Group #37510906 — subgroups

Subgroups — #37510906

Order 22912.

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

normal subgroups only

This group has 126 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	C₈×Dic₃₅₈	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₁₇₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	Dic₁₇₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	Dic₁₇₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	#26038214	C₂	1	1	yes	maximal, characteristic
6	C₁₆×Dic₁₇₉	C₂	1	1	yes	maximal, characteristic
7	#346×C₁₇₉	C₂	1	1	yes	maximal, characteristic
8	C₄×Dic₃₅₈	C₄	1	1	yes	characteristic
9	C₂×C₁₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₁₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₄⋊(C₁₇₉⋊C₈)	C₄	1	1	yes	characteristic
12	Dic₁₇₉⋊C₈	C₄	1	1	yes	characteristic
13	Dic₁₇₉⋊C₈	C₄	1	1	yes	characteristic
14	C₈×Dic₁₇₉	C₂²	1	1	yes	characteristic
15	C₈×Dic₁₇₉	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×C₁₇₉	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₁₇₉	C₂²	1	1	yes	characteristic
18	C₄×C₈×C₁₇₉	C₂²	1	1	yes	characteristic
19	C₂×Dic₃₅₈	C₈	1	1	yes	characteristic
20	C₂×C₁₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
22	C₄⋊Dic₁₇₉	C₈	1	1	yes	characteristic
23	Dic₁₇₉⋊C₄	C₈	1	1	yes	characteristic
24	Dic₁₇₉⋊C₄	C₈	1	1	yes	characteristic
25	C₄×Dic₁₇₉	C₂×C₄	1	1	yes	characteristic
26	C₄×Dic₁₇₉	C₂×C₄	1	1	yes	characteristic
27	C₁₇₉⋊C₁₆	—	2	1	no	—
28	C₁₇₉⋊C₁₆	—	2	1	no	—
29	C₂×C₈×C₁₇₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₇₉	C₂³	1	1	yes	characteristic
31	C₄²×C₁₇₉	C₂×C₄	1	1	yes	characteristic
32	C₁₆×C₁₇₉	—	2	1	no	—
33	C₁₆×C₁₇₉	Q₈	1	2	yes	complemented
34	Dic₃₅₈	—	2	2	no	—
35	C₂×Dic₁₇₉	C₂×C₈	1	1	yes	characteristic
36	C₂×Dic₁₇₉	C₂×C₈	1	1	yes	characteristic
37	C₂×Dic₁₇₉	C₂×C₈	1	1	yes	characteristic
38	C₂×Dic₁₇₉	C₂×C₈	1	1	yes	characteristic
39	C₁₇₉⋊C₈	—	2	1	no	—
40	C₁₇₉⋊C₈	—	2	1	no	—
41	C₂×C₄×C₁₇₉	C₂×C₈	1	1	yes	characteristic
42	C₂×C₄×C₁₇₉	C₂×C₈	1	1	yes	characteristic
43	C₂×C₄×C₁₇₉	C₂²×C₄	1	1	yes	characteristic
44	C₈×C₁₇₉	—	2	1	no	—
45	C₈×C₁₇₉	C₂×Q₈	1	1	yes	characteristic
46	C₈×C₁₇₉	#41	1	1	yes	characteristic
47	C₂²×C₁₇₉	C₂²×C₈	1	1	yes	characteristic
48	C₄×C₁₇₉	—	2	1	no	—
49	C₄×C₁₇₉	—	2	1	no	—
50	C₄×C₁₇₉	C₄×Q₈	1	1	yes	characteristic
51	C₄×C₁₇₉	#131	1	1	yes	characteristic
52	Dic₁₇₉	—	2	1	no	—
53	Dic₁₇₉	—	2	1	no	—
54	Dic₁₇₉	—	2	1	no	—
55	Dic₁₇₉	M₃₂	1	2	yes	—
56	C₂×C₁₇₉	C₈×Q₈	1	1	yes	characteristic
57	C₂×C₁₇₉	C₂×M₃₂	1	1	yes	characteristic
58	C₂×C₁₇₉	#504	1	1	yes	characteristic
59	C₁₇₉	#2183	1	1	yes	characteristic, complemented, Sylow
60	#2183	—	179	1	no	maximal, Sylow
61	C₈×Q₈	—	179	1	no	—
62	C₄⋊C₁₆	—	179	1	no	—
63	C₄⋊C₁₆	—	179	1	no	—
64	C₄⋊C₁₆	—	179	1	no	—
65	#346	—	179	1	no	—
66	#346	D₁₇₉	1	1	yes	characteristic
67	C₄×C₁₆	—	179	1	no	—
68	C₄×Q₈	—	179	1	no	—
69	C₂×C₁₆	—	179	1	no	—
70	C₂×C₁₆	—	179	1	no	—
71	C₂×C₁₆	C₂×D₁₇₉	1	1	yes	characteristic
72	C₂×C₁₆	C₂×D₁₇₉	1	1	yes	characteristic
73	C₄⋊C₈	—	179	1	no	—
74	C₄⋊C₈	—	179	1	no	—
75	C₄⋊C₈	—	179	1	no	—
76	C₄×C₈	—	179	1	no	—
77	C₄×C₈	—	179	1	no	—
78	C₄×C₈	C₂×D₁₇₉	1	1	yes	characteristic
79	C₂×Q₈	—	179	1	no	—
80	C₂×C₈	—	179	1	no	—
81	C₂×C₈	—	179	1	no	—
82	C₂×C₈	C₄×D₁₇₉	1	1	yes	characteristic
83	C₂×C₈	C₂²×D₁₇₉	1	1	yes	characteristic
84	C₄⋊C₄	—	179	1	no	—
85	C₄⋊C₄	—	179	1	no	—
86	C₄⋊C₄	—	179	1	no	—
87	C₄²	—	179	1	no	—
88	C₄²	—	179	1	no	—
89	C₄²	C₄×D₁₇₉	1	1	yes	characteristic
90	C₁₆	—	2	1	no	—
91	C₁₆	—	358	1	no	—
92	C₁₆	—	358	1	no	—
93	C₁₆	Dic₃₅₈	1	2	yes	complemented
94	Q₈	—	358	2	no	—
95	C₂×C₄	—	179	1	no	—
96	C₂×C₄	—	179	1	no	—
97	C₂×C₄	—	179	1	no	—
98	C₂×C₄	—	179	1	no	—
99	C₂×C₄	C₈×D₁₇₉	1	1	yes	characteristic
100	C₂×C₄	C₈×D₁₇₉	1	1	yes	characteristic
101	C₂×C₄	C₂×C₄×D₁₇₉	1	1	yes	characteristic
102	C₈	—	2	1	no	—
103	C₈	—	358	1	no	—
104	C₈	—	358	1	no	—
105	C₈	C₂×Dic₃₅₈	1	1	yes	characteristic
106	C₈	#15668259	1	1	yes	characteristic
107	C₂²	C₂×C₈×D₁₇₉	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	358	1	no	—
111	C₄	—	358	1	no	—
112	C₄	—	358	1	no	—
113	C₄	—	179	2	no	—
114	C₄	C₄×Dic₃₅₈	1	1	yes	characteristic
115	C₄	#19153119	1	1	yes	characteristic
116	C₂	C₈×Dic₃₅₈	1	1	yes	characteristic
117	C₂	M₃₂×D₁₇₉	1	1	yes	characteristic
118	C₂	#26038857	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈×Dic₃₅₈

