Group #23765112 — subgroups

Subgroups — #23765112

Order 10112.

232 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 93 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×#14579101	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14579095	C₂	1	1	yes	maximal, characteristic, complemented
3	#19098367	C₂	1	2	yes	maximal, complemented
4	#19098364	C₂	1	2	yes	maximal, complemented
5	C₂×#135×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₇₉	—	2	1	no	—
7	#14579101	—	2	2	no	—
8	C₂×C₇₉⋊C₁₆	C₂²	1	2	yes	—
9	#14579095	—	2	1	no	—
10	#14579095	C₂²	1	2	yes	—
11	C₂×#41×C₇₉	—	2	1	no	—
12	#135×C₇₉	—	2	2	no	—
13	#135×C₇₉	C₄	1	4	yes	complemented
14	C₂×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
15	C₂×SD₁₆×C₇₉	—	2	1	no	—
16	C₂×D₈×C₇₉	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
18	C₂²×Dic₇₉	—	4	1	no	—
19	C₄⋊Dic₇₉	—	2	2	no	—
20	C₄⋊Dic₇₉	—	4	1	no	—
21	C₇₉⋊C₁₆	—	2	2	no	—
22	#41×C₇₉	—	2	4	no	—
23	#41×C₇₉	—	4	2	no	—
24	C₂×Q₈×C₇₉	—	2	1	no	—
25	C₂×D₄×C₇₉	—	2	1	no	—
26	C₂×D₄×C₇₉	—	4	1	no	—
27	C₂²×C₄×C₇₉	—	4	1	no	—
28	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
29	Q₁₆×C₇₉	—	2	1	no	—
30	Q₁₆×C₇₉	C₂×C₄	1	2	yes	complemented
31	SD₁₆×C₇₉	—	2	4	no	—
32	D₈×C₇₉	—	2	1	no	—
33	D₈×C₇₉	C₂×C₄	1	2	yes	complemented
34	C₂×C₈×C₇₉	C₂×C₄	1	2	yes	—
35	C₂×C₈×C₇₉	D₄	1	2	yes	—
36	C₂×C₈×C₇₉	D₄	1	1	yes	characteristic
37	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
38	C₂³×C₇₉	—	4	1	no	—
39	C₂³×C₇₉	D₈	1	1	yes	characteristic
40	Q₈×C₇₉	—	2	2	no	—
41	Q₈×C₇₉	—	4	1	no	—
42	D₄×C₇₉	—	2	2	no	—
43	D₄×C₇₉	—	4	4	no	—
44	D₄×C₇₉	—	4	1	no	—
45	C₂×C₄×C₇₉	—	4	1	no	—
46	C₂×C₄×C₇₉	—	4	4	no	—
47	C₂×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
48	C₂×C₄×C₇₉	D₈	1	1	yes	characteristic
49	C₂×C₄×C₇₉	SD₁₆	1	2	yes	—
50	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
51	C₂×Dic₇₉	—	4	2	no	—
52	C₂×Dic₇₉	—	8	2	no	—
53	C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
54	C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
55	C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₇₉	—	2	1	no	—
57	C₂²×C₇₉	—	2	1	no	—
58	C₂²×C₇₉	—	4	2	no	—
59	C₂²×C₇₉	—	8	1	no	—
60	C₂²×C₇₉	—	8	1	no	—
61	C₂²×C₇₉	#102	1	2	yes	—
62	C₂²×C₇₉	C₂×D₈	1	1	yes	characteristic
63	C₄×C₇₉	—	4	2	no	—
64	C₄×C₇₉	#102	1	2	yes	—
65	C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
67	Dic₇₉	—	8	2	no	—
