Group #23762539 — subgroups

Subgroups — #23762539

Order 10112.

206 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 59 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	Q₈×Dic₁₅₈	C₂	1	1	yes	maximal, characteristic
2	#19098942	C₂	1	1	yes	maximal, characteristic
3	#19098855	C₂	1	1	yes	maximal, characteristic
4	#19098311	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊(C₇₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	#19098293	C₂	1	1	yes	maximal, characteristic
7	#477×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#11929614	—	2	2	no	—
9	#14579161	—	2	1	no	—
10	#14579161	C₂²	1	1	yes	characteristic
11	#14579160	—	2	1	no	—
12	#14579160	—	2	2	no	—
13	Q₈×Dic₇₉	—	2	1	no	—
14	C₄×Dic₁₅₈	—	2	1	no	—
15	C₄×Dic₁₅₈	C₂²	1	1	yes	characteristic
16	#14579094	—	2	1	no	—
17	C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
18	#14579084	—	2	1	no	—
19	C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
22	#107×C₇₉	—	2	1	no	—
23	C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
24	#103×C₇₉	—	2	1	no	—
25	C₂×Dic₁₅₈	—	2	2	no	—
26	C₂×Dic₁₅₈	D₄	1	1	yes	characteristic
27	C₂×Dic₁₅₈	D₄	1	2	yes	—
28	#11929614	—	4	4	no	—
29	C₂×C₇₉⋊C₈	—	2	1	no	—
30	C₂×C₇₉⋊C₈	—	2	1	no	—
31	C₄⋊Dic₇₉	—	2	2	no	—
32	C₄⋊Dic₇₉	—	2	1	no	—
33	C₄⋊Dic₇₉	—	2	1	no	—
34	C₄⋊Dic₇₉	D₄	1	1	yes	characteristic
35	Dic₇₉⋊C₄	—	2	2	no	—
36	Dic₇₉⋊C₄	—	2	1	no	—
37	C₄×Dic₇₉	—	2	2	no	—
38	C₄×Dic₇₉	—	2	1	no	—
39	C₂×Q₈×C₇₉	—	2	1	no	—
40	C₂×Q₈×C₇₉	D₄	1	1	yes	characteristic
41	C₂×C₈×C₇₉	—	2	1	no	—
42	C₄⋊C₄×C₇₉	—	2	1	no	—
43	C₄⋊C₄×C₇₉	—	2	1	no	—
44	C₄⋊C₄×C₇₉	—	2	1	no	—
45	C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
46	C₄²×C₇₉	—	2	1	no	—
47	C₄²×C₇₉	C₂³	1	1	yes	characteristic
48	Q₈×C₇₉	—	2	1	no	—
49	Q₈×C₇₉	—	4	2	no	—
50	Q₈×C₇₉	Q₁₆	1	2	yes	complemented
51	C₂×C₄×C₇₉	—	2	1	no	—
52	C₂×C₄×C₇₉	—	2	1	no	—
53	C₂×C₄×C₇₉	—	4	1	no	—
54	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
57	C₂×Dic₇₉	—	2	1	no	—
58	C₂×Dic₇₉	—	2	2	no	—
59	C₂×Dic₇₉	—	2	1	no	—
60	Dic₁₅₈	—	2	1	no	—
61	Dic₁₅₈	—	4	2	no	—
62	Dic₁₅₈	—	8	2	no	—
63	Dic₁₅₈	Q₁₆	1	2	yes	complemented
64	C₈×C₇₉	—	4	1	no	—
65	C₇₉⋊C₈	—	2	2	no	—
66	C₇₉⋊C₈	—	4	1	no	—
67	C₂²×C₇₉	#120	1	1	yes	characteristic
68	C₄×C₇₉	—	2	2	no	—
69	C₄×C₇₉	—	2	1	no	—
70	C₄×C₇₉	—	4	1	no	—
71	C₄×C₇₉	—	8	1	no	—
72	C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
73	C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
74	C₄×C₇₉	#137	1	1	yes	characteristic
