Group #35318764 — subgroups

Subgroups — #35318764

Order 21248.

204 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 69 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	#24909738	C₂	1	1	yes	maximal, characteristic
2	C₁₆×Dic₁₆₆	C₂	1	1	yes	maximal, characteristic
3	#24906858	C₂	1	2	yes	maximal
4	#24905519	C₂	1	2	yes	maximal
5	C₈₃×#1371	C₂	1	1	yes	maximal, characteristic
6	#19128106	C₂²	1	2	yes	—
7	#19127912	—	2	2	no	—
8	C₈×Dic₁₆₆	C₂²	1	1	yes	characteristic
9	#19127527	—	2	1	no	—
10	#19127527	—	2	1	no	—
11	C₄⋊(C₈₃⋊C₁₆)	C₂²	1	1	yes	characteristic
12	Dic₈₃⋊C₁₆	—	2	1	no	—
13	C₁₆×Dic₈₃	—	2	1	no	—
14	#19127448	—	2	2	no	—
15	C₄×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
16	#335×C₈₃	C₂²	1	2	yes	—
17	C₂×Dic₃₃₂	—	4	2	no	—
18	#14581792	—	2	2	no	—
19	C₄×Dic₁₆₆	D₄	1	1	yes	characteristic
20	C₂×C₈₃⋊C₁₆	—	2	1	no	—
21	#14581726	—	2	2	no	—
22	#14581726	—	2	2	no	—
23	C₄⋊(C₈₃⋊C₈)	D₄	1	1	yes	characteristic
24	Dic₈₃⋊C₈	—	2	1	no	—
25	#14581717	—	4	2	no	—
26	C₈×Dic₈₃	—	2	1	no	—
27	C₂×C₁₆×C₈₃	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₈₃	D₄	1	1	yes	characteristic
29	C₄⋊C₈×C₈₃	—	2	2	no	—
30	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
31	C₂×Dic₁₆₆	—	4	2	no	—
32	C₂×Dic₁₆₆	D₈	1	1	yes	characteristic
33	Dic₃₃₂	—	8	4	no	—
34	C₂×C₈₃⋊C₈	—	2	1	no	—
35	C₄⋊Dic₈₃	—	4	2	no	—
36	C₄⋊Dic₈₃	—	4	2	no	—
37	C₄⋊Dic₈₃	D₈	1	1	yes	characteristic
38	Dic₈₃⋊C₄	—	2	1	no	—
39	C₄×Dic₈₃	—	2	1	no	—
40	C₈₃⋊C₁₆	—	4	1	no	—
41	C₂×C₈×C₈₃	—	4	2	no	—
42	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
44	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
45	C₁₆×C₈₃	—	2	1	no	—
46	C₁₆×C₈₃	Q₁₆	1	2	yes	complemented
47	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
48	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
49	C₂×C₄×C₈₃	#135	1	1	yes	characteristic
50	C₂×Dic₈₃	—	2	1	no	—
51	C₂×Dic₈₃	—	2	1	no	—
52	C₂×Dic₈₃	—	8	2	no	—
53	Dic₁₆₆	—	2	1	no	—
54	Dic₁₆₆	—	8	2	no	—
55	Dic₁₆₆	—	8	2	no	—
56	Dic₁₆₆	D₁₆	1	2	yes	—
57	C₈×C₈₃	—	2	1	no	—
58	C₈×C₈₃	—	8	2	no	—
59	C₈×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
60	C₈×C₈₃	#137	1	1	yes	characteristic
61	C₈₃⋊C₈	—	4	1	no	—
62	C₂²×C₈₃	#466	1	1	yes	characteristic
63	C₄×C₈₃	—	2	1	no	—
64	C₄×C₈₃	#462	1	1	yes	characteristic
65	C₄×C₈₃	#473	1	1	yes	characteristic
66	C₄×C₈₃	C₂×D₁₆	1	1	yes	characteristic
67	C₄×C₈₃	#508	1	1	yes	characteristic
