Group #31796357 — subgroups

Subgroups — #31796357

Order 17536.

168 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 70 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	#22578407	C₂	1	1	yes	maximal, characteristic
2	#22578404	C₂	1	1	yes	maximal, characteristic
3	#22578265	C₂	1	2	yes	maximal, complemented
4	#22578264	C₂	1	2	yes	maximal, complemented
5	#22577455	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555520	—	2	1	no	—
7	C₂×#14555518	—	2	1	no	—
8	#17984155	—	2	1	no	—
9	C₄⋊#11764047	—	2	1	no	—
10	C₄⋊#11764047	C₂²	1	1	yes	characteristic
11	C₄⋊#11764047	C₂²	1	2	yes	—
12	C₄⋊#11764047	C₂²	1	1	yes	characteristic
13	C₄²×D₁₃₇	C₂²	1	1	yes	characteristic
14	C₄⋊#11764040	C₄	1	1	yes	characteristic
15	#17984000	C₄	1	2	yes	—
16	C₈×Dic₁₃₇	C₂²	1	2	yes	—
17	C₄⋊C₈×C₁₃₇	C₄	1	1	yes	characteristic
18	#14555520	—	4	2	no	—
19	#14555518	—	4	2	no	—
20	C₂×#11764047	—	2	2	no	—
21	C₂×#11764047	—	2	1	no	—
22	C₂×#11764047	—	2	1	no	—
23	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₁₃₇	D₄	1	1	yes	characteristic
25	C₂×C₄×D₁₃₇	Q₈	1	1	yes	characteristic
26	C₂×#11764040	C₂×C₄	1	2	yes	—
27	C₄×Dic₁₃₇	D₄	1	1	yes	characteristic
28	C₄×Dic₁₃₇	Q₈	1	1	yes	characteristic
29	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
30	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
31	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
32	#11764047	—	4	1	no	—
33	#11764047	—	4	1	no	—
34	#11764047	—	4	2	no	—
35	C₂²×D₁₃₇	C₄⋊C₄	1	1	yes	characteristic
36	C₂×Dic₁₃₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂×Dic₁₃₇	C₂×D₄	1	1	yes	characteristic
38	C₂×Dic₁₃₇	C₂×Q₈	1	1	yes	characteristic
39	C₄×D₁₃₇	—	2	1	no	—
40	C₄×D₁₃₇	—	2	1	no	—
41	C₄×D₁₃₇	—	4	1	no	—
42	C₄×D₁₃₇	SD₁₆	1	2	yes	—
43	C₄×D₁₃₇	SD₁₆	1	2	yes	—
44	#11764040	—	2	2	no	—
45	C₂×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
48	C₈×C₁₃₇	—	2	2	no	—
49	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×D₁₃₇	—	2	1	no	—
51	C₂×D₁₃₇	—	2	1	no	—
52	C₂×D₁₃₇	#106	1	2	yes	—
53	C₄×C₁₃₇	—	2	1	no	—
54	C₄×C₁₃₇	#106	1	2	yes	—
55	C₄×C₁₃₇	#131	1	1	yes	characteristic
