Group #23782356 — subgroups

Subgroups — #23782356

Order 10432.

214 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 97 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	#19104729	C₂	1	2	yes	maximal, complemented
2	#19104727	C₂	1	1	yes	maximal, characteristic, complemented
3	#19104725	C₂	1	2	yes	maximal, complemented
4	#19104720	C₂	1	2	yes	maximal
5	#19104711	C₂	1	1	yes	maximal, characteristic, complemented
6	#19104711	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×D₆₅₂	C₂	1	1	yes	maximal, characteristic
8	#19104693	C₂	1	1	yes	maximal, characteristic, complemented
9	#19104692	C₂	1	2	yes	maximal, complemented
10	C₄⋊C₄×D₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Q₈×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₆₅₂	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₁₆₃	C₂²	1	2	yes	—
14	C₂×C₄×D₁₆₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₁₆₃	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₁₆₃	C₂²	1	1	yes	characteristic
17	C₄⋊Dic₁₆₃	C₂²	1	1	yes	characteristic
18	C₄⋊Dic₁₆₃	C₂²	1	1	yes	characteristic
19	Dic₁₆₃⋊C₄	C₂²	1	2	yes	—
20	Dic₁₆₃⋊C₄	C₂²	1	2	yes	—
21	Dic₁₆₃⋊C₄	C₂²	1	2	yes	—
22	Dic₁₆₃⋊C₄	C₂²	1	2	yes	—
23	Dic₁₆₃⋊C₄	C₂²	1	2	yes	—
24	(C₂×D₁₆₃)⋊C₄	C₂²	1	1	yes	characteristic
25	(C₂×D₁₆₃)⋊C₄	C₂²	1	2	yes	—
26	(C₂×D₁₆₃)⋊C₄	C₂²	1	2	yes	—
27	(C₂×D₁₆₃)⋊C₄	C₂²	1	1	yes	characteristic
28	(C₂×D₁₆₃)⋊C₄	C₂²	1	2	yes	—
29	C₄×Dic₁₆₃	C₂²	1	2	yes	—
30	C₂×Q₈×C₁₆₃	C₂²	1	1	yes	characteristic
31	C₄⋊C₄×C₁₆₃	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×C₁₆₃	C₂²	1	2	yes	—
33	C₄²×C₁₆₃	C₂²	1	2	yes	—
34	C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
35	C₂²×D₁₆₃	C₂³	1	1	yes	characteristic
36	C₂²×D₁₆₃	C₂³	1	1	yes	characteristic
37	D₆₅₂	—	2	2	no	—
38	C₂×Dic₁₆₃	C₂³	1	1	yes	characteristic
39	C₂×Dic₁₆₃	C₂³	1	2	yes	—
40	C₂×Dic₁₆₃	C₂³	1	1	yes	characteristic
41	C₂×Dic₁₆₃	C₂³	1	2	yes	—
42	C₄×D₁₆₃	—	2	4	no	—
43	C₄×D₁₆₃	—	2	2	no	—
44	C₄×D₁₆₃	—	2	2	no	—
45	Q₈×C₁₆₃	—	2	2	no	—
46	C₂×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₁₆₃	C₂³	1	2	yes	—
48	C₂×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
49	C₂×C₄×C₁₆₃	C₂³	1	2	yes	—
50	C₂×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
51	C₂²×C₁₆₃	C₂⁴	1	1	yes	characteristic
52	C₂×D₁₆₃	—	2	1	no	—
53	C₂×D₁₆₃	—	2	1	no	—
54	C₂×D₁₆₃	—	2	1	no	—
55	C₂×D₁₆₃	—	2	1	no	—
56	C₂×D₁₆₃	—	2	1	no	—
57	C₂×D₁₆₃	#41	1	2	yes	—
