Group #34125419 — subgroups

Subgroups — C₇₉×#13730 #34125419

Order 20224.

232 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 86 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	C₂×Q₆₄×C₇₉	C₂	1	2	yes	maximal, complemented
2	C₂×SD₆₄×C₇₉	C₂	1	2	yes	maximal
3	C₇₉×#2249	C₂	1	1	yes	maximal, characteristic
4	C₇₉×#2247	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₉×#1397	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₃₂×C₇₉	—	2	1	no	—
7	C₂×Q₃₂×C₇₉	C₂²	1	2	yes	—
8	C₂×Q₃₂×C₇₉	C₂²	1	1	yes	characteristic
9	C₂×SD₃₂×C₇₉	—	2	1	no	—
10	C₂×D₁₆×C₇₉	C₂²	1	1	yes	characteristic
11	#495×C₇₉	—	2	1	no	—
12	#494×C₇₉	—	2	1	no	—
13	Q₆₄×C₇₉	—	2	4	no	—
14	SD₆₄×C₇₉	—	2	4	no	—
15	C₂×C₃₂×C₇₉	C₂²	1	2	yes	—
16	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₇₉	—	2	2	no	—
18	C₂×Q₁₆×C₇₉	—	2	1	no	—
19	C₂×Q₁₆×C₇₉	—	4	1	no	—
20	C₂×SD₁₆×C₇₉	—	4	1	no	—
21	C₂×D₈×C₇₉	—	2	1	no	—
22	C₄⋊Q₈×C₇₉	—	4	1	no	—
23	#124×C₇₉	—	4	1	no	—
24	Q₃₂×C₇₉	—	2	2	no	—
25	Q₃₂×C₇₉	—	2	2	no	—
26	Q₃₂×C₇₉	—	2	2	no	—
27	Q₃₂×C₇₉	—	4	2	no	—
28	SD₃₂×C₇₉	—	4	2	no	—
29	D₁₆×C₇₉	—	2	2	no	—
30	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
31	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
33	C₃₂×C₇₉	D₄	1	4	yes	—
34	C₂×Q₈×C₇₉	—	4	1	no	—
35	C₂×Q₈×C₇₉	—	4	2	no	—
36	C₂×D₄×C₇₉	—	4	1	no	—
37	Q₁₆×C₇₉	—	4	2	no	—
38	Q₁₆×C₇₉	—	4	1	no	—
39	Q₁₆×C₇₉	—	4	1	no	—
40	Q₁₆×C₇₉	—	4	2	no	—
41	Q₁₆×C₇₉	—	8	2	no	—
42	SD₁₆×C₇₉	—	8	2	no	—
43	D₈×C₇₉	—	4	1	no	—
44	D₈×C₇₉	—	4	1	no	—
45	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
46	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
47	C₄⋊C₄×C₇₉	—	8	2	no	—
48	C₂²⋊C₄×C₇₉	—	8	2	no	—
49	C₄²×C₇₉	D₈	1	1	yes	characteristic
50	C₁₆×C₇₉	D₈	1	2	yes	—
51	C₁₆×C₇₉	C₂×D₄	1	2	yes	—
52	C₂³×C₇₉	—	8	1	no	—
53	Q₈×C₇₉	—	8	1	no	—
54	Q₈×C₇₉	—	8	2	no	—
55	Q₈×C₇₉	—	8	1	no	—
56	Q₈×C₇₉	—	8	2	no	—
57	D₄×C₇₉	—	8	1	no	—
58	D₄×C₇₉	—	8	1	no	—
59	C₂×C₄×C₇₉	—	8	1	no	—
60	C₂×C₄×C₇₉	—	8	2	no	—
61	C₂×C₄×C₇₉	D₁₆	1	2	yes	—
62	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
63	C₈×C₇₉	D₁₆	1	2	yes	—
64	C₈×C₇₉	#127	1	1	yes	characteristic
65	C₈×C₇₉	C₂×D₈	1	1	yes	characteristic
66	C₂²×C₇₉	—	16	1	no	—
67	C₂²×C₇₉	—	16	2	no	—
68	C₂²×C₇₉	C₂×D₁₆	1	1	yes	characteristic
69	C₄×C₇₉	—	2	2	no	—
70	C₄×C₇₉	—	16	1	no	—
71	C₄×C₇₉	—	16	2	no	—
72	C₄×C₇₉	#493	1	1	yes	characteristic
