Group #35320786 — subgroups

Subgroups — #35320786

Order 21248.

310 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 155 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×C₄×C₈₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#24907627	C₂	1	1	yes	maximal, characteristic
3	#24905554	C₂	1	4	yes	maximal
4	C₈₃×#1846	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈₃⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#14581726	C₂²	1	2	yes	—
7	C₂×#14581724	—	2	1	no	—
8	#19127567	—	2	1	no	—
9	C₄×C₈₃⋊C₁₆	C₄	1	4	yes	complemented
10	C₂×#107×C₈₃	C₂²	1	2	yes	—
11	C₂×#106×C₈₃	—	2	1	no	—
12	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
13	#382×C₈₃	—	2	1	no	—
14	C₂×C₄⋊Dic₈₃	—	2	2	no	—
15	C₂×C₈₃⋊C₁₆	—	2	2	no	—
16	C₂×C₈₃⋊C₁₆	—	2	2	no	—
17	C₂×C₈₃⋊C₁₆	C₂×C₄	1	4	yes	—
18	#14581726	—	2	4	no	—
19	#14581724	—	4	2	no	—
20	C₈₃⋊#95	—	4	2	no	—
21	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
24	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
25	#107×C₈₃	—	2	4	no	—
26	#106×C₈₃	—	4	2	no	—
27	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₈₃	Q₈	1	2	yes	—
29	#95×C₈₃	—	4	2	no	—
30	C₂²×Dic₈₃	—	4	2	no	—
31	C₄⋊Dic₈₃	—	4	2	no	—
32	C₄⋊Dic₈₃	—	4	2	no	—
33	C₈₃⋊C₁₆	—	2	4	no	—
34	C₂²×C₄×C₈₃	—	4	2	no	—
35	C₂²×C₄×C₈₃	D₈	1	2	yes	—
36	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
38	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
39	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
40	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₈×C₈₃	#41	1	2	yes	—
43	C₂×C₈×C₈₃	#41	1	2	yes	—
44	C₄⋊C₄×C₈₃	—	4	2	no	—
45	C₄⋊C₄×C₈₃	—	4	2	no	—
46	C₄²×C₈₃	C₄⋊C₄	1	2	yes	—
47	C₄²×C₈₃	Q₁₆	1	2	yes	—
48	C₂³×C₈₃	C₂×D₈	1	1	yes	characteristic
49	C₂×C₄×C₈₃	—	2	4	no	—
50	C₂×C₄×C₈₃	—	8	2	no	—
51	C₂×C₄×C₈₃	—	8	4	no	—
52	C₂×C₄×C₈₃	#107	1	4	yes	—
53	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
55	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
56	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
58	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
59	C₂×Dic₈₃	—	8	2	no	—
60	C₂×Dic₈₃	—	8	4	no	—
61	C₈×C₈₃	#106	1	4	yes	—
62	C₈×C₈₃	#117	1	2	yes	—
63	C₈×C₈₃	#125	1	2	yes	—
64	C₂²×C₈₃	C₂×#107	1	1	yes	characteristic
65	C₂²×C₈₃	#486	1	1	yes	characteristic
66	C₂²×C₈₃	#486	1	1	yes	characteristic
67	C₂²×C₈₃	#508	1	4	yes	—
68	C₄×C₈₃	—	2	4	no	—
69	C₄×C₈₃	—	16	2	no	—
70	C₄×C₈₃	#382	1	1	yes	characteristic
71	C₄×C₈₃	C₂×#106	1	1	yes	characteristic
72	C₄×C₈₃	#487	1	1	yes	characteristic
73	C₄×C₈₃	#487	1	1	yes	characteristic
74	Dic₈₃	—	16	2	no	—
75	#12889	—	83	1	no	maximal, Sylow
76	C₂×C₈₃	#1834	1	1	yes	characteristic
77	C₂×C₈₃	#2158	1	2	yes	—
78	C₂×C₈₃	#2242	1	2	yes	—
79	C₂×C₈₃	#2242	1	2	yes	—
80	C₂×C₄×C₁₆	—	83	1	no	—
81	#1846	—	83	1	no	—
82	#1846	D₈₃	1	1	yes	characteristic
83	#1380	—	83	4	no	—
84	C₈₃	#12889	1	1	yes	characteristic, complemented, Sylow
85	C₂²×C₁₆	—	83	2	no	—
86	C₂×#107	—	83	2	no	—
87	C₂×#107	C₂×D₈₃	1	2	yes	—
88	C₂×#106	—	2	1	no	—
89	C₂×#106	—	166	1	no	—
90	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
91	#382	—	2	1	no	—
92	#382	—	166	1	no	—
93	C₄×C₁₆	—	83	4	no	—
94	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
95	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
96	C₂×C₄⋊C₄	—	2	2	no	—
97	C₂×C₄⋊C₄	—	166	2	no	—
98	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
99	C₂×C₁₆	—	83	4	no	—
100	C₂×C₁₆	—	166	2	no	—
101	C₂×C₁₆	—	166	2	no	—
102	#107	—	2	4	no	—
103	#107	—	166	4	no	—
104	#106	—	4	2	no	—
105	#106	—	332	2	no	—
106	C₄×C₈	Dic₁₆₆	1	2	yes	—
107	C₄×C₈	C₄×D₈₃	1	2	yes	—
108	#95	—	4	2	no	—
109	#95	—	332	2	no	—
110	C₂²×C₄	—	4	2	no	—
111	C₂²×C₄	—	332	2	no	—
112	C₂²×C₄	#13050575	1	2	yes	—
113	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
114	C₂×C₈	Dic₈₃⋊C₄	1	2	yes	—
115	C₂×C₈	#13050577	1	2	yes	—
116	C₂×C₈	#13050578	1	2	yes	—
117	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
118	C₂×C₈	C₂×Dic₁₆₆	1	1	yes	characteristic
119	C₂×C₈	#13050589	1	2	yes	—
120	C₂×C₈	#13050591	1	2	yes	—
121	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	—	332	2	no	—
124	C₄⋊C₄	—	332	2	no	—
125	C₄²	Dic₈₃⋊C₄	1	2	yes	—
126	C₄²	#13050580	1	2	yes	—
127	C₁₆	—	166	4	no	—
128	C₂³	C₂×#13050575	1	1	yes	characteristic
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	8	4	no	—
131	C₂×C₄	—	8	2	no	—
132	C₂×C₄	—	664	4	no	—
133	C₂×C₄	—	664	2	no	—
134	C₂×C₄	#14581725	1	4	yes	—
135	C₂×C₄	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
136	C₂×C₄	#14581779	1	1	yes	characteristic
137	C₂×C₄	#14581781	1	1	yes	characteristic
138	C₂×C₄	C₂×#13050577	1	1	yes	characteristic
139	C₂×C₄	C₂×#13050578	1	1	yes	characteristic
140	C₂×C₄	C₂×#13050580	1	1	yes	characteristic
141	C₈	#14581723	1	4	yes	—
142	C₈	#14581750	1	2	yes	—
143	C₈	#14581783	1	2	yes	—
144	C₂²	C₂×#14581725	1	1	yes	characteristic
145	C₂²	#19128047	1	1	yes	characteristic
146	C₂²	#19128049	1	1	yes	characteristic
147	C₂²	#19128131	1	4	yes	—
148	C₄	—	2	4	no	—
149	C₄	—	16	2	no	—
150	C₄	—	1328	2	no	—
151	C₄	#19127566	1	1	yes	characteristic
152	C₄	C₂×#14581723	1	1	yes	characteristic
153	C₄	#19128051	1	1	yes	characteristic
154	C₄	#19128053	1	1	yes	characteristic
155	C₂	#24907590	1	1	yes	characteristic
156	C₂	#24909245	1	2	yes	—
157	C₂	#24909694	1	2	yes	—
158	C₂	#24909697	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₈₃⋊C₁₆

