Group #34123728 — subgroups

Subgroups — C₇₉×#12039 #34123728

Order 20224.

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

normal subgroups only

This group has 242 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×#2169	C₂	1	2	yes	maximal, complemented
2	C₈×D₈×C₇₉	C₂	1	1	yes	maximal, characteristic
3	C₇₉×#1370	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₉×D₄⋊C₁₆	C₂	1	2	yes	maximal, complemented
5	C₈×C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×M₃₂×C₇₉	—	2	2	no	—
7	C₄×D₈×C₇₉	C₄	1	1	yes	characteristic
8	C₈×D₄×C₇₉	C₂²	1	2	yes	—
9	C₄⋊C₁₆×C₇₉	C₂²	1	2	yes	—
10	C₂²⋊C₁₆×C₇₉	—	2	2	no	—
11	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
12	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
13	#335×C₇₉	C₄	1	1	yes	characteristic
14	D₄⋊C₈×C₇₉	C₄	1	2	yes	—
15	C₈²×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×D₈×C₇₉	C₈	1	1	yes	characteristic
17	C₂²×C₈×C₇₉	—	2	2	no	—
18	C₄×D₄×C₇₉	C₂×C₄	1	2	yes	—
19	M₃₂×C₇₉	—	4	4	no	—
20	C₂×C₁₆×C₇₉	—	2	1	no	—
21	C₂×C₁₆×C₇₉	—	2	2	no	—
22	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
23	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
24	#107×C₇₉	C₈	1	1	yes	characteristic
25	C₄⋊C₈×C₇₉	C₂×C₄	1	2	yes	—
26	#102×C₇₉	C₈	1	2	yes	—
27	C₂²⋊C₈×C₇₉	—	2	2	no	—
28	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
31	C₂×D₄×C₇₉	C₂×C₈	1	2	yes	—
32	C₂²×C₄×C₇₉	—	2	2	no	—
33	D₈×C₇₉	—	2	2	no	—
34	C₂×C₈×C₇₉	—	2	2	no	—
35	C₂×C₈×C₇₉	—	2	1	no	—
36	C₂×C₈×C₇₉	—	4	2	no	—
37	C₂×C₈×C₇₉	—	4	2	no	—
38	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
39	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
40	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
42	C₄⋊C₄×C₇₉	C₂×C₈	1	2	yes	—
43	C₂²⋊C₄×C₇₉	—	2	2	no	—
44	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
45	C₁₆×C₇₉	—	2	2	no	—
46	C₁₆×C₇₉	—	2	1	no	—
47	C₁₆×C₇₉	—	4	2	no	—
48	C₁₆×C₇₉	D₈	1	2	yes	complemented
49	C₂³×C₇₉	—	2	2	no	—
50	D₄×C₇₉	—	2	2	no	—
51	D₄×C₇₉	—	2	2	no	—
52	C₂×C₄×C₇₉	—	2	2	no	—
53	C₂×C₄×C₇₉	—	4	2	no	—
54	C₂×C₄×C₇₉	—	4	2	no	—
55	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
56	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
57	C₂×C₄×C₇₉	#131	1	1	yes	characteristic
58	C₈×C₇₉	—	2	2	no	—
59	C₈×C₇₉	—	2	1	no	—
60	C₈×C₇₉	—	2	1	no	—
61	C₈×C₇₉	—	4	2	no	—
62	C₈×C₇₉	M₃₂	1	2	yes	complemented
63	C₈×C₇₉	C₂×D₈	1	1	yes	characteristic
64	C₈×C₇₉	#135	1	1	yes	characteristic
65	C₂²×C₇₉	—	4	2	no	—
66	C₂²×C₇₉	—	4	2	no	—
67	C₂²×C₇₉	—	4	2	no	—
68	C₂²×C₇₉	C₈×D₄	1	1	yes	characteristic
69	C₄×C₇₉	—	2	1	no	—
70	C₄×C₇₉	—	4	2	no	—
71	C₄×C₇₉	C₄×D₈	1	1	yes	characteristic
72	C₄×C₇₉	#443	1	1	yes	characteristic
73	C₄×C₇₉	C₂×M₃₂	1	1	yes	characteristic
74	C₄×C₇₉	#504	1	1	yes	characteristic