68	C₂×C₇₉	—	2	2	no	—
69	C₂×C₇₉	—	8	2	no	—
70	C₂×C₇₉	C₂×#102	1	1	yes	characteristic
71	C₂×C₇₉	#509	1	1	yes	characteristic
72	C₂×C₇₉	#510	1	1	yes	characteristic
73	#2139	—	79	1	no	maximal, Sylow
74	C₇₉	#2139	1	1	yes	characteristic, complemented, Sylow
75	C₂×#135	D₇₉	1	1	yes	characteristic
76	C₂×M₃₂	—	79	1	no	—
77	C₂×#107	—	79	1	no	—
78	#358	—	79	2	no	—
79	#357	—	79	2	no	—
80	C₂×#41	—	2	1	no	—
81	#135	—	2	2	no	—
82	#135	Dic₇₉	1	4	yes	complemented
83	C₂×Q₁₆	C₂×D₇₉	1	1	yes	characteristic
84	C₂×SD₁₆	—	2	1	no	—
85	C₂×D₈	C₂×D₇₉	1	1	yes	characteristic
86	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
87	C₂×C₄⋊C₄	—	158	1	no	—
88	M₃₂	—	158	2	no	—
89	C₂×C₁₆	—	79	2	no	—
90	#107	—	79	2	no	—
91	#107	—	158	1	no	—
92	#41	—	2	4	no	—
93	#41	—	4	2	no	—
94	C₂×Q₈	—	2	1	no	—
95	C₂×D₄	—	2	1	no	—
96	C₂×D₄	—	4	1	no	—
97	C₂²×C₄	—	4	1	no	—
98	C₂²×C₄	—	316	1	no	—
99	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
100	Q₁₆	—	2	1	no	—
101	Q₁₆	C₂×Dic₇₉	1	2	yes	complemented
102	SD₁₆	—	2	4	no	—
103	D₈	—	2	1	no	—
104	D₈	C₂×Dic₇₉	1	2	yes	complemented
105	C₂×C₈	C₂×Dic₇₉	1	2	yes	—
106	C₂×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
107	C₂×C₈	C₂²⋊D₇₉	1	2	yes	—
108	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
109	C₄⋊C₄	—	316	1	no	—
110	C₄⋊C₄	—	158	2	no	—
111	C₁₆	—	158	2	no	—
112	C₂³	—	4	1	no	—
113	C₂³	#11929609	1	1	yes	characteristic
114	Q₈	—	2	2	no	—
115	Q₈	—	4	1	no	—
116	D₄	—	2	2	no	—
117	D₄	—	4	4	no	—
118	D₄	—	4	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	4	no	—
121	C₂×C₄	—	316	2	no	—
122	C₂×C₄	—	632	2	no	—
123	C₂×C₄	C₂²⋊Dic₇₉	1	2	yes	—
124	C₂×C₄	#11929609	1	1	yes	characteristic
125	C₂×C₄	#11929612	1	2	yes	—
126	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
127	C₈	C₂²⋊Dic₇₉	1	2	yes	—
128	C₈	C₂²×Dic₇₉	1	1	yes	characteristic
129	C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	4	2	no	—
133	C₂²	—	8	1	no	—
134	C₂²	—	8	1	no	—
135	C₂²	#14579082	1	2	yes	—
136	C₂²	C₂×#11929609	1	1	yes	characteristic
137	C₄	—	4	2	no	—
138	C₄	—	632	2	no	—
139	C₄	#14579082	1	2	yes	—
140	C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
141	C₄	C₂×#11929612	1	1	yes	characteristic
142	C₂	—	2	2	no	—
143	C₂	—	8	2	no	—
144	C₂	C₂×#14579082	1	1	yes	characteristic
145	C₂	#19098998	1	1	yes	characteristic
146	C₂	#19099003	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579101

