Group #34123720 — subgroups

Subgroups — C₇₉×#12031 #34123720

Order 20224.

200 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 114 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×#1568	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	#432×C₇₉	—	2	1	no	—
6	#431×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	#335×C₇₉	—	2	1	no	—
13	#334×C₇₉	C₂²	1	2	yes	—
14	#334×C₇₉	C₂²	1	2	yes	—
15	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₇₉	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₇₉	C₂×C₄	1	4	yes	—
19	C₄⋊C₈×C₇₉	—	2	2	no	—
20	C₄⋊C₈×C₇₉	—	2	2	no	—
21	C₂²⋊C₈×C₇₉	—	4	1	no	—
22	C₂²⋊C₈×C₇₉	—	4	1	no	—
23	#97×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	C₂×C₈×C₇₉	—	2	2	no	—
33	C₂×C₈×C₇₉	—	4	1	no	—
34	C₂×C₈×C₇₉	—	4	2	no	—
35	C₂×C₈×C₇₉	—	4	2	no	—
36	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
40	C₄²×C₇₉	C₄⋊C₄	1	1	yes	characteristic
41	C₄²×C₇₉	SD₁₆	1	1	yes	characteristic
42	C₄²×C₇₉	SD₁₆	1	1	yes	characteristic
43	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
44	C₁₆×C₇₉	—	2	4	no	—
45	C₂³×C₇₉	#106	1	1	yes	characteristic
46	C₂×C₄×C₇₉	—	2	1	no	—
47	C₂×C₄×C₇₉	—	2	1	no	—
48	C₂×C₄×C₇₉	—	4	1	no	—
49	C₂×C₄×C₇₉	—	4	1	no	—
50	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
51	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
52	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
53	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
55	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
56	C₈×C₇₉	—	2	1	no	—
57	C₈×C₇₉	—	2	1	no	—
58	C₈×C₇₉	—	2	1	no	—
59	C₈×C₇₉	—	8	2	no	—
60	C₈×C₇₉	—	8	2	no	—
61	C₈×C₇₉	#131	1	2	yes	—
62	C₂²×C₇₉	—	2	1	no	—
63	C₂²×C₇₉	—	2	1	no	—
64	C₂²×C₇₉	—	2	1	no	—
65	C₂²×C₇₉	C₂×#106	1	1	yes	characteristic
66	C₄×C₇₉	—	2	1	no	—
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	#424	1	1	yes	characteristic
72	C₄×C₇₉	#430	1	1	yes	characteristic
73	#12031	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂×C₇₉	—	4	1	no	—
75	C₂×C₇₉	#1570	1	1	yes	characteristic
76	C₂×C₇₉	#2155	1	1	yes	characteristic
77	C₂×C₇₉	#2155	1	1	yes	characteristic
78	#2163	C₂×C₇₉	1	1	yes	characteristic
79	#1568	C₂×C₇₉	1	1	yes	characteristic
80	#1564	C₂×C₇₉	1	1	yes	characteristic
81	#1368	C₂×C₇₉	1	4	yes	complemented
82	C₇₉	#12031	1	1	yes	characteristic, direct product factor, Sylow
83	#432	—	2	1	no	—
84	#431	—	2	1	no	—
85	C₂×C₄×C₈	C₂²×C₇₉	1	1	yes	characteristic
86	C₄⋊C₁₆	C₄×C₇₉	1	2	yes	—
87	C₂²⋊C₁₆	C₄×C₇₉	1	2	yes	—
88	#346	C₂²×C₇₉	1	2	yes	—
89	#335	—	2	1	no	—
90	#335	—	2	1	no	—
91	#334	C₂²×C₇₉	1	2	yes	—
92	#334	C₂²×C₇₉	1	2	yes	—
93	C₂²×C₈	C₂×C₄×C₇₉	1	1	yes	characteristic
94	C₂²×C₈	Q₈×C₇₉	1	1	yes	characteristic
95	C₂×C₄²	D₄×C₇₉	1	1	yes	characteristic
96	C₂×C₁₆	C₂×C₄×C₇₉	1	4	yes	—
97	C₄⋊C₈	—	2	2	no	—
98	C₄⋊C₈	—	2	2	no	—
99	C₂²⋊C₈	—	4	1	no	—
100	C₂²⋊C₈	—	4	1	no	—
101	#97	—	4	1	no	—
102	C₄×C₈	—	4	1	no	—
103	C₄×C₈	C₂×C₄×C₇₉	1	1	yes	characteristic
104	C₄×C₈	D₄×C₇₉	1	1	yes	characteristic
105	C₄×C₈	Q₈×C₇₉	1	1	yes	characteristic
106	C₄×C₈	C₂³×C₇₉	1	1	yes	characteristic
107	C₂²×C₄	C₄⋊C₄×C₇₉	1	1	yes	characteristic
108	C₂²×C₄	SD₁₆×C₇₉	1	1	yes	characteristic
109	C₂²×C₄	SD₁₆×C₇₉	1	1	yes	characteristic
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	4	1	no	—
112	C₂×C₈	—	4	2	no	—
113	C₂×C₈	—	4	2	no	—
114	C₂×C₈	C₄⋊C₄×C₇₉	1	1	yes	characteristic
115	C₂×C₈	C₄⋊C₄×C₇₉	1	1	yes	characteristic
116	C₂×C₈	C₂²×C₄×C₇₉	1	1	yes	characteristic
117	C₂×C₈	C₂×Q₈×C₇₉	1	1	yes	characteristic
118	C₄²	C₄⋊C₄×C₇₉	1	1	yes	characteristic
119	C₄²	SD₁₆×C₇₉	1	1	yes	characteristic
120	C₄²	SD₁₆×C₇₉	1	1	yes	characteristic
121	C₄²	C₂×D₄×C₇₉	1	1	yes	characteristic
122	C₁₆	—	2	4	no	—
123	C₂³	#106×C₇₉	1	1	yes	characteristic
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	#106×C₇₉	1	1	yes	characteristic
129	C₂×C₄	#106×C₇₉	1	1	yes	characteristic
130	C₂×C₄	#106×C₇₉	1	1	yes	characteristic
131	C₂×C₄	C₂×C₄⋊C₄×C₇₉	1	1	yes	characteristic
132	C₂×C₄	C₂×SD₁₆×C₇₉	1	1	yes	characteristic
133	C₂×C₄	C₂×SD₁₆×C₇₉	1	1	yes	characteristic
134	C₈	—	2	1	no	—
135	C₈	—	2	1	no	—
136	C₈	—	2	1	no	—
137	C₈	—	8	2	no	—
138	C₈	—	8	2	no	—
139	C₈	#131×C₇₉	1	2	yes	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	C₂×#106×C₇₉	1	1	yes	characteristic
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	#424×C₇₉	1	1	yes	characteristic
150	C₄	#430×C₇₉	1	1	yes	characteristic
151	C₂	—	4	1	no	—
152	C₂	C₇₉×#1570	1	1	yes	characteristic
153	C₂	C₇₉×#2155	1	1	yes	characteristic
154	C₂	C₇₉×#2155	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₉×#2163

