Group #35225085 — subgroups

Subgroups — #35225085

Order 20864.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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	#23782222	C₂	1	1	yes	maximal, characteristic
2	#23782075	C₂	1	4	yes	maximal, complemented
3	#23782064	C₂	1	4	yes	maximal, complemented
4	#23781705	C₂	1	4	yes	maximal
5	C₄×#14579683	C₂	1	1	yes	maximal, characteristic
6	C₂×#125×C₁₆₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#14579683	C₂²	1	2	yes	—
8	#19104735	C₂²	1	2	yes	—
9	#19104727	C₂²	1	2	yes	—
10	#19104714	—	2	2	no	—
11	C₄×Dic₃₂₆	—	2	2	no	—
12	#19104694	C₂²	1	2	yes	—
13	#19104668	C₂²	1	8	yes	—
14	#19104666	C₂²	1	8	yes	—
15	#19104646	C₂²	1	2	yes	—
16	C₄×C₁₆₃⋊C₈	C₂²	1	2	yes	—
17	#125×C₁₆₃	—	2	2	no	—
18	#125×C₁₆₃	C₂²	1	2	yes	—
19	#125×C₁₆₃	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₁₆₃	—	2	2	no	—
21	C₂×C₄⋊C₄×C₁₆₃	C₂²	1	2	yes	—
22	C₂×C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
23	C₂×Dic₃₂₆	—	2	2	no	—
24	#14579683	Q₈	1	8	yes	complemented
25	C₂×C₁₆₃⋊C₈	C₂³	1	4	yes	—
26	C₄⋊Dic₁₆₃	—	2	2	no	—
27	C₄⋊Dic₁₆₃	C₂³	1	4	yes	—
28	Dic₁₆₃⋊C₄	—	2	4	no	—
29	C₂²⋊Dic₁₆₃	—	2	2	no	—
30	C₄×Dic₁₆₃	—	2	2	no	—
31	C₂²×C₄×C₁₆₃	—	2	2	no	—
32	C₂²×C₄×C₁₆₃	D₄	1	2	yes	—
33	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₁₆₃	—	2	4	no	—
35	C₄⋊C₄×C₁₆₃	—	2	4	no	—
36	C₄⋊C₄×C₁₆₃	—	2	2	no	—
37	C₄⋊C₄×C₁₆₃	C₂³	1	4	yes	—
38	C₄²×C₁₆₃	D₄	1	2	yes	—
39	C₄²×C₁₆₃	C₂³	1	1	yes	characteristic
40	C₄²×C₁₆₃	C₂³	1	1	yes	characteristic
41	Dic₃₂₆	—	4	4	no	—
42	C₂×Dic₁₆₃	—	2	4	no	—
43	C₁₆₃⋊C₈	C₂×Q₈	1	8	yes	—
44	C₂³×C₁₆₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₆₃	—	2	4	no	—
46	C₂×C₄×C₁₆₃	—	4	2	no	—
47	C₂×C₄×C₁₆₃	—	4	4	no	—
48	C₂×C₄×C₁₆₃	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₁₆₃	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₆₃	C₂×Q₈	1	4	yes	—
52	C₂×C₄×C₁₆₃	C₂⁴	1	1	yes	characteristic
53	C₂²×C₁₆₃	—	2	2	no	—
54	C₂²×C₁₆₃	C₄⋊Q₈	1	2	yes	—
55	C₂²×C₁₆₃	C₂²×D₄	1	1	yes	characteristic
56	C₄×C₁₆₃	—	2	2	no	—
57	C₄×C₁₆₃	—	2	2	no	—
58	C₄×C₁₆₃	—	4	4	no	—
59	C₄×C₁₆₃	C₄⋊Q₈	1	2	yes	—
60	C₄×C₁₆₃	C₂²×Q₈	1	2	yes	—
