Group #34181946 — subgroups

Subgroups — #34181946

Order 20224.

342 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 155 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	#23763552	C₂	1	1	yes	maximal, characteristic
2	#23763517	C₂	1	1	yes	maximal, characteristic
3	#23761235	C₂	1	4	yes	maximal
4	C₇₉×#1744	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
6	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
8	#19098318	—	2	2	no	—
9	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
10	C₄³×C₇₉	C₂²	1	1	yes	characteristic
11	#336×C₇₉	C₄	1	2	yes	—
12	#336×C₇₉	C₄	1	2	yes	—
13	C₂²×C₇₉⋊C₈	—	2	2	no	—
14	C₂²×C₇₉⋊C₈	—	2	2	no	—
15	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
16	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
17	C₄⋊(C₇₉⋊C₈)	—	4	4	no	—
18	C₂²×C₈×C₇₉	C₂×C₄	1	4	yes	—
19	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₇₉	D₄	1	2	yes	—
22	C₂×C₄²×C₇₉	Q₈	1	2	yes	—
23	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
24	C₄×C₈×C₇₉	—	2	2	no	—
25	C₄×C₈×C₇₉	—	2	2	no	—
26	C₂×C₇₉⋊C₈	—	4	2	no	—
27	C₂×C₇₉⋊C₈	—	4	2	no	—
28	C₂×C₇₉⋊C₈	—	4	4	no	—
29	C₂×C₇₉⋊C₈	—	4	2	no	—
30	C₂×C₇₉⋊C₈	—	4	2	no	—
31	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
32	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
33	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₈×C₇₉	—	2	4	no	—
37	C₂×C₈×C₇₉	—	2	4	no	—
38	C₂×C₈×C₇₉	—	2	4	no	—
39	C₄²×C₇₉	—	2	4	no	—
40	C₄²×C₇₉	—	2	2	no	—
41	C₄²×C₇₉	—	2	2	no	—
42	C₄²×C₇₉	—	2	2	no	—
43	C₄²×C₇₉	—	2	2	no	—
44	C₄²×C₇₉	#41	1	2	yes	—
45	C₄²×C₇₉	#41	1	2	yes	—
46	C₂³×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₇₉	—	2	2	no	—
48	C₂×C₄×C₇₉	—	2	2	no	—
49	C₂×C₄×C₇₉	—	2	2	no	—
50	C₂×C₄×C₇₉	—	2	2	no	—
51	C₂×C₄×C₇₉	—	2	2	no	—
52	C₂×C₄×C₇₉	—	2	2	no	—
53	C₂×C₄×C₇₉	#108	1	4	yes	—
54	C₂×C₄×C₇₉	#117	1	2	yes	—
55	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
56	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
57	C₂×C₄×C₇₉	#125	1	2	yes	—
58	C₂×C₄×C₇₉	#131	1	2	yes	—
59	C₂×C₄×C₇₉	#131	1	2	yes	—
60	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
61	C₂×C₄×C₇₉	#137	1	2	yes	—
62	C₈×C₇₉	—	4	4	no	—
63	C₇₉⋊C₈	—	8	2	no	—
64	C₇₉⋊C₈	—	8	2	no	—
65	C₂²×C₇₉	#382	1	1	yes	characteristic
66	C₂²×C₇₉	#424	1	1	yes	characteristic
67	C₂²×C₇₉	#424	1	1	yes	characteristic
68	C₂²×C₇₉	#428	1	2	yes	—
69	C₂²×C₇₉	C₂×#108	1	2	yes	—
70	C₄×C₇₉	—	2	4	no	—
71	C₄×C₇₉	—	2	2	no	—
72	C₄×C₇₉	—	4	2	no	—
73	C₄×C₇₉	#433	1	4	yes	—
74	C₄×C₇₉	#489	1	2	yes	—
75	C₄×C₇₉	#490	1	1	yes	characteristic
76	C₄×C₇₉	#491	1	1	yes	characteristic
77	#8789	—	79	1	no	maximal, Sylow
