Group #35316293 — subgroups

Subgroups — #35316293

Order 21248.

204 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 73 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	#24909350	C₂	1	1	yes	maximal, characteristic
2	#24909304	C₂	1	2	yes	maximal, complemented
3	#24905438	C₂	1	2	yes	maximal, complemented
4	#24905383	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₃×#1369	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128133	C₄	1	1	yes	characteristic
7	#19128119	—	2	2	no	—
8	#19127819	C₂²	1	2	yes	—
9	#19127531	C₄	1	1	yes	characteristic
10	#19127509	C₄	1	2	yes	—
11	#19127486	—	2	2	no	—
12	#19127477	C₂²	1	1	yes	characteristic
13	#19127422	C₂²	1	1	yes	characteristic
14	#364×C₈₃	C₂²	1	2	yes	—
15	#346×C₈₃	C₂²	1	1	yes	characteristic
16	#14581814	C₂×C₄	1	2	yes	—
17	#14581798	—	2	2	no	—
18	Dic₆₆₄	C₈	1	1	yes	characteristic, complemented
19	#14581737	C₈	1	2	yes	complemented
20	D₁₃₂₈	C₈	1	1	yes	characteristic, complemented
21	#14581731	—	4	2	no	—
22	C₂×C₈₃⋊C₁₆	D₄	1	2	yes	—
23	C₁₆×D₈₃	—	4	2	no	—
24	#14581728	C₂×C₄	1	2	yes	—
25	#14581720	—	2	2	no	—
26	C₄×C₈₃⋊C₈	—	2	1	no	—
27	M₃₂×C₈₃	—	2	2	no	—
28	C₂×C₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
31	#13050590	—	2	2	no	—
32	Dic₃₃₂	C₂×C₈	1	2	yes	—
33	#13050579	—	2	2	no	—
34	D₆₆₄	C₂×C₈	1	2	yes	—
35	#13050574	—	2	2	no	—
36	#13050573	—	4	2	no	—
37	C₂×C₈₃⋊C₈	—	4	1	no	—
38	C₈×D₈₃	—	4	2	no	—
39	C₈₃⋊C₁₆	—	2	2	no	—
40	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
42	C₄²×C₈₃	#41	1	1	yes	characteristic
43	C₁₆×C₈₃	—	2	1	no	—
44	C₁₆×C₈₃	—	4	2	no	—
45	C₁₆×C₈₃	C₂×C₈	1	1	yes	characteristic
46	C₁₆×C₈₃	C₂×C₈	1	1	yes	characteristic
47	C₂×C₄×C₈₃	—	2	1	no	—
48	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
49	C₂²⋊D₈₃	—	4	2	no	—
50	D₃₃₂	—	2	2	no	—
51	C₄×D₈₃	—	4	2	no	—
52	Dic₁₆₆	—	2	2	no	—
53	C₈×C₈₃	C₂²×C₈	1	1	yes	characteristic
54	C₈×C₈₃	#131	1	1	yes	characteristic
55	C₈×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
56	C₈₃⋊C₈	—	4	2	no	—
57	C₈₃⋊C₈	—	8	1	no	—
58	C₂²×C₈₃	#444	1	1	yes	characteristic
59	C₂×D₈₃	—	4	2	no	—
60	C₄×C₈₃	—	4	1	no	—
61	C₄×C₈₃	C₈×D₄	1	1	yes	characteristic
