Group #35213619 — subgroups

Subgroups — #35213619

Order 20864.

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

normal subgroups only

This group has 240 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	#23782039	C₂	1	1	yes	maximal, characteristic, complemented
2	#23781860	C₂	1	2	yes	maximal, complemented
3	#23781414	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄⋊(C₁₆₃⋊C₈)	C₂	1	2	yes	maximal, complemented
5	#555×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14579686	—	2	2	no	—
7	C₂×#14579684	—	2	2	no	—
8	#19104732	C₂²	1	1	yes	characteristic
9	C₄⋊(C₁₆₃⋊C₈)	C₂²	1	2	yes	—
10	#19104655	—	2	2	no	—
11	C₄×C₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₁₆₃	C₂²	1	1	yes	characteristic
13	#124×C₁₆₃	—	2	2	no	—
14	#121×C₁₆₃	—	2	2	no	—
15	C₄×D₄×C₁₆₃	C₂²	1	2	yes	—
16	#117×C₁₆₃	—	2	2	no	—
17	C₂×D₆₅₂	—	2	2	no	—
18	C₂×D₆₅₂	—	2	1	no	—
19	#14579686	—	4	4	no	—
20	#14579684	—	4	4	no	—
21	C₂×C₁₆₃⋊C₈	—	2	1	no	—
22	C₂×C₁₆₃⋊C₈	—	2	2	no	—
23	C₂×Q₈×C₁₆₃	D₄	1	2	yes	—
24	C₂×D₄×C₁₆₃	—	2	2	no	—
25	C₂×D₄×C₁₆₃	D₄	1	2	yes	—
26	C₂²×C₄×C₁₆₃	—	2	2	no	—
27	C₄⋊C₄×C₁₆₃	—	2	2	no	—
28	C₄⋊C₄×C₁₆₃	D₄	1	2	yes	—
29	C₂²⋊C₄×C₁₆₃	—	2	4	no	—
30	C₂²⋊C₄×C₁₆₃	—	2	2	no	—
31	C₄²×C₁₆₃	—	2	2	no	—
32	C₄²×C₁₆₃	C₂³	1	1	yes	characteristic
33	C₂²×D₁₆₃	—	4	1	no	—
34	D₆₅₂	—	4	2	no	—
35	D₆₅₂	—	4	2	no	—
36	D₆₅₂	—	4	2	no	—
37	C₁₆₃⋊C₈	—	2	2	no	—
38	C₁₆₃⋊C₈	—	4	2	no	—
39	C₂³×C₁₆₃	—	2	2	no	—
40	Q₈×C₁₆₃	—	4	2	no	—
41	D₄×C₁₆₃	—	2	2	no	—
42	D₄×C₁₆₃	—	2	2	no	—
43	D₄×C₁₆₃	—	8	2	no	—
44	C₂×C₄×C₁₆₃	—	2	2	no	—
45	C₂×C₄×C₁₆₃	—	2	2	no	—
46	C₂×C₄×C₁₆₃	—	4	2	no	—
47	C₂×C₄×C₁₆₃	—	4	2	no	—
48	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₆₃	C₂×D₄	1	2	yes	—
50	C₂²×C₁₆₃	—	4	2	no	—
51	C₂²×C₁₆₃	—	4	2	no	—
52	C₂²×C₁₆₃	—	4	2	no	—
53	C₂²×C₁₆₃	#120	1	1	yes	characteristic
54	C₂×D₁₆₃	—	8	2	no	—
55	C₂×D₁₆₃	—	8	1	no	—
56	C₄×C₁₆₃	—	2	1	no	—
57	C₄×C₁₆₃	—	4	2	no	—
58	C₄×C₁₆₃	—	4	2	no	—
59	C₄×C₁₆₃	#135	1	2	yes	—
60	C₄×C₁₆₃	AGL(1,ℤ/8)	1	2	yes	—
61	C₂×C₁₆₃	—	8	2	no	—
62	C₂×C₁₆₃	#449	1	2	yes	—
63	C₂×C₁₆₃	#453	1	1	yes	characteristic
64	D₁₆₃	—	16	1	no	—
65	C₁₆₃	#1624	1	1	yes	characteristic, complemented, Sylow
66	#1624	—	163	1	no	maximal, Sylow
67	#555	D₁₆₃	1	1	yes	characteristic, complemented
68	#492	—	163	1	no	—
69	#459	—	163	2	no	—
70	#333	—	163	1	no	—
71	D₄⋊C₈	—	163	2	no	—
72	C₂×SD₁₆	—	326	2	no	—
73	C₂×D₈	—	326	2	no	—
74	C₄⋊Q₈	C₂×D₁₆₃	1	1	yes	characteristic
75	#127	—	163	1	no	—
76	#124	—	2	2	no	—
77	#121	—	2	2	no	—
78	C₄×D₄	C₂×D₁₆₃	1	2	yes	—
79	#117	—	2	2	no	—
80	C₄⋊C₈	—	163	2	no	—
81	#102	—	326	2	no	—
82	C₄×C₈	—	163	1	no	—
83	C₂×Q₈	C₂²⋊D₁₆₃	1	2	yes	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	—	326	2	no	—
86	C₂×D₄	—	326	1	no	—
87	C₂×D₄	C₂²⋊D₁₆₃	1	2	yes	—
88	C₂²×C₄	—	2	2	no	—
89	SD₁₆	—	652	4	no	—
90	D₈	—	652	4	no	—
91	C₂×C₈	—	326	1	no	—
92	C₂×C₈	—	326	2	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	C₂²⋊D₁₆₃	1	2	yes	—
95	C₂²⋊C₄	—	2	4	no	—
96	C₂²⋊C₄	—	2	2	no	—
97	C₄²	—	2	2	no	—
98	C₄²	C₂²×D₁₆₃	1	1	yes	characteristic
99	C₂³	—	2	2	no	—
100	C₂³	—	652	1	no	—
101	Q₈	—	4	2	no	—
102	D₄	—	2	2	no	—
103	D₄	—	2	2	no	—
104	D₄	—	8	2	no	—
105	D₄	—	652	2	no	—
106	D₄	—	652	2	no	—
107	D₄	—	652	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	4	2	no	—
112	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
113	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	2	yes	—
114	C₈	—	326	2	no	—
115	C₈	—	652	2	no	—
116	C₂²	—	4	2	no	—
117	C₂²	—	4	2	no	—
118	C₂²	—	4	2	no	—
119	C₂²	—	1304	2	no	—
120	C₂²	—	1304	1	no	—
121	C₂²	#19104704	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	4	2	no	—
124	C₄	—	4	2	no	—
125	C₄	#19104755	1	2	yes	—
126	C₄	#19104760	1	2	yes	—
127	C₂	—	8	2	no	—
128	C₂	—	2608	1	no	—
129	C₂	#23781810	1	2	yes	—
130	C₂	#23781833	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23782039

