Group #37545520 — subgroups

Subgroups — #37545520

Order 23168.

476 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 117 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	#26042080	C₂	1	4	yes	maximal, complemented
2	#26041546	C₂	1	4	yes	maximal, complemented
3	#26041446	C₂	1	4	yes	maximal, complemented
4	#26041397	C₂	1	2	yes	maximal, complemented
5	#493×C₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#19153861	C₂²	1	4	yes	complemented
7	#19153842	C₂²	1	8	yes	complemented
8	#19153840	—	2	8	no	—
9	#19153838	C₂²	1	2	yes	—
10	#19153835	—	2	4	no	—
11	D₈×D₁₈₁	—	2	4	no	—
12	#19153822	C₂²	1	4	yes	complemented
13	#19153752	C₂²	1	2	yes	complemented
14	#19153744	C₂²	1	4	yes	complemented
15	#19153733	C₂²	1	4	yes	complemented
16	C₂×D₈×C₁₈₁	C₂²	1	4	yes	complemented
17	#127×C₁₈₁	C₂²	1	2	yes	complemented
18	C₄×C₈×C₁₈₁	C₂²	1	1	yes	characteristic, complemented
19	#15668449	C₂³	1	4	yes	—
20	#15668448	—	2	8	no	—
21	C₂×C₂²⋊D₁₈₁	—	4	4	no	—
22	D₄×D₁₈₁	—	2	8	no	—
23	Dic₇₂₄	D₄	1	2	yes	complemented
24	#15668438	D₄	1	4	yes	complemented
25	#15668437	—	2	8	no	—
26	D₁₄₄₈	D₄	1	2	yes	complemented
27	#15668434	—	2	8	no	—
28	#15668433	C₂³	1	4	yes	—
29	#15668432	—	2	4	no	—
30	C₈×D₁₈₁	—	2	4	no	—
31	C₂×D₄×C₁₈₁	—	4	2	no	—
32	C₂×D₄×C₁₈₁	C₂³	1	4	yes	—
33	D₈×C₁₈₁	—	2	4	no	—
34	D₈×C₁₈₁	—	2	4	no	—
35	C₂×C₈×C₁₈₁	C₂³	1	2	yes	—
36	C₄²×C₁₈₁	C₂³	1	1	yes	characteristic
37	C₂²×D₁₈₁	—	4	4	no	—
38	Dic₃₆₂	C₂×D₄	1	4	yes	—
39	D₇₂₄	C₂×D₄	1	4	yes	—
40	C₂²⋊D₁₈₁	—	2	4	no	—
41	C₂²⋊D₁₈₁	—	4	8	no	—
42	C₂×Dic₁₈₁	—	4	4	no	—
43	C₄×D₁₈₁	—	2	4	no	—
44	#13089678	—	2	4	no	—
45	C₂³×C₁₈₁	—	2	4	no	—
46	D₄×C₁₈₁	—	2	8	no	—
47	D₄×C₁₈₁	—	8	4	no	—
48	C₂×C₄×C₁₈₁	—	2	1	no	—
49	C₂×C₄×C₁₈₁	C₂⁴	1	1	yes	characteristic
50	C₈×C₁₈₁	C₂×D₄	1	2	yes	—
51	C₈×C₁₈₁	C₂×D₄	1	2	yes	—
52	C₂²×C₁₈₁	—	4	4	no	—
53	C₂²×C₁₈₁	—	8	4	no	—
54	C₂²×C₁₈₁	H(5,𝔽₂)	1	1	yes	characteristic
55	C₂×D₁₈₁	—	2	4	no	—
56	C₂×D₁₈₁	—	16	4	no	—
57	C₄×C₁₈₁	—	2	2	no	—
58	C₄×C₁₈₁	C₂²×D₄	1	2	yes	—
59	Dic₁₈₁	—	2	4	no	—
60	C₂×C₁₈₁	—	2	1	no	—
61	C₂×C₁₈₁	—	8	4	no	—
62	C₂×C₁₈₁	D₄²	1	1	yes	characteristic
63	D₁₈₁	—	4	4	no	—
64	C₁₈₁	#3292	1	1	yes	characteristic, complemented, Sylow
65	#3292	—	181	1	no	maximal, Sylow
66	#576	—	181	4	no	—
67	#493	D₁₈₁	1	1	yes	characteristic, complemented
68	#472	—	181	4	no	—
69	#453	—	181	4	no	—
70	#443	—	181	2	no	—
71	H(5,𝔽₂)	—	181	4	no	—
72	AGL(1,ℤ/8)	—	362	8	no	—
73	AGL(1,ℤ/8)	—	181	8	no	—
74	#135	—	362	4	no	—
75	#135	—	181	2	no	—
76	C₂×D₈	—	362	4	no	—
77	C₂×D₈	C₂×D₁₈₁	1	4	yes	complemented
78	#131	—	181	4	no	—
79	#127	C₂×D₁₈₁	1	2	yes	complemented
80	#108	—	181	2	no	—
81	#104	—	181	4	no	—
82	#100	—	181	4	no	—
83	C₄×C₈	C₂×D₁₈₁	1	1	yes	characteristic, complemented
84	#41	—	362	8	no	—
85	#41	—	181	4	no	—
86	C₂×D₄	—	4	2	no	—
87	C₂×D₄	—	362	8	no	—
88	C₂×D₄	—	724	4	no	—
89	C₂×D₄	C₂²×D₁₈₁	1	4	yes	—
90	Q₁₆	—	181	2	no	—
91	SD₁₆	—	362	8	no	—
92	SD₁₆	—	181	4	no	—
93	D₈	—	2	4	no	—
94	D₈	—	2	4	no	—
95	D₈	—	362	8	no	—
96	D₈	—	181	2	no	—
97	M₁₆	—	362	4	no	—
98	M₁₆	—	181	4	no	—
99	C₂×C₈	—	362	4	no	—
100	C₂×C₈	C₂²×D₁₈₁	1	2	yes	—
101	C₄²	C₂²×D₁₈₁	1	1	yes	characteristic
102	C₂³	—	2	4	no	—
103	C₂³	—	724	4	no	—
104	Q₈	—	181	4	no	—
105	D₄	—	2	8	no	—
106	D₄	—	8	4	no	—
107	D₄	—	362	4	no	—
108	D₄	—	181	4	no	—
109	D₄	—	724	8	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	362	4	no	—
112	C₂×C₄	—	724	4	no	—
113	C₂×C₄	C₂³×D₁₈₁	1	1	yes	characteristic
114	C₈	—	362	4	no	—
115	C₈	D₄×D₁₈₁	1	2	yes	—
116	C₈	D₄×D₁₈₁	1	2	yes	—
117	C₂²	—	4	4	no	—
118	C₂²	—	8	4	no	—
119	C₂²	—	2896	4	no	—
120	C₂²	—	362	4	no	—
121	C₂²	#19153861	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	362	4	no	—
124	C₄	C₂×D₄×D₁₈₁	1	2	yes	—
125	C₂	—	2	1	no	—
126	C₂	—	8	4	no	—
127	C₂	—	724	4	no	—
128	C₂	#26041914	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26042080