73	C₄×C₇₉	C₂×D₁₆	1	1	yes	characteristic
74	#13730	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂×C₇₉	—	32	1	no	—
76	C₂×C₇₉	#2246	1	1	yes	characteristic
77	C₂×C₇₉	#2264	1	2	yes	—
78	C₂×Q₆₄	C₂×C₇₉	1	2	yes	complemented
79	C₂×SD₆₄	C₂×C₇₉	1	2	yes	—
80	#2249	C₂×C₇₉	1	1	yes	characteristic
81	#2247	C₂×C₇₉	1	1	yes	characteristic, complemented
82	#1397	C₂×C₇₉	1	1	yes	characteristic, complemented
83	C₇₉	#13730	1	1	yes	characteristic, direct product factor, Sylow
84	C₂×Q₃₂	—	2	1	no	—
85	C₂×Q₃₂	C₂²×C₇₉	1	2	yes	—
86	C₂×Q₃₂	C₂²×C₇₉	1	1	yes	characteristic
87	C₂×SD₃₂	—	2	1	no	—
88	C₂×D₁₆	C₂²×C₇₉	1	1	yes	characteristic
89	#495	—	2	1	no	—
90	#494	—	2	1	no	—
91	Q₆₄	—	2	4	no	—
92	SD₆₄	—	2	4	no	—
93	C₂×C₃₂	C₂²×C₇₉	1	2	yes	—
94	C₄×C₁₆	C₂²×C₇₉	1	1	yes	characteristic
95	C₂×Q₁₆	—	2	2	no	—
96	C₂×Q₁₆	—	2	1	no	—
97	C₂×Q₁₆	—	4	1	no	—
98	C₂×SD₁₆	—	4	1	no	—
99	C₂×D₈	—	2	1	no	—
100	C₄⋊Q₈	—	4	1	no	—
101	#124	—	4	1	no	—
102	Q₃₂	—	2	2	no	—
103	Q₃₂	—	2	2	no	—
104	Q₃₂	—	2	2	no	—
105	Q₃₂	—	4	2	no	—
106	SD₃₂	—	4	2	no	—
107	D₁₆	—	2	2	no	—
108	C₂×C₁₆	D₄×C₇₉	1	1	yes	characteristic
109	C₂×C₁₆	C₂³×C₇₉	1	1	yes	characteristic
110	C₄×C₈	D₄×C₇₉	1	1	yes	characteristic
111	C₃₂	D₄×C₇₉	1	4	yes	—
112	C₂×Q₈	—	4	1	no	—
113	C₂×Q₈	—	4	2	no	—
114	C₂×D₄	—	4	1	no	—
115	Q₁₆	—	4	2	no	—
116	Q₁₆	—	4	1	no	—
117	Q₁₆	—	4	1	no	—
118	Q₁₆	—	4	2	no	—
119	Q₁₆	—	8	2	no	—
120	SD₁₆	—	8	2	no	—
121	D₈	—	4	1	no	—
122	D₈	—	4	1	no	—
123	C₂×C₈	D₈×C₇₉	1	1	yes	characteristic
124	C₂×C₈	C₂×D₄×C₇₉	1	1	yes	characteristic
125	C₄⋊C₄	—	8	2	no	—
126	C₂²⋊C₄	—	8	2	no	—
127	C₄²	D₈×C₇₉	1	1	yes	characteristic
128	C₁₆	D₈×C₇₉	1	2	yes	—
129	C₁₆	C₂×D₄×C₇₉	1	2	yes	—
130	C₂³	—	8	1	no	—
131	Q₈	—	8	1	no	—
132	Q₈	—	8	2	no	—
133	Q₈	—	8	1	no	—
134	Q₈	—	8	2	no	—
135	D₄	—	8	1	no	—
136	D₄	—	8	1	no	—
137	C₂×C₄	—	8	1	no	—
138	C₂×C₄	—	8	2	no	—
139	C₂×C₄	D₁₆×C₇₉	1	2	yes	—
140	C₂×C₄	C₂×D₈×C₇₉	1	1	yes	characteristic
141	C₈	D₁₆×C₇₉	1	2	yes	—
142	C₈	#127×C₇₉	1	1	yes	characteristic
143	C₈	C₂×D₈×C₇₉	1	1	yes	characteristic
144	C₂²	—	16	2	no	—
145	C₂²	—	16	1	no	—
146	C₂²	C₂×D₁₆×C₇₉	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	16	1	no	—
149	C₄	—	16	2	no	—
150	C₄	#493×C₇₉	1	1	yes	characteristic
151	C₄	C₂×D₁₆×C₇₉	1	1	yes	characteristic
152	C₂	—	32	1	no	—
153	C₂	C₇₉×#2246	1	1	yes	characteristic
154	C₂	C₇₉×#2264	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₆₄×C₇₉

