Group #34175652 — subgroups

Subgroups — #34175652

Order 20224.

268 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 47 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	#23761346	C₂	1	2	yes	maximal, complemented
2	C₇₉×#1957	C₂	1	1	yes	maximal, characteristic
3	#19098289	—	2	2	no	—
4	C₂×#127×C₇₉	C₄	1	1	yes	characteristic
5	C₂×#102×C₇₉	—	2	2	no	—
6	C₂×C₄×C₈×C₇₉	C₄	1	1	yes	characteristic
7	#384×C₇₉	C₂²	1	1	yes	characteristic
8	C₂²⋊(C₇₉⋊C₈)	—	4	2	no	—
9	C₂²×D₄×C₇₉	—	2	1	no	—
10	C₂²×D₄×C₇₉	—	4	1	no	—
11	C₂²×C₈×C₇₉	—	2	1	no	—
12	#127×C₇₉	—	2	1	no	—
13	#127×C₇₉	—	4	1	no	—
14	#127×C₇₉	C₈	1	2	yes	complemented
15	C₂×C₄⋊C₄×C₇₉	—	4	1	no	—
16	C₂×C₄⋊C₄×C₇₉	D₄	1	2	yes	—
17	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
18	#102×C₇₉	—	4	4	no	—
19	C₄×C₈×C₇₉	—	4	1	no	—
20	C₂×C₇₉⋊C₈	—	8	2	no	—
21	C₂⁴×C₇₉	—	4	1	no	—
22	C₂×D₄×C₇₉	—	2	2	no	—
23	C₂×D₄×C₇₉	—	4	2	no	—
24	C₂×D₄×C₇₉	—	4	2	no	—
25	C₂×D₄×C₇₉	—	4	2	no	—
26	C₂×D₄×C₇₉	—	4	2	no	—
27	C₂×D₄×C₇₉	—	4	1	no	—
28	C₂×D₄×C₇₉	—	8	2	no	—
29	C₂×D₄×C₇₉	—	8	2	no	—
30	C₂²×C₄×C₇₉	—	2	1	no	—
31	C₂²×C₄×C₇₉	—	2	2	no	—
32	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₇₉	—	2	2	no	—
34	C₂×C₈×C₇₉	—	4	1	no	—
35	C₂×C₈×C₇₉	—	4	1	no	—
36	C₄⋊C₄×C₇₉	—	4	2	no	—
37	C₄⋊C₄×C₇₉	—	8	2	no	—
38	C₄²×C₇₉	—	2	1	no	—
39	C₄²×C₇₉	C₂×C₈	1	1	yes	characteristic
40	C₄²×C₇₉	M₁₆	1	1	yes	characteristic
41	C₂³×C₇₉	—	4	2	no	—
42	C₂³×C₇₉	—	8	2	no	—
43	C₂³×C₇₉	—	8	2	no	—
44	C₂³×C₇₉	—	8	1	no	—
45	C₂³×C₇₉	—	8	1	no	—
46	C₂³×C₇₉	#99	1	1	yes	characteristic
47	D₄×C₇₉	—	4	4	no	—
48	D₄×C₇₉	—	8	2	no	—
49	D₄×C₇₉	—	8	4	no	—
50	D₄×C₇₉	—	8	4	no	—
51	C₂×C₄×C₇₉	—	2	2	no	—
52	C₂×C₄×C₇₉	—	2	2	no	—
53	C₂×C₄×C₇₉	—	2	1	no	—
54	C₂×C₄×C₇₉	—	2	1	no	—
55	C₂×C₄×C₇₉	—	2	1	no	—
56	C₂×C₄×C₇₉	—	2	1	no	—
57	C₂×C₄×C₇₉	—	4	2	no	—
58	C₂×C₄×C₇₉	—	8	2	no	—
59	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
60	C₂×C₄×C₇₉	#100	1	1	yes	characteristic
61	C₈×C₇₉	—	4	2	no	—
62	C₇₉⋊C₈	—	16	2	no	—
63	C₂²×C₇₉	—	2	1	no	—
64	C₂²×C₇₉	—	2	1	no	—
65	C₂²×C₇₉	—	8	4	no	—
66	C₂²×C₇₉	—	8	2	no	—
67	C₂²×C₇₉	—	16	2	no	—
68	C₂²×C₇₉	—	16	1	no	—
69	C₂²×C₇₉	—	16	1	no	—
70	C₂²×C₇₉	#323	1	1	yes	characteristic
71	C₂²×C₇₉	#353	1	1	yes	characteristic
72	C₂²×C₇₉	#355	1	1	yes	characteristic
73	C₄×C₇₉	—	2	2	no	—
74	C₄×C₇₉	—	2	2	no	—
75	C₄×C₇₉	—	2	1	no	—
76	C₄×C₇₉	—	2	1	no	—
77	C₄×C₇₉	—	8	2	no	—