62	C₄×C₈₃	#442	1	1	yes	characteristic
63	Dic₈₃	—	4	2	no	—
64	#12068	—	83	1	no	maximal, Sylow
65	C₂×C₈₃	—	2	1	no	—
66	C₂×C₈₃	#1580	1	1	yes	characteristic
67	D₈₃	—	8	2	no	—
68	#2179	—	83	1	no	—
69	#2171	—	83	2	no	—
70	#1369	D₈₃	1	1	yes	characteristic, complemented
71	#1336	—	83	2	no	—
72	#1313	—	83	1	no	—
73	C₈₃	#12068	1	1	yes	characteristic, complemented, Sylow
74	#508	—	83	1	no	—
75	#504	—	166	2	no	—
76	#443	—	83	2	no	—
77	#368	—	83	1	no	—
78	#364	C₂×D₈₃	1	2	yes	—
79	#359	—	83	2	no	—
80	#350	—	166	2	no	—
81	#346	—	83	1	no	—
82	#346	C₂×D₈₃	1	1	yes	characteristic
83	#322	—	83	1	no	—
84	#135	—	83	2	no	—
85	#131	—	166	2	no	—
86	Q₃₂	—	83	1	no	—
87	SD₃₂	—	83	2	no	—
88	D₁₆	—	83	1	no	—
89	M₃₂	—	2	2	no	—
90	M₃₂	—	332	2	no	—
91	C₂×C₁₆	—	332	2	no	—
92	C₂×C₁₆	—	83	2	no	—
93	C₂×C₁₆	C₄×D₈₃	1	1	yes	characteristic
94	C₂×C₁₆	C₄×D₈₃	1	1	yes	characteristic
95	#108	—	83	2	no	—
96	#104	—	166	2	no	—
97	C₄×C₈	—	166	1	no	—
98	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
99	#41	—	166	2	no	—
100	Q₁₆	—	83	2	no	—
101	SD₁₆	—	166	2	no	—
102	D₈	—	83	2	no	—
103	M₁₆	—	332	2	no	—
104	M₁₆	—	166	2	no	—
105	C₂×C₈	—	332	2	no	—
106	C₂×C₈	—	332	1	no	—
107	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
108	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
109	C₄²	#13050591	1	1	yes	characteristic
110	C₁₆	—	2	1	no	—
111	C₁₆	—	4	2	no	—
112	C₁₆	—	166	2	no	—
113	C₁₆	C₈×D₈₃	1	1	yes	characteristic
114	C₁₆	C₈×D₈₃	1	1	yes	characteristic
115	Q₈	—	166	2	no	—
116	D₄	—	332	2	no	—
117	D₄	—	166	2	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	332	2	no	—
120	C₂×C₄	#14581754	1	1	yes	characteristic
121	C₈	—	332	2	no	—
122	C₈	—	664	1	no	—
123	C₈	C₂×C₈×D₈₃	1	1	yes	characteristic
124	C₈	#14581800	1	1	yes	characteristic
125	C₈	#14581819	1	2	yes	—
126	C₂²	—	332	2	no	—
127	C₂²	#19127822	1	1	yes	characteristic
128	C₄	—	4	1	no	—
129	C₄	—	332	2	no	—
130	C₄	#19127766	1	1	yes	characteristic
131	C₄	#19127812	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	664	2	no	—
134	C₂	#24906286	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24909350

