Group #35318763 — subgroups

Subgroups — #35318763

Order 21248.

204 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 75 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	#24906857	C₂	1	2	yes	maximal
3	#24905535	C₂	1	1	yes	maximal, characteristic
4	#24905517	C₂	1	2	yes	maximal
5	Q₈×C₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic
6	#19128106	C₂²	1	2	yes	—
7	#19127911	—	2	2	no	—
8	#19127527	—	2	1	no	—
9	#19127527	—	2	1	no	—
10	#19127454	C₂²	1	2	yes	—
11	#19127446	—	2	2	no	—
12	C₈×Q₈×C₈₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₁₆×C₈₃	—	2	1	no	—
14	C₄⋊C₁₆×C₈₃	C₂²	1	1	yes	characteristic
15	C₄×C₁₆×C₈₃	—	2	1	no	—
16	C₄×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
17	C₂×#13050580	—	4	2	no	—
18	#14581792	—	2	2	no	—
19	#14581726	—	2	2	no	—
20	#14581726	—	2	2	no	—
21	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
22	#14581715	—	4	2	no	—
23	C₄×Q₈×C₈₃	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₈₃	—	2	1	no	—
25	C₂×C₁₆×C₈₃	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₈₃	D₄	1	1	yes	characteristic
27	C₄⋊C₈×C₈₃	—	2	1	no	—
28	C₄⋊C₈×C₈₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₈₃	—	2	1	no	—
30	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
31	C₂×Dic₁₆₆	—	4	2	no	—
32	#13050580	—	8	4	no	—
33	C₂×C₈₃⋊C₈	—	4	2	no	—
34	C₄⋊Dic₈₃	—	4	2	no	—
35	C₄⋊Dic₈₃	—	4	2	no	—
36	C₂×Q₈×C₈₃	D₈	1	1	yes	characteristic
37	C₂×C₈×C₈₃	—	2	1	no	—
38	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
40	C₄⋊C₄×C₈₃	—	2	1	no	—
41	C₄⋊C₄×C₈₃	D₈	1	1	yes	characteristic
42	C₄²×C₈₃	—	2	1	no	—
43	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
44	C₁₆×C₈₃	—	2	1	no	—
45	C₁₆×C₈₃	—	4	1	no	—
46	C₁₆×C₈₃	Q₁₆	1	2	yes	complemented
47	Q₈×C₈₃	—	2	1	no	—
48	Q₈×C₈₃	D₁₆	1	2	yes	—
49	C₂×C₄×C₈₃	—	2	1	no	—
50	C₂×C₄×C₈₃	—	2	1	no	—
51	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
52	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₈₃	#135	1	1	yes	characteristic
54	C₂×Dic₈₃	—	8	2	no	—
55	Dic₁₆₆	—	8	2	no	—
56	Dic₁₆₆	—	8	2	no	—
57	C₈×C₈₃	—	2	1	no	—
58	C₈×C₈₃	—	4	1	no	—
59	C₈×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
60	C₈×C₈₃	#137	1	1	yes	characteristic
61	C₈₃⋊C₈	—	8	2	no	—
62	C₂²×C₈₃	#466	1	1	yes	characteristic
63	C₄×C₈₃	—	2	2	no	—
64	C₄×C₈₃	—	2	1	no	—
65	C₄×C₈₃	—	4	1	no	—
66	C₄×C₈₃	#462	1	1	yes	characteristic