58	C₄×C₁₆₃	—	2	2	no	—
59	C₄×C₁₆₃	—	2	2	no	—
60	C₄×C₁₆₃	—	2	1	no	—
61	C₄×C₁₆₃	—	2	1	no	—
62	C₄×C₁₆₃	#41	1	2	yes	—
63	Dic₁₆₃	—	2	2	no	—
64	Dic₁₆₃	—	2	1	no	—
65	Dic₁₆₃	—	2	1	no	—
66	Dic₁₆₃	—	2	2	no	—
67	C₂×C₁₆₃	C₂×#41	1	1	yes	characteristic
68	C₂×C₁₆₃	C₂×#41	1	1	yes	characteristic
69	C₂×C₁₆₃	2₋¹⁺⁴	1	1	yes	characteristic
70	D₁₆₃	—	2	2	no	—
71	D₁₆₃	—	4	1	no	—
72	C₁₆₃	#552	1	1	yes	characteristic, complemented, Sylow
73	#552	—	163	1	no	maximal, Sylow
74	#126	—	163	2	no	—
75	#125	—	163	2	no	—
76	#125	—	163	1	no	—
77	#123	—	163	2	no	—
78	C₂²⋊Q₈	—	163	1	no	—
79	C₂²⋊Q₈	—	163	1	no	—
80	C₄×Q₈	D₁₆₃	1	1	yes	characteristic, complemented
81	C₄×D₄	—	163	1	no	—
82	#117	—	163	1	no	—
83	#117	—	163	2	no	—
84	C₂×C₄⋊C₄	—	163	1	no	—
85	C₂×Q₈	C₂×D₁₆₃	1	1	yes	characteristic
86	C₂×D₄	—	163	1	no	—
87	C₂²×C₄	—	163	2	no	—
88	C₂²×C₄	—	163	1	no	—
89	C₂²×C₄	—	163	1	no	—
90	C₄⋊C₄	—	163	1	no	—
91	C₄⋊C₄	—	163	2	no	—
92	C₄⋊C₄	—	163	2	no	—
93	C₄⋊C₄	—	163	1	no	—
94	C₄⋊C₄	—	163	2	no	—
95	C₄⋊C₄	—	163	1	no	—
96	C₄⋊C₄	—	163	2	no	—
97	C₄⋊C₄	—	163	2	no	—
98	C₄⋊C₄	C₂×D₁₆₃	1	1	yes	characteristic
99	C₄⋊C₄	C₂×D₁₆₃	1	2	yes	—
100	C₂²⋊C₄	—	163	1	no	—
101	C₂²⋊C₄	—	163	2	no	—
102	C₂²⋊C₄	—	163	2	no	—
103	C₂²⋊C₄	—	163	1	no	—
104	C₂²⋊C₄	—	163	2	no	—
105	C₄²	—	163	2	no	—
106	C₄²	C₂×D₁₆₃	1	2	yes	—
107	C₄²	C₂×D₁₆₃	1	1	yes	characteristic
108	C₂³	—	163	1	no	—
109	C₂³	—	163	1	no	—
110	Q₈	—	2	2	no	—
111	D₄	—	326	2	no	—
112	C₂×C₄	—	326	4	no	—
113	C₂×C₄	—	326	2	no	—
114	C₂×C₄	—	326	2	no	—
115	C₂×C₄	—	163	1	no	—
116	C₂×C₄	—	163	2	no	—
117	C₂×C₄	—	163	2	no	—
118	C₂×C₄	—	163	1	no	—
119	C₂×C₄	C₂²×D₁₆₃	1	2	yes	—
120	C₂×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
121	C₂×C₄	C₂²×D₁₆₃	1	2	yes	—
122	C₂×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
123	C₂×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
124	C₂²	—	326	1	no	—
125	C₂²	—	326	1	no	—
126	C₂²	—	326	1	no	—
127	C₂²	—	326	1	no	—
128	C₂²	—	326	1	no	—
129	C₂²	—	163	2	no	—
130	C₂²	C₂³×D₁₆₃	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	2	1	no	—
133	C₄	—	2	2	no	—
134	C₄	—	2	1	no	—
135	C₄	—	326	2	no	—
136	C₄	—	326	1	no	—
137	C₄	—	326	1	no	—
138	C₄	—	326	2	no	—
139	C₄	#14579699	1	2	yes	—
140	C₂	—	326	2	no	—
141	C₂	—	652	1	no	—
142	C₂	#41×D₁₆₃	1	1	yes	characteristic
143	C₂	C₂×#14579699	1	1	yes	characteristic
144	C₂	#19104783	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19104729

