Subgroups — #31736511

Order 17152.

1778 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22574021	C₂	1	2	yes	maximal, complemented
2	#22573132	C₂	1	4	yes	maximal, complemented
3	#22573121	C₂	1	4	yes	maximal, complemented
4	#22573101	C₂	1	2	yes	maximal, complemented
5	Q₈×Dic₂₆₈	C₂	1	2	yes	maximal, complemented
6	#22573077	C₂	1	2	yes	maximal, complemented
7	Q₈×D₅₃₆	C₂	1	2	yes	maximal, complemented
8	#22570748	C₂	1	2	yes	maximal, complemented
9	C₂×#16863801	C₂	1	1	yes	maximal, characteristic, complemented
10	#22570062	C₂	1	4	yes	maximal, complemented
11	#22570025	C₂	1	2	yes	maximal, complemented
12	#22569086	C₂	1	2	yes	maximal
13	C₄×#14524065	C₂	1	1	yes	maximal, characteristic
14	C₆₇×#2960	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#14524065	C₂²	1	1	yes	characteristic
16	Q₈×Dic₁₃₄	C₂²	1	4	yes	—
17	#16864124	C₂²	1	4	yes	—
18	#16864106	C₂²	1	4	yes	—
19	#16864104	—	2	4	no	—
20	Q₈×D₂₆₈	C₂²	1	4	yes	—
21	#16864059	—	2	4	no	—
22	#16864026	—	2	4	no	—
23	#16864024	C₂²	1	4	yes	—
24	#16863951	—	2	2	no	—
25	C₂×#14524043	C₂²	1	2	yes	—
26	C₄⋊Q₈×D₆₇	—	2	2	no	—
27	#16863876	C₂²	1	4	yes	—
28	C₄×#11761628	C₂²	1	2	yes	—
29	#16863805	C₂²	1	8	yes	—
30	#16863801	C₂²	1	4	yes	—
31	#16863801	C₂²	1	2	yes	—
32	#16863801	C₂²	1	2	yes	—
33	#16863801	C₂²	1	2	yes	—
34	#16863801	C₂²	1	2	yes	—
35	#16863798	C₂²	1	4	yes	—
36	#16863704	C₂²	1	8	yes	—
37	#16863697	C₂²	1	8	yes	—
38	#16863692	C₂²	1	4	yes	—
39	#16863692	C₂²	1	8	yes	—
40	#16863687	C₂²	1	4	yes	—
41	#16863682	C₂²	1	4	yes	—
42	#16863677	C₂²	1	4	yes	—
43	#16863677	C₂²	1	8	yes	—
44	#16863498	C₂²	1	8	yes	—
45	C₄×Dic₂₆₈	C₂²	1	1	yes	characteristic
46	C₄×Dic₂₆₈	C₂²	1	1	yes	characteristic
47	C₄×Dic₂₆₈	C₂²	1	4	yes	—
48	C₄×#11761617	C₂²	1	2	yes	—
49	C₄×#11761617	C₂²	1	2	yes	—
50	C₄×D₅₃₆	C₂²	1	1	yes	characteristic
51	C₄×D₅₃₆	C₂²	1	4	yes	—
52	C₄×D₅₃₆	C₂²	1	1	yes	characteristic
53	#16863438	C₂²	1	4	yes	—
54	#16863434	C₂²	1	1	yes	characteristic
55	C₂×#14523977	C₂²	1	2	yes	—
56	C₂×#14523977	C₂²	1	2	yes	—
57	#16863398	C₂²	1	4	yes	—
58	#16863393	C₂²	1	2	yes	—
59	#533×C₆₇	C₂²	1	1	yes	characteristic
60	#446×C₆₇	C₂²	1	2	yes	—
61	#446×C₆₇	C₂²	1	2	yes	—
62	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
63	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
64	#424×C₆₇	C₂²	1	2	yes	—
65	#423×C₆₇	C₂²	1	1	yes	characteristic
66	#405×C₆₇	C₂²	1	2	yes	—
67	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
68	C₂×#11761628	C₂³	1	2	yes	—
69	C₂²×Dic₁₃₄	—	2	2	no	—
70	C₂×Q₈×D₆₇	—	2	4	no	—
71	#14524065	Q₈	1	8	yes	complemented
72	C₂×Dic₂₆₈	C₂³	1	1	yes	characteristic
73	C₂×#11761617	C₂³	1	2	yes	—
