Subgroups — #22574108

Order 8576.

1040 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16864128	C₂	1	4	yes	maximal, complemented
2	#16864120	C₂	1	4	yes	maximal, complemented
3	#16864069	C₂	1	8	yes	maximal, complemented
4	#16864019	C₂	1	4	yes	maximal, complemented
5	#16863960	C₂	1	1	yes	maximal, characteristic
6	#16863949	C₂	1	1	yes	maximal, characteristic, complemented
7	#16863932	C₂	1	4	yes	maximal, complemented
8	C₂×#14524032	C₂	1	1	yes	maximal, characteristic, complemented
9	#16863874	C₂	1	1	yes	maximal, characteristic, complemented
10	#16863870	C₂	1	1	yes	maximal, characteristic, complemented
11	#41×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#536×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×D₂₆₈	C₂²	1	1	yes	characteristic
14	#14524042	C₂²	1	2	yes	complemented
15	#14524039	C₂²	1	2	yes	—
16	#14524038	C₂²	1	8	yes	complemented
17	#14524038	C₂²	1	8	yes	complemented
18	#14524034	C₂²	1	2	yes	complemented
19	#14524032	C₂²	1	2	yes	—
20	#14524032	C₂²	1	2	yes	—
21	#14524032	C₂²	1	4	yes	—
22	#14524032	C₂²	1	2	yes	—
23	#14524029	C₂²	1	8	yes	complemented
24	#14524027	C₂²	1	8	yes	complemented
25	#14524025	C₂²	1	8	yes	complemented
26	#14524023	C₂²	1	8	yes	complemented
27	#14524020	C₂²	1	4	yes	—
28	#14524017	C₂²	1	4	yes	—
29	#14524011	C₂²	1	4	yes	—
30	Q₈×Dic₆₇	C₂²	1	4	yes	complemented
31	Q₈×Dic₆₇	C₂²	1	2	yes	complemented
32	Q₈×Dic₆₇	C₂²	1	4	yes	complemented
33	C₄×Dic₁₃₄	C₂²	1	4	yes	complemented
34	C₄×D₂₆₈	C₂²	1	4	yes	—
35	D₄×Dic₆₇	C₂²	1	4	yes	—
36	D₄×Dic₆₇	C₂²	1	2	yes	complemented
37	#14524005	C₂²	1	4	yes	—
38	(C₂²⋊D₆₇)⋊C₄	C₂²	1	8	yes	complemented
39	#14524002	C₂²	1	4	yes	—
40	#14524002	C₂²	1	1	yes	characteristic
41	#14524002	C₂²	1	1	yes	characteristic
42	#14524002	C₂²	1	2	yes	complemented
43	#14524001	C₂²	1	8	yes	complemented
44	C₂×(C₂×D₆₇)⋊C₄	C₂²	1	4	yes	—
45	C₂²⋊C₄×D₆₇	C₂²	1	4	yes	—
46	C₂×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
47	C₂×C₄×Dic₆₇	C₂²	1	2	yes	complemented
48	C₂×#41×C₆₇	C₂²	1	1	yes	characteristic
49	C₂²×Q₈×C₆₇	C₂²	1	1	yes	characteristic
50	C₄⋊Q₈×C₆₇	C₂²	1	2	yes	complemented
51	#124×C₆₇	C₂²	1	2	yes	complemented
52	#123×C₆₇	C₂²	1	4	yes	—
53	C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	complemented
54	#117×C₆₇	C₂²	1	1	yes	characteristic
55	C₂³×D₆₇	C₂³	1	2	yes	—
56	C₂×Dic₁₃₄	C₂³	1	4	yes	—
57	C₂×D₂₆₈	—	2	4	no	—
58	C₂×D₂₆₈	C₂³	1	2	yes	—
59	C₂×D₂₆₈	C₂³	1	2	yes	—
60	C₂×C₂²⋊D₆₇	C₂³	1	8	yes	—
61	C₂²×Dic₆₇	C₂³	1	2	yes	—
62	C₂²×Dic₆₇	C₂³	1	4	yes	—
63	C₂×C₄×D₆₇	C₂³	1	8	yes	—
64	C₄⋊Dic₆₇	C₂³	1	4	yes	—
65	C₄⋊Dic₆₇	C₂³	1	2	yes	—
