Group #35304252 — subgroups

Subgroups — #35304252

Order 21248.

542 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 191 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×#19128081	C₂	1	1	yes	maximal, characteristic
2	#24908423	C₂	1	4	yes	maximal
3	C₂×#19127422	C₂	1	1	yes	maximal, characteristic
4	C₈₃×#1844	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₃₂	—	2	2	no	—
6	C₂×#14581792	C₂²	1	2	yes	—
7	#19128081	—	2	2	no	—
8	#19128081	C₄	1	4	yes	complemented
9	C₂×C₄×C₈₃⋊C₈	C₂²	1	2	yes	—
10	#19127463	—	2	8	no	—
11	#19127422	—	2	2	no	—
12	C₂×#106×C₈₃	—	2	2	no	—
13	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
14	#384×C₈₃	C₂²	1	2	yes	—
15	C₂²×Dic₁₆₆	—	2	2	no	—
16	C₂×Dic₃₃₂	—	2	8	no	—
17	C₂×Dic₃₃₂	—	4	4	no	—
18	C₂×Dic₃₃₂	—	4	4	no	—
19	C₂²×C₈₃⋊C₈	D₄	1	4	yes	—
20	#14581792	—	2	2	no	—
21	#14581792	—	2	4	no	—
22	#14581792	C₂×C₄	1	4	yes	—
23	C₂×C₄⋊Dic₈₃	—	2	4	no	—
24	C₄×C₈₃⋊C₈	—	2	2	no	—
25	C₄×C₈₃⋊C₈	—	2	2	no	—
26	C₂²×C₈×C₈₃	D₄	1	2	yes	—
27	C₂×C₄⋊C₄×C₈₃	—	2	4	no	—
28	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
29	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
30	#106×C₈₃	—	4	4	no	—
31	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₈₃	D₄	1	2	yes	—
33	C₂×Dic₁₆₆	—	2	4	no	—
34	C₂×Dic₁₆₆	—	4	4	no	—
35	C₂×Dic₁₆₆	—	4	4	no	—
36	C₂×Dic₁₆₆	—	4	2	no	—
37	C₂²×Dic₈₃	—	4	2	no	—
38	Dic₃₃₂	—	4	16	no	—
39	C₂×C₈₃⋊C₈	—	2	4	no	—
40	C₂×C₈₃⋊C₈	—	2	8	no	—
41	C₄⋊Dic₈₃	—	2	8	no	—
42	C₄⋊Dic₈₃	—	4	4	no	—
43	C₂²×C₄×C₈₃	—	4	2	no	—
44	C₂²×C₄×C₈₃	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₈₃	C₂²⋊C₄	1	4	yes	—
47	C₂×C₈×C₈₃	D₈	1	4	yes	—
48	C₂×C₈×C₈₃	SD₁₆	1	4	yes	—
49	C₄⋊C₄×C₈₃	—	4	4	no	—
50	C₄⋊C₄×C₈₃	—	4	8	no	—
51	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₈₃	#41	1	2	yes	—
54	C₂³×C₈₃	#127	1	1	yes	characteristic
55	C₂×C₄×C₈₃	—	8	2	no	—
56	C₂×C₄×C₈₃	—	8	4	no	—
57	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₈₃	C₄×D₄	1	2	yes	—
59	C₂×C₄×C₈₃	#121	1	2	yes	—
60	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
61	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
62	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
63	C₂×C₄×C₈₃	#137	1	8	yes	—
64	C₂×Dic₈₃	—	4	4	no	—
65	C₂×Dic₈₃	—	8	2	no	—
66	C₂×Dic₈₃	—	8	2	no	—
67	Dic₁₆₆	—	4	8	no	—
68	Dic₁₆₆	—	8	4	no	—
69	C₈×C₈₃	#102	1	8	yes	—
70	C₈₃⋊C₈	—	4	4	no	—
71	C₂²×C₈₃	#390	1	1	yes	characteristic
72	C₂²×C₈₃	#492	1	1	yes	characteristic
73	C₂²×C₈₃	#493	1	1	yes	characteristic
74	C₂²×C₈₃	#497	1	4	yes	—
75	C₄×C₈₃	—	16	2	no	—
76	C₄×C₈₃	C₂×#102	1	2	yes	—
77	C₄×C₈₃	#441	1	4	yes	—
78	C₄×C₈₃	#470	1	4	yes	—
79	C₄×C₈₃	#486	1	2	yes	—
80	Dic₈₃	—	8	4	no	—
81	#9839	—	83	1	no	maximal, Sylow
