Group #34120484 — subgroups

Subgroups — C₇₉×#8795 #34120484

Order 20224.

270 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 126 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	C₇₉×#1846	C₂	1	1	yes	maximal, characteristic
2	C₇₉×#1837	C₂	1	1	yes	maximal, characteristic
3	C₇₉×#1452	C₂	1	1	yes	maximal, characteristic
4	C₇₉×#1286	C₂	1	2	yes	maximal
5	C₇₉×#1286	C₂	1	2	yes	maximal
6	#431×C₇₉	C₂²	1	2	yes	—
7	C₂×#107×C₇₉	—	2	1	no	—
8	C₂×#106×C₇₉	—	2	1	no	—
9	#423×C₇₉	C₂²	1	1	yes	characteristic
10	#423×C₇₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
12	#382×C₇₉	C₂²	1	2	yes	—
13	#341×C₇₉	—	2	1	no	—
14	#341×C₇₉	—	2	1	no	—
15	#322×C₇₉	C₄	1	4	yes	complemented
16	C₂×M₁₆×C₇₉	—	2	1	no	—
17	C₂×M₁₆×C₇₉	—	2	1	no	—
18	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
21	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
22	#107×C₇₉	—	4	2	no	—
23	#106×C₇₉	—	4	2	no	—
24	C₄⋊C₈×C₇₉	—	2	2	no	—
25	C₄⋊C₈×C₇₉	—	2	2	no	—
26	C₄⋊C₈×C₇₉	—	2	2	no	—
27	C₂²⋊C₈×C₇₉	—	2	2	no	—
28	C₄×C₈×C₇₉	C₂×C₄	1	4	yes	—
29	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₇₉	Q₈	1	2	yes	—
31	#95×C₇₉	—	2	4	no	—
32	C₂²×C₄×C₇₉	—	4	2	no	—
33	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₉	#41	1	2	yes	—
35	M₁₆×C₇₉	—	4	1	no	—
36	M₁₆×C₇₉	—	4	1	no	—
37	M₁₆×C₇₉	—	4	1	no	—
38	M₁₆×C₇₉	—	4	1	no	—
39	C₂×C₈×C₇₉	—	2	4	no	—
40	C₂×C₈×C₇₉	—	2	1	no	—
41	C₂×C₈×C₇₉	—	2	1	no	—
42	C₂×C₈×C₇₉	—	2	1	no	—
43	C₂×C₈×C₇₉	—	2	1	no	—
44	C₂×C₈×C₇₉	—	4	1	no	—
45	C₂×C₈×C₇₉	—	4	1	no	—
46	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
47	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
48	C₄⋊C₄×C₇₉	—	4	2	no	—
49	C₄⋊C₄×C₇₉	—	4	2	no	—
50	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
52	C₄²×C₇₉	#41	1	2	yes	—
53	C₂³×C₇₉	#124	1	1	yes	characteristic
54	C₂×C₄×C₇₉	—	4	2	no	—
55	C₂×C₄×C₇₉	—	8	4	no	—
56	C₂×C₄×C₇₉	—	8	2	no	—
57	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₇₉	#117	1	2	yes	—
59	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
60	C₂×C₄×C₇₉	#125	1	2	yes	—
61	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
62	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
63	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
64	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
65	C₈×C₇₉	—	2	2	no	—
66	C₈×C₇₉	—	2	2	no	—
67	C₈×C₇₉	—	4	4	no	—
68	C₈×C₇₉	—	8	1	no	—
69	C₈×C₇₉	—	8	1	no	—
70	C₂²×C₇₉	—	2	2	no	—
71	C₂²×C₇₉	#382	1	1	yes	characteristic
72	C₂²×C₇₉	#490	1	1	yes	characteristic
73	C₂²×C₇₉	#491	1	1	yes	characteristic
74	C₄×C₇₉	—	2	2	no	—
75	C₄×C₇₉	—	2	2	no	—
76	C₄×C₇₉	—	16	2	no	—
77	C₄×C₇₉	#428	1	1	yes	characteristic
78	C₄×C₇₉	#428	1	1	yes	characteristic
79	C₄×C₇₉	#489	1	1	yes	characteristic
80	C₄×C₇₉	#489	1	1	yes	characteristic
81	#8795	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂×C₇₉	—	2	1	no	—
83	C₂×C₇₉	—	2	1	no	—
84	C₂×C₇₉	#1833	1	2	yes	—
85	C₂×C₇₉	#1836	1	1	yes	characteristic
86	#1846	C₂×C₇₉	1	1	yes	characteristic
87	#1837	C₂×C₇₉	1	1	yes	characteristic
88	#1452	C₂×C₇₉	1	1	yes	characteristic
89	#1286	C₂×C₇₉	1	2	yes	—
90	#1286	C₂×C₇₉	1	2	yes	—
91	C₇₉	#8795	1	1	yes	characteristic, direct product factor, Sylow
92	#431	C₂²×C₇₉	1	2	yes	—
93	C₂×#107	—	2	1	no	—
94	C₂×#106	—	2	1	no	—
95	#423	C₂²×C₇₉	1	1	yes	characteristic
96	#423	C₂²×C₇₉	1	1	yes	characteristic
97	C₂×C₄×C₈	C₂²×C₇₉	1	1	yes	characteristic
98	#382	C₂²×C₇₉	1	2	yes	—
99	#341	—	2	1	no	—
100	#341	—	2	1	no	—
101	#322	C₄×C₇₉	1	4	yes	complemented
102	C₂×M₁₆	—	2	1	no	—
103	C₂×M₁₆	—	2	1	no	—
104	C₂²×C₈	D₄×C₇₉	1	1	yes	characteristic
105	C₂²×C₈	D₄×C₇₉	1	1	yes	characteristic
106	C₂×C₄⋊C₄	—	2	2	no	—
107	C₂×C₄²	C₂³×C₇₉	1	1	yes	characteristic
108	#107	—	4	2	no	—
109	#106	—	4	2	no	—
110	C₄⋊C₈	—	2	2	no	—
111	C₄⋊C₈	—	2	2	no	—
112	C₄⋊C₈	—	2	2	no	—
113	C₂²⋊C₈	—	2	2	no	—
114	C₄×C₈	C₂×C₄×C₇₉	1	4	yes	—
115	C₄×C₈	C₂×C₄×C₇₉	1	2	yes	—
116	C₄×C₈	Q₈×C₇₉	1	2	yes	—
117	#95	—	2	4	no	—
118	C₂²×C₄	—	4	2	no	—
119	C₂²×C₄	C₂×D₄×C₇₉	1	1	yes	characteristic
120	C₂²×C₄	#41×C₇₉	1	2	yes	—
121	M₁₆	—	4	1	no	—
122	M₁₆	—	4	1	no	—
123	M₁₆	—	4	1	no	—
124	M₁₆	—	4	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	4	no	—
130	C₂×C₈	—	4	1	no	—
131	C₂×C₈	—	4	1	no	—
132	C₂×C₈	C₄⋊C₄×C₇₉	1	2	yes	—
133	C₂×C₈	C₄⋊C₄×C₇₉	1	2	yes	—
134	C₄⋊C₄	—	4	2	no	—
135	C₄⋊C₄	—	4	2	no	—
136	C₄²	C₂²×C₄×C₇₉	1	1	yes	characteristic
137	C₄²	C₂×Q₈×C₇₉	1	1	yes	characteristic
138	C₄²	#41×C₇₉	1	2	yes	—
139	C₂³	#124×C₇₉	1	1	yes	characteristic
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	8	4	no	—
142	C₂×C₄	—	8	2	no	—
143	C₂×C₄	C₂×C₄⋊C₄×C₇₉	1	1	yes	characteristic
144	C₂×C₄	#117×C₇₉	1	2	yes	—
145	C₂×C₄	#124×C₇₉	1	1	yes	characteristic
146	C₂×C₄	#125×C₇₉	1	2	yes	—
147	C₂×C₄	AGL(1,ℤ/8)×C₇₉	1	1	yes	characteristic
148	C₂×C₄	AGL(1,ℤ/8)×C₇₉	1	1	yes	characteristic
149	C₂×C₄	#137×C₇₉	1	1	yes	characteristic
150	C₂×C₄	#137×C₇₉	1	1	yes	characteristic
151	C₈	—	2	2	no	—
152	C₈	—	2	2	no	—
153	C₈	—	4	4	no	—
154	C₈	—	8	1	no	—
155	C₈	—	8	1	no	—
156	C₂²	—	2	2	no	—
157	C₂²	#382×C₇₉	1	1	yes	characteristic
158	C₂²	#490×C₇₉	1	1	yes	characteristic
159	C₂²	#491×C₇₉	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	16	2	no	—
163	C₄	#428×C₇₉	1	1	yes	characteristic
164	C₄	#428×C₇₉	1	1	yes	characteristic
165	C₄	#489×C₇₉	1	1	yes	characteristic
166	C₄	#489×C₇₉	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	C₇₉×#1833	1	2	yes	—
170	C₂	C₇₉×#1836	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₉×#1846