68	Dic₈₃	—	2	2	no	—
69	Dic₈₃	—	4	1	no	—
70	Dic₈₃	—	16	2	no	—
71	#12457	—	83	1	no	maximal, Sylow
72	C₂×C₈₃	#1674	1	1	yes	characteristic
73	C₂×C₈₃	#2215	1	1	yes	characteristic
74	C₂×C₈₃	#2222	1	1	yes	characteristic
75	#2252	—	83	1	no	—
76	Q₈×C₁₆	—	83	1	no	—
77	#1674	—	83	2	no	—
78	#1371	D₈₃	1	1	yes	characteristic
79	#1364	—	83	2	no	—
80	C₈₃	#12457	1	1	yes	characteristic, complemented, Sylow
81	#500	—	83	2	no	—
82	#462	—	166	2	no	—
83	C₈×Q₈	—	83	1	no	—
84	#366	—	166	1	no	—
85	#366	—	166	1	no	—
86	C₄⋊C₁₆	—	83	1	no	—
87	C₄⋊C₁₆	—	166	1	no	—
88	C₄×C₁₆	—	166	1	no	—
89	C₄×C₁₆	C₂×D₈₃	1	1	yes	characteristic
90	#335	C₂×D₈₃	1	2	yes	—
91	#332	—	166	2	no	—
92	C₂×Q₁₆	—	332	2	no	—
93	C₄⋊Q₈	—	166	2	no	—
94	C₄×Q₈	—	83	1	no	—
95	C₂×C₁₆	—	166	1	no	—
96	C₂×C₁₆	D₃₃₂	1	1	yes	characteristic
97	C₂×C₁₆	D₃₃₂	1	1	yes	characteristic
98	#107	—	166	2	no	—
99	#107	—	166	2	no	—
100	C₄⋊C₈	—	2	2	no	—
101	C₄⋊C₈	—	83	1	no	—
102	C₄⋊C₈	—	166	1	no	—
103	#103	—	332	2	no	—
104	C₄×C₈	—	166	1	no	—
105	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
106	C₂×Q₈	—	332	2	no	—
107	C₂×Q₈	—	83	1	no	—
108	Q₁₆	—	664	4	no	—
109	C₂×C₈	—	4	2	no	—
110	C₂×C₈	—	166	1	no	—
111	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
112	C₂×C₈	#13050591	1	1	yes	characteristic
113	C₄⋊C₄	—	332	2	no	—
114	C₄⋊C₄	—	332	2	no	—
115	C₄⋊C₄	—	83	1	no	—
116	C₄⋊C₄	—	166	1	no	—
117	C₄²	—	166	1	no	—
118	C₄²	D₄×D₈₃	1	1	yes	characteristic
119	C₁₆	—	2	1	no	—
120	C₁₆	—	332	1	no	—
121	C₁₆	Dic₃₃₂	1	2	yes	complemented
122	Q₈	—	83	2	no	—
123	Q₈	—	664	2	no	—
124	Q₈	—	664	2	no	—
125	Q₈	—	166	1	no	—
126	C₂×C₄	—	664	2	no	—
127	C₂×C₄	—	166	1	no	—
128	C₂×C₄	—	166	1	no	—
129	C₂×C₄	#14581767	1	1	yes	characteristic
130	C₂×C₄	D₈×D₈₃	1	1	yes	characteristic
131	C₂×C₄	#14581815	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	8	2	no	—
134	C₈	—	332	1	no	—
135	C₈	C₂×Dic₃₃₂	1	1	yes	characteristic
136	C₈	#14581825	1	1	yes	characteristic
137	C₂²	#19127938	1	1	yes	characteristic
138	C₄	—	2	1	no	—
139	C₄	—	1328	2	no	—
140	C₄	—	332	1	no	—
141	C₄	—	166	2	no	—
142	C₄	#19127912	1	1	yes	characteristic
143	C₄	#19127977	1	1	yes	characteristic
144	C₄	D₁₆×D₈₃	1	1	yes	characteristic
145	C₄	#19128134	1	1	yes	characteristic
146	C₂	#24906858	1	1	yes	characteristic
147	C₂	#24909550	1	1	yes	characteristic
148	C₂	#24909589	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24909738