C₂

yes

maximal, complemented

#19104727

C₂

yes

maximal, characteristic, complemented

#19104725

C₂

yes

maximal, complemented

#19104720

C₂

yes

maximal

#19104711

C₂

yes

maximal, characteristic, complemented

#19104711

C₂

yes

maximal, characteristic, complemented

C₄×D₆₅₂

C₂

yes

maximal, characteristic

#19104693

C₂

yes

maximal, characteristic, complemented

#19104692

C₂

yes

maximal, complemented

C₄⋊C₄×D₁₆₃

C₂

yes

maximal, characteristic, complemented

C₄×Q₈×C₁₆₃

C₂

yes

maximal, characteristic, complemented

C₂×D₆₅₂

C₂²

yes

characteristic

C₂×C₄×D₁₆₃

C₂²

yes

—

C₂×C₄×D₁₆₃

C₂²

yes

characteristic

C₂×C₄×D₁₆₃

C₂²

yes

characteristic

C₄⋊Dic₁₆₃

C₂²

yes

characteristic

C₄⋊Dic₁₆₃

C₂²

yes

characteristic

C₄⋊Dic₁₆₃

C₂²

yes

characteristic

Dic₁₆₃⋊C₄

C₂²

yes

—

Dic₁₆₃⋊C₄

C₂²

yes

—

Dic₁₆₃⋊C₄

C₂²

yes

—

Dic₁₆₃⋊C₄

C₂²

yes

—

Dic₁₆₃⋊C₄

C₂²

yes

—

(C₂×D₁₆₃)⋊C₄

C₂²

yes

characteristic

(C₂×D₁₆₃)⋊C₄

C₂²

yes

—

(C₂×D₁₆₃)⋊C₄

C₂²

yes

—

(C₂×D₁₆₃)⋊C₄

C₂²

yes

characteristic

(C₂×D₁₆₃)⋊C₄

C₂²

yes

—

C₄×Dic₁₆₃

C₂²

yes

—

C₂×Q₈×C₁₆₃

C₂²

yes

characteristic

C₄⋊C₄×C₁₆₃

C₂²

yes

characteristic

C₄⋊C₄×C₁₆₃

C₂²

yes

—

C₄²×C₁₆₃

C₂²

yes

—

C₄²×C₁₆₃

C₂²

yes

characteristic

C₂²×D₁₆₃

C₂³

yes

characteristic

C₂²×D₁₆₃

C₂³

yes

characteristic

D₆₅₂

—

C₂×Dic₁₆₃

C₂³

yes

characteristic

C₂×Dic₁₆₃

C₂³

yes

—

C₂×Dic₁₆₃

C₂³

yes

characteristic

C₂×Dic₁₆₃

C₂³

yes

—

C₄×D₁₆₃

—

C₄×D₁₆₃

—

C₄×D₁₆₃

—

Q₈×C₁₆₃

—

C₂×C₄×C₁₆₃

C₂³

yes

characteristic

C₂×C₄×C₁₆₃

C₂³

yes

—

C₂×C₄×C₁₆₃

C₂³

yes

characteristic

C₂×C₄×C₁₆₃

C₂³

yes

—

C₂×C₄×C₁₆₃

C₂³

yes

characteristic

C₂²×C₁₆₃

C₂⁴

yes

characteristic

C₂×D₁₆₃

—

C₂×D₁₆₃

—

C₂×D₁₆₃

—

C₂×D₁₆₃

—

C₂×D₁₆₃

—

C₂×D₁₆₃

#41

yes

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

#41

yes

—

Dic₁₆₃

—

Dic₁₆₃

—

Dic₁₆₃

—

Dic₁₆₃

—

C₂×C₁₆₃

C₂×#41

yes

characteristic

C₂×C₁₆₃

C₂×#41

yes

characteristic

C₂×C₁₆₃

2₋¹⁺⁴

yes

characteristic

D₁₆₃

—

D₁₆₃

—

C₁₆₃

#552

yes

characteristic, complemented, Sylow

#552

—

163

maximal, Sylow

#126

—

163

—

#125

—

163

—

#125

—

163

—

#123

—

163

—

C₂²⋊Q₈

—

163

—

C₂²⋊Q₈

—

163

—

C₄×Q₈

D₁₆₃

yes

characteristic, complemented

C₄×D₄

—

163

—

#117

—

163

—

#117

—

163

—

C₂×C₄⋊C₄

—

163

—

C₂×Q₈

C₂×D₁₆₃

yes

characteristic

C₂×D₄

—

163

—

C₂²×C₄

—

163

—

C₂²×C₄

—

163

—

C₂²×C₄

—

163