74	C₂×D₅₃₆	C₂³	1	1	yes	characteristic
75	#14524043	—	2	2	no	—
76	#14524043	—	2	2	no	—
77	#14524043	C₂³	1	8	yes	—
78	#14524043	C₂³	1	4	yes	—
79	#14524043	C₂³	1	4	yes	—
80	#14524042	—	2	4	no	—
81	#14524042	—	2	8	no	—
82	#14524042	—	2	4	no	—
83	#14524041	—	2	4	no	—
84	#14524035	C₂³	1	8	yes	—
85	#14524034	—	2	4	no	—
86	#14524033	—	2	8	no	—
87	#14524023	—	2	8	no	—
88	#14524023	—	2	4	no	—
89	#14524021	—	2	8	no	—
90	#14524020	—	2	8	no	—
91	#14524019	—	2	4	no	—
92	Q₈×Dic₆₇	—	2	4	no	—
93	C₄×Dic₁₃₄	C₂³	1	8	yes	—
94	C₄×Dic₁₃₄	C₂³	1	2	yes	—
95	C₄×Dic₁₃₄	C₂³	1	2	yes	—
96	#14524008	—	2	4	no	—
97	C₄×D₂₆₈	C₂³	1	8	yes	—
98	C₄×D₂₆₈	C₂³	1	2	yes	—
99	C₄×D₂₆₈	C₂³	1	2	yes	—
100	C₄×C₂²⋊D₆₇	—	2	2	no	—
101	C₄×C₂²⋊D₆₇	—	2	2	no	—
102	(C₂²⋊D₆₇)⋊C₄	—	2	8	no	—
103	#14524002	C₂³	1	2	yes	—
104	#14524001	—	2	4	no	—
105	#14524000	—	2	2	no	—
106	C₂×C₄⋊Dic₆₇	—	2	2	no	—
107	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
108	C₄⋊C₄×D₆₇	—	2	4	no	—
109	C₄⋊C₄×D₆₇	—	2	4	no	—
110	C₄²×D₆₇	—	2	2	no	—
111	#14523977	C₂³	1	1	yes	characteristic
112	#14523977	C₂³	1	4	yes	—
113	#14523977	C₂³	1	1	yes	characteristic
114	#14523977	C₂³	1	4	yes	—
115	#14523977	C₂³	1	4	yes	—
116	#14523977	C₂³	1	4	yes	—
117	#14523977	C₂³	1	8	yes	—
118	#14523975	C₂³	1	8	yes	—
119	#14523975	C₂³	1	2	yes	—
120	#14523968	C₂³	1	8	yes	—
121	#14523968	C₂³	1	2	yes	—
122	#14523968	C₂³	1	2	yes	—
123	#14523965	C₂³	1	8	yes	—
124	#14523965	C₂³	1	2	yes	—
125	#14523965	C₂³	1	2	yes	—
126	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
127	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
128	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
129	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
130	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
131	#125×C₆₇	D₄	1	2	yes	—
132	C₂²⋊Q₈×C₆₇	D₄	1	2	yes	—
133	C₂²⋊Q₈×C₆₇	D₄	1	2	yes	—
134	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
135	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
136	#117×C₆₇	C₂³	1	2	yes	—
137	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
138	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
139	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
140	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
141	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
142	#97×C₆₇	C₂³	1	4	yes	—
143	C₄×C₈×C₆₇	C₂³	1	4	yes	—
144	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
145	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
146	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
147	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
148	#11761628	—	2	4	no	—
