Group #14558155 — subgroups

Subgroups — #14558155

Order 2368.

216 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 95 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	#11924535	C₂	1	1	yes	maximal, characteristic, complemented
2	#11924530	C₂	1	2	yes	maximal, complemented
3	#11924525	C₂	1	1	yes	maximal, characteristic, complemented
4	#11924524	C₂	1	2	yes	maximal, complemented
5	#11924521	C₂	1	2	yes	maximal, complemented
6	#11924514	C₂	1	2	yes	maximal, complemented
7	#11924512	C₂	1	1	yes	maximal, characteristic, complemented
8	#11924511	C₂	1	1	yes	maximal, characteristic
9	C₄×D₁₄₈	C₂	1	1	yes	maximal, characteristic, complemented
10	#11924493	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×C₃₇	C₂²	1	1	yes	characteristic
13	C₂×D₁₄₈	C₂²	1	1	yes	characteristic
14	C₂×D₁₄₈	C₂²	1	2	yes	—
15	C₂×C₄×D₃₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₃₇	C₂²	1	2	yes	—
17	C₂×Dic₇₄	C₂²	1	2	yes	—
18	C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
20	C₄²×C₃₇	C₂²	1	1	yes	characteristic
21	C₄²×C₃₇	C₂²	1	2	yes	—
22	#10596938	C₂²	1	1	yes	characteristic
23	#10596938	C₂²	1	1	yes	characteristic
24	#10596938	C₂²	1	2	yes	—
25	#10596938	C₂²	1	2	yes	—
26	#10596938	C₂²	1	2	yes	—
27	#10596938	C₂²	1	2	yes	—
28	#10596938	C₂²	1	2	yes	—
29	C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
30	#10596936	C₂²	1	1	yes	characteristic
31	#10596936	C₂²	1	1	yes	characteristic
32	#10596936	C₂²	1	2	yes	—
33	#10596936	C₂²	1	2	yes	—
34	C₄×Dic₃₇	C₂²	1	1	yes	characteristic
35	C₂²×D₃₇	C₂³	1	1	yes	characteristic
36	C₂²×D₃₇	C₂³	1	2	yes	—
37	Q₈×C₃₇	—	2	2	no	—
38	C₂×C₄×C₃₇	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₃₇	C₂³	1	2	yes	—
40	C₂×C₄×C₃₇	C₂³	1	2	yes	—
41	C₂×C₄×C₃₇	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₃₇	C₂³	1	1	yes	characteristic
43	C₂×Dic₃₇	C₂³	1	1	yes	characteristic
44	C₂×Dic₃₇	C₂³	1	1	yes	characteristic
45	C₂×Dic₃₇	C₂³	1	1	yes	characteristic
46	C₂×Dic₃₇	C₂³	1	2	yes	—
47	D₁₄₈	—	2	2	no	—
48	D₁₄₈	—	4	2	no	—
49	C₄×D₃₇	—	2	2	no	—
50	C₄×D₃₇	—	4	2	no	—
51	Dic₇₄	—	4	2	no	—
52	C₂²×C₃₇	C₂⁴	1	1	yes	characteristic
53	C₂×D₃₇	—	2	1	no	—
54	C₂×D₃₇	—	2	2	no	—
55	C₂×D₃₇	—	2	2	no	—
56	C₂×D₃₇	—	2	1	no	—
57	C₂×D₃₇	—	2	1	no	—
58	C₂×D₃₇	—	2	2	no	—
59	C₄×C₃₇	—	2	2	no	—
60	C₄×C₃₇	—	2	2	no	—
61	C₄×C₃₇	—	2	1	no	—