C₂

yes

maximal, characteristic

C₄⋊(C₁₇₉⋊C₁₆)

C₂

yes

maximal, characteristic

Dic₁₇₉⋊C₁₆

C₂

yes

maximal, characteristic

Dic₁₇₉⋊C₁₆

C₂

yes

maximal, characteristic

#26038214

C₂

yes

maximal, characteristic

C₁₆×Dic₁₇₉

C₂

yes

maximal, characteristic

#346×C₁₇₉

C₂

yes

maximal, characteristic

C₄×Dic₃₅₈

C₄

yes

characteristic

C₂×C₁₇₉⋊C₁₆

C₂²

yes

characteristic

C₂×C₁₇₉⋊C₁₆

C₂²

yes

characteristic

C₄⋊(C₁₇₉⋊C₈)

C₄

yes

characteristic

Dic₁₇₉⋊C₈

C₄

yes

characteristic

Dic₁₇₉⋊C₈

C₄

yes

characteristic

C₈×Dic₁₇₉

C₂²

yes

characteristic

C₈×Dic₁₇₉

C₂²

yes

characteristic

C₂×C₁₆×C₁₇₉

C₂²

yes

characteristic

C₂×C₁₆×C₁₇₉

C₂²

yes

characteristic

C₄×C₈×C₁₇₉

C₂²

yes

characteristic

C₂×Dic₃₅₈

C₈

yes

characteristic

C₂×C₁₇₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₇₉⋊C₈

C₂×C₄

yes

characteristic

C₄⋊Dic₁₇₉

C₈

yes

characteristic

Dic₁₇₉⋊C₄

C₈

yes

characteristic

Dic₁₇₉⋊C₄

C₈

yes

characteristic

C₄×Dic₁₇₉

C₂×C₄

yes

characteristic

C₄×Dic₁₇₉

C₂×C₄

yes

characteristic

C₁₇₉⋊C₁₆

—

C₁₇₉⋊C₁₆

—

C₂×C₈×C₁₇₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₇₉

C₂³

yes

characteristic

C₄²×C₁₇₉

C₂×C₄

yes

characteristic

C₁₆×C₁₇₉

—

C₁₆×C₁₇₉

Q₈

yes

complemented

Dic₃₅₈

—

C₂×Dic₁₇₉

C₂×C₈

yes

characteristic

C₂×Dic₁₇₉

C₂×C₈

yes

characteristic

C₂×Dic₁₇₉

C₂×C₈

yes

characteristic

C₂×Dic₁₇₉

C₂×C₈

yes

characteristic

C₁₇₉⋊C₈

—

C₁₇₉⋊C₈

—

C₂×C₄×C₁₇₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₇₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₇₉

