Group #31796908 — subgroups

Subgroups — #31796908

Order 17536.

184 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 66 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	#22578418	C₂	1	2	yes	maximal
2	C₂×#17984158	C₂	1	1	yes	maximal, characteristic
3	#22578366	C₂	1	1	yes	maximal, characteristic
4	#22578286	C₂	1	2	yes	maximal
5	#108×D₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555520	C₂²	1	2	yes	—
7	D₁₃₇⋊M₁₆	C₂²	1	1	yes	characteristic
8	#17984167	C₂²	1	1	yes	characteristic
9	#17984159	—	2	2	no	—
10	#17984158	—	2	2	no	—
11	#17984158	—	2	1	no	—
12	#17984158	—	2	1	no	—
13	#17984142	—	2	1	no	—
14	C₈×#10587739	—	2	1	no	—
15	M₁₆×D₁₃₇	C₂²	1	2	yes	—
16	C₂×C₈×D₁₃₇	C₂²	1	1	yes	characteristic
17	#17984024	C₄	1	1	yes	characteristic, complemented
18	#17984023	C₄	1	2	yes	complemented
19	#108×C₁₃₇	C₄	1	1	yes	characteristic, complemented
20	#14555520	—	2	2	no	—
21	#14555520	—	2	2	no	—
22	#14555520	D₄	1	2	yes	—
23	#14555519	—	2	1	no	—
24	#14555518	—	2	1	no	—
25	C₂×#11764047	—	2	2	no	—
26	C₄⋊#10587739	—	2	1	no	—
27	C₂²⋊#10587739	Q₈	1	2	yes	—
28	C₄×#10587739	—	2	1	no	—
29	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
30	#14555494	C₂×C₄	1	2	yes	—
31	#14555493	—	2	2	no	—
32	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
33	C₈×D₁₃₇	—	2	2	no	—
34	C₈×D₁₃₇	D₄	1	1	yes	characteristic
35	C₈×D₁₃₇	D₄	1	1	yes	characteristic
36	C₈×D₁₃₇	Q₈	1	1	yes	characteristic
37	C₈×D₁₃₇	Q₈	1	1	yes	characteristic
38	M₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
39	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
40	C₂×#10587739	—	2	2	no	—
41	#11764047	—	2	2	no	—
42	#11764047	—	4	2	no	—
43	C₂²×D₁₃₇	C₂×Q₈	1	1	yes	characteristic
44	C₂×Dic₁₃₇	#41	1	1	yes	characteristic
45	C₄×D₁₃₇	C₂×D₄	1	1	yes	characteristic
46	C₄×D₁₃₇	C₂×D₄	1	1	yes	characteristic
47	C₄×D₁₃₇	C₂×Q₈	1	1	yes	characteristic
48	C₄×D₁₃₇	#41	1	1	yes	characteristic
49	#11764040	—	2	2	no	—
50	#11764040	C₄⋊C₄	1	1	yes	characteristic
51	#11764040	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
53	C₈×C₁₃₇	—	2	2	no	—
54	C₈×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
55	C₈×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₁₃₇	C₄×Q₈	1	1	yes	characteristic
57	C₂×D₁₃₇	—	4	1	no	—