149	#11761628	C₂×Q₈	1	8	yes	complemented
150	C₂×Dic₁₃₄	—	2	4	no	—
151	C₂×Dic₁₃₄	—	2	8	no	—
152	C₂×Dic₁₃₄	—	2	4	no	—
153	C₂×Dic₁₃₄	—	8	4	no	—
154	C₂×Dic₁₃₄	C₂⁴	1	2	yes	—
155	Q₈×D₆₇	—	8	8	no	—
156	C₂×D₂₆₈	C₂⁴	1	2	yes	—
157	C₂×C₂²⋊D₆₇	—	2	2	no	—
158	C₂²×Dic₆₇	—	2	4	no	—
159	C₂×C₄×D₆₇	—	2	2	no	—
160	C₂×C₄×D₆₇	—	2	2	no	—
161	C₂×C₄×D₆₇	—	2	2	no	—
162	C₂×C₄×D₆₇	—	2	8	no	—
163	Dic₂₆₈	C₂×Q₈	1	4	yes	complemented
164	#11761617	C₂×Q₈	1	8	yes	complemented
165	D₅₃₆	C₂×Q₈	1	4	yes	complemented
166	C₄⋊Dic₆₇	—	2	4	no	—
167	C₄⋊Dic₆₇	—	2	4	no	—
168	C₄⋊Dic₆₇	—	2	4	no	—
169	C₄⋊Dic₆₇	—	2	8	no	—
170	C₄⋊Dic₆₇	C₂⁴	1	2	yes	—
171	C₄⋊Dic₆₇	C₂⁴	1	8	yes	—
172	C₄⋊Dic₆₇	C₂⁴	1	2	yes	—
173	Dic₆₇⋊C₄	—	2	8	no	—
174	Dic₆₇⋊C₄	—	2	8	no	—
175	Dic₆₇⋊C₄	—	2	2	no	—
176	Dic₆₇⋊C₄	—	2	2	no	—
177	Dic₆₇⋊C₄	—	2	8	no	—
178	C₂²⋊Dic₆₇	—	2	2	no	—
179	(C₂×D₆₇)⋊C₄	—	2	2	no	—
180	(C₂×D₆₇)⋊C₄	—	2	2	no	—
181	(C₂×D₆₇)⋊C₄	—	2	8	no	—
182	C₄×Dic₆₇	—	2	2	no	—
183	C₄×Dic₆₇	—	2	2	no	—
184	C₄×Dic₆₇	—	2	2	no	—
185	C₄×Dic₆₇	—	2	8	no	—
186	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
187	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
188	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
189	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
190	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
191	M₁₆×C₆₇	—	2	4	no	—
192	C₂×C₈×C₆₇	—	2	2	no	—
193	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	complemented
194	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
195	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
196	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
197	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
198	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
199	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
200	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
201	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
202	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
203	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
204	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
205	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
206	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
207	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
208	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
209	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
210	C₄²×C₆₇	C₂⁴	1	4	yes	—
211	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
212	C₂²×D₆₇	—	2	2	no	—
213	Q₈×C₆₇	—	2	4	no	—
214	Q₈×C₆₇	—	2	4	no	—
215	C₂×C₄×C₆₇	—	2	2	no	—
216	C₂×C₄×C₆₇	—	2	2	no	—
