Group #31789860 — subgroups

Subgroups — #31789860

Order 17536.

394 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 197 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	#22577475	C₂	1	8	yes	maximal, complemented
2	#22577460	C₂	1	2	yes	maximal, complemented
3	#22577370	C₂	1	4	yes	maximal, complemented
4	#532×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14555494	C₂²	1	4	yes	—
6	C₂²×#11764040	C₂²	1	4	yes	—
7	C₄⋊#11764040	C₂²	1	4	yes	—
8	C₂²⋊#11764040	C₂²	1	4	yes	—
9	C₂²⋊#11764040	C₂²	1	8	yes	—
10	#17983999	C₂²	1	4	yes	—
11	C₂×#41×C₁₃₇	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₁₃₇	C₄	1	1	yes	characteristic
13	#127×C₁₃₇	C₄	1	1	yes	characteristic
14	#124×C₁₃₇	C₄	1	2	yes	—
15	#121×C₁₃₇	C₄	1	4	yes	—
16	C₄×D₄×C₁₃₇	C₂²	1	4	yes	—
17	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
18	#14555494	—	4	4	no	—
19	C₂×#11764040	—	4	4	no	—
20	C₂×#11764040	C₂³	1	8	yes	—
21	#41×C₁₃₇	—	4	2	no	—
22	#41×C₁₃₇	—	4	2	no	—
23	C₂×Q₈×C₁₃₇	C₂×C₄	1	2	yes	—
24	C₂×D₄×C₁₃₇	C₂×C₄	1	4	yes	—
25	C₂×D₄×C₁₃₇	C₂×C₄	1	4	yes	—
26	C₂×D₄×C₁₃₇	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₁₃₇	C₂³	1	4	yes	—
30	C₄⋊C₄×C₁₃₇	C₂×C₄	1	4	yes	—
31	C₂²⋊C₄×C₁₃₇	C₂×C₄	1	8	yes	—
32	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₁₃₇	C₂³	1	2	yes	—
35	#11764040	—	2	8	no	—
36	C₂³×C₁₃₇	C₂²×C₄	1	4	yes	—
37	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
38	Q₈×C₁₃₇	—	4	2	no	—
39	D₄×C₁₃₇	—	4	4	no	—
40	D₄×C₁₃₇	—	4	2	no	—
41	D₄×C₁₃₇	—	4	4	no	—
42	C₂×C₄×C₁₃₇	—	2	2	no	—
43	C₂×C₄×C₁₃₇	—	2	4	no	—
44	C₂×C₄×C₁₃₇	—	2	4	no	—
45	C₂×C₄×C₁₃₇	—	4	2	no	—
46	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
48	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
50	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
51	C₂²×C₁₃₇	—	2	1	no	—
52	C₂²×C₁₃₇	—	2	4	no	—
53	C₂²×C₁₃₇	—	2	2	no	—
54	C₂²×C₁₃₇	—	2	8	no	—
55	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
56	C₄×C₁₃₇	—	2	1	no	—
57	C₄×C₁₃₇	—	2	4	no	—
58	C₄×C₁₃₇	—	2	2	no	—
59	C₄×C₁₃₇	—	2	2	no	—
60	C₄×C₁₃₇	H(5,𝔽₂)	1	2	yes	—
61	C₂×C₁₃₇	—	4	1	no	—
62	C₂×C₁₃₇	—	4	4	no	—
63	C₂×C₁₃₇	#518	1	1	yes	characteristic
64	C₂×C₁₃₇	#568	1	2	yes	—
65	C₁₃₇	#2980	1	1	yes	characteristic, complemented, Sylow
66	#2980	—	137	1	no	maximal, Sylow
67	#532	D₁₃₇	1	1	yes	characteristic
68	#435	—	137	8	no	—
69	#432	—	137	2	no	—
70	#408	—	137	4	no	—
71	C₂×#41	C₂×D₁₃₇	1	2	yes	—
72	C₂×M₁₆	—	137	4	no	—
73	C₂²×C₈	—	137	4	no	—
74	C₄⋊Q₈	Dic₁₃₇	1	1	yes	characteristic
75	#127	Dic₁₃₇	1	1	yes	characteristic
76	#124	Dic₁₃₇	1	2	yes	—
77	#121	Dic₁₃₇	1	4	yes	—
78	C₄×D₄	C₂×D₁₃₇	1	4	yes	—
79	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
80	C₄⋊C₈	—	137	4	no	—
81	C₂²⋊C₈	—	137	4	no	—
82	C₂²⋊C₈	—	137	8	no	—
83	#97	—	137	4	no	—
84	#41	—	4	2	no	—
85	#41	—	4	2	no	—
86	C₂×Q₈	C₂×Dic₁₃₇	1	2	yes	—
87	C₂×D₄	C₂×Dic₁₃₇	1	4	yes	—
88	C₂×D₄	C₂×Dic₁₃₇	1	4	yes	—
89	C₂×D₄	C₂×Dic₁₃₇	1	2	yes	—
90	C₂²×C₄	C₂×Dic₁₃₇	1	2	yes	—
91	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
92	C₂²×C₄	C₂²×D₁₃₇	1	4	yes	—
93	M₁₆	—	548	4	no	—
94	C₂×C₈	—	548	4	no	—
95	C₂×C₈	—	137	8	no	—
96	C₄⋊C₄	C₂×Dic₁₃₇	1	4	yes	—
97	C₂²⋊C₄	C₂×Dic₁₃₇	1	8	yes	—
98	C₄²	C₂×Dic₁₃₇	1	1	yes	characteristic
99	C₄²	C₂×Dic₁₃₇	1	1	yes	characteristic
100	C₄²	C₂²×D₁₃₇	1	2	yes	—
101	C₂³	C₂²×Dic₁₃₇	1	4	yes	—
102	C₂³	C₂²×Dic₁₃₇	1	1	yes	characteristic
103	Q₈	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	4	4	no	—
106	D₄	—	4	4	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	C₂²×Dic₁₃₇	1	1	yes	characteristic
112	C₂×C₄	C₂²×Dic₁₃₇	1	2	yes	—
113	C₂×C₄	C₂²×Dic₁₃₇	1	2	yes	—
114	C₂×C₄	C₂²×Dic₁₃₇	1	4	yes	—
115	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
116	C₈	—	274	8	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	4	no	—
119	C₂²	—	2	2	no	—
120	C₂²	—	2	8	no	—
121	C₂²	C₂³×Dic₁₃₇	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	2	4	no	—
124	C₄	—	2	2	no	—
125	C₄	—	2	2	no	—
126	C₄	#17984133	1	2	yes	—
127	C₂	—	4	1	no	—
128	C₂	—	4	4	no	—
129	C₂	#22577878	1	1	yes	characteristic
130	C₂	#22578170	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22577475