C₂

yes

maximal, characteristic

C₁₆×Dic₁₆₆

C₂

yes

maximal, characteristic

#24906858

C₂

yes

maximal

#24905519

C₂

yes

maximal

C₈₃×#1371

C₂

yes

maximal, characteristic

#19128106

C₂²

yes

—

#19127912

—

C₈×Dic₁₆₆

C₂²

yes

characteristic

#19127527

—

#19127527

—

C₄⋊(C₈₃⋊C₁₆)

C₂²

yes

characteristic

Dic₈₃⋊C₁₆

—

C₁₆×Dic₈₃

—

#19127448

—

C₄×C₁₆×C₈₃

C₂²

yes

characteristic

#335×C₈₃

C₂²

yes

—

C₂×Dic₃₃₂

—

#14581792

—

C₄×Dic₁₆₆

D₄

yes

characteristic

C₂×C₈₃⋊C₁₆

—

#14581726

—

#14581726

—

C₄⋊(C₈₃⋊C₈)

D₄

yes

characteristic

Dic₈₃⋊C₈

—

#14581717

—

C₈×Dic₈₃

—

C₂×C₁₆×C₈₃

D₄

yes

characteristic

C₂×C₁₆×C₈₃

D₄

yes

characteristic

C₄⋊C₈×C₈₃

—

C₄×C₈×C₈₃

C₂³

yes

characteristic

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

D₈

yes

characteristic

Dic₃₃₂

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

D₈

yes

characteristic

Dic₈₃⋊C₄

—

C₄×Dic₈₃

—

C₈₃⋊C₁₆

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂×D₄

yes

characteristic

C₂×C₈×C₈₃

#41

yes

characteristic

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₁₆×C₈₃

—

C₁₆×C₈₃

Q₁₆

yes

complemented

C₂×C₄×C₈₃

#121

yes

characteristic

C₂×C₄×C₈₃

C₂×D₈

yes

characteristic

C₂×C₄×C₈₃

#135

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

D₁₆

yes

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

C₂×Q₁₆

yes

characteristic

C₈×C₈₃

#137

yes

characteristic

C₈₃⋊C₈

—

C₂²×C₈₃

#466

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

C₂×D₁₆

yes

characteristic

C₄×C₈₃

#508

yes

characteristic

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

Q₈×C₁₆

—

#1674

—

#1371

D₈₃

yes

characteristic

#1364

—

C₈₃

#12457

yes

characteristic, complemented, Sylow

#500

—

#462

—

166

—

C₈×Q₈

—

#366

—

166

—

#366

—

166

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

166

—

C₄×C₁₆

—

166

—

C₄×C₁₆

C₂×D₈₃

yes

characteristic

#335

C₂×D₈₃

yes

—

#332

—

166

—

C₂×Q₁₆

—

332

—

C₄⋊Q₈

—

166

—

C₄×Q₈

—

C₂×C₁₆

—

166

—

C₂×C₁₆

D₃₃₂

yes

characteristic

C₂×C₁₆

D₃₃₂

yes

characteristic

#107

—

166

—

#107

—

166

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

C₄⋊C₈

—

166

—

103

#103

—

332

—

104

C₄×C₈

—

166

—

105

C₄×C₈

C₂²×D₈₃

yes

characteristic

106

C₂×Q₈

—

332

—

107

C₂×Q₈

—

108

Q₁₆

—

664

—

109

C₂×C₈

—

110

C₂×C₈

—

166

—

111

C₂×C₈

C₂×D₃₃₂

yes

characteristic

112

C₂×C₈

#13050591

yes

characteristic

113

C₄⋊C₄

—

332

—

114

C₄⋊C₄

—

332

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

166

—

117

C₄²

—

166

—

118

C₄²

D₄×D₈₃

yes

characteristic

119

C₁₆

—

120

C₁₆

—

332

—

121

C₁₆

Dic₃₃₂

yes

complemented

122

Q₈

—

123

Q₈

—

664

—

124

Q₈

—

664

—

125

Q₈

—

166

—

126

C₂×C₄

—

664

—

127

C₂×C₄

—

166

—

128

C₂×C₄

—

166

—

129

C₂×C₄

#14581767

yes

characteristic

130

C₂×C₄

D₈×D₈₃

yes

characteristic

131

C₂×C₄

#14581815

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

332

—

135

C₈

C₂×Dic₃₃₂

yes

characteristic

136

C₈

#14581825

yes

characteristic

137

C₂²

#19127938

yes

characteristic

138

C₄

—

139

C₄

—

1328

—

140

C₄

—

332

—

141

C₄

—

166

—

142

C₄

#19127912

yes

characteristic

143

C₄

#19127977

yes

characteristic

144

C₄

D₁₆×D₈₃

yes

characteristic

145

C₄

#19128134

yes

characteristic

146

C₂

#24906858

yes

characteristic

147

C₂

#24909550

yes

characteristic

148

C₂

#24909589

yes

characteristic

	subgroup	G/H
center	C₂²	#19127938
commutator, cosocle	C₄×C₈×C₈₃	C₂³
Frattini	C₄×C₈	C₂²×D₈₃
Fitting	C₈₃×#1371	C₂
socle, prime radical	C₂²×C₈₃	#466
hypercenter	#1371	D₈₃
hypocenter	C₈₃	#12457
LCS	G ▷ C₄×C₈×C₈₃ ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1371

subgroup

G/H

center

C₂²

#19127938

commutator, cosocle

C₄×C₈×C₈₃

C₂³

Frattini

C₄×C₈

C₂²×D₈₃

Fitting

C₈₃×#1371

C₂

C₂²×C₈₃

D₈₃

C₈₃

G ▷ C₄×C₈×C₈₃ ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1371

N	H ≅ G/N	type
46. C₁₆×C₈₃	Q₁₆	semidirect
80. C₈₃	#12457	semidirect
121. C₁₆	Dic₃₃₂	semidirect

H ≅ G/N

type

46. C₁₆×C₈₃

Q₁₆

semidirect

80. C₈₃

#12457

semidirect

121. C₁₆

Dic₃₃₂

semidirect

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

subgroup	index
#24909738	2
C₁₆×Dic₁₆₆	2
#24906858	2
#24905519	2
C₈₃×#1371	2
#12457	83

subgroup

index

C₁₆×Dic₁₆₆

C₈₃×#1371

Subgroups — #35318764

Conjugacy classes of subgroups

Complements

Maximal subgroups