Group #35320823 — subgroups

Subgroups — #35320823

Order 21248.

342 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 159 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×#19127521	C₂	1	4	yes	maximal
2	#24909219	C₂	1	1	yes	maximal, characteristic
3	#24907149	C₂	1	1	yes	maximal, characteristic
4	C₈₃×#2150	C₂	1	1	yes	maximal, characteristic
5	C₂×#14581732	C₂²	1	2	yes	—
6	C₂×#14581703	C₄	1	1	yes	characteristic
7	C₂×#14581702	C₄	1	2	yes	—
8	C₂×C₈×Dic₈₃	C₂²	1	2	yes	—
9	#19127547	C₄	1	1	yes	characteristic
10	#19127521	—	2	8	no	—
11	C₄⋊(C₈₃⋊C₁₆)	—	2	2	no	—
12	C₂×M₃₂×C₈₃	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆×C₈₃	—	2	2	no	—
15	C₂²×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
16	C₂×C₄×Dic₈₃	C₂×C₄	1	2	yes	—
17	#14581732	—	2	4	no	—
18	C₂×C₈₃⋊C₁₆	—	2	2	no	—
19	#14581703	C₈	1	4	yes	—
20	#14581702	—	2	4	no	—
21	C₈×Dic₈₃	D₄	1	4	yes	—
22	C₈×Dic₈₃	Q₈	1	4	yes	—
23	C₈₃⋊#95	—	2	2	no	—
24	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
27	M₃₂×C₈₃	—	2	4	no	—
28	C₂×C₁₆×C₈₃	—	2	2	no	—
29	C₄×C₈×C₈₃	D₄	1	2	yes	—
30	C₄×C₈×C₈₃	Q₈	1	2	yes	—
31	C₂²×Dic₈₃	—	2	2	no	—
32	C₂×C₈₃⋊C₈	C₄⋊C₄	1	4	yes	—
33	C₂×C₈₃⋊C₈	C₂×C₈	1	4	yes	—
34	C₂×C₈₃⋊C₈	M₁₆	1	4	yes	—
35	C₄×Dic₈₃	—	2	2	no	—
36	C₄×Dic₈₃	C₄⋊C₄	1	4	yes	—
37	C₈₃⋊C₁₆	—	4	2	no	—
38	C₂²×C₄×C₈₃	M₁₆	1	2	yes	—
39	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₈₃	—	2	2	no	—
41	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
42	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
43	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₈₃	C₂×Q₈	1	2	yes	—
45	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
46	C₄²×C₈₃	C₄⋊C₄	1	2	yes	—
47	C₄²×C₈₃	C₂×C₈	1	2	yes	—
48	C₁₆×C₈₃	—	4	2	no	—
49	C₂³×C₈₃	C₂×M₁₆	1	1	yes	characteristic
50	C₂×C₄×C₈₃	—	4	2	no	—
51	C₂×C₄×C₈₃	C₄⋊C₈	1	4	yes	—
52	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	2	yes	—
53	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄×C₈₃	C₂×M₁₆	1	2	yes	—
56	C₂×Dic₈₃	—	2	4	no	—
57	C₂×Dic₈₃	—	4	4	no	—
58	C₈×C₈₃	—	2	2	no	—
59	C₈×C₈₃	#127	1	1	yes	characteristic
60	C₈×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
61	C₈×C₈₃	C₄⋊Q₈	1	2	yes	—
62	C₈₃⋊C₈	C₄⋊C₈	1	8	yes	—
63	C₂²×C₈₃	—	2	2	no	—
64	C₂²×C₈₃	C₂×C₄⋊C₈	1	1	yes	characteristic
65	C₂²×C₈₃	#423	1	2	yes	—
66	C₄×C₈₃	—	2	4	no	—
67	C₄×C₈₃	#384	1	1	yes	characteristic
68	C₄×C₈₃	C₂×C₄⋊C₈	1	2	yes	—
69	C₄×C₈₃	#423	1	1	yes	characteristic
70	Dic₈₃	—	4	4	no	—
