Group #34208256 — subgroups

Subgroups — #34208256

Order 20224.

298 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 109 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⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#23764131	C₂	1	2	yes	maximal
3	#23761492	C₂	1	1	yes	maximal, characteristic
4	#23761491	C₂	1	2	yes	maximal
5	C₂×#366×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#14579095	C₂²	1	2	yes	—
8	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
9	#19098459	—	2	2	no	—
10	Dic₇₉⋊C₁₆	—	2	2	no	—
11	#19098325	—	2	2	no	—
12	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
13	C₂×#107×C₇₉	C₂²	1	2	yes	—
14	#366×C₇₉	—	2	2	no	—
15	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
16	C₂×C₄⋊Dic₇₉	—	2	2	no	—
17	C₂×Dic₇₉⋊C₄	—	4	2	no	—
18	C₂×C₄×Dic₇₉	D₄	1	1	yes	characteristic
19	C₂×C₇₉⋊C₁₆	—	2	1	no	—
20	C₂×C₇₉⋊C₁₆	—	2	2	no	—
21	#14579095	—	2	2	no	—
22	#14579095	Q₈	1	4	yes	—
23	C₈×Dic₇₉	—	2	1	no	—
24	C₈×Dic₇₉	—	2	1	no	—
25	C₇₉⋊#95	—	4	2	no	—
26	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
28	C₂×C₁₆×C₇₉	—	2	2	no	—
29	C₂×C₁₆×C₇₉	D₄	1	2	yes	—
30	#107×C₇₉	—	2	2	no	—
31	#107×C₇₉	—	2	2	no	—
32	C₂²×Dic₇₉	—	4	2	no	—
33	C₂²×Dic₇₉	D₈	1	1	yes	characteristic
34	C₂²×Dic₇₉	D₈	1	1	yes	characteristic
35	C₂×C₇₉⋊C₈	—	2	1	no	—
36	C₂×C₇₉⋊C₈	—	2	2	no	—
37	C₄⋊Dic₇₉	—	2	4	no	—
38	C₄⋊Dic₇₉	—	4	2	no	—
39	Dic₇₉⋊C₄	—	8	4	no	—
40	C₄×Dic₇₉	—	2	1	no	—
41	C₄×Dic₇₉	—	2	1	no	—
42	C₇₉⋊C₁₆	—	4	1	no	—
43	C₂²×C₄×C₇₉	—	4	2	no	—
44	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
47	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
48	C₂×C₈×C₇₉	#41	1	2	yes	—
49	C₄⋊C₄×C₇₉	—	4	2	no	—
50	C₄⋊C₄×C₇₉	—	4	2	no	—
51	C₁₆×C₇₉	—	2	2	no	—
52	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₇₉	—	8	2	no	—
54	C₂×C₄×C₇₉	—	8	2	no	—
55	C₂×C₄×C₇₉	—	8	2	no	—
56	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
57	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
58	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
59	C₂×C₄×C₇₉	#137	1	2	yes	—
60	C₂×Dic₇₉	—	2	1	no	—
61	C₂×Dic₇₉	—	2	1	no	—
62	C₂×Dic₇₉	—	2	1	no	—
63	C₂×Dic₇₉	—	2	2	no	—
64	C₂×Dic₇₉	—	4	4	no	—
65	C₂×Dic₇₉	—	8	2	no	—
66	C₂×Dic₇₉	—	8	2	no	—
67	C₂×Dic₇₉	D₁₆	1	2	yes	—
68	C₈×C₇₉	#123	1	1	yes	characteristic
69	C₈×C₇₉	#125	1	2	yes	—
70	C₈×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
71	C₇₉⋊C₈	—	4	1	no	—
72	C₂²×C₇₉	#466	1	1	yes	characteristic
73	C₂²×C₇₉	#474	1	2	yes	—
74	C₂²×C₇₉	C₂×D₁₆	1	1	yes	characteristic
75	C₂²×C₇₉	#508	1	1	yes	characteristic
76	C₂²×C₇₉	#510	1	2	yes	—
77	C₄×C₇₉	—	16	2	no	—
78	C₄×C₇₉	#398	1	1	yes	characteristic
79	C₄×C₇₉	#470	1	1	yes	characteristic
80	C₄×C₇₉	#479	1	2	yes	—
81	Dic₇₉	—	2	2	no	—
82	Dic₇₉	—	4	1	no	—
83	Dic₇₉	—	8	4	no	—
84	#13514	—	79	1	no	maximal, Sylow
