Group #35215697 — subgroups

Subgroups — #35215697

Order 20864.

246 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 69 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	#23782296	C₂	1	1	yes	maximal, characteristic
2	#23781674	C₂	1	2	yes	maximal
3	#23781434	C₂	1	1	yes	maximal, characteristic
4	#23781425	C₂	1	2	yes	maximal
5	#391×C₁₆₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579683	C₂²	1	2	yes	—
7	#19104734	—	2	2	no	—
8	#19104726	—	2	2	no	—
9	#19104714	—	2	2	no	—
10	#19104694	C₂²	1	2	yes	—
11	#19104659	—	2	2	no	—
12	#19104659	—	2	2	no	—
13	C₂²×Q₈×C₁₆₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₁₆₃	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
16	#95×C₁₆₃	—	2	2	no	—
17	C₂×Dic₃₂₆	—	2	2	no	—
18	#14579683	—	2	2	no	—
19	#14579683	—	2	2	no	—
20	C₂×C₁₆₃⋊C₈	—	2	2	no	—
21	C₄⋊Dic₁₆₃	—	2	2	no	—
22	C₄⋊Dic₁₆₃	D₄	1	2	yes	—
23	C₄⋊Dic₁₆₃	D₄	1	2	yes	—
24	Dic₁₆₃⋊C₄	—	2	4	no	—
25	C₂²⋊Dic₁₆₃	—	2	2	no	—
26	C₄×Dic₁₆₃	—	2	2	no	—
27	C₂×Q₈×C₁₆₃	—	2	1	no	—
28	C₂×Q₈×C₁₆₃	—	2	2	no	—
29	C₂×Q₈×C₁₆₃	—	2	2	no	—
30	C₂×Q₈×C₁₆₃	Q₈	1	2	yes	complemented
31	C₂²×C₄×C₁₆₃	—	2	1	no	—
32	C₂²×C₄×C₁₆₃	—	2	1	no	—
33	C₂²×C₄×C₁₆₃	D₄	1	2	yes	—
34	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₁₆₃	—	2	2	no	—
36	C₄²×C₁₆₃	—	2	2	no	—
37	Dic₃₂₆	—	8	2	no	—
38	C₂×Dic₁₆₃	—	2	2	no	—
39	C₂×Dic₁₆₃	—	2	2	no	—
40	C₁₆₃⋊C₈	—	4	2	no	—
41	C₂³×C₁₆₃	C₂×D₄	1	1	yes	characteristic
42	Q₈×C₁₆₃	—	2	4	no	—
43	Q₈×C₁₆₃	—	4	2	no	—
44	C₂×C₄×C₁₆₃	—	2	2	no	—
45	C₂×C₄×C₁₆₃	—	2	2	no	—
46	C₂×C₄×C₁₆₃	—	2	2	no	—
47	C₂×C₄×C₁₆₃	—	4	1	no	—
48	C₂×C₄×C₁₆₃	—	4	1	no	—
49	C₂×C₄×C₁₆₃	—	4	1	no	—
50	C₂×C₄×C₁₆₃	—	4	1	no	—
51	C₂×C₄×C₁₆₃	—	4	2	no	—
52	C₂×C₄×C₁₆₃	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₁₆₃	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄×C₁₆₃	#41	1	2	yes	—
55	C₂×C₄×C₁₆₃	#41	1	1	yes	characteristic
56	C₂²×C₁₆₃	—	2	1	no	—
57	C₂²×C₁₆₃	—	2	1	no	—
58	C₂²×C₁₆₃	#120	1	1	yes	characteristic
59	C₂²×C₁₆₃	C₂²⋊Q₈	1	1	yes	characteristic
60	C₂²×C₁₆₃	#124	1	1	yes	characteristic
61	C₄×C₁₆₃	—	4	2	no	—
62	C₄×C₁₆₃	—	4	2	no	—
63	C₄×C₁₆₃	—	4	2	no	—
64	C₄×C₁₆₃	C₂²⋊Q₈	1	2	yes	—
65	C₄×C₁₆₃	#124	1	2	yes	—
