Group #37508623 — subgroups

Subgroups — #37508623

Order 22912.

198 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 65 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	#26038835	C₂	1	1	yes	maximal, characteristic, complemented
2	#26038813	C₂	1	1	yes	maximal, characteristic
3	#26038812	C₂	1	1	yes	maximal, characteristic
4	#26038789	C₂	1	1	yes	maximal, characteristic, complemented
5	#26038787	C₂	1	1	yes	maximal, characteristic
6	#26038160	C₂	1	1	yes	maximal, characteristic, complemented
7	#500×C₁₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668248	—	2	2	no	—
9	C₂×#15668246	—	2	2	no	—
10	#19153113	C₂²	1	1	yes	characteristic
11	#19153110	C₂²	1	1	yes	characteristic
12	#19153044	—	2	2	no	—
13	#19153037	—	2	2	no	—
14	#19153033	—	2	2	no	—
15	C₄×C₁₇₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₁₇₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₁₇₉	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₁₇₉	C₂²	1	1	yes	characteristic
19	#107×C₁₇₉	—	2	2	no	—
20	C₄×C₈×C₁₇₉	C₂²	1	1	yes	characteristic
21	C₂×Dic₃₅₈	—	2	1	no	—
22	C₂×D₇₁₆	—	2	1	no	—
23	C₂×D₇₁₆	—	2	1	no	—
24	C₂×D₇₁₆	—	2	1	no	—
25	#15668248	—	4	4	no	—
26	#15668246	—	4	4	no	—
27	C₂×C₁₇₉⋊C₈	D₄	1	2	yes	—
28	C₂×C₁₇₉⋊C₈	D₄	1	2	yes	—
29	C₄⋊Dic₁₇₉	—	2	1	no	—
30	C₄⋊Dic₁₇₉	—	2	1	no	—
31	C₂×Q₈×C₁₇₉	—	2	1	no	—
32	C₂×Q₈×C₁₇₉	—	2	1	no	—
33	C₂×C₈×C₁₇₉	D₄	1	2	yes	—
34	C₄⋊C₄×C₁₇₉	—	2	1	no	—
35	C₄⋊C₄×C₁₇₉	—	2	1	no	—
36	C₄⋊C₄×C₁₇₉	—	2	1	no	—
37	C₄⋊C₄×C₁₇₉	—	2	1	no	—
38	C₄²×C₁₇₉	C₂³	1	1	yes	characteristic
39	C₂²×D₁₇₉	—	4	1	no	—
40	Dic₃₅₈	—	4	2	no	—
41	D₇₁₆	—	4	2	no	—
42	D₇₁₆	—	4	2	no	—
43	D₇₁₆	—	4	2	no	—
44	C₂×Dic₁₇₉	—	4	1	no	—
45	C₁₇₉⋊C₈	—	2	2	no	—
46	C₁₇₉⋊C₈	—	2	2	no	—
47	Q₈×C₁₇₉	—	4	2	no	—
48	Q₈×C₁₇₉	—	4	2	no	—
49	C₂×C₄×C₁₇₉	—	4	1	no	—
50	C₂×C₄×C₁₇₉	—	4	1	no	—
51	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
54	C₈×C₁₇₉	SD₁₆	1	4	yes	complemented
55	C₂²×C₁₇₉	#127	1	1	yes	characteristic
56	C₂×D₁₇₉	—	8	1	no	—
57	C₂×D₁₇₉	—	8	1	no	—
58	C₂×D₁₇₉	—	8	1	no	—
59	C₄×C₁₇₉	—	8	1	no	—
60	C₄×C₁₇₉	—	8	1	no	—
61	C₄×C₁₇₉	C₂×SD₁₆	1	2	yes	—
62	C₄×C₁₇₉	AGL(1,ℤ/8)	1	2	yes	—
63	C₄×C₁₇₉	#137	1	2	yes	—
64	Dic₁₇₉	—	8	1	no	—
65	C₂×C₁₇₉	#492	1	1	yes	characteristic
66	C₂×C₁₇₉	#496	1	1	yes	characteristic
67	C₂×C₁₇₉	#497	1	1	yes	characteristic
68	D₁₇₉	—	16	1	no	—
69	C₁₇₉	#1714	1	1	yes	characteristic, complemented, Sylow
70	#1714	—	179	1	no	maximal, Sylow
71	#500	D₁₇₉	1	1	yes	characteristic, complemented
72	#498	—	179	1	no	—
73	#492	—	179	1	no	—
74	#492	—	179	1	no	—
75	#487	—	179	1	no	—
76	#486	—	179	1	no	—
77	#322	—	179	1	no	—
78	C₂×SD₁₆	—	358	2	no	—
79	C₂×SD₁₆	—	358	2	no	—
80	C₄⋊Q₈	—	179	1	no	—
81	C₄⋊Q₈	C₂×D₁₇₉	1	1	yes	characteristic
82	C₄⋊Q₈	C₂×D₁₇₉	1	1	yes	characteristic
83	#127	—	179	1	no	—
84	#107	—	2	2	no	—
85	#106	—	358	2	no	—
86	#103	—	358	2	no	—
87	#102	—	358	2	no	—
88	C₄×C₈	—	179	1	no	—
89	C₄×C₈	—	179	1	no	—
90	C₄×C₈	C₂×D₁₇₉	1	1	yes	characteristic
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	2	1	no	—
93	C₂×Q₈	—	358	1	no	—
94	C₂×D₄	—	358	1	no	—
95	C₂×D₄	—	358	1	no	—
96	C₂×D₄	—	358	1	no	—
97	SD₁₆	—	716	4	no	—
98	SD₁₆	—	716	4	no	—
99	C₂×C₈	—	179	2	no	—
100	C₂×C₈	—	179	2	no	—
101	C₂×C₈	C₂²⋊D₁₇₉	1	2	yes	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	358	1	no	—
107	C₄⋊C₄	—	358	1	no	—
108	C₄²	C₂²×D₁₇₉	1	1	yes	characteristic
109	C₂³	—	716	1	no	—
110	Q₈	—	4	2	no	—
111	Q₈	—	4	2	no	—
112	Q₈	—	716	2	no	—
113	D₄	—	716	2	no	—
114	D₄	—	716	2	no	—
115	D₄	—	716	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	716	1	no	—
119	C₂×C₄	D₄×D₁₇₉	1	1	yes	characteristic
120	C₂×C₄	D₄×D₁₇₉	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₁₇₉	1	1	yes	characteristic
122	C₈	—	358	2	no	—
123	C₈	—	358	2	no	—
124	C₈	#15668246	1	4	yes	complemented
125	C₂²	—	1432	1	no	—
126	C₂²	—	1432	1	no	—
127	C₂²	—	1432	1	no	—
128	C₂²	#19153109	1	1	yes	characteristic
129	C₄	—	8	1	no	—
130	C₄	—	8	1	no	—
131	C₄	—	1432	1	no	—
132	C₄	C₂×#15668246	1	2	yes	—
133	C₄	#19153137	1	2	yes	—
134	C₄	#19153145	1	2	yes	—
135	C₂	—	2864	1	no	—
136	C₂	#26038813	1	1	yes	characteristic
137	C₂	#26038827	1	1	yes	characteristic
138	C₂	#26038832	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26038835

