Group #31791295 — subgroups

Subgroups — C₂×#22577806 #31791295

Order 17536.

424 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 197 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×#17984088	C₂	1	1	yes	maximal, characteristic
2	#22577806	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17984021	C₂	1	4	yes	maximal
4	C₂×C₄×#11764040	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₇₄	—	2	1	no	—
7	C₂²×#11764040	C₂²	1	2	yes	—
8	#17984088	—	2	2	no	—
9	#17984088	C₂²	1	4	yes	—
10	C₂×C₄⋊Dic₁₃₇	—	2	1	no	—
11	C₂×C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
12	#17984021	C₂²	1	16	yes	—
13	C₄×#11764040	C₂²	1	4	yes	—
14	C₂²×Q₈×C₁₃₇	—	2	1	no	—
15	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
16	C₄⋊Q₈×C₁₃₇	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
18	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
19	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
20	C₂×Dic₂₇₄	—	2	4	no	—
21	C₂×Dic₂₇₄	—	4	4	no	—
22	C₂²×Dic₁₃₇	—	2	2	no	—
23	C₂×#11764040	Q₈	1	8	yes	complemented
24	C₂×#11764040	C₂³	1	4	yes	—
25	C₄⋊Dic₁₃₇	—	2	4	no	—
26	C₄⋊Dic₁₃₇	—	2	2	no	—
27	C₄⋊Dic₁₃₇	C₂³	1	4	yes	—
28	C₂×Q₈×C₁₃₇	—	2	4	no	—
29	C₂×Q₈×C₁₃₇	—	4	4	no	—
30	C₂²×C₄×C₁₃₇	—	2	2	no	—
31	C₂²×C₄×C₁₃₇	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₃₇	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₁₃₇	—	2	4	no	—
35	C₄⋊C₄×C₁₃₇	—	2	2	no	—
36	C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
37	C₄²×C₁₃₇	D₄	1	2	yes	—
38	C₄²×C₁₃₇	C₂³	1	2	yes	—
39	Dic₂₇₄	—	4	8	no	—
40	C₂×Dic₁₃₇	—	2	4	no	—
41	C₂×Dic₁₃₇	—	4	4	no	—
42	#11764040	C₂×Q₈	1	8	yes	complemented
43	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
44	Q₈×C₁₃₇	—	4	8	no	—
45	C₂×C₄×C₁₃₇	—	2	4	no	—
46	C₂×C₄×C₁₃₇	—	4	4	no	—
47	C₂×C₄×C₁₃₇	D₈	1	4	yes	—
48	C₂×C₄×C₁₃₇	Q₁₆	1	4	yes	—
49	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₁₃₇	C₂×Q₈	1	4	yes	—
53	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
54	C₂²×C₁₃₇	C₄⋊Q₈	1	2	yes	—
55	C₂²×C₁₃₇	C₂×D₈	1	2	yes	—
56	C₂²×C₁₃₇	C₂×Q₁₆	1	2	yes	—
57	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₁₃₇	—	4	4	no	—
59	C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
60	C₄×C₁₃₇	C₂×D₈	1	4	yes	—
61	C₄×C₁₃₇	C₂×Q₁₆	1	4	yes	—
62	C₄×C₁₃₇	C₂²×Q₈	1	2	yes	—
63	Dic₁₃₇	—	4	4	no	—
64	C₂×C₁₃₇	#500	1	4	yes	complemented
65	C₂×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₁₃₇	C₂²×D₈	1	1	yes	characteristic
67	C₂×C₁₃₇	C₂²×Q₁₆	1	1	yes	characteristic
68	C₁₃₇	C₂×#500	1	1	yes	characteristic, complemented, Sylow
69	C₂×#500	—	137	1	no	maximal, Sylow
70	C₂×C₄⋊Q₈	—	137	1	no	—
71	C₂×C₄⋊Q₈	D₁₃₇	1	1	yes	characteristic
72	#500	—	137	8	no	—
73	C₂×#107	—	137	4	no	—
74	C₂×C₄×C₈	—	137	1	no	—
75	C₂²×Q₈	—	2	1	no	—
76	C₂²×Q₈	—	274	1	no	—
77	C₂²×C₈	—	137	2	no	—
78	C₄⋊Q₈	—	2	2	no	—
79	C₄⋊Q₈	—	274	2	no	—
80	C₄⋊Q₈	—	137	4	no	—
81	C₄⋊Q₈	C₂×D₁₃₇	1	4	yes	—
82	C₂×C₄⋊C₄	—	2	1	no	—
83	C₂×C₄⋊C₄	—	274	1	no	—
84	C₂×C₄⋊C₄	—	137	2	no	—
85	C₂×C₄⋊C₄	C₂×D₁₃₇	1	2	yes	—
86	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
87	#107	—	137	16	no	—
88	C₄×C₈	—	137	4	no	—
89	C₂×Q₈	—	2	4	no	—
90	C₂×Q₈	—	4	4	no	—
91	C₂×Q₈	—	274	4	no	—
92	C₂×Q₈	—	548	4	no	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	274	2	no	—
95	C₂²×C₄	C₂²⋊D₁₃₇	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₁₃₇	1	1	yes	characteristic
97	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
98	C₂×C₈	—	137	8	no	—
99	C₂×C₈	—	137	4	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	274	4	no	—
103	C₄⋊C₄	—	274	2	no	—
104	C₄⋊C₄	—	137	4	no	—
105	C₄⋊C₄	C₂²×D₁₃₇	1	4	yes	—
106	C₄²	C₂²⋊D₁₃₇	1	2	yes	—
107	C₄²	C₂²×D₁₃₇	1	2	yes	—
108	C₂³	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
109	Q₈	—	4	8	no	—
110	Q₈	—	548	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	4	4	no	—
113	C₂×C₄	—	274	4	no	—
114	C₂×C₄	—	548	4	no	—
115	C₂×C₄	#14555495	1	4	yes	—
116	C₂×C₄	#14555500	1	4	yes	—
117	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
118	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	2	yes	—
119	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	2	yes	—
120	C₂×C₄	Q₈×D₁₃₇	1	4	yes	—
121	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
122	C₈	—	137	8	no	—
123	C₂²	#17984086	1	2	yes	—
124	C₂²	C₂×#14555495	1	2	yes	—
125	C₂²	C₂×#14555500	1	2	yes	—
126	C₂²	C₂²×C₂²⋊D₁₃₇	1	1	yes	characteristic
127	C₄	—	4	4	no	—
128	C₄	—	548	4	no	—
129	C₄	#17984086	1	2	yes	—
130	C₄	C₂×#14555495	1	4	yes	—
131	C₄	C₂×#14555500	1	4	yes	—
132	C₄	C₂×Q₈×D₁₃₇	1	2	yes	—
133	C₂	#22577806	1	4	yes	direct product factor
134	C₂	C₂×#17984086	1	1	yes	characteristic
135	C₂	C₂²×#14555495	1	1	yes	characteristic
136	C₂	C₂²×#14555500	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984088

