Group #37510447 — subgroups

Subgroups — #37510447

Order 22912.

198 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 63 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×#19153045	C₂	1	1	yes	maximal, characteristic, complemented
2	#26038441	C₂	1	2	yes	maximal
3	#26038405	C₂	1	1	yes	maximal, characteristic
4	#26038207	C₂	1	2	yes	maximal, complemented
5	#341×C₁₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15668259	C₂²	1	1	yes	characteristic
7	C₂×#15668243	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₇₉	C₂²	1	2	yes	—
9	#19153045	—	2	2	no	—
10	#19153038	—	2	2	no	—
11	#19153035	—	2	2	no	—
12	(C₂×D₁₇₉)⋊C₈	—	2	2	no	—
13	C₂×M₁₆×C₁₇₉	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₇₉	C₂²	1	1	yes	characteristic
15	#15668259	—	2	2	no	—
16	#15668259	—	2	2	no	—
17	C₂×Dic₃₅₈	D₄	1	1	yes	characteristic
18	C₂×D₇₁₆	D₄	1	1	yes	characteristic
19	C₂×C₂²⋊D₁₇₉	—	2	1	no	—
20	C₂²×Dic₁₇₉	—	2	2	no	—
21	C₂×C₄×D₁₇₉	—	2	1	no	—
22	#15668243	—	2	2	no	—
23	C₂×C₁₇₉⋊C₈	—	2	1	no	—
24	C₄⋊Dic₁₇₉	—	2	2	no	—
25	C₄⋊Dic₁₇₉	—	2	2	no	—
26	C₂²×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
27	M₁₆×C₁₇₉	—	2	2	no	—
28	M₁₆×C₁₇₉	—	2	2	no	—
29	C₂×C₈×C₁₇₉	—	2	2	no	—
30	C₂×C₈×C₁₇₉	—	2	2	no	—
31	C₂×C₈×C₁₇₉	D₄	1	2	yes	—
32	C₂²×D₁₇₉	—	2	1	no	—
33	Dic₃₅₈	—	2	2	no	—
34	D₇₁₆	—	2	2	no	—
35	C₂²⋊D₁₇₉	—	4	2	no	—
36	C₂×Dic₁₇₉	—	2	1	no	—
37	C₂×Dic₁₇₉	—	4	2	no	—
38	C₂×Dic₁₇₉	—	8	2	no	—
39	C₄×D₁₇₉	—	4	1	no	—
40	C₄×D₁₇₉	—	4	1	no	—
41	C₁₇₉⋊C₈	—	4	1	no	—
42	C₂³×C₁₇₉	#41	1	1	yes	characteristic
43	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₇₉	#41	1	2	yes	—
46	C₂×C₄×C₁₇₉	#41	1	2	yes	—
47	C₈×C₁₇₉	—	2	2	no	—
48	C₈×C₁₇₉	—	4	2	no	—
49	C₂²×C₁₇₉	—	2	2	no	—
50	C₂²×C₁₇₉	#121	1	1	yes	characteristic
51	C₂²×C₁₇₉	#126	1	2	yes	—
52	C₂×D₁₇₉	—	4	1	no	—
53	C₂×D₁₇₉	—	4	2	no	—
54	C₄×C₁₇₉	#123	1	2	yes	—
55	C₄×C₁₇₉	#123	1	1	yes	characteristic
56	C₄×C₁₇₉	#124	1	1	yes	characteristic
