Group #34124115 — subgroups

Subgroups — C₇₉×#12426 #34124115

Order 20224.

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

normal subgroups only

This group has 246 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	C₇₉×#2208	C₂	1	2	yes	maximal
2	C₇₉×#1713	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×D₈×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₇₉×#1370	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₉×#1362	C₂	1	2	yes	maximal, complemented
6	C₂×SD₃₂×C₇₉	—	2	2	no	—
7	#500×C₇₉	—	2	1	no	—
8	#500×C₇₉	—	2	1	no	—
9	#494×C₇₉	C₂²	1	2	yes	—
10	C₄×D₈×C₇₉	C₂²	1	1	yes	characteristic
11	C₈×D₄×C₇₉	—	2	1	no	—
12	C₄⋊C₁₆×C₇₉	C₂²	1	2	yes	—
13	#358×C₇₉	—	2	2	no	—
14	#335×C₇₉	C₂²	1	1	yes	characteristic
15	D₄⋊C₈×C₇₉	—	2	1	no	—
16	C₈²×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₇₉	—	2	2	no	—
18	C₂×Q₁₆×C₇₉	—	2	2	no	—
19	C₂×D₈×C₇₉	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₇₉	—	4	1	no	—
21	C₄⋊Q₈×C₇₉	—	2	2	no	—
22	C₄×D₄×C₇₉	—	2	1	no	—
23	SD₃₂×C₇₉	—	4	4	no	—
24	C₂×C₁₆×C₇₉	—	2	2	no	—
25	#107×C₇₉	—	4	2	no	—
26	#107×C₇₉	—	4	1	no	—
27	#107×C₇₉	—	4	1	no	—
28	#107×C₇₉	D₄	1	1	yes	characteristic
29	C₄⋊C₈×C₇₉	—	2	1	no	—
30	#102×C₇₉	—	2	1	no	—
31	C₂²⋊C₈×C₇₉	—	4	1	no	—
32	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
33	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
34	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
35	C₂×Q₈×C₇₉	—	4	2	no	—
36	C₂×D₄×C₇₉	—	2	1	no	—
37	C₂²×C₄×C₇₉	—	4	1	no	—
38	Q₁₆×C₇₉	—	4	2	no	—
39	Q₁₆×C₇₉	—	4	2	no	—
40	Q₁₆×C₇₉	—	8	2	no	—
41	D₈×C₇₉	—	2	1	no	—
42	D₈×C₇₉	D₈	1	2	yes	—
43	C₂×C₈×C₇₉	—	2	1	no	—
44	C₂×C₈×C₇₉	—	4	1	no	—
45	C₂×C₈×C₇₉	—	8	2	no	—
46	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
47	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
48	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
50	C₄⋊C₄×C₇₉	—	2	1	no	—
51	C₄⋊C₄×C₇₉	—	4	2	no	—
52	C₄⋊C₄×C₇₉	—	4	2	no	—
53	C₂²⋊C₄×C₇₉	—	4	1	no	—
54	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
55	C₁₆×C₇₉	—	4	2	no	—
56	C₂³×C₇₉	—	4	1	no	—
57	Q₈×C₇₉	—	8	2	no	—
58	Q₈×C₇₉	—	8	2	no	—
59	D₄×C₇₉	—	2	2	no	—
60	D₄×C₇₉	—	4	1	no	—
61	C₂×C₄×C₇₉	—	4	1	no	—
62	C₂×C₄×C₇₉	—	8	1	no	—
63	C₂×C₄×C₇₉	—	8	1	no	—
64	C₂×C₄×C₇₉	—	8	2	no	—
65	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
66	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
68	C₈×C₇₉	—	2	2	no	—
69	C₈×C₇₉	—	2	1	no	—
70	C₈×C₇₉	—	2	1	no	—
71	C₈×C₇₉	—	8	1	no	—
72	C₈×C₇₉	SD₃₂	1	2	yes	complemented
73	C₈×C₇₉	C₂×D₈	1	1	yes	characteristic
74	C₈×C₇₉	#135	1	1	yes	characteristic
75	C₂²×C₇₉	—	4	2	no	—
76	C₂²×C₇₉	—	8	1	no	—
77	C₂²×C₇₉	—	8	1	no	—
78	C₂²×C₇₉	#459	1	1	yes	characteristic
79	C₄×C₇₉	—	2	1	no	—
80	C₄×C₇₉	—	8	1	no	—
81	C₄×C₇₉	—	16	2	no	—
82	C₄×C₇₉	#466	1	1	yes	characteristic
83	C₄×C₇₉	#472	1	1	yes	characteristic
84	C₄×C₇₉	C₂×SD₃₂	1	1	yes	characteristic
85	C₄×C₇₉	#510	1	1	yes	characteristic