85	C₂×C₇₉	#2086	1	1	yes	characteristic
86	C₂×C₇₉	#2215	1	1	yes	characteristic
87	C₂×C₇₉	#2219	1	1	yes	characteristic
88	C₂×C₇₉	#2224	1	2	yes	—
89	C₂×C₇₉	#2229	1	2	yes	—
90	C₂×#366	D₇₉	1	1	yes	characteristic
91	C₂×C₄⋊C₁₆	—	79	1	no	—
92	#1942	—	79	2	no	—
93	#1380	—	79	1	no	—
94	#1380	—	79	2	no	—
95	C₇₉	#13514	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₁₆	—	79	1	no	—
97	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
98	C₂×#107	—	79	2	no	—
99	C₂×#107	C₂×D₇₉	1	2	yes	—
100	C₂×C₄×C₈	—	79	1	no	—
101	#389	—	158	2	no	—
102	#366	—	2	2	no	—
103	C₄⋊C₁₆	—	158	2	no	—
104	#340	—	158	2	no	—
105	C₂²×C₈	—	79	1	no	—
106	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
107	C₂×C₄⋊C₄	—	2	2	no	—
108	C₂×C₄⋊C₄	—	316	2	no	—
109	C₂×C₄⋊C₄	—	158	2	no	—
110	C₂×C₄²	—	79	1	no	—
111	C₂×C₁₆	—	2	2	no	—
112	C₂×C₁₆	—	158	1	no	—
113	C₂×C₁₆	—	158	2	no	—
114	C₂×C₁₆	C₂²⋊D₇₉	1	2	yes	—
115	#107	—	2	2	no	—
116	#107	—	2	2	no	—
117	#107	—	79	4	no	—
118	#107	—	158	2	no	—
119	C₄×C₈	—	158	1	no	—
120	C₄×C₈	—	158	1	no	—
121	#95	—	316	2	no	—
122	C₂²×C₄	—	4	2	no	—
123	C₂²×C₄	—	316	2	no	—
124	C₂²×C₄	—	79	1	no	—
125	C₂²×C₄	—	79	1	no	—
126	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
127	C₂×C₈	—	158	1	no	—
128	C₂×C₈	—	158	2	no	—
129	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
130	C₂×C₈	Q₈×D₇₉	1	2	yes	—
131	C₂×C₈	#11929623	1	1	yes	characteristic
132	C₂×C₈	#11929624	1	2	yes	—
133	C₄⋊C₄	—	4	2	no	—
134	C₄⋊C₄	—	4	2	no	—
135	C₄⋊C₄	—	316	2	no	—
136	C₄⋊C₄	—	632	4	no	—
137	C₄⋊C₄	—	158	4	no	—
138	C₄²	—	158	1	no	—
139	C₄²	—	158	1	no	—
140	C₁₆	—	2	2	no	—
141	C₁₆	—	316	1	no	—
142	C₂³	D₈×D₇₉	1	1	yes	characteristic
143	C₂×C₄	—	8	2	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	—	8	2	no	—
146	C₂×C₄	—	316	4	no	—
147	C₂×C₄	—	79	2	no	—
148	C₂×C₄	—	632	2	no	—
149	C₂×C₄	—	632	2	no	—
150	C₂×C₄	—	158	2	no	—
151	C₂×C₄	—	158	1	no	—
152	C₂×C₄	—	158	1	no	—
153	C₂×C₄	—	158	1	no	—
154	C₂×C₄	#14579132	1	1	yes	characteristic
155	C₂×C₄	#14579139	1	2	yes	—
156	C₂×C₄	#14579180	1	1	yes	characteristic
157	C₂×C₄	#14579193	1	2	yes	—
158	C₈	—	316	1	no	—
159	C₈	#14579144	1	1	yes	characteristic
160	C₈	#14579151	1	2	yes	—
161	C₈	#14579159	1	1	yes	characteristic
162	C₂²	#19098796	1	1	yes	characteristic
163	C₂²	#19098842	1	2	yes	—
164	C₂²	D₁₆×D₇₉	1	1	yes	characteristic
165	C₂²	#19098992	1	1	yes	characteristic
166	C₂²	#19099005	1	2	yes	—
167	C₄	—	16	2	no	—
168	C₄	—	316	1	no	—
169	C₄	—	632	4	no	—
170	C₄	—	158	2	no	—
171	C₄	#19098494	1	1	yes	characteristic
172	C₄	#19098818	1	1	yes	characteristic
173	C₄	#19098869	1	2	yes	—
174	C₂	#23764889	1	1	yes	characteristic
175	C₂	#23765483	1	1	yes	characteristic
176	C₂	#23765505	1	1	yes	characteristic
177	C₂	#23765536	1	2	yes	—
178	C₂	#23765563	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₇₉⋊C₁₆

