Group #35222912 — subgroups

Subgroups — #35222912

Order 20864.

504 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 181 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×#19104734	C₂	1	2	yes	maximal
2	#23781569	C₂	1	8	yes	maximal
3	#23781531	C₂	1	1	yes	maximal, characteristic
4	#23781530	C₂	1	2	yes	maximal
5	C₄²×Dic₁₆₃	C₂	1	1	yes	maximal, characteristic
6	#382×C₁₆₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₃₂₆	C₂²	1	4	yes	—
8	#19104734	—	2	8	no	—
9	C₂×C₄⋊Dic₁₆₃	C₂²	1	2	yes	—
10	C₂×Dic₁₆₃⋊C₄	C₂²	1	4	yes	—
11	C₂×C₄×Dic₁₆₃	C₂²	1	4	yes	—
12	C₂×C₄×Dic₁₆₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₁₆₃	C₂²	1	1	yes	characteristic
14	C₁₆₃⋊#95	C₂²	1	8	yes	—
15	C₁₆₃⋊#95	C₂²	1	4	yes	—
16	C₂×C₄⋊C₄×C₁₆₃	C₂²	1	2	yes	—
17	C₂×C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
18	#95×C₁₆₃	C₂²	1	4	yes	—
19	C₂×Dic₃₂₆	—	2	8	no	—
20	C₂×Dic₃₂₆	—	2	16	no	—
21	C₂²×Dic₁₆₃	C₂³	1	4	yes	—
22	C₂²×Dic₁₆₃	C₂³	1	2	yes	—
23	C₂²×Dic₁₆₃	C₂³	1	2	yes	—
24	C₄⋊Dic₁₆₃	—	2	4	no	—
25	Dic₁₆₃⋊C₄	—	2	8	no	—
26	C₄×Dic₁₆₃	—	2	8	no	—
27	C₄×Dic₁₆₃	D₄	1	4	yes	—
28	C₄×Dic₁₆₃	Q₈	1	4	yes	complemented
29	C₂²×C₄×C₁₆₃	C₂³	1	4	yes	—
30	C₂²×C₄×C₁₆₃	C₂³	1	2	yes	—
31	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₁₆₃	—	2	4	no	—
33	C₄²×C₁₆₃	Q₈	1	4	yes	complemented
34	Dic₃₂₆	—	4	16	no	—
35	C₂×Dic₁₆₃	—	2	4	no	—
36	C₂×Dic₁₆₃	—	2	4	no	—
37	C₂×Dic₁₆₃	—	2	8	no	—
38	C₂×Dic₁₆₃	C₂×D₄	1	4	yes	—
39	C₂×Dic₁₆₃	C₂×Q₈	1	4	yes	—
40	C₂×Dic₁₆₃	#41	1	8	yes	—
41	C₂³×C₁₆₃	C₂⁴	1	1	yes	characteristic
42	C₂×C₄×C₁₆₃	—	2	4	no	—
43	C₂×C₄×C₁₆₃	—	2	4	no	—
44	C₂×C₄×C₁₆₃	—	2	8	no	—
45	C₂×C₄×C₁₆₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₆₃	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₁₆₃	C₂×Q₈	1	4	yes	—
48	C₂×C₄×C₁₆₃	C₂×Q₈	1	2	yes	—
49	C₂²×C₁₆₃	C₂²×D₄	1	1	yes	characteristic
50	C₂²×C₁₆₃	C₂²×Q₈	1	1	yes	characteristic
51	C₂²×C₁₆₃	C₂²×Q₈	1	1	yes	characteristic
52	C₂²×C₁₆₃	C₂×#41	1	4	yes	—
53	C₄×C₁₆₃	—	2	4	no	—
54	C₄×C₁₆₃	—	4	4	no	—
