Group #35316256 — subgroups

Subgroups — #35316256

Order 21248.

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

normal subgroups only

This group has 200 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	#24909371	C₂	1	2	yes	maximal
2	C₈×Dic₃₃₂	C₂	1	1	yes	maximal, characteristic
3	#24905527	C₂	1	1	yes	maximal, characteristic
4	#24905443	C₂	1	2	yes	maximal
5	C₈₃×#1312	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₁₆₆	C₂²	1	2	yes	—
7	C₄×Dic₃₃₂	C₄	1	1	yes	characteristic
8	C₄⋊(C₈₃⋊C₁₆)	C₂²	1	2	yes	—
9	Dic₈₃⋊C₁₆	—	2	2	no	—
10	#19127476	—	2	2	no	—
11	#19127454	C₄	1	1	yes	characteristic
12	#19127433	C₄	1	2	yes	—
13	C₄×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
15	C₈²×C₈₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₃₃₂	C₈	1	1	yes	characteristic
17	C₄×Dic₁₆₆	C₂×C₄	1	2	yes	—
18	C₂×C₈₃⋊C₁₆	—	2	2	no	—
19	#14581726	C₈	1	1	yes	characteristic
20	C₄⋊(C₈₃⋊C₈)	C₂×C₄	1	2	yes	—
21	Dic₈₃⋊C₈	—	2	2	no	—
22	#14581717	C₈	1	2	yes	—
23	C₈×Dic₈₃	—	2	2	no	—
24	C₂×C₁₆×C₈₃	—	2	1	no	—
25	C₂×C₁₆×C₈₃	D₄	1	2	yes	—
26	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₆₆	C₂×C₈	1	2	yes	—
30	Dic₃₃₂	—	2	2	no	—
31	C₂×C₈₃⋊C₈	—	2	2	no	—
32	C₄⋊Dic₈₃	C₂×C₈	1	2	yes	—
33	Dic₈₃⋊C₄	—	2	2	no	—
34	C₄×Dic₈₃	—	2	2	no	—
35	C₈₃⋊C₁₆	—	4	2	no	—
36	C₂×C₈×C₈₃	—	2	1	no	—
37	C₂×C₈×C₈₃	C₂×C₈	1	1	yes	characteristic
38	C₂×C₈×C₈₃	C₂×C₈	1	1	yes	characteristic
39	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
41	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
42	C₁₆×C₈₃	—	2	2	no	—
43	C₁₆×C₈₃	—	4	1	no	—
44	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
45	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
46	C₂×C₄×C₈₃	#131	1	1	yes	characteristic
47	C₂×Dic₈₃	—	2	2	no	—
48	C₂×Dic₈₃	—	2	2	no	—
49	Dic₁₆₆	—	2	2	no	—
50	Dic₁₆₆	—	2	2	no	—
51	C₈×C₈₃	—	2	1	no	—
52	C₈×C₈₃	—	2	1	no	—
53	C₈×C₈₃	—	4	1	no	—
54	C₈×C₈₃	M₃₂	1	2	yes	—
55	C₈×C₈₃	#137	1	2	yes	—
56	C₈₃⋊C₈	—	4	2	no	—
57	C₂²×C₈₃	C₈×D₄	1	1	yes	characteristic
58	C₄×C₈₃	—	2	1	no	—
59	C₄×C₈₃	#441	1	1	yes	characteristic
60	C₄×C₈₃	#444	1	1	yes	characteristic
61	C₄×C₈₃	C₂×M₃₂	1	1	yes	characteristic
62	C₄×C₈₃	#504	1	1	yes	characteristic
63	Dic₈₃	—	4	2	no	—
64	Dic₈₃	—	4	2	no	—
65	#12061	—	83	1	no	maximal, Sylow
