Group #34201665 — subgroups

Subgroups — #34201665

Order 20224.

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

normal subgroups only

This group has 366 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, complemented
2	#23765473	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×#1709	C₂	1	1	yes	maximal, characteristic, complemented
6	#19099376	—	2	2	no	—
7	#19098999	—	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	#19098376	—	2	2	no	—
13	#494×C₇₉	C₂²	1	1	yes	characteristic
14	#493×C₇₉	C₂²	1	1	yes	characteristic
15	#486×C₇₉	—	2	2	no	—
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	#14579105	—	4	2	no	—
25	#14579102	—	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×D₈×C₇₉	—	2	2	no	—
32	C₄⋊Q₈×C₇₉	D₄	1	2	yes	—
33	#127×C₇₉	—	2	1	no	—
34	#102×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×D₄×C₇₉	—	4	2	no	—
52	C₂×D₄×C₇₉	—	4	1	no	—
53	Q₁₆×C₇₉	—	4	2	no	—
54	Q₁₆×C₇₉	—	4	2	no	—
55	D₈×C₇₉	—	4	2	no	—
56	D₈×C₇₉	—	4	2	no	—
57	C₂×C₈×C₇₉	—	2	2	no	—
58	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
59	C₄⋊C₄×C₇₉	—	4	2	no	—
60	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
61	C₂³×C₇₉	—	8	1	no	—
62	Q₈×C₇₉	—	4	4	no	—
63	D₄×C₇₉	—	8	4	no	—
64	D₄×C₇₉	—	8	2	no	—
65	C₂×C₄×C₇₉	—	2	1	no	—
66	C₂×C₄×C₇₉	—	4	2	no	—
67	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
68	C₂²⋊D₇₉	—	4	2	no	—
69	D₃₁₆	—	2	2	no	—
70	D₃₁₆	—	8	2	no	—
71	C₄×D₇₉	—	4	2	no	—
72	C₄×D₇₉	—	8	4	no	—
73	Dic₁₅₈	—	2	2	no	—
74	Dic₁₅₈	—	8	2	no	—
75	C₈×C₇₉	—	4	2	no	—
76	C₈×C₇₉	C₂×D₈	1	2	yes	—
77	C₈×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
78	C₇₉⋊C₈	—	4	2	no	—
79	C₂²×C₇₉	—	16	2	no	—
80	C₂²×C₇₉	—	16	1	no	—
81	C₂²×C₇₉	#453	1	1	yes	characteristic
82	C₂×D₇₉	—	4	2	no	—
83	C₄×C₇₉	—	2	2	no	—
84	C₄×C₇₉	—	8	2	no	—
85	C₄×C₇₉	#447	1	2	yes	—
86	Dic₇₉	—	4	2	no	—
87	#12311	—	79	1	no	maximal, Sylow
88	C₂×C₇₉	—	2	1	no	—
89	C₂×C₇₉	—	32	1	no	—
90	C₂×C₇₉	#1619	1	1	yes	characteristic
91	D₇₉	—	8	2	no	—
92	#3293	—	79	1	no	—
93	#2213	—	79	2	no	—
94	#1709	D₇₉	1	1	yes	characteristic, complemented
95	#1358	—	79	1	no	—
96	#1335	—	79	2	no	—
