Group #35320825 — subgroups

Subgroups — #35320825

Order 21248.

278 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 119 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	#24909229	C₂	1	1	yes	maximal, characteristic
2	#24909228	C₂	1	2	yes	maximal
3	#24909221	C₂	1	2	yes	maximal
4	#24907218	C₂	1	1	yes	maximal, characteristic
5	C₈₃×#2151	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581732	C₂²	1	2	yes	—
7	#19127656	C₂²	1	2	yes	—
8	#19127522	—	2	2	no	—
9	#19127521	—	2	4	no	—
10	Dic₈₃⋊C₁₆	—	2	4	no	—
11	C₈₃⋊#338	C₄	1	1	yes	characteristic
12	#19127458	C₄	1	1	yes	characteristic
13	#19127457	C₄	1	2	yes	—
14	C₂×M₃₂×C₈₃	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
16	#405×C₈₃	C₂²	1	1	yes	characteristic
17	C₄⋊C₁₆×C₈₃	—	2	2	no	—
18	C₂×#13050574	C₂×C₄	1	2	yes	—
19	#14581751	C₂×C₄	1	2	yes	—
20	#14581732	—	2	2	no	—
21	#14581732	—	2	2	no	—
22	C₂×C₈₃⋊C₁₆	—	2	2	no	—
23	#14581702	Q₈	1	4	yes	—
24	C₈×Dic₈₃	D₄	1	4	yes	—
25	C₂×M₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
27	#117×C₈₃	C₂×C₄	1	1	yes	characteristic
28	M₃₂×C₈₃	—	2	2	no	—
29	C₂×C₁₆×C₈₃	—	2	1	no	—
30	C₂×C₁₆×C₈₃	—	2	2	no	—
31	C₂×C₁₆×C₈₃	—	2	1	no	—
32	#97×C₈₃	Q₈	1	2	yes	—
33	C₄×C₈×C₈₃	D₄	1	2	yes	—
34	#13050574	—	2	2	no	—
35	#13050574	—	2	2	no	—
36	C₂×C₈₃⋊C₈	C₄⋊C₄	1	4	yes	—
37	C₄⋊Dic₈₃	—	2	2	no	—
38	C₂²⋊Dic₈₃	—	2	2	no	—
39	C₄×Dic₈₃	C₄⋊C₄	1	4	yes	—
40	C₈₃⋊C₁₆	—	4	2	no	—
41	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
42	M₁₆×C₈₃	—	2	2	no	—
43	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
45	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₈₃	C₂×Q₈	1	2	yes	—
47	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
48	C₄⋊C₄×C₈₃	—	2	1	no	—
49	C₂²⋊C₄×C₈₃	—	2	1	no	—
50	C₄²×C₈₃	C₄⋊C₄	1	2	yes	—
51	C₁₆×C₈₃	—	4	1	no	—
52	C₁₆×C₈₃	—	4	1	no	—
53	C₂³×C₈₃	#131	1	1	yes	characteristic
54	C₂×C₄×C₈₃	—	2	2	no	—
55	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	2	yes	—
56	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₈₃	#131	1	1	yes	characteristic
58	C₂×C₄×C₈₃	#131	1	2	yes	—
59	C₂×Dic₈₃	—	2	4	no	—
60	C₈×C₈₃	—	2	2	no	—
61	C₈×C₈₃	#127	1	1	yes	characteristic
62	C₈×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
63	C₈×C₈₃	C₄⋊Q₈	1	2	yes	—
64	C₈₃⋊C₈	—	2	4	no	—
65	C₂²×C₈₃	—	4	1	no	—
66	C₂²×C₈₃	#424	1	2	yes	—
67	C₂²×C₈₃	#424	1	1	yes	characteristic
68	C₄×C₈₃	—	4	2	no	—
69	C₄×C₈₃	#384	1	1	yes	characteristic
70	C₄×C₈₃	#424	1	2	yes	—
71	C₄×C₈₃	#424	1	1	yes	characteristic
