Group #35325401 — subgroups

Subgroups — #35325401

Order 21248.

230 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 79 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	C₂×#19127541	C₂	1	4	yes	maximal
2	#24909716	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈₃⋊C₃₂	C₂	1	1	yes	maximal, characteristic
4	C₈₃×#2247	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₆₆₄	—	2	1	no	—
6	C₂×Dic₆₆₄	C₂²	1	2	yes	—
7	#19128081	—	2	1	no	—
8	#19127541	—	2	8	no	—
9	C₂×C₈₃⋊C₃₂	C₂²	1	2	yes	—
10	C₂×Q₃₂×C₈₃	—	2	1	no	—
11	C₂×Q₃₂×C₈₃	C₂²	1	2	yes	—
12	#494×C₈₃	—	2	1	no	—
13	C₄×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
14	C₂×Dic₃₃₂	—	2	2	no	—
15	C₂×Dic₃₃₂	—	4	1	no	—
16	#14581792	—	4	1	no	—
17	Dic₆₆₄	—	2	4	no	—
18	Dic₆₆₄	—	4	2	no	—
19	C₈₃⋊C₃₂	D₄	1	4	yes	—
20	C₂×Q₁₆×C₈₃	—	2	2	no	—
21	C₂×Q₁₆×C₈₃	—	4	1	no	—
22	C₄⋊Q₈×C₈₃	—	4	1	no	—
23	Q₃₂×C₈₃	—	2	4	no	—
24	Q₃₂×C₈₃	—	4	2	no	—
25	C₂×C₁₆×C₈₃	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₈₃	C₂³	1	1	yes	characteristic
27	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
28	C₂×Dic₁₆₆	—	4	2	no	—
29	Dic₃₃₂	—	4	2	no	—
30	Dic₃₃₂	—	4	2	no	—
31	Dic₃₃₂	—	8	2	no	—
32	C₄⋊Dic₈₃	—	8	2	no	—
33	C₂×Q₈×C₈₃	—	4	2	no	—
34	Q₁₆×C₈₃	—	4	2	no	—
35	Q₁₆×C₈₃	—	4	2	no	—
36	Q₁₆×C₈₃	—	8	2	no	—
37	C₂×C₈×C₈₃	D₈	1	1	yes	characteristic
38	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
39	C₄⋊C₄×C₈₃	—	8	2	no	—
40	C₄²×C₈₃	D₈	1	1	yes	characteristic
41	C₁₆×C₈₃	D₈	1	2	yes	—
42	C₁₆×C₈₃	C₂×D₄	1	2	yes	—
43	Q₈×C₈₃	—	8	2	no	—
44	Q₈×C₈₃	—	8	2	no	—
45	C₂×C₄×C₈₃	—	8	2	no	—
46	C₂×C₄×C₈₃	D₁₆	1	2	yes	—
47	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
48	C₂×Dic₈₃	—	8	2	no	—
49	Dic₁₆₆	—	8	2	no	—
50	Dic₁₆₆	—	8	2	no	—
51	C₈×C₈₃	D₁₆	1	2	yes	—
52	C₈×C₈₃	#127	1	1	yes	characteristic
53	C₈×C₈₃	C₂×D₈	1	1	yes	characteristic
54	C₂²×C₈₃	C₂×D₁₆	1	1	yes	characteristic
55	C₄×C₈₃	—	16	2	no	—
56	C₄×C₈₃	Q₆₄	1	4	yes	complemented
57	C₄×C₈₃	#493	1	1	yes	characteristic
58	C₄×C₈₃	C₂×D₁₆	1	1	yes	characteristic
59	Dic₈₃	—	16	2	no	—
60	#13726	—	83	1	no	maximal, Sylow
61	C₂×C₈₃	#2246	1	1	yes	characteristic
62	C₂×C₈₃	C₂×Q₆₄	1	2	yes	—
63	C₂×Q₆₄	—	83	4	no	—
64	#2247	—	83	1	no	—
65	#2247	D₈₃	1	1	yes	characteristic
66	C₄×C₃₂	—	83	1	no	—
67	C₈₃	#13726	1	1	yes	characteristic, complemented, Sylow
