Group #31796597 — subgroups

Subgroups — #31796597

Order 17536.

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

normal subgroups only

This group has 224 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×#17984142	C₂	1	1	yes	maximal, characteristic
4	#22578273	C₂	1	2	yes	maximal
5	#22577438	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	#17984142	C₄	1	4	yes	complemented
12	C₄×#11764047	—	2	2	no	—
13	C₂×C₈×D₁₃₇	C₂²	1	1	yes	characteristic
14	#17984046	C₂²	1	2	yes	—
15	#17984021	C₄	1	2	yes	complemented
16	#17984020	C₄	1	1	yes	characteristic, complemented
17	#106×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	2	yes	—
25	C₄×#10587739	—	2	1	no	—
26	C₄×#10587739	C₂×C₄	1	2	yes	—
27	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
28	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
29	C₈×D₁₃₇	C₂×C₄	1	2	yes	—
30	C₈×D₁₃₇	C₂×C₄	1	2	yes	—
31	C₄⋊Dic₁₃₇	C₂×C₄	1	2	yes	—
32	#14555487	—	2	2	no	—
33	C₄×Dic₁₃₇	—	2	2	no	—
34	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
35	C₄⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
36	C₂×#10587739	—	2	2	no	—
37	C₂×#10587739	—	4	2	no	—
38	#11764047	—	2	4	no	—
39	#11764047	C₄⋊C₄	1	4	yes	—
40	C₂²×D₁₃₇	#41	1	1	yes	characteristic
41	C₂×Dic₁₃₇	—	2	2	no	—
42	C₂×Dic₁₃₇	C₂×D₄	1	1	yes	characteristic
43	C₄×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
44	C₄×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
45	C₄×D₁₃₇	C₂×Q₈	1	1	yes	characteristic
46	C₄×D₁₃₇	#41	1	1	yes	characteristic
47	#11764040	C₄²	1	2	yes	—
48	C₂×C₄×C₁₃₇	—	2	2	no	—
49	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
50	C₈×C₁₃₇	C₄²	1	2	yes	—
51	C₂²×C₁₃₇	C₄×D₄	1	1	yes	characteristic
52	C₂×D₁₃₇	—	2	2	no	—
53	C₂×D₁₃₇	#117	1	1	yes	characteristic
54	C₂×D₁₃₇	C₄×Q₈	1	1	yes	characteristic
55	#10587739	—	4	2	no	—
56	C₄×C₁₃₇	—	4	2	no	—
57	C₄×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
58	C₄×C₁₃₇	C₄×Q₈	1	1	yes	characteristic
59	Dic₁₃₇	—	4	2	no	—
60	Dic₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
61	Dic₁₃₇	C₄×D₄	1	1	yes	characteristic
62	C₂×C₁₃₇	C₄×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₁₃₇	#444	1	2	yes	—
64	D₁₃₇	—	2	1	no	—
65	D₁₃₇	—	2	1	no	—
66	C₁₃₇	#1773	1	1	yes	characteristic, complemented, Sylow
67	#1773	—	137	1	no	maximal, Sylow
68	C₂×#108	—	137	1	no	—
69	#427	—	137	1	no	—
70	#405	—	137	2	no	—
71	C₂×#97	—	137	1	no	—
72	#339	—	137	2	no	—
73	C₂×M₁₆	—	137	2	no	—
74	C₂²×C₈	—	137	1	no	—
75	C₂²×C₈	—	137	1	no	—
76	#117	—	137	2	no	—
77	C₂×C₄²	—	137	1	no	—
78	#108	—	137	4	no	—
79	#107	—	137	2	no	—
80	#106	—	137	1	no	—
81	#106	#10587739	1	1	yes	characteristic, complemented
82	#97	—	274	2	no	—
83	#97	—	137	4	no	—
84	C₄×C₈	—	274	2	no	—
85	C₂²×C₄	—	274	1	no	—
86	C₂²×C₄	—	137	1	no	—
87	M₁₆	—	274	2	no	—
88	M₁₆	—	137	4	no	—
89	C₂×C₈	—	274	2	no	—
90	C₂×C₈	—	137	2	no	—
91	C₂×C₈	—	137	2	no	—
92	C₂×C₈	—	137	2	no	—
93	C₂×C₈	—	137	1	no	—
94	C₂×C₈	—	137	2	no	—
95	C₂×C₈	—	137	2	no	—
96	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
97	C₄⋊C₄	—	137	2	no	—
98	C₄⋊C₄	C₂×#10587739	1	2	yes	—
99	C₂²⋊C₄	—	274	2	no	—
100	C₄²	—	274	2	no	—
101	C₄²	—	274	1	no	—
102	C₄²	—	137	2	no	—
103	C₂³	—	137	1	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	274	2	no	—
106	C₂×C₄	—	274	2	no	—
107	C₂×C₄	—	548	2	no	—
108	C₂×C₄	—	137	1	no	—
109	C₂×C₄	—	137	1	no	—
110	C₂×C₄	—	137	1	no	—
111	C₂×C₄	—	137	1	no	—
112	C₂×C₄	—	137	1	no	—
113	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
114	C₈	—	274	4	no	—
115	C₈	—	137	4	no	—
116	C₈	—	137	2	no	—
117	C₈	C₄×#10587739	1	2	yes	—
118	C₂²	—	274	2	no	—
119	C₂²	—	137	1	no	—
120	C₂²	—	137	1	no	—
121	C₂²	D₄×#10587739	1	1	yes	characteristic
122	C₄	—	4	2	no	—
123	C₄	—	548	2	no	—
124	C₄	—	548	2	no	—
125	C₄	—	137	1	no	—
126	C₄	—	137	1	no	—
127	C₄	C₂×C₄×#10587739	1	1	yes	characteristic
128	C₄	Q₈×#10587739	1	1	yes	characteristic
129	C₂	—	274	1	no	—
130	C₂	—	274	1	no	—
131	C₂	C₄⋊C₄×#10587739	1	1	yes	characteristic
132	C₂	#22578448	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984159

