Subgroups — #22573601

Order 8576.

1040 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16864020	C₂	1	4	yes	maximal, complemented
2	#16864009	C₂	1	2	yes	maximal, complemented
3	#16864002	C₂	1	2	yes	maximal, complemented
4	#16863995	C₂	1	4	yes	maximal, complemented
5	#16863986	C₂	1	4	yes	maximal, complemented
6	#16863946	C₂	1	1	yes	maximal, characteristic, complemented
7	#16863933	C₂	1	8	yes	maximal, complemented
8	#16863908	C₂	1	2	yes	maximal, complemented
9	#16863864	C₂	1	2	yes	maximal, complemented
10	C₂×#14524000	C₂	1	1	yes	maximal, characteristic, complemented
11	#533×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11761628	C₂²	1	2	yes	—
13	C₂²×C₄×D₆₇	C₂²	1	1	yes	characteristic
14	#14524043	C₂²	1	1	yes	characteristic, complemented
15	#14524040	C₂²	1	1	yes	characteristic, complemented
16	#14524038	C₂²	1	8	yes	—
17	#14524038	C₂²	1	8	yes	complemented
18	#14524034	C₂²	1	8	yes	complemented
19	#14524032	C₂²	1	8	yes	—
20	#14524031	C₂²	1	2	yes	complemented
21	#14524028	C₂²	1	8	yes	—
22	#14524028	C₂²	1	8	yes	complemented
23	#14524027	C₂²	1	4	yes	—
24	#14524023	C₂²	1	4	yes	—
25	#14524019	C₂²	1	8	yes	complemented
26	#14524018	C₂²	1	4	yes	—
27	#14524018	C₂²	1	8	yes	—
28	#14524017	C₂²	1	4	yes	—
29	#14524013	C₂²	1	4	yes	—
30	#14524008	C₂²	1	4	yes	complemented
31	C₄×D₂₆₈	C₂²	1	8	yes	—
32	C₄×D₂₆₈	C₂²	1	4	yes	complemented
33	#14524005	C₂²	1	4	yes	complemented
34	(C₂²⋊D₆₇)⋊C₄	C₂²	1	8	yes	—
35	#14524000	C₂²	1	2	yes	—
36	#14524000	C₂²	1	2	yes	—
37	#14524000	C₂²	1	4	yes	—
38	#14524000	C₂²	1	4	yes	complemented
39	C₂×Dic₆₇⋊C₄	C₂²	1	2	yes	—
40	C₂×(C₂×D₆₇)⋊C₄	C₂²	1	2	yes	—
41	C₂×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
42	C₄²×D₆₇	C₂²	1	4	yes	complemented
43	C₄⋊Q₈×C₆₇	C₂²	1	2	yes	complemented
44	#125×C₆₇	C₂²	1	2	yes	complemented
45	C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	—
46	C₄×Q₈×C₆₇	C₂²	1	4	yes	complemented
47	#117×C₆₇	C₂²	1	2	yes	—
48	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
49	C₂³×D₆₇	C₂³	1	1	yes	characteristic
50	#11761628	—	2	8	no	—
51	C₂×Dic₁₃₄	C₂³	1	2	yes	—
52	C₂×D₂₆₈	C₂³	1	2	yes	—
53	C₂×D₂₆₈	C₂³	1	8	yes	—
54	C₂×D₂₆₈	C₂³	1	4	yes	—
55	C₂×C₂²⋊D₆₇	C₂³	1	4	yes	—
56	C₂²×Dic₆₇	C₂³	1	1	yes	characteristic
57	C₂²×Dic₆₇	C₂³	1	2	yes	—
58	C₂×C₄×D₆₇	—	2	4	no	—
59	C₂×C₄×D₆₇	C₂³	1	4	yes	—
60	C₂×C₄×D₆₇	C₂³	1	2	yes	—