C₂

yes

maximal, characteristic, complemented

C₂×#14579095

C₂

yes

maximal, characteristic, complemented

#19098367

C₂

yes

maximal, complemented

#19098364

C₂

yes

maximal, complemented

C₂×#135×C₇₉

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₇₉

—

#14579101

—

C₂×C₇₉⋊C₁₆

C₂²

yes

—

#14579095

—

#14579095

C₂²

yes

—

C₂×#41×C₇₉

—

#135×C₇₉

—

#135×C₇₉

C₄

yes

complemented

C₂×Q₁₆×C₇₉

C₂²

yes

characteristic

C₂×SD₁₆×C₇₉

—

C₂×D₈×C₇₉

C₂²

yes

characteristic

C₂²×C₈×C₇₉

C₂²

yes

characteristic

C₂²×Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₇₉⋊C₁₆

—

#41×C₇₉

—

#41×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

D₄

yes

characteristic

Q₁₆×C₇₉

—

Q₁₆×C₇₉

C₂×C₄

yes

complemented

SD₁₆×C₇₉

—

D₈×C₇₉

—

D₈×C₇₉

C₂×C₄

yes

complemented

C₂×C₈×C₇₉

C₂×C₄

yes

—

C₂×C₈×C₇₉

D₄

yes

—

C₂×C₈×C₇₉

D₄

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

characteristic

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

C₂²⋊C₄

yes

—

C₂×C₄×C₇₉

D₈

yes

characteristic

C₂×C₄×C₇₉

SD₁₆

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

C₂²⋊C₄

yes

—

C₈×C₇₉

C₂²×C₄

yes

characteristic

C₈×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#102

yes

—

C₂²×C₇₉

C₂×D₈

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

#102

yes

—

C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₉

C₂×SD₁₆

yes

characteristic

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

C₂×#135

D₇₉

yes

characteristic

C₂×M₃₂

—

C₂×#107

—

#358

—

#357

—

C₂×#41

—

#135

—

#135

Dic₇₉

yes

complemented

C₂×Q₁₆

C₂×D₇₉

yes

characteristic

C₂×SD₁₆

—

C₂×D₈

C₂×D₇₉

yes

characteristic

C₂²×C₈

C₂×D₇₉

yes

characteristic

C₂×C₄⋊C₄

—

158

—

M₃₂

—

158

—

C₂×C₁₆

—

#107

—

#107

—

158

—

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

316

—

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

100

Q₁₆

—

101

Q₁₆

C₂×Dic₇₉

yes

complemented

102

SD₁₆

—

103

D₈

—

104

D₈

C₂×Dic₇₉

yes

complemented

105

C₂×C₈

C₂×Dic₇₉

yes

—

106

C₂×C₈

C₂²⋊D₇₉

yes

characteristic

107

C₂×C₈

C₂²⋊D₇₉

yes

—

108

C₂×C₈

C₂²×D₇₉

yes

characteristic

109

C₄⋊C₄

—

316

—

110

C₄⋊C₄

—

158

—

111

C₁₆

—

158

—

112

C₂³

—

113

C₂³

#11929609

yes

characteristic

114

Q₈

—

115

Q₈

—

116

D₄

—

117

D₄

—

118

D₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

316

—

122

C₂×C₄

—

632

—

123

C₂×C₄

C₂²⋊Dic₇₉

yes

—

124

C₂×C₄

#11929609

yes

characteristic

125

C₂×C₄

#11929612

yes

—

126

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

127

C₈

C₂²⋊Dic₇₉

yes

—

128

C₈

C₂²×Dic₇₉

yes

characteristic

129

C₈

C₂×C₂²⋊D₇₉

yes

characteristic

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

#14579082

yes

—

136

C₂²

C₂×#11929609

yes

characteristic

137

C₄

—

138

C₄

—

632

—

139

C₄

#14579082

yes

—

140

C₄

C₂×C₂²⋊Dic₇₉

yes

characteristic

141

C₄

C₂×#11929612

yes

characteristic

142

C₂

—

143

C₂

—

144

C₂

C₂×#14579082

yes

characteristic

145

C₂

#19098998

yes

characteristic

146

C₂

#19099003

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11929609
commutator	C₈×C₇₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₇₉
Fitting, prime radical	C₂×#135×C₇₉	C₂
socle	C₂²×C₇₉	C₂×D₈
cosocle	C₂×C₈×C₇₉	C₂³
hypercenter	C₂×#135	D₇₉
hypocenter	C₇₉	#2139
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#135

subgroup

G/H

center

C₂²

C₂×#11929609

commutator

C₈×C₇₉

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₇₉

Fitting, prime radical

C₂×#135×C₇₉

C₂

socle

C₂²×C₇₉

C₂×D₈

cosocle

C₂×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#135

N	H ≅ G/N	type
1. C₂×#14579101	C₂	semidirect
2. C₂×#14579095	C₂	semidirect
3. #19098367	C₂	semidirect
4. #19098364	C₂	semidirect
13. #135×C₇₉	C₄	semidirect
30. Q₁₆×C₇₉	C₂×C₄	semidirect
33. D₈×C₇₉	C₂×C₄	semidirect
74. C₇₉	#2139	semidirect
82. #135	Dic₇₉	semidirect
101. Q₁₆	C₂×Dic₇₉	semidirect
104. D₈	C₂×Dic₇₉	semidirect

H ≅ G/N

type

1. C₂×#14579101

C₂

semidirect

2. C₂×#14579095

C₂

semidirect

3. #19098367

C₂

semidirect

4. #19098364

C₂

semidirect

13. #135×C₇₉

C₄

semidirect

30. Q₁₆×C₇₉

C₂×C₄

semidirect

33. D₈×C₇₉

C₂×C₄

semidirect

74. C₇₉

#2139

semidirect

82. #135

Dic₇₉

semidirect

101. Q₁₆

C₂×Dic₇₉

semidirect

104. D₈

C₂×Dic₇₉

semidirect

Subgroups — #23765112

Conjugacy classes of subgroups

Complements