Group #34193081 — subgroups

Subgroups — #34193081

Order 20224.

294 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 85 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	#23764220	C₂	1	1	yes	maximal, characteristic
2	#23763541	C₂	1	1	yes	maximal, characteristic
3	#23762071	C₂	1	2	yes	maximal
4	#23761259	C₂	1	2	yes	maximal
5	C₇₉×#1281	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579161	C₂²	1	1	yes	characteristic
7	#19098587	C₂²	1	2	yes	—
8	C₂×#14579086	—	2	2	no	—
9	#19098429	C₂²	1	1	yes	characteristic
10	#19098328	—	2	2	no	—
11	#19098308	—	2	4	no	—
12	#431×C₇₉	C₂²	1	2	yes	—
13	C₂×#97×C₇₉	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₅₈	—	2	1	no	—
15	C₂×#11929608	—	2	2	no	—
16	#14579161	—	2	2	no	—
17	#14579161	—	2	1	no	—
18	#14579161	D₄	1	2	yes	—
19	C₂×C₄⋊Dic₇₉	—	2	2	no	—
20	C₂×C₄⋊Dic₇₉	—	2	1	no	—
21	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
22	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
23	#14579086	—	4	4	no	—
24	C₇₉⋊#95	—	2	2	no	—
25	C₂²×C₈×C₇₉	D₄	1	2	yes	—
26	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
27	C₄⋊C₈×C₇₉	—	2	2	no	—
28	C₂²⋊C₈×C₇₉	—	2	2	no	—
29	#97×C₇₉	—	2	2	no	—
30	C₄×C₈×C₇₉	—	2	2	no	—
31	C₂×Dic₁₅₈	—	2	2	no	—
32	C₂×Dic₁₅₈	—	4	1	no	—
33	C₂×Dic₁₅₈	—	4	2	no	—
34	C₂×Dic₁₅₈	—	4	2	no	—
35	C₂²×Dic₇₉	—	4	1	no	—
36	C₂²×Dic₇₉	—	4	1	no	—
37	#11929608	—	4	2	no	—
38	#11929608	—	4	2	no	—
39	C₂×C₇₉⋊C₈	—	4	2	no	—
40	C₄⋊Dic₇₉	—	2	4	no	—
41	C₄⋊Dic₇₉	—	4	2	no	—
42	C₄⋊Dic₇₉	—	4	2	no	—
43	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₇₉	#41	1	2	yes	—
45	C₂×C₈×C₇₉	—	2	2	no	—
46	C₂×C₈×C₇₉	—	2	2	no	—
47	C₂×C₈×C₇₉	—	2	2	no	—
48	C₂×C₈×C₇₉	—	4	2	no	—
49	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
50	C₄²×C₇₉	#41	1	1	yes	characteristic
51	C₄²×C₇₉	#41	1	2	yes	—
52	C₂³×C₇₉	#124	1	1	yes	characteristic
53	C₂×C₄×C₇₉	—	4	2	no	—
54	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
55	C₂×C₄×C₇₉	#123	1	2	yes	—
56	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
57	C₂×C₄×C₇₉	#126	1	2	yes	—
58	C₂×C₄×C₇₉	#135	1	2	yes	—
59	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
60	C₂×Dic₇₉	—	4	2	no	—
61	C₂×Dic₇₉	—	8	1	no	—
62	C₂×Dic₇₉	—	8	1	no	—
63	C₂×Dic₇₉	—	8	2	no	—
64	C₂×Dic₇₉	—	8	1	no	—
65	Dic₁₅₈	—	4	4	no	—
66	Dic₁₅₈	—	8	2	no	—
67	C₈×C₇₉	—	4	2	no	—
68	C₈×C₇₉	—	8	2	no	—
69	C₇₉⋊C₈	—	8	2	no	—
70	C₂²×C₇₉	—	2	2	no	—
71	C₂²×C₇₉	#397	1	1	yes	characteristic
72	C₂²×C₇₉	#488	1	1	yes	characteristic
73	C₂²×C₇₉	#490	1	1	yes	characteristic
74	C₄×C₇₉	—	2	2	no	—
75	C₄×C₇₉	—	2	2	no	—
76	C₄×C₇₉	#463	1	2	yes	—
77	C₄×C₇₉	#482	1	2	yes	—
78	Dic₇₉	—	8	2	no	—
79	Dic₇₉	—	16	1	no	—
80	#10635	—	79	1	no	maximal, Sylow
81	C₂×C₇₉	—	2	1	no	—
82	C₂×C₇₉	—	2	1	no	—
83	C₂×C₇₉	#2073	1	1	yes	characteristic
84	C₂×C₇₉	#2082	1	2	yes	—
85	#1960	—	79	1	no	—
86	#1838	—	79	1	no	—
87	#1546	—	79	2	no	—
88	#1286	—	79	2	no	—
89	#1281	D₇₉	1	1	yes	characteristic
90	C₇₉	#10635	1	1	yes	characteristic, complemented, Sylow
91	C₂×C₄⋊Q₈	—	79	1	no	—
92	#431	C₂×D₇₉	1	2	yes	—
93	#423	—	79	2	no	—
94	C₂×#103	—	158	2	no	—
95	C₂×#97	C₂×D₇₉	1	1	yes	characteristic
96	#382	—	79	1	no	—
97	#341	—	158	2	no	—
98	#332	—	158	4	no	—
99	C₂²×Q₈	—	158	1	no	—
100	C₂×M₁₆	—	158	2	no	—
101	C₂²×C₈	D₃₁₆	1	2	yes	—
102	C₄⋊Q₈	—	79	2	no	—
103	C₄⋊Q₈	—	158	2	no	—
104	C₄⋊Q₈	—	158	1	no	—
105	C₂×C₄⋊C₄	—	158	1	no	—
106	C₂×C₄⋊C₄	—	158	2	no	—
107	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
108	C₄⋊C₈	—	2	2	no	—
109	C₄⋊C₈	—	158	2	no	—
110	C₄⋊C₈	—	158	2	no	—
111	#103	—	316	4	no	—
112	C₂²⋊C₈	—	2	2	no	—
113	#97	—	2	2	no	—
114	C₄×C₈	—	2	2	no	—
115	#95	—	158	2	no	—
116	C₂×Q₈	—	316	1	no	—
117	C₂×Q₈	—	316	2	no	—
118	C₂×Q₈	—	316	2	no	—
119	C₂×Q₈	—	158	2	no	—
120	C₂²×C₄	—	316	1	no	—
121	C₂²×C₄	—	316	1	no	—
122	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
123	C₂²×C₄	#11929624	1	2	yes	—
124	M₁₆	—	316	2	no	—
125	M₁₆	—	316	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	—	316	2	no	—
131	C₄⋊C₄	—	316	2	no	—
132	C₄⋊C₄	—	316	2	no	—
133	C₄⋊C₄	—	158	4	no	—
134	C₄²	D₄×D₇₉	1	1	yes	characteristic
135	C₄²	#11929623	1	2	yes	—
136	C₄²	#11929625	1	1	yes	characteristic
137	C₂³	#14579149	1	1	yes	characteristic
138	Q₈	—	316	4	no	—
139	Q₈	—	632	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	316	2	no	—
142	C₂×C₄	—	632	1	no	—
143	C₂×C₄	—	632	1	no	—
144	C₂×C₄	—	632	2	no	—
145	C₂×C₄	—	632	1	no	—
146	C₂×C₄	#14579136	1	1	yes	characteristic
147	C₂×C₄	#14579143	1	2	yes	—
148	C₂×C₄	#14579145	1	1	yes	characteristic
149	C₂×C₄	#14579156	1	2	yes	—
150	C₂×C₄	#14579183	1	2	yes	—
151	C₂×C₄	#14579189	1	2	yes	—
152	C₈	—	4	2	no	—
153	C₈	—	8	2	no	—
154	C₈	—	632	2	no	—
155	C₂²	—	2	2	no	—
156	C₂²	#19098490	1	1	yes	characteristic
157	C₂²	#19098918	1	1	yes	characteristic
158	C₂²	#19098928	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	—	632	2	no	—
162	C₄	—	1264	1	no	—
163	C₄	#19098779	1	2	yes	—
164	C₄	#19098888	1	2	yes	—
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	#23764815	1	1	yes	characteristic
168	C₂	#23764866	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23764220

