Group #37509821 — subgroups

Subgroups — #37509821

Order 22912.

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

normal subgroups only

This group has 308 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×#19153037	C₂	1	4	yes	maximal
2	C₂×#19153024	C₂	1	1	yes	maximal, characteristic
3	#26038305	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Q₈×C₁₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₇₉⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₇₉	C₂²	1	2	yes	—
7	#19153037	—	2	8	no	—
8	#19153024	—	2	2	no	—
9	C₁₇₉⋊#95	—	2	2	no	—
10	C₂²×Q₈×C₁₇₉	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₁₇₉	—	2	2	no	—
12	C₄⋊Q₈×C₁₇₉	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₁₇₉	—	2	2	no	—
14	C₂×C₄²×C₁₇₉	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₇₉	—	2	2	no	—
16	C₂×C₁₇₉⋊C₈	—	2	2	no	—
17	C₂×C₁₇₉⋊C₈	—	2	2	no	—
18	C₂×C₁₇₉⋊C₈	D₄	1	4	yes	—
19	C₄⋊Dic₁₇₉	—	2	4	no	—
20	C₂×Q₈×C₁₇₉	—	2	2	no	—
21	C₂×Q₈×C₁₇₉	—	2	4	no	—
22	C₂×Q₈×C₁₇₉	—	2	4	no	—
23	C₂×Q₈×C₁₇₉	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₁₇₉	—	2	2	no	—
25	C₂²×C₄×C₁₇₉	D₄	1	2	yes	—
26	C₂²×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₁₇₉	—	2	8	no	—
28	C₄²×C₁₇₉	C₂×C₄	1	2	yes	—
29	C₄²×C₁₇₉	D₄	1	2	yes	—
30	C₂×Dic₁₇₉	—	4	2	no	—
31	C₂×Dic₁₇₉	—	4	4	no	—
32	C₁₇₉⋊C₈	—	2	4	no	—
33	C₂³×C₁₇₉	C₂×D₄	1	1	yes	characteristic
34	Q₈×C₁₇₉	—	2	8	no	—
35	Q₈×C₁₇₉	—	4	4	no	—
36	C₂×C₄×C₁₇₉	—	2	4	no	—
37	C₂×C₄×C₁₇₉	—	2	4	no	—
38	C₂×C₄×C₁₇₉	—	4	2	no	—
39	C₂×C₄×C₁₇₉	—	4	2	no	—
40	C₂×C₄×C₁₇₉	C₂²⋊C₄	1	4	yes	—
41	C₂×C₄×C₁₇₉	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₇₉	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₇₉	#41	1	2	yes	—
45	C₂²×C₁₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₁₇₉	#124	1	1	yes	characteristic
47	C₂²×C₁₇₉	#127	1	1	yes	characteristic
48	C₂²×C₁₇₉	#137	1	4	yes	—
49	C₄×C₁₇₉	—	2	4	no	—
