Subgroups — #42119416

Order 27584.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28352177	C₂	1	1	yes	maximal, characteristic
2	#28352112	C₂	1	2	yes	maximal, complemented
3	#28352109	C₂	1	2	yes	maximal
4	C₄×C₄₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#125×C₄₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₈₆₂	C₂²	1	1	yes	characteristic
7	C₂×D₁₇₂₄	C₂²	1	1	yes	characteristic
8	C₂×C₄₃₁⋊C₈	C₂²	1	2	yes	—
9	(C₂×D₄₃₁)⋊C₄	—	2	2	no	—
10	C₄⋊C₄×C₄₃₁	—	2	2	no	—
11	C₄⋊C₄×C₄₃₁	C₂²	1	2	yes	—
12	C₄²×C₄₃₁	C₂²	1	1	yes	characteristic
13	C₂²×D₄₃₁	—	2	1	no	—
14	Dic₈₆₂	—	2	2	no	—
15	D₁₇₂₄	—	2	2	no	—
16	C₂×Dic₄₃₁	—	2	1	no	—
17	C₄₃₁⋊C₈	—	2	2	no	—
18	C₂×C₄×C₄₃₁	—	2	2	no	—
19	C₂×C₄×C₄₃₁	D₄	1	2	yes	—
20	C₂×C₄×C₄₃₁	C₂³	1	1	yes	characteristic
21	C₂²×C₄₃₁	C₂×D₄	1	1	yes	characteristic
22	C₂×D₄₃₁	—	4	1	no	—
23	C₂×D₄₃₁	—	4	2	no	—
24	C₄×C₄₃₁	—	2	2	no	—
25	C₄×C₄₃₁	—	4	2	no	—
26	C₄×C₄₃₁	#41	1	2	yes	—
27	Dic₄₃₁	—	4	1	no	—
28	C₂×C₄₃₁	#124	1	1	yes	characteristic
29	C₂×C₄₃₁	#135	1	2	yes	—
30	D₄₃₁	—	8	1	no	—
31	C₄₃₁	#488	1	1	yes	characteristic, complemented, Sylow
32	#488	—	431	1	no	maximal, Sylow
33	#125	D₄₃₁	1	1	yes	characteristic, complemented
34	#124	—	431	1	no	—
35	#103	—	431	2	no	—
36	#102	—	431	2	no	—
37	C₄×C₈	—	431	1	no	—
38	C₂×Q₈	—	431	1	no	—
39	C₂×D₄	—	431	1	no	—
40	C₂×C₈	—	431	2	no	—
41	C₄⋊C₄	—	2	2	no	—
42	C₄⋊C₄	C₂×D₄₃₁	1	2	yes	—
43	C₂²⋊C₄	—	862	2	no	—
44	C₄²	C₂×D₄₃₁	1	1	yes	characteristic
45	C₂³	—	862	1	no	—
46	Q₈	—	862	2	no	—
47	D₄	—	862	2	no	—
48	C₂×C₄	—	2	2	no	—
49	C₂×C₄	—	862	1	no	—
50	C₂×C₄	C₂²⋊D₄₃₁	1	2	yes	—
51	C₂×C₄	C₂²×D₄₃₁	1	1	yes	characteristic
52	C₈	—	862	2	no	—
53	C₂²	—	1724	1	no	—
54	C₂²	—	1724	2	no	—
55	C₂²	C₂×C₂²⋊D₄₃₁	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	—	4	2	no	—
58	C₄	—	1724	1	no	—
59	C₄	#20293801	1	2	yes	—
60	C₂	—	3448	1	no	—
61	C₂	#28352178	1	1	yes	characteristic
62	C₂	#28352209	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₄₃₁
commutator, cosocle	C₂×C₄×C₄₃₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₃₁
Fitting	#125×C₄₃₁	C₂
socle	C₂²×C₄₃₁	C₂×D₄
prime radical	C₂×D₁₇₂₄	C₂²
hypercenter	#125	D₄₃₁
hypocenter	C₄₃₁	#488
LCS	G ▷ C₂×C₄×C₄₃₁ ▷ C₂×C₄₃₁ ▷ C₄₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
2. #28352112	C₂	semidirect
4. C₄×C₄₃₁⋊C₈	C₂	semidirect
5. #125×C₄₃₁	C₂	semidirect
31. C₄₃₁	#488	semidirect
33. #125	D₄₃₁	semidirect

Maximal subgroups

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

subgroup	index
#28352177	2
#28352112	2
#28352109	2
C₄×C₄₃₁⋊C₈	2
#125×C₄₃₁	2
#488	431