217	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
218	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
219	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
220	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
221	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
224	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
225	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
226	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
227	C₂²⋊D₆₇	—	2	8	no	—
228	C₂×Dic₆₇	—	2	2	no	—
229	C₂×Dic₆₇	—	2	2	no	—
230	C₂×Dic₆₇	—	2	2	no	—
231	C₂×Dic₆₇	—	2	8	no	—
232	C₂×Dic₆₇	—	8	4	no	—
233	D₂₆₈	—	2	2	no	—
234	D₂₆₈	C₂²×Q₈	1	4	yes	—
235	C₄×D₆₇	—	2	8	no	—
236	C₄×D₆₇	—	4	4	no	—
237	C₄×D₆₇	—	4	4	no	—
238	C₄×D₆₇	—	8	8	no	—
239	Dic₁₃₄	—	2	2	no	—
240	Dic₁₃₄	—	4	8	no	—
241	Dic₁₃₄	—	8	8	no	—
242	Dic₁₃₄	—	8	4	no	—
243	Dic₁₃₄	C₂²×Q₈	1	4	yes	—
244	C₈×C₆₇	—	2	1	no	—
245	C₈×C₆₇	—	2	1	no	—
246	C₈×C₆₇	—	2	4	no	—
247	C₈×C₆₇	C₂²×Q₈	1	2	yes	—
248	C₈×C₆₇	C₂²×Q₈	1	2	yes	—
249	C₂²×C₆₇	—	2	1	no	—
250	C₂²×C₆₇	—	2	1	no	—
251	C₂²×C₆₇	D₄×Q₈	1	2	yes	—
252	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
253	C₂×D₆₇	—	2	4	no	—
254	C₂×D₆₇	—	4	2	no	—
255	C₂×D₆₇	—	4	2	no	—
256	C₄×C₆₇	—	2	1	no	—
257	C₄×C₆₇	—	2	4	no	—
258	C₄×C₆₇	—	2	1	no	—
259	C₄×C₆₇	—	2	4	no	—
260	C₄×C₆₇	D₄×Q₈	1	2	yes	—
261	C₄×C₆₇	#576	1	2	yes	—
262	C₄×C₆₇	#578	1	2	yes	—
263	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
264	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
265	Dic₆₇	—	2	4	no	—
266	Dic₆₇	—	4	8	no	—
267	Dic₆₇	—	4	2	no	—
268	Dic₆₇	—	4	2	no	—
269	#61373	—	67	1	no	maximal, Sylow
270	C₂×C₆₇	—	2	2	no	—
271	C₂×C₆₇	C₂×D₄×Q₈	1	1	yes	characteristic
272	C₂×C₆₇	C₂×#576	1	1	yes	characteristic
273	C₂×C₆₇	C₂×#578	1	1	yes	characteristic
274	D₆₇	—	4	4	no	—
275	#3503	—	67	2	no	—
276	#3386	—	67	4	no	—
277	#3385	—	67	4	no	—
278	#3383	—	67	2	no	—
279	Q₈×Q₁₆	—	67	2	no	—
280	#3380	—	67	2	no	—
281	Q₈×D₈	—	67	2	no	—
282	#3164	—	67	2	no	—
283	C₂×#500	—	67	1	no	—
284	#3083	—	67	4	no	—
285	#3079	—	67	2	no	—
286	#2960	D₆₇	1	1	yes	characteristic, complemented
287	#2951	—	67	2	no	—
288	C₄×#135	—	67	1	no	—
289	C₆₇	#61373	1	1	yes	characteristic, complemented, Sylow
290	C₂×#135	—	67	1	no	—
291	Q₈²	—	67	4	no	—
292	#557	—	67	4	no	—
293	#554	—	67	4	no	—
294	#554	—	134	4	no	—
295	D₄×Q₈	—	67	4	no	—
296	D₄×Q₈	—	134	4	no	—
297	#544	—	67	4	no	—
298	#544	—	134	4	no	—
299	#533	—	134	2	no	—
300	#533	C₂×D₆₇	1	1	yes	characteristic
301	C₂×C₄⋊Q₈	—	67	2	no	—
302	C₂×C₄⋊Q₈	—	134	2	no	—
303	#520	—	67	4	no	—
304	C₄×#41	—	67	2	no	—
305	#501	—	67	8	no	—
306	#500	—	67	4	no	—
307	#500	—	67	2	no	—
