Group #34193290 — subgroups

Subgroups — #34193290

Order 20224.

316 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 101 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	#23764381	C₂	1	1	yes	maximal, characteristic
2	#23763554	C₂	1	1	yes	maximal, characteristic
3	#23762175	C₂	1	2	yes	maximal
4	#23761804	C₂	1	2	yes	maximal
5	C₇₉×#1274	C₂	1	1	yes	maximal, characteristic
6	#19099120	C₂²	1	1	yes	characteristic
7	#19098587	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
9	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
10	#19098323	—	2	2	no	—
11	#19098316	—	2	4	no	—
12	#19098295	—	2	4	no	—
13	#431×C₇₉	C₂²	1	2	yes	—
14	#376×C₇₉	C₂²	1	1	yes	characteristic
15	C₂×#11929608	—	2	1	no	—
16	C₂²×C₇₉⋊C₈	—	2	2	no	—
17	#14579161	Q₈	1	2	yes	—
18	#14579153	—	2	1	no	—
19	#14579140	—	2	2	no	—
20	C₄×Dic₁₅₈	—	2	2	no	—
21	#14579120	—	2	1	no	—
22	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
23	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
24	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
25	C₄⋊(C₇₉⋊C₈)	—	4	4	no	—
26	C₂×C₄²×C₇₉	D₄	1	2	yes	—
27	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₇₉	—	2	2	no	—
29	C₂²⋊C₈×C₇₉	—	2	2	no	—
30	C₄×C₈×C₇₉	D₄	1	4	yes	—
31	#95×C₇₉	—	2	2	no	—
32	C₂×Dic₁₅₈	—	2	2	no	—
33	#11929608	—	4	2	no	—
34	C₂×C₇₉⋊C₈	—	4	2	no	—
35	C₂×C₇₉⋊C₈	—	4	1	no	—
36	C₂×C₇₉⋊C₈	—	4	4	no	—
37	C₄⋊Dic₇₉	—	2	2	no	—
38	C₄⋊Dic₇₉	—	2	2	no	—
39	Dic₇₉⋊C₄	—	4	2	no	—
40	C₂²⋊Dic₇₉	—	4	1	no	—
41	C₄×Dic₇₉	—	4	1	no	—
42	C₂²×C₄×C₇₉	—	2	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	4	no	—
46	C₄²×C₇₉	—	2	2	no	—
47	C₄²×C₇₉	Q₁₆	1	4	yes	—
48	C₄²×C₇₉	C₂×D₄	1	2	yes	—
49	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₇₉	#41	1	1	yes	characteristic
51	C₂³×C₇₉	#123	1	1	yes	characteristic
52	C₂×C₄×C₇₉	—	2	4	no	—
53	C₂×C₄×C₇₉	—	4	2	no	—
54	C₂×C₄×C₇₉	—	4	4	no	—
55	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
56	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
57	C₂×C₄×C₇₉	#123	1	2	yes	—
58	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
59	C₂×C₄×C₇₉	C₂×Q₁₆	1	2	yes	—
60	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
61	C₂×Dic₇₉	—	4	2	no	—
62	Dic₁₅₈	—	4	4	no	—
63	C₈×C₇₉	—	4	4	no	—
64	C₇₉⋊C₈	—	8	2	no	—
65	C₇₉⋊C₈	—	8	1	no	—
66	C₂²×C₇₉	—	2	2	no	—
67	C₂²×C₇₉	#399	1	1	yes	characteristic
68	C₂²×C₇₉	#484	1	2	yes	—
69	C₄×C₇₉	—	2	2	no	—
70	C₄×C₇₉	—	2	2	no	—
71	C₄×C₇₉	—	4	4	no	—
72	C₄×C₇₉	#477	1	2	yes	—
