Group #23763273 — subgroups

Subgroups — #23763273

Order 10112.

272 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 125 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	C₂×Dic₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#14579072	C₂	1	1	yes	maximal, characteristic
3	C₂×#14579071	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×Dic₇₉	C₂	1	1	yes	maximal, characteristic
5	C₄×Dic₇₉⋊C₄	C₂	1	1	yes	maximal, characteristic
6	#19098317	C₂	1	1	yes	maximal, characteristic
7	#336×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×Dic₇₉⋊C₄	C₄	1	2	yes	—
11	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
13	Dic₇₉⋊C₈	C₄	1	4	yes	complemented
14	#14579072	—	2	2	no	—
15	#14579071	C₄	1	4	yes	complemented
16	C₈×Dic₇₉	—	2	2	no	—
17	C₇₉⋊#95	C₄	1	2	yes	—
18	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
21	C₂²×Dic₇₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×Dic₇₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₇₉	C₂×C₄	1	2	yes	—
24	C₂×C₇₉⋊C₈	—	2	1	no	—
25	C₂×C₇₉⋊C₈	—	2	1	no	—
26	C₂×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
27	C₂×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
28	C₂×C₇₉⋊C₈	D₄	1	2	yes	—
29	C₂×C₇₉⋊C₈	Q₈	1	2	yes	—
30	Dic₇₉⋊C₄	—	2	4	no	—
31	C₄×Dic₇₉	—	2	1	no	—
32	C₄×Dic₇₉	—	2	1	no	—
33	C₄×Dic₇₉	C₂×C₄	1	2	yes	—
34	C₄×Dic₇₉	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₇₉	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₇₉	—	2	1	no	—
38	C₂×C₈×C₇₉	—	2	1	no	—
39	C₂×C₈×C₇₉	—	2	1	no	—
40	C₂×C₈×C₇₉	—	2	1	no	—
41	C₂×C₈×C₇₉	C₂×C₄	1	2	yes	—
42	C₂×C₈×C₇₉	C₂×C₄	1	2	yes	—
43	C₄²×C₇₉	—	2	1	no	—
44	C₄²×C₇₉	—	2	1	no	—
45	C₂³×C₇₉	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₇₉	—	2	2	no	—
47	C₂×C₄×C₇₉	—	2	2	no	—
48	C₂×C₄×C₇₉	—	2	2	no	—
49	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
54	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
55	C₂×Dic₇₉	—	2	2	no	—
56	C₂×Dic₇₉	—	2	2	no	—
57	C₂×Dic₇₉	—	2	2	no	—
58	C₂×Dic₇₉	—	2	2	no	—
59	C₂×Dic₇₉	C₄²	1	2	yes	—
60	C₂×Dic₇₉	C₄²	1	2	yes	—
61	C₂×Dic₇₉	M₁₆	1	4	yes	—
62	C₈×C₇₉	—	2	2	no	—
63	C₈×C₇₉	—	2	2	no	—
64	C₇₉⋊C₈	—	2	2	no	—
65	C₇₉⋊C₈	C₄⋊C₄	1	4	yes	complemented
66	C₂²×C₇₉	C₂×C₄²	1	1	yes	characteristic
67	C₂²×C₇₉	C₄×D₄	1	1	yes	characteristic
68	C₂²×C₇₉	C₄×Q₈	1	1	yes	characteristic
69	C₂²×C₇₉	C₂×M₁₆	1	2	yes	—
70	C₂²×C₇₉	#131	1	2	yes	—
71	C₄×C₇₉	—	4	2	no	—
72	C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₄×C₇₉	#117	1	1	yes	characteristic
74	C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
75	C₄×C₇₉	C₄×Q₈	1	1	yes	characteristic
76	Dic₇₉	—	2	2	no	—
77	Dic₇₉	—	2	2	no	—
78	Dic₇₉	—	2	4	no	—
79	C₂×C₇₉	C₄×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₇₉	C₄×M₁₆	1	1	yes	characteristic
81	C₂×C₇₉	#405	1	1	yes	characteristic
82	C₂×C₇₉	#435	1	2	yes	—
83	C₂×C₇₉	#446	1	2	yes	—
84	#1771	—	79	1	no	maximal, Sylow
85	C₇₉	#1771	1	1	yes	characteristic, complemented, Sylow
86	C₂×C₄⋊C₈	—	79	1	no	—
87	C₂×#97	—	79	1	no	—
88	C₂×#97	—	79	1	no	—
89	C₂×C₄×C₈	—	79	1	no	—
90	C₄×C₄⋊C₄	—	79	1	no	—
91	#336	—	79	1	no	—
92	#336	D₇₉	1	1	yes	characteristic
93	C₂²×C₈	—	79	1	no	—
94	C₂²×C₈	—	79	1	no	—
95	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
96	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
97	C₂×C₄⋊C₄	—	79	2	no	—
98	C₂×C₄²	—	79	1	no	—
99	C₂×C₄²	—	79	1	no	—
100	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
101	C₄⋊C₈	—	79	4	no	—
102	#97	—	79	4	no	—
103	#97	—	158	2	no	—
104	C₄×C₈	—	158	2	no	—
105	#95	—	79	2	no	—
106	C₂²×C₄	—	79	1	no	—
107	C₂²×C₄	—	79	2	no	—
108	C₂²×C₄	—	79	1	no	—
109	C₂²×C₄	C₄×D₇₉	1	2	yes	—
110	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	79	2	no	—
116	C₂×C₈	—	79	2	no	—
117	C₂×C₈	—	79	2	no	—
118	C₂×C₈	—	79	2	no	—
119	C₂×C₈	—	158	1	no	—
120	C₂×C₈	—	158	1	no	—
121	C₂×C₈	C₄×D₇₉	1	2	yes	—
122	C₂×C₈	C₄×D₇₉	1	2	yes	—
123	C₄⋊C₄	—	158	4	no	—
124	C₄²	—	2	1	no	—
125	C₄²	—	2	1	no	—
126	C₄²	—	79	2	no	—
127	C₄²	—	79	2	no	—
128	C₄²	—	158	1	no	—
129	C₄²	—	158	1	no	—
130	C₂³	C₂×C₄×D₇₉	1	1	yes	characteristic
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	79	2	no	—
135	C₂×C₄	—	79	2	no	—
136	C₂×C₄	—	79	4	no	—
137	C₂×C₄	—	158	2	no	—
138	C₂×C₄	—	158	2	no	—
139	C₂×C₄	—	158	2	no	—
140	C₂×C₄	—	158	2	no	—
141	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
142	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
143	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
144	C₂×C₄	Q₈×D₇₉	1	1	yes	characteristic
145	C₂×C₄	#11929623	1	1	yes	characteristic
146	C₂×C₄	#11929623	1	1	yes	characteristic
147	C₈	—	2	2	no	—
148	C₈	—	2	2	no	—
149	C₈	—	79	4	no	—
150	C₈	—	158	2	no	—
151	C₂²	C₄²×D₇₉	1	1	yes	characteristic
152	C₂²	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
153	C₂²	#14579126	1	1	yes	characteristic
154	C₂²	M₁₆×D₇₉	1	2	yes	—
155	C₂²	#14579167	1	2	yes	—
156	C₄	—	4	2	no	—
157	C₄	—	158	4	no	—
158	C₄	—	158	2	no	—
159	C₄	—	158	2	no	—
160	C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
161	C₄	#14579117	1	1	yes	characteristic
162	C₄	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
163	C₄	#14579126	1	1	yes	characteristic
164	C₂	#19098414	1	1	yes	characteristic
165	C₂	#19098515	1	1	yes	characteristic
166	C₂	#19098517	1	1	yes	characteristic
167	C₂	#19098637	1	2	yes	—
168	C₂	#19098692	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₇₉⋊C₈