58	C₂×D₁₃₇	C₂²⋊Q₈	1	1	yes	characteristic
59	C₂×D₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
60	#10587739	—	4	2	no	—
61	C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
63	Dic₁₃₇	#121	1	1	yes	characteristic
64	Dic₁₃₇	#125	1	1	yes	characteristic
65	C₂×C₁₃₇	—	2	1	no	—
66	C₂×C₁₃₇	#389	1	1	yes	characteristic
67	D₁₃₇	—	2	1	no	—
68	D₁₃₇	—	2	1	no	—
69	C₁₃₇	#1953	1	1	yes	characteristic, complemented, Sylow
70	#1953	—	137	1	no	maximal, Sylow
71	#430	—	137	2	no	—
72	C₂×#108	—	137	1	no	—
73	C₂×#108	—	137	1	no	—
74	#405	—	137	1	no	—
75	#344	—	137	2	no	—
76	C₂×M₁₆	—	137	2	no	—
77	C₂×M₁₆	—	137	1	no	—
78	C₂×M₁₆	—	137	2	no	—
79	C₂²×C₈	—	137	1	no	—
80	#117	—	137	1	no	—
81	#108	—	274	2	no	—
82	#108	—	274	2	no	—
83	#108	—	274	1	no	—
84	#108	—	274	1	no	—
85	#108	—	137	2	no	—
86	#108	—	137	1	no	—
87	#108	#10587739	1	1	yes	characteristic, complemented
88	#97	—	274	1	no	—
89	C₄×C₈	—	274	1	no	—
90	C₂²×C₄	—	137	1	no	—
91	M₁₆	—	274	2	no	—
92	M₁₆	—	274	1	no	—
93	M₁₆	—	274	2	no	—
94	M₁₆	—	274	2	no	—
95	M₁₆	—	137	2	no	—
96	M₁₆	—	137	2	no	—
97	M₁₆	C₂×#10587739	1	2	yes	—
98	C₂×C₈	—	274	2	no	—
99	C₂×C₈	—	274	1	no	—
100	C₂×C₈	—	274	2	no	—
101	C₂×C₈	—	137	1	no	—
102	C₂×C₈	—	137	1	no	—
103	C₂×C₈	—	137	1	no	—
104	C₂×C₈	—	137	1	no	—
105	C₂×C₈	—	137	1	no	—
106	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
107	C₄⋊C₄	—	274	1	no	—
108	C₂²⋊C₄	—	137	2	no	—
109	C₄²	—	274	1	no	—
110	C₂³	—	137	1	no	—
111	C₂×C₄	—	274	2	no	—
112	C₂×C₄	—	137	1	no	—
113	C₂×C₄	—	137	1	no	—
114	C₂×C₄	—	137	1	no	—
115	C₂×C₄	—	137	1	no	—
116	C₂×C₄	—	137	1	no	—
117	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
118	C₈	—	2	2	no	—
119	C₈	—	274	2	no	—
120	C₈	—	274	2	no	—
121	C₈	—	548	2	no	—
122	C₈	—	137	1	no	—
123	C₈	—	137	1	no	—
124	C₈	C₄⋊#10587739	1	1	yes	characteristic
125	C₈	C₄⋊#10587739	1	1	yes	characteristic
126	C₂²	—	548	1	no	—
127	C₂²	—	137	1	no	—
128	C₂²	—	137	1	no	—
129	C₂²	Q₈×#10587739	1	1	yes	characteristic
130	C₄	—	548	2	no	—
131	C₄	—	137	1	no	—
132	C₄	—	137	1	no	—
133	C₄	C₂×C₄⋊#10587739	1	1	yes	characteristic
134	C₄	D₄×#10587739	1	1	yes	characteristic
135	C₂	—	2	1	no	—
136	C₂	—	274	1	no	—
137	C₂	—	274	1	no	—
138	C₂	#22578354	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22578418

