Group #35216231 — subgroups

Subgroups — #35216231

Order 20864.

230 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 73 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	#23781649	C₂	1	2	yes	maximal
2	#23781464	C₂	1	4	yes	maximal, complemented
3	#536×C₁₆₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579683	—	2	2	no	—
5	#19104654	C₂²	1	4	yes	—
6	#19104652	—	2	2	no	—
7	C₂×#41×C₁₆₃	C₂²	1	1	yes	characteristic
8	C₂²×Q₈×C₁₆₃	C₄	1	1	yes	characteristic
9	C₄⋊Q₈×C₁₆₃	C₂²	1	2	yes	—
10	#124×C₁₆₃	C₄	1	2	yes	—
11	#123×C₁₆₃	—	2	2	no	—
12	C₂²⋊Q₈×C₁₆₃	—	2	2	no	—
13	#117×C₁₆₃	C₄	1	1	yes	characteristic
14	#14579683	—	2	4	no	—
15	#14579683	—	4	2	no	—
16	C₂×C₁₆₃⋊C₈	—	4	2	no	—
17	#41×C₁₆₃	—	4	2	no	—
18	C₂×Q₈×C₁₆₃	—	2	1	no	—
19	C₂×Q₈×C₁₆₃	—	4	2	no	—
20	C₂×Q₈×C₁₆₃	C₂×C₄	1	2	yes	—
21	C₂×Q₈×C₁₆₃	C₂³	1	1	yes	characteristic
22	C₂×D₄×C₁₆₃	C₂×C₄	1	1	yes	characteristic
23	C₂×D₄×C₁₆₃	D₄	1	2	yes	—
24	C₂²×C₄×C₁₆₃	—	2	1	no	—
25	C₂²×C₄×C₁₆₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₆₃	D₄	1	2	yes	—
27	C₄⋊C₄×C₁₆₃	—	2	1	no	—
28	C₄⋊C₄×C₁₆₃	—	2	4	no	—
29	C₂²⋊C₄×C₁₆₃	—	2	1	no	—
30	C₂²⋊C₄×C₁₆₃	—	2	4	no	—
31	C₄²×C₁₆₃	C₂×C₄	1	2	yes	—
32	C₁₆₃⋊C₈	—	4	4	no	—
33	C₂³×C₁₆₃	C₂²⋊C₄	1	2	yes	—
34	C₂³×C₁₆₃	C₂²⋊C₄	1	1	yes	characteristic
35	Q₈×C₁₆₃	—	4	1	no	—
36	Q₈×C₁₆₃	—	4	2	no	—
37	Q₈×C₁₆₃	—	4	2	no	—
38	D₄×C₁₆₃	—	4	1	no	—
39	D₄×C₁₆₃	—	4	2	no	—
40	C₂×C₄×C₁₆₃	—	2	2	no	—
41	C₂×C₄×C₁₆₃	—	2	2	no	—
42	C₂×C₄×C₁₆₃	—	2	2	no	—
43	C₂×C₄×C₁₆₃	—	2	2	no	—
44	C₂×C₄×C₁₆₃	—	2	2	no	—
45	C₂×C₄×C₁₆₃	—	4	2	no	—
46	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₆₃	C₂×D₄	1	2	yes	—
49	C₂²×C₁₆₃	—	2	1	no	—
50	C₂²×C₁₆₃	—	2	2	no	—
51	C₂²×C₁₆₃	—	4	2	no	—
52	C₂²×C₁₆₃	—	4	1	no	—
53	C₂²×C₁₆₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₄×C₁₆₃	—	2	1	no	—
55	C₄×C₁₆₃	—	2	2	no	—
56	C₄×C₁₆₃	—	4	2	no	—