C₂

yes

maximal, characteristic, complemented

#26038813

C₂

yes

maximal, characteristic

#26038812

C₂

yes

maximal, characteristic

#26038789

C₂

yes

maximal, characteristic, complemented

#26038787

C₂

yes

maximal, characteristic

#26038160

C₂

yes

maximal, characteristic, complemented

#500×C₁₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#15668248

—

C₂×#15668246

—

#19153113

C₂²

yes

characteristic

#19153110

C₂²

yes

—

—

—

C₄×C₁₇₉⋊C₈

C₂²

yes

characteristic

C₄×C₁₇₉⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₁₇₉

C₂²

yes

characteristic

C₄⋊Q₈×C₁₇₉

C₂²

yes

characteristic

#107×C₁₇₉

—

C₄×C₈×C₁₇₉

C₂²

yes

characteristic

C₂×Dic₃₅₈

—

C₂×D₇₁₆

—

C₂×D₇₁₆

—

C₂×D₇₁₆

—

#15668248

—

#15668246

—

C₂×C₁₇₉⋊C₈

D₄

yes

—

C₂×C₁₇₉⋊C₈

D₄

yes

—

C₄⋊Dic₁₇₉

—

C₄⋊Dic₁₇₉

—

C₂×Q₈×C₁₇₉

—

C₂×Q₈×C₁₇₉

—

C₂×C₈×C₁₇₉

D₄

yes

—

C₄⋊C₄×C₁₇₉

—

C₄⋊C₄×C₁₇₉

—

C₄⋊C₄×C₁₇₉

—

C₄⋊C₄×C₁₇₉

—

C₄²×C₁₇₉

C₂³

yes

characteristic

C₂²×D₁₇₉

—

Dic₃₅₈

—

D₇₁₆

—

D₇₁₆

—

D₇₁₆

—

C₂×Dic₁₇₉

—

C₁₇₉⋊C₈

—

C₁₇₉⋊C₈

—

Q₈×C₁₇₉

—

Q₈×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₈×C₁₇₉

SD₁₆

yes

complemented

C₂²×C₁₇₉

#127

yes

characteristic

C₂×D₁₇₉

—

C₂×D₁₇₉

—

C₂×D₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

C₂×SD₁₆

yes

—

C₄×C₁₇₉

AGL(1,ℤ/8)

