Group #35320770 — subgroups

Subgroups — #35320770

Order 21248.

234 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 103 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	#24907218	C₂	1	1	yes	maximal, characteristic
3	#24905551	C₂	1	2	yes	maximal
4	#24905550	C₂	1	2	yes	maximal
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	C₈₃⋊#338	C₄	1	1	yes	characteristic
10	#19127458	C₄	1	1	yes	characteristic
11	#19127457	C₄	1	2	yes	—
12	C₂×M₃₂×C₈₃	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
14	#405×C₈₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₁₆×C₈₃	—	2	2	no	—
16	C₂×#13050574	C₂×C₄	1	2	yes	—
17	#14581751	C₂×C₄	1	2	yes	—
18	#14581732	—	2	2	no	—
19	#14581732	—	2	2	no	—
20	C₂×C₈₃⋊C₁₆	—	2	2	no	—
21	#14581702	—	2	2	no	—
22	C₈×Dic₈₃	—	2	2	no	—
23	C₂×M₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
25	#117×C₈₃	C₂×C₄	1	1	yes	characteristic
26	M₃₂×C₈₃	—	2	2	no	—
27	C₂×C₁₆×C₈₃	—	2	1	no	—
28	C₂×C₁₆×C₈₃	—	2	1	no	—
29	C₂×C₁₆×C₈₃	—	2	2	no	—
30	#97×C₈₃	Q₈	1	2	yes	—
31	C₄×C₈×C₈₃	D₄	1	2	yes	—
32	#13050574	—	2	2	no	—
33	#13050574	—	2	2	no	—
34	C₂×C₈₃⋊C₈	—	2	2	no	—
35	C₄⋊Dic₈₃	—	2	2	no	—
36	C₂²⋊Dic₈₃	—	2	2	no	—
37	C₄×Dic₈₃	—	2	2	no	—
38	C₈₃⋊C₁₆	—	4	2	no	—
39	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
40	M₁₆×C₈₃	—	2	2	no	—
41	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
42	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₈₃	#41	1	2	yes	—
45	C₂×C₈×C₈₃	#41	1	2	yes	—
46	C₄⋊C₄×C₈₃	—	2	1	no	—
47	C₂²⋊C₄×C₈₃	—	2	1	no	—
48	C₄²×C₈₃	C₄⋊C₄	1	2	yes	—
49	C₁₆×C₈₃	—	4	1	no	—
50	C₁₆×C₈₃	—	4	1	no	—
51	C₂³×C₈₃	#131	1	1	yes	characteristic
52	C₂×C₄×C₈₃	—	2	2	no	—
53	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₈₃	#117	1	2	yes	—
55	C₂×C₄×C₈₃	#131	1	1	yes	characteristic
56	C₂×C₄×C₈₃	#131	1	2	yes	—
57	C₂×Dic₈₃	—	4	2	no	—
58	C₈×C₈₃	—	2	2	no	—
59	C₈×C₈₃	#124	1	1	yes	characteristic
60	C₈×C₈₃	#124	1	1	yes	characteristic
61	C₈×C₈₃	#125	1	2	yes	—
62	C₈₃⋊C₈	—	4	2	no	—
63	C₂²×C₈₃	—	4	1	no	—
64	C₂²×C₈₃	#424	1	1	yes	characteristic
65	C₂²×C₈₃	#433	1	2	yes	—
66	C₄×C₈₃	—	4	2	no	—
67	C₄×C₈₃	#382	1	1	yes	characteristic
68	C₄×C₈₃	#424	1	1	yes	characteristic
69	C₄×C₈₃	#433	1	2	yes	—
70	Dic₈₃	—	8	2	no	—
71	#12885	—	83	1	no	maximal, Sylow
72	C₂×C₈₃	—	2	2	no	—
73	C₂×C₈₃	—	2	1	no	—
74	C₂×C₈₃	#1832	1	1	yes	characteristic
