Group #34123718 — subgroups

Subgroups — C₇₉×#12029 #34123718

Order 20224.

208 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 118 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×#2163	C₂	1	1	yes	maximal, characteristic
2	C₇₉×#1566	C₂	1	1	yes	maximal, characteristic
3	C₇₉×#1564	C₂	1	1	yes	maximal, characteristic
4	C₇₉×#1368	C₂	1	4	yes	maximal, complemented
5	#431×C₇₉	—	2	1	no	—
6	#423×C₇₉	—	2	1	no	—
7	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
8	C₄⋊C₁₆×C₇₉	C₄	1	2	yes	—
9	C₂²⋊C₁₆×C₇₉	C₄	1	2	yes	—
10	#346×C₇₉	C₂²	1	2	yes	—
11	#335×C₇₉	—	2	1	no	—
12	#334×C₇₉	—	2	1	no	—
13	#334×C₇₉	C₂²	1	2	yes	—
14	#334×C₇₉	C₂²	1	2	yes	—
15	C₂×M₁₆×C₇₉	—	4	1	no	—
16	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₇₉	Q₈	1	1	yes	characteristic
18	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
19	C₂×C₁₆×C₇₉	C₂×C₄	1	4	yes	—
20	C₄⋊C₈×C₇₉	—	2	2	no	—
21	C₄⋊C₈×C₇₉	—	2	2	no	—
22	C₄⋊C₈×C₇₉	—	4	1	no	—
23	C₂²⋊C₈×C₇₉	—	4	1	no	—
24	C₄×C₈×C₇₉	—	4	1	no	—
25	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
27	C₄×C₈×C₇₉	Q₈	1	1	yes	characteristic
28	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
31	C₂²×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
32	M₁₆×C₇₉	—	8	2	no	—
33	C₂×C₈×C₇₉	—	2	2	no	—
34	C₂×C₈×C₇₉	—	4	1	no	—
35	C₂×C₈×C₇₉	—	4	2	no	—
36	C₂×C₈×C₇₉	—	4	2	no	—
37	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
41	C₄²×C₇₉	C₄⋊C₄	1	1	yes	characteristic
42	C₄²×C₇₉	SD₁₆	1	1	yes	characteristic
43	C₄²×C₇₉	SD₁₆	1	1	yes	characteristic
44	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
45	C₁₆×C₇₉	—	2	4	no	—
46	C₂³×C₇₉	#106	1	1	yes	characteristic
47	C₂×C₄×C₇₉	—	2	2	no	—
48	C₂×C₄×C₇₉	—	2	1	no	—
49	C₂×C₄×C₇₉	—	2	1	no	—
50	C₂×C₄×C₇₉	—	4	1	no	—
51	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
52	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
53	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
54	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
56	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
57	C₈×C₇₉	—	2	1	no	—
58	C₈×C₇₉	—	2	1	no	—
59	C₈×C₇₉	—	2	1	no	—
60	C₈×C₇₉	—	8	2	no	—
61	C₈×C₇₉	—	8	2	no	—
62	C₈×C₇₉	#131	1	2	yes	—
63	C₂²×C₇₉	—	2	1	no	—
64	C₂²×C₇₉	—	2	1	no	—
65	C₂²×C₇₉	—	2	1	no	—
66	C₂²×C₇₉	C₂×#106	1	1	yes	characteristic
67	C₄×C₇₉	—	2	1	no	—
68	C₄×C₇₉	—	2	1	no	—
69	C₄×C₇₉	—	2	1	no	—
70	C₄×C₇₉	—	2	1	no	—
71	C₄×C₇₉	#365	1	2	yes	—
72	C₄×C₇₉	#424	1	1	yes	characteristic
73	C₄×C₇₉	#430	1	1	yes	characteristic
74	#12029	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂×C₇₉	—	4	1	no	—
