Group #37510269 — subgroups

Subgroups — #37510269

Order 22912.

256 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 67 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	C₂²×Dic₇₁₆	C₂	1	1	yes	maximal, characteristic
2	#26038465	C₂	1	1	yes	maximal, characteristic
3	#26038449	C₂	1	2	yes	maximal
4	C₂×#19153031	C₂	1	2	yes	maximal
5	#341×C₁₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₅₈	C₂²	1	2	yes	—
7	C₂×Dic₇₁₆	—	2	4	no	—
8	C₂×Dic₇₁₆	—	2	2	no	—
9	C₂×#15668243	C₂²	1	1	yes	characteristic
10	#19153072	C₂²	1	1	yes	characteristic
11	Dic₁₇₉⋊C₈	—	2	2	no	—
12	#19153038	—	2	4	no	—
13	#19153031	—	2	4	no	—
14	C₂×M₁₆×C₁₇₉	C₂²	1	2	yes	—
15	C₂²×C₈×C₁₇₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₃₅₈	—	2	2	no	—
17	C₂×Dic₃₅₈	—	2	2	no	—
18	C₂×Dic₃₅₈	—	2	2	no	—
19	C₂×Dic₃₅₈	—	2	2	no	—
20	C₂×Dic₃₅₈	—	2	2	no	—
21	C₂×Dic₃₅₈	D₄	1	4	yes	—
22	C₂²×Dic₁₇₉	—	2	2	no	—
23	Dic₇₁₆	—	2	8	no	—
24	#15668243	—	2	2	no	—
25	C₂×C₁₇₉⋊C₈	—	2	1	no	—
26	C₄⋊Dic₁₇₉	D₄	1	2	yes	—
27	C₂²⋊Dic₁₇₉	—	2	1	no	—
28	C₄×Dic₁₇₉	—	2	1	no	—
29	C₂²×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
30	M₁₆×C₁₇₉	—	2	2	no	—
31	M₁₆×C₁₇₉	—	2	2	no	—
32	C₂×C₈×C₁₇₉	—	2	2	no	—
33	C₂×C₈×C₁₇₉	—	2	2	no	—
34	C₂×C₈×C₁₇₉	D₄	1	2	yes	—
35	Dic₃₅₈	—	2	4	no	—
36	Dic₃₅₈	—	2	4	no	—
37	Dic₃₅₈	—	4	2	no	—
38	Dic₃₅₈	—	4	2	no	—
39	C₂×Dic₁₇₉	—	2	2	no	—
40	C₂×Dic₁₇₉	—	2	4	no	—
41	C₂×Dic₁₇₉	—	4	2	no	—
42	C₂×Dic₁₇₉	—	4	2	no	—
43	C₁₇₉⋊C₈	—	4	1	no	—
44	C₂³×C₁₇₉	#41	1	1	yes	characteristic
45	C₂×C₄×C₁₇₉	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₇₉	#41	1	2	yes	—
49	C₈×C₁₇₉	—	2	2	no	—
50	C₈×C₁₇₉	—	4	2	no	—
51	C₂²×C₁₇₉	—	2	2	no	—
52	C₂²×C₁₇₉	#121	1	1	yes	characteristic
53	C₂²×C₁₇₉	#123	1	2	yes	—
54	C₄×C₁₇₉	#120	1	1	yes	characteristic
55	C₄×C₁₇₉	#121	1	2	yes	—
56	C₄×C₁₇₉	#124	1	1	yes	characteristic
57	Dic₁₇₉	—	4	4	no	—