61	Dic₁₆₃	—	4	4	no	—
62	C₂×C₁₆₃	—	2	2	no	—
63	C₂×C₁₆₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₁₆₃	#575	1	2	yes	—
65	C₁₆₃	#3166	1	1	yes	characteristic, complemented, Sylow
66	#3166	—	163	1	no	maximal, Sylow
67	#533	—	163	1	no	—
68	C₂×#125	D₁₆₃	1	1	yes	characteristic
69	#501	—	163	4	no	—
70	#499	—	163	4	no	—
71	#428	—	163	4	no	—
72	C₄×M₁₆	—	163	1	no	—
73	C₂×M₁₆	—	163	2	no	—
74	C₄⋊Q₈	—	163	2	no	—
75	#125	—	2	2	no	—
76	#125	—	163	2	no	—
77	#125	C₂×D₁₆₃	1	2	yes	—
78	#125	C₂×D₁₆₃	1	2	yes	—
79	C₂²⋊Q₈	—	326	2	no	—
80	C₄×Q₈	—	326	2	no	—
81	#117	—	163	2	no	—
82	C₂×C₄⋊C₄	—	2	2	no	—
83	C₂×C₄⋊C₄	C₂×D₁₆₃	1	2	yes	—
84	C₂×C₄²	C₂×D₁₆₃	1	1	yes	characteristic
85	#107	—	163	8	no	—
86	#106	—	163	8	no	—
87	#97	—	163	2	no	—
88	C₄×C₈	—	163	2	no	—
89	C₂×Q₈	—	326	2	no	—
90	C₂²×C₄	—	2	2	no	—
91	C₂²×C₄	C₂²⋊D₁₆₃	1	2	yes	—
92	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
93	M₁₆	—	163	8	no	—
94	C₂×C₈	—	163	4	no	—
95	C₄⋊C₄	—	2	4	no	—
96	C₄⋊C₄	—	2	4	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	326	4	no	—
99	C₄⋊C₄	—	326	2	no	—
100	C₄⋊C₄	—	163	4	no	—
101	C₄⋊C₄	C₂²×D₁₆₃	1	4	yes	—
102	C₂²⋊C₄	—	326	2	no	—
103	C₄²	—	326	2	no	—
104	C₄²	C₂²⋊D₁₆₃	1	2	yes	—
105	C₄²	C₂²×D₁₆₃	1	1	yes	characteristic
106	C₄²	C₂²×D₁₆₃	1	1	yes	characteristic
107	C₂³	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
108	Q₈	—	652	4	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	4	4	no	—
112	C₂×C₄	—	326	4	no	—
113	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	2	yes	—
114	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	2	yes	—
115	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
116	C₂×C₄	Q₈×D₁₆₃	1	4	yes	—
117	C₂×C₄	C₂³×D₁₆₃	1	1	yes	characteristic
118	C₈	—	163	8	no	—
119	C₂²	—	2	2	no	—
120	C₂²	#19104733	1	2	yes	—
121	C₂²	C₂²×C₂²⋊D₁₆₃	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	4	4	no	—
125	C₄	—	652	4	no	—
126	C₄	#19104733	1	2	yes	—
127	C₄	C₂×Q₈×D₁₆₃	1	2	yes	—
128	C₂	—	2	2	no	—
129	C₂	C₂×#19104733	1	1	yes	characteristic
130	C₂	#23782463	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23782222

