Group #23762817 — subgroups

Subgroups — #23762817

Order 10112.

160 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 53 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	#19098910	C₂	1	1	yes	maximal, characteristic
2	#19098859	C₂	1	1	yes	maximal, characteristic, complemented
3	#19098764	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₁₅₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#19098311	C₂	1	1	yes	maximal, characteristic, complemented
6	#19098306	C₂	1	1	yes	maximal, characteristic
7	#334×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11929613	—	2	1	no	—
9	#14579161	C₂²	1	1	yes	characteristic
10	#14579158	C₂²	1	1	yes	characteristic
11	C₄×Dic₁₅₈	C₂²	1	1	yes	characteristic
12	#14579095	—	2	1	no	—
13	C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
14	Dic₇₉⋊C₈	—	2	1	no	—
15	#14579086	—	2	1	no	—
16	#14579083	—	2	1	no	—
17	#14579083	—	2	1	no	—
18	#14579083	—	2	1	no	—
19	C₈×Dic₇₉	—	2	1	no	—
20	C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₅₈	—	2	1	no	—
24	C₂×Dic₁₅₈	D₄	1	1	yes	characteristic
25	C₂×D₃₁₆	—	2	1	no	—
26	C₂×D₃₁₆	—	2	1	no	—
27	C₂×D₃₁₆	—	2	1	no	—
28	#11929613	—	4	2	no	—
29	C₂×C₇₉⋊C₈	—	2	1	no	—
30	C₄⋊Dic₇₉	—	2	1	no	—
31	C₄⋊Dic₇₉	—	2	1	no	—
32	C₄⋊Dic₇₉	D₄	1	1	yes	characteristic
33	Dic₇₉⋊C₄	—	2	1	no	—
34	C₄×Dic₇₉	—	2	1	no	—
35	C₂×C₈×C₇₉	—	2	1	no	—
36	C₂×C₈×C₇₉	—	2	1	no	—
37	C₂×C₈×C₇₉	D₄	1	1	yes	characteristic
38	C₂×C₈×C₇₉	D₄	1	1	yes	characteristic
39	C₄²×C₇₉	C₂³	1	1	yes	characteristic
40	C₂²×D₇₉	—	4	1	no	—
41	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
44	C₂×Dic₇₉	—	2	1	no	—
45	C₂×Dic₇₉	—	2	1	no	—
46	C₂×Dic₇₉	—	4	1	no	—
47	D₃₁₆	—	4	1	no	—
48	D₃₁₆	—	4	1	no	—
49	D₃₁₆	—	4	1	no	—
50	D₃₁₆	—	4	1	no	—
51	D₃₁₆	—	8	1	no	—
52	Dic₁₅₈	—	2	1	no	—
53	Dic₁₅₈	—	8	1	no	—
54	Dic₁₅₈	SD₁₆	1	2	yes	—
55	C₈×C₇₉	—	2	1	no	—
56	C₈×C₇₉	—	2	1	no	—
57	C₈×C₇₉	—	4	1	no	—
58	C₈×C₇₉	—	4	1	no	—
59	C₇₉⋊C₈	—	4	1	no	—
60	C₂²×C₇₉	#121	1	1	yes	characteristic
61	C₂×D₇₉	—	8	1	no	—
62	C₂×D₇₉	—	8	1	no	—
63	C₂×D₇₉	—	8	1	no	—
64	C₄×C₇₉	—	2	1	no	—
65	C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
66	C₄×C₇₉	#135	1	1	yes	characteristic
67	C₄×C₇₉	#135	1	1	yes	characteristic
68	C₄×C₇₉	#137	1	1	yes	characteristic
69	Dic₇₉	—	2	2	no	—
70	Dic₇₉	—	4	1	no	—
71	Dic₇₉	—	8	1	no	—
72	C₂×C₇₉	#464	1	1	yes	characteristic
