Group #34183221 — subgroups

Subgroups — #34183221

Order 20224.

376 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 95 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	#23773922	C₂	1	1	yes	maximal, characteristic
2	#23772506	C₂	1	1	yes	maximal, characteristic
3	#23761874	C₂	1	2	yes	maximal
4	#23761300	C₂	1	2	yes	maximal
5	C₇₉×#1544	C₂	1	1	yes	maximal, characteristic
6	#19099120	C₂²	1	1	yes	characteristic
7	#19098973	C₄	1	4	yes	—
8	#19098969	—	2	2	no	—
9	#19098694	—	2	4	no	—
10	#19098624	—	2	2	no	—
11	#19098620	C₂²	1	2	yes	—
12	#19098599	—	2	2	no	—
13	C₂×#14579072	C₂²	1	1	yes	characteristic
14	#19098295	—	2	4	no	—
15	#533×C₇₉	C₂²	1	1	yes	characteristic
16	#431×C₇₉	C₂²	1	2	yes	—
17	#332×C₇₉	—	2	2	no	—
18	C₂²×C₇₉⋊C₈	—	2	1	no	—
19	#14579161	C₂×C₄	1	2	yes	—
20	#14579153	—	2	1	no	—
21	#14579140	—	2	2	no	—
22	C₄×Dic₁₅₈	—	2	2	no	—
23	#14579120	—	2	1	no	—
24	#14579094	—	2	4	no	—
25	#14579092	—	2	4	no	—
26	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
27	C₄⋊(C₇₉⋊C₈)	D₄	1	4	yes	—
28	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
29	#14579072	—	2	2	no	—
30	#14579072	—	2	2	no	—
31	#14579072	—	2	1	no	—
32	#14579072	C₂×C₄	1	2	yes	—
33	C₄×C₇₉⋊C₈	—	2	2	no	—
34	C₄⋊Q₈×C₇₉	D₄	1	2	yes	—
35	#125×C₇₉	C₂×C₄	1	2	yes	—
36	C₂²⋊Q₈×C₇₉	—	2	2	no	—
37	C₄×Q₈×C₇₉	—	2	2	no	—
38	#117×C₇₉	D₄	1	2	yes	—
39	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₇₉	—	2	2	no	—
41	C₂²⋊C₈×C₇₉	—	2	2	no	—
42	C₄×C₈×C₇₉	—	2	2	no	—
43	C₂×Dic₁₅₈	—	2	2	no	—
44	C₂×C₇₉⋊C₈	—	2	4	no	—
45	C₂×C₇₉⋊C₈	—	2	2	no	—
46	C₂×C₇₉⋊C₈	—	2	4	no	—
47	C₄⋊Dic₇₉	—	2	2	no	—
48	C₄⋊Dic₇₉	—	2	2	no	—
49	Dic₇₉⋊C₄	—	4	2	no	—
50	C₂²⋊Dic₇₉	—	4	1	no	—
51	C₄×Dic₇₉	—	4	1	no	—
52	C₂×Q₈×C₇₉	—	2	2	no	—
53	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₇₉	#41	1	2	yes	—
55	C₂×C₈×C₇₉	—	4	2	no	—
56	C₄⋊C₄×C₇₉	—	2	4	no	—
57	C₄⋊C₄×C₇₉	—	2	2	no	—
58	C₄⋊C₄×C₇₉	C₂²⋊C₄	1	4	yes	—
59	C₂²⋊C₄×C₇₉	—	2	2	no	—
60	C₄²×C₇₉	—	2	2	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	—	8	2	no	—
66	C₂×C₄×C₇₉	—	2	2	no	—
67	C₂×C₄×C₇₉	—	2	4	no	—
68	C₂×C₄×C₇₉	—	4	2	no	—
69	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
71	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
72	C₂×C₄×C₇₉	#121	1	2	yes	—
73	C₂×Dic₇₉	—	4	2	no	—
74	Dic₁₅₈	—	4	4	no	—
75	C₈×C₇₉	—	8	2	no	—
76	C₇₉⋊C₈	—	2	4	no	—
77	C₇₉⋊C₈	—	4	4	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	#456	1	2	yes	—
86	Dic₇₉	—	8	2	no	—
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	—	79	1	no	—