C₂

yes

maximal, complemented

C₂×SD₆₄×C₇₉

C₂

yes

maximal

C₇₉×#2249

C₂

yes

maximal, characteristic

C₇₉×#2247

C₂

yes

maximal, characteristic, complemented

C₇₉×#1397

C₂

yes

maximal, characteristic, complemented

C₂×Q₃₂×C₇₉

—

C₂×Q₃₂×C₇₉

C₂²

yes

—

C₂×Q₃₂×C₇₉

C₂²

yes

characteristic

C₂×SD₃₂×C₇₉

—

C₂×D₁₆×C₇₉

C₂²

yes

characteristic

#495×C₇₉

—

#494×C₇₉

—

Q₆₄×C₇₉

—

SD₆₄×C₇₉

—

C₂×C₃₂×C₇₉

C₂²

yes

—

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

C₂×Q₁₆×C₇₉

—

C₂×Q₁₆×C₇₉

—

C₂×Q₁₆×C₇₉

—

C₂×SD₁₆×C₇₉

—

C₂×D₈×C₇₉

—

C₄⋊Q₈×C₇₉

—

#124×C₇₉

—

Q₃₂×C₇₉

—

Q₃₂×C₇₉

—

Q₃₂×C₇₉

—

Q₃₂×C₇₉

—

SD₃₂×C₇₉

—

D₁₆×C₇₉

—

C₂×C₁₆×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

C₂³

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₃₂×C₇₉

D₄

yes

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×D₄×C₇₉

—

Q₁₆×C₇₉

—

Q₁₆×C₇₉

—

Q₁₆×C₇₉

—

Q₁₆×C₇₉

—

Q₁₆×C₇₉

—

SD₁₆×C₇₉

—

D₈×C₇₉

—

D₈×C₇₉

—

C₂×C₈×C₇₉

D₈

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

D₈

yes

characteristic

C₁₆×C₇₉

D₈

yes

—

C₁₆×C₇₉

C₂×D₄

yes

—

C₂³×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

D₁₆

yes

—

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₈×C₇₉

D₁₆

yes

—

C₈×C₇₉

#127

yes

characteristic

C₈×C₇₉

C₂×D₈

yes

characteristic

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×D₁₆

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#493

yes

characteristic

C₄×C₇₉

C₂×D₁₆

yes

characteristic

#13730

C₇₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₉

—

C₂×C₇₉

#2246

yes

characteristic

C₂×C₇₉

#2264

yes

—

C₂×Q₆₄

C₂×C₇₉

yes

complemented

C₂×SD₆₄

C₂×C₇₉

yes

—

#2249

C₂×C₇₉

yes

characteristic

#2247

C₂×C₇₉

yes

characteristic, complemented

#1397

C₂×C₇₉

yes

characteristic, complemented

C₇₉

#13730

yes

characteristic, direct product factor, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

