Group #31796976 — subgroups

Subgroups — #31796976

Order 17536.

244 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 70 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	#22578407	C₂	1	1	yes	maximal, characteristic
2	#22578339	C₂	1	1	yes	maximal, characteristic
3	#22578273	C₂	1	4	yes	maximal
4	#22577959	C₂	1	1	yes	maximal, characteristic
5	C₂×#14555520	C₂²	1	2	yes	—
6	C₂×C₄×#10587739	C₂²	1	2	yes	—
7	C₄⋊#11764047	—	2	2	no	—
8	(C₂×#10587739)⋊C₄	—	2	2	no	—
9	#17984087	C₄	1	2	yes	complemented
10	#17984080	C₄	1	1	yes	characteristic, complemented
11	#17984066	—	2	2	no	—
12	#17984053	—	2	2	no	—
13	#17984046	C₂²	1	2	yes	—
14	C₄²×D₁₃₇	C₂²	1	1	yes	characteristic
15	#125×C₁₃₇	C₄	1	1	yes	characteristic, complemented
16	C₂²×#10587739	—	2	2	no	—
17	#14555520	—	2	4	no	—
18	C₂×#11764047	—	2	2	no	—
19	C₄×#10587739	—	2	2	no	—
20	C₄×#10587739	Q₈	1	4	yes	complemented
21	C₂×Dic₂₇₄	—	2	2	no	—
22	C₂×C₄×D₁₃₇	D₄	1	2	yes	—
23	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
24	C₄⋊Dic₁₃₇	C₂×C₄	1	2	yes	—
25	#14555489	—	2	2	no	—
26	#14555489	—	2	2	no	—
27	#14555487	—	2	2	no	—
28	C₄×Dic₁₃₇	—	2	2	no	—
29	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₁₃₇	D₄	1	2	yes	—
31	C₄⋊C₄×C₁₃₇	—	2	2	no	—
32	C₄⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
33	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
34	C₂×#10587739	—	2	4	no	—
35	C₂×#10587739	—	4	4	no	—
36	#11764047	—	4	2	no	—
37	C₂²×D₁₃₇	C₂×D₄	1	1	yes	characteristic
38	Dic₂₇₄	—	4	4	no	—
39	C₂×Dic₁₃₇	—	2	2	no	—
40	C₂×Dic₁₃₇	C₂²⋊C₄	1	2	yes	—
41	C₂×Dic₁₃₇	C₂×D₄	1	1	yes	characteristic
42	C₄×D₁₃₇	—	4	2	no	—
43	C₄×D₁₃₇	C₂×Q₈	1	2	yes	—
44	C₄×D₁₃₇	#41	1	2	yes	—
45	C₂×C₄×C₁₃₇	—	2	2	no	—
46	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂×D₁₃₇	—	2	2	no	—
50	C₂×D₁₃₇	C₂²⋊Q₈	1	2	yes	—
51	#10587739	—	4	4	no	—
52	C₄×C₁₃₇	—	2	2	no	—
53	C₄×C₁₃₇	—	4	2	no	—
54	C₄×C₁₃₇	C₄×Q₈	1	2	yes	—
55	Dic₁₃₇	—	2	2	no	—
56	Dic₁₃₇	—	4	2	no	—
57	Dic₁₃₇	#124	1	1	yes	characteristic
58	Dic₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×C₁₃₇	#391	1	1	yes	characteristic
60	C₂×C₁₃₇	#421	1	2	yes	—
61	D₁₃₇	—	2	2	no	—
62	C₁₃₇	#1995	1	1	yes	characteristic, complemented, Sylow
63	#1995	—	137	1	no	maximal, Sylow
64	#533	—	137	1	no	—
65	#423	—	137	1	no	—
66	#376	—	137	1	no	—
67	#339	—	137	4	no	—
68	C₂×M₁₆	—	137	2	no	—
69	C₄⋊Q₈	—	137	2	no	—
70	#125	—	137	1	no	—
71	#125	#10587739	1	1	yes	characteristic, complemented
72	C₂²⋊Q₈	—	274	2	no	—
73	C₄×Q₈	—	274	2	no	—
74	#117	—	137	2	no	—
75	C₂×C₄²	—	137	2	no	—
76	C₂×C₄²	—	137	1	no	—
77	C₄⋊C₈	—	274	2	no	—
78	#95	—	274	2	no	—
79	C₂×Q₈	—	274	2	no	—
80	C₂²×C₄	—	274	2	no	—
81	C₂²×C₄	—	137	2	no	—
82	C₂²×C₄	—	137	1	no	—
83	M₁₆	—	274	4	no	—
84	C₂×C₈	—	274	2	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	—	274	2	no	—
87	C₄⋊C₄	—	274	2	no	—
88	C₄⋊C₄	—	137	2	no	—
89	C₄⋊C₄	C₂×#10587739	1	2	yes	—
90	C₂²⋊C₄	—	274	2	no	—
91	C₄²	—	274	2	no	—
92	C₄²	—	274	2	no	—
93	C₄²	—	137	4	no	—
94	C₄²	—	137	2	no	—
95	C₄²	—	137	1	no	—
96	C₄²	C₂×#10587739	1	1	yes	characteristic
97	C₂³	—	137	1	no	—
98	Q₈	—	548	4	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	274	4	no	—
101	C₂×C₄	—	274	2	no	—
102	C₂×C₄	—	548	2	no	—
103	C₂×C₄	—	548	4	no	—
104	C₂×C₄	—	137	2	no	—
105	C₂×C₄	—	137	1	no	—
106	C₂×C₄	—	137	2	no	—
107	C₂×C₄	—	137	2	no	—
108	C₂×C₄	C₂²⋊#10587739	1	2	yes	—
109	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
110	C₈	—	548	2	no	—
111	C₂²	—	274	2	no	—
112	C₂²	—	137	2	no	—
113	C₂²	C₂×C₂²⋊#10587739	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	2	no	—
116	C₄	—	274	2	no	—
117	C₄	—	548	4	no	—
118	C₄	—	548	2	no	—
119	C₄	—	137	1	no	—
120	C₄	—	137	1	no	—
121	C₄	Q₈×#10587739	1	2	yes	—
122	C₂	—	274	2	no	—
123	C₂	(C₂×#10587739)⋊Q₈	1	1	yes	characteristic
124	C₂	#22578400	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22578407

