Group #31796593 — subgroups

Subgroups — #31796593

Order 17536.

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

normal subgroups only

This group has 232 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×#17984159	C₂	1	1	yes	maximal, characteristic
2	#22578368	C₂	1	2	yes	maximal
3	C₂×C₈×#10587739	C₂	1	1	yes	maximal, characteristic
4	#22578273	C₂	1	2	yes	maximal
5	#22577440	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555520	C₂²	1	2	yes	—
7	C₂×#14555518	C₂²	1	1	yes	characteristic
8	C₂×C₄×#10587739	C₂²	1	1	yes	characteristic
9	#17984159	C₄	1	4	yes	complemented
10	#17984143	—	2	2	no	—
11	C₄×#11764047	—	2	2	no	—
12	C₈×#10587739	C₄	1	4	yes	complemented
13	C₂×C₈×D₁₃₇	C₂²	1	1	yes	characteristic
14	#17984046	C₂²	1	2	yes	—
15	#17984022	C₄	1	1	yes	characteristic, complemented
16	#17984019	C₄	1	2	yes	complemented
17	#107×C₁₃₇	C₄	1	1	yes	characteristic, complemented
18	C₂²×#10587739	—	2	1	no	—
19	#14555520	—	2	2	no	—
20	#14555520	C₂×C₄	1	4	yes	—
21	#14555518	C₂×C₄	1	2	yes	—
22	#14555518	Q₈	1	2	yes	—
23	C₂×#11764047	—	2	2	no	—
24	C₂×#11764047	D₄	1	1	yes	characteristic
25	C₂×#11764047	D₄	1	1	yes	characteristic
26	C₄×#10587739	—	2	1	no	—
27	C₄×#10587739	C₂×C₄	1	2	yes	—
28	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
29	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
30	C₈×D₁₃₇	C₂×C₄	1	2	yes	—
31	C₈×D₁₃₇	C₂×C₄	1	2	yes	—
32	C₄⋊Dic₁₃₇	C₂×C₄	1	2	yes	—
33	#14555487	—	2	2	no	—
34	C₄×Dic₁₃₇	—	2	2	no	—
35	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
36	C₄⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
37	C₂×#10587739	—	2	2	no	—
38	C₂×#10587739	—	2	2	no	—
39	C₂×#10587739	—	2	2	no	—
40	#11764047	—	2	4	no	—
41	#11764047	C₄⋊C₄	1	2	yes	—
42	#11764047	C₄⋊C₄	1	2	yes	—
43	C₂²×D₁₃₇	#41	1	1	yes	characteristic
44	C₂×Dic₁₃₇	—	2	2	no	—
45	C₂×Dic₁₃₇	C₂×D₄	1	1	yes	characteristic
46	C₄×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
47	C₄×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
48	C₄×D₁₃₇	C₂×Q₈	1	1	yes	characteristic
49	C₄×D₁₃₇	#41	1	1	yes	characteristic
50	#11764040	C₄²	1	2	yes	—
51	C₂×C₄×C₁₃₇	—	2	2	no	—
52	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
53	C₈×C₁₃₇	C₄²	1	2	yes	—
54	C₂²×C₁₃₇	C₄×D₄	1	1	yes	characteristic
55	C₂×D₁₃₇	—	2	1	no	—
56	C₂×D₁₃₇	—	2	1	no	—
57	C₂×D₁₃₇	#117	1	1	yes	characteristic
58	C₂×D₁₃₇	C₄×Q₈	1	1	yes	characteristic
59	#10587739	—	2	4	no	—
