Group #34175367 — subgroups

Subgroups — #34175367

Order 20224.

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

normal subgroups only

This group has 206 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	C₂×C₄⋊(C₇₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	#23762159	C₂	1	1	yes	maximal, characteristic
3	C₇₉×#1567	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₇₉⋊C₁₆	C₄	1	2	yes	—
5	#19098587	—	2	1	no	—
6	C₄⋊(C₇₉⋊C₁₆)	—	2	1	no	—
7	C₄⋊(C₇₉⋊C₁₆)	—	2	1	no	—
8	#19098316	—	2	1	no	—
9	#19098316	C₄	1	2	yes	—
10	#423×C₇₉	—	2	1	no	—
11	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
12	#335×C₇₉	—	2	1	no	—
13	#335×C₇₉	C₄	1	2	yes	—
14	C₂×#11929608	—	4	1	no	—
15	C₂×C₇₉⋊C₁₆	—	2	2	no	—
16	C₂×C₇₉⋊C₁₆	—	4	2	no	—
17	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
18	C₄⋊(C₇₉⋊C₈)	—	4	1	no	—
19	C₂×M₁₆×C₇₉	—	4	1	no	—
20	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₇₉	Q₈	1	1	yes	characteristic
22	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
23	C₄⋊C₈×C₇₉	—	2	2	no	—
24	C₄⋊C₈×C₇₉	—	4	1	no	—
25	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
28	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
29	#11929608	—	8	1	no	—
30	#11929608	—	8	1	no	—
31	C₂×C₇₉⋊C₈	—	4	2	no	—
32	C₇₉⋊C₁₆	—	4	2	no	—
33	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
35	C₂²×C₄×C₇₉	Q₁₆	1	1	yes	characteristic
36	M₁₆×C₇₉	—	8	1	no	—
37	M₁₆×C₇₉	—	8	1	no	—
38	C₂×C₈×C₇₉	—	2	1	no	—
39	C₂×C₈×C₇₉	—	2	1	no	—
40	C₂×C₈×C₇₉	—	4	1	no	—
41	C₂×C₈×C₇₉	—	4	2	no	—
42	C₂×C₈×C₇₉	C₄²	1	1	yes	characteristic
43	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
46	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₄²×C₇₉	SD₁₆	1	1	yes	characteristic
49	C₄²×C₇₉	SD₁₆	1	1	yes	characteristic
50	C₂³×C₇₉	#107	1	1	yes	characteristic
51	C₂×C₄×C₇₉	—	2	1	no	—
52	C₂×C₄×C₇₉	—	2	1	no	—
53	C₂×C₄×C₇₉	—	4	1	no	—
54	C₂×C₄×C₇₉	#95	1	1	yes	characteristic
55	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
56	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
57	C₂×C₄×C₇₉	#103	1	1	yes	characteristic
58	C₂×C₄×C₇₉	#103	1	1	yes	characteristic
59	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
60	C₂×C₄×C₇₉	D₁₆	1	1	yes	characteristic
61	C₂×C₄×C₇₉	SD₃₂	1	1	yes	characteristic
62	C₂×C₄×C₇₉	SD₃₂	1	1	yes	characteristic
63	C₂×C₄×C₇₉	Q₃₂	1	1	yes	characteristic
64	C₈×C₇₉	—	2	1	no	—
65	C₈×C₇₉	—	2	1	no	—
66	C₈×C₇₉	—	2	1	no	—
67	C₈×C₇₉	—	8	2	no	—
68	C₈×C₇₉	#100	1	1	yes	characteristic
69	C₈×C₇₉	#101	1	1	yes	characteristic
70	C₇₉⋊C₈	—	8	2	no	—
71	C₂²×C₇₉	—	2	1	no	—
72	C₂²×C₇₉	—	2	1	no	—
73	C₂²×C₇₉	#340	1	1	yes	characteristic
74	C₂²×C₇₉	#366	1	1	yes	characteristic
75	C₂²×C₇₉	#367	1	1	yes	characteristic
76	C₄×C₇₉	—	2	1	no	—
77	C₄×C₇₉	—	2	1	no	—
78	C₄×C₇₉	—	2	1	no	—
79	C₄×C₇₉	#338	1	1	yes	characteristic
80	C₄×C₇₉	#343	1	1	yes	characteristic
81	C₄×C₇₉	#357	1	1	yes	characteristic
82	C₄×C₇₉	#357	1	1	yes	characteristic
83	C₄×C₇₉	#358	1	1	yes	characteristic
84	C₄×C₇₉	#358	1	1	yes	characteristic
85	#7241	—	79	1	no	maximal, Sylow
86	C₂×C₇₉	—	2	1	no	—
87	C₂×C₇₉	—	2	1	no	—
88	C₂×C₇₉	#1292	1	1	yes	characteristic
89	C₂×C₇₉	#1380	1	1	yes	characteristic