73	C₂×C₇₉	#465	1	1	yes	characteristic
74	C₂×C₇₉	#472	1	1	yes	characteristic
75	D₇₉	—	16	1	no	—
76	#1678	—	79	1	no	maximal, Sylow
77	C₇₉	#1678	1	1	yes	characteristic, complemented, Sylow
78	#486	—	79	1	no	—
79	#477	—	79	1	no	—
80	#460	—	79	1	no	—
81	C₈×Q₈	—	79	1	no	—
82	#334	D₇₉	1	1	yes	characteristic, complemented
83	#333	—	79	1	no	—
84	#331	—	79	1	no	—
85	C₂×SD₁₆	—	158	1	no	—
86	C₄⋊Q₈	—	79	1	no	—
87	#127	—	79	1	no	—
88	C₄×Q₈	—	79	1	no	—
89	#107	—	158	1	no	—
90	C₄⋊C₈	—	79	1	no	—
91	C₄⋊C₈	—	158	1	no	—
92	C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
93	C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
94	#103	—	158	1	no	—
95	#102	—	158	1	no	—
96	#102	—	158	1	no	—
97	#102	—	158	1	no	—
98	C₄×C₈	—	158	1	no	—
99	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
100	C₂×Q₈	—	79	1	no	—
101	C₂×Q₈	—	158	1	no	—
102	C₂×D₄	—	158	1	no	—
103	C₂×D₄	—	158	1	no	—
104	C₂×D₄	—	158	1	no	—
105	SD₁₆	—	316	2	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	158	1	no	—
109	C₂×C₈	D₃₁₆	1	1	yes	characteristic
110	C₂×C₈	D₃₁₆	1	1	yes	characteristic
111	C₄⋊C₄	—	79	1	no	—
112	C₄⋊C₄	—	158	1	no	—
113	C₄⋊C₄	—	158	1	no	—
114	C₄⋊C₄	—	158	1	no	—
115	C₄²	—	158	1	no	—
116	C₄²	C₂²×D₇₉	1	1	yes	characteristic
117	C₂³	—	316	1	no	—
118	Q₈	—	79	2	no	—
119	Q₈	—	632	1	no	—
120	Q₈	—	158	1	no	—
121	D₄	—	316	1	no	—
122	D₄	—	316	1	no	—
123	D₄	—	316	1	no	—
124	D₄	—	316	1	no	—
125	D₄	—	632	1	no	—
126	C₂×C₄	—	316	1	no	—
127	C₂×C₄	—	158	1	no	—
128	C₂×C₄	—	158	1	no	—
129	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
130	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
131	C₂×C₄	#11929625	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	2	1	no	—
134	C₈	—	4	1	no	—
135	C₈	—	4	1	no	—
136	C₈	—	316	1	no	—
137	C₂²	—	632	1	no	—
138	C₂²	—	632	1	no	—
139	C₂²	—	632	1	no	—
140	C₂²	#14579136	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	316	1	no	—
143	C₄	—	632	1	no	—
144	C₄	—	158	2	no	—
145	C₄	SD₁₆×D₇₉	1	1	yes	characteristic
146	C₄	#14579182	1	1	yes	characteristic
147	C₄	#14579183	1	1	yes	characteristic
148	C₄	#14579194	1	1	yes	characteristic
149	C₂	—	1264	1	no	—
150	C₂	#19098786	1	1	yes	characteristic
151	C₂	#19098793	1	1	yes	characteristic
152	C₂	#19098834	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19098910

