Group #37509801 — subgroups

Subgroups — #37509801

Order 22912.

332 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 137 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×#19153036	C₂	1	4	yes	maximal
3	C₂×C₄×C₁₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#384×C₁₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₅₈	C₂²	1	2	yes	—
6	C₂²×C₁₇₉⋊C₈	C₂²	1	2	yes	—
7	#19153113	—	2	2	no	—
8	#19153113	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₁₇₉	—	2	2	no	—
10	#19153036	—	2	8	no	—
11	C₄×C₁₇₉⋊C₈	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₇₉	—	2	2	no	—
13	C₂×C₄⋊C₄×C₁₇₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₇₉	C₂²	1	1	yes	characteristic
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	C₂×C₁₇₉⋊C₈	—	2	2	no	—
21	C₂×C₁₇₉⋊C₈	—	2	2	no	—
22	C₂×C₁₇₉⋊C₈	D₄	1	4	yes	—
23	C₄⋊Dic₁₇₉	—	2	8	no	—
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	4	no	—
28	C₄⋊C₄×C₁₇₉	—	4	4	no	—
29	C₄²×C₁₇₉	C₂×C₄	1	2	yes	—
30	C₄²×C₁₇₉	D₄	1	2	yes	—
31	Dic₃₅₈	—	2	8	no	—
32	Dic₃₅₈	—	4	4	no	—
33	C₂×Dic₁₇₉	—	2	4	no	—
34	C₂×Dic₁₇₉	—	4	2	no	—
35	C₂×Dic₁₇₉	—	4	2	no	—
36	C₁₇₉⋊C₈	—	2	4	no	—
37	C₂³×C₁₇₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₇₉	—	4	2	no	—
39	C₂×C₄×C₁₇₉	—	4	4	no	—
40	C₂×C₄×C₁₇₉	C₂²⋊C₄	1	4	yes	—
41	C₂×C₄×C₁₇₉	SD₁₆	1	4	yes	—
42	C₂×C₄×C₁₇₉	Q₁₆	1	4	yes	—
43	C₂×C₄×C₁₇₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₇₉	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₇₉	#41	1	2	yes	—
47	C₂²×C₁₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₁₇₉	#124	1	1	yes	characteristic
49	C₂²×C₁₇₉	#127	1	1	yes	characteristic
50	C₂²×C₁₇₉	C₂×SD₁₆	1	2	yes	—
51	C₂²×C₁₇₉	C₂×Q₁₆	1	2	yes	—
52	C₄×C₁₇₉	—	8	2	no	—
53	C₄×C₁₇₉	#103	1	8	yes	complemented
54	C₄×C₁₇₉	C₄×D₄	1	2	yes	—
55	C₄×C₁₇₉	#121	1	2	yes	—
56	Dic₁₇₉	—	4	4	no	—
57	C₂×C₁₇₉	#390	1	1	yes	characteristic
58	C₂×C₁₇₉	C₂×#103	1	2	yes	—
59	C₂×C₁₇₉	#487	1	2	yes	—
60	C₂×C₁₇₉	#492	1	1	yes	characteristic
