Group #23765311 — subgroups

Subgroups — #23765311

Order 10112.

266 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 57 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×D₆₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14579102	C₂	1	2	yes	maximal, complemented
3	#19098363	C₂	1	2	yes	maximal, complemented
4	C₂²⋊(C₇₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
5	#466×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₁₆	—	2	1	no	—
7	C₂×D₆₃₂	—	2	1	no	—
8	C₂×D₆₃₂	—	2	4	no	—
9	C₂×D₆₃₂	C₂²	1	2	yes	—
10	#14579102	—	2	4	no	—
11	C₂×C₇₉⋊C₁₆	C₂²	1	2	yes	complemented
12	C₂×D₈×C₇₉	C₂²	1	1	yes	characteristic, complemented
13	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
14	#121×C₇₉	—	2	1	no	—
15	#107×C₇₉	C₂²	1	1	yes	characteristic, complemented
16	#102×C₇₉	—	2	1	no	—
17	C₂³×D₇₉	—	4	1	no	—
18	C₂×D₃₁₆	—	2	4	no	—
19	C₂×D₃₁₆	—	2	2	no	—
20	C₂×D₃₁₆	—	4	2	no	—
21	C₂×D₃₁₆	—	4	1	no	—
22	D₆₃₂	—	2	4	no	—
23	D₆₃₂	—	2	2	no	—
24	D₆₃₂	D₄	1	4	yes	complemented
25	C₇₉⋊C₁₆	—	2	2	no	—
26	C₂×D₄×C₇₉	—	2	1	no	—
27	C₂×D₄×C₇₉	—	4	1	no	—
28	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
29	D₈×C₇₉	—	2	2	no	—
30	C₂×C₈×C₇₉	—	2	2	no	—
31	C₂×C₈×C₇₉	D₄	1	1	yes	characteristic
32	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₇₉	—	2	1	no	—
34	C₂²⋊C₄×C₇₉	—	4	1	no	—
35	C₂³×C₇₉	—	4	1	no	—
36	C₂³×C₇₉	D₈	1	1	yes	characteristic
37	C₂²×D₇₉	—	4	4	no	—
38	C₂²×D₇₉	—	4	2	no	—
39	C₂²×D₇₉	—	8	2	no	—
40	C₂²×D₇₉	—	8	2	no	—
41	D₄×C₇₉	—	4	1	no	—
42	D₄×C₇₉	—	4	1	no	—
43	D₄×C₇₉	—	8	2	no	—
44	C₂×C₄×C₇₉	—	2	1	no	—
45	C₂×C₄×C₇₉	—	2	1	no	—
46	C₂×C₄×C₇₉	—	4	1	no	—
47	C₂×C₄×C₇₉	D₈	1	1	yes	characteristic
48	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
49	D₃₁₆	—	2	4	no	—
50	D₃₁₆	—	4	4	no	—
51	D₃₁₆	—	4	2	no	—
52	C₈×C₇₉	—	2	1	no	—
53	C₈×C₇₉	C₂×D₄	1	2	yes	—
54	C₂²×C₇₉	—	2	1	no	—
55	C₂²×C₇₉	—	2	1	no	—
56	C₂²×C₇₉	—	8	1	no	—
57	C₂²×C₇₉	—	8	1	no	—
58	C₂²×C₇₉	—	8	1	no	—
59	C₂²×C₇₉	D₁₆	1	2	yes	complemented
60	C₂²×C₇₉	C₂×D₈	1	1	yes	characteristic
61	C₂×D₇₉	—	4	4	no	—
62	C₂×D₇₉	—	8	8	no	—
63	C₂×D₇₉	—	8	2	no	—
64	C₂×D₇₉	—	8	2	no	—
65	C₄×C₇₉	—	2	1	no	—
66	C₄×C₇₉	—	8	1	no	—
67	C₄×C₇₉	#120	1	1	yes	characteristic
68	C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
69	C₂×C₇₉	—	2	2	no	—
