Group #34123729 — subgroups

Subgroups — C₈×D₁₆×C₇₉ #34123729

Order 20224.

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

normal subgroups only

This group has 258 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	C₄×D₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
2	C₈×D₈×C₇₉	C₂	1	2	yes	maximal, complemented
3	C₇₉×#1371	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₉×#1331	C₂	1	2	yes	maximal, complemented
5	C₈×C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₆×C₇₉	C₄	1	1	yes	characteristic
7	C₄×D₈×C₇₉	C₂²	1	2	yes	—
8	C₈×D₄×C₇₉	—	2	2	no	—
9	#366×C₇₉	C₄	1	1	yes	characteristic
10	#357×C₇₉	C₄	1	2	yes	—
11	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
12	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
13	#335×C₇₉	C₂²	1	2	yes	—
14	D₄⋊C₈×C₇₉	—	2	2	no	—
15	C₈²×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×D₈×C₇₉	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₇₉	—	4	2	no	—
18	C₄×D₄×C₇₉	—	2	2	no	—
19	D₁₆×C₇₉	C₈	1	4	yes	direct product factor
20	C₂×C₁₆×C₇₉	—	2	1	no	—
21	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
23	#107×C₇₉	C₂×C₄	1	2	yes	—
24	C₄⋊C₈×C₇₉	—	2	2	no	—
25	#102×C₇₉	—	2	2	no	—
26	C₂²⋊C₈×C₇₉	—	4	2	no	—
27	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
28	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
29	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
30	C₂×D₄×C₇₉	—	2	2	no	—
31	C₂²×C₄×C₇₉	—	4	2	no	—
32	D₈×C₇₉	—	2	2	no	—
33	D₈×C₇₉	C₂×C₈	1	4	yes	complemented
34	C₂×C₈×C₇₉	—	2	1	no	—
35	C₂×C₈×C₇₉	—	4	2	no	—
36	C₂×C₈×C₇₉	—	8	4	no	—
37	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
38	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
39	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
41	C₄⋊C₄×C₇₉	—	2	2	no	—
42	C₂²⋊C₄×C₇₉	—	4	2	no	—
43	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
44	C₁₆×C₇₉	—	2	2	no	—
45	C₁₆×C₇₉	—	2	1	no	—
46	C₁₆×C₇₉	C₂×C₈	1	2	yes	complemented
47	C₂³×C₇₉	—	4	2	no	—
48	D₄×C₇₉	—	2	4	no	—
49	D₄×C₇₉	—	4	2	no	—
50	C₂×C₄×C₇₉	—	4	2	no	—
51	C₂×C₄×C₇₉	—	8	2	no	—
52	C₂×C₄×C₇₉	—	8	2	no	—
53	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
54	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
56	C₈×C₇₉	—	2	2	no	—
57	C₈×C₇₉	—	2	1	no	—
58	C₈×C₇₉	—	2	1	no	—
59	C₈×C₇₉	—	8	2	no	—
60	C₈×C₇₉	D₁₆	1	2	yes	direct product factor
61	C₈×C₇₉	C₂²×C₈	1	1	yes	characteristic
62	C₈×C₇₉	#131	1	1	yes	characteristic
63	C₂²×C₇₉	—	4	4	no	—
64	C₂²×C₇₉	—	8	2	no	—
65	C₂²×C₇₉	—	8	2	no	—
66	C₂²×C₇₉	C₄×D₈	1	1	yes	characteristic
67	C₄×C₇₉	—	2	1	no	—
68	C₄×C₇₉	—	8	2	no	—
69	C₄×C₇₉	C₈×D₄	1	1	yes	characteristic
70	C₄×C₇₉	#443	1	1	yes	characteristic
71	C₄×C₇₉	C₂×D₁₆	1	1	yes	characteristic
72	C₄×C₇₉	#508	1	1	yes	characteristic
