Group #31791104 — subgroups

Subgroups — C₂×#22577757 #31791104

Order 17536.

412 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 149 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×#17984080	C₂	1	1	yes	maximal, characteristic, complemented
2	#22577757	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17984012	C₂	1	2	yes	maximal, complemented
4	C₂×#17984009	C₂	1	2	yes	maximal
5	C₂×C₄×#11764040	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#124×C₁₃₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11764040	C₂²	1	2	yes	—
8	#17984080	—	2	2	no	—
9	#17984080	C₂²	1	4	yes	complemented
10	C₂×C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
11	C₂×#14555489	—	2	2	no	—
12	#17984012	C₂²	1	8	yes	complemented
13	#17984009	C₂²	1	8	yes	—
14	C₄×#11764040	C₂²	1	4	yes	complemented
15	C₂²×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₂²×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
17	#124×C₁₃₇	—	2	2	no	—
18	#124×C₁₃₇	C₂²	1	4	yes	—
19	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
20	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
21	C₂²×Dic₁₃₇	—	2	2	no	—
22	C₂×#11764040	—	2	4	no	—
23	C₂×#11764040	C₂³	1	4	yes	—
24	C₄⋊Dic₁₃₇	—	2	2	no	—
25	C₄⋊Dic₁₃₇	C₂³	1	4	yes	—
26	#14555489	—	2	8	no	—
27	C₂⁴×C₁₃₇	—	2	1	no	—
28	C₂×Q₈×C₁₃₇	—	2	1	no	—
29	C₂×Q₈×C₁₃₇	—	2	4	no	—
30	C₂×Q₈×C₁₃₇	C₂³	1	2	yes	—
31	C₂×D₄×C₁₃₇	—	2	1	no	—
32	C₂×D₄×C₁₃₇	—	2	4	no	—
33	C₂×D₄×C₁₃₇	C₂³	1	2	yes	—
34	C₂²×C₄×C₁₃₇	—	2	1	no	—
35	C₂²×C₄×C₁₃₇	D₄	1	2	yes	—
36	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
37	C₂²⋊C₄×C₁₃₇	—	2	8	no	—
38	C₄²×C₁₃₇	D₄	1	2	yes	—
39	C₄²×C₁₃₇	C₂³	1	2	yes	—
40	C₂×Dic₁₃₇	—	2	4	no	—
41	C₂×Dic₁₃₇	—	4	4	no	—
42	#11764040	—	2	4	no	—
43	C₂³×C₁₃₇	—	2	2	no	—
44	C₂³×C₁₃₇	—	4	2	no	—
45	C₂³×C₁₃₇	—	4	4	no	—
46	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
47	Q₈×C₁₃₇	—	2	4	no	—
48	Q₈×C₁₃₇	—	4	2	no	—
49	D₄×C₁₃₇	—	2	4	no	—
50	D₄×C₁₃₇	—	4	2	no	—
51	C₂×C₄×C₁₃₇	—	2	4	no	—
52	C₂×C₄×C₁₃₇	—	2	2	no	—
53	C₂×C₄×C₁₃₇	—	4	2	no	—
54	C₂×C₄×C₁₃₇	C₂×D₄	1	4	yes	—
55	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₃₇	#41	1	4	yes	—
57	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
58	C₂²×C₁₃₇	—	4	2	no	—
59	C₂²×C₁₃₇	—	4	4	no	—
60	C₂²×C₁₃₇	—	8	4	no	—
61	C₂²×C₁₃₇	#124	1	2	yes	—
62	C₂²×C₁₃₇	#135	1	4	yes	—
63	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
64	C₄×C₁₃₇	—	2	4	no	—
