Group #35319137 — subgroups

Subgroups — #35319137

Order 21248.

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

normal subgroups only

This group has 316 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
2	#24909716	C₂	1	2	yes	maximal, complemented
3	#24907081	C₂	1	1	yes	maximal, characteristic, complemented
4	#24907069	C₂	1	1	yes	maximal, characteristic
5	C₈₃×#1311	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₆₆₄	—	2	4	no	—
7	C₂×#14581737	—	2	4	no	—
8	#19128106	—	2	2	no	—
9	#19128081	C₂²	1	1	yes	characteristic
10	#19128081	C₂²	1	2	yes	—
11	#19128078	C₂²	1	1	yes	characteristic
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×Dic₃₃₂	—	2	2	no	—
17	C₂×Dic₃₃₂	—	2	2	no	—
18	C₂×D₆₆₄	—	2	2	no	—
19	#14581792	—	2	2	no	—
20	#14581792	—	2	1	no	—
21	#14581789	—	2	1	no	—
22	Dic₆₆₄	—	4	8	no	—
23	#14581737	—	4	8	no	—
24	#14581726	—	4	4	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	2	no	—
30	C₂×Dic₁₆₆	—	4	1	no	—
31	C₂×D₃₃₂	—	4	1	no	—
32	C₂×D₃₃₂	—	4	2	no	—
33	Dic₃₃₂	—	4	4	no	—
34	Dic₃₃₂	—	4	4	no	—
35	Dic₃₃₂	—	4	4	no	—
36	D₆₆₄	—	4	4	no	—
37	C₄⋊Dic₈₃	—	4	2	no	—
38	C₄⋊Dic₈₃	—	4	4	no	—
39	C₂×C₈×C₈₃	D₈	1	4	yes	—
40	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
41	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₈₃	D₈	1	8	yes	—
43	C₂²×D₈₃	—	8	1	no	—
44	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
45	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
46	C₂×Dic₈₃	—	8	1	no	—
47	C₂×Dic₈₃	—	8	2	no	—
48	D₃₃₂	—	8	2	no	—
49	D₃₃₂	—	8	4	no	—
50	Dic₁₆₆	—	8	2	no	—
51	Dic₁₆₆	—	8	2	no	—
52	Dic₁₆₆	—	8	2	no	—
53	C₈×C₈₃	—	2	4	no	—
54	C₈×C₈₃	C₂×D₈	1	4	yes	—
55	C₂²×C₈₃	#493	1	1	yes	characteristic
56	C₂×D₈₃	—	16	1	no	—
57	C₂×D₈₃	—	16	2	no	—
58	C₄×C₈₃	#493	1	2	yes	—
59	C₄×C₈₃	#510	1	4	yes	—
60	Dic₈₃	—	16	2	no	—
61	Dic₈₃	—	16	1	no	—
62	#12520	—	83	1	no	maximal, Sylow
63	C₂×C₈₃	#1712	1	1	yes	characteristic
64	C₂×C₈₃	#2251	1	2	yes	—
65	D₈₃	—	32	1	no	—
66	#2248	—	83	2	no	—
67	#2247	—	83	2	no	—
68	#1713	—	83	1	no	—
69	#1709	—	83	1	no	—
70	#1311	D₈₃	1	1	yes	characteristic, complemented
71	C₈₃	#12520	1	1	yes	characteristic, complemented, Sylow
72	C₂×Q₃₂	—	166	4	no	—
73	C₂×SD₃₂	—	166	4	no	—
74	#500	—	166	2	no	—
75	#494	—	83	1	no	—
76	#494	—	83	2	no	—
77	#493	—	83	1	no	—
78	#486	—	166	2	no	—
79	C₄×C₁₆	C₂×D₈₃	1	2	yes	—
80	C₈²	C₂×D₈₃	1	1	yes	characteristic
81	C₂×Q₁₆	—	166	2	no	—
82	C₂×Q₁₆	—	166	2	no	—
83	C₂×Q₁₆	—	166	2	no	—
84	C₂×D₈	—	166	2	no	—
85	C₄⋊Q₈	—	166	1	no	—
86	C₄⋊Q₈	—	166	2	no	—
87	#127	—	166	1	no	—
88	Q₃₂	—	332	8	no	—
89	SD₃₂	—	332	8	no	—
90	C₂×C₁₆	D₃₃₂	1	4	yes	—
91	#107	—	332	4	no	—
92	#102	—	332	4	no	—
93	C₄×C₈	D₃₃₂	1	2	yes	—
94	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
95	C₂×Q₈	—	332	1	no	—
96	C₂×Q₈	—	332	2	no	—
97	C₂×D₄	—	332	1	no	—
98	C₂×D₄	—	332	2	no	—
99	Q₁₆	—	332	4	no	—
100	Q₁₆	—	332	4	no	—
101	Q₁₆	—	332	4	no	—
102	D₈	—	332	4	no	—
103	C₂×C₈	D₆₆₄	1	4	yes	—
104	C₂×C₈	C₂×D₃₃₂	1	2	yes	—
105	C₄⋊C₄	—	332	2	no	—
106	C₄⋊C₄	—	332	4	no	—
107	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
108	C₁₆	D₆₆₄	1	8	yes	—
109	C₂³	—	664	1	no	—
110	Q₈	—	664	2	no	—
111	Q₈	—	664	2	no	—
112	Q₈	—	664	2	no	—
113	D₄	—	664	2	no	—
114	D₄	—	664	4	no	—
115	C₂×C₄	—	664	1	no	—
116	C₂×C₄	—	664	2	no	—
117	C₂×C₄	#14581789	1	1	yes	characteristic
118	C₂×C₄	C₂×D₆₆₄	1	2	yes	—
119	C₈	—	2	4	no	—
120	C₈	C₂×D₆₆₄	1	4	yes	—
121	C₂²	—	1328	1	no	—
122	C₂²	—	1328	2	no	—
123	C₂²	#19128078	1	1	yes	characteristic
124	C₄	—	1328	1	no	—
125	C₄	—	1328	2	no	—
126	C₄	#19128078	1	2	yes	—
127	C₄	#19128144	1	4	yes	—
128	C₂	—	2656	1	no	—
129	C₂	#24907078	1	1	yes	characteristic
130	C₂	#24909734	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24909719