73	C₄×C₇₉	#494	1	1	yes	characteristic
74	C₄×C₇₉	#496	1	1	yes	characteristic
75	Dic₇₉	—	8	2	no	—
76	#10668	—	79	1	no	maximal, Sylow
77	C₂×C₇₉	—	2	2	no	—
78	C₂×C₇₉	#2095	1	1	yes	characteristic
79	C₂×C₇₉	#2098	1	2	yes	—
80	#1995	—	79	1	no	—
81	#1843	—	79	1	no	—
82	#1571	—	79	2	no	—
83	#1481	—	79	2	no	—
84	#1274	D₇₉	1	1	yes	characteristic
85	C₇₉	#10668	1	1	yes	characteristic, complemented, Sylow
86	#533	—	79	1	no	—
87	#431	C₂×D₇₉	1	2	yes	—
88	#423	—	79	1	no	—
89	C₂×C₄⋊C₈	—	79	2	no	—
90	C₂×C₄⋊C₈	—	158	2	no	—
91	#376	C₂×D₇₉	1	1	yes	characteristic
92	#339	—	158	2	no	—
93	#335	—	158	4	no	—
94	Q₈⋊C₈	—	158	4	no	—
95	C₂×M₁₆	—	158	1	no	—
96	C₂²×C₈	—	158	2	no	—
97	C₄⋊Q₈	—	79	2	no	—
98	#125	—	158	1	no	—
99	C₂²⋊Q₈	—	158	2	no	—
100	C₄×Q₈	—	158	2	no	—
101	#117	—	158	1	no	—
102	C₂×C₄²	D₃₁₆	1	2	yes	—
103	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	316	4	no	—
106	C₄⋊C₈	—	158	2	no	—
107	C₄⋊C₈	—	158	2	no	—
108	C₄⋊C₈	—	158	2	no	—
109	C₂²⋊C₈	—	2	2	no	—
110	C₄×C₈	D₃₁₆	1	4	yes	—
111	#95	—	2	2	no	—
112	C₂×Q₈	—	158	2	no	—
113	C₂²×C₄	—	2	2	no	—
114	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
115	C₂²×C₄	#11929623	1	2	yes	—
116	M₁₆	—	316	2	no	—
117	C₂×C₈	—	2	4	no	—
118	C₂×C₈	—	316	2	no	—
119	C₂×C₈	—	316	1	no	—
120	C₂×C₈	—	316	4	no	—
121	C₄⋊C₄	—	316	2	no	—
122	C₄⋊C₄	—	158	2	no	—
123	C₄⋊C₄	—	158	2	no	—
124	C₂²⋊C₄	—	316	1	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	316	1	no	—
127	C₄²	Dic₃₁₆	1	4	yes	—
128	C₄²	C₂×D₃₁₆	1	2	yes	—
129	C₄²	Q₈×D₇₉	1	1	yes	characteristic
130	C₄²	#11929623	1	1	yes	characteristic
131	C₂³	#14579145	1	1	yes	characteristic
132	Q₈	—	316	4	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	4	4	no	—
136	C₂×C₄	—	316	2	no	—
137	C₂×C₄	#14579141	1	2	yes	—
138	C₂×C₄	#14579141	1	1	yes	characteristic
139	C₂×C₄	#14579145	1	2	yes	—
140	C₂×C₄	#14579158	1	1	yes	characteristic
141	C₂×C₄	C₂×Dic₃₁₆	1	2	yes	—
142	C₂×C₄	#14579189	1	2	yes	—
143	C₈	—	4	4	no	—
144	C₈	—	632	2	no	—
145	C₈	—	632	1	no	—
146	C₂²	—	2	2	no	—
147	C₂²	#19098500	1	1	yes	characteristic
148	C₂²	#19098899	1	2	yes	—
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	4	4	no	—
152	C₄	—	632	2	no	—
153	C₄	#19098859	1	2	yes	—
154	C₄	#19098943	1	1	yes	characteristic
155	C₄	#19098952	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	#23764937	1	1	yes	characteristic
158	C₂	#23764954	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23764381

