Group #35316294 — subgroups

Subgroups — #35316294

Order 21248.

204 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 89 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	#24909305	C₂	1	2	yes	maximal, complemented
2	#24905531	C₂	1	1	yes	maximal, characteristic, complemented
3	#24905439	C₂	1	2	yes	maximal, complemented
4	#24905383	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₃×#2179	C₂	1	1	yes	maximal, characteristic
6	#19128118	—	2	2	no	—
7	#19127522	C₂²	1	2	yes	—
8	#19127485	—	2	2	no	—
9	#19127477	C₂²	1	1	yes	characteristic
10	#19127422	C₂²	1	1	yes	characteristic
11	#508×C₈₃	C₄	1	1	yes	characteristic
12	#443×C₈₃	C₂²	1	2	yes	—
13	#368×C₈₃	C₄	1	1	yes	characteristic
14	#359×C₈₃	C₄	1	2	yes	—
15	#346×C₈₃	C₂²	1	1	yes	characteristic
16	#14581732	—	2	2	no	—
17	#14581732	—	4	2	no	—
18	C₂×C₈₃⋊C₁₆	—	4	2	no	—
19	C₂×C₈₃⋊C₁₆	D₄	1	2	yes	—
20	C₄×C₈₃⋊C₈	—	2	1	no	—
21	#135×C₈₃	C₂×C₄	1	2	yes	—
22	#131×C₈₃	—	2	2	no	—
23	Q₃₂×C₈₃	C₈	1	1	yes	characteristic, complemented
24	SD₃₂×C₈₃	C₈	1	2	yes	complemented
25	D₁₆×C₈₃	C₈	1	1	yes	characteristic, complemented
26	C₂×C₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
28	#108×C₈₃	C₂×C₄	1	2	yes	—
29	#104×C₈₃	—	2	2	no	—
30	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
31	C₂×C₈₃⋊C₈	—	4	1	no	—
32	C₈₃⋊C₁₆	—	2	2	no	—
33	C₈₃⋊C₁₆	—	4	2	no	—
34	#41×C₈₃	—	2	2	no	—
35	Q₁₆×C₈₃	C₂×C₈	1	2	yes	—
36	SD₁₆×C₈₃	—	2	2	no	—
37	D₈×C₈₃	C₂×C₈	1	2	yes	—
38	M₁₆×C₈₃	—	2	2	no	—
39	M₁₆×C₈₃	—	4	2	no	—
40	C₂×C₈×C₈₃	—	4	2	no	—
41	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
43	C₄²×C₈₃	#41	1	1	yes	characteristic
44	C₁₆×C₈₃	—	2	1	no	—
45	C₁₆×C₈₃	C₂×C₈	1	1	yes	characteristic
46	C₁₆×C₈₃	C₂×C₈	1	1	yes	characteristic
47	Q₈×C₈₃	—	2	2	no	—
48	D₄×C₈₃	—	2	2	no	—
49	D₄×C₈₃	—	4	2	no	—
50	C₂×C₄×C₈₃	—	2	1	no	—
51	C₂×C₄×C₈₃	—	4	2	no	—
52	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
53	C₈×C₈₃	—	4	2	no	—
54	C₈×C₈₃	C₂²×C₈	1	1	yes	characteristic
55	C₈×C₈₃	#131	1	1	yes	characteristic
56	C₈×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
57	C₈₃⋊C₈	—	8	1	no	—
58	C₂²×C₈₃	—	4	2	no	—
59	C₂²×C₈₃	#444	1	1	yes	characteristic
60	C₄×C₈₃	—	4	1	no	—
