Group #37509982 — subgroups

Subgroups — #37509982

Order 22912.

244 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 97 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	#26038463	C₂	1	1	yes	maximal, characteristic
2	#26038285	C₂	1	1	yes	maximal, characteristic
3	#26038198	C₂	1	4	yes	maximal
4	#533×C₁₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#15668243	C₂²	1	2	yes	—
6	C₂×C₄×Dic₁₇₉	C₂²	1	2	yes	—
7	C₄⋊(C₁₇₉⋊C₈)	—	2	2	no	—
8	C₁₇₉⋊#95	—	2	2	no	—
9	C₄⋊Q₈×C₁₇₉	C₄	1	2	yes	complemented
10	#125×C₁₇₉	C₄	1	2	yes	complemented
11	C₂²⋊Q₈×C₁₇₉	—	2	2	no	—
12	C₄×Q₈×C₁₇₉	—	2	2	no	—
13	#117×C₁₇₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₇₉	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₇₉	—	2	2	no	—
16	#15668243	—	2	4	no	—
17	C₂×C₁₇₉⋊C₈	—	2	2	no	—
18	C₄×Dic₁₇₉	—	2	2	no	—
19	C₄×Dic₁₇₉	Q₈	1	4	yes	complemented
20	C₂×Q₈×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	4	no	—
24	C₄⋊C₄×C₁₇₉	—	2	2	no	—
25	C₄⋊C₄×C₁₇₉	C₂×C₄	1	4	yes	—
26	C₂²⋊C₄×C₁₇₉	—	2	2	no	—
27	C₄²×C₁₇₉	—	2	2	no	—
28	C₄²×C₁₇₉	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₁₇₉	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₁₇₉	D₄	1	2	yes	—
31	C₂×Dic₁₇₉	—	2	4	no	—
32	C₂×Dic₁₇₉	—	4	4	no	—
33	C₁₇₉⋊C₈	—	4	2	no	—
34	C₂³×C₁₇₉	C₂×D₄	1	1	yes	characteristic
35	Q₈×C₁₇₉	—	4	4	no	—
36	C₂×C₄×C₁₇₉	—	2	4	no	—
37	C₂×C₄×C₁₇₉	—	4	2	no	—
38	C₂×C₄×C₁₇₉	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₇₉	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₁₇₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₇₉	C₂×Q₈	1	2	yes	—
43	C₂×C₄×C₁₇₉	#41	1	2	yes	—
44	C₂²×C₁₇₉	—	2	2	no	—
45	C₂²×C₁₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₁₇₉	C₂²⋊Q₈	1	2	yes	—
47	C₄×C₁₇₉	—	2	2	no	—
48	C₄×C₁₇₉	—	2	2	no	—
49	C₄×C₁₇₉	—	4	4	no	—
50	C₄×C₁₇₉	C₄×Q₈	1	2	yes	—
51	C₄×C₁₇₉	#124	1	1	yes	characteristic
52	C₄×C₁₇₉	C₄⋊Q₈	1	1	yes	characteristic
