Subgroups — #22558513

Order 8576.

214 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#11761611	C₂	1	2	yes	maximal
2	C₄×#11761611	C₂	1	1	yes	maximal, characteristic
3	#16863118	C₂	1	2	yes	maximal, complemented
4	C₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#322×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11761611	C₄	1	4	yes	—
7	C₂×#11761611	C₄	1	2	yes	—
8	C₄²×D₆₇	C₂²	1	1	yes	characteristic
9	#14523953	C₄	1	4	yes	—
10	#14523953	C₄	1	2	yes	—
11	C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
12	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄×D₆₇	C₂×C₄	1	2	yes	—
16	C₂×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
17	#11761611	—	2	4	no	—
18	#11761611	—	2	8	no	—
19	C₂×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
20	C₂×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
21	C₄×Dic₆₇	C₂×C₄	1	2	yes	—
22	C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₆₇	C₂×C₄	1	4	yes	—
24	C₂×C₈×C₆₇	C₂×C₄	1	2	yes	—
25	C₄²×C₆₇	C₂³	1	1	yes	characteristic
26	C₂²×D₆₇	C₄²	1	1	yes	characteristic
27	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
28	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
29	C₂×Dic₆₇	C₄²	1	1	yes	characteristic
30	C₂×Dic₆₇	C₄²	1	2	yes	—
31	C₄×D₆₇	—	2	4	no	—
32	C₄×D₆₇	—	2	2	no	—
33	C₈×C₆₇	—	2	4	no	—
34	C₈×C₆₇	M₁₆	1	4	yes	complemented
35	C₆₇⋊C₈	—	2	2	no	—
36	C₆₇⋊C₈	M₁₆	1	8	yes	complemented
37	C₂²×C₆₇	C₂×C₄²	1	1	yes	characteristic
38	C₂×D₆₇	—	2	1	no	—
39	C₂×D₆₇	—	2	2	no	—
40	C₄×C₆₇	C₂×M₁₆	1	4	yes	—
41	C₄×C₆₇	C₂×M₁₆	1	2	yes	—
42	Dic₆₇	—	2	1	no	—
43	Dic₆₇	—	2	2	no	—
44	C₂×C₆₇	C₄×M₁₆	1	1	yes	characteristic
45	C₂×C₆₇	C₄×M₁₆	1	2	yes	—
46	D₆₇	—	4	1	no	—
47	#1455	—	67	1	no	maximal, Sylow
48	C₆₇	#1455	1	1	yes	characteristic, complemented, Sylow
49	C₄×M₁₆	—	67	2	no	—
50	C₄×M₁₆	—	67	1	no	—
51	#322	—	67	2	no	—
52	#322	D₆₇	1	1	yes	characteristic, complemented
53	C₈²	—	67	1	no	—
54	C₂×M₁₆	—	67	4	no	—
55	C₂×M₁₆	—	67	2	no	—
56	C₂×C₄²	—	67	1	no	—
57	#97	—	67	4	no	—
58	#97	—	67	2	no	—
59	C₄×C₈	—	67	2	no	—
60	C₄×C₈	—	67	1	no	—
61	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
62	C₄×C₈	C₂×D₆₇	1	2	yes	—
63	C₂²×C₄	—	67	2	no	—
64	C₂²×C₄	—	67	1	no	—
65	M₁₆	—	134	8	no	—
66	M₁₆	—	134	4	no	—
67	C₂×C₈	—	67	4	no	—
68	C₂×C₈	—	67	2	no	—
69	C₂×C₈	C₄×D₆₇	1	4	yes	—
70	C₂×C₈	C₄×D₆₇	1	2	yes	—
71	C₄²	—	67	2	no	—
72	C₄²	—	67	1	no	—
73	C₄²	C₂²×D₆₇	1	1	yes	characteristic
74	C₂³	—	67	1	no	—
75	C₂×C₄	—	67	1	no	—
76	C₂×C₄	—	67	2	no	—
77	C₂×C₄	—	134	4	no	—
78	C₂×C₄	—	134	2	no	—
79	C₂×C₄	C₂×C₄×D₆₇	1	2	yes	—
80	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
81	C₈	—	2	4	no	—
82	C₈	—	67	8	no	—
83	C₈	—	134	2	no	—
84	C₈	#11761611	1	4	yes	complemented
85	C₂²	—	134	1	no	—
86	C₂²	—	134	2	no	—
87	C₂²	C₄²×D₆₇	1	1	yes	characteristic
88	C₄	—	134	1	no	—
89	C₄	—	134	2	no	—
90	C₄	M₁₆×D₆₇	1	4	yes	—
91	C₄	C₂×#11761611	1	2	yes	—
92	C₂	—	268	1	no	—
93	C₂	C₄×#11761611	1	1	yes	characteristic
94	C₂	#16863348	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₆₇
commutator, socle	C₂²×C₆₇	C₂×C₄²
Fitting	#322×C₆₇	C₂
hypercenter	#322	D₆₇
hypocenter	C₆₇	#1455
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄² ◁ #322

Complements

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

N	H ≅ G/N	type
3. #16863118	C₂	semidirect
4. C₈×C₆₇⋊C₈	C₂	semidirect
5. #322×C₆₇	C₂	semidirect
34. C₈×C₆₇	M₁₆	semidirect
36. C₆₇⋊C₈	M₁₆	semidirect
48. C₆₇	#1455	semidirect
52. #322	D₆₇	semidirect
84. C₈	#11761611	semidirect