70	C₂×C₇₉	—	16	1	no	—
71	C₂×C₇₉	#447	1	1	yes	characteristic
72	C₂×C₇₉	C₂×D₁₆	1	1	yes	characteristic
73	C₂×C₇₉	#509	1	1	yes	characteristic
74	D₇₉	—	8	4	no	—
75	#2184	—	79	1	no	maximal, Sylow
76	C₇₉	#2184	1	1	yes	characteristic, complemented, Sylow
77	C₂²×D₈	—	79	1	no	—
78	C₂×D₁₆	—	79	2	no	—
79	#466	D₇₉	1	1	yes	characteristic, complemented
80	#357	—	79	2	no	—
81	C₂²⋊C₁₆	—	79	1	no	—
82	C₂²×D₄	—	158	1	no	—
83	C₂×D₈	—	79	2	no	—
84	C₂×D₈	—	158	4	no	—
85	C₂×D₈	—	158	1	no	—
86	C₂×D₈	C₂×D₇₉	1	1	yes	characteristic, complemented
87	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
88	#121	—	2	1	no	—
89	D₁₆	—	158	4	no	—
90	C₂×C₁₆	—	79	2	no	—
91	#107	C₂×D₇₉	1	1	yes	characteristic, complemented
92	#102	—	2	1	no	—
93	C₂⁴	—	316	1	no	—
94	C₂×D₄	—	2	1	no	—
95	C₂×D₄	—	4	1	no	—
96	C₂×D₄	—	316	2	no	—
97	C₂×D₄	—	316	1	no	—
98	C₂×D₄	—	158	4	no	—
99	C₂×D₄	—	158	2	no	—
100	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
101	D₈	—	2	2	no	—
102	D₈	—	79	4	no	—
103	D₈	—	158	4	no	—
104	D₈	—	158	2	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
107	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
108	C₄⋊C₄	—	2	1	no	—
109	C₂²⋊C₄	—	4	1	no	—
110	C₁₆	—	158	2	no	—
111	C₂³	—	4	1	no	—
112	C₂³	—	316	4	no	—
113	C₂³	—	316	2	no	—
114	C₂³	—	632	2	no	—
115	C₂³	—	632	2	no	—
116	C₂³	#11929609	1	1	yes	characteristic
117	D₄	—	4	1	no	—
118	D₄	—	4	1	no	—
119	D₄	—	8	2	no	—
120	D₄	—	316	4	no	—
121	D₄	—	316	2	no	—
122	D₄	—	158	4	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	#11929609	1	1	yes	characteristic
127	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
128	C₈	—	2	1	no	—
129	C₈	D₄×D₇₉	1	2	yes	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	8	1	no	—
133	C₂²	—	8	1	no	—
134	C₂²	—	8	1	no	—
135	C₂²	—	316	4	no	—
136	C₂²	—	632	8	no	—
137	C₂²	—	632	2	no	—
138	C₂²	—	632	2	no	—
139	C₂²	#14579102	1	2	yes	complemented
140	C₂²	C₂×#11929609	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	8	1	no	—
143	C₄	#14579129	1	1	yes	characteristic
144	C₄	#14579186	1	1	yes	characteristic
145	C₂	—	2	2	no	—
146	C₂	—	16	1	no	—
147	C₂	—	632	4	no	—
148	C₂	#19098696	1	1	yes	characteristic
149	C₂	C₂×#14579102	1	1	yes	characteristic
150	C₂	#19098998	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₆₃₂