61	C₂×C₄×D₆₇	C₂³	1	2	yes	—
62	C₂×C₄×D₆₇	C₂³	1	8	yes	—
63	C₄⋊Dic₆₇	C₂³	1	8	yes	—
64	C₄⋊Dic₆₇	C₂³	1	4	yes	—
65	Dic₆₇⋊C₄	C₂³	1	4	yes	—
66	Dic₆₇⋊C₄	C₂³	1	4	yes	—
67	Dic₆₇⋊C₄	C₂³	1	8	yes	—
68	(C₂×D₆₇)⋊C₄	C₂³	1	4	yes	—
69	(C₂×D₆₇)⋊C₄	C₂³	1	4	yes	—
70	(C₂×D₆₇)⋊C₄	C₂³	1	8	yes	—
71	(C₂×D₆₇)⋊C₄	C₂³	1	8	yes	—
72	(C₂×D₆₇)⋊C₄	C₂³	1	8	yes	—
73	(C₂×D₆₇)⋊C₄	C₂³	1	8	yes	—
74	C₄×Dic₆₇	C₂³	1	2	yes	—
75	C₄×Dic₆₇	C₂³	1	8	yes	—
76	C₄×Dic₆₇	C₂³	1	2	yes	—
77	C₂×Q₈×C₆₇	C₂³	1	4	yes	—
78	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
79	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
81	C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
82	C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
83	C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
84	C₄²×C₆₇	C₂³	1	2	yes	—
85	C₄²×C₆₇	C₂³	1	1	yes	characteristic
86	C₄²×C₆₇	C₂³	1	1	yes	characteristic
87	C₄²×C₆₇	C₂³	1	4	yes	—
88	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
89	C₂²×D₆₇	—	2	2	no	—
90	C₂²×D₆₇	—	4	2	no	—
91	C₂²×D₆₇	C₂⁴	1	2	yes	—
92	C₂²×D₆₇	C₂⁴	1	4	yes	—
93	Q₈×C₆₇	—	4	4	no	—
94	C₂×C₄×C₆₇	—	4	2	no	—
95	C₂×C₄×C₆₇	#41	1	4	yes	—
96	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
97	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
98	C₂×C₄×C₆₇	C₂⁴	1	8	yes	—
99	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
100	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
101	C₂²⋊D₆₇	—	2	8	no	—
102	C₂×Dic₆₇	—	2	2	no	—
103	C₂×Dic₆₇	—	4	2	no	—
104	C₂×Dic₆₇	C₂⁴	1	4	yes	—
105	C₂×Dic₆₇	C₂⁴	1	4	yes	—
106	C₂×Dic₆₇	C₂⁴	1	2	yes	—
107	D₂₆₈	—	2	4	no	—
108	D₂₆₈	—	4	8	no	—
109	D₂₆₈	—	4	4	no	—
110	C₄×D₆₇	—	2	4	no	—
111	C₄×D₆₇	—	2	4	no	—
112	C₄×D₆₇	—	2	8	no	—
113	C₄×D₆₇	—	4	8	no	—
114	Dic₁₃₄	—	2	4	no	—
115	C₂²×C₆₇	—	2	2	no	—
116	C₂²×C₆₇	C₂×#41	1	2	yes	—
117	C₂²×C₆₇	C₂⁵	1	1	yes	characteristic
118	C₂×D₆₇	—	2	2	no	—
119	C₂×D₆₇	—	2	4	no	—
120	C₂×D₆₇	—	2	8	no	—
121	C₂×D₆₇	—	2	4	no	—
122	C₂×D₆₇	—	4	4	no	—
123	C₄×C₆₇	—	2	8	no	—
124	C₄×C₆₇	—	2	2	no	—
125	C₄×C₆₇	—	2	2	no	—
126	C₄×C₆₇	C₂×#41	1	2	yes	—
127	C₄×C₆₇	C₂×#41	1	2	yes	—
128	Dic₆₇	—	2	4	no	—
129	Dic₆₇	—	2	2	no	—
130	Dic₆₇	—	2	4	no	—
131	C₂×C₆₇	—	2	2	no	—
132	C₂×C₆₇	C₂²×#41	1	1	yes	characteristic
133	C₂×C₆₇	#585	1	2	yes	—
134	D₆₇	—	4	4	no	—
135	D₆₇	—	4	2	no	—
136	#3455	—	67	1	no	maximal, Sylow
137	C₆₇	#3455	1	1	yes	characteristic, complemented, Sylow
138	#542	—	67	4	no	—