82	C₂×C₈₃	#1716	1	2	yes	—
83	C₂×C₈₃	#1722	1	2	yes	—
84	C₂×C₈₃	#1957	1	1	yes	characteristic
85	C₂×C₈₃	#1971	1	2	yes	—
86	C₂×#494	—	83	1	no	—
87	#1990	—	83	4	no	—
88	#1844	D₈₃	1	1	yes	characteristic
89	C₂×#322	—	83	1	no	—
90	C₈₃	#9839	1	1	yes	characteristic, complemented, Sylow
91	C₂²×Q₁₆	—	166	2	no	—
92	C₂×C₄⋊Q₈	—	83	2	no	—
93	#494	—	83	4	no	—
94	#494	—	166	2	no	—
95	C₂×#106	—	2	2	no	—
96	C₂×C₄×C₈	—	83	2	no	—
97	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
98	#384	C₂×D₈₃	1	2	yes	—
99	#340	—	166	8	no	—
100	#322	—	166	2	no	—
101	C₂²×Q₈	—	166	2	no	—
102	C₂×Q₁₆	—	332	4	no	—
103	C₂×Q₁₆	—	332	4	no	—
104	C₂×Q₁₆	—	166	8	no	—
105	C₂²×C₈	—	83	4	no	—
106	C₂²×C₈	C₂²⋊D₈₃	1	2	yes	—
107	C₄⋊Q₈	—	83	4	no	—
108	C₄⋊Q₈	—	166	2	no	—
109	C₄⋊Q₈	—	166	4	no	—
110	C₂×C₄⋊C₄	—	2	4	no	—
111	C₂×C₄⋊C₄	—	2	2	no	—
112	C₂×C₄⋊C₄	—	166	4	no	—
113	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
114	#106	—	4	4	no	—
115	C₄×C₈	—	166	2	no	—
116	C₄×C₈	—	166	2	no	—
117	C₄×C₈	C₄×D₈₃	1	2	yes	—
118	C₄×C₈	D₃₃₂	1	2	yes	—
119	C₂×Q₈	—	332	2	no	—
120	C₂×Q₈	—	332	4	no	—
121	C₂×Q₈	—	332	4	no	—
122	C₂×Q₈	—	166	4	no	—
123	C₂²×C₄	—	4	2	no	—
124	C₂²×C₄	—	332	2	no	—
125	C₂²×C₄	D₄×D₈₃	1	2	yes	—
126	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
127	Q₁₆	—	332	16	no	—
128	C₂×C₈	—	166	4	no	—
129	C₂×C₈	—	166	8	no	—
130	C₂×C₈	(C₂×D₈₃)⋊C₄	1	4	yes	—
131	C₂×C₈	#13050575	1	4	yes	—
132	C₂×C₈	#13050577	1	4	yes	—
133	C₄⋊C₄	—	4	4	no	—
134	C₄⋊C₄	—	4	8	no	—
135	C₄⋊C₄	—	332	4	no	—
136	C₄⋊C₄	—	166	8	no	—
137	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
138	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
139	C₄²	#13050591	1	2	yes	—
140	C₂³	#14581788	1	1	yes	characteristic
141	Q₈	—	332	8	no	—
142	Q₈	—	664	4	no	—
143	C₂×C₄	—	8	2	no	—
144	C₂×C₄	—	8	4	no	—
145	C₂×C₄	—	332	4	no	—
146	C₂×C₄	—	664	2	no	—
147	C₂×C₄	—	664	2	no	—
148	C₂×C₄	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
149	C₂×C₄	#14581754	1	2	yes	—
150	C₂×C₄	#14581767	1	2	yes	—
151	C₂×C₄	#14581781	1	1	yes	characteristic
152	C₂×C₄	C₂×#13050575	1	2	yes	—
153	C₂×C₄	C₂×#13050577	1	2	yes	—
154	C₂×C₄	#14581821	1	8	yes	—
155	C₈	—	332	4	no	—
156	C₈	#14581712	1	8	yes	—
157	C₂²	#19127602	1	1	yes	characteristic
158	C₂²	#19128075	1	1	yes	characteristic
159	C₂²	#19128077	1	1	yes	characteristic
160	C₂²	#19128092	1	4	yes	—
161	C₄	—	16	2	no	—
162	C₄	—	664	4	no	—
163	C₄	C₂×#14581712	1	2	yes	—
164	C₄	#19127807	1	4	yes	—
165	C₄	#19127960	1	4	yes	—
166	C₄	#19128049	1	2	yes	—
167	C₂	#24907097	1	2	yes	—
168	C₂	#24907132	1	2	yes	—
169	C₂	#24908268	1	1	yes	characteristic
170	C₂	#24908332	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19128081

