Group #34201672 — subgroups

Subgroups — #34201672

Order 20224.

372 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 67 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	#23765481	C₂	1	2	yes	maximal, complemented
2	#23763005	C₂	1	1	yes	maximal, characteristic, complemented
3	#23761437	C₂	1	1	yes	maximal, characteristic, complemented
4	#23761373	C₂	1	2	yes	maximal, complemented
5	C₇₉×#3292	C₂	1	1	yes	maximal, characteristic
6	#19099003	—	2	2	no	—
7	#19098943	C₂²	1	1	yes	characteristic, complemented
8	#19098909	—	2	2	no	—
9	#19098384	C₂²	1	2	yes	complemented
10	#19098378	—	2	2	no	—
11	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic, complemented
12	#576×C₇₉	—	2	2	no	—
13	#493×C₇₉	C₂²	1	1	yes	characteristic
14	#472×C₇₉	—	2	2	no	—
15	#453×C₇₉	—	2	2	no	—
16	#443×C₇₉	C₂²	1	2	yes	—
17	C₂×Dic₃₁₆	—	2	2	no	—
18	#14579161	—	2	1	no	—
19	#14579107	—	4	2	no	—
20	#14579105	—	4	2	no	—
21	#14579101	—	2	2	no	—
22	#14579082	—	4	4	no	—
23	C₄×C₇₉⋊C₈	—	2	1	no	—
24	H(5,𝔽₂)×C₇₉	—	2	2	no	—
25	AGL(1,ℤ/8)×C₇₉	—	2	4	no	—
26	AGL(1,ℤ/8)×C₇₉	—	4	4	no	—
27	#135×C₇₉	—	4	2	no	—
28	#135×C₇₉	D₄	1	2	yes	—
29	C₂×D₈×C₇₉	—	2	2	no	—
30	C₂×D₈×C₇₉	—	4	2	no	—
31	#131×C₇₉	—	2	2	no	—
32	#127×C₇₉	D₄	1	2	yes	—
33	#108×C₇₉	D₄	1	2	yes	—
34	#104×C₇₉	—	2	2	no	—
35	#100×C₇₉	—	2	2	no	—
36	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
37	C₂×Dic₁₅₈	—	4	1	no	—
38	Dic₃₁₆	—	4	2	no	—
39	Dic₃₁₆	—	4	2	no	—
40	C₂×C₇₉⋊C₈	—	2	2	no	—
41	C₄⋊Dic₇₉	—	4	2	no	—
42	C₇₉⋊C₁₆	—	4	2	no	—
43	#41×C₇₉	—	2	2	no	—
44	#41×C₇₉	—	4	4	no	—
45	C₂×D₄×C₇₉	—	2	2	no	—
46	C₂×D₄×C₇₉	—	4	2	no	—
47	C₂×D₄×C₇₉	—	4	4	no	—
48	C₂×D₄×C₇₉	—	8	2	no	—
49	Q₁₆×C₇₉	D₈	1	2	yes	—
50	SD₁₆×C₇₉	—	2	2	no	—
51	SD₁₆×C₇₉	—	4	4	no	—
52	D₈×C₇₉	—	4	4	no	—
53	D₈×C₇₉	—	4	2	no	—
54	D₈×C₇₉	—	4	2	no	—
55	D₈×C₇₉	D₈	1	2	yes	—
56	M₁₆×C₇₉	—	2	2	no	—
57	M₁₆×C₇₉	—	4	2	no	—
58	C₂×C₈×C₇₉	—	4	2	no	—
59	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
60	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
61	C₂³×C₇₉	—	4	2	no	—
62	C₂³×C₇₉	—	8	2	no	—
63	Q₈×C₇₉	—	2	2	no	—
64	D₄×C₇₉	—	2	2	no	—
65	D₄×C₇₉	—	4	2	no	—
66	D₄×C₇₉	—	4	4	no	—
67	D₄×C₇₉	—	8	4	no	—
68	D₄×C₇₉	—	8	4	no	—
69	C₂×C₄×C₇₉	—	2	1	no	—
70	C₂×C₄×C₇₉	—	4	2	no	—
71	C₂×C₄×C₇₉	—	8	2	no	—
72	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
73	C₂×Dic₇₉	—	8	1	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	—	4	2	no	—
80	C₂²×C₇₉	—	8	2	no	—
81	C₂²×C₇₉	—	16	2	no	—
82	C₂²×C₇₉	—	32	2	no	—
83	C₂²×C₇₉	#453	1	1	yes	characteristic