yes

—

C₄×C₁₇₉

#137

yes

—

Dic₁₇₉

—

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

D₁₇₉

—

C₁₇₉

#1714

yes

characteristic, complemented, Sylow

#1714

—

179

maximal, Sylow

#500

D₁₇₉

yes

characteristic, complemented

#498

—

179

—

#492

—

179

—

#492

—

179

—

#487

—

179

—

#486

—

179

—

#322

—

179

—

C₂×SD₁₆

—

358

—

C₂×SD₁₆

—

358

—

C₄⋊Q₈

—

179

—

C₄⋊Q₈

C₂×D₁₇₉

yes

characteristic

C₄⋊Q₈

C₂×D₁₇₉

yes

characteristic

#127

—

179

—

#107

—

#106

—

358

—

#103

—

358

—

#102

—

358

—

C₄×C₈

—

179

—

C₄×C₈

—

179

—

C₄×C₈

C₂×D₁₇₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

358

—

C₂×D₄

—

358

—

C₂×D₄

—

358

—

C₂×D₄

—

358

—

SD₁₆

—

716

—

SD₁₆

—

716

—

C₂×C₈

—

179

—

100

C₂×C₈

—

179

—

101

C₂×C₈

C₂²⋊D₁₇₉

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

358

—

107

C₄⋊C₄

—

358

—

108

C₄²

C₂²×D₁₇₉

yes

characteristic

109

C₂³

—

716

—

110

Q₈

—

111

Q₈

—

112

Q₈

—

716

—

113

D₄

—

716

—

114

D₄

—

716

—

115

D₄

—

716

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

716

—

119

C₂×C₄

D₄×D₁₇₉

yes

characteristic

120

C₂×C₄

D₄×D₁₇₉

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₁₇₉

yes

characteristic

122

C₈

—

358

—

123

C₈

—

358

—

124

C₈

#15668246

yes

complemented

125

C₂²

—

1432

—

126

C₂²

—

1432

—

127

C₂²

—

1432

—

128

C₂²

#19153109

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

1432

—

132

C₄

C₂×#15668246

yes

—

133

C₄

#19153137

yes

—

134

C₄

#19153145

yes

—

135

C₂

—

2864

—

136

C₂

#26038813

yes

characteristic

137

C₂

#26038827

yes

characteristic

138

C₂

#26038832

yes

characteristic

	subgroup	G/H
center	C₂²	#19153109
commutator, cosocle	C₄²×C₁₇₉	C₂³
Frattini	C₄²	C₂²×D₁₇₉
Fitting	#500×C₁₇₉	C₂
socle	C₂²×C₁₇₉	#127
prime radical	#19153110	C₂²
hypercenter	#500	D₁₇₉
hypocenter	C₁₇₉	#1714
LCS	G ▷ C₄²×C₁₇₉ ▷ C₂²×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

#19153109

commutator, cosocle

C₄²×C₁₇₉

C₂³

Frattini

C₄²

C₂²×D₁₇₉

Fitting

#500×C₁₇₉

C₂

C₂²×C₁₇₉

C₂²

D₁₇₉

C₁₇₉

G ▷ C₄²×C₁₇₉ ▷ C₂²×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
1. #26038835	C₂	semidirect
4. #26038789	C₂	semidirect
6. #26038160	C₂	semidirect
7. #500×C₁₇₉	C₂	semidirect
54. C₈×C₁₇₉	SD₁₆	semidirect
69. C₁₇₉	#1714	semidirect
71. #500	D₁₇₉	semidirect
124. C₈	#15668246	semidirect

H ≅ G/N

type

1. #26038835

C₂

semidirect

4. #26038789

C₂

semidirect

6. #26038160

C₂

semidirect

7. #500×C₁₇₉

C₂

semidirect

54. C₈×C₁₇₉

SD₁₆

semidirect

69. C₁₇₉

#1714

semidirect

71. #500

D₁₇₉

semidirect

124. C₈

#15668246

semidirect

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

subgroup	index
#26038835	2
#26038813	2
#26038812	2
#26038789	2
#26038787	2
#26038160	2
#500×C₁₇₉	2
#1714	179

subgroup

index

#500×C₁₇₉

179

Subgroups — #37508623

Conjugacy classes of subgroups

Complements

Maximal subgroups