Group #23781831 — subgroups

Subgroups — #23781831

Order 10432.

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

normal subgroups only

This group has 168 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	#19104781	C₂	1	1	yes	maximal, characteristic, complemented
2	#19104761	C₂	1	1	yes	maximal, characteristic, complemented
3	#19104731	C₂	1	1	yes	maximal, characteristic, complemented
4	#19104662	C₂	1	1	yes	maximal, characteristic, complemented
5	#19104661	C₂	1	1	yes	maximal, characteristic, complemented
6	#19104651	C₂	1	1	yes	maximal, characteristic, complemented
7	AGL(1,ℤ/8)×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#14579700	—	2	1	no	—
9	#14579699	—	2	1	no	—
10	#14579699	C₂²	1	1	yes	characteristic
11	C₂×D₆₅₂	—	2	1	no	—
12	C₂×D₆₅₂	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊D₁₆₃	—	2	2	no	—
14	D₄×D₁₆₃	—	2	2	no	—
15	D₄×D₁₆₃	—	2	1	no	—
16	#14579687	—	2	1	no	—
17	#14579684	—	2	1	no	—
18	#14579683	C₂²	1	1	yes	characteristic, complemented
19	C₄×Dic₁₆₃	C₂²	1	1	yes	characteristic, complemented
20	#41×C₁₆₃	C₂²	1	1	yes	characteristic
21	C₂×D₄×C₁₆₃	C₂²	1	1	yes	characteristic, complemented
22	SD₁₆×C₁₆₃	—	2	1	no	—
23	D₈×C₁₆₃	—	2	1	no	—
24	M₁₆×C₁₆₃	C₂²	1	1	yes	characteristic, complemented
25	C₂²×D₁₆₃	—	2	2	no	—
26	C₂²×D₁₆₃	D₄	1	2	yes	—
27	Dic₃₂₆	D₄	1	1	yes	characteristic, complemented
28	D₆₅₂	—	2	2	no	—
29	D₆₅₂	—	2	1	no	—
30	D₆₅₂	—	2	1	no	—
31	D₆₅₂	D₄	1	1	yes	characteristic, complemented
32	C₂²⋊D₁₆₃	—	2	2	no	—
33	C₂²⋊D₁₆₃	—	2	1	no	—
34	C₂²⋊D₁₆₃	—	4	4	no	—
35	C₂×Dic₁₆₃	—	2	1	no	—
36	C₄×D₁₆₃	—	2	2	no	—
37	C₄×D₁₆₃	—	2	1	no	—
38	C₁₆₃⋊C₈	—	2	1	no	—
39	C₂³×C₁₆₃	—	2	1	no	—
40	Q₈×C₁₆₃	D₄	1	1	yes	characteristic, complemented
41	D₄×C₁₆₃	—	2	1	no	—
42	D₄×C₁₆₃	—	2	1	no	—
43	D₄×C₁₆₃	—	2	1	no	—
44	D₄×C₁₆₃	D₄	1	1	yes	characteristic, complemented
45	C₂×C₄×C₁₆₃	—	2	1	no	—
46	C₂×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
47	C₈×C₁₆₃	—	2	1	no	—
48	C₂²×C₁₆₃	—	2	1	no	—
49	C₂²×C₁₆₃	—	4	2	no	—
50	C₂²×C₁₆₃	—	4	1	no	—
51	C₂²×C₁₆₃	C₂×D₄	1	1	yes	characteristic
52	C₂×D₁₆₃	—	2	2	no	—
53	C₂×D₁₆₃	—	2	1	no	—
54	C₂×D₁₆₃	—	4	2	no	—
55	C₂×D₁₆₃	—	8	2	no	—
56	C₄×C₁₆₃	—	2	1	no	—