73	C₈×D₁₆	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂×C₇₉	—	8	4	no	—
75	C₂×C₇₉	C₈×D₈	1	1	yes	characteristic
76	C₂×C₇₉	C₄×D₁₆	1	1	yes	characteristic
77	C₂×C₇₉	#2178	1	1	yes	characteristic
78	C₄×D₁₆	C₂×C₇₉	1	1	yes	characteristic
79	C₈×D₈	C₂×C₇₉	1	2	yes	complemented
80	#1371	C₂×C₇₉	1	1	yes	characteristic, complemented
81	#1331	C₂×C₇₉	1	2	yes	complemented
82	C₈×C₁₆	C₂×C₇₉	1	1	yes	characteristic, complemented
83	C₇₉	C₈×D₁₆	1	1	yes	characteristic, direct product factor, Sylow
84	C₂×D₁₆	C₄×C₇₉	1	1	yes	characteristic
85	C₄×D₈	C₂²×C₇₉	1	2	yes	—
86	C₈×D₄	—	2	2	no	—
87	#366	C₄×C₇₉	1	1	yes	characteristic
88	#357	C₄×C₇₉	1	2	yes	—
89	C₄×C₁₆	C₂²×C₇₉	1	1	yes	characteristic
90	C₄×C₁₆	C₂²×C₇₉	1	1	yes	characteristic
91	#335	C₂²×C₇₉	1	2	yes	—
92	D₄⋊C₈	—	2	2	no	—
93	C₈²	C₂²×C₇₉	1	1	yes	characteristic
94	C₂×D₈	C₂×C₄×C₇₉	1	2	yes	—
95	C₂²×C₈	—	4	2	no	—
96	C₄×D₄	—	2	2	no	—
97	D₁₆	C₈×C₇₉	1	4	yes	direct product factor
98	C₂×C₁₆	—	2	1	no	—
99	C₂×C₁₆	C₂×C₄×C₇₉	1	1	yes	characteristic
100	C₂×C₁₆	C₂×C₄×C₇₉	1	1	yes	characteristic
101	#107	C₂×C₄×C₇₉	1	2	yes	—
102	C₄⋊C₈	—	2	2	no	—
103	#102	—	2	2	no	—
104	C₂²⋊C₈	—	4	2	no	—
105	C₄×C₈	D₄×C₇₉	1	1	yes	characteristic
106	C₄×C₈	D₄×C₇₉	1	1	yes	characteristic
107	C₄×C₈	C₂³×C₇₉	1	1	yes	characteristic
108	C₂×D₄	—	2	2	no	—
109	C₂²×C₄	—	4	2	no	—
110	D₈	—	2	2	no	—
111	D₈	C₂×C₈×C₇₉	1	4	yes	complemented
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	4	2	no	—
114	C₂×C₈	—	8	4	no	—
115	C₂×C₈	D₈×C₇₉	1	1	yes	characteristic
116	C₂×C₈	D₈×C₇₉	1	1	yes	characteristic
117	C₂×C₈	C₂²×C₄×C₇₉	1	1	yes	characteristic
118	C₂×C₈	#41×C₇₉	1	1	yes	characteristic
119	C₄⋊C₄	—	2	2	no	—
120	C₂²⋊C₄	—	4	2	no	—
121	C₄²	C₂×D₄×C₇₉	1	1	yes	characteristic
122	C₁₆	—	2	2	no	—
123	C₁₆	—	2	1	no	—
124	C₁₆	C₂×C₈×C₇₉	1	2	yes	complemented
125	C₂³	—	4	2	no	—
126	D₄	—	2	4	no	—
127	D₄	—	4	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	8	2	no	—
130	C₂×C₄	—	8	2	no	—
131	C₂×C₄	C₄×D₄×C₇₉	1	1	yes	characteristic
132	C₂×C₄	C₂×D₈×C₇₉	1	1	yes	characteristic
133	C₂×C₄	#135×C₇₉	1	1	yes	characteristic
134	C₈	—	2	2	no	—
135	C₈	—	2	1	no	—
136	C₈	—	2	1	no	—
137	C₈	—	8	2	no	—
138	C₈	D₁₆×C₇₉	1	2	yes	direct product factor
139	C₈	C₂²×C₈×C₇₉	1	1	yes	characteristic
140	C₈	#131×C₇₉	1	1	yes	characteristic
141	C₂²	—	4	4	no	—
142	C₂²	—	8	2	no	—
143	C₂²	—	8	2	no	—
144	C₂²	C₄×D₈×C₇₉	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	8	2	no	—
147	C₄	C₈×D₄×C₇₉	1	1	yes	characteristic
148	C₄	#443×C₇₉	1	1	yes	characteristic
149	C₄	C₂×D₁₆×C₇₉	1	1	yes	characteristic
150	C₄	#508×C₇₉	1	1	yes	characteristic
151	C₂	—	8	4	no	—
152	C₂	C₈×D₈×C₇₉	1	1	yes	characteristic
153	C₂	C₄×D₁₆×C₇₉	1	1	yes	characteristic
154	C₂	C₇₉×#2178	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₁₆×C₇₉

