Group #31789438 — subgroups

Subgroups — #31789438

Order 17536.

384 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 173 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	C₄×#14555510	C₂	1	1	yes	maximal, characteristic
2	#22577465	C₂	1	4	yes	maximal, complemented
3	#22577458	C₂	1	4	yes	maximal, complemented
4	#22577429	C₂	1	2	yes	maximal, complemented
5	#22577369	C₂	1	2	yes	maximal
6	C₄×#14555494	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#97×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14555510	C₂²	1	1	yes	characteristic
9	C₂×#14555494	C₂²	1	2	yes	—
10	C₄×Dic₂₇₄	C₄	1	2	yes	—
11	C₄×D₅₄₈	C₄	1	2	yes	—
12	C₄×C₂²⋊D₁₃₇	C₄	1	4	yes	—
13	#17984047	C₂²	1	1	yes	characteristic
14	#17984045	C₂²	1	2	yes	—
15	C₄²×D₁₃₇	C₂²	1	2	yes	—
16	#17984017	C₂²	1	8	yes	—
17	#17984000	C₂²	1	8	yes	—
18	#17983999	C₂²	1	2	yes	—
19	C₄×#11764040	C₂²	1	2	yes	—
20	C₂²×C₈×C₁₃₇	C₂²	1	2	yes	—
21	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
22	#97×C₁₃₇	C₂²	1	2	yes	—
23	#97×C₁₃₇	C₂²	1	2	yes	—
24	#14555510	—	2	4	no	—
25	C₂×Dic₂₇₄	C₂×C₄	1	1	yes	characteristic
26	C₂×D₅₄₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₂²⋊D₁₃₇	C₂×C₄	1	2	yes	—
28	C₂×C₄×D₁₃₇	C₂×C₄	1	4	yes	—
29	C₂×C₄×D₁₃₇	C₂³	1	2	yes	—
30	#14555494	—	2	4	no	—
31	C₂×#11764040	C₂³	1	4	yes	—
32	C₄⋊Dic₁₃₇	C₂×C₄	1	2	yes	—
33	#14555489	C₂×C₄	1	4	yes	—
34	C₂²⋊Dic₁₃₇	C₂×C₄	1	2	yes	—
35	#14555487	C₂×C₄	1	4	yes	—
36	C₄×Dic₁₃₇	C₂×C₄	1	4	yes	—
37	C₄×Dic₁₃₇	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
40	C₂×C₈×C₁₃₇	—	2	4	no	—
41	C₂×C₈×C₁₃₇	C₂³	1	4	yes	—
42	C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
43	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
44	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
45	C₂²×D₁₃₇	C₂²×C₄	1	2	yes	—
46	Dic₂₇₄	—	2	2	no	—
47	D₅₄₈	—	2	2	no	—
48	C₂²⋊D₁₃₇	—	2	4	no	—
49	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
50	C₂×Dic₁₃₇	C₂²×C₄	1	4	yes	—
51	C₄×D₁₃₇	—	2	4	no	—
52	C₄×D₁₃₇	—	4	4	no	—
53	#11764040	—	2	4	no	—
54	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₃₇	—	4	2	no	—
56	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₃₇	#41	1	4	yes	—
60	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
61	C₈×C₁₃₇	—	2	4	no	—
62	C₂²×C₁₃₇	—	2	2	no	—
63	C₂²×C₁₃₇	#117	1	2	yes	—
64	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
65	C₂×D₁₃₇	—	2	2	no	—
66	C₂×D₁₃₇	—	2	4	no	—
67	C₄×C₁₃₇	—	2	2	no	—
68	C₄×C₁₃₇	—	2	2	no	—
69	C₄×C₁₃₇	#117	1	2	yes	—