67	C₄×C₈₃	#473	1	1	yes	characteristic
68	C₄×C₈₃	C₂×D₁₆	1	1	yes	characteristic
69	C₄×C₈₃	#508	1	1	yes	characteristic
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	D₈₃	1	1	yes	characteristic
77	#1674	—	83	2	no	—
78	#1371	—	83	1	no	—
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	C₂×D₈₃	1	1	yes	characteristic
84	#366	—	166	1	no	—
85	#366	—	166	1	no	—
86	C₄⋊C₁₆	—	2	1	no	—
87	C₄⋊C₁₆	C₂×D₈₃	1	1	yes	characteristic
88	C₄×C₁₆	—	2	1	no	—
89	C₄×C₁₆	C₂×D₈₃	1	1	yes	characteristic
90	#335	—	83	2	no	—
91	#332	—	166	2	no	—
92	C₂×Q₁₆	—	332	2	no	—
93	C₄⋊Q₈	—	166	2	no	—
94	C₄×Q₈	D₃₃₂	1	1	yes	characteristic
95	C₂×C₁₆	—	2	1	no	—
96	C₂×C₁₆	C₂²⋊D₈₃	1	1	yes	characteristic
97	C₂×C₁₆	C₂²⋊D₈₃	1	1	yes	characteristic
98	#107	—	166	2	no	—
99	#107	—	166	2	no	—
100	C₄⋊C₈	—	2	1	no	—
101	C₄⋊C₈	—	166	2	no	—
102	C₄⋊C₈	D₃₃₂	1	1	yes	characteristic
103	#103	—	332	2	no	—
104	C₄×C₈	—	2	1	no	—
105	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
106	C₂×Q₈	—	332	2	no	—
107	C₂×Q₈	D₆₆₄	1	1	yes	characteristic
108	Q₁₆	—	664	4	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	332	2	no	—
111	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
112	C₂×C₈	#13050590	1	1	yes	characteristic
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	332	2	no	—
115	C₄⋊C₄	—	332	2	no	—
116	C₄⋊C₄	D₆₆₄	1	1	yes	characteristic
117	C₄²	—	2	1	no	—
118	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
119	C₁₆	—	2	1	no	—
120	C₁₆	—	4	1	no	—
121	C₁₆	#13050580	1	2	yes	complemented
122	Q₈	—	2	1	no	—
123	Q₈	—	664	2	no	—
124	Q₈	—	664	2	no	—
125	Q₈	D₁₃₂₈	1	2	yes	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	664	2	no	—
129	C₂×C₄	#14581766	1	1	yes	characteristic
130	C₂×C₄	C₂×D₆₆₄	1	1	yes	characteristic
131	C₂×C₄	#14581814	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	664	2	no	—
135	C₈	C₂×#13050580	1	1	yes	characteristic
136	C₈	#14581822	1	1	yes	characteristic
137	C₂²	#19127937	1	1	yes	characteristic
138	C₄	—	2	1	no	—
139	C₄	—	2	2	no	—
140	C₄	—	4	1	no	—
141	C₄	—	1328	2	no	—
142	C₄	#19127911	1	1	yes	characteristic
143	C₄	#19127976	1	1	yes	characteristic
144	C₄	C₂×D₁₃₂₈	1	1	yes	characteristic
145	C₄	#19128133	1	1	yes	characteristic
146	C₂	#24906857	1	1	yes	characteristic
147	C₂	#24909549	1	1	yes	characteristic
148	C₂	#24909588	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24909738