92	#2987	—	79	1	no	—
93	#1544	D₇₉	1	1	yes	characteristic
94	#1497	—	79	2	no	—
95	#1302	—	79	2	no	—
96	C₇₉	#9006	1	1	yes	characteristic, complemented, Sylow
97	#533	—	79	1	no	—
98	#533	C₂×D₇₉	1	1	yes	characteristic
99	#501	—	79	4	no	—
100	#501	—	158	2	no	—
101	#446	—	158	4	no	—
102	#433	—	158	2	no	—
103	#432	—	79	2	no	—
104	#431	C₂×D₇₉	1	2	yes	—
105	#427	—	158	2	no	—
106	C₂×#97	—	79	1	no	—
107	#332	—	2	2	no	—
108	Q₈⋊C₈	—	158	4	no	—
109	C₂²×C₈	—	158	1	no	—
110	C₄⋊Q₈	—	79	2	no	—
111	C₄⋊Q₈	C₂²⋊D₇₉	1	2	yes	—
112	#125	—	158	1	no	—
113	#125	C₄×D₇₉	1	2	yes	—
114	C₂²⋊Q₈	—	2	2	no	—
115	C₂²⋊Q₈	—	158	2	no	—
116	C₄×Q₈	—	2	2	no	—
117	C₄×Q₈	—	158	2	no	—
118	#117	—	158	1	no	—
119	#117	D₃₁₆	1	2	yes	—
120	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
121	#107	—	158	4	no	—
122	#106	—	158	4	no	—
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	79	4	no	—
125	C₄⋊C₈	—	158	4	no	—
126	C₂²⋊C₈	—	2	2	no	—
127	C₂²⋊C₈	—	158	2	no	—
128	#97	—	79	2	no	—
129	#97	—	158	2	no	—
130	#97	—	158	2	no	—
131	#97	—	158	1	no	—
132	C₄×C₈	—	2	2	no	—
133	C₄×C₈	—	158	2	no	—
134	C₂×Q₈	—	2	2	no	—
135	C₂×Q₈	—	158	2	no	—
136	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
137	C₂²×C₄	#11929623	1	2	yes	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	158	4	no	—
140	C₂×C₈	—	158	4	no	—
141	C₂×C₈	—	158	2	no	—
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	—	2	2	no	—
144	C₄⋊C₄	—	316	2	no	—
145	C₄⋊C₄	—	158	2	no	—
146	C₄⋊C₄	—	158	2	no	—
147	C₄⋊C₄	(C₂×D₇₉)⋊C₄	1	4	yes	—
148	C₂²⋊C₄	—	2	2	no	—
149	C₂²⋊C₄	—	316	1	no	—
150	C₄²	—	2	2	no	—
151	C₄²	—	316	1	no	—
152	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
153	C₄²	D₄×D₇₉	1	2	yes	—
154	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
155	C₂³	#14579145	1	1	yes	characteristic
156	Q₈	—	8	2	no	—
157	Q₈	—	316	4	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	316	2	no	—
162	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
163	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	2	yes	—
164	C₂×C₄	#14579131	1	1	yes	characteristic
165	C₂×C₄	#14579133	1	2	yes	—
166	C₈	—	8	2	no	—
167	C₈	—	316	4	no	—
168	C₈	—	158	4	no	—
169	C₂²	—	2	2	no	—
170	C₂²	—	2	1	no	—
171	C₂²	#19098440	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	—	632	2	no	—
177	C₄	#19098748	1	2	yes	—
178	C₂	—	4	1	no	—
179	C₂	#23763760	1	2	yes	—
180	C₂	#23763791	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23773922

