Subgroups — #22569841

Order 8576.

496 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14524073	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#11761612	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14524022	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863652	C₂	1	2	yes	maximal, complemented
5	#16863640	C₂	1	2	yes	maximal, complemented
6	#16863635	C₂	1	2	yes	maximal, complemented
7	#16863623	C₂	1	2	yes	maximal, complemented
8	#16863439	C₂	1	1	yes	maximal, characteristic, complemented
9	#16863404	C₂	1	1	yes	maximal, characteristic, complemented
10	#16863397	C₂	1	1	yes	maximal, characteristic, complemented
11	#525×C₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₁₃₄	C₂²	1	1	yes	characteristic
13	#14524073	—	2	4	no	—
14	C₂×#11761618	C₂²	1	2	yes	complemented
15	C₂×#11761616	C₂²	1	2	yes	complemented
16	C₂×#11761612	—	2	4	no	—
17	C₂×#11761612	C₂²	1	2	yes	—
18	C₂×#11761612	C₂²	1	1	yes	characteristic
19	C₂×#11761612	C₂²	1	1	yes	characteristic
20	C₂²×C₆₇⋊C₈	C₂²	1	2	yes	—
21	#14524022	—	2	1	no	—
22	#14524022	—	2	1	no	—
23	#14524022	C₂²	1	2	yes	—
24	#14524022	C₂²	1	2	yes	—
25	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
26	C₂×Dic₆₇⋊C₄	—	2	1	no	—
27	C₂×C₂²⋊Dic₆₇	—	2	1	no	—
28	#14523976	C₂²	1	2	yes	complemented
29	#14523974	C₂²	1	2	yes	complemented
30	#14523967	C₂²	1	2	yes	complemented
31	#14523966	C₂²	1	2	yes	complemented
32	#14523966	C₂²	1	2	yes	complemented
33	#14523964	C₂²	1	2	yes	complemented
34	C₂×#41×C₆₇	C₂²	1	1	yes	characteristic
35	C₂³×C₄×C₆₇	C₂²	1	1	yes	characteristic
36	#123×C₆₇	—	2	1	no	—
37	C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
38	C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
39	#121×C₆₇	C₂²	1	1	yes	characteristic
40	#121×C₆₇	C₂²	1	1	yes	characteristic
41	#120×C₆₇	—	2	1	no	—
42	C₄×D₄×C₆₇	—	2	1	no	—
43	C₄×D₄×C₆₇	—	2	1	no	—
44	#117×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×Dic₁₃₄	—	2	2	no	—
46	C₂×Dic₁₃₄	—	2	2	no	—
47	C₂×Dic₁₃₄	—	2	1	no	—
48	C₂×Dic₁₃₄	C₂³	1	2	yes	—
49	C₂²×Dic₆₇	—	2	1	no	—
50	C₂²×Dic₆₇	—	2	1	no	—
51	#11761618	—	2	4	no	—
52	#11761616	—	2	4	no	—
53	#11761612	—	2	2	no	—
54	#11761612	—	2	4	no	—
55	#11761612	D₄	1	4	yes	complemented
56	C₂×C₆₇⋊C₈	—	2	4	no	—
57	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
58	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
59	C₄⋊Dic₆₇	—	2	1	no	—
60	C₄⋊Dic₆₇	C₂³	1	2	yes	—
61	Dic₆₇⋊C₄	—	2	2	no	—
62	Dic₆₇⋊C₄	—	2	2	no	—
63	C₂²⋊Dic₆₇	—	2	2	no	—
64	C₂²⋊Dic₆₇	—	2	2	no	—
65	C₂⁴×C₆₇	D₄	1	1	yes	characteristic
66	#41×C₆₇	—	2	2	no	—
