Group #23782260 — subgroups

Subgroups — #23782260

Order 10432.

218 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 95 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	#19104728	C₂	1	2	yes	maximal, complemented
2	#19104725	C₂	1	2	yes	maximal, complemented
3	#19104716	C₂	1	2	yes	maximal, complemented
4	#19104716	C₂	1	2	yes	maximal, complemented
5	#19104714	C₂	1	1	yes	maximal, characteristic, complemented
6	#19104712	C₂	1	2	yes	maximal, complemented
7	D₄×Dic₁₆₃	C₂	1	2	yes	maximal, complemented
8	C₂×Dic₁₆₃⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
9	#121×C₁₆₃	C₂	1	1	yes	maximal, characteristic
10	C₂×Dic₃₂₆	C₂²	1	1	yes	characteristic
11	C₂²×Dic₁₆₃	C₂²	1	1	yes	characteristic
12	C₂²×Dic₁₆₃	C₂²	1	1	yes	characteristic
13	C₂²×Dic₁₆₃	C₂²	1	2	yes	—
14	C₄⋊Dic₁₆₃	C₂²	1	2	yes	—
15	C₄⋊Dic₁₆₃	C₂²	1	1	yes	characteristic
16	Dic₁₆₃⋊C₄	C₂²	1	2	yes	—
17	Dic₁₆₃⋊C₄	C₂²	1	2	yes	—
18	Dic₁₆₃⋊C₄	C₂²	1	2	yes	—
19	Dic₁₆₃⋊C₄	C₂²	1	2	yes	—
20	Dic₁₆₃⋊C₄	C₂²	1	2	yes	—
21	C₂²⋊Dic₁₆₃	C₂²	1	2	yes	—
22	C₂²⋊Dic₁₆₃	C₂²	1	2	yes	—
23	C₂²⋊Dic₁₆₃	C₂²	1	2	yes	—
24	C₂²⋊Dic₁₆₃	C₂²	1	2	yes	—
25	C₄×Dic₁₆₃	C₂²	1	2	yes	—
26	C₂×D₄×C₁₆₃	C₂²	1	2	yes	—
27	C₂×D₄×C₁₆₃	C₂²	1	1	yes	characteristic
28	C₂²×C₄×C₁₆₃	C₂²	1	1	yes	characteristic
29	C₄⋊C₄×C₁₆₃	C₂²	1	1	yes	characteristic
30	C₂²⋊C₄×C₁₆₃	C₂²	1	2	yes	—
31	Dic₃₂₆	—	4	1	no	—
32	C₂×Dic₁₆₃	—	2	2	no	—
33	C₂×Dic₁₆₃	—	4	1	no	—
34	C₂×Dic₁₆₃	—	4	2	no	—
35	C₂×Dic₁₆₃	C₂³	1	2	yes	—
36	C₂×Dic₁₆₃	C₂³	1	2	yes	—
37	C₂×Dic₁₆₃	C₂³	1	2	yes	—
38	C₂×Dic₁₆₃	C₂³	1	2	yes	—
39	C₂³×C₁₆₃	C₂³	1	2	yes	—
40	C₂³×C₁₆₃	C₂³	1	1	yes	characteristic
41	D₄×C₁₆₃	—	2	4	no	—
42	D₄×C₁₆₃	—	4	1	no	—
43	C₂×C₄×C₁₆₃	—	4	1	no	—
44	C₂×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
45	C₂×C₄×C₁₆₃	C₂³	1	2	yes	—
46	C₂×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
47	C₂²×C₁₆₃	—	2	1	no	—
48	C₂²×C₁₆₃	—	2	1	no	—
49	C₂²×C₁₆₃	—	2	2	no	—
50	C₂²×C₁₆₃	—	2	2	no	—
51	C₂²×C₁₆₃	—	2	2	no	—
52	C₂²×C₁₆₃	#41	1	2	yes	—
53	C₂²×C₁₆₃	C₂⁴	1	1	yes	characteristic
54	C₄×C₁₆₃	—	2	2	no	—
