Group #37509970 — subgroups

Subgroups — #37509970

Order 22912.

260 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 113 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×#19153024	C₂	1	1	yes	maximal, characteristic
2	#26038309	C₂	1	1	yes	maximal, characteristic
3	#26038201	C₂	1	4	yes	maximal
4	C₂×#125×C₁₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₇₉⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₇₉	—	2	2	no	—
7	C₂×C₄⋊Dic₁₇₉	C₂²	1	2	yes	—
8	#19153024	—	2	2	no	—
9	#125×C₁₇₉	—	2	2	no	—
10	#125×C₁₇₉	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₄×C₁₇₉	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₇₉	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₇₉	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₇₉	—	2	2	no	—
15	C₂×C₁₇₉⋊C₈	—	2	2	no	—
16	C₂×C₁₇₉⋊C₈	—	2	2	no	—
17	C₂×C₁₇₉⋊C₈	Q₈	1	4	yes	—
18	C₄⋊Dic₁₇₉	—	2	4	no	—
19	C₄⋊Dic₁₇₉	—	4	4	no	—
20	C₂²×C₄×C₁₇₉	—	2	2	no	—
21	C₂²×C₄×C₁₇₉	D₄	1	2	yes	—
22	C₂²×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₁₇₉	—	2	8	no	—
24	C₄⋊C₄×C₁₇₉	—	2	2	no	—
25	C₄⋊C₄×C₁₇₉	C₂×C₄	1	4	yes	—
26	C₄²×C₁₇₉	C₂×C₄	1	2	yes	—
27	C₄²×C₁₇₉	D₄	1	2	yes	—
28	C₂×Dic₁₇₉	—	4	2	no	—
29	C₂×Dic₁₇₉	—	4	4	no	—
30	C₁₇₉⋊C₈	—	2	4	no	—
31	C₂³×C₁₇₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₇₉	—	2	2	no	—
33	C₂×C₄×C₁₇₉	—	2	2	no	—
34	C₂×C₄×C₁₇₉	—	2	4	no	—
35	C₂×C₄×C₁₇₉	—	4	2	no	—
36	C₂×C₄×C₁₇₉	—	4	2	no	—
37	C₂×C₄×C₁₇₉	C₂²⋊C₄	1	4	yes	—
38	C₂×C₄×C₁₇₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₇₉	C₂×Q₈	1	2	yes	—
41	C₂×C₄×C₁₇₉	#41	1	2	yes	—
42	C₂²×C₁₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₁₇₉	#124	1	1	yes	characteristic
44	C₂²×C₁₇₉	C₄⋊Q₈	1	1	yes	characteristic
45	C₂²×C₁₇₉	AGL(1,ℤ/8)	1	2	yes	—
46	C₂²×C₁₇₉	#137	1	2	yes	—
47	C₄×C₁₇₉	—	2	4	no	—
48	C₄×C₁₇₉	—	4	4	no	—
49	C₄×C₁₇₉	C₄×Q₈	1	2	yes	—
50	C₄×C₁₇₉	C₂²⋊Q₈	1	2	yes	—
51	Dic₁₇₉	—	8	2	no	—
52	C₂×C₁₇₉	#391	1	1	yes	characteristic
53	C₂×C₁₇₉	#417	1	2	yes	—
54	C₂×C₁₇₉	#490	1	1	yes	characteristic
55	C₂×C₁₇₉	#491	1	1	yes	characteristic
56	C₂×C₁₇₉	#501	1	2	yes	—
57	C₁₇₉	#1992	1	1	yes	characteristic, complemented, Sylow
58	#1992	—	179	1	no	maximal, Sylow
59	C₂×#125	D₁₇₉	1	1	yes	characteristic
60	C₂×#97	—	179	1	no	—
61	#384	—	179	1	no	—
62	#340	—	179	4	no	—
63	C₂²×C₈	—	179	2	no	—
64	#125	—	2	2	no	—
65	#125	Dic₁₇₉	1	4	yes	complemented
66	C₂×C₄⋊C₄	—	2	2	no	—
67	C₂×C₄⋊C₄	—	358	2	no	—
68	C₂×C₄⋊C₄	—	179	2	no	—
69	C₂×C₄⋊C₄	C₂×D₁₇₉	1	2	yes	—
70	C₂×C₄²	C₂×D₁₇₉	1	1	yes	characteristic
71	#97	—	358	2	no	—
72	C₂²×C₄	—	2	2	no	—
73	C₂²×C₄	—	358	2	no	—
74	C₂²×C₄	C₂²⋊D₁₇₉	1	2	yes	—
75	C₂²×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
76	C₂×C₈	—	358	2	no	—
77	C₂×C₈	—	358	2	no	—
78	C₂×C₈	—	179	4	no	—
79	C₄⋊C₄	—	2	8	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	358	4	no	—
82	C₄⋊C₄	—	716	4	no	—
83	C₄⋊C₄	C₂×Dic₁₇₉	1	4	yes	—
84	C₄²	C₂×Dic₁₇₉	1	2	yes	—
85	C₄²	C₂²⋊D₁₇₉	1	2	yes	—
86	C₂³	C₂×C₂²⋊D₁₇₉	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	4	2	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	—	716	2	no	—
93	C₂×C₄	—	716	4	no	—
94	C₂×C₄	C₂²⋊Dic₁₇₉	1	4	yes	—
95	C₂×C₄	C₂²×Dic₁₇₉	1	1	yes	characteristic
96	C₂×C₄	C₂×C₂²⋊D₁₇₉	1	1	yes	characteristic
97	C₂×C₄	Q₈×D₁₇₉	1	2	yes	—
98	C₂×C₄	#15668260	1	2	yes	—
99	C₈	—	358	4	no	—
100	C₂²	C₂×C₂²⋊Dic₁₇₉	1	1	yes	characteristic
101	C₂²	#19153102	1	1	yes	characteristic
102	C₂²	#19153111	1	1	yes	characteristic
103	C₂²	#19153138	1	2	yes	—
104	C₂²	#19153143	1	2	yes	—
105	C₄	—	2	4	no	—
106	C₄	—	4	4	no	—
107	C₄	—	1432	2	no	—
108	C₄	Q₈×Dic₁₇₉	1	2	yes	—
109	C₄	#19153089	1	2	yes	—
110	C₂	#26038346	1	1	yes	characteristic
111	C₂	#26038440	1	2	yes	—
112	C₂	#26038805	1	1	yes	characteristic
113	C₂	#26038809	1	1	yes	characteristic
114	C₂	#26038849	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19153024