C₂

yes

maximal, characteristic

#23782075

C₂

yes

maximal, complemented

#23782064

C₂

yes

maximal, complemented

#23781705

C₂

yes

maximal

C₄×#14579683

C₂

yes

maximal, characteristic

C₂×#125×C₁₆₃

C₂

yes

maximal, characteristic

C₂×#14579683

C₂²

yes

—

#19104735

C₂²

yes

—

#19104727

C₂²

yes

—

#19104714

—

C₄×Dic₃₂₆

—

#19104694

C₂²

yes

—

#19104668

C₂²

yes

—

#19104666

C₂²

yes

—

#19104646

C₂²

yes

—

C₄×C₁₆₃⋊C₈

C₂²

yes

—

#125×C₁₆₃

—

#125×C₁₆₃

C₂²

yes

—

#125×C₁₆₃

C₂²

yes

—

C₂×C₄⋊C₄×C₁₆₃

—

C₂×C₄⋊C₄×C₁₆₃

C₂²

yes

—

C₂×C₄²×C₁₆₃

C₂²

yes

characteristic

C₂×Dic₃₂₆

—

#14579683

Q₈

yes

complemented

C₂×C₁₆₃⋊C₈

C₂³

yes

—

C₄⋊Dic₁₆₃

—

C₄⋊Dic₁₆₃

C₂³

yes

—

Dic₁₆₃⋊C₄

—

C₂²⋊Dic₁₆₃

—

C₄×Dic₁₆₃

—

C₂²×C₄×C₁₆₃

—

C₂²×C₄×C₁₆₃

D₄

yes

—

C₂²×C₄×C₁₆₃

C₂³

yes

characteristic

C₄⋊C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

C₂³

yes

—

C₄²×C₁₆₃

D₄

yes

—

C₄²×C₁₆₃

C₂³

yes

characteristic

C₄²×C₁₆₃

C₂³

yes

characteristic

Dic₃₂₆

—

C₂×Dic₁₆₃

—

C₁₆₃⋊C₈

C₂×Q₈

yes

—

C₂³×C₁₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂×Q₈

yes

—

C₂×C₄×C₁₆₃

C₂⁴

yes

characteristic

C₂²×C₁₆₃

—

C₂²×C₁₆₃

C₄⋊Q₈

yes

—

C₂²×C₁₆₃

C₂²×D₄

yes

characteristic

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

C₄⋊Q₈

yes

—

C₄×C₁₆₃

C₂²×Q₈

yes

—

Dic₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₆₃

#575

yes

—

C₁₆₃

#3166

yes

characteristic, complemented, Sylow

#3166

—

163

maximal, Sylow

#533

—

163

—

C₂×#125

D₁₆₃

yes

characteristic

#501

—

163

—

#499

—

163

—

#428

—

163

—

C₄×M₁₆

—

163

—

C₂×M₁₆

—

163

—

C₄⋊Q₈

—

163

—

#125

—

#125

—

163

—

#125

C₂×D₁₆₃

yes

—

#125

C₂×D₁₆₃

yes

—

C₂²⋊Q₈

—

326

—

C₄×Q₈

—

326

—

#117

—

163

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₆₃

yes

—

C₂×C₄²

C₂×D₁₆₃

yes

characteristic

#107

—

163

—

#106

—

163

—

#97

—

163

—

C₄×C₈

—

163

—

C₂×Q₈

—

326

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₆₃

yes

—

C₂²×C₄

C₂²×D₁₆₃

yes

characteristic

M₁₆

—

163

—

C₂×C₈

—

163

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

326

—

C₄⋊C₄

—

326

—

100

C₄⋊C₄

—

163

—

101

C₄⋊C₄

C₂²×D₁₆₃

yes

—

102

C₂²⋊C₄

—

326

—

103

C₄²

—

326

—

104

C₄²

C₂²⋊D₁₆₃

yes

—

105

C₄²

C₂²×D₁₆₃

yes

characteristic

106

C₄²

C₂²×D₁₆₃

yes

characteristic

107

C₂³

C₂×C₂²⋊D₁₆₃

yes

characteristic

108

Q₈

—

652

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

326

—

113

C₂×C₄

C₂×C₂²⋊D₁₆₃

yes

—

114

C₂×C₄

C₂×C₂²⋊D₁₆₃

yes

—

115

C₂×C₄

C₂×C₂²⋊D₁₆₃

yes

characteristic

116

C₂×C₄

Q₈×D₁₆₃

yes

—

117

C₂×C₄

C₂³×D₁₆₃

yes

characteristic

118

C₈

—

163

—

119

C₂²

—

120

C₂²

#19104733

yes

—

121

C₂²

C₂²×C₂²⋊D₁₆₃

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

652

—

126

C₄

#19104733

yes

—

127

C₄

C₂×Q₈×D₁₆₃

yes

—

128

C₂

—

129

C₂

C₂×#19104733

yes

characteristic

130

C₂

#23782463

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₆₃
commutator, cosocle	C₂×C₄×C₁₆₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₆₃
Fitting	C₂×#125×C₁₆₃	C₂
socle	C₂²×C₁₆₃	C₂²×D₄
prime radical	C₂³×C₁₆₃	C₂×D₄
hypercenter	C₂×#125	D₁₆₃
hypocenter	C₁₆₃	#3166
LCS	G ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₆₃

commutator, cosocle

C₂×C₄×C₁₆₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₆₃

Fitting

C₂×#125×C₁₆₃

C₂

socle

C₂²×C₁₆₃

C₂²×D₄

prime radical

C₂³×C₁₆₃

C₂×D₄

D₁₆₃

C₁₆₃

G ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #23782075	C₂	semidirect
3. #23782064	C₂	semidirect
24. #14579683	Q₈	semidirect
65. C₁₆₃	#3166	semidirect

H ≅ G/N

type

2. #23782075

C₂

semidirect

3. #23782064

C₂

semidirect

24. #14579683

Q₈

semidirect

65. C₁₆₃

#3166

semidirect

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

subgroup	index
#23782222	2
#23782075	2
#23782064	2
#23781705	2
C₄×#14579683	2
C₂×#125×C₁₆₃	2
#3166	163

subgroup

index

C₂×#125×C₁₆₃

163

Subgroups — #35225085

Conjugacy classes of subgroups

Complements

Maximal subgroups