Group #34181920 — subgroups

Subgroups — #34181920

Order 20224.

262 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 99 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	#23763153	C₂	1	1	yes	maximal, characteristic
2	#23762146	C₂	1	2	yes	maximal
3	#23761245	C₂	1	1	yes	maximal, characteristic
4	#23761228	C₂	1	2	yes	maximal
5	C₇₉×#1281	C₂	1	1	yes	maximal, characteristic
6	#19098616	C₂²	1	2	yes	—
7	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
8	#19098409	C₂²	1	1	yes	characteristic
9	#19098317	C₄	1	2	yes	—
10	#19098317	C₄	1	2	yes	—
11	#19098315	—	2	2	no	—
12	#19098313	—	2	2	no	—
13	#431×C₇₉	C₂²	1	2	yes	—
14	C₂×#97×C₇₉	C₂²	1	1	yes	characteristic
15	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
16	C₂×C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
18	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
19	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
20	C₄×C₇₉⋊C₈	—	2	2	no	—
21	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
22	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
23	C₄×C₇₉⋊C₈	Q₈	1	2	yes	—
24	C₇₉⋊#95	—	2	2	no	—
25	C₂²×C₈×C₇₉	C₂×C₄	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	C₂²×Dic₇₉	—	2	1	no	—
32	C₂²×Dic₇₉	—	2	1	no	—
33	C₂×C₇₉⋊C₈	—	2	2	no	—
34	C₂×C₇₉⋊C₈	—	2	2	no	—
35	C₂×C₇₉⋊C₈	—	4	2	no	—
36	C₂×C₇₉⋊C₈	C₄⋊C₄	1	2	yes	—
37	C₂×C₇₉⋊C₈	C₄⋊C₄	1	2	yes	—
38	C₄×Dic₇₉	—	2	2	no	—
39	C₄×Dic₇₉	—	2	2	no	—
40	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₇₉	#41	1	2	yes	—
42	C₂×C₈×C₇₉	—	2	2	no	—
43	C₂×C₈×C₇₉	—	2	2	no	—
44	C₂×C₈×C₇₉	—	2	2	no	—
45	C₂×C₈×C₇₉	—	4	2	no	—
46	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
48	C₄²×C₇₉	#41	1	2	yes	—
49	C₂³×C₇₉	#117	1	1	yes	characteristic
50	C₂×C₄×C₇₉	—	4	2	no	—
51	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
53	C₂×C₄×C₇₉	#124	1	2	yes	—
54	C₂×C₄×C₇₉	#125	1	2	yes	—
55	C₂×C₄×C₇₉	#131	1	2	yes	—
56	C₂×C₄×C₇₉	#131	1	2	yes	—
57	C₂×Dic₇₉	—	4	2	no	—
58	C₂×Dic₇₉	—	4	2	no	—
59	C₂×Dic₇₉	—	4	1	no	—
60	C₂×Dic₇₉	—	4	1	no	—
61	C₈×C₇₉	—	4	2	no	—
62	C₈×C₇₉	—	8	2	no	—
63	C₇₉⋊C₈	—	2	2	no	—
64	C₇₉⋊C₈	—	2	2	no	—
65	C₇₉⋊C₈	—	8	2	no	—
66	C₂²×C₇₉	—	2	2	no	—
67	C₂²×C₇₉	#382	1	1	yes	characteristic
68	C₂²×C₇₉	#433	1	1	yes	characteristic
69	C₂²×C₇₉	#433	1	1	yes	characteristic
70	C₄×C₇₉	—	2	2	no	—
71	C₄×C₇₉	—	2	2	no	—
72	C₄×C₇₉	#424	1	2	yes	—
73	C₄×C₇₉	#433	1	2	yes	—
74	Dic₇₉	—	8	1	no	—
