Group #18011323 — subgroups

Subgroups — #18011323

Order 4736.

484 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 162 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	C₂×#11924624	C₂	1	2	yes	maximal, complemented
2	#14558563	C₂	1	8	yes	maximal, complemented
3	#14558538	C₂	1	1	yes	maximal, characteristic, complemented
4	#14558508	C₂	1	2	yes	maximal, complemented
5	#14558492	C₂	1	1	yes	maximal, characteristic, complemented
6	#14558084	C₂	1	1	yes	maximal, characteristic
7	#11924624	—	2	8	no	—
8	C₂×#10596953	C₂²	1	2	yes	—
9	C₂×#10596953	C₂²	1	4	yes	—
10	C₂×D₃₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×D₃₇⋊C₈	C₂²	1	4	yes	—
12	C₂×D₃₇⋊C₈	C₂²	1	1	yes	characteristic
13	#11924603	C₂²	1	4	yes	—
14	#11924593	C₂²	1	8	yes	—
15	#11924591	C₂²	1	4	yes	—
16	C₂×#10596966	C₂²	1	2	yes	—
17	#11924533	C₄	1	1	yes	characteristic
18	#11924525	C₄	1	2	yes	—
19	#11924511	C₄	1	4	yes	—
20	#11924498	C₂²	1	4	yes	—
21	C₄²×D₃₇	C₂²	1	1	yes	characteristic
22	C₄⋊Q₈×C₃₇	C₄	1	1	yes	characteristic
23	C₂×Q₈×C₃₇	C₂×C₄	1	2	yes	—
24	#10596966	—	2	4	no	—
25	#10596966	—	2	4	no	—
26	C₂×D₁₄₈	C₂×C₄	1	4	yes	—
27	C₂×D₁₄₈	C₂×C₄	1	2	yes	—
28	C₂×D₁₄₈	C₂×C₄	1	4	yes	—
29	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
30	C₂×C₄×D₃₇	C₂³	1	4	yes	—
31	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
32	C₂×#64438	C₂³	1	2	yes	—
33	C₂×#64438	C₂³	1	4	yes	—
34	C₂×#64438	C₂³	1	2	yes	—
35	#10596953	—	2	4	no	—
36	#10596953	—	2	8	no	—
37	D₃₇⋊C₈	—	2	2	no	—
38	D₃₇⋊C₈	—	2	8	no	—
39	D₃₇⋊C₈	D₄	1	4	yes	complemented
40	C₄⋊C₄×C₃₇	C₂×C₄	1	4	yes	—
41	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
42	#10596938	C₂×C₄	1	8	yes	—
43	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
44	C₄×Dic₃₇	C₂³	1	2	yes	—
45	C₂²×D₃₇	C₂²×C₄	1	4	yes	—
46	C₂²×D₃₇	C₂²×C₄	1	1	yes	characteristic
47	Q₈×C₃₇	—	2	4	no	—
48	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₃₇	C₂²×C₄	1	4	yes	—
50	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
52	C₂×Dic₃₇	C₂⁴	1	1	yes	characteristic
53	D₁₄₈	—	2	8	no	—
54	D₁₄₈	—	2	4	no	—
55	D₁₄₈	—	4	4	no	—
56	C₄×D₃₇	—	2	4	no	—
57	C₄×D₃₇	—	2	4	no	—
58	C₄×D₃₇	—	4	2	no	—
59	C₄×D₃₇	C₂×D₄	1	2	yes	—
60	C₄×D₃₇	#41	1	2	yes	—
61	#64438	—	2	4	no	—
62	#64438	—	2	2	no	—
63	#64438	C₂×D₄	1	4	yes	—
64	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
65	C₂×D₃₇	—	2	2	no	—
