Group #18006395 — subgroups

Subgroups — #18006395

Order 4736.

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	#14558334	C₂	1	4	yes	maximal, complemented
2	#14557800	C₂	1	4	yes	maximal, complemented
3	#14557700	C₂	1	4	yes	maximal, complemented
4	#14557651	C₂	1	2	yes	maximal, complemented
5	C₃₇×#493	C₂	1	1	yes	maximal, characteristic, complemented
6	#11924581	C₂²	1	4	yes	complemented
7	#11924562	C₂²	1	8	yes	complemented
8	#11924560	—	2	8	no	—
9	#11924558	C₂²	1	2	yes	—
10	#11924555	—	2	4	no	—
11	D₈×D₃₇	—	2	4	no	—
12	#11924542	C₂²	1	4	yes	complemented
13	#11924472	C₂²	1	2	yes	complemented
14	#11924464	C₂²	1	4	yes	complemented
15	#11924453	C₂²	1	4	yes	complemented
16	C₂×D₈×C₃₇	C₂²	1	4	yes	complemented
17	#127×C₃₇	C₂²	1	2	yes	complemented
18	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic, complemented
19	C₂×D₄×C₃₇	—	4	2	no	—
20	C₂×D₄×C₃₇	C₂³	1	4	yes	—
21	C₂×C₂²⋊D₃₇	—	4	4	no	—
22	#10596964	—	2	8	no	—
23	D₄×D₃₇	—	2	8	no	—
24	#10596962	C₂³	1	4	yes	—
25	D₈×C₃₇	—	2	4	no	—
26	D₈×C₃₇	—	2	4	no	—
27	C₂×C₈×C₃₇	C₂³	1	2	yes	—
28	C₄²×C₃₇	C₂³	1	1	yes	characteristic
29	#10596940	—	2	8	no	—
30	#10596939	—	2	8	no	—
31	#10596934	C₂³	1	4	yes	—
32	Dic₁₄₈	D₄	1	2	yes	complemented
33	D₂₉₆	D₄	1	2	yes	complemented
34	#10596930	D₄	1	4	yes	complemented
35	#10596929	—	2	4	no	—
36	C₈×D₃₇	—	2	4	no	—
37	C₂³×C₃₇	—	2	4	no	—
38	C₂²×D₃₇	—	4	4	no	—
39	D₄×C₃₇	—	2	8	no	—
40	D₄×C₃₇	—	8	4	no	—
41	C₂×C₄×C₃₇	—	2	1	no	—
42	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
43	C₂²⋊D₃₇	—	2	4	no	—
44	C₂²⋊D₃₇	—	4	8	no	—
45	C₂×Dic₃₇	—	4	4	no	—
46	D₁₄₈	C₂×D₄	1	4	yes	—
47	C₄×D₃₇	—	2	4	no	—
48	Dic₇₄	C₂×D₄	1	4	yes	—
49	C₈×C₃₇	C₂×D₄	1	2	yes	—
50	C₈×C₃₇	C₂×D₄	1	2	yes	—
51	#64436	—	2	4	no	—
52	C₂²×C₃₇	—	4	4	no	—
53	C₂²×C₃₇	—	8	4	no	—
54	C₂²×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
55	C₂×D₃₇	—	2	4	no	—
56	C₂×D₃₇	—	16	4	no	—
57	C₄×C₃₇	—	2	2	no	—
58	C₄×C₃₇	C₂²×D₄	1	2	yes	—
59	Dic₃₇	—	2	4	no	—
60	#3292	—	37	1	no	maximal, Sylow
61	C₂×C₃₇	—	2	1	no	—
62	C₂×C₃₇	—	8	4	no	—
63	C₂×C₃₇	D₄²	1	1	yes	characteristic
64	D₃₇	—	4	4	no	—
65	#576	—	37	4	no	—
66	#493	D₃₇	1	1	yes	characteristic, complemented
67	#472	—	37	4	no	—
68	#453	—	37	4	no	—
69	#443	—	37	2	no	—
70	C₃₇	#3292	1	1	yes	characteristic, complemented, Sylow
71	H(5,𝔽₂)	—	37	4	no	—
72	AGL(1,ℤ/8)	—	37	8	no	—
73	AGL(1,ℤ/8)	—	74	8	no	—
74	#135	—	37	2	no	—
75	#135	—	74	4	no	—
76	C₂×D₈	—	74	4	no	—
77	C₂×D₈	C₂×D₃₇	1	4	yes	complemented
78	#131	—	37	4	no	—
79	#127	C₂×D₃₇	1	2	yes	complemented
80	#108	—	37	2	no	—
81	#104	—	37	4	no	—
82	#100	—	37	4	no	—
83	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic, complemented
84	#41	—	37	4	no	—
85	#41	—	74	8	no	—
86	C₂×D₄	—	4	2	no	—
87	C₂×D₄	—	74	8	no	—
88	C₂×D₄	—	148	4	no	—
89	C₂×D₄	C₂²×D₃₇	1	4	yes	—
90	Q₁₆	—	37	2	no	—
91	SD₁₆	—	37	4	no	—
92	SD₁₆	—	74	8	no	—
93	D₈	—	2	4	no	—
94	D₈	—	2	4	no	—
95	D₈	—	37	2	no	—
96	D₈	—	74	8	no	—
97	M₁₆	—	37	4	no	—
98	M₁₆	—	74	4	no	—
99	C₂×C₈	—	74	4	no	—
100	C₂×C₈	C₂²×D₃₇	1	2	yes	—
101	C₄²	C₂²×D₃₇	1	1	yes	characteristic
102	C₂³	—	2	4	no	—
103	C₂³	—	148	4	no	—
104	Q₈	—	37	4	no	—
105	D₄	—	2	8	no	—
106	D₄	—	8	4	no	—
107	D₄	—	37	4	no	—
108	D₄	—	74	4	no	—
109	D₄	—	148	8	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	74	4	no	—
112	C₂×C₄	—	148	4	no	—
113	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
114	C₈	—	74	4	no	—
115	C₈	D₄×D₃₇	1	2	yes	—
116	C₈	D₄×D₃₇	1	2	yes	—
117	C₂²	—	4	4	no	—
118	C₂²	—	8	4	no	—
119	C₂²	—	74	4	no	—
120	C₂²	—	592	4	no	—
121	C₂²	#11924581	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	74	4	no	—
124	C₄	C₂×D₄×D₃₇	1	2	yes	—
125	C₂	—	2	1	no	—
126	C₂	—	8	4	no	—
127	C₂	—	148	4	no	—
128	C₂	#14558168	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558334