C₂

yes

maximal, characteristic

#24906857

C₂

yes

maximal

#24905535

C₂

yes

maximal, characteristic

#24905517

C₂

yes

maximal

Q₈×C₁₆×C₈₃

C₂

yes

maximal, characteristic

#19128106

C₂²

yes

—

—

—

—

C₂²

yes

—

#19127446

—

C₈×Q₈×C₈₃

C₂²

yes

characteristic

C₄⋊C₁₆×C₈₃

—

C₄⋊C₁₆×C₈₃

C₂²

yes

characteristic

C₄×C₁₆×C₈₃

—

C₄×C₁₆×C₈₃

C₂²

yes

—

—

—

—

C₄⋊(C₈₃⋊C₈)

—

#14581715

—

C₄×Q₈×C₈₃

D₄

yes

characteristic

C₂×C₁₆×C₈₃

—

C₂×C₁₆×C₈₃

D₄

yes

characteristic

C₂×C₁₆×C₈₃

D₄

yes

characteristic

C₄⋊C₈×C₈₃

—

C₄⋊C₈×C₈₃

D₄

yes

characteristic

C₄×C₈×C₈₃

—

C₄×C₈×C₈₃

C₂³

yes

characteristic

C₂×Dic₁₆₆

—

#13050580

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₂×Q₈×C₈₃

D₈

yes

characteristic

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂×D₄

yes

characteristic

C₂×C₈×C₈₃

#41

yes

characteristic

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

D₈

yes

characteristic

C₄²×C₈₃

—

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₁₆×C₈₃

—

C₁₆×C₈₃

—

C₁₆×C₈₃

Q₁₆

yes

complemented

Q₈×C₈₃

—

Q₈×C₈₃

D₁₆

yes

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

#121

yes

characteristic

C₂×C₄×C₈₃

C₂×D₈

yes

characteristic

C₂×C₄×C₈₃

#135

yes

characteristic

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

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

—

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

C₂×D₁₆

yes

characteristic

C₄×C₈₃

#508

yes

characteristic

Dic₈₃

—

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

Q₈×C₁₆

D₈₃

yes

—

—

—

C₈₃

#12457

yes

characteristic, complemented, Sylow

#500

—

#462

—

166

—

C₈×Q₈

C₂×D₈₃

yes

characteristic

#366

—

166

—

#366

—

166

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×D₈₃

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

C₂×D₈₃

yes

characteristic

#335

—

#332

—

166

—

C₂×Q₁₆

—

332

—

C₄⋊Q₈

—

166

—

C₄×Q₈

D₃₃₂

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₈₃

yes

characteristic

C₂×C₁₆

C₂²⋊D₈₃

yes

characteristic

#107

—

166

—

#107

—

166

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

166

—

102

C₄⋊C₈

D₃₃₂

yes

characteristic

103

#103

—

332

—

104

C₄×C₈

—

105

C₄×C₈

C₂²×D₈₃

yes

characteristic

106

C₂×Q₈

—

332

—

107

C₂×Q₈

D₆₆₄

yes

characteristic

108

Q₁₆

—

664

—

109

C₂×C₈

—

110

C₂×C₈

—

332

—

111

C₂×C₈

C₂×C₂²⋊D₈₃

yes

characteristic

112

C₂×C₈

#13050590

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

332

—

115

C₄⋊C₄

—

332

—

116

C₄⋊C₄

D₆₆₄

yes

characteristic

117

C₄²

—

118

C₄²

C₂×D₃₃₂

yes

characteristic

119

C₁₆

—

120

C₁₆

—

121

C₁₆

#13050580

yes

complemented

122

Q₈

—

123

Q₈

—

664

—

124

Q₈

—

664

—

125

Q₈

D₁₃₂₈

yes

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

664

—

129

C₂×C₄

#14581766

yes

characteristic

130

C₂×C₄

C₂×D₆₆₄

yes

characteristic

131

C₂×C₄

#14581814

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

664

—

135

C₈

C₂×#13050580

yes

characteristic

136

C₈

#14581822

yes

characteristic

137

C₂²

#19127937

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

1328

—

142

C₄

#19127911

yes

characteristic

143

C₄

#19127976

yes

characteristic

144

C₄

C₂×D₁₃₂₈

yes

characteristic

145

C₄

#19128133

yes

characteristic

146

C₂

#24906857

yes

characteristic

147

C₂

#24909549

yes

characteristic

148

C₂

#24909588

yes

characteristic

	subgroup	G/H
center	C₂²	#19127937
commutator, cosocle	C₄×C₈×C₈₃	C₂³
Frattini	C₄×C₈	C₂²×D₈₃
Fitting	Q₈×C₁₆×C₈₃	C₂
socle, prime radical	C₂²×C₈₃	#466
hypercenter	Q₈×C₁₆	D₈₃
hypocenter	C₈₃	#12457
LCS	G ▷ C₄×C₈×C₈₃ ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

subgroup

G/H

center

C₂²

#19127937

commutator, cosocle

C₄×C₈×C₈₃

C₂³

Frattini

C₄×C₈

C₂²×D₈₃

Fitting

Q₈×C₁₆×C₈₃

C₂

socle, prime radical

C₂²×C₈₃

#466

hypercenter

Q₈×C₁₆

D₈₃

hypocenter

C₈₃

#12457

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

N	H ≅ G/N	type
46. C₁₆×C₈₃	Q₁₆	semidirect
80. C₈₃	#12457	semidirect
121. C₁₆	#13050580	semidirect

H ≅ G/N

type

46. C₁₆×C₈₃

Q₁₆

semidirect

80. C₈₃

#12457

semidirect

121. C₁₆

#13050580

semidirect

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

subgroup	index
#24909738	2
#24906857	2
#24905535	2
#24905517	2
Q₈×C₁₆×C₈₃	2
#12457	83

subgroup

index

Q₈×C₁₆×C₈₃

Subgroups — #35318763

Conjugacy classes of subgroups

Complements

Maximal subgroups