C₂

yes

maximal, characteristic

C₇₉×#1837

C₂

yes

maximal, characteristic

C₇₉×#1452

C₂

yes

maximal, characteristic

C₇₉×#1286

C₂

yes

maximal

C₇₉×#1286

C₂

yes

maximal

#431×C₇₉

C₂²

yes

—

C₂×#107×C₇₉

—

C₂×#106×C₇₉

—

#423×C₇₉

C₂²

yes

characteristic

#423×C₇₉

C₂²

yes

characteristic

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#382×C₇₉

C₂²

yes

—

#341×C₇₉

—

#341×C₇₉

—

#322×C₇₉

C₄

yes

complemented

C₂×M₁₆×C₇₉

—

C₂×M₁₆×C₇₉

—

C₂²×C₈×C₇₉

D₄

yes

characteristic

C₂²×C₈×C₇₉

D₄

yes

characteristic

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

#107×C₇₉

—

#106×C₇₉

—

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

—

C₂²⋊C₈×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

Q₈

yes

—

#95×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

#41

yes

—

M₁₆×C₇₉

—

M₁₆×C₇₉

—

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×Q₈

yes

characteristic

C₄²×C₇₉

#41

yes

—

C₂³×C₇₉

#124

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

#117

yes

—

C₂×C₄×C₇₉

#124

yes

characteristic

C₂×C₄×C₇₉

#125

yes

—

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₇₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#1833

