Subgroups — #31774151

Order 17344.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₁₆₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16864637	C₂	1	1	yes	maximal, characteristic, complemented
3	#22576135	C₂	1	2	yes	maximal, complemented
4	#22576080	C₂	1	2	yes	maximal, complemented
5	#106×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₀₈₄	—	2	2	no	—
7	C₂×D₁₀₈₄	C₂²	1	2	yes	—
8	C₂×C₄×D₂₇₁	C₂²	1	1	yes	characteristic
9	D₂₁₆₈	—	2	2	no	—
10	#16864637	—	2	2	no	—
11	C₂×C₂₇₁⋊C₈	C₂²	1	1	yes	characteristic
12	(C₂×D₂₇₁)⋊C₄	—	2	2	no	—
13	C₂×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₂₇₁	C₂²	1	2	yes	—
15	C₂²×D₂₇₁	—	2	2	no	—
16	C₂²×D₂₇₁	D₄	1	1	yes	characteristic
17	D₁₀₈₄	—	2	2	no	—
18	D₁₀₈₄	—	2	2	no	—
19	D₁₀₈₄	—	4	4	no	—
20	C₂×Dic₂₇₁	D₄	1	1	yes	characteristic
21	C₄×D₂₇₁	—	2	1	no	—
22	C₄×D₂₇₁	—	2	1	no	—
23	C₂₇₁⋊C₈	—	2	1	no	—
24	C₂×C₄×C₂₇₁	—	2	2	no	—
25	C₂×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
26	C₈×C₂₇₁	D₄	1	2	yes	complemented
27	C₂²×C₂₇₁	C₂×D₄	1	1	yes	characteristic
28	C₂×D₂₇₁	—	2	1	no	—
29	C₂×D₂₇₁	—	2	2	no	—
30	C₂×D₂₇₁	—	4	2	no	—
31	C₂×D₂₇₁	—	4	2	no	—
32	C₂×D₂₇₁	—	4	2	no	—
33	C₄×C₂₇₁	—	4	2	no	—
34	C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
35	C₄×C₂₇₁	#41	1	1	yes	characteristic
36	Dic₂₇₁	—	2	1	no	—
37	C₂×C₂₇₁	#121	1	1	yes	characteristic
38	C₂×C₂₇₁	AGL(1,ℤ/8)	1	2	yes	—
39	D₂₇₁	—	4	1	no	—
40	D₂₇₁	—	8	2	no	—
41	C₂₇₁	#469	1	1	yes	characteristic, complemented, Sylow
42	#469	—	271	1	no	maximal, Sylow
43	C₂×D₈	—	271	1	no	—
44	C₂×M₁₆	—	271	1	no	—
45	#121	—	271	2	no	—
46	#106	D₂₇₁	1	1	yes	characteristic, complemented
47	#102	—	271	2	no	—
48	C₂×D₄	—	271	2	no	—
49	C₂×D₄	—	542	2	no	—
50	C₂²×C₄	—	271	1	no	—
51	D₈	—	542	2	no	—
52	M₁₆	—	542	2	no	—
53	C₂×C₈	—	271	1	no	—
54	C₂×C₈	C₂×D₂₇₁	1	1	yes	characteristic
55	C₄⋊C₄	C₂×D₂₇₁	1	2	yes	—
56	C₂²⋊C₄	—	542	2	no	—
57	C₂³	—	271	1	no	—
58	C₂³	—	542	2	no	—
59	D₄	—	542	2	no	—
60	D₄	—	542	2	no	—
61	D₄	—	1084	4	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	271	1	no	—
64	C₂×C₄	—	542	1	no	—
65	C₂×C₄	—	542	1	no	—
66	C₂×C₄	C₂²×D₂₇₁	1	1	yes	characteristic
67	C₈	—	542	1	no	—
68	C₈	D₁₀₈₄	1	2	yes	complemented
69	C₂²	—	542	1	no	—
70	C₂²	—	542	2	no	—
71	C₂²	—	1084	2	no	—
72	C₂²	—	1084	2	no	—
73	C₂²	—	1084	2	no	—
74	C₂²	D₄×D₂₇₁	1	1	yes	characteristic
75	C₄	—	4	2	no	—
76	C₄	—	542	1	no	—
77	C₄	C₂×D₁₀₈₄	1	1	yes	characteristic
78	C₄	#16864655	1	1	yes	characteristic
79	C₂	—	1084	1	no	—
80	C₂	—	2168	2	no	—
81	C₂	#22576135	1	1	yes	characteristic
82	C₂	#22576187	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₂₇₁
commutator, cosocle	C₂×C₄×C₂₇₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₇₁
Fitting	#106×C₂₇₁	C₂
socle	C₂²×C₂₇₁	C₂×D₄
hypercenter	#106	D₂₇₁
hypocenter	C₂₇₁	#469
LCS	G ▷ C₂×C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
1. C₂×D₂₁₆₈	C₂	semidirect
2. C₂×#16864637	C₂	semidirect
3. #22576135	C₂	semidirect
4. #22576080	C₂	semidirect
5. #106×C₂₇₁	C₂	semidirect
26. C₈×C₂₇₁	D₄	semidirect
41. C₂₇₁	#469	semidirect
46. #106	D₂₇₁	semidirect
68. C₈	D₁₀₈₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₁₆₈	2
C₂×#16864637	2
#22576135	2
#22576080	2
#106×C₂₇₁	2
#469	271