Group #31789856 — subgroups

Subgroups — #31789856

Order 17536.

410 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 205 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	#22577481	C₂	1	4	yes	maximal, complemented
2	D₄×#11764040	C₂	1	4	yes	maximal, complemented
3	#22577456	C₂	1	2	yes	maximal, complemented
4	#22577370	C₂	1	4	yes	maximal, complemented
5	#532×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555494	C₂²	1	4	yes	—
7	C₂²×#11764040	C₂²	1	4	yes	—
8	C₄⋊#11764040	C₂²	1	2	yes	—
9	C₄⋊#11764040	C₂²	1	2	yes	—
10	C₂²⋊#11764040	C₂²	1	4	yes	—
11	C₂²⋊#11764040	C₂²	1	8	yes	—
12	C₄×#11764040	C₂²	1	4	yes	—
13	C₂×#41×C₁₃₇	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₁₃₇	C₄	1	1	yes	characteristic
15	#127×C₁₃₇	C₄	1	1	yes	characteristic
16	#124×C₁₃₇	C₄	1	2	yes	—
17	#121×C₁₃₇	C₄	1	4	yes	—
18	C₄×D₄×C₁₃₇	C₂²	1	4	yes	—
19	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
20	#14555494	—	4	4	no	—
21	C₂×#11764040	—	4	4	no	—
22	C₂×#11764040	C₂³	1	4	yes	—
23	C₂×#11764040	C₂³	1	4	yes	—
24	#41×C₁₃₇	—	4	2	no	—
25	#41×C₁₃₇	—	4	2	no	—
26	C₂×Q₈×C₁₃₇	C₂×C₄	1	2	yes	—
27	C₂×D₄×C₁₃₇	C₂×C₄	1	4	yes	—
28	C₂×D₄×C₁₃₇	C₂×C₄	1	4	yes	—
29	C₂×D₄×C₁₃₇	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₁₃₇	C₂³	1	4	yes	—
33	C₄⋊C₄×C₁₃₇	C₂×C₄	1	4	yes	—
34	C₂²⋊C₄×C₁₃₇	C₂×C₄	1	8	yes	—
35	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₁₃₇	C₂³	1	2	yes	—
38	#11764040	—	2	4	no	—
39	#11764040	—	2	4	no	—
40	C₂³×C₁₃₇	C₂²×C₄	1	4	yes	—
41	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
42	Q₈×C₁₃₇	—	4	2	no	—
43	D₄×C₁₃₇	—	2	8	no	—
44	D₄×C₁₃₇	—	4	2	no	—
45	D₄×C₁₃₇	—	4	4	no	—
46	C₂×C₄×C₁₃₇	—	2	4	no	—
47	C₂×C₄×C₁₃₇	—	2	1	no	—
48	C₂×C₄×C₁₃₇	—	2	1	no	—
49	C₂×C₄×C₁₃₇	—	2	4	no	—
50	C₂×C₄×C₁₃₇	—	2	4	no	—
51	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
53	C₂×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
54	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
56	C₂²×C₁₃₇	—	2	1	no	—
57	C₂²×C₁₃₇	—	2	4	no	—
58	C₂²×C₁₃₇	—	2	2	no	—
59	C₂²×C₁₃₇	—	2	8	no	—
60	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
61	C₄×C₁₃₇	—	2	1	no	—
62	C₄×C₁₃₇	—	2	4	no	—
63	C₄×C₁₃₇	—	2	2	no	—
64	C₄×C₁₃₇	#131	1	4	yes	—
65	C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
66	C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
67	C₂×C₁₃₇	—	4	1	no	—
68	C₂×C₁₃₇	—	4	4	no	—
69	C₂×C₁₃₇	#518	1	1	yes	characteristic
70	C₂×C₁₃₇	C₂×#131	1	2	yes	—
71	C₁₃₇	#2979	1	1	yes	characteristic, complemented, Sylow
72	#2979	—	137	1	no	maximal, Sylow
73	#532	D₁₃₇	1	1	yes	characteristic
74	#436	—	137	4	no	—
75	C₈×D₄	—	137	4	no	—
76	#431	—	137	2	no	—
77	#408	—	137	4	no	—
78	C₂×#41	C₂×D₁₃₇	1	2	yes	—
79	C₂×M₁₆	—	137	4	no	—
80	C₂²×C₈	—	137	4	no	—
81	C₄⋊Q₈	Dic₁₃₇	1	1	yes	characteristic
82	#127	Dic₁₃₇	1	1	yes	characteristic
83	#124	Dic₁₃₇	1	2	yes	—
84	#121	Dic₁₃₇	1	4	yes	—
85	C₄×D₄	C₂×D₁₃₇	1	4	yes	—
86	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
87	C₄⋊C₈	—	137	2	no	—
88	C₄⋊C₈	—	137	2	no	—
89	C₂²⋊C₈	—	137	4	no	—
90	C₂²⋊C₈	—	137	8	no	—
91	C₄×C₈	—	137	4	no	—
92	#41	—	4	2	no	—
93	#41	—	4	2	no	—
94	C₂×Q₈	C₂×Dic₁₃₇	1	2	yes	—
95	C₂×D₄	C₂×Dic₁₃₇	1	4	yes	—
96	C₂×D₄	C₂×Dic₁₃₇	1	4	yes	—
97	C₂×D₄	C₂×Dic₁₃₇	1	2	yes	—
98	C₂²×C₄	C₂×Dic₁₃₇	1	2	yes	—
99	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
100	C₂²×C₄	C₂²×D₁₃₇	1	4	yes	—
101	M₁₆	—	548	4	no	—
102	C₂×C₈	—	548	4	no	—
103	C₂×C₈	—	137	4	no	—
104	C₂×C₈	—	137	4	no	—
105	C₄⋊C₄	C₂×Dic₁₃₇	1	4	yes	—
106	C₂²⋊C₄	C₂×Dic₁₃₇	1	8	yes	—
107	C₄²	C₂×Dic₁₃₇	1	1	yes	characteristic
108	C₄²	C₂×Dic₁₃₇	1	1	yes	characteristic
109	C₄²	C₂²×D₁₃₇	1	2	yes	—
110	C₂³	C₂²×Dic₁₃₇	1	4	yes	—
111	C₂³	C₂²×Dic₁₃₇	1	1	yes	characteristic
112	Q₈	—	4	2	no	—
113	D₄	—	2	8	no	—
114	D₄	—	4	2	no	—
115	D₄	—	4	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	C₂²×Dic₁₃₇	1	1	yes	characteristic
122	C₂×C₄	C₂²×Dic₁₃₇	1	2	yes	—
123	C₂×C₄	C₂²×Dic₁₃₇	1	4	yes	—
124	C₂×C₄	C₂²×Dic₁₃₇	1	2	yes	—
125	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
126	C₈	—	274	4	no	—
127	C₈	—	274	4	no	—
128	C₂²	—	2	1	no	—
129	C₂²	—	2	4	no	—
130	C₂²	—	2	2	no	—
131	C₂²	—	2	8	no	—
132	C₂²	C₂³×Dic₁₃₇	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	4	no	—
135	C₄	—	2	2	no	—
136	C₄	#17984095	1	4	yes	—
137	C₄	#17984133	1	1	yes	characteristic
138	C₄	#17984133	1	1	yes	characteristic
139	C₂	—	4	1	no	—
140	C₂	—	4	4	no	—
141	C₂	#22577878	1	1	yes	characteristic
142	C₂	C₂×#17984095	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22577481

