Group #31797019 — subgroups

Subgroups — #31797019

Order 17536.

228 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 90 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	C₂²⋊#14555513	C₂	1	4	yes	maximal, complemented
4	C₂×#17984095	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	#17984095	—	2	2	no	—
10	#17984095	—	2	2	no	—
11	C₂×#14555494	C₄	1	2	yes	—
12	C₂×#14555494	C₄	1	1	yes	characteristic
13	C₂²×#11764040	C₂²	1	2	yes	—
14	C₂²×#11764040	C₂²	1	1	yes	characteristic
15	C₂×#41×C₁₃₇	C₄	1	1	yes	characteristic
16	#14555513	—	2	2	no	—
17	#14555513	—	2	2	no	—
18	#14555494	—	2	4	no	—
19	#14555494	—	2	2	no	—
20	C₂×#11764040	—	2	2	no	—
21	C₂×#11764040	—	2	2	no	—
22	C₂×#11764040	C₂×C₄	1	2	yes	—
23	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
24	C₂×#11764040	D₄	1	2	yes	—
25	C₂×#11764040	D₄	1	2	yes	—
26	C₂×#11764040	C₂³	1	1	yes	characteristic
27	#41×C₁₃₇	—	2	2	no	—
28	#41×C₁₃₇	—	2	2	no	—
29	C₂×Q₈×C₁₃₇	C₈	1	1	yes	characteristic
30	C₂×D₄×C₁₃₇	C₈	1	2	yes	—
31	C₂×D₄×C₁₃₇	C₈	1	1	yes	characteristic
32	C₂²×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
34	#11764040	—	2	2	no	—
35	#11764040	C₂²⋊C₄	1	2	yes	—
36	#11764040	C₂×D₄	1	1	yes	characteristic
37	#11764040	C₂×D₄	1	1	yes	characteristic
38	C₂³×C₁₃₇	C₂×C₈	1	2	yes	—
39	C₂³×C₁₃₇	C₂×C₈	1	1	yes	characteristic
40	Q₈×C₁₃₇	—	2	2	no	—
41	D₄×C₁₃₇	—	2	4	no	—
42	D₄×C₁₃₇	—	2	2	no	—
43	C₂×C₄×C₁₃₇	—	2	2	no	—
44	C₂×C₄×C₁₃₇	—	2	2	no	—
45	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₁₃₇	C₂×C₈	1	2	yes	—
47	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
48	C₂×C₄×C₁₃₇	M₁₆	1	2	yes	—
49	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₁₃₇	—	2	2	no	—
51	C₂²×C₁₃₇	—	2	2	no	—
52	C₂²×C₁₃₇	—	2	4	no	—
53	C₂²×C₁₃₇	C₂²⋊C₈	1	2	yes	—
54	C₂²×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
55	C₄×C₁₃₇	—	2	2	no	—
56	C₄×C₁₃₇	C₂²⋊C₈	1	2	yes	—
57	C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₄×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
59	C₂×C₁₃₇	—	2	2	no	—
60	C₂×C₁₃₇	—	4	2	no	—
61	C₂×C₁₃₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
62	C₂×C₁₃₇	#504	1	2	yes	—
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	#10587739	1	1	yes	characteristic
70	#131	—	274	2	no	—
71	#131	—	274	2	no	—
72	C₂×M₁₆	—	137	1	no	—
73	C₂×M₁₆	—	137	2	no	—
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	—	2	2	no	—
81	#41	—	2	2	no	—
82	C₂×Q₈	#11764047	1	1	yes	characteristic
83	C₂×D₄	#11764047	1	2	yes	—
84	C₂×D₄	#11764047	1	1	yes	characteristic
85	C₂²×C₄	C₂×#10587739	1	2	yes	—
86	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
87	M₁₆	—	274	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	2	no	—
94	C₂×C₈	—	137	1	no	—
95	C₂×C₈	—	137	1	no	—
96	C₁₆	—	274	2	no	—
97	C₁₆	—	274	2	no	—
98	C₂³	C₂×#11764047	1	2	yes	—
99	C₂³	C₂×#11764047	1	1	yes	characteristic
100	Q₈	—	2	2	no	—
101	D₄	—	2	4	no	—
102	D₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	C₂²⋊#10587739	1	2	yes	—
106	C₂×C₄	C₂×#11764047	1	2	yes	—
107	C₂×C₄	C₂×#11764047	1	1	yes	characteristic
108	C₂×C₄	#14555519	1	2	yes	—
109	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
110	C₈	—	274	2	no	—
111	C₈	—	137	1	no	—
112	C₈	—	137	1	no	—
113	C₈	—	137	2	no	—
114	C₂²	—	2	2	no	—
115	C₂²	—	2	4	no	—
116	C₂²	—	2	2	no	—
117	C₂²	C₂²⋊#11764047	1	2	yes	—
118	C₂²	C₂²×#11764047	1	1	yes	characteristic
119	C₄	—	2	2	no	—
120	C₄	C₂²⋊#11764047	1	2	yes	—
121	C₄	C₂×C₂²⋊#10587739	1	1	yes	characteristic
122	C₄	C₂×#14555519	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	4	2	no	—
125	C₂	C₂×C₂²⋊#11764047	1	1	yes	characteristic
126	C₂	#22578461	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