61	C₄×C₈₃	—	4	2	no	—
62	C₄×C₈₃	C₈×D₄	1	1	yes	characteristic
63	C₄×C₈₃	#442	1	1	yes	characteristic
64	#12068	—	83	1	no	maximal, Sylow
65	C₂×C₈₃	—	2	1	no	—
66	C₂×C₈₃	—	8	2	no	—
67	C₂×C₈₃	#1580	1	1	yes	characteristic
68	#2179	D₈₃	1	1	yes	characteristic
69	#2171	—	83	2	no	—
70	#1369	—	83	1	no	—
71	#1336	—	83	2	no	—
72	#1313	—	83	1	no	—
73	C₈₃	#12068	1	1	yes	characteristic, complemented, Sylow
74	#508	Dic₈₃	1	1	yes	characteristic
75	#504	—	166	2	no	—
76	#443	C₂×D₈₃	1	2	yes	—
77	#368	Dic₈₃	1	1	yes	characteristic
78	#364	—	83	2	no	—
79	#359	Dic₈₃	1	2	yes	—
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	C₂×Dic₈₃	1	2	yes	—
85	#131	—	2	2	no	—
86	Q₃₂	C₈₃⋊C₈	1	1	yes	characteristic, complemented
87	SD₃₂	C₈₃⋊C₈	1	2	yes	complemented
88	D₁₆	C₈₃⋊C₈	1	1	yes	characteristic, complemented
89	M₃₂	—	332	2	no	—
90	M₃₂	—	166	2	no	—
91	C₂×C₁₆	—	332	2	no	—
92	C₂×C₁₆	—	83	2	no	—
93	C₂×C₁₆	C₂×Dic₈₃	1	1	yes	characteristic
94	C₂×C₁₆	C₂×Dic₈₃	1	1	yes	characteristic
95	#108	C₂×Dic₈₃	1	2	yes	—
96	#104	—	2	2	no	—
97	C₄×C₈	—	166	1	no	—
98	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
99	#41	—	2	2	no	—
100	Q₁₆	C₂×C₈₃⋊C₈	1	2	yes	—
101	SD₁₆	—	2	2	no	—
102	D₈	C₂×C₈₃⋊C₈	1	2	yes	—
103	M₁₆	—	2	2	no	—
104	M₁₆	—	4	2	no	—
105	C₂×C₈	—	4	2	no	—
106	C₂×C₈	—	332	1	no	—
107	C₂×C₈	C₂²×Dic₈₃	1	1	yes	characteristic
108	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
109	C₄²	#13050589	1	1	yes	characteristic
110	C₁₆	—	2	1	no	—
111	C₁₆	—	332	2	no	—
112	C₁₆	—	166	2	no	—
113	C₁₆	C₂×C₈₃⋊C₈	1	1	yes	characteristic
114	C₁₆	C₂×C₈₃⋊C₈	1	1	yes	characteristic
115	Q₈	—	2	2	no	—
116	D₄	—	2	2	no	—
117	D₄	—	4	2	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	D₄×Dic₈₃	1	1	yes	characteristic
121	C₈	—	4	2	no	—
122	C₈	—	664	1	no	—
123	C₈	C₂²×C₈₃⋊C₈	1	1	yes	characteristic
124	C₈	#14581799	1	1	yes	characteristic
125	C₈	#14581816	1	2	yes	—
126	C₂²	—	4	2	no	—
127	C₂²	#19127823	1	1	yes	characteristic
128	C₄	—	4	1	no	—
129	C₄	—	4	2	no	—
130	C₄	D₄×C₈₃⋊C₈	1	1	yes	characteristic
131	C₄	#19127813	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	8	2	no	—
134	C₂	#24906287	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24909305