C₂

yes

maximal, characteristic

#19098859

C₂

yes

maximal, characteristic, complemented

#19098764

C₂

yes

maximal, characteristic

C₈×Dic₁₅₈

C₂

yes

maximal, characteristic, complemented

#19098311

C₂

yes

maximal, characteristic, complemented

#19098306

C₂

yes

maximal, characteristic

#334×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#11929613

—

#14579161

C₂²

yes

characteristic

#14579158

C₂²

yes

characteristic

C₄×Dic₁₅₈

C₂²

yes

characteristic

#14579095

—

C₄⋊(C₇₉⋊C₈)

C₂²

yes

characteristic

Dic₇₉⋊C₈

—

—

—

—

—

C₈×Dic₇₉

—

C₄⋊C₈×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

C₂²

yes

characteristic

C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

D₄

yes

characteristic

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

#11929613

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

D₄

yes

characteristic

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

D₄

yes

characteristic

C₂×C₈×C₇₉

D₄

yes

characteristic

C₄²×C₇₉

C₂³

yes

characteristic

C₂²×D₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

#41

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

SD₁₆

yes

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

#121

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×SD₁₆

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

D₇₉

—

#1678

—

maximal, Sylow

C₇₉

#1678

yes

characteristic, complemented, Sylow

#486

—

#477

—

#460

—

C₈×Q₈

—

#334

D₇₉

yes

characteristic, complemented

#333

—

#331

—

C₂×SD₁₆

—

158

—

C₄⋊Q₈

—

#127

—

C₄×Q₈

—

#107

—

158

—

C₄⋊C₈

—

C₄⋊C₈

—

158

—

C₄⋊C₈

C₂×D₇₉

yes

characteristic

C₄⋊C₈

C₂×D₇₉

yes

characteristic

#103

—

158

—

#102

—

158

—

#102

—

158

—

#102

—

158

—

C₄×C₈

—

158

—

C₄×C₈

C₂×D₇₉

yes

characteristic

100

C₂×Q₈

—

101

C₂×Q₈

—

158

—

102

C₂×D₄

—

158

—

103

C₂×D₄

—

158

—

104

C₂×D₄

—

158

—

105

SD₁₆

—

316

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

158

—

109

C₂×C₈

D₃₁₆

yes

characteristic

110

C₂×C₈

D₃₁₆

yes

characteristic

111

C₄⋊C₄

—

112

C₄⋊C₄

—

158

—

113

C₄⋊C₄

—

158

—

114

C₄⋊C₄

—

158

—

115

C₄²

—

158

—

116

C₄²

C₂²×D₇₉

yes

characteristic

117

C₂³

—

316

—

118

Q₈

—

119

Q₈

—

632

—

120

Q₈

—

158

—

121

D₄

—

316

—

122

D₄

—

316

—

123

D₄

—

316

—

124

D₄

—

316

—

125

D₄

—

632

—

126

C₂×C₄

—

316

—

127

C₂×C₄

—

158

—

128

C₂×C₄

—

158

—

129

C₂×C₄

D₄×D₇₉

yes

characteristic

130

C₂×C₄

C₂×D₃₁₆

yes

characteristic

131

C₂×C₄

#11929625

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

316

—

137

C₂²

—

632

—

138

C₂²

—

632

—

139

C₂²

—

632

—

140

C₂²

#14579136

yes

characteristic

141

C₄

—

142

C₄

—

316

—

143

C₄

—

632

—

144

C₄

—

158

—

145

C₄

SD₁₆×D₇₉

yes

characteristic

146

C₄

#14579182

yes

characteristic

147

C₄

#14579183

yes

characteristic

148

C₄

#14579194

yes

characteristic

149

C₂

—

1264

—

150

C₂

#19098786

yes

characteristic

151

C₂

#19098793

yes

characteristic

152

C₂

#19098834

yes

characteristic

	subgroup	G/H
center	C₂²	#14579136
commutator, cosocle	C₄²×C₇₉	C₂³
Frattini	C₄²	C₂²×D₇₉
Fitting	#334×C₇₉	C₂
socle	C₂²×C₇₉	#121
prime radical	#14579158	C₂²
hypercenter	#334	D₇₉
hypocenter	C₇₉	#1678
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

subgroup

G/H

center

C₂²

#14579136

commutator, cosocle

C₄²×C₇₉

C₂³

Frattini

C₄²

C₂²×D₇₉

Fitting

#334×C₇₉

C₂

C₂²×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #334

N	H ≅ G/N	type
2. #19098859	C₂	semidirect
4. C₈×Dic₁₅₈	C₂	semidirect
5. #19098311	C₂	semidirect
7. #334×C₇₉	C₂	semidirect
77. C₇₉	#1678	semidirect
82. #334	D₇₉	semidirect

H ≅ G/N

type

2. #19098859

C₂

semidirect

4. C₈×Dic₁₅₈

C₂

semidirect

5. #19098311

C₂

semidirect

7. #334×C₇₉

C₂

semidirect

77. C₇₉

#1678

semidirect

82. #334

D₇₉

semidirect

Subgroups — #23762817

Conjugacy classes of subgroups

Complements