C₂

yes

maximal, characteristic

#26038309

C₂

yes

maximal, characteristic

#26038201

C₂

yes

maximal

C₂×#125×C₁₇₉

C₂

yes

maximal, characteristic

C₂²×C₁₇₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₇₉

—

C₂×C₄⋊Dic₁₇₉

C₂²

yes

—

#19153024

—

#125×C₁₇₉

—

#125×C₁₇₉

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₇₉

—

C₂×C₄⋊C₄×C₁₇₉

C₂²

yes

—

C₂×C₄²×C₁₇₉

C₂²

yes

characteristic

C₂²×Dic₁₇₉

—

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

Q₈

yes

—

C₄⋊Dic₁₇₉

—

C₄⋊Dic₁₇₉

—

C₂²×C₄×C₁₇₉

—

C₂²×C₄×C₁₇₉

D₄

yes

—

C₂²×C₄×C₁₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₇₉

—

C₄⋊C₄×C₁₇₉

—

C₄⋊C₄×C₁₇₉

C₂×C₄

yes

—

C₄²×C₁₇₉

C₂×C₄

yes

—

C₄²×C₁₇₉

D₄

yes

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₁₇₉⋊C₈

—

C₂³×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂×Q₈

yes

—

C₂×C₄×C₁₇₉

#41

yes

—

C₂²×C₁₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₇₉

#124

yes

characteristic

C₂²×C₁₇₉

C₄⋊Q₈

yes

characteristic

C₂²×C₁₇₉

AGL(1,ℤ/8)