65	C₄×C₁₃₇	—	4	2	no	—
66	C₄×C₁₃₇	#124	1	2	yes	—
67	C₄×C₁₃₇	C₂×#41	1	2	yes	—
68	Dic₁₃₇	—	4	4	no	—
69	C₂×C₁₃₇	—	8	2	no	—
70	C₂×C₁₃₇	#488	1	4	yes	complemented
71	C₂×C₁₃₇	C₂×#124	1	1	yes	characteristic
72	C₂×C₁₃₇	C₂×#135	1	2	yes	—
73	C₁₃₇	C₂×#488	1	1	yes	characteristic, complemented, Sylow
74	C₂×#488	—	137	1	no	maximal, Sylow
75	C₂×#125	—	137	1	no	—
76	C₂×#124	D₁₃₇	1	1	yes	characteristic
77	#488	—	137	8	no	—
78	C₂×#103	—	137	2	no	—
79	C₂×#102	—	137	2	no	—
80	C₂×C₄×C₈	—	137	1	no	—
81	C₂²×Q₈	C₂×D₁₃₇	1	1	yes	characteristic
82	C₂²×D₄	C₂×D₁₃₇	1	1	yes	characteristic
83	C₂²×C₈	—	137	2	no	—
84	#125	—	274	2	no	—
85	#125	—	137	4	no	—
86	#124	—	2	2	no	—
87	#124	C₂×D₁₃₇	1	4	yes	—
88	C₂×C₄⋊C₄	—	274	2	no	—
89	C₂×C₄⋊C₄	—	137	2	no	—
90	C₂×C₂²⋊C₄	—	2	2	no	—
91	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
92	#103	—	137	8	no	—
93	#102	—	137	8	no	—
94	C₄×C₈	—	137	4	no	—
95	C₂⁴	—	2	1	no	—
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	2	4	no	—
98	C₂×Q₈	C₂²×D₁₃₇	1	2	yes	—
99	C₂×D₄	—	2	1	no	—
100	C₂×D₄	—	2	4	no	—
101	C₂×D₄	C₂²×D₁₃₇	1	2	yes	—
102	C₂²×C₄	—	2	1	no	—
103	C₂²×C₄	—	274	2	no	—
104	C₂²×C₄	C₂²⋊D₁₃₇	1	2	yes	—
105	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
106	C₂×C₈	—	274	4	no	—
107	C₂×C₈	—	137	4	no	—
108	C₄⋊C₄	—	274	8	no	—
109	C₄⋊C₄	—	274	2	no	—
110	C₄⋊C₄	—	137	4	no	—
111	C₂²⋊C₄	—	2	8	no	—
112	C₄²	C₂²⋊D₁₃₇	1	2	yes	—
113	C₄²	C₂²×D₁₃₇	1	2	yes	—
114	C₂³	—	2	2	no	—
115	C₂³	—	4	2	no	—
116	C₂³	—	4	4	no	—
117	C₂³	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
118	Q₈	—	2	4	no	—
119	Q₈	—	4	2	no	—
120	D₄	—	2	4	no	—
121	D₄	—	4	2	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	274	4	no	—
126	C₂×C₄	—	548	4	no	—
127	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	4	yes	—
128	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
129	C₂×C₄	#14555509	1	4	yes	—
130	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
131	C₈	—	274	4	no	—
132	C₂²	—	4	4	no	—
133	C₂²	—	4	2	no	—
134	C₂²	—	8	4	no	—
135	C₂²	#17984075	1	2	yes	—
136	C₂²	#17984108	1	4	yes	—
137	C₂²	C₂²×C₂²⋊D₁₃₇	1	1	yes	characteristic
138	C₄	—	2	4	no	—
139	C₄	—	4	2	no	—
140	C₄	—	548	4	no	—
141	C₄	#17984075	1	2	yes	—
142	C₄	C₂×#14555509	1	2	yes	—
143	C₂	—	8	2	no	—
144	C₂	#22577757	1	4	yes	direct product factor
145	C₂	C₂×#17984075	1	1	yes	characteristic
146	C₂	C₂×#17984108	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984080