78	C₂×C₇₉	#1573	1	2	yes	—
79	C₂×C₇₉	#1574	1	2	yes	—
80	C₂×C₇₉	#1832	1	1	yes	characteristic
81	C₂×C₇₉	#1836	1	1	yes	characteristic
82	C₂×C₇₉	#1837	1	1	yes	characteristic
83	#1842	—	79	1	no	—
84	#1831	—	79	1	no	—
85	#1744	D₇₉	1	1	yes	characteristic
86	#1277	—	79	4	no	—
87	C₇₉	#8789	1	1	yes	characteristic, complemented, Sylow
88	C₂×C₄⋊C₈	—	79	2	no	—
89	C₂×C₄⋊C₈	—	79	2	no	—
90	C₂×C₄⋊C₈	—	158	2	no	—
91	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
92	C₄³	C₂×D₇₉	1	1	yes	characteristic
93	#336	—	158	2	no	—
94	#336	Dic₇₉	1	2	yes	—
95	#336	Dic₇₉	1	2	yes	—
96	C₂²×C₈	—	158	2	no	—
97	C₂²×C₈	—	158	2	no	—
98	C₂²×C₈	C₂×Dic₇₉	1	4	yes	—
99	C₂×C₄²	Dic₁₅₈	1	2	yes	—
100	C₂×C₄²	D₃₁₆	1	2	yes	—
101	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
102	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
103	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
104	C₄⋊C₈	—	316	4	no	—
105	C₄⋊C₈	—	158	4	no	—
106	C₄⋊C₈	—	158	4	no	—
107	C₄×C₈	—	2	2	no	—
108	C₄×C₈	—	2	2	no	—
109	C₂²×C₄	C₄⋊Dic₇₉	1	2	yes	—
110	C₂²×C₄	C₄⋊Dic₇₉	1	2	yes	—
111	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
112	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
113	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	2	4	no	—
116	C₂×C₈	—	2	4	no	—
117	C₂×C₈	—	316	2	no	—
118	C₂×C₈	—	316	2	no	—
119	C₂×C₈	—	316	4	no	—
120	C₂×C₈	—	316	2	no	—
121	C₂×C₈	—	316	2	no	—
122	C₄²	—	2	4	no	—
123	C₄²	—	2	2	no	—
124	C₄²	—	2	2	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	2	2	no	—
127	C₄²	#11929624	1	2	yes	—
128	C₄²	#11929624	1	2	yes	—
129	C₂³	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	#14579097	1	4	yes	—
137	C₂×C₄	#14579120	1	2	yes	—
138	C₂×C₄	#14579149	1	1	yes	characteristic
139	C₂×C₄	#14579149	1	1	yes	characteristic
140	C₂×C₄	#14579153	1	2	yes	—
141	C₂×C₄	#14579168	1	2	yes	—
142	C₂×C₄	#14579168	1	2	yes	—
143	C₂×C₄	#14579189	1	2	yes	—
144	C₂×C₄	#14579194	1	2	yes	—
145	C₈	—	4	4	no	—
146	C₈	—	632	2	no	—
147	C₈	—	632	2	no	—
148	C₂²	#19098429	1	1	yes	characteristic
149	C₂²	#19098590	1	1	yes	characteristic
150	C₂²	#19098590	1	1	yes	characteristic
151	C₂²	#19098605	1	2	yes	—
152	C₂²	C₂×#14579097	1	2	yes	—
153	C₄	—	2	4	no	—
154	C₄	—	2	2	no	—
155	C₄	—	4	2	no	—
156	C₄	#19098624	1	4	yes	—
157	C₄	#19098923	1	2	yes	—
158	C₄	#19098928	1	1	yes	characteristic
159	C₄	#19098932	1	1	yes	characteristic
160	C₂	#23762185	1	2	yes	—
161	C₂	#23762189	1	2	yes	—
162	C₂	#23763520	1	1	yes	characteristic
163	C₂	#23763534	1	1	yes	characteristic
164	C₂	#23763538	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23763552

