Group #35315967 — subgroups

Subgroups — #35315967

Order 21248.

220 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 99 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	#364×D₈₃	C₂	1	2	yes	maximal
2	#24909056	C₂	1	1	yes	maximal, characteristic
3	#24905531	C₂	1	1	yes	maximal, characteristic, complemented
4	#24905530	C₂	1	2	yes	maximal, complemented
5	C₈₃×#1369	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14581731	C₄	1	1	yes	characteristic
7	C₂×#14581731	C₄	1	1	yes	characteristic
8	M₃₂×D₈₃	—	2	2	no	—
9	C₄×C₈×D₈₃	C₂²	1	1	yes	characteristic
10	#19127522	C₂²	1	2	yes	—
11	#19127521	—	2	2	no	—
12	#19127477	C₂²	1	1	yes	characteristic
13	C₁₆×Dic₈₃	C₄	1	1	yes	characteristic
14	C₁₆×Dic₈₃	C₄	1	1	yes	characteristic
15	#364×C₈₃	C₂²	1	2	yes	—
16	#346×C₈₃	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₈₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×D₈₃	D₄	1	1	yes	characteristic
19	C₄²×D₈₃	Q₈	1	1	yes	characteristic
20	#14581732	—	2	2	no	—
21	#14581731	—	2	2	no	—
22	#14581731	—	4	2	no	—
23	#14581731	C₈	1	4	yes	—
24	C₂×C₈₃⋊C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈₃⋊C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₁₆×D₈₃	—	4	2	no	—
27	C₄×C₈₃⋊C₈	Q₈	1	1	yes	characteristic
28	C₈×Dic₈₃	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₈₃	D₄	1	1	yes	characteristic
30	M₃₂×C₈₃	—	2	2	no	—
31	C₂×C₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
34	C₂×C₄×D₈₃	—	2	1	no	—
35	C₂×C₄×D₈₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈₃⋊C₈	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈₃⋊C₈	C₄⋊C₄	1	1	yes	characteristic
38	C₈×D₈₃	C₂×C₈	1	2	yes	—
39	C₈×D₈₃	M₁₆	1	2	yes	—
40	C₈×D₈₃	SD₁₆	1	2	yes	—
41	C₈×D₈₃	SD₁₆	1	2	yes	—
42	C₄×Dic₈₃	—	2	1	no	—
43	C₄×Dic₈₃	C₄⋊C₄	1	1	yes	characteristic
44	C₈₃⋊C₁₆	—	2	1	no	—
45	C₈₃⋊C₁₆	—	4	2	no	—
46	C₈₃⋊C₁₆	C₂×C₈	1	2	yes	—
47	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
50	C₁₆×C₈₃	—	2	1	no	—
51	C₁₆×C₈₃	—	4	2	no	—
52	C₁₆×C₈₃	C₂×C₈	1	2	yes	—
53	C₂²×D₈₃	#108	1	1	yes	characteristic
54	C₂×C₄×C₈₃	—	2	1	no	—
55	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×Dic₈₃	—	2	1	no	—
57	C₂×Dic₈₃	#108	1	1	yes	characteristic
58	C₄×D₈₃	—	4	2	no	—
59	C₄×D₈₃	C₄⋊C₈	1	2	yes	—
60	C₄×D₈₃	#106	1	2	yes	—
61	C₈×C₈₃	C₂²×C₈	1	1	yes	characteristic
62	C₈×C₈₃	C₂×M₁₆	1	1	yes	characteristic
63	C₈×C₈₃	C₂×SD₁₆	1	2	yes	—
64	C₈₃⋊C₈	C₄⋊C₈	1	2	yes	—
65	C₈₃⋊C₈	#106	1	1	yes	characteristic
66	C₈₃⋊C₈	#106	1	1	yes	characteristic
67	C₂²×C₈₃	C₂×#108	1	1	yes	characteristic
