Group #35319132 — subgroups

Subgroups — #35319132

Order 21248.

360 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 99 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	#24909719	C₂	1	2	yes	maximal, complemented
2	#24909713	C₂	1	2	yes	maximal, complemented
3	#24907078	C₂	1	1	yes	maximal, characteristic, complemented
4	#24907069	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₃×#1311	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14581737	—	2	4	no	—
7	C₂×D₁₃₂₈	—	2	4	no	—
8	#19128081	C₂²	1	1	yes	characteristic
9	#19128078	—	2	2	no	—
10	#19128078	C₂²	1	1	yes	characteristic
11	#19128078	C₂²	1	2	yes	—
12	#19128048	—	2	2	no	—
13	C₄×C₁₆×C₈₃	C₂²	1	2	yes	—
14	C₈²×C₈₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₃₂	—	2	2	no	—
16	C₂×D₆₆₄	—	2	2	no	—
17	C₂×D₆₆₄	—	2	2	no	—
18	C₂×D₆₆₄	—	2	2	no	—
19	C₂×D₆₆₄	—	4	4	no	—
20	#14581792	—	2	1	no	—
21	#14581789	—	2	2	no	—
22	#14581789	—	2	1	no	—
23	#14581737	—	4	8	no	—
24	D₁₃₂₈	—	4	8	no	—
25	#14581714	—	4	4	no	—
26	C₂×C₁₆×C₈₃	D₄	1	4	yes	—
27	C₄×C₈×C₈₃	D₄	1	2	yes	—
28	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₆₆	—	4	1	no	—
30	C₂×D₃₃₂	—	4	2	no	—
31	C₂×D₃₃₂	—	4	1	no	—
32	C₂×D₃₃₂	—	4	4	no	—
33	C₂×D₃₃₂	—	4	2	no	—
34	Dic₃₃₂	—	4	4	no	—
35	D₆₆₄	—	4	4	no	—
36	D₆₆₄	—	4	4	no	—
37	D₆₆₄	—	4	4	no	—
38	D₆₆₄	—	8	8	no	—
39	C₄⋊Dic₈₃	—	4	2	no	—
40	C₂×C₈×C₈₃	D₈	1	4	yes	—
41	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
42	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
43	C₁₆×C₈₃	D₈	1	8	yes	complemented
44	C₂²×D₈₃	—	8	2	no	—
45	C₂²×D₈₃	—	8	1	no	—
46	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
47	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
48	C₂×Dic₈₃	—	8	1	no	—
49	D₃₃₂	—	8	2	no	—
50	D₃₃₂	—	8	2	no	—
51	D₃₃₂	—	8	2	no	—
52	D₃₃₂	—	8	8	no	—
53	D₃₃₂	—	8	4	no	—
54	Dic₁₆₆	—	8	2	no	—
55	C₈×C₈₃	—	2	4	no	—
56	C₈×C₈₃	C₂×D₈	1	4	yes	—
57	C₂²×C₈₃	#493	1	1	yes	characteristic
58	C₂×D₈₃	—	16	1	no	—
59	C₂×D₈₃	—	16	2	no	—
60	C₂×D₈₃	—	16	4	no	—
61	C₂×D₈₃	—	16	2	no	—
62	C₄×C₈₃	#493	1	2	yes	—
