Group #31797723 — subgroups

Subgroups — #31797723

Order 17536.

394 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 168 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	Q₈×#11764047	C₂	1	2	yes	maximal, complemented
2	#22578437	C₂	1	6	yes	maximal, complemented
3	#22578422	C₂	1	3	yes	maximal, complemented
4	#22578407	C₂	1	3	yes	maximal, complemented
5	C₄×#14555511	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555520	C₂²	1	6	yes	—
7	C₄⋊#11764047	C₂²	1	6	yes	—
8	C₄⋊#11764047	C₂²	1	3	yes	—
9	C₄⋊#11764047	C₂²	1	1	yes	characteristic
10	#17984145	C₂²	1	6	yes	—
11	C₄×#11764047	C₂²	1	6	yes	—
12	C₂×#14555511	C₄	1	1	yes	characteristic
13	Q₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
14	C₄×D₅₄₈	C₄	1	3	yes	—
15	#17984050	C₂²	1	3	yes	—
16	#17984046	C₄	1	3	yes	—
17	C₄²×D₁₃₇	C₂²	1	3	yes	—
18	C₄×Q₈×C₁₃₇	C₄	1	1	yes	characteristic
19	#14555520	—	4	6	no	—
20	C₂×#11764047	C₂³	1	2	yes	—
21	C₂×#11764047	C₂³	1	6	yes	—
22	#14555511	—	2	4	no	—
23	C₂×D₅₄₈	C₂×C₄	1	3	yes	—
24	C₂×C₄×D₁₃₇	C₂×C₄	1	3	yes	—
25	C₂×C₄×D₁₃₇	C₂×C₄	1	3	yes	—
26	C₂×C₄×D₁₃₇	C₂³	1	3	yes	—
27	C₄⋊Dic₁₃₇	C₂×C₄	1	3	yes	—
28	#14555487	C₂×C₄	1	6	yes	—
29	C₄×Dic₁₃₇	C₂×C₄	1	3	yes	—
30	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
31	C₄×Dic₁₃₇	C₂³	1	3	yes	—
32	C₂×Q₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
33	C₄⋊C₄×C₁₃₇	C₂×C₄	1	3	yes	—
34	C₄²×C₁₃₇	C₂×C₄	1	3	yes	—
35	#11764047	—	2	6	no	—
36	#11764047	—	2	2	no	—
37	C₂²×D₁₃₇	C₂²×C₄	1	3	yes	—
38	D₅₄₈	—	2	6	no	—
39	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×Dic₁₃₇	C₂²×C₄	1	3	yes	—
41	C₂×Dic₁₃₇	C₂⁴	1	1	yes	characteristic
42	C₄×D₁₃₇	—	2	6	no	—
43	C₄×D₁₃₇	—	2	6	no	—
44	C₄×D₁₃₇	—	2	3	no	—
45	C₄×D₁₃₇	—	2	3	no	—
46	Q₈×C₁₃₇	M₁₆	1	4	yes	complemented
47	C₂×C₄×C₁₃₇	C₂²×C₄	1	3	yes	—
48	C₂×C₄×C₁₃₇	C₂²×C₄	1	3	yes	—
49	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
51	C₂×D₁₃₇	—	2	3	no	—
52	C₂×D₁₃₇	—	2	3	no	—
53	C₂×D₁₃₇	—	2	3	no	—
54	C₄×C₁₃₇	—	2	3	no	—
55	C₄×C₁₃₇	C₂×M₁₆	1	6	yes	—
56	C₄×C₁₃₇	#131	1	2	yes	—
57	Dic₁₃₇	—	2	1	no	—
58	Dic₁₃₇	—	2	3	no	—
59	Dic₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
60	Dic₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
61	C₂×C₁₃₇	#520	1	1	yes	characteristic
62	C₂×C₁₃₇	C₂²×M₁₆	1	1	yes	characteristic
63	C₂×C₁₃₇	C₂×#131	1	1	yes	characteristic
64	D₁₃₇	—	4	3	no	—
65	C₁₃₇	#2991	1	1	yes	characteristic, complemented, Sylow
66	#2991	—	137	1	no	maximal, Sylow
67	C₄×#41	—	137	1	no	—
68	C₈×Q₈	—	137	2	no	—
69	#436	—	137	6	no	—
70	#431	—	137	3	no	—
71	#423	—	137	3	no	—
72	C₂×#41	—	137	1	no	—
73	C₂×M₁₆	—	137	6	no	—
74	C₄×Q₈	—	137	1	no	—
75	C₄×Q₈	#10587739	1	1	yes	characteristic
76	C₄×D₄	—	137	3	no	—
77	C₄×D₄	—	137	3	no	—
78	#117	—	137	3	no	—
79	C₂×C₄²	—	137	3	no	—
80	C₄⋊C₈	—	137	3	no	—
81	C₄⋊C₈	—	137	6	no	—
82	C₄⋊C₈	—	137	1	no	—
83	C₂²⋊C₈	—	137	6	no	—
84	C₄×C₈	—	137	6	no	—
85	#41	—	274	4	no	—
86	C₂×Q₈	C₂×#10587739	1	1	yes	characteristic
87	C₂×D₄	—	137	3	no	—
88	C₂²×C₄	—	137	3	no	—
89	C₂²×C₄	—	137	3	no	—
90	C₂²×C₄	—	137	3	no	—
91	M₁₆	—	548	6	no	—
92	C₂×C₈	—	137	2	no	—
93	C₂×C₈	—	137	6	no	—
94	C₄⋊C₄	—	137	3	no	—
95	C₄⋊C₄	C₂×#10587739	1	3	yes	—
96	C₂²⋊C₄	—	137	6	no	—
97	C₄²	—	137	3	no	—
98	C₄²	—	137	3	no	—
99	C₄²	—	137	1	no	—
100	C₄²	C₂×#10587739	1	3	yes	—
101	C₂³	—	137	3	no	—
102	Q₈	#14555520	1	4	yes	complemented
103	D₄	—	274	6	no	—
104	C₂×C₄	—	274	3	no	—
105	C₂×C₄	—	274	3	no	—
106	C₂×C₄	—	274	6	no	—
107	C₂×C₄	—	274	6	no	—
108	C₂×C₄	—	137	1	no	—
109	C₂×C₄	—	137	3	no	—
110	C₂×C₄	—	137	1	no	—
111	C₂×C₄	C₂²×#10587739	1	3	yes	—
112	C₂×C₄	C₂²×#10587739	1	3	yes	—
113	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
114	C₈	—	274	6	no	—
115	C₈	—	274	2	no	—
116	C₂²	—	274	3	no	—
117	C₂²	—	274	3	no	—
118	C₂²	—	274	3	no	—
119	C₂²	C₂³×#10587739	1	1	yes	characteristic
120	C₄	—	2	3	no	—
121	C₄	—	274	1	no	—
122	C₄	—	274	3	no	—
123	C₄	—	137	1	no	—
124	C₄	—	137	1	no	—
125	C₄	C₂×#14555520	1	6	yes	—
126	C₄	#17984176	1	2	yes	—
127	C₂	—	548	3	no	—
128	C₂	#22578472	1	1	yes	characteristic
129	C₂	C₂²×#14555520	1	1	yes	characteristic
130	C₂	C₂×#17984176	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#11764047

