Group #31791114 — subgroups

Subgroups — #31791114

Order 17536.

400 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 141 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	#22577953	C₂	1	1	yes	maximal, characteristic
2	#22577760	C₂	1	4	yes	maximal, complemented
3	#22577755	C₂	1	2	yes	maximal, complemented
4	#22577751	C₂	1	2	yes	maximal, complemented
5	#22577397	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×#11764040	C₂	1	1	yes	maximal, characteristic, complemented
7	#533×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14555510	C₂²	1	2	yes	—
9	C₂²×#11764040	C₂²	1	2	yes	—
10	#17984088	C₂²	1	1	yes	characteristic
11	#17984085	C₂²	1	1	yes	characteristic
12	#17984076	C₂²	1	2	yes	—
13	#17984062	—	2	2	no	—
14	C₄×D₅₄₈	—	2	2	no	—
15	#17984011	C₂²	1	8	yes	—
16	#17984008	C₂²	1	8	yes	—
17	C₄×#11764040	C₂²	1	2	yes	—
18	C₄×#11764040	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
20	#125×C₁₃₇	C₂²	1	2	yes	—
21	C₂²⋊Q₈×C₁₃₇	—	2	2	no	—
22	C₄×Q₈×C₁₃₇	—	2	2	no	—
23	#117×C₁₃₇	C₂²	1	2	yes	—
24	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
25	#14555510	—	2	8	no	—
26	C₂×Dic₂₇₄	C₂³	1	2	yes	—
27	C₂×D₅₄₈	—	2	2	no	—
28	C₂×D₅₄₈	C₂³	1	2	yes	—
29	C₂×C₂²⋊D₁₃₇	—	2	2	no	—
30	C₂×C₄×D₁₃₇	—	2	2	no	—
31	C₂×#11764040	—	2	4	no	—
32	C₂×#11764040	C₂³	1	4	yes	—
33	C₄⋊Dic₁₃₇	—	2	2	no	—
34	#14555487	—	2	4	no	—
35	C₂×Q₈×C₁₃₇	—	2	2	no	—
36	C₂²×C₄×C₁₃₇	D₄	1	2	yes	—
37	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₁₃₇	—	2	4	no	—
39	C₄⋊C₄×C₁₃₇	—	2	2	no	—
40	C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
41	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
42	C₄²×C₁₃₇	—	2	2	no	—
43	C₄²×C₁₃₇	D₄	1	2	yes	—
44	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
45	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
46	C₂²×D₁₃₇	—	2	2	no	—
47	Dic₂₇₄	—	2	4	no	—
48	D₅₄₈	—	2	4	no	—
49	D₅₄₈	—	4	4	no	—
50	C₂²⋊D₁₃₇	—	4	4	no	—
51	C₂×Dic₁₃₇	—	2	2	no	—
52	C₄×D₁₃₇	—	4	4	no	—
53	#11764040	—	2	4	no	—
54	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
55	Q₈×C₁₃₇	—	4	4	no	—
56	C₂×C₄×C₁₃₇	—	2	4	no	—
57	C₂×C₄×C₁₃₇	—	2	4	no	—
58	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₁₃₇	#41	1	4	yes	—
62	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
63	C₂²×C₁₃₇	—	2	2	no	—
64	C₂²×C₁₃₇	#124	1	2	yes	—
65	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
66	C₂×D₁₃₇	—	4	2	no	—
67	C₂×D₁₃₇	—	4	4	no	—
68	C₄×C₁₃₇	—	2	2	no	—
69	C₄×C₁₃₇	—	4	4	no	—
70	C₄×C₁₃₇	#124	1	2	yes	—
71	C₄×C₁₃₇	#135	1	4	yes	—
72	C₄×C₁₃₇	C₂×#41	1	2	yes	—
73	Dic₁₃₇	—	4	2	no	—
74	C₂×C₁₃₇	—	2	2	no	—
75	C₂×C₁₃₇	C₂×#124	1	1	yes	characteristic
76	C₂×C₁₃₇	C₂×#135	1	2	yes	—
77	D₁₃₇	—	8	2	no	—
78	C₁₃₇	#3131	1	1	yes	characteristic, complemented, Sylow
79	#3131	—	137	1	no	maximal, Sylow
80	#533	D₁₃₇	1	1	yes	characteristic, complemented
81	#532	—	137	1	no	—
82	#488	—	137	4	no	—
83	#487	—	137	2	no	—
84	#486	—	137	2	no	—
85	#416	—	137	4	no	—
86	C₂×C₄×C₈	—	137	1	no	—
87	C₂×#41	—	137	2	no	—
88	C₂²×C₈	—	137	2	no	—
89	C₄⋊Q₈	—	137	1	no	—
90	C₄⋊Q₈	C₂×D₁₃₇	1	2	yes	—
91	#127	—	137	1	no	—
92	#125	C₂×D₁₃₇	1	2	yes	—
93	#124	—	137	2	no	—
94	C₂²⋊Q₈	—	2	2	no	—
95	#121	—	274	2	no	—
96	C₄×Q₈	—	2	2	no	—
97	C₄×D₄	—	274	2	no	—
98	#117	C₂×D₁₃₇	1	2	yes	—
99	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
100	#103	—	137	8	no	—
101	#102	—	137	8	no	—
102	C₄×C₈	—	137	2	no	—
103	C₄×C₈	—	137	2	no	—
104	#41	—	274	8	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	—	137	2	no	—
107	C₂×D₄	—	274	2	no	—
108	C₂×D₄	—	274	2	no	—
109	C₂×D₄	—	137	2	no	—
110	C₂²×C₄	—	274	2	no	—
111	C₂²×C₄	C₂²⋊D₁₃₇	1	2	yes	—
112	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
113	C₂×C₈	—	274	4	no	—
114	C₂×C₈	—	137	4	no	—
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	274	2	no	—
118	C₄⋊C₄	C₂²×D₁₃₇	1	4	yes	—
119	C₂²⋊C₄	—	2	2	no	—
120	C₂²⋊C₄	—	274	4	no	—
121	C₄²	—	2	2	no	—
122	C₄²	C₂²⋊D₁₃₇	1	2	yes	—
123	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
124	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
125	C₂³	—	274	2	no	—
126	C₂³	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
127	Q₈	—	4	4	no	—
128	Q₈	—	274	4	no	—
129	D₄	—	274	4	no	—
130	D₄	—	548	4	no	—
131	D₄	—	548	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	274	2	no	—
135	C₂×C₄	—	548	4	no	—
136	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	2	yes	—
137	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	2	yes	—
138	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
139	C₂×C₄	#14555511	1	4	yes	—
140	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
141	C₈	—	274	4	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	548	2	no	—
144	C₂²	—	548	4	no	—
145	C₂²	#17984077	1	2	yes	—
146	C₂²	C₂²×C₂²⋊D₁₃₇	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	4	4	no	—
149	C₄	—	548	2	no	—
150	C₄	#17984077	1	2	yes	—
151	C₄	#17984108	1	4	yes	—
152	C₄	C₂×#14555511	1	2	yes	—
153	C₂	—	2	2	no	—
154	C₂	—	1096	2	no	—
155	C₂	C₂×#17984077	1	1	yes	characteristic
156	C₂	C₂×#17984108	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22577953

