Group #23762560 — subgroups

Subgroups — #23762560

Order 10112.

188 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 55 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	#19099291	C₂	1	1	yes	maximal, characteristic
2	#19098914	C₂	1	1	yes	maximal, characteristic
3	#19098859	C₂	1	1	yes	maximal, characteristic
4	#19098858	C₂	1	1	yes	maximal, characteristic
5	#19098308	C₂	1	1	yes	maximal, characteristic
6	#19098295	C₂	1	1	yes	maximal, characteristic
7	Q₈⋊C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
8	#14579161	C₂²	1	1	yes	characteristic
9	#14579161	C₂²	1	1	yes	characteristic
10	#14579159	—	2	1	no	—
11	#14579153	—	2	1	no	—
12	#14579151	—	2	2	no	—
13	C₄×Dic₁₅₈	—	2	1	no	—
14	C₄×Dic₁₅₈	C₂²	1	1	yes	characteristic
15	#14579126	—	2	1	no	—
16	#14579095	—	2	1	no	—
17	#14579094	—	2	1	no	—
18	C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
19	#14579086	—	2	1	no	—
20	#14579086	—	2	2	no	—
21	#14579085	—	2	1	no	—
22	C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
24	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
25	C₂×Dic₁₅₈	—	2	1	no	—
26	C₂×Dic₁₅₈	—	2	1	no	—
27	C₂×Dic₁₅₈	D₄	1	1	yes	characteristic
28	C₂×C₇₉⋊C₈	—	2	1	no	—
29	C₄⋊Dic₇₉	—	2	1	no	—
30	C₄⋊Dic₇₉	—	2	1	no	—
31	C₄⋊Dic₇₉	—	2	1	no	—
32	C₄⋊Dic₇₉	D₄	1	2	yes	—
33	C₄⋊Dic₇₉	D₄	1	1	yes	characteristic
34	Dic₇₉⋊C₄	—	2	2	no	—
35	Dic₇₉⋊C₄	—	2	2	no	—
36	Dic₇₉⋊C₄	—	2	2	no	—
37	Dic₇₉⋊C₄	—	2	1	no	—
38	C₄×Dic₇₉	—	2	2	no	—
39	C₄×Dic₇₉	—	2	1	no	—
40	C₂×Q₈×C₇₉	D₄	1	1	yes	characteristic
41	C₂×C₈×C₇₉	—	2	1	no	—
42	C₂×C₈×C₇₉	—	2	1	no	—
43	C₄⋊C₄×C₇₉	—	2	1	no	—
44	C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
45	C₄²×C₇₉	—	2	1	no	—
46	C₄²×C₇₉	C₂³	1	1	yes	characteristic
47	Q₈×C₇₉	—	2	1	no	—
48	Q₈×C₇₉	—	2	1	no	—
49	C₂×C₄×C₇₉	—	2	1	no	—
50	C₂×C₄×C₇₉	—	2	1	no	—
51	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
54	C₂×Dic₇₉	—	2	1	no	—
55	C₂×Dic₇₉	—	2	1	no	—
56	C₂×Dic₇₉	—	2	2	no	—
57	C₂×Dic₇₉	—	4	1	no	—
58	Dic₁₅₈	—	2	1	no	—
59	Dic₁₅₈	—	4	1	no	—
60	Dic₁₅₈	—	4	1	no	—
61	Dic₁₅₈	—	4	2	no	—
62	Dic₁₅₈	Q₁₆	1	2	yes	—
63	C₈×C₇₉	—	4	1	no	—
64	C₈×C₇₉	—	4	1	no	—
65	C₇₉⋊C₈	—	4	1	no	—
66	C₂²×C₇₉	#120	1	1	yes	characteristic
67	C₄×C₇₉	—	2	1	no	—
68	C₄×C₇₉	—	4	1	no	—
69	C₄×C₇₉	—	4	1	no	—
70	C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
71	C₄×C₇₉	#135	1	1	yes	characteristic
72	C₄×C₇₉	#137	1	1	yes	characteristic
73	C₄×C₇₉	#137	1	1	yes	characteristic
74	Dic₇₉	—	2	2	no	—
75	Dic₇₉	—	4	2	no	—
76	Dic₇₉	—	4	1	no	—
