Subgroups — #42119270

Order 27584.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₇₂₄	C₂	1	2	yes	maximal
2	#28352189	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₈₆₂	C₂	1	1	yes	maximal, characteristic
4	#28352114	C₂	1	2	yes	maximal
5	C₄⋊C₈×C₄₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₈₆₂	C₂²	1	2	yes	—
7	C₂×Dic₈₆₂	C₂²	1	1	yes	characteristic
8	Dic₁₇₂₄	—	2	4	no	—
9	C₄⋊Dic₄₃₁	—	2	1	no	—
10	C₄⋊Dic₄₃₁	—	2	1	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	Dic₈₆₂	—	2	1	no	—
17	Dic₈₆₂	—	2	2	no	—
18	Dic₈₆₂	—	2	2	no	—
19	Dic₈₆₂	D₄	1	2	yes	—
20	C₂×Dic₄₃₁	—	2	1	no	—
21	C₂×Dic₄₃₁	—	2	1	no	—
22	C₂×Dic₄₃₁	—	2	2	no	—
23	C₂×C₄×C₄₃₁	D₄	1	1	yes	characteristic
24	C₂×C₄×C₄₃₁	D₄	1	1	yes	characteristic
25	C₂×C₄×C₄₃₁	C₂³	1	1	yes	characteristic
26	C₈×C₄₃₁	—	2	2	no	—
27	C₂²×C₄₃₁	C₂×D₄	1	1	yes	characteristic
28	C₄×C₄₃₁	—	2	1	no	—
29	C₄×C₄₃₁	Q₁₆	1	2	yes	complemented
30	C₄×C₄₃₁	C₂×D₄	1	1	yes	characteristic
31	C₄×C₄₃₁	#41	1	1	yes	characteristic
32	Dic₄₃₁	—	2	2	no	—
33	Dic₄₃₁	—	4	2	no	—
34	Dic₄₃₁	—	4	1	no	—
35	C₂×C₄₃₁	#121	1	1	yes	characteristic
36	C₂×C₄₃₁	C₂×Q₁₆	1	1	yes	characteristic
37	C₂×C₄₃₁	#137	1	1	yes	characteristic
38	C₄₃₁	#462	1	1	yes	characteristic, complemented, Sylow
39	#462	—	431	1	no	maximal, Sylow
40	C₂×Q₁₆	—	431	2	no	—
41	C₄⋊Q₈	—	431	1	no	—
42	C₄×Q₈	—	431	1	no	—
43	C₄⋊C₈	D₄₃₁	1	1	yes	characteristic
44	#103	—	431	2	no	—
45	C₂×Q₈	—	431	2	no	—
46	C₂×Q₈	—	431	1	no	—
47	Q₁₆	—	862	4	no	—
48	C₂×C₈	C₂×D₄₃₁	1	2	yes	—
49	C₄⋊C₄	—	862	1	no	—
50	C₄⋊C₄	—	862	1	no	—
51	C₄⋊C₄	—	862	1	no	—
52	C₄⋊C₄	—	431	1	no	—
53	C₄²	—	862	1	no	—
54	C₄²	C₂×D₄₃₁	1	1	yes	characteristic
55	Q₈	—	862	1	no	—
56	Q₈	—	862	2	no	—
57	Q₈	—	862	2	no	—
58	Q₈	—	431	2	no	—
59	C₂×C₄	—	862	1	no	—
60	C₂×C₄	—	862	1	no	—
61	C₂×C₄	—	862	2	no	—
62	C₂×C₄	D₁₇₂₄	1	1	yes	characteristic
63	C₂×C₄	D₁₇₂₄	1	1	yes	characteristic
64	C₂×C₄	C₂²×D₄₃₁	1	1	yes	characteristic
65	C₈	—	2	2	no	—
66	C₂²	C₂×D₁₇₂₄	1	1	yes	characteristic
67	C₄	—	2	1	no	—
68	C₄	—	862	2	no	—
69	C₄	—	1724	2	no	—
70	C₄	—	1724	1	no	—
71	C₄	Dic₁₇₂₄	1	2	yes	complemented
72	C₄	D₄×D₄₃₁	1	1	yes	characteristic
73	C₄	#20293801	1	1	yes	characteristic
74	C₂	#28352164	1	1	yes	characteristic
75	C₂	C₂×Dic₁₇₂₄	1	1	yes	characteristic
76	C₂	#28352222	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, prime radical	C₂²×C₄₃₁	C₂×D₄
hypercenter	C₄⋊C₈	D₄₃₁
hypocenter	C₄₃₁	#462
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
29. C₄×C₄₃₁	Q₁₆	semidirect
38. C₄₃₁	#462	semidirect
71. C₄	Dic₁₇₂₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₇₂₄	2
#28352189	2
C₄×Dic₈₆₂	2
#28352114	2
C₄⋊C₈×C₄₃₁	2
#462	431