Group #34174871 — subgroups

Subgroups — #34174871

Order 20224.

352 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 107 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	#23763380	C₂	1	2	yes	maximal
2	C₇₉×#1817	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
4	C₂×C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
5	#19098318	C₄	1	4	yes	—
6	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊C₈×C₇₉	C₄	1	2	yes	—
8	#336×C₇₉	—	2	2	no	—
9	C₂²×C₇₉⋊C₈	—	2	4	no	—
10	C₄⋊(C₇₉⋊C₈)	—	4	4	no	—
11	C₂²⋊(C₇₉⋊C₈)	—	4	2	no	—
12	C₂²⋊(C₇₉⋊C₈)	—	4	2	no	—
13	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
14	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
15	C₂³×C₄×C₇₉	Q₈	1	1	yes	characteristic
16	C₂²×C₈×C₇₉	—	2	2	no	—
17	C₂×C₄²×C₇₉	—	4	2	no	—
18	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₇₉	D₄	1	2	yes	—
20	C₂²⋊C₈×C₇₉	—	4	2	no	—
21	C₂×C₇₉⋊C₈	—	4	4	no	—
22	C₂×C₇₉⋊C₈	—	4	4	no	—
23	C₂×C₇₉⋊C₈	—	4	4	no	—
24	C₂⁴×C₇₉	C₄⋊C₄	1	1	yes	characteristic
25	C₂²×C₄×C₇₉	—	2	2	no	—
26	C₂²×C₄×C₇₉	—	2	4	no	—
27	C₂²×C₄×C₇₉	—	4	1	no	—
28	C₂²×C₄×C₇₉	—	4	1	no	—
29	C₂²×C₄×C₇₉	—	4	2	no	—
30	C₂²×C₄×C₇₉	—	4	1	no	—
31	C₂²×C₄×C₇₉	C₄²	1	1	yes	characteristic
32	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
34	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
36	C₂²×C₄×C₇₉	SD₁₆	1	2	yes	—
37	C₂²×C₄×C₇₉	Q₁₆	1	1	yes	characteristic
38	C₂×C₈×C₇₉	—	4	4	no	—
39	C₂×C₈×C₇₉	—	4	2	no	—
40	C₄²×C₇₉	—	2	4	no	—
41	C₄²×C₇₉	—	4	2	no	—
42	C₂³×C₇₉	—	2	1	no	—
43	C₂³×C₇₉	—	2	2	no	—
44	C₂³×C₇₉	—	2	1	no	—
45	C₂³×C₇₉	—	2	2	no	—
46	C₂³×C₇₉	#95	1	1	yes	characteristic
47	C₂³×C₇₉	#106	1	1	yes	characteristic
48	C₂³×C₇₉	#107	1	1	yes	characteristic
49	C₂×C₄×C₇₉	—	2	2	no	—
50	C₂×C₄×C₇₉	—	2	2	no	—
51	C₂×C₄×C₇₉	—	2	2	no	—
52	C₂×C₄×C₇₉	—	2	2	no	—
53	C₂×C₄×C₇₉	—	2	2	no	—
54	C₂×C₄×C₇₉	—	2	1	no	—
55	C₂×C₄×C₇₉	—	2	2	no	—
56	C₂×C₄×C₇₉	—	4	1	no	—
57	C₂×C₄×C₇₉	—	4	2	no	—
58	C₂×C₄×C₇₉	—	4	1	no	—
59	C₂×C₄×C₇₉	—	4	4	no	—
60	C₂×C₄×C₇₉	—	4	4	no	—
61	C₂×C₄×C₇₉	#99	1	2	yes	—
62	C₂×C₄×C₇₉	#102	1	2	yes	—
63	C₂×C₄×C₇₉	#103	1	2	yes	—
64	C₂×C₄×C₇₉	#108	1	4	yes	—
65	C₈×C₇₉	—	8	2	no	—
66	C₇₉⋊C₈	—	8	4	no	—
67	C₂²×C₇₉	—	2	1	no	—
68	C₂²×C₇₉	—	2	1	no	—
69	C₂²×C₇₉	—	2	2	no	—
70	C₂²×C₇₉	—	2	2	no	—
71	C₂²×C₇₉	—	4	2	no	—
72	C₂²×C₇₉	—	4	1	no	—
73	C₂²×C₇₉	—	4	1	no	—
74	C₂²×C₇₉	H(3,ℤ/4)	1	1	yes	characteristic
75	C₂²×C₇₉	#340	1	1	yes	characteristic
76	C₂²×C₇₉	#341	1	2	yes	—
77	C₂²×C₇₉	#342	1	1	yes	characteristic
78	C₂²×C₇₉	#344	1	2	yes	—
79	C₄×C₇₉	—	2	2	no	—
80	C₄×C₇₉	—	2	2	no	—
81	C₄×C₇₉	—	2	4	no	—
82	C₄×C₇₉	—	2	2	no	—
83	C₄×C₇₉	—	4	1	no	—
84	#7044	—	79	1	no	maximal, Sylow