86	#12426	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂×C₇₉	—	8	2	no	—
88	C₂×C₇₉	#1667	1	1	yes	characteristic
89	C₂×C₇₉	#2208	1	1	yes	characteristic
90	C₂×C₇₉	#2214	1	1	yes	characteristic
91	#2208	C₂×C₇₉	1	2	yes	—
92	#1713	C₂×C₇₉	1	1	yes	characteristic, complemented
93	C₈×D₈	C₂×C₇₉	1	1	yes	characteristic
94	#1370	C₂×C₇₉	1	1	yes	characteristic, complemented
95	#1362	C₂×C₇₉	1	2	yes	complemented
96	C₇₉	#12426	1	1	yes	characteristic, direct product factor, Sylow
97	C₂×SD₃₂	—	2	2	no	—
98	#500	—	2	1	no	—
99	#500	—	2	1	no	—
100	#494	C₂²×C₇₉	1	2	yes	—
101	C₄×D₈	C₂²×C₇₉	1	1	yes	characteristic
102	C₈×D₄	—	2	1	no	—
103	C₄⋊C₁₆	C₂²×C₇₉	1	2	yes	—
104	#358	—	2	2	no	—
105	#335	C₂²×C₇₉	1	1	yes	characteristic
106	D₄⋊C₈	—	2	1	no	—
107	C₈²	C₂²×C₇₉	1	1	yes	characteristic
108	C₂×Q₁₆	—	2	2	no	—
109	C₂×Q₁₆	—	2	2	no	—
110	C₂×D₈	D₄×C₇₉	1	1	yes	characteristic
111	C₂²×C₈	—	4	1	no	—
112	C₄⋊Q₈	—	2	2	no	—
113	C₄×D₄	—	2	1	no	—
114	SD₃₂	—	4	4	no	—
115	C₂×C₁₆	—	2	2	no	—
116	#107	—	4	2	no	—
117	#107	—	4	1	no	—
118	#107	—	4	1	no	—
119	#107	D₄×C₇₉	1	1	yes	characteristic
120	C₄⋊C₈	—	2	1	no	—
121	#102	—	2	1	no	—
122	C₂²⋊C₈	—	4	1	no	—
123	C₄×C₈	D₄×C₇₉	1	1	yes	characteristic
124	C₄×C₈	D₄×C₇₉	1	1	yes	characteristic
125	C₄×C₈	C₂³×C₇₉	1	1	yes	characteristic
126	C₂×Q₈	—	4	2	no	—
127	C₂×D₄	—	2	1	no	—
128	C₂²×C₄	—	4	1	no	—
129	Q₁₆	—	4	2	no	—
130	Q₁₆	—	4	2	no	—
131	Q₁₆	—	8	2	no	—
132	D₈	—	2	1	no	—
133	D₈	D₈×C₇₉	1	2	yes	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	4	1	no	—
136	C₂×C₈	—	8	2	no	—
137	C₂×C₈	D₈×C₇₉	1	1	yes	characteristic
138	C₂×C₈	D₈×C₇₉	1	1	yes	characteristic
139	C₂×C₈	C₂×D₄×C₇₉	1	1	yes	characteristic
140	C₂×C₈	#41×C₇₉	1	1	yes	characteristic
141	C₄⋊C₄	—	2	1	no	—
142	C₄⋊C₄	—	4	2	no	—
143	C₄⋊C₄	—	4	2	no	—
144	C₂²⋊C₄	—	4	1	no	—
145	C₄²	C₂×D₄×C₇₉	1	1	yes	characteristic
146	C₁₆	—	4	2	no	—
147	C₂³	—	4	1	no	—
148	Q₈	—	8	2	no	—
149	Q₈	—	8	2	no	—
150	D₄	—	2	2	no	—
151	D₄	—	4	1	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	—	8	1	no	—
154	C₂×C₄	—	8	1	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	#121×C₇₉	1	1	yes	characteristic
157	C₂×C₄	C₂×D₈×C₇₉	1	1	yes	characteristic
158	C₂×C₄	AGL(1,ℤ/8)×C₇₉	1	1	yes	characteristic
159	C₈	—	2	2	no	—
160	C₈	—	2	1	no	—
161	C₈	—	2	1	no	—
162	C₈	—	8	1	no	—
163	C₈	SD₃₂×C₇₉	1	2	yes	complemented
164	C₈	C₂×D₈×C₇₉	1	1	yes	characteristic
165	C₈	#135×C₇₉	1	1	yes	characteristic
166	C₂²	—	4	2	no	—
167	C₂²	—	8	1	no	—
168	C₂²	—	8	1	no	—
169	C₂²	#459×C₇₉	1	1	yes	characteristic
170	C₄	—	2	1	no	—
171	C₄	—	8	1	no	—
172	C₄	—	16	2	no	—
173	C₄	#466×C₇₉	1	1	yes	characteristic
174	C₄	#472×C₇₉	1	1	yes	characteristic
175	C₄	C₂×SD₃₂×C₇₉	1	1	yes	characteristic
176	C₄	#510×C₇₉	1	1	yes	characteristic
177	C₂	—	8	2	no	—
178	C₂	C₇₉×#1667	1	1	yes	characteristic
179	C₂	C₇₉×#2208	1	1	yes	characteristic
180	C₂	C₇₉×#2214	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₉×#2208