yes

—

C₂×C₇₉

#1836

yes

characteristic

#1846

C₂×C₇₉

yes

characteristic

#1837

C₂×C₇₉

yes

characteristic

#1452

C₂×C₇₉

yes

characteristic

#1286

C₂×C₇₉

yes

—

#1286

C₂×C₇₉

yes

—

C₇₉

#8795

yes

characteristic, direct product factor, Sylow

#431

C₂²×C₇₉

yes

—

C₂×#107

—

C₂×#106

—

#423

C₂²×C₇₉

yes

characteristic

#423

C₂²×C₇₉

yes

characteristic

C₂×C₄×C₈

C₂²×C₇₉

yes

characteristic

#382

C₂²×C₇₉

yes

—

#341

—

100

#341

—

101

#322

C₄×C₇₉

yes

complemented

102

C₂×M₁₆

—

103

C₂×M₁₆

—

104

C₂²×C₈

D₄×C₇₉

yes

characteristic

105

C₂²×C₈

D₄×C₇₉

yes

characteristic

106

C₂×C₄⋊C₄

—

107

C₂×C₄²

C₂³×C₇₉

yes

characteristic

108

#107

—

109

#106

—

110

C₄⋊C₈

—

111

C₄⋊C₈

—

112

C₄⋊C₈

—

113

C₂²⋊C₈

—

114

C₄×C₈

C₂×C₄×C₇₉

yes

—

115

C₄×C₈

C₂×C₄×C₇₉

yes

—

116

C₄×C₈

Q₈×C₇₉

yes

—

117

#95

—

118

C₂²×C₄

—

119

C₂²×C₄

C₂×D₄×C₇₉

yes

characteristic

120

C₂²×C₄

#41×C₇₉

yes

—

121

M₁₆

—

122

M₁₆

—

123

M₁₆

—

124

M₁₆

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

C₄⋊C₄×C₇₉

yes

—

133

C₂×C₈

C₄⋊C₄×C₇₉

yes

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄²

C₂²×C₄×C₇₉

yes

characteristic

137

C₄²

C₂×Q₈×C₇₉

yes

characteristic

138

C₄²

#41×C₇₉

yes

—

139

C₂³

#124×C₇₉

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

C₂×C₄⋊C₄×C₇₉

yes

characteristic

144

C₂×C₄

#117×C₇₉

yes

—

145

C₂×C₄

#124×C₇₉

yes

characteristic

146

C₂×C₄

#125×C₇₉

yes

—

147

C₂×C₄

AGL(1,ℤ/8)×C₇₉

yes

characteristic

148

C₂×C₄

AGL(1,ℤ/8)×C₇₉

yes

characteristic

149

C₂×C₄

#137×C₇₉

yes

characteristic

150

C₂×C₄

#137×C₇₉

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

—

155

C₈

—

156

C₂²

—

157

C₂²

#382×C₇₉

yes

characteristic

158

C₂²

#490×C₇₉

yes

characteristic

159

C₂²

#491×C₇₉

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

#428×C₇₉

yes

characteristic

164

C₄

#428×C₇₉

yes

characteristic

165

C₄

#489×C₇₉

yes

characteristic

166

C₄

#489×C₇₉

yes

characteristic

167

C₂

—

168

C₂

—

169

C₂

C₇₉×#1833

yes

—

170

C₂

C₇₉×#1836

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₇₉	#382
commutator	C₄²	C₂²×C₄×C₇₉
Frattini, cosocle	C₂×C₄²	C₂³×C₇₉
prime radical	C₂³×C₇₉	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂×C₄²×C₇₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₇₉

#382

commutator

C₄²

C₂²×C₄×C₇₉

Frattini, cosocle

C₂×C₄²

C₂³×C₇₉

C₂³×C₇₉

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂²×C₇₉ ◁ C₂×C₄²×C₇₉ ◁ G

N	H ≅ G/N	type
15. #322×C₇₉	C₄	semidirect
81. #8795	C₇₉	direct (N × H)
91. C₇₉	#8795	direct (N × H)
101. #322	C₄×C₇₉	semidirect

H ≅ G/N

type

15. #322×C₇₉

C₄

semidirect

81. #8795

C₇₉

direct (N × H)

91. C₇₉

#8795

direct (N × H)

101. #322

C₄×C₇₉

semidirect

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

subgroup	index
C₇₉×#1846	2
C₇₉×#1837	2
C₇₉×#1452	2
C₇₉×#1286	2
#8795	79

subgroup

index

C₇₉×#1846

C₇₉×#1837

C₇₉×#1452

C₇₉×#1286

#8795

Subgroups — C₇₉×#8795 #34120484

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C79×#8795 #34120484

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₉×#8795 #34120484