66	C₂×D₃₇	—	2	4	no	—
67	C₂×D₃₇	—	2	8	no	—
68	C₂×D₃₇	C₄×D₄	1	2	yes	—
69	C₄×C₃₇	—	2	4	no	—
70	C₄×C₃₇	—	2	2	no	—
71	C₄×C₃₇	C₄×D₄	1	2	yes	—
72	Dic₃₇	—	2	2	no	—
73	Dic₃₇	C₂²×D₄	1	1	yes	characteristic
74	Dic₃₇	C₂×#41	1	1	yes	characteristic
75	#2932	—	37	1	no	maximal, Sylow
76	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
77	C₂×C₃₇	#568	1	2	yes	—
78	D₃₇	—	2	2	no	—
79	D₃₇	—	4	4	no	—
80	C₂×#131	—	37	2	no	—
81	#532	—	37	1	no	—
82	#435	—	37	8	no	—
83	#423	—	37	1	no	—
84	#408	—	37	2	no	—
85	C₂×#97	—	37	1	no	—
86	C₃₇	#2932	1	1	yes	characteristic, complemented, Sylow
87	C₂×#41	—	37	2	no	—
88	#131	—	74	8	no	—
89	C₂×M₁₆	—	37	2	no	—
90	C₂×M₁₆	—	37	4	no	—
91	C₂²×C₈	—	37	1	no	—
92	C₂²×C₈	—	37	4	no	—
93	C₂²×C₈	—	37	1	no	—
94	C₄⋊Q₈	#3866	1	1	yes	characteristic
95	#127	—	37	1	no	—
96	#124	—	37	2	no	—
97	#121	—	37	4	no	—
98	C₄×D₄	—	37	4	no	—
99	C₂×C₄²	—	37	1	no	—
100	C₄⋊C₈	—	37	4	no	—
101	C₂²⋊C₈	—	37	8	no	—
102	#97	—	37	4	no	—
103	#41	—	74	4	no	—
104	#41	—	74	4	no	—
105	C₂×Q₈	C₂×#3866	1	2	yes	—
106	C₂×D₄	—	37	4	no	—
107	C₂×D₄	—	37	2	no	—
108	C₂×D₄	—	37	4	no	—
109	C₂²×C₄	—	37	1	no	—
110	C₂²×C₄	—	37	4	no	—
111	C₂²×C₄	—	37	2	no	—
112	M₁₆	—	74	8	no	—
113	M₁₆	—	74	4	no	—
114	C₂×C₈	—	37	4	no	—
115	C₂×C₈	—	37	2	no	—
116	C₂×C₈	—	37	2	no	—
117	C₂×C₈	—	37	4	no	—
118	C₂×C₈	—	74	8	no	—
119	C₂×C₈	—	74	2	no	—
120	C₄⋊C₄	C₂×#3866	1	4	yes	—
121	C₂²⋊C₄	—	37	8	no	—
122	C₄²	—	37	1	no	—
123	C₄²	—	37	2	no	—
124	C₄²	C₂×#3866	1	1	yes	characteristic
125	C₂³	—	37	1	no	—
126	C₂³	—	37	4	no	—
127	Q₈	—	2	4	no	—
128	D₄	—	74	8	no	—
129	D₄	—	74	4	no	—
130	D₄	—	148	4	no	—
131	C₂×C₄	—	37	1	no	—
132	C₂×C₄	—	37	2	no	—
133	C₂×C₄	—	37	2	no	—
134	C₂×C₄	—	37	2	no	—
135	C₂×C₄	—	74	4	no	—
136	C₂×C₄	—	74	4	no	—
137	C₂×C₄	—	148	2	no	—
138	C₂×C₄	C₂²×#3866	1	2	yes	—
139	C₂×C₄	C₂²×#3866	1	4	yes	—
140	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
141	C₈	—	37	4	no	—
142	C₈	—	74	4	no	—
143	C₈	—	74	2	no	—
144	C₂²	—	37	2	no	—
145	C₂²	—	74	2	no	—
146	C₂²	—	74	4	no	—
147	C₂²	—	74	8	no	—
148	C₂²	C₂³×#3866	1	1	yes	characteristic
149	C₄	—	2	4	no	—
150	C₄	—	2	2	no	—
151	C₄	—	37	1	no	—
152	C₄	—	37	1	no	—
153	C₄	—	74	2	no	—
154	C₄	D₄×#3866	1	2	yes	—
155	C₂	—	74	2	no	—
156	C₂	—	148	4	no	—
157	C₂	C₂×D₄×#3866	1	1	yes	characteristic
158	C₂	#14558613	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924624