139	#541	—	67	2	no	—
140	#540	—	67	2	no	—
141	#539	—	67	4	no	—
142	#538	—	67	4	no	—
143	#533	D₆₇	1	1	yes	characteristic, complemented
144	#532	—	67	1	no	—
145	#529	—	67	8	no	—
146	#525	—	67	2	no	—
147	C₄×#41	—	67	2	no	—
148	C₂×#117	—	67	1	no	—
149	C₂×#41	—	67	2	no	—
150	C₂³×C₄	—	67	1	no	—
151	C₄⋊Q₈	—	67	1	no	—
152	C₄⋊Q₈	C₂×D₆₇	1	2	yes	complemented
153	#127	—	67	1	no	—
154	#126	—	67	8	no	—
155	#126	—	67	8	no	—
156	#125	—	67	8	no	—
157	#125	C₂×D₆₇	1	2	yes	complemented
158	#124	—	67	8	no	—
159	#124	—	67	2	no	—
160	#123	—	67	8	no	—
161	#123	—	67	8	no	—
162	#123	—	67	4	no	—
163	C₂²⋊Q₈	—	67	8	no	—
164	C₂²⋊Q₈	—	67	4	no	—
165	C₂²⋊Q₈	C₂×D₆₇	1	4	yes	—
166	#121	—	67	4	no	—
167	#121	—	67	4	no	—
168	#121	—	67	8	no	—
169	#120	—	67	4	no	—
170	C₄×Q₈	—	67	4	no	—
171	C₄×Q₈	C₂×D₆₇	1	4	yes	complemented
172	C₄×D₄	—	67	4	no	—
173	C₄×D₄	—	67	8	no	—
174	C₄×D₄	—	67	8	no	—
175	C₄×D₄	—	67	4	no	—
176	#117	—	67	4	no	—
177	#117	—	67	2	no	—
178	#117	—	67	2	no	—
179	#117	—	67	4	no	—
180	#117	C₂×D₆₇	1	2	yes	—
181	C₂×C₄⋊C₄	—	67	2	no	—
182	C₂×C₂²⋊C₄	—	67	2	no	—
183	C₂×C₄²	—	67	1	no	—
184	C₂×C₄²	—	67	4	no	—
185	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
186	C₂⁴	—	67	1	no	—
187	#41	—	134	8	no	—
188	C₂×Q₈	—	67	2	no	—
189	C₂×Q₈	C₂²×D₆₇	1	4	yes	—
190	C₂×D₄	—	67	4	no	—
191	C₂×D₄	—	67	8	no	—
192	C₂×D₄	—	67	4	no	—
193	C₂×D₄	—	67	2	no	—
194	C₂²×C₄	—	67	2	no	—
195	C₂²×C₄	—	67	8	no	—
196	C₂²×C₄	—	67	1	no	—
197	C₂²×C₄	—	67	2	no	—
198	C₂²×C₄	—	67	4	no	—
199	C₂²×C₄	—	67	2	no	—
200	C₂²×C₄	—	134	4	no	—
201	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
202	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
203	C₄⋊C₄	—	67	4	no	—
204	C₄⋊C₄	—	67	4	no	—
205	C₄⋊C₄	—	67	8	no	—
206	C₄⋊C₄	—	67	4	no	—
207	C₄⋊C₄	—	67	8	no	—
208	C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
209	C₄⋊C₄	C₂²×D₆₇	1	8	yes	—
210	C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
211	C₂²⋊C₄	—	67	4	no	—
212	C₂²⋊C₄	—	67	4	no	—
213	C₂²⋊C₄	—	67	8	no	—
214	C₂²⋊C₄	—	67	8	no	—
215	C₂²⋊C₄	—	67	8	no	—
216	C₂²⋊C₄	—	67	8	no	—
217	C₂²⋊C₄	C₂²×D₆₇	1	4	yes	—
218	C₄²	—	67	2	no	—
219	C₄²	—	67	8	no	—
220	C₄²	—	67	2	no	—
221	C₄²	C₂²×D₆₇	1	1	yes	characteristic
222	C₄²	C₂²×D₆₇	1	1	yes	characteristic
223	C₄²	C₂²×D₆₇	1	4	yes	—
224	C₄²	C₂²×D₆₇	1	2	yes	—
225	C₂³	—	268	2	no	—
226	C₂³	—	67	2	no	—
227	C₂³	—	67	4	no	—
228	C₂³	—	134	2	no	—