C₂

yes

maximal, complemented

#22578437

C₂

yes

maximal, complemented

#22578422

C₂

yes

maximal, complemented

#22578407

C₂

yes

maximal, complemented

C₄×#14555511

C₂

yes

maximal, characteristic

C₂×#14555520

C₂²

yes

—

C₄⋊#11764047

C₂²

yes

—

C₄⋊#11764047

C₂²

yes

—

C₄⋊#11764047

C₂²

yes

characteristic

#17984145

C₂²

yes

—

C₄×#11764047

C₂²

yes

—

C₂×#14555511

C₄

yes

characteristic

Q₈×Dic₁₃₇

C₂²

yes

characteristic

C₄×D₅₄₈

C₄

yes

—

#17984050

C₂²

yes

—

#17984046

C₄

yes

—

C₄²×D₁₃₇

C₂²

yes

—

C₄×Q₈×C₁₃₇

C₄

yes

characteristic

#14555520

—

C₂×#11764047

C₂³

yes

—

C₂×#11764047

C₂³

yes

—

#14555511

—

C₂×D₅₄₈

C₂×C₄

yes

—

C₂×C₄×D₁₃₇

C₂×C₄

yes

—

C₂×C₄×D₁₃₇

C₂×C₄

yes

—

C₂×C₄×D₁₃₇

C₂³

yes

—

C₄⋊Dic₁₃₇

C₂×C₄

yes

—

#14555487

C₂×C₄

yes

—

C₄×Dic₁₃₇

C₂×C₄

yes

—

C₄×Dic₁₃₇

C₂³

yes

characteristic

C₄×Dic₁₃₇

C₂³

yes

—

C₂×Q₈×C₁₃₇

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₃₇

C₂×C₄

yes

—

C₄²×C₁₃₇

C₂×C₄

yes

—

#11764047

—

#11764047

—

C₂²×D₁₃₇

C₂²×C₄

yes

—

D₅₄₈

—

C₂×Dic₁₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₁₃₇

C₂²×C₄

yes

—

C₂×Dic₁₃₇

C₂⁴

yes

characteristic

C₄×D₁₃₇

—

C₄×D₁₃₇

—

C₄×D₁₃₇

—

C₄×D₁₃₇

—

Q₈×C₁₃₇

M₁₆

yes

complemented

C₂×C₄×C₁₃₇

C₂²×C₄

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₂²×C₁₃₇

C₂³×C₄

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂×M₁₆

yes

—

C₄×C₁₃₇

#131

yes

—

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

H(5,𝔽₂)