58	Dic₁₇₉	—	4	2	no	—
59	C₂×C₁₇₉	—	2	2	no	—
60	C₂×C₁₇₉	#394	1	1	yes	characteristic
61	C₂×C₁₇₉	#473	1	2	yes	—
62	C₁₇₉	#2047	1	1	yes	characteristic, complemented, Sylow
63	#2047	—	179	1	no	maximal, Sylow
64	C₂²×Q₁₆	—	179	1	no	—
65	#424	—	179	1	no	—
66	#419	—	179	2	no	—
67	C₂×#101	—	179	2	no	—
68	#341	D₁₇₉	1	1	yes	characteristic
69	C₂²×Q₈	—	179	2	no	—
70	C₂×Q₁₆	—	358	4	no	—
71	C₂×Q₁₆	—	358	2	no	—
72	C₂×M₁₆	—	179	1	no	—
73	C₂×M₁₆	C₂×D₁₇₉	1	2	yes	—
74	C₂²×C₈	C₂×D₁₇₉	1	1	yes	characteristic
75	#117	—	179	1	no	—
76	C₄⋊C₈	—	358	2	no	—
77	#103	—	358	4	no	—
78	#101	—	358	4	no	—
79	C₂×Q₈	—	358	2	no	—
80	C₂×Q₈	—	358	2	no	—
81	C₂×Q₈	—	358	2	no	—
82	C₂×Q₈	—	358	2	no	—
83	C₂×Q₈	—	358	2	no	—
84	C₂×Q₈	—	179	4	no	—
85	C₂²×C₄	—	358	2	no	—
86	C₂²×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
87	Q₁₆	—	358	8	no	—
88	M₁₆	—	2	2	no	—
89	M₁₆	—	2	2	no	—
90	M₁₆	—	358	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	358	1	no	—
94	C₂×C₈	D₇₁₆	1	2	yes	—
95	C₄⋊C₄	—	179	2	no	—
96	C₂²⋊C₄	—	358	1	no	—
97	C₄²	—	358	1	no	—
98	C₂³	#15668260	1	1	yes	characteristic
99	Q₈	—	358	4	no	—
100	Q₈	—	358	4	no	—
101	Q₈	—	716	2	no	—
102	Q₈	—	716	2	no	—
103	C₂×C₄	—	358	2	no	—
104	C₂×C₄	—	358	4	no	—
105	C₂×C₄	—	716	2	no	—
106	C₂×C₄	—	716	2	no	—
107	C₂×C₄	D₄×D₁₇₉	1	2	yes	—
108	C₂×C₄	D₄×D₁₇₉	1	1	yes	characteristic
109	C₂×C₄	C₂×D₇₁₆	1	1	yes	characteristic
110	C₂×C₄	#15668259	1	2	yes	—
111	C₈	—	2	2	no	—
112	C₈	—	4	2	no	—
113	C₈	—	716	1	no	—
114	C₂²	—	2	2	no	—
115	C₂²	#19153088	1	1	yes	characteristic
116	C₂²	#19153098	1	2	yes	—
117	C₄	—	716	4	no	—
118	C₄	—	716	2	no	—
119	C₄	#19153083	1	1	yes	characteristic
120	C₄	#19153086	1	2	yes	—
121	C₄	#19153101	1	1	yes	characteristic
122	C₂	—	2	2	no	—
123	C₂	#26038357	1	1	yes	characteristic
124	C₂	#26038715	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₇₁₆