63	C₄×C₈₃	#509	1	4	yes	—
64	Dic₈₃	—	16	1	no	—
65	#12519	—	83	1	no	maximal, Sylow
66	C₂×C₈₃	#1712	1	1	yes	characteristic
67	C₂×C₈₃	#2250	1	2	yes	—
68	D₈₃	—	32	2	no	—
69	D₈₃	—	32	1	no	—
70	#2248	—	83	2	no	—
71	#2246	—	83	2	no	—
72	#1712	—	83	1	no	—
73	#1709	—	83	1	no	—
74	#1311	D₈₃	1	1	yes	characteristic, complemented
75	C₈₃	#12519	1	1	yes	characteristic, complemented, Sylow
76	C₂×SD₃₂	—	166	4	no	—
77	C₂×D₁₆	—	166	4	no	—
78	#494	—	83	1	no	—
79	#493	—	83	1	no	—
80	#493	—	83	2	no	—
81	#493	—	166	2	no	—
82	#486	—	166	2	no	—
83	C₄×C₁₆	C₂×D₈₃	1	2	yes	—
84	C₈²	C₂×D₈₃	1	1	yes	characteristic
85	C₂×Q₁₆	—	166	2	no	—
86	C₂×D₈	—	332	4	no	—
87	C₂×D₈	—	166	2	no	—
88	C₂×D₈	—	166	2	no	—
89	C₂×D₈	—	166	2	no	—
90	C₄⋊Q₈	—	166	1	no	—
91	#127	—	166	1	no	—
92	#127	—	166	2	no	—
93	SD₃₂	—	332	8	no	—
94	D₁₆	—	332	8	no	—
95	C₂×C₁₆	D₃₃₂	1	4	yes	—
96	#102	—	332	4	no	—
97	C₄×C₈	D₃₃₂	1	2	yes	—
98	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
99	C₂×Q₈	—	332	1	no	—
100	C₂×D₄	—	332	1	no	—
101	C₂×D₄	—	332	2	no	—
102	C₂×D₄	—	332	2	no	—
103	C₂×D₄	—	332	4	no	—
104	Q₁₆	—	332	4	no	—
105	D₈	—	332	4	no	—
106	D₈	—	332	4	no	—
107	D₈	—	332	4	no	—
108	D₈	—	664	8	no	—
109	C₂×C₈	D₆₆₄	1	4	yes	—
110	C₂×C₈	C₂×D₃₃₂	1	2	yes	—
111	C₄⋊C₄	—	332	2	no	—
112	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
113	C₁₆	D₆₆₄	1	8	yes	complemented
114	C₂³	—	664	1	no	—
115	C₂³	—	664	2	no	—
116	Q₈	—	664	2	no	—
117	D₄	—	664	2	no	—
118	D₄	—	664	2	no	—
119	D₄	—	664	2	no	—
120	D₄	—	664	8	no	—
121	D₄	—	664	4	no	—
122	C₂×C₄	—	664	1	no	—
123	C₂×C₄	#14581789	1	1	yes	characteristic
124	C₂×C₄	C₂×D₆₆₄	1	2	yes	—
125	C₈	—	2	4	no	—
126	C₈	C₂×D₆₆₄	1	4	yes	—
127	C₂²	—	1328	1	no	—
128	C₂²	—	1328	2	no	—
129	C₂²	—	1328	4	no	—
130	C₂²	—	1328	2	no	—
131	C₂²	#19128078	1	1	yes	characteristic
132	C₄	—	1328	1	no	—
133	C₄	#19128078	1	2	yes	—
134	C₄	#19128139	1	4	yes	—
135	C₂	—	2656	2	no	—
136	C₂	—	2656	1	no	—
137	C₂	#24907078	1	1	yes	characteristic
138	C₂	#24909729	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24909719