55	C₄×C₁₆₃	C₄⋊Q₈	1	4	yes	—
56	Dic₁₆₃	—	2	4	no	—
57	Dic₁₆₃	—	4	4	no	—
58	Dic₁₆₃	#124	1	8	yes	—
59	C₂×C₁₆₃	C₂×#124	1	2	yes	—
60	C₂×C₁₆₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
61	C₂×C₁₆₃	#533	1	2	yes	—
62	C₂×C₁₆₃	#533	1	2	yes	—
63	C₁₆₃	#2862	1	1	yes	characteristic, complemented, Sylow
64	#2862	—	163	1	no	maximal, Sylow
65	C₂×C₄⋊Q₈	—	163	2	no	—
66	#391	—	163	8	no	—
67	#382	—	163	2	no	—
68	#382	—	163	1	no	—
69	#382	D₁₆₃	1	1	yes	characteristic
70	C₄³	—	163	1	no	—
71	C₂²×Q₈	—	163	4	no	—
72	C₄⋊Q₈	—	326	8	no	—
73	C₂×C₄⋊C₄	—	163	2	no	—
74	C₂×C₄⋊C₄	—	163	4	no	—
75	C₂×C₄⋊C₄	C₂×D₁₆₃	1	2	yes	—
76	C₂×C₄²	—	163	4	no	—
77	C₂×C₄²	—	163	1	no	—
78	C₂×C₄²	—	163	1	no	—
79	C₂×C₄²	C₂×D₁₆₃	1	1	yes	characteristic
80	#95	—	163	4	no	—
81	#95	—	163	8	no	—
82	#95	C₂×D₁₆₃	1	4	yes	—
83	C₂×Q₈	—	326	8	no	—
84	C₂×Q₈	—	326	16	no	—
85	C₂²×C₄	—	163	4	no	—
86	C₂²×C₄	—	163	2	no	—
87	C₂²×C₄	—	163	2	no	—
88	C₂²×C₄	C₂²×D₁₆₃	1	4	yes	—
89	C₂²×C₄	C₂²×D₁₆₃	1	2	yes	—
90	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	326	8	no	—
93	C₄⋊C₄	—	326	4	no	—
94	C₄²	—	326	8	no	—
95	C₄²	—	163	4	no	—
96	C₄²	—	163	4	no	—
97	C₄²	Dic₃₂₆	1	4	yes	complemented
98	C₂³	C₂³×D₁₆₃	1	1	yes	characteristic
99	Q₈	—	652	16	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	8	no	—
103	C₂×C₄	—	326	4	no	—
104	C₂×C₄	—	326	4	no	—
105	C₂×C₄	—	326	8	no	—
106	C₂×C₄	—	163	4	no	—
107	C₂×C₄	—	163	4	no	—
108	C₂×C₄	—	163	8	no	—
109	C₂×C₄	D₄×D₁₆₃	1	2	yes	—
110	C₂×C₄	Q₈×D₁₆₃	1	2	yes	—
111	C₂×C₄	C₂×Dic₃₂₆	1	4	yes	—
112	C₂×C₄	C₂×Dic₃₂₆	1	2	yes	—
113	C₂²	C₂×D₄×D₁₆₃	1	1	yes	characteristic
114	C₂²	C₂×Q₈×D₁₆₃	1	1	yes	characteristic
115	C₂²	C₂²×Dic₃₂₆	1	1	yes	characteristic
116	C₂²	#41×D₁₆₃	1	4	yes	—
117	C₄	—	2	4	no	—
118	C₄	—	4	4	no	—
119	C₄	—	326	4	no	—
120	C₄	—	652	4	no	—
121	C₄	—	163	8	no	—
122	C₄	#19104734	1	4	yes	—
123	C₂	#124×D₁₆₃	1	2	yes	—
124	C₂	C₂×#19104734	1	1	yes	characteristic
125	C₂	#23782218	1	2	yes	—
126	C₂	#23782221	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19104734