57	C₄×C₁₆₃	—	4	2	no	—
58	C₄×C₁₆₃	#99	1	2	yes	—
59	C₂×C₁₆₃	—	2	1	no	—
60	C₂×C₁₆₃	—	4	1	no	—
61	C₂×C₁₆₃	—	4	2	no	—
62	C₂×C₁₆₃	C₂×#99	1	1	yes	characteristic
63	C₁₆₃	#2135	1	1	yes	characteristic, complemented, Sylow
64	#2135	—	163	1	no	maximal, Sylow
65	#536	D₁₆₃	1	1	yes	characteristic
66	#413	—	163	2	no	—
67	#356	—	163	4	no	—
68	C₂×#41	C₂×D₁₆₃	1	1	yes	characteristic
69	C₂²×Q₈	Dic₁₆₃	1	1	yes	characteristic
70	C₂×M₁₆	—	326	2	no	—
71	C₄⋊Q₈	C₂×D₁₆₃	1	2	yes	—
72	#124	Dic₁₆₃	1	2	yes	—
73	#123	—	2	2	no	—
74	C₂²⋊Q₈	—	2	2	no	—
75	#117	Dic₁₆₃	1	1	yes	characteristic
76	#101	—	163	4	no	—
77	#100	—	326	2	no	—
78	#41	—	4	2	no	—
79	C₂×Q₈	—	2	1	no	—
80	C₂×Q₈	—	4	2	no	—
81	C₂×Q₈	C₂×Dic₁₆₃	1	2	yes	—
82	C₂×Q₈	C₂²×D₁₆₃	1	1	yes	characteristic
83	C₂×D₄	C₂×Dic₁₆₃	1	1	yes	characteristic
84	C₂×D₄	C₂²⋊D₁₆₃	1	2	yes	—
85	C₂²×C₄	—	2	1	no	—
86	C₂²×C₄	C₂×Dic₁₆₃	1	1	yes	characteristic
87	C₂²×C₄	C₂²⋊D₁₆₃	1	2	yes	—
88	M₁₆	—	326	4	no	—
89	M₁₆	—	652	2	no	—
90	C₂×C₈	—	652	2	no	—
91	C₄⋊C₄	—	2	1	no	—
92	C₄⋊C₄	—	2	4	no	—
93	C₂²⋊C₄	—	2	1	no	—
94	C₂²⋊C₄	—	2	4	no	—
95	C₄²	C₂×Dic₁₆₃	1	2	yes	—
96	C₂³	C₂²⋊Dic₁₆₃	1	2	yes	—
97	C₂³	C₂²⋊Dic₁₆₃	1	1	yes	characteristic
98	Q₈	—	4	1	no	—
99	Q₈	—	4	2	no	—
100	Q₈	—	4	2	no	—
101	D₄	—	4	1	no	—
102	D₄	—	4	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	C₂²⋊Dic₁₆₃	1	1	yes	characteristic
110	C₂×C₄	C₂²×Dic₁₆₃	1	1	yes	characteristic
111	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	2	yes	—
112	C₈	—	652	4	no	—
113	C₂²	—	2	1	no	—
114	C₂²	—	2	2	no	—
115	C₂²	—	4	2	no	—
116	C₂²	—	4	1	no	—
117	C₂²	C₂×C₂²⋊Dic₁₆₃	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	1	no	—
120	C₄	—	4	2	no	—
121	C₄	—	4	2	no	—
122	C₄	#19104650	1	2	yes	—
123	C₂	—	2	1	no	—
124	C₂	—	4	1	no	—
125	C₂	—	4	2	no	—
126	C₂	C₂×#19104650	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23781649

