Group #31780294 — subgroups

Subgroups — #31780294

Order 17536.

146 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 51 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	#22577799	C₂	1	1	yes	maximal, characteristic
2	#22577699	C₂	1	1	yes	maximal, characteristic
3	#22577610	C₂	1	1	yes	maximal, characteristic
4	#22577530	C₂	1	1	yes	maximal, characteristic
5	#22577155	C₂	1	1	yes	maximal, characteristic
6	#22577151	C₂	1	1	yes	maximal, characteristic
7	#332×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#14555500	—	2	1	no	—
9	#17984088	C₂²	1	1	yes	characteristic
10	C₄×Dic₂₇₄	C₂²	1	1	yes	characteristic
11	#17984022	—	2	1	no	—
12	#17984019	—	2	1	no	—
13	#17984019	—	2	1	no	—
14	C₄⋊#11764040	C₂²	1	1	yes	characteristic
15	#17984017	—	2	1	no	—
16	#17984013	—	2	1	no	—
17	#17984012	—	2	1	no	—
18	#17983998	—	2	1	no	—
19	C₄×#11764040	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
23	C₂×Dic₂₇₄	—	2	1	no	—
24	C₂×Dic₂₇₄	D₄	1	1	yes	characteristic
25	#14555500	—	4	2	no	—
26	C₂×#11764040	—	2	1	no	—
27	C₂×#11764040	D₄	1	1	yes	characteristic
28	C₂×#11764040	D₄	1	1	yes	characteristic
29	C₄⋊Dic₁₃₇	—	2	1	no	—
30	C₄⋊Dic₁₃₇	—	2	1	no	—
31	C₄⋊Dic₁₃₇	D₄	1	1	yes	characteristic
32	#14555489	—	2	1	no	—
33	C₄×Dic₁₃₇	—	2	1	no	—
34	C₂×Q₈×C₁₃₇	—	2	1	no	—
35	C₂×C₈×C₁₃₇	—	2	1	no	—
36	C₂×C₈×C₁₃₇	—	2	1	no	—
37	C₄⋊C₄×C₁₃₇	—	2	1	no	—
38	C₄⋊C₄×C₁₃₇	—	2	1	no	—
39	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
40	Dic₂₇₄	—	2	1	no	—
41	Dic₂₇₄	—	2	1	no	—
42	Dic₂₇₄	—	8	1	no	—
43	C₂×Dic₁₃₇	—	2	1	no	—
44	C₂×Dic₁₃₇	—	2	1	no	—
45	C₂×Dic₁₃₇	—	4	1	no	—
46	#11764040	—	2	1	no	—
47	#11764040	—	4	1	no	—
48	#11764040	Q₁₆	1	2	yes	complemented
49	Q₈×C₁₃₇	—	4	1	no	—
50	Q₈×C₁₃₇	—	4	1	no	—
51	C₂×C₄×C₁₃₇	—	4	1	no	—
52	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₃₇	#41	1	1	yes	characteristic
55	C₈×C₁₃₇	—	4	1	no	—
56	C₈×C₁₃₇	—	4	1	no	—
57	C₂²×C₁₃₇	#121	1	1	yes	characteristic
58	C₄×C₁₃₇	—	2	1	no	—
59	C₄×C₁₃₇	—	8	1	no	—
60	C₄×C₁₃₇	C₂×Q₁₆	1	1	yes	characteristic
61	C₄×C₁₃₇	AGL(1,ℤ/8)	1	1	yes	characteristic
62	C₄×C₁₃₇	#137	1	1	yes	characteristic
63	C₄×C₁₃₇	#137	1	1	yes	characteristic