55	C₄×C₁₆₃	—	2	1	no	—
56	C₄×C₁₆₃	—	2	1	no	—
57	Dic₁₆₃	—	2	2	no	—
58	Dic₁₆₃	—	2	2	no	—
59	Dic₁₆₃	—	2	2	no	—
60	Dic₁₆₃	—	2	2	no	—
61	C₂×C₁₆₃	—	2	2	no	—
62	C₂×C₁₆₃	—	4	2	no	—
63	C₂×C₁₆₃	C₂×#41	1	1	yes	characteristic
64	C₂×C₁₆₃	H(5,𝔽₂)	1	1	yes	characteristic
65	C₂×C₁₆₃	2₋¹⁺⁴	1	1	yes	characteristic
66	C₁₆₃	#539	1	1	yes	characteristic, complemented, Sylow
67	#539	—	163	1	no	maximal, Sylow
68	#126	—	163	2	no	—
69	#125	—	163	2	no	—
70	#123	—	163	2	no	—
71	#123	—	163	2	no	—
72	C₂²⋊Q₈	—	163	1	no	—
73	C₂²⋊Q₈	—	163	2	no	—
74	#121	D₁₆₃	1	1	yes	characteristic
75	C₄×D₄	—	163	2	no	—
76	C₂×C₄⋊C₄	—	163	1	no	—
77	C₂×Q₈	—	163	1	no	—
78	C₂×D₄	C₂×D₁₆₃	1	2	yes	—
79	C₂×D₄	C₂×D₁₆₃	1	1	yes	characteristic
80	C₂²×C₄	—	163	1	no	—
81	C₂²×C₄	—	163	1	no	—
82	C₂²×C₄	—	163	2	no	—
83	C₂²×C₄	C₂×D₁₆₃	1	1	yes	characteristic
84	C₄⋊C₄	—	163	2	no	—
85	C₄⋊C₄	—	163	2	no	—
86	C₄⋊C₄	—	163	2	no	—
87	C₄⋊C₄	—	163	2	no	—
88	C₄⋊C₄	—	163	2	no	—
89	C₄⋊C₄	—	163	1	no	—
90	C₄⋊C₄	—	163	2	no	—
91	C₄⋊C₄	C₂×D₁₆₃	1	1	yes	characteristic
92	C₂²⋊C₄	—	163	2	no	—
93	C₂²⋊C₄	—	163	2	no	—
94	C₂²⋊C₄	—	163	2	no	—
95	C₂²⋊C₄	—	163	2	no	—
96	C₂²⋊C₄	C₂×D₁₆₃	1	2	yes	—
97	C₄²	—	163	2	no	—
98	C₂³	C₂²×D₁₆₃	1	2	yes	—
99	C₂³	C₂²×D₁₆₃	1	1	yes	characteristic
100	Q₈	—	652	1	no	—
101	D₄	—	2	4	no	—
102	D₄	—	4	1	no	—
103	C₂×C₄	—	4	1	no	—
104	C₂×C₄	—	326	2	no	—
105	C₂×C₄	—	652	1	no	—
106	C₂×C₄	—	652	2	no	—
107	C₂×C₄	—	163	2	no	—
108	C₂×C₄	—	163	2	no	—
109	C₂×C₄	—	163	2	no	—
110	C₂×C₄	—	163	2	no	—
111	C₂×C₄	C₂²×D₁₆₃	1	2	yes	—
112	C₂×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
114	C₂²	—	2	1	no	—
115	C₂²	—	2	1	no	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	2	no	—
119	C₂²	#14579698	1	2	yes	—
120	C₂²	C₂³×D₁₆₃	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	—	326	2	no	—
125	C₄	—	326	2	no	—
126	C₄	—	326	2	no	—
127	C₄	—	326	2	no	—
128	C₂	—	2	2	no	—
129	C₂	—	4	2	no	—
130	C₂	C₂×#14579698	1	1	yes	characteristic
131	C₂	#19104780	1	1	yes	characteristic
132	C₂	#19104782	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19104728

