Group #14558277 — subgroups

Subgroups — #14558277

Order 2368.

220 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 91 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	#11924528	C₂	1	1	yes	maximal, characteristic, complemented
2	#11924524	C₂	1	1	yes	maximal, characteristic, complemented
3	#11924520	C₂	1	2	yes	maximal, complemented
4	#11924518	C₂	1	2	yes	maximal, complemented
5	#11924513	C₂	1	2	yes	maximal, complemented
6	#11924512	C₂	1	2	yes	maximal, complemented
7	#11924509	C₂	1	2	yes	maximal, complemented
8	#11924508	C₂	1	2	yes	maximal, complemented
9	#124×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₈×C₃₇	C₂²	1	1	yes	characteristic
11	C₂×D₄×C₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₃₇	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₃₇	C₂²	1	2	yes	—
14	C₂²×Dic₃₇	C₂²	1	2	yes	—
15	C₂×D₁₄₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₃₇	C₂²	1	2	yes	—
17	C₂×Dic₇₄	C₂²	1	1	yes	characteristic
18	C₂²⋊C₄×C₃₇	C₂²	1	2	yes	—
19	C₂²⋊C₄×C₃₇	C₂²	1	2	yes	—
20	C₄²×C₃₇	C₂²	1	1	yes	characteristic
21	C₂²⋊Dic₃₇	C₂²	1	2	yes	—
22	#10596938	C₂²	1	2	yes	—
23	#10596938	C₂²	1	2	yes	—
24	#10596938	C₂²	1	2	yes	—
25	C₄⋊Dic₃₇	C₂²	1	2	yes	—
26	C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
27	C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
28	#10596936	C₂²	1	2	yes	—
29	#10596936	C₂²	1	2	yes	—
30	#10596936	C₂²	1	2	yes	—
31	C₂³×C₃₇	C₂³	1	2	yes	—
32	C₂²×D₃₇	C₂³	1	2	yes	—
33	Q₈×C₃₇	—	4	1	no	—
34	D₄×C₃₇	—	4	1	no	—
35	C₂×C₄×C₃₇	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₃₇	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₃₇	C₂³	1	2	yes	—
38	C₂×C₄×C₃₇	C₂³	1	1	yes	characteristic
39	C₂²⋊D₃₇	—	4	2	no	—
40	C₂²⋊D₃₇	—	4	2	no	—
41	C₂×Dic₃₇	—	4	2	no	—
42	C₂×Dic₃₇	C₂³	1	2	yes	—
43	C₂×Dic₃₇	C₂³	1	2	yes	—
44	C₂×Dic₃₇	C₂³	1	2	yes	—
45	D₁₄₈	—	4	1	no	—
46	C₄×D₃₇	—	4	2	no	—
47	Dic₇₄	—	4	1	no	—
48	C₂²×C₃₇	—	2	2	no	—
49	C₂²×C₃₇	—	2	2	no	—
50	C₂²×C₃₇	—	2	2	no	—
51	C₂²×C₃₇	C₂⁴	1	1	yes	characteristic
52	C₂×D₃₇	—	2	2	no	—
53	C₂×D₃₇	—	2	2	no	—
54	C₂×D₃₇	—	2	2	no	—