C₂

yes

maximal, complemented

#26041546

C₂

yes

maximal, complemented

#26041446

C₂

yes

maximal, complemented

#26041397

C₂

yes

maximal, complemented

#493×C₁₈₁

C₂

yes

maximal, characteristic, complemented

#19153861

C₂²

yes

complemented

#19153842

C₂²

yes

complemented

#19153840

—

#19153838

C₂²

yes

—

#19153835

—

D₈×D₁₈₁

—

#19153822

C₂²

yes

complemented

#19153752

C₂²

yes

complemented

#19153744

C₂²

yes

complemented

#19153733

C₂²

yes

complemented

C₂×D₈×C₁₈₁

C₂²

yes

complemented

#127×C₁₈₁

C₂²

yes

complemented

C₄×C₈×C₁₈₁

C₂²

yes

characteristic, complemented

#15668449

C₂³

yes

—

#15668448

—

C₂×C₂²⋊D₁₈₁

—

D₄×D₁₈₁

—

Dic₇₂₄

D₄

yes

complemented

#15668438

D₄

yes

complemented

#15668437

—

D₁₄₄₈

D₄

yes

complemented

#15668434

—

#15668433

C₂³

yes

—

#15668432

—

C₈×D₁₈₁

—

C₂×D₄×C₁₈₁

—

C₂×D₄×C₁₈₁

C₂³

yes

—

D₈×C₁₈₁

—

D₈×C₁₈₁

—

C₂×C₈×C₁₈₁

C₂³

yes

—

C₄²×C₁₈₁

C₂³

yes

characteristic

C₂²×D₁₈₁

—

Dic₃₆₂

C₂×D₄

yes

—

D₇₂₄

C₂×D₄

yes

—

C₂²⋊D₁₈₁

—

C₂²⋊D₁₈₁

—

C₂×Dic₁₈₁

—

C₄×D₁₈₁

—

#13089678

—

C₂³×C₁₈₁

—

D₄×C₁₈₁

—

D₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

C₂⁴

yes

characteristic

C₈×C₁₈₁

C₂×D₄

yes

—

C₈×C₁₈₁

C₂×D₄

yes

—

C₂²×C₁₈₁

—

C₂²×C₁₈₁

—

C₂²×C₁₈₁

H(5,𝔽₂)

yes

characteristic

C₂×D₁₈₁

—

C₂×D₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

C₂²×D₄

yes

—

Dic₁₈₁

—

C₂×C₁₈₁

—

C₂×C₁₈₁

—

C₂×C₁₈₁

D₄²

yes

characteristic

D₁₈₁

—

C₁₈₁

#3292

yes

characteristic, complemented, Sylow

#3292

—

181

maximal, Sylow

#576

—

181

—

#493

D₁₈₁

yes

characteristic, complemented

#472

—

181

—

#453

—

181

—

#443

—

181

—

H(5,𝔽₂)

—

181

—

AGL(1,ℤ/8)

—

362

—

AGL(1,ℤ/8)

