Group #35216100 — subgroups

Subgroups — #35216100

Order 20864.

214 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 87 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	#23781563	C₂	1	1	yes	maximal, characteristic
2	#23781535	C₂	1	1	yes	maximal, characteristic
3	#23781428	C₂	1	2	yes	maximal
4	#23781428	C₂	1	2	yes	maximal
5	#336×C₁₆₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₁₆₃	—	2	1	no	—
7	C₂×C₄⋊Dic₁₆₃	—	2	1	no	—
8	C₂×C₄⋊Dic₁₆₃	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₁₆₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₆₃	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₆₃⋊C₄	—	2	1	no	—
12	C₁₆₃⋊#95	—	2	1	no	—
13	C₂²×C₈×C₁₆₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₆₃	—	2	1	no	—
16	C₂²×Dic₁₆₃	—	2	1	no	—
17	C₂²×Dic₁₆₃	—	2	2	no	—
18	C₄⋊Dic₁₆₃	—	2	1	no	—
19	C₄⋊Dic₁₆₃	—	2	2	no	—
20	C₄⋊Dic₁₆₃	—	2	4	no	—
21	C₄⋊Dic₁₆₃	—	4	2	no	—
22	C₄⋊Dic₁₆₃	—	4	2	no	—
23	C₄⋊Dic₁₆₃	Q₈	1	2	yes	—
24	Dic₁₆₃⋊C₄	—	4	2	no	—
25	C₂²×C₄×C₁₆₃	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₆₃	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₁₆₃	—	2	2	no	—
29	C₂×C₈×C₁₆₃	—	2	2	no	—
30	C₂×C₈×C₁₆₃	—	2	2	no	—
31	C₄²×C₁₆₃	—	2	2	no	—
32	C₂×Dic₁₆₃	—	2	2	no	—
33	C₂×Dic₁₆₃	—	4	1	no	—
34	C₂×Dic₁₆₃	—	4	1	no	—
35	C₂×Dic₁₆₃	—	4	1	no	—
36	C₂×Dic₁₆₃	—	4	2	no	—
37	C₂×Dic₁₆₃	—	4	1	no	—
38	C₂×Dic₁₆₃	—	4	2	no	—
39	C₂×Dic₁₆₃	—	4	2	no	—
40	C₂×Dic₁₆₃	—	4	1	no	—
41	C₂³×C₁₆₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₆₃	—	2	1	no	—
43	C₂×C₄×C₁₆₃	—	2	1	no	—
44	C₂×C₄×C₁₆₃	—	2	1	no	—
45	C₂×C₄×C₁₆₃	—	2	1	no	—
46	C₂×C₄×C₁₆₃	SD₁₆	1	2	yes	—
47	C₂×C₄×C₁₆₃	Q₁₆	1	2	yes	—
48	C₂×C₄×C₁₆₃	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₄×C₁₆₃	#41	1	2	yes	—
50	C₂×C₄×C₁₆₃	#41	1	1	yes	characteristic
51	C₂×C₄×C₁₆₃	#41	1	2	yes	—
52	C₈×C₁₆₃	—	4	2	no	—
53	C₂²×C₁₆₃	C₂²⋊Q₈	1	1	yes	characteristic
54	C₂²×C₁₆₃	#123	1	1	yes	characteristic
55	C₂²×C₁₆₃	#124	1	1	yes	characteristic
56	C₂²×C₁₆₃	C₂×SD₁₆	1	1	yes	characteristic
57	C₂²×C₁₆₃	C₂×Q₁₆	1	1	yes	characteristic
58	C₂²×C₁₆₃	AGL(1,ℤ/8)	1	1	yes	characteristic
59	C₂²×C₁₆₃	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₄×C₁₆₃	—	2	2	no	—
61	C₄×C₁₆₃	—	2	2	no	—
62	C₄×C₁₆₃	#125	1	2	yes	—
63	C₄×C₁₆₃	#126	1	2	yes	—
64	Dic₁₆₃	—	4	2	no	—
65	Dic₁₆₃	—	8	1	no	—
66	Dic₁₆₃	—	8	2	no	—
67	C₂×C₁₆₃	#400	1	1	yes	characteristic
68	C₂×C₁₆₃	#476	1	1	yes	characteristic
69	C₂×C₁₆₃	#477	1	1	yes	characteristic
70	C₂×C₁₆₃	#481	1	1	yes	characteristic
71	C₂×C₁₆₃	#484	1	1	yes	characteristic
72	C₂×C₁₆₃	#487	1	1	yes	characteristic
73	C₂×C₁₆₃	#490	1	1	yes	characteristic
74	C₁₆₃	#2100	1	1	yes	characteristic, complemented, Sylow
75	#2100	—	163	1	no	maximal, Sylow
76	#389	—	163	1	no	—
77	#384	—	163	1	no	—
78	#340	—	163	2	no	—
79	#340	—	163	2	no	—
80	#336	D₁₆₃	1	1	yes	characteristic
81	C₂²×C₈	C₂×D₁₆₃	1	2	yes	—
82	C₂×C₄⋊C₄	—	326	1	no	—
83	C₂×C₄⋊C₄	—	326	1	no	—
84	C₂×C₄⋊C₄	—	326	1	no	—
85	C₂×C₄⋊C₄	—	163	2	no	—
86	C₂×C₄⋊C₄	—	163	1	no	—
87	C₂×C₄⋊C₄	—	163	1	no	—
88	C₂×C₄²	C₂×D₁₆₃	1	1	yes	characteristic
89	#95	—	326	1	no	—
90	C₂²×C₄	—	326	1	no	—
91	C₂²×C₄	—	326	1	no	—
92	C₂²×C₄	—	326	2	no	—
93	C₂²×C₄	D₆₅₂	1	1	yes	characteristic
94	C₂²×C₄	D₆₅₂	1	1	yes	characteristic
95	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₄⋊C₄	—	326	1	no	—
100	C₄⋊C₄	—	326	2	no	—
101	C₄⋊C₄	—	326	4	no	—
102	C₄⋊C₄	—	652	2	no	—
103	C₄⋊C₄	—	652	2	no	—
104	C₄⋊C₄	—	652	2	no	—
105	C₄⋊C₄	—	163	2	no	—
106	C₄²	—	2	2	no	—
107	C₂³	C₂×D₆₅₂	1	1	yes	characteristic
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	326	2	no	—
113	C₂×C₄	—	652	1	no	—
114	C₂×C₄	—	652	2	no	—
115	C₂×C₄	—	652	1	no	—
116	C₂×C₄	—	652	2	no	—
117	C₂×C₄	—	652	2	no	—
118	C₂×C₄	—	652	1	no	—
119	C₂×C₄	—	652	1	no	—
120	C₂×C₄	—	652	1	no	—
121	C₂×C₄	#14579688	1	2	yes	—
122	C₂×C₄	Dic₆₅₂	1	2	yes	—
123	C₂×C₄	Q₈×D₁₆₃	1	1	yes	characteristic
124	C₂×C₄	#14579698	1	2	yes	—
125	C₂×C₄	#14579698	1	1	yes	characteristic
126	C₂×C₄	#14579699	1	2	yes	—
127	C₈	—	4	2	no	—
128	C₂²	#19104715	1	1	yes	characteristic
129	C₂²	#19104719	1	1	yes	characteristic
130	C₂²	#19104723	1	1	yes	characteristic
131	C₂²	C₂×#14579688	1	1	yes	characteristic
132	C₂²	C₂×Dic₆₅₂	1	1	yes	characteristic
133	C₂²	#19104763	1	1	yes	characteristic
134	C₂²	#19104763	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	2	2	no	—
137	C₄	—	1304	1	no	—
138	C₄	—	1304	2	no	—
139	C₄	—	652	2	no	—
140	C₄	#19104725	1	2	yes	—
141	C₄	#19104728	1	2	yes	—
142	C₂	#23781605	1	1	yes	characteristic
143	C₂	#23781956	1	1	yes	characteristic
144	C₂	#23781961	1	1	yes	characteristic
145	C₂	#23781984	1	1	yes	characteristic
146	C₂	#23782001	1	1	yes	characteristic
147	C₂	#23782016	1	1	yes	characteristic
148	C₂	#23782030	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23781563