85	C₂×C₇₉	—	4	1	no	—
86	C₂×C₇₉	—	4	1	no	—
87	C₂×C₇₉	#1287	1	2	yes	—
88	C₂×C₇₉	#1289	1	1	yes	characteristic
89	C₂×C₇₉	#1298	1	1	yes	characteristic
90	C₂×C₇₉	#1301	1	2	yes	—
91	C₂×C₇₉	#1306	1	1	yes	characteristic
92	#1817	D₇₉	1	1	yes	characteristic
93	#1797	—	79	2	no	—
94	C₇₉	#7044	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
96	C₂×C₄⋊C₈	—	158	2	no	—
97	C₂×C₂²⋊C₈	—	158	2	no	—
98	C₂×C₂²⋊C₈	Dic₇₉	1	2	yes	—
99	#336	—	2	2	no	—
100	#336	—	79	4	no	—
101	C₂³×C₄	Dic₁₅₈	1	1	yes	characteristic
102	C₂³×C₄	D₃₁₆	1	1	yes	characteristic
103	C₂³×C₄	C₂×Dic₇₉	1	1	yes	characteristic
104	C₂²×C₈	—	2	2	no	—
105	C₂²×C₈	—	158	4	no	—
106	C₂×C₄²	—	4	2	no	—
107	C₂×C₄²	C₄×D₇₉	1	2	yes	—
108	C₂×C₄²	C₂²⋊D₇₉	1	2	yes	—
109	C₄⋊C₈	—	316	4	no	—
110	C₂²⋊C₈	—	4	2	no	—
111	C₂²⋊C₈	—	316	2	no	—
112	C₂²⋊C₈	—	316	2	no	—
113	C₂⁴	C₄⋊Dic₇₉	1	1	yes	characteristic
114	C₂²×C₄	—	2	2	no	—
115	C₂²×C₄	—	2	4	no	—
116	C₂²×C₄	—	4	1	no	—
117	C₂²×C₄	—	4	1	no	—
118	C₂²×C₄	—	4	1	no	—
119	C₂²×C₄	—	4	2	no	—
120	C₂²×C₄	C₄×Dic₇₉	1	1	yes	characteristic
121	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
122	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
123	C₂²×C₄	Dic₇₉⋊C₄	1	2	yes	—
124	C₂²×C₄	D₆₃₂	1	1	yes	characteristic
125	C₂²×C₄	#11929613	1	2	yes	—
126	C₂²×C₄	Dic₃₁₆	1	1	yes	characteristic
127	C₂×C₈	—	4	4	no	—
128	C₂×C₈	—	4	2	no	—
129	C₂×C₈	—	316	4	no	—
130	C₂×C₈	—	316	4	no	—
131	C₂×C₈	—	316	4	no	—
132	C₄²	—	2	4	no	—
133	C₄²	—	4	2	no	—
134	C₂³	—	2	1	no	—
135	C₂³	—	2	2	no	—
136	C₂³	—	2	1	no	—
137	C₂³	—	2	2	no	—
138	C₂³	C₇₉⋊#95	1	1	yes	characteristic
139	C₂³	#14579093	1	1	yes	characteristic
140	C₂³	#14579095	1	1	yes	characteristic
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	4	no	—
152	C₂×C₄	—	4	4	no	—
153	C₂×C₄	#14579076	1	2	yes	—
154	C₂×C₄	#14579083	1	2	yes	—
155	C₂×C₄	#14579086	1	2	yes	—
156	C₂×C₄	#14579096	1	4	yes	—
157	C₈	—	8	2	no	—
158	C₈	—	632	4	no	—
159	C₂²	—	2	1	no	—
160	C₂²	—	2	1	no	—
161	C₂²	—	2	2	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	4	2	no	—
164	C₂²	—	4	1	no	—
165	C₂²	—	4	1	no	—
166	C₂²	#19098320	1	1	yes	characteristic
167	C₂²	#19098326	1	1	yes	characteristic
168	C₂²	#19098327	1	2	yes	—
169	C₂²	#19098330	1	1	yes	characteristic
170	C₂²	#19098333	1	2	yes	—
171	C₄	—	2	2	no	—
172	C₄	—	2	2	no	—
173	C₄	—	2	4	no	—
174	C₄	—	2	2	no	—
175	C₄	—	4	1	no	—
176	C₂	—	4	1	no	—
177	C₂	—	4	1	no	—
178	C₂	#23761263	1	2	yes	—
179	C₂	#23761268	1	1	yes	characteristic
180	C₂	#23761287	1	1	yes	characteristic
181	C₂	#23761296	1	2	yes	—
182	C₂	#23761308	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23763380