55	C₄×C₃₇	—	2	1	no	—
56	C₄×C₃₇	—	2	2	no	—
57	C₄×C₃₇	—	2	1	no	—
58	C₄×C₃₇	—	2	1	no	—
59	Dic₃₇	—	2	2	no	—
60	Dic₃₇	—	2	2	no	—
61	Dic₃₇	—	2	2	no	—
62	C₂×C₃₇	—	4	2	no	—
63	C₂×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
64	C₂×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
65	C₂×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
66	D₃₇	—	4	2	no	—
67	#562	—	37	1	no	maximal, Sylow
68	C₃₇	#562	1	1	yes	characteristic, complemented, Sylow
69	#125	—	37	1	no	—
70	#124	—	37	1	no	—
71	#124	D₃₇	1	1	yes	characteristic, complemented
72	#123	—	37	2	no	—
73	#123	—	37	2	no	—
74	C₂²⋊Q₈	—	37	2	no	—
75	C₂²⋊Q₈	—	37	2	no	—
76	#121	—	37	2	no	—
77	#121	—	37	2	no	—
78	C₂×Q₈	—	37	1	no	—
79	C₂×Q₈	C₂×D₃₇	1	1	yes	characteristic
80	C₂×D₄	—	37	2	no	—
81	C₂×D₄	—	37	2	no	—
82	C₂×D₄	—	37	1	no	—
83	C₂×D₄	C₂×D₃₇	1	1	yes	characteristic
84	C₂²×C₄	—	37	2	no	—
85	C₂²×C₄	—	37	2	no	—
86	C₄⋊C₄	—	37	2	no	—
87	C₄⋊C₄	—	37	2	no	—
88	C₄⋊C₄	—	37	2	no	—
89	C₄⋊C₄	—	37	2	no	—
90	C₄⋊C₄	—	37	1	no	—
91	C₄⋊C₄	—	37	1	no	—
92	C₂²⋊C₄	—	37	2	no	—
93	C₂²⋊C₄	—	37	2	no	—
94	C₂²⋊C₄	—	37	2	no	—
95	C₂²⋊C₄	—	37	2	no	—
96	C₂²⋊C₄	C₂×D₃₇	1	2	yes	—
97	C₂²⋊C₄	C₂×D₃₇	1	2	yes	—
98	C₄²	C₂×D₃₇	1	1	yes	characteristic
99	C₂³	—	37	2	no	—
100	C₂³	C₂²×D₃₇	1	2	yes	—
101	Q₈	—	4	1	no	—
102	Q₈	—	148	1	no	—
103	D₄	—	4	1	no	—
104	D₄	—	148	1	no	—
105	D₄	—	148	2	no	—
106	D₄	—	148	2	no	—
107	C₂×C₄	—	37	2	no	—
108	C₂×C₄	—	37	2	no	—
109	C₂×C₄	—	37	2	no	—
110	C₂×C₄	—	148	2	no	—
111	C₂×C₄	—	148	2	no	—
112	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
114	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
115	C₂×C₄	C₂²×D₃₇	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	74	2	no	—
120	C₂²	—	74	2	no	—
121	C₂²	—	74	2	no	—
122	C₂²	C₂³×D₃₇	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	—	74	2	no	—
128	C₄	—	74	2	no	—
129	C₄	—	74	2	no	—
130	C₂	—	4	2	no	—
131	C₂	—	148	2	no	—
132	C₂	#11924581	1	1	yes	characteristic
133	C₂	#11924582	1	1	yes	characteristic
134	C₂	#11924583	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11924528