75	C₄×C₇₉	#137	1	1	yes	characteristic
76	Dic₇₉	—	2	2	no	—
77	Dic₇₉	—	4	2	no	—
78	Dic₇₉	—	4	1	no	—
79	C₂×C₇₉	#451	1	1	yes	characteristic
80	C₂×C₇₉	#451	1	1	yes	characteristic
81	C₂×C₇₉	#453	1	1	yes	characteristic
82	#1633	—	79	1	no	maximal, Sylow
83	C₇₉	#1633	1	1	yes	characteristic, complemented, Sylow
84	Q₈²	—	79	1	no	—
85	#494	—	79	1	no	—
86	#477	—	79	1	no	—
87	#477	D₇₉	1	1	yes	characteristic
88	#333	—	79	1	no	—
89	Q₈⋊C₈	—	79	1	no	—
90	Q₈⋊C₈	—	79	1	no	—
91	C₂×Q₁₆	—	158	2	no	—
92	C₄⋊Q₈	—	79	1	no	—
93	C₄⋊Q₈	—	158	1	no	—
94	C₄⋊Q₈	—	158	2	no	—
95	C₄⋊Q₈	—	158	1	no	—
96	C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
97	C₄×Q₈	—	79	1	no	—
98	C₄×Q₈	—	158	1	no	—
99	C₄×Q₈	—	158	1	no	—
100	C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
101	#107	—	2	1	no	—
102	#107	—	158	1	no	—
103	C₄⋊C₈	—	79	1	no	—
104	C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
105	#103	—	2	1	no	—
106	#103	—	158	1	no	—
107	C₄×C₈	—	79	1	no	—
108	C₂×Q₈	—	2	1	no	—
109	C₂×Q₈	—	79	2	no	—
110	C₂×Q₈	—	79	1	no	—
111	C₂×Q₈	—	158	2	no	—
112	C₂×Q₈	C₂²⋊D₇₉	1	1	yes	characteristic
113	Q₁₆	—	316	4	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	158	1	no	—
116	C₂×C₈	—	158	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	79	1	no	—
121	C₄⋊C₄	—	158	2	no	—
122	C₄⋊C₄	—	158	2	no	—
123	C₄⋊C₄	—	158	1	no	—
124	C₄⋊C₄	—	158	1	no	—
125	C₄⋊C₄	—	158	1	no	—
126	C₄⋊C₄	C₂²⋊D₇₉	1	1	yes	characteristic
127	C₄²	—	2	1	no	—
128	C₄²	—	158	2	no	—
129	C₄²	—	158	1	no	—
130	C₄²	C₂²×D₇₉	1	1	yes	characteristic
131	Q₈	—	2	1	no	—
132	Q₈	—	4	2	no	—
133	Q₈	—	316	2	no	—
134	Q₈	—	79	2	no	—
135	Q₈	—	632	2	no	—
136	Q₈	—	158	1	no	—
137	Q₈	#11929614	1	2	yes	complemented
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	158	1	no	—
142	C₂×C₄	—	158	2	no	—
143	C₂×C₄	—	158	1	no	—
144	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
145	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
147	C₈	—	4	1	no	—
148	C₈	—	316	1	no	—
149	C₈	—	158	2	no	—
150	C₂²	#14579129	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	2	1	no	—
153	C₄	—	4	1	no	—
154	C₄	—	8	1	no	—
155	C₄	—	316	2	no	—
156	C₄	—	316	1	no	—
157	C₄	—	158	2	no	—
158	C₄	Q₁₆×D₇₉	1	1	yes	characteristic
159	C₄	C₂×#11929614	1	1	yes	characteristic
160	C₄	#14579191	1	1	yes	characteristic
161	C₄	#14579193	1	1	yes	characteristic
162	C₂	#19098717	1	1	yes	characteristic
163	C₂	#19098718	1	1	yes	characteristic
164	C₂	#19098729	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₅₈