61	C₂×C₁₇₉	#494	1	1	yes	characteristic
62	C₁₇₉	#1956	1	1	yes	characteristic, complemented, Sylow
63	#1956	—	179	1	no	maximal, Sylow
64	C₂×C₄⋊Q₈	—	179	1	no	—
65	C₂×#103	—	179	4	no	—
66	C₂×C₄×C₈	—	179	1	no	—
67	#384	D₁₇₉	1	1	yes	characteristic
68	C₂²×Q₈	—	179	2	no	—
69	C₂²×C₈	—	179	2	no	—
70	C₄⋊Q₈	—	358	2	no	—
71	C₄⋊Q₈	—	179	4	no	—
72	C₂×C₄⋊C₄	—	2	2	no	—
73	C₂×C₄⋊C₄	—	358	2	no	—
74	C₂×C₄⋊C₄	C₂×D₁₇₉	1	2	yes	—
75	C₂×C₄²	C₂×D₁₇₉	1	1	yes	characteristic
76	#103	—	358	8	no	—
77	C₄×C₈	—	358	2	no	—
78	C₂×Q₈	—	358	2	no	—
79	C₂×Q₈	—	358	4	no	—
80	C₂×Q₈	—	358	4	no	—
81	C₂×Q₈	—	179	4	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	358	2	no	—
84	C₂²×C₄	C₂²⋊D₁₇₉	1	2	yes	—
85	C₂²×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
86	C₂×C₈	—	358	2	no	—
87	C₂×C₈	—	358	2	no	—
88	C₂×C₈	—	179	4	no	—
89	C₄⋊C₄	—	2	4	no	—
90	C₄⋊C₄	—	4	4	no	—
91	C₄⋊C₄	—	358	8	no	—
92	C₄²	C₄×D₁₇₉	1	2	yes	—
93	C₄²	D₇₁₆	1	2	yes	—
94	C₂³	C₂×C₂²⋊D₁₇₉	1	1	yes	characteristic
95	Q₈	—	358	8	no	—
96	Q₈	—	716	4	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	4	4	no	—
99	C₂×C₄	—	358	4	no	—
100	C₂×C₄	—	716	2	no	—
101	C₂×C₄	—	716	2	no	—
102	C₂×C₄	(C₂×D₁₇₉)⋊C₄	1	4	yes	—
103	C₂×C₄	#15668247	1	4	yes	—
104	C₂×C₄	#15668249	1	4	yes	—
105	C₂×C₄	C₂×C₄×D₁₇₉	1	1	yes	characteristic
106	C₂×C₄	D₄×D₁₇₉	1	2	yes	—
107	C₂×C₄	C₂×D₇₁₆	1	1	yes	characteristic
108	C₂×C₄	#15668260	1	2	yes	—
109	C₈	—	358	4	no	—
110	C₂²	C₂×(C₂×D₁₇₉)⋊C₄	1	1	yes	characteristic
111	C₂²	#19153102	1	1	yes	characteristic
112	C₂²	#19153109	1	1	yes	characteristic
113	C₂²	C₂×#15668247	1	2	yes	—
114	C₂²	C₂×#15668249	1	2	yes	—
115	C₄	—	8	2	no	—
116	C₄	—	716	4	no	—
117	C₄	#19153036	1	8	yes	complemented
118	C₄	#19153075	1	2	yes	—
119	C₄	#19153088	1	2	yes	—
120	C₂	#26038340	1	1	yes	characteristic
121	C₂	C₂×#19153036	1	2	yes	—
122	C₂	#26038791	1	2	yes	—
123	C₂	#26038811	1	1	yes	characteristic
124	C₂	#26038818	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19153113