C₂

yes

maximal

C₇₉×#1713

C₂

yes

maximal, characteristic, complemented

C₈×D₈×C₇₉

C₂

yes

maximal, characteristic

C₇₉×#1370

C₂

yes

maximal, characteristic, complemented

C₇₉×#1362

C₂

yes

maximal, complemented

C₂×SD₃₂×C₇₉

—

#500×C₇₉

—

#500×C₇₉

—

#494×C₇₉

C₂²

yes

—

C₄×D₈×C₇₉

C₂²

yes

characteristic

C₈×D₄×C₇₉

—

C₄⋊C₁₆×C₇₉

C₂²

yes

—

#358×C₇₉

—

#335×C₇₉

C₂²

yes

characteristic

D₄⋊C₈×C₇₉

—

C₈²×C₇₉

C₂²

yes

characteristic

C₂×Q₁₆×C₇₉

—

C₂×Q₁₆×C₇₉

—

C₂×D₈×C₇₉

D₄

yes

characteristic

C₂²×C₈×C₇₉

—

C₄⋊Q₈×C₇₉

—

C₄×D₄×C₇₉

—

SD₃₂×C₇₉

—

C₂×C₁₆×C₇₉

—

#107×C₇₉

—

#107×C₇₉

—

#107×C₇₉

—

#107×C₇₉

D₄

yes

characteristic

C₄⋊C₈×C₇₉

—

#102×C₇₉

—

C₂²⋊C₈×C₇₉

—

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×Q₈×C₇₉

—

C₂×D₄×C₇₉

—

C₂²×C₄×C₇₉

—

Q₁₆×C₇₉

—

Q₁₆×C₇₉

—

Q₁₆×C₇₉

—

D₈×C₇₉

—

D₈×C₇₉

D₈

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

D₈

yes

characteristic

C₂×C₈×C₇₉

D₈

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

#41

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₁₆×C₇₉

—

C₂³×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#121

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

AGL(1,ℤ/8)