C₂

yes

maximal

C₇₉×#1817

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₇₉⋊C₈)

—

C₂×C₂²⋊(C₇₉⋊C₈)

—

#19098318

C₄

yes

—

C₂²×C₄²×C₇₉

C₂²

yes

characteristic

C₂×C₂²⋊C₈×C₇₉

C₄

yes

—

#336×C₇₉

—

C₂²×C₇₉⋊C₈

—

C₄⋊(C₇₉⋊C₈)

—

C₂²⋊(C₇₉⋊C₈)

—

C₂²⋊(C₇₉⋊C₈)

—

C₂³×C₄×C₇₉

C₂×C₄

yes

characteristic

C₂³×C₄×C₇₉

D₄

yes

characteristic

C₂³×C₄×C₇₉

Q₈

yes

characteristic

C₂²×C₈×C₇₉

—

C₂×C₄²×C₇₉

—

C₂×C₄²×C₇₉

C₂×C₄

yes

—

C₂×C₄²×C₇₉

D₄

yes

—

C₂²⋊C₈×C₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂⁴×C₇₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₄²

yes

characteristic

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₉

C₄⋊C₄

yes

—

C₂²×C₄×C₇₉

D₈

yes

characteristic

C₂²×C₄×C₇₉

SD₁₆

yes

—

C₂²×C₄×C₇₉

Q₁₆

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

yes

C₂³×C₇₉

yes

C₂³×C₇₉

yes

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#99

yes

—

C₂×C₄×C₇₉

#102

yes

—

C₂×C₄×C₇₉

#103

yes

—

C₂×C₄×C₇₉

#108

yes

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

—

C₂²×C₇₉

#342

yes

characteristic

C₂²×C₇₉

#344

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

#7044

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#1287

yes

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₂×C₇₉

#1306

yes

characteristic

#1817

D₇₉

yes

characteristic

#1797

—

C₇₉

#7044

yes

characteristic, complemented, Sylow

C₂²×C₄²

C₂×D₇₉

yes

characteristic

C₂×C₄⋊C₈

—

158

—

C₂×C₂²⋊C₈

—

158

—

C₂×C₂²⋊C₈

Dic₇₉

yes

—

#336

—

100

#336

—

101

C₂³×C₄

Dic₁₅₈

yes

characteristic

102

C₂³×C₄

D₃₁₆

yes

characteristic

103

C₂³×C₄

C₂×Dic₇₉

yes

characteristic

104

C₂²×C₈

—

105

C₂²×C₈

—

158

—

106

C₂×C₄²

—

107

C₂×C₄²

C₄×D₇₉

yes

—

108

C₂×C₄²

C₂²⋊D₇₉

yes

—

109

C₄⋊C₈

—

316

—

110

C₂²⋊C₈

—

111

C₂²⋊C₈

—

316

—

112

C₂²⋊C₈

—

316

—

113

C₂⁴

C₄⋊Dic₇₉

yes

characteristic

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

C₄×Dic₇₉

yes

characteristic

121

C₂²×C₄