C₂

yes

maximal, characteristic

#22578339

C₂

yes

maximal, characteristic

#22578273

C₂

yes

maximal

#22577959

C₂

yes

maximal, characteristic

C₂×#14555520

C₂²

yes

—

C₂×C₄×#10587739

C₂²

yes

—

C₄⋊#11764047

—

(C₂×#10587739)⋊C₄

—

#17984087

C₄

yes

complemented

#17984080

C₄

yes

characteristic, complemented

#17984066

—

#17984053

—

#17984046

C₂²

yes

—

C₄²×D₁₃₇

C₂²

yes

characteristic

#125×C₁₃₇

C₄

yes

characteristic, complemented

—

—

—

—

Q₈

yes

complemented

C₂×Dic₂₇₄

—

C₂×C₄×D₁₃₇

D₄

yes

—

C₂×C₄×D₁₃₇

C₂³

yes

characteristic

C₄⋊Dic₁₃₇

C₂×C₄

yes

—

#14555489

—

#14555489

—

#14555487

—

C₄×Dic₁₃₇

—

C₄×Dic₁₃₇

C₂×C₄

yes

characteristic

C₄×Dic₁₃₇

D₄

yes

—

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

C₂×C₄

yes

—

C₄²×C₁₃₇

C₂×C₄

yes

—

—

—

C₂²×D₁₃₇

C₂×D₄

yes

characteristic

Dic₂₇₄

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

C₂²⋊C₄

yes

—

C₂×Dic₁₃₇

C₂×D₄

yes

characteristic

C₄×D₁₃₇

—

C₄×D₁₃₇

C₂×Q₈

yes

—

C₄×D₁₃₇

#41

yes

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₂²×C₁₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