60	C₄×C₁₃₇	—	4	2	no	—
61	C₄×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
62	C₄×C₁₃₇	C₄×Q₈	1	1	yes	characteristic
63	Dic₁₃₇	—	4	2	no	—
64	Dic₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
65	Dic₁₃₇	C₄×D₄	1	1	yes	characteristic
66	C₂×C₁₃₇	C₄×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₁₃₇	#443	1	1	yes	characteristic
68	C₂×C₁₃₇	#443	1	1	yes	characteristic
69	D₁₃₇	—	2	1	no	—
70	D₁₃₇	—	2	1	no	—
71	C₁₃₇	#1772	1	1	yes	characteristic, complemented, Sylow
72	#1772	—	137	1	no	maximal, Sylow
73	C₂×#108	—	137	1	no	—
74	#427	—	137	1	no	—
75	#405	—	137	2	no	—
76	C₂×C₄×C₈	—	137	1	no	—
77	#339	—	137	2	no	—
78	C₂×M₁₆	—	137	2	no	—
79	C₂²×C₈	—	137	1	no	—
80	C₂²×C₈	—	137	1	no	—
81	#117	—	137	2	no	—
82	C₂×C₄²	—	137	1	no	—
83	#108	—	137	4	no	—
84	#107	—	137	1	no	—
85	#107	#10587739	1	1	yes	characteristic, complemented
86	#106	—	137	2	no	—
87	#97	—	274	2	no	—
88	C₄×C₈	—	274	2	no	—
89	C₄×C₈	—	137	4	no	—
90	C₂²×C₄	—	274	1	no	—
91	C₂²×C₄	—	137	1	no	—
92	M₁₆	—	274	2	no	—
93	M₁₆	—	137	4	no	—
94	C₂×C₈	—	274	2	no	—
95	C₂×C₈	—	137	1	no	—
96	C₂×C₈	—	137	2	no	—
97	C₂×C₈	—	137	1	no	—
98	C₂×C₈	—	137	2	no	—
99	C₂×C₈	—	137	1	no	—
100	C₂×C₈	—	137	2	no	—
101	C₂×C₈	—	137	2	no	—
102	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
103	C₄⋊C₄	—	137	2	no	—
104	C₄⋊C₄	C₂×#10587739	1	2	yes	—
105	C₂²⋊C₄	—	274	2	no	—
106	C₄²	—	274	2	no	—
107	C₄²	—	274	1	no	—
108	C₄²	—	137	2	no	—
109	C₂³	—	137	1	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	274	2	no	—
112	C₂×C₄	—	274	2	no	—
113	C₂×C₄	—	274	2	no	—
114	C₂×C₄	—	274	2	no	—
115	C₂×C₄	—	137	1	no	—
116	C₂×C₄	—	137	1	no	—
117	C₂×C₄	—	137	1	no	—
118	C₂×C₄	—	137	1	no	—
119	C₂×C₄	—	137	1	no	—
120	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
121	C₈	—	274	4	no	—
122	C₈	—	137	2	no	—
123	C₈	—	137	2	no	—
124	C₈	—	137	2	no	—
125	C₈	C₄×#10587739	1	2	yes	—
126	C₂²	—	274	1	no	—
127	C₂²	—	274	1	no	—
128	C₂²	—	137	1	no	—
129	C₂²	—	137	1	no	—
130	C₂²	D₄×#10587739	1	1	yes	characteristic
131	C₄	—	4	2	no	—
132	C₄	—	274	4	no	—
133	C₄	—	548	2	no	—
134	C₄	—	137	1	no	—
135	C₄	—	137	1	no	—
136	C₄	C₂×C₄×#10587739	1	1	yes	characteristic
137	C₄	Q₈×#10587739	1	1	yes	characteristic
138	C₂	—	274	1	no	—
139	C₂	—	274	1	no	—
140	C₂	C₄⋊C₄×#10587739	1	1	yes	characteristic
141	C₂	#22578445	1	1	yes	characteristic
142	C₂	#22578447	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984159

