Group #31791303 — subgroups

Subgroups — #31791303

Order 17536.

356 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 173 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577960	C₂	1	2	yes	maximal
2	#22577808	C₂	1	2	yes	maximal, complemented
3	#22577805	C₂	1	4	yes	maximal, complemented
4	#22577801	C₂	1	2	yes	maximal, complemented
5	#22577441	C₂	1	4	yes	maximal
6	C₂×C₄×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
7	#17984088	C₂²	1	4	yes	—
8	#17984080	C₂²	1	4	yes	—
9	#17984067	—	2	4	no	—
10	C₄×Dic₂₇₄	—	2	4	no	—
11	#17984047	C₂²	1	4	yes	—
12	#17984022	C₂²	1	8	yes	—
13	#17984020	C₂²	1	8	yes	—
14	C₂²×C₈×C₁₃₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
18	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
19	C₂×Dic₂₇₄	—	2	4	no	—
20	C₄⋊Dic₁₃₇	—	2	4	no	—
21	C₄⋊Dic₁₃₇	C₂³	1	8	yes	—
22	#14555489	—	2	8	no	—
23	C₂²⋊Dic₁₃₇	—	2	4	no	—
24	C₄×Dic₁₃₇	—	2	4	no	—
25	C₂²×C₄×C₁₃₇	D₄	1	2	yes	—
26	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₁₃₇	Q₈	1	8	yes	complemented
28	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
29	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
30	C₄²×C₁₃₇	D₄	1	2	yes	—
31	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
32	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
33	Dic₂₇₄	—	4	8	no	—
34	C₂×Dic₁₃₇	—	2	8	no	—
35	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₃₇	—	2	4	no	—
37	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
38	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₃₇	C₂×Q₈	1	4	yes	—
41	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
42	C₈×C₁₃₇	C₂×Q₈	1	4	yes	—
43	C₈×C₁₃₇	C₂×Q₈	1	4	yes	—
44	C₂²×C₁₃₇	—	2	2	no	—
45	C₂²×C₁₃₇	C₄⋊Q₈	1	2	yes	—
46	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
47	C₄×C₁₃₇	—	2	2	no	—
48	C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
49	C₄×C₁₃₇	#135	1	4	yes	—
50	C₄×C₁₃₇	C₂²×Q₈	1	2	yes	—
51	Dic₁₃₇	—	4	8	no	—
52	C₂×C₁₃₇	—	2	2	no	—
53	C₂×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
54	C₂×C₁₃₇	C₂×#135	1	2	yes	—
55	C₁₃₇	#3160	1	1	yes	characteristic, complemented, Sylow
56	#3160	—	137	1	no	maximal, Sylow
57	#533	—	137	2	no	—
58	#500	—	137	2	no	—
59	#499	—	137	4	no	—
60	#498	—	137	2	no	—
61	#427	—	137	4	no	—
62	C₂×C₄×C₈	D₁₃₇	1	1	yes	characteristic
63	C₂²×C₈	C₂×D₁₃₇	1	2	yes	—
64	C₄⋊Q₈	—	137	4	no	—
65	#125	—	137	4	no	—
66	C₂²⋊Q₈	—	274	4	no	—
67	C₄×Q₈	—	274	4	no	—
68	#117	—	137	4	no	—
69	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
70	#107	—	137	8	no	—
71	#106	—	137	8	no	—
72	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic
73	C₄×C₈	C₂×D₁₃₇	1	2	yes	—
74	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic
75	C₂×Q₈	—	274	4	no	—
76	C₂²×C₄	D₅₄₈	1	2	yes	—
77	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
78	C₂×C₈	Dic₂₇₄	1	8	yes	complemented
79	C₂×C₈	C₂²×D₁₃₇	1	2	yes	—
80	C₂×C₈	C₂²×D₁₃₇	1	2	yes	—
81	C₄⋊C₄	—	274	8	no	—
82	C₄⋊C₄	—	274	4	no	—
83	C₄⋊C₄	—	137	8	no	—
84	C₂²⋊C₄	—	274	4	no	—
85	C₄²	—	274	4	no	—
86	C₄²	D₅₄₈	1	2	yes	—
87	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
88	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
89	C₂³	C₂×D₅₄₈	1	1	yes	characteristic
90	Q₈	—	548	8	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	274	8	no	—
93	C₂×C₄	C₂×D₅₄₈	1	2	yes	—
94	C₂×C₄	C₂×D₅₄₈	1	2	yes	—
95	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
96	C₂×C₄	C₂×Dic₂₇₄	1	4	yes	—
97	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
98	C₈	C₂×Dic₂₇₄	1	4	yes	—
99	C₈	C₂×Dic₂₇₄	1	4	yes	—
100	C₂²	—	2	2	no	—
101	C₂²	#17984088	1	2	yes	—
102	C₂²	C₂²×D₅₄₈	1	1	yes	characteristic
103	C₄	—	2	2	no	—
104	C₄	—	548	8	no	—
105	C₄	#17984088	1	2	yes	—
106	C₄	#17984110	1	4	yes	—
107	C₄	C₂²×Dic₂₇₄	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	C₂×#17984088	1	1	yes	characteristic
110	C₂	C₂×#17984110	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22577960