C₂

yes

maximal, characteristic

#22577806

C₂

yes

maximal, direct product factor

C₂×#17984021

C₂

yes

maximal

C₂×C₄×#11764040

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₃₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₇₄

—

C₂²×#11764040

C₂²

yes

—

#17984088

—

#17984088

C₂²

yes

—

C₂×C₄⋊Dic₁₃₇

—

C₂×C₄⋊Dic₁₃₇

C₂²

yes

—

#17984021

C₂²

yes

—

C₄×#11764040

C₂²

yes

—

C₂²×Q₈×C₁₃₇

—

C₄⋊Q₈×C₁₃₇

—

C₄⋊Q₈×C₁₃₇

C₂²

yes

—

C₂×C₄⋊C₄×C₁₃₇

—

C₂×C₄⋊C₄×C₁₃₇

C₂²

yes

—

C₂×C₄²×C₁₃₇

C₂²

yes

characteristic

C₂×Dic₂₇₄

—

C₂×Dic₂₇₄

—

C₂²×Dic₁₃₇

—

C₂×#11764040

Q₈

yes

complemented

C₂×#11764040

C₂³

yes

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

C₂³

yes

—

C₂×Q₈×C₁₃₇

—

C₂×Q₈×C₁₃₇

—

C₂²×C₄×C₁₃₇

—

C₂²×C₄×C₁₃₇

D₄

yes

characteristic

C₂²×C₄×C₁₃₇

D₄

yes

characteristic

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

C₂³

yes

—

C₄²×C₁₃₇

D₄

yes

—

C₄²×C₁₃₇

C₂³

yes

—

Dic₂₇₄

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

#11764040

C₂×Q₈

yes

complemented

C₂³×C₁₃₇

C₂×D₄

yes

characteristic

Q₈×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

D₈

yes

—

C₂×C₄×C₁₃₇

Q₁₆

yes

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×D₄

yes

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

—

C₂×C₄×C₁₃₇

C₂×Q₈

yes

—

C₂×C₄×C₁₃₇

C₂⁴

yes

characteristic

C₂²×C₁₃₇

C₄⋊Q₈

yes

—

C₂²×C₁₃₇

C₂×D₈

yes

—

C₂²×C₁₃₇

C₂×Q₁₆

yes

—

C₂²×C₁₃₇

C₂²×D₄

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

C₄⋊Q₈

yes

—

C₄×C₁₃₇

C₂×D₈

yes

—

C₄×C₁₃₇

C₂×Q₁₆

yes

—

C₄×C₁₃₇

C₂²×Q₈

yes

—

Dic₁₃₇

—

C₂×C₁₃₇

#500

yes

complemented

C₂×C₁₃₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₃₇

C₂²×D₈

yes

characteristic

C₂×C₁₃₇

C₂²×Q₁₆

yes

characteristic

C₁₃₇

C₂×#500

yes

characteristic, complemented, Sylow

C₂×#500

—

137

maximal, Sylow

C₂×C₄⋊Q₈

—

137

—

C₂×C₄⋊Q₈

D₁₃₇

yes

characteristic

#500

—

137

—

C₂×#107

—

137

—

C₂×C₄×C₈

—

137

—

C₂²×Q₈

—

C₂²×Q₈

—

274

—

C₂²×C₈

—

137

—

C₄⋊Q₈

—

C₄⋊Q₈

—

274

—

C₄⋊Q₈

—

137

—

C₄⋊Q₈

C₂×D₁₃₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

274

—

C₂×C₄⋊C₄

—

137

—

C₂×C₄⋊C₄

C₂×D₁₃₇

yes

—

C₂×C₄²

C₂×D₁₃₇

yes

characteristic

#107

—

137

—

C₄×C₈

—

137

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

274

—

C₂×Q₈

—

548

—

C₂²×C₄

—

C₂²×C₄

—

274

—

C₂²×C₄

C₂²⋊D₁₃₇

yes

characteristic

C₂²×C₄

C₂²⋊D₁₃₇

yes

characteristic

C₂²×C₄

C₂²×D₁₃₇

yes