75	#12039	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂×C₇₉	—	8	2	no	—
77	C₂×C₇₉	C₈×D₈	1	1	yes	characteristic
78	C₂×C₇₉	#2169	1	1	yes	characteristic
79	C₂×C₇₉	#2170	1	1	yes	characteristic
80	#2169	C₂×C₇₉	1	2	yes	complemented
81	C₈×D₈	C₂×C₇₉	1	1	yes	characteristic
82	#1370	C₂×C₇₉	1	1	yes	characteristic, complemented
83	D₄⋊C₁₆	C₂×C₇₉	1	2	yes	complemented
84	C₈×C₁₆	C₂×C₇₉	1	1	yes	characteristic, complemented
85	C₇₉	#12039	1	1	yes	characteristic, direct product factor, Sylow
86	C₂×M₃₂	—	2	2	no	—
87	C₄×D₈	C₄×C₇₉	1	1	yes	characteristic
88	C₈×D₄	C₂²×C₇₉	1	2	yes	—
89	C₄⋊C₁₆	C₂²×C₇₉	1	2	yes	—
90	C₂²⋊C₁₆	—	2	2	no	—
91	C₄×C₁₆	C₂²×C₇₉	1	1	yes	characteristic
92	C₄×C₁₆	C₂²×C₇₉	1	1	yes	characteristic
93	#335	C₄×C₇₉	1	1	yes	characteristic
94	D₄⋊C₈	C₄×C₇₉	1	2	yes	—
95	C₈²	C₂²×C₇₉	1	1	yes	characteristic
96	C₂×D₈	C₈×C₇₉	1	1	yes	characteristic
97	C₂²×C₈	—	2	2	no	—
98	C₄×D₄	C₂×C₄×C₇₉	1	2	yes	—
99	M₃₂	—	4	4	no	—
100	C₂×C₁₆	—	2	1	no	—
101	C₂×C₁₆	—	2	2	no	—
102	C₂×C₁₆	D₄×C₇₉	1	1	yes	characteristic
103	C₂×C₁₆	D₄×C₇₉	1	1	yes	characteristic
104	#107	C₈×C₇₉	1	1	yes	characteristic
105	C₄⋊C₈	C₂×C₄×C₇₉	1	2	yes	—
106	#102	C₈×C₇₉	1	2	yes	—
107	C₂²⋊C₈	—	2	2	no	—
108	C₄×C₈	C₂×C₄×C₇₉	1	1	yes	characteristic
109	C₄×C₈	C₂×C₄×C₇₉	1	1	yes	characteristic
110	C₄×C₈	C₂³×C₇₉	1	1	yes	characteristic
111	C₂×D₄	C₂×C₈×C₇₉	1	2	yes	—
112	C₂²×C₄	—	2	2	no	—
113	D₈	—	2	2	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	4	2	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	C₂×C₈×C₇₉	1	1	yes	characteristic
119	C₂×C₈	C₂×C₈×C₇₉	1	1	yes	characteristic
120	C₂×C₈	C₂×D₄×C₇₉	1	1	yes	characteristic
121	C₂×C₈	#41×C₇₉	1	1	yes	characteristic
122	C₄⋊C₄	C₂×C₈×C₇₉	1	2	yes	—
123	C₂²⋊C₄	—	2	2	no	—
124	C₄²	C₂²×C₄×C₇₉	1	1	yes	characteristic
125	C₁₆	—	2	2	no	—
126	C₁₆	—	2	1	no	—
127	C₁₆	—	4	2	no	—
128	C₁₆	D₈×C₇₉	1	2	yes	complemented
129	C₂³	—	2	2	no	—
130	D₄	—	2	2	no	—
131	D₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	C₄×D₄×C₇₉	1	1	yes	characteristic
136	C₂×C₄	C₂²×C₈×C₇₉	1	1	yes	characteristic
137	C₂×C₄	#131×C₇₉	1	1	yes	characteristic
138	C₈	—	2	2	no	—
139	C₈	—	2	1	no	—
140	C₈	—	2	1	no	—
141	C₈	—	4	2	no	—
142	C₈	M₃₂×C₇₉	1	2	yes	complemented
143	C₈	C₂×D₈×C₇₉	1	1	yes	characteristic
144	C₈	#135×C₇₉	1	1	yes	characteristic
145	C₂²	—	4	2	no	—
146	C₂²	—	4	2	no	—
147	C₂²	—	4	2	no	—
148	C₂²	C₈×D₄×C₇₉	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	4	2	no	—
151	C₄	C₄×D₈×C₇₉	1	1	yes	characteristic
152	C₄	#443×C₇₉	1	1	yes	characteristic
153	C₄	C₂×M₃₂×C₇₉	1	1	yes	characteristic
154	C₄	#504×C₇₉	1	1	yes	characteristic
155	C₂	—	8	2	no	—
156	C₂	C₈×D₈×C₇₉	1	1	yes	characteristic
157	C₂	C₇₉×#2169	1	1	yes	characteristic
158	C₂	C₇₉×#2170	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₉×#2169