66	C₂×C₈₃	#1579	1	1	yes	characteristic
67	C₂×C₈₃	#2169	1	1	yes	characteristic
68	C₂×C₈₃	#2171	1	1	yes	characteristic
69	#2183	—	83	2	no	—
70	C₈×Q₁₆	—	83	1	no	—
71	#1367	—	83	1	no	—
72	Q₈⋊C₁₆	—	83	2	no	—
73	#1312	D₈₃	1	1	yes	characteristic
74	C₈₃	#12061	1	1	yes	characteristic, complemented, Sylow
75	C₈×Q₈	—	83	2	no	—
76	C₄×Q₁₆	—	83	1	no	—
77	C₄⋊C₁₆	—	83	2	no	—
78	C₄⋊C₁₆	—	166	2	no	—
79	#346	—	166	2	no	—
80	C₄×C₁₆	C₂×D₈₃	1	1	yes	characteristic
81	C₄×C₁₆	C₂×D₈₃	1	1	yes	characteristic
82	#335	—	83	1	no	—
83	Q₈⋊C₈	—	83	2	no	—
84	C₈²	C₂×D₈₃	1	1	yes	characteristic
85	C₂×Q₁₆	—	83	1	no	—
86	C₄×Q₈	—	83	2	no	—
87	C₂×C₁₆	—	2	1	no	—
88	C₂×C₁₆	—	166	2	no	—
89	C₂×C₁₆	D₃₃₂	1	2	yes	—
90	#107	—	83	1	no	—
91	C₄⋊C₈	—	83	2	no	—
92	C₄⋊C₈	—	166	2	no	—
93	#103	—	83	2	no	—
94	C₄×C₈	—	166	2	no	—
95	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
96	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
97	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
98	C₂×Q₈	—	83	2	no	—
99	Q₁₆	—	166	2	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	166	2	no	—
102	C₂×C₈	C₈×D₈₃	1	1	yes	characteristic
103	C₂×C₈	C₈×D₈₃	1	1	yes	characteristic
104	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
105	C₂×C₈	#13050590	1	1	yes	characteristic
106	C₄⋊C₄	—	83	2	no	—
107	C₄⋊C₄	—	166	2	no	—
108	C₄²	—	166	2	no	—
109	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
110	C₁₆	—	2	2	no	—
111	C₁₆	—	4	1	no	—
112	C₁₆	—	332	2	no	—
113	Q₈	—	166	2	no	—
114	Q₈	—	166	2	no	—
115	C₂×C₄	—	166	2	no	—
116	C₂×C₄	—	166	2	no	—
117	C₂×C₄	C₄×D₃₃₂	1	1	yes	characteristic
118	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
119	C₂×C₄	#14581798	1	1	yes	characteristic
120	C₈	—	2	1	no	—
121	C₈	—	2	1	no	—
122	C₈	—	4	1	no	—
123	C₈	—	332	2	no	—
124	C₈	#14581731	1	2	yes	—
125	C₈	#14581825	1	2	yes	—
126	C₂²	C₈×D₃₃₂	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	332	2	no	—
129	C₄	—	332	2	no	—
130	C₄	#19127809	1	1	yes	characteristic
131	C₄	#19127824	1	1	yes	characteristic
132	C₄	C₂×#14581731	1	1	yes	characteristic
133	C₄	#19128117	1	1	yes	characteristic
134	C₂	#24906283	1	1	yes	characteristic
135	C₂	#24909296	1	1	yes	characteristic
136	C₂	#24909306	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24909371

