Subgroups — #42119330

Order 27584.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28352217	C₂	1	2	yes	maximal, complemented
2	C₂×D₃₄₄₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#28352195	C₂	1	1	yes	maximal, characteristic, complemented
4	#28352105	C₂	1	2	yes	maximal, complemented
5	#108×C₄₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#20293800	C₂²	1	1	yes	characteristic
7	C₂×D₁₇₂₄	C₂²	1	2	yes	—
8	#20293789	—	2	2	no	—
9	D₃₄₄₈	—	2	2	no	—
10	D₃₄₄₈	—	2	1	no	—
11	D₃₄₄₈	—	2	1	no	—
12	#20293784	C₂²	1	1	yes	characteristic
13	#20293783	—	2	1	no	—
14	C₈×D₄₃₁	—	2	1	no	—
15	M₁₆×C₄₃₁	C₂²	1	2	yes	—
16	C₂×C₈×C₄₃₁	C₂²	1	1	yes	characteristic
17	C₂²×D₄₃₁	—	2	2	no	—
18	Dic₈₆₂	D₄	1	1	yes	characteristic
19	D₁₇₂₄	—	2	2	no	—
20	D₁₇₂₄	—	2	2	no	—
21	D₁₇₂₄	D₄	1	1	yes	characteristic
22	C₂²⋊D₄₃₁	—	2	1	no	—
23	C₄×D₄₃₁	—	2	1	no	—
24	C₄₃₁⋊C₈	—	2	1	no	—
25	C₂×C₄×C₄₃₁	C₂³	1	1	yes	characteristic
26	C₈×C₄₃₁	—	2	2	no	—
27	C₈×C₄₃₁	D₄	1	1	yes	characteristic
28	C₈×C₄₃₁	D₄	1	1	yes	characteristic
29	C₂²×C₄₃₁	#41	1	1	yes	characteristic
30	C₂×D₄₃₁	—	2	1	no	—
31	C₂×D₄₃₁	—	4	2	no	—
32	C₂×D₄₃₁	—	8	2	no	—
33	C₄×C₄₃₁	C₂×D₄	1	1	yes	characteristic
34	C₄×C₄₃₁	C₂×D₄	1	1	yes	characteristic
35	Dic₄₃₁	—	2	1	no	—
36	C₂×C₄₃₁	—	2	1	no	—
37	C₂×C₄₃₁	#121	1	1	yes	characteristic
38	D₄₃₁	—	4	1	no	—
39	D₄₃₁	—	8	2	no	—
40	C₄₃₁	#472	1	1	yes	characteristic, complemented, Sylow
41	#472	—	431	1	no	maximal, Sylow
42	AGL(1,ℤ/8)	—	431	2	no	—
43	C₂×D₈	—	431	1	no	—
44	#131	—	431	1	no	—
45	#108	D₄₃₁	1	1	yes	characteristic, complemented
46	#100	—	431	2	no	—
47	#41	—	431	1	no	—
48	C₂×D₄	—	431	2	no	—
49	SD₁₆	—	862	2	no	—
50	D₈	—	862	2	no	—
51	D₈	—	862	1	no	—
52	D₈	—	862	1	no	—
53	M₁₆	—	862	1	no	—
54	M₁₆	—	431	1	no	—
55	M₁₆	C₂×D₄₃₁	1	2	yes	—
56	C₂×C₈	—	862	1	no	—
57	C₂×C₈	C₂×D₄₃₁	1	1	yes	characteristic
58	C₂³	—	862	2	no	—
59	Q₈	—	431	1	no	—
60	D₄	—	862	1	no	—
61	D₄	—	862	2	no	—
62	D₄	—	862	2	no	—
63	D₄	—	431	1	no	—
64	C₂×C₄	—	862	1	no	—
65	C₂×C₄	C₂²×D₄₃₁	1	1	yes	characteristic
66	C₈	—	2	2	no	—
67	C₈	—	862	1	no	—
68	C₈	D₁₇₂₄	1	1	yes	characteristic
69	C₈	D₁₇₂₄	1	1	yes	characteristic
70	C₂²	—	862	1	no	—
71	C₂²	—	3448	2	no	—
72	C₂²	—	1724	2	no	—
73	C₂²	#20293801	1	1	yes	characteristic
74	C₄	—	862	1	no	—
75	C₄	D₄×D₄₃₁	1	1	yes	characteristic
76	C₄	C₂×D₁₇₂₄	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	—	3448	2	no	—
79	C₂	—	1724	1	no	—
80	C₂	#28352164	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#28352164
commutator, cosocle	C₂×C₄×C₄₃₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₃₁
Fitting	#108×C₄₃₁	C₂
socle	C₂×C₄₃₁	#121
hypercenter	#108	D₄₃₁
hypocenter	C₄₃₁	#472
LCS	G ▷ C₂×C₄×C₄₃₁ ▷ C₂×C₄₃₁ ▷ C₄₃₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #108

Complements

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

N	H ≅ G/N	type
1. #28352217	C₂	semidirect
2. C₂×D₃₄₄₈	C₂	semidirect
3. #28352195	C₂	semidirect
4. #28352105	C₂	semidirect
5. #108×C₄₃₁	C₂	semidirect
40. C₄₃₁	#472	semidirect
45. #108	D₄₃₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#28352217	2
C₂×D₃₄₄₈	2
#28352195	2
#28352105	2
#108×C₄₃₁	2
#472	431