Group #23781851 — subgroups

Subgroups — #23781851

Order 10432.

198 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 39 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	#19104780	C₂	1	1	yes	maximal, characteristic, complemented
2	#19104704	C₂	1	2	yes	maximal, complemented
3	#19104650	C₂	1	1	yes	maximal, characteristic, complemented
4	#19104649	C₂	1	2	yes	maximal, complemented
5	#120×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#14579699	—	2	1	no	—
7	#14579698	—	2	2	no	—
8	C₂×C₂²⋊D₁₆₃	—	2	2	no	—
9	C₂×C₂²⋊D₁₆₃	—	2	1	no	—
10	C₂×C₂²⋊D₁₆₃	C₂²	1	2	yes	—
11	D₄×D₁₆₃	—	2	2	no	—
12	C₂²⋊Dic₁₆₃	—	2	2	no	—
13	C₂²⋊Dic₁₆₃	C₂²	1	2	yes	complemented
14	C₂⁴×C₁₆₃	C₂²	1	1	yes	characteristic, complemented
15	C₂×D₄×C₁₆₃	—	2	1	no	—
16	C₂×D₄×C₁₆₃	C₂²	1	1	yes	characteristic
17	C₂²⋊C₄×C₁₆₃	—	2	1	no	—
18	C₂²⋊C₄×C₁₆₃	C₂²	1	1	yes	characteristic, complemented
19	C₂²×D₁₆₃	—	2	1	no	—
20	C₂²×D₁₆₃	D₄	1	2	yes	complemented
21	Dic₃₂₆	—	2	1	no	—
22	D₆₅₂	—	2	1	no	—
23	C₂²⋊D₁₆₃	—	2	2	no	—
24	C₂²⋊D₁₆₃	—	2	2	no	—
25	C₂²⋊D₁₆₃	—	2	2	no	—
26	C₂²⋊D₁₆₃	—	4	4	no	—
27	C₂×Dic₁₆₃	—	2	1	no	—
28	C₂×Dic₁₆₃	—	2	2	no	—
29	C₂×Dic₁₆₃	D₄	1	2	yes	complemented
30	C₄×D₁₆₃	—	2	2	no	—
31	C₂³×C₁₆₃	—	2	1	no	—
32	C₂³×C₁₆₃	—	2	2	no	—
33	C₂³×C₁₆₃	—	4	2	no	—
34	C₂³×C₁₆₃	D₄	1	1	yes	characteristic, complemented
35	C₂³×C₁₆₃	C₂³	1	1	yes	characteristic
36	D₄×C₁₆₃	—	2	2	no	—
37	D₄×C₁₆₃	—	4	2	no	—
38	C₂×C₄×C₁₆₃	—	2	1	no	—
39	C₂×C₄×C₁₆₃	D₄	1	1	yes	characteristic, complemented
40	C₂²×C₁₆₃	—	2	2	no	—
41	C₂²×C₁₆₃	—	2	4	no	—
42	C₂²×C₁₆₃	—	2	1	no	—
43	C₂²×C₁₆₃	—	4	1	no	—
44	C₂²×C₁₆₃	—	4	1	no	—
45	C₂²×C₁₆₃	—	4	2	no	—
46	C₂²×C₁₆₃	—	4	1	no	—
47	C₂²×C₁₆₃	—	4	1	no	—
48	C₂²×C₁₆₃	C₂×D₄	1	2	yes	—
49	C₂²×C₁₆₃	C₂×D₄	1	1	yes	characteristic
50	C₂×D₁₆₃	—	2	2	no	—
51	C₂×D₁₆₃	—	4	2	no	—
52	C₂×D₁₆₃	—	8	1	no	—
53	C₄×C₁₆₃	—	2	1	no	—
54	C₄×C₁₆₃	—	4	1	no	—
55	Dic₁₆₃	—	2	2	no	—
56	Dic₁₆₃	—	4	2	no	—
57	C₂×C₁₆₃	—	2	2	no	—
58	C₂×C₁₆₃	—	2	1	no	—
59	C₂×C₁₆₃	—	4	1	no	—
60	C₂×C₁₆₃	—	4	2	no	—
61	C₂×C₁₆₃	#120	1	1	yes	characteristic
62	D₁₆₃	—	4	2	no	—
63	C₁₆₃	B(4,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
64	B(4,𝔽₂)	—	163	1	no	maximal, Sylow
65	H(5,𝔽₂)	—	163	1	no	—
66	#120	—	163	2	no	—
67	#120	D₁₆₃	1	1	yes	characteristic, complemented
68	#99	—	163	1	no	—
69	#99	—	163	2	no	—
70	C₂⁴	C₂×D₁₆₃	1	1	yes	characteristic, complemented
71	#41	—	326	2	no	—
72	#41	—	326	1	no	—
73	C₂×D₄	—	2	1	no	—
74	C₂×D₄	—	326	2	no	—
75	C₂×D₄	—	326	1	no	—
76	C₂×D₄	—	326	2	no	—
77	C₂×D₄	—	163	2	no	—
78	C₂×D₄	C₂×D₁₆₃	1	1	yes	characteristic
79	C₂²⋊C₄	—	2	1	no	—
80	C₂²⋊C₄	—	326	2	no	—
81	C₂²⋊C₄	—	163	2	no	—
82	C₂²⋊C₄	C₂×D₁₆₃	1	1	yes	characteristic, complemented
83	C₂³	—	2	2	no	—
84	C₂³	—	2	1	no	—
85	C₂³	—	4	2	no	—
86	C₂³	—	326	1	no	—
87	C₂³	—	163	2	no	—
88	C₂³	C₂²⋊D₁₆₃	1	1	yes	characteristic, complemented
89	C₂³	C₂²×D₁₆₃	1	1	yes	characteristic
90	Q₈	—	326	1	no	—
91	D₄	—	2	2	no	—
92	D₄	—	4	2	no	—
93	D₄	—	326	1	no	—
94	D₄	—	326	2	no	—
95	D₄	—	326	2	no	—
96	D₄	—	326	2	no	—
97	D₄	—	652	4	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	326	2	no	—
100	C₂×C₄	—	326	1	no	—
101	C₂×C₄	—	326	2	no	—
102	C₂×C₄	—	163	2	no	—
103	C₂×C₄	C₂²⋊D₁₆₃	1	1	yes	characteristic, complemented
104	C₂²	—	2	2	no	—
105	C₂²	—	2	4	no	—
106	C₂²	—	2	1	no	—
107	C₂²	—	4	1	no	—
108	C₂²	—	4	1	no	—
109	C₂²	—	4	1	no	—
110	C₂²	—	4	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	1304	1	no	—
113	C₂²	—	326	2	no	—
114	C₂²	—	652	2	no	—
115	C₂²	D₄×D₁₆₃	1	2	yes	—
116	C₂²	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
117	C₄	—	2	1	no	—
118	C₄	—	4	1	no	—
119	C₄	—	326	2	no	—
120	C₄	—	652	2	no	—
121	C₂	—	2	1	no	—
122	C₂	—	2	2	no	—
123	C₂	—	4	2	no	—
124	C₂	—	4	1	no	—
125	C₂	—	652	2	no	—
126	C₂	#19104703	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19104780