68	C₂×D₈₃	—	4	1	no	—
69	C₂×D₈₃	#334	1	2	yes	—
70	C₄×C₈₃	—	4	1	no	—
71	C₄×C₈₃	C₂×C₄⋊C₈	1	1	yes	characteristic
72	C₄×C₈₃	C₂×#106	1	1	yes	characteristic
73	Dic₈₃	—	2	2	no	—
74	Dic₈₃	#334	1	2	yes	—
75	#12002	—	83	1	no	maximal, Sylow
76	C₂×C₈₃	—	2	1	no	—
77	C₂×C₈₃	C₂×#334	1	1	yes	characteristic
78	D₈₃	—	2	2	no	—
79	C₂×#364	—	83	2	no	—
80	C₄×M₃₂	—	83	1	no	—
81	#1369	—	83	2	no	—
82	#1369	—	83	1	no	—
83	#1369	D₈₃	1	1	yes	characteristic, complemented
84	C₈₃	#12002	1	1	yes	characteristic, complemented, Sylow
85	C₂×M₃₂	—	83	1	no	—
86	C₂×M₃₂	—	83	1	no	—
87	C₂×M₃₂	—	166	2	no	—
88	C₂×C₄×C₈	—	83	1	no	—
89	#364	—	83	2	no	—
90	#364	—	166	2	no	—
91	#364	C₂×D₈₃	1	2	yes	—
92	#346	—	83	1	no	—
93	#346	C₂×D₈₃	1	1	yes	characteristic
94	C₄×C₁₆	—	83	1	no	—
95	C₄×C₁₆	—	83	1	no	—
96	C₂²×C₈	—	83	1	no	—
97	C₂²×C₈	—	83	1	no	—
98	C₂×C₄²	—	83	1	no	—
99	M₃₂	—	2	2	no	—
100	M₃₂	—	332	2	no	—
101	M₃₂	—	83	4	no	—
102	M₃₂	—	166	2	no	—
103	M₃₂	—	166	2	no	—
104	C₂×C₁₆	—	332	2	no	—
105	C₂×C₁₆	—	83	1	no	—
106	C₂×C₁₆	—	83	1	no	—
107	C₂×C₁₆	C₄×D₈₃	1	1	yes	characteristic
108	C₂×C₁₆	C₄×D₈₃	1	1	yes	characteristic
109	C₄×C₈	—	83	1	no	—
110	C₄×C₈	—	83	1	no	—
111	C₄×C₈	—	83	1	no	—
112	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
113	C₂²×C₄	—	83	1	no	—
114	C₂²×C₄	—	166	1	no	—
115	C₂×C₈	—	83	1	no	—
116	C₂×C₈	—	83	1	no	—
117	C₂×C₈	—	83	2	no	—
118	C₂×C₈	—	83	2	no	—
119	C₂×C₈	—	83	2	no	—
120	C₂×C₈	—	83	2	no	—
121	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
122	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
123	C₄²	—	83	1	no	—
124	C₄²	—	166	1	no	—
125	C₄²	Q₈×D₈₃	1	1	yes	characteristic
126	C₁₆	—	2	1	no	—
127	C₁₆	—	4	2	no	—
128	C₁₆	—	332	2	no	—
129	C₁₆	—	83	2	no	—
130	C₁₆	—	166	1	no	—
131	C₁₆	C₈×D₈₃	1	2	yes	—
132	C₂³	—	83	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	332	2	no	—
135	C₂×C₄	—	83	1	no	—
136	C₂×C₄	—	83	2	no	—
137	C₂×C₄	—	83	2	no	—
138	C₂×C₄	—	166	1	no	—
139	C₂×C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
140	C₈	—	83	2	no	—
141	C₈	—	83	1	no	—
142	C₈	—	83	1	no	—
143	C₈	C₂×C₈×D₈₃	1	1	yes	characteristic
144	C₈	M₁₆×D₈₃	1	1	yes	characteristic
145	C₈	SD₁₆×D₈₃	1	2	yes	—
146	C₂²	—	332	1	no	—
147	C₂²	—	83	2	no	—
148	C₂²	#108×D₈₃	1	1	yes	characteristic
149	C₄	—	4	1	no	—
150	C₄	—	83	2	no	—
151	C₄	—	166	2	no	—
152	C₄	C₄⋊C₈×D₈₃	1	1	yes	characteristic
153	C₄	#106×D₈₃	1	1	yes	characteristic
154	C₂	—	2	1	no	—
155	C₂	—	166	2	no	—
156	C₂	#334×D₈₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#364×D₈₃

