Group #34201675 — subgroups

Subgroups — #34201675

Order 20224.

356 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 59 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	#23775204	C₂	1	1	yes	maximal, characteristic
2	#23765478	C₂	1	2	yes	maximal, complemented
3	#23761439	C₂	1	1	yes	maximal, characteristic, complemented
4	#23761374	C₂	1	2	yes	maximal, complemented
5	C₇₉×#1713	C₂	1	1	yes	maximal, characteristic, complemented
6	#19099376	—	2	2	no	—
7	#19099004	—	2	2	no	—
8	#19098835	—	2	2	no	—
9	#19098745	—	2	2	no	—
10	#19098681	C₂²	1	2	yes	—
11	#19098384	C₂²	1	2	yes	—
12	#19098379	—	2	2	no	—
13	#500×C₇₉	—	2	2	no	—
14	#494×C₇₉	C₂²	1	1	yes	characteristic
15	#494×C₇₉	C₂²	1	1	yes	characteristic
16	C₈²×C₇₉	C₂²	1	1	yes	characteristic
17	#14579209	—	2	2	no	—
18	#14579193	—	4	4	no	—
19	#14579192	—	2	4	no	—
20	#14579184	—	4	2	no	—
21	#14579183	D₄	1	2	yes	—
22	Q₁₆×D₇₉	—	4	2	no	—
23	#14579167	—	2	2	no	—
24	#14579107	—	4	2	no	—
25	#14579104	—	4	2	no	—
26	#14579101	—	2	2	no	—
27	#14579097	D₄	1	2	yes	—
28	#14579089	—	2	2	no	—
29	#14579080	—	2	2	no	—
30	C₂×Q₁₆×C₇₉	—	2	2	no	—
31	C₂×Q₁₆×C₇₉	—	2	2	no	—
32	C₄⋊Q₈×C₇₉	—	2	1	no	—
33	C₄⋊Q₈×C₇₉	D₄	1	2	yes	—
34	#107×C₇₉	—	4	4	no	—
35	C₄×C₈×C₇₉	—	2	1	no	—
36	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
37	#11929625	—	4	4	no	—
38	#11929624	—	2	2	no	—
39	#11929624	—	8	2	no	—
40	Q₈×D₇₉	—	4	4	no	—
41	Dic₃₁₆	D₈	1	2	yes	—
42	#11929614	—	4	4	no	—
43	#11929613	—	2	2	no	—
44	#11929611	—	4	4	no	—
45	D₆₃₂	D₈	1	2	yes	—
46	#11929608	—	2	2	no	—
47	#11929607	—	4	2	no	—
48	C₈×D₇₉	—	4	2	no	—
49	C₇₉⋊C₁₆	—	4	2	no	—
50	C₂×Q₈×C₇₉	—	2	2	no	—
51	C₂×Q₈×C₇₉	—	4	1	no	—
52	Q₁₆×C₇₉	—	4	2	no	—
53	Q₁₆×C₇₉	—	4	2	no	—
54	Q₁₆×C₇₉	—	4	2	no	—
55	Q₁₆×C₇₉	—	4	2	no	—
56	C₂×C₈×C₇₉	—	2	2	no	—
57	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
58	C₄⋊C₄×C₇₉	—	4	2	no	—
59	C₄⋊C₄×C₇₉	—	4	2	no	—
60	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
61	Q₈×C₇₉	—	4	4	no	—
62	Q₈×C₇₉	—	8	2	no	—
63	C₂×C₄×C₇₉	—	2	1	no	—
64	C₂×C₄×C₇₉	—	4	2	no	—
65	C₂×C₄×C₇₉	—	8	1	no	—
66	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
67	C₂²⋊D₇₉	—	4	2	no	—
68	D₃₁₆	—	2	2	no	—
69	D₃₁₆	—	8	2	no	—
70	C₄×D₇₉	—	4	2	no	—
71	C₄×D₇₉	—	8	4	no	—
72	Dic₁₅₈	—	2	2	no	—
73	Dic₁₅₈	—	8	2	no	—
74	C₈×C₇₉	—	2	4	no	—
75	C₈×C₇₉	C₂×D₈	1	2	yes	—
76	C₈×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
77	C₇₉⋊C₈	—	4	2	no	—
78	C₂²×C₇₉	#453	1	1	yes	characteristic
79	C₂×D₇₉	—	4	2	no	—
80	C₄×C₇₉	—	2	2	no	—
81	C₄×C₇₉	—	8	2	no	—
82	C₄×C₇₉	—	16	1	no	—
83	C₄×C₇₉	#447	1	2	yes	—
84	Dic₇₉	—	4	2	no	—
85	#12313	—	79	1	no	maximal, Sylow
86	C₂×C₇₉	—	2	1	no	—
87	C₂×C₇₉	#1619	1	1	yes	characteristic
88	D₇₉	—	8	2	no	—
89	#3293	—	79	1	no	—
90	#2214	—	79	2	no	—
91	#1713	D₇₉	1	1	yes	characteristic, complemented
92	#1358	—	79	1	no	—
93	#1335	—	79	2	no	—
94	C₇₉	#12313	1	1	yes	characteristic, complemented, Sylow
95	#578	—	158	2	no	—
96	#510	—	158	2	no	—
97	#500	—	2	2	no	—
98	#494	C₂×D₇₉	1	1	yes	characteristic
99	#494	C₂×D₇₉	1	1	yes	characteristic
100	#473	—	158	2	no	—
101	#456	—	158	2	no	—
102	#443	—	79	2	no	—
103	#364	—	79	2	no	—
104	#362	—	158	2	no	—
105	C₈²	C₂×D₇₉	1	1	yes	characteristic
106	2₋¹⁺⁴	—	158	2	no	—
107	#137	—	316	4	no	—
108	#137	—	158	4	no	—
109	#135	—	316	2	no	—
110	#135	—	79	2	no	—
111	C₂×Q₁₆	—	2	2	no	—
112	C₂×Q₁₆	—	2	2	no	—
113	C₂×Q₁₆	—	316	2	no	—
114	#131	—	158	2	no	—
115	C₄⋊Q₈	—	2	1	no	—
116	C₄⋊Q₈	C₂²⋊D₇₉	1	2	yes	—
117	Q₃₂	—	316	2	no	—
118	SD₃₂	—	316	2	no	—
119	M₃₂	—	158	2	no	—
120	#108	—	79	2	no	—
121	#107	—	4	4	no	—
122	#104	—	158	2	no	—
123	#101	—	158	2	no	—
124	C₄×C₈	—	2	1	no	—
125	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
126	#41	—	316	4	no	—
127	#41	—	632	2	no	—
128	#41	—	158	2	no	—
129	C₂×Q₈	—	2	2	no	—
130	C₂×Q₈	—	4	1	no	—
131	C₂×Q₈	—	316	4	no	—
132	Q₁₆	—	4	2	no	—
133	Q₁₆	—	4	2	no	—
134	Q₁₆	—	4	2	no	—
135	Q₁₆	—	4	2	no	—
136	Q₁₆	—	316	4	no	—
137	Q₁₆	—	79	2	no	—
138	SD₁₆	—	316	4	no	—
139	SD₁₆	—	158	2	no	—
140	D₈	—	79	2	no	—
141	M₁₆	—	316	2	no	—
142	M₁₆	—	158	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	316	2	no	—
145	C₂×C₈	D₄×D₇₉	1	2	yes	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
149	C₁₆	—	316	2	no	—
150	Q₈	—	4	4	no	—
151	Q₈	—	8	2	no	—
152	Q₈	—	632	2	no	—
153	Q₈	—	158	2	no	—
154	D₄	—	316	2	no	—
155	D₄	—	632	2	no	—
156	D₄	—	158	2	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	8	1	no	—
160	C₂×C₄	—	316	2	no	—
161	C₂×C₄	—	632	4	no	—
162	C₂×C₄	#14579129	1	1	yes	characteristic
163	C₈	—	2	4	no	—
164	C₈	—	316	2	no	—
165	C₈	D₈×D₇₉	1	2	yes	—
166	C₈	#14579185	1	2	yes	—
167	C₂²	—	316	2	no	—
168	C₂²	#19098730	1	1	yes	characteristic
169	C₄	—	2	2	no	—
170	C₄	—	8	2	no	—
171	C₄	—	16	1	no	—
172	C₄	—	316	2	no	—
173	C₄	#19098695	1	2	yes	—
174	C₂	—	2	1	no	—
175	C₂	—	632	2	no	—
176	C₂	#23762452	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23775204

