Group #23759320 — subgroups

Subgroups — C₇₉×#1698 #23759320

Order 10112.

156 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 64 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	#486×C₇₉	C₂	1	1	yes	maximal, characteristic
2	#477×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#460×C₇₉	C₂	1	1	yes	maximal, characteristic
4	#446×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#335×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#333×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#331×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂×SD₁₆×C₇₉	—	2	1	no	—
9	C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
10	#127×C₇₉	C₂²	1	1	yes	characteristic
11	C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
12	#107×C₇₉	—	2	1	no	—
13	C₄⋊C₈×C₇₉	—	2	1	no	—
14	C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
17	#103×C₇₉	—	2	1	no	—
18	#102×C₇₉	—	2	1	no	—
19	#102×C₇₉	—	2	1	no	—
20	#102×C₇₉	—	2	1	no	—
21	#97×C₇₉	—	2	1	no	—
22	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₇₉	—	2	1	no	—
24	C₂×Q₈×C₇₉	D₄	1	1	yes	characteristic
25	C₂×D₄×C₇₉	—	2	1	no	—
26	C₂×D₄×C₇₉	—	2	1	no	—
27	C₂×D₄×C₇₉	—	2	1	no	—
28	SD₁₆×C₇₉	—	4	2	no	—
29	C₂×C₈×C₇₉	—	2	1	no	—
30	C₂×C₈×C₇₉	—	2	1	no	—
31	C₂×C₈×C₇₉	—	2	1	no	—
32	C₂×C₈×C₇₉	D₄	1	1	yes	characteristic
33	C₂×C₈×C₇₉	D₄	1	1	yes	characteristic
34	C₄⋊C₄×C₇₉	—	2	1	no	—
35	C₄⋊C₄×C₇₉	—	2	1	no	—
36	C₄⋊C₄×C₇₉	—	2	1	no	—
37	C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
38	C₄²×C₇₉	—	2	1	no	—
39	C₄²×C₇₉	C₂³	1	1	yes	characteristic
40	C₂³×C₇₉	—	4	1	no	—
41	Q₈×C₇₉	—	2	1	no	—
42	Q₈×C₇₉	—	2	1	no	—
43	Q₈×C₇₉	—	8	1	no	—
44	D₄×C₇₉	—	4	1	no	—
45	D₄×C₇₉	—	4	1	no	—
46	D₄×C₇₉	—	4	1	no	—
47	D₄×C₇₉	—	4	1	no	—
48	D₄×C₇₉	—	8	1	no	—
49	C₂×C₄×C₇₉	—	2	1	no	—
50	C₂×C₄×C₇₉	—	2	1	no	—
51	C₂×C₄×C₇₉	—	4	1	no	—
52	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
55	C₈×C₇₉	—	2	1	no	—
56	C₈×C₇₉	—	2	1	no	—
57	C₈×C₇₉	—	4	1	no	—
58	C₈×C₇₉	—	4	1	no	—
59	C₈×C₇₉	—	4	1	no	—
60	C₂²×C₇₉	—	8	1	no	—
61	C₂²×C₇₉	—	8	1	no	—
62	C₂²×C₇₉	—	8	1	no	—
63	C₂²×C₇₉	#121	1	1	yes	characteristic
64	C₄×C₇₉	—	2	1	no	—
65	C₄×C₇₉	—	4	1	no	—
66	C₄×C₇₉	—	4	1	no	—
67	C₄×C₇₉	—	8	1	no	—
68	C₄×C₇₉	#135	1	1	yes	characteristic
69	C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
70	C₄×C₇₉	#137	1	1	yes	characteristic
71	C₄×C₇₉	#137	1	1	yes	characteristic
72	C₂×C₇₉	—	16	1	no	—