66	Dic₁₆₃	—	4	2	no	—
67	Dic₁₆₃	—	4	2	no	—
68	C₂×C₁₆₃	—	2	1	no	—
69	C₂×C₁₆₃	—	2	1	no	—
70	C₂×C₁₆₃	#393	1	1	yes	characteristic
71	C₂×C₁₆₃	#455	1	1	yes	characteristic
72	C₂×C₁₆₃	#456	1	1	yes	characteristic
73	C₁₆₃	#2028	1	1	yes	characteristic, complemented, Sylow
74	#2028	—	163	1	no	maximal, Sylow
75	#544	—	163	1	no	—
76	#419	—	163	2	no	—
77	#391	D₁₆₃	1	1	yes	characteristic
78	#343	—	163	1	no	—
79	#339	—	163	2	no	—
80	C₂²×Q₈	C₂×D₁₆₃	1	1	yes	characteristic
81	C₂×M₁₆	—	163	2	no	—
82	C₄⋊Q₈	—	326	2	no	—
83	#125	—	326	2	no	—
84	C₂²⋊Q₈	—	326	2	no	—
85	#117	—	163	2	no	—
86	C₂×C₄⋊C₄	C₂×D₁₆₃	1	1	yes	characteristic
87	C₂×C₄²	C₂×D₁₆₃	1	1	yes	characteristic
88	#103	—	326	2	no	—
89	#103	—	326	2	no	—
90	#95	—	2	2	no	—
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	—	326	2	no	—
95	C₂×Q₈	Dic₃₂₆	1	2	yes	complemented
96	C₂²×C₄	—	2	1	no	—
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	C₂²⋊D₁₆₃	1	2	yes	—
99	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
100	M₁₆	—	326	2	no	—
101	M₁₆	—	326	2	no	—
102	C₂×C₈	—	326	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	326	2	no	—
105	C₄⋊C₄	—	326	4	no	—
106	C₄⋊C₄	—	163	2	no	—
107	C₄⋊C₄	—	163	2	no	—
108	C₂²⋊C₄	—	326	2	no	—
109	C₄²	—	2	2	no	—
110	C₄²	—	326	2	no	—
111	C₂³	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
112	Q₈	—	2	4	no	—
113	Q₈	—	4	2	no	—
114	Q₈	—	1304	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	326	2	no	—
124	C₂×C₄	—	326	2	no	—
125	C₂×C₄	D₄×D₁₆₃	1	2	yes	—
126	C₂×C₄	C₂×Dic₃₂₆	1	1	yes	characteristic
127	C₂×C₄	#14579698	1	2	yes	—
128	C₂×C₄	#14579699	1	1	yes	characteristic
129	C₈	—	652	2	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	#19104703	1	1	yes	characteristic
133	C₂²	#19104714	1	1	yes	characteristic
134	C₂²	#19104722	1	1	yes	characteristic
135	C₄	—	4	2	no	—
136	C₄	—	4	2	no	—
137	C₄	—	4	2	no	—
138	C₄	—	652	2	no	—
139	C₄	—	652	2	no	—
140	C₄	#19104712	1	2	yes	—
141	C₄	#19104721	1	2	yes	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	#23781577	1	1	yes	characteristic
145	C₂	#23781842	1	1	yes	characteristic
146	C₂	#23781847	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23782296

