Group #35210414 — subgroups

Subgroups — #1656×C₁₆₃ #35210414

Order 20864.

230 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 60 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	#562×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#496×C₁₆₃	C₂	1	2	yes	maximal, complemented
3	#490×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#330×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#329×C₁₆₃	C₂	1	2	yes	maximal, complemented
6	C₂×SD₁₆×C₁₆₃	—	2	2	no	—
7	C₂×D₈×C₁₆₃	—	2	2	no	—
8	#127×C₁₆₃	C₂²	1	1	yes	characteristic
9	#125×C₁₆₃	C₂²	1	1	yes	characteristic
10	#124×C₁₆₃	C₂²	1	2	yes	—
11	#123×C₁₆₃	—	2	2	no	—
12	C₂²⋊Q₈×C₁₆₃	—	2	2	no	—
13	#121×C₁₆₃	—	2	2	no	—
14	#102×C₁₆₃	—	2	2	no	—
15	#97×C₁₆₃	C₂²	1	1	yes	characteristic
16	#97×C₁₆₃	C₂²	1	2	yes	—
17	C₂×Q₈×C₁₆₃	D₄	1	2	yes	—
18	C₂×D₄×C₁₆₃	—	2	2	no	—
19	C₂×D₄×C₁₆₃	—	2	1	no	—
20	C₂×D₄×C₁₆₃	—	2	2	no	—
21	C₂×D₄×C₁₆₃	D₄	1	2	yes	—
22	C₂²×C₄×C₁₆₃	—	2	2	no	—
23	SD₁₆×C₁₆₃	—	4	4	no	—
24	D₈×C₁₆₃	—	4	4	no	—
25	C₂×C₈×C₁₆₃	—	2	2	no	—
26	C₂×C₈×C₁₆₃	—	2	1	no	—
27	C₄⋊C₄×C₁₆₃	—	2	2	no	—
28	C₄⋊C₄×C₁₆₃	—	2	2	no	—
29	C₄⋊C₄×C₁₆₃	D₄	1	2	yes	—
30	C₂²⋊C₄×C₁₆₃	—	2	2	no	—
31	C₂²⋊C₄×C₁₆₃	—	2	2	no	—
32	C₂²⋊C₄×C₁₆₃	—	2	2	no	—
33	C₄²×C₁₆₃	C₂³	1	1	yes	characteristic
34	C₂³×C₁₆₃	—	2	2	no	—
35	C₂³×C₁₆₃	—	4	1	no	—
36	Q₈×C₁₆₃	—	4	2	no	—
37	D₄×C₁₆₃	—	4	2	no	—
38	D₄×C₁₆₃	—	4	2	no	—
39	D₄×C₁₆₃	—	4	2	no	—
40	D₄×C₁₆₃	—	4	2	no	—
41	D₄×C₁₆₃	—	8	2	no	—
42	C₂×C₄×C₁₆₃	—	2	2	no	—
43	C₂×C₄×C₁₆₃	—	2	2	no	—
44	C₂×C₄×C₁₆₃	—	8	2	no	—
45	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₆₃	C₂×D₄	1	2	yes	—
47	C₈×C₁₆₃	—	2	4	no	—
48	C₈×C₁₆₃	—	4	1	no	—
49	C₂²×C₁₆₃	—	4	2	no	—
50	C₂²×C₁₆₃	—	4	2	no	—
51	C₂²×C₁₆₃	—	4	2	no	—
52	C₂²×C₁₆₃	—	8	2	no	—
53	C₂²×C₁₆₃	—	8	1	no	—
54	C₂²×C₁₆₃	#120	1	1	yes	characteristic
55	C₄×C₁₆₃	—	2	2	no	—
56	C₄×C₁₆₃	—	2	1	no	—
57	C₄×C₁₆₃	—	4	2	no	—
58	C₄×C₁₆₃	—	4	2	no	—
