Group #23759752 — subgroups

Subgroups — C₂×#355×C₇₉ #23759752

Order 10112.

260 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 88 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×#124×C₇₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#101×C₇₉	C₂	1	2	yes	maximal, complemented
3	#355×C₇₉	C₂	1	4	yes	maximal, direct product factor
4	C₂²×Q₈×C₇₉	C₂²	1	1	yes	characteristic
5	C₂²×D₄×C₇₉	C₄	1	1	yes	characteristic
6	C₂×M₁₆×C₇₉	—	2	2	no	—
7	#124×C₇₉	—	2	2	no	—
8	#124×C₇₉	C₄	1	2	yes	—
9	#124×C₇₉	C₂²	1	2	yes	—
10	C₂×C₂²⋊C₄×C₇₉	—	2	2	no	—
11	C₂×C₄²×C₇₉	C₄	1	1	yes	characteristic
12	#101×C₇₉	—	2	2	no	—
13	#101×C₇₉	C₂²	1	4	yes	complemented
14	C₂⁴×C₇₉	—	2	1	no	—
15	C₂×Q₈×C₇₉	—	4	1	no	—
16	C₂×Q₈×C₇₉	—	4	1	no	—
17	C₂×Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
18	C₂×Q₈×C₇₉	D₄	1	2	yes	—
19	C₂×Q₈×C₇₉	C₂³	1	1	yes	characteristic
20	C₂×D₄×C₇₉	—	2	1	no	—
21	C₂×D₄×C₇₉	—	4	1	no	—
22	C₂×D₄×C₇₉	—	4	1	no	—
23	C₂×D₄×C₇₉	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₇₉	—	2	1	no	—
25	C₂²×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₇₉	D₄	1	2	yes	—
27	M₁₆×C₇₉	—	2	4	no	—
28	M₁₆×C₇₉	—	4	2	no	—
29	C₂×C₈×C₇₉	—	4	2	no	—
30	C₂²⋊C₄×C₇₉	—	2	4	no	—
31	C₂²⋊C₄×C₇₉	—	2	4	no	—
32	C₄²×C₇₉	—	2	1	no	—
33	C₄²×C₇₉	C₂×C₄	1	2	yes	—
34	C₂³×C₇₉	—	2	2	no	—
35	C₂³×C₇₉	—	4	4	no	—
36	C₂³×C₇₉	—	4	1	no	—
37	C₂³×C₇₉	—	4	1	no	—
38	C₂³×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
39	Q₈×C₇₉	—	4	1	no	—
40	Q₈×C₇₉	—	4	2	no	—
41	Q₈×C₇₉	—	4	1	no	—
42	D₄×C₇₉	—	4	2	no	—
43	D₄×C₇₉	—	4	2	no	—
44	C₂×C₄×C₇₉	—	2	2	no	—
45	C₂×C₄×C₇₉	—	2	1	no	—
46	C₂×C₄×C₇₉	—	2	2	no	—
47	C₂×C₄×C₇₉	—	2	1	no	—
48	C₂×C₄×C₇₉	—	2	2	no	—
49	C₂×C₄×C₇₉	—	4	2	no	—
50	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
52	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
54	C₈×C₇₉	—	4	4	no	—
55	C₂²×C₇₉	—	2	2	no	—
56	C₂²×C₇₉	—	4	4	no	—
57	C₂²×C₇₉	—	4	2	no	—
58	C₂²×C₇₉	—	8	2	no	—
59	C₂²×C₇₉	—	8	2	no	—
60	C₂²×C₇₉	#99	1	1	yes	characteristic
61	C₂²×C₇₉	#99	1	1	yes	characteristic
62	C₂²×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₄×C₇₉	—	2	1	no	—
64	C₄×C₇₉	—	2	2	no	—
65	C₄×C₇₉	—	2	1	no	—
66	C₄×C₇₉	—	2	2	no	—