84	C₄×C₇₉	—	2	2	no	—
85	C₄×C₇₉	—	4	2	no	—
86	C₄×C₇₉	#447	1	2	yes	—
87	Dic₇₉	—	16	1	no	—
88	#12312	—	79	1	no	maximal, Sylow
89	C₂×C₇₉	—	2	1	no	—
90	C₂×C₇₉	—	8	2	no	—
91	C₂×C₇₉	—	16	2	no	—
92	C₂×C₇₉	#1619	1	1	yes	characteristic
93	#3292	D₇₉	1	1	yes	characteristic
94	#2214	—	79	2	no	—
95	#1709	—	79	1	no	—
96	#1357	—	79	1	no	—
97	#1335	—	79	2	no	—
98	C₇₉	#12312	1	1	yes	characteristic, complemented, Sylow
99	#576	—	2	2	no	—
100	#510	—	158	2	no	—
101	#494	—	79	1	no	—
102	#493	C₂×D₇₉	1	1	yes	characteristic
103	#486	—	158	2	no	—
104	#472	—	2	2	no	—
105	#453	—	2	2	no	—
106	#443	C₂×D₇₉	1	2	yes	—
107	#364	—	79	2	no	—
108	#362	—	158	2	no	—
109	C₈²	—	79	1	no	—
110	H(5,𝔽₂)	—	2	2	no	—
111	AGL(1,ℤ/8)	—	2	4	no	—
112	AGL(1,ℤ/8)	—	4	4	no	—
113	#135	—	4	2	no	—
114	#135	C₂²⋊D₇₉	1	2	yes	—
115	C₂×Q₁₆	—	158	2	no	—
116	C₂×D₈	—	2	2	no	—
117	C₂×D₈	—	4	2	no	—
118	#131	—	2	2	no	—
119	C₄⋊Q₈	—	158	1	no	—
120	#127	C₂²⋊D₇₉	1	2	yes	—
121	Q₃₂	—	316	2	no	—
122	SD₃₂	—	316	2	no	—
123	M₃₂	—	158	2	no	—
124	#108	C₂²⋊D₇₉	1	2	yes	—
125	#104	—	2	2	no	—
126	#102	—	316	4	no	—
127	#100	—	2	2	no	—
128	C₄×C₈	—	158	1	no	—
129	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
130	#41	—	2	2	no	—
131	#41	—	4	4	no	—
132	C₂×Q₈	—	316	1	no	—
133	C₂×D₄	—	2	2	no	—
134	C₂×D₄	—	4	2	no	—
135	C₂×D₄	—	4	4	no	—
136	C₂×D₄	—	8	2	no	—
137	Q₁₆	—	316	2	no	—
138	Q₁₆	—	316	2	no	—
139	Q₁₆	#11929609	1	2	yes	—
140	SD₁₆	—	2	2	no	—
141	SD₁₆	—	4	4	no	—
142	D₈	—	4	4	no	—
143	D₈	—	4	2	no	—
144	D₈	—	4	2	no	—
145	D₈	#11929609	1	2	yes	—
146	M₁₆	—	2	2	no	—
147	M₁₆	—	4	2	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	158	2	no	—
150	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
151	C₄⋊C₄	—	316	2	no	—
152	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
153	C₁₆	—	316	2	no	—
154	C₂³	—	4	2	no	—
155	C₂³	—	8	2	no	—
156	Q₈	—	2	2	no	—
157	Q₈	—	632	2	no	—
158	D₄	—	2	2	no	—
159	D₄	—	4	2	no	—
160	D₄	—	4	4	no	—
161	D₄	—	8	4	no	—
162	D₄	—	8	4	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	—	632	1	no	—
167	C₂×C₄	#14579130	1	1	yes	characteristic
168	C₈	—	4	2	no	—
169	C₈	—	316	2	no	—
170	C₈	C₂×#11929609	1	2	yes	—
171	C₈	#14579187	1	2	yes	—
172	C₂²	—	4	2	no	—
173	C₂²	—	8	2	no	—
174	C₂²	—	16	2	no	—
175	C₂²	—	32	2	no	—
176	C₂²	#19098731	1	1	yes	characteristic
177	C₄	—	2	2	no	—
178	C₄	—	4	2	no	—
179	C₄	—	1264	1	no	—
180	C₄	#19098697	1	2	yes	—
181	C₂	—	2	1	no	—
182	C₂	—	8	2	no	—
183	C₂	—	16	2	no	—
184	C₂	#23762454	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765481