72	Dic₈₃	—	4	4	no	—
73	#12902	—	83	1	no	maximal, Sylow
74	C₂×C₈₃	—	2	2	no	—
75	C₂×C₈₃	—	2	1	no	—
76	C₂×C₈₃	#1843	1	1	yes	characteristic
77	#2153	—	83	2	no	—
78	#2153	—	83	1	no	—
79	#2151	—	83	2	no	—
80	#2151	D₈₃	1	1	yes	characteristic
81	#1747	—	83	1	no	—
82	C₈₃	#12902	1	1	yes	characteristic, complemented, Sylow
83	C₂×M₃₂	—	83	2	no	—
84	C₂×M₃₂	C₂×D₈₃	1	1	yes	characteristic
85	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
86	#405	—	83	2	no	—
87	#405	C₂×D₈₃	1	1	yes	characteristic
88	#364	—	166	4	no	—
89	#364	—	166	2	no	—
90	C₄⋊C₁₆	—	2	2	no	—
91	C₄⋊C₁₆	—	166	4	no	—
92	#338	—	83	1	no	—
93	H(3,ℤ/4)	—	83	1	no	—
94	H(3,ℤ/4)	—	83	2	no	—
95	C₂×M₁₆	—	83	2	no	—
96	C₂×M₁₆	C₄×D₈₃	1	1	yes	characteristic
97	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
98	#117	—	83	2	no	—
99	#117	C₄×D₈₃	1	1	yes	characteristic
100	M₃₂	—	2	2	no	—
101	M₃₂	—	166	2	no	—
102	M₃₂	—	166	2	no	—
103	C₂×C₁₆	—	2	2	no	—
104	C₂×C₁₆	—	2	1	no	—
105	C₂×C₁₆	—	2	1	no	—
106	C₂×C₁₆	—	166	2	no	—
107	#97	—	83	4	no	—
108	#97	Dic₁₆₆	1	2	yes	—
109	C₄×C₈	—	83	4	no	—
110	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
111	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
112	M₁₆	—	2	2	no	—
113	M₁₆	—	166	2	no	—
114	M₁₆	—	166	2	no	—
115	C₂×C₈	—	83	4	no	—
116	C₂×C₈	Dic₈₃⋊C₄	1	2	yes	—
117	C₂×C₈	D₄×D₈₃	1	2	yes	—
118	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
119	C₂×C₈	Q₈×D₈₃	1	2	yes	—
120	C₂×C₈	C₂×Dic₁₆₆	1	1	yes	characteristic
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	166	2	no	—
123	C₂²⋊C₄	—	2	1	no	—
124	C₂²⋊C₄	—	166	2	no	—
125	C₄²	—	83	4	no	—
126	C₄²	Dic₈₃⋊C₄	1	2	yes	—
127	C₁₆	—	4	1	no	—
128	C₁₆	—	4	1	no	—
129	C₁₆	—	332	2	no	—
130	C₂³	#14581798	1	1	yes	characteristic
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	166	4	no	—
133	C₂×C₄	C₄⋊C₄×D₈₃	1	2	yes	—
134	C₂×C₄	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
135	C₂×C₄	#14581798	1	1	yes	characteristic
136	C₂×C₄	#14581800	1	2	yes	—
137	C₈	—	2	2	no	—
138	C₈	—	166	4	no	—
139	C₈	#14581788	1	1	yes	characteristic
140	C₈	#14581790	1	1	yes	characteristic
141	C₈	#14581791	1	2	yes	—
142	C₂²	—	4	1	no	—
143	C₂²	#19127726	1	2	yes	—
144	C₂²	#19127727	1	1	yes	characteristic
145	C₄	—	4	2	no	—
146	C₄	—	332	4	no	—
147	C₄	#19127570	1	1	yes	characteristic
148	C₄	#19127726	1	2	yes	—
149	C₄	#19127729	1	1	yes	characteristic
150	C₂	—	2	1	no	—
151	C₂	—	2	2	no	—
152	C₂	#24907616	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24909229

