Subgroups — #31774108

Order 17344.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₀₈₄	C₂	1	2	yes	maximal
2	#22576160	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₅₄₂	C₂	1	1	yes	maximal, characteristic
4	#22576085	C₂	1	2	yes	maximal
5	C₄⋊C₈×C₂₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₄₂	C₂²	1	2	yes	—
7	C₂×Dic₅₄₂	C₂²	1	1	yes	characteristic
8	Dic₁₀₈₄	—	2	4	no	—
9	C₄⋊Dic₂₇₁	—	2	1	no	—
10	C₄⋊Dic₂₇₁	—	2	1	no	—
11	C₄⋊Dic₂₇₁	C₂²	1	1	yes	characteristic
12	Dic₂₇₁⋊C₄	—	2	1	no	—
13	C₄×Dic₂₇₁	—	2	1	no	—
14	C₂×C₈×C₂₇₁	C₂²	1	2	yes	—
15	C₄²×C₂₇₁	C₂²	1	1	yes	characteristic
16	Dic₅₄₂	—	2	1	no	—
17	Dic₅₄₂	—	2	2	no	—
18	Dic₅₄₂	—	2	2	no	—
19	Dic₅₄₂	D₄	1	2	yes	—
20	C₂×Dic₂₇₁	—	2	1	no	—
21	C₂×Dic₂₇₁	—	2	1	no	—
22	C₂×Dic₂₇₁	—	2	2	no	—
23	C₂×C₄×C₂₇₁	D₄	1	1	yes	characteristic
24	C₂×C₄×C₂₇₁	D₄	1	1	yes	characteristic
25	C₂×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
26	C₈×C₂₇₁	—	2	2	no	—
27	C₂²×C₂₇₁	C₂×D₄	1	1	yes	characteristic
28	C₄×C₂₇₁	—	2	1	no	—
29	C₄×C₂₇₁	Q₁₆	1	2	yes	complemented
30	C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
31	C₄×C₂₇₁	#41	1	1	yes	characteristic
32	Dic₂₇₁	—	2	2	no	—
33	Dic₂₇₁	—	4	2	no	—
34	Dic₂₇₁	—	4	1	no	—
35	C₂×C₂₇₁	#121	1	1	yes	characteristic
36	C₂×C₂₇₁	C₂×Q₁₆	1	1	yes	characteristic
37	C₂×C₂₇₁	#137	1	1	yes	characteristic
38	C₂₇₁	#462	1	1	yes	characteristic, complemented, Sylow
39	#462	—	271	1	no	maximal, Sylow
40	C₂×Q₁₆	—	271	2	no	—
41	C₄⋊Q₈	—	271	1	no	—
42	C₄×Q₈	—	271	1	no	—
43	C₄⋊C₈	D₂₇₁	1	1	yes	characteristic
44	#103	—	271	2	no	—
45	C₂×Q₈	—	271	1	no	—
46	C₂×Q₈	—	271	2	no	—
47	Q₁₆	—	542	4	no	—
48	C₂×C₈	C₂×D₂₇₁	1	2	yes	—
49	C₄⋊C₄	—	271	1	no	—
50	C₄⋊C₄	—	542	1	no	—
51	C₄⋊C₄	—	542	1	no	—
52	C₄⋊C₄	—	542	1	no	—
53	C₄²	—	542	1	no	—
54	C₄²	C₂×D₂₇₁	1	1	yes	characteristic
55	Q₈	—	271	2	no	—
56	Q₈	—	542	1	no	—
57	Q₈	—	542	2	no	—
58	Q₈	—	542	2	no	—
59	C₂×C₄	—	542	1	no	—
60	C₂×C₄	—	542	1	no	—
61	C₂×C₄	—	542	2	no	—
62	C₂×C₄	D₁₀₈₄	1	1	yes	characteristic
63	C₂×C₄	D₁₀₈₄	1	1	yes	characteristic
64	C₂×C₄	C₂²×D₂₇₁	1	1	yes	characteristic
65	C₈	—	2	2	no	—
66	C₂²	C₂×D₁₀₈₄	1	1	yes	characteristic
67	C₄	—	2	1	no	—
68	C₄	—	542	2	no	—
69	C₄	—	1084	2	no	—
70	C₄	—	1084	1	no	—
71	C₄	Dic₁₀₈₄	1	2	yes	complemented
72	C₄	D₄×D₂₇₁	1	1	yes	characteristic
73	C₄	#16864655	1	1	yes	characteristic
74	C₂	#22576135	1	1	yes	characteristic
75	C₂	C₂×Dic₁₀₈₄	1	1	yes	characteristic
76	C₂	#22576193	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₀₈₄
commutator, cosocle	C₂×C₄×C₂₇₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₇₁
Fitting	C₄⋊C₈×C₂₇₁	C₂
socle, prime radical	C₂²×C₂₇₁	C₂×D₄
hypercenter	C₄⋊C₈	D₂₇₁
hypocenter	C₂₇₁	#462
LCS	G ▷ C₂×C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
29. C₄×C₂₇₁	Q₁₆	semidirect
38. C₂₇₁	#462	semidirect
71. C₄	Dic₁₀₈₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₀₈₄	2
#22576160	2
C₄×Dic₅₄₂	2
#22576085	2
C₄⋊C₈×C₂₇₁	2
#462	271