—

181

—

#135

—

362

—

#135

—

181

—

C₂×D₈

—

362

—

C₂×D₈

C₂×D₁₈₁

yes

complemented

#131

—

181

—

#127

C₂×D₁₈₁

yes

complemented

#108

—

181

—

#104

—

181

—

#100

—

181

—

C₄×C₈

C₂×D₁₈₁

yes

characteristic, complemented

#41

—

362

—

#41

—

181

—

C₂×D₄

—

C₂×D₄

—

362

—

C₂×D₄

—

724

—

C₂×D₄

C₂²×D₁₈₁

yes

—

Q₁₆

—

181

—

SD₁₆

—

362

—

SD₁₆

—

181

—

D₈

—

D₈

—

D₈

—

362

—

D₈

—

181

—

M₁₆

—

362

—

M₁₆

—

181

—

C₂×C₈

—

362

—

100

C₂×C₈

C₂²×D₁₈₁

yes

—

101

C₄²

C₂²×D₁₈₁

yes

characteristic

102

C₂³

—

103

C₂³

—

724

—

104

Q₈

—

181

—

105

D₄

—

106

D₄

—

107

D₄

—

362

—

108

D₄

—

181

—

109

D₄

—

724

—

110

C₂×C₄

—

111

C₂×C₄

—

362

—

112

C₂×C₄

—

724

—

113

C₂×C₄

C₂³×D₁₈₁

yes

characteristic

114

C₈

—

362

—

115

C₈

D₄×D₁₈₁

yes

—

116

C₈

D₄×D₁₈₁

yes

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

2896

—

120

C₂²

—

362

—

121

C₂²

#19153861

yes

characteristic

122

C₄

—

123

C₄

—

362

—

124

C₄

C₂×D₄×D₁₈₁

yes

—

125

C₂

—

126

C₂

—

127

C₂

—

724

—

128

C₂

#26041914

yes

characteristic

	subgroup	G/H
center	C₂	#26041914
commutator, cosocle	C₂×C₄×C₁₈₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₈₁
Fitting	#493×C₁₈₁	C₂
socle	C₂×C₁₈₁	D₄²
hypercenter	#493	D₁₈₁
hypocenter	C₁₈₁	#3292
LCS	G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #493

subgroup

G/H

center

C₂

#26041914

commutator, cosocle

C₂×C₄×C₁₈₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₈₁

Fitting

#493×C₁₈₁

C₂

socle

C₂×C₁₈₁

D₄²

D₁₈₁

C₁₈₁

G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #493

N	H ≅ G/N	type
1. #26042080	C₂	semidirect
2. #26041546	C₂	semidirect
3. #26041446	C₂	semidirect
4. #26041397	C₂	semidirect
5. #493×C₁₈₁	C₂	semidirect
6. #19153861	C₂²	semidirect
7. #19153842	C₂²	semidirect
12. #19153822	C₂²	semidirect
13. #19153752	C₂²	semidirect
14. #19153744	C₂²	semidirect
15. #19153733	C₂²	semidirect
16. C₂×D₈×C₁₈₁	C₂²	semidirect
17. #127×C₁₈₁	C₂²	semidirect
18. C₄×C₈×C₁₈₁	C₂²	semidirect
23. Dic₇₂₄	D₄	semidirect
24. #15668438	D₄	semidirect
26. D₁₄₄₈	D₄	semidirect
64. C₁₈₁	#3292	semidirect
67. #493	D₁₈₁	semidirect
77. C₂×D₈	C₂×D₁₈₁	semidirect
79. #127	C₂×D₁₈₁	semidirect
83. C₄×C₈	C₂×D₁₈₁	semidirect

H ≅ G/N

type

1. #26042080

C₂

semidirect

2. #26041546

C₂

semidirect

3. #26041446

C₂

semidirect

4. #26041397

C₂

semidirect

5. #493×C₁₈₁

C₂

semidirect

6. #19153861

C₂²

semidirect

7. #19153842

C₂²

semidirect

12. #19153822

C₂²

semidirect

13. #19153752

C₂²

semidirect

14. #19153744

C₂²

semidirect

15. #19153733

C₂²

semidirect

16. C₂×D₈×C₁₈₁

C₂²

semidirect

17. #127×C₁₈₁

C₂²

semidirect

18. C₄×C₈×C₁₈₁

C₂²

semidirect

23. Dic₇₂₄

D₄

semidirect

24. #15668438

D₄

semidirect

26. D₁₄₄₈

D₄

semidirect

64. C₁₈₁

#3292

semidirect

67. #493

D₁₈₁

semidirect

77. C₂×D₈

C₂×D₁₈₁

semidirect

79. #127

C₂×D₁₈₁

semidirect

83. C₄×C₈

C₂×D₁₈₁

semidirect

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

subgroup	index
#26042080	2
#26041546	2
#26041446	2
#26041397	2
#493×C₁₈₁	2
#3292	181

subgroup

index

#493×C₁₈₁

181

Subgroups — #37545520

Conjugacy classes of subgroups

Complements

Maximal subgroups