57	Dic₁₇₉	—	4	1	no	—
58	Dic₁₇₉	—	8	2	no	—
59	C₂×C₁₇₉	—	2	2	no	—
60	C₂×C₁₇₉	#397	1	1	yes	characteristic
61	C₂×C₁₇₉	#471	1	2	yes	—
62	D₁₇₉	—	8	1	no	—
63	C₁₇₉	#2079	1	1	yes	characteristic, complemented, Sylow
64	#2079	—	179	1	no	maximal, Sylow
65	C₂×#106	—	179	1	no	—
66	#417	—	179	2	no	—
67	#408	—	179	1	no	—
68	#343	—	179	2	no	—
69	#341	D₁₇₉	1	1	yes	characteristic, complemented
70	C₂×#41	—	179	1	no	—
71	C₂×M₁₆	—	179	1	no	—
72	C₂×M₁₆	C₂×D₁₇₉	1	2	yes	—
73	C₂²×C₈	C₂×D₁₇₉	1	1	yes	characteristic
74	C₂×C₄⋊C₄	—	179	2	no	—
75	#106	—	358	2	no	—
76	#103	—	358	2	no	—
77	#102	—	358	2	no	—
78	C₂²⋊C₈	—	358	2	no	—
79	#41	—	358	2	no	—
80	#41	—	358	2	no	—
81	C₂×Q₈	—	179	1	no	—
82	C₂×D₄	—	358	1	no	—
83	C₂×D₄	—	179	1	no	—
84	C₂²×C₄	—	358	1	no	—
85	C₂²×C₄	—	358	2	no	—
86	C₂²×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
87	M₁₆	—	2	2	no	—
88	M₁₆	—	2	2	no	—
89	M₁₆	—	358	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	358	1	no	—
93	C₂×C₈	D₇₁₆	1	2	yes	—
94	C₄⋊C₄	—	358	2	no	—
95	C₄⋊C₄	—	358	2	no	—
96	C₂³	—	358	1	no	—
97	C₂³	#15668260	1	1	yes	characteristic
98	Q₈	—	358	2	no	—
99	D₄	—	358	2	no	—
100	D₄	—	716	2	no	—
101	C₂×C₄	—	358	1	no	—
102	C₂×C₄	—	1432	2	no	—
103	C₂×C₄	—	716	2	no	—
104	C₂×C₄	—	716	1	no	—
105	C₂×C₄	—	716	1	no	—
106	C₂×C₄	D₄×D₁₇₉	1	1	yes	characteristic
107	C₂×C₄	C₂×D₇₁₆	1	1	yes	characteristic
108	C₂×C₄	#15668258	1	2	yes	—
109	C₂×C₄	#15668259	1	2	yes	—
110	C₈	—	2	2	no	—
111	C₈	—	4	2	no	—
112	C₈	—	716	1	no	—
113	C₂²	—	2	2	no	—
114	C₂²	—	716	1	no	—
115	C₂²	—	716	2	no	—
116	C₂²	#19153088	1	1	yes	characteristic
117	C₂²	#19153108	1	2	yes	—
118	C₄	—	1432	2	no	—
119	C₄	—	716	1	no	—
120	C₄	#19153095	1	2	yes	—
121	C₄	#19153097	1	1	yes	characteristic
122	C₄	#19153101	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	1432	1	no	—
125	C₂	#26038366	1	1	yes	characteristic
126	C₂	#26038703	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19153045