C₂

yes

maximal, characteristic

#23781535

C₂

yes

maximal, characteristic

#23781428

C₂

yes

maximal

#23781428

C₂

yes

maximal

#336×C₁₆₃

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₁₆₃

—

C₂×C₄⋊Dic₁₆₃

—

C₂×C₄⋊Dic₁₆₃

C₂²

yes

—

C₂×C₄⋊Dic₁₆₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₆₃

C₂²

yes

characteristic

C₂×Dic₁₆₃⋊C₄

—

C₁₆₃⋊#95

—

C₂²×C₈×C₁₆₃

C₂²

yes

—

C₂×C₄²×C₁₆₃

C₂²

yes

characteristic

C₂²×Dic₁₆₃

—

C₂²×Dic₁₆₃

—

C₂²×Dic₁₆₃

—

C₄⋊Dic₁₆₃

—

C₄⋊Dic₁₆₃

—

C₄⋊Dic₁₆₃

—

C₄⋊Dic₁₆₃

—

C₄⋊Dic₁₆₃

—

C₄⋊Dic₁₆₃

Q₈

yes

—

Dic₁₆₃⋊C₄

—

C₂²×C₄×C₁₆₃

D₄

yes

characteristic

C₂²×C₄×C₁₆₃

D₄

yes

characteristic

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

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂³×C₁₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