C₂

yes

maximal, characteristic

#23763554

C₂

yes

maximal, characteristic

#23762175

C₂

yes

maximal

#23761804

C₂

yes

maximal

C₇₉×#1274

C₂

yes

maximal, characteristic

#19099120

C₂²

yes

characteristic

#19098587

C₂²

yes

characteristic

C₂×C₄⋊(C₇₉⋊C₈)

—

C₂×C₄⋊(C₇₉⋊C₈)

C₂²

yes

—

#19098323

—

#19098316

—

#19098295

—

#431×C₇₉

C₂²

yes

—

#376×C₇₉

C₂²

yes

characteristic

C₂×#11929608

—

C₂²×C₇₉⋊C₈

—

#14579161

Q₈

yes

—

#14579153

—

#14579140

—

C₄×Dic₁₅₈

—

#14579120

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₂×C₄²×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄⋊C₈×C₇₉

—

C₂²⋊C₈×C₇₉

—

C₄×C₈×C₇₉

D₄

yes

—

#95×C₇₉

—

C₂×Dic₁₅₈

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

#41

yes

—

C₂×C₈×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

Q₁₆

yes

—

C₄²×C₇₉

C₂×D₄

yes

—

C₄²×C₇₉

C₂×Q₈

yes

C₄²×C₇₉

yes

C₂³×C₇₉

yes

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

#123

yes

—

C₂×C₄×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₁₆

yes

—

C₂×C₄×C₇₉

AGL(1,ℤ/8)

yes

—

C₂×Dic₇₉

—

Dic₁₅₈

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

#399

yes

characteristic

C₂²×C₇₉

#484

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#477

yes

—

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

#10668

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

#2095

yes

characteristic

C₂×C₇₉

#2098

yes

—

—

—

—

—

D₇₉

yes

characteristic

C₇₉

#10668

yes

characteristic, complemented, Sylow

#533

—

#431

C₂×D₇₉

yes

—

#423

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

158

—

#376

C₂×D₇₉

yes

characteristic

#339

—

158

—

#335

—

158

—

Q₈⋊C₈

—

158

—

C₂×M₁₆

—

158

—

C₂²×C₈

—

158

—

C₄⋊Q₈

—

#125

—

158

—

C₂²⋊Q₈

—

158

—

100

C₄×Q₈

—

158

—

101

#117

—

158

—

102

C₂×C₄²

D₃₁₆

yes

—

103

C₂×C₄²

C₂²×D₇₉

yes

characteristic

104

C₄⋊C₈

—

105

C₄⋊C₈

—

316

—

106

C₄⋊C₈

—

158

—

107

C₄⋊C₈

—

158

—

108

C₄⋊C₈

—

158

—

109

C₂²⋊C₈

—

110

C₄×C₈

D₃₁₆

yes

—

111

#95

—

112

C₂×Q₈

—

158

—

113

C₂²×C₄

—

114

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

115

C₂²×C₄

#11929623

yes

—

116

M₁₆

—

316

—

117

C₂×C₈

—

118

C₂×C₈

—

316

—

119

C₂×C₈

—

316

—

120

C₂×C₈

—

316

—

121

C₄⋊C₄

—

316

—

122

C₄⋊C₄

—

158

—

123

C₄⋊C₄

—

158

—

124

C₂²⋊C₄

—

316

—

125

C₄²

—

126

C₄²

—

316

—

127

C₄²

Dic₃₁₆

yes

—

128

C₄²

C₂×D₃₁₆

yes

—

129

C₄²

Q₈×D₇₉

yes

characteristic

130

C₄²

#11929623

yes

characteristic

131

C₂³

#14579145

yes

characteristic

132

Q₈

—

316

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

316

—

137

C₂×C₄

#14579141

yes

—

138

C₂×C₄

#14579141

yes

characteristic

139

C₂×C₄

#14579145

yes

—

140

C₂×C₄

#14579158

yes

characteristic

141

C₂×C₄

C₂×Dic₃₁₆

yes

—

142

C₂×C₄

#14579189

yes

—

143

C₈

—

144

C₈

—

632

—

145

C₈

—

632

—

146

C₂²

—

147

C₂²

#19098500

yes

characteristic

148

C₂²

#19098899

yes

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

632

—

153

C₄

#19098859

yes

—

154

C₄

#19098943

yes

characteristic

155

C₄

#19098952

yes

characteristic

156

C₂

—

157

C₂

#23764937

yes

characteristic

158

C₂

#23764954

yes

—

	subgroup	G/H
center	C₂×C₄	#14579158
commutator, cosocle	C₂×C₄²×C₇₉	C₂³
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1274	C₂
socle	C₂²×C₇₉	#399
prime radical	C₂³×C₇₉	#123
hypercenter	#1274	D₇₉
hypocenter	C₇₉	#10668
LCS	G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

subgroup

G/H

center

C₂×C₄

#14579158

commutator, cosocle

C₂×C₄²×C₇₉

C₂³

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1274

C₂

C₂²×C₇₉

C₂³×C₇₉

D₇₉

C₇₉

G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

N	H ≅ G/N	type
85. C₇₉	#10668	semidirect

H ≅ G/N

type

85. C₇₉

#10668

semidirect

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

subgroup	index
#23764381	2
#23763554	2
#23762175	2
#23761804	2
C₇₉×#1274	2
#10668	79

subgroup

index

C₇₉×#1274

Subgroups — #34193290

Conjugacy classes of subgroups

Complements

Maximal subgroups