(C₂×D₇₉)⋊C₄

yes

—

122

C₂²×C₄

C₂²⋊Dic₇₉

yes

characteristic

123

C₂²×C₄

Dic₇₉⋊C₄

yes

—

124

C₂²×C₄

D₆₃₂

yes

characteristic

125

C₂²×C₄

#11929613

yes

—

126

C₂²×C₄

Dic₃₁₆

yes

characteristic

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

316

—

130

C₂×C₈

—

316

—

131

C₂×C₈

—

316

—

132

C₄²

—

133

C₄²

—

134

C₂³

—

135

C₂³

—

136

C₂³

—

137

C₂³

—

138

C₂³

C₇₉⋊#95

yes

characteristic

139

C₂³

#14579093

yes

characteristic

140

C₂³

#14579095

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

#14579076

yes

—

154

C₂×C₄

#14579083

yes

—

155

C₂×C₄

#14579086

yes

—

156

C₂×C₄

#14579096

yes

—

157

C₈

—

158

C₈

—

632

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

#19098320

yes

characteristic

167

C₂²

#19098326

yes

characteristic

168

C₂²

#19098327

yes

—

169

C₂²

#19098330

yes

characteristic

170

C₂²

#19098333

yes

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₂

—

177

C₂

—

178

C₂

#23761263

yes

—

179

C₂

#23761268

yes

characteristic

180

C₂

#23761287

yes

characteristic

181

C₂

#23761296

yes

—

182

C₂

#23761308

yes

characteristic

	subgroup	G/H
center	C₂³	C₇₉⋊#95
commutator	C₂²×C₄×C₇₉	C₄²
Frattini	C₂²×C₄²	C₂×D₇₉
Fitting	C₇₉×#1817	C₂
socle	C₂³×C₇₉	#95
prime radical	C₂⁴×C₇₉	C₄⋊C₄
cosocle	C₂²×C₄²×C₇₉	C₂²
hypercenter	#1817	D₇₉
hypocenter	C₇₉	#7044
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1817

subgroup

G/H

center

C₂³

C₇₉⋊#95

commutator

C₂²×C₄×C₇₉

C₄²

Frattini

C₂²×C₄²

C₂×D₇₉

Fitting

C₇₉×#1817

C₂

socle

C₂³×C₇₉

#95

prime radical

C₂⁴×C₇₉

C₄⋊C₄

cosocle

C₂²×C₄²×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1817

N	H ≅ G/N	type
94. C₇₉	#7044	semidirect

H ≅ G/N

type

94. C₇₉

#7044

semidirect

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

subgroup	index
#23763380	2
C₇₉×#1817	2
#7044	79

subgroup

index

#23763380

C₇₉×#1817

#7044

Subgroups — #34174871

Conjugacy classes of subgroups

Complements

Maximal subgroups