Group #31790195 — subgroups

Subgroups — #31790195

Order 17536.

716 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 161 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	#22577974	C₂	1	3	yes	maximal, complemented
2	#22577593	C₂	1	8	yes	maximal, complemented
3	#22577377	C₂	1	3	yes	maximal, complemented
4	#585×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₇₄	C₂²	1	1	yes	characteristic, complemented
6	#17984086	—	2	3	no	—
7	#17984075	—	2	3	no	—
8	#17984069	C₂²	1	12	yes	complemented
9	#17984067	—	2	6	no	—
10	#17984047	C₂²	1	3	yes	complemented
11	#17984003	C₂²	1	12	yes	complemented
12	2₋¹⁺⁴×C₁₃₇	—	2	3	no	—
13	H(5,𝔽₂)×C₁₃₇	—	2	3	no	—
14	H(5,𝔽₂)×C₁₃₇	C₂²	1	4	yes	—
15	C₂×#41×C₁₃₇	—	2	6	no	—
16	C₂×#41×C₁₃₇	C₂²	1	3	yes	—
17	C₂×Dic₂₇₄	—	2	3	no	—
18	C₂×Dic₂₇₄	—	2	3	no	—
19	C₂²×Dic₁₃₇	C₂³	1	2	yes	—
20	C₄⋊Dic₁₃₇	—	2	3	no	—
21	#14555489	—	2	12	no	—
22	C₂²⋊Dic₁₃₇	—	2	12	no	—
23	C₂²⋊Dic₁₃₇	C₂³	1	6	yes	—
24	C₄×Dic₁₃₇	—	2	3	no	—
25	#41×C₁₃₇	—	2	4	no	—
26	#41×C₁₃₇	—	2	12	no	—
27	#41×C₁₃₇	—	2	6	no	—
28	#41×C₁₃₇	—	2	12	no	—
29	#41×C₁₃₇	—	2	6	no	—
30	C₂×Q₈×C₁₃₇	—	2	6	no	—
31	C₂×Q₈×C₁₃₇	D₄	1	3	yes	—
32	C₂×D₄×C₁₃₇	—	2	6	no	—
33	C₂×D₄×C₁₃₇	—	2	12	no	—
34	C₂×D₄×C₁₃₇	D₄	1	3	yes	—
35	C₂×D₄×C₁₃₇	C₂³	1	6	yes	—
36	C₂²×C₄×C₁₃₇	—	2	4	no	—
37	C₂²×C₄×C₁₃₇	D₄	1	6	yes	—
38	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
39	Dic₂₇₄	—	4	1	no	—
40	Dic₂₇₄	—	4	3	no	—
41	C₂×Dic₁₃₇	—	2	6	no	—
42	C₂×Dic₁₃₇	—	2	6	no	—
43	C₂³×C₁₃₇	—	2	4	no	—
44	C₂³×C₁₃₇	C₂×D₄	1	6	yes	—
45	C₂³×C₁₃₇	C₂⁴	1	1	yes	characteristic
46	Q₈×C₁₃₇	—	2	4	no	—
47	Q₈×C₁₃₇	—	2	6	no	—
48	D₄×C₁₃₇	—	2	12	no	—
49	D₄×C₁₃₇	—	2	12	no	—
50	D₄×C₁₃₇	—	2	6	no	—
51	C₂×C₄×C₁₃₇	—	2	12	no	—
52	C₂×C₄×C₁₃₇	—	2	6	no	—
53	C₂×C₄×C₁₃₇	—	2	6	no	—
54	C₂×C₄×C₁₃₇	C₂×D₄	1	6	yes	—
55	C₂×C₄×C₁₃₇	C₂×D₄	1	3	yes	—
56	C₂×C₄×C₁₃₇	C₂×D₄	1	3	yes	—
57	C₂²×C₁₃₇	—	2	6	no	—
58	C₂²×C₁₃₇	—	4	1	no	—
59	C₂²×C₁₃₇	—	4	6	no	—
60	C₂²×C₁₃₇	—	8	4	no	—
61	C₂²×C₁₃₇	C₂²×D₄	1	3	yes	—
62	C₄×C₁₃₇	—	2	6	no	—
63	C₄×C₁₃₇	#120	1	1	yes	characteristic
64	C₄×C₁₃₇	#120	1	3	yes	—