C₂²⋊#14555513

C₂

yes

maximal, complemented

C₂×#17984095

C₂

yes

maximal, characteristic

#17984162

—

C₂×#14555513

—

C₂×#14555513

C₂²

yes

—

C₂×#14555513

C₂²

yes

—

#17984095

—

#17984095

—

C₂×#14555494

C₄

yes

—

C₂×#14555494

C₄

yes

characteristic

C₂²×#11764040

C₂²

yes

—

C₂²×#11764040

C₂²

yes

characteristic

C₂×#41×C₁₃₇

C₄

yes

—

—

—

—

—

—

C₂×C₄

yes

—

C₂×#11764040

C₂×C₄

yes

characteristic

C₂×#11764040

D₄

yes

—

C₂×#11764040

D₄

yes

—

C₂×#11764040

C₂³

yes

characteristic

#41×C₁₃₇

—

#41×C₁₃₇

—

C₂×Q₈×C₁₃₇

C₈

yes

characteristic

C₂×D₄×C₁₃₇

C₈

yes

—

C₂×D₄×C₁₃₇

C₈

yes

characteristic

C₂²×C₄×C₁₃₇

C₂×C₄

yes

—

C₂²×C₄×C₁₃₇

C₂×C₄

yes

characteristic

#11764040

—

#11764040

C₂²⋊C₄

yes

—

#11764040

C₂×D₄

yes

characteristic

#11764040

C₂×D₄

yes

characteristic

C₂³×C₁₃₇

C₂×C₈

yes

—

C₂³×C₁₃₇

C₂×C₈

yes

characteristic

Q₈×C₁₃₇

—

D₄×C₁₃₇

—

D₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃₇

C₂×C₈

yes

—

C₂×C₄×C₁₃₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₃₇

M₁₆

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₂²⋊C₈

yes

—

C₂²×C₁₃₇

C₂²×C₈

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂²⋊C₈

yes

—

C₄×C₁₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₃₇

C₂×M₁₆

yes

characteristic

C₂×C₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₁₃₇

#504

yes

—

C₁₃₇

#2113

yes

characteristic, complemented, Sylow

#2113

—

137

maximal, Sylow

C₂×#131

—

137

—

C₂×M₃₂

—

137

—

C₂²×C₁₆

—

137

—

C₂²⋊C₁₆

—

137

—

C₂×#41

#10587739

yes

characteristic

#131

—

274

—

#131

—

274

—

C₂×M₁₆

—

137

—

C₂×M₁₆

—

137

—

C₂²×C₈

—

137

—

C₂²×C₈

—

137

—

M₃₂

—

274

—

C₂×C₁₆

—

274

—

C₂×C₁₆

—

137

—

C₂×C₁₆

—

137

—

#41

—

#41

—

C₂×Q₈

#11764047

yes

characteristic

C₂×D₄

#11764047

yes

—

C₂×D₄

#11764047

yes

characteristic

C₂²×C₄

C₂×#10587739

yes

—

C₂²×C₄

C₂×#10587739

yes

characteristic

M₁₆

—

274

—

M₁₆

—

274

—

C₂×C₈

—

274

—

C₂×C₈

—

274

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₁₆

—

274

—

C₁₆

—

274

—

C₂³

C₂×#11764047

yes

—

C₂³

C₂×#11764047

yes

characteristic

100

Q₈

—

101

D₄

—

102

D₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂²⋊#10587739

yes

—

106

C₂×C₄

C₂×#11764047

yes

—

107

C₂×C₄

C₂×#11764047

yes

characteristic

108

C₂×C₄

#14555519

yes

—

109

C₂×C₄

C₂²×#10587739

yes

characteristic

110

C₈

—

274

—

111

C₈

—

137

—

112

C₈

—

137

—

113

C₈

—

137

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

C₂²⋊#11764047

yes

—

118

C₂²

C₂²×#11764047

yes

characteristic

119

C₄

—

120

C₄

C₂²⋊#11764047

yes

—

121

C₄

C₂×C₂²⋊#10587739

yes

characteristic

122

C₄

C₂×#14555519

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

C₂×C₂²⋊#11764047

yes

characteristic

126

C₂

#22578461

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10587739
commutator, socle	C₂²×C₁₃₇	C₂²×C₈
Fitting, prime radical	C₂×#41×C₁₃₇	C₄
cosocle	C₂×#11764040	C₂³
hypercenter	C₂×#41	#10587739
hypocenter	C₁₃₇	#2113
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#41

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#10587739

commutator, socle

C₂²×C₁₃₇

C₂²×C₈

Fitting, prime radical

C₂×#41×C₁₃₇

C₄

C₂³

C₂×#41

C₁₃₇

G ▷ C₂²×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂×C₄ ◁ C₂×#41

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

H ≅ G/N

type

1. C₂×#17984162

C₂

semidirect

2. C₂²×#14555513

C₂

semidirect

3. C₂²⋊#14555513

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
C₂²⋊#14555513	2
C₂×#17984095	2
#2113	137

subgroup

index

137

Subgroups — #31797019

Conjugacy classes of subgroups

Complements

Maximal subgroups