66	C₄⋊Dic₆₇	C₂³	1	2	yes	—
67	C₄⋊Dic₆₇	C₂³	1	8	yes	—
68	C₄⋊Dic₆₇	C₂³	1	4	yes	—
69	Dic₆₇⋊C₄	C₂³	1	4	yes	—
70	Dic₆₇⋊C₄	C₂³	1	4	yes	—
71	C₂²⋊Dic₆₇	C₂³	1	2	yes	—
72	C₂²⋊Dic₆₇	C₂³	1	2	yes	—
73	C₂²⋊Dic₆₇	C₂³	1	4	yes	—
74	(C₂×D₆₇)⋊C₄	C₂³	1	8	yes	—
75	(C₂×D₆₇)⋊C₄	C₂³	1	8	yes	—
76	(C₂×D₆₇)⋊C₄	C₂³	1	8	yes	—
77	(C₂×D₆₇)⋊C₄	C₂³	1	8	yes	—
78	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
79	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
80	C₄×Dic₆₇	C₂³	1	2	yes	—
81	C₄×Dic₆₇	C₂³	1	4	yes	—
82	C₄×Dic₆₇	C₂³	1	2	yes	—
83	C₄×Dic₆₇	C₂³	1	8	yes	—
84	C₄×Dic₆₇	C₂³	1	2	yes	—
85	#41×C₆₇	—	2	4	no	—
86	C₂×Q₈×C₆₇	—	2	4	no	—
87	C₂×Q₈×C₆₇	C₂³	1	2	yes	—
88	C₂×Q₈×C₆₇	C₂³	1	2	yes	—
89	C₂×Q₈×C₆₇	C₂³	1	1	yes	characteristic
90	C₂×D₄×C₆₇	C₂³	1	2	yes	—
91	C₂×D₄×C₆₇	C₂³	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
93	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
94	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
95	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
96	C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
97	C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
98	C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
99	C₄²×C₆₇	C₂³	1	2	yes	—
100	C₂³×C₆₇	C₂⁴	1	2	yes	—
101	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
102	C₂²×D₆₇	—	2	4	no	—
103	C₂²×D₆₇	—	4	4	no	—
104	C₂²×D₆₇	C₂⁴	1	4	yes	—
105	Q₈×C₆₇	—	2	4	no	—
106	Q₈×C₆₇	—	2	4	no	—
107	Q₈×C₆₇	—	2	2	no	—
108	D₄×C₆₇	—	2	4	no	—
109	D₄×C₆₇	—	2	2	no	—
110	C₂×C₄×C₆₇	—	2	4	no	—
111	C₂×C₄×C₆₇	—	2	4	no	—
112	C₂×C₄×C₆₇	#41	1	4	yes	—
113	C₂×C₄×C₆₇	C₂⁴	1	4	yes	—
114	C₂×C₄×C₆₇	C₂⁴	1	4	yes	—
115	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
116	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
118	C₂²⋊D₆₇	—	4	8	no	—
119	C₂×Dic₆₇	—	4	2	no	—
120	C₂×Dic₆₇	—	4	4	no	—
121	C₂×Dic₆₇	C₂⁴	1	2	yes	—
122	C₂×Dic₆₇	C₂⁴	1	4	yes	—
123	C₂×Dic₆₇	C₂⁴	1	4	yes	—
124	C₂×Dic₆₇	C₂⁴	1	2	yes	—
125	D₂₆₈	—	2	4	no	—
126	D₂₆₈	—	2	4	no	—
127	C₄×D₆₇	—	4	8	no	—
128	Dic₁₃₄	—	4	4	no	—
129	C₂²×C₆₇	—	2	2	no	—
130	C₂²×C₆₇	—	2	2	no	—
131	C₂²×C₆₇	—	2	4	no	—
132	C₂²×C₆₇	C₂×#41	1	2	yes	—
133	C₂²×C₆₇	C₂⁵	1	1	yes	characteristic
134	C₂×D₆₇	—	2	4	no	—
135	C₂×D₆₇	—	2	8	no	—
136	C₂×D₆₇	—	8	4	no	—
137	C₄×C₆₇	—	2	2	no	—
138	C₄×C₆₇	—	2	4	no	—
139	C₄×C₆₇	—	2	4	no	—
140	C₄×C₆₇	C₂×#41	1	2	yes	—
141	C₄×C₆₇	C₂×#41	1	2	yes	—
142	Dic₆₇	—	2	2	no	—
143	Dic₆₇	—	2	4	no	—
144	Dic₆₇	—	2	4	no	—
145	Dic₆₇	—	2	2	no	—
146	C₂×C₆₇	—	2	2	no	—
147	C₂×C₆₇	—	4	2	no	—
148	C₂×C₆₇	C₂²×#41	1	1	yes	characteristic
149	C₂×C₆₇	#585	1	2	yes	—
150	D₆₇	—	4	4	no	—
151	#3510	—	67	1	no	maximal, Sylow