C₂

yes

maximal, characteristic

#23763541

C₂

yes

maximal, characteristic

#23762071

C₂

yes

maximal

#23761259

C₂

yes

maximal

C₇₉×#1281

C₂

yes

maximal, characteristic

C₂×#14579161

C₂²

yes

characteristic

#19098587

C₂²

yes

—

C₂×#14579086

—

#19098429

C₂²

yes

characteristic

#19098328

—

#19098308

—

#431×C₇₉

C₂²

yes

—

C₂×#97×C₇₉

C₂²

yes

characteristic

C₂²×Dic₁₅₈

—

—

—

—

D₄

yes

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

#14579086

—

C₇₉⋊#95

—

C₂²×C₈×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄⋊C₈×C₇₉

—

C₂²⋊C₈×C₇₉

—

#97×C₇₉

—

C₄×C₈×C₇₉

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

#41

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄²×C₇₉

C₂×D₄

yes

C₄²×C₇₉

yes

C₄²×C₇₉

yes

—

C₂³×C₇₉

#124

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#121

yes

characteristic

C₂×C₄×C₇₉

#123

yes

—

C₂×C₄×C₇₉

#123

yes

characteristic

C₂×C₄×C₇₉

#126

yes

—

C₂×C₄×C₇₉

#135

yes

—

C₂×C₄×C₇₉

AGL(1,ℤ/8)