70	C₄×C₁₃₇	C₂×#41	1	2	yes	—
71	Dic₁₃₇	—	2	2	no	—
72	Dic₁₃₇	—	2	4	no	—
73	C₂×C₁₃₇	—	2	2	no	—
74	C₂×C₁₃₇	C₂×#117	1	1	yes	characteristic
75	C₂×C₁₃₇	#568	1	2	yes	—
76	D₁₃₇	—	4	2	no	—
77	C₁₃₇	#2923	1	1	yes	characteristic, complemented, Sylow
78	#2923	—	137	1	no	maximal, Sylow
79	C₄×#41	—	137	1	no	—
80	#433	—	137	4	no	—
81	#432	—	137	4	no	—
82	#424	—	137	2	no	—
83	#408	—	137	2	no	—
84	C₄×M₁₆	—	137	1	no	—
85	C₂×#97	D₁₃₇	1	1	yes	characteristic, complemented
86	C₂×#41	—	137	1	no	—
87	C₂×M₁₆	—	137	2	no	—
88	C₂²×C₈	C₂×D₁₃₇	1	2	yes	—
89	C₄×Q₈	—	137	2	no	—
90	C₄×D₄	—	137	4	no	—
91	C₄×D₄	—	137	2	no	—
92	#117	—	137	1	no	—
93	#117	—	137	2	no	—
94	C₂×C₄²	—	137	2	no	—
95	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
96	C₄⋊C₈	—	137	8	no	—
97	C₂²⋊C₈	—	137	8	no	—
98	#97	—	137	2	no	—
99	#97	C₂×D₁₃₇	1	2	yes	—
100	#97	C₂×D₁₃₇	1	2	yes	—
101	C₄×C₈	—	137	2	no	—
102	#41	—	274	4	no	—
103	C₂×Q₈	—	137	1	no	—
104	C₂×D₄	—	137	2	no	—
105	C₂×D₄	—	137	1	no	—
106	C₂²×C₄	—	137	2	no	—
107	C₂²×C₄	—	137	4	no	—
108	C₂²×C₄	C₄×D₁₃₇	1	2	yes	—
109	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
110	M₁₆	—	274	4	no	—
111	C₂×C₈	—	2	4	no	—
112	C₂×C₈	—	137	4	no	—
113	C₂×C₈	C₂²×D₁₃₇	1	4	yes	—
114	C₄⋊C₄	—	137	4	no	—
115	C₄⋊C₄	—	137	2	no	—
116	C₂²⋊C₄	—	137	2	no	—
117	C₂²⋊C₄	—	137	4	no	—
118	C₄²	—	137	4	no	—
119	C₄²	—	137	2	no	—
120	C₄²	C₄×D₁₃₇	1	2	yes	—
121	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
122	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
123	C₂³	—	137	2	no	—
124	C₂³	C₂×C₄×D₁₃₇	1	1	yes	characteristic
125	Q₈	—	274	2	no	—
126	D₄	—	274	4	no	—
127	D₄	—	274	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	274	4	no	—
130	C₂×C₄	—	548	4	no	—
131	C₂×C₄	—	137	4	no	—
132	C₂×C₄	—	137	2	no	—
133	C₂×C₄	C₂×C₄×D₁₃₇	1	2	yes	—
134	C₂×C₄	C₂×C₄×D₁₃₇	1	2	yes	—
135	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
136	C₂×C₄	#14555510	1	4	yes	—
137	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
138	C₈	—	2	4	no	—
139	C₈	—	274	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	274	2	no	—
142	C₂²	—	274	4	no	—
143	C₂²	#17984045	1	2	yes	—
144	C₂²	C₂²×C₄×D₁₃₇	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	274	2	no	—
148	C₄	—	274	4	no	—
149	C₄	#17984045	1	2	yes	—
150	C₄	C₂×#14555510	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	548	2	no	—
153	C₂	C₂×#17984045	1	1	yes	characteristic
154	C₂	#22578169	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14555510