C₂

yes

maximal, complemented

#24905531

C₂

yes

maximal, characteristic, complemented

#24905439

C₂

yes

maximal, complemented

#24905383

C₂

yes

maximal, characteristic, complemented

C₈₃×#2179

C₂

yes

maximal, characteristic

#19128118

—

#19127522

C₂²

yes

—

#19127485

—

#19127477

C₂²

yes

characteristic

#19127422

C₂²

yes

characteristic

#508×C₈₃

C₄

yes

characteristic

#443×C₈₃

C₂²

yes

—

#368×C₈₃

C₄

yes

characteristic

#359×C₈₃

C₄

yes

—

#346×C₈₃

C₂²

yes

characteristic

#14581732

—

#14581732

—

C₂×C₈₃⋊C₁₆

—

C₂×C₈₃⋊C₁₆

D₄

yes

—

C₄×C₈₃⋊C₈

—

#135×C₈₃

C₂×C₄

yes

—

#131×C₈₃

—

Q₃₂×C₈₃

C₈

yes

characteristic, complemented

SD₃₂×C₈₃

C₈

yes

complemented

D₁₆×C₈₃

C₈

yes

characteristic, complemented

C₂×C₁₆×C₈₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₃

C₂×C₄

yes

characteristic

#108×C₈₃

C₂×C₄

yes

—

#104×C₈₃

—

C₄×C₈×C₈₃

C₂³

yes

characteristic

C₂×C₈₃⋊C₈

—

C₈₃⋊C₁₆

—

C₈₃⋊C₁₆

—

#41×C₈₃

—

Q₁₆×C₈₃

C₂×C₈

yes

—

SD₁₆×C₈₃

—

D₈×C₈₃

C₂×C₈

yes

—

M₁₆×C₈₃

—

M₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₃

C₂×D₄

yes

characteristic

C₄²×C₈₃

#41

yes

characteristic

C₁₆×C₈₃

—

C₁₆×C₈₃

C₂×C₈

yes

characteristic

C₁₆×C₈₃

C₂×C₈

yes

characteristic

Q₈×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₄×D₄

yes

characteristic

C₈×C₈₃

—

C₈×C₈₃

C₂²×C₈

yes

C₈×C₈₃

yes

C₈×C₈₃

yes

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

#444

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₈×D₄

yes

C₄×C₈₃

yes

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

#1580

yes

characteristic

#2179

D₈₃

yes

—

—

—

—

C₈₃

#12068

yes

characteristic, complemented, Sylow

#508

Dic₈₃

yes

characteristic

#504

—

166

—

#443

C₂×D₈₃

yes

—

#368

Dic₈₃

yes

characteristic

#364

—

#359

Dic₈₃

yes

—

#350

—

166

—

#346

—

#346

C₂×D₈₃

yes

characteristic

#322

—

#135

C₂×Dic₈₃

yes

—

#131

—

Q₃₂

C₈₃⋊C₈

yes

characteristic, complemented

SD₃₂

C₈₃⋊C₈

yes

complemented

D₁₆

C₈₃⋊C₈

yes

characteristic, complemented

M₃₂

—

332

—

M₃₂

—

166

—

C₂×C₁₆

—

332

—

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₈₃

yes

characteristic

C₂×C₁₆

C₂×Dic₈₃

yes

characteristic

#108

C₂×Dic₈₃

yes

—

#104

—

C₄×C₈

—

166

—

C₄×C₈

C₂²×D₈₃

yes

characteristic

#41

—

100

Q₁₆

C₂×C₈₃⋊C₈

yes

—

101

SD₁₆

—

102

D₈

C₂×C₈₃⋊C₈

yes

—

103

M₁₆

—

104

M₁₆

—

105

C₂×C₈

—

106

C₂×C₈

—

332

—

107

C₂×C₈

C₂²×Dic₈₃

yes

characteristic

108

C₂×C₈

D₄×D₈₃

yes

characteristic

109

C₄²

#13050589

yes

characteristic

110

C₁₆

—

111

C₁₆

—

332

—

112

C₁₆

—

166

—

113

C₁₆

C₂×C₈₃⋊C₈

yes

characteristic

114

C₁₆

C₂×C₈₃⋊C₈

yes

characteristic

115

Q₈

—

116

D₄

—

117

D₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

D₄×Dic₈₃

yes

characteristic

121

C₈

—

122

C₈

—

664

—

123

C₈

C₂²×C₈₃⋊C₈

yes

characteristic

124

C₈

#14581799

yes

characteristic

125

C₈

#14581816

yes

—

126

C₂²

—

127

C₂²

#19127823

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

D₄×C₈₃⋊C₈

yes

characteristic

131

C₄

#19127813

yes

characteristic

132

C₂

—

133

C₂

—

134

C₂

#24906287

yes

characteristic

	subgroup	G/H
center	C₄	#19127813
commutator	C₈×C₈₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₈₃
Fitting	C₈₃×#2179	C₂
socle	C₂×C₈₃	#1580
prime radical	#508×C₈₃	C₄
cosocle	C₄×C₈×C₈₃	C₂³
hypercenter	#2179	D₈₃
hypocenter	C₈₃	#12068
LCS	G ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #2179

subgroup

G/H

center

C₄

#19127813

commutator

C₈×C₈₃

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₈₃

Fitting

C₈₃×#2179

C₂

socle

C₂×C₈₃

#1580

prime radical

#508×C₈₃

C₄

cosocle

C₄×C₈×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #2179

N	H ≅ G/N	type
1. #24909305	C₂	semidirect
2. #24905531	C₂	semidirect
3. #24905439	C₂	semidirect
4. #24905383	C₂	semidirect
23. Q₃₂×C₈₃	C₈	semidirect
24. SD₃₂×C₈₃	C₈	semidirect
25. D₁₆×C₈₃	C₈	semidirect
73. C₈₃	#12068	semidirect
86. Q₃₂	C₈₃⋊C₈	semidirect
87. SD₃₂	C₈₃⋊C₈	semidirect
88. D₁₆	C₈₃⋊C₈	semidirect

H ≅ G/N

type

1. #24909305

C₂

semidirect

2. #24905531

C₂

semidirect

3. #24905439

C₂

semidirect

4. #24905383

C₂

semidirect

23. Q₃₂×C₈₃

C₈

semidirect

24. SD₃₂×C₈₃

C₈

semidirect

25. D₁₆×C₈₃

C₈

semidirect

73. C₈₃

#12068

semidirect

86. Q₃₂

C₈₃⋊C₈

semidirect

87. SD₃₂

C₈₃⋊C₈

semidirect

88. D₁₆

C₈₃⋊C₈

semidirect

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

subgroup	index
#24909305	2
#24905531	2
#24905439	2
#24905383	2
C₈₃×#2179	2
#12068	83

subgroup

index

C₈₃×#2179

Subgroups — #35316294

Conjugacy classes of subgroups

Complements

Maximal subgroups