67	C₄×C₇₉	—	4	2	no	—
68	C₂×C₇₉	—	2	1	no	—
69	C₂×C₇₉	—	2	1	no	—
70	C₂×C₇₉	—	8	2	no	—
71	C₂×C₇₉	#355	1	2	yes	direct product factor
72	C₂×C₇₉	C₂×#99	1	1	yes	characteristic
73	C₂×#355	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₇₉	C₂×#355	1	1	yes	characteristic, direct product factor, Sylow
75	C₂×#124	C₂×C₇₉	1	1	yes	characteristic
76	C₂×#101	C₂×C₇₉	1	2	yes	complemented
77	#355	C₂×C₇₉	1	4	yes	direct product factor
78	C₂²×Q₈	C₂²×C₇₉	1	1	yes	characteristic
79	C₂²×D₄	C₄×C₇₉	1	1	yes	characteristic
80	C₂×M₁₆	—	2	2	no	—
81	#124	—	2	2	no	—
82	#124	C₄×C₇₉	1	2	yes	—
83	#124	C₂²×C₇₉	1	2	yes	—
84	C₂×C₂²⋊C₄	—	2	2	no	—
85	C₂×C₄²	C₄×C₇₉	1	1	yes	characteristic
86	#101	—	2	2	no	—
87	#101	C₂²×C₇₉	1	4	yes	complemented
88	C₂⁴	—	2	1	no	—
89	C₂×Q₈	—	4	1	no	—
90	C₂×Q₈	—	4	1	no	—
91	C₂×Q₈	C₂×C₄×C₇₉	1	1	yes	characteristic
92	C₂×Q₈	D₄×C₇₉	1	2	yes	—
93	C₂×Q₈	C₂³×C₇₉	1	1	yes	characteristic
94	C₂×D₄	—	2	1	no	—
95	C₂×D₄	—	4	1	no	—
96	C₂×D₄	—	4	1	no	—
97	C₂×D₄	C₂×C₄×C₇₉	1	2	yes	—
98	C₂²×C₄	—	2	1	no	—
99	C₂²×C₄	C₂×C₄×C₇₉	1	1	yes	characteristic
100	C₂²×C₄	D₄×C₇₉	1	2	yes	—
101	M₁₆	—	2	4	no	—
102	M₁₆	—	4	2	no	—
103	C₂×C₈	—	4	2	no	—
104	C₂²⋊C₄	—	2	4	no	—
105	C₂²⋊C₄	—	2	4	no	—
106	C₄²	—	2	1	no	—
107	C₄²	C₂×C₄×C₇₉	1	2	yes	—
108	C₂³	—	2	2	no	—
109	C₂³	—	4	4	no	—
110	C₂³	—	4	1	no	—
111	C₂³	—	4	1	no	—
112	C₂³	C₂²⋊C₄×C₇₉	1	1	yes	characteristic
113	Q₈	—	4	1	no	—
114	Q₈	—	4	2	no	—
115	Q₈	—	4	1	no	—
116	D₄	—	4	2	no	—
117	D₄	—	4	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	C₂²⋊C₄×C₇₉	1	1	yes	characteristic
125	C₂×C₄	C₂²⋊C₄×C₇₉	1	2	yes	—
126	C₂×C₄	C₂²×C₄×C₇₉	1	1	yes	characteristic
127	C₂×C₄	C₂×D₄×C₇₉	1	2	yes	—
128	C₈	—	4	4	no	—
129	C₂²	—	2	2	no	—
130	C₂²	—	4	4	no	—
131	C₂²	—	4	2	no	—
132	C₂²	—	8	2	no	—
133	C₂²	—	8	2	no	—
134	C₂²	#99×C₇₉	1	1	yes	characteristic
135	C₂²	#99×C₇₉	1	1	yes	characteristic
136	C₂²	C₂×C₂²⋊C₄×C₇₉	1	1	yes	characteristic
137	C₄	—	2	2	no	—
138	C₄	—	2	1	no	—
139	C₄	—	2	2	no	—
140	C₄	—	2	1	no	—
141	C₄	—	4	2	no	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	—	8	2	no	—
145	C₂	#355×C₇₉	1	2	yes	direct product factor
146	C₂	C₂×#99×C₇₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#124×C₇₉