C₂

yes

maximal, characteristic

#24908423

C₂

yes

maximal

C₂×#19127422

C₂

yes

maximal, characteristic

C₈₃×#1844

C₂

yes

maximal, characteristic

C₂²×Dic₃₃₂

—

C₂×#14581792

C₂²

yes

—

#19128081

—

#19128081

C₄

yes

complemented

C₂×C₄×C₈₃⋊C₈

C₂²

yes

—

#19127463

—

#19127422

—

C₂×#106×C₈₃

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

#384×C₈₃

C₂²

yes

—

C₂²×Dic₁₆₆

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂²×C₈₃⋊C₈

D₄

yes

—

#14581792

—

#14581792

—

#14581792

C₂×C₄

yes

—

C₂×C₄⋊Dic₈₃

—

C₄×C₈₃⋊C₈

—

C₄×C₈₃⋊C₈

—

C₂²×C₈×C₈₃

D₄

yes

—

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄²×C₈₃

C₂³

yes

characteristic

#106×C₈₃

—

C₄×C₈×C₈₃

C₂×C₄

yes

—

C₄×C₈×C₈₃

D₄

yes

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂²×Dic₈₃

—

Dic₃₃₂

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂×D₄

yes

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₈×C₈₃

C₂²⋊C₄

yes

—

C₂×C₈×C₈₃

D₈

yes

—

C₂×C₈×C₈₃

SD₁₆

yes

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

C₂²×C₄

yes

characteristic

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₄²×C₈₃

#41

yes

—

C₂³×C₈₃

#127

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₄×D₄

yes

—

C₂×C₄×C₈₃

#121

yes

—

C₂×C₄×C₈₃

#124

yes

characteristic

C₂×C₄×C₈₃

C₂×D₈

yes

—

C₂×C₄×C₈₃

C₂×SD₁₆

yes

—

C₂×C₄×C₈₃

#137

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

#102

yes

—

C₈₃⋊C₈

—

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

—

C₄×C₈₃

—

C₄×C₈₃

C₂×#102

yes

—

C₄×C₈₃

#441

yes

—

C₄×C₈₃

#470

yes

—

C₄×C₈₃

#486

yes

—

Dic₈₃

—

#9839

—

maximal, Sylow

C₂×C₈₃

#1716

yes

—

C₂×C₈₃

#1722

yes

—

C₂×C₈₃

#1957

yes

characteristic

C₂×C₈₃

#1971

yes

—

C₂×#494

—

#1990

—

#1844

D₈₃

yes

characteristic

C₂×#322

—

C₈₃

#9839

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

166

—

C₂×C₄⋊Q₈

—

#494

—

#494

—

166

—

C₂×#106

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

#384

C₂×D₈₃

yes

—

#340

—

166

—

100

#322

—

166

—

101

C₂²×Q₈

—

166

—

102

C₂×Q₁₆

—

332

—

103

C₂×Q₁₆

—

332

—

104

C₂×Q₁₆

—

166

—

105

C₂²×C₈

—

106

C₂²×C₈

C₂²⋊D₈₃

yes

—

107

C₄⋊Q₈

—

108

C₄⋊Q₈

—

166

—

109

C₄⋊Q₈

—

166

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

166

—

113

C₂×C₄²

C₂²×D₈₃

yes

characteristic

114

#106

—

115

C₄×C₈

—

166

—

116

C₄×C₈

—

166

—