67	#41×C₆₇	—	2	2	no	—
68	C₂×Q₈×C₆₇	C₂³	1	1	yes	characteristic
69	C₂×D₄×C₆₇	—	2	1	no	—
70	C₂×D₄×C₆₇	—	2	1	no	—
71	C₂×D₄×C₆₇	—	2	1	no	—
72	C₂×D₄×C₆₇	C₂³	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	—	2	2	no	—
74	C₂²×C₄×C₆₇	—	2	2	no	—
75	C₂²×C₄×C₆₇	—	2	1	no	—
76	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₆₇	—	2	1	no	—
83	C₄⋊C₄×C₆₇	—	2	1	no	—
84	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
85	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
86	C₂²⋊C₄×C₆₇	—	2	1	no	—
87	C₂²⋊C₄×C₆₇	—	2	1	no	—
88	C₂²⋊C₄×C₆₇	—	2	1	no	—
89	C₂²⋊C₄×C₆₇	—	2	1	no	—
90	C₂²⋊C₄×C₆₇	—	2	1	no	—
91	C₄²×C₆₇	—	2	1	no	—
92	C₂³×C₆₇	—	2	2	no	—
93	C₂³×C₆₇	—	2	2	no	—
94	C₂³×C₆₇	—	2	1	no	—
95	C₂³×C₆₇	—	4	1	no	—
96	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
97	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
99	Q₈×C₆₇	—	2	1	no	—
100	Q₈×C₆₇	—	2	1	no	—
101	D₄×C₆₇	—	2	1	no	—
102	D₄×C₆₇	—	2	1	no	—
103	D₄×C₆₇	—	4	2	no	—
104	D₄×C₆₇	—	4	2	no	—
105	D₄×C₆₇	—	4	2	no	—
106	C₂×C₄×C₆₇	—	2	2	no	—
107	C₂×C₄×C₆₇	—	2	2	no	—
108	C₂×C₄×C₆₇	—	2	2	no	—
109	C₂×C₄×C₆₇	—	2	1	no	—
110	C₂×C₄×C₆₇	—	2	1	no	—
111	C₂×C₄×C₆₇	—	2	1	no	—
112	C₂×C₄×C₆₇	—	2	1	no	—
113	C₂×C₄×C₆₇	—	2	1	no	—
114	C₂×C₄×C₆₇	—	2	1	no	—
115	C₂×C₄×C₆₇	—	2	1	no	—
116	C₂×C₄×C₆₇	—	4	2	no	—
117	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
118	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	#41	1	2	yes	—
122	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
123	C₂×Dic₆₇	—	2	2	no	—
124	C₂×Dic₆₇	—	2	2	no	—
125	C₂×Dic₆₇	—	4	2	no	—
126	C₂×Dic₆₇	—	4	2	no	—
127	Dic₁₃₄	—	2	2	no	—
128	Dic₁₃₄	—	2	2	no	—
129	Dic₁₃₄	—	4	2	no	—
130	C₆₇⋊C₈	—	2	2	no	—
131	C₆₇⋊C₈	C₂×D₄	1	4	yes	—
132	C₂²×C₆₇	—	2	1	no	—
133	C₂²×C₆₇	—	2	1	no	—
134	C₂²×C₆₇	—	2	1	no	—
135	C₂²×C₆₇	—	2	1	no	—
136	C₂²×C₆₇	—	2	1	no	—
137	C₂²×C₆₇	—	2	1	no	—
138	C₂²×C₆₇	—	2	1	no	—
139	C₂²×C₆₇	—	4	4	no	—
140	C₂²×C₆₇	—	4	1	no	—
141	C₂²×C₆₇	—	4	1	no	—
142	C₂²×C₆₇	—	4	1	no	—
143	C₂²×C₆₇	#121	1	2	yes	—
144	C₂²×C₆₇	#137	1	2	yes	complemented
145	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
146	C₄×C₆₇	—	2	1	no	—
147	C₄×C₆₇	—	2	1	no	—
148	C₄×C₆₇	—	4	1	no	—
149	C₄×C₆₇	—	4	1	no	—
150	C₄×C₆₇	—	4	1	no	—
151	C₄×C₆₇	#121	1	2	yes	—
152	C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
153	C₄×C₆₇	C₂×#41	1	1	yes	characteristic
154	Dic₆₇	—	4	2	no	—
155	Dic₆₇	—	4	2	no	—
156	C₂×C₆₇	—	2	2	no	—
157	C₂×C₆₇	—	2	2	no	—
158	C₂×C₆₇	—	4	1	no	—
159	C₂×C₆₇	—	8	1	no	—
160	C₂×C₆₇	C₂×#121	1	1	yes	characteristic
161	C₂×C₆₇	C₂×#137	1	1	yes	characteristic
