Group #31797021 — subgroups

Subgroups — #31797021

Order 17536.

228 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 82 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×#17984162	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#14555513	C₂	1	1	yes	maximal, characteristic, complemented
3	#22578298	C₂	1	4	yes	maximal, complemented
4	C₂×#17984096	C₂	1	1	yes	maximal, characteristic
5	#17984162	—	2	2	no	—
6	C₂×#14555513	—	2	2	no	—
7	C₂×#14555513	C₂²	1	2	yes	—
8	C₂×#14555513	C₂²	1	2	yes	—
9	C₂×#14555510	C₄	1	1	yes	characteristic
10	#17984096	—	2	2	no	—
11	#17984096	—	2	2	no	—
12	C₂×#14555493	C₄	1	2	yes	—
13	C₂²×#11764040	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₁₃₇	C₂²	1	2	yes	—
15	C₂×M₁₆×C₁₃₇	C₄	1	1	yes	characteristic
16	#14555513	—	2	2	no	—
17	#14555513	—	2	2	no	—
18	#14555510	—	2	2	no	—
19	#14555510	—	2	2	no	—
20	C₂×Dic₂₇₄	C₈	1	1	yes	characteristic
21	C₂×D₅₄₈	C₈	1	1	yes	characteristic
22	C₂×C₂²⋊D₁₃₇	C₈	1	2	yes	—
23	C₂×C₄×D₁₃₇	C₂×C₄	1	2	yes	—
24	#14555493	—	2	4	no	—
25	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
26	C₂×#11764040	D₄	1	2	yes	—
27	C₂×#11764040	D₄	1	2	yes	—
28	C₂×#11764040	C₂³	1	1	yes	characteristic
29	C₈×D₁₃₇	—	2	2	no	—
30	C₈×D₁₃₇	—	2	2	no	—
31	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
32	M₁₆×C₁₃₇	—	2	2	no	—
33	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
34	C₂²×D₁₃₇	C₂×C₈	1	2	yes	—
35	Dic₂₇₄	—	2	2	no	—
36	D₅₄₈	—	2	2	no	—
37	C₂²⋊D₁₃₇	—	2	4	no	—
38	C₂×Dic₁₃₇	C₂×C₈	1	2	yes	—
39	C₄×D₁₃₇	—	2	2	no	—
40	C₄×D₁₃₇	—	2	2	no	—
41	#11764040	C₂²⋊C₄	1	2	yes	—
42	#11764040	C₂×D₄	1	1	yes	characteristic
43	#11764040	C₂×D₄	1	1	yes	characteristic
44	C₂³×C₁₃₇	C₂×C₈	1	1	yes	characteristic
45	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
47	C₂×C₄×C₁₃₇	M₁₆	1	2	yes	—
48	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
49	C₈×C₁₃₇	—	2	2	no	—
50	C₂²×C₁₃₇	—	2	2	no	—
51	C₂²×C₁₃₇	C₂²⋊C₈	1	2	yes	—
52	C₂²×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
53	C₂×D₁₃₇	—	2	2	no	—
54	C₂×D₁₃₇	—	2	4	no	—
55	C₄×C₁₃₇	C₂²⋊C₈	1	2	yes	—
56	C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₄×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
58	Dic₁₃₇	—	2	2	no	—
59	C₂×C₁₃₇	—	2	2	no	—
60	C₂×C₁₃₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
61	C₂×C₁₃₇	#504	1	2	yes	—
62	D₁₃₇	—	4	2	no	—
63	C₁₃₇	#2113	1	1	yes	characteristic, complemented, Sylow
64	#2113	—	137	1	no	maximal, Sylow
65	C₂×#131	—	137	1	no	—
66	C₂×M₃₂	—	137	1	no	—
67	C₂²×C₁₆	—	137	1	no	—
68	C₂²⋊C₁₆	—	137	4	no	—
69	C₂×#41	—	137	1	no	—
70	#131	—	274	2	no	—
71	#131	—	274	2	no	—
72	C₂×M₁₆	—	137	2	no	—
73	C₂×M₁₆	#10587739	1	1	yes	characteristic
74	C₂²×C₈	—	137	2	no	—
75	C₂²×C₈	—	137	1	no	—
76	M₃₂	—	274	2	no	—
77	C₂×C₁₆	—	274	2	no	—
78	C₂×C₁₆	—	137	2	no	—
79	C₂×C₁₆	—	137	2	no	—
80	#41	—	274	2	no	—
81	#41	—	274	2	no	—
82	C₂×Q₈	—	137	1	no	—
83	C₂×D₄	—	137	2	no	—
84	C₂×D₄	—	137	1	no	—
85	C₂²×C₄	—	137	2	no	—
86	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
87	M₁₆	—	2	2	no	—
88	M₁₆	—	274	4	no	—
89	C₂×C₈	—	274	2	no	—
90	C₂×C₈	—	274	2	no	—
91	C₂×C₈	—	137	2	no	—
92	C₂×C₈	—	137	2	no	—
93	C₂×C₈	—	137	1	no	—
94	C₂×C₈	—	137	1	no	—
95	C₂×C₈	C₂×#10587739	1	2	yes	—
96	C₁₆	—	274	2	no	—
97	C₁₆	—	274	2	no	—
98	C₂³	—	137	2	no	—
99	C₂³	#14555518	1	1	yes	characteristic
100	Q₈	—	274	2	no	—
101	D₄	—	274	4	no	—
102	D₄	—	274	2	no	—
103	C₂×C₄	—	274	2	no	—
104	C₂×C₄	—	274	2	no	—
105	C₂×C₄	—	137	2	no	—
106	C₂×C₄	C₂²⋊#10587739	1	2	yes	—
107	C₂×C₄	#14555518	1	1	yes	characteristic
108	C₂×C₄	#14555520	1	2	yes	—
109	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
110	C₈	—	2	2	no	—
111	C₈	—	137	1	no	—
112	C₈	—	137	1	no	—
113	C₈	—	137	2	no	—
114	C₂²	—	2	2	no	—
115	C₂²	—	274	4	no	—
116	C₂²	—	274	2	no	—
117	C₂²	#17984145	1	2	yes	—
118	C₂²	C₂×#14555518	1	1	yes	characteristic
119	C₄	—	274	2	no	—
120	C₄	#17984145	1	2	yes	—
121	C₄	C₂×C₂²⋊#10587739	1	1	yes	characteristic
122	C₄	C₂×#14555520	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	548	2	no	—
125	C₂	C₂×#17984145	1	1	yes	characteristic
126	C₂	#22578462	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984162