C₂

yes

maximal, characteristic

#24909228

C₂

yes

maximal

#24909221

C₂

yes

maximal

#24907218

C₂

yes

maximal, characteristic

C₈₃×#2151

C₂

yes

maximal, characteristic

C₂×#14581732

C₂²

yes

—

#19127656

C₂²

yes

—

#19127522

—

#19127521

—

Dic₈₃⋊C₁₆

—

C₈₃⋊#338

C₄

yes

characteristic

#19127458

C₄

yes

characteristic

#19127457

C₄

yes

—

C₂×M₃₂×C₈₃

C₂²

yes

characteristic

C₂²×C₁₆×C₈₃

C₂²

yes

characteristic

#405×C₈₃

C₂²

yes

characteristic

C₄⋊C₁₆×C₈₃

—

C₂×#13050574

C₂×C₄

yes

—

#14581751

C₂×C₄

yes

—

#14581732

—

#14581732

—

C₂×C₈₃⋊C₁₆

—

#14581702

Q₈

yes

—

C₈×Dic₈₃

D₄

yes

—

C₂×M₁₆×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₈₃

C₂³

yes

characteristic

#117×C₈₃

C₂×C₄

yes

characteristic

M₃₂×C₈₃

—

C₂×C₁₆×C₈₃

—

C₂×C₁₆×C₈₃

—

C₂×C₁₆×C₈₃

—

#97×C₈₃

Q₈

yes

—

C₄×C₈×C₈₃

D₄

yes

—

#13050574

—

#13050574

—

C₂×C₈₃⋊C₈

C₄⋊C₄

yes

—

C₄⋊Dic₈₃

—

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

C₄⋊C₄

yes

—

C₈₃⋊C₁₆

—

C₂²×C₄×C₈₃

C₂²×C₄

yes

characteristic

M₁₆×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

C₄⋊C₄×C₈₃

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

C₄⋊C₄

yes

—

C₁₆×C₈₃

—

C₁₆×C₈₃

—

C₂³×C₈₃

#131

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

—

C₂×Dic₈₃

—

C₈×C₈₃

—

C₈×C₈₃

#127

yes

characteristic

C₈×C₈₃

C₄⋊Q₈

yes

characteristic

C₈×C₈₃

C₄⋊Q₈

yes

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

#424

yes

—

C₂²×C₈₃

#424

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

#384

yes

characteristic

C₄×C₈₃

#424

yes

—

C₄×C₈₃

#424

yes

characteristic

Dic₈₃

—

#12902

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

yes

—

—

—

D₈₃

yes

characteristic

#1747

—

C₈₃

#12902

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

C₂×D₈₃

yes

characteristic

C₂²×C₁₆

C₂×D₈₃

yes

characteristic

#405

—

#405

C₂×D₈₃

yes

characteristic

#364

—

166

—

#364

—

166

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

166

—

#338

—

H(3,ℤ/4)

—

H(3,ℤ/4)

—

C₂×M₁₆

—

C₂×M₁₆

C₄×D₈₃

yes

characteristic

C₂²×C₈

C₂²×D₈₃

yes

characteristic

#117

—

#117

C₄×D₈₃

yes

characteristic

100

M₃₂

—

101

M₃₂

—

166

—

102

M₃₂

—

166

—

103

C₂×C₁₆

—

104

C₂×C₁₆

—

105

C₂×C₁₆

—

106

C₂×C₁₆

—

166

—

107

#97

—

108

#97

Dic₁₆₆

yes

—

109

C₄×C₈

—

110

C₄×C₈

C₂²⋊D₈₃

yes

—

111

C₂²×C₄

C₂×C₄×D₈₃

yes

characteristic

112

M₁₆

—

113

M₁₆

—

166

—

114

M₁₆

—

166

—

115

C₂×C₈

—

116

C₂×C₈

Dic₈₃⋊C₄

yes

—

117

C₂×C₈

D₄×D₈₃

yes

—

118

C₂×C₈

C₂×C₂²⋊D₈₃

yes

characteristic

119

C₂×C₈

Q₈×D₈₃

yes

—

120

C₂×C₈

C₂×Dic₁₆₆

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

166

—

123

C₂²⋊C₄

—

124

C₂²⋊C₄

—

166

—

125

C₄²

—

126

C₄²

Dic₈₃⋊C₄

yes

—

127

C₁₆

—

128

C₁₆

—

129

C₁₆

—

332

—

130

C₂³

#14581798

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

166

—

133

C₂×C₄

C₄⋊C₄×D₈₃

yes

—

134

C₂×C₄

C₂×Dic₈₃⋊C₄

yes

characteristic

135

C₂×C₄

#14581798

yes

characteristic

136

C₂×C₄

#14581800

yes

—

137

C₈

—

138

C₈

—

166

—

139

C₈

#14581788

yes

characteristic

140

C₈

#14581790

yes

characteristic

141

C₈

#14581791

yes

—

142

C₂²

—

143

C₂²

#19127726

yes

—

144

C₂²

#19127727

yes

characteristic

145

C₄

—

146

C₄

—

332

—

147

C₄

#19127570

yes

characteristic

148

C₄

#19127726

yes

—

149

C₄

#19127729

yes

characteristic

150

C₂

—

151

C₂

—

152

C₂

#24907616

yes

characteristic

	subgroup	G/H
center	C₈	#14581788
commutator	C₂²×C₄×C₈₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#2151	C₂
socle	C₂×C₈₃	#1843
prime radical	C₂³×C₈₃	#131
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	#2151	D₈₃
hypocenter	C₈₃	#12902
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2151

subgroup

G/H

center

C₈

#14581788

commutator

C₂²×C₄×C₈₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₈₃

Fitting

C₈₃×#2151

C₂

C₂×C₈₃

C₂³×C₈₃

C₂²×C₈×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2151

N	H ≅ G/N	type
82. C₈₃	#12902	semidirect

H ≅ G/N

type

82. C₈₃

#12902

semidirect

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

subgroup	index
#24909229	2
#24909228	2
#24909221	2
#24907218	2
C₈₃×#2151	2
#12902	83

subgroup

index

C₈₃×#2151

Subgroups — #35320825

Conjugacy classes of subgroups

Complements

Maximal subgroups