yes

characteristic

Dic₁₃₇

2₋¹⁺⁴

yes

characteristic

C₂×C₁₃₇

#520

yes

characteristic

C₂×C₁₃₇

C₂²×M₁₆

yes

characteristic

C₂×C₁₃₇

C₂×#131

yes

characteristic

D₁₃₇

—

C₁₃₇

#2991

yes

characteristic, complemented, Sylow

#2991

—

137

maximal, Sylow

C₄×#41

—

137

—

C₈×Q₈

—

137

—

#436

—

137

—

#431

—

137

—

#423

—

137

—

C₂×#41

—

137

—

C₂×M₁₆

—

137

—

C₄×Q₈

—

137

—

C₄×Q₈

#10587739

yes

characteristic

C₄×D₄

—

137

—

C₄×D₄

—

137

—

#117

—

137

—

C₂×C₄²

—

137

—

C₄⋊C₈

—

137

—

C₄⋊C₈

—

137

—

C₄⋊C₈

—

137

—

C₂²⋊C₈

—

137

—

C₄×C₈

—

137

—

#41

—

274

—

C₂×Q₈

C₂×#10587739

yes

characteristic

C₂×D₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

M₁₆

—

548

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₄⋊C₄

—

137

—

C₄⋊C₄

C₂×#10587739

yes

—

C₂²⋊C₄

—

137

—

C₄²

—

137

—

C₄²

—

137

—

C₄²

—

137

—

100

C₄²

C₂×#10587739

yes

—

101

C₂³

—

137

—

102

Q₈

#14555520

yes

complemented

103

D₄

—

274

—

104

C₂×C₄

—

274

—

105

C₂×C₄

—

274

—

106

C₂×C₄

—

274

—

107

C₂×C₄

—

274

—

108

C₂×C₄

—

137

—

109

C₂×C₄

—

137

—

110

C₂×C₄

—

137

—

111

C₂×C₄

C₂²×#10587739

yes

—

112

C₂×C₄

C₂²×#10587739

yes

—

113

C₂×C₄

C₂²×#10587739

yes

characteristic

114

C₈

—

274

—

115

C₈

—

274

—

116

C₂²

—

274

—

117

C₂²

—

274

—

118

C₂²

—

274

—

119

C₂²

C₂³×#10587739

yes

characteristic

120

C₄

—

121

C₄

—

274

—

122

C₄

—

274

—

123

C₄

—

137

—

124

C₄

—

137

—

125

C₄

C₂×#14555520

yes

—

126

C₄

#17984176

yes

—

127

C₂

—

548

—

128

C₂

#22578472

yes

characteristic

129

C₂

C₂²×#14555520

yes

characteristic

130

C₂

C₂×#17984176

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10587739
commutator, socle	C₂²×C₁₃₇	C₂³×C₄
Fitting	C₄×Q₈×C₁₃₇	C₄
prime radical	C₂×#14555511	C₄
cosocle	C₂×Dic₁₃₇	C₂⁴
hypercenter	C₄×Q₈	#10587739
hypocenter	C₁₃₇	#2991
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×#10587739

commutator, socle

C₂²×C₁₃₇

C₂³×C₄

Fitting

C₄×Q₈×C₁₃₇

C₄

prime radical

C₂×#14555511

C₄

cosocle

C₂×Dic₁₃₇

C₂⁴

C₄×Q₈

C₁₃₇

G ▷ C₂²×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
1. Q₈×#11764047	C₂	semidirect
2. #22578437	C₂	semidirect
3. #22578422	C₂	semidirect
4. #22578407	C₂	semidirect
46. Q₈×C₁₃₇	M₁₆	semidirect
65. C₁₃₇	#2991	semidirect
102. Q₈	#14555520	semidirect

H ≅ G/N

type

1. Q₈×#11764047

C₂

semidirect

2. #22578437

C₂

semidirect

3. #22578422

C₂

semidirect

4. #22578407

C₂

semidirect

46. Q₈×C₁₃₇

M₁₆

semidirect

65. C₁₃₇

#2991

semidirect

102. Q₈

#14555520

semidirect

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

subgroup	index
Q₈×#11764047	2
#22578437	2
#22578422	2
#22578407	2
C₄×#14555511	2
#2991	137

subgroup

index

137

Subgroups — #31797723

Conjugacy classes of subgroups

Complements

Maximal subgroups