Group #37509820 — subgroups

Subgroups — #37509820

Order 22912.

308 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 109 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×#19153113	C₂	1	1	yes	maximal, characteristic
2	C₂×#19153038	C₂	1	4	yes	maximal
3	#26038305	C₂	1	1	yes	maximal, characteristic
4	C₂×#97×C₁₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₅₈	C₂²	1	2	yes	—
6	#19153113	—	2	2	no	—
7	#19153113	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₁₇₉	—	2	2	no	—
9	C₂×C₄⋊Dic₁₇₉	C₂²	1	2	yes	—
10	#19153038	—	2	8	no	—
11	C₁₇₉⋊#95	—	2	2	no	—
12	C₂²×C₈×C₁₇₉	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₇₉	C₂²	1	1	yes	characteristic
14	#97×C₁₇₉	—	2	2	no	—
15	C₂×Dic₃₅₈	—	2	2	no	—
16	C₂×Dic₃₅₈	—	2	4	no	—
17	C₂×Dic₃₅₈	—	2	4	no	—
18	C₂×Dic₃₅₈	C₂×C₄	1	4	yes	—
19	C₂²×Dic₁₇₉	—	2	2	no	—
20	C₂²×Dic₁₇₉	—	2	2	no	—
21	C₄⋊Dic₁₇₉	—	2	8	no	—
22	C₄⋊Dic₁₇₉	—	2	4	no	—
23	C₂²×C₄×C₁₇₉	D₄	1	2	yes	—
24	C₂²×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₁₇₉	—	2	2	no	—
26	C₂×C₈×C₁₇₉	—	2	2	no	—
27	C₂×C₈×C₁₇₉	D₄	1	4	yes	—
28	C₄²×C₁₇₉	C₂×C₄	1	2	yes	—
29	C₄²×C₁₇₉	D₄	1	2	yes	—
30	Dic₃₅₈	—	2	8	no	—
31	Dic₃₅₈	—	4	4	no	—
32	C₂×Dic₁₇₉	—	2	4	no	—
33	C₂×Dic₁₇₉	—	4	2	no	—
34	C₂×Dic₁₇₉	—	4	2	no	—
35	C₂×Dic₁₇₉	—	4	2	no	—
36	C₂×Dic₁₇₉	—	4	4	no	—
37	C₂³×C₁₇₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₇₉	—	2	4	no	—
39	C₂×C₄×C₁₇₉	C₂²⋊C₄	1	4	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×D₄	1	2	yes	—
43	C₂×C₄×C₁₇₉	#41	1	2	yes	—
44	C₈×C₁₇₉	—	2	4	no	—
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	C₄×D₄	1	2	yes	—
51	C₄×C₁₇₉	#121	1	2	yes	—
52	Dic₁₇₉	—	4	4	no	—
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	—	179	1	no	—
61	C₂×#103	—	179	4	no	—
62	C₂×#97	D₁₇₉	1	1	yes	characteristic
63	#382	—	179	1	no	—
64	C₂²×Q₈	—	179	2	no	—
65	C₂²×C₈	C₂×D₁₇₉	1	2	yes	—
66	C₄⋊Q₈	—	358	2	no	—
67	C₄⋊Q₈	—	179	4	no	—
68	C₂×C₄⋊C₄	—	358	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	—	2	2	no	—
73	#95	—	358	2	no	—
74	C₂×Q₈	—	358	2	no	—
75	C₂×Q₈	—	358	4	no	—
76	C₂×Q₈	—	358	4	no	—
77	C₂×Q₈	—	179	4	no	—
78	C₂²×C₄	—	358	2	no	—
79	C₂²×C₄	—	358	2	no	—
80	C₂²×C₄	D₇₁₆	1	2	yes	—
81	C₂²×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	D₇₁₆	1	4	yes	—
85	C₄⋊C₄	—	358	8	no	—
86	C₄⋊C₄	—	358	4	no	—
87	C₄²	C₄×D₁₇₉	1	2	yes	—
88	C₄²	C₂²⋊D₁₇₉	1	2	yes	—
89	C₂³	C₂×D₇₁₆	1	1	yes	characteristic
90	Q₈	—	358	8	no	—
91	Q₈	—	716	4	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	358	4	no	—
94	C₂×C₄	—	716	2	no	—
95	C₂×C₄	—	716	4	no	—
96	C₂×C₄	—	716	2	no	—
97	C₂×C₄	—	716	2	no	—
98	C₂×C₄	(C₂×D₁₇₉)⋊C₄	1	4	yes	—
99	C₂×C₄	C₂×C₄×D₁₇₉	1	1	yes	characteristic
100	C₂×C₄	C₂×C₂²⋊D₁₇₉	1	1	yes	characteristic
101	C₂×C₄	C₂×D₇₁₆	1	2	yes	—
102	C₂×C₄	#15668259	1	2	yes	—
103	C₈	—	2	4	no	—
104	C₂²	C₂×(C₂×D₁₇₉)⋊C₄	1	1	yes	characteristic
105	C₂²	#19153101	1	1	yes	characteristic
106	C₂²	#19153110	1	1	yes	characteristic
107	C₂²	#19153146	1	4	yes	—
108	C₄	—	2	4	no	—
109	C₄	—	1432	2	no	—
110	C₄	—	716	4	no	—
111	C₄	C₄×D₇₁₆	1	2	yes	—
112	C₄	#19153087	1	2	yes	—
113	C₂	#26038341	1	1	yes	characteristic
114	C₂	#26038449	1	2	yes	—
115	C₂	#26038808	1	2	yes	—
116	C₂	#26038833	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19153113

