Group #31796402 — subgroups

Subgroups — #31796402

Order 17536.

152 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 58 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	#22578454	C₂	1	1	yes	maximal, characteristic
2	Q₈×#11764047	C₂	1	1	yes	maximal, characteristic
3	#22578267	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊#11764047	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊#11764047	C₂	1	1	yes	maximal, characteristic
6	#22578245	C₂	1	1	yes	maximal, characteristic
7	Q₁₆×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic
8	#17984155	—	2	1	no	—
9	C₄⋊#11764047	—	2	1	no	—
10	C₄⋊#11764047	C₂²	1	1	yes	characteristic
11	C₄⋊#11764047	C₂²	1	1	yes	characteristic
12	#17984143	—	2	1	no	—
13	C₄×#11764047	—	2	1	no	—
14	C₄×#11764047	C₂²	1	1	yes	characteristic
15	C₄×#11764047	C₂²	1	1	yes	characteristic
16	Q₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
17	Q₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
18	#17984022	C₄	1	1	yes	characteristic
19	#17984012	C₄	1	1	yes	characteristic
20	#17984012	C₄	1	1	yes	characteristic
21	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
22	C₂×Q₁₆×C₁₃₇	C₄	1	1	yes	characteristic
23	C₂×#11764047	—	2	1	no	—
24	C₂×#11764047	—	2	1	no	—
25	C₂×#11764047	—	2	1	no	—
26	C₂×#11764047	D₄	1	1	yes	characteristic
27	C₂×#11764047	D₄	1	1	yes	characteristic
28	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
29	C₄⋊Dic₁₃₇	—	2	1	no	—
30	C₄⋊Dic₁₃₇	—	2	1	no	—
31	C₄⋊Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
32	C₄⋊Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
33	C₄×Dic₁₃₇	—	2	1	no	—
34	C₄×Dic₁₃₇	—	2	1	no	—
35	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
36	C₂×Q₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
37	C₂×Q₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
38	Q₁₆×C₁₃₇	—	2	2	no	—
39	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
40	#11764047	—	2	1	no	—
41	#11764047	—	4	1	no	—
42	#11764047	—	4	1	no	—
43	#11764047	—	4	1	no	—
44	#11764047	Q₁₆	1	2	yes	complemented
45	C₂×Dic₁₃₇	—	2	1	no	—
46	C₂×Dic₁₃₇	—	2	1	no	—
47	C₂×Dic₁₃₇	C₂×D₄	1	1	yes	characteristic
48	C₂×Dic₁₃₇	#41	1	1	yes	characteristic
49	#11764040	—	2	1	no	—
50	Q₈×C₁₃₇	—	2	1	no	—
51	Q₈×C₁₃₇	—	2	1	no	—
52	Q₈×C₁₃₇	—	2	1	no	—
53	Q₈×C₁₃₇	M₁₆	1	2	yes	—
54	C₂×C₄×C₁₃₇	—	2	1	no	—
55	C₂×C₄×C₁₃₇	—	2	1	no	—
56	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