152	C₆₇	#3510	1	1	yes	characteristic, complemented, Sylow
153	#559	—	67	4	no	—
154	#556	—	67	4	no	—
155	#551	—	67	8	no	—
156	#542	—	67	4	no	—
157	#536	D₆₇	1	1	yes	characteristic, complemented
158	#535	—	67	1	no	—
159	#533	—	67	1	no	—
160	#529	—	67	4	no	—
161	C₂×#124	—	67	1	no	—
162	#519	—	67	1	no	—
163	#518	—	67	1	no	—
164	C₄×#41	—	67	1	no	—
165	C₂×#41	C₂×D₆₇	1	1	yes	characteristic
166	C₂²×Q₈	C₂×D₆₇	1	1	yes	characteristic
167	C₂²×D₄	—	67	1	no	—
168	C₄⋊Q₈	—	67	2	no	—
169	C₄⋊Q₈	C₂×D₆₇	1	2	yes	complemented
170	#127	—	67	2	no	—
171	#126	—	67	8	no	—
172	#126	—	67	8	no	—
173	#125	—	67	2	no	—
174	#124	—	67	2	no	—
175	#124	—	67	2	no	—
176	#124	—	67	4	no	—
177	#124	—	67	2	no	—
178	#124	—	67	8	no	—
179	#124	C₂×D₆₇	1	2	yes	complemented
180	#123	—	67	8	no	—
181	#123	—	67	8	no	—
182	#123	C₂×D₆₇	1	4	yes	—
183	C₂²⋊Q₈	—	67	8	no	—
184	C₂²⋊Q₈	—	67	4	no	—
185	C₂²⋊Q₈	C₂×D₆₇	1	4	yes	complemented
186	#121	—	67	4	no	—
187	#120	—	67	4	no	—
188	C₄×Q₈	—	67	2	no	—
189	C₄×Q₈	—	67	4	no	—
190	C₄×Q₈	—	67	4	no	—
191	C₄×Q₈	—	67	4	no	—
192	C₄×D₄	—	67	4	no	—
193	C₄×D₄	—	67	2	no	—
194	C₄×D₄	—	67	4	no	—
195	C₄×D₄	—	67	8	no	—
196	C₄×D₄	—	67	4	no	—
197	#117	—	67	1	no	—
198	#117	—	67	2	no	—
199	#117	—	67	1	no	—
200	#117	—	67	4	no	—
201	#117	—	67	8	no	—
202	#117	C₂×D₆₇	1	1	yes	characteristic
203	C₂×C₂²⋊C₄	—	67	4	no	—
204	C₂×C₂²⋊C₄	—	67	4	no	—
205	C₂×C₄²	—	67	1	no	—
206	C₂×C₄²	—	67	2	no	—
207	C₂⁴	—	67	2	no	—
208	#41	—	2	4	no	—
209	C₂×Q₈	—	2	4	no	—
210	C₂×Q₈	—	67	4	no	—
211	C₂×Q₈	C₂²×D₆₇	1	2	yes	—
212	C₂×Q₈	C₂²×D₆₇	1	2	yes	—
213	C₂×Q₈	C₂²×D₆₇	1	1	yes	characteristic
214	C₂×D₄	—	67	8	no	—
215	C₂×D₄	—	67	2	no	—
216	C₂×D₄	—	67	2	no	—
217	C₂×D₄	—	134	4	no	—
218	C₂×D₄	C₂²×D₆₇	1	2	yes	—
219	C₂×D₄	C₂²×D₆₇	1	1	yes	characteristic
220	C₂²×C₄	—	67	2	no	—
221	C₂²×C₄	—	67	4	no	—
222	C₂²×C₄	—	67	8	no	—
223	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
224	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
225	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
226	C₄⋊C₄	—	67	4	no	—
227	C₄⋊C₄	—	67	2	no	—
228	C₄⋊C₄	—	67	4	no	—
229	C₄⋊C₄	—	67	2	no	—
230	C₄⋊C₄	—	67	8	no	—
231	C₄⋊C₄	—	67	4	no	—
232	C₄⋊C₄	—	67	4	no	—
233	C₄⋊C₄	C₂²×D₆₇	1	8	yes	—
234	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
235	C₂²⋊C₄	—	67	2	no	—
236	C₂²⋊C₄	—	67	2	no	—
237	C₂²⋊C₄	—	67	4	no	—
238	C₂²⋊C₄	—	67	8	no	—
239	C₂²⋊C₄	—	67	8	no	—
240	C₂²⋊C₄	—	67	8	no	—
241	C₂²⋊C₄	—	67	8	no	—
242	C₂²⋊C₄	C₂²×D₆₇	1	2	yes	—
243	C₂²⋊C₄	C₂²×D₆₇	1	8	yes	—
244	C₄²	—	67	1	no	—
245	C₄²	—	67	2	no	—
246	C₄²	—	67	4	no	—
247	C₄²	—	67	1	no	—
248	C₄²	—	67	2	no	—