C₂

yes

maximal, complemented

#14557800

C₂

yes

maximal, complemented

#14557700

C₂

yes

maximal, complemented

#14557651

C₂

yes

maximal, complemented

C₃₇×#493

C₂

yes

maximal, characteristic, complemented

#11924581

C₂²

yes

complemented

#11924562

C₂²

yes

complemented

#11924560

—

#11924558

C₂²

yes

—

#11924555

—

D₈×D₃₇

—

#11924542

C₂²

yes

complemented

#11924472

C₂²

yes

complemented

#11924464

C₂²

yes

complemented

#11924453

C₂²

yes

complemented

C₂×D₈×C₃₇

C₂²

yes

complemented

#127×C₃₇

C₂²

yes

complemented

C₄×C₈×C₃₇

C₂²

yes

characteristic, complemented

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

C₂³

yes

—

C₂×C₂²⋊D₃₇

—

#10596964

—

D₄×D₃₇

—

#10596962

C₂³

yes

—

D₈×C₃₇

—

D₈×C₃₇

—

C₂×C₈×C₃₇

C₂³

yes

—

C₄²×C₃₇

C₂³

yes

—

—

C₂³

yes

—

Dic₁₄₈

D₄

yes

complemented

D₂₉₆

D₄

yes

complemented

#10596930

D₄

yes

complemented

#10596929

—

C₈×D₃₇

—

C₂³×C₃₇

—

C₂²×D₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂⁴

yes

characteristic

C₂²⋊D₃₇

—

C₂²⋊D₃₇

—

C₂×Dic₃₇

—

D₁₄₈

C₂×D₄

yes

—

C₄×D₃₇

—

Dic₇₄

C₂×D₄

yes

—

C₈×C₃₇

C₂×D₄

yes

—

C₈×C₃₇

C₂×D₄

yes

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

H(5,𝔽₂)