C₂

yes

maximal

#24909056

C₂

yes

maximal, characteristic

#24905531

C₂

yes

maximal, characteristic, complemented

#24905530

C₂

yes

maximal, complemented

C₈₃×#1369

C₂

yes

maximal, characteristic, complemented

C₂×#14581731

C₄

yes

characteristic

C₂×#14581731

C₄

yes

characteristic

M₃₂×D₈₃

—

C₄×C₈×D₈₃

C₂²

yes

characteristic

#19127522

C₂²

yes

—

#19127521

—

#19127477

C₂²

yes

characteristic

C₁₆×Dic₈₃

C₄

yes

characteristic

C₁₆×Dic₈₃

C₄

yes

characteristic

#364×C₈₃

C₂²

yes

—

#346×C₈₃

C₂²

yes

characteristic

C₂×C₈×D₈₃

C₂×C₄

yes

characteristic

C₂×C₈×D₈₃

D₄

yes

characteristic

C₄²×D₈₃

Q₈

yes

—

—

—

C₈

yes

—

C₂×C₈₃⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₈₃⋊C₁₆

C₂×C₄

yes

characteristic

C₁₆×D₈₃

—

C₄×C₈₃⋊C₈

Q₈

yes

characteristic

C₈×Dic₈₃

C₂×C₄

yes

characteristic

C₈×Dic₈₃

D₄

yes

characteristic

M₃₂×C₈₃

—

C₂×C₁₆×C₈₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₃

C₂×C₄

yes

characteristic

C₄×C₈×C₈₃

C₂³

yes

characteristic

C₂×C₄×D₈₃

—

C₂×C₄×D₈₃

C₄⋊C₄

yes

characteristic

C₂×C₈₃⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₈₃⋊C₈

C₄⋊C₄

yes

characteristic

C₈×D₈₃

C₂×C₈

yes

—

C₈×D₈₃

M₁₆

yes

—

C₈×D₈₃

SD₁₆

yes

—

C₈×D₈₃

SD₁₆

yes

—

C₄×Dic₈₃

—

C₄×Dic₈₃

C₄⋊C₄

yes

characteristic

C₈₃⋊C₁₆

—

C₈₃⋊C₁₆

—

C₈₃⋊C₁₆

C₂×C₈

yes

—

C₂×C₈×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₃

C₂×D₄

yes

characteristic

C₄²×C₈₃

C₂×Q₈

yes

characteristic

C₁₆×C₈₃

—

C₁₆×C₈₃

—

C₁₆×C₈₃

C₂×C₈

yes

—

C₂²×D₈₃

#108

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

#108

yes

characteristic

C₄×D₈₃

—

C₄×D₈₃

C₄⋊C₈

yes

—

C₄×D₈₃

#106

yes

—

C₈×C₈₃

C₂²×C₈

yes

characteristic

C₈×C₈₃

C₂×M₁₆

yes

characteristic

C₈×C₈₃

C₂×SD₁₆

yes

—

C₈₃⋊C₈

C₄⋊C₈

yes

—

C₈₃⋊C₈

yes

C₈₃⋊C₈

yes

C₂²×C₈₃

yes

C₂×D₈₃

—

C₂×D₈₃

#334

yes

—

C₄×C₈₃

—

C₄×C₈₃

C₂×C₄⋊C₈

yes

characteristic

C₄×C₈₃

C₂×#106

yes

characteristic

Dic₈₃

—

Dic₈₃

#334

yes

—

#12002

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

C₂×#334

yes

characteristic

D₈₃

—

C₂×#364

—

C₄×M₃₂

—

#1369

—

#1369

—

#1369

D₈₃

yes

characteristic, complemented

C₈₃

#12002

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

—

C₂×M₃₂

—

166

—

C₂×C₄×C₈

—

#364

—

#364

—

166

—

#364

C₂×D₈₃

yes

—

#346

—

#346