C₂

yes

maximal, characteristic

#23765478

C₂

yes

maximal, complemented

#23761439

C₂

yes

maximal, characteristic, complemented

#23761374

C₂

yes

maximal, complemented

C₇₉×#1713

C₂

yes

maximal, characteristic, complemented

—

—

—

—

C₂²

yes

—

#19098384

C₂²

yes

—

#19098379

—

#500×C₇₉

—

#494×C₇₉

C₂²

yes

characteristic

#494×C₇₉

C₂²

yes

characteristic

C₈²×C₇₉

C₂²

yes

—

—

—

—

D₄

yes

—

Q₁₆×D₇₉

—

—

—

—

—

D₄

yes

—

#14579089

—

#14579080

—

C₂×Q₁₆×C₇₉

—

C₂×Q₁₆×C₇₉

—

C₄⋊Q₈×C₇₉

—

C₄⋊Q₈×C₇₉

D₄

yes

—

#107×C₇₉

—

C₄×C₈×C₇₉

—

C₄×C₈×C₇₉

C₂³

yes

—

—

—

Q₈×D₇₉

—

Dic₃₁₆

D₈

yes

—

#11929614

—

#11929613

—

#11929611

—

D₆₃₂

D₈

yes

—

#11929608

—

#11929607

—

C₈×D₇₉

—

C₇₉⋊C₁₆

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

Q₁₆×C₇₉

—

Q₁₆×C₇₉

—

Q₁₆×C₇₉

—

Q₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂×D₄

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂×D₄

yes

characteristic

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#120

yes

characteristic

C₂²⋊D₇₉

—

D₃₁₆

—

D₃₁₆

—

C₄×D₇₉

—

C₄×D₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

C₂×D₈

yes

—

C₈×C₇₉

AGL(1,ℤ/8)