C₂

yes

maximal, characteristic

#24907627

C₂

yes

maximal, characteristic

#24905554

C₂

yes

maximal

C₈₃×#1846

C₂

yes

maximal, characteristic

C₂²×C₈₃⋊C₁₆

C₂²

yes

—

C₂×#14581726

C₂²

yes

—

C₂×#14581724

—

#19127567

—

C₄×C₈₃⋊C₁₆

C₄

yes

complemented

C₂×#107×C₈₃

C₂²

yes

—

C₂×#106×C₈₃

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

#382×C₈₃

—

C₂×C₄⋊Dic₈₃

—

C₂×C₈₃⋊C₁₆

—

C₂×C₈₃⋊C₁₆

—

C₂×C₈₃⋊C₁₆

C₂×C₄

yes

—

#14581726

—

#14581724

—

C₈₃⋊#95

—

C₂²×C₈×C₈₃

D₄

yes

characteristic

C₂²×C₈×C₈₃

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄²×C₈₃

D₄

yes

characteristic

#107×C₈₃

—

#106×C₈₃

—

C₄×C₈×C₈₃

C₂×C₄

yes

—

C₄×C₈×C₈₃

Q₈

yes

—

#95×C₈₃

—

C₂²×Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₈₃⋊C₁₆

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

D₈

yes

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₈×C₈₃

C₄⋊C₄

yes

—

C₂×C₈×C₈₃

SD₁₆

yes

—

C₂×C₈×C₈₃

SD₁₆

yes

—

C₂×C₈×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₈₃

#41

yes

—

C₂×C₈×C₈₃

#41

yes

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

C₄⋊C₄

yes

—

C₄²×C₈₃

Q₁₆

yes

—

C₂³×C₈₃

C₂×D₈

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

#107

yes

—

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₈₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₈₃