C₂

yes

maximal, characteristic, complemented

#11924524

C₂

yes

maximal, characteristic, complemented

#11924520

C₂

yes

maximal, complemented

#11924518

C₂

yes

maximal, complemented

#11924513

C₂

yes

maximal, complemented

#11924512

C₂

yes

maximal, complemented

#11924509

C₂

yes

maximal, complemented

#11924508

C₂

yes

maximal, complemented

#124×C₃₇

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₃₇

C₂²

yes

characteristic

C₂×D₄×C₃₇

C₂²

yes

characteristic

C₂×C₂²⋊D₃₇

C₂²

yes

—

C₂×C₂²⋊D₃₇

C₂²

yes

—

C₂²×Dic₃₇

C₂²

yes

—

C₂×D₁₄₈

C₂²

yes

characteristic

C₂×C₄×D₃₇

C₂²

yes

—

C₂×Dic₇₄

C₂²

yes

characteristic

C₂²⋊C₄×C₃₇

C₂²

yes

—

C₂²⋊C₄×C₃₇

C₂²

yes

—

C₄²×C₃₇

C₂²

yes

characteristic

C₂²⋊Dic₃₇

C₂²

yes

—

#10596938

C₂²

yes

—

#10596938

C₂²

yes

—

#10596938

C₂²

yes

—

C₄⋊Dic₃₇

C₂²

yes

—

C₄⋊Dic₃₇

C₂²

yes

characteristic

C₄⋊Dic₃₇

C₂²

yes

characteristic

#10596936

C₂²

yes

—

#10596936

C₂²

yes

—

#10596936

C₂²

yes

—

C₂³×C₃₇

C₂³

yes

—

C₂²×D₃₇

C₂³

yes

—

Q₈×C₃₇

—

D₄×C₃₇

—

C₂×C₄×C₃₇

C₂³

yes

characteristic

C₂×C₄×C₃₇

C₂³

yes

characteristic

C₂×C₄×C₃₇

C₂³

yes

—

C₂×C₄×C₃₇

C₂³

yes

characteristic

C₂²⋊D₃₇

—

C₂²⋊D₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂³

yes

—

C₂×Dic₃₇

C₂³

yes

—

C₂×Dic₃₇

C₂³

yes

—

D₁₄₈

—

C₄×D₃₇

—

Dic₇₄

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂⁴

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

C₂×C₃₇

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

2₋¹⁺⁴

yes

characteristic

D₃₇

—

#562

—

maximal, Sylow

C₃₇

#562

yes

characteristic, complemented, Sylow

#125

—

#124

—

#124

D₃₇

yes

characteristic, complemented

#123

—

#123

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

#121

—

#121

—

C₂×Q₈

—

C₂×Q₈

C₂×D₃₇

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₃₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₃₇

yes

—

C₂²⋊C₄

C₂×D₃₇

yes

—

C₄²

C₂×D₃₇

yes

characteristic

C₂³

—

100

C₂³

C₂²×D₃₇

yes

—

101

Q₈

—

102

Q₈

—

148

—

103

D₄

—

104

D₄

—

148

—

105

D₄

—

148

—

106

D₄

—

148

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

148

—

111

C₂×C₄

—

148

—

112

C₂×C₄

C₂²×D₃₇

yes

characteristic

113

C₂×C₄

C₂²×D₃₇

yes

characteristic

114

C₂×C₄

C₂²×D₃₇

yes

characteristic

115

C₂×C₄

C₂²×D₃₇

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

C₂³×D₃₇

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₂

—

131

C₂

—

148

—

132

C₂

#11924581

yes

characteristic

133

C₂

#11924582

yes

characteristic

134

C₂

#11924583

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₇
commutator, socle, cosocle	C₂²×C₃₇	C₂⁴
Fitting	#124×C₃₇	C₂
hypercenter	#124	D₃₇
hypocenter	C₃₇	#562
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₇

commutator, socle, cosocle

C₂²×C₃₇

C₂⁴

Fitting

#124×C₃₇

C₂

D₃₇

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. #11924528	C₂	semidirect
2. #11924524	C₂	semidirect
3. #11924520	C₂	semidirect
4. #11924518	C₂	semidirect
5. #11924513	C₂	semidirect
6. #11924512	C₂	semidirect
7. #11924509	C₂	semidirect
8. #11924508	C₂	semidirect
9. #124×C₃₇	C₂	semidirect
68. C₃₇	#562	semidirect
71. #124	D₃₇	semidirect

H ≅ G/N

type

1. #11924528

C₂

semidirect

2. #11924524

C₂

semidirect

3. #11924520

C₂

semidirect

4. #11924518

C₂

semidirect

5. #11924513

C₂

semidirect

6. #11924512

C₂

semidirect

7. #11924509

C₂

semidirect

8. #11924508

C₂

semidirect

9. #124×C₃₇

C₂

semidirect

68. C₃₇

#562

semidirect

71. #124

D₃₇

semidirect

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

subgroup	index
#11924528	2
#11924524	2
#11924520	2
#11924518	2
#11924513	2
#11924512	2
#11924509	2
#11924508	2
#124×C₃₇	2
#562	37

subgroup

index

#124×C₃₇

Subgroups — #14558277

Conjugacy classes of subgroups

Complements

Maximal subgroups