50	C₄×C₁₇₉	—	4	4	no	—
51	C₄×C₁₇₉	C₄×D₄	1	2	yes	—
52	C₄×C₁₇₉	#121	1	2	yes	—
53	Dic₁₇₉	—	8	2	no	—
54	C₂×C₁₇₉	#390	1	1	yes	characteristic
55	C₂×C₁₇₉	#419	1	2	yes	—
56	C₂×C₁₇₉	#491	1	2	yes	—
57	C₂×C₁₇₉	#497	1	2	yes	—
58	C₁₇₉	#1960	1	1	yes	characteristic, complemented, Sylow
59	#1960	—	179	1	no	maximal, Sylow
60	C₂×C₄⋊Q₈	D₁₇₉	1	1	yes	characteristic
61	C₂×#103	—	179	4	no	—
62	C₂×#97	—	179	1	no	—
63	#382	—	179	1	no	—
64	C₂²×Q₈	C₂×D₁₇₉	1	2	yes	—
65	C₂²×C₈	—	179	2	no	—
66	C₄⋊Q₈	—	2	2	no	—
67	C₄⋊Q₈	Dic₁₇₉	1	4	yes	complemented
68	C₂×C₄⋊C₄	—	2	2	no	—
69	C₂×C₄⋊C₄	—	179	2	no	—
70	C₂×C₄²	C₂×D₁₇₉	1	1	yes	characteristic
71	#103	—	358	8	no	—
72	#97	—	358	2	no	—
73	#95	—	358	2	no	—
74	C₂×Q₈	—	2	2	no	—
75	C₂×Q₈	—	2	4	no	—
76	C₂×Q₈	—	2	4	no	—
77	C₂×Q₈	C₂×Dic₁₇₉	1	4	yes	—
78	C₂²×C₄	—	2	2	no	—
79	C₂²×C₄	—	358	2	no	—
80	C₂²×C₄	C₂²⋊D₁₇₉	1	2	yes	—
81	C₂²×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
82	C₂×C₈	—	358	2	no	—
83	C₂×C₈	—	358	2	no	—
84	C₂×C₈	—	179	4	no	—
85	C₄⋊C₄	—	2	8	no	—
86	C₄⋊C₄	—	358	4	no	—
87	C₄²	C₂×Dic₁₇₉	1	2	yes	—
88	C₄²	C₂²⋊D₁₇₉	1	2	yes	—
89	C₂³	C₂×C₂²⋊D₁₇₉	1	1	yes	characteristic
90	Q₈	—	2	8	no	—
91	Q₈	—	4	4	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	716	2	no	—
97	C₂×C₄	—	716	4	no	—
98	C₂×C₄	C₂²⋊Dic₁₇₉	1	4	yes	—
99	C₂×C₄	C₂²×Dic₁₇₉	1	1	yes	characteristic
100	C₂×C₄	D₄×D₁₇₉	1	2	yes	—
101	C₂×C₄	C₂×C₂²⋊D₁₇₉	1	1	yes	characteristic
102	C₂×C₄	#15668258	1	2	yes	—
103	C₈	—	358	4	no	—
104	C₂²	C₂×C₂²⋊Dic₁₇₉	1	1	yes	characteristic
105	C₂²	#19153100	1	1	yes	characteristic
106	C₂²	#19153109	1	1	yes	characteristic
107	C₂²	#19153144	1	4	yes	—
108	C₄	—	2	4	no	—
109	C₄	—	4	4	no	—
110	C₄	—	1432	2	no	—
111	C₄	D₄×Dic₁₇₉	1	2	yes	—
112	C₄	#19153084	1	2	yes	—
113	C₂	#26038342	1	1	yes	characteristic
114	C₂	#26038448	1	2	yes	—
115	C₂	#26038807	1	2	yes	—
116	C₂	#26038831	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19153037