C₂×Q₁₆

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₈×C₈₃

#106

yes

—

C₈×C₈₃

#117

yes

—

C₈×C₈₃

#125

yes

—

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

Dic₈₃

—

—

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₂×C₈₃

#2242

yes

—

C₂×C₈₃

#2242

yes

—

C₂×C₄×C₁₆

—

#1846

—

#1846

D₈₃

yes

characteristic

#1380

—

C₈₃

#12889

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂×#107

—

C₂×#107

C₂×D₈₃

yes

—

C₂×#106

—

C₂×#106

—

166

—

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

#382

—

#382

—

166

—

C₄×C₁₆

—

C₂²×C₈

C₂²⋊D₈₃

yes

characteristic

C₂²×C₈

C₂²×D₈₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

166

—

C₂×C₄²

C₂²⋊D₈₃

yes

characteristic

C₂×C₁₆

—

100

C₂×C₁₆

—

166

—

101

C₂×C₁₆

—

166

—

102

#107

—

103

#107

—

166

—

104

#106

—

105

#106

—

332

—

106

C₄×C₈

Dic₁₆₆

yes

—

107

C₄×C₈

C₄×D₈₃

yes

—

108

#95

—

109

#95

—

332

—

110

C₂²×C₄

—

111

C₂²×C₄

—

332

—

112

C₂²×C₄

#13050575

yes

—

113

C₂²×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

114

C₂×C₈

Dic₈₃⋊C₄

yes

—

115

C₂×C₈

#13050577

yes

—

116

C₂×C₈

#13050578

yes

—

117

C₂×C₈

C₂×C₄×D₈₃

yes

characteristic

118

C₂×C₈

C₂×Dic₁₆₆

yes

characteristic

119

C₂×C₈

#13050589

yes

—

120

C₂×C₈

#13050591

yes

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

332

—

124

C₄⋊C₄

—

332

—

125

C₄²

Dic₈₃⋊C₄

yes

—

126

C₄²

#13050580

yes

—

127

C₁₆

—

166

—

128

C₂³

C₂×#13050575

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

664

—

133

C₂×C₄

—

664

—

134

C₂×C₄

#14581725

yes

—

135

C₂×C₄

C₂×Dic₈₃⋊C₄

yes

characteristic

136

C₂×C₄

#14581779

yes

characteristic

137

C₂×C₄

#14581781

yes

characteristic

138

C₂×C₄

C₂×#13050577

yes

characteristic

139

C₂×C₄

C₂×#13050578

yes

characteristic

140

C₂×C₄

C₂×#13050580

yes

characteristic

141

C₈

#14581723

yes

—

142

C₈

#14581750

yes

—

143

C₈

#14581783

yes

—

144

C₂²

C₂×#14581725

yes

characteristic

145

C₂²

#19128047

yes

characteristic

146

C₂²

#19128049

yes

characteristic

147

C₂²

#19128131

yes

—

148

C₄

—

149

C₄

—

150

C₄

—

1328

—

151

C₄

#19127566

yes

characteristic

152

C₄

C₂×#14581723

yes

characteristic

153

C₄

#19128051

yes

characteristic

154

C₄

#19128053

yes

characteristic

155

C₂

#24907590

yes

characteristic

156

C₂

#24909245

yes

—

157

C₂

#24909694

yes

—

158

C₂

#24909697

yes

—

	subgroup	G/H
center	C₂³	C₂×#13050575
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#1846	C₂
socle, prime radical	C₂³×C₈₃	C₂×D₈
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	#1846	D₈₃
hypocenter	C₈₃	#12889
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

subgroup

G/H

center

C₂³

C₂×#13050575

commutator

C₂×C₈×C₈₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₈₃

Fitting

C₈₃×#1846

C₂

socle, prime radical

C₂³×C₈₃

C₂×D₈

cosocle

C₂²×C₈×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

N	H ≅ G/N	type
9. C₄×C₈₃⋊C₁₆	C₄	semidirect
84. C₈₃	#12889	semidirect

H ≅ G/N

type

9. C₄×C₈₃⋊C₁₆

C₄

semidirect

84. C₈₃

#12889

semidirect

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

subgroup	index
C₂×C₄×C₈₃⋊C₁₆	2
#24907627	2
#24905554	2
C₈₃×#1846	2
#12889	83

subgroup

index

C₂×C₄×C₈₃⋊C₁₆

#24907627

#24905554

C₈₃×#1846

#12889

Subgroups — #35320786

Conjugacy classes of subgroups

Complements

Maximal subgroups