C₂

yes

maximal

C₈×Dic₃₃₂

C₂

yes

maximal, characteristic

#24905527

C₂

yes

maximal, characteristic

#24905443

C₂

yes

maximal

C₈₃×#1312

C₂

yes

maximal, characteristic

C₈×Dic₁₆₆

C₂²

yes

—

C₄×Dic₃₃₂

C₄

yes

characteristic

C₄⋊(C₈₃⋊C₁₆)

C₂²

yes

—

Dic₈₃⋊C₁₆

—

#19127476

—

#19127454

C₄

yes

characteristic

#19127433

C₄

yes

—

C₄×C₁₆×C₈₃

C₂²

yes

characteristic

C₄×C₁₆×C₈₃

C₂²

yes

characteristic

C₈²×C₈₃

C₂²

yes

characteristic

C₂×Dic₃₃₂

C₈

yes

characteristic

C₄×Dic₁₆₆

C₂×C₄

yes

—

C₂×C₈₃⋊C₁₆

—

#14581726

C₈

yes

characteristic

C₄⋊(C₈₃⋊C₈)

C₂×C₄

yes

—

Dic₈₃⋊C₈

—

#14581717

C₈

yes

—

C₈×Dic₈₃

—

C₂×C₁₆×C₈₃

—

C₂×C₁₆×C₈₃

D₄

yes

—

C₄×C₈×C₈₃

C₂×C₄

yes

characteristic

C₄×C₈×C₈₃

C₂×C₄

yes

characteristic

C₄×C₈×C₈₃

C₂³

yes

characteristic

C₂×Dic₁₆₆

C₂×C₈

yes

—

Dic₃₃₂

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

C₂×C₈

yes

—

Dic₈₃⋊C₄

—

C₄×Dic₈₃

—

C₈₃⋊C₁₆

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂×C₈

yes

characteristic

C₂×C₈×C₈₃

C₂×C₈

yes

characteristic

C₂×C₈×C₈₃

C₂×D₄

yes

characteristic

C₂×C₈×C₈₃

#41

yes

characteristic

C₄²×C₈₃

C₂²×C₄

yes

characteristic

C₁₆×C₈₃

—

C₁₆×C₈₃

—

C₂×C₄×C₈₃

C₄×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₈₃

#131

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

M₃₂

yes

—

C₈×C₈₃

#137

yes

—

C₈₃⋊C₈

—

C₂²×C₈₃

C₈×D₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

C₂×M₃₂

yes

characteristic

C₄×C₈₃

#504

yes

characteristic

Dic₈₃

—

Dic₈₃

—

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈×Q₁₆

—

#1367

—

Q₈⋊C₁₆

—

#1312

D₈₃

yes

characteristic

C₈₃

#12061

yes

characteristic, complemented, Sylow

C₈×Q₈

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

166

—

#346

—

166

—

C₄×C₁₆

C₂×D₈₃

yes

characteristic

C₄×C₁₆

C₂×D₈₃

yes

characteristic

#335

—

Q₈⋊C₈

—

C₈²

C₂×D₈₃

yes

characteristic

C₂×Q₁₆

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

166

—

C₂×C₁₆

D₃₃₂

yes

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

166

—

#103

—

C₄×C₈

—

166

—

C₄×C₈

C₄×D₈₃

yes

characteristic

C₄×C₈

C₄×D₈₃

yes

characteristic

C₄×C₈

C₂²×D₈₃

yes

characteristic

C₂×Q₈

—

Q₁₆

—

166

—

100

C₂×C₈

—

101

C₂×C₈

—

166

—

102

C₂×C₈

C₈×D₈₃

yes

characteristic

103

C₂×C₈

C₈×D₈₃

yes

characteristic

104

C₂×C₈

C₂×D₃₃₂

yes

characteristic

105

C₂×C₈

#13050590

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

166

—

108

C₄²

—

166

—

109

C₄²

C₂×C₄×D₈₃

yes

characteristic

110

C₁₆

—

111

C₁₆

—

112

C₁₆

—

332

—

113

Q₈

—

166

—

114

Q₈

—

166

—

115

C₂×C₄

—

166

—

116

C₂×C₄

—

166

—

117

C₂×C₄

C₄×D₃₃₂

yes

characteristic

118

C₂×C₄

C₂×C₈×D₈₃

yes

characteristic

119

C₂×C₄

#14581798

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

332

—

124

C₈

#14581731

yes

—

125

C₈

#14581825

yes

—

126

C₂²

C₈×D₃₃₂

yes

characteristic

127

C₄

—

128

C₄

—

332

—

129

C₄

—

332

—

130

C₄

#19127809

yes

characteristic

131

C₄

#19127824

yes

characteristic

132

C₄

C₂×#14581731

yes

characteristic

133

C₄

#19128117

yes

characteristic

134

C₂

#24906283

yes

characteristic

135

C₂

#24909296

yes

characteristic

136

C₂

#24909306

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₃₃₂
commutator	C₂×C₄×C₈₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₈₃
Fitting	C₈₃×#1312	C₂
socle, prime radical	C₂²×C₈₃	C₈×D₄
cosocle	C₄×C₈×C₈₃	C₂³
hypercenter	#1312	D₈₃
hypocenter	C₈₃	#12061
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ #1312

subgroup

G/H

center

C₂×C₈

C₂×D₃₃₂

commutator

C₂×C₄×C₈₃

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₈₃

Fitting

C₈₃×#1312

C₂

socle, prime radical

C₂²×C₈₃

C₈×D₄

cosocle

C₄×C₈×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ #1312

N	H ≅ G/N	type
74. C₈₃	#12061	semidirect

H ≅ G/N

type

74. C₈₃

#12061

semidirect

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

subgroup	index
#24909371	2
C₈×Dic₃₃₂	2
#24905527	2
#24905443	2
C₈₃×#1312	2
#12061	83

subgroup

index

C₈×Dic₃₃₂

C₈₃×#1312

Subgroups — #35316256

Conjugacy classes of subgroups

Complements

Maximal subgroups