C₂

yes

maximal

C₂×#19153024

C₂

yes

maximal, characteristic

#26038305

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₇₉

C₂

yes

maximal, characteristic

C₂²×C₁₇₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₇₉

C₂²

yes

—

#19153037

—

#19153024

—

C₁₇₉⋊#95

—

C₂²×Q₈×C₁₇₉

C₂²

yes

—

C₄⋊Q₈×C₁₇₉

—

C₄⋊Q₈×C₁₇₉

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₇₉

—

C₂×C₄²×C₁₇₉

C₂²

yes

characteristic

C₂²×Dic₁₇₉

—

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

D₄

yes

—

C₄⋊Dic₁₇₉

—

C₂×Q₈×C₁₇₉

—

C₂×Q₈×C₁₇₉

—

C₂×Q₈×C₁₇₉

—

C₂×Q₈×C₁₇₉

C₂×C₄

yes

—

C₂²×C₄×C₁₇₉

—

C₂²×C₄×C₁₇₉

D₄

yes

—

C₂²×C₄×C₁₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₇₉

—

C₄²×C₁₇₉

C₂×C₄

yes

—

C₄²×C₁₇₉

D₄

yes

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₁₇₉⋊C₈

—

C₂³×C₁₇₉

C₂×D₄

yes

characteristic

Q₈×C₁₇₉

—

Q₈×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

—

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

#41

yes

—

C₂²×C₁₇₉

C₂×C₂²⋊C₄

yes

C₂²×C₁₇₉

yes

C₂²×C₁₇₉

yes

C₂²×C₁₇₉

yes

—

C₄×C₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

C₄×D₄

yes

—

C₄×C₁₇₉

#121

yes

—

Dic₁₇₉

—

C₂×C₁₇₉

#390

yes

characteristic

C₂×C₁₇₉

#419

yes

—

C₂×C₁₇₉

#491

yes

—

C₂×C₁₇₉

#497

yes

—

C₁₇₉

#1960

yes

characteristic, complemented, Sylow

#1960

—

179

maximal, Sylow

C₂×C₄⋊Q₈

D₁₇₉

yes

characteristic

C₂×#103

—

179

—

C₂×#97

—

179

—

#382

—

179

—

C₂²×Q₈

C₂×D₁₇₉

yes

—

C₂²×C₈

—

179

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₁₇₉

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

179

—

C₂×C₄²

C₂×D₁₇₉

yes

characteristic

#103

—

358

—

#97

—

358

—

#95

—

358

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₁₇₉

yes

—

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

—

358

—

C₄²

C₂×Dic₁₇₉

yes

—

C₄²

C₂²⋊D₁₇₉

yes

—

C₂³

C₂×C₂²⋊D₁₇₉

yes

characteristic

Q₈

—

Q₈

—

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

100

C₂×C₄

D₄×D₁₇₉

yes

—

101

C₂×C₄

C₂×C₂²⋊D₁₇₉

yes

characteristic

102

C₂×C₄

#15668258

yes

—

103

C₈

—

358

—

104

C₂²

C₂×C₂²⋊Dic₁₇₉

yes

characteristic

105

C₂²

#19153100

yes

characteristic

106

C₂²

#19153109

yes

characteristic

107

C₂²

#19153144

yes

—

108

C₄

—

109

C₄

—

110

C₄

—

1432

—

111

C₄

D₄×Dic₁₇₉

yes

—

112

C₄

#19153084

yes

—

113

C₂

#26038342

yes

characteristic

114

C₂

#26038448

yes

—

115

C₂

#26038807

yes

—

116

C₂

#26038831

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₇₉
commutator	C₂×C₄×C₁₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇₉
Fitting	C₂×C₄⋊Q₈×C₁₇₉	C₂
socle, prime radical	C₂³×C₁₇₉	C₂×D₄
cosocle	C₂²×C₄×C₁₇₉	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₇₉
hypocenter	C₁₇₉	#1960
LCS	G ▷ C₂×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₇₉

commutator

C₂×C₄×C₁₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇₉

Fitting

C₂×C₄⋊Q₈×C₁₇₉

C₂

socle, prime radical

C₂³×C₁₇₉

C₂×D₄

cosocle

C₂²×C₄×C₁₇₉

C₂³

hypercenter

C₂×C₄⋊Q₈

D₁₇₉

hypocenter

C₁₇₉

#1960

LCS

G ▷ C₂×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
12. C₄⋊Q₈×C₁₇₉	C₄	semidirect
58. C₁₇₉	#1960	semidirect
67. C₄⋊Q₈	Dic₁₇₉	semidirect

H ≅ G/N

type

12. C₄⋊Q₈×C₁₇₉

C₄

semidirect

58. C₁₇₉

#1960

semidirect

67. C₄⋊Q₈

Dic₁₇₉

semidirect

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

subgroup	index
C₂×#19153037	2
C₂×#19153024	2
#26038305	2
C₂×C₄⋊Q₈×C₁₇₉	2
#1960	179

subgroup

index

C₂×C₄⋊Q₈×C₁₇₉

179

Subgroups — #37509821

Conjugacy classes of subgroups

Complements

Maximal subgroups