C₂

yes

maximal, characteristic

C₇₉×#1568

C₂

yes

maximal, characteristic

C₇₉×#1564

C₂

yes

maximal, characteristic

C₇₉×#1368

C₂

yes

maximal, complemented

#432×C₇₉

—

#431×C₇₉

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₉

C₄

yes

—

C₂²⋊C₁₆×C₇₉

C₄

yes

—

#346×C₇₉

C₂²

yes

—

#335×C₇₉

—

#335×C₇₉

—

#334×C₇₉

C₂²

yes

—

#334×C₇₉

C₂²

yes

—

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

—

C₂²⋊C₈×C₇₉

—

C₂²⋊C₈×C₇₉

—

#97×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

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

—

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

#1568

C₂×C₇₉

yes

characteristic

#1564

C₂×C₇₉

yes

characteristic

#1368

C₂×C₇₉

yes

complemented

C₇₉

#12031

yes

characteristic, direct product factor, Sylow

#432

—

#431

—

C₂×C₄×C₈

C₂²×C₇₉

yes

characteristic

C₄⋊C₁₆

C₄×C₇₉

yes

—

C₂²⋊C₁₆

C₄×C₇₉

yes

—

#346

C₂²×C₇₉

yes

—

#335

—

#335

—

#334

C₂²×C₇₉

yes

—

#334

C₂²×C₇₉

yes

—

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

—

C₄⋊C₈

—

C₂²⋊C₈

—

100

C₂²⋊C₈

—

101

#97

—

102

C₄×C₈

—

103

C₄×C₈

C₂×C₄×C₇₉

yes

characteristic

104

C₄×C₈

D₄×C₇₉

yes

characteristic

105

C₄×C₈

Q₈×C₇₉

yes

characteristic

106

C₄×C₈

C₂³×C₇₉

yes

characteristic

107

C₂²×C₄

C₄⋊C₄×C₇₉

yes

characteristic

108

C₂²×C₄

SD₁₆×C₇₉

yes

characteristic

109

C₂²×C₄

SD₁₆×C₇₉

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

C₄⋊C₄×C₇₉

yes

characteristic

115

C₂×C₈

C₄⋊C₄×C₇₉

yes

characteristic

116

C₂×C₈

C₂²×C₄×C₇₉

yes

characteristic

117

C₂×C₈

C₂×Q₈×C₇₉

yes

characteristic

118

C₄²

C₄⋊C₄×C₇₉

yes

characteristic

119

C₄²

SD₁₆×C₇₉

yes

characteristic

120

C₄²

SD₁₆×C₇₉

yes

characteristic

121

C₄²

C₂×D₄×C₇₉

yes

characteristic

122

C₁₆

—

123

C₂³

#106×C₇₉

yes

characteristic

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

#106×C₇₉

yes

characteristic

129

C₂×C₄

#106×C₇₉

yes

characteristic

130

C₂×C₄

#106×C₇₉

yes

characteristic

131

C₂×C₄

C₂×C₄⋊C₄×C₇₉

yes

characteristic

132

C₂×C₄

C₂×SD₁₆×C₇₉

yes

characteristic

133

C₂×C₄

C₂×SD₁₆×C₇₉

yes

characteristic

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

#131×C₇₉

yes

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

C₂×#106×C₇₉

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

#424×C₇₉

yes

characteristic

150

C₄

#430×C₇₉

yes

characteristic

151

C₂

—

152

C₂

C₇₉×#1570

yes

characteristic

153

C₂

C₇₉×#2155

yes

characteristic

154

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
73. #12031	C₇₉	direct (N × H)
81. #1368	C₂×C₇₉	semidirect
82. C₇₉	#12031	direct (N × H)

H ≅ G/N

type

4. C₇₉×#1368

C₂

semidirect

73. #12031

C₇₉

direct (N × H)

81. #1368

C₂×C₇₉

semidirect

82. C₇₉

#12031

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×#1568	2
C₇₉×#1564	2
C₇₉×#1368	2
#12031	79

subgroup

index

C₇₉×#2163

C₇₉×#1568

C₇₉×#1564

C₇₉×#1368

#12031

Subgroups — C₇₉×#12031 #34123720

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C79×#12031 #34123720

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₉×#12031 #34123720