C₂

yes

maximal, characteristic

#22577465

C₂

yes

maximal, complemented

#22577458

C₂

yes

maximal, complemented

#22577429

C₂

yes

maximal, complemented

#22577369

C₂

yes

maximal

C₄×#14555494

C₂

yes

maximal, characteristic, complemented

C₂×#97×C₁₃₇

C₂

yes

maximal, characteristic, complemented

C₂×#14555510

C₂²

yes

characteristic

C₂×#14555494

C₂²

yes

—

C₄×Dic₂₇₄

C₄

yes

—

C₄×D₅₄₈

C₄

yes

—

C₄×C₂²⋊D₁₃₇

C₄

yes

—

#17984047

C₂²

yes

characteristic

#17984045

C₂²

yes

—

C₄²×D₁₃₇

C₂²

yes

—

#17984017

C₂²

yes

—

#17984000

C₂²

yes

—

#17983999

C₂²

yes

—

C₄×#11764040

C₂²

yes

—

C₂²×C₈×C₁₃₇

C₂²

yes

—

C₂×C₄²×C₁₃₇

C₂²

yes

characteristic

#97×C₁₃₇

C₂²

yes

—

#97×C₁₃₇

C₂²

yes

—

#14555510

—

C₂×Dic₂₇₄

C₂×C₄

yes

characteristic

C₂×D₅₄₈

C₂×C₄

yes

characteristic

C₂×C₂²⋊D₁₃₇

C₂×C₄

yes

—

C₂×C₄×D₁₃₇

C₂×C₄

yes

—

C₂×C₄×D₁₃₇

C₂³

yes

—

#14555494

—

C₂×#11764040

C₂³

yes

—

C₄⋊Dic₁₃₇

C₂×C₄

yes

—

#14555489

C₂×C₄

yes

—

C₂²⋊Dic₁₃₇

C₂×C₄

yes

—

#14555487

C₂×C₄

yes

—

C₄×Dic₁₃₇

C₂×C₄

yes

—

C₄×Dic₁₃₇

C₂³

yes

—

C₂²×C₄×C₁₃₇

C₂×C₄

yes

—

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

C₂³

yes

—

C₄²×C₁₃₇

C₂×C₄

yes

—

C₄²×C₁₃₇

C₂³

yes

characteristic

C₄²×C₁₃₇

C₂³

yes

characteristic

C₂²×D₁₃₇

C₂²×C₄

yes

—

Dic₂₇₄

—

D₅₄₈

—

C₂²⋊D₁₃₇

—

C₂×Dic₁₃₇

C₂²×C₄

yes

—

C₂×Dic₁₃₇

C₂²×C₄

yes

—

C₄×D₁₃₇

—

C₄×D₁₃₇

—

#11764040

—

C₂³×C₁₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₇

#41

yes

—

C₂×C₄×C₁₃₇

C₂⁴

yes

characteristic

C₈×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

#117

yes

—

C₂²×C₁₃₇

C₂³×C₄

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

#117

yes

—

C₄×C₁₃₇

C₂×#41

yes

—

Dic₁₃₇

—

Dic₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

C₂×#117

yes

characteristic

C₂×C₁₃₇

#568

yes

—

D₁₃₇

—

C₁₃₇

#2923

yes

characteristic, complemented, Sylow

#2923

—

137

maximal, Sylow

C₄×#41

—

137

—

#433

—

137

—

#432

—

137

—

#424

—

137

—

#408

—

137

—

C₄×M₁₆

—

137

—

C₂×#97

D₁₃₇

yes

characteristic, complemented

C₂×#41

—

137

—

C₂×M₁₆

—

137

—

C₂²×C₈

C₂×D₁₃₇

yes

—

C₄×Q₈

—

137

—

C₄×D₄

—

137

—

C₄×D₄

—

137

—

#117

—

137

—

#117

—

137

—

C₂×C₄²

—

137

—

C₂×C₄²

C₂×D₁₃₇

yes

characteristic

C₄⋊C₈

—

137

—

C₂²⋊C₈

—

137

—

#97

—

137

—

#97

C₂×D₁₃₇

yes

—

100

#97

C₂×D₁₃₇

yes

—

101

C₄×C₈

—

137

—

102

#41

—

274

—

103

C₂×Q₈

—

137

—

104

C₂×D₄

—

137