53	Dic₁₇₉	—	4	4	no	—
54	C₂×C₁₇₉	—	2	2	no	—
55	C₂×C₁₇₉	#391	1	1	yes	characteristic
56	C₂×C₁₇₉	#421	1	2	yes	—
57	C₁₇₉	#1995	1	1	yes	characteristic, complemented, Sylow
58	#1995	—	179	1	no	maximal, Sylow
59	#533	D₁₇₉	1	1	yes	characteristic
60	#423	—	179	1	no	—
61	#376	—	179	1	no	—
62	#339	—	179	4	no	—
63	C₂×M₁₆	—	179	2	no	—
64	C₄⋊Q₈	Dic₁₇₉	1	2	yes	complemented
65	#125	Dic₁₇₉	1	2	yes	complemented
66	C₂²⋊Q₈	—	2	2	no	—
67	C₄×Q₈	—	2	2	no	—
68	#117	C₂×D₁₇₉	1	2	yes	—
69	C₂×C₄²	—	179	2	no	—
70	C₂×C₄²	C₂×D₁₇₉	1	1	yes	characteristic
71	C₄⋊C₈	—	358	2	no	—
72	#95	—	358	2	no	—
73	C₂×Q₈	—	2	2	no	—
74	C₂²×C₄	—	358	2	no	—
75	C₂²×C₄	C₂²⋊D₁₇₉	1	2	yes	—
76	C₂²×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
77	M₁₆	—	358	4	no	—
78	C₂×C₈	—	358	2	no	—
79	C₄⋊C₄	—	2	4	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	C₂×Dic₁₇₉	1	4	yes	—
82	C₂²⋊C₄	—	2	2	no	—
83	C₄²	—	2	2	no	—
84	C₄²	—	358	2	no	—
85	C₄²	—	179	4	no	—
86	C₄²	C₂×Dic₁₇₉	1	1	yes	characteristic
87	C₄²	C₂×Dic₁₇₉	1	1	yes	characteristic
88	C₄²	C₂²⋊D₁₇₉	1	2	yes	—
89	C₂³	C₂×C₂²⋊D₁₇₉	1	1	yes	characteristic
90	Q₈	—	4	4	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	4	2	no	—
93	C₂×C₄	—	358	4	no	—
94	C₂×C₄	—	716	4	no	—
95	C₂×C₄	C₂²⋊Dic₁₇₉	1	2	yes	—
96	C₂×C₄	C₂²⋊Dic₁₇₉	1	2	yes	—
97	C₂×C₄	C₂²×Dic₁₇₉	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₁₇₉	1	1	yes	characteristic
99	C₂×C₄	Q₈×D₁₇₉	1	2	yes	—
100	C₂×C₄	#15668260	1	2	yes	—
101	C₈	—	716	2	no	—
102	C₂²	—	2	2	no	—
103	C₂²	C₂×C₂²⋊Dic₁₇₉	1	1	yes	characteristic
104	C₂²	#19153089	1	2	yes	—
105	C₄	—	2	2	no	—
106	C₄	—	2	2	no	—
107	C₄	—	4	4	no	—
108	C₄	—	716	4	no	—
109	C₄	Q₈×Dic₁₇₉	1	2	yes	—
110	C₄	#19153102	1	1	yes	characteristic
111	C₄	#19153111	1	1	yes	characteristic
112	C₂	—	2	2	no	—
113	C₂	#26038346	1	1	yes	characteristic
114	C₂	#26038456	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26038463