yes

—

C₂²×C₁₇₉

#137

yes

—

C₄×C₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

C₄×Q₈

yes

—

C₄×C₁₇₉

C₂²⋊Q₈

yes

—

Dic₁₇₉

—

C₂×C₁₇₉

#391

yes

characteristic

C₂×C₁₇₉

#417

yes

—

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

—

C₁₇₉

#1992

yes

characteristic, complemented, Sylow

#1992

—

179

maximal, Sylow

C₂×#125

D₁₇₉

yes

characteristic

C₂×#97

—

179

—

#384

—

179

—

#340

—

179

—

C₂²×C₈

—

179

—

#125

—

#125

Dic₁₇₉

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

358

—

C₂×C₄⋊C₄

—

179

—

C₂×C₄⋊C₄

C₂×D₁₇₉

yes

—

C₂×C₄²

C₂×D₁₇₉

yes

characteristic

#97

—

358

—

C₂²×C₄

—

C₂²×C₄

—

358

—

C₂²×C₄

C₂²⋊D₁₇₉

yes

—

C₂²×C₄

C₂²×D₁₇₉

yes

characteristic

C₂×C₈

—

358

—

C₂×C₈

—

358

—

C₂×C₈

—

179

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

358

—

C₄⋊C₄

—

716

—

C₄⋊C₄

C₂×Dic₁₇₉

yes

—

C₄²

C₂×Dic₁₇₉

yes

—

C₄²

C₂²⋊D₁₇₉

yes

—

C₂³

C₂×C₂²⋊D₁₇₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

716

—

C₂×C₄

—

716

—

C₂×C₄

C₂²⋊Dic₁₇₉

yes

—

C₂×C₄

C₂²×Dic₁₇₉

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₇₉

yes

characteristic

C₂×C₄

Q₈×D₁₇₉

yes

—

C₂×C₄

#15668260

yes

—

C₈

—

358

—

100

C₂²

C₂×C₂²⋊Dic₁₇₉

yes

characteristic

101

C₂²

#19153102

yes

characteristic

102

C₂²

#19153111

yes

characteristic

103

C₂²

#19153138

yes

—

104

C₂²

#19153143

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

1432

—

108

C₄

Q₈×Dic₁₇₉

yes

—

109

C₄

#19153089

yes

—

110

C₂

#26038346

yes

characteristic

111

C₂

#26038440

yes

—

112

C₂

#26038805

yes

characteristic

113

C₂

#26038809

yes

characteristic

114

C₂

#26038849

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₇₉
commutator	C₂×C₄×C₁₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇₉
Fitting	C₂×#125×C₁₇₉	C₂
socle, prime radical	C₂³×C₁₇₉	C₂×D₄
cosocle	C₂²×C₄×C₁₇₉	C₂³
hypercenter	C₂×#125	D₁₇₉
hypocenter	C₁₇₉	#1992
LCS	G ▷ C₂×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₇₉

commutator

C₂×C₄×C₁₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇₉

Fitting

C₂×#125×C₁₇₉

C₂

socle, prime radical

C₂³×C₁₇₉

C₂×D₄

cosocle

C₂²×C₄×C₁₇₉

C₂³

D₁₇₉

C₁₇₉

G ▷ C₂×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
10. #125×C₁₇₉	C₄	semidirect
57. C₁₇₉	#1992	semidirect
65. #125	Dic₁₇₉	semidirect

H ≅ G/N

type

10. #125×C₁₇₉

C₄

semidirect

57. C₁₇₉

#1992

semidirect

65. #125

Dic₁₇₉

semidirect

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

subgroup	index
C₂×#19153024	2
#26038309	2
#26038201	2
C₂×#125×C₁₇₉	2
#1992	179

subgroup

index

C₂×#125×C₁₇₉

179

Subgroups — #37509970

Conjugacy classes of subgroups

Complements

Maximal subgroups