97	C₇₉	#12311	1	1	yes	characteristic, complemented, Sylow
98	#578	—	158	2	no	—
99	#509	—	158	2	no	—
100	#494	C₂×D₇₉	1	1	yes	characteristic
101	#493	C₂×D₇₉	1	1	yes	characteristic
102	#486	—	2	2	no	—
103	#473	—	158	2	no	—
104	#456	—	158	2	no	—
105	#443	—	79	2	no	—
106	#364	—	79	2	no	—
107	#361	—	158	2	no	—
108	C₈²	C₂×D₇₉	1	1	yes	characteristic
109	2₋¹⁺⁴	—	158	2	no	—
110	#137	—	316	4	no	—
111	#137	—	158	4	no	—
112	#135	—	316	2	no	—
113	#135	—	79	2	no	—
114	C₂×Q₁₆	—	2	2	no	—
115	C₂×Q₁₆	—	316	2	no	—
116	C₂×D₈	—	2	2	no	—
117	#131	—	158	2	no	—
118	C₄⋊Q₈	C₂²⋊D₇₉	1	2	yes	—
119	#127	—	2	1	no	—
120	SD₃₂	—	316	2	no	—
121	D₁₆	—	316	2	no	—
122	M₃₂	—	158	2	no	—
123	#108	—	79	2	no	—
124	#104	—	158	2	no	—
125	#102	—	4	4	no	—
126	#101	—	158	2	no	—
127	C₄×C₈	—	2	1	no	—
128	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
129	#41	—	316	4	no	—
130	#41	—	632	2	no	—
131	#41	—	158	2	no	—
132	C₂×Q₈	—	2	2	no	—
133	C₂×Q₈	—	316	4	no	—
134	C₂×D₄	—	4	2	no	—
135	C₂×D₄	—	4	1	no	—
136	Q₁₆	—	4	2	no	—
137	Q₁₆	—	4	2	no	—
138	Q₁₆	—	316	4	no	—
139	Q₁₆	—	79	2	no	—
140	SD₁₆	—	316	4	no	—
141	SD₁₆	—	158	2	no	—
142	D₈	—	4	2	no	—
143	D₈	—	4	2	no	—
144	D₈	—	79	2	no	—
145	M₁₆	—	316	2	no	—
146	M₁₆	—	158	2	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	316	2	no	—
149	C₂×C₈	D₄×D₇₉	1	2	yes	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
152	C₁₆	—	316	2	no	—
153	C₂³	—	8	1	no	—
154	Q₈	—	4	4	no	—
155	Q₈	—	632	2	no	—
156	Q₈	—	158	2	no	—
157	D₄	—	8	4	no	—
158	D₄	—	8	2	no	—
159	D₄	—	316	2	no	—
160	D₄	—	632	2	no	—
161	D₄	—	158	2	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	316	2	no	—
165	C₂×C₄	—	632	4	no	—
166	C₂×C₄	#14579129	1	1	yes	characteristic
167	C₈	—	4	2	no	—
168	C₈	—	316	2	no	—
169	C₈	D₈×D₇₉	1	2	yes	—
170	C₈	#14579185	1	2	yes	—
171	C₂²	—	16	2	no	—
172	C₂²	—	16	1	no	—
173	C₂²	—	316	2	no	—
174	C₂²	#19098730	1	1	yes	characteristic
175	C₄	—	2	2	no	—
176	C₄	—	8	2	no	—
177	C₄	—	316	2	no	—
178	C₄	#19098695	1	2	yes	—
179	C₂	—	2	1	no	—
180	C₂	—	32	1	no	—
181	C₂	—	632	2	no	—
182	C₂	#23762452	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23775204

