Group #37511244 — subgroups

Subgroups — #37511244

Order 22912.

132 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 47 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	#26038840	C₂	1	1	yes	maximal, characteristic
2	#26038817	C₂	1	1	yes	maximal, characteristic
3	#26038244	C₂	1	1	yes	maximal, characteristic
4	#26038243	C₂	1	1	yes	maximal, characteristic
5	#26038242	C₂	1	1	yes	maximal, characteristic
6	#26038214	C₂	1	1	yes	maximal, characteristic
7	#358×C₁₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₇₁₆	C₂²	1	1	yes	characteristic
9	C₂×#15668249	—	2	1	no	—
10	#19153111	—	2	1	no	—
11	#19153104	—	2	1	no	—
12	C₂×C₁₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
13	#19153047	C₂²	1	1	yes	characteristic
14	#19153044	—	2	1	no	—
15	C₈×Dic₁₇₉	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₁₇₉	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₁₇₉	C₂²	1	1	yes	characteristic
18	#107×C₁₇₉	C₂²	1	1	yes	characteristic
19	C₂×Dic₃₅₈	—	2	1	no	—
20	Dic₇₁₆	—	2	1	no	—
21	Dic₇₁₆	—	2	1	no	—
22	#15668249	—	4	2	no	—
23	C₂×C₁₇₉⋊C₈	D₄	1	1	yes	characteristic
24	C₄⋊Dic₁₇₉	—	2	1	no	—
25	C₄⋊Dic₁₇₉	—	4	1	no	—
26	Dic₁₇₉⋊C₄	—	4	1	no	—
27	Dic₁₇₉⋊C₄	—	4	1	no	—
28	Dic₁₇₉⋊C₄	—	4	1	no	—
29	C₄×Dic₁₇₉	D₄	1	1	yes	characteristic
30	C₁₇₉⋊C₁₆	—	2	1	no	—
31	C₂×Q₈×C₁₇₉	—	2	1	no	—
32	Q₁₆×C₁₇₉	—	2	1	no	—
33	Q₁₆×C₁₇₉	—	2	1	no	—
34	C₂×C₈×C₁₇₉	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₁₇₉	—	2	1	no	—
36	C₁₆×C₁₇₉	—	2	1	no	—
37	Dic₃₅₈	—	4	1	no	—
38	Dic₃₅₈	—	4	1	no	—
39	C₂×Dic₁₇₉	—	4	1	no	—
40	C₂×Dic₁₇₉	—	4	1	no	—
41	C₂×Dic₁₇₉	D₈	1	1	yes	characteristic
42	C₂×Dic₁₇₉	D₈	1	1	yes	characteristic
43	C₁₇₉⋊C₈	SD₁₆	1	2	yes	—
44	Q₈×C₁₇₉	—	4	1	no	—
45	Q₈×C₁₇₉	—	4	1	no	—
46	C₂×C₄×C₁₇₉	—	4	1	no	—
47	C₂×C₄×C₁₇₉	—	4	1	no	—
48	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
49	C₈×C₁₇₉	#41	1	1	yes	characteristic
50	C₈×C₁₇₉	#41	1	1	yes	characteristic
51	C₂²×C₁₇₉	C₂×D₈	1	1	yes	characteristic
52	C₄×C₁₇₉	—	8	1	no	—
53	C₄×C₁₇₉	—	8	1	no	—