C₂

yes

maximal, characteristic, complemented

C₂²×#14555513

C₂

yes

maximal, characteristic, complemented

#22578298

C₂

yes

maximal, complemented

C₂×#17984096

C₂

yes

maximal, characteristic

#17984162

—

C₂×#14555513

—

C₂×#14555513

C₂²

yes

—

C₂×#14555513

C₂²

yes

—

C₂×#14555510

C₄

yes

—

—

C₄

yes

—

C₂²×#11764040

C₂²

yes

characteristic

C₂×C₈×D₁₃₇

C₂²

yes

—

C₂×M₁₆×C₁₃₇

C₄

yes

—

—

—

—

C₂×Dic₂₇₄

C₈

yes

characteristic

C₂×D₅₄₈

C₈

yes

characteristic

C₂×C₂²⋊D₁₃₇

C₈

yes

—

C₂×C₄×D₁₃₇

C₂×C₄

yes

—

#14555493

—

C₂×#11764040

C₂×C₄

yes

characteristic

C₂×#11764040

D₄

yes

—

C₂×#11764040

D₄

yes

—

C₂×#11764040

C₂³

yes

characteristic

C₈×D₁₃₇

—

C₈×D₁₃₇

—

C₂²×C₄×C₁₃₇

C₂×C₄

yes

characteristic

M₁₆×C₁₃₇

—

C₂×C₈×C₁₃₇

C₂×C₄

yes

—

C₂²×D₁₃₇

C₂×C₈

yes

—

Dic₂₇₄

—

D₅₄₈

—

C₂²⋊D₁₃₇

—

C₂×Dic₁₃₇

C₂×C₈

yes

—

C₄×D₁₃₇

—

C₄×D₁₃₇

—

#11764040

C₂²⋊C₄

yes

—

#11764040

C₂×D₄

yes

characteristic

#11764040

C₂×D₄

yes

characteristic

C₂³×C₁₃₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₃₇

M₁₆

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₈×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₂²⋊C₈