yes

characteristic

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

SD₃₂

yes

complemented

C₈×C₇₉

C₂×D₈

yes

characteristic

C₈×C₇₉

#135

yes

characteristic

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#459

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

C₂×SD₃₂

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

—

#1713

C₂×C₇₉

yes

characteristic, complemented

C₈×D₈

C₂×C₇₉

yes

characteristic

#1370

C₂×C₇₉

yes

characteristic, complemented

#1362

C₂×C₇₉

yes

complemented

C₇₉

#12426

yes

characteristic, direct product factor, Sylow

C₂×SD₃₂

—

#500

—

#500

—

100

#494

C₂²×C₇₉

yes

—

101

C₄×D₈

C₂²×C₇₉

yes

characteristic

102

C₈×D₄

—

103

C₄⋊C₁₆

C₂²×C₇₉

yes

—

104

#358

—

105

#335

C₂²×C₇₉

yes

characteristic

106

D₄⋊C₈

—

107

C₈²

C₂²×C₇₉

yes

characteristic

108

C₂×Q₁₆

—

109

C₂×Q₁₆

—

110

C₂×D₈

D₄×C₇₉

yes

characteristic

111

C₂²×C₈

—

112

C₄⋊Q₈

—

113

C₄×D₄

—

114

SD₃₂

—

115

C₂×C₁₆

—

116

#107

—

117

#107

—

118

#107

—

119

#107

D₄×C₇₉

yes

characteristic

120

C₄⋊C₈

—

121

#102

—

122

C₂²⋊C₈

—

123

C₄×C₈

D₄×C₇₉

yes

characteristic

124

C₄×C₈

D₄×C₇₉

yes

characteristic

125

C₄×C₈

C₂³×C₇₉

yes

characteristic

126

C₂×Q₈

—

127

C₂×D₄

—

128

C₂²×C₄

—

129

Q₁₆

—

130

Q₁₆

—

131

Q₁₆

—

132

D₈

—

133

D₈

D₈×C₇₉

yes

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

D₈×C₇₉

yes

characteristic

138

C₂×C₈

D₈×C₇₉

yes

characteristic

139

C₂×C₈

C₂×D₄×C₇₉

yes

characteristic

140

C₂×C₈

#41×C₇₉

yes

characteristic

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₂²⋊C₄

—

145

C₄²

C₂×D₄×C₇₉

yes

characteristic

146

C₁₆

—

147

C₂³

—

148

Q₈

—

149

Q₈

—

150

D₄

—

151

D₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

#121×C₇₉

yes

characteristic

157

C₂×C₄

C₂×D₈×C₇₉

yes

characteristic

158

C₂×C₄

AGL(1,ℤ/8)×C₇₉

yes

characteristic

159

C₈

—

160

C₈

—

161

C₈

—

162

C₈

—

163

C₈

SD₃₂×C₇₉

yes

complemented

164

C₈

C₂×D₈×C₇₉

yes

characteristic

165

C₈

#135×C₇₉

yes

characteristic

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

#459×C₇₉

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

#466×C₇₉

yes

characteristic

174

C₄

#472×C₇₉

yes

characteristic

175

C₄

C₂×SD₃₂×C₇₉

yes

characteristic

176

C₄

#510×C₇₉

yes

characteristic

177

C₂

—

178

C₂

C₇₉×#1667

yes

characteristic

179

C₂

C₇₉×#2208

yes

characteristic

180

C₂

C₇₉×#2214

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₇₉	#459
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇₉
prime radical	C₂×D₈×C₇₉	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₄²×C₇₉ ◁ C₈²×C₇₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₇₉

#459

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₇₉

prime radical

C₂×D₈×C₇₉

D₄

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₇₉ ◁ C₄²×C₇₉ ◁ C₈²×C₇₉ ◁ G

N	H ≅ G/N	type
2. C₇₉×#1713	C₂	semidirect
4. C₇₉×#1370	C₂	semidirect
5. C₇₉×#1362	C₂	semidirect
72. C₈×C₇₉	SD₃₂	semidirect
86. #12426	C₇₉	direct (N × H)
92. #1713	C₂×C₇₉	semidirect
94. #1370	C₂×C₇₉	semidirect
95. #1362	C₂×C₇₉	semidirect
96. C₇₉	#12426	direct (N × H)
163. C₈	SD₃₂×C₇₉	semidirect

H ≅ G/N

type

2. C₇₉×#1713

C₂

semidirect

4. C₇₉×#1370

C₂

semidirect

5. C₇₉×#1362

C₂

semidirect

72. C₈×C₇₉

SD₃₂

semidirect

86. #12426

C₇₉

direct (N × H)

92. #1713

C₂×C₇₉

semidirect

94. #1370

C₂×C₇₉

semidirect

95. #1362

C₂×C₇₉

semidirect

96. C₇₉

#12426

direct (N × H)

163. C₈

SD₃₂×C₇₉

semidirect

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

subgroup	index
C₇₉×#2208	2
C₇₉×#1713	2
C₈×D₈×C₇₉	2
C₇₉×#1370	2
C₇₉×#1362	2
#12426	79

subgroup

index

C₇₉×#2208

C₇₉×#1713

C₈×D₈×C₇₉

C₇₉×#1370

C₇₉×#1362

#12426

Subgroups — C₇₉×#12426 #34124115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C79×#12426 #34124115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₉×#12426 #34124115