62	C₄×C₃₇	—	2	1	no	—
63	C₄×C₃₇	#41	1	2	yes	—
64	Dic₃₇	—	2	1	no	—
65	Dic₃₇	—	2	1	no	—
66	Dic₃₇	—	2	2	no	—
67	Dic₃₇	—	2	1	no	—
68	C₂×C₃₇	C₂×#41	1	1	yes	characteristic
69	C₂×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
70	C₂×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
71	D₃₇	—	4	1	no	—
72	D₃₇	—	4	2	no	—
73	#542	—	37	1	no	maximal, Sylow
74	C₃₇	#542	1	1	yes	characteristic, complemented, Sylow
75	C₄⋊Q₈	—	37	1	no	—
76	#126	—	37	2	no	—
77	#124	—	37	1	no	—
78	#124	—	37	2	no	—
79	#123	—	37	2	no	—
80	C₂²⋊Q₈	—	37	1	no	—
81	C₂²⋊Q₈	—	37	2	no	—
82	#121	—	37	1	no	—
83	C₄×Q₈	D₃₇	1	1	yes	characteristic, complemented
84	C₄×D₄	—	37	1	no	—
85	#117	—	37	1	no	—
86	C₂×Q₈	—	37	2	no	—
87	C₂×Q₈	C₂×D₃₇	1	1	yes	characteristic
88	C₂×D₄	—	37	1	no	—
89	C₂×D₄	—	37	2	no	—
90	C₂²×C₄	—	37	1	no	—
91	C₂²×C₄	—	37	2	no	—
92	C₄⋊C₄	—	37	1	no	—
93	C₄⋊C₄	—	37	1	no	—
94	C₄⋊C₄	—	37	2	no	—
95	C₄⋊C₄	—	37	2	no	—
96	C₄⋊C₄	—	37	1	no	—
97	C₄⋊C₄	C₂×D₃₇	1	1	yes	characteristic
98	C₄⋊C₄	C₂×D₃₇	1	2	yes	—
99	C₂²⋊C₄	—	37	1	no	—
100	C₂²⋊C₄	—	37	1	no	—
101	C₂²⋊C₄	—	37	2	no	—
102	C₂²⋊C₄	—	37	2	no	—
103	C₂²⋊C₄	—	37	2	no	—
104	C₂²⋊C₄	—	37	2	no	—
105	C₂²⋊C₄	—	37	2	no	—
106	C₄²	—	37	1	no	—
107	C₄²	C₂×D₃₇	1	1	yes	characteristic
108	C₄²	C₂×D₃₇	1	2	yes	—
109	C₂³	—	37	1	no	—
110	C₂³	—	37	2	no	—
111	Q₈	—	2	2	no	—
112	Q₈	—	148	2	no	—
113	D₄	—	74	2	no	—
114	D₄	—	148	2	no	—
115	C₂×C₄	—	37	1	no	—
116	C₂×C₄	—	37	1	no	—
117	C₂×C₄	—	37	1	no	—
118	C₂×C₄	—	37	2	no	—
119	C₂×C₄	—	74	2	no	—
120	C₂×C₄	—	148	2	no	—
121	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
122	C₂×C₄	C₂²×D₃₇	1	2	yes	—
123	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
124	C₂×C₄	C₂²×D₃₇	1	2	yes	—
125	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
126	C₂²	—	74	1	no	—
127	C₂²	—	74	2	no	—
128	C₂²	—	74	2	no	—
129	C₂²	—	74	1	no	—
130	C₂²	—	74	1	no	—
131	C₂²	—	74	2	no	—
132	C₂²	C₂³×D₃₇	1	1	yes	characteristic
133	C₄	—	2	2	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	2	no	—
136	C₄	—	2	1	no	—
137	C₄	—	74	1	no	—
138	C₄	—	74	2	no	—
139	C₄	—	74	1	no	—
140	C₄	—	74	1	no	—
141	C₄	#10596962	1	2	yes	—
142	C₂	—	148	1	no	—
143	C₂	—	148	2	no	—
144	C₂	C₂×#10596962	1	1	yes	characteristic
145	C₂	#11924582	1	1	yes	characteristic
146	C₂	#11924584	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11924535