C₂

yes

maximal, characteristic

#23763517

C₂

yes

maximal, characteristic

#23761235

C₂

yes

maximal

C₇₉×#1744

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₇₉⋊C₈)

—

C₂×C₄⋊(C₇₉⋊C₈)

C₂²

yes

—

C₂×C₄⋊(C₇₉⋊C₈)

C₂²

yes

—

#19098318

—

C₂×C₄×C₈×C₇₉

C₂²

yes

—

C₄³×C₇₉

C₂²

yes

characteristic

#336×C₇₉

C₄

yes

—

#336×C₇₉

C₄

yes

—

C₂²×C₇₉⋊C₈

—

C₂²×C₇₉⋊C₈

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₂²×C₈×C₇₉

C₂×C₄

yes

—

C₂×C₄²×C₇₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

Q₈

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄×C₈×C₇₉

—

C₄×C₈×C₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂²×C₄×C₇₉

C₄⋊C₄

yes

—

C₂²×C₄×C₇₉

C₄⋊C₄

yes

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

#41

yes

—

C₄²×C₇₉

#41

yes

—

C₂³×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#108

yes

—

C₂×C₄×C₇₉

#117

yes

—

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

—

C₂×C₄×C₇₉

#131

yes

—

C₂×C₄×C₇₉

#131

yes

—

C₂×C₄×C₇₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇₉

#137

yes

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

—

C₂²×C₇₉

C₂×#108

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#433

yes

—

C₄×C₇₉

#489

yes

—

C₄×C₇₉

yes

C₄×C₇₉

yes

—

C₂×C₇₉

yes

—

C₂×C₇₉

#1574

yes

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

—

D₇₉

yes

characteristic

#1277

—

C₇₉

#8789

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

158

—

C₂×C₄×C₈

C₂×D₇₉

yes

—

C₄³

C₂×D₇₉

yes

characteristic

#336

—

158

—

#336

Dic₇₉

yes

—

#336

Dic₇₉

yes

—

C₂²×C₈

—

158

—

C₂²×C₈

—

158

—

C₂²×C₈

C₂×Dic₇₉

yes

—

C₂×C₄²

Dic₁₅₈

yes

—

100

C₂×C₄²

D₃₁₆

yes

—

101

C₂×C₄²

C₂×Dic₇₉

yes

characteristic

102

C₂×C₄²

C₂×Dic₇₉

yes

characteristic

103

C₂×C₄²

C₂²×D₇₉

yes

characteristic

104

C₄⋊C₈

—

316

—

105

C₄⋊C₈

—

158

—

106

C₄⋊C₈

—

158

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₂²×C₄

C₄⋊Dic₇₉

yes

—

110

C₂²×C₄

C₄⋊Dic₇₉

yes

—

111

C₂²×C₄

C₂²×Dic₇₉

yes

characteristic

112

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

113

C₂²×C₄

C₂×Dic₁₅₈

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

316

—

118

C₂×C₈

—

316

—

119

C₂×C₈

—

316

—

120

C₂×C₈

—

316

—

121

C₂×C₈

—

316

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

#11929624

yes

—

128

C₄²

#11929624

yes

—

129

C₂³

C₂×C₄⋊Dic₇₉

yes

characteristic

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

#14579097

yes

—

137

C₂×C₄

#14579120

yes

—

138

C₂×C₄

#14579149

yes

characteristic

139

C₂×C₄

#14579149

yes

characteristic

140

C₂×C₄

#14579153

yes

—

141

C₂×C₄

#14579168

yes

—

142

C₂×C₄

#14579168

yes

—

143

C₂×C₄

#14579189

yes

—

144

C₂×C₄

#14579194

yes

—

145

C₈

—

146

C₈

—

632

—

147

C₈

—

632

—

148

C₂²

#19098429

yes

characteristic

149

C₂²

#19098590

yes

characteristic

150

C₂²

#19098590

yes

characteristic

151

C₂²

#19098605

yes

—

152

C₂²

C₂×#14579097

yes

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

#19098624

yes

—

157

C₄

#19098923

yes

—

158

C₄

#19098928

yes

characteristic

159

C₄

#19098932

yes

characteristic

160

C₂

#23762185

yes

—

161

C₂

#23762189

yes

—

162

C₂

#23763520

yes

characteristic

163

C₂

#23763534

yes

characteristic

164

C₂

#23763538

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₁₆
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1744	C₂
socle, prime radical	C₂³×C₇₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1744	D₇₉
hypocenter	C₇₉	#8789
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

subgroup

G/H

center

C₂²×C₄

C₂×D₃₁₆

commutator

C₂²×C₄×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1744

C₂

socle, prime radical

C₂³×C₇₉

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

N	H ≅ G/N	type
87. C₇₉	#8789	semidirect

H ≅ G/N

type

87. C₇₉

#8789

semidirect

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

subgroup	index
#23763552	2
#23763517	2
#23761235	2
C₇₉×#1744	2
#8789	79

subgroup

index

C₇₉×#1744

Subgroups — #34181946

Conjugacy classes of subgroups

Complements

Maximal subgroups