C₂

yes

maximal, characteristic

#24909304

C₂

yes

maximal, complemented

#24905438

C₂

yes

maximal, complemented

#24905383

C₂

yes

maximal, characteristic, complemented

C₈₃×#1369

C₂

yes

maximal, characteristic, complemented

#19128133

C₄

yes

characteristic

#19128119

—

#19127819

C₂²

yes

—

#19127531

C₄

yes

characteristic

#19127509

C₄

yes

—

#19127486

—

#19127477

C₂²

yes

characteristic

#19127422

C₂²

yes

characteristic

#364×C₈₃

C₂²

yes

—

#346×C₈₃

C₂²

yes

characteristic

#14581814

C₂×C₄

yes

—

#14581798

—

Dic₆₆₄

C₈

yes

characteristic, complemented

#14581737

C₈

yes

complemented

D₁₃₂₈

C₈

yes

characteristic, complemented

#14581731

—

C₂×C₈₃⋊C₁₆

D₄

yes

—

C₁₆×D₈₃

—

#14581728

C₂×C₄

yes

—

#14581720

—

C₄×C₈₃⋊C₈

—

M₃₂×C₈₃

—

C₂×C₁₆×C₈₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₃

C₂×C₄

yes

characteristic

C₄×C₈×C₈₃

C₂³

yes

characteristic

#13050590

—

Dic₃₃₂

C₂×C₈

yes

—

#13050579

—

D₆₆₄

C₂×C₈

yes

—

#13050574

—

#13050573

—

C₂×C₈₃⋊C₈

—

C₈×D₈₃

—

C₈₃⋊C₁₆

—

C₂×C₈×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₃

C₂×D₄

yes

characteristic

C₄²×C₈₃

#41

yes

characteristic

C₁₆×C₈₃

—

C₁₆×C₈₃

—

C₁₆×C₈₃

C₂×C₈

yes

characteristic

C₁₆×C₈₃

C₂×C₈

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₄×D₄

yes

characteristic

C₂²⋊D₈₃

—

D₃₃₂

—

C₄×D₈₃

—

Dic₁₆₆

—

C₈×C₈₃

C₂²×C₈

yes

C₈×C₈₃

yes

C₈×C₈₃

yes

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

#444

yes

characteristic

C₂×D₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₈×D₄

yes

characteristic

C₄×C₈₃

#442

yes

characteristic

Dic₈₃

—

#12068

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

#1580

yes

characteristic

D₈₃

—

#2179

—

#2171

—

#1369

D₈₃

yes

characteristic, complemented

#1336

—

#1313

—

C₈₃

#12068

yes

characteristic, complemented, Sylow

#508

—

#504

—

166

—

#443

—

#368

—

#364

C₂×D₈₃

yes

—

#359

—

#350

—

166

—

#346

—

#346

C₂×D₈₃

yes

—

—

—

166

—

Q₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

M₃₂

—

332

—

C₂×C₁₆

—

332

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₈₃

yes

characteristic

C₂×C₁₆

C₄×D₈₃

yes

characteristic

#108

—

#104

—

166

—

C₄×C₈

—

166

—

C₄×C₈

C₂²×D₈₃

yes

characteristic

#41

—

166

—

100

Q₁₆

—

101

SD₁₆

—

166

—

102

D₈

—

103

M₁₆

—

332

—

104

M₁₆

—

166

—

105

C₂×C₈

—

332

—

106

C₂×C₈

—

332

—

107

C₂×C₈

C₂×C₄×D₈₃

yes

characteristic

108

C₂×C₈

D₄×D₈₃

yes

characteristic

109

C₄²

#13050591

yes

characteristic

110

C₁₆

—

111

C₁₆

—

112

C₁₆

—

166

—

113

C₁₆

C₈×D₈₃

yes

characteristic

114

C₁₆

C₈×D₈₃

yes

characteristic

115

Q₈

—

166

—

116

D₄

—

332

—

117

D₄

—

166

—

118

C₂×C₄

—

119

C₂×C₄

—

332

—

120

C₂×C₄

#14581754

yes

characteristic

121

C₈

—

332

—

122

C₈

—

664

—

123

C₈

C₂×C₈×D₈₃

yes

characteristic

124

C₈

#14581800

yes

characteristic

125

C₈

#14581819

yes

—

126

C₂²

—

332

—

127

C₂²

#19127822

yes

characteristic

128

C₄

—

129

C₄

—

332

—

130

C₄

#19127766

yes

characteristic

131

C₄

#19127812

yes

characteristic

132

C₂

—

133

C₂

—

664

—

134

C₂

#24906286

yes

characteristic

	subgroup	G/H
center	C₄	#19127812
commutator	C₈×C₈₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₈₃
Fitting	C₈₃×#1369	C₂
socle	C₂×C₈₃	#1580
prime radical	#19128133	C₄
cosocle	C₄×C₈×C₈₃	C₂³
hypercenter	#1369	D₈₃
hypocenter	C₈₃	#12068
LCS	G ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1369

subgroup

G/H

center

C₄

#19127812

commutator

C₈×C₈₃

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₈₃

Fitting

C₈₃×#1369

C₂

C₂×C₈₃

C₄

C₄×C₈×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1369

N	H ≅ G/N	type
2. #24909304	C₂	semidirect
3. #24905438	C₂	semidirect
4. #24905383	C₂	semidirect
5. C₈₃×#1369	C₂	semidirect
18. Dic₆₆₄	C₈	semidirect
19. #14581737	C₈	semidirect
20. D₁₃₂₈	C₈	semidirect
70. #1369	D₈₃	semidirect
73. C₈₃	#12068	semidirect

H ≅ G/N

type

2. #24909304

C₂

semidirect

3. #24905438

C₂

semidirect

4. #24905383

C₂

semidirect

5. C₈₃×#1369

C₂

semidirect

18. Dic₆₆₄

C₈

semidirect

19. #14581737

C₈

semidirect

20. D₁₃₂₈

C₈

semidirect

70. #1369

D₈₃

semidirect

73. C₈₃

#12068

semidirect

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

subgroup	index
#24909350	2
#24909304	2
#24905438	2
#24905383	2
C₈₃×#1369	2
#12068	83

subgroup

index

C₈₃×#1369

Subgroups — #35316293

Conjugacy classes of subgroups

Complements

Maximal subgroups