C₂²⋊Q₈

yes

—

#10587739

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₄×Q₈

yes

—

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

#124

yes

characteristic

Dic₁₃₇

C₄⋊Q₈

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

—

D₁₃₇

—

C₁₃₇

#1995

yes

characteristic, complemented, Sylow

#1995

—

137

maximal, Sylow

#533

—

137

—

#423

—

137

—

#376

—

137

—

#339

—

137

—

C₂×M₁₆

—

137

—

C₄⋊Q₈

—

137

—

#125

—

137

—

#125

#10587739

yes

characteristic, complemented

C₂²⋊Q₈

—

274

—

C₄×Q₈

—

274

—

#117

—

137

—

C₂×C₄²

—

137

—

C₂×C₄²

—

137

—

C₄⋊C₈

—

274

—

#95

—

274

—

C₂×Q₈

—

274

—

C₂²×C₄

—

274

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

M₁₆

—

274

—

C₂×C₈

—

274

—

C₄⋊C₄

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

137

—

C₄⋊C₄

C₂×#10587739

yes

—

C₂²⋊C₄

—

274

—

C₄²

—

274

—

C₄²

—

274

—

C₄²

—

137

—

C₄²

—

137

—

C₄²

—

137

—

C₄²

C₂×#10587739

yes

characteristic

C₂³

—

137

—

Q₈

—

548

—

C₂×C₄

—

100

C₂×C₄

—

274

—

101

C₂×C₄

—

274

—

102

C₂×C₄

—

548

—

103

C₂×C₄

—

548

—

104

C₂×C₄

—

137

—

105

C₂×C₄

—

137

—

106

C₂×C₄

—

137

—

107

C₂×C₄

—

137

—

108

C₂×C₄

C₂²⋊#10587739

yes

—

109

C₂×C₄

C₂²×#10587739

yes

characteristic

110

C₈

—

548

—

111

C₂²

—

274

—

112

C₂²

—

137

—

113

C₂²

C₂×C₂²⋊#10587739

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

274

—

117

C₄

—

548

—

118

C₄

—

548

—

119

C₄

—

137

—

120

C₄

—

137

—

121

C₄

Q₈×#10587739

yes

—

122

C₂

—

274

—

123

C₂

(C₂×#10587739)⋊Q₈

yes

characteristic

124

C₂

#22578400

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#10587739
commutator	C₂×C₄×C₁₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#10587739
Fitting	#125×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₃₇	C₂×D₄
cosocle	C₂×C₄×D₁₃₇	C₂³
hypercenter	#125	#10587739
hypocenter	C₁₃₇	#1995
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

subgroup

G/H

center

C₂²

C₂×C₂²⋊#10587739

commutator

C₂×C₄×C₁₃₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#10587739

Fitting

#125×C₁₃₇

C₄

socle

C₂²×C₁₃₇

C₂×C₂²⋊C₄

prime radical

C₂²×D₁₃₇

C₂×D₄

cosocle

C₂×C₄×D₁₃₇

C₂³

C₁₃₇

G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #125

N	H ≅ G/N	type
9. #17984087	C₄	semidirect
10. #17984080	C₄	semidirect
15. #125×C₁₃₇	C₄	semidirect
20. C₄×#10587739	Q₈	semidirect
62. C₁₃₇	#1995	semidirect
71. #125	#10587739	semidirect

H ≅ G/N

type

9. #17984087

C₄

semidirect

10. #17984080

C₄

semidirect

15. #125×C₁₃₇

C₄

semidirect

20. C₄×#10587739

Q₈

semidirect

62. C₁₃₇

#1995

semidirect

71. #125

#10587739

semidirect

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

subgroup	index
#22578407	2
#22578339	2
#22578273	2
#22577959	2
#1995	137

subgroup

index

137

Subgroups — #31796976

Conjugacy classes of subgroups

Complements

Maximal subgroups