162	C₂×C₆₇	#575	1	1	yes	characteristic
163	#3056	—	67	1	no	maximal, Sylow
164	C₆₇	#3056	1	1	yes	characteristic, complemented, Sylow
165	C₂×#137	—	67	1	no	—
166	C₂²×M₁₆	—	67	1	no	—
167	#525	D₆₇	1	1	yes	characteristic
168	C₂×C₂²⋊Q₈	—	67	1	no	—
169	#470	—	67	2	no	—
170	#468	—	67	2	no	—
171	#467	—	67	2	no	—
172	#465	—	67	2	no	—
173	#428	—	67	1	no	—
174	#419	—	67	1	no	—
175	#417	—	67	1	no	—
176	C₂×#41	C₂×D₆₇	1	1	yes	characteristic
177	C₂²×Q₈	—	67	1	no	—
178	C₂³×C₄	C₂×D₆₇	1	1	yes	characteristic
179	#137	—	134	4	no	—
180	C₂×Q₁₆	—	67	2	no	—
181	C₂×SD₁₆	—	67	2	no	—
182	C₂×M₁₆	—	67	1	no	—
183	C₂×M₁₆	—	67	1	no	—
184	C₂×M₁₆	—	67	2	no	—
185	C₂×M₁₆	—	134	4	no	—
186	C₂²×C₈	—	67	2	no	—
187	#123	—	2	1	no	—
188	C₂²⋊Q₈	—	67	2	no	—
189	C₂²⋊Q₈	—	67	2	no	—
190	C₂²⋊Q₈	—	134	1	no	—
191	C₂²⋊Q₈	—	134	1	no	—
192	C₂²⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
193	C₂²⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
194	#121	C₂×D₆₇	1	1	yes	characteristic
195	#121	C₂×D₆₇	1	1	yes	characteristic
196	#120	—	2	1	no	—
197	C₄×D₄	—	2	1	no	—
198	C₄×D₄	—	2	1	no	—
199	#117	C₂×D₆₇	1	1	yes	characteristic
200	C₂×C₄⋊C₄	—	67	1	no	—
201	C₂×C₄⋊C₄	—	134	1	no	—
202	C₂×C₂²⋊C₄	—	134	1	no	—
203	#107	—	67	2	no	—
204	#106	—	67	2	no	—
205	#103	—	67	2	no	—
206	#103	—	67	2	no	—
207	#103	—	67	2	no	—
208	#102	—	67	2	no	—
209	C₂⁴	C₂²⋊D₆₇	1	1	yes	characteristic
210	#41	—	2	2	no	—
211	#41	—	2	2	no	—
212	C₂×Q₈	—	67	2	no	—
213	C₂×Q₈	—	134	2	no	—
214	C₂×Q₈	—	134	2	no	—
215	C₂×Q₈	—	134	1	no	—
216	C₂×Q₈	C₂²×D₆₇	1	1	yes	characteristic
217	C₂×D₄	—	2	1	no	—
218	C₂×D₄	—	2	1	no	—
219	C₂×D₄	—	2	1	no	—
220	C₂×D₄	C₂²×D₆₇	1	1	yes	characteristic
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	134	1	no	—
225	C₂²×C₄	—	134	1	no	—
226	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
227	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
228	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
229	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
230	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
231	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
232	Q₁₆	—	134	4	no	—
233	SD₁₆	—	134	4	no	—
234	M₁₆	—	67	4	no	—
235	M₁₆	—	134	4	no	—
236	M₁₆	—	134	2	no	—
237	C₂×C₈	—	67	2	no	—
238	C₂×C₈	—	67	2	no	—
239	C₂×C₈	—	134	4	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	67	2	no	—
243	C₄⋊C₄	—	134	1	no	—
244	C₄⋊C₄	—	134	2	no	—
245	C₄⋊C₄	—	134	2	no	—
246	C₄⋊C₄	C₂²×D₆₇	1	1	yes	characteristic
247	C₄⋊C₄	C₂²×D₆₇	1	1	yes	characteristic
248	C₂²⋊C₄	—	2	1	no	—
249	C₂²⋊C₄	—	2	1	no	—
250	C₂²⋊C₄	—	2	1	no	—
251	C₂²⋊C₄	—	2	1	no	—