C₂

yes

maximal, complemented

#23763005

C₂

yes

maximal, characteristic, complemented

#23761437

C₂

yes

maximal, characteristic, complemented

#23761373

C₂

yes

maximal, complemented

C₇₉×#3292

C₂

yes

maximal, characteristic

#19099003

—

#19098943

C₂²

yes

characteristic, complemented

#19098909

—

#19098384

C₂²

yes

complemented

#19098378

—

C₈×C₇₉⋊C₈

C₂²

yes

characteristic, complemented

#576×C₇₉

—

#493×C₇₉

C₂²

yes

characteristic

#472×C₇₉

—

#453×C₇₉

—

#443×C₇₉

C₂²

yes

—

C₂×Dic₃₁₆

—

—

—

—

—

—

C₄×C₇₉⋊C₈

—

H(5,𝔽₂)×C₇₉

—

AGL(1,ℤ/8)×C₇₉

—

AGL(1,ℤ/8)×C₇₉

—

#135×C₇₉

—

#135×C₇₉

D₄

yes

—

C₂×D₈×C₇₉

—

C₂×D₈×C₇₉

—

#131×C₇₉

—

#127×C₇₉

D₄

yes

—

#108×C₇₉

D₄

yes

—

#104×C₇₉

—

#100×C₇₉

—

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×Dic₁₅₈

—

Dic₃₁₆

—

Dic₃₁₆

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₇₉⋊C₁₆

—

#41×C₇₉

—

#41×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

Q₁₆×C₇₉

D₈

yes

—

SD₁₆×C₇₉

—

SD₁₆×C₇₉

—

D₈×C₇₉

—

D₈×C₇₉

—

D₈×C₇₉

—

D₈×C₇₉

D₈

yes

—

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂×D₄

yes

—

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₂³×C₇₉

—

C₂³×C₇₉

—

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#120

yes

characteristic

C₂×Dic₇₉

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

C₂×D₈

yes

—

C₈×C₇₉

AGL(1,ℤ/8)

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#453

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#447

yes

—

Dic₇₉

—

#12312

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#1619

yes

characteristic

#3292

D₇₉

yes

—

—

—

—

C₇₉

#12312

yes

characteristic, complemented, Sylow

#576

—

100

#510

—

158

—

101

#494

—

102

#493

C₂×D₇₉

yes

characteristic

103

#486

—

158

—

104

#472

—

105

#453

—

106

#443

C₂×D₇₉

yes

—

107

#364

—

108

#362

—

158

—

109

C₈²

—

110

H(5,𝔽₂)

—

111

AGL(1,ℤ/8)

—

112

AGL(1,ℤ/8)