C₂

yes

maximal, characteristic

C₈×D₈×C₇₉

C₂

yes

maximal, complemented

C₇₉×#1371

C₂

yes

maximal, characteristic, complemented

C₇₉×#1331

C₂

yes

maximal, complemented

C₈×C₁₆×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×D₁₆×C₇₉

C₄

yes

characteristic

C₄×D₈×C₇₉

C₂²

yes

—

C₈×D₄×C₇₉

—

#366×C₇₉

C₄

yes

characteristic

#357×C₇₉

C₄

yes

—

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

#335×C₇₉

C₂²

yes

—

D₄⋊C₈×C₇₉

—

C₈²×C₇₉

C₂²

yes

characteristic

C₂×D₈×C₇₉

C₂×C₄

yes

—

C₂²×C₈×C₇₉

—

C₄×D₄×C₇₉

—

D₁₆×C₇₉

C₈

yes

direct product factor

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₉

C₂×C₄

yes

characteristic

#107×C₇₉

C₂×C₄

yes

—

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×D₄×C₇₉

—

C₂²×C₄×C₇₉

—

D₈×C₇₉

—

D₈×C₇₉

C₂×C₈

yes

complemented

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

D₈

yes

characteristic

C₂×C₈×C₇₉

D₈

yes

characteristic

C₂×C₈×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

#41

yes

characteristic

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₁₆×C₇₉

—

C₁₆×C₇₉

—

C₁₆×C₇₉

C₂×C₈

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

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

#135

yes

characteristic

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

D₁₆

yes

direct product factor

C₈×C₇₉

C₂²×C₈

yes

characteristic

C₈×C₇₉

#131

yes

characteristic

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₄×D₈

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₈×D₄

yes

characteristic

C₄×C₇₉

#443

yes

characteristic

C₄×C₇₉

C₂×D₁₆

yes

characteristic

C₄×C₇₉

#508

yes

characteristic

C₈×D₁₆

C₇₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₉

—

C₂×C₇₉

C₈×D₈

yes

characteristic

C₂×C₇₉

C₄×D₁₆

yes

characteristic

C₂×C₇₉

#2178

yes

characteristic

C₄×D₁₆

C₂×C₇₉

yes

characteristic

C₈×D₈

C₂×C₇₉

yes

complemented

#1371

C₂×C₇₉

yes

characteristic, complemented

#1331

C₂×C₇₉

yes

complemented

C₈×C₁₆

C₂×C₇₉

yes

characteristic, complemented

C₇₉

C₈×D₁₆

yes

characteristic, direct product factor, Sylow

C₂×D₁₆

C₄×C₇₉

yes

characteristic

C₄×D₈

C₂²×C₇₉

yes

—

C₈×D₄

—

#366

C₄×C₇₉

yes

characteristic

#357

C₄×C₇₉

yes

—

C₄×C₁₆

C₂²×C₇₉

yes

characteristic

C₄×C₁₆

C₂²×C₇₉

yes

characteristic

#335

C₂²×C₇₉

yes

—

D₄⋊C₈

—

C₈²

C₂²×C₇₉

yes

characteristic

C₂×D₈

C₂×C₄×C₇₉

yes

—

C₂²×C₈

—

C₄×D₄

—

D₁₆

C₈×C₇₉

yes

direct product factor

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄×C₇₉

yes

characteristic

100

C₂×C₁₆

C₂×C₄×C₇₉

yes

characteristic

101

#107

C₂×C₄×C₇₉

yes

—

102

C₄⋊C₈

—

103

#102

—

104

C₂²⋊C₈

—

105

C₄×C₈

D₄×C₇₉

yes

characteristic

106

C₄×C₈

D₄×C₇₉

yes

characteristic

107

C₄×C₈

C₂³×C₇₉

yes

characteristic

108

C₂×D₄

—

109

C₂²×C₄

—

110

D₈

—

111

D₈

C₂×C₈×C₇₉

yes

complemented

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

D₈×C₇₉

yes

characteristic

116

C₂×C₈

D₈×C₇₉

yes

characteristic

117

C₂×C₈