76	C₂×C₇₉	#1570	1	1	yes	characteristic
77	C₂×C₇₉	C₂×#365	1	1	yes	characteristic
78	C₂×C₇₉	#2155	1	1	yes	characteristic
79	#2163	C₂×C₇₉	1	1	yes	characteristic
80	#1566	C₂×C₇₉	1	1	yes	characteristic
81	#1564	C₂×C₇₉	1	1	yes	characteristic
82	#1368	C₂×C₇₉	1	4	yes	complemented
83	C₇₉	#12029	1	1	yes	characteristic, direct product factor, Sylow
84	#431	—	2	1	no	—
85	#423	—	2	1	no	—
86	C₂×C₄×C₈	C₂²×C₇₉	1	1	yes	characteristic
87	C₄⋊C₁₆	C₄×C₇₉	1	2	yes	—
88	C₂²⋊C₁₆	C₄×C₇₉	1	2	yes	—
89	#346	C₂²×C₇₉	1	2	yes	—
90	#335	—	2	1	no	—
91	#334	—	2	1	no	—
92	#334	C₂²×C₇₉	1	2	yes	—
93	#334	C₂²×C₇₉	1	2	yes	—
94	C₂×M₁₆	—	4	1	no	—
95	C₂²×C₈	C₂×C₄×C₇₉	1	1	yes	characteristic
96	C₂²×C₈	Q₈×C₇₉	1	1	yes	characteristic
97	C₂×C₄²	D₄×C₇₉	1	1	yes	characteristic
98	C₂×C₁₆	C₂×C₄×C₇₉	1	4	yes	—
99	C₄⋊C₈	—	2	2	no	—
100	C₄⋊C₈	—	2	2	no	—
101	C₄⋊C₈	—	4	1	no	—
102	C₂²⋊C₈	—	4	1	no	—
103	C₄×C₈	—	4	1	no	—
104	C₄×C₈	C₂×C₄×C₇₉	1	1	yes	characteristic
105	C₄×C₈	D₄×C₇₉	1	1	yes	characteristic
106	C₄×C₈	Q₈×C₇₉	1	1	yes	characteristic
107	C₄×C₈	C₂³×C₇₉	1	1	yes	characteristic
108	C₂²×C₄	C₄⋊C₄×C₇₉	1	1	yes	characteristic
109	C₂²×C₄	SD₁₆×C₇₉	1	1	yes	characteristic
110	C₂²×C₄	SD₁₆×C₇₉	1	1	yes	characteristic
111	M₁₆	—	8	2	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	4	1	no	—
114	C₂×C₈	—	4	2	no	—
115	C₂×C₈	—	4	2	no	—
116	C₂×C₈	C₄⋊C₄×C₇₉	1	1	yes	characteristic
117	C₂×C₈	C₄⋊C₄×C₇₉	1	1	yes	characteristic
118	C₂×C₈	C₂²×C₄×C₇₉	1	1	yes	characteristic
119	C₂×C₈	C₂×Q₈×C₇₉	1	1	yes	characteristic
120	C₄²	C₄⋊C₄×C₇₉	1	1	yes	characteristic
121	C₄²	SD₁₆×C₇₉	1	1	yes	characteristic
122	C₄²	SD₁₆×C₇₉	1	1	yes	characteristic
123	C₄²	C₂×D₄×C₇₉	1	1	yes	characteristic
124	C₁₆	—	2	4	no	—
125	C₂³	#106×C₇₉	1	1	yes	characteristic
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	#106×C₇₉	1	1	yes	characteristic
131	C₂×C₄	#106×C₇₉	1	1	yes	characteristic
132	C₂×C₄	#106×C₇₉	1	1	yes	characteristic
133	C₂×C₄	C₂×C₄⋊C₄×C₇₉	1	1	yes	characteristic
134	C₂×C₄	C₂×SD₁₆×C₇₉	1	1	yes	characteristic
135	C₂×C₄	C₂×SD₁₆×C₇₉	1	1	yes	characteristic
136	C₈	—	2	1	no	—
137	C₈	—	2	1	no	—
138	C₈	—	2	1	no	—
139	C₈	—	8	2	no	—
140	C₈	—	8	2	no	—
141	C₈	#131×C₇₉	1	2	yes	—
142	C₂²	—	2	1	no	—
143	C₂²	—	2	1	no	—
144	C₂²	—	2	1	no	—
145	C₂²	C₂×#106×C₇₉	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	#365×C₇₉	1	2	yes	—
151	C₄	#424×C₇₉	1	1	yes	characteristic
152	C₄	#430×C₇₉	1	1	yes	characteristic
153	C₂	—	4	1	no	—
154	C₂	C₇₉×#1570	1	1	yes	characteristic
155	C₂	C₂×#365×C₇₉	1	1	yes	characteristic
156	C₂	C₇₉×#2155	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₉×#2163