C₂

yes

maximal, characteristic

#22578368

C₂

yes

maximal

C₂×#17984142

C₂

yes

maximal, characteristic

#22578273

C₂

yes

maximal

#22577438

C₂

yes

maximal, characteristic

C₂×#14555520

C₂²

yes

—

C₂×#14555518

C₂²

yes

characteristic

C₂×C₄×#10587739

C₂²

yes

characteristic

#17984159

C₄

yes

complemented

#17984143

—

#17984142

C₄

yes

complemented

C₄×#11764047

—

C₂×C₈×D₁₃₇

C₂²

yes

characteristic

#17984046

C₂²

yes

—

#17984021

C₄

yes

complemented

#17984020

C₄

yes

characteristic, complemented

#106×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

—

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

—

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

#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

characteristic

C₂×C₁₃₇

#444

yes

—

D₁₃₇

—

D₁₃₇

—

C₁₃₇

#1773

yes

characteristic, complemented, Sylow

#1773

—

137

maximal, Sylow

C₂×#108

—

137

—

#427

—

137

—

#405

—

137

—

C₂×#97

—

137

—

#339

—

137

—

C₂×M₁₆

—

137

—

C₂²×C₈

—

137

—

C₂²×C₈

—

137

—

#117

—

137

—

C₂×C₄²

—

137

—

#108

—

137

—

#107

—

137

—

#106

—

137

—

#106

#10587739

yes

characteristic, complemented

#97

—

274

—

#97

—

137

—

C₄×C₈

—

274

—

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

—

C₂×C₈

—

137

—

C₂×C₈

C₂×#10587739

yes

characteristic

C₄⋊C₄

—

137

—

C₄⋊C₄

C₂×#10587739

yes

—

C₂²⋊C₄

—

274

—

100

C₄²

—

274

—

101

C₄²

—

274

—

102

C₄²

—

137

—

103

C₂³

—

137

—

104

C₂×C₄

—

105

C₂×C₄

—

274

—

106

C₂×C₄

—

274

—

107

C₂×C₄

—

548

—

108

C₂×C₄

—

137

—

109

C₂×C₄

—

137

—

110

C₂×C₄

—

137

—

111

C₂×C₄

—

137

—

112

C₂×C₄

—

137

—

113

C₂×C₄

C₂²×#10587739

yes

characteristic

114

C₈

—

274

—

115

C₈

—

137

—

116

C₈

—

137

—

117

C₈

C₄×#10587739

yes

—

118

C₂²

—

274

—

119

C₂²

—

137

—

120

C₂²

—

137

—

121

C₂²

D₄×#10587739

yes

characteristic

122

C₄

—

123

C₄

—

548

—

124

C₄

—

548

—

125

C₄

—

137

—

126

C₄

—

137

—

127

C₄

C₂×C₄×#10587739

yes

characteristic

128

C₄

Q₈×#10587739

yes

characteristic

129

C₂

—

274

—

130

C₂

—

274

—

131

C₂

C₄⋊C₄×#10587739

yes

characteristic

132

C₂

#22578448

yes

—

	subgroup	G/H
center	C₂²	D₄×#10587739
commutator	C₄×C₁₃₇	C₂×C₄²
Frattini	C₂×C₄	C₂²×#10587739
Fitting	#106×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₄×D₄
prime radical	C₂²×D₁₃₇	#41
cosocle	C₂×C₄×D₁₃₇	C₂³
hypercenter	#106	#10587739
hypocenter	C₁₃₇	#1773
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

subgroup

G/H

center

C₂²

D₄×#10587739

commutator

C₄×C₁₃₇

C₂×C₄²

Frattini

C₂×C₄

C₂²×#10587739

Fitting

#106×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 ◁ #106

N	H ≅ G/N	type
9. #17984159	C₄	semidirect
11. #17984142	C₄	semidirect
15. #17984021	C₄	semidirect
16. #17984020	C₄	semidirect
17. #106×C₁₃₇	C₄	semidirect
66. C₁₃₇	#1773	semidirect
81. #106	#10587739	semidirect

H ≅ G/N

type

9. #17984159

C₄

semidirect

11. #17984142

C₄

semidirect

15. #17984021

C₄

semidirect

16. #17984020

C₄

semidirect

17. #106×C₁₃₇

C₄

semidirect

66. C₁₃₇

#1773

semidirect

81. #106

#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×#17984142	2
#22578273	2
#22577438	2
#1773	137

subgroup

index

137

Subgroups — #31796597

Conjugacy classes of subgroups

Complements

Maximal subgroups