C₂

yes

maximal, characteristic

C₂×#14579072

C₂

yes

maximal, characteristic

C₂×#14579071

C₂

yes

maximal, characteristic

C₂×C₈×Dic₇₉

C₂

yes

maximal, characteristic

C₄×Dic₇₉⋊C₄

C₂

yes

maximal, characteristic

#19098317

C₂

yes

maximal, characteristic

#336×C₇₉

C₂

yes

maximal, characteristic

C₂²×C₇₉⋊C₈

C₂²

yes

characteristic

C₂²×C₇₉⋊C₈

C₂²

yes

characteristic

C₂×Dic₇₉⋊C₄

C₄

yes

—

C₂×C₄×Dic₇₉

C₂²

yes

characteristic

C₂×C₄×Dic₇₉

C₂²

yes

characteristic

Dic₇₉⋊C₈

C₄

yes

complemented

#14579072

—

#14579071

C₄

yes

complemented

C₈×Dic₇₉

—

C₇₉⋊#95

C₄

yes

—

C₂²×C₈×C₇₉

C₂²

yes

characteristic

C₂²×C₈×C₇₉

C₂²

yes

characteristic

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₂²×Dic₇₉

C₂×C₄

yes

characteristic

C₂²×Dic₇₉

C₂×C₄

yes

characteristic

C₂²×Dic₇₉

C₂×C₄

yes

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₂×C₄

yes

—

C₂×C₇₉⋊C₈

C₂×C₄

yes

—

C₂×C₇₉⋊C₈

D₄

yes

—

C₂×C₇₉⋊C₈

Q₈

yes

—

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

C₂×C₄

yes

—

C₄×Dic₇₉

C₂×C₄

yes

—

C₂²×C₄×C₇₉

C₂×C₄

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂×C₄

yes

—

C₂×C₈×C₇₉

C₂×C₄

yes

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₂³×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₈

