Group #37535798 — subgroups

Subgroups — #37535798

Order 23168.

170 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 47 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	#26041717	C₂	1	2	yes	maximal, complemented
2	#26041656	C₂	1	1	yes	maximal, characteristic, complemented
3	#26041397	C₂	1	1	yes	maximal, characteristic, complemented
4	#26041093	C₂	1	2	yes	maximal, complemented
5	#364×C₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#19153838	C₂²	1	1	yes	characteristic
7	C₂×D₁₄₄₈	—	2	1	no	—
8	C₂×D₁₄₄₈	C₂²	1	2	yes	—
9	#19153822	—	2	1	no	—
10	#19153813	—	2	1	no	—
11	#19153761	—	2	2	no	—
12	D₂₈₉₆	—	2	2	no	—
13	#19153752	C₂²	1	1	yes	characteristic
14	#19153744	—	2	1	no	—
15	M₃₂×C₁₈₁	C₂²	1	2	yes	—
16	C₄×C₈×C₁₈₁	C₂²	1	1	yes	characteristic
17	#15668449	—	2	1	no	—
18	C₂×D₇₂₄	—	2	2	no	—
19	C₂×D₇₂₄	—	4	2	no	—
20	Dic₇₂₄	D₄	1	1	yes	characteristic, complemented
21	#15668438	—	2	1	no	—
22	D₁₄₄₈	—	2	2	no	—
23	D₁₄₄₈	—	2	2	no	—
24	D₁₄₄₈	—	4	1	no	—
25	D₁₄₄₈	—	4	1	no	—
26	D₁₄₄₈	D₄	1	1	yes	characteristic, complemented
27	#15668433	—	2	1	no	—
28	#15668432	—	4	1	no	—
29	C₈×D₁₈₁	—	4	1	no	—
30	C₂×C₈×C₁₈₁	D₄	1	1	yes	characteristic
31	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
32	C₄²×C₁₈₁	D₄	1	1	yes	characteristic
33	C₁₆×C₁₈₁	—	2	2	no	—
34	C₂²×D₁₈₁	—	4	2	no	—
35	Dic₃₆₂	—	2	1	no	—
36	D₇₂₄	—	2	1	no	—
37	D₇₂₄	—	4	2	no	—
38	D₇₂₄	—	4	2	no	—
39	D₇₂₄	—	8	4	no	—
40	C₂²⋊D₁₈₁	—	4	1	no	—
41	C₄×D₁₈₁	—	4	1	no	—
42	#13089678	—	4	1	no	—
43	C₂×C₄×C₁₈₁	—	2	1	no	—
44	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
45	C₈×C₁₈₁	D₈	1	1	yes	characteristic
46	C₈×C₁₈₁	D₈	1	1	yes	characteristic
47	C₈×C₁₈₁	C₂×D₄	1	1	yes	characteristic
48	C₈×C₁₈₁	#41	1	1	yes	characteristic
49	C₂²×C₁₈₁	AGL(1,ℤ/8)	1	1	yes	characteristic
50	C₂×D₁₈₁	—	4	1	no	—
51	C₂×D₁₈₁	—	8	4	no	—
52	C₂×D₁₈₁	—	8	2	no	—
53	C₄×C₁₈₁	—	2	2	no	—
54	C₄×C₁₈₁	#121	1	1	yes	characteristic
55	C₄×C₁₈₁	C₂×D₈	1	1	yes	characteristic
56	Dic₁₈₁	—	4	1	no	—
57	C₂×C₁₈₁	—	2	1	no	—
58	C₂×C₁₈₁	#459	1	1	yes	characteristic
59	D₁₈₁	—	8	1	no	—
60	D₁₈₁	—	16	2	no	—
61	C₁₈₁	#2213	1	1	yes	characteristic, complemented, Sylow
62	#2213	—	181	1	no	maximal, Sylow
63	#509	—	181	2	no	—
64	#493	—	181	1	no	—
65	#443	—	181	1	no	—
66	#364	D₁₈₁	1	1	yes	characteristic, complemented
67	#361	—	181	2	no	—
68	#135	—	181	1	no	—
69	C₂×D₈	—	362	1	no	—
70	C₂×D₈	—	181	2	no	—
71	#131	—	362	1	no	—
72	#127	—	362	1	no	—
73	SD₃₂	—	362	2	no	—
74	D₁₆	—	362	2	no	—
75	M₃₂	C₂×D₁₈₁	1	2	yes	—
76	#108	—	181	1	no	—
77	#104	—	362	1	no	—
78	C₄×C₈	C₂×D₁₈₁	1	1	yes	characteristic
79	#41	—	362	1	no	—
80	C₂×D₄	—	362	2	no	—
81	C₂×D₄	—	724	2	no	—
82	Q₁₆	—	181	1	no	—
83	SD₁₆	—	362	1	no	—
84	D₈	—	362	2	no	—
85	D₈	—	362	2	no	—
86	D₈	—	181	1	no	—
87	D₈	—	724	1	no	—
88	D₈	—	724	1	no	—
89	M₁₆	—	362	1	no	—
90	M₁₆	—	724	1	no	—
91	C₂×C₈	—	724	1	no	—
92	C₂×C₈	D₇₂₄	1	1	yes	characteristic
93	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
94	C₄²	D₇₂₄	1	1	yes	characteristic
95	C₁₆	—	2	2	no	—
96	C₂³	—	724	2	no	—
97	Q₈	—	362	1	no	—
98	D₄	—	362	1	no	—
99	D₄	—	1448	4	no	—
100	D₄	—	724	1	no	—
101	D₄	—	724	2	no	—
102	D₄	—	724	2	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	724	1	no	—
105	C₂×C₄	C₂×D₇₂₄	1	1	yes	characteristic
106	C₈	—	724	1	no	—
107	C₈	D₁₄₄₈	1	1	yes	characteristic
108	C₈	D₁₄₄₈	1	1	yes	characteristic
109	C₈	D₄×D₁₈₁	1	1	yes	characteristic
110	C₈	#15668450	1	1	yes	characteristic
111	C₂²	—	1448	4	no	—
112	C₂²	—	1448	2	no	—
113	C₂²	—	724	1	no	—
114	C₂²	#19153844	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	724	1	no	—
117	C₄	#19153791	1	1	yes	characteristic
118	C₄	C₂×D₁₄₄₈	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2896	2	no	—
121	C₂	—	1448	1	no	—
122	C₂	#26041475	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26041717

