Group #35319138 — subgroups

Subgroups — #35319138

Order 21248.

316 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 87 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	#24909718	C₂	1	2	yes	maximal
2	#24909715	C₂	1	2	yes	maximal, complemented
3	#24907081	C₂	1	1	yes	maximal, characteristic, complemented
4	#24905380	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₃×#1709	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581738	—	2	4	no	—
7	C₂×#14581736	—	2	4	no	—
8	#19128106	—	2	2	no	—
9	#19128081	C₂²	1	2	yes	—
10	C₄×C₈₃⋊C₁₆	C₂²	1	2	yes	—
11	#494×C₈₃	C₂²	1	1	yes	characteristic
12	#493×C₈₃	C₂²	1	1	yes	characteristic
13	#486×C₈₃	—	2	2	no	—
14	C₈²×C₈₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₃₂	—	2	2	no	—
16	C₂×Dic₃₃₂	—	2	2	no	—
17	#14581792	—	2	2	no	—
18	#14581738	—	4	8	no	—
19	#14581736	—	4	8	no	—
20	C₂×C₈₃⋊C₁₆	D₄	1	4	yes	—
21	#14581726	—	4	4	no	—
22	C₂×Q₁₆×C₈₃	—	2	2	no	—
23	C₂×D₈×C₈₃	—	2	2	no	—
24	C₄⋊Q₈×C₈₃	—	2	1	no	—
25	#127×C₈₃	—	2	1	no	—
26	#102×C₈₃	—	4	4	no	—
27	C₄×C₈×C₈₃	D₄	1	2	yes	—
28	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₆₆	—	4	2	no	—
30	Dic₃₃₂	—	4	4	no	—
31	Dic₃₃₂	—	4	4	no	—
32	C₄⋊Dic₈₃	—	4	4	no	—
33	C₈₃⋊C₁₆	D₈	1	8	yes	—
34	C₂×Q₈×C₈₃	—	4	1	no	—
35	C₂×D₄×C₈₃	—	4	1	no	—
36	C₂×D₄×C₈₃	—	4	2	no	—
37	Q₁₆×C₈₃	—	4	4	no	—
38	D₈×C₈₃	—	4	4	no	—
39	C₂×C₈×C₈₃	D₈	1	4	yes	—
40	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
41	C₄⋊C₄×C₈₃	—	4	2	no	—
42	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
43	C₂³×C₈₃	—	8	1	no	—
44	Q₈×C₈₃	—	8	2	no	—
45	D₄×C₈₃	—	8	2	no	—
46	D₄×C₈₃	—	8	4	no	—
47	C₂×C₄×C₈₃	—	8	1	no	—
48	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
49	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
50	C₂×Dic₈₃	—	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	—	16	1	no	—
56	C₂²×C₈₃	—	16	2	no	—
57	C₂²×C₈₃	#493	1	1	yes	characteristic
58	C₄×C₈₃	—	16	1	no	—
59	C₄×C₈₃	#493	1	2	yes	—
60	C₄×C₈₃	#510	1	4	yes	—
61	Dic₈₃	—	16	2	no	—
62	#12520	—	83	1	no	maximal, Sylow
63	C₂×C₈₃	—	32	1	no	—
64	C₂×C₈₃	#1712	1	1	yes	characteristic
65	C₂×C₈₃	#2251	1	2	yes	—
66	#2248	—	83	2	no	—
67	#2247	—	83	2	no	—
68	#1713	—	83	1	no	—
69	#1709	D₈₃	1	1	yes	characteristic
70	#1311	—	83	1	no	—
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	2	no	—
76	#494	C₂×D₈₃	1	1	yes	characteristic
77	#493	C₂×D₈₃	1	1	yes	characteristic
78	#486	—	2	2	no	—
79	C₄×C₁₆	—	83	2	no	—
80	C₈²	C₂×D₈₃	1	1	yes	characteristic
81	C₂×Q₁₆	—	2	2	no	—
82	C₂×Q₁₆	—	166	2	no	—
83	C₂×Q₁₆	—	166	2	no	—
84	C₂×D₈	—	2	2	no	—
85	C₄⋊Q₈	—	2	1	no	—
86	C₄⋊Q₈	—	166	2	no	—
87	#127	—	2	1	no	—
88	Q₃₂	—	332	8	no	—
89	SD₃₂	—	332	8	no	—
90	C₂×C₁₆	—	83	4	no	—
91	#107	—	332	4	no	—
92	#102	—	4	4	no	—
93	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
94	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
95	C₂×Q₈	—	4	1	no	—
96	C₂×Q₈	—	332	2	no	—
97	C₂×D₄	—	4	1	no	—
98	C₂×D₄	—	4	2	no	—
99	Q₁₆	—	4	4	no	—
100	Q₁₆	—	332	4	no	—
101	Q₁₆	—	332	4	no	—
102	D₈	—	4	4	no	—
103	C₂×C₈	#13050575	1	4	yes	—
104	C₂×C₈	D₄×D₈₃	1	2	yes	—
105	C₄⋊C₄	—	4	2	no	—
106	C₄⋊C₄	—	332	4	no	—
107	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
108	C₁₆	—	83	8	no	—
109	C₂³	—	8	1	no	—
110	Q₈	—	8	2	no	—
111	Q₈	—	664	2	no	—
112	Q₈	—	664	2	no	—
113	D₄	—	8	2	no	—
114	D₄	—	8	4	no	—
115	C₂×C₄	—	8	1	no	—
116	C₂×C₄	—	664	2	no	—
117	C₂×C₄	#14581788	1	1	yes	characteristic
118	C₂×C₄	C₂×#13050575	1	2	yes	—
119	C₈	—	2	4	no	—
120	C₈	D₈×D₈₃	1	4	yes	—
121	C₂²	—	16	1	no	—
122	C₂²	—	16	2	no	—
123	C₂²	#19128077	1	1	yes	characteristic
124	C₄	—	16	1	no	—
125	C₄	—	1328	2	no	—
126	C₄	#19128076	1	2	yes	—
127	C₄	#19128141	1	4	yes	—
128	C₂	—	32	1	no	—
129	C₂	#24907077	1	1	yes	characteristic
130	C₂	#24909735	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24909718