71	#12901	—	83	1	no	maximal, Sylow
72	C₂×C₈₃	—	2	2	no	—
73	C₂×C₈₃	#1842	1	1	yes	characteristic
74	C₂×C₈₃	#2153	1	2	yes	—
75	C₂×#364	—	83	4	no	—
76	#2150	—	83	1	no	—
77	#2150	D₈₃	1	1	yes	characteristic
78	#1726	—	83	1	no	—
79	C₈₃	#12901	1	1	yes	characteristic, complemented, Sylow
80	C₂×M₃₂	—	83	2	no	—
81	C₂×M₃₂	C₂×D₈₃	1	2	yes	—
82	C₂×#97	—	83	1	no	—
83	C₂×#97	—	83	2	no	—
84	C₂×C₄×C₈	—	83	2	no	—
85	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
86	#376	—	83	1	no	—
87	#364	—	166	8	no	—
88	C₄⋊C₁₆	—	2	2	no	—
89	C₄⋊C₁₆	—	166	2	no	—
90	C₂²×C₈	—	83	2	no	—
91	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
92	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
93	C₂×C₄²	—	83	2	no	—
94	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
95	M₃₂	—	2	4	no	—
96	M₃₂	—	166	4	no	—
97	C₂×C₁₆	—	2	2	no	—
98	C₂×C₁₆	—	166	2	no	—
99	#97	—	83	4	no	—
100	#97	—	166	4	no	—
101	C₄×C₈	—	83	4	no	—
102	C₄×C₈	—	83	4	no	—
103	C₄×C₈	Dic₁₆₆	1	2	yes	—
104	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
105	#95	—	166	2	no	—
106	C₂²×C₄	—	166	2	no	—
107	C₂²×C₄	#13050573	1	2	yes	—
108	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	83	4	no	—
111	C₂×C₈	—	83	4	no	—
112	C₂×C₈	—	83	4	no	—
113	C₂×C₈	Dic₈₃⋊C₄	1	2	yes	—
114	C₂×C₈	D₄×D₈₃	1	2	yes	—
115	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
116	C₂×C₈	Q₈×D₈₃	1	2	yes	—
117	C₂×C₈	C₂×Dic₁₆₆	1	1	yes	characteristic
118	C₄²	—	83	4	no	—
119	C₄²	—	166	2	no	—
120	C₄²	Dic₈₃⋊C₄	1	2	yes	—
121	C₄²	C₈×D₈₃	1	2	yes	—
122	C₁₆	—	4	2	no	—
123	C₁₆	—	332	2	no	—
124	C₂³	C₂×#13050573	1	1	yes	characteristic
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	332	4	no	—
127	C₂×C₄	—	166	4	no	—
128	C₂×C₄	Dic₈₃⋊C₈	1	4	yes	—
129	C₂×C₄	C₄⋊C₄×D₈₃	1	2	yes	—
130	C₂×C₄	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
131	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
132	C₂×C₄	M₁₆×D₈₃	1	2	yes	—
133	C₈	—	2	2	no	—
134	C₈	—	83	8	no	—
135	C₈	#14581788	1	1	yes	characteristic
136	C₈	#14581790	1	1	yes	characteristic
137	C₈	#14581791	1	2	yes	—
138	C₂²	—	2	2	no	—
139	C₂²	C₂×Dic₈₃⋊C₈	1	1	yes	characteristic
140	C₂²	#19127723	1	2	yes	—
141	C₄	—	2	4	no	—
142	C₄	—	332	4	no	—
143	C₄	#19127570	1	1	yes	characteristic
144	C₄	C₄⋊C₈×D₈₃	1	2	yes	—
145	C₄	#19127724	1	1	yes	characteristic
146	C₂	—	2	2	no	—
147	C₂	#24907614	1	1	yes	characteristic
148	C₂	#24909228	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19127521