75	#2153	—	83	1	no	—
76	#2151	D₈₃	1	1	yes	characteristic
77	#1747	—	83	1	no	—
78	#1379	—	83	2	no	—
79	#1378	—	83	2	no	—
80	C₈₃	#12885	1	1	yes	characteristic, complemented, Sylow
81	C₂×M₃₂	—	83	2	no	—
82	C₂×M₃₂	C₂×D₈₃	1	1	yes	characteristic
83	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
84	#405	—	83	2	no	—
85	#405	C₂×D₈₃	1	1	yes	characteristic
86	#364	—	166	2	no	—
87	C₄⋊C₁₆	—	2	2	no	—
88	#338	—	83	1	no	—
89	H(3,ℤ/4)	—	83	1	no	—
90	H(3,ℤ/4)	—	83	2	no	—
91	C₂×M₁₆	—	83	2	no	—
92	C₂×M₁₆	C₄×D₈₃	1	1	yes	characteristic
93	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
94	#117	—	83	2	no	—
95	#117	C₄×D₈₃	1	1	yes	characteristic
96	M₃₂	—	2	2	no	—
97	M₃₂	—	166	2	no	—
98	M₃₂	—	166	2	no	—
99	C₂×C₁₆	—	2	1	no	—
100	C₂×C₁₆	—	2	1	no	—
101	C₂×C₁₆	—	2	2	no	—
102	C₂×C₁₆	—	166	2	no	—
103	#97	—	166	2	no	—
104	#97	Dic₁₆₆	1	2	yes	—
105	C₄×C₈	—	166	2	no	—
106	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
107	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
108	M₁₆	—	2	2	no	—
109	M₁₆	—	166	2	no	—
110	M₁₆	—	166	2	no	—
111	C₂×C₈	—	166	2	no	—
112	C₂×C₈	Dic₈₃⋊C₄	1	2	yes	—
113	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
114	C₂×C₈	C₂×Dic₁₆₆	1	1	yes	characteristic
115	C₂×C₈	#13050589	1	2	yes	—
116	C₂×C₈	#13050591	1	2	yes	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	166	2	no	—
119	C₂²⋊C₄	—	2	1	no	—
120	C₂²⋊C₄	—	166	2	no	—
121	C₄²	—	166	2	no	—
122	C₄²	Dic₈₃⋊C₄	1	2	yes	—
123	C₁₆	—	4	1	no	—
124	C₁₆	—	4	1	no	—
125	C₁₆	—	332	2	no	—
126	C₂³	#14581798	1	1	yes	characteristic
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	332	2	no	—
129	C₂×C₄	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
130	C₂×C₄	#14581750	1	2	yes	—
131	C₂×C₄	#14581798	1	1	yes	characteristic
132	C₂×C₄	#14581800	1	2	yes	—
133	C₈	—	2	2	no	—
134	C₈	—	332	2	no	—
135	C₈	#14581779	1	1	yes	characteristic
136	C₈	#14581781	1	1	yes	characteristic
137	C₈	#14581783	1	2	yes	—
138	C₂²	—	4	1	no	—
139	C₂²	#19127727	1	1	yes	characteristic
140	C₂²	#19127761	1	2	yes	—
141	C₄	—	4	2	no	—
142	C₄	—	664	2	no	—
143	C₄	#19127566	1	1	yes	characteristic
144	C₄	#19127729	1	1	yes	characteristic
145	C₄	#19127761	1	2	yes	—
146	C₂	—	2	1	no	—
147	C₂	—	2	2	no	—
148	C₂	#24907582	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24909229