308	#500	—	67	2	no	—
309	#500	—	67	2	no	—
310	#500	—	67	2	no	—
311	#499	—	67	4	no	—
312	#479	—	67	8	no	—
313	#478	—	67	8	no	—
314	#477	—	67	4	no	—
315	#477	—	67	8	no	—
316	#476	—	67	4	no	—
317	#475	—	67	4	no	—
318	#474	—	67	4	no	—
319	#474	—	67	8	no	—
320	#446	C₂×D₆₇	1	2	yes	—
321	#446	C₂×D₆₇	1	2	yes	—
322	C₈×Q₈	C₂×D₆₇	1	2	yes	—
323	C₈×Q₈	C₂×D₆₇	1	2	yes	—
324	#440	—	67	8	no	—
325	C₄×Q₁₆	—	67	1	no	—
326	C₄×Q₁₆	—	67	4	no	—
327	C₄×Q₁₆	—	67	1	no	—
328	C₄×SD₁₆	—	67	2	no	—
329	C₄×SD₁₆	—	67	2	no	—
330	C₄×D₈	—	67	1	no	—
331	C₄×D₈	—	67	4	no	—
332	C₄×D₈	—	67	1	no	—
333	#428	—	67	4	no	—
334	#427	—	67	1	no	—
335	C₂×#107	—	67	2	no	—
336	C₂×#107	—	67	2	no	—
337	#424	C₂×D₆₇	1	2	yes	—
338	#423	C₂×D₆₇	1	1	yes	characteristic
339	#417	—	67	4	no	—
340	#416	—	67	2	no	—
341	#405	C₂×D₆₇	1	2	yes	—
342	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
343	C₂×#41	—	67	2	no	—
344	C₂²×Q₈	—	134	4	no	—
345	C₂²×Q₈	—	134	2	no	—
346	#135	—	67	8	no	—
347	C₂×Q₁₆	—	67	1	no	—
348	C₂×SD₁₆	—	67	2	no	—
349	C₂×D₈	—	67	1	no	—
350	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
351	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
352	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
353	C₄⋊Q₈	—	67	4	no	—
354	C₄⋊Q₈	—	67	4	no	—
355	C₄⋊Q₈	—	67	8	no	—
356	C₄⋊Q₈	—	134	4	no	—
357	C₄⋊Q₈	—	134	4	no	—
358	C₄⋊Q₈	—	134	4	no	—
359	C₄⋊Q₈	—	134	8	no	—
360	C₄⋊Q₈	—	134	2	no	—
361	C₄⋊Q₈	—	134	2	no	—
362	C₄⋊Q₈	D₂₆₈	1	1	yes	characteristic
363	C₄⋊Q₈	D₂₆₈	1	1	yes	characteristic
364	#125	—	67	8	no	—
365	#125	—	134	4	no	—
366	#125	—	134	8	no	—
367	#125	D₂₆₈	1	2	yes	—
368	C₂²⋊Q₈	—	134	4	no	—
369	C₂²⋊Q₈	—	134	4	no	—
370	C₂²⋊Q₈	—	134	8	no	—
371	C₂²⋊Q₈	—	134	8	no	—
372	C₂²⋊Q₈	—	134	8	no	—
373	C₂²⋊Q₈	D₂₆₈	1	2	yes	—
374	C₂²⋊Q₈	D₂₆₈	1	2	yes	—
375	C₄×Q₈	—	67	2	no	—
376	C₄×Q₈	—	67	2	no	—
377	C₄×Q₈	—	67	8	no	—
378	C₄×Q₈	—	134	4	no	—
379	C₄×Q₈	—	134	4	no	—
380	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
381	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
382	C₄×D₄	—	67	2	no	—
383	C₄×D₄	—	67	2	no	—
384	C₄×D₄	—	67	8	no	—
385	C₄×D₄	—	134	2	no	—
386	C₄×D₄	—	134	2	no	—
387	C₄×D₄	—	134	8	no	—
388	#117	—	67	2	no	—
389	#117	—	134	4	no	—
390	#117	—	134	2	no	—
391	#117	C₂²×D₆₇	1	2	yes	—
392	C₂×C₄⋊C₄	—	67	4	no	—
393	C₂×C₄⋊C₄	—	134	2	no	—
394	C₂×C₄⋊C₄	—	134	4	no	—
395	C₂×C₄⋊C₄	—	134	4	no	—
396	C₂×C₄²	—	134	2	no	—
397	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
398	#107	—	67	4	no	—
399	#107	—	67	4	no	—
400	#107	—	67	4	no	—
401	#107	—	67	4	no	—
402	#107	—	67	1	no	—
403	#107	—	67	1	no	—
404	#107	—	67	8	no	—
405	#106	—	67	2	no	—
406	#106	—	67	8	no	—