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#463

yes

—

C₄×C₇₉

#482

yes

—

Dic₇₉

—

Dic₇₉

—

#10635

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#2073

yes

characteristic

C₂×C₇₉

#2082

yes

—

—

—

—

—

D₇₉

yes

characteristic

C₇₉

#10635

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

#431

C₂×D₇₉

yes

—

#423

—

C₂×#103

—

158

—

C₂×#97

C₂×D₇₉

yes

characteristic

#382

—

#341

—

158

—

#332

—

158

—

C₂²×Q₈

—

158

—

100

C₂×M₁₆

—

158

—

101

C₂²×C₈

D₃₁₆

yes

—

102

C₄⋊Q₈

—

103

C₄⋊Q₈

—

158

—

104

C₄⋊Q₈

—

158

—

105

C₂×C₄⋊C₄

—

158

—

106

C₂×C₄⋊C₄

—

158

—

107

C₂×C₄²

C₂²×D₇₉

yes

characteristic

108

C₄⋊C₈

—

109

C₄⋊C₈

—

158

—

110

C₄⋊C₈

—

158

—

111

#103

—

316

—

112

C₂²⋊C₈

—

113

#97

—

114

C₄×C₈

—

115

#95

—

158

—

116

C₂×Q₈

—

316

—

117

C₂×Q₈

—

316

—

118

C₂×Q₈

—

316

—

119

C₂×Q₈

—

158

—

120

C₂²×C₄

—

316

—

121

C₂²×C₄

—

316

—

122

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

123

C₂²×C₄

#11929624

yes

—

124

M₁₆

—

316

—

125

M₁₆

—

316

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

316

—

131

C₄⋊C₄

—

316

—

132

C₄⋊C₄

—

316

—

133

C₄⋊C₄

—

158

—

134

C₄²

D₄×D₇₉

yes

characteristic

135

C₄²

#11929623

yes

—

136

C₄²

#11929625

yes

characteristic

137

C₂³

#14579149

yes

characteristic

138

Q₈

—

316

—

139

Q₈

—

632

—

140

C₂×C₄

—

141

C₂×C₄

—

316

—

142

C₂×C₄

—

632

—

143

C₂×C₄

—

632

—

144

C₂×C₄

—

632

—

145

C₂×C₄

—

632

—

146

C₂×C₄

#14579136

yes

characteristic

147

C₂×C₄

#14579143

yes

—

148

C₂×C₄

#14579145

yes

characteristic

149

C₂×C₄

#14579156

yes

—

150

C₂×C₄

#14579183

yes

—

151

C₂×C₄

#14579189

yes

—

152

C₈

—

153

C₈

—

154

C₈

—

632

—

155

C₂²

—

156

C₂²

#19098490

yes

characteristic

157

C₂²

#19098918

yes

characteristic

158

C₂²

#19098928

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

632

—

162

C₄

—

1264

—

163

C₄

#19098779

yes

—

164

C₄

#19098888

yes

—

165

C₂

—

166

C₂

—

167

C₂

#23764815

yes

characteristic

168

C₂

#23764866

yes

—

	subgroup	G/H
center	C₂²	#19098490
commutator, cosocle	C₂×C₄²×C₇₉	C₂³
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1281	C₂
socle	C₂²×C₇₉	#397
prime radical	C₂³×C₇₉	#124
hypercenter	#1281	D₇₉
hypocenter	C₇₉	#10635
LCS	G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1281

subgroup

G/H

center

C₂²

#19098490

commutator, cosocle

C₂×C₄²×C₇₉

C₂³

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1281

C₂

C₂²×C₇₉

C₂³×C₇₉

D₇₉

C₇₉

G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1281

N	H ≅ G/N	type
90. C₇₉	#10635	semidirect

H ≅ G/N

type

90. C₇₉

#10635

semidirect

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

subgroup	index
#23764220	2
#23763541	2
#23762071	2
#23761259	2
C₇₉×#1281	2
#10635	79

subgroup

index

C₇₉×#1281

Subgroups — #34193081

Conjugacy classes of subgroups

Complements

Maximal subgroups