—

C₄⋊C₄

—

163

—

C₄⋊C₄

—

163

—

C₄⋊C₄

—

163

—

C₄⋊C₄

—

163

—

C₄⋊C₄

—

163

—

C₄⋊C₄

—

163

—

C₄⋊C₄

—

163

—

C₄⋊C₄

—

163

—

C₄⋊C₄

C₂×D₁₆₃

yes

characteristic

C₄⋊C₄

C₂×D₁₆₃

yes

—

100

C₂²⋊C₄

—

163

—

101

C₂²⋊C₄

—

163

—

102

C₂²⋊C₄

—

163

—

103

C₂²⋊C₄

—

163

—

104

C₂²⋊C₄

—

163

—

105

C₄²

—

163

—

106

C₄²

C₂×D₁₆₃

yes

—

107

C₄²

C₂×D₁₆₃

yes

characteristic

108

C₂³

—

163

—

109

C₂³

—

163

—

110

Q₈

—

111

D₄

—

326

—

112

C₂×C₄

—

326

—

113

C₂×C₄

—

326

—

114

C₂×C₄

—

326

—

115

C₂×C₄

—

163

—

116

C₂×C₄

—

163

—

117

C₂×C₄

—

163

—

118

C₂×C₄

—

163

—

119

C₂×C₄

C₂²×D₁₆₃

yes

—

120

C₂×C₄

C₂²×D₁₆₃

yes

characteristic

121

C₂×C₄

C₂²×D₁₆₃

yes

—

122

C₂×C₄

C₂²×D₁₆₃

yes

characteristic

123

C₂×C₄

C₂²×D₁₆₃

yes

characteristic

124

C₂²

—

326

—

125

C₂²

—

326

—

126

C₂²

—

326

—

127

C₂²

—

326

—

128

C₂²

—

326

—

129

C₂²

—

163

—

130

C₂²

C₂³×D₁₆₃

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

326

—

136

C₄

—

326

—

137

C₄

—

326

—

138

C₄

—

326

—

139

C₄

#14579699

yes

—

140

C₂

—

326

—

141

C₂

—

652

—

142

C₂

#41×D₁₆₃

yes

characteristic

143

C₂

C₂×#14579699

yes

characteristic

144

C₂

#19104783

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₆₃
commutator, socle, cosocle	C₂²×C₁₆₃	C₂⁴
Fitting	C₄×Q₈×C₁₆₃	C₂
prime radical	C₂×D₆₅₂	C₂²
hypercenter	C₄×Q₈	D₁₆₃
hypocenter	C₁₆₃	#552
LCS	G ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₆₃

commutator, socle, cosocle

C₂²×C₁₆₃

C₂⁴

Fitting

C₄×Q₈×C₁₆₃

C₂

prime radical

C₂×D₆₅₂

C₂²

hypercenter

C₄×Q₈

D₁₆₃

C₁₆₃

G ▷ C₂²×C₁₆₃ ▷ C₁₆₃

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
1. #19104729	C₂	semidirect
2. #19104727	C₂	semidirect
3. #19104725	C₂	semidirect
5. #19104711	C₂	semidirect
6. #19104711	C₂	semidirect
8. #19104693	C₂	semidirect
9. #19104692	C₂	semidirect
10. C₄⋊C₄×D₁₆₃	C₂	semidirect
11. C₄×Q₈×C₁₆₃	C₂	semidirect
72. C₁₆₃	#552	semidirect
80. C₄×Q₈	D₁₆₃	semidirect

H ≅ G/N

type

1. #19104729

C₂

semidirect

2. #19104727

C₂

semidirect

3. #19104725

C₂

semidirect

5. #19104711

C₂

semidirect

6. #19104711

C₂

semidirect

8. #19104693

C₂

semidirect

9. #19104692

C₂

semidirect

10. C₄⋊C₄×D₁₆₃

C₂

semidirect

11. C₄×Q₈×C₁₆₃

C₂

semidirect

72. C₁₆₃

#552

semidirect

80. C₄×Q₈

D₁₆₃

semidirect

Subgroups — #23782356

Conjugacy classes of subgroups

Complements