Group #34183224 — subgroups

Subgroups — #34183224

Order 20224.

376 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 103 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	#23772506	C₂	1	1	yes	maximal, characteristic
2	#23762065	C₂	1	1	yes	maximal, characteristic
3	#23761873	C₂	1	2	yes	maximal
4	#23761299	C₂	1	2	yes	maximal
5	C₇₉×#3162	C₂	1	1	yes	maximal, characteristic
6	#19098694	—	2	4	no	—
7	#19098624	—	2	2	no	—
8	#19098620	C₂²	1	2	yes	—
9	#19098616	C₂²	1	2	yes	—
10	#19098309	—	2	2	no	—
11	Q₈⋊(C₇₉⋊C₈)	—	2	4	no	—
12	#533×C₇₉	C₂²	1	1	yes	characteristic
13	#533×C₇₉	C₂²	1	1	yes	characteristic
14	#501×C₇₉	—	2	2	no	—
15	#501×C₇₉	C₄	1	4	yes	—
16	#427×C₇₉	—	2	2	no	—
17	C₂×#97×C₇₉	C₂²	1	1	yes	characteristic
18	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
19	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
20	C₄⋊(C₇₉⋊C₈)	D₄	1	4	yes	—
21	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
22	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
23	#14579072	—	2	2	no	—
24	#14579072	—	2	2	no	—
25	C₄×C₇₉⋊C₈	—	2	2	no	—
26	C₄×C₇₉⋊C₈	—	2	2	no	—
27	C₂²×C₈×C₇₉	—	2	1	no	—
28	C₄⋊Q₈×C₇₉	C₂×C₄	1	2	yes	—
29	C₄⋊Q₈×C₇₉	D₄	1	2	yes	—
30	#125×C₇₉	—	2	1	no	—
31	#125×C₇₉	C₂×C₄	1	2	yes	—
32	C₂²⋊Q₈×C₇₉	—	2	2	no	—
33	C₂²⋊Q₈×C₇₉	—	2	2	no	—
34	C₄×Q₈×C₇₉	—	2	2	no	—
35	C₄×Q₈×C₇₉	—	2	2	no	—
36	#117×C₇₉	—	2	1	no	—
37	#117×C₇₉	D₄	1	2	yes	—
38	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
39	#107×C₇₉	—	2	4	no	—
40	#106×C₇₉	—	2	4	no	—
41	#97×C₇₉	—	2	1	no	—
42	#97×C₇₉	C₂×C₄	1	2	yes	—
43	C₂×C₇₉⋊C₈	—	2	4	no	—
44	C₂×C₇₉⋊C₈	—	4	2	no	—
45	C₂×Q₈×C₇₉	—	2	2	no	—
46	C₂×Q₈×C₇₉	—	2	2	no	—
47	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₇₉	#41	1	2	yes	—
49	C₂×C₈×C₇₉	—	2	4	no	—
50	C₂×C₈×C₇₉	—	2	2	no	—
51	C₄⋊C₄×C₇₉	—	2	4	no	—
52	C₄⋊C₄×C₇₉	—	2	2	no	—
53	C₄⋊C₄×C₇₉	—	2	2	no	—
54	C₄⋊C₄×C₇₉	—	2	2	no	—
55	C₄⋊C₄×C₇₉	—	4	2	no	—
56	C₄⋊C₄×C₇₉	C₂²⋊C₄	1	4	yes	—
57	C₂²⋊C₄×C₇₉	—	2	2	no	—
58	C₂²⋊C₄×C₇₉	—	4	1	no	—
59	C₄²×C₇₉	—	2	2	no	—
60	C₄²×C₇₉	—	4	1	no	—
61	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₇₉	C₂×D₄	1	2	yes	—
64	C₂³×C₇₉	#123	1	1	yes	characteristic
65	Q₈×C₇₉	—	4	4	no	—
66	Q₈×C₇₉	—	8	2	no	—
67	C₂×C₄×C₇₉	—	2	2	no	—
68	C₂×C₄×C₇₉	—	2	4	no	—
69	C₂×C₄×C₇₉	—	4	2	no	—
70	C₂×C₄×C₇₉	—	4	2	no	—
71	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
73	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
74	C₂×C₄×C₇₉	#121	1	2	yes	—
75	C₈×C₇₉	—	2	4	no	—
76	C₇₉⋊C₈	—	4	4	no	—
77	C₇₉⋊C₈	—	8	2	no	—
78	C₂²×C₇₉	—	2	2	no	—
79	C₂²×C₇₉	—	2	1	no	—
80	C₂²×C₇₉	#386	1	1	yes	characteristic
81	C₄×C₇₉	—	2	2	no	—
82	C₄×C₇₉	—	2	2	no	—
