Group #23759809 — subgroups

Subgroups — C₇₉×#2187 #23759809

Order 10112.

256 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 72 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	C₂²×D₈×C₇₉	C₂	1	1	yes	maximal, characteristic
2	C₂×SD₃₂×C₇₉	C₂	1	2	yes	maximal, complemented
3	#467×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#357×C₇₉	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄×C₇₉	—	2	1	no	—
7	C₂×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic, complemented
8	C₂×D₈×C₇₉	—	2	4	no	—
9	C₂×D₈×C₇₉	—	2	1	no	—
10	C₂×D₈×C₇₉	C₂²	1	2	yes	—
11	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
12	C₂²⋊Q₈×C₇₉	—	2	1	no	—
13	SD₃₂×C₇₉	—	2	4	no	—
14	C₂×C₁₆×C₇₉	C₂²	1	2	yes	complemented
15	#107×C₇₉	C₂²	1	1	yes	characteristic, complemented
16	#103×C₇₉	—	2	1	no	—
17	C₂⁴×C₇₉	—	4	1	no	—
18	C₂×Q₈×C₇₉	—	2	1	no	—
19	C₂×D₄×C₇₉	—	2	4	no	—
20	C₂×D₄×C₇₉	—	2	2	no	—
21	C₂×D₄×C₇₉	—	4	2	no	—
22	C₂×D₄×C₇₉	—	4	1	no	—
23	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
24	Q₁₆×C₇₉	—	2	2	no	—
25	D₈×C₇₉	—	2	4	no	—
26	D₈×C₇₉	—	2	2	no	—
27	D₈×C₇₉	D₄	1	4	yes	—
28	C₂×C₈×C₇₉	—	2	2	no	—
29	C₂×C₈×C₇₉	D₄	1	1	yes	characteristic
30	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₇₉	—	2	1	no	—
32	C₄⋊C₄×C₇₉	—	4	1	no	—
33	C₂²⋊C₄×C₇₉	—	4	1	no	—
34	C₁₆×C₇₉	—	2	2	no	—
35	C₂³×C₇₉	—	4	4	no	—
36	C₂³×C₇₉	—	4	2	no	—
37	C₂³×C₇₉	—	8	2	no	—
38	C₂³×C₇₉	—	8	2	no	—
39	C₂³×C₇₉	D₈	1	1	yes	characteristic
40	Q₈×C₇₉	—	4	1	no	—
41	Q₈×C₇₉	—	4	1	no	—
42	D₄×C₇₉	—	2	4	no	—
43	D₄×C₇₉	—	4	4	no	—
44	D₄×C₇₉	—	4	2	no	—
45	C₂×C₄×C₇₉	—	2	1	no	—
46	C₂×C₄×C₇₉	—	2	1	no	—
47	C₂×C₄×C₇₉	—	4	1	no	—
48	C₂×C₄×C₇₉	—	4	1	no	—
49	C₂×C₄×C₇₉	D₈	1	1	yes	characteristic
50	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
51	C₈×C₇₉	—	2	1	no	—
52	C₈×C₇₉	C₂×D₄	1	2	yes	—
53	C₂²×C₇₉	—	2	1	no	—
54	C₂²×C₇₉	—	2	1	no	—
55	C₂²×C₇₉	—	4	4	no	—
56	C₂²×C₇₉	—	8	8	no	—
57	C₂²×C₇₉	—	8	2	no	—
58	C₂²×C₇₉	—	8	2	no	—
59	C₂²×C₇₉	SD₃₂	1	2	yes	complemented
60	C₂²×C₇₉	C₂×D₈	1	1	yes	characteristic
61	C₄×C₇₉	—	2	1	no	—
62	C₄×C₇₉	—	8	1	no	—
63	C₄×C₇₉	—	8	1	no	—
64	C₄×C₇₉	#120	1	1	yes	characteristic
65	C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₂×C₇₉	—	2	2	no	—
67	C₂×C₇₉	—	8	4	no	—