C₂

yes

maximal, characteristic, complemented

#23781860

C₂

yes

maximal, complemented

#23781414

C₂

yes

maximal, characteristic, complemented

D₄⋊(C₁₆₃⋊C₈)

C₂

yes

maximal, complemented

#555×C₁₆₃

C₂

yes

maximal, characteristic, complemented

C₂×#14579686

—

C₂×#14579684

—

#19104732

C₂²

yes

characteristic

C₄⋊(C₁₆₃⋊C₈)

C₂²

yes

—

#19104655

—

C₄×C₁₆₃⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₁₆₃

C₂²

yes

characteristic

#124×C₁₆₃

—

#121×C₁₆₃

—

C₄×D₄×C₁₆₃

C₂²

yes

—

#117×C₁₆₃

—

C₂×D₆₅₂

—

C₂×D₆₅₂

—

#14579686

—

#14579684

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

—

C₂×Q₈×C₁₆₃

D₄

yes

—

C₂×D₄×C₁₆₃

—

C₂×D₄×C₁₆₃

D₄

yes

—

C₂²×C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

D₄

yes

—

C₂²⋊C₄×C₁₆₃

—

C₂²⋊C₄×C₁₆₃

—

C₄²×C₁₆₃

—

C₄²×C₁₆₃

C₂³

yes

characteristic

C₂²×D₁₆₃

—

D₆₅₂

—

D₆₅₂

—

D₆₅₂

—

C₁₆₃⋊C₈

—

C₁₆₃⋊C₈

—

C₂³×C₁₆₃

—

Q₈×C₁₆₃

—

D₄×C₁₆₃

—

D₄×C₁₆₃

—

D₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂×D₄

yes

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

#120

yes

characteristic

C₂×D₁₆₃

—

C₂×D₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

#135

yes

—

C₄×C₁₆₃

AGL(1,ℤ/8)