83	C₄×C₇₉	—	2	1	no	—
84	C₄×C₇₉	—	4	4	no	—
85	C₄×C₇₉	—	8	2	no	—
86	C₄×C₇₉	#456	1	2	yes	—
87	#9006	—	79	1	no	maximal, Sylow
88	C₂×C₇₉	—	4	1	no	—
89	C₂×C₇₉	#1881	1	2	yes	—
90	C₂×C₇₉	#1886	1	1	yes	characteristic
91	#3162	D₇₉	1	1	yes	characteristic
92	#2987	—	79	1	no	—
93	#1544	—	79	1	no	—
94	#1497	—	79	2	no	—
95	#1302	—	79	2	no	—
96	C₇₉	#9006	1	1	yes	characteristic, complemented, Sylow
97	#533	C₂×D₇₉	1	1	yes	characteristic
98	#533	C₂×D₇₉	1	1	yes	characteristic
99	#501	—	2	2	no	—
100	#501	Dic₇₉	1	4	yes	—
101	#446	—	158	4	no	—
102	#433	—	158	2	no	—
103	#432	—	79	2	no	—
104	#431	—	79	2	no	—
105	#427	—	2	2	no	—
106	C₂×#97	C₂×D₇₉	1	1	yes	characteristic
107	#332	—	158	2	no	—
108	Q₈⋊C₈	—	158	4	no	—
109	C₂²×C₈	—	2	1	no	—
110	C₄⋊Q₈	C₂×Dic₇₉	1	2	yes	—
111	C₄⋊Q₈	C₂²⋊D₇₉	1	2	yes	—
112	#125	—	2	1	no	—
113	#125	C₂×Dic₇₉	1	2	yes	—
114	C₂²⋊Q₈	—	2	2	no	—
115	C₂²⋊Q₈	—	2	2	no	—
116	C₄×Q₈	—	2	2	no	—
117	C₄×Q₈	—	2	2	no	—
118	#117	—	2	1	no	—
119	#117	C₂²⋊D₇₉	1	2	yes	—
120	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
121	#107	—	2	4	no	—
122	#106	—	2	4	no	—
123	C₄⋊C₈	—	79	4	no	—
124	C₄⋊C₈	—	158	4	no	—
125	C₄⋊C₈	—	158	2	no	—
126	C₂²⋊C₈	—	158	2	no	—
127	C₂²⋊C₈	—	158	2	no	—
128	#97	—	2	1	no	—
129	#97	—	158	2	no	—
130	#97	—	158	2	no	—
131	#97	C₂×Dic₇₉	1	2	yes	—
132	C₄×C₈	—	158	2	no	—
133	C₄×C₈	—	158	2	no	—
134	C₂×Q₈	—	2	2	no	—
135	C₂×Q₈	—	2	2	no	—
136	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
137	C₂²×C₄	#11929623	1	2	yes	—
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	316	2	no	—
141	C₂×C₈	—	158	4	no	—
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	—	2	2	no	—
144	C₄⋊C₄	—	2	2	no	—
145	C₄⋊C₄	—	2	2	no	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	C₂²⋊Dic₇₉	1	4	yes	—
148	C₂²⋊C₄	—	2	2	no	—
149	C₂²⋊C₄	—	4	1	no	—
150	C₄²	—	2	2	no	—
151	C₄²	—	4	1	no	—
152	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
153	C₄²	D₄×D₇₉	1	2	yes	—
154	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
155	C₂³	#14579142	1	1	yes	characteristic
156	Q₈	—	4	4	no	—
157	Q₈	—	8	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
163	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
164	C₂×C₄	#14579129	1	1	yes	characteristic
165	C₂×C₄	#14579133	1	2	yes	—
166	C₈	—	2	4	no	—
167	C₈	—	316	4	no	—
168	C₈	—	632	2	no	—
169	C₂²	—	2	2	no	—
170	C₂²	—	2	1	no	—
171	C₂²	#19098443	1	1	yes	characteristic
172	C₄	—	2	2	no	—
173	C₄	—	2	2	no	—
174	C₄	—	2	1	no	—
175	C₄	—	4	4	no	—
176	C₄	—	8	2	no	—
177	C₄	#19098745	1	2	yes	—
178	C₂	—	4	1	no	—
179	C₂	#23763764	1	2	yes	—
180	C₂	#23763794	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23772506