C₂²×C₄×C₇₉

yes

characteristic

118

C₂×C₈

#41×C₇₉

yes

characteristic

119

C₄⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

C₂×D₄×C₇₉

yes

characteristic

122

C₁₆

—

123

C₁₆

—

124

C₁₆

C₂×C₈×C₇₉

yes

complemented

125

C₂³

—

126

D₄

—

127

D₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

C₄×D₄×C₇₉

yes

characteristic

132

C₂×C₄

C₂×D₈×C₇₉

yes

characteristic

133

C₂×C₄

#135×C₇₉

yes

characteristic

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

D₁₆×C₇₉

yes

direct product factor

139

C₈

C₂²×C₈×C₇₉

yes

characteristic

140

C₈

#131×C₇₉

yes

characteristic

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

C₄×D₈×C₇₉

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

C₈×D₄×C₇₉

yes

characteristic

148

C₄

#443×C₇₉

yes

characteristic

149

C₄

C₂×D₁₆×C₇₉

yes

characteristic

150

C₄

#508×C₇₉

yes

characteristic

151

C₂

—

152

C₂

C₈×D₈×C₇₉

yes

characteristic

153

C₂

C₄×D₁₆×C₇₉

yes

characteristic

154

C₂

C₇₉×#2178

yes

characteristic

	subgroup	G/H
center	C₂×C₈×C₇₉	D₈
commutator	C₈	C₂²×C₈×C₇₉
Frattini, cosocle	C₄×C₈	C₂³×C₇₉
socle	C₂²×C₇₉	C₄×D₈
prime radical	C₂×D₁₆×C₇₉	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₇₉ ◁ C₄×C₈×C₇₉ ◁ C₈²×C₇₉ ◁ G

subgroup

G/H

center

C₂×C₈×C₇₉

D₈

commutator

C₈

C₂²×C₈×C₇₉

Frattini, cosocle

C₄×C₈

C₂³×C₇₉

socle

C₂²×C₇₉

C₄×D₈

prime radical

C₂×D₁₆×C₇₉

C₄

LCS

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₈×C₇₉ ◁ C₄×C₈×C₇₉ ◁ C₈²×C₇₉ ◁ G

N	H ≅ G/N	type
2. C₈×D₈×C₇₉	C₂	semidirect
3. C₇₉×#1371	C₂	semidirect
4. C₇₉×#1331	C₂	semidirect
5. C₈×C₁₆×C₇₉	C₂	semidirect
19. D₁₆×C₇₉	C₈	direct (N × H)
33. D₈×C₇₉	C₂×C₈	semidirect
46. C₁₆×C₇₉	C₂×C₈	semidirect
60. C₈×C₇₉	D₁₆	direct (N × H)
73. C₈×D₁₆	C₇₉	direct (N × H)
79. C₈×D₈	C₂×C₇₉	semidirect
80. #1371	C₂×C₇₉	semidirect
81. #1331	C₂×C₇₉	semidirect
82. C₈×C₁₆	C₂×C₇₉	semidirect
83. C₇₉	C₈×D₁₆	direct (N × H)
97. D₁₆	C₈×C₇₉	direct (N × H)
111. D₈	C₂×C₈×C₇₉	semidirect
124. C₁₆	C₂×C₈×C₇₉	semidirect
138. C₈	D₁₆×C₇₉	direct (N × H)

H ≅ G/N

type

2. C₈×D₈×C₇₉

C₂

semidirect

3. C₇₉×#1371

C₂

semidirect

4. C₇₉×#1331

C₂

semidirect

5. C₈×C₁₆×C₇₉

C₂

semidirect

19. D₁₆×C₇₉

C₈

direct (N × H)

33. D₈×C₇₉

C₂×C₈

semidirect

46. C₁₆×C₇₉

C₂×C₈

semidirect

60. C₈×C₇₉

D₁₆

direct (N × H)

73. C₈×D₁₆

C₇₉

direct (N × H)

79. C₈×D₈

C₂×C₇₉

semidirect

80. #1371

C₂×C₇₉

semidirect

81. #1331

C₂×C₇₉

semidirect

82. C₈×C₁₆

C₂×C₇₉

semidirect

83. C₇₉

C₈×D₁₆

direct (N × H)

97. D₁₆

C₈×C₇₉

direct (N × H)

111. D₈

C₂×C₈×C₇₉

semidirect

124. C₁₆

C₂×C₈×C₇₉

semidirect

138. C₈

D₁₆×C₇₉

direct (N × H)

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

subgroup	index
C₄×D₁₆×C₇₉	2
C₈×D₈×C₇₉	2
C₇₉×#1371	2
C₇₉×#1331	2
C₈×C₁₆×C₇₉	2
C₈×D₁₆	79

subgroup

index

C₄×D₁₆×C₇₉

C₈×D₈×C₇₉

C₇₉×#1371

C₇₉×#1331

C₈×C₁₆×C₇₉

C₈×D₁₆

Subgroups — C₈×D₁₆×C₇₉ #34123729

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C8×D16×C79 #34123729

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×D₁₆×C₇₉ #34123729