Group #37507914 — subgroups

Subgroups — #37507914

Order 22912.

176 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 71 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	#26038517	C₂	1	2	yes	maximal, complemented
2	#26038190	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄⋊(C₁₇₉⋊C₈)	C₂	1	2	yes	maximal, complemented
4	#26038160	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×D₈×C₁₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668243	—	2	2	no	—
7	C₄⋊(C₁₇₉⋊C₈)	C₂²	1	2	yes	—
8	C₂²⋊(C₁₇₉⋊C₈)	—	2	2	no	—
9	C₄×C₁₇₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₄×C₁₇₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×D₈×C₁₇₉	C₄	1	1	yes	characteristic
12	C₄×D₄×C₁₇₉	C₂²	1	2	yes	—
13	#107×C₁₇₉	C₄	1	1	yes	characteristic
14	#102×C₁₇₉	C₄	1	2	yes	—
15	C₄×C₈×C₁₇₉	C₂²	1	1	yes	characteristic
16	#15668243	—	4	4	no	—
17	C₂×C₁₇₉⋊C₈	—	2	2	no	—
18	C₂×C₁₇₉⋊C₈	—	2	1	no	—
19	C₂×C₁₇₉⋊C₈	D₄	1	2	yes	—
20	C₂×D₄×C₁₇₉	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₁₇₉	—	2	2	no	—
22	D₈×C₁₇₉	—	2	2	no	—
23	C₂×C₈×C₁₇₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₁₇₉	C₂×C₄	1	1	yes	characteristic
25	C₄⋊C₄×C₁₇₉	C₂×C₄	1	2	yes	—
26	C₂²⋊C₄×C₁₇₉	—	2	2	no	—
27	C₄²×C₁₇₉	C₂³	1	1	yes	characteristic
28	C₁₇₉⋊C₈	—	2	2	no	—
29	C₁₇₉⋊C₈	—	4	1	no	—
30	C₁₇₉⋊C₈	—	4	2	no	—
31	C₂³×C₁₇₉	—	2	2	no	—
32	D₄×C₁₇₉	—	2	2	no	—
33	D₄×C₁₇₉	—	2	2	no	—
34	C₂×C₄×C₁₇₉	—	2	2	no	—
35	C₂×C₄×C₁₇₉	—	4	2	no	—
36	C₂×C₄×C₁₇₉	—	4	2	no	—
37	C₂×C₄×C₁₇₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₇₉	#41	1	1	yes	characteristic
40	C₈×C₁₇₉	—	2	1	no	—
41	C₈×C₁₇₉	M₁₆	1	2	yes	complemented
42	C₂²×C₁₇₉	—	4	2	no	—
43	C₂²×C₁₇₉	—	4	2	no	—
44	C₂²×C₁₇₉	—	4	2	no	—
45	C₂²×C₁₇₉	C₄×D₄	1	1	yes	characteristic
46	C₄×C₁₇₉	—	2	1	no	—
47	C₄×C₁₇₉	—	4	2	no	—
48	C₄×C₁₇₉	C₂×M₁₆	1	1	yes	characteristic