C₂²×C₇₉

yes

—

C₂×Q₃₂

C₂²×C₇₉

yes

characteristic

C₂×SD₃₂

—

C₂×D₁₆

C₂²×C₇₉

yes

characteristic

#495

—

#494

—

Q₆₄

—

SD₆₄

—

C₂×C₃₂

C₂²×C₇₉

yes

—

C₄×C₁₆

C₂²×C₇₉

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

100

C₄⋊Q₈

—

101

#124

—

102

Q₃₂

—

103

Q₃₂

—

104

Q₃₂

—

105

Q₃₂

—

106

SD₃₂

—

107

D₁₆

—

108

C₂×C₁₆

D₄×C₇₉

yes

characteristic

109

C₂×C₁₆

C₂³×C₇₉

yes

characteristic

110

C₄×C₈

D₄×C₇₉

yes

characteristic

111

C₃₂

D₄×C₇₉

yes

—

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂×D₄

—

115

Q₁₆

—

116

Q₁₆

—

117

Q₁₆

—

118

Q₁₆

—

119

Q₁₆

—

120

SD₁₆

—

121

D₈

—

122

D₈

—

123

C₂×C₈

D₈×C₇₉

yes

characteristic

124

C₂×C₈

C₂×D₄×C₇₉

yes

characteristic

125

C₄⋊C₄

—

126

C₂²⋊C₄

—

127

C₄²

D₈×C₇₉

yes

characteristic

128

C₁₆

D₈×C₇₉

yes

—

129

C₁₆

C₂×D₄×C₇₉

yes

—

130

C₂³

—

131

Q₈

—

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

D₄

—

136

D₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

D₁₆×C₇₉

yes

—

140

C₂×C₄

C₂×D₈×C₇₉

yes

characteristic

141

C₈

D₁₆×C₇₉

yes

—

142

C₈

#127×C₇₉

yes

characteristic

143

C₈

C₂×D₈×C₇₉

yes

characteristic

144

C₂²

—

145

C₂²

—

146

C₂²

C₂×D₁₆×C₇₉

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

#493×C₇₉

yes

characteristic

151

C₄

C₂×D₁₆×C₇₉

yes

characteristic

152

C₂

—

153

C₂

C₇₉×#2246

yes

characteristic

154

C₂

C₇₉×#2264

yes

—

	subgroup	G/H
center, socle	C₂²×C₇₉	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₇₉
prime radical	C₂×D₁₆×C₇₉	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₄²×C₇₉ ◁ C₄×C₈×C₇₉ ◁ C₄×C₁₆×C₇₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₇₉

C₂×D₁₆

commutator, Frattini, cosocle

C₂×C₁₆

C₂³×C₇₉

prime radical

C₂×D₁₆×C₇₉

C₂²

LCS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₇₉ ◁ C₄²×C₇₉ ◁ C₄×C₈×C₇₉ ◁ C₄×C₁₆×C₇₉ ◁ G

N	H ≅ G/N	type
1. C₂×Q₆₄×C₇₉	C₂	semidirect
4. C₇₉×#2247	C₂	semidirect
5. C₇₉×#1397	C₂	semidirect
74. #13730	C₇₉	direct (N × H)
78. C₂×Q₆₄	C₂×C₇₉	semidirect
81. #2247	C₂×C₇₉	semidirect
82. #1397	C₂×C₇₉	semidirect
83. C₇₉	#13730	direct (N × H)

H ≅ G/N

type

1. C₂×Q₆₄×C₇₉

C₂

semidirect

4. C₇₉×#2247

C₂

semidirect

5. C₇₉×#1397

C₂

semidirect

74. #13730

C₇₉

direct (N × H)

78. C₂×Q₆₄

C₂×C₇₉

semidirect

81. #2247

C₂×C₇₉

semidirect

82. #1397

C₂×C₇₉

semidirect

83. C₇₉

#13730

direct (N × H)

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

subgroup	index
C₂×Q₆₄×C₇₉	2
C₂×SD₆₄×C₇₉	2
C₇₉×#2249	2
C₇₉×#2247	2
C₇₉×#1397	2
#13730	79

subgroup

index

C₂×Q₆₄×C₇₉

C₂×SD₆₄×C₇₉

C₇₉×#2249

C₇₉×#2247

C₇₉×#1397

#13730

Subgroups — C₇₉×#13730 #34125419

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C79×#13730 #34125419

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₉×#13730 #34125419