252	C₂²⋊C₄	—	2	1	no	—
253	C₂²⋊C₄	—	134	2	no	—
254	C₂²⋊C₄	—	134	2	no	—
255	C₄²	—	2	1	no	—
256	C₂³	—	2	2	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	1	no	—
259	C₂³	—	4	1	no	—
260	C₂³	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
261	C₂³	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
262	C₂³	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
263	Q₈	—	2	1	no	—
264	Q₈	—	2	1	no	—
265	Q₈	—	268	2	no	—
266	Q₈	—	134	2	no	—
267	Q₈	—	134	2	no	—
268	D₄	—	2	1	no	—
269	D₄	—	2	1	no	—
270	D₄	—	4	2	no	—
271	D₄	—	4	2	no	—
272	D₄	—	4	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	268	2	no	—
285	C₂×C₄	—	268	2	no	—
286	C₂×C₄	—	134	2	no	—
287	C₂×C₄	—	134	2	no	—
288	C₂×C₄	D₄×D₆₇	1	2	yes	—
289	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
290	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
291	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
292	C₂×C₄	#11761627	1	2	yes	—
293	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
294	C₈	—	67	4	no	—
295	C₈	—	134	2	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	4	4	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	#14524015	1	2	yes	—
308	C₂²	#14524073	1	2	yes	complemented
309	C₂²	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	268	2	no	—
316	C₄	—	268	2	no	—
317	C₄	#14524015	1	2	yes	—
318	C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
319	C₄	C₂×#11761627	1	1	yes	characteristic
320	C₂	—	2	2	no	—
321	C₂	—	2	2	no	—
322	C₂	—	4	1	no	—
323	C₂	—	8	1	no	—
324	C₂	C₂×#14524015	1	1	yes	characteristic
325	C₂	C₂×#14524073	1	1	yes	characteristic
326	C₂	#16864194	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	#525×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	#525	D₆₇
hypocenter	C₆₇	#3056
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #525

Complements

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

N	H ≅ G/N	type
1. C₂×#14524073	C₂	semidirect
2. C₂²×#11761612	C₂	semidirect
3. C₂×#14524022	C₂	semidirect
4. #16863652	C₂	semidirect
5. #16863640	C₂	semidirect
6. #16863635	C₂	semidirect
7. #16863623	C₂	semidirect
8. #16863439	C₂	semidirect
9. #16863404	C₂	semidirect
10. #16863397	C₂	semidirect
14. C₂×#11761618	C₂²	semidirect
15. C₂×#11761616	C₂²	semidirect
28. #14523976	C₂²	semidirect
29. #14523974	C₂²	semidirect
30. #14523967	C₂²	semidirect
31. #14523966	C₂²	semidirect
32. #14523966	C₂²	semidirect
33. #14523964	C₂²	semidirect
55. #11761612	D₄	semidirect
144. C₂²×C₆₇	#137	semidirect
164. C₆₇	#3056	semidirect
308. C₂²	#14524073	semidirect