C₂

yes

maximal, characteristic

#26038465

C₂

yes

maximal, characteristic

#26038449

C₂

yes

maximal

C₂×#19153031

C₂

yes

maximal

#341×C₁₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₃₅₈

C₂²

yes

—

C₂×Dic₇₁₆

—

C₂×Dic₇₁₆

—

C₂×#15668243

C₂²

yes

characteristic

#19153072

C₂²

yes

characteristic

Dic₁₇₉⋊C₈

—

#19153038

—

#19153031

—

C₂×M₁₆×C₁₇₉

C₂²

yes

—

C₂²×C₈×C₁₇₉

C₂²

yes

characteristic

C₂×Dic₃₅₈

—

C₂×Dic₃₅₈

—

C₂×Dic₃₅₈

—

C₂×Dic₃₅₈

—

C₂×Dic₃₅₈

—

C₂×Dic₃₅₈

D₄

yes

—

C₂²×Dic₁₇₉

—

Dic₇₁₆

—

#15668243

—

C₂×C₁₇₉⋊C₈

—

C₄⋊Dic₁₇₉

D₄

yes

—

C₂²⋊Dic₁₇₉

—

C₄×Dic₁₇₉

—

C₂²×C₄×C₁₇₉

C₂³

yes

characteristic

M₁₆×C₁₇₉

—

M₁₆×C₁₇₉

—

C₂×C₈×C₁₇₉

—

C₂×C₈×C₁₇₉

—

C₂×C₈×C₁₇₉

D₄

yes

—

Dic₃₅₈

—

Dic₃₅₈

—

Dic₃₅₈

—

Dic₃₅₈

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₁₇₉⋊C₈

—

C₂³×C₁₇₉

#41

yes

characteristic

C₂×C₄×C₁₇₉

C₂×D₄

yes

—

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

#41

yes

—

C₈×C₁₇₉

—

C₈×C₁₇₉

—

C₂²×C₁₇₉

—

C₂²×C₁₇₉

#121

yes

characteristic

C₂²×C₁₇₉

#123

yes

—

C₄×C₁₇₉

#120

yes

characteristic

C₄×C₁₇₉

#121

yes

—

C₄×C₁₇₉

#124

yes

characteristic

Dic₁₇₉

—

Dic₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

#394

yes

characteristic

C₂×C₁₇₉

#473

yes

—

C₁₇₉

#2047

yes

characteristic, complemented, Sylow

#2047

—

179

maximal, Sylow

C₂²×Q₁₆

—

179

—

#424

—

179

—

#419

—

179

—

C₂×#101

—

179

—

#341

D₁₇₉

yes

characteristic

C₂²×Q₈

—

179

—

C₂×Q₁₆

—

358

—

C₂×Q₁₆

—

358

—

C₂×M₁₆

—

179

—

C₂×M₁₆

C₂×D₁₇₉

yes

—

C₂²×C₈

C₂×D₁₇₉

yes

characteristic

#117

—

179

—

C₄⋊C₈

—

358

—

#103

—

358

—

#101

—

358

—

C₂×Q₈

—

358

—

C₂×Q₈

—

358

—

C₂×Q₈

—

358

—

C₂×Q₈

—

358

—

C₂×Q₈

—

358

—

C₂×Q₈

—

179

—

C₂²×C₄

—

358

—

C₂²×C₄

C₂²×D₁₇₉

yes

characteristic

Q₁₆

—

358

—

M₁₆

—

M₁₆

—

M₁₆

—

358

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

358

—

C₂×C₈

D₇₁₆

yes

—

C₄⋊C₄

—

179

—

C₂²⋊C₄

—

358

—

C₄²

—

358

—

C₂³

#15668260

yes

characteristic

Q₈

—

358

—

100

Q₈

—

358

—

101

Q₈

—

716

—

102

Q₈

—

716

—

103

C₂×C₄

—

358

—

104

C₂×C₄

—

358

—

105

C₂×C₄

—

716

—

106

C₂×C₄

—

716

—

107

C₂×C₄

D₄×D₁₇₉

yes

—

108

C₂×C₄

D₄×D₁₇₉

yes

characteristic

109

C₂×C₄

C₂×D₇₁₆

yes

characteristic

110

C₂×C₄

#15668259

yes

—

111

C₈

—

112

C₈

—

113

C₈

—

716

—

114

C₂²

—

115

C₂²

#19153088

yes

characteristic

116

C₂²

#19153098

yes

—

117

C₄

—

716

—

118

C₄

—

716

—

119

C₄

#19153083

yes

characteristic

120

C₄

#19153086

yes

—

121

C₄

#19153101

yes

characteristic

122

C₂

—

123

C₂

#26038357

yes

characteristic

124

C₂

#26038715

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	C₂³×C₁₇₉	#41
hypercenter	#341	D₁₇₉
hypocenter	C₁₇₉	#2047
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

C₂³×C₁₇₉

D₁₇₉

C₁₇₉

G ▷ C₂²×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
62. C₁₇₉	#2047	semidirect

H ≅ G/N

type

62. C₁₇₉

#2047

semidirect

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

subgroup	index
C₂²×Dic₇₁₆	2
#26038465	2
#26038449	2
C₂×#19153031	2
#341×C₁₇₉	2
#2047	179

subgroup

index

C₂²×Dic₇₁₆

#341×C₁₇₉

179

Subgroups — #37510269

Conjugacy classes of subgroups

Complements

Maximal subgroups