59	C₂×C₁₆₃	—	8	2	no	—
60	C₂×C₁₆₃	—	16	1	no	—
61	C₂×C₁₆₃	#453	1	2	yes	—
62	C₂×C₁₆₃	#454	1	1	yes	characteristic
63	C₁₆₃	#1656	1	1	yes	characteristic, direct product factor, Sylow
64	#1656	C₁₆₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	#562	C₂×C₁₆₃	1	1	yes	characteristic, complemented
66	#496	C₂×C₁₆₃	1	2	yes	complemented
67	#490	C₂×C₁₆₃	1	1	yes	characteristic, complemented
68	#330	C₂×C₁₆₃	1	1	yes	characteristic, complemented
69	#329	C₂×C₁₆₃	1	2	yes	complemented
70	C₂×SD₁₆	—	2	2	no	—
71	C₂×D₈	—	2	2	no	—
72	#127	C₂²×C₁₆₃	1	1	yes	characteristic
73	#125	C₂²×C₁₆₃	1	1	yes	characteristic
74	#124	C₂²×C₁₆₃	1	2	yes	—
75	#123	—	2	2	no	—
76	C₂²⋊Q₈	—	2	2	no	—
77	#121	—	2	2	no	—
78	#102	—	2	2	no	—
79	#97	C₂²×C₁₆₃	1	1	yes	characteristic
80	#97	C₂²×C₁₆₃	1	2	yes	—
81	C₂×Q₈	D₄×C₁₆₃	1	2	yes	—
82	C₂×D₄	—	2	2	no	—
83	C₂×D₄	—	2	1	no	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	D₄×C₁₆₃	1	2	yes	—
86	C₂²×C₄	—	2	2	no	—
87	SD₁₆	—	4	4	no	—
88	D₈	—	4	4	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	1	no	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	D₄×C₁₆₃	1	2	yes	—
94	C₂²⋊C₄	—	2	2	no	—
95	C₂²⋊C₄	—	2	2	no	—
96	C₂²⋊C₄	—	2	2	no	—
97	C₄²	C₂³×C₁₆₃	1	1	yes	characteristic
98	C₂³	—	2	2	no	—
99	C₂³	—	4	1	no	—
100	Q₈	—	4	2	no	—
101	D₄	—	4	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	8	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	8	2	no	—
109	C₂×C₄	C₂×D₄×C₁₆₃	1	1	yes	characteristic
110	C₂×C₄	C₂×D₄×C₁₆₃	1	2	yes	—
111	C₈	—	2	4	no	—
112	C₈	—	4	1	no	—
113	C₂²	—	4	2	no	—
114	C₂²	—	4	2	no	—
115	C₂²	—	4	2	no	—
116	C₂²	—	8	2	no	—
117	C₂²	—	8	1	no	—
118	C₂²	#120×C₁₆₃	1	1	yes	characteristic
119	C₄	—	2	2	no	—
120	C₄	—	2	1	no	—
121	C₄	—	4	2	no	—
122	C₄	—	4	2	no	—
123	C₂	—	8	2	no	—
124	C₂	—	16	1	no	—
125	C₂	#453×C₁₆₃	1	2	yes	—
126	C₂	#454×C₁₆₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#562×C₁₆₃