C₂

yes

maximal, characteristic

C₂×#19153038

C₂

yes

maximal

#26038305

C₂

yes

maximal, characteristic

C₂×#97×C₁₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₃₅₈

C₂²

yes

—

#19153113

—

#19153113

C₄

yes

complemented

C₂×C₄⋊Dic₁₇₉

—

C₂×C₄⋊Dic₁₇₉

C₂²

yes

—

#19153038

—

C₁₇₉⋊#95

—

C₂²×C₈×C₁₇₉

C₂²

yes

—

C₂×C₄²×C₁₇₉

C₂²

yes

characteristic

#97×C₁₇₉

—

C₂×Dic₃₅₈

—

C₂×Dic₃₅₈

—

C₂×Dic₃₅₈

—

C₂×Dic₃₅₈

C₂×C₄

yes

—

C₂²×Dic₁₇₉

—

C₂²×Dic₁₇₉

—

C₄⋊Dic₁₇₉

—

C₄⋊Dic₁₇₉

—

C₂²×C₄×C₁₇₉

D₄

yes

—

C₂²×C₄×C₁₇₉

C₂³

yes

characteristic

C₂×C₈×C₁₇₉

—

C₂×C₈×C₁₇₉

—

C₂×C₈×C₁₇₉

D₄

yes

—

C₄²×C₁₇₉

C₂×C₄

yes

—

C₄²×C₁₇₉

D₄

yes

—

Dic₃₅₈

—

Dic₃₅₈

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₂³×C₁₇₉

C₂×D₄

yes

characteristic

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×D₄

yes

—

C₂×C₄×C₁₇₉

#41

yes

—

C₈×C₁₇₉

—

C₂²×C₁₇₉

C₂×C₂²⋊C₄

yes

C₂²×C₁₇₉

yes

C₂²×C₁₇₉

yes

C₂²×C₁₇₉

yes

—

C₄×C₁₇₉

—

C₄×C₁₇₉

C₄×D₄

yes

—

C₄×C₁₇₉

#121

yes

—

Dic₁₇₉

—

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

—

179

—

C₂×#103

—

179

—

C₂×#97

D₁₇₉

yes

characteristic

#382

—

179

—

C₂²×Q₈

—

179

—

C₂²×C₈

C₂×D₁₇₉

yes

—

C₄⋊Q₈

—

358

—

C₄⋊Q₈

—

179

—

C₂×C₄⋊C₄

—

358

—

C₂×C₄⋊C₄

—

179

—

C₂×C₄²

C₂×D₁₇₉

yes

characteristic

#103

—

358

—

#97

—

#95

—

358

—

C₂×Q₈

—

358

—

C₂×Q₈

—

358

—

C₂×Q₈

—

358

—

C₂×Q₈

—

179

—

C₂²×C₄

—

358

—

C₂²×C₄

—

358

—

C₂²×C₄

D₇₁₆

yes

—

C₂²×C₄

C₂²×D₁₇₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₇₁₆

yes

—

C₄⋊C₄

—

358

—

C₄⋊C₄

—

358

—

C₄²

C₄×D₁₇₉

yes

—

C₄²

C₂²⋊D₁₇₉

yes

—

C₂³

C₂×D₇₁₆

yes

characteristic

Q₈

—

358

—

Q₈

—

716

—

C₂×C₄

—

C₂×C₄

—

358

—

C₂×C₄

—

716

—

C₂×C₄

—

716

—

C₂×C₄

—

716

—

C₂×C₄

—

716

—

C₂×C₄

(C₂×D₁₇₉)⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₁₇₉

yes

characteristic

100

C₂×C₄

C₂×C₂²⋊D₁₇₉

yes

characteristic

101

C₂×C₄

C₂×D₇₁₆

yes

—

102

C₂×C₄

#15668259

yes

—

103

C₈

—

104

C₂²

C₂×(C₂×D₁₇₉)⋊C₄

yes

characteristic

105

C₂²

#19153101

yes

characteristic

106

C₂²

#19153110

yes

characteristic

107

C₂²

#19153146

yes

—

108

C₄

—

109

C₄

—

1432

—

110

C₄

—

716

—

111

C₄

C₄×D₇₁₆

yes

—

112

C₄

#19153087

yes

—

113

C₂

#26038341

yes

characteristic

114

C₂

#26038449

yes

—

115

C₂

#26038808

yes

—

116

C₂

#26038833

yes

—

	subgroup	G/H
center	C₂³	C₂×D₇₁₆
commutator	C₂×C₄×C₁₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇₉
Fitting	C₂×#97×C₁₇₉	C₂
socle, prime radical	C₂³×C₁₇₉	C₂×D₄
cosocle	C₂²×C₄×C₁₇₉	C₂³
hypercenter	C₂×#97	D₁₇₉
hypocenter	C₁₇₉	#1960
LCS	G ▷ C₂×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₇₁₆

commutator

C₂×C₄×C₁₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇₉

Fitting

C₂×#97×C₁₇₉

C₂

socle, prime radical

C₂³×C₁₇₉

C₂×D₄

cosocle

C₂²×C₄×C₁₇₉

C₂³

hypercenter

C₂×#97

D₁₇₉

hypocenter

C₁₇₉

#1960

LCS

G ▷ C₂×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
7. #19153113	C₄	semidirect
58. C₁₇₉	#1960	semidirect

H ≅ G/N

type

7. #19153113

C₄

semidirect

58. C₁₇₉

#1960

semidirect

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

subgroup	index
C₂×#19153113	2
C₂×#19153038	2
#26038305	2
C₂×#97×C₁₇₉	2
#1960	179

subgroup

index

C₂×#97×C₁₇₉

179

Subgroups — #37509820

Conjugacy classes of subgroups

Complements

Maximal subgroups