57	C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
58	C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
59	Dic₁₆₃	—	2	2	no	—
60	Dic₁₆₃	—	2	1	no	—
61	C₂×C₁₆₃	—	2	1	no	—
62	C₂×C₁₆₃	—	4	1	no	—
63	C₂×C₁₆₃	—	8	1	no	—
64	C₂×C₁₆₃	#120	1	1	yes	characteristic
65	D₁₆₃	—	4	2	no	—
66	D₁₆₃	—	4	1	no	—
67	C₁₆₃	#453	1	1	yes	characteristic, complemented, Sylow
68	#453	—	163	1	no	maximal, Sylow
69	H(5,𝔽₂)	—	163	1	no	—
70	AGL(1,ℤ/8)	—	163	1	no	—
71	AGL(1,ℤ/8)	D₁₆₃	1	1	yes	characteristic, complemented
72	#127	—	163	1	no	—
73	#104	—	163	1	no	—
74	#104	—	163	1	no	—
75	#100	—	163	1	no	—
76	#41	—	326	1	no	—
77	#41	—	326	1	no	—
78	#41	—	163	1	no	—
79	#41	C₂×D₁₆₃	1	1	yes	characteristic
80	C₂×D₄	—	326	1	no	—
81	C₂×D₄	—	326	2	no	—
82	C₂×D₄	—	326	1	no	—
83	C₂×D₄	—	326	2	no	—
84	C₂×D₄	—	163	1	no	—
85	C₂×D₄	C₂×D₁₆₃	1	1	yes	characteristic, complemented
86	SD₁₆	—	2	1	no	—
87	SD₁₆	—	326	1	no	—
88	D₈	—	2	1	no	—
89	D₈	—	326	1	no	—
90	M₁₆	—	163	1	no	—
91	M₁₆	C₂×D₁₆₃	1	1	yes	characteristic, complemented
92	C₄²	—	163	1	no	—
93	C₂³	—	2	1	no	—
94	C₂³	—	326	2	no	—
95	C₂³	—	163	2	no	—
96	Q₈	—	163	1	no	—
97	Q₈	C₂²⋊D₁₆₃	1	1	yes	characteristic, complemented
98	D₄	—	2	1	no	—
99	D₄	—	2	1	no	—
100	D₄	—	2	1	no	—
101	D₄	—	326	2	no	—
102	D₄	—	326	2	no	—
103	D₄	—	326	1	no	—
104	D₄	—	326	1	no	—
105	D₄	—	326	1	no	—
106	D₄	—	652	4	no	—
107	D₄	—	163	1	no	—
108	D₄	C₂²⋊D₁₆₃	1	1	yes	characteristic, complemented
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	326	2	no	—
111	C₂×C₄	—	326	1	no	—
112	C₂×C₄	—	326	1	no	—
113	C₂×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
114	C₈	—	2	1	no	—
115	C₈	—	326	1	no	—
116	C₂²	—	2	1	no	—
117	C₂²	—	4	2	no	—
118	C₂²	—	4	1	no	—
119	C₂²	—	1304	2	no	—
120	C₂²	—	326	2	no	—
121	C₂²	—	326	1	no	—
122	C₂²	—	652	2	no	—
123	C₂²	D₄×D₁₆₃	1	1	yes	characteristic
124	C₄	—	2	1	no	—
125	C₄	—	326	2	no	—
126	C₄	—	326	1	no	—
127	C₄	D₄×D₁₆₃	1	1	yes	characteristic
128	C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
129	C₂	—	2	1	no	—
130	C₂	—	4	1	no	—
131	C₂	—	8	1	no	—
132	C₂	—	652	1	no	—
133	C₂	—	652	2	no	—
134	C₂	#19104703	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19104781