C₂

yes

maximal

#22577808

C₂

yes

maximal, complemented

#22577805

C₂

yes

maximal, complemented

#22577801

C₂

yes

maximal, complemented

#22577441

C₂

yes

maximal

C₂×C₄×C₈×C₁₃₇

C₂

yes

maximal, characteristic

#17984088

C₂²

yes

—

#17984080

C₂²

yes

—

#17984067

—

C₄×Dic₂₇₄

—

#17984047

C₂²

yes

—

#17984022

C₂²

yes

—

#17984020

C₂²

yes

—

C₂²×C₈×C₁₃₇

C₂²

yes

—

C₂×C₄²×C₁₃₇

C₂²

yes

characteristic

C₄×C₈×C₁₃₇

C₂²

yes

characteristic

C₄×C₈×C₁₃₇

C₂²

yes

—

C₄×C₈×C₁₃₇

C₂²

yes

characteristic

C₂×Dic₂₇₄

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

C₂³

yes

—

#14555489

—

C₂²⋊Dic₁₃₇

—

C₄×Dic₁₃₇

—

C₂²×C₄×C₁₃₇

D₄

yes

—

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

C₂×C₈×C₁₃₇

Q₈

yes

complemented

C₂×C₈×C₁₃₇

C₂³

yes

—

C₂×C₈×C₁₃₇

C₂³

yes

—

C₄²×C₁₃₇

D₄

yes

—

C₄²×C₁₃₇

C₂³

yes

characteristic

C₄²×C₁₃₇

C₂³

yes

characteristic

Dic₂₇₄

—

C₂×Dic₁₃₇

—

C₂³×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×Q₈

yes

—

C₂×C₄×C₁₃₇

C₂⁴

yes

characteristic

C₈×C₁₃₇

C₂×Q₈

yes

—

C₈×C₁₃₇

C₂×Q₈

yes

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₄⋊Q₈

yes

—

C₂²×C₁₃₇

C₂²×D₄

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

C₄⋊Q₈

yes

—

C₄×C₁₃₇

#135

yes

—

C₄×C₁₃₇

C₂²×Q₈

yes

—

Dic₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₃₇

C₂×#135

yes

—

C₁₃₇

#3160

yes

characteristic, complemented, Sylow

#3160

—

137

maximal, Sylow

#533

—

137

—

#500

—

137

—

#499

—

137

—

#498

—

137

—

#427

—

137

—

C₂×C₄×C₈

D₁₃₇

yes

characteristic

C₂²×C₈

C₂×D₁₃₇

yes

—

C₄⋊Q₈

—

137

—

#125

—

137

—

C₂²⋊Q₈

—

274

—

C₄×Q₈

—

274

—

#117

—

137

—

C₂×C₄²

C₂×D₁₃₇

yes

characteristic

#107

—

137

—

#106

—

137

—

C₄×C₈

C₂×D₁₃₇

yes

characteristic

C₄×C₈

C₂×D₁₃₇

yes

—

C₄×C₈

C₂×D₁₃₇

yes

characteristic

C₂×Q₈

—

274

—

C₂²×C₄

D₅₄₈

yes

—

C₂²×C₄

C₂²×D₁₃₇

yes

characteristic

C₂×C₈

Dic₂₇₄

yes

complemented

C₂×C₈

C₂²×D₁₃₇

yes

—

C₂×C₈

C₂²×D₁₃₇