C₂

yes

maximal, characteristic

#26038285

C₂

yes

maximal, characteristic

#26038198

C₂

yes

maximal

#533×C₁₇₉

C₂

yes

maximal, characteristic

C₂×#15668243

C₂²

yes

—

C₂×C₄×Dic₁₇₉

C₂²

yes

—

C₄⋊(C₁₇₉⋊C₈)

—

C₁₇₉⋊#95

—

C₄⋊Q₈×C₁₇₉

C₄

yes

complemented

#125×C₁₇₉

C₄

yes

complemented

C₂²⋊Q₈×C₁₇₉

—

C₄×Q₈×C₁₇₉

—

#117×C₁₇₉

C₂²

yes

—

C₂×C₄²×C₁₇₉

C₂²

yes

characteristic

C₂²×Dic₁₇₉

—

#15668243

—

C₂×C₁₇₉⋊C₈

—

C₄×Dic₁₇₉

—

C₄×Dic₁₇₉

Q₈

yes

complemented

C₂×Q₈×C₁₇₉

—

C₂²×C₄×C₁₇₉

D₄

yes

—

C₂²×C₄×C₁₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₇₉

—

C₄⋊C₄×C₁₇₉

—

C₄⋊C₄×C₁₇₉

C₂×C₄

yes

—

C₂²⋊C₄×C₁₇₉

—

C₄²×C₁₇₉

—

C₄²×C₁₇₉

C₂×C₄

yes

characteristic

C₄²×C₁₇₉

C₂×C₄

yes

characteristic

C₄²×C₁₇₉

D₄

yes

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₁₇₉⋊C₈

—

C₂³×C₁₇₉

C₂×D₄

yes

characteristic

Q₈×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

C₂²⋊C₄

yes

—

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

—

C₂²×C₁₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₇₉

C₂²⋊Q₈

yes

—

C₄×C₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

C₄×Q₈

yes

—

C₄×C₁₇₉

#124

yes

characteristic

C₄×C₁₇₉

C₄⋊Q₈

yes

characteristic

Dic₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

#391

yes

characteristic

C₂×C₁₇₉

#421

yes

—

C₁₇₉

#1995

yes

characteristic, complemented, Sylow

#1995

—

179

maximal, Sylow

#533

D₁₇₉

yes

characteristic

#423

—

179

—

#376

—

179

—

#339

—

179

—

C₂×M₁₆

—

179

—

C₄⋊Q₈

Dic₁₇₉

yes

complemented

#125

Dic₁₇₉

yes

complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₁₇₉

yes

—

C₂×C₄²

—

179

—

C₂×C₄²

C₂×D₁₇₉

yes

characteristic

C₄⋊C₈

—

358

—

#95

—

358

—

C₂×Q₈

—

C₂²×C₄

—

358

—

C₂²×C₄

C₂²⋊D₁₇₉

yes

—

C₂²×C₄

C₂²×D₁₇₉

yes

characteristic

M₁₆

—

358

—

C₂×C₈

—

358

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₁₇₉

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

358

—

C₄²

—

179

—

C₄²

C₂×Dic₁₇₉

yes

characteristic

C₄²

C₂×Dic₁₇₉

yes

characteristic

C₄²

C₂²⋊D₁₇₉

yes

—

C₂³

C₂×C₂²⋊D₁₇₉

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

358

—

C₂×C₄

—

716

—

C₂×C₄

C₂²⋊Dic₁₇₉

yes

—

C₂×C₄

C₂²⋊Dic₁₇₉

yes

—

C₂×C₄

C₂²×Dic₁₇₉

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₇₉

yes

characteristic

C₂×C₄

Q₈×D₁₇₉

yes

—

100

C₂×C₄

#15668260

yes

—

101

C₈

—

716

—

102

C₂²

—

103

C₂²

C₂×C₂²⋊Dic₁₇₉

yes

characteristic

104

C₂²

#19153089

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

716

—

109

C₄

Q₈×Dic₁₇₉

yes

—

110

C₄

#19153102

yes

characteristic

111

C₄

#19153111

yes

characteristic

112

C₂

—

113

C₂

#26038346

yes

characteristic

114

C₂

#26038456

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₇₉
commutator	C₂×C₄×C₁₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇₉
Fitting	#533×C₁₇₉	C₂
socle	C₂²×C₁₇₉	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₇₉	C₂×D₄
cosocle	C₂²×C₄×C₁₇₉	C₂³
hypercenter	#533	D₁₇₉
hypocenter	C₁₇₉	#1995
LCS	G ▷ C₂×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₇₉

commutator

C₂×C₄×C₁₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇₉

Fitting

#533×C₁₇₉

C₂

socle

C₂²×C₁₇₉

C₂×C₂²⋊C₄

prime radical

C₂³×C₁₇₉

C₂×D₄

cosocle

C₂²×C₄×C₁₇₉

C₂³

D₁₇₉

C₁₇₉

G ▷ C₂×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
9. C₄⋊Q₈×C₁₇₉	C₄	semidirect
10. #125×C₁₇₉	C₄	semidirect
19. C₄×Dic₁₇₉	Q₈	semidirect
57. C₁₇₉	#1995	semidirect
64. C₄⋊Q₈	Dic₁₇₉	semidirect
65. #125	Dic₁₇₉	semidirect

H ≅ G/N

type

9. C₄⋊Q₈×C₁₇₉

C₄

semidirect

10. #125×C₁₇₉

C₄

semidirect

19. C₄×Dic₁₇₉

Q₈

semidirect

57. C₁₇₉

#1995

semidirect

64. C₄⋊Q₈

Dic₁₇₉

semidirect

65. #125

Dic₁₇₉

semidirect

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

subgroup	index
#26038463	2
#26038285	2
#26038198	2
#533×C₁₇₉	2
#1995	179

subgroup

index

#533×C₁₇₉

179

Subgroups — #37509982

Conjugacy classes of subgroups

Complements

Maximal subgroups