C₂

yes

maximal, characteristic

#22578368

C₂

yes

maximal

C₂×C₈×#10587739

C₂

yes

maximal, characteristic

#22578273

C₂

yes

maximal

#22577440

C₂

yes

maximal, characteristic

C₂×#14555520

C₂²

yes

—

C₂×#14555518

C₂²

yes

characteristic

C₂×C₄×#10587739

C₂²

yes

characteristic

#17984159

C₄

yes

—

—

C₄

yes

complemented

C₂×C₈×D₁₃₇

C₂²

yes

characteristic

#17984046

C₂²

yes

—

#17984022

C₄

yes

characteristic, complemented

#17984019

C₄

yes

complemented

#107×C₁₃₇

C₄

yes

characteristic, complemented

C₂²×#10587739

—

#14555520

—

#14555520

C₂×C₄

yes

—

#14555518

C₂×C₄

yes

—

#14555518

Q₈

yes

—

C₂×#11764047

—

C₂×#11764047

D₄

yes

characteristic

C₂×#11764047

D₄

yes

characteristic

C₄×#10587739

—

C₄×#10587739

C₂×C₄

yes

—

C₂×C₄×D₁₃₇

C₂³

yes

characteristic

C₂×#11764040

C₂×C₄

yes

characteristic

C₈×D₁₃₇

C₂×C₄

yes

—

C₈×D₁₃₇

C₂×C₄

yes

—

C₄⋊Dic₁₃₇

C₂×C₄

yes

—

#14555487

—

C₄×Dic₁₃₇

—

C₂×C₈×C₁₃₇

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₃₇

C₂×C₄

yes

—

—

—

—

—

C₄⋊C₄

yes

—

#11764047

C₄⋊C₄

yes

—

C₂²×D₁₃₇

#41

yes

characteristic

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

C₂×D₄

yes

characteristic

C₄×D₁₃₇

C₂²×C₄

yes

characteristic

C₄×D₁₃₇

C₂²×C₄

yes

characteristic

C₄×D₁₃₇

C₂×Q₈

yes

C₄×D₁₃₇

yes

C₄²

yes

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₈×C₁₃₇

C₄²

yes

—

C₂²×C₁₃₇

C₄×D₄

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

#117

yes

characteristic

C₂×D₁₃₇

C₄×Q₈

yes

characteristic

#10587739

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂×C₄²

yes

characteristic

C₄×C₁₃₇

C₄×Q₈

yes

characteristic

Dic₁₃₇

—

Dic₁₃₇

C₂×C₄⋊C₄

yes

characteristic

Dic₁₃₇

C₄×D₄

yes

characteristic

C₂×C₁₃₇

C₄×C₄⋊C₄

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

D₁₃₇

—

D₁₃₇

—

C₁₃₇

#1772

yes

characteristic, complemented, Sylow

#1772

—

137

maximal, Sylow

C₂×#108

—

137

—

#427

—

137

—

#405

—

137

—

C₂×C₄×C₈

—

137

—

#339

—

137

—

C₂×M₁₆

—

137

—

C₂²×C₈

—

137

—

C₂²×C₈

—

137

—

#117

—

137

—

C₂×C₄²

—

137

—

#108

—

137

—

#107

—

137

—

#107

#10587739

yes

characteristic, complemented

#106

—

137

—

#97

—

274

—

C₄×C₈

—

274

—

C₄×C₈

—

137

—

C₂²×C₄

—

274

—

C₂²×C₄

—

137

—

M₁₆

—

274

—

M₁₆

—

137

—

C₂×C₈

—

274

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

100

C₂×C₈

—

137

—

101

C₂×C₈

—

137

—

102

C₂×C₈

C₂×#10587739

yes

characteristic

103

C₄⋊C₄

—

137

—

104

C₄⋊C₄

C₂×#10587739

yes

—

105

C₂²⋊C₄

—

274

—

106

C₄²

—

274

—

107

C₄²

—

274

—

108

C₄²

—

137

—

109

C₂³

—

137

—

110

C₂×C₄

—

111

C₂×C₄

—

274

—

112

C₂×C₄

—

274

—

113

C₂×C₄

—

274

—

114

C₂×C₄

—

274

—

115

C₂×C₄

—

137

—

116

C₂×C₄

—

137

—

117

C₂×C₄

—

137

—

118

C₂×C₄

—

137

—

119

C₂×C₄

—

137

—

120

C₂×C₄

C₂²×#10587739

yes

characteristic

121

C₈

—

274

—

122

C₈

—

137

—

123

C₈

—

137

—

124

C₈

—

137

—

125

C₈

C₄×#10587739

yes

—

126

C₂²

—

274

—

127

C₂²

—

274

—

128

C₂²

—

137

—

129

C₂²

—

137

—

130

C₂²

D₄×#10587739

yes

characteristic

131

C₄

—

132

C₄

—

274

—

133

C₄

—

548

—

134

C₄

—

137

—

135

C₄

—

137

—

136

C₄

C₂×C₄×#10587739

yes

characteristic

137

C₄

Q₈×#10587739

yes

characteristic

138

C₂

—

274

—

139

C₂

—

274

—

140

C₂

C₄⋊C₄×#10587739

yes

characteristic

141

C₂

#22578445

yes

characteristic

142

C₂

#22578447

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×#10587739
commutator	C₄×C₁₃₇	C₂×C₄²
Frattini	C₂×C₄	C₂²×#10587739
Fitting	#107×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₄×D₄
prime radical	C₂²×D₁₃₇	#41
cosocle	C₂×C₄×D₁₃₇	C₂³
hypercenter	#107	#10587739
hypocenter	C₁₃₇	#1772
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

subgroup

G/H

center

C₂²

D₄×#10587739

commutator

C₄×C₁₃₇

C₂×C₄²

Frattini

C₂×C₄

C₂²×#10587739

Fitting

#107×C₁₃₇

C₄

socle

C₂²×C₁₃₇

C₄×D₄

prime radical

C₂²×D₁₃₇

#41

cosocle

C₂×C₄×D₁₃₇

C₂³

C₁₃₇

G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

N	H ≅ G/N	type
9. #17984159	C₄	semidirect
12. C₈×#10587739	C₄	semidirect
15. #17984022	C₄	semidirect
16. #17984019	C₄	semidirect
17. #107×C₁₃₇	C₄	semidirect
71. C₁₃₇	#1772	semidirect
85. #107	#10587739	semidirect

H ≅ G/N

type

9. #17984159

C₄

semidirect

12. C₈×#10587739

C₄

semidirect

15. #17984022

C₄

semidirect

16. #17984019

C₄

semidirect

17. #107×C₁₃₇

C₄

semidirect

71. C₁₃₇

#1772

semidirect

85. #107

#10587739

semidirect

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

subgroup	index
C₂×#17984159	2
#22578368	2
C₂×C₈×#10587739	2
#22578273	2
#22577440	2
#1772	137

subgroup

index

137

Subgroups — #31796593

Conjugacy classes of subgroups

Complements

Maximal subgroups