C₂×D₈₃

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

100

M₃₂

—

332

—

101

M₃₂

—

102

M₃₂

—

166

—

103

M₃₂

—

166

—

104

C₂×C₁₆

—

332

—

105

C₂×C₁₆

—

106

C₂×C₁₆

—

107

C₂×C₁₆

C₄×D₈₃

yes

characteristic

108

C₂×C₁₆

C₄×D₈₃

yes

characteristic

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

C₂²×D₈₃

yes

characteristic

113

C₂²×C₄

—

114

C₂²×C₄

—

166

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

C₂×C₄×D₈₃

yes

characteristic

122

C₂×C₈

D₄×D₈₃

yes

characteristic

123

C₄²

—

124

C₄²

—

166

—

125

C₄²

Q₈×D₈₃

yes

characteristic

126

C₁₆

—

127

C₁₆

—

128

C₁₆

—

332

—

129

C₁₆

—

130

C₁₆

—

166

—

131

C₁₆

C₈×D₈₃

yes

—

132

C₂³

—

133

C₂×C₄

—

134

C₂×C₄

—

332

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

166

—

139

C₂×C₄

C₄⋊C₄×D₈₃

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

C₂×C₈×D₈₃

yes

characteristic

144

C₈

M₁₆×D₈₃

yes

characteristic

145

C₈

SD₁₆×D₈₃

yes

—

146

C₂²

—

332

—

147

C₂²

—

148

C₂²

#108×D₈₃

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

166

—

152

C₄

C₄⋊C₈×D₈₃

yes

characteristic

153

C₄

#106×D₈₃

yes

characteristic

154

C₂

—

155

C₂

—

166

—

156

C₂

#334×D₈₃

yes

characteristic

	subgroup	G/H
center	C₄	#106×D₈₃
commutator	C₈×C₈₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₈₃
Fitting	C₈₃×#1369	C₂
socle	C₂×C₈₃	C₂×#334
prime radical	C₂²×D₈₃	#108
cosocle	C₄×C₈×C₈₃	C₂³
hypercenter	#1369	D₈₃
hypocenter	C₈₃	#12002
LCS	G ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1369

subgroup

G/H

center

C₄

#106×D₈₃

commutator

C₈×C₈₃

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₈₃

Fitting

C₈₃×#1369

C₂

C₂×C₈₃

C₂²×D₈₃

C₄×C₈×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1369

N	H ≅ G/N	type
3. #24905531	C₂	semidirect
4. #24905530	C₂	semidirect
5. C₈₃×#1369	C₂	semidirect
83. #1369	D₈₃	semidirect
84. C₈₃	#12002	semidirect

H ≅ G/N

type

3. #24905531

C₂

semidirect

4. #24905530

C₂

semidirect

5. C₈₃×#1369

C₂

semidirect

83. #1369

D₈₃

semidirect

84. C₈₃

#12002

semidirect

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

subgroup	index
#364×D₈₃	2
#24909056	2
#24905531	2
#24905530	2
C₈₃×#1369	2
#12002	83

subgroup

index

#364×D₈₃

C₈₃×#1369

Subgroups — #35315967

Conjugacy classes of subgroups

Complements

Maximal subgroups