Subgroups — #22558366

Order 8576.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16863193	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863191	C₂	1	1	yes	maximal, characteristic, complemented
3	#496×C₆₇	C₂	1	1	yes	maximal, characteristic
4	#14523962	—	2	1	no	—
5	#14523958	—	2	1	no	—
6	C₂×SD₁₆×C₆₇	—	2	1	no	—
7	C₂×D₈×C₆₇	—	2	1	no	—
8	#127×C₆₇	C₄	1	1	yes	characteristic, complemented
9	#124×C₆₇	C₂²	1	1	yes	characteristic
10	#97×C₆₇	C₄	1	1	yes	characteristic, complemented
11	#11761612	—	4	1	no	—
12	C₂²⋊Dic₆₇	—	4	1	no	—
13	C₂×Q₈×C₆₇	D₄	1	1	yes	characteristic
14	C₂×D₄×C₆₇	—	2	1	no	—
15	C₂×D₄×C₆₇	—	4	1	no	—
16	C₂×D₄×C₆₇	D₄	1	1	yes	characteristic
17	SD₁₆×C₆₇	—	4	2	no	—
18	D₈×C₆₇	—	4	2	no	—
19	C₂×C₈×C₆₇	—	2	1	no	—
20	C₂²⋊C₄×C₆₇	—	4	1	no	—
21	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₆₇	—	2	1	no	—
23	C₂³×C₆₇	—	4	1	no	—
24	Q₈×C₆₇	—	4	1	no	—
25	D₄×C₆₇	—	4	1	no	—
26	D₄×C₆₇	—	4	1	no	—
27	D₄×C₆₇	—	4	1	no	—
28	D₄×C₆₇	—	8	2	no	—
29	C₂×C₄×C₆₇	—	2	1	no	—
30	C₂×C₄×C₆₇	—	2	1	no	—
31	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×Dic₆₇	—	8	1	no	—
33	C₈×C₆₇	—	2	2	no	—
34	C₆₇⋊C₈	—	8	1	no	—
35	C₂²×C₆₇	—	4	1	no	—
36	C₂²×C₆₇	—	8	1	no	—
37	C₂²×C₆₇	—	8	2	no	—
38	C₂²×C₆₇	—	8	1	no	—
39	C₂²×C₆₇	#99	1	1	yes	characteristic
40	C₄×C₆₇	—	2	1	no	—
41	C₄×C₆₇	—	4	1	no	—
42	C₄×C₆₇	—	4	1	no	—
43	Dic₆₇	—	16	1	no	—
44	C₂×C₆₇	—	2	1	no	—
45	C₂×C₆₇	—	8	1	no	—
46	C₂×C₆₇	—	16	1	no	—
47	C₂×C₆₇	#353	1	1	yes	characteristic
48	#1406	—	67	1	no	maximal, Sylow
49	C₆₇	#1406	1	1	yes	characteristic, complemented, Sylow
50	#496	D₆₇	1	1	yes	characteristic
51	#355	—	67	1	no	—
52	#354	—	67	1	no	—
53	C₂×SD₁₆	—	2	1	no	—
54	C₂×D₈	—	2	1	no	—
55	#127	Dic₆₇	1	1	yes	characteristic, complemented
56	#124	C₂×D₆₇	1	1	yes	characteristic
57	#101	—	134	1	no	—
58	#99	—	134	1	no	—
59	#97	Dic₆₇	1	1	yes	characteristic, complemented
60	C₂×Q₈	C₂²⋊D₆₇	1	1	yes	characteristic
61	C₂×D₄	—	2	1	no	—
62	C₂×D₄	—	4	1	no	—
63	C₂×D₄	C₂²⋊D₆₇	1	1	yes	characteristic
64	SD₁₆	—	4	2	no	—
65	D₈	—	4	2	no	—
66	M₁₆	—	268	1	no	—
67	C₂×C₈	—	2	1	no	—
68	C₂²⋊C₄	—	4	1	no	—
69	C₂²⋊C₄	—	268	1	no	—
70	C₄²	C₂×Dic₆₇	1	1	yes	characteristic
71	C₂³	—	2	1	no	—
72	C₂³	—	4	1	no	—
73	Q₈	—	4	1	no	—
74	D₄	—	4	1	no	—
75	D₄	—	4	1	no	—
76	D₄	—	4	1	no	—
77	D₄	—	8	2	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	2	1	no	—
80	C₂×C₄	—	536	1	no	—
81	C₂×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
82	C₈	—	2	2	no	—
83	C₈	—	536	1	no	—
84	C₂²	—	4	1	no	—
85	C₂²	—	8	1	no	—
86	C₂²	—	8	2	no	—
87	C₂²	—	8	1	no	—
88	C₂²	#14523958	1	1	yes	characteristic
89	C₄	—	2	1	no	—
90	C₄	—	4	1	no	—
91	C₄	—	4	1	no	—
92	C₄	—	1072	1	no	—
93	C₂	—	2	1	no	—
94	C₂	—	8	1	no	—
95	C₂	—	16	1	no	—
96	C₂	#16863189	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16863189
commutator	C₄²×C₆₇	C₂×C₄
Frattini	#124	C₂×D₆₇
Fitting	#496×C₆₇	C₂
socle	C₂×C₆₇	#353
hypercenter	#496	D₆₇
hypocenter	C₆₇	#1406
LCS	G ▷ C₄²×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #124 ◁ #496

Complements

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

N	H ≅ G/N	type
1. #16863193	C₂	semidirect
2. #16863191	C₂	semidirect
8. #127×C₆₇	C₄	semidirect
10. #97×C₆₇	C₄	semidirect
49. C₆₇	#1406	semidirect
55. #127	Dic₆₇	semidirect
59. #97	Dic₆₇	semidirect