65	Dic₁₃₇	—	4	6	no	—
66	Dic₁₃₇	—	4	2	no	—
67	C₂×C₁₃₇	—	2	3	no	—
68	C₂×C₁₃₇	—	4	6	no	—
69	C₂×C₁₃₇	C₂×#120	1	1	yes	characteristic
70	C₁₃₇	#3028	1	1	yes	characteristic, complemented, Sylow
71	#3028	—	137	1	no	maximal, Sylow
72	#585	D₁₃₇	1	1	yes	characteristic
73	#536	—	137	3	no	—
74	#458	—	137	8	no	—
75	#410	—	137	3	no	—
76	2₋¹⁺⁴	—	2	3	no	—
77	H(5,𝔽₂)	—	2	3	no	—
78	H(5,𝔽₂)	C₂×D₁₃₇	1	4	yes	—
79	C₂×#41	—	2	6	no	—
80	C₂×#41	C₂×D₁₃₇	1	3	yes	—
81	C₂²×Q₈	—	137	1	no	—
82	C₄⋊Q₈	—	274	3	no	—
83	#124	—	274	3	no	—
84	#123	—	137	12	no	—
85	C₂²⋊Q₈	—	274	6	no	—
86	#117	—	137	3	no	—
87	#99	—	137	12	no	—
88	#41	—	2	4	no	—
89	#41	—	2	12	no	—
90	#41	—	2	6	no	—
91	#41	—	2	12	no	—
92	#41	—	2	6	no	—
93	C₂×Q₈	—	2	6	no	—
94	C₂×Q₈	—	274	3	no	—
95	C₂×Q₈	—	274	3	no	—
96	C₂×Q₈	C₂²⋊D₁₃₇	1	3	yes	—
97	C₂×D₄	—	2	6	no	—
98	C₂×D₄	—	2	12	no	—
99	C₂×D₄	C₂²⋊D₁₃₇	1	3	yes	—
100	C₂×D₄	C₂²×D₁₃₇	1	6	yes	—
101	C₂²×C₄	—	2	4	no	—
102	C₂²×C₄	—	137	2	no	—
103	C₂²×C₄	C₂²⋊D₁₃₇	1	6	yes	—
104	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
105	C₄⋊C₄	—	274	12	no	—
106	C₄⋊C₄	—	274	3	no	—
107	C₂²⋊C₄	—	274	12	no	—
108	C₂²⋊C₄	—	137	6	no	—
109	C₄²	—	274	3	no	—
110	C₂³	—	2	4	no	—
111	C₂³	C₂×C₂²⋊D₁₃₇	1	6	yes	—
112	C₂³	C₂³×D₁₃₇	1	1	yes	characteristic
113	Q₈	—	2	4	no	—
114	Q₈	—	2	6	no	—
115	Q₈	—	548	1	no	—
116	Q₈	—	548	3	no	—
117	D₄	—	2	12	no	—
118	D₄	—	2	12	no	—
119	D₄	—	2	6	no	—
120	C₂×C₄	—	2	12	no	—
121	C₂×C₄	—	2	6	no	—
122	C₂×C₄	—	2	6	no	—
123	C₂×C₄	—	274	6	no	—
124	C₂×C₄	—	274	6	no	—
125	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	6	yes	—
126	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	3	yes	—
127	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	3	yes	—
128	C₂²	—	2	6	no	—
129	C₂²	—	4	1	no	—
130	C₂²	—	4	6	no	—
131	C₂²	—	8	4	no	—
132	C₂²	C₂²×C₂²⋊D₁₃₇	1	3	yes	—
133	C₄	—	2	6	no	—
134	C₄	—	548	6	no	—
135	C₄	—	548	2	no	—
136	C₄	#17984057	1	1	yes	characteristic
137	C₄	#17984057	1	3	yes	—
138	C₂	—	2	3	no	—
139	C₂	—	4	6	no	—
140	C₂	C₂×#17984057	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22577974