407	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
408	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
409	C₄⋊C₈	C₂²×D₆₇	1	4	yes	—
410	C₄⋊C₈	C₂²×D₆₇	1	4	yes	—
411	#103	—	67	2	no	—
412	#103	—	67	2	no	—
413	#103	—	67	8	no	—
414	#102	—	67	2	no	—
415	#102	—	67	2	no	—
416	#102	—	67	8	no	—
417	#97	C₂²×D₆₇	1	4	yes	—
418	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
419	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
420	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
421	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
422	C₄×C₈	C₂²×D₆₇	1	4	yes	—
423	#41	—	67	8	no	—
424	#41	—	134	4	no	—
425	C₂×Q₈	—	536	4	no	—
426	C₂×Q₈	—	536	8	no	—
427	C₂×Q₈	—	67	2	no	—
428	C₂×Q₈	—	134	4	no	—
429	C₂×Q₈	—	134	8	no	—
430	C₂×Q₈	—	134	4	no	—
431	C₂×Q₈	C₂×D₂₆₈	1	2	yes	—
432	C₂×Q₈	C₂×D₂₆₈	1	2	yes	—
433	C₂×D₄	—	67	2	no	—
434	C₂×D₄	—	134	2	no	—
435	C₂²×C₄	—	134	2	no	—
436	C₂²×C₄	—	134	2	no	—
437	C₂²×C₄	—	134	4	no	—
438	C₂²×C₄	—	134	8	no	—
439	C₂²×C₄	—	134	2	no	—
440	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
441	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
442	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
443	Q₁₆	—	67	4	no	—
444	SD₁₆	—	67	8	no	—
445	D₈	—	67	4	no	—
446	M₁₆	—	2	4	no	—
447	C₂×C₈	—	2	2	no	—
448	C₂×C₈	Q₈×D₆₇	1	4	yes	complemented
449	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
450	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
451	C₂×C₈	C₂³×D₆₇	1	4	yes	—
452	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
453	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
454	C₄⋊C₄	—	67	2	no	—
455	C₄⋊C₄	—	67	8	no	—
456	C₄⋊C₄	—	67	2	no	—
457	C₄⋊C₄	—	134	4	no	—
458	C₄⋊C₄	—	134	4	no	—
459	C₄⋊C₄	—	134	8	no	—
460	C₄⋊C₄	—	134	8	no	—
461	C₄⋊C₄	—	134	4	no	—
462	C₄⋊C₄	—	134	2	no	—
463	C₄⋊C₄	—	134	2	no	—
464	C₄⋊C₄	—	134	8	no	—
465	C₄⋊C₄	—	134	8	no	—
466	C₄⋊C₄	C₂×D₂₆₈	1	2	yes	—
467	C₄⋊C₄	C₂×D₂₆₈	1	2	yes	—
468	C₄⋊C₄	C₂×D₂₆₈	1	2	yes	—
469	C₄⋊C₄	C₂×D₂₆₈	1	2	yes	—
470	C₄⋊C₄	C₂×D₂₆₈	1	4	yes	—
471	C₄⋊C₄	C₂×D₂₆₈	1	4	yes	—
472	C₂²⋊C₄	—	134	2	no	—
473	C₂²⋊C₄	—	134	2	no	—
474	C₂²⋊C₄	—	134	2	no	—
475	C₂²⋊C₄	—	134	8	no	—
476	C₂²⋊C₄	C₂×D₂₆₈	1	4	yes	—
477	C₄²	—	134	2	no	—
478	C₄²	—	134	2	no	—
479	C₄²	—	134	2	no	—
480	C₄²	—	134	8	no	—
481	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
482	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
483	C₄²	C₂³×D₆₇	1	1	yes	characteristic
484	C₄²	C₂³×D₆₇	1	1	yes	characteristic
485	C₄²	C₂³×D₆₇	1	4	yes	—
486	C₂³	—	134	2	no	—
487	C₂³	C₂²×D₂₆₈	1	1	yes	characteristic
488	Q₈	—	2	4	no	—
489	Q₈	—	2	4	no	—
490	Q₈	—	268	8	no	—