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

C₄²

yes

—

C₂×Dic₇₉

C₄²

yes

—

C₂×Dic₇₉

M₁₆

yes

—

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

C₄⋊C₄

yes

complemented

C₂²×C₇₉

C₂×C₄²

yes

characteristic

C₂²×C₇₉

C₄×D₄

yes

characteristic

C₂²×C₇₉

C₄×Q₈

yes

characteristic

C₂²×C₇₉

C₂×M₁₆

yes

—

C₂²×C₇₉

#131

yes

—

C₄×C₇₉

—

C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₉

#117

yes

characteristic

C₄×C₇₉

C₄×D₄

yes

characteristic

C₄×C₇₉

C₄×Q₈

yes

characteristic

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

C₄×C₄⋊C₄

yes

characteristic

C₂×C₇₉

C₄×M₁₆

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₂×C₇₉

#446

yes

—

#1771

—

maximal, Sylow

C₇₉

#1771

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₄×C₄⋊C₄

—

#336

—

#336

D₇₉

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₇₉

yes

characteristic

C₂²×C₈

C₂×D₇₉

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

100

C₂×C₄²

C₂×D₇₉

yes

characteristic

101

C₄⋊C₈

—

102

#97

—

103

#97

—

158

—

104

C₄×C₈

—

158

—

105

#95

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

C₄×D₇₉

yes

—

110

C₂²×C₄

C₂²×D₇₉

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

158

—

120

C₂×C₈

—

158

—

121

C₂×C₈

C₄×D₇₉

yes

—

122

C₂×C₈

C₄×D₇₉

yes

—

123

C₄⋊C₄

—

158

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₄²

—

158

—

129

C₄²

—

158

—

130

C₂³

C₂×C₄×D₇₉

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

158

—

138

C₂×C₄

—

158

—

139

C₂×C₄

—

158

—

140

C₂×C₄

—

158

—

141

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

142

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

143

C₂×C₄

D₄×D₇₉

yes

characteristic

144

C₂×C₄

Q₈×D₇₉

yes

characteristic

145

C₂×C₄

#11929623

yes

characteristic

146

C₂×C₄

#11929623

yes

characteristic

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

—

158

—

151

C₂²

C₄²×D₇₉

yes

characteristic

152

C₂²

(C₂²⋊D₇₉)⋊C₄

yes

characteristic

153

C₂²

#14579126

yes

characteristic

154

C₂²

M₁₆×D₇₉

yes

—

155

C₂²

#14579167

yes

—

156

C₄

—

157

C₄

—

158

—

158

C₄

—

158

—

159

C₄

—

158

—

160

C₄

C₄⋊C₄×D₇₉

yes

characteristic

161

C₄

#14579117

yes

characteristic

162

C₄

(C₂²⋊D₇₉)⋊C₄

yes

characteristic

163

C₄

#14579126

yes

characteristic

164

C₂

#19098414

yes

characteristic

165

C₂

#19098515

yes

characteristic

166

C₂

#19098517

yes

characteristic

167

C₂

#19098637

yes

—

168

C₂

#19098692

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₇₉
commutator	C₂²×C₇₉	C₂×C₄²
Fitting	#336×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂²×C₄
cosocle	C₂²×C₄×C₇₉	C₂³
hypercenter	#336	D₇₉
hypocenter	C₇₉	#1771
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₇₉

commutator

C₂²×C₇₉

C₂×C₄²

Fitting

#336×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

C₂²×C₄

cosocle

C₂²×C₄×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂²×C₇₉ ▷ C₇₉

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
13. Dic₇₉⋊C₈	C₄	semidirect
15. #14579071	C₄	semidirect
65. C₇₉⋊C₈	C₄⋊C₄	semidirect
85. C₇₉	#1771	semidirect

H ≅ G/N

type

13. Dic₇₉⋊C₈

C₄

semidirect

15. #14579071

C₄

semidirect

65. C₇₉⋊C₈

C₄⋊C₄

semidirect

85. C₇₉

#1771

semidirect

Subgroups — #23763273

Conjugacy classes of subgroups

Complements