404	C₂×Q₈	—	134	2	no	—
405	C₂×Q₈	—	134	8	no	—
406	C₂×Q₈	C₂×D₂₆₈	1	2	yes	—
407	C₂×Q₈	C₂×D₂₆₈	1	2	yes	—
408	C₂×D₄	—	67	8	no	—
409	C₂×D₄	—	67	4	no	—
410	C₂×D₄	—	134	2	no	—
411	C₂×D₄	—	134	4	no	—
412	C₂²×C₄	—	536	4	no	—
413	C₂²×C₄	—	134	4	no	—
414	C₂²×C₄	—	134	2	no	—
415	C₂²×C₄	—	134	8	no	—
416	C₂²×C₄	—	134	8	no	—
417	C₂²×C₄	—	134	4	no	—
418	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
419	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
420	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
421	D₈	—	67	8	no	—
422	D₈	—	67	8	no	—
423	M₁₆	—	2	4	no	—
424	C₂×C₈	—	2	2	no	—
425	C₂×C₈	Q₈×D₆₇	1	4	yes	complemented
426	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
427	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
428	C₂×C₈	C₂³×D₆₇	1	4	yes	—
429	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
430	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
431	C₄⋊C₄	—	67	4	no	—
432	C₄⋊C₄	—	67	4	no	—
433	C₄⋊C₄	—	67	4	no	—
434	C₄⋊C₄	—	134	2	no	—
435	C₄⋊C₄	—	134	4	no	—
436	C₄⋊C₄	—	134	2	no	—
437	C₄⋊C₄	—	134	8	no	—
438	C₄⋊C₄	—	134	8	no	—
439	C₄⋊C₄	—	134	8	no	—
440	C₄⋊C₄	—	134	4	no	—
441	C₄⋊C₄	—	134	2	no	—
442	C₄⋊C₄	C₂×D₂₆₈	1	4	yes	—
443	C₄⋊C₄	C₂×D₂₆₈	1	2	yes	—
444	C₄⋊C₄	C₂×D₂₆₈	1	2	yes	—
445	C₄⋊C₄	C₂×D₂₆₈	1	2	yes	—
446	C₄⋊C₄	C₂×D₂₆₈	1	4	yes	—
447	C₄⋊C₄	C₂×D₂₆₈	1	2	yes	—
448	C₂²⋊C₄	—	134	4	no	—
449	C₂²⋊C₄	—	134	8	no	—
450	C₂²⋊C₄	—	134	8	no	—
451	C₂²⋊C₄	—	134	4	no	—
452	C₂²⋊C₄	—	134	4	no	—
453	C₂²⋊C₄	C₂×D₂₆₈	1	4	yes	—
454	C₄²	—	134	4	no	—
455	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
456	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
457	C₄²	C₂³×D₆₇	1	1	yes	characteristic
458	C₄²	C₂³×D₆₇	1	1	yes	characteristic
459	C₄²	C₂³×D₆₇	1	4	yes	—
460	C₂³	—	536	2	no	—
461	C₂³	—	536	2	no	—
462	C₂³	—	134	4	no	—
463	C₂³	—	134	8	no	—
464	C₂³	C₂²×D₂₆₈	1	1	yes	characteristic
465	Q₈	—	2	4	no	—
466	Q₈	—	4	2	no	—
467	Q₈	—	268	4	no	—
468	Q₈	—	536	8	no	—
469	Q₈	—	536	2	no	—
470	D₄	—	67	8	no	—
471	D₄	—	134	8	no	—
472	D₄	—	134	4	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	4	1	no	—
475	C₂×C₄	—	268	8	no	—
476	C₂×C₄	—	536	4	no	—
477	C₂×C₄	—	536	8	no	—
478	C₂×C₄	—	536	8	no	—
479	C₂×C₄	—	536	2	no	—
480	C₂×C₄	—	134	4	no	—
481	C₂×C₄	—	134	4	no	—
482	C₂×C₄	—	134	4	no	—
483	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
484	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
485	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
486	C₂×C₄	C₂²×D₂₆₈	1	4	yes	—
487	C₂×C₄	C₂²×D₂₆₈	1	4	yes	—
488	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
489	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
490	C₂×C₄	C₂×Q₈×D₆₇	1	2	yes	—
491	C₂×C₄	#14524090	1	2	yes	—
492	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
493	C₈	—	2	1	no	—
494	C₈	—	2	1	no	—
495	C₈	—	2	4	no	—
496	C₈	C₂×Q₈×D₆₇	1	2	yes	—
497	C₈	C₂×Q₈×D₆₇	1	2	yes	—
498	C₂²	—	2	1	no	—
499	C₂²	—	2	1	no	—
500	C₂²	—	268	4	no	—
501	C₂²	—	268	4	no	—
502	C₂²	—	536	8	no	—
503	C₂²	—	134	8	no	—
504	C₂²	Q₈×D₂₆₈	1	2	yes	—
505	C₂²	C₂³×D₂₆₈	1	1	yes	characteristic
506	C₄	—	2	1	no	—
507	C₄	—	2	4	no	—
508	C₄	—	2	1	no	—
509	C₄	—	2	1	no	—
510	C₄	—	2	4	no	—
511	C₄	—	268	4	no	—
512	C₄	—	268	4	no	—
513	C₄	—	268	4	no	—
514	C₄	Q₈×D₂₆₈	1	2	yes	—
515	C₄	#16864201	1	2	yes	—
516	C₄	C₂²×Q₈×D₆₇	1	1	yes	characteristic
517	C₄	C₂×#14524090	1	1	yes	characteristic
518	C₂	—	2	2	no	—
519	C₂	—	268	8	no	—
520	C₂	C₂×Q₈×D₂₆₈	1	1	yes	characteristic
521	C₂	C₂×#16864201	1	1	yes	characteristic
522	C₂	#22574983	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×D₂₆₈
commutator, cosocle	C₂×C₄×C₆₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₆₇×#2960	C₂
socle	C₂²×C₆₇	C₂³×D₄
prime radical	C₂²×D₅₃₆	C₂²
hypercenter	#2960	D₆₇
hypocenter	C₆₇	#61370
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ #533 ◁ #2960