C₂

yes

maximal, characteristic, complemented

#19104761

C₂

yes

maximal, characteristic, complemented

#19104731

C₂

yes

maximal, characteristic, complemented

#19104662

C₂

yes

maximal, characteristic, complemented

#19104661

C₂

yes

maximal, characteristic, complemented

#19104651

C₂

yes

maximal, characteristic, complemented

AGL(1,ℤ/8)×C₁₆₃

C₂

yes

maximal, characteristic, complemented

#14579700

—

#14579699

—

#14579699

C₂²

yes

characteristic

C₂×D₆₅₂

—

C₂×D₆₅₂

C₂²

yes

characteristic

C₂×C₂²⋊D₁₆₃

—

D₄×D₁₆₃

—

D₄×D₁₆₃

—

#14579687

—

#14579684

—

#14579683

C₂²

yes

characteristic, complemented

C₄×Dic₁₆₃

C₂²

yes

characteristic, complemented

#41×C₁₆₃

C₂²

yes

characteristic

C₂×D₄×C₁₆₃

C₂²

yes

characteristic, complemented

SD₁₆×C₁₆₃

—

D₈×C₁₆₃

—

M₁₆×C₁₆₃

C₂²

yes

characteristic, complemented

C₂²×D₁₆₃

—

C₂²×D₁₆₃

D₄

yes

—

Dic₃₂₆

D₄

yes

characteristic, complemented

D₆₅₂

—

D₆₅₂

—

D₆₅₂

—

D₆₅₂

D₄

yes

characteristic, complemented

C₂²⋊D₁₆₃

—

C₂²⋊D₁₆₃

—

C₂²⋊D₁₆₃

—

C₂×Dic₁₆₃

—

C₄×D₁₆₃

—

C₄×D₁₆₃

—

C₁₆₃⋊C₈

—

C₂³×C₁₆₃

—

Q₈×C₁₆₃

D₄

yes

characteristic, complemented

D₄×C₁₆₃

—

D₄×C₁₆₃

—

D₄×C₁₆₃

—

D₄×C₁₆₃

D₄

yes

characteristic, complemented

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₂³

yes

characteristic

C₈×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

C₂×D₄

yes

characteristic

C₂×D₁₆₃

—

C₂×D₁₆₃

—

C₂×D₁₆₃

—

C₂×D₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

C₂×D₄

yes

characteristic

C₄×C₁₆₃

C₂×D₄

yes

characteristic

Dic₁₆₃

—

Dic₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

#120

yes

characteristic

D₁₆₃

—

D₁₆₃

—

C₁₆₃

#453

yes

characteristic, complemented, Sylow

#453

—

163

maximal, Sylow

H(5,𝔽₂)

—

163

—

AGL(1,ℤ/8)

—

163

—

AGL(1,ℤ/8)