57	C₈×C₁₃₇	—	2	1	no	—
58	C₂²×C₁₃₇	C₄×D₄	1	1	yes	characteristic
59	C₄×C₁₃₇	—	2	2	no	—
60	C₄×C₁₃₇	—	4	1	no	—
61	C₄×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
62	C₄×C₁₃₇	#131	1	1	yes	characteristic
63	Dic₁₃₇	—	2	1	no	—
64	Dic₁₃₇	—	4	1	no	—
65	Dic₁₃₇	—	4	1	no	—
66	Dic₁₃₇	C₂×Q₁₆	1	1	yes	characteristic
67	Dic₁₃₇	#135	1	1	yes	characteristic
68	C₂×C₁₃₇	#435	1	1	yes	characteristic
69	C₂×C₁₃₇	C₄×Q₁₆	1	1	yes	characteristic
70	C₂×C₁₃₇	#444	1	1	yes	characteristic
71	C₁₃₇	#1584	1	1	yes	characteristic, complemented, Sylow
72	#1584	—	137	1	no	maximal, Sylow
73	#446	—	137	1	no	—
74	C₈×Q₈	—	137	1	no	—
75	C₄×Q₁₆	—	137	1	no	—
76	#335	—	137	1	no	—
77	Q₈⋊C₈	—	137	1	no	—
78	Q₈⋊C₈	—	137	1	no	—
79	#322	—	137	1	no	—
80	C₂×Q₁₆	#10587739	1	1	yes	characteristic
81	C₄×Q₈	—	137	1	no	—
82	C₄×Q₈	—	137	1	no	—
83	#107	—	137	1	no	—
84	C₄⋊C₈	—	274	1	no	—
85	C₄⋊C₈	—	274	1	no	—
86	C₄⋊C₈	—	137	1	no	—
87	C₄⋊C₈	—	137	1	no	—
88	#103	—	137	1	no	—
89	#103	—	137	1	no	—
90	#97	—	274	1	no	—
91	C₄×C₈	—	274	1	no	—
92	C₄×C₈	—	137	1	no	—
93	C₄×C₈	—	137	1	no	—
94	C₄×C₈	—	137	1	no	—
95	C₂×Q₈	C₂×#10587739	1	1	yes	characteristic
96	C₂×Q₈	C₂×#10587739	1	1	yes	characteristic
97	Q₁₆	—	2	2	no	—
98	C₂×C₈	—	274	1	no	—
99	C₂×C₈	—	274	1	no	—
100	C₂×C₈	—	274	1	no	—
101	C₂×C₈	—	137	1	no	—
102	C₂×C₈	—	137	1	no	—
103	C₂×C₈	—	137	1	no	—
104	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
105	C₄⋊C₄	—	274	1	no	—
106	C₄⋊C₄	—	274	1	no	—
107	C₄⋊C₄	—	137	1	no	—
108	C₄⋊C₄	—	137	1	no	—
109	C₄²	—	274	1	no	—
110	C₄²	—	274	1	no	—
111	C₄²	—	137	1	no	—
112	Q₈	—	2	1	no	—
113	Q₈	—	2	1	no	—
114	Q₈	—	2	1	no	—
115	Q₈	#14555519	1	2	yes	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	274	1	no	—
119	C₂×C₄	—	274	1	no	—
120	C₂×C₄	—	137	1	no	—
121	C₂×C₄	—	137	1	no	—
122	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	274	1	no	—
125	C₈	—	274	1	no	—
126	C₈	—	548	1	no	—
127	C₈	—	548	1	no	—
128	C₈	—	548	1	no	—
129	C₈	—	137	2	no	—
130	C₂²	D₄×#10587739	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	4	1	no	—
133	C₄	—	274	1	no	—
134	C₄	—	548	1	no	—
135	C₄	—	548	1	no	—
136	C₄	—	137	1	no	—
137	C₄	—	137	1	no	—
138	C₄	C₂×#14555519	1	1	yes	characteristic
139	C₄	#17984175	1	1	yes	characteristic
140	C₂	#22578434	1	1	yes	characteristic
141	C₂	Q₁₆×#10587739	1	1	yes	characteristic
142	C₂	#22578449	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22578454