C₂

yes

maximal, complemented

D₄×#11764040

C₂

yes

maximal, complemented

#22577456

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

—

C₄⋊#11764040

C₂²

yes

—

C₂²⋊#11764040

C₂²

yes

—

C₂²⋊#11764040

C₂²

yes

—

C₄×#11764040

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

—

C₂×#11764040

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

—

#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

—

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

#131

yes

—

C₄×C₁₃₇

yes

C₄×C₁₃₇

yes

C₂×C₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

#518

yes

characteristic

C₂×C₁₃₇

C₂×#131

yes

—

C₁₃₇

#2979

yes

characteristic, complemented, Sylow

#2979

—

137

maximal, Sylow

#532

D₁₃₇

yes

characteristic

#436

—

137

—

C₈×D₄

—

137

—

#431

—

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

—

C₄⋊C₈

—

137

—

C₂²⋊C₈

—

137

—

C₂²⋊C₈

—

137

—

C₄×C₈

—

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

100

C₂²×C₄

C₂²×D₁₃₇

yes

—

101

M₁₆

—

548

—

102

C₂×C₈

—

548

—

103

C₂×C₈

—

137

—

104

C₂×C₈

—

137

—

105

C₄⋊C₄

C₂×Dic₁₃₇

yes

—

106

C₂²⋊C₄

C₂×Dic₁₃₇

yes

—

107

C₄²

C₂×Dic₁₃₇

yes

characteristic

108

C₄²

C₂×Dic₁₃₇

yes

characteristic

109

C₄²

C₂²×D₁₃₇

yes

—

110

C₂³

C₂²×Dic₁₃₇

yes

—

111

C₂³

C₂²×Dic₁₃₇

yes

characteristic

112

Q₈

—

113

D₄

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

C₂²×Dic₁₃₇

yes

characteristic

122

C₂×C₄

C₂²×Dic₁₃₇

yes

—

123

C₂×C₄

C₂²×Dic₁₃₇

yes

—

124

C₂×C₄

C₂²×Dic₁₃₇

yes

—

125

C₂×C₄

C₂³×D₁₃₇

yes

characteristic

126

C₈

—

274

—

127

C₈

—

274

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂³×Dic₁₃₇

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#17984095

yes

—

137

C₄

#17984133

yes

characteristic

138

C₄

#17984133

yes

characteristic

139

C₂

—

140

C₂

—

141

C₂

#22577878

yes

characteristic

142

C₂

C₂×#17984095

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	#2979
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. #22577481	C₂	semidirect
2. D₄×#11764040	C₂	semidirect
3. #22577456	C₂	semidirect
4. #22577370	C₂	semidirect
71. C₁₃₇	#2979	semidirect

H ≅ G/N

type

1. #22577481

C₂

semidirect

2. D₄×#11764040

C₂

semidirect

3. #22577456

C₂

semidirect

4. #22577370

C₂

semidirect

71. C₁₃₇

#2979

semidirect

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

subgroup	index
#22577481	2
D₄×#11764040	2
#22577456	2
#22577370	2
#532×C₁₃₇	2
#2979	137

subgroup

index

#532×C₁₃₇

137

Subgroups — #31789856

Conjugacy classes of subgroups

Complements

Maximal subgroups