77	Dic₇₉	—	8	1	no	—
78	C₂×C₇₉	#451	1	1	yes	characteristic
79	C₂×C₇₉	#452	1	1	yes	characteristic
80	C₂×C₇₉	#454	1	1	yes	characteristic
81	#1636	—	79	1	no	maximal, Sylow
82	C₇₉	#1636	1	1	yes	characteristic, complemented, Sylow
83	#557	—	79	1	no	—
84	#487	—	79	1	no	—
85	#477	—	79	1	no	—
86	#477	—	79	1	no	—
87	#332	—	79	1	no	—
88	Q₈⋊C₈	—	79	1	no	—
89	Q₈⋊C₈	D₇₉	1	1	yes	characteristic
90	C₄⋊Q₈	—	79	1	no	—
91	C₄⋊Q₈	—	79	1	no	—
92	C₄⋊Q₈	—	158	1	no	—
93	#125	—	158	1	no	—
94	#125	—	158	2	no	—
95	C₄×Q₈	—	79	1	no	—
96	C₄×Q₈	—	158	1	no	—
97	C₄×Q₈	—	158	1	no	—
98	C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
99	#107	—	158	1	no	—
100	#107	—	158	1	no	—
101	C₄⋊C₈	—	79	1	no	—
102	C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
103	#103	—	158	2	no	—
104	#103	—	158	1	no	—
105	#103	—	158	1	no	—
106	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
107	C₂×Q₈	—	79	1	no	—
108	C₂×Q₈	—	158	1	no	—
109	C₂×Q₈	—	158	1	no	—
110	C₂×Q₈	D₃₁₆	1	1	yes	characteristic
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	158	1	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	79	2	no	—
116	C₄⋊C₄	—	79	1	no	—
117	C₄⋊C₄	—	158	2	no	—
118	C₄⋊C₄	—	158	2	no	—
119	C₄⋊C₄	—	158	2	no	—
120	C₄⋊C₄	—	158	1	no	—
121	C₄⋊C₄	—	158	1	no	—
122	C₄⋊C₄	—	158	1	no	—
123	C₄⋊C₄	—	158	1	no	—
124	C₄⋊C₄	D₃₁₆	1	1	yes	characteristic
125	C₄²	—	2	1	no	—
126	C₄²	—	158	2	no	—
127	C₄²	—	158	1	no	—
128	C₄²	C₂²×D₇₉	1	1	yes	characteristic
129	Q₈	—	2	1	no	—
130	Q₈	—	2	1	no	—
131	Q₈	—	316	2	no	—
132	Q₈	—	316	1	no	—
133	Q₈	—	316	1	no	—
134	Q₈	—	79	2	no	—
135	Q₈	—	158	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	316	1	no	—
139	C₂×C₄	—	158	1	no	—
140	C₂×C₄	—	158	2	no	—
141	C₂×C₄	—	158	1	no	—
142	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
143	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
144	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
145	C₈	—	4	1	no	—
146	C₈	—	4	1	no	—
147	C₈	—	316	1	no	—
148	C₂²	#14579131	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	4	1	no	—
151	C₄	—	4	1	no	—
152	C₄	—	316	2	no	—
153	C₄	—	316	1	no	—
154	C₄	—	632	1	no	—
155	C₄	—	158	2	no	—
156	C₄	Q₁₆×D₇₉	1	1	yes	characteristic
157	C₄	#14579183	1	1	yes	characteristic
158	C₄	#14579190	1	1	yes	characteristic
159	C₄	#14579194	1	1	yes	characteristic
160	C₂	#19098720	1	1	yes	characteristic
161	C₂	#19098727	1	1	yes	characteristic
162	C₂	#19098737	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19099291