—

105

C₂×D₄

—

137

—

106

C₂²×C₄

—

137

—

107

C₂²×C₄

—

137

—

108

C₂²×C₄

C₄×D₁₃₇

yes

—

109

C₂²×C₄

C₂²×D₁₃₇

yes

characteristic

110

M₁₆

—

274

—

111

C₂×C₈

—

112

C₂×C₈

—

137

—

113

C₂×C₈

C₂²×D₁₃₇

yes

—

114

C₄⋊C₄

—

137

—

115

C₄⋊C₄

—

137

—

116

C₂²⋊C₄

—

137

—

117

C₂²⋊C₄

—

137

—

118

C₄²

—

137

—

119

C₄²

—

137

—

120

C₄²

C₄×D₁₃₇

yes

—

121

C₄²

C₂²×D₁₃₇

yes

characteristic

122

C₄²

C₂²×D₁₃₇

yes

characteristic

123

C₂³

—

137

—

124

C₂³

C₂×C₄×D₁₃₇

yes

characteristic

125

Q₈

—

274

—

126

D₄

—

274

—

127

D₄

—

274

—

128

C₂×C₄

—

129

C₂×C₄

—

274

—

130

C₂×C₄

—

548

—

131

C₂×C₄

—

137

—

132

C₂×C₄

—

137

—

133

C₂×C₄

C₂×C₄×D₁₃₇

yes

—

134

C₂×C₄

C₂×C₄×D₁₃₇

yes

—

135

C₂×C₄

C₂×C₄×D₁₃₇

yes

characteristic

136

C₂×C₄

#14555510

yes

—

137

C₂×C₄

C₂³×D₁₃₇

yes

characteristic

138

C₈

—

139

C₈

—

274

—

140

C₂²

—

141

C₂²

—

274

—

142

C₂²

—

274

—

143

C₂²

#17984045

yes

—

144

C₂²

C₂²×C₄×D₁₃₇

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

274

—

148

C₄

—

274

—

149

C₄

#17984045

yes

—

150

C₄

C₂×#14555510

yes

—

151

C₂

—

152

C₂

—

548

—

153

C₂

C₂×#17984045

yes

characteristic

154

C₂

#22578169

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃₇
commutator, socle	C₂²×C₁₃₇	C₂³×C₄
Fitting	C₂×#97×C₁₃₇	C₂
prime radical	C₂×#14555510	C₂²
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	C₂×#97	D₁₃₇
hypocenter	C₁₃₇	#2923
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₃₇

commutator, socle

C₂²×C₁₃₇

C₂³×C₄

Fitting

C₂×#97×C₁₃₇

C₂

prime radical

C₂×#14555510

C₂²

cosocle

C₂×C₄×C₁₃₇

C₂⁴

hypercenter

C₂×#97

D₁₃₇

C₁₃₇

G ▷ C₂²×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂×C₄ ◁ C₂×#97

N	H ≅ G/N	type
2. #22577465	C₂	semidirect
3. #22577458	C₂	semidirect
4. #22577429	C₂	semidirect
6. C₄×#14555494	C₂	semidirect
7. C₂×#97×C₁₃₇	C₂	semidirect
77. C₁₃₇	#2923	semidirect
85. C₂×#97	D₁₃₇	semidirect

H ≅ G/N

type

2. #22577465

C₂

semidirect

3. #22577458

C₂

semidirect

4. #22577429

C₂

semidirect

6. C₄×#14555494

C₂

semidirect

7. C₂×#97×C₁₃₇

C₂

semidirect

77. C₁₃₇

#2923

semidirect

85. C₂×#97

D₁₃₇

semidirect

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

subgroup	index
C₄×#14555510	2
#22577465	2
#22577458	2
#22577429	2
#22577369	2
C₄×#14555494	2
C₂×#97×C₁₃₇	2
#2923	137

subgroup

index

C₂×#97×C₁₃₇

137

Subgroups — #31789438

Conjugacy classes of subgroups

Complements

Maximal subgroups