C₂

yes

maximal, complemented

#22577593

C₂

yes

maximal, complemented

#22577377

C₂

yes

maximal, complemented

#585×C₁₃₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₇₄

C₂²

yes

characteristic, complemented

#17984086

—

#17984075

—

#17984069

C₂²

yes

complemented

#17984067

—

#17984047

C₂²

yes

complemented

#17984003

C₂²

yes

complemented

2₋¹⁺⁴×C₁₃₇

—

H(5,𝔽₂)×C₁₃₇

—

H(5,𝔽₂)×C₁₃₇

C₂²

yes

—

C₂×#41×C₁₃₇

—

C₂×#41×C₁₃₇

C₂²

yes

—

C₂×Dic₂₇₄

—

C₂×Dic₂₇₄

—

C₂²×Dic₁₃₇

C₂³

yes

—

C₄⋊Dic₁₃₇

—

#14555489

—

C₂²⋊Dic₁₃₇

—

C₂²⋊Dic₁₃₇

C₂³

yes

—

C₄×Dic₁₃₇

—

#41×C₁₃₇

—

#41×C₁₃₇

—

#41×C₁₃₇

—

#41×C₁₃₇

—

#41×C₁₃₇

—

C₂×Q₈×C₁₃₇

—

C₂×Q₈×C₁₃₇

D₄

yes

—

C₂×D₄×C₁₃₇

—

C₂×D₄×C₁₃₇

—

C₂×D₄×C₁₃₇

D₄

yes

—

C₂×D₄×C₁₃₇

C₂³

yes

—

C₂²×C₄×C₁₃₇

—

C₂²×C₄×C₁₃₇

D₄

yes

—

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

Dic₂₇₄

—

Dic₂₇₄

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂³×C₁₃₇

—

C₂³×C₁₃₇

C₂×D₄

yes

—

C₂³×C₁₃₇

C₂⁴

yes

characteristic

Q₈×C₁₃₇

—

Q₈×C₁₃₇

—

D₄×C₁₃₇

—

D₄×C₁₃₇

—

D₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₂²×D₄

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

#120

yes

characteristic

C₄×C₁₃₇

#120

yes

—

Dic₁₃₇

—

Dic₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

C₂×#120

yes

characteristic

C₁₃₇

#3028

yes

characteristic, complemented, Sylow

#3028

—

137

maximal, Sylow

#585

D₁₃₇

yes

characteristic

#536

—

137

—

#458

—

137

—

#410

—

137

—

2₋¹⁺⁴

—

H(5,𝔽₂)

—

H(5,𝔽₂)

C₂×D₁₃₇

yes

—

C₂×#41

—

C₂×#41

C₂×D₁₃₇

yes

—

C₂²×Q₈

—

137

—

C₄⋊Q₈

—

274

—

#124

—

274

—

#123

—

137

—

C₂²⋊Q₈

—

274

—

#117

—

137

—

#99

—

137

—

—

—

—

—

—

C₂×Q₈

—

C₂×Q₈

—

274

—

C₂×Q₈

—

274

—

C₂×Q₈

C₂²⋊D₁₃₇

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₁₃₇

yes

—

100

C₂×D₄

C₂²×D₁₃₇

yes

—

101

C₂²×C₄

—

102

C₂²×C₄

—

137

—

103

C₂²×C₄

C₂²⋊D₁₃₇

yes

—

104

C₂²×C₄

C₂²×D₁₃₇

yes

characteristic

105

C₄⋊C₄

—

274

—

106

C₄⋊C₄

—

274

—

107

C₂²⋊C₄

—

274

—

108

C₂²⋊C₄

—

137

—

109

C₄²

—

274

—

110

C₂³

—

111

C₂³

C₂×C₂²⋊D₁₃₇

yes

—

112

C₂³

C₂³×D₁₃₇

yes

characteristic

113

Q₈

—

114

Q₈

—

115

Q₈

—

548

—

116

Q₈

—

548

—

117

D₄

—

118

D₄

—

119

D₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

274

—

124

C₂×C₄

—

274

—

125

C₂×C₄

C₂×C₂²⋊D₁₃₇

yes

—

126

C₂×C₄

C₂×C₂²⋊D₁₃₇

yes

—

127

C₂×C₄

C₂×C₂²⋊D₁₃₇

yes

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂²×C₂²⋊D₁₃₇

yes

—

133

C₄

—

134

C₄

—

548

—

135

C₄

—

548

—

136

C₄

#17984057

yes

characteristic

137

C₄

#17984057

yes

—

138

C₂

—

139

C₂

—

140

C₂

C₂×#17984057

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#17984057
commutator, cosocle	C₂³×C₁₃₇	C₂⁴
Frattini	C₂³	C₂³×D₁₃₇
Fitting, prime radical	#585×C₁₃₇	C₂
socle	C₂×C₁₃₇	C₂×#120
hypercenter	#585	D₁₃₇
hypocenter	C₁₃₇	#3028
LCS	G ▷ C₂³×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #585

subgroup

G/H

center

C₂

C₂×#17984057

commutator, cosocle

C₂³×C₁₃₇

C₂⁴

Frattini

C₂³

C₂³×D₁₃₇

Fitting, prime radical

#585×C₁₃₇

C₂

C₂×C₁₃₇

D₁₃₇

C₁₃₇

G ▷ C₂³×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #585

N	H ≅ G/N	type
1. #22577974	C₂	semidirect
2. #22577593	C₂	semidirect
3. #22577377	C₂	semidirect
5. C₂²×Dic₂₇₄	C₂²	semidirect
8. #17984069	C₂²	semidirect
10. #17984047	C₂²	semidirect
11. #17984003	C₂²	semidirect
70. C₁₃₇	#3028	semidirect

H ≅ G/N

type

1. #22577974

C₂

semidirect

2. #22577593

C₂

semidirect

3. #22577377

C₂

semidirect

5. C₂²×Dic₂₇₄

C₂²

semidirect

8. #17984069

C₂²

semidirect

10. #17984047

C₂²

semidirect

11. #17984003

C₂²

semidirect

70. C₁₃₇

#3028

semidirect

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

subgroup	index
#22577974	2
#22577593	2
#22577377	2
#585×C₁₃₇	2
#3028	137

subgroup

index

#585×C₁₃₇

137

Subgroups — #31790195

Conjugacy classes of subgroups

Complements

Maximal subgroups