Group #35214654 — subgroups

Subgroups — #35214654

Order 20864.

216 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₂×Dic₁₆₃⋊C₈	C₂	1	2	yes	maximal
2	#23781511	C₂	1	1	yes	maximal, characteristic
3	#23781420	C₂	1	1	yes	maximal, characteristic
4	#23781419	C₂	1	2	yes	maximal
5	#336×C₁₆₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₆₃⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×Dic₁₆₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₆₃	C₂²	1	1	yes	characteristic
9	Dic₁₆₃⋊C₈	—	2	4	no	—
10	C₁₆₃⋊#95	C₄	1	2	yes	—
11	C₁₆₃⋊#95	C₄	1	2	yes	—
12	C₂²×C₈×C₁₆₃	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₆₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×Dic₁₆₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₁₆₃	C₂×C₄	1	2	yes	—
17	C₂×C₁₆₃⋊C₈	—	2	2	no	—
18	C₂×C₁₆₃⋊C₈	—	2	2	no	—
19	C₂×C₁₆₃⋊C₈	—	2	2	no	—
20	C₄×Dic₁₆₃	—	2	2	no	—
21	C₄×Dic₁₆₃	D₄	1	2	yes	—
22	C₄×Dic₁₆₃	Q₈	1	2	yes	—
23	C₂²×C₄×C₁₆₃	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₁₆₃	—	2	2	no	—
26	C₂×C₈×C₁₆₃	—	2	2	no	—
27	C₂×C₈×C₁₆₃	—	2	2	no	—
28	C₄²×C₁₆₃	—	2	2	no	—
29	C₂×Dic₁₆₃	—	2	2	no	—
30	C₂×Dic₁₆₃	—	2	2	no	—
31	C₂×Dic₁₆₃	—	2	1	no	—
32	C₂×Dic₁₆₃	—	2	1	no	—
33	C₂×Dic₁₆₃	—	2	2	no	—
34	C₂×Dic₁₆₃	—	2	2	no	—
35	C₂×Dic₁₆₃	C₄⋊C₄	1	4	yes	—
36	C₁₆₃⋊C₈	—	4	2	no	—
37	C₂³×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₆₃	—	2	2	no	—
39	C₂×C₄×C₁₆₃	—	2	2	no	—
40	C₂×C₄×C₁₆₃	—	2	2	no	—
41	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₆₃	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₁₆₃	#41	1	2	yes	—
44	C₂×C₄×C₁₆₃	#41	1	2	yes	—
45	C₈×C₁₆₃	—	4	2	no	—
46	C₂²×C₁₆₃	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₁₆₃	#117	1	1	yes	characteristic
48	C₂²×C₁₆₃	#117	1	1	yes	characteristic
49	C₂²×C₁₆₃	#131	1	2	yes	—
50	C₂²×C₁₆₃	#131	1	2	yes	—
51	C₄×C₁₆₃	—	4	2	no	—
52	C₄×C₁₆₃	#124	1	2	yes	—
53	C₄×C₁₆₃	#125	1	2	yes	—
54	Dic₁₆₃	—	2	4	no	—