characteristic

C₂×C₈

—

137

—

C₂×C₈

—

137

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

274

—

103

C₄⋊C₄

—

274

—

104

C₄⋊C₄

—

137

—

105

C₄⋊C₄

C₂²×D₁₃₇

yes

—

106

C₄²

C₂²⋊D₁₃₇

yes

—

107

C₄²

C₂²×D₁₃₇

yes

—

108

C₂³

C₂×C₂²⋊D₁₃₇

yes

characteristic

109

Q₈

—

110

Q₈

—

548

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

274

—

114

C₂×C₄

—

548

—

115

C₂×C₄

#14555495

yes

—

116

C₂×C₄

#14555500

yes

—

117

C₂×C₄

C₂×C₂²⋊D₁₃₇

yes

characteristic

118

C₂×C₄

C₂×C₂²⋊D₁₃₇

yes

—

119

C₂×C₄

C₂×C₂²⋊D₁₃₇

yes

—

120

C₂×C₄

Q₈×D₁₃₇

yes

—

121

C₂×C₄

C₂³×D₁₃₇

yes

characteristic

122

C₈

—

137

—

123

C₂²

#17984086

yes

—

124

C₂²

C₂×#14555495

yes

—

125

C₂²

C₂×#14555500

yes

—

126

C₂²

C₂²×C₂²⋊D₁₃₇

yes

characteristic

127

C₄

—

128

C₄

—

548

—

129

C₄

#17984086

yes

—

130

C₄

C₂×#14555495

yes

—

131

C₄

C₂×#14555500

yes

—

132

C₄

C₂×Q₈×D₁₃₇

yes

—

133

C₂

#22577806

yes

direct product factor

134

C₂

C₂×#17984086

yes

characteristic

135

C₂

C₂²×#14555495

yes

characteristic

136

C₂

C₂²×#14555500

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₃₇
commutator, cosocle	C₂×C₄×C₁₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting	C₂×C₄⋊Q₈×C₁₃₇	C₂
socle, prime radical	C₂³×C₁₃₇	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₁₃₇
hypocenter	C₁₃₇	C₂×#500
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₃₇

commutator, cosocle

C₂×C₄×C₁₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₇

Fitting

C₂×C₄⋊Q₈×C₁₃₇

C₂

socle, prime radical

C₂³×C₁₃₇

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₁₃₇

hypocenter

C₁₃₇

C₂×#500

LCS

G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #22577806	C₂	direct (N × H)
23. C₂×#11764040	Q₈	semidirect
42. #11764040	C₂×Q₈	semidirect
64. C₂×C₁₃₇	#500	semidirect
68. C₁₃₇	C₂×#500	semidirect
133. C₂	#22577806	direct (N × H)

H ≅ G/N

type

2. #22577806

C₂

direct (N × H)

23. C₂×#11764040

Q₈

semidirect

42. #11764040

C₂×Q₈

semidirect

64. C₂×C₁₃₇

#500

semidirect

68. C₁₃₇

C₂×#500

semidirect

133. C₂

#22577806

direct (N × H)

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

subgroup	index
C₂×#17984088	2
#22577806	2
C₂×#17984021	2
C₂×C₄×#11764040	2
C₂×C₄⋊Q₈×C₁₃₇	2
C₂×#500	137

subgroup

index

C₂×C₄⋊Q₈×C₁₃₇

137

Subgroups — C₂×#22577806 #31791295

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#22577806 #31791295

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22577806 #31791295