C₂

yes

maximal, characteristic, complemented

#496×C₁₆₃

C₂

yes

maximal, complemented

#490×C₁₆₃

C₂

yes

maximal, characteristic, complemented

#330×C₁₆₃

C₂

yes

maximal, characteristic, complemented

#329×C₁₆₃

C₂

yes

maximal, complemented

C₂×SD₁₆×C₁₆₃

—

C₂×D₈×C₁₆₃

—

#127×C₁₆₃

C₂²

yes

characteristic

#125×C₁₆₃

C₂²

yes

characteristic

#124×C₁₆₃

C₂²

yes

—

#123×C₁₆₃

—

C₂²⋊Q₈×C₁₆₃

—

#121×C₁₆₃

—

#102×C₁₆₃

—

#97×C₁₆₃

C₂²

yes

characteristic

#97×C₁₆₃

C₂²

yes

—

C₂×Q₈×C₁₆₃

D₄

yes

—

C₂×D₄×C₁₆₃

—

C₂×D₄×C₁₆₃

—

C₂×D₄×C₁₆₃

—

C₂×D₄×C₁₆₃

D₄

yes

—

C₂²×C₄×C₁₆₃

—

SD₁₆×C₁₆₃

—

D₈×C₁₆₃

—

C₂×C₈×C₁₆₃

—

C₂×C₈×C₁₆₃

—

C₄⋊C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

D₄

yes

—

C₂²⋊C₄×C₁₆₃

—

C₂²⋊C₄×C₁₆₃

—

C₂²⋊C₄×C₁₆₃

—

C₄²×C₁₆₃

C₂³

yes

characteristic

C₂³×C₁₆₃

—

C₂³×C₁₆₃

—

Q₈×C₁₆₃

—

D₄×C₁₆₃

—

D₄×C₁₆₃

—

D₄×C₁₆₃

—

D₄×C₁₆₃

—

D₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂×D₄

yes

—

C₈×C₁₆₃

—

C₈×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

#120

yes

characteristic

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

#453

yes

—

C₂×C₁₆₃

#454

yes

characteristic

C₁₆₃

#1656

yes

characteristic, direct product factor, Sylow

#1656

C₁₆₃

yes

maximal, characteristic, direct product factor, Sylow

#562

C₂×C₁₆₃

yes

characteristic, complemented

#496

C₂×C₁₆₃

yes

complemented

#490

C₂×C₁₆₃

yes

characteristic, complemented

#330

C₂×C₁₆₃

yes

characteristic, complemented

#329

C₂×C₁₆₃

yes

complemented

C₂×SD₁₆

—

C₂×D₈

—

#127

C₂²×C₁₆₃

yes

characteristic

#125

C₂²×C₁₆₃

yes

characteristic

#124

C₂²×C₁₆₃

yes

—

#123

—

C₂²⋊Q₈

—

#121

—

#102

—

#97

C₂²×C₁₆₃

yes

characteristic

#97

C₂²×C₁₆₃

yes

—

C₂×Q₈

D₄×C₁₆₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

D₄×C₁₆₃

yes

—

C₂²×C₄

—

SD₁₆

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₄×C₁₆₃

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂³×C₁₆₃

yes

characteristic

C₂³

—

C₂³

—

100

Q₈

—

101

D₄

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂×D₄×C₁₆₃

yes

characteristic

110

C₂×C₄

C₂×D₄×C₁₆₃

yes

—

111

C₈

—

112

C₈

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

#120×C₁₆₃

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₂

—

124

C₂

—

125

C₂

#453×C₁₆₃

yes

—

126

C₂

#454×C₁₆₃

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₆₃	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₁₆₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆₃ ◁ C₄²×C₁₆₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₆₃

#120

commutator, Frattini, cosocle

C₄²

C₂³×C₁₆₃

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₁₆₃ ◁ C₄²×C₁₆₃ ◁ G

N	H ≅ G/N	type
1. #562×C₁₆₃	C₂	semidirect
2. #496×C₁₆₃	C₂	semidirect
3. #490×C₁₆₃	C₂	semidirect
4. #330×C₁₆₃	C₂	semidirect
5. #329×C₁₆₃	C₂	semidirect
63. C₁₆₃	#1656	direct (N × H)
64. #1656	C₁₆₃	direct (N × H)
65. #562	C₂×C₁₆₃	semidirect
66. #496	C₂×C₁₆₃	semidirect
67. #490	C₂×C₁₆₃	semidirect
68. #330	C₂×C₁₆₃	semidirect
69. #329	C₂×C₁₆₃	semidirect

H ≅ G/N

type

1. #562×C₁₆₃

C₂

semidirect

2. #496×C₁₆₃

C₂

semidirect

3. #490×C₁₆₃

C₂

semidirect

4. #330×C₁₆₃

C₂

semidirect

5. #329×C₁₆₃

C₂

semidirect

63. C₁₆₃

#1656

direct (N × H)

64. #1656

C₁₆₃

direct (N × H)

65. #562

C₂×C₁₆₃

semidirect

66. #496

C₂×C₁₆₃

semidirect

67. #490

C₂×C₁₆₃

semidirect

68. #330

C₂×C₁₆₃

semidirect

69. #329

C₂×C₁₆₃

semidirect

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

subgroup	index
#562×C₁₆₃	2
#496×C₁₆₃	2
#490×C₁₆₃	2
#330×C₁₆₃	2
#329×C₁₆₃	2
#1656	163

subgroup

index

#562×C₁₆₃

#496×C₁₆₃

#490×C₁₆₃

#330×C₁₆₃

#329×C₁₆₃

#1656

163

Subgroups — #1656×C₁₆₃ #35210414

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1656×C163 #35210414

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1656×C₁₆₃ #35210414