56	C₄×C₁₃₇	#131	1	1	yes	characteristic
57	Dic₁₃₇	—	2	1	no	—
58	Dic₁₃₇	—	2	1	no	—
59	Dic₁₃₇	C₂×SD₁₆	1	1	yes	characteristic
60	Dic₁₃₇	C₂×SD₁₆	1	1	yes	characteristic
61	C₂×C₁₃₇	#424	1	1	yes	characteristic
62	C₂×C₁₃₇	C₂×#106	1	1	yes	characteristic
63	C₂×C₁₃₇	#430	1	1	yes	characteristic
64	D₁₃₇	—	2	2	no	—
65	C₁₃₇	#1570	1	1	yes	characteristic, complemented, Sylow
66	#1570	—	137	1	no	maximal, Sylow
67	#431	—	137	1	no	—
68	#423	—	137	1	no	—
69	C₂×C₄⋊C₈	—	137	1	no	—
70	#334	—	137	2	no	—
71	#334	—	137	2	no	—
72	C₂×M₁₆	—	274	1	no	—
73	C₂²×C₈	—	274	1	no	—
74	C₂×C₄²	—	137	1	no	—
75	C₄⋊C₈	—	274	1	no	—
76	C₄⋊C₈	—	274	1	no	—
77	C₄⋊C₈	—	137	1	no	—
78	C₄⋊C₈	—	137	1	no	—
79	C₄⋊C₈	—	137	2	no	—
80	C₄⋊C₈	—	137	1	no	—
81	C₄⋊C₈	#10587739	1	1	yes	characteristic
82	C₂²⋊C₈	—	137	2	no	—
83	C₄×C₈	—	137	2	no	—
84	C₂²×C₄	—	137	1	no	—
85	C₂²×C₄	—	137	1	no	—
86	C₂²×C₄	—	137	1	no	—
87	M₁₆	—	548	2	no	—
88	C₂×C₈	—	274	1	no	—
89	C₂×C₈	—	274	1	no	—
90	C₂×C₈	—	274	2	no	—
91	C₂×C₈	—	548	2	no	—
92	C₂×C₈	—	137	2	no	—
93	C₂×C₈	C₂×#10587739	1	2	yes	—
94	C₄²	—	137	1	no	—
95	C₄²	—	137	1	no	—
96	C₄²	—	137	1	no	—
97	C₄²	C₂×#10587739	1	1	yes	characteristic
98	C₂³	—	137	1	no	—
99	C₂×C₄	—	274	1	no	—
100	C₂×C₄	—	274	1	no	—
101	C₂×C₄	—	548	1	no	—
102	C₂×C₄	—	137	1	no	—
103	C₂×C₄	—	137	1	no	—
104	C₂×C₄	—	137	1	no	—
105	C₂×C₄	—	137	2	no	—
106	C₂×C₄	—	137	2	no	—
107	C₂×C₄	C₄⋊#10587739	1	1	yes	characteristic
108	C₂×C₄	C₄⋊#10587739	1	1	yes	characteristic
109	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
110	C₈	—	2	2	no	—
111	C₈	—	274	2	no	—
112	C₈	—	548	1	no	—
113	C₈	—	548	1	no	—
114	C₈	—	548	2	no	—
115	C₂²	—	274	1	no	—
116	C₂²	—	274	1	no	—
117	C₂²	—	137	2	no	—
118	C₂²	C₂×C₄⋊#10587739	1	1	yes	characteristic
119	C₄	—	2	1	no	—
120	C₄	—	274	1	no	—
121	C₄	—	274	1	no	—
122	C₄	—	137	1	no	—
123	C₄	—	137	1	no	—
124	C₄	#17984156	1	2	yes	—
125	C₄	#17984175	1	1	yes	characteristic
126	C₄	#17984176	1	1	yes	characteristic
127	C₂	—	274	2	no	—
128	C₂	#22578410	1	1	yes	characteristic
129	C₂	C₂×#17984156	1	1	yes	characteristic
130	C₂	#22578418	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22578407