C₂

yes

maximal

#24909716

C₂

yes

maximal, complemented

#24907081

C₂

yes

maximal, characteristic, complemented

#24907069

C₂

yes

maximal, characteristic

C₈₃×#1311

C₂

yes

maximal, characteristic, complemented

C₂×Dic₆₆₄

—

C₂×#14581737

—

#19128106

—

#19128081

C₂²

yes

characteristic

#19128081

C₂²

yes

—

#19128078

C₂²

yes

characteristic

#19128048

—

C₄×C₁₆×C₈₃

C₂²

yes

—

C₈²×C₈₃

C₂²

yes

characteristic

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂×D₆₆₄

—

#14581792

—

#14581792

—

#14581789

—

Dic₆₆₄

—

#14581737

—

#14581726

—

#14581714

—

C₂×C₁₆×C₈₃

D₄

yes

—

C₄×C₈×C₈₃

D₄

yes

—

C₄×C₈×C₈₃

C₂³

yes

characteristic

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

Dic₃₃₂

—

Dic₃₃₂

—

Dic₃₃₂

—

D₆₆₄

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₂×C₈×C₈₃

D₈

yes

—

C₂×C₈×C₈₃

C₂×D₄

yes

—

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₁₆×C₈₃

D₈

yes

—

C₂²×D₈₃

—

C₂×C₄×C₈₃

#127

yes

characteristic

C₂×C₄×C₈₃

C₂×D₈

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

D₃₃₂

—

D₃₃₂

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

C₂×D₈

yes

—

C₂²×C₈₃

#493

yes

characteristic

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

#493

yes

—

C₄×C₈₃

#510

yes

—

Dic₈₃

—

Dic₈₃

—

—

C₂×C₈₃

yes

C₂×C₈₃

yes

—

D₈₃

—

—

—

—

—

D₈₃

yes

characteristic, complemented

C₈₃

#12520

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

166

—

C₂×SD₃₂

—

166

—

#500

—

166

—

—

—

—

—

166

—

C₄×C₁₆

C₂×D₈₃

yes

—

C₈²

C₂×D₈₃

yes

characteristic

C₂×Q₁₆

—

166

—

C₂×Q₁₆

—

166

—

C₂×Q₁₆

—

166

—

C₂×D₈

—

166

—

C₄⋊Q₈

—

166

—

C₄⋊Q₈

—

166

—

#127

—

166

—

Q₃₂

—

332

—

SD₃₂

—

332

—

C₂×C₁₆

D₃₃₂

yes

—

#107

—

332

—

#102

—

332

—

C₄×C₈

D₃₃₂

yes

—

C₄×C₈

C₂²×D₈₃

yes

characteristic

C₂×Q₈

—

332

—

C₂×Q₈

—

332

—

C₂×D₄

—

332

—

C₂×D₄

—

332

—

Q₁₆

—

332

—

100

Q₁₆

—

332

—

101

Q₁₆

—

332

—

102

D₈

—

332

—

103

C₂×C₈

D₆₆₄

yes

—

104

C₂×C₈

C₂×D₃₃₂

yes

—

105

C₄⋊C₄

—

332

—

106

C₄⋊C₄

—

332

—

107

C₄²

C₂×D₃₃₂

yes

characteristic

108

C₁₆

D₆₆₄

yes

—

109

C₂³

—

664

—

110

Q₈

—

664

—

111

Q₈

—

664

—

112

Q₈

—

664

—

113

D₄

—

664

—

114

D₄

—

664

—

115

C₂×C₄

—

664

—

116

C₂×C₄

—

664

—

117

C₂×C₄

#14581789

yes

characteristic

118

C₂×C₄

C₂×D₆₆₄

yes

—

119

C₈

—

120

C₈

C₂×D₆₆₄

yes

—

121

C₂²

—

1328

—

122

C₂²

—

1328

—

123

C₂²

#19128078

yes

characteristic

124

C₄

—

1328

—

125

C₄

—

1328

—

126

C₄

#19128078

yes

—

127

C₄

#19128144

yes

—

128

C₂

—

2656

—

129

C₂

#24907078

yes

characteristic

130

C₂

#24909734

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
prime radical	#19128078	C₂²
hypercenter	#1311	D₈₃
hypocenter	C₈₃	#12520
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

C₂²

D₈₃

C₈₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

N	H ≅ G/N	type
2. #24909716	C₂	semidirect
3. #24907081	C₂	semidirect
5. C₈₃×#1311	C₂	semidirect
70. #1311	D₈₃	semidirect
71. C₈₃	#12520	semidirect

H ≅ G/N

type

2. #24909716

C₂

semidirect

3. #24907081

C₂

semidirect

5. C₈₃×#1311

C₂

semidirect

70. #1311

D₈₃

semidirect

71. C₈₃

#12520

semidirect

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

subgroup	index
#24909719	2
#24909716	2
#24907081	2
#24907069	2
C₈₃×#1311	2
#12520	83

subgroup

index

C₈₃×#1311

Subgroups — #35319137

Conjugacy classes of subgroups

Complements

Maximal subgroups