C₂

yes

maximal, characteristic, complemented

#23765473

C₂

yes

maximal, complemented

#23761439

C₂

yes

maximal, characteristic, complemented

#23761374

C₂

yes

maximal, complemented

C₇₉×#1709

C₂

yes

maximal, characteristic, complemented

—

—

—

—

C₂²

yes

—

#19098384

C₂²

yes

—

#19098376

—

#494×C₇₉

C₂²

yes

characteristic

#493×C₇₉

C₂²

yes

characteristic

#486×C₇₉

—

C₈²×C₇₉

C₂²

yes

—

—

—

—

D₄

yes

—

Q₁₆×D₇₉

—

—

—

—

—

D₄

yes

—

#14579089

—

#14579080

—

C₂×Q₁₆×C₇₉

—

C₂×D₈×C₇₉

—

C₄⋊Q₈×C₇₉

D₄

yes

—

#127×C₇₉

—

#102×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×D₄×C₇₉

—

C₂×D₄×C₇₉

—

Q₁₆×C₇₉

—

Q₁₆×C₇₉

—

D₈×C₇₉

—

D₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂×D₄

yes

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₂³×C₇₉

—

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×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

—

C₂²×C₇₉

—

C₂²×C₇₉

#453

yes

characteristic

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#447

yes

—

Dic₇₉

—

#12311

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#1619

yes

characteristic

D₇₉

—

#3293

—

#2213

—

#1709

D₇₉

yes

characteristic, complemented

#1358

—

#1335

—

C₇₉

#12311

yes

characteristic, complemented, Sylow

#578

—

158

—

#509

—

158

—

100

#494

C₂×D₇₉

yes

characteristic

101

#493

C₂×D₇₉

yes

characteristic

102

#486

—

103

#473

—

158

—

104

#456

—

158

—

105

#443

—

106

#364

—

107

#361

—

158

—

108

C₈²

C₂×D₇₉

yes

characteristic

109

2₋¹⁺⁴

—

158

—

110

#137

—

316

—

111

#137

—

158

—

112

#135

—

316

—

113

#135

—

114

C₂×Q₁₆

—

115

C₂×Q₁₆

—

316

—

116

C₂×D₈

—

117

#131

—

158

—

118

C₄⋊Q₈

C₂²⋊D₇₉

yes

—

119

#127

—

120

SD₃₂

—

316

—

121

D₁₆

—

316

—

122

M₃₂

—

158

—

123

#108

—

124

#104

—

158

—

125

#102

—

126

#101

—

158

—

127

C₄×C₈

—

128

C₄×C₈

C₂²×D₇₉

yes

characteristic

129

#41

—

316

—

130

#41

—

632

—

131

#41

—

158

—

132

C₂×Q₈

—

133

C₂×Q₈

—

316

—

134

C₂×D₄

—

135

C₂×D₄

—

136

Q₁₆

—

137

Q₁₆

—

138

Q₁₆

—

316

—

139

Q₁₆

—

140

SD₁₆

—

316

—

141

SD₁₆

—

158

—

142

D₈

—

143

D₈

—

144

D₈

—

145

M₁₆

—

316

—

146

M₁₆

—

158

—

147

C₂×C₈

—

148

C₂×C₈

—

316

—

149

C₂×C₈

D₄×D₇₉

yes

—

150

C₄⋊C₄

—

151

C₄²

C₂×C₂²⋊D₇₉

yes

characteristic

152

C₁₆

—

316

—

153

C₂³

—

154

Q₈

—

155

Q₈

—

632

—

156

Q₈

—

158

—

157

D₄

—

158

D₄

—

159

D₄

—

316

—

160

D₄

—

632

—

161

D₄

—

158

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

316

—

165

C₂×C₄

—

632

—

166

C₂×C₄

#14579129

yes

characteristic

167

C₈

—

168

C₈

—

316

—

169

C₈

D₈×D₇₉

yes

—

170

C₈

#14579185

yes

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

316

—

174

C₂²

#19098730

yes

characteristic

175

C₄

—

176

C₄

—

177

C₄

—

316

—

178

C₄

#19098695

yes

—

179

C₂

—

180

C₂

—

181

C₂

—

632

—

182

C₂

#23762452

yes

characteristic

	subgroup	G/H
center	C₂	#23762452
commutator, cosocle	C₄×C₈×C₇₉	C₂³
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₇₉×#1709	C₂
socle	C₂×C₇₉	#1619
hypercenter	#1709	D₇₉
hypocenter	C₇₉	#12311
LCS	G ▷ C₄×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1709

subgroup

G/H

center

C₂

#23762452

commutator, cosocle

C₄×C₈×C₇₉

C₂³

Frattini

C₄×C₈

C₂²×D₇₉

Fitting

C₇₉×#1709

C₂

C₂×C₇₉

D₇₉

C₇₉

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

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1709

N	H ≅ G/N	type
1. #23775204	C₂	semidirect
2. #23765473	C₂	semidirect
3. #23761439	C₂	semidirect
4. #23761374	C₂	semidirect
5. C₇₉×#1709	C₂	semidirect
94. #1709	D₇₉	semidirect
97. C₇₉	#12311	semidirect

H ≅ G/N

type

1. #23775204

C₂

semidirect

2. #23765473

C₂

semidirect

3. #23761439

C₂

semidirect

4. #23761374

C₂

semidirect

5. C₇₉×#1709

C₂

semidirect

94. #1709

D₇₉

semidirect

97. C₇₉

#12311

semidirect

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

subgroup	index
#23775204	2
#23765473	2
#23761439	2
#23761374	2
C₇₉×#1709	2
#12311	79

subgroup

index

C₇₉×#1709

Subgroups — #34201665

Conjugacy classes of subgroups

Complements

Maximal subgroups