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂²×D₄

yes

—

Dic₃₇

—

#3292

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

D₄²

yes

characteristic

D₃₇

—

#576

—

#493

D₃₇

yes

characteristic, complemented

#472

—

#453

—

#443

—

C₃₇

#3292

yes

characteristic, complemented, Sylow

—

—

—

—

—

C₂×D₈

—

C₂×D₈

C₂×D₃₇

yes

complemented

#131

—

#127

C₂×D₃₇

yes

—

—

—

C₄×C₈

C₂×D₃₇

yes

characteristic, complemented

#41

—

#41

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

148

—

C₂×D₄

C₂²×D₃₇

yes

—

Q₁₆

—

SD₁₆

—

SD₁₆

—

D₈

—

D₈

—

D₈

—

D₈

—

M₁₆

—

M₁₆

—

C₂×C₈

—

100

C₂×C₈

C₂²×D₃₇

yes

—

101

C₄²

C₂²×D₃₇

yes

characteristic

102

C₂³

—

103

C₂³

—

148

—

104

Q₈

—

105

D₄

—

106

D₄

—

107

D₄

—

108

D₄

—

109

D₄

—

148

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

148

—

113

C₂×C₄

C₂³×D₃₇

yes

characteristic

114

C₈

—

115

C₈

D₄×D₃₇

yes

—

116

C₈

D₄×D₃₇

yes

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

592

—

121

C₂²

#11924581

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

C₂×D₄×D₃₇

yes

—

125

C₂

—

126

C₂

—

127

C₂

—

148

—

128

C₂

#14558168

yes

characteristic

	subgroup	G/H
center	C₂	#14558168
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₃₇×#493	C₂
socle	C₂×C₃₇	D₄²
hypercenter	#493	D₃₇
hypocenter	C₃₇	#3292
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #493

subgroup

G/H

center

C₂

#14558168

commutator, cosocle

C₂×C₄×C₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₇

Fitting

C₃₇×#493

C₂

socle

C₂×C₃₇

D₄²

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #493

N	H ≅ G/N	type
1. #14558334	C₂	semidirect
2. #14557800	C₂	semidirect
3. #14557700	C₂	semidirect
4. #14557651	C₂	semidirect
5. C₃₇×#493	C₂	semidirect
6. #11924581	C₂²	semidirect
7. #11924562	C₂²	semidirect
12. #11924542	C₂²	semidirect
13. #11924472	C₂²	semidirect
14. #11924464	C₂²	semidirect
15. #11924453	C₂²	semidirect
16. C₂×D₈×C₃₇	C₂²	semidirect
17. #127×C₃₇	C₂²	semidirect
18. C₄×C₈×C₃₇	C₂²	semidirect
32. Dic₁₄₈	D₄	semidirect
33. D₂₉₆	D₄	semidirect
34. #10596930	D₄	semidirect
66. #493	D₃₇	semidirect
70. C₃₇	#3292	semidirect
77. C₂×D₈	C₂×D₃₇	semidirect
79. #127	C₂×D₃₇	semidirect
83. C₄×C₈	C₂×D₃₇	semidirect

H ≅ G/N

type

1. #14558334

C₂

semidirect

2. #14557800

C₂

semidirect

3. #14557700

C₂

semidirect

4. #14557651

C₂

semidirect

5. C₃₇×#493

C₂

semidirect

6. #11924581

C₂²

semidirect

7. #11924562

C₂²

semidirect

12. #11924542

C₂²

semidirect

13. #11924472

C₂²

semidirect

14. #11924464

C₂²

semidirect

15. #11924453

C₂²

semidirect

16. C₂×D₈×C₃₇

C₂²

semidirect

17. #127×C₃₇

C₂²

semidirect

18. C₄×C₈×C₃₇

C₂²

semidirect

32. Dic₁₄₈

D₄

semidirect

33. D₂₉₆

D₄

semidirect

34. #10596930

D₄

semidirect

66. #493

D₃₇

semidirect

70. C₃₇

#3292

semidirect

77. C₂×D₈

C₂×D₃₇

semidirect

79. #127

C₂×D₃₇

semidirect

83. C₄×C₈

C₂×D₃₇

semidirect

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

subgroup	index
#14558334	2
#14557800	2
#14557700	2
#14557651	2
C₃₇×#493	2
#3292	37

subgroup

index

C₃₇×#493

Subgroups — #18006395

Conjugacy classes of subgroups

Complements

Maximal subgroups