C₂

yes

maximal

C₂×#17984158

C₂

yes

maximal, characteristic

#22578366

C₂

yes

maximal, characteristic

#22578286

C₂

yes

maximal

#108×D₁₃₇

C₂

yes

maximal, characteristic

C₂×#14555520

C₂²

yes

—

D₁₃₇⋊M₁₆

C₂²

yes

characteristic

#17984167

C₂²

yes

—

—

—

—

—

—

M₁₆×D₁₃₇

C₂²

yes

—

C₂×C₈×D₁₃₇

C₂²

yes

characteristic

#17984024

C₄

yes

characteristic, complemented

#17984023

C₄

yes

complemented

#108×C₁₃₇

C₄

yes

characteristic, complemented

#14555520

—

#14555520

—

#14555520

D₄

yes

—

—

—

—

—

Q₈

yes

—

C₄×#10587739

—

C₂×C₄×D₁₃₇

C₂³

yes

characteristic

#14555494

C₂×C₄

yes

—

#14555493

—

C₂×#11764040

C₂×C₄

yes

characteristic

C₈×D₁₃₇

—

C₈×D₁₃₇

D₄

yes

characteristic

C₈×D₁₃₇

D₄

yes

characteristic

C₈×D₁₃₇

Q₈

yes

characteristic

C₈×D₁₃₇

Q₈

yes

characteristic

M₁₆×C₁₃₇

C₂×C₄

yes

—

C₂×C₈×C₁₃₇

C₂×C₄

yes

—

—

—

C₂²×D₁₃₇

C₂×Q₈

yes

characteristic

C₂×Dic₁₃₇

#41

yes

characteristic

C₄×D₁₃₇

C₂×D₄

yes

characteristic

C₄×D₁₃₇

C₂×D₄

yes

characteristic

C₄×D₁₃₇

C₂×Q₈

yes

C₄×D₁₃₇

yes

—

C₄⋊C₄

yes

characteristic

#11764040

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₈×C₁₃₇

—

C₈×C₁₃₇

C₄⋊C₄

yes

characteristic

C₈×C₁₃₇

C₄⋊C₄

yes

characteristic

C₂²×C₁₃₇

C₄×Q₈

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

C₂²⋊Q₈

yes

characteristic

C₂×D₁₃₇

C₄⋊Q₈

yes

characteristic

#10587739

—

C₄×C₁₃₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₃₇

C₄×D₄

yes

Dic₁₃₇

yes

Dic₁₃₇

yes

C₂×C₁₃₇

—

C₂×C₁₃₇

#389

yes

characteristic

D₁₃₇

—

D₁₃₇

—

C₁₃₇

#1953

yes

characteristic, complemented, Sylow

#1953

—

137

maximal, Sylow

#430

—

137

—

C₂×#108

—

137

—

C₂×#108

—

137

—

#405

—

137

—

#344

—

137

—

C₂×M₁₆

—

137

—

C₂×M₁₆

—

137

—

C₂×M₁₆

—

137

—

C₂²×C₈

—

137

—

#117

—

137

—

#108

—

274

—

#108

—

274

—

#108

—

274

—

#108

—

274

—

#108

—

137

—

#108

—

137

—

#108

#10587739

yes

characteristic, complemented

#97

—

274

—

C₄×C₈

—

274

—

C₂²×C₄

—

137

—

M₁₆

—

274

—

M₁₆

—

274

—

M₁₆

—

274

—

M₁₆

—

274

—

M₁₆

—

137

—

M₁₆

—

137

—

M₁₆

C₂×#10587739

yes

—

C₂×C₈

—

274

—

C₂×C₈

—

274

—

100

C₂×C₈

—

274

—

101

C₂×C₈

—

137

—

102

C₂×C₈

—

137

—

103

C₂×C₈

—

137

—

104

C₂×C₈

—

137

—

105

C₂×C₈

—

137

—

106

C₂×C₈

C₂×#10587739

yes

characteristic

107

C₄⋊C₄

—

274

—

108

C₂²⋊C₄

—

137

—

109

C₄²

—

274

—

110

C₂³

—

137

—

111

C₂×C₄

—

274

—

112

C₂×C₄

—

137

—

113

C₂×C₄

—

137

—

114

C₂×C₄

—

137

—

115

C₂×C₄

—

137

—

116

C₂×C₄

—

137

—

117

C₂×C₄

C₂²×#10587739

yes

characteristic

118

C₈

—

119

C₈

—

274

—

120

C₈

—

274

—

121

C₈

—

548

—

122

C₈

—

137

—

123

C₈

—

137

—

124

C₈

C₄⋊#10587739

yes

characteristic

125

C₈

C₄⋊#10587739

yes

characteristic

126

C₂²

—

548

—

127

C₂²

—

137

—

128

C₂²

—

137

—

129

C₂²

Q₈×#10587739

yes

characteristic

130

C₄

—

548

—

131

C₄

—

137

—

132

C₄

—

137

—

133

C₄

C₂×C₄⋊#10587739

yes

characteristic

134

C₄

D₄×#10587739

yes

characteristic

135

C₂

—

136

C₂

—

274

—

137

C₂

—

274

—

138

C₂

#22578354

yes

characteristic

	subgroup	G/H
center	C₂	#22578354
commutator	C₂×C₄×C₁₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#10587739
Fitting	#108×C₁₃₇	C₄
socle	C₂×C₁₃₇	#389
prime radical	C₂²×D₁₃₇	C₂×Q₈
cosocle	C₂×C₄×D₁₃₇	C₂³
hypercenter	#108	#10587739
hypocenter	C₁₃₇	#1953
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #108

subgroup

G/H

center

C₂

#22578354

commutator

C₂×C₄×C₁₃₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#10587739

Fitting

#108×C₁₃₇

C₄

socle

C₂×C₁₃₇

#389

prime radical

C₂²×D₁₃₇

C₂×Q₈

cosocle

C₂×C₄×D₁₃₇

C₂³

C₁₃₇

G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #108

N	H ≅ G/N	type
17. #17984024	C₄	semidirect
18. #17984023	C₄	semidirect
19. #108×C₁₃₇	C₄	semidirect
69. C₁₃₇	#1953	semidirect
87. #108	#10587739	semidirect

H ≅ G/N

type

17. #17984024

C₄

semidirect

18. #17984023

C₄

semidirect

19. #108×C₁₃₇

C₄

semidirect

69. C₁₃₇

#1953

semidirect

87. #108

#10587739

semidirect

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

subgroup	index
#22578418	2
C₂×#17984158	2
#22578366	2
#22578286	2
#108×D₁₃₇	2
#1953	137

subgroup

index

#108×D₁₃₇

137

Subgroups — #31796908

Conjugacy classes of subgroups

Complements

Maximal subgroups