C₂

yes

maximal, characteristic

#19098942

C₂

yes

maximal, characteristic

#19098855

C₂

yes

maximal, characteristic

#19098311

C₂

yes

maximal, characteristic

Q₈⋊(C₇₉⋊C₈)

C₂

yes

maximal, characteristic

#19098293

C₂

yes

maximal, characteristic

#477×C₇₉

C₂

yes

maximal, characteristic

C₂×#11929614

—

#14579161

—

#14579161

C₂²

yes

characteristic

#14579160

—

#14579160

—

Q₈×Dic₇₉

—

C₄×Dic₁₅₈

—

C₄×Dic₁₅₈

C₂²

yes

characteristic

#14579094

—

C₄⋊(C₇₉⋊C₈)

C₂²

yes

characteristic

#14579084

—

C₄×C₇₉⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₇₉

C₂²

yes

characteristic

C₄×Q₈×C₇₉

C₂²

yes

characteristic

#107×C₇₉

—

C₄⋊C₈×C₇₉

C₂²

yes

characteristic

#103×C₇₉

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

D₄

yes

characteristic

C₂×Dic₁₅₈

D₄

yes

—

#11929614

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

D₄

yes

characteristic

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

D₄

yes

characteristic

C₂×C₈×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

D₄

yes

characteristic

C₄²×C₇₉

—

C₄²×C₇₉

C₂³

yes

characteristic

Q₈×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

Q₁₆

yes

complemented

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

Q₁₆

yes

complemented

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

#120

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×Q₁₆

yes

characteristic

C₄×C₇₉

C₂×Q₁₆

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

Q₈²

—

#494

—

#477

—

#477

D₇₉

yes

characteristic

#333

—

Q₈⋊C₈

—

Q₈⋊C₈

—

C₂×Q₁₆

—

158

—

C₄⋊Q₈

—

C₄⋊Q₈

—

158

—

C₄⋊Q₈

—

158

—

C₄⋊Q₈

—

158

—

C₄⋊Q₈

C₂×D₇₉

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

158

—

C₄×Q₈

—

158

—

100

C₄×Q₈

C₂×D₇₉

yes

characteristic

101

#107

—

102

#107

—

158

—

103

C₄⋊C₈

—

104

C₄⋊C₈

C₂×D₇₉

yes

characteristic

105

#103

—

106

#103

—

158

—

107

C₄×C₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

158

—

112

C₂×Q₈

C₂²⋊D₇₉

yes

characteristic

113

Q₁₆

—

316

—

114

C₂×C₈

—

115

C₂×C₈

—

158

—

116

C₂×C₈

—

158

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

158

—

122

C₄⋊C₄

—

158

—

123

C₄⋊C₄

—

158

—

124

C₄⋊C₄

—

158

—

125

C₄⋊C₄

—

158

—

126

C₄⋊C₄

C₂²⋊D₇₉

yes

characteristic

127

C₄²

—

128

C₄²

—

158

—

129

C₄²

—

158

—

130

C₄²

C₂²×D₇₉

yes

characteristic

131

Q₈

—

132

Q₈

—

133

Q₈

—

316

—

134

Q₈

—

135

Q₈

—

632

—

136

Q₈

—

158

—

137

Q₈

#11929614

yes

complemented

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

158

—

142

C₂×C₄

—

158

—

143

C₂×C₄

—

158

—

144

C₂×C₄

D₄×D₇₉

yes

characteristic

145

C₂×C₄

D₄×D₇₉

yes

characteristic

146

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

147

C₈

—

148

C₈

—

316

—

149

C₈

—

158

—

150

C₂²

#14579129

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

316

—

156

C₄

—

316

—

157

C₄

—

158

—

158

C₄

Q₁₆×D₇₉

yes

characteristic

159

C₄

C₂×#11929614

yes

characteristic

160

C₄

#14579191

yes

characteristic

161

C₄

#14579193

yes

characteristic

162

C₂

#19098717

yes

characteristic

163

C₂

#19098718

yes

characteristic

164

C₂

#19098729

yes

characteristic

	subgroup	G/H
center	C₂²	#14579129
commutator, cosocle	C₄²×C₇₉	C₂³
Frattini	C₄²	C₂²×D₇₉
Fitting	#477×C₇₉	C₂
socle, prime radical	C₂²×C₇₉	#120
hypercenter	#477	D₇₉
hypocenter	C₇₉	#1633
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #477

subgroup

G/H

center

C₂²

#14579129

commutator, cosocle

C₄²×C₇₉

C₂³

Frattini

C₄²

C₂²×D₇₉

Fitting

#477×C₇₉

C₂

C₂²×C₇₉

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #477

N	H ≅ G/N	type
50. Q₈×C₇₉	Q₁₆	semidirect
63. Dic₁₅₈	Q₁₆	semidirect
83. C₇₉	#1633	semidirect
137. Q₈	#11929614	semidirect

H ≅ G/N

type

50. Q₈×C₇₉

Q₁₆

semidirect

63. Dic₁₅₈

Q₁₆

semidirect

83. C₇₉

#1633

semidirect

137. Q₈

#11929614

semidirect

Subgroups — #23762539

Conjugacy classes of subgroups

Complements