64	Dic₁₃₇	—	4	1	no	—
65	Dic₁₃₇	—	4	1	no	—
66	Dic₁₃₇	—	8	1	no	—
67	C₂×C₁₃₇	#462	1	1	yes	characteristic
68	C₂×C₁₃₇	#468	1	1	yes	characteristic
69	C₂×C₁₃₇	#471	1	1	yes	characteristic
70	C₁₃₇	#1692	1	1	yes	characteristic, complemented, Sylow
71	#1692	—	137	1	no	maximal, Sylow
72	#498	—	137	1	no	—
73	#477	—	137	1	no	—
74	#462	—	137	1	no	—
75	#446	—	137	1	no	—
76	#335	—	137	1	no	—
77	#333	—	137	1	no	—
78	#332	D₁₃₇	1	1	yes	characteristic
79	C₂×Q₁₆	—	274	1	no	—
80	C₄⋊Q₈	—	137	1	no	—
81	C₄⋊Q₈	C₂×D₁₃₇	1	1	yes	characteristic
82	C₄×Q₈	—	137	1	no	—
83	#107	—	274	1	no	—
84	#106	—	274	1	no	—
85	#106	—	274	1	no	—
86	C₄⋊C₈	—	274	1	no	—
87	C₄⋊C₈	—	137	1	no	—
88	C₄⋊C₈	C₂×D₁₃₇	1	1	yes	characteristic
89	C₄⋊C₈	C₂×D₁₃₇	1	1	yes	characteristic
90	#103	—	274	1	no	—
91	#103	—	274	1	no	—
92	#97	—	274	1	no	—
93	C₄×C₈	—	137	1	no	—
94	C₂×Q₈	—	2	1	no	—
95	C₂×Q₈	—	274	1	no	—
96	C₂×Q₈	—	137	1	no	—
97	Q₁₆	—	548	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	274	1	no	—
101	C₂×C₈	—	137	1	no	—
102	C₂×C₈	—	137	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	274	1	no	—
106	C₄⋊C₄	—	274	1	no	—
107	C₄⋊C₄	—	274	1	no	—
108	C₄⋊C₄	—	137	1	no	—
109	C₄²	—	274	1	no	—
110	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
111	Q₈	—	4	1	no	—
112	Q₈	—	4	1	no	—
113	Q₈	—	274	1	no	—
114	Q₈	—	274	1	no	—
115	Q₈	—	1096	1	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	274	1	no	—
118	C₂×C₄	—	274	1	no	—
119	C₂×C₄	—	548	1	no	—
120	C₂×C₄	D₄×D₁₃₇	1	1	yes	characteristic
121	C₂×C₄	D₄×D₁₃₇	1	1	yes	characteristic
122	C₂×C₄	#14555510	1	1	yes	characteristic
123	C₈	—	4	1	no	—
124	C₈	—	4	1	no	—
125	C₈	—	274	1	no	—
126	C₈	—	548	1	no	—
127	C₈	—	137	2	no	—
128	C₂²	#17984061	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	8	1	no	—
131	C₄	—	548	1	no	—
132	C₄	—	548	1	no	—
133	C₄	—	1096	1	no	—
134	C₄	Q₁₆×D₁₃₇	1	1	yes	characteristic
135	C₄	#17984112	1	1	yes	characteristic
136	C₄	#17984117	1	1	yes	characteristic
137	C₄	#17984117	1	1	yes	characteristic
138	C₂	#22577612	1	1	yes	characteristic
139	C₂	#22577648	1	1	yes	characteristic
140	C₂	#22577669	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22577799