75	Dic₇₉	—	8	1	no	—
76	#8782	—	79	1	no	maximal, Sylow
77	C₂×C₇₉	—	2	1	no	—
78	C₂×C₇₉	—	2	1	no	—
79	C₂×C₇₉	#1832	1	1	yes	characteristic
80	C₂×C₇₉	#1833	1	2	yes	—
81	#1746	—	79	1	no	—
82	#1564	—	79	2	no	—
83	#1281	—	79	1	no	—
84	#1281	D₇₉	1	1	yes	characteristic
85	#1274	—	79	2	no	—
86	C₇₉	#8782	1	1	yes	characteristic, complemented, Sylow
87	#431	—	79	2	no	—
88	#431	C₂×D₇₉	1	2	yes	—
89	C₂×#97	C₂×D₇₉	1	1	yes	characteristic
90	C₂×C₄×C₈	—	79	1	no	—
91	#376	—	79	1	no	—
92	#336	—	79	2	no	—
93	#336	—	79	2	no	—
94	#335	—	158	2	no	—
95	#334	—	158	2	no	—
96	C₂²×C₈	—	79	2	no	—
97	C₂²×C₈	C₄×D₇₉	1	2	yes	—
98	C₂×C₄²	—	79	1	no	—
99	C₂×C₄²	—	79	1	no	—
100	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
101	C₄⋊C₈	—	2	2	no	—
102	C₄⋊C₈	—	158	2	no	—
103	C₂²⋊C₈	—	2	2	no	—
104	C₂²⋊C₈	—	158	2	no	—
105	#97	—	2	2	no	—
106	C₄×C₈	—	2	2	no	—
107	C₄×C₈	—	79	1	no	—
108	C₄×C₈	—	79	1	no	—
109	C₄×C₈	—	79	2	no	—
110	C₄×C₈	—	158	2	no	—
111	#95	—	158	2	no	—
112	C₂²×C₄	—	158	1	no	—
113	C₂²×C₄	—	158	1	no	—
114	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
115	C₂²×C₄	#11929624	1	2	yes	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	4	2	no	—
120	C₂×C₈	—	316	2	no	—
121	C₂×C₈	—	79	2	no	—
122	C₂×C₈	—	79	2	no	—
123	C₂×C₈	—	158	2	no	—
124	C₂×C₈	—	158	2	no	—
125	C₄²	—	158	2	no	—
126	C₄²	—	158	2	no	—
127	C₄²	D₄×D₇₉	1	1	yes	characteristic
128	C₄²	Q₈×D₇₉	1	1	yes	characteristic
129	C₄²	#11929623	1	2	yes	—
130	C₂³	#14579118	1	1	yes	characteristic
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	316	2	no	—
133	C₂×C₄	—	316	2	no	—
134	C₂×C₄	—	316	1	no	—
135	C₂×C₄	—	316	1	no	—
136	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
137	C₂×C₄	#14579117	1	1	yes	characteristic
138	C₂×C₄	#14579147	1	2	yes	—
139	C₂×C₄	#14579151	1	2	yes	—
140	C₂×C₄	#14579167	1	2	yes	—
141	C₂×C₄	#14579169	1	2	yes	—
142	C₈	—	4	2	no	—
143	C₈	—	8	2	no	—
144	C₈	—	632	2	no	—
145	C₈	—	158	2	no	—
146	C₈	—	158	2	no	—
147	C₂²	—	2	2	no	—
148	C₂²	#19098427	1	1	yes	characteristic
149	C₂²	#19098622	1	1	yes	characteristic
150	C₂²	#19098625	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	2	2	no	—
153	C₄	—	632	1	no	—
154	C₄	—	632	1	no	—
155	C₄	#19098588	1	2	yes	—
156	C₄	#19098621	1	2	yes	—
157	C₂	—	2	1	no	—
158	C₂	—	2	1	no	—
159	C₂	#23763518	1	1	yes	characteristic
160	C₂	#23763521	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23763153