C₂

yes

maximal, characteristic

C₇₉×#1566

C₂

yes

maximal, characteristic

C₇₉×#1564

C₂

yes

maximal, characteristic

C₇₉×#1368

C₂

yes

maximal, complemented

#431×C₇₉

—

#423×C₇₉

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₉

C₄

yes

—

C₂²⋊C₁₆×C₇₉

C₄

yes

—

#346×C₇₉

C₂²

yes

—

#335×C₇₉

—

#334×C₇₉

—

#334×C₇₉

C₂²

yes

—

#334×C₇₉

C₂²

yes

—

C₂×M₁₆×C₇₉

—

C₂²×C₈×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₉

Q₈

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

C₂×C₄

yes

—

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

—

C₂²⋊C₈×C₇₉

—

C₄×C₈×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

Q₈

yes

characteristic

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂²×C₄×C₇₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₇₉

SD₁₆

yes

characteristic

C₂²×C₄×C₇₉

SD₁₆

yes

characteristic

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

C₂×Q₈

yes

characteristic

C₄²×C₇₉

C₄⋊C₄

yes

characteristic

C₄²×C₇₉

SD₁₆

yes

characteristic

C₄²×C₇₉

SD₁₆

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₁₆×C₇₉

—

C₂³×C₇₉

#106

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₉

C₂×SD₁₆

yes

characteristic

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

#131

yes

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×#106

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#365

yes

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₇₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

characteristic

#1566

C₂×C₇₉

yes

characteristic

#1564

C₂×C₇₉

yes

characteristic

#1368

C₂×C₇₉

yes

complemented

C₇₉

#12029

yes

characteristic, direct product factor, Sylow

#431

—

#423

—

C₂×C₄×C₈

C₂²×C₇₉

yes

characteristic

C₄⋊C₁₆

C₄×C₇₉

yes

—

C₂²⋊C₁₆

C₄×C₇₉

yes

—

#346

C₂²×C₇₉

yes

—

#335

—

#334

—

#334

C₂²×C₇₉

yes

—

#334

C₂²×C₇₉

yes

—

C₂×M₁₆

—

C₂²×C₈

C₂×C₄×C₇₉

yes

characteristic

C₂²×C₈

Q₈×C₇₉

yes

characteristic

C₂×C₄²

D₄×C₇₉

yes

characteristic

C₂×C₁₆

C₂×C₄×C₇₉

yes

—

C₄⋊C₈

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

C₂²⋊C₈

—

103

C₄×C₈

—

104

C₄×C₈

C₂×C₄×C₇₉

yes

characteristic

105

C₄×C₈

D₄×C₇₉

yes

characteristic

106

C₄×C₈

Q₈×C₇₉

yes

characteristic

107

C₄×C₈

C₂³×C₇₉

yes

characteristic

108

C₂²×C₄

C₄⋊C₄×C₇₉

yes

characteristic

109

C₂²×C₄

SD₁₆×C₇₉

yes

characteristic

110

C₂²×C₄

SD₁₆×C₇₉

yes

characteristic

111

M₁₆

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

C₄⋊C₄×C₇₉

yes

characteristic

117

C₂×C₈

C₄⋊C₄×C₇₉

yes

characteristic

118

C₂×C₈

C₂²×C₄×C₇₉

yes

characteristic

119

C₂×C₈

C₂×Q₈×C₇₉

yes

characteristic

120

C₄²

C₄⋊C₄×C₇₉

yes

characteristic

121

C₄²

SD₁₆×C₇₉

yes

characteristic

122

C₄²

SD₁₆×C₇₉

yes

characteristic

123

C₄²

C₂×D₄×C₇₉

yes

characteristic

124

C₁₆

—

125

C₂³

#106×C₇₉

yes

characteristic

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

#106×C₇₉

yes

characteristic

131

C₂×C₄

#106×C₇₉

yes

characteristic

132

C₂×C₄

#106×C₇₉

yes

characteristic

133

C₂×C₄

C₂×C₄⋊C₄×C₇₉

yes

characteristic

134

C₂×C₄

C₂×SD₁₆×C₇₉

yes

characteristic

135

C₂×C₄

C₂×SD₁₆×C₇₉

yes

characteristic

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

#131×C₇₉

yes

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

C₂×#106×C₇₉

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

#365×C₇₉

yes

—

151

C₄

#424×C₇₉

yes

characteristic

152

C₄

#430×C₇₉

yes

characteristic

153

C₂

—

154

C₂

C₇₉×#1570

yes

characteristic

155

C₂

C₂×#365×C₇₉

yes

characteristic

156

C₂

C₇₉×#2155

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₇₉	C₂×#106
commutator	C₂×C₈	C₂²×C₄×C₇₉
Frattini, cosocle	C₄×C₈	C₂³×C₇₉
prime radical	C₂³×C₇₉	#106
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂×C₄²×C₇₉ ◁ C₂×C₄×C₈×C₇₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₇₉

C₂×#106

commutator

C₂×C₈

C₂²×C₄×C₇₉

Frattini, cosocle

C₄×C₈

C₂³×C₇₉

prime radical

C₂³×C₇₉

#106

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₇₉ ◁ C₂×C₄²×C₇₉ ◁ C₂×C₄×C₈×C₇₉ ◁ G

N	H ≅ G/N	type
4. C₇₉×#1368	C₂	semidirect
74. #12029	C₇₉	direct (N × H)
82. #1368	C₂×C₇₉	semidirect
83. C₇₉	#12029	direct (N × H)

H ≅ G/N

type

4. C₇₉×#1368

C₂

semidirect

74. #12029

C₇₉

direct (N × H)

82. #1368

C₂×C₇₉

semidirect

83. C₇₉

#12029

direct (N × H)

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

subgroup	index
C₇₉×#2163	2
C₇₉×#1566	2
C₇₉×#1564	2
C₇₉×#1368	2
#12029	79

subgroup

index

C₇₉×#2163

C₇₉×#1566

C₇₉×#1564

C₇₉×#1368

#12029

Subgroups — C₇₉×#12029 #34123718

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C79×#12029 #34123718

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₉×#12029 #34123718