Subgroups — #22562452

Order 8576.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16863773	C₂	1	1	yes	maximal, characteristic
2	#16863196	C₂	1	4	yes	maximal, complemented
3	#16863172	C₂	1	1	yes	maximal, characteristic, complemented
4	#499×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₅₃₆	—	2	1	no	—
6	C₂×D₅₃₆	C₂²	1	2	yes	—
7	#14524040	—	2	1	no	—
8	C₂×C₆₇⋊C₁₆	C₂²	1	2	yes	—
9	#125×C₆₇	—	2	1	no	—
10	#107×C₆₇	C₂²	1	2	yes	—
11	#106×C₆₇	—	2	1	no	—
12	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×D₂₆₈	—	2	2	no	—
14	C₂×D₂₆₈	—	4	2	no	—
15	D₅₃₆	—	2	4	no	—
16	D₅₃₆	—	4	2	no	—
17	C₆₇⋊C₁₆	—	2	2	no	—
18	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
19	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₆₇	—	2	2	no	—
21	C₄⋊C₄×C₆₇	—	4	2	no	—
22	C₄²×C₆₇	D₄	1	1	yes	characteristic
23	C₂²×D₆₇	—	4	2	no	—
24	C₂×C₄×C₆₇	—	4	2	no	—
25	C₂×C₄×C₆₇	D₈	1	2	yes	—
26	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
27	D₂₆₈	—	4	2	no	—
28	D₂₆₈	—	4	2	no	—
29	D₂₆₈	—	8	4	no	—
30	C₈×C₆₇	SD₁₆	1	2	yes	—
31	C₈×C₆₇	#41	1	2	yes	—
32	C₂²×C₆₇	C₂×D₈	1	1	yes	characteristic
33	C₂×D₆₇	—	8	2	no	—
34	C₂×D₆₇	—	8	4	no	—
35	C₄×C₆₇	—	2	2	no	—
36	C₄×C₆₇	—	8	2	no	—
37	C₄×C₆₇	#124	1	1	yes	characteristic
38	C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
39	C₂×C₆₇	#486	1	1	yes	characteristic
40	C₂×C₆₇	#509	1	2	yes	—
41	D₆₇	—	16	2	no	—
42	#2244	—	67	1	no	maximal, Sylow
43	C₆₇	#2244	1	1	yes	characteristic, complemented, Sylow
44	#499	D₆₇	1	1	yes	characteristic, complemented
45	#493	—	67	1	no	—
46	#357	—	67	4	no	—
47	#346	—	67	1	no	—
48	C₂×D₈	—	67	2	no	—
49	C₂×D₈	—	134	1	no	—
50	#127	—	134	1	no	—
51	#125	—	2	1	no	—
52	C₂×C₁₆	—	67	2	no	—
53	#107	C₂×D₆₇	1	2	yes	—
54	#106	—	2	1	no	—
55	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
56	C₂×D₄	—	268	2	no	—
57	C₂×D₄	—	134	2	no	—
58	D₈	—	268	2	no	—
59	D₈	—	134	4	no	—
60	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
61	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
62	C₄⋊C₄	—	2	2	no	—
63	C₄⋊C₄	—	4	2	no	—
64	C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
65	C₁₆	—	134	2	no	—
66	C₂³	—	268	2	no	—
67	D₄	—	268	2	no	—
68	D₄	—	268	2	no	—
69	D₄	—	536	4	no	—
70	C₂×C₄	—	4	2	no	—
71	C₂×C₄	#11761613	1	2	yes	—
72	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
73	C₈	#11761615	1	2	yes	—
74	C₈	#11761629	1	2	yes	—
75	C₂²	—	536	2	no	—
76	C₂²	—	536	4	no	—
77	C₂²	C₂×#11761613	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	8	2	no	—
80	C₄	#14524032	1	1	yes	characteristic
81	C₄	C₂×#11761615	1	1	yes	characteristic
82	C₂	—	1072	2	no	—
83	C₂	#16863744	1	1	yes	characteristic
84	C₂	#16863831	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#11761613
commutator	C₂×C₈×C₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	#499×C₆₇	C₂
socle	C₂²×C₆₇	C₂×D₈
hypercenter	#499	D₆₇
hypocenter	C₆₇	#2244
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #499

Complements

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

N	H ≅ G/N	type
2. #16863196	C₂	semidirect
3. #16863172	C₂	semidirect
4. #499×C₆₇	C₂	semidirect
43. C₆₇	#2244	semidirect
44. #499	D₆₇	semidirect