73	C₂×C₇₉	#464	1	1	yes	characteristic
74	C₂×C₇₉	#468	1	1	yes	characteristic
75	C₂×C₇₉	#472	1	1	yes	characteristic
76	#1698	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₇₉	#1698	1	1	yes	characteristic, direct product factor, Sylow
78	#486	C₂×C₇₉	1	1	yes	characteristic
79	#477	C₂×C₇₉	1	1	yes	characteristic, complemented
80	#460	C₂×C₇₉	1	1	yes	characteristic
81	#446	C₂×C₇₉	1	1	yes	characteristic, complemented
82	#335	C₂×C₇₉	1	1	yes	characteristic, complemented
83	#333	C₂×C₇₉	1	1	yes	characteristic, complemented
84	#331	C₂×C₇₉	1	1	yes	characteristic
85	C₂×SD₁₆	—	2	1	no	—
86	C₄⋊Q₈	C₂²×C₇₉	1	1	yes	characteristic
87	#127	C₂²×C₇₉	1	1	yes	characteristic
88	C₄×Q₈	C₂²×C₇₉	1	1	yes	characteristic
89	#107	—	2	1	no	—
90	C₄⋊C₈	—	2	1	no	—
91	C₄⋊C₈	C₂²×C₇₉	1	1	yes	characteristic
92	C₄⋊C₈	C₂²×C₇₉	1	1	yes	characteristic
93	C₄⋊C₈	C₂²×C₇₉	1	1	yes	characteristic
94	#103	—	2	1	no	—
95	#102	—	2	1	no	—
96	#102	—	2	1	no	—
97	#102	—	2	1	no	—
98	#97	—	2	1	no	—
99	C₄×C₈	C₂²×C₇₉	1	1	yes	characteristic
100	C₂×Q₈	—	2	1	no	—
101	C₂×Q₈	D₄×C₇₉	1	1	yes	characteristic
102	C₂×D₄	—	2	1	no	—
103	C₂×D₄	—	2	1	no	—
104	C₂×D₄	—	2	1	no	—
105	SD₁₆	—	4	2	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	D₄×C₇₉	1	1	yes	characteristic
110	C₂×C₈	D₄×C₇₉	1	1	yes	characteristic
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	D₄×C₇₉	1	1	yes	characteristic
115	C₄²	—	2	1	no	—
116	C₄²	C₂³×C₇₉	1	1	yes	characteristic
117	C₂³	—	4	1	no	—
118	Q₈	—	2	1	no	—
119	Q₈	—	2	1	no	—
120	Q₈	—	8	1	no	—
121	D₄	—	4	1	no	—
122	D₄	—	4	1	no	—
123	D₄	—	4	1	no	—
124	D₄	—	4	1	no	—
125	D₄	—	8	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	C₂×D₄×C₇₉	1	1	yes	characteristic
130	C₂×C₄	C₂×D₄×C₇₉	1	1	yes	characteristic
131	C₂×C₄	#41×C₇₉	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	2	1	no	—
134	C₈	—	4	1	no	—
135	C₈	—	4	1	no	—
136	C₈	—	4	1	no	—
137	C₂²	—	8	1	no	—
138	C₂²	—	8	1	no	—
139	C₂²	—	8	1	no	—
140	C₂²	#121×C₇₉	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	4	1	no	—
143	C₄	—	4	1	no	—
144	C₄	—	8	1	no	—
145	C₄	#135×C₇₉	1	1	yes	characteristic
146	C₄	AGL(1,ℤ/8)×C₇₉	1	1	yes	characteristic
147	C₄	#137×C₇₉	1	1	yes	characteristic
148	C₄	#137×C₇₉	1	1	yes	characteristic
149	C₂	—	16	1	no	—
150	C₂	#464×C₇₉	1	1	yes	characteristic
151	C₂	#468×C₇₉	1	1	yes	characteristic
152	C₂	#472×C₇₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#486×C₇₉