C₂

yes

maximal, characteristic, complemented

#22577757

C₂

yes

maximal, direct product factor

C₂×#17984012

C₂

yes

maximal, complemented

C₂×#17984009

C₂

yes

maximal

C₂×C₄×#11764040

C₂

yes

maximal, characteristic, complemented

C₂×#124×C₁₃₇

C₂

yes

maximal, characteristic

C₂²×#11764040

C₂²

yes

—

#17984080

—

#17984080

C₂²

yes

complemented

C₂×C₄⋊Dic₁₃₇

C₂²

yes

—

C₂×#14555489

—

#17984012

C₂²

yes

complemented

#17984009

C₂²

yes

—

C₄×#11764040

C₂²

yes

complemented

C₂²×Q₈×C₁₃₇

C₂²

yes

characteristic

C₂²×D₄×C₁₃₇

C₂²

yes

characteristic

#124×C₁₃₇

—

#124×C₁₃₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₁₃₇

—

C₂×C₄²×C₁₃₇

C₂²

yes

characteristic

C₂²×Dic₁₃₇

—

C₂×#11764040

—

C₂×#11764040

C₂³

yes

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

C₂³

yes

—

#14555489

—

C₂⁴×C₁₃₇

—

C₂×Q₈×C₁₃₇

—

C₂×Q₈×C₁₃₇

—

C₂×Q₈×C₁₃₇

C₂³

yes

—

C₂×D₄×C₁₃₇

—

C₂×D₄×C₁₃₇

—

C₂×D₄×C₁₃₇

C₂³

yes

—

C₂²×C₄×C₁₃₇

—

C₂²×C₄×C₁₃₇

D₄

yes

—

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

C₂²⋊C₄×C₁₃₇

—

C₄²×C₁₃₇

D₄

yes

—

C₄²×C₁₃₇

C₂³

yes

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

#11764040

—

C₂³×C₁₃₇

—

C₂³×C₁₃₇

—

C₂³×C₁₃₇

—

C₂³×C₁₃₇

C₂×D₄

yes

characteristic

Q₈×C₁₃₇

—

Q₈×C₁₃₇

—

D₄×C₁₃₇

—

D₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

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

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

#124

yes

—

C₂²×C₁₃₇

#135

yes

—

C₂²×C₁₃₇

C₂²×D₄

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

#124

yes

—

C₄×C₁₃₇

C₂×#41

yes

—

Dic₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

—

C₁₃₇

C₂×#488

yes

characteristic, complemented, Sylow

C₂×#488

—

137

maximal, Sylow

C₂×#125

—

137

—

C₂×#124

D₁₃₇

yes

characteristic

#488

—

137

—

C₂×#103

—

137

—

C₂×#102

—

137

—

C₂×C₄×C₈

—

137

—

C₂²×Q₈

C₂×D₁₃₇

yes

characteristic

C₂²×D₄

C₂×D₁₃₇

yes

characteristic

C₂²×C₈

—

137

—

#125

—

274

—

#125

—

137

—

#124

—

#124

C₂×D₁₃₇

yes

—

C₂×C₄⋊C₄

—

274

—

C₂×C₄⋊C₄

—

137

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₁₃₇

yes

characteristic

#103

—

137

—

#102

—

137

—

C₄×C₈

—

137

—

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²×D₁₃₇

yes

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

C₂²×D₁₃₇

yes

—

102

C₂²×C₄

—

103

C₂²×C₄

—

274

—

104

C₂²×C₄

C₂²⋊D₁₃₇

yes

—

105

C₂²×C₄

C₂²×D₁₃₇

yes

characteristic

106

C₂×C₈

—

274

—

107

C₂×C₈

—

137

—

108

C₄⋊C₄

—

274

—

109

C₄⋊C₄

—

274

—

110

C₄⋊C₄

—

137

—

111

C₂²⋊C₄

—

112

C₄²

C₂²⋊D₁₃₇

yes

—

113

C₄²

C₂²×D₁₃₇

yes

—

114

C₂³

—

115

C₂³

—

116

C₂³

—

117

C₂³

C₂×C₂²⋊D₁₃₇

yes

characteristic

118

Q₈

—

119

Q₈

—

120

D₄

—

121

D₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

274

—

126

C₂×C₄

—

548

—

127

C₂×C₄

C₂×C₂²⋊D₁₃₇

yes

—

128

C₂×C₄

C₂×C₂²⋊D₁₃₇

yes

characteristic

129

C₂×C₄

#14555509

yes

—

130

C₂×C₄

C₂³×D₁₃₇

yes

characteristic

131

C₈

—

274

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

#17984075

yes

—

136

C₂²

#17984108

yes

—

137

C₂²

C₂²×C₂²⋊D₁₃₇

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

548

—

141

C₄

#17984075

yes

—

142

C₄

C₂×#14555509

yes

—

143

C₂

—

144

C₂

#22577757

yes

direct product factor

145

C₂

C₂×#17984075

yes

characteristic

146

C₂

C₂×#17984108

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₃₇
commutator, cosocle	C₂×C₄×C₁₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting	C₂×#124×C₁₃₇	C₂
socle	C₂³×C₁₃₇	C₂×D₄
prime radical	C₂²×D₄×C₁₃₇	C₂²
hypercenter	C₂×#124	D₁₃₇
hypocenter	C₁₃₇	C₂×#488
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#124

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₃₇

commutator, cosocle

C₂×C₄×C₁₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₇

Fitting

C₂×#124×C₁₃₇

C₂

socle

C₂³×C₁₃₇

C₂×D₄

prime radical

C₂²×D₄×C₁₃₇

C₂²

D₁₃₇

C₁₃₇

G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#124

N	H ≅ G/N	type
1. C₂×#17984080	C₂	semidirect
2. #22577757	C₂	direct (N × H)
3. C₂×#17984012	C₂	semidirect
5. C₂×C₄×#11764040	C₂	semidirect
9. #17984080	C₂²	semidirect
12. #17984012	C₂²	semidirect
14. C₄×#11764040	C₂²	semidirect
70. C₂×C₁₃₇	#488	semidirect
73. C₁₃₇	C₂×#488	semidirect
144. C₂	#22577757	direct (N × H)

H ≅ G/N

type

1. C₂×#17984080

C₂

semidirect

2. #22577757

C₂

direct (N × H)

3. C₂×#17984012

C₂

semidirect

5. C₂×C₄×#11764040

C₂

semidirect

9. #17984080

C₂²

semidirect

12. #17984012

C₂²

semidirect

14. C₄×#11764040

C₂²

semidirect

70. C₂×C₁₃₇

#488

semidirect

73. C₁₃₇

C₂×#488

semidirect

144. C₂

#22577757

direct (N × H)

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

subgroup	index
C₂×#17984080	2
#22577757	2
C₂×#17984012	2
C₂×#17984009	2
C₂×C₄×#11764040	2
C₂×#124×C₁₃₇	2
C₂×#488	137

subgroup

index

C₂×#124×C₁₃₇

137

Subgroups — C₂×#22577757 #31791104

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#22577757 #31791104

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22577757 #31791104