C₂

yes

maximal

#23781464

C₂

yes

maximal, complemented

#536×C₁₆₃

C₂

yes

maximal, characteristic

C₂×#14579683

—

#19104654

C₂²

yes

—

#19104652

—

C₂×#41×C₁₆₃

C₂²

yes

characteristic

C₂²×Q₈×C₁₆₃

C₄

yes

characteristic

C₄⋊Q₈×C₁₆₃

C₂²

yes

—

#124×C₁₆₃

C₄

yes

—

#123×C₁₆₃

—

C₂²⋊Q₈×C₁₆₃

—

#117×C₁₆₃

C₄

yes

characteristic

#14579683

—

#14579683

—

C₂×C₁₆₃⋊C₈

—

#41×C₁₆₃

—

C₂×Q₈×C₁₆₃

—

C₂×Q₈×C₁₆₃

—

C₂×Q₈×C₁₆₃

C₂×C₄

yes

—

C₂×Q₈×C₁₆₃

C₂³

yes

characteristic

C₂×D₄×C₁₆₃

C₂×C₄

yes

characteristic

C₂×D₄×C₁₆₃

D₄

yes

—

C₂²×C₄×C₁₆₃

—

C₂²×C₄×C₁₆₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₆₃

D₄

yes

—

C₄⋊C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

—

C₂²⋊C₄×C₁₆₃

—

C₂²⋊C₄×C₁₆₃

—

C₄²×C₁₆₃

C₂×C₄

yes

—

C₁₆₃⋊C₈

—

C₂³×C₁₆₃

C₂²⋊C₄

yes

—

C₂³×C₁₆₃

C₂²⋊C₄

yes

characteristic

Q₈×C₁₆₃

—

Q₈×C₁₆₃

—

Q₈×C₁₆₃

—

D₄×C₁₆₃

—

D₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂×D₄

yes

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

#99

yes

—

C₂×C₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

C₂×#99

yes

characteristic

C₁₆₃

#2135

yes

characteristic, complemented, Sylow

#2135

—

163

maximal, Sylow

#536

D₁₆₃

yes

characteristic

#413

—

163

—

#356

—

163

—

C₂×#41

C₂×D₁₆₃

yes

characteristic

C₂²×Q₈

Dic₁₆₃

yes

characteristic

C₂×M₁₆

—

326

—

C₄⋊Q₈

C₂×D₁₆₃

yes

—

#124

Dic₁₆₃

yes

—

#123

—

C₂²⋊Q₈

—

#117

Dic₁₆₃

yes

characteristic

#101

—

163

—

#100

—

326

—

#41

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₁₆₃

yes

—

C₂×Q₈

C₂²×D₁₆₃

yes

characteristic

C₂×D₄

C₂×Dic₁₆₃

yes

characteristic

C₂×D₄

C₂²⋊D₁₆₃

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₁₆₃

yes

characteristic

C₂²×C₄

C₂²⋊D₁₆₃

yes

—

M₁₆

—

326

—

M₁₆

—

652

—

C₂×C₈

—

652

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×Dic₁₆₃

yes

—

C₂³

C₂²⋊Dic₁₆₃

yes

—

C₂³

C₂²⋊Dic₁₆₃

yes

characteristic

Q₈

—

Q₈

—

100

Q₈

—

101

D₄

—

102

D₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂²⋊Dic₁₆₃

yes

characteristic

110

C₂×C₄

C₂²×Dic₁₆₃

yes

characteristic

111

C₂×C₄

C₂×C₂²⋊D₁₆₃

yes

—

112

C₈

—

652

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

C₂×C₂²⋊Dic₁₆₃

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

#19104650

yes

—

123

C₂

—

124

C₂

—

125

C₂

—

126

C₂

C₂×#19104650

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#19104650
commutator	C₂×C₄×C₁₆₃	C₂²×C₄
Frattini	C₂×Q₈	C₂²×D₁₆₃
Fitting	#536×C₁₆₃	C₂
socle	C₂×C₁₆₃	C₂×#99
prime radical	C₂×#41×C₁₆₃	C₂²
cosocle	C₂×Q₈×C₁₆₃	C₂³
hypercenter	#536	D₁₆₃
hypocenter	C₁₆₃	#2135
LCS	G ▷ C₂×C₄×C₁₆₃ ▷ C₂²×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #536

subgroup

G/H

center

C₂

C₂×#19104650

commutator

C₂×C₄×C₁₆₃

C₂²×C₄

Frattini

C₂×Q₈

C₂²×D₁₆₃

Fitting

#536×C₁₆₃

C₂

socle

C₂×C₁₆₃

C₂×#99

prime radical

C₂×#41×C₁₆₃

C₂²

cosocle

C₂×Q₈×C₁₆₃

C₂³

D₁₆₃

C₁₆₃

G ▷ C₂×C₄×C₁₆₃ ▷ C₂²×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #536

N	H ≅ G/N	type
2. #23781464	C₂	semidirect
63. C₁₆₃	#2135	semidirect

H ≅ G/N

type

2. #23781464

C₂

semidirect

63. C₁₆₃

#2135

semidirect

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

subgroup	index
#23781649	2
#23781464	2
#536×C₁₆₃	2
#2135	163

subgroup

index

#23781649

#23781464

#536×C₁₆₃

#2135

163

Subgroups — #35216231

Conjugacy classes of subgroups

Complements

Maximal subgroups