Complements

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

N	H ≅ G/N	type
1. #22574010	C₂	semidirect
2. #22573169	C₂	semidirect
3. #22573110	C₂	semidirect
4. #22573077	C₂	semidirect
5. #22573077	C₂	semidirect
6. Q₈×D₅₃₆	C₂	semidirect
7. #22570748	C₂	semidirect
8. #22570723	C₂	semidirect
9. #22570041	C₂	semidirect
10. #22570008	C₂	semidirect
13. C₆₇×#2960	C₂	semidirect
16. #16864106	C₂²	semidirect
17. #16864106	C₂²	semidirect
19. Q₈×D₂₆₈	C₂²	semidirect
27. #16863805	C₂²	semidirect
28. #16863801	C₂²	semidirect
29. #16863801	C₂²	semidirect
30. #16863801	C₂²	semidirect
31. #16863798	C₂²	semidirect
32. #16863798	C₂²	semidirect
33. #16863798	C₂²	semidirect
34. #16863794	C₂²	semidirect
35. #16863697	C₂²	semidirect
36. #16863697	C₂²	semidirect
37. #16863697	C₂²	semidirect
38. #16863687	C₂²	semidirect
39. #16863687	C₂²	semidirect
40. #16863677	C₂²	semidirect
41. #16863677	C₂²	semidirect
56. #446×C₆₇	C₂²	semidirect
57. #446×C₆₇	C₂²	semidirect
58. C₈×Q₈×C₆₇	C₂²	semidirect
59. C₈×Q₈×C₆₇	C₂²	semidirect
67. C₂×D₅₃₆	Q₈	semidirect
146. C₂×D₂₆₈	C₂×Q₈	semidirect
154. D₅₃₆	C₂×Q₈	semidirect
155. D₅₃₆	C₂×Q₈	semidirect
180. C₂×C₈×C₆₇	C₂×Q₈	semidirect
272. #2960	D₆₇	semidirect
275. C₆₇	#61370	semidirect
305. #446	C₂×D₆₇	semidirect
306. #446	C₂×D₆₇	semidirect
307. C₈×Q₈	C₂×D₆₇	semidirect
308. C₈×Q₈	C₂×D₆₇	semidirect
425. C₂×C₈	Q₈×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22574010	2
#22573169	2
#22573110	2
#22573077	2
Q₈×D₅₃₆	2
#22570748	2
#22570723	2
#22570041	2
#22570008	2
#22569062	2
C₂×C₄×D₅₃₆	2
C₆₇×#2960	2
#61370	67