C₂

yes

maximal, characteristic

Q₈×#11764047

C₂

yes

maximal, characteristic

#22578267

C₂

yes

maximal, characteristic

Q₈⋊#11764047

C₂

yes

maximal, characteristic

Q₈⋊#11764047

C₂

yes

maximal, characteristic

#22578245

C₂

yes

maximal, characteristic

Q₁₆×Dic₁₃₇

C₂

yes

maximal, characteristic

#17984155

—

C₄⋊#11764047

—

C₄⋊#11764047

C₂²

yes

characteristic

C₄⋊#11764047

C₂²

yes

—

—

C₂²

yes

characteristic

C₄×#11764047

C₂²

yes

characteristic

Q₈×Dic₁₃₇

C₂²

yes

characteristic

Q₈×Dic₁₃₇

C₂²

yes

characteristic

#17984022

C₄

yes

characteristic

#17984012

C₄

yes

characteristic

#17984012

C₄

yes

characteristic

C₈×Dic₁₃₇

C₂²

yes

characteristic

C₂×Q₁₆×C₁₃₇

C₄

yes

—

—

—

D₄

yes

characteristic

C₂×#11764047

D₄

yes

characteristic

C₂×#11764040

C₂×C₄

yes

characteristic

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

C₂×C₄

yes

characteristic

C₄⋊Dic₁₃₇

C₂×C₄

yes

characteristic

C₄×Dic₁₃₇

—

C₄×Dic₁₃₇

—

C₄×Dic₁₃₇

C₂³

yes

characteristic

C₂×Q₈×C₁₃₇

C₂×C₄

yes

characteristic

C₂×Q₈×C₁₃₇

C₂×C₄

yes

characteristic

Q₁₆×C₁₃₇

—

C₂×C₈×C₁₃₇

C₂×C₄

yes

—

—

—

—

Q₁₆

yes

complemented

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

C₂×D₄

yes

C₂×Dic₁₃₇

yes

—

Q₈×C₁₃₇

—

Q₈×C₁₃₇

—

Q₈×C₁₃₇

—

Q₈×C₁₃₇

M₁₆

yes

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₈×C₁₃₇

—

C₂²×C₁₃₇

C₄×D₄

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂×M₁₆

yes

characteristic

C₄×C₁₃₇

#131

yes

characteristic

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

C₂×Q₁₆

yes

Dic₁₃₇

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

C₄×Q₁₆

yes

C₂×C₁₃₇

yes

C₁₃₇

yes

characteristic, complemented, Sylow

#1584

—

137

maximal, Sylow

#446

—

137

—

C₈×Q₈

—

137

—

C₄×Q₁₆

—

137

—

#335

—

137

—

Q₈⋊C₈

—

137

—

Q₈⋊C₈

—

137

—

#322

—

137

—

C₂×Q₁₆

#10587739

yes

characteristic

C₄×Q₈

—

137

—

C₄×Q₈

—

137

—

#107

—

137

—

C₄⋊C₈

—

274

—

C₄⋊C₈

—

274

—

C₄⋊C₈

—

137

—

C₄⋊C₈

—

137

—

#103

—

137

—

#103

—

137

—

#97

—

274

—

C₄×C₈

—

274

—

C₄×C₈

—

137

—

C₄×C₈

—

137

—

C₄×C₈

—

137

—

C₂×Q₈

yes

C₂×Q₈

yes

Q₁₆

—

C₂×C₈

—

274

—

C₂×C₈

—

274

—

100

C₂×C₈

—

274

—

101

C₂×C₈

—

137

—

102

C₂×C₈

—

137

—

103

C₂×C₈

—

137

—

104

C₂×C₈

C₂×#10587739

yes

characteristic

105

C₄⋊C₄

—

274

—

106

C₄⋊C₄

—

274

—

107

C₄⋊C₄

—

137

—

108

C₄⋊C₄

—

137

—

109

C₄²

—

274

—

110

C₄²

—

274

—

111

C₄²

—

137

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

Q₈

#14555519

yes

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

274

—

119

C₂×C₄

—

274

—

120

C₂×C₄

—

137

—

121

C₂×C₄

—

137

—

122

C₂×C₄

C₂²×#10587739

yes

characteristic

123

C₈

—

124

C₈

—

274

—

125

C₈

—

274

—

126

C₈

—

548

—

127

C₈

—

548

—

128

C₈

—

548

—

129

C₈

—

137

—

130

C₂²

D₄×#10587739

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

274

—

134

C₄

—

548

—

135

C₄

—

548

—

136

C₄

—

137

—

137

C₄

—

137

—

138

C₄

C₂×#14555519

yes

characteristic

139

C₄

#17984175

yes

characteristic

140

C₂

#22578434

yes

characteristic

141

C₂

Q₁₆×#10587739

yes

characteristic

142

C₂

#22578449

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×#10587739
commutator	C₂×C₄×C₁₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#10587739
Fitting	C₂×Q₁₆×C₁₃₇	C₄
socle, prime radical	C₂²×C₁₃₇	C₄×D₄
cosocle	C₄×Dic₁₃₇	C₂³
hypercenter	C₂×Q₁₆	#10587739
hypocenter	C₁₃₇	#1584
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

subgroup

G/H

center

C₂²

D₄×#10587739

commutator

C₂×C₄×C₁₃₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#10587739

Fitting

C₂×Q₁₆×C₁₃₇

C₄

socle, prime radical

C₂²×C₁₃₇

C₄×D₄

cosocle

C₄×Dic₁₃₇

C₂³

C₂×Q₁₆

C₁₃₇

G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

N	H ≅ G/N	type
44. #11764047	Q₁₆	semidirect
71. C₁₃₇	#1584	semidirect

H ≅ G/N

type

44. #11764047

Q₁₆

semidirect

71. C₁₃₇

#1584

semidirect

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

subgroup	index
#22578454	2
Q₈×#11764047	2
#22578267	2
Q₈⋊#11764047	2
#22578245	2
Q₁₆×Dic₁₃₇	2
#1584	137

subgroup

index

Q₁₆×Dic₁₃₇

137

Subgroups — #31796402

Conjugacy classes of subgroups

Complements

Maximal subgroups