C₂

yes

maximal, characteristic, complemented

#19104704

C₂

yes

maximal, complemented

#19104650

C₂

yes

maximal, characteristic, complemented

#19104649

C₂

yes

maximal, complemented

#120×C₁₆₃

C₂

yes

maximal, characteristic, complemented

#14579699

—

#14579698

—

C₂×C₂²⋊D₁₆₃

—

C₂×C₂²⋊D₁₆₃

—

C₂×C₂²⋊D₁₆₃

C₂²

yes

—

D₄×D₁₆₃

—

C₂²⋊Dic₁₆₃

—

C₂²⋊Dic₁₆₃

C₂²

yes

complemented

C₂⁴×C₁₆₃

C₂²

yes

characteristic, complemented

C₂×D₄×C₁₆₃

—

C₂×D₄×C₁₆₃

C₂²

yes

characteristic

C₂²⋊C₄×C₁₆₃

—

C₂²⋊C₄×C₁₆₃

C₂²

yes

characteristic, complemented

C₂²×D₁₆₃

—

C₂²×D₁₆₃

D₄

yes

complemented

Dic₃₂₆

—

D₆₅₂

—

C₂²⋊D₁₆₃

—

C₂²⋊D₁₆₃

—

C₂²⋊D₁₆₃

—

C₂²⋊D₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

D₄

yes

complemented

C₄×D₁₆₃

—

C₂³×C₁₆₃

—

C₂³×C₁₆₃

—

C₂³×C₁₆₃

—

C₂³×C₁₆₃

D₄

yes

characteristic, complemented

C₂³×C₁₆₃

C₂³

yes

characteristic

D₄×C₁₆₃

—

D₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

D₄

yes

characteristic, complemented

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

C₂×D₄

yes

—

C₂²×C₁₆₃

C₂×D₄

yes

characteristic

C₂×D₁₆₃

—

C₂×D₁₆₃

—

C₂×D₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

Dic₁₆₃

—

Dic₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

#120

yes

characteristic

D₁₆₃

—

C₁₆₃

B(4,𝔽₂)

yes

characteristic, complemented, Sylow

B(4,𝔽₂)

—

163

maximal, Sylow

H(5,𝔽₂)