C₂

yes

maximal

#24909219

C₂

yes

maximal, characteristic

#24907149

C₂

yes

maximal, characteristic

C₈₃×#2150

C₂

yes

maximal, characteristic

C₂×#14581732

C₂²

yes

—

C₂×#14581703

C₄

yes

characteristic

C₂×#14581702

C₄

yes

—

C₂×C₈×Dic₈₃

C₂²

yes

—

#19127547

C₄

yes

characteristic

#19127521

—

C₄⋊(C₈₃⋊C₁₆)

—

C₂×M₃₂×C₈₃

C₂²

yes

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

C₄⋊C₁₆×C₈₃

—

C₂²×C₈₃⋊C₈

C₂×C₄

yes

—

C₂×C₄×Dic₈₃

C₂×C₄

yes

—

#14581732

—

C₂×C₈₃⋊C₁₆

—

#14581703

C₈

yes

—

#14581702

—

C₈×Dic₈₃

D₄

yes

—

C₈×Dic₈₃

Q₈

yes

—

C₈₃⋊#95

—

C₂²×C₈×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₈₃

C₂³

yes

characteristic

C₂×C₄²×C₈₃

C₂×C₄

yes

characteristic

M₃₂×C₈₃

—

C₂×C₁₆×C₈₃

—

C₄×C₈×C₈₃

D₄

yes

—

C₄×C₈×C₈₃

Q₈

yes

—

C₂²×Dic₈₃

—

C₂×C₈₃⋊C₈

C₄⋊C₄

yes

—

C₂×C₈₃⋊C₈

C₂×C₈

yes

—

C₂×C₈₃⋊C₈

M₁₆

yes

—

C₄×Dic₈₃

—

C₄×Dic₈₃

C₄⋊C₄

yes

—

C₈₃⋊C₁₆

—

C₂²×C₄×C₈₃

M₁₆

yes

—

C₂²×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₄⋊C₄

yes

—

C₂×C₈×C₈₃

C₂×D₄

yes

—

C₂×C₈×C₈₃

C₂×D₄

yes

characteristic

C₂×C₈×C₈₃

C₂×Q₈

yes

—

C₂×C₈×C₈₃

C₂×Q₈

yes

characteristic

C₄²×C₈₃

C₄⋊C₄

yes

—

C₄²×C₈₃

C₂×C₈

yes

—

C₁₆×C₈₃

—

C₂³×C₈₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₄⋊C₈

yes

—

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₈₃

C₂×M₁₆

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₈×C₈₃

—

C₈×C₈₃

#127

yes

characteristic

C₈×C₈₃

C₄⋊Q₈

yes

characteristic

C₈×C₈₃

C₄⋊Q₈

yes

—

C₈₃⋊C₈

C₄⋊C₈

yes

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₈₃

#423

yes

—

C₄×C₈₃

—

C₄×C₈₃

#384

yes

characteristic

C₄×C₈₃

C₂×C₄⋊C₈

yes

—

C₄×C₈₃

#423

yes

characteristic

Dic₈₃

—

#12901

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

#1842

yes

characteristic

C₂×C₈₃

#2153

yes

—

C₂×#364

—

#2150

—

#2150

D₈₃

yes

characteristic

#1726

—

C₈₃

#12901

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

C₂×D₈₃

yes

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

#376

—

#364

—

166

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

166

—

C₂²×C₈

—

C₂²×C₈

C₄×D₈₃

yes

characteristic

C₂²×C₈

C₂²×D₈₃

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₄×D₈₃

yes

characteristic

M₃₂

—

M₃₂

—

166

—

C₂×C₁₆

—

C₂×C₁₆

—

166

—

#97

—

100

#97

—

166

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

Dic₁₆₆

yes

—

104

C₄×C₈

C₂²⋊D₈₃

yes

—

105

#95

—

166

—

106

C₂²×C₄

—

166

—

107

C₂²×C₄

#13050573

yes

—

108

C₂²×C₄

C₂×C₄×D₈₃

yes

characteristic

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

Dic₈₃⋊C₄

yes

—

114

C₂×C₈

D₄×D₈₃

yes

—

115

C₂×C₈

C₂×C₂²⋊D₈₃

yes

characteristic

116

C₂×C₈

Q₈×D₈₃

yes

—

117

C₂×C₈

C₂×Dic₁₆₆

yes

characteristic

118

C₄²

—

119

C₄²

—

166

—

120

C₄²

Dic₈₃⋊C₄

yes

—

121

C₄²

C₈×D₈₃

yes

—

122

C₁₆

—

123

C₁₆

—

332

—

124

C₂³

C₂×#13050573

yes

characteristic

125

C₂×C₄

—

126

C₂×C₄

—

332

—

127

C₂×C₄

—

166

—

128

C₂×C₄

Dic₈₃⋊C₈

yes

—

129

C₂×C₄

C₄⋊C₄×D₈₃

yes

—

130

C₂×C₄

C₂×Dic₈₃⋊C₄

yes

characteristic

131

C₂×C₄

C₂×C₈×D₈₃

yes

characteristic

132

C₂×C₄

M₁₆×D₈₃

yes

—

133

C₈

—

134

C₈

—

135

C₈

#14581788

yes

characteristic

136

C₈

#14581790

yes

characteristic

137

C₈

#14581791

yes

—

138

C₂²

—

139

C₂²

C₂×Dic₈₃⋊C₈

yes

characteristic

140

C₂²

#19127723

yes

—

141

C₄

—

142

C₄

—

332

—

143

C₄

#19127570

yes

characteristic

144

C₄

C₄⋊C₈×D₈₃

yes

—

145

C₄

#19127724

yes

characteristic

146

C₂

—

147

C₂

#24907614

yes

characteristic

148

C₂

#24909228

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₈₃
commutator	C₂×C₄×C₈₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#2150	C₂
socle	C₂²×C₈₃	C₂×C₄⋊C₈
prime radical	C₂³×C₈₃	C₂×M₁₆
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	#2150	D₈₃
hypocenter	C₈₃	#12901
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂×C₈

C₂×C₂²⋊D₈₃

commutator

C₂×C₄×C₈₃

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₈₃

Fitting

C₈₃×#2150

C₂

socle

C₂²×C₈₃

C₂×C₄⋊C₈

prime radical

C₂³×C₈₃

C₂×M₁₆

cosocle

C₂²×C₈×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
79. C₈₃	#12901	semidirect

H ≅ G/N

type

79. C₈₃

#12901

semidirect

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

subgroup	index
C₂×#19127521	2
#24909219	2
#24907149	2
C₈₃×#2150	2
#12901	83

subgroup

index

C₈₃×#2150

Subgroups — #35320823

Conjugacy classes of subgroups

Complements

Maximal subgroups