D₁₆₃

yes

characteristic, complemented

#127

—

163

—

#104

—

163

—

#104

—

163

—

#100

—

163

—

#41

—

326

—

#41

—

326

—

#41

—

163

—

#41

C₂×D₁₆₃

yes

characteristic

C₂×D₄

—

326

—

C₂×D₄

—

326

—

C₂×D₄

—

326

—

C₂×D₄

—

326

—

C₂×D₄

—

163

—

C₂×D₄

C₂×D₁₆₃

yes

characteristic, complemented

SD₁₆

—

SD₁₆

—

326

—

D₈

—

D₈

—

326

—

M₁₆

—

163

—

M₁₆

C₂×D₁₆₃

yes

characteristic, complemented

C₄²

—

163

—

C₂³

—

C₂³

—

326

—

C₂³

—

163

—

Q₈

—

163

—

Q₈

C₂²⋊D₁₆₃

yes

characteristic, complemented

D₄

—

D₄

—

100

D₄

—

101

D₄

—

326

—

102

D₄

—

326

—

103

D₄

—

326

—

104

D₄

—

326

—

105

D₄

—

326

—

106

D₄

—

652

—

107

D₄

—

163

—

108

D₄

C₂²⋊D₁₆₃

yes

characteristic, complemented

109

C₂×C₄

—

110

C₂×C₄

—

326

—

111

C₂×C₄

—

326

—

112

C₂×C₄

—

326

—

113

C₂×C₄

C₂²×D₁₆₃

yes

characteristic

114

C₈

—

115

C₈

—

326

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

1304

—

120

C₂²

—

326

—

121

C₂²

—

326

—

122

C₂²

—

652

—

123

C₂²

D₄×D₁₆₃

yes

characteristic

124

C₄

—

125

C₄

—

326

—

126

C₄

—

326

—

127

C₄

D₄×D₁₆₃

yes

characteristic

128

C₄

C₂×C₂²⋊D₁₆₃

yes

characteristic

129

C₂

—

130

C₂

—

131

C₂

—

132

C₂

—

652

—

133

C₂

—

652

—

134

C₂

#19104703

yes

characteristic

	subgroup	G/H
center	C₂	#19104703
commutator, cosocle	C₂×C₄×C₁₆₃	C₂³
Frattini	C₂×C₄	C₂²×D₁₆₃
Fitting	AGL(1,ℤ/8)×C₁₆₃	C₂
socle	C₂×C₁₆₃	#120
hypercenter	AGL(1,ℤ/8)	D₁₆₃
hypocenter	C₁₆₃	#453
LCS	G ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ AGL(1,ℤ/8)

subgroup

G/H

center

C₂

#19104703

commutator, cosocle

C₂×C₄×C₁₆₃

C₂³

Frattini

C₂×C₄

C₂²×D₁₆₃

Fitting

AGL(1,ℤ/8)×C₁₆₃

C₂

C₂×C₁₆₃

D₁₆₃

C₁₆₃

G ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ AGL(1,ℤ/8)

N	H ≅ G/N	type
1. #19104781	C₂	semidirect
2. #19104761	C₂	semidirect
3. #19104731	C₂	semidirect
4. #19104662	C₂	semidirect
5. #19104661	C₂	semidirect
6. #19104651	C₂	semidirect
7. AGL(1,ℤ/8)×C₁₆₃	C₂	semidirect
18. #14579683	C₂²	semidirect
19. C₄×Dic₁₆₃	C₂²	semidirect
21. C₂×D₄×C₁₆₃	C₂²	semidirect
24. M₁₆×C₁₆₃	C₂²	semidirect
27. Dic₃₂₆	D₄	semidirect
31. D₆₅₂	D₄	semidirect
40. Q₈×C₁₆₃	D₄	semidirect
44. D₄×C₁₆₃	D₄	semidirect
67. C₁₆₃	#453	semidirect
71. AGL(1,ℤ/8)	D₁₆₃	semidirect
85. C₂×D₄	C₂×D₁₆₃	semidirect
91. M₁₆	C₂×D₁₆₃	semidirect
97. Q₈	C₂²⋊D₁₆₃	semidirect
108. D₄	C₂²⋊D₁₆₃	semidirect

H ≅ G/N

type

1. #19104781

C₂

semidirect

2. #19104761

C₂

semidirect

3. #19104731

C₂

semidirect

4. #19104662

C₂

semidirect

5. #19104661

C₂

semidirect

6. #19104651

C₂

semidirect

7. AGL(1,ℤ/8)×C₁₆₃

C₂

semidirect

18. #14579683

C₂²

semidirect

19. C₄×Dic₁₆₃

C₂²

semidirect

21. C₂×D₄×C₁₆₃

C₂²

semidirect

24. M₁₆×C₁₆₃

C₂²

semidirect

27. Dic₃₂₆

D₄

semidirect

31. D₆₅₂

D₄

semidirect

40. Q₈×C₁₆₃

D₄

semidirect

44. D₄×C₁₆₃

D₄

semidirect

67. C₁₆₃

#453

semidirect

71. AGL(1,ℤ/8)

D₁₆₃

semidirect

85. C₂×D₄

C₂×D₁₆₃

semidirect

91. M₁₆

C₂×D₁₆₃

semidirect

97. Q₈

C₂²⋊D₁₆₃

semidirect

108. D₄

C₂²⋊D₁₆₃

semidirect

Subgroups — #23781831

Conjugacy classes of subgroups

Complements