90	C₂×C₇₉	#1385	1	1	yes	characteristic
91	C₂×C₄⋊C₁₆	—	79	1	no	—
92	#1567	—	79	1	no	—
93	#1567	D₇₉	1	1	yes	characteristic
94	C₇₉	#7241	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₁₆	—	79	2	no	—
96	#423	—	2	1	no	—
97	#423	—	158	1	no	—
98	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
99	C₄⋊C₁₆	—	158	1	no	—
100	C₄⋊C₁₆	—	158	1	no	—
101	#335	—	2	1	no	—
102	#335	—	79	2	no	—
103	#335	—	158	1	no	—
104	#335	Dic₇₉	1	2	yes	—
105	C₂×M₁₆	—	4	1	no	—
106	C₂×M₁₆	—	316	1	no	—
107	C₂²×C₈	Dic₁₅₈	1	1	yes	characteristic
108	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
109	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
110	C₂×C₁₆	—	316	2	no	—
111	C₂×C₁₆	—	158	2	no	—
112	C₄⋊C₈	—	2	2	no	—
113	C₄⋊C₈	—	4	1	no	—
114	C₄⋊C₈	—	316	1	no	—
115	C₄⋊C₈	—	158	2	no	—
116	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
117	C₄×C₈	D₃₁₆	1	1	yes	characteristic
118	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
119	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
120	C₂²×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
121	C₂²×C₄	#11929609	1	1	yes	characteristic
122	C₂²×C₄	#11929614	1	1	yes	characteristic
123	M₁₆	—	8	1	no	—
124	M₁₆	—	8	1	no	—
125	M₁₆	—	632	1	no	—
126	M₁₆	—	632	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	4	1	no	—
130	C₂×C₈	—	4	2	no	—
131	C₂×C₈	—	316	2	no	—
132	C₂×C₈	C₄×Dic₇₉	1	1	yes	characteristic
133	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
134	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
135	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
136	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
137	C₄²	C₂²⋊Dic₇₉	1	1	yes	characteristic
138	C₄²	#11929611	1	1	yes	characteristic
139	C₄²	#11929612	1	1	yes	characteristic
140	C₁₆	—	316	2	no	—
141	C₂³	#14579094	1	1	yes	characteristic
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	4	1	no	—
145	C₂×C₄	C₇₉⋊#95	1	1	yes	characteristic
146	C₂×C₄	#14579081	1	1	yes	characteristic
147	C₂×C₄	#14579082	1	1	yes	characteristic
148	C₂×C₄	#14579084	1	1	yes	characteristic
149	C₂×C₄	#14579085	1	1	yes	characteristic
150	C₂×C₄	#14579092	1	1	yes	characteristic
151	C₂×C₄	#14579102	1	1	yes	characteristic
152	C₂×C₄	#14579104	1	1	yes	characteristic
153	C₂×C₄	#14579105	1	1	yes	characteristic
154	C₂×C₄	#14579107	1	1	yes	characteristic
155	C₈	—	2	1	no	—
156	C₈	—	2	1	no	—
157	C₈	—	2	1	no	—
158	C₈	—	8	2	no	—
159	C₈	—	632	2	no	—
160	C₈	#14579077	1	1	yes	characteristic
161	C₈	#14579079	1	1	yes	characteristic
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	#19098325	1	1	yes	characteristic
165	C₂²	#19098388	1	1	yes	characteristic
166	C₂²	#19098390	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	C₇₉⋊#338	1	1	yes	characteristic
171	C₄	#19098331	1	1	yes	characteristic
172	C₄	#19098363	1	1	yes	characteristic
173	C₄	#19098364	1	1	yes	characteristic
174	C₄	#19098366	1	1	yes	characteristic
175	C₄	#19098367	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	2	1	no	—
178	C₂	#23761274	1	1	yes	characteristic
179	C₂	#23761491	1	1	yes	characteristic
180	C₂	#23761503	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₇₉⋊C₁₆)