55	Dic₁₆₃	—	4	1	no	—
56	Dic₁₆₃	—	4	1	no	—
57	C₂×C₁₆₃	#382	1	1	yes	characteristic
58	C₂×C₁₆₃	#424	1	2	yes	—
59	C₂×C₁₆₃	#433	1	2	yes	—
60	C₂×C₁₆₃	#433	1	1	yes	characteristic
61	C₂×C₁₆₃	#433	1	1	yes	characteristic
62	C₁₆₃	#1832	1	1	yes	characteristic, complemented, Sylow
63	#1832	—	163	1	no	maximal, Sylow
64	C₂×C₄⋊C₈	—	163	2	no	—
65	#376	—	163	1	no	—
66	#336	—	163	2	no	—
67	#336	—	163	1	no	—
68	#336	D₁₆₃	1	1	yes	characteristic
69	C₂²×C₈	—	163	2	no	—
70	C₂²×C₈	C₂×D₁₆₃	1	2	yes	—
71	C₂×C₄²	—	163	1	no	—
72	C₂×C₄²	—	163	1	no	—
73	C₂×C₄²	C₂×D₁₆₃	1	1	yes	characteristic
74	C₄⋊C₈	—	326	4	no	—
75	#95	—	163	2	no	—
76	#95	—	163	2	no	—
77	C₂²×C₄	—	163	1	no	—
78	C₂²×C₄	—	163	1	no	—
79	C₂²×C₄	—	163	2	no	—
80	C₂²×C₄	C₄×D₁₆₃	1	2	yes	—
81	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	326	2	no	—
86	C₂×C₈	—	326	2	no	—
87	C₂×C₈	—	326	2	no	—
88	C₄²	—	2	2	no	—
89	C₄²	—	326	2	no	—
90	C₄²	—	163	2	no	—
91	C₄²	—	163	2	no	—
92	C₂³	C₂×C₄×D₁₆₃	1	1	yes	characteristic
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	326	2	no	—
97	C₂×C₄	—	326	2	no	—
98	C₂×C₄	—	326	1	no	—
99	C₂×C₄	—	326	1	no	—
100	C₂×C₄	—	326	2	no	—
101	C₂×C₄	—	326	2	no	—
102	C₂×C₄	—	163	4	no	—
103	C₂×C₄	D₄×D₁₆₃	1	1	yes	characteristic
104	C₂×C₄	Q₈×D₁₆₃	1	1	yes	characteristic
105	C₂×C₄	#14579698	1	2	yes	—
106	C₂×C₄	#14579699	1	2	yes	—
107	C₈	—	4	2	no	—
108	C₈	—	652	2	no	—
109	C₂²	C₄⋊C₄×D₁₆₃	1	1	yes	characteristic
110	C₂²	#19104691	1	1	yes	characteristic
111	C₂²	#19104692	1	1	yes	characteristic
112	C₂²	#19104741	1	2	yes	—
113	C₂²	#19104743	1	2	yes	—
114	C₄	—	4	2	no	—
115	C₄	—	326	4	no	—
116	C₄	—	652	1	no	—
117	C₄	—	652	1	no	—
118	C₄	#19104721	1	2	yes	—
119	C₄	#19104725	1	2	yes	—
120	C₂	#23781529	1	1	yes	characteristic
121	C₂	#23781690	1	2	yes	—
122	C₂	#23781723	1	2	yes	—
123	C₂	#23781724	1	1	yes	characteristic
124	C₂	#23781727	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₁₆₃⋊C₈