C₂

yes

maximal, complemented

#22577460

C₂

yes

maximal, complemented

#22577370

C₂

yes

maximal, complemented

#532×C₁₃₇

C₂

yes

maximal, characteristic

C₂×#14555494

C₂²

yes

—

C₂²×#11764040

C₂²

yes

—

C₄⋊#11764040

C₂²

yes

—

C₂²⋊#11764040

C₂²

yes

—

C₂²⋊#11764040

C₂²

yes

—

#17983999

C₂²

yes

—

C₂×#41×C₁₃₇

C₂²

yes

—

C₄⋊Q₈×C₁₃₇

C₄

yes

characteristic

#127×C₁₃₇

C₄

yes

characteristic

#124×C₁₃₇

C₄

yes

—

#121×C₁₃₇

C₄

yes

—

C₄×D₄×C₁₃₇

C₂²

yes

—

C₂×C₄²×C₁₃₇

C₂²

yes

—

—

C₂³

yes

—

#41×C₁₃₇

—

#41×C₁₃₇

—

C₂×Q₈×C₁₃₇

C₂×C₄

yes

—

C₂×D₄×C₁₃₇

C₂×C₄

yes

—

C₂×D₄×C₁₃₇

C₂×C₄

yes

—

C₂×D₄×C₁₃₇

C₂×C₄

yes

—

C₂²×C₄×C₁₃₇

C₂×C₄

yes

—

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

C₂²×C₄×C₁₃₇

C₂³

yes

—

C₄⋊C₄×C₁₃₇

C₂×C₄

yes

—

C₂²⋊C₄×C₁₃₇

C₂×C₄

yes

—

C₄²×C₁₃₇

C₂×C₄

yes

characteristic

C₄²×C₁₃₇

C₂×C₄

yes

characteristic

C₄²×C₁₃₇

C₂³

yes

—

#11764040

—

C₂³×C₁₃₇

C₂²×C₄

yes

—

C₂³×C₁₃₇

C₂²×C₄

yes

characteristic

Q₈×C₁₃₇

—

D₄×C₁₃₇

—

D₄×C₁₃₇

—

D₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂²×C₄

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

—

C₂×C₄×C₁₃₇

C₂⁴

yes

characteristic

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₂³×C₄

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

H(5,𝔽₂)