117

C₄×C₈

C₄×D₈₃

yes

—

118

C₄×C₈

D₃₃₂

yes

—

119

C₂×Q₈

—

332

—

120

C₂×Q₈

—

332

—

121

C₂×Q₈

—

332

—

122

C₂×Q₈

—

166

—

123

C₂²×C₄

—

124

C₂²×C₄

—

332

—

125

C₂²×C₄

D₄×D₈₃

yes

—

126

C₂²×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

127

Q₁₆

—

332

—

128

C₂×C₈

—

166

—

129

C₂×C₈

—

166

—

130

C₂×C₈

(C₂×D₈₃)⋊C₄

yes

—

131

C₂×C₈

#13050575

yes

—

132

C₂×C₈

#13050577

yes

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

332

—

136

C₄⋊C₄

—

166

—

137

C₄²

C₂×C₄×D₈₃

yes

characteristic

138

C₄²

C₂×D₃₃₂

yes

characteristic

139

C₄²

#13050591

yes

—

140

C₂³

#14581788

yes

characteristic

141

Q₈

—

332

—

142

Q₈

—

664

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

332

—

146

C₂×C₄

—

664

—

147

C₂×C₄

—

664

—

148

C₂×C₄

C₂×(C₂×D₈₃)⋊C₄

yes

characteristic

149

C₂×C₄

#14581754

yes

—

150

C₂×C₄

#14581767

yes

—

151

C₂×C₄

#14581781

yes

characteristic

152

C₂×C₄

C₂×#13050575

yes

—

153

C₂×C₄

C₂×#13050577

yes

—

154

C₂×C₄

#14581821

yes

—

155

C₈

—

332

—

156

C₈

#14581712

yes

—

157

C₂²

#19127602

yes

characteristic

158

C₂²

#19128075

yes

characteristic

159

C₂²

#19128077

yes

characteristic

160

C₂²

#19128092

yes

—

161

C₄

—

162

C₄

—

664

—

163

C₄

C₂×#14581712

yes

—

164

C₄

#19127807

yes

—

165

C₄

#19127960

yes

—

166

C₄

#19128049

yes

—

167

C₂

#24907097

yes

—

168

C₂

#24907132

yes

—

169

C₂

#24908268

yes

characteristic

170

C₂

#24908332

yes

—

	subgroup	G/H
center	C₂³	#14581788
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1844	C₂
socle, prime radical	C₂³×C₈₃	#127
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1844	D₈₃
hypocenter	C₈₃	#9839
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

subgroup

G/H

center

C₂³

#14581788

commutator

C₄²×C₈₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₈₃×#1844

C₂

socle, prime radical

C₂³×C₈₃

#127

cosocle

C₂×C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

N	H ≅ G/N	type
8. #19128081	C₄	semidirect
90. C₈₃	#9839	semidirect

H ≅ G/N

type

8. #19128081

C₄

semidirect

90. C₈₃

#9839

semidirect

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

subgroup	index
C₂×#19128081	2
#24908423	2
C₂×#19127422	2
C₈₃×#1844	2
#9839	83

subgroup

index

C₈₃×#1844

Subgroups — #35304252

Conjugacy classes of subgroups

Complements

Maximal subgroups