C₂

yes

maximal, characteristic

#23764131

C₂

yes

maximal

#23761492

C₂

yes

maximal, characteristic

#23761491

C₂

yes

maximal

C₂×#366×C₇₉

C₂

yes

maximal, characteristic

C₂²×C₇₉⋊C₁₆

C₂²

yes

characteristic

C₂×#14579095

C₂²

yes

—

C₂×C₈×Dic₇₉

C₂²

yes

characteristic

#19098459

—

Dic₇₉⋊C₁₆

—

#19098325

—

C₂²×C₁₆×C₇₉

C₂²

yes

characteristic

C₂×#107×C₇₉

C₂²

yes

—

#366×C₇₉

—

C₂²×C₇₉⋊C₈

D₄

yes

characteristic

C₂×C₄⋊Dic₇₉

—

C₂×Dic₇₉⋊C₄

—

C₂×C₄×Dic₇₉

D₄

yes

characteristic

C₂×C₇₉⋊C₁₆

—

C₂×C₇₉⋊C₁₆

—

#14579095

—

#14579095

Q₈

yes

—

C₈×Dic₇₉

—

C₈×Dic₇₉

—

C₇₉⋊#95

—

C₂²×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₇₉

—

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

D₄

yes

—

#107×C₇₉

—

#107×C₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

D₈

yes

characteristic

C₂²×Dic₇₉

D₈

yes

characteristic

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂×Q₈

yes

—

C₂×C₈×C₇₉

#41

yes

characteristic

C₂×C₈×C₇₉

#41

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₁₆×C₇₉

—

C₂³×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#121

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₉

#135

yes

characteristic

C₂×C₄×C₇₉

#137

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

D₁₆

yes

—

C₈×C₇₉

#123

yes

characteristic

C₈×C₇₉

#125

yes

—

C₈×C₇₉

C₄⋊Q₈

yes

characteristic

C₇₉⋊C₈

—

C₂²×C₇₉

#466

yes

characteristic

C₂²×C₇₉

#474

yes

—

C₂²×C₇₉

C₂×D₁₆

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₂×C₇₉

#2229

yes

—

C₂×#366

D₇₉

yes

characteristic

C₂×C₄⋊C₁₆

—

#1942

—

#1380

—

#1380

—

C₇₉

#13514

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₇₉

yes

characteristic

C₂×#107

—

C₂×#107

C₂×D₇₉

yes

—

100

C₂×C₄×C₈

—

101

#389

—

158

—

102

#366

—

103

C₄⋊C₁₆

—

158

—

104

#340

—

158

—

105

C₂²×C₈

—

106

C₂²×C₈

C₂²×D₇₉

yes

characteristic

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

316

—

109

C₂×C₄⋊C₄

—

158

—

110

C₂×C₄²

—

111

C₂×C₁₆

—

112

C₂×C₁₆

—

158

—

113

C₂×C₁₆

—

158

—

114

C₂×C₁₆

C₂²⋊D₇₉

yes

—

115

#107

—

116

#107

—

117

#107

—

118

#107

—

158

—

119

C₄×C₈

—

158

—

120

C₄×C₈

—

158

—

121

#95

—

316

—

122

C₂²×C₄

—

123

C₂²×C₄

—

316

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

D₄×D₇₉

yes

characteristic

127

C₂×C₈