49	C₄×C₁₇₉	#131	1	1	yes	characteristic
50	C₄×C₁₇₉	AGL(1,ℤ/8)	1	2	yes	—
51	C₂×C₁₇₉	—	8	2	no	—
52	C₂×C₁₇₉	#436	1	1	yes	characteristic
53	C₂×C₁₇₉	#442	1	1	yes	characteristic
54	C₂×C₁₇₉	#444	1	1	yes	characteristic
55	C₁₇₉	#1594	1	1	yes	characteristic, complemented, Sylow
56	#1594	—	179	1	no	maximal, Sylow
57	C₄×D₈	D₁₇₉	1	1	yes	characteristic
58	#436	—	179	2	no	—
59	#334	—	179	1	no	—
60	D₄⋊C₈	—	179	2	no	—
61	#322	—	179	1	no	—
62	C₂×D₈	Dic₁₇₉	1	1	yes	characteristic
63	C₂×M₁₆	—	358	2	no	—
64	C₄×D₄	C₂×D₁₇₉	1	2	yes	—
65	#107	Dic₁₇₉	1	1	yes	characteristic
66	C₄⋊C₈	—	179	2	no	—
67	#102	Dic₁₇₉	1	2	yes	—
68	C₂²⋊C₈	—	358	2	no	—
69	C₄×C₈	—	179	1	no	—
70	C₄×C₈	—	179	1	no	—
71	C₄×C₈	C₂×D₁₇₉	1	1	yes	characteristic
72	C₂×D₄	C₂×Dic₁₇₉	1	2	yes	—
73	C₂²×C₄	—	2	2	no	—
74	D₈	—	2	2	no	—
75	M₁₆	—	716	4	no	—
76	C₂×C₈	—	358	2	no	—
77	C₂×C₈	—	358	1	no	—
78	C₂×C₈	—	179	2	no	—
79	C₂×C₈	C₂×Dic₁₇₉	1	1	yes	characteristic
80	C₂×C₈	C₂×Dic₁₇₉	1	1	yes	characteristic
81	C₄⋊C₄	C₂×Dic₁₇₉	1	2	yes	—
82	C₂²⋊C₄	—	2	2	no	—
83	C₄²	C₂²×D₁₇₉	1	1	yes	characteristic
84	C₂³	—	2	2	no	—
85	D₄	—	2	2	no	—
86	D₄	—	2	2	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	C₂²×Dic₁₇₉	1	1	yes	characteristic
91	C₂×C₄	D₄×D₁₇₉	1	1	yes	characteristic
92	C₂×C₄	#15668258	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	358	2	no	—
95	C₈	—	716	1	no	—
96	C₈	—	716	2	no	—
97	C₈	#15668243	1	2	yes	complemented
98	C₂²	—	4	2	no	—
99	C₂²	—	4	2	no	—
100	C₂²	—	4	2	no	—
101	C₂²	D₄×Dic₁₇₉	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	4	2	no	—
104	C₄	C₂×#15668243	1	1	yes	characteristic
105	C₄	#19153120	1	1	yes	characteristic
106	C₄	#19153137	1	2	yes	—
107	C₂	—	8	2	no	—
108	C₂	#26038517	1	1	yes	characteristic
109	C₂	#26038551	1	1	yes	characteristic
110	C₂	#26038561	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26038517