yes

—

C₂²×C₁₃₇

C₂²×C₈

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₄×C₁₃₇

C₂²⋊C₈

yes

—

C₄×C₁₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₃₇

C₂×M₁₆

yes

characteristic

Dic₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₁₃₇

#504

yes

—

D₁₃₇

—

C₁₃₇

#2113

yes

characteristic, complemented, Sylow

#2113

—

137

maximal, Sylow

C₂×#131

—

137

—

C₂×M₃₂

—

137

—

C₂²×C₁₆

—

137

—

C₂²⋊C₁₆

—

137

—

C₂×#41

—

137

—

#131

—

274

—

#131

—

274

—

C₂×M₁₆

—

137

—

C₂×M₁₆

#10587739

yes

characteristic

C₂²×C₈

—

137

—

C₂²×C₈

—

137

—

M₃₂

—

274

—

C₂×C₁₆

—

274

—

C₂×C₁₆

—

137

—

C₂×C₁₆

—

137

—

#41

—

274

—

#41

—

274

—

C₂×Q₈

—

137

—

C₂×D₄

—

137

—

C₂×D₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

C₂×#10587739

yes

characteristic

M₁₆

—

M₁₆

—

274

—

C₂×C₈

—

274

—

C₂×C₈

—

274

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

C₂×#10587739

yes

—

C₁₆

—

274

—

C₁₆

—

274

—

C₂³

—

137

—

C₂³

#14555518

yes

characteristic

100

Q₈

—

274

—

101

D₄

—

274

—

102

D₄

—

274

—

103

C₂×C₄

—

274

—

104

C₂×C₄

—

274

—

105

C₂×C₄

—

137

—

106

C₂×C₄

C₂²⋊#10587739

yes

—

107

C₂×C₄

#14555518

yes

characteristic

108

C₂×C₄

#14555520

yes

—

109

C₂×C₄

C₂²×#10587739

yes

characteristic

110

C₈

—

111

C₈

—

137

—

112

C₈

—

137

—

113

C₈

—

137

—

114

C₂²

—

115

C₂²

—

274

—

116

C₂²

—

274

—

117

C₂²

#17984145

yes

—

118

C₂²

C₂×#14555518

yes

characteristic

119

C₄

—

274

—

120

C₄

#17984145

yes

—

121

C₄

C₂×C₂²⋊#10587739

yes

characteristic

122

C₄

C₂×#14555520

yes

characteristic

123

C₂

—

124

C₂

—

548

—

125

C₂

C₂×#17984145

yes

characteristic

126

C₂

#22578462

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10587739
commutator, socle	C₂²×C₁₃₇	C₂²×C₈
Fitting	C₂×M₁₆×C₁₃₇	C₄
prime radical	C₂×#14555510	C₄
cosocle	C₂×#11764040	C₂³
hypercenter	C₂×M₁₆	#10587739
hypocenter	C₁₃₇	#2113
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#10587739

commutator, socle

C₂²×C₁₃₇

C₂²×C₈

Fitting

C₂×M₁₆×C₁₃₇

C₄

C₄

C₂³

C₂×M₁₆

C₁₃₇

G ▷ C₂²×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
1. C₂×#17984162	C₂	semidirect
2. C₂²×#14555513	C₂	semidirect
3. #22578298	C₂	semidirect
63. C₁₃₇	#2113	semidirect

H ≅ G/N

type

1. C₂×#17984162

C₂

semidirect

2. C₂²×#14555513

C₂

semidirect

3. #22578298

C₂

semidirect

63. C₁₃₇

#2113

semidirect

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

subgroup	index
C₂×#17984162	2
C₂²×#14555513	2
#22578298	2
C₂×#17984096	2
#2113	137

subgroup

index

137

Subgroups — #31797021

Conjugacy classes of subgroups

Complements

Maximal subgroups