229	C₂³	C₂³×D₆₇	1	1	yes	characteristic
230	Q₈	—	4	4	no	—
231	Q₈	—	134	4	no	—
232	D₄	—	268	4	no	—
233	D₄	—	268	8	no	—
234	D₄	—	134	8	no	—
235	D₄	—	134	4	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	268	8	no	—
238	C₂×C₄	—	268	2	no	—
239	C₂×C₄	—	67	4	no	—
240	C₂×C₄	—	67	4	no	—
241	C₂×C₄	—	67	2	no	—
242	C₂×C₄	—	134	2	no	—
243	C₂×C₄	—	134	4	no	—
244	C₂×C₄	—	134	4	no	—
245	C₂×C₄	—	134	8	no	—
246	C₂×C₄	#11761629	1	4	yes	—
247	C₂×C₄	C₂³×D₆₇	1	8	yes	—
248	C₂×C₄	C₂³×D₆₇	1	2	yes	—
249	C₂×C₄	C₂³×D₆₇	1	2	yes	—
250	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
251	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
252	C₂²	—	2	2	no	—
253	C₂²	—	268	4	no	—
254	C₂²	—	134	2	no	—
255	C₂²	—	134	4	no	—
256	C₂²	—	134	8	no	—
257	C₂²	—	134	4	no	—
258	C₂²	C₂×#11761629	1	2	yes	—
259	C₂²	C₂⁴×D₆₇	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	2	no	—
262	C₄	—	2	8	no	—
263	C₄	—	134	2	no	—
264	C₄	—	134	4	no	—
265	C₄	—	134	4	no	—
266	C₄	C₂×#11761629	1	2	yes	—
267	C₄	C₂×#11761629	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	268	2	no	—
270	C₂	—	268	4	no	—
271	C₂	C₂²×#11761629	1	1	yes	characteristic
272	C₂	#16864232	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂⁵
Frattini	C₂²	C₂⁴×D₆₇
Fitting	#533×C₆₇	C₂
hypercenter	#533	D₆₇
hypocenter	C₆₇	#3455
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ #533

Complements

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

N	H ≅ G/N	type
1. #16864020	C₂	semidirect
2. #16864009	C₂	semidirect
3. #16864002	C₂	semidirect
4. #16863995	C₂	semidirect
5. #16863986	C₂	semidirect
6. #16863946	C₂	semidirect
7. #16863933	C₂	semidirect
8. #16863908	C₂	semidirect
9. #16863864	C₂	semidirect
10. C₂×#14524000	C₂	semidirect
11. #533×C₆₇	C₂	semidirect
14. #14524043	C₂²	semidirect
15. #14524040	C₂²	semidirect
17. #14524038	C₂²	semidirect
18. #14524034	C₂²	semidirect
20. #14524031	C₂²	semidirect
22. #14524028	C₂²	semidirect
25. #14524019	C₂²	semidirect
30. #14524008	C₂²	semidirect
32. C₄×D₂₆₈	C₂²	semidirect
33. #14524005	C₂²	semidirect
38. #14524000	C₂²	semidirect
42. C₄²×D₆₇	C₂²	semidirect
43. C₄⋊Q₈×C₆₇	C₂²	semidirect
44. #125×C₆₇	C₂²	semidirect
46. C₄×Q₈×C₆₇	C₂²	semidirect
137. C₆₇	#3455	semidirect
143. #533	D₆₇	semidirect
152. C₄⋊Q₈	C₂×D₆₇	semidirect
157. #125	C₂×D₆₇	semidirect
171. C₄×Q₈	C₂×D₆₇	semidirect