54	C₄×C₁₇₉	#124	1	1	yes	characteristic
55	C₄×C₁₇₉	C₂×SD₁₆	1	1	yes	characteristic
56	Dic₁₇₉	—	2	1	no	—
57	Dic₁₇₉	—	2	1	no	—
58	Dic₁₇₉	—	8	1	no	—
59	Dic₁₇₉	—	8	1	no	—
60	C₂×C₁₇₉	#486	1	1	yes	characteristic
61	C₂×C₁₇₉	#510	1	1	yes	characteristic
62	C₂×C₁₇₉	#510	1	1	yes	characteristic
63	C₁₇₉	#2245	1	1	yes	characteristic, complemented, Sylow
64	#2245	—	179	1	no	maximal, Sylow
65	#499	—	179	1	no	—
66	#494	—	179	1	no	—
67	#358	—	179	1	no	—
68	#358	—	179	1	no	—
69	#358	—	179	1	no	—
70	#358	D₁₇₉	1	1	yes	characteristic
71	#346	—	179	1	no	—
72	C₂×Q₁₆	—	358	1	no	—
73	C₂×Q₁₆	—	179	1	no	—
74	C₂×Q₁₆	C₂×D₁₇₉	1	1	yes	characteristic
75	C₄⋊Q₈	—	358	1	no	—
76	#125	—	358	1	no	—
77	C₂×C₁₆	—	179	1	no	—
78	C₂×C₁₆	C₂×D₁₇₉	1	1	yes	characteristic
79	#107	—	179	1	no	—
80	#107	C₂×D₁₇₉	1	1	yes	characteristic
81	#106	—	358	1	no	—
82	C₄×C₈	—	179	1	no	—
83	C₂×Q₈	—	2	1	no	—
84	C₂×Q₈	—	358	1	no	—
85	Q₁₆	—	2	1	no	—
86	Q₁₆	—	2	1	no	—
87	Q₁₆	—	358	1	no	—
88	Q₁₆	—	358	1	no	—
89	Q₁₆	—	716	2	no	—
90	C₂×C₈	—	179	1	no	—
91	C₂×C₈	C₂²×D₁₇₉	1	1	yes	characteristic
92	C₄⋊C₄	—	2	1	no	—
93	C₄⋊C₄	—	358	1	no	—
94	C₄⋊C₄	—	716	1	no	—
95	C₄⋊C₄	—	716	1	no	—
96	C₄⋊C₄	—	716	1	no	—
97	C₄⋊C₄	—	716	1	no	—
98	C₄²	—	179	1	no	—
99	C₁₆	—	2	1	no	—
100	C₁₆	—	358	1	no	—
101	Q₈	—	4	1	no	—
102	Q₈	—	4	1	no	—
103	Q₈	—	716	1	no	—
104	Q₈	—	716	1	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	179	1	no	—
108	C₂×C₄	—	179	1	no	—
109	C₂×C₄	—	716	1	no	—
110	C₂×C₄	—	716	1	no	—
111	C₂×C₄	D₄×D₁₇₉	1	1	yes	characteristic
112	C₈	—	179	2	no	—
113	C₈	#15668258	1	1	yes	characteristic
114	C₈	#15668259	1	1	yes	characteristic
115	C₂²	D₈×D₁₇₉	1	1	yes	characteristic
116	C₄	—	8	1	no	—
117	C₄	—	8	1	no	—
118	C₄	—	358	1	no	—
119	C₄	—	358	1	no	—
120	C₄	—	1432	1	no	—
121	C₄	—	1432	1	no	—
122	C₄	#19153099	1	1	yes	characteristic
123	C₄	SD₁₆×D₁₇₉	1	1	yes	characteristic
124	C₂	#26038784	1	1	yes	characteristic
125	C₂	#26038878	1	1	yes	characteristic
126	C₂	#26038881	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26038840