C₂

yes

maximal, characteristic, complemented

#26038441

C₂

yes

maximal

#26038405

C₂

yes

maximal, characteristic

#26038207

C₂

yes

maximal, complemented

#341×C₁₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#15668259

C₂²

yes

characteristic

C₂×#15668243

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇₉

C₂²

yes

—

#19153045

—

#19153038

—

#19153035

—

(C₂×D₁₇₉)⋊C₈

—

C₂×M₁₆×C₁₇₉

C₂²

yes

—

C₂²×C₈×C₁₇₉

C₂²

yes

characteristic

#15668259

—

#15668259

—

C₂×Dic₃₅₈

D₄

yes

characteristic

C₂×D₇₁₆

D₄

yes

characteristic

C₂×C₂²⋊D₁₇₉

—

C₂²×Dic₁₇₉

—

C₂×C₄×D₁₇₉

—

#15668243

—

C₂×C₁₇₉⋊C₈

—

C₄⋊Dic₁₇₉

—

C₄⋊Dic₁₇₉

—

C₂²×C₄×C₁₇₉

C₂³

yes

characteristic

M₁₆×C₁₇₉

—

M₁₆×C₁₇₉

—

C₂×C₈×C₁₇₉

—

C₂×C₈×C₁₇₉

—

C₂×C₈×C₁₇₉

D₄

yes

—

C₂²×D₁₇₉

—

Dic₃₅₈

—

D₇₁₆

—

C₂²⋊D₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₄×D₁₇₉

—

C₄×D₁₇₉

—

C₁₇₉⋊C₈

—

C₂³×C₁₇₉

#41

yes

characteristic

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

#41

yes

—

C₂×C₄×C₁₇₉

#41

yes

—

C₈×C₁₇₉

—

C₈×C₁₇₉

—

C₂²×C₁₇₉

—

C₂²×C₁₇₉

#121

yes

characteristic

C₂²×C₁₇₉

#126

yes

—

C₂×D₁₇₉

—

C₂×D₁₇₉

—

C₄×C₁₇₉

#123

yes

—

C₄×C₁₇₉

yes

C₄×C₁₇₉

yes

Dic₁₇₉

—

Dic₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

#397

yes

characteristic

C₂×C₁₇₉

#471

yes

—

D₁₇₉

—

C₁₇₉

#2079

yes

characteristic, complemented, Sylow

#2079

—

179

maximal, Sylow

C₂×#106

—

179

—

#417

—

179

—

#408

—

179

—

#343

—

179

—

#341

D₁₇₉

yes

characteristic, complemented

C₂×#41

—

179

—

C₂×M₁₆

—

179

—

C₂×M₁₆

C₂×D₁₇₉

yes

—

C₂²×C₈

C₂×D₁₇₉

yes

characteristic

C₂×C₄⋊C₄

—

179

—

#106

—

358

—

#103

—

358

—

#102

—

358

—

C₂²⋊C₈

—

358

—

#41

—

358

—

#41

—

358

—

C₂×Q₈

—

179

—

C₂×D₄

—

358

—

C₂×D₄

—

179

—

C₂²×C₄

—

358

—

C₂²×C₄

—

358

—

C₂²×C₄

C₂²×D₁₇₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

358

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

358

—

C₂×C₈

D₇₁₆

yes

—

C₄⋊C₄

—

358

—

C₄⋊C₄

—

358

—

C₂³

—

358

—

C₂³

#15668260

yes

characteristic

Q₈

—

358

—

D₄

—

358

—

100

D₄

—

716

—

101

C₂×C₄

—

358

—

102

C₂×C₄

—

1432

—

103

C₂×C₄

—

716

—

104

C₂×C₄

—

716

—

105

C₂×C₄

—

716

—

106

C₂×C₄

D₄×D₁₇₉

yes

characteristic

107

C₂×C₄

C₂×D₇₁₆

yes

characteristic

108

C₂×C₄

#15668258

yes

—

109

C₂×C₄

#15668259

yes

—

110

C₈

—

111

C₈

—

112

C₈

—

716

—

113

C₂²

—

114

C₂²

—

716

—

115

C₂²

—

716

—

116

C₂²

#19153088

yes

characteristic

117

C₂²

#19153108

yes

—

118

C₄

—

1432

—

119

C₄

—

716

—

120

C₄

#19153095

yes

—

121

C₄

#19153097

yes

characteristic

122

C₄

#19153101

yes

characteristic

123

C₂

—

124

C₂

—

1432

—

125

C₂

#26038366

yes

characteristic

126

C₂

#26038703

yes

—

	subgroup	G/H
center	C₂²	#19153088
commutator, cosocle	C₂²×C₄×C₁₇₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇₉
Fitting	#341×C₁₇₉	C₂
socle	C₂²×C₁₇₉	#121
prime radical	C₂×#15668259	C₂²
hypercenter	#341	D₁₇₉
hypocenter	C₁₇₉	#2079
LCS	G ▷ C₂²×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂²

#19153088

commutator, cosocle

C₂²×C₄×C₁₇₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₇₉

Fitting

#341×C₁₇₉

C₂

C₂²×C₁₇₉

C₂²

D₁₇₉

C₁₇₉

G ▷ C₂²×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
1. C₂×#19153045	C₂	semidirect
4. #26038207	C₂	semidirect
5. #341×C₁₇₉	C₂	semidirect
63. C₁₇₉	#2079	semidirect
69. #341	D₁₇₉	semidirect

H ≅ G/N

type

1. C₂×#19153045

C₂

semidirect

4. #26038207

C₂

semidirect

5. #341×C₁₇₉

C₂

semidirect

63. C₁₇₉

#2079

semidirect

69. #341

D₁₇₉

semidirect

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

subgroup	index
C₂×#19153045	2
#26038441	2
#26038405	2
#26038207	2
#341×C₁₇₉	2
#2079	179

subgroup

index

#341×C₁₇₉

179

Subgroups — #37510447

Conjugacy classes of subgroups

Complements

Maximal subgroups