—

158

—

128

C₂×C₈

—

158

—

129

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

130

C₂×C₈

Q₈×D₇₉

yes

—

131

C₂×C₈

#11929623

yes

characteristic

132

C₂×C₈

#11929624

yes

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

316

—

136

C₄⋊C₄

—

632

—

137

C₄⋊C₄

—

158

—

138

C₄²

—

158

—

139

C₄²

—

158

—

140

C₁₆

—

141

C₁₆

—

316

—

142

C₂³

D₈×D₇₉

yes

characteristic

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

316

—

147

C₂×C₄

—

148

C₂×C₄

—

632

—

149

C₂×C₄

—

632

—

150

C₂×C₄

—

158

—

151

C₂×C₄

—

158

—

152

C₂×C₄

—

158

—

153

C₂×C₄

—

158

—

154

C₂×C₄

#14579132

yes

characteristic

155

C₂×C₄

#14579139

yes

—

156

C₂×C₄

#14579180

yes

characteristic

157

C₂×C₄

#14579193

yes

—

158

C₈

—

316

—

159

C₈

#14579144

yes

characteristic

160

C₈

#14579151

yes

—

161

C₈

#14579159

yes

characteristic

162

C₂²

#19098796

yes

characteristic

163

C₂²

#19098842

yes

—

164

C₂²

D₁₆×D₇₉

yes

characteristic

165

C₂²

#19098992

yes

characteristic

166

C₂²

#19099005

yes

—

167

C₄

—

168

C₄

—

316

—

169

C₄

—

632

—

170

C₄

—

158

—

171

C₄

#19098494

yes

characteristic

172

C₄

#19098818

yes

characteristic

173

C₄

#19098869

yes

—

174

C₂

#23764889

yes

characteristic

175

C₂

#23765483

yes

characteristic

176

C₂

#23765505

yes

characteristic

177

C₂

#23765536

yes

—

178

C₂

#23765563

yes

—

	subgroup	G/H
center	C₂³	D₈×D₇₉
commutator, cosocle	C₂²×C₈×C₇₉	C₂³
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×#366×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₈
hypercenter	C₂×#366	D₇₉
hypocenter	C₇₉	#13514
LCS	G ▷ C₂²×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

subgroup

G/H

center

C₂³

D₈×D₇₉

commutator, cosocle

C₂²×C₈×C₇₉

C₂³

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₂×#366×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

C₂×D₈

D₇₉

C₇₉

G ▷ C₂²×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

N	H ≅ G/N	type
95. C₇₉	#13514	semidirect

H ≅ G/N

type

95. C₇₉

#13514

semidirect

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

subgroup	index
C₂×Dic₇₉⋊C₁₆	2
#23764131	2
#23761492	2
#23761491	2
C₂×#366×C₇₉	2
#13514	79

subgroup

index

C₂×Dic₇₉⋊C₁₆

C₂×#366×C₇₉

Subgroups — #34208256

Conjugacy classes of subgroups

Complements

Maximal subgroups