68	C₂×C₇₉	#447	1	1	yes	characteristic
69	C₂×C₇₉	C₂×SD₃₂	1	1	yes	characteristic
70	C₂×C₇₉	#509	1	1	yes	characteristic
71	#2187	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₇₉	#2187	1	1	yes	characteristic, direct product factor, Sylow
73	C₂²×D₈	C₂×C₇₉	1	1	yes	characteristic
74	C₂×SD₃₂	C₂×C₇₉	1	2	yes	complemented
75	#467	C₂×C₇₉	1	1	yes	characteristic, complemented
76	#357	C₂×C₇₉	1	2	yes	complemented
77	C₂²⋊C₁₆	C₂×C₇₉	1	1	yes	characteristic, complemented
78	C₂²×D₄	—	2	1	no	—
79	C₂×Q₁₆	C₂²×C₇₉	1	1	yes	characteristic, complemented
80	C₂×D₈	—	2	4	no	—
81	C₂×D₈	—	2	1	no	—
82	C₂×D₈	C₂²×C₇₉	1	2	yes	—
83	C₂²×C₈	C₂²×C₇₉	1	1	yes	characteristic
84	C₂²⋊Q₈	—	2	1	no	—
85	SD₃₂	—	2	4	no	—
86	C₂×C₁₆	C₂²×C₇₉	1	2	yes	complemented
87	#107	C₂²×C₇₉	1	1	yes	characteristic, complemented
88	#103	—	2	1	no	—
89	C₂⁴	—	4	1	no	—
90	C₂×Q₈	—	2	1	no	—
91	C₂×D₄	—	2	4	no	—
92	C₂×D₄	—	2	2	no	—
93	C₂×D₄	—	4	2	no	—
94	C₂×D₄	—	4	1	no	—
95	C₂²×C₄	D₄×C₇₉	1	1	yes	characteristic
96	Q₁₆	—	2	2	no	—
97	D₈	—	2	4	no	—
98	D₈	—	2	2	no	—
99	D₈	D₄×C₇₉	1	4	yes	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	D₄×C₇₉	1	1	yes	characteristic
102	C₂×C₈	C₂³×C₇₉	1	1	yes	characteristic
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	4	1	no	—
105	C₂²⋊C₄	—	4	1	no	—
106	C₁₆	—	2	2	no	—
107	C₂³	—	4	4	no	—
108	C₂³	—	4	2	no	—
109	C₂³	—	8	2	no	—
110	C₂³	—	8	2	no	—
111	C₂³	D₈×C₇₉	1	1	yes	characteristic
112	Q₈	—	4	1	no	—
113	Q₈	—	4	1	no	—
114	D₄	—	2	4	no	—
115	D₄	—	4	4	no	—
116	D₄	—	4	2	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	D₈×C₇₉	1	1	yes	characteristic
122	C₂×C₄	C₂×D₄×C₇₉	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	C₂×D₄×C₇₉	1	2	yes	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	—	4	4	no	—
128	C₂²	—	8	8	no	—
129	C₂²	—	8	2	no	—
130	C₂²	—	8	2	no	—
131	C₂²	SD₃₂×C₇₉	1	2	yes	complemented
132	C₂²	C₂×D₈×C₇₉	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	8	1	no	—
135	C₄	—	8	1	no	—
136	C₄	#120×C₇₉	1	1	yes	characteristic
137	C₄	AGL(1,ℤ/8)×C₇₉	1	1	yes	characteristic
138	C₂	—	2	2	no	—
139	C₂	—	8	4	no	—
140	C₂	#447×C₇₉	1	1	yes	characteristic
141	C₂	C₂×SD₃₂×C₇₉	1	1	yes	characteristic
142	C₂	#509×C₇₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₈×C₇₉