C₂

yes

maximal, characteristic

#22578404

C₂

yes

maximal, characteristic

#22578265

C₂

yes

maximal, complemented

#22578264

C₂

yes

maximal, complemented

#22577455

C₂

yes

maximal, characteristic

—

—

—

—

C₂²

yes

characteristic

C₄⋊#11764047

C₂²

yes

—

C₄⋊#11764047

C₂²

yes

characteristic

C₄²×D₁₃₇

C₂²

yes

characteristic

C₄⋊#11764040

C₄

yes

characteristic

#17984000

C₄

yes

—

C₈×Dic₁₃₇

C₂²

yes

—

C₄⋊C₈×C₁₃₇

C₄

yes

—

—

—

—

—

C₂×C₄×D₁₃₇

C₂×C₄

yes

characteristic

C₂×C₄×D₁₃₇

D₄

yes

characteristic

C₂×C₄×D₁₃₇

Q₈

yes

characteristic

C₂×#11764040

C₂×C₄

yes

—

C₄×Dic₁₃₇

D₄

yes

characteristic

C₄×Dic₁₃₇

Q₈

yes

characteristic

C₄×Dic₁₃₇

C₂³

yes

characteristic

C₂×C₈×C₁₃₇

C₂×C₄

yes

—

C₄²×C₁₃₇

C₂×C₄

yes

—

—

—

C₂²×D₁₃₇

C₄⋊C₄

yes

characteristic

C₂×Dic₁₃₇

C₄⋊C₄

yes

characteristic

C₂×Dic₁₃₇

C₂×D₄

yes

characteristic

C₂×Dic₁₃₇

C₂×Q₈

yes

characteristic

C₄×D₁₃₇

—

C₄×D₁₃₇

—

C₄×D₁₃₇

—

C₄×D₁₃₇

SD₁₆

yes

—

C₄×D₁₃₇

SD₁₆

yes

—

#11764040

—

C₂×C₄×C₁₃₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₈×C₁₃₇

—

C₂²×C₁₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

#106

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

#106

yes

—

C₄×C₁₃₇

yes

C₄×C₁₃₇

yes

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

C₂×SD₁₆

yes

characteristic

Dic₁₃₇

C₂×SD₁₆

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

D₁₃₇

—

C₁₃₇

#1570

yes

characteristic, complemented, Sylow

#1570

—

137

maximal, Sylow

#431

—

137

—

#423

—

137

—

C₂×C₄⋊C₈

—

137

—

#334

—

137

—

#334

—

137

—

C₂×M₁₆

—

274

—

C₂²×C₈

—

274

—

C₂×C₄²

—

137

—

C₄⋊C₈

—

274

—

C₄⋊C₈

—

274

—

C₄⋊C₈

—

137

—

C₄⋊C₈

—

137

—

C₄⋊C₈

—

137

—

C₄⋊C₈

—

137

—

C₄⋊C₈

#10587739

yes

characteristic

C₂²⋊C₈

—

137

—

C₄×C₈

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

M₁₆

—

548

—

C₂×C₈

—

274

—

C₂×C₈

—

274

—

C₂×C₈

—

274

—

C₂×C₈

—

548

—

C₂×C₈

—

137

—

C₂×C₈

C₂×#10587739

yes

—

C₄²

—

137

—

C₄²

—

137

—

C₄²

—

137

—

C₄²

C₂×#10587739

yes

characteristic

C₂³

—

137

—

C₂×C₄

—

274

—

100

C₂×C₄

—

274

—

101

C₂×C₄

—

548

—

102

C₂×C₄

—

137

—

103

C₂×C₄

—

137

—

104

C₂×C₄

—

137

—

105

C₂×C₄

—

137

—

106

C₂×C₄

—

137

—

107

C₂×C₄

C₄⋊#10587739

yes

characteristic

108

C₂×C₄

C₄⋊#10587739

yes

characteristic

109

C₂×C₄

C₂²×#10587739

yes

characteristic

110

C₈

—

111

C₈

—

274

—

112

C₈

—

548

—

113

C₈

—

548

—

114

C₈

—

548

—

115

C₂²

—

274

—

116

C₂²

—

274

—

117

C₂²

—

137

—

118

C₂²

C₂×C₄⋊#10587739

yes

characteristic

119

C₄

—

120

C₄

—

274

—

121

C₄

—

274

—

122

C₄

—

137

—

123

C₄

—

137

—

124

C₄

#17984156

yes

—

125

C₄

#17984175

yes

characteristic

126

C₄

#17984176

yes

characteristic

127

C₂

—

274

—

128

C₂

#22578410

yes

characteristic

129

C₂

C₂×#17984156

yes

characteristic

130

C₂

#22578418

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#10587739
commutator	C₂×C₄×C₁₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#10587739
Fitting	C₄⋊C₈×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₂×C₄⋊C₄
prime radical	C₂²×D₁₃₇	C₄⋊C₄
cosocle	C₄×Dic₁₃₇	C₂³
hypercenter	C₄⋊C₈	#10587739
hypocenter	C₁₃₇	#1570
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

subgroup

G/H

center

C₂²

C₂×C₄⋊#10587739

commutator

C₂×C₄×C₁₃₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#10587739

Fitting

C₄⋊C₈×C₁₃₇

C₄

socle

C₂²×C₁₃₇

C₂×C₄⋊C₄

prime radical

C₂²×D₁₃₇

C₄⋊C₄

cosocle

C₄×Dic₁₃₇

C₂³

C₄⋊C₈

C₁₃₇

G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

N	H ≅ G/N	type
3. #22578265	C₂	semidirect
4. #22578264	C₂	semidirect
65. C₁₃₇	#1570	semidirect

H ≅ G/N

type

3. #22578265

C₂

semidirect

4. #22578264

C₂

semidirect

65. C₁₃₇

#1570

semidirect

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

subgroup	index
#22578407	2
#22578404	2
#22578265	2
#22578264	2
#22577455	2
#1570	137

subgroup

index

137

Subgroups — #31796357

Conjugacy classes of subgroups

Complements

Maximal subgroups