C₂

yes

maximal, complemented

#26041656

C₂

yes

maximal, characteristic, complemented

#26041397

C₂

yes

maximal, characteristic, complemented

#26041093

C₂

yes

maximal, complemented

#364×C₁₈₁

C₂

yes

maximal, characteristic, complemented

#19153838

C₂²

yes

characteristic

C₂×D₁₄₄₈

—

C₂×D₁₄₄₈

C₂²

yes

—

#19153822

—

#19153813

—

#19153761

—

D₂₈₉₆

—

#19153752

C₂²

yes

characteristic

#19153744

—

M₃₂×C₁₈₁

C₂²

yes

—

C₄×C₈×C₁₈₁

C₂²

yes

characteristic

#15668449

—

C₂×D₇₂₄

—

C₂×D₇₂₄

—

Dic₇₂₄

D₄

yes

characteristic, complemented

#15668438

—

D₁₄₄₈

—

D₁₄₄₈

—

D₁₄₄₈

—

D₁₄₄₈

—

D₁₄₄₈

D₄

yes

characteristic, complemented

#15668433

—

#15668432

—

C₈×D₁₈₁

—

C₂×C₈×C₁₈₁

D₄

yes

characteristic

C₂×C₈×C₁₈₁

C₂³

yes

characteristic

C₄²×C₁₈₁

D₄

yes

characteristic

C₁₆×C₁₈₁

—

C₂²×D₁₈₁

—

Dic₃₆₂

—

D₇₂₄

—

D₇₂₄

—

D₇₂₄

—

D₇₂₄

—

C₂²⋊D₁₈₁

—

C₄×D₁₈₁

—

#13089678

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

C₂×D₄

yes

characteristic

C₈×C₁₈₁

D₈

yes

characteristic

C₈×C₁₈₁

D₈

yes

characteristic

C₈×C₁₈₁

C₂×D₄

yes

C₈×C₁₈₁

yes

C₂²×C₁₈₁

yes

C₂×D₁₈₁

—

C₂×D₁₈₁

—

C₂×D₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

#121

yes

characteristic

C₄×C₁₈₁

C₂×D₈

yes

characteristic

Dic₁₈₁

—

C₂×C₁₈₁

—

C₂×C₁₈₁

#459

yes

characteristic

D₁₈₁

—

D₁₈₁

—

C₁₈₁

#2213

yes

characteristic, complemented, Sylow

#2213

—

181

maximal, Sylow

#509

—

181

—

#493

—

181

—

#443

—

181

—

#364

D₁₈₁

yes

characteristic, complemented

#361

—

181

—

#135

—

181

—

C₂×D₈

—

362

—

C₂×D₈

—

181

—

#131

—

362

—

#127

—

362

—

SD₃₂

—

362

—

D₁₆

—

362

—

M₃₂

C₂×D₁₈₁

yes

—

#108

—

181

—

#104

—

362

—

C₄×C₈

C₂×D₁₈₁

yes

characteristic

#41

—

362

—

C₂×D₄

—

362

—

C₂×D₄

—

724

—

Q₁₆

—

181

—

SD₁₆

—

362

—

D₈

—

362

—

D₈

—

362

—

D₈

—

181

—

D₈

—

724

—

D₈

—

724

—

M₁₆

—

362

—

M₁₆

—

724

—

C₂×C₈

—

724

—

C₂×C₈

D₇₂₄

yes

characteristic

C₂×C₈