C₂

yes

maximal, characteristic

#477×C₇₉

C₂

yes

maximal, characteristic, complemented

#460×C₇₉

C₂

yes

maximal, characteristic

#446×C₇₉

C₂

yes

maximal, characteristic, complemented

#335×C₇₉

C₂

yes

maximal, characteristic, complemented

#333×C₇₉

C₂

yes

maximal, characteristic, complemented

#331×C₇₉

C₂

yes

maximal, characteristic

C₂×SD₁₆×C₇₉

—

C₄⋊Q₈×C₇₉

C₂²

yes

characteristic

#127×C₇₉

C₂²

yes

characteristic

C₄×Q₈×C₇₉

C₂²

yes

characteristic

#107×C₇₉

—

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

C₂²

yes

characteristic

#103×C₇₉

—

#102×C₇₉

—

#102×C₇₉

—

#102×C₇₉

—

#97×C₇₉

—

C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

D₄

yes

characteristic

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

SD₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

D₄

yes

characteristic

C₂×C₈×C₇₉

D₄

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

D₄

yes

characteristic

C₄²×C₇₉

—

C₄²×C₇₉

C₂³

yes

characteristic

C₂³×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

#41

yes

characteristic

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#121

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₇₉

yes

maximal, characteristic, direct product factor, Sylow

C₇₉

#1698

yes

characteristic, direct product factor, Sylow

#486

C₂×C₇₉

yes

characteristic

#477

C₂×C₇₉

yes

characteristic, complemented

#460

C₂×C₇₉

yes

characteristic

#446

C₂×C₇₉

yes

characteristic, complemented

#335

C₂×C₇₉

yes

characteristic, complemented

#333

C₂×C₇₉

yes

characteristic, complemented

#331

C₂×C₇₉

yes

characteristic

C₂×SD₁₆

—

C₄⋊Q₈

C₂²×C₇₉

yes

characteristic

#127

C₂²×C₇₉

yes

characteristic

C₄×Q₈

C₂²×C₇₉

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

C₂²×C₇₉

yes

characteristic

C₄⋊C₈

C₂²×C₇₉

yes

characteristic

C₄⋊C₈

C₂²×C₇₉

yes

—

—

—

—

—

C₄×C₈

C₂²×C₇₉

yes

characteristic

100

C₂×Q₈

—

101

C₂×Q₈

D₄×C₇₉

yes

characteristic

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

—

105

SD₁₆

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

D₄×C₇₉

yes

characteristic

110

C₂×C₈

D₄×C₇₉

yes

characteristic

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

D₄×C₇₉

yes

characteristic

115

C₄²

—

116

C₄²

C₂³×C₇₉

yes

characteristic

117

C₂³

—

118

Q₈

—

119

Q₈

—

120

Q₈

—

121

D₄

—

122

D₄

—

123

D₄

—

124

D₄

—

125

D₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂×D₄×C₇₉

yes

characteristic

130

C₂×C₄

C₂×D₄×C₇₉

yes

characteristic

131

C₂×C₄

#41×C₇₉

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

#121×C₇₉

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

#135×C₇₉

yes

characteristic

146

C₄

AGL(1,ℤ/8)×C₇₉

yes

characteristic

147

C₄

#137×C₇₉

yes

characteristic

148

C₄

#137×C₇₉

yes

characteristic

149

C₂

—

150

C₂

#464×C₇₉

yes

characteristic

151

C₂

#468×C₇₉

yes

characteristic

152

C₂

#472×C₇₉

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₇₉	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₇₉
prime radical	#127×C₇₉	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₄²×C₇₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₇₉

#121

commutator, Frattini, cosocle

C₄²

C₂³×C₇₉

prime radical

#127×C₇₉

C₂²

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₇₉ ◁ C₄²×C₇₉ ◁ G

N	H ≅ G/N	type
2. #477×C₇₉	C₂	semidirect
4. #446×C₇₉	C₂	semidirect
5. #335×C₇₉	C₂	semidirect
6. #333×C₇₉	C₂	semidirect
76. #1698	C₇₉	direct (N × H)
77. C₇₉	#1698	direct (N × H)
79. #477	C₂×C₇₉	semidirect
81. #446	C₂×C₇₉	semidirect
82. #335	C₂×C₇₉	semidirect
83. #333	C₂×C₇₉	semidirect

H ≅ G/N

type

2. #477×C₇₉

C₂

semidirect

4. #446×C₇₉

C₂

semidirect

5. #335×C₇₉

C₂

semidirect

6. #333×C₇₉

C₂

semidirect

76. #1698

C₇₉

direct (N × H)

77. C₇₉

#1698

direct (N × H)

79. #477

C₂×C₇₉

semidirect

81. #446

C₂×C₇₉

semidirect

82. #335

C₂×C₇₉

semidirect

83. #333

C₂×C₇₉

semidirect

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

subgroup	index
#486×C₇₉	2
#477×C₇₉	2
#460×C₇₉	2
#446×C₇₉	2
#335×C₇₉	2
#333×C₇₉	2
#331×C₇₉	2
#1698	79

subgroup

index

#486×C₇₉

#477×C₇₉

#460×C₇₉

#446×C₇₉

#335×C₇₉

#333×C₇₉

#331×C₇₉

#1698

Subgroups — C₇₉×#1698 #23759320

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C79×#1698 #23759320

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₉×#1698 #23759320