C₂

yes

maximal, complemented

C₈×D₈×C₇₉

C₂

yes

maximal, characteristic

C₇₉×#1370

C₂

yes

maximal, characteristic, complemented

C₇₉×D₄⋊C₁₆

C₂

yes

maximal, complemented

C₈×C₁₆×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×M₃₂×C₇₉

—

C₄×D₈×C₇₉

C₄

yes

characteristic

C₈×D₄×C₇₉

C₂²

yes

—

C₄⋊C₁₆×C₇₉

C₂²

yes

—

C₂²⋊C₁₆×C₇₉

—

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

#335×C₇₉

C₄

yes

characteristic

D₄⋊C₈×C₇₉

C₄

yes

—

C₈²×C₇₉

C₂²

yes

characteristic

C₂×D₈×C₇₉

C₈

yes

characteristic

C₂²×C₈×C₇₉

—

C₄×D₄×C₇₉

C₂×C₄

yes

—

M₃₂×C₇₉

—

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

D₄

yes

characteristic

#107×C₇₉

C₈

yes

characteristic

C₄⋊C₈×C₇₉

C₂×C₄

yes

—

#102×C₇₉

C₈

yes

—

C₂²⋊C₈×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×D₄×C₇₉

C₂×C₈

yes

—

C₂²×C₄×C₇₉

—

D₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂×C₈

yes

characteristic

C₂×C₈×C₇₉

C₂×C₈

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

#41

yes

characteristic

C₄⋊C₄×C₇₉

C₂×C₈

yes

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₁₆×C₇₉

—

C₁₆×C₇₉

—

C₁₆×C₇₉

—

C₁₆×C₇₉

D₈

yes

complemented

C₂³×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₄×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₉