C₂

yes

maximal

#24909715

C₂

yes

maximal, complemented

#24907081

C₂

yes

maximal, characteristic, complemented

#24905380

C₂

yes

maximal, characteristic, complemented

C₈₃×#1709

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

C₄×C₈₃⋊C₁₆

C₂²

yes

—

#494×C₈₃

C₂²

yes

characteristic

#493×C₈₃

C₂²

yes

characteristic

#486×C₈₃

—

C₈²×C₈₃

C₂²

yes

characteristic

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

#14581792

—

#14581738

—

#14581736

—

C₂×C₈₃⋊C₁₆

D₄

yes

—

#14581726

—

C₂×Q₁₆×C₈₃

—

C₂×D₈×C₈₃

—

C₄⋊Q₈×C₈₃

—

#127×C₈₃

—

#102×C₈₃

—

C₄×C₈×C₈₃

D₄

yes

—

C₄×C₈×C₈₃

C₂³

yes

characteristic

C₂×Dic₁₆₆

—

Dic₃₃₂

—

Dic₃₃₂

—

C₄⋊Dic₈₃

—

C₈₃⋊C₁₆

D₈

yes

—

C₂×Q₈×C₈₃

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

—

Q₁₆×C₈₃

—

D₈×C₈₃

—

C₂×C₈×C₈₃

D₈

yes

—

C₂×C₈×C₈₃

C₂×D₄

yes

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₂³×C₈₃

—

Q₈×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

#127

yes

characteristic

C₂×C₄×C₈₃

C₂×D₈

yes

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

C₂×D₈

yes

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#493

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

#493

yes

—

C₄×C₈₃

#510

yes

—

Dic₈₃

—

#12520

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

#1712

yes

characteristic

C₂×C₈₃

#2251

yes

—

—

—

—

D₈₃

yes

characteristic

#1311

—

C₈₃

#12520

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

166

—

C₂×SD₃₂

—

166

—

#500

—

166

—

#494

—

#494

C₂×D₈₃

yes

characteristic

#493

C₂×D₈₃

yes

characteristic

#486

—

C₄×C₁₆

—

C₈²

C₂×D₈₃

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

166

—

C₂×Q₁₆

—

166

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

166

—

#127

—

Q₃₂

—

332

—

SD₃₂

—

332

—

C₂×C₁₆

—

#107

—

332

—

#102

—

C₄×C₈

C₂²⋊D₈₃

yes

—

C₄×C₈

C₂²×D₈₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

332

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

100

Q₁₆

—

332

—

101

Q₁₆

—

332

—

102

D₈

—

103

C₂×C₈

#13050575

yes

—

104

C₂×C₈

D₄×D₈₃

yes

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

332

—

107

C₄²

C₂×C₂²⋊D₈₃

yes

characteristic

108

C₁₆

—

109

C₂³

—

110

Q₈

—

111

Q₈

—

664

—

112

Q₈

—

664

—

113

D₄

—

114

D₄

—

115

C₂×C₄

—

116

C₂×C₄

—

664

—

117

C₂×C₄

#14581788

yes

characteristic

118

C₂×C₄

C₂×#13050575

yes

—

119

C₈

—

120

C₈

D₈×D₈₃

yes

—

121

C₂²

—

122

C₂²

—

123

C₂²

#19128077

yes

characteristic

124

C₄

—

125

C₄

—

1328

—

126

C₄

#19128076

yes

—

127

C₄

#19128141

yes

—

128

C₂

—

129

C₂

#24907077

yes

characteristic

130

C₂

#24909735

yes

—

	subgroup	G/H
center	C₂²	#19128077
commutator, cosocle	C₄×C₈×C₈₃	C₂³
Frattini	C₄×C₈	C₂²×D₈₃
Fitting	C₈₃×#1709	C₂
socle	C₂²×C₈₃	#493
prime radical	#493×C₈₃	C₂²
hypercenter	#1709	D₈₃
hypocenter	C₈₃	#12520
LCS	G ▷ C₄×C₈×C₈₃ ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

subgroup

G/H

center

C₂²

#19128077

commutator, cosocle

C₄×C₈×C₈₃

C₂³

Frattini

C₄×C₈

C₂²×D₈₃

Fitting

C₈₃×#1709

C₂

socle

C₂²×C₈₃

#493

prime radical

#493×C₈₃

C₂²

D₈₃

C₈₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

N	H ≅ G/N	type
2. #24909715	C₂	semidirect
3. #24907081	C₂	semidirect
4. #24905380	C₂	semidirect
71. C₈₃	#12520	semidirect

H ≅ G/N

type

2. #24909715

C₂

semidirect

3. #24907081

C₂

semidirect

4. #24905380

C₂

semidirect

71. C₈₃

#12520

semidirect

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

subgroup	index
#24909718	2
#24909715	2
#24907081	2
#24905380	2
C₈₃×#1709	2
#12520	83

subgroup

index

C₈₃×#1709

Subgroups — #35319138

Conjugacy classes of subgroups

Complements

Maximal subgroups