C₂

yes

maximal, complemented

#19104725

C₂

yes

maximal, complemented

#19104716

C₂

yes

maximal, complemented

#19104716

C₂

yes

maximal, complemented

#19104714

C₂

yes

maximal, characteristic, complemented

#19104712

C₂

yes

maximal, complemented

D₄×Dic₁₆₃

C₂

yes

maximal, complemented

C₂×Dic₁₆₃⋊C₄

C₂

yes

maximal, characteristic, complemented

#121×C₁₆₃

C₂

yes

maximal, characteristic

C₂×Dic₃₂₆

C₂²

yes

characteristic

C₂²×Dic₁₆₃

C₂²

yes

characteristic

C₂²×Dic₁₆₃

C₂²

yes

characteristic

C₂²×Dic₁₆₃

C₂²

yes

—

C₄⋊Dic₁₆₃

C₂²

yes

—

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

—

C₂²⋊Dic₁₆₃

C₂²

yes

—

C₂²⋊Dic₁₆₃

C₂²

yes

—

C₂²⋊Dic₁₆₃

C₂²

yes

—

C₂²⋊Dic₁₆₃

C₂²

yes

—

C₄×Dic₁₆₃

C₂²

yes

—

C₂×D₄×C₁₆₃

C₂²

yes

—

C₂×D₄×C₁₆₃

C₂²

yes

characteristic

C₂²×C₄×C₁₆₃

C₂²

yes

characteristic

C₄⋊C₄×C₁₆₃

C₂²

yes

characteristic

C₂²⋊C₄×C₁₆₃

C₂²

yes

—

Dic₃₂₆

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

C₂³

yes

—

C₂×Dic₁₆₃

C₂³

yes

—

C₂×Dic₁₆₃

C₂³

yes

—

C₂×Dic₁₆₃

C₂³

yes

—

C₂³×C₁₆₃

C₂³

yes

—

C₂³×C₁₆₃

C₂³

yes

characteristic

D₄×C₁₆₃

—

D₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₂³

yes

characteristic

C₂×C₄×C₁₆₃

C₂³

yes

—

C₂×C₄×C₁₆₃

C₂³

yes

characteristic

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

#41

yes

—

C₂²×C₁₆₃

C₂⁴

yes

characteristic

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

Dic₁₆₃

—

Dic₁₆₃

—

Dic₁₆₃

—

Dic₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

C₂×#41

yes

characteristic

C₂×C₁₆₃

H(5,𝔽₂)

yes

characteristic

C₂×C₁₆₃

2₋¹⁺⁴

yes

characteristic

C₁₆₃

#539

yes

characteristic, complemented, Sylow

#539

—

163

maximal, Sylow

#126

—

163

—

#125

—

163

—

#123

—

163

—

#123

—

163

—

C₂²⋊Q₈

—

163

—

C₂²⋊Q₈

—

163

—

#121

D₁₆₃

yes

characteristic

C₄×D₄

—

163

—

C₂×C₄⋊C₄

—

163

—

C₂×Q₈

—

163

—

C₂×D₄

C₂×D₁₆₃

yes

—

C₂×D₄

C₂×D₁₆₃

yes

characteristic

C₂²×C₄

—

163

—

C₂²×C₄

—

163

—

C₂²×C₄

—

163

—

C₂²×C₄

C₂×D₁₆₃

yes

characteristic

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

C₂²⋊C₄

—

163

—

C₂²⋊C₄

—

163

—

C₂²⋊C₄

—

163

—

C₂²⋊C₄

—

163

—

C₂²⋊C₄

C₂×D₁₆₃

yes

—

C₄²

—

163

—

C₂³

C₂²×D₁₆₃

yes

—

C₂³

C₂²×D₁₆₃

yes

characteristic

100

Q₈

—

652

—

101

D₄

—

102

D₄

—

103

C₂×C₄

—

104

C₂×C₄

—

326

—

105

C₂×C₄

—

652

—

106

C₂×C₄

—

652

—

107

C₂×C₄

—

163

—

108

C₂×C₄

—

163

—

109

C₂×C₄

—

163

—

110

C₂×C₄

—

163

—

111

C₂×C₄

C₂²×D₁₆₃

yes

—

112

C₂×C₄

C₂²×D₁₆₃

yes

characteristic

113

C₂×C₄

C₂²×D₁₆₃

yes

characteristic

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

#14579698

yes

—

120

C₂²

C₂³×D₁₆₃

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

326

—

125

C₄

—

326

—

126

C₄

—

326

—

127

C₄

—

326

—

128

C₂

—

129

C₂

—

130

C₂

C₂×#14579698

yes

characteristic

131

C₂

#19104780

yes

characteristic

132

C₂

#19104782

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₆₃
commutator, socle, cosocle	C₂²×C₁₆₃	C₂⁴
Fitting, prime radical	#121×C₁₆₃	C₂
hypercenter	#121	D₁₆₃
hypocenter	C₁₆₃	#539
LCS	G ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂² ◁ #121

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₆₃

commutator, socle, cosocle

C₂²×C₁₆₃

C₂⁴

Fitting, prime radical

#121×C₁₆₃

C₂

D₁₆₃

C₁₆₃

G ▷ C₂²×C₁₆₃ ▷ C₁₆₃

C₁ ◁ C₂² ◁ #121

N	H ≅ G/N	type
1. #19104728	C₂	semidirect
2. #19104725	C₂	semidirect
3. #19104716	C₂	semidirect
4. #19104716	C₂	semidirect
5. #19104714	C₂	semidirect
6. #19104712	C₂	semidirect
7. D₄×Dic₁₆₃	C₂	semidirect
8. C₂×Dic₁₆₃⋊C₄	C₂	semidirect
66. C₁₆₃	#539	semidirect

H ≅ G/N

type

1. #19104728

C₂

semidirect

2. #19104725

C₂

semidirect

3. #19104716

C₂

semidirect

4. #19104716

C₂

semidirect

5. #19104714

C₂

semidirect

6. #19104712

C₂

semidirect

7. D₄×Dic₁₆₃

C₂

semidirect

8. C₂×Dic₁₆₃⋊C₄

C₂

semidirect

66. C₁₆₃

#539

semidirect

Subgroups — #23782260

Conjugacy classes of subgroups

Complements