C₂

yes

maximal, characteristic, complemented

C₂×#14579102

C₂

yes

maximal, complemented

#19098363

C₂

yes

maximal, complemented

C₂²⋊(C₇₉⋊C₁₆)

C₂

yes

maximal, characteristic, complemented

#466×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₃₁₆

—

C₂×D₆₃₂

—

C₂×D₆₃₂

—

C₂×D₆₃₂

C₂²

yes

—

#14579102

—

C₂×C₇₉⋊C₁₆

C₂²

yes

complemented

C₂×D₈×C₇₉

C₂²

yes

characteristic, complemented

C₂²×C₈×C₇₉

C₂²

yes

characteristic

#121×C₇₉

—

#107×C₇₉

C₂²

yes

characteristic, complemented

#102×C₇₉

—

C₂³×D₇₉

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

D₆₃₂

—

D₆₃₂

—

D₆₃₂

D₄

yes

complemented

C₇₉⋊C₁₆

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂²×C₄×C₇₉

D₄

yes

characteristic

D₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

D₄

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

D₈

yes

characteristic

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂²×D₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

D₈

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

C₈×C₇₉

—

C₈×C₇₉

C₂×D₄

yes

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

D₁₆

yes

complemented

C₂²×C₇₉

C₂×D₈

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#447

yes

characteristic

C₂×C₇₉

C₂×D₁₆

yes

characteristic

C₂×C₇₉

#509

yes

characteristic

D₇₉

—

#2184

—

maximal, Sylow

C₇₉

#2184

yes

characteristic, complemented, Sylow

C₂²×D₈

—

C₂×D₁₆

—

#466

D₇₉

yes

characteristic, complemented

#357

—

C₂²⋊C₁₆

—

C₂²×D₄

—

158

—

C₂×D₈

—

C₂×D₈

—

158

—

C₂×D₈

—

158

—

C₂×D₈

C₂×D₇₉

yes

characteristic, complemented

C₂²×C₈

C₂×D₇₉

yes

characteristic

#121

—

D₁₆

—

158

—

C₂×C₁₆

—

#107

C₂×D₇₉

yes

characteristic, complemented

#102

—

C₂⁴

—

316

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

316

—

C₂×D₄

—

316

—

C₂×D₄

—

158

—

C₂×D₄

—

158

—

100

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

101

D₈

—

102

D₈

—

103

D₈

—

158

—

104

D₈

—

158

—

105

C₂×C₈

—

106

C₂×C₈

C₂²⋊D₇₉

yes

characteristic

107

C₂×C₈

C₂²×D₇₉

yes

characteristic

108

C₄⋊C₄

—

109

C₂²⋊C₄

—

110

C₁₆

—

158

—

111

C₂³

—

112

C₂³

—

316

—

113

C₂³

—

316

—

114

C₂³

—

632

—

115

C₂³

—

632

—

116

C₂³

#11929609

yes

characteristic

117

D₄

—

118

D₄

—

119

D₄

—

120

D₄

—

316

—

121

D₄

—

316

—

122

D₄

—

158

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

#11929609

yes

characteristic

127

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

128

C₈

—

129

C₈

D₄×D₇₉

yes

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

316

—

136

C₂²

—

632

—

137

C₂²

—

632

—

138

C₂²

—

632

—

139

C₂²

#14579102

yes

complemented

140

C₂²

C₂×#11929609

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

#14579129

yes

characteristic

144

C₄

#14579186

yes

characteristic

145

C₂

—

146

C₂

—

147

C₂

—

632

—

148

C₂

#19098696

yes

characteristic

149

C₂

C₂×#14579102

yes

characteristic

150

C₂

#19098998

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11929609
commutator, cosocle	C₂×C₈×C₇₉	C₂³
Frattini	C₂×C₈	C₂²×D₇₉
Fitting	#466×C₇₉	C₂
socle	C₂²×C₇₉	C₂×D₈
hypercenter	#466	D₇₉
hypocenter	C₇₉	#2184
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #466

subgroup

G/H

center

C₂²

C₂×#11929609

commutator, cosocle

C₂×C₈×C₇₉

C₂³

Frattini

C₂×C₈

C₂²×D₇₉

Fitting

#466×C₇₉

C₂

socle

C₂²×C₇₉

C₂×D₈

D₇₉

C₇₉

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #466

N	H ≅ G/N	type
1. C₂²×D₆₃₂	C₂	semidirect
2. C₂×#14579102	C₂	semidirect
3. #19098363	C₂	semidirect
4. C₂²⋊(C₇₉⋊C₁₆)	C₂	semidirect
5. #466×C₇₉	C₂	semidirect
11. C₂×C₇₉⋊C₁₆	C₂²	semidirect
12. C₂×D₈×C₇₉	C₂²	semidirect
15. #107×C₇₉	C₂²	semidirect
24. D₆₃₂	D₄	semidirect
59. C₂²×C₇₉	D₁₆	semidirect
76. C₇₉	#2184	semidirect
79. #466	D₇₉	semidirect
86. C₂×D₈	C₂×D₇₉	semidirect
91. #107	C₂×D₇₉	semidirect
139. C₂²	#14579102	semidirect

H ≅ G/N

type

1. C₂²×D₆₃₂

C₂

semidirect

2. C₂×#14579102

C₂

semidirect

3. #19098363

C₂

semidirect

4. C₂²⋊(C₇₉⋊C₁₆)

C₂

semidirect

5. #466×C₇₉

C₂

semidirect

11. C₂×C₇₉⋊C₁₆

C₂²

semidirect

12. C₂×D₈×C₇₉

C₂²

semidirect

15. #107×C₇₉

C₂²

semidirect

24. D₆₃₂

D₄

semidirect

59. C₂²×C₇₉

D₁₆

semidirect

76. C₇₉

#2184

semidirect

79. #466

D₇₉

semidirect

86. C₂×D₈

C₂×D₇₉

semidirect

91. #107

C₂×D₇₉

semidirect

139. C₂²

#14579102

semidirect

Subgroups — #23765311

Conjugacy classes of subgroups

Complements