78	#7354	—	79	1	no	maximal, Sylow
79	C₂×C₇₉	—	2	1	no	—
80	C₂×C₇₉	—	2	1	no	—
81	C₂×C₇₉	—	16	2	no	—
82	C₂×C₇₉	#1318	1	1	yes	characteristic
83	C₂×C₇₉	#1408	1	1	yes	characteristic
84	C₂×C₇₉	#1410	1	1	yes	characteristic
85	#1957	D₇₉	1	1	yes	characteristic
86	#1325	—	79	2	no	—
87	C₇₉	#7354	1	1	yes	characteristic, complemented, Sylow
88	C₂×#127	Dic₇₉	1	1	yes	characteristic
89	C₂×#102	—	2	2	no	—
90	C₂×C₄×C₈	Dic₇₉	1	1	yes	characteristic
91	#384	C₂×D₇₉	1	1	yes	characteristic
92	#324	—	158	2	no	—
93	C₂²×D₄	—	2	1	no	—
94	C₂²×D₄	—	4	1	no	—
95	C₂²×C₈	—	2	1	no	—
96	#127	—	2	1	no	—
97	#127	—	4	1	no	—
98	#127	C₇₉⋊C₈	1	2	yes	complemented
99	C₂×C₄⋊C₄	—	4	1	no	—
100	C₂×C₄⋊C₄	C₂²⋊D₇₉	1	2	yes	—
101	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
102	#102	—	4	4	no	—
103	C₂²⋊C₈	—	316	2	no	—
104	C₄×C₈	—	4	1	no	—
105	C₂⁴	—	4	1	no	—
106	C₂×D₄	—	2	2	no	—
107	C₂×D₄	—	4	2	no	—
108	C₂×D₄	—	4	2	no	—
109	C₂×D₄	—	4	2	no	—
110	C₂×D₄	—	4	2	no	—
111	C₂×D₄	—	4	1	no	—
112	C₂×D₄	—	8	2	no	—
113	C₂×D₄	—	8	2	no	—
114	C₂²×C₄	—	2	1	no	—
115	C₂²×C₄	—	2	2	no	—
116	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	4	1	no	—
119	C₂×C₈	—	4	1	no	—
120	C₂×C₈	—	632	2	no	—
121	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	8	2	no	—
123	C₄²	—	2	1	no	—
124	C₄²	C₂×C₇₉⋊C₈	1	1	yes	characteristic
125	C₄²	#11929608	1	1	yes	characteristic
126	C₂³	—	4	2	no	—
127	C₂³	—	8	2	no	—
128	C₂³	—	8	2	no	—
129	C₂³	—	8	1	no	—
130	C₂³	—	8	1	no	—
131	C₂³	#14579076	1	1	yes	characteristic
132	D₄	—	4	4	no	—
133	D₄	—	8	2	no	—
134	D₄	—	8	4	no	—
135	D₄	—	8	4	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	8	2	no	—
144	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
145	C₂×C₄	#14579078	1	1	yes	characteristic
146	C₈	—	4	2	no	—
147	C₈	—	1264	2	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	8	4	no	—
151	C₂²	—	8	2	no	—
152	C₂²	—	16	2	no	—
153	C₂²	—	16	1	no	—
154	C₂²	—	16	1	no	—
155	C₂²	#19098287	1	1	yes	characteristic
156	C₂²	#19098356	1	1	yes	characteristic
157	C₂²	#19098360	1	1	yes	characteristic
158	C₄	—	2	2	no	—
159	C₄	—	2	2	no	—
160	C₄	—	2	1	no	—
161	C₄	—	2	1	no	—
162	C₄	—	8	2	no	—
163	C₂	—	2	1	no	—
164	C₂	—	2	1	no	—
165	C₂	—	16	2	no	—
166	C₂	#23761332	1	1	yes	characteristic
167	C₂	#23761563	1	1	yes	characteristic
168	C₂	#23761567	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23761346