C₂

yes

maximal, characteristic

C₂×#101×C₇₉

C₂

yes

maximal, complemented

#355×C₇₉

C₂

yes

maximal, direct product factor

C₂²×Q₈×C₇₉

C₂²

yes

characteristic

C₂²×D₄×C₇₉

C₄

yes

characteristic

C₂×M₁₆×C₇₉

—

#124×C₇₉

—

#124×C₇₉

C₄

yes

—

#124×C₇₉

C₂²

yes

—

C₂×C₂²⋊C₄×C₇₉

—

C₂×C₄²×C₇₉

C₄

yes

characteristic

#101×C₇₉

—

#101×C₇₉

C₂²

yes

complemented

C₂⁴×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

C₂×C₄

yes

characteristic

C₂×Q₈×C₇₉

D₄

yes

—

C₂×Q₈×C₇₉

C₂³

yes

characteristic

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

C₂×C₄

yes

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₉

D₄

yes

—

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂×C₄

yes

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

C₂²⋊C₄

yes

characteristic

Q₈×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊C₄

yes

—

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₈×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#355

yes

direct product factor

C₂×C₇₉

C₂×#99

yes

characteristic

C₂×#355

C₇₉

yes

maximal, characteristic, direct product factor, Sylow

C₇₉

C₂×#355

yes

characteristic, direct product factor, Sylow

C₂×#124

C₂×C₇₉

yes

characteristic

C₂×#101

C₂×C₇₉

yes

complemented

#355

C₂×C₇₉

yes

direct product factor

C₂²×Q₈

C₂²×C₇₉

yes

characteristic

C₂²×D₄

C₄×C₇₉

yes

characteristic

C₂×M₁₆

—

#124

—

#124

C₄×C₇₉

yes

—

#124

C₂²×C₇₉

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₄×C₇₉

yes

characteristic

#101

—

#101

C₂²×C₇₉

yes

complemented

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×C₄×C₇₉

yes

characteristic

C₂×Q₈

D₄×C₇₉

yes

—

C₂×Q₈

C₂³×C₇₉

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×C₄×C₇₉

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×C₇₉

yes

characteristic

100

C₂²×C₄

D₄×C₇₉

yes

—

101

M₁₆

—

102

M₁₆

—

103

C₂×C₈

—

104

C₂²⋊C₄

—

105

C₂²⋊C₄

—

106

C₄²

—

107

C₄²

C₂×C₄×C₇₉

yes

—

108

C₂³

—

109

C₂³

—

110

C₂³

—

111

C₂³

—

112

C₂³

C₂²⋊C₄×C₇₉

yes

characteristic

113

Q₈

—

114

Q₈

—

115

Q₈

—

116

D₄

—

117

D₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

C₂²⋊C₄×C₇₉

yes

characteristic

125

C₂×C₄

C₂²⋊C₄×C₇₉

yes

—

126

C₂×C₄

C₂²×C₄×C₇₉

yes

characteristic

127

C₂×C₄

C₂×D₄×C₇₉

yes

—

128

C₈

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

#99×C₇₉

yes

characteristic

135

C₂²

#99×C₇₉

yes

characteristic

136

C₂²

C₂×C₂²⋊C₄×C₇₉

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₂

—

143

C₂

—

144

C₂

—

145

C₂

#355×C₇₉

yes

direct product factor

146

C₂

C₂×#99×C₇₉

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₇₉	#99
commutator	C₂×C₄	C₂²×C₄×C₇₉
Frattini, cosocle	C₂×Q₈	C₂³×C₇₉
prime radical	C₂²×D₄×C₇₉	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂³×C₇₉ ◁ C₂²×Q₈×C₇₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₇₉

#99

commutator

C₂×C₄

C₂²×C₄×C₇₉

Frattini, cosocle

C₂×Q₈

C₂³×C₇₉

prime radical

C₂²×D₄×C₇₉

C₄

LCS

G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₇₉ ◁ C₂³×C₇₉ ◁ C₂²×Q₈×C₇₉ ◁ G

N	H ≅ G/N	type
2. C₂×#101×C₇₉	C₂	semidirect
3. #355×C₇₉	C₂	direct (N × H)
13. #101×C₇₉	C₂²	semidirect
71. C₂×C₇₉	#355	direct (N × H)
73. C₂×#355	C₇₉	direct (N × H)
74. C₇₉	C₂×#355	direct (N × H)
76. C₂×#101	C₂×C₇₉	semidirect
77. #355	C₂×C₇₉	direct (N × H)
87. #101	C₂²×C₇₉	semidirect
145. C₂	#355×C₇₉	direct (N × H)

H ≅ G/N

type

2. C₂×#101×C₇₉

C₂

semidirect

3. #355×C₇₉

C₂

direct (N × H)

13. #101×C₇₉

C₂²

semidirect

71. C₂×C₇₉

#355

direct (N × H)

73. C₂×#355

C₇₉

direct (N × H)

74. C₇₉

C₂×#355

direct (N × H)

76. C₂×#101

C₂×C₇₉

semidirect

77. #355

C₂×C₇₉

direct (N × H)

87. #101

C₂²×C₇₉

semidirect

145. C₂

#355×C₇₉

direct (N × H)

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

subgroup	index
C₂×#124×C₇₉	2
C₂×#101×C₇₉	2
#355×C₇₉	2
C₂×#355	79

subgroup

index

C₂×#124×C₇₉

C₂×#101×C₇₉

#355×C₇₉

C₂×#355

Subgroups — C₂×#355×C₇₉ #23759752

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#355×C79 #23759752

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#355×C₇₉ #23759752