C₂

yes

maximal, characteristic

#23762159

C₂

yes

maximal, characteristic

C₇₉×#1567

C₂

yes

maximal, characteristic

C₂²×C₇₉⋊C₁₆

C₄

yes

—

#19098587

—

C₄⋊(C₇₉⋊C₁₆)

—

C₄⋊(C₇₉⋊C₁₆)

—

#19098316

—

#19098316

C₄

yes

—

#423×C₇₉

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#335×C₇₉

—

#335×C₇₉

C₄

yes

—

C₂×#11929608

—

C₂×C₇₉⋊C₁₆

—

C₂×C₇₉⋊C₁₆

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₂×M₁₆×C₇₉

—

C₂²×C₈×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₉

Q₈

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

characteristic

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₇₉

D₈

yes

characteristic

C₂²×C₄×C₇₉

Q₁₆

yes

characteristic

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₄²

yes

characteristic

C₂×C₈×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₉

C₄⋊C₄

yes

characteristic

C₄²×C₇₉

C₂²⋊C₄

yes

characteristic

C₄²×C₇₉

C₂²⋊C₄

yes

characteristic

C₄²×C₇₉

SD₁₆

yes

characteristic

C₄²×C₇₉

SD₁₆

yes

characteristic

C₂³×C₇₉

#107

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

D₁₆

yes

characteristic

C₂×C₄×C₇₉

SD₃₂

yes

characteristic

C₂×C₄×C₇₉

SD₃₂

yes

characteristic

C₂×C₄×C₇₉

Q₃₂

yes

characteristic

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

yes

C₈×C₇₉

yes

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₄⋊C₁₆

—

#1567

—

#1567

D₇₉

yes

characteristic

C₇₉

#7241

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

#423

—

#423

—

158

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

C₄⋊C₁₆

—

158

—

100

C₄⋊C₁₆

—

158

—

101

#335

—

102

#335

—

103

#335

—

158

—

104

#335

Dic₇₉

yes

—

105

C₂×M₁₆

—

106

C₂×M₁₆

—

316

—

107

C₂²×C₈

Dic₁₅₈

yes

characteristic

108

C₂²×C₈

C₄×D₇₉

yes

characteristic

109

C₂×C₄²

C₂²⋊D₇₉

yes

characteristic

110

C₂×C₁₆

—

316

—

111

C₂×C₁₆

—

158

—

112

C₄⋊C₈

—

113

C₄⋊C₈

—

114

C₄⋊C₈

—

316

—

115

C₄⋊C₈

—

158

—

116

C₄×C₈

C₄×D₇₉

yes

characteristic

117

C₄×C₈

D₃₁₆

yes

characteristic

118

C₄×C₈

C₂×Dic₇₉

yes

characteristic

119

C₄×C₈

C₂²⋊D₇₉

yes

characteristic

120

C₂²×C₄

Dic₇₉⋊C₄

yes

characteristic

121

C₂²×C₄

#11929609

yes

characteristic

122

C₂²×C₄

#11929614

yes

characteristic

123

M₁₆

—

124

M₁₆

—

125

M₁₆

—

632

—

126

M₁₆

—

632

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

316

—

132

C₂×C₈

C₄×Dic₇₉

yes

characteristic

133

C₂×C₈

(C₂×D₇₉)⋊C₄

yes

characteristic

134

C₂×C₈

Dic₇₉⋊C₄

yes

characteristic

135

C₂×C₈

C₄⋊Dic₇₉

yes

characteristic

136

C₄²

(C₂×D₇₉)⋊C₄

yes

characteristic

137

C₄²

C₂²⋊Dic₇₉

yes

characteristic

138

C₄²

#11929611

yes

characteristic

139

C₄²

#11929612

yes

characteristic

140

C₁₆

—

316

—

141

C₂³

#14579094

yes

characteristic

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

C₇₉⋊#95

yes

characteristic

146

C₂×C₄

#14579081

yes

characteristic

147

C₂×C₄

#14579082

yes

characteristic

148

C₂×C₄

#14579084

yes

characteristic

149

C₂×C₄

#14579085

yes

characteristic

150

C₂×C₄

#14579092

yes

characteristic

151

C₂×C₄

#14579102

yes

characteristic

152

C₂×C₄

#14579104

yes

characteristic

153

C₂×C₄

#14579105

yes

characteristic

154

C₂×C₄

#14579107

yes

characteristic

155

C₈

—

156

C₈

—

157

C₈

—

158

C₈

—

159

C₈

—

632

—

160

C₈

#14579077

yes

characteristic

161

C₈

#14579079

yes

characteristic

162

C₂²

—

163

C₂²

—

164

C₂²

#19098325

yes

characteristic

165

C₂²

#19098388

yes

characteristic

166

C₂²

#19098390

yes

characteristic

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

C₇₉⋊#338

yes

characteristic

171

C₄

#19098331

yes

characteristic

172

C₄

#19098363

yes

characteristic

173

C₄

#19098364

yes

characteristic

174

C₄

#19098366

yes

characteristic

175

C₄

#19098367

yes

characteristic

176

C₂

—

177

C₂

—

178

C₂

#23761274

yes

characteristic

179

C₂

#23761491

yes

characteristic

180

C₂

#23761503

yes

characteristic

	subgroup	G/H
center	C₂²	#19098325
commutator	C₂×C₈×C₇₉	C₄²
Frattini	C₂×C₄×C₈	C₂×D₇₉
Fitting	C₇₉×#1567	C₂
socle	C₂²×C₇₉	#340
prime radical	C₂³×C₇₉	#107
cosocle	C₂×C₄×C₈×C₇₉	C₂²
hypercenter	#1567	D₇₉
hypocenter	C₇₉	#7241
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1567

subgroup

G/H

center

C₂²

#19098325

commutator

C₂×C₈×C₇₉

C₄²

Frattini

C₂×C₄×C₈

C₂×D₇₉

Fitting

C₇₉×#1567

C₂

C₂²×C₇₉

C₂³×C₇₉

C₂×C₄×C₈×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1567

N	H ≅ G/N	type
94. C₇₉	#7241	semidirect

H ≅ G/N

type

94. C₇₉

#7241

semidirect

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

subgroup	index
C₂×C₄⋊(C₇₉⋊C₁₆)	2
#23762159	2
C₇₉×#1567	2
#7241	79

subgroup

index

C₂×C₄⋊(C₇₉⋊C₁₆)

#23762159

C₇₉×#1567

#7241

Subgroups — #34175367

Conjugacy classes of subgroups

Complements

Maximal subgroups