C₂

yes

maximal, complemented

C₇₉×#1957

C₂

yes

maximal, characteristic

#19098289

—

C₂×#127×C₇₉

C₄

yes

characteristic

C₂×#102×C₇₉

—

C₂×C₄×C₈×C₇₉

C₄

yes

characteristic

#384×C₇₉

C₂²

yes

characteristic

C₂²⋊(C₇₉⋊C₈)

—

C₂²×D₄×C₇₉

—

C₂²×D₄×C₇₉

—

C₂²×C₈×C₇₉

—

#127×C₇₉

—

#127×C₇₉

—

#127×C₇₉

C₈

yes

complemented

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

C₂×C₄

yes

characteristic

#102×C₇₉

—

C₄×C₈×C₇₉

—

C₂×C₇₉⋊C₈

—

C₂⁴×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂×C₈

yes

characteristic

C₄²×C₇₉

M₁₆

yes

characteristic

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

#99

yes

characteristic

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×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

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₉

#100

yes

characteristic

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

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

#7354

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

D₇₉

yes

characteristic

#1325

—

C₇₉

#7354

yes

characteristic, complemented, Sylow

C₂×#127

Dic₇₉

yes

characteristic

C₂×#102

—

C₂×C₄×C₈

Dic₇₉

yes

characteristic

#384

C₂×D₇₉

yes

characteristic

#324

—

158

—

C₂²×D₄

—

C₂²×D₄

—

C₂²×C₈

—

#127

—

#127

—

#127

C₇₉⋊C₈

yes

complemented

C₂×C₄⋊C₄

—

100

C₂×C₄⋊C₄

C₂²⋊D₇₉

yes

—

101

C₂×C₄²

C₂×Dic₇₉

yes

characteristic

102

#102

—

103

C₂²⋊C₈

—

316

—

104

C₄×C₈

—

105

C₂⁴

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

C₂²⋊Dic₇₉

yes

characteristic

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

632

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄²

—

124

C₄²

C₂×C₇₉⋊C₈

yes

characteristic

125

C₄²

#11929608

yes

characteristic

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

—

130

C₂³

—

131

C₂³

#14579076

yes

characteristic

132

D₄

—

133

D₄

—

134

D₄

—

135

D₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

C₂²⋊(C₇₉⋊C₈)

yes

characteristic

145

C₂×C₄

#14579078

yes

characteristic

146

C₈

—

147

C₈

—

1264

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

#19098287

yes

characteristic

156

C₂²

#19098356

yes

characteristic

157

C₂²

#19098360

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₂

—

164

C₂

—

165

C₂

—

166

C₂

#23761332

yes

characteristic

167

C₂

#23761563

yes

characteristic

168

C₂

#23761567

yes

characteristic

	subgroup	G/H
center	C₂²	#19098356
commutator	C₄²×C₇₉	C₂×C₈
Frattini	#384	C₂×D₇₉
Fitting	C₇₉×#1957	C₂
socle	C₂²×C₇₉	#353
prime radical	C₂×#127×C₇₉	C₄
cosocle	#384×C₇₉	C₂²
hypercenter	#1957	D₇₉
hypocenter	C₇₉	#7354
LCS	G ▷ C₄²×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #384 ◁ #1957

subgroup

G/H

center

C₂²

#19098356

commutator

C₄²×C₇₉

C₂×C₈

Frattini

#384

C₂×D₇₉

Fitting

C₇₉×#1957

C₂

socle

C₂²×C₇₉

#353

prime radical

C₂×#127×C₇₉

C₄

cosocle

#384×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #384 ◁ #1957

N	H ≅ G/N	type
1. #23761346	C₂	semidirect
14. #127×C₇₉	C₈	semidirect
87. C₇₉	#7354	semidirect
98. #127	C₇₉⋊C₈	semidirect

H ≅ G/N

type

1. #23761346

C₂

semidirect

14. #127×C₇₉

C₈

semidirect

87. C₇₉

#7354

semidirect

98. #127

C₇₉⋊C₈

semidirect

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

subgroup	index
#23761346	2
C₇₉×#1957	2
#7354	79

subgroup

index

#23761346

C₇₉×#1957

#7354

Subgroups — #34175652

Conjugacy classes of subgroups

Complements

Maximal subgroups