C₂

yes

maximal, characteristic

#22577760

C₂

yes

maximal, complemented

#22577755

C₂

yes

maximal, complemented

#22577751

C₂

yes

maximal, complemented

#22577397

C₂

yes

maximal, complemented

C₂×C₄×#11764040

C₂

yes

maximal, characteristic, complemented

#533×C₁₃₇

C₂

yes

maximal, characteristic, complemented

C₂×#14555510

C₂²

yes

—

C₂²×#11764040

C₂²

yes

—

#17984088

C₂²

yes

characteristic

#17984085

C₂²

yes

characteristic

#17984076

C₂²

yes

—

#17984062

—

C₄×D₅₄₈

—

#17984011

C₂²

yes

—

#17984008

C₂²

yes

—

C₄×#11764040

C₂²

yes

—

C₄×#11764040

C₂²

yes

—

C₄⋊Q₈×C₁₃₇

C₂²

yes

—

#125×C₁₃₇

C₂²

yes

—

C₂²⋊Q₈×C₁₃₇

—

C₄×Q₈×C₁₃₇

—

#117×C₁₃₇

C₂²

yes

—

C₂×C₄²×C₁₃₇

C₂²

yes

characteristic

#14555510

—

C₂×Dic₂₇₄

C₂³

yes

—

C₂×D₅₄₈

—

C₂×D₅₄₈

C₂³

yes

—

C₂×C₂²⋊D₁₃₇

—

C₂×C₄×D₁₃₇

—

C₂×#11764040

—

C₂×#11764040

C₂³

yes

—

C₄⋊Dic₁₃₇

—

#14555487

—

C₂×Q₈×C₁₃₇

—

C₂²×C₄×C₁₃₇

D₄

yes

—

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

C₂³

yes

—

C₂²⋊C₄×C₁₃₇

—

C₄²×C₁₃₇

—

C₄²×C₁₃₇

D₄

yes

—

C₄²×C₁₃₇

C₂³

yes

characteristic

C₄²×C₁₃₇

C₂³

yes

characteristic

C₂²×D₁₃₇

—

Dic₂₇₄

—

D₅₄₈

—

D₅₄₈

—

C₂²⋊D₁₃₇

—

C₂×Dic₁₃₇

—

C₄×D₁₃₇

—

#11764040

—

C₂³×C₁₃₇

C₂×D₄

yes

characteristic

Q₈×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

#41

yes

—

C₂×C₄×C₁₃₇

C₂⁴

yes

characteristic

C₂²×C₁₃₇

—

C₂²×C₁₃₇

#124

yes

—

C₂²×C₁₃₇

C₂²×D₄

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

#124

yes

—

C₄×C₁₃₇

#135

yes

—

C₄×C₁₃₇

C₂×#41

yes

—

Dic₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

C₂×#124

yes

characteristic

C₂×C₁₃₇

C₂×#135

yes

—

D₁₃₇

—

C₁₃₇

#3131

yes

characteristic, complemented, Sylow

#3131

—

137

maximal, Sylow

#533

D₁₃₇

yes

characteristic, complemented

#532

—

137

—

#488

—

137

—

#487

—

137

—

#486

—

137

—

#416

—

137

—

C₂×C₄×C₈

—

137

—

C₂×#41

—

137

—

C₂²×C₈

—

137

—

C₄⋊Q₈

—

137

—

C₄⋊Q₈

C₂×D₁₃₇

yes

—

#127

—

137

—

#125

C₂×D₁₃₇

yes

—

#124

—

137

—

C₂²⋊Q₈

—

#121

—

274

—

C₄×Q₈

—

C₄×D₄

—

274

—

#117

C₂×D₁₃₇

yes

—

C₂×C₄²

C₂×D₁₃₇

yes

characteristic

100

#103

—

137

—

101

#102

—

137

—

102

C₄×C₈

—

137

—

103

C₄×C₈

—

137

—

104

#41

—

274

—

105

C₂×Q₈

—

106

C₂×Q₈

—

137

—

107

C₂×D₄

—

274

—

108

C₂×D₄

—

274

—

109

C₂×D₄

—

137

—

110

C₂²×C₄

—

274

—

111

C₂²×C₄

C₂²⋊D₁₃₇

yes

—

112