C₂

yes

maximal, characteristic

#26038817

C₂

yes

maximal, characteristic

#26038244

C₂

yes

maximal, characteristic

#26038243

C₂

yes

maximal, characteristic

#26038242

C₂

yes

maximal, characteristic

#26038214

C₂

yes

maximal, characteristic

#358×C₁₇₉

C₂

yes

maximal, characteristic

C₂×Dic₇₁₆

C₂²

yes

—

—

—

C₂×C₁₇₉⋊C₁₆

C₂²

yes

characteristic

#19153047

C₂²

yes

characteristic

#19153044

—

C₈×Dic₁₇₉

C₂²

yes

characteristic

C₂×Q₁₆×C₁₇₉

C₂²

yes

characteristic

C₂×C₁₆×C₁₇₉

C₂²

yes

characteristic

#107×C₁₇₉

C₂²

yes

characteristic

C₂×Dic₃₅₈

—

Dic₇₁₆

—

Dic₇₁₆

—

#15668249

—

C₂×C₁₇₉⋊C₈

D₄

yes

characteristic

C₄⋊Dic₁₇₉

—

C₄⋊Dic₁₇₉

—

Dic₁₇₉⋊C₄

—

Dic₁₇₉⋊C₄

—

Dic₁₇₉⋊C₄

—

C₄×Dic₁₇₉

D₄

yes

characteristic

C₁₇₉⋊C₁₆

—

C₂×Q₈×C₁₇₉

—

Q₁₆×C₁₇₉

—

Q₁₆×C₁₇₉

—

C₂×C₈×C₁₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₇₉

—

C₁₆×C₁₇₉

—

Dic₃₅₈

—

Dic₃₅₈

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

D₈

yes

characteristic

C₂×Dic₁₇₉

D₈

yes

characteristic

C₁₇₉⋊C₈

SD₁₆

yes

—

Q₈×C₁₇₉

—

Q₈×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

C₂×D₄

yes

C₈×C₁₇₉

yes

C₈×C₁₇₉

yes

C₂²×C₁₇₉

C₂×D₈

yes

characteristic

C₄×C₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

#124

yes

characteristic

C₄×C₁₇₉

C₂×SD₁₆

yes

characteristic

Dic₁₇₉

—

Dic₁₇₉

—

Dic₁₇₉

—

Dic₁₇₉

—

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

C₁₇₉

yes

characteristic, complemented, Sylow

#2245

—

179

maximal, Sylow

#499

—

179

—

#494

—

179

—

#358

—

179

—

#358

—

179

—

#358

—

179

—

#358

D₁₇₉

yes

characteristic

#346

—

179

—

C₂×Q₁₆

—

358

—

C₂×Q₁₆

—

179

—

C₂×Q₁₆

C₂×D₁₇₉

yes

characteristic

C₄⋊Q₈

—

358

—

#125

—

358

—

C₂×C₁₆

—

179

—

C₂×C₁₆

C₂×D₁₇₉

yes

characteristic

#107

—

179

—

#107

C₂×D₁₇₉

yes

characteristic

#106

—

358

—

C₄×C₈

—

179

—

C₂×Q₈

—

C₂×Q₈

—

358

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

358

—

Q₁₆

—

358

—

Q₁₆

—

716

—

C₂×C₈

—

179

—

C₂×C₈

C₂²×D₁₇₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

358

—

C₄⋊C₄

—

716

—

C₄⋊C₄

—

716

—

C₄⋊C₄

—

716

—

C₄⋊C₄

—

716

—

C₄²

—

179

—

C₁₆

—

100

C₁₆

—

358

—

101

Q₈

—

102

Q₈

—

103

Q₈

—

716

—

104

Q₈

—

716

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

179

—

108

C₂×C₄

—

179

—

109

C₂×C₄

—

716

—

110

C₂×C₄

—

716

—

111

C₂×C₄

D₄×D₁₇₉

yes

characteristic

112

C₈

—

179

—

113

C₈

#15668258

yes

characteristic

114

C₈

#15668259

yes

characteristic

115

C₂²

D₈×D₁₇₉

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

358

—

119

C₄

—

358

—

120

C₄

—

1432

—

121

C₄

—

1432

—

122

C₄

#19153099

yes

characteristic

123

C₄

SD₁₆×D₁₇₉

yes

characteristic

124

C₂

#26038784

yes

characteristic

125

C₂

#26038878

yes

characteristic

126

C₂

#26038881

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₇₉
commutator, cosocle	C₂×C₈×C₁₇₉	C₂³
Frattini	C₂×C₈	C₂²×D₁₇₉
Fitting	#358×C₁₇₉	C₂
socle, prime radical	C₂²×C₁₇₉	C₂×D₈
hypercenter	#358	D₁₇₉
hypocenter	C₁₇₉	#2245
LCS	G ▷ C₂×C₈×C₁₇₉ ▷ C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₁₇₉

commutator, cosocle

C₂×C₈×C₁₇₉

C₂³

Frattini

C₂×C₈

C₂²×D₁₇₉

Fitting

#358×C₁₇₉

C₂

socle, prime radical

C₂²×C₁₇₉

C₂×D₈

D₁₇₉

C₁₇₉

G ▷ C₂×C₈×C₁₇₉ ▷ C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
63. C₁₇₉	#2245	semidirect

H ≅ G/N

type

63. C₁₇₉

#2245

semidirect

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

subgroup	index
#26038840	2
#26038817	2
#26038244	2
#26038243	2
#26038242	2
#26038214	2
#358×C₁₇₉	2
#2245	179

subgroup

index

#358×C₁₇₉

179

Subgroups — #37511244

Conjugacy classes of subgroups

Complements

Maximal subgroups