yes

—

C₂×C₁₆₃

—

C₂×C₁₆₃

#449

yes

—

C₂×C₁₆₃

#453

yes

characteristic

D₁₆₃

—

C₁₆₃

#1624

yes

characteristic, complemented, Sylow

#1624

—

163

maximal, Sylow

#555

D₁₆₃

yes

characteristic, complemented

#492

—

163

—

#459

—

163

—

#333

—

163

—

D₄⋊C₈

—

163

—

C₂×SD₁₆

—

326

—

C₂×D₈

—

326

—

C₄⋊Q₈

C₂×D₁₆₃

yes

characteristic

#127

—

163

—

#124

—

#121

—

C₄×D₄

C₂×D₁₆₃

yes

—

#117

—

C₄⋊C₈

—

163

—

#102

—

326

—

C₄×C₈

—

163

—

C₂×Q₈

C₂²⋊D₁₆₃

yes

—

C₂×D₄

—

C₂×D₄

—

326

—

C₂×D₄

—

326

—

C₂×D₄

C₂²⋊D₁₆₃

yes

—

C₂²×C₄

—

SD₁₆

—

652

—

D₈

—

652

—

C₂×C₈

—

326

—

C₂×C₈

—

326

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊D₁₆₃

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×D₁₆₃

yes

characteristic

C₂³

—

100

C₂³

—

652

—

101

Q₈

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

652

—

106

D₄

—

652

—

107

D₄

—

652

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₂×C₂²⋊D₁₆₃

yes

characteristic

113

C₂×C₄

C₂×C₂²⋊D₁₆₃

yes

—

114

C₈

—

326

—

115

C₈

—

652

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

1304

—

120

C₂²

—

1304

—

121

C₂²

#19104704

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

#19104755

yes

—

126

C₄

#19104760

yes

—

127

C₂

—

128

C₂

—

2608

—

129

C₂

#23781810

yes

—

130

C₂

#23781833

yes

characteristic

	subgroup	G/H
center	C₂²	#19104704
commutator, cosocle	C₄²×C₁₆₃	C₂³
Frattini	C₄²	C₂²×D₁₆₃
Fitting	#555×C₁₆₃	C₂
socle	C₂²×C₁₆₃	#120
hypercenter	#555	D₁₆₃
hypocenter	C₁₆₃	#1624
LCS	G ▷ C₄²×C₁₆₃ ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #555

subgroup

G/H

center

C₂²

#19104704

commutator, cosocle

C₄²×C₁₆₃

C₂³

Frattini

C₄²

C₂²×D₁₆₃

Fitting

#555×C₁₆₃

C₂

C₂²×C₁₆₃

D₁₆₃

C₁₆₃

G ▷ C₄²×C₁₆₃ ▷ C₂²×C₁₆₃ ▷ C₁₆₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #555

N	H ≅ G/N	type
1. #23782039	C₂	semidirect
2. #23781860	C₂	semidirect
3. #23781414	C₂	semidirect
4. D₄⋊(C₁₆₃⋊C₈)	C₂	semidirect
5. #555×C₁₆₃	C₂	semidirect
65. C₁₆₃	#1624	semidirect
67. #555	D₁₆₃	semidirect

H ≅ G/N

type

1. #23782039

C₂

semidirect

2. #23781860

C₂

semidirect

3. #23781414

C₂

semidirect

4. D₄⋊(C₁₆₃⋊C₈)

C₂

semidirect

5. #555×C₁₆₃

C₂

semidirect

65. C₁₆₃

#1624

semidirect

67. #555

D₁₆₃

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#23782039	2
#23781860	2
#23781414	2
D₄⋊(C₁₆₃⋊C₈)	2
#555×C₁₆₃	2
#1624	163

subgroup

index

#23782039

#23781860

#23781414

D₄⋊(C₁₆₃⋊C₈)

#555×C₁₆₃

#1624

163

Subgroups — #35213619

Conjugacy classes of subgroups

Complements

Maximal subgroups