C₂²×C₄

C₂²×D₁₃₇

yes

characteristic

113

C₂×C₈

—

274

—

114

C₂×C₈

—

137

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

274

—

118

C₄⋊C₄

C₂²×D₁₃₇

yes

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

274

—

121

C₄²

—

122

C₄²

C₂²⋊D₁₃₇

yes

—

123

C₄²

C₂²×D₁₃₇

yes

characteristic

124

C₄²

C₂²×D₁₃₇

yes

characteristic

125

C₂³

—

274

—

126

C₂³

C₂×C₂²⋊D₁₃₇

yes

characteristic

127

Q₈

—

128

Q₈

—

274

—

129

D₄

—

274

—

130

D₄

—

548

—

131

D₄

—

548

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

274

—

135

C₂×C₄

—

548

—

136

C₂×C₄

C₂×C₂²⋊D₁₃₇

yes

—

137

C₂×C₄

C₂×C₂²⋊D₁₃₇

yes

—

138

C₂×C₄

C₂×C₂²⋊D₁₃₇

yes

characteristic

139

C₂×C₄

#14555511

yes

—

140

C₂×C₄

C₂³×D₁₃₇

yes

characteristic

141

C₈

—

274

—

142

C₂²

—

143

C₂²

—

548

—

144

C₂²

—

548

—

145

C₂²

#17984077

yes

—

146

C₂²

C₂²×C₂²⋊D₁₃₇

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

548

—

150

C₄

#17984077

yes

—

151

C₄

#17984108

yes

—

152

C₄

C₂×#14555511

yes

—

153

C₂

—

154

C₂

—

1096

—

155

C₂

C₂×#17984077

yes

characteristic

156

C₂

C₂×#17984108

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₃₇
commutator, cosocle	C₂×C₄×C₁₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting	#533×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂²×D₄
prime radical	#22577953	C₂
hypercenter	#533	D₁₃₇
hypocenter	C₁₃₇	#3131
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₃₇

commutator, cosocle

C₂×C₄×C₁₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₇

Fitting

#533×C₁₃₇

C₂

socle

C₂²×C₁₃₇

C₂²×D₄

C₂

D₁₃₇

C₁₃₇

G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #22577760	C₂	semidirect
3. #22577755	C₂	semidirect
4. #22577751	C₂	semidirect
5. #22577397	C₂	semidirect
6. C₂×C₄×#11764040	C₂	semidirect
7. #533×C₁₃₇	C₂	semidirect
78. C₁₃₇	#3131	semidirect
80. #533	D₁₃₇	semidirect

H ≅ G/N

type

2. #22577760

C₂

semidirect

3. #22577755

C₂

semidirect

4. #22577751

C₂

semidirect

5. #22577397

C₂

semidirect

6. C₂×C₄×#11764040

C₂

semidirect

7. #533×C₁₃₇

C₂

semidirect

78. C₁₃₇

#3131

semidirect

80. #533

D₁₃₇

semidirect

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

subgroup	index
#22577953	2
#22577760	2
#22577755	2
#22577751	2
#22577397	2
C₂×C₄×#11764040	2
#533×C₁₃₇	2
#3131	137

subgroup

index

#533×C₁₃₇

137

Subgroups — #31791114

Conjugacy classes of subgroups

Complements

Maximal subgroups