Subgroups — #22562004

Order 8576.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14523979	C₂	1	2	yes	maximal
2	#16863226	C₂	1	4	yes	maximal, complemented
3	C₂×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#11761612	—	2	2	no	—
5	#14523979	—	2	2	no	—
6	#14523979	C₂²	1	4	yes	—
7	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
8	M₃₂×C₆₇	C₄	1	4	yes	—
9	C₂×C₁₆×C₆₇	C₂²	1	2	yes	—
10	#11761612	—	2	4	no	—
11	#11761612	—	4	2	no	—
12	C₂×C₆₇⋊C₈	—	4	2	no	—
13	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
14	C₂×C₈×C₆₇	C₂×C₄	1	2	yes	—
15	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
16	C₂×C₈×C₆₇	Q₈	1	2	yes	—
17	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
18	C₁₆×C₆₇	C₂×C₄	1	4	yes	—
19	C₂³×C₆₇	Q₁₆	1	1	yes	characteristic
20	C₂×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
21	C₂×C₄×C₆₇	D₈	1	2	yes	—
22	C₂×C₄×C₆₇	Q₁₆	1	1	yes	characteristic
23	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
24	C₈×C₆₇	C₄⋊C₄	1	2	yes	—
25	C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
26	C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
27	C₆₇⋊C₈	—	4	4	no	—
28	C₂²×C₆₇	—	4	1	no	—
29	C₂²×C₆₇	#107	1	2	yes	—
30	C₂²×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
31	C₄×C₆₇	#107	1	2	yes	—
32	C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
33	C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
34	C₂×C₆₇	—	2	2	no	—
35	C₂×C₆₇	—	2	1	no	—
36	C₂×C₆₇	C₂×#107	1	1	yes	characteristic
37	#2161	—	67	1	no	maximal, Sylow
38	C₆₇	#2161	1	1	yes	characteristic, complemented, Sylow
39	C₂×M₃₂	D₆₇	1	1	yes	characteristic
40	C₂×#108	—	67	2	no	—
41	#368	—	67	4	no	—
42	C₂×M₁₆	—	134	2	no	—
43	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
44	M₃₂	Dic₆₇	1	4	yes	—
45	C₂×C₁₆	C₂×D₆₇	1	2	yes	—
46	#108	—	67	4	no	—
47	#108	—	134	2	no	—
48	C₂²×C₄	D₂₆₈	1	1	yes	characteristic
49	M₁₆	—	268	2	no	—
50	M₁₆	—	134	4	no	—
51	C₂×C₈	—	268	2	no	—
52	C₂×C₈	Dic₁₃₄	1	2	yes	—
53	C₂×C₈	D₂₆₈	1	1	yes	characteristic
54	C₂×C₈	C₂×Dic₆₇	1	2	yes	—
55	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
56	C₁₆	C₂×Dic₆₇	1	4	yes	—
57	C₂³	Dic₂₆₈	1	1	yes	characteristic
58	C₂×C₄	C₄⋊Dic₆₇	1	2	yes	—
59	C₂×C₄	D₅₃₆	1	2	yes	—
60	C₂×C₄	Dic₂₆₈	1	1	yes	characteristic
61	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
62	C₈	—	268	4	no	—
63	C₈	C₄⋊Dic₆₇	1	2	yes	—
64	C₈	C₂²×Dic₆₇	1	1	yes	characteristic
65	C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
66	C₂²	—	4	1	no	—
67	C₂²	#14523977	1	2	yes	—
68	C₂²	C₂×Dic₂₆₈	1	1	yes	characteristic
69	C₄	#14523977	1	2	yes	—
70	C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
71	C₄	C₂×D₅₃₆	1	1	yes	characteristic
72	C₂	—	2	2	no	—
73	C₂	—	2	1	no	—
74	C₂	C₂×#14523977	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₅₃₆
commutator	C₈×C₆₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	C₂×M₃₂×C₆₇	C₂
socle	C₂×C₆₇	C₂×#107
hypercenter	C₂×M₃₂	D₆₇
hypocenter	C₆₇	#2161
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
2. #16863226	C₂	semidirect
38. C₆₇	#2161	semidirect