249	C₄²	—	67	8	no	—
250	C₄²	—	67	2	no	—
251	C₄²	C₂²×D₆₇	1	2	yes	—
252	C₂³	—	268	4	no	—
253	C₂³	—	67	4	no	—
254	C₂³	—	134	4	no	—
255	C₂³	C₂³×D₆₇	1	2	yes	—
256	C₂³	C₂³×D₆₇	1	1	yes	characteristic
257	Q₈	—	2	4	no	—
258	Q₈	—	2	4	no	—
259	Q₈	—	2	2	no	—
260	Q₈	—	268	4	no	—
261	D₄	—	2	4	no	—
262	D₄	—	2	2	no	—
263	D₄	—	268	8	no	—
264	D₄	—	134	4	no	—
265	D₄	—	134	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	268	8	no	—
269	C₂×C₄	—	268	4	no	—
270	C₂×C₄	—	268	2	no	—
271	C₂×C₄	—	67	2	no	—
272	C₂×C₄	—	67	4	no	—
273	C₂×C₄	—	67	4	no	—
274	C₂×C₄	—	67	2	no	—
275	C₂×C₄	#11761627	1	4	yes	—
276	C₂×C₄	C₂³×D₆₇	1	4	yes	—
277	C₂×C₄	C₂³×D₆₇	1	2	yes	—
278	C₂×C₄	C₂³×D₆₇	1	4	yes	—
279	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
280	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	536	4	no	—
285	C₂²	—	134	4	no	—
286	C₂²	—	134	8	no	—
287	C₂²	C₂×#11761627	1	2	yes	—
288	C₂²	C₂⁴×D₆₇	1	1	yes	characteristic
289	C₄	—	2	2	no	—
290	C₄	—	2	4	no	—
291	C₄	—	2	4	no	—
292	C₄	—	134	4	no	—
293	C₄	—	134	2	no	—
294	C₄	—	134	2	no	—
295	C₄	—	134	4	no	—
296	C₄	C₂×#11761627	1	2	yes	—
297	C₄	C₂×#11761627	1	2	yes	—
298	C₂	—	2	2	no	—
299	C₂	—	4	2	no	—
300	C₂	—	268	4	no	—
301	C₂	C₂²×#11761627	1	1	yes	characteristic
302	C₂	#16864233	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₆₇
commutator, socle	C₂²×C₆₇	C₂⁵
Fitting	#536×C₆₇	C₂
hypercenter	#536	D₆₇
hypocenter	C₆₇	#3510
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ #536

Complements

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

N	H ≅ G/N	type
1. #16864128	C₂	semidirect
2. #16864120	C₂	semidirect
3. #16864069	C₂	semidirect
4. #16864019	C₂	semidirect
6. #16863949	C₂	semidirect
7. #16863932	C₂	semidirect
8. C₂×#14524032	C₂	semidirect
9. #16863874	C₂	semidirect
10. #16863870	C₂	semidirect
11. #41×Dic₆₇	C₂	semidirect
12. #536×C₆₇	C₂	semidirect
14. #14524042	C₂²	semidirect
16. #14524038	C₂²	semidirect
17. #14524038	C₂²	semidirect
18. #14524034	C₂²	semidirect
23. #14524029	C₂²	semidirect
24. #14524027	C₂²	semidirect
25. #14524025	C₂²	semidirect
26. #14524023	C₂²	semidirect
30. Q₈×Dic₆₇	C₂²	semidirect
31. Q₈×Dic₆₇	C₂²	semidirect
32. Q₈×Dic₆₇	C₂²	semidirect
33. C₄×Dic₁₃₄	C₂²	semidirect
36. D₄×Dic₆₇	C₂²	semidirect
38. (C₂²⋊D₆₇)⋊C₄	C₂²	semidirect
42. #14524002	C₂²	semidirect
43. #14524001	C₂²	semidirect
47. C₂×C₄×Dic₆₇	C₂²	semidirect
50. C₄⋊Q₈×C₆₇	C₂²	semidirect
51. #124×C₆₇	C₂²	semidirect
53. C₂²⋊Q₈×C₆₇	C₂²	semidirect
152. C₆₇	#3510	semidirect
157. #536	D₆₇	semidirect
169. C₄⋊Q₈	C₂×D₆₇	semidirect
179. #124	C₂×D₆₇	semidirect
185. C₂²⋊Q₈	C₂×D₆₇	semidirect