C₂

yes

maximal, characteristic

C₂×SD₃₂×C₇₉

C₂

yes

maximal, complemented

#467×C₇₉

C₂

yes

maximal, characteristic, complemented

#357×C₇₉

C₂

yes

maximal, complemented

C₂²⋊C₁₆×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₇₉

—

C₂×Q₁₆×C₇₉

C₂²

yes

characteristic, complemented

C₂×D₈×C₇₉

—

C₂×D₈×C₇₉

—

C₂×D₈×C₇₉

C₂²

yes

—

C₂²×C₈×C₇₉

C₂²

yes

characteristic

C₂²⋊Q₈×C₇₉

—

SD₃₂×C₇₉

—

C₂×C₁₆×C₇₉

C₂²

yes

complemented

#107×C₇₉

C₂²

yes

characteristic, complemented

#103×C₇₉

—

C₂⁴×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂²×C₄×C₇₉

D₄

yes

characteristic

Q₁₆×C₇₉

—

D₈×C₇₉

—

D₈×C₇₉

—

D₈×C₇₉

D₄

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

D₄

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₁₆×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

D₈

yes

characteristic

Q₈×C₇₉

—

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

D₈

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₈×C₇₉

—

C₈×C₇₉

C₂×D₄

yes

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

SD₃₂

yes

complemented

C₂²×C₇₉

C₂×D₈

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#447

yes

characteristic

C₂×C₇₉

C₂×SD₃₂

yes

C₂×C₇₉

yes

C₇₉

yes

maximal, characteristic, direct product factor, Sylow

C₇₉

#2187

yes

characteristic, direct product factor, Sylow

C₂²×D₈

C₂×C₇₉

yes

characteristic

C₂×SD₃₂

C₂×C₇₉

yes

complemented

#467

C₂×C₇₉

yes

characteristic, complemented

#357

C₂×C₇₉

yes

complemented

C₂²⋊C₁₆

C₂×C₇₉

yes

characteristic, complemented

C₂²×D₄

—

C₂×Q₁₆

C₂²×C₇₉

yes

characteristic, complemented

C₂×D₈

—

C₂×D₈

—

C₂×D₈

C₂²×C₇₉

yes

—

C₂²×C₈

C₂²×C₇₉

yes

characteristic

C₂²⋊Q₈

—

SD₃₂

—

C₂×C₁₆

C₂²×C₇₉

yes

complemented

#107

C₂²×C₇₉

yes

characteristic, complemented

#103

—

C₂⁴

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

D₄×C₇₉

yes

characteristic

Q₁₆

—

D₈

—

D₈

—

D₈

D₄×C₇₉

yes

—

100

C₂×C₈

—

101

C₂×C₈

D₄×C₇₉

yes

characteristic

102

C₂×C₈

C₂³×C₇₉

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₂²⋊C₄

—

106

C₁₆

—

107

C₂³

—

108

C₂³

—

109

C₂³

—

110

C₂³

—

111

C₂³

D₈×C₇₉

yes

characteristic

112

Q₈

—

113

Q₈

—

114

D₄

—

115

D₄

—

116

D₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

D₈×C₇₉

yes

characteristic

122

C₂×C₄

C₂×D₄×C₇₉

yes

characteristic

123

C₈

—

124

C₈

C₂×D₄×C₇₉

yes

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

SD₃₂×C₇₉

yes

complemented

132

C₂²

C₂×D₈×C₇₉

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#120×C₇₉

yes

characteristic

137

C₄

AGL(1,ℤ/8)×C₇₉

yes

characteristic

138

C₂

—

139

C₂

—

140

C₂

#447×C₇₉

yes

characteristic

141

C₂

C₂×SD₃₂×C₇₉

yes

characteristic

142

C₂

#509×C₇₉

yes

characteristic

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

subgroup

G/H

center, socle

C₂²×C₇₉

C₂×D₈

commutator, Frattini, cosocle

C₂×C₈

C₂³×C₇₉

prime radical

C₂²×D₈×C₇₉

C₂

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

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

N	H ≅ G/N	type
2. C₂×SD₃₂×C₇₉	C₂	semidirect
3. #467×C₇₉	C₂	semidirect
4. #357×C₇₉	C₂	semidirect
5. C₂²⋊C₁₆×C₇₉	C₂	semidirect
7. C₂×Q₁₆×C₇₉	C₂²	semidirect
14. C₂×C₁₆×C₇₉	C₂²	semidirect
15. #107×C₇₉	C₂²	semidirect
59. C₂²×C₇₉	SD₃₂	semidirect
71. #2187	C₇₉	direct (N × H)
72. C₇₉	#2187	direct (N × H)
74. C₂×SD₃₂	C₂×C₇₉	semidirect
75. #467	C₂×C₇₉	semidirect
76. #357	C₂×C₇₉	semidirect
77. C₂²⋊C₁₆	C₂×C₇₉	semidirect
79. C₂×Q₁₆	C₂²×C₇₉	semidirect
86. C₂×C₁₆	C₂²×C₇₉	semidirect
87. #107	C₂²×C₇₉	semidirect
131. C₂²	SD₃₂×C₇₉	semidirect

H ≅ G/N

type

2. C₂×SD₃₂×C₇₉

C₂

semidirect

3. #467×C₇₉

C₂

semidirect

4. #357×C₇₉

C₂

semidirect

5. C₂²⋊C₁₆×C₇₉

C₂

semidirect

7. C₂×Q₁₆×C₇₉

C₂²

semidirect

14. C₂×C₁₆×C₇₉

C₂²

semidirect

15. #107×C₇₉

C₂²

semidirect

59. C₂²×C₇₉

SD₃₂

semidirect

71. #2187

C₇₉

direct (N × H)

72. C₇₉

#2187

direct (N × H)

74. C₂×SD₃₂

C₂×C₇₉

semidirect

75. #467

C₂×C₇₉

semidirect

76. #357

C₂×C₇₉

semidirect

77. C₂²⋊C₁₆

C₂×C₇₉

semidirect

79. C₂×Q₁₆

C₂²×C₇₉

semidirect

86. C₂×C₁₆

C₂²×C₇₉

semidirect

87. #107

C₂²×C₇₉

semidirect

131. C₂²

SD₃₂×C₇₉

semidirect

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

subgroup	index
C₂²×D₈×C₇₉	2
C₂×SD₃₂×C₇₉	2
#467×C₇₉	2
#357×C₇₉	2
C₂²⋊C₁₆×C₇₉	2
#2187	79

subgroup

index

C₂²×D₈×C₇₉

C₂×SD₃₂×C₇₉

#467×C₇₉

#357×C₇₉

C₂²⋊C₁₆×C₇₉

#2187

Subgroups — C₇₉×#2187 #23759809

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C79×#2187 #23759809

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₉×#2187 #23759809