—

163

—

#120

—

163

—

#120

D₁₆₃

yes

characteristic, complemented

#99

—

163

—

#99

—

163

—

C₂⁴

C₂×D₁₆₃

yes

characteristic, complemented

#41

—

326

—

#41

—

326

—

C₂×D₄

—

C₂×D₄

—

326

—

C₂×D₄

—

326

—

C₂×D₄

—

326

—

C₂×D₄

—

163

—

C₂×D₄

C₂×D₁₆₃

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

326

—

C₂²⋊C₄

—

163

—

C₂²⋊C₄

C₂×D₁₆₃

yes

characteristic, complemented

C₂³

—

C₂³

—

C₂³

—

C₂³

—

326

—

C₂³

—

163

—

C₂³

C₂²⋊D₁₆₃

yes

characteristic, complemented

C₂³

C₂²×D₁₆₃

yes

characteristic

Q₈

—

326

—

D₄

—

D₄

—

D₄

—

326

—

D₄

—

326

—

D₄

—

326

—

D₄

—

326

—

D₄

—

652

—

C₂×C₄

—

C₂×C₄

—

326

—

100

C₂×C₄

—

326

—

101

C₂×C₄

—

326

—

102

C₂×C₄

—

163

—

103

C₂×C₄

C₂²⋊D₁₆₃

yes

characteristic, complemented

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

1304

—

113

C₂²

—

326

—

114

C₂²

—

652

—

115

C₂²

D₄×D₁₆₃

yes

—

116

C₂²

C₂×C₂²⋊D₁₆₃

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

326

—

120

C₄

—

652

—

121

C₂

—

122

C₂

—

123

C₂

—

124

C₂

—

125

C₂

—

652

—

126

C₂

#19104703

yes

characteristic

	subgroup	G/H
center	C₂	#19104703
commutator, cosocle	C₂³×C₁₆₃	C₂³
Frattini	C₂³	C₂²×D₁₆₃
Fitting	#120×C₁₆₃	C₂
socle	C₂×C₁₆₃	#120
hypercenter	#120	D₁₆₃
hypocenter	C₁₆₃	B(4,𝔽₂)
LCS	G ▷ C₂³×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ #120

subgroup

G/H

center

C₂

#19104703

commutator, cosocle

C₂³×C₁₆₃

C₂³

Frattini

C₂³

C₂²×D₁₆₃

Fitting

#120×C₁₆₃

C₂

C₂×C₁₆₃

D₁₆₃

C₁₆₃

G ▷ C₂³×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃

UCS

C₁ ◁ C₂ ◁ C₂³ ◁ #120

N	H ≅ G/N	type
1. #19104780	C₂	semidirect
2. #19104704	C₂	semidirect
3. #19104650	C₂	semidirect
4. #19104649	C₂	semidirect
5. #120×C₁₆₃	C₂	semidirect
13. C₂²⋊Dic₁₆₃	C₂²	semidirect
14. C₂⁴×C₁₆₃	C₂²	semidirect
18. C₂²⋊C₄×C₁₆₃	C₂²	semidirect
20. C₂²×D₁₆₃	D₄	semidirect
29. C₂×Dic₁₆₃	D₄	semidirect
34. C₂³×C₁₆₃	D₄	semidirect
39. C₂×C₄×C₁₆₃	D₄	semidirect
63. C₁₆₃	B(4,𝔽₂)	semidirect
67. #120	D₁₆₃	semidirect
70. C₂⁴	C₂×D₁₆₃	semidirect
82. C₂²⋊C₄	C₂×D₁₆₃	semidirect
88. C₂³	C₂²⋊D₁₆₃	semidirect
103. C₂×C₄	C₂²⋊D₁₆₃	semidirect

H ≅ G/N

type

1. #19104780

C₂

semidirect

2. #19104704

C₂

semidirect

3. #19104650

C₂

semidirect

4. #19104649

C₂

semidirect

5. #120×C₁₆₃

C₂

semidirect

13. C₂²⋊Dic₁₆₃

C₂²

semidirect

14. C₂⁴×C₁₆₃

C₂²

semidirect

18. C₂²⋊C₄×C₁₆₃

C₂²

semidirect

20. C₂²×D₁₆₃

D₄

semidirect

29. C₂×Dic₁₆₃

D₄

semidirect

34. C₂³×C₁₆₃

D₄

semidirect

39. C₂×C₄×C₁₆₃

D₄

semidirect

63. C₁₆₃

B(4,𝔽₂)

semidirect

67. #120

D₁₆₃

semidirect

70. C₂⁴

C₂×D₁₆₃

semidirect

82. C₂²⋊C₄

C₂×D₁₆₃

semidirect

88. C₂³

C₂²⋊D₁₆₃

semidirect

103. C₂×C₄

C₂²⋊D₁₆₃

semidirect

Subgroups — #23781851

Conjugacy classes of subgroups

Complements