C₂

yes

maximal, characteristic

#23762065

C₂

yes

maximal, characteristic

#23761873

C₂

yes

maximal

#23761299

C₂

yes

maximal

C₇₉×#3162

C₂

yes

maximal, characteristic

#19098694

—

#19098624

—

#19098620

C₂²

yes

—

#19098616

C₂²

yes

—

#19098309

—

Q₈⋊(C₇₉⋊C₈)

—

#533×C₇₉

C₂²

yes

characteristic

#533×C₇₉

C₂²

yes

characteristic

#501×C₇₉

—

#501×C₇₉

C₄

yes

—

#427×C₇₉

—

C₂×#97×C₇₉

C₂²

yes

characteristic

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

D₄

yes

—

C₂²⋊(C₇₉⋊C₈)

—

C₂²⋊(C₇₉⋊C₈)

—

#14579072

—

#14579072

—

C₄×C₇₉⋊C₈

—

C₄×C₇₉⋊C₈

—

C₂²×C₈×C₇₉

—

C₄⋊Q₈×C₇₉

C₂×C₄

yes

—

C₄⋊Q₈×C₇₉

D₄

yes

—

#125×C₇₉

—

#125×C₇₉

C₂×C₄

yes

—

C₂²⋊Q₈×C₇₉

—

C₂²⋊Q₈×C₇₉

—

C₄×Q₈×C₇₉

—

C₄×Q₈×C₇₉

—

#117×C₇₉

—

#117×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

#107×C₇₉

—

#106×C₇₉

—

#97×C₇₉

—

#97×C₇₉

C₂×C₄

yes

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

#41

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

C₂²⋊C₄

yes

—

C₂²⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

—

C₂³×C₇₉

#123

yes

characteristic

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄×D₄

yes

—

C₂×C₄×C₇₉

#120

yes

characteristic

C₂×C₄×C₇₉

#121

yes

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#386

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#456

yes

—

#9006

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

#1881

yes

—

C₂×C₇₉

#1886

yes

characteristic

#3162

D₇₉

yes

—

—

—

—

C₇₉

#9006

yes

characteristic, complemented, Sylow

#533

C₂×D₇₉

yes

characteristic

#533

C₂×D₇₉

yes

characteristic

#501

—

100

#501

Dic₇₉

yes

—

101

#446

—

158

—

102

#433

—

158

—

103

#432

—

104

#431

—

105

#427

—

106

C₂×#97

C₂×D₇₉

yes

characteristic

107

#332

—

158

—

108

Q₈⋊C₈

—

158

—

109

C₂²×C₈

—

110

C₄⋊Q₈

C₂×Dic₇₉

yes

—

111

C₄⋊Q₈

C₂²⋊D₇₉

yes

—

112

#125

—

113

#125

C₂×Dic₇₉

yes

—

114

C₂²⋊Q₈

—

115

C₂²⋊Q₈

—

116

C₄×Q₈

—

117

C₄×Q₈

—

118

#117

—

119

#117

C₂²⋊D₇₉

yes

—

120

C₂×C₄²

C₂²×D₇₉

yes

characteristic

121

#107

—

122

#106

—

123

C₄⋊C₈

—

124

C₄⋊C₈

—

158

—

125

C₄⋊C₈

—

158

—

126

C₂²⋊C₈

—

158

—

127

C₂²⋊C₈

—

158

—

128

#97

—

129

#97

—

158

—

130

#97

—

158

—

131

#97

C₂×Dic₇₉

yes

—

132

C₄×C₈

—

158

—

133

C₄×C₈

—

158

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

137

C₂²×C₄

#11929623

yes

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

316

—

141

C₂×C₈

—

158

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

C₂²⋊Dic₇₉

yes

—

148

C₂²⋊C₄

—

149

C₂²⋊C₄

—

150

C₄²

—

151

C₄²

—

152

C₄²

C₂²×Dic₇₉

yes

characteristic

153

C₄²

D₄×D₇₉

yes

—

154

C₄²

C₂×C₂²⋊D₇₉

yes

characteristic

155

C₂³

#14579142

yes

characteristic

156

Q₈

—

157

Q₈

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

C₂×C₂²⋊Dic₇₉

yes

characteristic

163

C₂×C₄

D₄×Dic₇₉

yes

—

164

C₂×C₄

#14579129

yes

characteristic

165

C₂×C₄

#14579133

yes

—

166

C₈

—

167

C₈

—

316

—

168

C₈

—

632

—

169

C₂²

—

170

C₂²

—

171

C₂²

#19098443

yes

characteristic

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

#19098745

yes

—

178

C₂

—

179

C₂

#23763764

yes

—

180

C₂

#23763794

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14579129
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#3162	C₂
socle	C₂²×C₇₉	#386
prime radical	C₂³×C₇₉	#123
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#3162	D₇₉
hypocenter	C₇₉	#9006
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3162

subgroup

G/H

center

C₂×C₄

#14579129

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#3162

C₂

C₂²×C₇₉

C₂³×C₇₉

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3162

N	H ≅ G/N	type
96. C₇₉	#9006	semidirect

H ≅ G/N

type

96. C₇₉

#9006

semidirect

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

subgroup	index
#23772506	2
#23762065	2
#23761873	2
#23761299	2
C₇₉×#3162	2
#9006	79

subgroup

index

C₇₉×#3162

Subgroups — #34183224

Conjugacy classes of subgroups

Complements

Maximal subgroups