C₂

yes

maximal, complemented

#14558563

C₂

yes

maximal, complemented

#14558538

C₂

yes

maximal, characteristic, complemented

#14558508

C₂

yes

maximal, complemented

#14558492

C₂

yes

maximal, characteristic, complemented

#14558084

C₂

yes

maximal, characteristic

#11924624

—

C₂×#10596953

C₂²

yes

—

C₂×#10596953

C₂²

yes

—

C₂×D₃₇⋊C₈

C₂²

yes

characteristic

C₂×D₃₇⋊C₈

C₂²

yes

—

C₂×D₃₇⋊C₈

C₂²

yes

characteristic

#11924603

C₂²

yes

—

#11924593

C₂²

yes

—

#11924591

C₂²

yes

—

C₂×#10596966

C₂²

yes

—

#11924533

C₄

yes

characteristic

#11924525

C₄

yes

—

#11924511

C₄

yes

—

#11924498

C₂²

yes

—

C₄²×D₃₇

C₂²

yes

characteristic

C₄⋊Q₈×C₃₇

C₄

yes

characteristic

C₂×Q₈×C₃₇

C₂×C₄

yes

—

#10596966

—

#10596966

—

C₂×D₁₄₈

C₂×C₄

yes

—

C₂×D₁₄₈

C₂×C₄

yes

—

C₂×D₁₄₈

C₂×C₄

yes

—

C₂×C₄×D₃₇

C₂×C₄

yes

—

C₂×C₄×D₃₇

C₂³

yes

—

C₂×C₄×D₃₇

C₂³

yes

characteristic

C₂×#64438

C₂³

yes

—

C₂×#64438

C₂³

yes

—

C₂×#64438

C₂³

yes

—

#10596953

—

#10596953

—

D₃₇⋊C₈

—

D₃₇⋊C₈

—

D₃₇⋊C₈

D₄

yes

complemented

C₄⋊C₄×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

C₂×C₄

yes

characteristic

#10596938

C₂×C₄

yes

—

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

C₂³

yes

—

C₂²×D₃₇

C₂²×C₄

yes

—

C₂²×D₃₇

C₂²×C₄

yes

characteristic

Q₈×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

C₂²×C₄

yes

—

C₂×Dic₃₇

C₂⁴

yes

characteristic

D₁₄₈

—

D₁₄₈

—

D₁₄₈

—

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

C₂×D₄

yes

—

C₄×D₃₇

#41

yes

—

#64438

—

#64438

—

#64438

C₂×D₄

yes

—

C₂²×C₃₇

C₂³×C₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

C₄×D₄

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₄×D₄

yes

—

Dic₃₇

—

Dic₃₇

C₂²×D₄

yes

characteristic

Dic₃₇

C₂×#41

yes

characteristic

#2932

—

maximal, Sylow

C₂×C₃₇

C₂×C₄×D₄

yes

characteristic

C₂×C₃₇

#568

yes

—

D₃₇

—

D₃₇

—

—

—

—

—

—

C₂×#97

—

C₃₇

#2932

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

yes

—

—

—

C₄×D₄

—

C₂×C₄²

—

100

C₄⋊C₈

—

101

C₂²⋊C₈

—

102

#97

—

103

#41

—

104

#41

—

105

C₂×Q₈

C₂×#3866

yes

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

M₁₆

—

113

M₁₆

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₄⋊C₄

C₂×#3866

yes

—

121

C₂²⋊C₄

—

122

C₄²

—

123

C₄²

—

124

C₄²

C₂×#3866

yes

characteristic

125

C₂³

—

126

C₂³

—

127

Q₈

—

128

D₄

—

129

D₄

—

130

D₄

—

148

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

148

—

138

C₂×C₄

C₂²×#3866

yes

—

139

C₂×C₄

C₂²×#3866

yes

—

140

C₂×C₄

C₂²×#3866

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

C₂³×#3866

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

D₄×#3866

yes

—

155

C₂

—

156

C₂

—

148

—

157

C₂

C₂×D₄×#3866

yes

characteristic

158

C₂

#14558613

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×#3866
commutator, socle	C₂²×C₃₇	C₂³×C₄
Fitting	C₄⋊Q₈×C₃₇	C₄
prime radical	#14558084	C₂
cosocle	C₂×Dic₃₇	C₂⁴
hypercenter	C₄⋊Q₈	#3866
hypocenter	C₃₇	#2932
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×#3866

commutator, socle

C₂²×C₃₇

C₂³×C₄

Fitting

C₄⋊Q₈×C₃₇

C₄

prime radical

#14558084

C₂

cosocle

C₂×Dic₃₇

C₂⁴

C₄⋊Q₈

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. C₂×#11924624	C₂	semidirect
2. #14558563	C₂	semidirect
3. #14558538	C₂	semidirect
4. #14558508	C₂	semidirect
5. #14558492	C₂	semidirect
39. D₃₇⋊C₈	D₄	semidirect
86. C₃₇	#2932	semidirect

H ≅ G/N

type

1. C₂×#11924624

C₂

semidirect

2. #14558563

C₂

semidirect

3. #14558538

C₂

semidirect

4. #14558508

C₂

semidirect

5. #14558492

C₂

semidirect

39. D₃₇⋊C₈

D₄

semidirect

86. C₃₇

#2932

semidirect

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

subgroup	index
C₂×#11924624	2
#14558563	2
#14558538	2
#14558508	2
#14558492	2
#14558084	2
#2932	37

subgroup

index

Subgroups — #18011323

Conjugacy classes of subgroups

Complements

Maximal subgroups