C₂

yes

maximal, characteristic

#23772506

C₂

yes

maximal, characteristic

#23761874

C₂

yes

maximal

#23761300

C₂

yes

maximal

C₇₉×#1544

C₂

yes

maximal, characteristic

#19099120

C₂²

yes

characteristic

#19098973

C₄

yes

—

—

—

—

C₂²

yes

—

#19098599

—

C₂×#14579072

C₂²

yes

characteristic

#19098295

—

#533×C₇₉

C₂²

yes

characteristic

#431×C₇₉

C₂²

yes

—

#332×C₇₉

—

C₂²×C₇₉⋊C₈

—

#14579161

C₂×C₄

yes

—

#14579153

—

#14579140

—

C₄×Dic₁₅₈

—

#14579120

—

#14579094

—

#14579092

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

D₄

yes

—

C₂²⋊(C₇₉⋊C₈)

—

—

—

—

C₂×C₄

yes

—

C₄×C₇₉⋊C₈

—

C₄⋊Q₈×C₇₉

D₄

yes

—

#125×C₇₉

C₂×C₄

yes

—

C₂²⋊Q₈×C₇₉

—

C₄×Q₈×C₇₉

—

#117×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄⋊C₈×C₇₉

—

C₂²⋊C₈×C₇₉

—

C₄×C₈×C₇₉

—

C₂×Dic₁₅₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₂×Q₈×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

#41

yes

—

C₂×C₈×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

C₂²⋊C₄

yes

—

C₂²⋊C₄×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

—

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

—

C₂×Dic₇₉

—

Dic₁₅₈

—

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

#456

yes

—

Dic₇₉

—

#9006

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

#1881

yes

—

C₂×C₇₉

yes

—

—

D₇₉

yes

characteristic

#1497

—

#1302

—

C₇₉

#9006

yes

characteristic, complemented, Sylow

#533

—

#533

C₂×D₇₉

yes

characteristic

#501

—

100

#501

—

158

—

101

#446

—

158

—

102

#433

—

158

—

103

#432

—

104

#431

C₂×D₇₉

yes

—

105

#427

—

158

—

106

C₂×#97

—

107

#332

—

108

Q₈⋊C₈

—

158

—

109

C₂²×C₈

—

158

—

110

C₄⋊Q₈

—

111

C₄⋊Q₈

C₂²⋊D₇₉

yes

—

112

#125

—

158

—

113

#125

C₄×D₇₉

yes

—

114

C₂²⋊Q₈

—

115

C₂²⋊Q₈

—

158

—

116

C₄×Q₈

—

117

C₄×Q₈

—

158

—

118

#117

—

158

—

119

#117

D₃₁₆

yes

—

120

C₂×C₄²

C₂²×D₇₉

yes

characteristic

121

#107

—

158

—

122

#106

—

158

—

123

C₄⋊C₈

—

124

C₄⋊C₈

—

125

C₄⋊C₈

—

158

—

126

C₂²⋊C₈

—

127

C₂²⋊C₈

—

158

—

128

#97

—

129

#97

—

158

—

130

#97

—

158

—

131

#97

—

158

—

132

C₄×C₈

—

133

C₄×C₈

—

158

—

134

C₂×Q₈

—

135

C₂×Q₈

—

158

—

136

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

137

C₂²×C₄

#11929623

yes

—

138

C₂×C₈

—

139

C₂×C₈

—

158

—

140

C₂×C₈

—

158

—

141

C₂×C₈

—

158

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

316

—

145

C₄⋊C₄

—

158

—

146

C₄⋊C₄

—

158

—

147

C₄⋊C₄

(C₂×D₇₉)⋊C₄

yes

—

148

C₂²⋊C₄

—

149

C₂²⋊C₄

—

316

—

150

C₄²

—

151

C₄²

—

316

—

152

C₄²

C₂×C₄×D₇₉

yes

characteristic

153

C₄²

D₄×D₇₉

yes

—

154

C₄²

C₂×C₂²⋊D₇₉

yes

characteristic

155

C₂³

#14579145

yes

characteristic

156

Q₈

—

157

Q₈

—

316

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

316

—

162

C₂×C₄

C₂×(C₂×D₇₉)⋊C₄

yes

characteristic

163

C₂×C₄

(C₂²⋊D₇₉)⋊C₄

yes

—

164

C₂×C₄

#14579131

yes

characteristic

165

C₂×C₄

#14579133

yes

—

166

C₈

—

167

C₈

—

316

—

168

C₈

—

158

—

169

C₂²

—

170

C₂²

—

171

C₂²

#19098440

yes

characteristic

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

632

—

177

C₄

#19098748

yes

—

178

C₂

—

179

C₂

#23763760

yes

—

180

C₂

#23763791

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14579131
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1544	C₂
socle	C₂²×C₇₉	#386
prime radical	C₂³×C₇₉	#123
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1544	D₇₉
hypocenter	C₇₉	#9006
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1544

subgroup

G/H

center

C₂×C₄

#14579131

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1544

C₂

C₂²×C₇₉

C₂³×C₇₉

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1544

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
#23773922	2
#23772506	2
#23761874	2
#23761300	2
C₇₉×#1544	2
#9006	79

subgroup

index

C₇₉×#1544

Subgroups — #34183221

Conjugacy classes of subgroups

Complements

Maximal subgroups