C₂

yes

maximal, characteristic

#23781674

C₂

yes

maximal

#23781434

C₂

yes

maximal, characteristic

#23781425

C₂

yes

maximal

#391×C₁₆₃

C₂

yes

maximal, characteristic

C₂×#14579683

C₂²

yes

—

—

—

—

C₂²

yes

—

#19104659

—

#19104659

—

C₂²×Q₈×C₁₆₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₆₃

C₂²

yes

characteristic

C₂×C₄²×C₁₆₃

C₂²

yes

characteristic

#95×C₁₆₃

—

C₂×Dic₃₂₆

—

#14579683

—

#14579683

—

C₂×C₁₆₃⋊C₈

—

C₄⋊Dic₁₆₃

—

C₄⋊Dic₁₆₃

D₄

yes

—

C₄⋊Dic₁₆₃

D₄

yes

—

Dic₁₆₃⋊C₄

—

C₂²⋊Dic₁₆₃

—

C₄×Dic₁₆₃

—

C₂×Q₈×C₁₆₃

—

C₂×Q₈×C₁₆₃

—

C₂×Q₈×C₁₆₃

—

C₂×Q₈×C₁₆₃

Q₈

yes

complemented

C₂²×C₄×C₁₆₃

—

C₂²×C₄×C₁₆₃

—

C₂²×C₄×C₁₆₃

D₄

yes

—

C₂²×C₄×C₁₆₃

C₂³

yes

characteristic

C₄⋊C₄×C₁₆₃

—

C₄²×C₁₆₃

—

Dic₃₂₆

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₁₆₃⋊C₈

—

C₂³×C₁₆₃

C₂×D₄

yes

characteristic

Q₈×C₁₆₃

—

Q₈×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

—

C₂×C₄×C₁₆₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₆₃

#41

yes

—

C₂×C₄×C₁₆₃

#41

yes

characteristic

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

#120

yes

characteristic

C₂²×C₁₆₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₆₃

#124

yes

characteristic

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

C₂²⋊Q₈

yes

—

C₄×C₁₆₃

#124

yes

—

Dic₁₆₃

—

Dic₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

C₁₆₃

yes

characteristic, complemented, Sylow

#2028

—

163

maximal, Sylow

#544

—

163

—

#419

—

163

—

#391

D₁₆₃

yes

characteristic

#343

—

163

—

#339

—

163

—

C₂²×Q₈

C₂×D₁₆₃

yes

characteristic

C₂×M₁₆

—

163

—

C₄⋊Q₈

—

326

—

#125

—

326

—

C₂²⋊Q₈

—

326

—

#117

—

163

—

C₂×C₄⋊C₄

C₂×D₁₆₃

yes

characteristic

C₂×C₄²

C₂×D₁₆₃

yes

characteristic

#103

—

326

—

#103

—

326

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

326

—

C₂×Q₈

Dic₃₂₆

yes

complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₆₃

yes

—

C₂²×C₄

C₂²×D₁₆₃

yes

characteristic

100

M₁₆

—

326

—

101

M₁₆

—

326

—

102

C₂×C₈

—

326

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

326

—

105

C₄⋊C₄

—

326

—

106

C₄⋊C₄

—

163

—

107

C₄⋊C₄

—

163

—

108

C₂²⋊C₄

—

326

—

109

C₄²

—

110

C₄²

—

326

—

111

C₂³

C₂×C₂²⋊D₁₆₃

yes

characteristic

112

Q₈

—

113

Q₈

—

114

Q₈

—

1304

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

326

—

124

C₂×C₄

—

326

—

125

C₂×C₄

D₄×D₁₆₃

yes

—

126

C₂×C₄

C₂×Dic₃₂₆

yes

characteristic

127

C₂×C₄

#14579698

yes

—

128

C₂×C₄

#14579699

yes

characteristic

129

C₈

—

652

—

130

C₂²

—

131

C₂²

—

132

C₂²

#19104703

yes

characteristic

133

C₂²

#19104714

yes

characteristic

134

C₂²

#19104722

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

652

—

139

C₄

—

652

—

140

C₄

#19104712

yes

—

141

C₄

#19104721

yes

—

142

C₂

—

143

C₂

—

144

C₂

#23781577

yes

characteristic

145

C₂

#23781842

yes

characteristic

146

C₂

#23781847

yes

characteristic

	subgroup	G/H
center	C₂²	#19104703
commutator, cosocle	C₂²×C₄×C₁₆₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₆₃
Fitting	#391×C₁₆₃	C₂
socle	C₂²×C₁₆₃	#120
prime radical	C₂³×C₁₆₃	C₂×D₄
hypercenter	#391	D₁₆₃
hypocenter	C₁₆₃	#2028
LCS	G ▷ C₂²×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #391

subgroup

G/H

center

C₂²

#19104703

commutator, cosocle

C₂²×C₄×C₁₆₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₆₃

Fitting

#391×C₁₆₃

C₂

socle

C₂²×C₁₆₃

#120

prime radical

C₂³×C₁₆₃

C₂×D₄

D₁₆₃

C₁₆₃

G ▷ C₂²×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #391

N	H ≅ G/N	type
30. C₂×Q₈×C₁₆₃	Q₈	semidirect
73. C₁₆₃	#2028	semidirect
95. C₂×Q₈	Dic₃₂₆	semidirect

H ≅ G/N

type

30. C₂×Q₈×C₁₆₃

Q₈

semidirect

73. C₁₆₃

#2028

semidirect

95. C₂×Q₈

Dic₃₂₆

semidirect

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

subgroup	index
#23782296	2
#23781674	2
#23781434	2
#23781425	2
#391×C₁₆₃	2
#2028	163

subgroup

index

#391×C₁₆₃

163

Subgroups — #35215697

Conjugacy classes of subgroups

Complements

Maximal subgroups