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

#453

yes

characteristic

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#447

yes

—

Dic₇₉

—

#12313

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

#1619

yes

characteristic

D₇₉

—

#3293

—

#2214

—

#1713

D₇₉

yes

characteristic, complemented

#1358

—

#1335

—

C₇₉

#12313

yes

characteristic, complemented, Sylow

#578

—

158

—

#510

—

158

—

#500

—

#494

C₂×D₇₉

yes

characteristic

#494

C₂×D₇₉

yes

characteristic

100

#473

—

158

—

101

#456

—

158

—

102

#443

—

103

#364

—

104

#362

—

158

—

105

C₈²

C₂×D₇₉

yes

characteristic

106

2₋¹⁺⁴

—

158

—

107

#137

—

316

—

108

#137

—

158

—

109

#135

—

316

—

110

#135

—

111

C₂×Q₁₆

—

112

C₂×Q₁₆

—

113

C₂×Q₁₆

—

316

—

114

#131

—

158

—

115

C₄⋊Q₈

—

116

C₄⋊Q₈

C₂²⋊D₇₉

yes

—

117

Q₃₂

—

316

—

118

SD₃₂

—

316

—

119

M₃₂

—

158

—

120

#108

—

121

#107

—

122

#104

—

158

—

123

#101

—

158

—

124

C₄×C₈

—

125

C₄×C₈

C₂²×D₇₉

yes

characteristic

126

#41

—

316

—

127

#41

—

632

—

128

#41

—

158

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

316

—

132

Q₁₆

—

133

Q₁₆

—

134

Q₁₆

—

135

Q₁₆

—

136

Q₁₆

—

316

—

137

Q₁₆

—

138

SD₁₆

—

316

—

139

SD₁₆

—

158

—

140

D₈

—

141

M₁₆

—

316

—

142

M₁₆

—

158

—

143

C₂×C₈

—

144

C₂×C₈

—

316

—

145

C₂×C₈

D₄×D₇₉

yes

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄²

C₂×C₂²⋊D₇₉

yes

characteristic

149

C₁₆

—

316

—

150

Q₈

—

151

Q₈

—

152

Q₈

—

632

—

153

Q₈

—

158

—

154

D₄

—

316

—

155

D₄

—

632

—

156

D₄

—

158

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

316

—

161

C₂×C₄

—

632

—

162

C₂×C₄

#14579129

yes

characteristic

163

C₈

—

164

C₈

—

316

—

165

C₈

D₈×D₇₉

yes

—

166

C₈

#14579185

yes

—

167

C₂²

—

316

—

168

C₂²

#19098730

yes

characteristic

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

316

—

173

C₄

#19098695

yes

—

174

C₂

—

175

C₂

—

632

—

176

C₂

#23762452

yes

characteristic

	subgroup	G/H
center	C₂	#23762452
commutator, cosocle	C₄×C₈×C₇₉	C₂³
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₇₉×#1713	C₂
socle	C₂×C₇₉	#1619
prime radical	#23775204	C₂
hypercenter	#1713	D₇₉
hypocenter	C₇₉	#12313
LCS	G ▷ C₄×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1713

subgroup

G/H

center

C₂

#23762452

commutator, cosocle

C₄×C₈×C₇₉

C₂³

Frattini

C₄×C₈

C₂²×D₇₉

Fitting

C₇₉×#1713

C₂

C₂×C₇₉

C₂

D₇₉

C₇₉

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

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1713

N	H ≅ G/N	type
2. #23765478	C₂	semidirect
3. #23761439	C₂	semidirect
4. #23761374	C₂	semidirect
5. C₇₉×#1713	C₂	semidirect
91. #1713	D₇₉	semidirect
94. C₇₉	#12313	semidirect

H ≅ G/N

type

2. #23765478

C₂

semidirect

3. #23761439

C₂

semidirect

4. #23761374

C₂

semidirect

5. C₇₉×#1713

C₂

semidirect

91. #1713

D₇₉

semidirect

94. C₇₉

#12313

semidirect

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

subgroup	index
#23775204	2
#23765478	2
#23761439	2
#23761374	2
C₇₉×#1713	2
#12313	79

subgroup

index

C₇₉×#1713

Subgroups — #34201675

Conjugacy classes of subgroups

Complements

Maximal subgroups