C₂

yes

maximal

#23781511

C₂

yes

maximal, characteristic

#23781420

C₂

yes

maximal, characteristic

#23781419

C₂

yes

maximal

#336×C₁₆₃

C₂

yes

maximal, characteristic

C₂²×C₁₆₃⋊C₈

C₂²

yes

—

C₂×C₄×Dic₁₆₃

C₂²

yes

characteristic

C₂×C₄×Dic₁₆₃

C₂²

yes

characteristic

Dic₁₆₃⋊C₈

—

C₁₆₃⋊#95

C₄

yes

—

C₁₆₃⋊#95

C₄

yes

—

C₂²×C₈×C₁₆₃

C₂²

yes

—

C₂×C₄²×C₁₆₃

C₂²

yes

characteristic

C₂²×Dic₁₆₃

C₂×C₄

yes

characteristic

C₂²×Dic₁₆₃

C₂×C₄

yes

characteristic

C₂²×Dic₁₆₃

C₂×C₄

yes

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

—

C₄×Dic₁₆₃

—

C₄×Dic₁₆₃

D₄

yes

—

C₄×Dic₁₆₃

Q₈

yes

—

C₂²×C₄×C₁₆₃

C₂×C₄

yes

—

C₂²×C₄×C₁₆₃

C₂³

yes

characteristic

C₂×C₈×C₁₆₃

—

C₂×C₈×C₁₆₃

—

C₂×C₈×C₁₆₃

—

C₄²×C₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

C₄⋊C₄

yes

—

C₁₆₃⋊C₈

—

C₂³×C₁₆₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₆₃

#41

yes

—

C₂×C₄×C₁₆₃

#41

yes

—

C₈×C₁₆₃

—

C₂²×C₁₆₃

C₂×C₄⋊C₄

yes

C₂²×C₁₆₃

yes

C₂²×C₁₆₃

yes

C₂²×C₁₆₃

yes

—

C₂²×C₁₆₃

#131

yes

—

C₄×C₁₆₃

—

C₄×C₁₆₃

#124

yes

—

C₄×C₁₆₃

#125

yes

—

Dic₁₆₃

—

Dic₁₆₃

—

Dic₁₆₃

—

C₂×C₁₆₃

#382

yes

characteristic

C₂×C₁₆₃

#424

yes

—

C₂×C₁₆₃

#433

yes

—

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

C₁₆₃

yes

characteristic, complemented, Sylow

#1832

—

163

maximal, Sylow

C₂×C₄⋊C₈

—

163

—

#376

—

163

—

#336

—

163

—

#336

—

163

—

#336

D₁₆₃

yes

characteristic

C₂²×C₈

—

163

—

C₂²×C₈

C₂×D₁₆₃

yes

—

C₂×C₄²

—

163

—

C₂×C₄²

—

163

—

C₂×C₄²

C₂×D₁₆₃

yes

characteristic

C₄⋊C₈

—

326

—

#95

—

163

—

#95

—

163

—

C₂²×C₄

—

163

—

C₂²×C₄

—

163

—

C₂²×C₄

—

163

—

C₂²×C₄

C₄×D₁₆₃

yes

—

C₂²×C₄

C₂²×D₁₆₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

326

—

C₂×C₈

—

326

—

C₂×C₈

—

326

—

C₄²

—

C₄²

—

326

—

C₄²

—

163

—

C₄²

—

163

—

C₂³

C₂×C₄×D₁₆₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

326

—

C₂×C₄

—

326

—

C₂×C₄

—

326

—

C₂×C₄

—

326

—

100

C₂×C₄

—

326

—

101

C₂×C₄

—

326

—

102

C₂×C₄

—

163

—

103

C₂×C₄

D₄×D₁₆₃

yes

characteristic

104

C₂×C₄

Q₈×D₁₆₃

yes

characteristic

105

C₂×C₄

#14579698

yes

—

106

C₂×C₄

#14579699

yes

—

107

C₈

—

108

C₈

—

652

—

109

C₂²

C₄⋊C₄×D₁₆₃

yes

characteristic

110

C₂²

#19104691

yes

characteristic

111

C₂²

#19104692

yes

characteristic

112

C₂²

#19104741

yes

—

113

C₂²

#19104743

yes

—

114

C₄

—

115

C₄

—

326

—

116

C₄

—

652

—

117

C₄

—

652

—

118

C₄

#19104721

yes

—

119

C₄

#19104725

yes

—

120

C₂

#23781529

yes

characteristic

121

C₂

#23781690

yes

—

122

C₂

#23781723

yes

—

123

C₂

#23781724

yes

characteristic

124

C₂

#23781727

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₆₃
commutator, socle, prime radical	C₂³×C₁₆₃	C₂²×C₄
Fitting	#336×C₁₆₃	C₂
cosocle	C₂²×C₄×C₁₆₃	C₂³
hypercenter	#336	D₁₆₃
hypocenter	C₁₆₃	#1832
LCS	G ▷ C₂³×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₆₃

commutator, socle, prime radical

C₂³×C₁₆₃

C₂²×C₄

Fitting

#336×C₁₆₃

C₂

cosocle

C₂²×C₄×C₁₆₃

C₂³

D₁₆₃

C₁₆₃

G ▷ C₂³×C₁₆₃ ▷ C₁₆₃

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
62. C₁₆₃	#1832	semidirect

H ≅ G/N

type

62. C₁₆₃

#1832

semidirect

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

subgroup	index
C₂×Dic₁₆₃⋊C₈	2
#23781511	2
#23781420	2
#23781419	2
#336×C₁₆₃	2
#1832	163

subgroup

index

C₂×Dic₁₆₃⋊C₈

#336×C₁₆₃

163

Subgroups — #35214654

Conjugacy classes of subgroups

Complements

Maximal subgroups