SD₁₆

yes

—

C₂×C₄×C₁₆₃

Q₁₆

yes

—

C₂×C₄×C₁₆₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₆₃

#41

yes

—

C₂×C₄×C₁₆₃

#41

yes

characteristic

C₂×C₄×C₁₆₃

#41

yes

—

C₈×C₁₆₃

—

C₂²×C₁₆₃

C₂²⋊Q₈

yes

C₂²×C₁₆₃

yes

C₂²×C₁₆₃

yes

C₂²×C₁₆₃

C₂×SD₁₆

yes

characteristic

C₂²×C₁₆₃

C₂×Q₁₆

yes

C₂²×C₁₆₃

yes

C₂²×C₁₆₃

yes

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

#125

yes

—

C₄×C₁₆₃

#126

yes

—

Dic₁₆₃

—

Dic₁₆₃

—

Dic₁₆₃

—

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

C₁₆₃

yes

characteristic, complemented, Sylow

#2100

—

163

maximal, Sylow

#389

—

163

—

#384

—

163

—

#340

—

163

—

#340

—

163

—

#336

D₁₆₃

yes

characteristic

C₂²×C₈

C₂×D₁₆₃

yes

—

C₂×C₄⋊C₄

—

326

—

C₂×C₄⋊C₄

—

326

—

C₂×C₄⋊C₄

—

326

—

C₂×C₄⋊C₄

—

163

—

C₂×C₄⋊C₄

—

163

—

C₂×C₄⋊C₄

—

163

—

C₂×C₄²

C₂×D₁₆₃

yes

characteristic

#95

—

326

—

C₂²×C₄

—

326

—

C₂²×C₄

—

326

—

C₂²×C₄

—

326

—

C₂²×C₄

D₆₅₂

yes

characteristic

C₂²×C₄

D₆₅₂

yes

characteristic

C₂²×C₄

C₂²×D₁₆₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

326

—

100

C₄⋊C₄

—

326

—

101

C₄⋊C₄

—

326

—

102

C₄⋊C₄

—

652

—

103

C₄⋊C₄

—

652

—

104

C₄⋊C₄

—

652

—

105

C₄⋊C₄

—

163

—

106

C₄²

—

107

C₂³

C₂×D₆₅₂

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

326

—

113

C₂×C₄

—

652

—

114

C₂×C₄

—

652

—

115

C₂×C₄

—

652

—

116

C₂×C₄

—

652

—

117

C₂×C₄

—

652

—

118

C₂×C₄

—

652

—

119

C₂×C₄

—

652

—

120

C₂×C₄

—

652

—

121

C₂×C₄

#14579688

yes

—

122

C₂×C₄

Dic₆₅₂

yes

—

123

C₂×C₄

Q₈×D₁₆₃

yes

characteristic

124

C₂×C₄

#14579698

yes

—

125

C₂×C₄

#14579698

yes

characteristic

126

C₂×C₄

#14579699

yes

—

127

C₈

—

128

C₂²

#19104715

yes

characteristic

129

C₂²

#19104719

yes

characteristic

130

C₂²

#19104723

yes

characteristic

131

C₂²

C₂×#14579688

yes

characteristic

132

C₂²

C₂×Dic₆₅₂

yes

characteristic

133

C₂²

#19104763

yes

characteristic

134

C₂²

#19104763

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

1304

—

138

C₄

—

1304

—

139

C₄

—

652

—

140

C₄

#19104725

yes

—

141

C₄

#19104728

yes

—

142

C₂

#23781605

yes

characteristic

143

C₂

#23781956

yes

characteristic

144

C₂

#23781961

yes

characteristic

145

C₂

#23781984

yes

characteristic

146

C₂

#23782001

yes

characteristic

147

C₂

#23782016

yes

characteristic

148

C₂

#23782030

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₆₅₂
commutator, cosocle	C₂²×C₄×C₁₆₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₆₃
Fitting	#336×C₁₆₃	C₂
socle, prime radical	C₂³×C₁₆₃	C₂×D₄
hypercenter	#336	D₁₆₃
hypocenter	C₁₆₃	#2100
LCS	G ▷ C₂²×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₆₅₂

commutator, cosocle

C₂²×C₄×C₁₆₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₆₃

Fitting

#336×C₁₆₃

C₂

socle, prime radical

C₂³×C₁₆₃

C₂×D₄

D₁₆₃

C₁₆₃

G ▷ C₂²×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
74. C₁₆₃	#2100	semidirect

H ≅ G/N

type

74. C₁₆₃

#2100

semidirect

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

subgroup	index
#23781563	2
#23781535	2
#23781428	2
#336×C₁₆₃	2
#2100	163

subgroup

index

#336×C₁₆₃

163

Subgroups — #35216100

Conjugacy classes of subgroups

Complements

Maximal subgroups