—

113

#135

—

114

#135

C₂²⋊D₇₉

yes

—

115

C₂×Q₁₆

—

158

—

116

C₂×D₈

—

117

C₂×D₈

—

118

#131

—

119

C₄⋊Q₈

—

158

—

120

#127

C₂²⋊D₇₉

yes

—

121

Q₃₂

—

316

—

122

SD₃₂

—

316

—

123

M₃₂

—

158

—

124

#108

C₂²⋊D₇₉

yes

—

125

#104

—

126

#102

—

316

—

127

#100

—

128

C₄×C₈

—

158

—

129

C₄×C₈

C₂²×D₇₉

yes

characteristic

130

#41

—

131

#41

—

132

C₂×Q₈

—

316

—

133

C₂×D₄

—

134

C₂×D₄

—

135

C₂×D₄

—

136

C₂×D₄

—

137

Q₁₆

—

316

—

138

Q₁₆

—

316

—

139

Q₁₆

#11929609

yes

—

140

SD₁₆

—

141

SD₁₆

—

142

D₈

—

143

D₈

—

144

D₈

—

145

D₈

#11929609

yes

—

146

M₁₆

—

147

M₁₆

—

148

C₂×C₈

—

149

C₂×C₈

—

158

—

150

C₂×C₈

C₂×C₂²⋊D₇₉

yes

—

151

C₄⋊C₄

—

316

—

152

C₄²

C₂×C₂²⋊D₇₉

yes

characteristic

153

C₁₆

—

316

—

154

C₂³

—

155

C₂³

—

156

Q₈

—

157

Q₈

—

632

—

158

D₄

—

159

D₄

—

160

D₄

—

161

D₄

—

162

D₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

632

—

167

C₂×C₄

#14579130

yes

characteristic

168

C₈

—

169

C₈

—

316

—

170

C₈

C₂×#11929609

yes

—

171

C₈

#14579187

yes

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

—

176

C₂²

#19098731

yes

characteristic

177

C₄

—

178

C₄

—

179

C₄

—

1264

—

180

C₄

#19098697

yes

—

181

C₂

—

182

C₂

—

183

C₂

—

184

C₂

#23762454

yes

characteristic

	subgroup	G/H
center	C₂	#23762454
commutator, cosocle	C₄×C₈×C₇₉	C₂³
Frattini	C₄×C₈	C₂²×D₇₉
Fitting, prime radical	C₇₉×#3292	C₂
socle	C₂×C₇₉	#1619
hypercenter	#3292	D₇₉
hypocenter	C₇₉	#12312
LCS	G ▷ C₄×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #3292

subgroup

G/H

center

C₂

#23762454

commutator, cosocle

C₄×C₈×C₇₉

C₂³

Frattini

C₄×C₈

C₂²×D₇₉

Fitting, prime radical

C₇₉×#3292

C₂

C₂×C₇₉

D₇₉

C₇₉

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

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #3292

N	H ≅ G/N	type
1. #23765481	C₂	semidirect
2. #23763005	C₂	semidirect
3. #23761437	C₂	semidirect
4. #23761373	C₂	semidirect
7. #19098943	C₂²	semidirect
9. #19098384	C₂²	semidirect
11. C₈×C₇₉⋊C₈	C₂²	semidirect
98. C₇₉	#12312	semidirect

H ≅ G/N

type

1. #23765481

C₂

semidirect

2. #23763005

C₂

semidirect

3. #23761437

C₂

semidirect

4. #23761373

C₂

semidirect

7. #19098943

C₂²

semidirect

9. #19098384

C₂²

semidirect

11. C₈×C₇₉⋊C₈

C₂²

semidirect

98. C₇₉

#12312

semidirect

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

subgroup	index
#23765481	2
#23763005	2
#23761437	2
#23761373	2
C₇₉×#3292	2
#12312	79

subgroup

index

C₇₉×#3292

Subgroups — #34201672

Conjugacy classes of subgroups

Complements

Maximal subgroups