C₂

yes

maximal, complemented

#26038190

C₂

yes

maximal, characteristic, complemented

D₄⋊(C₁₇₉⋊C₈)

C₂

yes

maximal, complemented

#26038160

C₂

yes

maximal, characteristic, complemented

C₄×D₈×C₁₇₉

C₂

yes

maximal, characteristic

C₂×#15668243

—

C₄⋊(C₁₇₉⋊C₈)

C₂²

yes

—

C₂²⋊(C₁₇₉⋊C₈)

—

C₄×C₁₇₉⋊C₈

C₂²

yes

characteristic

C₄×C₁₇₉⋊C₈

C₂²

yes

characteristic

C₂×D₈×C₁₇₉

C₄

yes

characteristic

C₄×D₄×C₁₇₉

C₂²

yes

—

#107×C₁₇₉

C₄

yes

characteristic

#102×C₁₇₉

C₄

yes

—

C₄×C₈×C₁₇₉

C₂²

yes

characteristic

#15668243

—

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

D₄

yes

—

C₂×D₄×C₁₇₉

C₂×C₄

yes

—

C₂²×C₄×C₁₇₉

—

D₈×C₁₇₉

—

C₂×C₈×C₁₇₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₇₉

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₇₉

C₂×C₄

yes

—

C₂²⋊C₄×C₁₇₉

—

C₄²×C₁₇₉

C₂³

yes

characteristic

C₁₇₉⋊C₈

—

C₁₇₉⋊C₈

—

C₁₇₉⋊C₈

—

C₂³×C₁₇₉

—

D₄×C₁₇₉

—

D₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

#41

yes

characteristic

C₈×C₁₇₉

—

C₈×C₁₇₉

M₁₆

yes

complemented

C₂²×C₁₇₉

—

C₂²×C₁₇₉

—

C₂²×C₁₇₉

—

C₂²×C₁₇₉

C₄×D₄

yes

characteristic

C₄×C₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

C₂×M₁₆

yes

C₄×C₁₇₉

yes

C₄×C₁₇₉

yes

—

C₂×C₁₇₉

—

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

C₁₇₉

yes

characteristic, complemented, Sylow

#1594

—

179

maximal, Sylow

C₄×D₈

D₁₇₉

yes

characteristic

#436

—

179

—

#334

—

179

—

D₄⋊C₈

—

179

—

#322

—

179

—

C₂×D₈

Dic₁₇₉

yes

characteristic

C₂×M₁₆

—

358

—

C₄×D₄

C₂×D₁₇₉

yes

—

#107

Dic₁₇₉

yes

characteristic

C₄⋊C₈

—

179

—

#102

Dic₁₇₉

yes

—

C₂²⋊C₈

—

358

—

C₄×C₈

—

179

—

C₄×C₈

—

179

—

C₄×C₈

C₂×D₁₇₉

yes

characteristic

C₂×D₄

C₂×Dic₁₇₉

yes

—

C₂²×C₄

—

D₈

—

M₁₆

—

716

—

C₂×C₈

—

358

—

C₂×C₈

—

358

—

C₂×C₈

—

179

—

C₂×C₈

C₂×Dic₁₇₉

yes

characteristic

C₂×C₈

C₂×Dic₁₇₉

yes

characteristic

C₄⋊C₄

C₂×Dic₁₇₉

yes

—

C₂²⋊C₄

—

C₄²

C₂²×D₁₇₉

yes

characteristic

C₂³

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×Dic₁₇₉

yes

characteristic

C₂×C₄

D₄×D₁₇₉

yes

characteristic

C₂×C₄

#15668258

yes

characteristic

C₈

—

C₈

—

358

—

C₈

—

716

—

C₈

—

716

—

C₈

#15668243

yes

complemented

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

D₄×Dic₁₇₉

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

C₂×#15668243

yes

characteristic

105

C₄

#19153120

yes

characteristic

106

C₄

#19153137

yes

—

107

C₂

—

108

C₂

#26038517

yes

characteristic

109

C₂

#26038551

yes

characteristic

110

C₂

#26038561

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₇₉
commutator	C₂×C₄×C₁₇₉	C₂²×C₄
Frattini	C₄²	C₂²×D₁₇₉
Fitting	C₄×D₈×C₁₇₉	C₂
socle	C₂²×C₁₇₉	C₄×D₄
prime radical	C₂×D₈×C₁₇₉	C₄
cosocle	C₄²×C₁₇₉	C₂³
hypercenter	C₄×D₈	D₁₇₉
hypocenter	C₁₇₉	#1594
LCS	G ▷ C₂×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

subgroup

G/H

center

C₂×C₄

D₄×D₁₇₉

commutator

C₂×C₄×C₁₇₉

C₂²×C₄

Frattini

C₄²

C₂²×D₁₇₉

Fitting

C₄×D₈×C₁₇₉

C₂

socle

C₂²×C₁₇₉

C₄×D₄

prime radical

C₂×D₈×C₁₇₉

C₄

cosocle

C₄²×C₁₇₉

C₂³

hypercenter

C₄×D₈

D₁₇₉

hypocenter

C₁₇₉

#1594

LCS

G ▷ C₂×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

N	H ≅ G/N	type
1. #26038517	C₂	semidirect
2. #26038190	C₂	semidirect
3. D₄⋊(C₁₇₉⋊C₈)	C₂	semidirect
4. #26038160	C₂	semidirect
41. C₈×C₁₇₉	M₁₆	semidirect
55. C₁₇₉	#1594	semidirect
97. C₈	#15668243	semidirect

H ≅ G/N

type

1. #26038517

C₂

semidirect

2. #26038190

C₂

semidirect

3. D₄⋊(C₁₇₉⋊C₈)

C₂

semidirect

4. #26038160

C₂

semidirect

41. C₈×C₁₇₉

M₁₆

semidirect

55. C₁₇₉

#1594

semidirect

97. C₈

#15668243

semidirect

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

subgroup	index
#26038517	2
#26038190	2
D₄⋊(C₁₇₉⋊C₈)	2
#26038160	2
C₄×D₈×C₁₇₉	2
#1594	179

subgroup

index

D₄⋊(C₁₇₉⋊C₈)

C₄×D₈×C₁₇₉

179

Subgroups — #37507914

Conjugacy classes of subgroups

Complements

Maximal subgroups