C₂

yes

maximal

#23781569

C₂

yes

maximal

#23781531

C₂

yes

maximal, characteristic

#23781530

C₂

yes

maximal

C₄²×Dic₁₆₃

C₂

yes

maximal, characteristic

#382×C₁₆₃

C₂

yes

maximal, characteristic

C₂²×Dic₃₂₆

C₂²

yes

—

#19104734

—

C₂×C₄⋊Dic₁₆₃

C₂²

yes

—

C₂×Dic₁₆₃⋊C₄

C₂²

yes

—

C₂×C₄×Dic₁₆₃

C₂²

yes

—

C₂×C₄×Dic₁₆₃

C₂²

yes

characteristic

C₂×C₄×Dic₁₆₃

C₂²

yes

characteristic

C₁₆₃⋊#95

C₂²

yes

—

C₁₆₃⋊#95

C₂²

yes

—

C₂×C₄⋊C₄×C₁₆₃

C₂²

yes

—

C₂×C₄²×C₁₆₃

C₂²

yes

characteristic

#95×C₁₆₃

C₂²

yes

—

C₂×Dic₃₂₆

—

C₂×Dic₃₂₆

—

C₂²×Dic₁₆₃

C₂³

yes

—

C₂²×Dic₁₆₃

C₂³

yes

—

C₂²×Dic₁₆₃

C₂³

yes

—

C₄⋊Dic₁₆₃

—

Dic₁₆₃⋊C₄

—

C₄×Dic₁₆₃

—

C₄×Dic₁₆₃

D₄

yes

—

C₄×Dic₁₆₃

Q₈

yes

complemented

C₂²×C₄×C₁₆₃

C₂³

yes

—

C₂²×C₄×C₁₆₃

C₂³

yes

—

C₂²×C₄×C₁₆₃

C₂³

yes

characteristic

C₄⋊C₄×C₁₆₃

—

C₄²×C₁₆₃

Q₈

yes

complemented

Dic₃₂₆

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

C₂×D₄

yes

—

C₂×Dic₁₆₃

C₂×Q₈

yes

—

C₂×Dic₁₆₃

#41

yes

—

C₂³×C₁₆₃

C₂⁴

yes

characteristic

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

—

C₂×C₄×C₁₆₃

C₂×Q₈

yes

—

C₂×C₄×C₁₆₃

C₂×Q₈

yes

—

C₂×C₄×C₁₆₃

C₂×Q₈

yes

—

C₂²×C₁₆₃

C₂²×D₄

yes

characteristic

C₂²×C₁₆₃

C₂²×Q₈

yes

characteristic

C₂²×C₁₆₃

C₂²×Q₈

yes

characteristic

C₂²×C₁₆₃

C₂×#41

yes

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

C₄⋊Q₈

yes

—

Dic₁₆₃

—

Dic₁₆₃

—

Dic₁₆₃

#124

yes

—

C₂×C₁₆₃

C₂×#124

yes

—

C₂×C₁₆₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₆₃

#533

yes

—

C₂×C₁₆₃

#533

yes

—

C₁₆₃

#2862

yes

characteristic, complemented, Sylow

#2862

—

163

maximal, Sylow

C₂×C₄⋊Q₈

—

163

—

#391

—

163

—

#382

—

163

—

#382

—

163

—

#382

D₁₆₃

yes

characteristic

C₄³

—

163

—

C₂²×Q₈

—

163

—

C₄⋊Q₈

—

326

—

C₂×C₄⋊C₄

—

163

—

C₂×C₄⋊C₄

—

163

—

C₂×C₄⋊C₄

C₂×D₁₆₃

yes

—

C₂×C₄²

—

163

—

C₂×C₄²

—

163

—

C₂×C₄²

—

163

—

C₂×C₄²

C₂×D₁₆₃

yes

characteristic

#95

—

163

—

#95

—

163

—

#95

C₂×D₁₆₃

yes

—

C₂×Q₈

—

326

—

C₂×Q₈

—

326

—

C₂²×C₄

—

163

—

C₂²×C₄

—

163

—

C₂²×C₄

—

163

—

C₂²×C₄

C₂²×D₁₆₃

yes

—

C₂²×C₄

C₂²×D₁₆₃

yes

—

C₂²×C₄

C₂²×D₁₆₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

326

—

C₄⋊C₄

—

326

—

C₄²

—

326

—

C₄²

—

163

—

C₄²

—

163

—

C₄²

Dic₃₂₆

yes

complemented

C₂³

C₂³×D₁₆₃

yes

characteristic

Q₈

—

652

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

326

—

104

C₂×C₄

—

326

—

105

C₂×C₄

—

326

—

106

C₂×C₄

—

163

—

107

C₂×C₄

—

163

—

108

C₂×C₄

—

163

—

109

C₂×C₄

D₄×D₁₆₃

yes

—

110

C₂×C₄

Q₈×D₁₆₃

yes

—

111

C₂×C₄

C₂×Dic₃₂₆

yes

—

112

C₂×C₄

C₂×Dic₃₂₆

yes

—

113

C₂²

C₂×D₄×D₁₆₃

yes

characteristic

114

C₂²

C₂×Q₈×D₁₆₃

yes

characteristic

115

C₂²

C₂²×Dic₃₂₆

yes

characteristic

116

C₂²

#41×D₁₆₃

yes

—

117

C₄

—

118

C₄

—

119

C₄

—

326

—

120

C₄

—

652

—

121

C₄

—

163

—

122

C₄

#19104734

yes

—

123

C₂

#124×D₁₆₃

yes

—

124

C₂

C₂×#19104734

yes

characteristic

125

C₂

#23782218

yes

—

126

C₂

#23782221

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₆₃
commutator, socle, prime radical, cosocle	C₂³×C₁₆₃	C₂⁴
Fitting	#382×C₁₆₃	C₂
hypercenter	#382	D₁₆₃
hypocenter	C₁₆₃	#2862
LCS	G ▷ C₂³×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂³ ◁ #382

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₆₃

commutator, socle, prime radical, cosocle

C₂³×C₁₆₃

C₂⁴

Fitting

#382×C₁₆₃

C₂

D₁₆₃

C₁₆₃

G ▷ C₂³×C₁₆₃ ▷ C₁₆₃

C₁ ◁ C₂³ ◁ #382

N	H ≅ G/N	type
28. C₄×Dic₁₆₃	Q₈	semidirect
33. C₄²×C₁₆₃	Q₈	semidirect
63. C₁₆₃	#2862	semidirect
97. C₄²	Dic₃₂₆	semidirect

H ≅ G/N

type

28. C₄×Dic₁₆₃

Q₈

semidirect

33. C₄²×C₁₆₃

Q₈

semidirect

63. C₁₆₃

#2862

semidirect

97. C₄²

Dic₃₂₆

semidirect

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

subgroup	index
C₂×#19104734	2
#23781569	2
#23781531	2
#23781530	2
C₄²×Dic₁₆₃	2
#382×C₁₆₃	2
#2862	163

subgroup

index

C₄²×Dic₁₆₃

#382×C₁₆₃

#2862

163

Subgroups — #35222912

Conjugacy classes of subgroups

Complements

Maximal subgroups