C₂

yes

maximal, characteristic

#24907218

C₂

yes

maximal, characteristic

#24905551

C₂

yes

maximal

#24905550

C₂

yes

maximal

C₈₃×#2151

C₂

yes

maximal, characteristic

C₂×#14581732

C₂²

yes

—

#19127656

C₂²

yes

—

#19127522

—

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

—

C₈×Dic₈₃

—

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⋊Dic₈₃

—

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

—

C₈₃⋊C₁₆

—

C₂²×C₄×C₈₃

C₂²×C₄

yes

characteristic

M₁₆×C₈₃

—

C₂×C₈×C₈₃

C₄⋊C₄

yes

—

C₂×C₈×C₈₃

C₂×D₄

yes

characteristic

C₂×C₈×C₈₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₈₃

#41

yes

—

C₂×C₈×C₈₃

#41

yes

—

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

characteristic

C₂×C₄×C₈₃

#117

yes

—

C₂×C₄×C₈₃

#131

yes

characteristic

C₂×C₄×C₈₃

#131

yes

—

C₂×Dic₈₃

—

C₈×C₈₃

—

C₈×C₈₃

yes

C₈×C₈₃

yes

C₈×C₈₃

yes

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

#424

yes

characteristic

C₂²×C₈₃

#433

yes

—

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

—

Dic₈₃

—

#12885

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

yes

—

D₈₃

yes

—

—

—

C₈₃

#12885

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

—

C₄⋊C₁₆

—

#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

M₃₂

—

M₃₂

—

166

—

M₃₂

—

166

—

C₂×C₁₆

—

100

C₂×C₁₆

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

166

—

103

#97

—

166

—

104

#97

Dic₁₆₆

yes

—

105

C₄×C₈

—

166

—

106

C₄×C₈

C₂²⋊D₈₃

yes

—

107

C₂²×C₄

C₂×C₄×D₈₃

yes

characteristic

108

M₁₆

—

109

M₁₆

—

166

—

110

M₁₆

—

166

—

111

C₂×C₈

—

166

—

112

C₂×C₈

Dic₈₃⋊C₄

yes

—

113

C₂×C₈

C₂×C₂²⋊D₈₃

yes

characteristic

114

C₂×C₈

C₂×Dic₁₆₆

yes

characteristic

115

C₂×C₈

#13050589

yes

—

116

C₂×C₈

#13050591

yes

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

166

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

166

—

121

C₄²

—

166

—

122

C₄²

Dic₈₃⋊C₄

yes

—

123

C₁₆

—

124

C₁₆

—

125

C₁₆

—

332

—

126

C₂³

#14581798

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

332

—

129

C₂×C₄

C₂×Dic₈₃⋊C₄

yes

characteristic

130

C₂×C₄

#14581750

yes

—

131

C₂×C₄

#14581798

yes

characteristic

132

C₂×C₄

#14581800

yes

—

133

C₈

—

134

C₈

—

332

—

135

C₈

#14581779

yes

characteristic

136

C₈

#14581781

yes

characteristic

137

C₈

#14581783

yes

—

138

C₂²

—

139

C₂²

#19127727

yes

characteristic

140

C₂²

#19127761

yes

—

141

C₄

—

142

C₄

—

664

—

143

C₄

#19127566

yes

characteristic

144

C₄

#19127729

yes

characteristic

145

C₄

#19127761

yes

—

146

C₂

—

147

C₂

—

148

C₂

#24907582

yes

characteristic

	subgroup	G/H
center	C₈	#14581781
commutator	C₂²×C₄×C₈₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#2151	C₂
socle	C₂×C₈₃	#1832
prime radical	C₂³×C₈₃	#131
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	#2151	D₈₃
hypocenter	C₈₃	#12885
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2151

subgroup

G/H

center

C₈

#14581781

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
80. C₈₃	#12885	semidirect

H ≅ G/N

type

80. C₈₃

#12885

semidirect

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

subgroup	index
#24909229	2
#24907218	2
#24905551	2
#24905550	2
C₈₃×#2151	2
#12885	83

subgroup

index

C₈₃×#2151

Subgroups — #35320770

Conjugacy classes of subgroups

Complements

Maximal subgroups