C₂

yes

maximal, characteristic

#19098914

C₂

yes

maximal, characteristic

#19098859

C₂

yes

maximal, characteristic

#19098858

C₂

yes

maximal, characteristic

#19098308

C₂

yes

maximal, characteristic

#19098295

C₂

yes

maximal, characteristic

Q₈⋊C₈×C₇₉

C₂

yes

maximal, characteristic

#14579161

C₂²

yes

characteristic

#14579161

C₂²

yes

—

—

—

C₄×Dic₁₅₈

—

C₄×Dic₁₅₈

C₂²

yes

—

—

—

C₄⋊(C₇₉⋊C₈)

C₂²

yes

—

—

—

C₄×Q₈×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

C₂²

yes

characteristic

C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

D₄

yes

characteristic

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

D₄

yes

—

C₄⋊Dic₇₉

D₄

yes

characteristic

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₂×Q₈×C₇₉

D₄

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

D₄

yes

characteristic

C₄²×C₇₉

—

C₄²×C₇₉

C₂³

yes

characteristic

Q₈×C₇₉

—

Q₈×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

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

Q₁₆

yes

—

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

#120

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×Q₁₆

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

—

—

—

—

—

Q₈⋊C₈

—

Q₈⋊C₈

D₇₉

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

158

—

#125

—

158

—

#125

—

158

—

C₄×Q₈

—

C₄×Q₈

—

158

—

C₄×Q₈

—

158

—

C₄×Q₈

C₂×D₇₉

yes

characteristic

#107

—

158

—

100

#107

—

158

—

101

C₄⋊C₈

—

102

C₄⋊C₈

C₂×D₇₉

yes

characteristic

103

#103

—

158

—

104

#103

—

158

—

105

#103

—

158

—

106

C₄×C₈

C₂×D₇₉

yes

characteristic

107

C₂×Q₈

—

108

C₂×Q₈

—

158

—

109

C₂×Q₈

—

158

—

110

C₂×Q₈

D₃₁₆

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

158

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

158

—

118

C₄⋊C₄

—

158

—

119

C₄⋊C₄

—

158

—

120

C₄⋊C₄

—

158

—

121

C₄⋊C₄

—

158

—

122

C₄⋊C₄

—

158

—

123

C₄⋊C₄

—

158

—

124

C₄⋊C₄

D₃₁₆

yes

characteristic

125

C₄²

—

126

C₄²

—

158

—

127

C₄²

—

158

—

128

C₄²

C₂²×D₇₉

yes

characteristic

129

Q₈

—

130

Q₈

—

131

Q₈

—

316

—

132

Q₈

—

316

—

133

Q₈

—

316

—

134

Q₈

—

135

Q₈

—

158

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

316

—

139

C₂×C₄

—

158

—

140

C₂×C₄

—

158

—

141

C₂×C₄

—

158

—

142

C₂×C₄

D₄×D₇₉

yes

characteristic

143

C₂×C₄

D₄×D₇₉

yes

characteristic

144

C₂×C₄

C₂×D₃₁₆

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

316

—

148

C₂²

#14579131

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

316

—

153

C₄

—

316

—

154

C₄

—

632

—

155

C₄

—

158

—

156

C₄

Q₁₆×D₇₉

yes

characteristic

157

C₄

#14579183

yes

characteristic

158

C₄

#14579190

yes

characteristic

159

C₄

#14579194

yes

characteristic

160

C₂

#19098720

yes

characteristic

161

C₂

#19098727

yes

characteristic

162

C₂

#19098737

yes

characteristic

	subgroup	G/H
center	C₂²	#14579131
commutator, cosocle	C₄²×C₇₉	C₂³
Frattini	C₄²	C₂²×D₇₉
Fitting	Q₈⋊C₈×C₇₉	C₂
socle, prime radical	C₂²×C₇₉	#120
hypercenter	Q₈⋊C₈	D₇₉
hypocenter	C₇₉	#1636
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

subgroup

G/H

center

C₂²

#14579131

commutator, cosocle

C₄²×C₇₉

C₂³

Frattini

C₄²

C₂²×D₇₉

Fitting

Q₈⋊C₈×C₇₉

C₂

socle, prime radical

C₂²×C₇₉

#120

hypercenter

Q₈⋊C₈

D₇₉

hypocenter

C₇₉

#1636

LCS

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

N	H ≅ G/N	type
82. C₇₉	#1636	semidirect

H ≅ G/N

type

82. C₇₉

#1636

semidirect

Subgroups — #23762560

Conjugacy classes of subgroups

Complements