C₂

yes

maximal, characteristic, complemented

#11924530

C₂

yes

maximal, complemented

#11924525

C₂

yes

maximal, characteristic, complemented

#11924524

C₂

yes

maximal, complemented

#11924521

C₂

yes

maximal, complemented

#11924514

C₂

yes

maximal, complemented

#11924512

C₂

yes

maximal, characteristic, complemented

#11924511

C₂

yes

maximal, characteristic

C₄×D₁₄₈

C₂

yes

maximal, characteristic, complemented

#11924493

C₂

yes

maximal, characteristic, complemented

C₄×Q₈×C₃₇

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₃₇

C₂²

yes

characteristic

C₂×D₁₄₈

C₂²

yes

characteristic

C₂×D₁₄₈

C₂²

yes

—

C₂×C₄×D₃₇

C₂²

yes

characteristic

C₂×C₄×D₃₇

C₂²

yes

—

C₂×Dic₇₄

C₂²

yes

—

C₄⋊C₄×C₃₇

C₂²

yes

characteristic

C₄⋊C₄×C₃₇

C₂²

yes

—

C₄²×C₃₇

C₂²

yes

characteristic

C₄²×C₃₇

C₂²

yes

—

#10596938

C₂²

yes

characteristic

#10596938

C₂²

yes

characteristic

#10596938

C₂²

yes

—

#10596938

C₂²

yes

—

#10596938

C₂²

yes

—

#10596938

C₂²

yes

—

#10596938

C₂²

yes

—

C₄⋊Dic₃₇

C₂²

yes

characteristic

#10596936

C₂²

yes

characteristic

#10596936

C₂²

yes

characteristic

#10596936

C₂²

yes

—

#10596936

C₂²

yes

—

C₄×Dic₃₇

C₂²

yes

characteristic

C₂²×D₃₇

C₂³

yes

characteristic

C₂²×D₃₇

C₂³

yes

—

Q₈×C₃₇

—

C₂×C₄×C₃₇

C₂³

yes

characteristic

C₂×C₄×C₃₇

C₂³

yes

—

C₂×C₄×C₃₇

C₂³

yes

—

C₂×C₄×C₃₇

C₂³

yes

characteristic

C₂×C₄×C₃₇

C₂³

yes

characteristic

C₂×Dic₃₇

C₂³

yes

characteristic

C₂×Dic₃₇

C₂³

yes

characteristic

C₂×Dic₃₇

C₂³

yes

characteristic

C₂×Dic₃₇

C₂³

yes

—

D₁₄₈

—

D₁₄₈

—

C₄×D₃₇

—

C₄×D₃₇

—

Dic₇₄

—

C₂²×C₃₇

C₂⁴

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#41

yes

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

C₂×C₃₇

C₂×#41

yes

characteristic

C₂×C₃₇

H(5,𝔽₂)

yes

characteristic

C₂×C₃₇

2₋¹⁺⁴

yes

characteristic

D₃₇

—

D₃₇

—

#542

—

maximal, Sylow

C₃₇

#542

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

#121

—

C₄×Q₈

D₃₇

yes

characteristic, complemented

C₄×D₄

—

#117

—

C₂×Q₈

—

C₂×Q₈

C₂×D₃₇

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₃₇

yes

characteristic

C₄⋊C₄

C₂×D₃₇

yes

—

C₂²⋊C₄

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

—

104

C₂²⋊C₄

—

105

C₂²⋊C₄

—

106

C₄²

—

107

C₄²

C₂×D₃₇

yes

characteristic

108

C₄²

C₂×D₃₇

yes

—

109

C₂³

—

110

C₂³

—

111

Q₈

—

112

Q₈

—

148

—

113

D₄

—

114

D₄

—

148

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

148

—

121

C₂×C₄

C₂²×D₃₇

yes

characteristic

122

C₂×C₄

C₂²×D₃₇

yes

—

123

C₂×C₄

C₂²×D₃₇

yes

characteristic

124

C₂×C₄

C₂²×D₃₇

yes

—

125

C₂×C₄

C₂²×D₃₇

yes

characteristic

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂³×D₃₇

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#10596962

yes

—

142

C₂

—

148

—

143

C₂

—

148

—

144

C₂

C₂×#10596962

yes

characteristic

145

C₂

#11924582

yes

characteristic

146

C₂

#11924584

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₇
commutator, socle, cosocle	C₂²×C₃₇	C₂⁴
Fitting	C₄×Q₈×C₃₇	C₂
prime radical	#11924511	C₂
hypercenter	C₄×Q₈	D₃₇
hypocenter	C₃₇	#542
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₇

commutator, socle, cosocle

C₂²×C₃₇

C₂⁴

Fitting

C₄×Q₈×C₃₇

C₂

prime radical

#11924511

C₂

hypercenter

C₄×Q₈

D₃₇

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
1. #11924535	C₂	semidirect
2. #11924530	C₂	semidirect
3. #11924525	C₂	semidirect
4. #11924524	C₂	semidirect
5. #11924521	C₂	semidirect
6. #11924514	C₂	semidirect
7. #11924512	C₂	semidirect
9. C₄×D₁₄₈	C₂	semidirect
10. #11924493	C₂	semidirect
11. C₄×Q₈×C₃₇	C₂	semidirect
74. C₃₇	#542	semidirect
83. C₄×Q₈	D₃₇	semidirect

H ≅ G/N

type

1. #11924535

C₂

semidirect

2. #11924530

C₂

semidirect

3. #11924525

C₂

semidirect

4. #11924524

C₂

semidirect

5. #11924521

C₂

semidirect

6. #11924514

C₂

semidirect

7. #11924512

C₂

semidirect

9. C₄×D₁₄₈

C₂

semidirect

10. #11924493

C₂

semidirect

11. C₄×Q₈×C₃₇

C₂

semidirect

74. C₃₇

#542

semidirect

83. C₄×Q₈

D₃₇

semidirect

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

subgroup	index
#11924535	2
#11924530	2
#11924525	2
#11924524	2
#11924521	2
#11924514	2
#11924512	2
#11924511	2
C₄×D₁₄₈	2
#11924493	2
C₄×Q₈×C₃₇	2
#542	37

subgroup

index

C₄×D₁₄₈

C₄×Q₈×C₃₇

Subgroups — #14558155

Conjugacy classes of subgroups

Complements

Maximal subgroups