Subgroups — #42119260

Order 27584.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20293789	C₂	1	2	yes	maximal, complemented
2	#28352186	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₈₆₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#28352111	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₈×C₄₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₈₆₂	C₂²	1	1	yes	characteristic
7	C₂×D₁₇₂₄	—	2	1	no	—
8	C₂×D₁₇₂₄	—	2	1	no	—
9	C₂×D₁₇₂₄	C₂²	1	2	yes	—
10	#20293789	—	2	4	no	—
11	C₄⋊Dic₄₃₁	C₂²	1	1	yes	characteristic
12	Dic₄₃₁⋊C₄	—	2	1	no	—
13	C₄×Dic₄₃₁	—	2	1	no	—
14	C₂×C₈×C₄₃₁	C₂²	1	2	yes	—
15	C₄²×C₄₃₁	C₂²	1	1	yes	characteristic
16	C₂²×D₄₃₁	—	2	2	no	—
17	Dic₈₆₂	—	2	1	no	—
18	Dic₈₆₂	D₄	1	2	yes	complemented
19	D₁₇₂₄	—	2	2	no	—
20	D₁₇₂₄	—	2	2	no	—
21	D₁₇₂₄	—	4	2	no	—
22	D₁₇₂₄	—	4	2	no	—
23	C₂×Dic₄₃₁	—	2	1	no	—
24	C₂×Dic₄₃₁	—	2	1	no	—
25	C₂×C₄×C₄₃₁	D₄	1	1	yes	characteristic
26	C₂×C₄×C₄₃₁	D₄	1	1	yes	characteristic
27	C₂×C₄×C₄₃₁	C₂³	1	1	yes	characteristic
28	C₈×C₄₃₁	—	2	2	no	—
29	C₂²×C₄₃₁	C₂×D₄	1	1	yes	characteristic
30	C₂×D₄₃₁	—	4	2	no	—
31	C₂×D₄₃₁	—	4	2	no	—
32	C₂×D₄₃₁	—	4	2	no	—
33	C₄×C₄₃₁	—	2	1	no	—
34	C₄×C₄₃₁	SD₁₆	1	2	yes	complemented
35	C₄×C₄₃₁	C₂×D₄	1	1	yes	characteristic
36	C₄×C₄₃₁	#41	1	1	yes	characteristic
37	Dic₄₃₁	—	2	2	no	—
38	Dic₄₃₁	—	4	1	no	—
39	C₂×C₄₃₁	#121	1	1	yes	characteristic
40	C₂×C₄₃₁	C₂×SD₁₆	1	1	yes	characteristic
41	C₂×C₄₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
42	D₄₃₁	—	8	2	no	—
43	C₄₃₁	#460	1	1	yes	characteristic, complemented, Sylow
44	#460	—	431	1	no	maximal, Sylow
45	C₂×SD₁₆	—	431	2	no	—
46	#127	—	431	1	no	—
47	C₄×Q₈	—	431	1	no	—
48	C₄⋊C₈	D₄₃₁	1	1	yes	characteristic, complemented
49	#102	—	431	2	no	—
50	C₂×Q₈	—	431	1	no	—
51	C₂×D₄	—	862	1	no	—
52	C₂×D₄	—	862	1	no	—
53	C₂×D₄	—	431	2	no	—
54	SD₁₆	—	862	4	no	—
55	C₂×C₈	C₂×D₄₃₁	1	2	yes	—
56	C₄⋊C₄	—	862	1	no	—
57	C₄⋊C₄	—	431	1	no	—
58	C₄²	—	862	1	no	—
59	C₄²	C₂×D₄₃₁	1	1	yes	characteristic
60	C₂³	—	862	2	no	—
61	Q₈	—	862	1	no	—
62	Q₈	—	431	2	no	—
63	D₄	—	862	2	no	—
64	D₄	—	862	2	no	—
65	D₄	—	1724	2	no	—
66	D₄	—	1724	2	no	—
67	C₂×C₄	—	862	1	no	—
68	C₂×C₄	—	862	1	no	—
69	C₂×C₄	D₁₇₂₄	1	1	yes	characteristic
70	C₂×C₄	D₁₇₂₄	1	1	yes	characteristic
71	C₂×C₄	C₂²×D₄₃₁	1	1	yes	characteristic
72	C₈	—	2	2	no	—
73	C₂²	—	1724	2	no	—
74	C₂²	—	1724	2	no	—
75	C₂²	—	1724	2	no	—
76	C₂²	C₂×D₁₇₂₄	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	862	2	no	—
79	C₄	—	1724	1	no	—
80	C₄	#20293789	1	2	yes	complemented
81	C₄	D₄×D₄₃₁	1	1	yes	characteristic
82	C₄	#20293801	1	1	yes	characteristic
83	C₂	—	3448	2	no	—
84	C₂	#28352164	1	1	yes	characteristic
85	C₂	C₂×#20293789	1	1	yes	characteristic
86	C₂	#28352217	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₇₂₄
commutator, cosocle	C₂×C₄×C₄₃₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₃₁
Fitting	C₄⋊C₈×C₄₃₁	C₂
socle	C₂²×C₄₃₁	C₂×D₄
prime radical	#28352186	C₂
hypercenter	C₄⋊C₈	D₄₃₁
hypocenter	C₄₃₁	#460
LCS	G ▷ C₂×C₄×C₄₃₁ ▷ C₂×C₄₃₁ ▷ C₄₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#20293789	C₂	semidirect
3. C₄×Dic₈₆₂	C₂	semidirect
4. #28352111	C₂	semidirect
5. C₄⋊C₈×C₄₃₁	C₂	semidirect
18. Dic₈₆₂	D₄	semidirect
34. C₄×C₄₃₁	SD₁₆	semidirect
43. C₄₃₁	#460	semidirect
48. C₄⋊C₈	D₄₃₁	semidirect
80. C₄	#20293789	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20293789	2
#28352186	2
C₄×Dic₈₆₂	2
#28352111	2
C₄⋊C₈×C₄₃₁	2
#460	431