C₂

yes

maximal, complemented

#24909713

C₂

yes

maximal, complemented

#24907078

C₂

yes

maximal, characteristic, complemented

#24907069

C₂

yes

maximal, characteristic, complemented

C₈₃×#1311

C₂

yes

maximal, characteristic, complemented

C₂×#14581737

—

C₂×D₁₃₂₈

—

#19128081

C₂²

yes

characteristic

#19128078

—

#19128078

C₂²

yes

characteristic

#19128078

C₂²

yes

—

#19128048

—

C₄×C₁₆×C₈₃

C₂²

yes

—

C₈²×C₈₃

C₂²

yes

characteristic

C₂×Dic₃₃₂

—

C₂×D₆₆₄

—

C₂×D₆₆₄

—

C₂×D₆₆₄

—

C₂×D₆₆₄

—

—

—

—

—

D₁₃₂₈

—

#14581714

—

C₂×C₁₆×C₈₃

D₄

yes

—

C₄×C₈×C₈₃

D₄

yes

—

C₄×C₈×C₈₃

C₂³

yes

characteristic

C₂×Dic₁₆₆

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

Dic₃₃₂

—

D₆₆₄

—

D₆₆₄

—

D₆₆₄

—

D₆₆₄

—

C₄⋊Dic₈₃

—

C₂×C₈×C₈₃

D₈

yes

—

C₂×C₈×C₈₃

C₂×D₄

yes

—

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₁₆×C₈₃

D₈

yes

complemented

C₂²×D₈₃

—

C₂²×D₈₃

—

C₂×C₄×C₈₃

#127

yes

characteristic

C₂×C₄×C₈₃

C₂×D₈

yes

—

C₂×Dic₈₃

—

D₃₃₂

—

D₃₃₂

—

D₃₃₂

—

D₃₃₂

—

D₃₃₂

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

C₂×D₈

yes

—

C₂²×C₈₃

#493

yes

characteristic

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

#493

yes

—

C₄×C₈₃

#509

yes

—

Dic₈₃

—

—

C₂×C₈₃

yes

C₂×C₈₃

yes

—

D₈₃

—

D₈₃

—

—

—

—

—

D₈₃

yes

characteristic, complemented

C₈₃

#12519

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

166

—

C₂×D₁₆

—

166

—

—

—

—

—

166

—

#486

—

166

—

C₄×C₁₆

C₂×D₈₃

yes

—

C₈²

C₂×D₈₃

yes

characteristic

C₂×Q₁₆

—

166

—

C₂×D₈

—

332

—

C₂×D₈

—

166

—

C₂×D₈

—

166

—

C₂×D₈

—

166

—

C₄⋊Q₈

—

166

—

#127

—

166

—

#127

—

166

—

SD₃₂

—

332

—

D₁₆

—

332

—

C₂×C₁₆

D₃₃₂

yes

—

#102

—

332

—

C₄×C₈

D₃₃₂

yes

—

C₄×C₈

C₂²×D₈₃

yes

characteristic

C₂×Q₈

—

332

—

100

C₂×D₄

—

332

—

101

C₂×D₄

—

332

—

102

C₂×D₄

—

332

—

103

C₂×D₄

—

332

—

104

Q₁₆

—

332

—

105

D₈

—

332

—

106

D₈

—

332

—

107

D₈

—

332

—

108

D₈

—

664

—

109

C₂×C₈

D₆₆₄

yes

—

110

C₂×C₈

C₂×D₃₃₂

yes

—

111

C₄⋊C₄

—

332

—

112

C₄²

C₂×D₃₃₂

yes

characteristic

113

C₁₆

D₆₆₄

yes

complemented

114

C₂³

—

664

—

115

C₂³

—

664

—

116

Q₈

—

664

—

117

D₄

—

664

—

118

D₄

—

664

—

119

D₄

—

664

—

120

D₄

—

664

—

121

D₄

—

664

—

122

C₂×C₄

—

664

—

123

C₂×C₄

#14581789

yes

characteristic

124

C₂×C₄

C₂×D₆₆₄

yes

—

125

C₈

—

126

C₈

C₂×D₆₆₄

yes

—

127

C₂²

—

1328

—

128

C₂²

—

1328

—

129

C₂²

—

1328

—

130

C₂²

—

1328

—

131

C₂²

#19128078

yes

characteristic

132

C₄

—

1328

—

133

C₄

#19128078

yes

—

134

C₄

#19128139

yes

—

135

C₂

—

2656

—

136

C₂

—

2656

—

137

C₂

#24907078

yes

characteristic

138

C₂

#24909729

yes

—

	subgroup	G/H
center	C₂²	#19128078
commutator, cosocle	C₄×C₈×C₈₃	C₂³
Frattini	C₄×C₈	C₂²×D₈₃
Fitting	C₈₃×#1311	C₂
socle	C₂²×C₈₃	#493
hypercenter	#1311	D₈₃
hypocenter	C₈₃	#12519
LCS	G ▷ C₄×C₈×C₈₃ ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂²

#19128078

commutator, cosocle

C₄×C₈×C₈₃

C₂³

Frattini

C₄×C₈

C₂²×D₈₃

Fitting

C₈₃×#1311

C₂

C₂²×C₈₃

D₈₃

C₈₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

N	H ≅ G/N	type
1. #24909719	C₂	semidirect
2. #24909713	C₂	semidirect
3. #24907078	C₂	semidirect
4. #24907069	C₂	semidirect
5. C₈₃×#1311	C₂	semidirect
43. C₁₆×C₈₃	D₈	semidirect
74. #1311	D₈₃	semidirect
75. C₈₃	#12519	semidirect
113. C₁₆	D₆₆₄	semidirect

H ≅ G/N

type

1. #24909719

C₂

semidirect

2. #24909713

C₂

semidirect

3. #24907078

C₂

semidirect

4. #24907069

C₂

semidirect

5. C₈₃×#1311

C₂

semidirect

43. C₁₆×C₈₃

D₈

semidirect

74. #1311

D₈₃

semidirect

75. C₈₃

#12519

semidirect

113. C₁₆

D₆₆₄

semidirect

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

subgroup	index
#24909719	2
#24909713	2
#24907078	2
#24907069	2
C₈₃×#1311	2
#12519	83

subgroup

index

C₈₃×#1311

Subgroups — #35319132

Conjugacy classes of subgroups

Complements

Maximal subgroups