C₂

yes

maximal, characteristic

#22577699

C₂

yes

maximal, characteristic

#22577610

C₂

yes

maximal, characteristic

#22577530

C₂

yes

maximal, characteristic

#22577155

C₂

yes

maximal, characteristic

#22577151

C₂

yes

maximal, characteristic

#332×C₁₃₇

C₂

yes

maximal, characteristic

C₂×#14555500

—

#17984088

C₂²

yes

characteristic

C₄×Dic₂₇₄

C₂²

yes

—

—

—

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄⋊Q₈×C₁₃₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₇

C₂²

yes

characteristic

C₂×Dic₂₇₄

—

C₂×Dic₂₇₄

D₄

yes

—

—

D₄

yes

characteristic

C₂×#11764040

D₄

yes

characteristic

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

D₄

yes

characteristic

#14555489

—

C₄×Dic₁₃₇

—

C₂×Q₈×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

—

C₄²×C₁₃₇

C₂³

yes

characteristic

Dic₂₇₄

—

Dic₂₇₄

—

Dic₂₇₄

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

#11764040

—

#11764040

—

#11764040

Q₁₆

yes

complemented

Q₈×C₁₃₇

—

Q₈×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

#41

yes

characteristic

C₈×C₁₃₇

—

C₈×C₁₃₇

—

C₂²×C₁₃₇

#121

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂×Q₁₆

yes

C₄×C₁₃₇

yes

C₄×C₁₃₇

yes

C₄×C₁₃₇

yes

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

—

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₁₃₇

yes

characteristic, complemented, Sylow

#1692

—

137

maximal, Sylow

#498

—

137

—

#477

—

137

—

#462

—

137

—

#446

—

137

—

#335

—

137

—

#333

—

137

—

#332

D₁₃₇

yes

characteristic

C₂×Q₁₆

—

274

—

C₄⋊Q₈

—

137

—

C₄⋊Q₈

C₂×D₁₃₇

yes

characteristic

C₄×Q₈

—

137

—

#107

—

274

—

#106

—

274

—

#106

—

274

—

C₄⋊C₈

—

274

—

C₄⋊C₈

—

137

—

C₄⋊C₈

C₂×D₁₃₇

yes

characteristic

C₄⋊C₈

C₂×D₁₃₇

yes

characteristic

#103

—

274

—

#103

—

274

—

#97

—

274

—

C₄×C₈

—

137

—

C₂×Q₈

—

C₂×Q₈

—

274

—

C₂×Q₈

—

137

—

Q₁₆

—

548

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

274

—

101

C₂×C₈

—

137

—

102

C₂×C₈

—

137

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

274

—

106

C₄⋊C₄

—

274

—

107

C₄⋊C₄

—

274

—

108

C₄⋊C₄

—

137

—

109

C₄²

—

274

—

110

C₄²

C₂²×D₁₃₇

yes

characteristic

111

Q₈

—

112

Q₈

—

113

Q₈

—

274

—

114

Q₈

—

274

—

115

Q₈

—

1096

—

116

C₂×C₄

—

117

C₂×C₄

—

274

—

118

C₂×C₄

—

274

—

119

C₂×C₄

—

548

—

120

C₂×C₄

D₄×D₁₃₇

yes

characteristic

121

C₂×C₄

D₄×D₁₃₇

yes

characteristic

122

C₂×C₄

#14555510

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

274

—

126

C₈

—

548

—

127

C₈

—

137

—

128

C₂²

#17984061

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

548

—

132

C₄

—

548

—

133

C₄

—

1096

—

134

C₄

Q₁₆×D₁₃₇

yes

characteristic

135

C₄

#17984112

yes

characteristic

136

C₄

#17984117

yes

characteristic

137

C₄

#17984117

yes

characteristic

138

C₂

#22577612

yes

characteristic

139

C₂

#22577648

yes

characteristic

140

C₂

#22577669

yes

characteristic

	subgroup	G/H
center	C₂²	#17984061
commutator, cosocle	C₄²×C₁₃₇	C₂³
Frattini	C₄²	C₂²×D₁₃₇
Fitting	#332×C₁₃₇	C₂
socle, prime radical	C₂²×C₁₃₇	#121
hypercenter	#332	D₁₃₇
hypocenter	C₁₃₇	#1692
LCS	G ▷ C₄²×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

#17984061

commutator, cosocle

C₄²×C₁₃₇

C₂³

Frattini

C₄²

C₂²×D₁₃₇

Fitting

#332×C₁₃₇

C₂

C₂²×C₁₃₇

D₁₃₇

C₁₃₇

G ▷ C₄²×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
48. #11764040	Q₁₆	semidirect
70. C₁₃₇	#1692	semidirect

H ≅ G/N

type

48. #11764040

Q₁₆

semidirect

70. C₁₃₇

#1692

semidirect

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

subgroup	index
#22577799	2
#22577699	2
#22577610	2
#22577530	2
#22577155	2
#22577151	2
#332×C₁₃₇	2
#1692	137

subgroup

index

#332×C₁₃₇

137

Subgroups — #31780294

Conjugacy classes of subgroups

Complements

Maximal subgroups