C₂²×D₁₈₁

yes

characteristic

C₄²

D₇₂₄

yes

characteristic

C₁₆

—

C₂³

—

724

—

Q₈

—

362

—

D₄

—

362

—

D₄

—

1448

—

100

D₄

—

724

—

101

D₄

—

724

—

102

D₄

—

724

—

103

C₂×C₄

—

104

C₂×C₄

—

724

—

105

C₂×C₄

C₂×D₇₂₄

yes

characteristic

106

C₈

—

724

—

107

C₈

D₁₄₄₈

yes

characteristic

108

C₈

D₁₄₄₈

yes

characteristic

109

C₈

D₄×D₁₈₁

yes

characteristic

110

C₈

#15668450

yes

characteristic

111

C₂²

—

1448

—

112

C₂²

—

1448

—

113

C₂²

—

724

—

114

C₂²

#19153844

yes

characteristic

115

C₄

—

116

C₄

—

724

—

117

C₄

#19153791

yes

characteristic

118

C₄

C₂×D₁₄₄₈

yes

characteristic

119

C₂

—

120

C₂

—

2896

—

121

C₂

—

1448

—

122

C₂

#26041475

yes

characteristic

	subgroup	G/H
center	C₂	#26041475
commutator, cosocle	C₂×C₈×C₁₈₁	C₂³
Frattini	C₂×C₈	C₂²×D₁₈₁
Fitting	#364×C₁₈₁	C₂
socle	C₂×C₁₈₁	#459
hypercenter	#364	D₁₈₁
hypocenter	C₁₈₁	#2213
LCS	G ▷ C₂×C₈×C₁₈₁ ▷ C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

subgroup

G/H

center

C₂

#26041475

commutator, cosocle

C₂×C₈×C₁₈₁

C₂³

Frattini

C₂×C₈

C₂²×D₁₈₁

Fitting

#364×C₁₈₁

C₂

C₂×C₁₈₁

D₁₈₁

C₁₈₁

G ▷ C₂×C₈×C₁₈₁ ▷ C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

N	H ≅ G/N	type
1. #26041717	C₂	semidirect
2. #26041656	C₂	semidirect
3. #26041397	C₂	semidirect
4. #26041093	C₂	semidirect
5. #364×C₁₈₁	C₂	semidirect
20. Dic₇₂₄	D₄	semidirect
26. D₁₄₄₈	D₄	semidirect
61. C₁₈₁	#2213	semidirect
66. #364	D₁₈₁	semidirect

H ≅ G/N

type

1. #26041717

C₂

semidirect

2. #26041656

C₂

semidirect

3. #26041397

C₂

semidirect

4. #26041093

C₂

semidirect

5. #364×C₁₈₁

C₂

semidirect

20. Dic₇₂₄

D₄

semidirect

26. D₁₄₄₈

D₄

semidirect

61. C₁₈₁

#2213

semidirect

66. #364

D₁₈₁

semidirect

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

subgroup	index
#26041717	2
#26041656	2
#26041397	2
#26041093	2
#364×C₁₈₁	2
#2213	181

subgroup

index

#364×C₁₈₁

181

Subgroups — #37535798

Conjugacy classes of subgroups

Complements

Maximal subgroups