#131

yes

characteristic

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

M₃₂

yes

complemented

C₈×C₇₉

C₂×D₈

yes

characteristic

C₈×C₇₉

#135

yes

characteristic

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₈×D₄

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₄×D₈

yes

characteristic

C₄×C₇₉

#443

yes

characteristic

C₄×C₇₉

C₂×M₃₂

yes

C₄×C₇₉

yes

C₇₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₉

—

C₂×C₇₉

C₈×D₈

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

complemented

C₈×D₈

C₂×C₇₉

yes

characteristic

#1370

C₂×C₇₉

yes

characteristic, complemented

D₄⋊C₁₆

C₂×C₇₉

yes

complemented

C₈×C₁₆

C₂×C₇₉

yes

characteristic, complemented

C₇₉

#12039

yes

characteristic, direct product factor, Sylow

C₂×M₃₂

—

C₄×D₈

C₄×C₇₉

yes

characteristic

C₈×D₄

C₂²×C₇₉

yes

—

C₄⋊C₁₆

C₂²×C₇₉

yes

—

C₂²⋊C₁₆

—

C₄×C₁₆

C₂²×C₇₉

yes

characteristic

C₄×C₁₆

C₂²×C₇₉

yes

characteristic

#335

C₄×C₇₉

yes

characteristic

D₄⋊C₈

C₄×C₇₉

yes

—

C₈²

C₂²×C₇₉

yes

characteristic

C₂×D₈

C₈×C₇₉

yes

characteristic

C₂²×C₈

—

C₄×D₄

C₂×C₄×C₇₉

yes

—

M₃₂

—

100

C₂×C₁₆

—

101

C₂×C₁₆

—

102

C₂×C₁₆

D₄×C₇₉

yes

characteristic

103

C₂×C₁₆

D₄×C₇₉

yes

characteristic

104

#107

C₈×C₇₉

yes

characteristic

105

C₄⋊C₈

C₂×C₄×C₇₉

yes

—

106

#102

C₈×C₇₉

yes

—

107

C₂²⋊C₈

—

108

C₄×C₈

C₂×C₄×C₇₉

yes

characteristic

109

C₄×C₈

C₂×C₄×C₇₉

yes

characteristic

110

C₄×C₈

C₂³×C₇₉

yes

characteristic

111

C₂×D₄

C₂×C₈×C₇₉

yes

—

112

C₂²×C₄

—

113

D₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

C₂×C₈×C₇₉

yes

characteristic

119

C₂×C₈

C₂×C₈×C₇₉

yes

characteristic

120

C₂×C₈

C₂×D₄×C₇₉

yes

characteristic

121

C₂×C₈

#41×C₇₉

yes

characteristic

122

C₄⋊C₄

C₂×C₈×C₇₉

yes

—

123

C₂²⋊C₄

—

124

C₄²

C₂²×C₄×C₇₉

yes

characteristic

125

C₁₆

—

126

C₁₆

—

127

C₁₆

—

128

C₁₆

D₈×C₇₉

yes

complemented

129

C₂³

—

130

D₄

—

131

D₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

C₄×D₄×C₇₉

yes

characteristic

136

C₂×C₄

C₂²×C₈×C₇₉

yes

characteristic

137

C₂×C₄

#131×C₇₉

yes

characteristic

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

M₃₂×C₇₉

yes

complemented

143

C₈

C₂×D₈×C₇₉

yes

characteristic

144

C₈

#135×C₇₉

yes

characteristic

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

C₈×D₄×C₇₉

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

C₄×D₈×C₇₉

yes

characteristic

152

C₄

#443×C₇₉

yes

characteristic

153

C₄

C₂×M₃₂×C₇₉

yes

characteristic

154

C₄

#504×C₇₉

yes

characteristic

155

C₂

—

156

C₂

C₈×D₈×C₇₉

yes

characteristic

157

C₂

C₇₉×#2169

yes

characteristic

158

C₂

C₇₉×#2170

yes

characteristic

	subgroup	G/H
center	C₂×C₈×C₇₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₇₉
Frattini, cosocle	C₄×C₈	C₂³×C₇₉
socle	C₂²×C₇₉	C₈×D₄
prime radical	C₂×D₈×C₇₉	C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₇₉ ◁ C₄×C₁₆×C₇₉ ◁ G

subgroup

G/H

center

C₂×C₈×C₇₉

C₂×D₄

commutator

C₂×C₄

C₂²×C₈×C₇₉

Frattini, cosocle

C₄×C₈

C₂³×C₇₉

socle

C₂²×C₇₉

C₈×D₄

prime radical

C₂×D₈×C₇₉

C₈

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₈×C₇₉ ◁ C₄×C₁₆×C₇₉ ◁ G

N	H ≅ G/N	type
1. C₇₉×#2169	C₂	semidirect
3. C₇₉×#1370	C₂	semidirect
4. C₇₉×D₄⋊C₁₆	C₂	semidirect
5. C₈×C₁₆×C₇₉	C₂	semidirect
48. C₁₆×C₇₉	D₈	semidirect
62. C₈×C₇₉	M₃₂	semidirect
75. #12039	C₇₉	direct (N × H)
80. #2169	C₂×C₇₉	semidirect
82. #1370	C₂×C₇₉	semidirect
83. D₄⋊C₁₆	C₂×C₇₉	semidirect
84. C₈×C₁₆	C₂×C₇₉	semidirect
85. C₇₉	#12039	direct (N × H)
128. C₁₆	D₈×C₇₉	semidirect
142. C₈	M₃₂×C₇₉	semidirect

H ≅ G/N

type

1. C₇₉×#2169

C₂

semidirect

3. C₇₉×#1370

C₂

semidirect

4. C₇₉×D₄⋊C₁₆

C₂

semidirect

5. C₈×C₁₆×C₇₉

C₂

semidirect

48. C₁₆×C₇₉

D₈

semidirect

62. C₈×C₇₉

M₃₂

semidirect

75. #12039

C₇₉

direct (N × H)

80. #2169

C₂×C₇₉

semidirect

82. #1370

C₂×C₇₉

semidirect

83. D₄⋊C₁₆

C₂×C₇₉

semidirect

84. C₈×C₁₆

C₂×C₇₉

semidirect

85. C₇₉

#12039

direct (N × H)

128. C₁₆

D₈×C₇₉

semidirect

142. C₈

M₃₂×C₇₉

semidirect

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

subgroup	index
C₇₉×#2169	2
C₈×D₈×C₇₉	2
C₇₉×#1370	2
C₇₉×D₄⋊C₁₆	2
C₈×C₁₆×C₇₉	2
#12039	79

subgroup

index

C₇₉×#2169

C₈×D₈×C₇₉

C₇₉×#1370

C₇₉×D₄⋊C₁₆

C₈×C₁₆×C₇₉

#12039

Subgroups — C₇₉×#12039 #34123728

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C79×#12039 #34123728

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₉×#12039 #34123728