yes

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

137

—

C₂²⋊C₄

—

274

—

C₄²

—

274

—

C₄²

D₅₄₈

yes

—

C₄²

C₂²×D₁₃₇

yes

characteristic

C₄²

C₂²×D₁₃₇

yes

characteristic

C₂³

C₂×D₅₄₈

yes

characteristic

Q₈

—

548

—

C₂×C₄

—

C₂×C₄

—

274

—

C₂×C₄

C₂×D₅₄₈

yes

—

C₂×C₄

C₂×D₅₄₈

yes

—

C₂×C₄

C₂×D₅₄₈

yes

characteristic

C₂×C₄

C₂×Dic₂₇₄

yes

—

C₂×C₄

C₂³×D₁₃₇

yes

characteristic

C₈

C₂×Dic₂₇₄

yes

—

C₈

C₂×Dic₂₇₄

yes

—

100

C₂²

—

101

C₂²

#17984088

yes

—

102

C₂²

C₂²×D₅₄₈

yes

characteristic

103

C₄

—

104

C₄

—

548

—

105

C₄

#17984088

yes

—

106

C₄

#17984110

yes

—

107

C₄

C₂²×Dic₂₇₄

yes

—

108

C₂

—

109

C₂

C₂×#17984088

yes

characteristic

110

C₂

C₂×#17984110

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₅₄₈
commutator, cosocle	C₂×C₄×C₁₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting	C₂×C₄×C₈×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂²×D₄
prime radical	C₂³×C₁₃₇	C₂×D₄
hypercenter	C₂×C₄×C₈	D₁₃₇
hypocenter	C₁₃₇	#3160
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×D₅₄₈

commutator, cosocle

C₂×C₄×C₁₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₇

Fitting

C₂×C₄×C₈×C₁₃₇

C₂

socle

C₂²×C₁₃₇

C₂²×D₄

prime radical

C₂³×C₁₃₇

C₂×D₄

hypercenter

C₂×C₄×C₈

D₁₃₇

hypocenter

C₁₃₇

#3160

LCS

G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #22577808	C₂	semidirect
3. #22577805	C₂	semidirect
4. #22577801	C₂	semidirect
27. C₂×C₈×C₁₃₇	Q₈	semidirect
55. C₁₃₇	#3160	semidirect
78. C₂×C₈	Dic₂₇₄	semidirect

H ≅ G/N

type

2. #22577808

C₂

semidirect

3. #22577805

C₂

semidirect

4. #22577801

C₂

semidirect

27. C₂×C₈×C₁₃₇

Q₈

semidirect

55. C₁₃₇

#3160

semidirect

78. C₂×C₈

Dic₂₇₄

semidirect

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

subgroup	index
#22577960	2
#22577808	2
#22577805	2
#22577801	2
#22577441	2
C₂×C₄×C₈×C₁₃₇	2
#3160	137

subgroup

index

C₂×C₄×C₈×C₁₃₇

137

Subgroups — #31791303

Conjugacy classes of subgroups

Complements

Maximal subgroups