C₂

yes

maximal, characteristic

#23762146

C₂

yes

maximal

#23761245

C₂

yes

maximal, characteristic

#23761228

C₂

yes

maximal

C₇₉×#1281

C₂

yes

maximal, characteristic

#19098616

C₂²

yes

—

C₂×C₄×C₇₉⋊C₈

C₂²

yes

characteristic

#19098409

C₂²

yes

characteristic

#19098317

C₄

yes

—

#19098317

C₄

yes

—

#19098315

—

#19098313

—

#431×C₇₉

C₂²

yes

—

C₂×#97×C₇₉

C₂²

yes

characteristic

C₂²×C₇₉⋊C₈

C₂×C₄

yes

—

C₂×C₄×Dic₇₉

C₂×C₄

yes

characteristic

C₂×C₄×Dic₇₉

C₂×C₄

yes

characteristic

C₄⋊(C₇₉⋊C₈)

—

C₂²⋊(C₇₉⋊C₈)

—

C₄×C₇₉⋊C₈

—

C₄×C₇₉⋊C₈

D₄

yes

characteristic

C₄×C₇₉⋊C₈

D₄

yes

characteristic

C₄×C₇₉⋊C₈

Q₈

yes

—

C₇₉⋊#95

—

C₂²×C₈×C₇₉

C₂×C₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄⋊C₈×C₇₉

—

C₂²⋊C₈×C₇₉

—

#97×C₇₉

—

C₄×C₈×C₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₄⋊C₄

yes

—

C₂×C₇₉⋊C₈

C₄⋊C₄

yes

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₉

#41

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

C₂×Q₈

yes

characteristic

C₄²×C₇₉

#41

yes

—

C₂³×C₇₉

#117

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

—

C₂×C₄×C₇₉

#125

yes

—

C₂×C₄×C₇₉

#131

yes

—

C₂×C₄×C₇₉

#131

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#424

yes

—

C₄×C₇₉

#433

yes

—

Dic₇₉

—

Dic₇₉

—

#8782

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#1832

yes

characteristic

C₂×C₇₉

#1833

yes

—

—

—

—

D₇₉

yes

characteristic

#1274

—

C₇₉

#8782

yes

characteristic, complemented, Sylow

#431

—

#431

C₂×D₇₉

yes

—

C₂×#97

C₂×D₇₉

yes

characteristic

C₂×C₄×C₈

—

—

—

—

—

158

—

#334

—

158

—

C₂²×C₈

—

C₂²×C₈

C₄×D₇₉

yes

—

C₂×C₄²

—

C₂×C₄²

—

100

C₂×C₄²

C₂²×D₇₉

yes

characteristic

101

C₄⋊C₈

—

102

C₄⋊C₈

—

158

—

103

C₂²⋊C₈

—

104

C₂²⋊C₈

—

158

—

105

#97

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

—

158

—

111

#95

—

158

—

112

C₂²×C₄

—

158

—

113

C₂²×C₄

—

158

—

114

C₂²×C₄

C₂×C₄×D₇₉

yes

characteristic

115

C₂²×C₄

#11929624

yes

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

316

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

158

—

124

C₂×C₈

—

158

—

125

C₄²

—

158

—

126

C₄²

—

158

—

127

C₄²

D₄×D₇₉

yes

characteristic

128

C₄²

Q₈×D₇₉

yes

characteristic

129

C₄²

#11929623

yes

—

130

C₂³

#14579118

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

316

—

133

C₂×C₄

—

316

—

134

C₂×C₄

—

316

—

135

C₂×C₄

—

316

—

136

C₂×C₄

C₄⋊C₄×D₇₉

yes

characteristic

137

C₂×C₄

#14579117

yes

characteristic

138

C₂×C₄

#14579147

yes

—

139

C₂×C₄

#14579151

yes

—

140

C₂×C₄

#14579167

yes

—

141

C₂×C₄

#14579169

yes

—

142

C₈

—

143

C₈

—

144

C₈

—

632

—

145

C₈

—

158

—

146

C₈

—

158

—

147

C₂²

—

148

C₂²

#19098427

yes

characteristic

149

C₂²

#19098622

yes

characteristic

150

C₂²

#19098625

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

632

—

154

C₄

—

632

—

155

C₄

#19098588

yes

—

156

C₄

#19098621

yes

—

157

C₂

—

158

C₂

—

159

C₂

#23763518

yes

characteristic

160

C₂

#23763521

yes

—

	subgroup	G/H
center	C₂×C₄	#14579117
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1281	C₂
socle	C₂²×C₇₉	#382
prime radical	C₂³×C₇₉	#117
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1281	D₇₉
hypocenter	C₇₉	#8782
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1281

subgroup

G/H

center

C₂×C₄

#14579117

commutator

C₂²×C₄×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1281

C₂

C₂²×C₇₉

C₂³×C₇₉

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1281

N	H ≅ G/N	type
86. C₇₉	#8782	semidirect

H ≅ G/N

type

86. C₇₉

#8782

semidirect

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

subgroup	index
#23763153	2
#23762146	2
#23761245	2
#23761228	2
C₇₉×#1281	2
#8782	79

subgroup

index

C₇₉×#1281

Subgroups — #34181920

Conjugacy classes of subgroups

Complements

Maximal subgroups