yes

—

C₂×C₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

#518

yes

characteristic

C₂×C₁₃₇

#568

yes

—

C₁₃₇

#2980

yes

characteristic, complemented, Sylow

#2980

—

137

maximal, Sylow

#532

D₁₃₇

yes

characteristic

#435

—

137

—

#432

—

137

—

#408

—

137

—

C₂×#41

C₂×D₁₃₇

yes

—

C₂×M₁₆

—

137

—

C₂²×C₈

—

137

—

C₄⋊Q₈

Dic₁₃₇

yes

characteristic

#127

Dic₁₃₇

yes

characteristic

#124

Dic₁₃₇

yes

—

#121

Dic₁₃₇

yes

—

C₄×D₄

C₂×D₁₃₇

yes

—

C₂×C₄²

C₂×D₁₃₇

yes

characteristic

C₄⋊C₈

—

137

—

C₂²⋊C₈

—

137

—

C₂²⋊C₈

—

137

—

#97

—

137

—

#41

—

#41

—

C₂×Q₈

C₂×Dic₁₃₇

yes

—

C₂×D₄

C₂×Dic₁₃₇

yes

—

C₂×D₄

C₂×Dic₁₃₇

yes

—

C₂×D₄

C₂×Dic₁₃₇

yes

—

C₂²×C₄

C₂×Dic₁₃₇

yes

—

C₂²×C₄

C₂²×D₁₃₇

yes

characteristic

C₂²×C₄

C₂²×D₁₃₇

yes

—

M₁₆

—

548

—

C₂×C₈

—

548

—

C₂×C₈

—

137

—

C₄⋊C₄

C₂×Dic₁₃₇

yes

—

C₂²⋊C₄

C₂×Dic₁₃₇

yes

—

C₄²

C₂×Dic₁₃₇

yes

characteristic

C₄²

C₂×Dic₁₃₇

yes

characteristic

100

C₄²

C₂²×D₁₃₇

yes

—

101

C₂³

C₂²×Dic₁₃₇

yes

—

102

C₂³

C₂²×Dic₁₃₇

yes

characteristic

103

Q₈

—

104

D₄

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂²×Dic₁₃₇

yes

characteristic

112

C₂×C₄

C₂²×Dic₁₃₇

yes

—

113

C₂×C₄

C₂²×Dic₁₃₇

yes

—

114

C₂×C₄

C₂²×Dic₁₃₇

yes

—

115

C₂×C₄

C₂³×D₁₃₇

yes

characteristic

116

C₈

—

274

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂³×Dic₁₃₇

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#17984133

yes

—

127

C₂

—

128

C₂

—

129

C₂

#22577878

yes

characteristic

130

C₂

#22578170

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃₇
commutator, socle	C₂²×C₁₃₇	C₂³×C₄
Fitting, prime radical	#532×C₁₃₇	C₂
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	#532	D₁₃₇
hypocenter	C₁₃₇	#2980
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ #532

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₃₇

commutator, socle

C₂²×C₁₃₇

C₂³×C₄

Fitting, prime radical

#532×C₁₃₇

C₂

cosocle

C₂×C₄×C₁₃₇

C₂⁴

D₁₃₇

C₁₃₇

G ▷ C₂²×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂×C₄ ◁ #532

N	H ≅ G/N	type
1. #22577475	C₂	semidirect
2. #22577460	C₂	semidirect
3. #22577370	C₂	semidirect
65. C₁₃₇	#2980	semidirect

H ≅ G/N

type

1. #22577475

C₂

semidirect

2. #22577460

C₂

semidirect

3. #22577370

C₂

semidirect

65. C₁₃₇

#2980

semidirect

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

subgroup	index
#22577475	2
#22577460	2
#22577370	2
#532×C₁₃₇	2
#2980	137

subgroup

index

#532×C₁₃₇

137

Subgroups — #31789860

Conjugacy classes of subgroups

Complements

Maximal subgroups