C₂

yes

maximal, characteristic

C₂×#19153036

C₂

yes

maximal

C₂×C₄×C₁₇₉⋊C₈

C₂

yes

maximal, characteristic

#384×C₁₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₃₅₈

C₂²

yes

—

C₂²×C₁₇₉⋊C₈

C₂²

yes

—

#19153113

—

#19153113

C₄

yes

complemented

C₂×C₄⋊Dic₁₇₉

—

#19153036

—

C₄×C₁₇₉⋊C₈

—

C₂×C₄⋊C₄×C₁₇₉

—

C₂×C₄⋊C₄×C₁₇₉

C₂²

yes

—

C₂×C₄²×C₁₇₉

C₂²

yes

characteristic

C₂×Dic₃₅₈

—

C₂×Dic₃₅₈

—

C₂×Dic₃₅₈

—

C₂×Dic₃₅₈

C₂×C₄

yes

—

C₂²×Dic₁₇₉

—

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

D₄

yes

—

C₄⋊Dic₁₇₉

—

C₂²×C₄×C₁₇₉

—

C₂²×C₄×C₁₇₉

D₄

yes

—

C₂²×C₄×C₁₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₇₉

—

C₄⋊C₄×C₁₇₉

—

C₄²×C₁₇₉

C₂×C₄

yes

—

C₄²×C₁₇₉

D₄

yes

—

Dic₃₅₈

—

Dic₃₅₈

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₁₇₉⋊C₈

—

C₂³×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇₉

SD₁₆

yes

—

C₂×C₄×C₁₇₉

Q₁₆

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

C₂×SD₁₆

yes

—

C₂²×C₁₇₉

C₂×Q₁₆

yes

—

C₄×C₁₇₉

—

C₄×C₁₇₉

#103

yes

complemented

C₄×C₁₇₉

C₄×D₄

yes

—

C₄×C₁₇₉

#121

yes

—

Dic₁₇₉

—

C₂×C₁₇₉

#390

yes

characteristic

C₂×C₁₇₉

C₂×#103

yes

—

C₂×C₁₇₉

#487

yes

—

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

C₁₇₉

yes

characteristic, complemented, Sylow

#1956

—

179

maximal, Sylow

C₂×C₄⋊Q₈

—

179

—

C₂×#103

—

179

—

C₂×C₄×C₈

—

179

—

#384

D₁₇₉

yes

characteristic

C₂²×Q₈

—

179

—

C₂²×C₈

—

179

—

C₄⋊Q₈

—

358

—

C₄⋊Q₈

—

179

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

358

—

C₂×C₄⋊C₄

C₂×D₁₇₉

yes

—

C₂×C₄²

C₂×D₁₇₉

yes

characteristic

#103

—

358

—

C₄×C₈

—

358

—

C₂×Q₈

—

358

—

C₂×Q₈

—

358

—

C₂×Q₈

—

358

—

C₂×Q₈

—

179

—

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

—

C₄²

C₄×D₁₇₉

yes

—

C₄²

D₇₁₆

yes

—

C₂³

C₂×C₂²⋊D₁₇₉

yes

characteristic

Q₈

—

358

—

Q₈

—

716

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

358

—

100

C₂×C₄

—

716

—

101

C₂×C₄

—

716

—

102

C₂×C₄

(C₂×D₁₇₉)⋊C₄

yes

—

103

C₂×C₄

#15668247

yes

—

104

C₂×C₄

#15668249

yes

—

105

C₂×C₄

C₂×C₄×D₁₇₉

yes

characteristic

106

C₂×C₄

D₄×D₁₇₉

yes

—

107

C₂×C₄

C₂×D₇₁₆

yes

characteristic

108

C₂×C₄

#15668260

yes

—

109

C₈

—

358

—

110

C₂²

C₂×(C₂×D₁₇₉)⋊C₄

yes

characteristic

111

C₂²

#19153102

yes

characteristic

112

C₂²

#19153109

yes

characteristic

113

C₂²

C₂×#15668247

yes

—

114

C₂²

C₂×#15668249

yes

—

115

C₄

—

116

C₄

—

716

—

117

C₄

#19153036

yes

complemented

118

C₄

#19153075

yes

—

119

C₄

#19153088

yes

—

120

C₂

#26038340

yes

characteristic

121

C₂

C₂×#19153036

yes

—

122

C₂

#26038791

yes

—

123

C₂

#26038811

yes

characteristic

124

C₂

#26038818

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₇₉
commutator	C₂×C₄×C₁₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇₉
Fitting	#384×C₁₇₉	C₂
socle, prime radical	C₂³×C₁₇₉	C₂×D₄
cosocle	C₂²×C₄×C₁₇₉	C₂³
hypercenter	#384	D₁₇₉
hypocenter	C₁₇₉	#1956
LCS	G ▷ C₂×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₇₉

commutator

C₂×C₄×C₁₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇₉

Fitting

#384×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 ◁ #384

N	H ≅ G/N	type
8. #19153113	C₄	semidirect
53. C₄×C₁₇₉	#103	semidirect
62. C₁₇₉	#1956	semidirect
117. C₄	#19153036	semidirect

H ≅ G/N

type

8. #19153113

C₄

semidirect

53. C₄×C₁₇₉

#103

semidirect

62. C₁₇₉

#1956

semidirect

117. C₄

#19153036

semidirect

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

subgroup	index
C₂×#19153113	2
C₂×#19153036	2
C₂×C₄×C₁₇₉⋊C₈	2
#384×C₁₇₉	2
#1956	179

subgroup

index

C₂×#19153113

C₂×#19153036

C₂×C₄×C₁₇₉⋊C₈

#384×C₁₇₉

#1956

179

Subgroups — #37509801

Conjugacy classes of subgroups

Complements

Maximal subgroups