C₂²×C₄

yes

characteristic

C₈×C₁₇₉

—

C₈×C₁₇₉

C₂×Q₈

yes

characteristic

C₈×C₁₇₉

#41

yes

characteristic

C₂²×C₁₇₉

C₂²×C₈

yes

characteristic

C₄×C₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

C₄×Q₈

yes

characteristic

C₄×C₁₇₉

#131

yes

characteristic

Dic₁₇₉

—

Dic₁₇₉

—

Dic₁₇₉

—

Dic₁₇₉

M₃₂

yes

—

C₂×C₁₇₉

C₈×Q₈

yes

characteristic

C₂×C₁₇₉

C₂×M₃₂

yes

C₂×C₁₇₉

yes

C₁₇₉

yes

characteristic, complemented, Sylow

#2183

—

179

maximal, Sylow

C₈×Q₈

—

179

—

C₄⋊C₁₆

—

179

—

C₄⋊C₁₆

—

179

—

C₄⋊C₁₆

—

179

—

#346

—

179

—

#346

D₁₇₉

yes

characteristic

C₄×C₁₆

—

179

—

C₄×Q₈

—

179

—

C₂×C₁₆

—

179

—

C₂×C₁₆

—

179

—

C₂×C₁₆

C₂×D₁₇₉

yes

characteristic

C₂×C₁₆

C₂×D₁₇₉

yes

characteristic

C₄⋊C₈

—

179

—

C₄⋊C₈

—

179

—

C₄⋊C₈

—

179

—

C₄×C₈

—

179

—

C₄×C₈

—

179

—

C₄×C₈

C₂×D₁₇₉

yes

characteristic

C₂×Q₈

—

179

—

C₂×C₈

—

179

—

C₂×C₈

—

179

—

C₂×C₈

C₄×D₁₇₉

yes

characteristic

C₂×C₈

C₂²×D₁₇₉

yes

characteristic

C₄⋊C₄

—

179

—

C₄⋊C₄

—

179

—

C₄⋊C₄

—

179

—

C₄²

—

179

—

C₄²

—

179

—

C₄²

C₄×D₁₇₉

yes

characteristic

C₁₆

—

C₁₆

—

358

—

C₁₆

—

358

—

C₁₆

Dic₃₅₈

yes

complemented

Q₈

—

358

—

C₂×C₄

—

179

—

C₂×C₄

—

179

—

C₂×C₄

—

179

—

C₂×C₄

—

179

—

C₂×C₄

C₈×D₁₇₉

yes

characteristic

100

C₂×C₄

C₈×D₁₇₉

yes

characteristic

101

C₂×C₄

C₂×C₄×D₁₇₉

yes

characteristic

102

C₈

—

103

C₈

—

358

—

104

C₈

—

358

—

105

C₈

C₂×Dic₃₅₈

yes

characteristic

106

C₈

#15668259

yes

characteristic

107

C₂²

C₂×C₈×D₁₇₉

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

358

—

111

C₄

—

358

—

112

C₄

—

358

—

113

C₄

—

179

—

114

C₄

C₄×Dic₃₅₈

yes

characteristic

115

C₄

#19153119

yes

characteristic

116

C₂

C₈×Dic₃₅₈

yes

characteristic

117

C₂

M₃₂×D₁₇₉

yes

characteristic

118

C₂

#26038857

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₇₉
commutator, socle, prime radical	C₂²×C₁₇₉	C₂²×C₈
Fitting	#346×C₁₇₉	C₂
cosocle	C₂×C₈×C₁₇₉	C₂³
hypercenter	#346	D₁₇₉
hypocenter	C₁₇₉	#2183
LCS	G ▷ C₂²×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂×C₈ ◁ #346

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₁₇₉

commutator, socle, prime radical

C₂²×C₁₇₉

C₂²×C₈

Fitting

#346×C₁₇₉

C₂

cosocle

C₂×C₈×C₁₇₉

C₂³

D₁₇₉

C₁₇₉

G ▷ C₂²×C₁₇₉ ▷ C₁₇₉

C₁ ◁ C₂×C₈ ◁ #346

N	H ≅ G/N	type
33. C₁₆×C₁₇₉	Q₈	semidirect
59. C₁₇₉	#2183	semidirect
93. C₁₆	Dic₃₅₈	semidirect

H ≅ G/N

type

33. C₁₆×C₁₇₉

Q₈

semidirect

59. C₁₇₉

#2183

semidirect

93. C₁₆

Dic₃₅₈

semidirect

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

subgroup	index
C₈×Dic₃₅₈	2
C₄⋊(C₁₇₉⋊C₁₆)	2
Dic₁₇₉⋊C₁₆	2
#26038214	2
C₁₆×Dic₁₇₉	2
#346×C₁₇₉	2
#2183	179

subgroup

index

C₈×Dic₃₅₈

C₄⋊(C₁₇₉⋊C₁₆)

Dic₁₇₉⋊C₁₆

#26038214

C₁₆×Dic₁₇₉

#346×C₁₇₉

#2183

179

Subgroups — #37510906

Conjugacy classes of subgroups

Complements

Maximal subgroups