491	Q₈	—	536	4	no	—
492	Q₈	—	536	8	no	—
493	Q₈	—	67	4	no	—
494	Q₈	—	134	2	no	—
495	D₄	—	67	4	no	—
496	D₄	—	134	8	no	—
497	D₄	—	134	2	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	268	4	no	—
501	C₂×C₄	—	268	4	no	—
502	C₂×C₄	—	536	8	no	—
503	C₂×C₄	—	536	4	no	—
504	C₂×C₄	—	134	2	no	—
505	C₂×C₄	—	134	2	no	—
506	C₂×C₄	—	134	8	no	—
507	C₂×C₄	—	134	2	no	—
508	C₂×C₄	—	134	8	no	—
509	C₂×C₄	C₂²×D₂₆₈	1	4	yes	—
510	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
511	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
512	C₂×C₄	C₂²×D₂₆₈	1	4	yes	—
513	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
514	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
515	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
516	C₂×C₄	C₂×Q₈×D₆₇	1	2	yes	—
517	C₂×C₄	#14524090	1	2	yes	—
518	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
519	C₈	—	2	1	no	—
520	C₈	—	2	1	no	—
521	C₈	—	2	4	no	—
522	C₈	C₂×Q₈×D₆₇	1	2	yes	—
523	C₈	C₂×Q₈×D₆₇	1	2	yes	—
524	C₂²	—	2	1	no	—
525	C₂²	—	2	1	no	—
526	C₂²	—	268	2	no	—
527	C₂²	—	268	2	no	—
528	C₂²	—	134	4	no	—
529	C₂²	Q₈×D₂₆₈	1	2	yes	—
530	C₂²	C₂³×D₂₆₈	1	1	yes	characteristic
531	C₄	—	2	1	no	—
532	C₄	—	2	4	no	—
533	C₄	—	2	1	no	—
534	C₄	—	2	4	no	—
535	C₄	—	268	2	no	—
536	C₄	—	268	8	no	—
537	C₄	—	268	2	no	—
538	C₄	—	134	4	no	—
539	C₄	Q₈×D₂₆₈	1	2	yes	—
540	C₄	#16864201	1	2	yes	—
541	C₄	#16864213	1	2	yes	—
542	C₄	C₂²×Q₈×D₆₇	1	1	yes	characteristic
543	C₄	C₂×#14524090	1	1	yes	characteristic
544	C₂	—	2	2	no	—
545	C₂	—	268	4	no	—
546	C₂	C₂×Q₈×D₂₆₈	1	1	yes	characteristic
547	C₂	C₂×#16864201	1	1	yes	characteristic
548	C₂	C₂×#16864213	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	C₂×#14524065	C₂²
hypercenter	#2960	D₆₇
hypocenter	C₆₇	#61373
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. #22574021	C₂	semidirect
2. #22573132	C₂	semidirect
3. #22573121	C₂	semidirect
4. #22573101	C₂	semidirect
5. Q₈×Dic₂₆₈	C₂	semidirect
6. #22573077	C₂	semidirect
7. Q₈×D₅₃₆	C₂	semidirect
8. #22570748	C₂	semidirect
9. C₂×#16863801	C₂	semidirect
10. #22570062	C₂	semidirect
11. #22570025	C₂	semidirect
14. C₆₇×#2960	C₂	semidirect
71. #14524065	Q₈	semidirect
149. #11761628	C₂×Q₈	semidirect
163. Dic₂₆₈	C₂×Q₈	semidirect
164. #11761617	C₂×Q₈	semidirect
165. D₅₃₆	C₂×Q₈	semidirect
193. C₂×C₈×C₆₇	C₂×Q₈	semidirect
286. #2960	D₆₇	semidirect
289. C₆₇	#61373	semidirect
448. 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
#22574021	2
#22573132	2
#22573121	2
#22573101	2
Q₈×Dic₂₆₈	2
#22573077	2
Q₈×D₅₃₆	2
#22570748	2
C₂×#16863801	2
#22570062	2
#22570025	2
#22569086	2
C₄×#14524065	2
C₆₇×#2960	2
#61373	67