68	C₂×Q₃₂	—	2	1	no	—
69	C₂×Q₃₂	—	83	2	no	—
70	C₂×Q₃₂	—	166	1	no	—
71	C₂×Q₃₂	C₂×D₈₃	1	2	yes	—
72	#494	—	2	1	no	—
73	#494	—	166	1	no	—
74	Q₆₄	—	166	8	no	—
75	C₂×C₃₂	—	83	2	no	—
76	C₄×C₁₆	C₂×D₈₃	1	1	yes	characteristic
77	C₂×Q₁₆	—	2	2	no	—
78	C₂×Q₁₆	—	4	1	no	—
79	C₂×Q₁₆	—	332	1	no	—
80	C₂×Q₁₆	—	166	2	no	—
81	C₄⋊Q₈	—	4	1	no	—
82	C₄⋊Q₈	—	332	1	no	—
83	Q₃₂	—	2	4	no	—
84	Q₃₂	—	4	2	no	—
85	Q₃₂	—	332	2	no	—
86	Q₃₂	—	166	4	no	—
87	C₂×C₁₆	C₂²⋊D₈₃	1	1	yes	characteristic
88	C₂×C₁₆	C₂²×D₈₃	1	1	yes	characteristic
89	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
90	C₃₂	—	83	4	no	—
91	C₂×Q₈	—	4	2	no	—
92	C₂×Q₈	—	332	2	no	—
93	Q₁₆	—	4	2	no	—
94	Q₁₆	—	4	2	no	—
95	Q₁₆	—	8	2	no	—
96	Q₁₆	—	332	2	no	—
97	Q₁₆	—	332	2	no	—
98	Q₁₆	—	664	2	no	—
99	C₂×C₈	#13050575	1	1	yes	characteristic
100	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
101	C₄⋊C₄	—	8	2	no	—
102	C₄⋊C₄	—	664	2	no	—
103	C₄²	#13050575	1	1	yes	characteristic
104	C₁₆	#13050575	1	2	yes	—
105	C₁₆	D₄×D₈₃	1	2	yes	—
106	Q₈	—	8	2	no	—
107	Q₈	—	8	2	no	—
108	Q₈	—	664	2	no	—
109	Q₈	—	664	2	no	—
110	C₂×C₄	—	8	2	no	—
111	C₂×C₄	—	664	2	no	—
112	C₂×C₄	#14581733	1	2	yes	—
113	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
114	C₈	#14581733	1	2	yes	—
115	C₈	#14581788	1	1	yes	characteristic
116	C₈	C₂×#13050575	1	1	yes	characteristic
117	C₂²	C₂×#14581733	1	1	yes	characteristic
118	C₄	—	16	2	no	—
119	C₄	—	1328	2	no	—
120	C₄	#19127541	1	4	yes	complemented
121	C₄	#19128077	1	1	yes	characteristic
122	C₄	C₂×#14581733	1	1	yes	characteristic
123	C₂	#24909712	1	1	yes	characteristic
124	C₂	C₂×#19127541	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19127541

C₂

yes

maximal

#24909716

C₂

yes

maximal, characteristic

C₄×C₈₃⋊C₃₂

C₂

yes

maximal, characteristic

C₈₃×#2247

C₂

yes

maximal, characteristic

C₂×Dic₆₆₄

—

C₂×Dic₆₆₄

C₂²

yes

—

#19128081

—

#19127541

—

C₂×C₈₃⋊C₃₂

C₂²

yes

—

C₂×Q₃₂×C₈₃

—

C₂×Q₃₂×C₈₃

C₂²

yes

—

#494×C₈₃

—

C₄×C₁₆×C₈₃

C₂²

yes

characteristic

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

#14581792

—

Dic₆₆₄

—

Dic₆₆₄

—

C₈₃⋊C₃₂

D₄

yes

—

C₂×Q₁₆×C₈₃

—

C₂×Q₁₆×C₈₃

—

C₄⋊Q₈×C₈₃

—

Q₃₂×C₈₃

—

Q₃₂×C₈₃

—

C₂×C₁₆×C₈₃

D₄

yes

characteristic

C₂×C₁₆×C₈₃

C₂³

yes

characteristic

C₄×C₈×C₈₃

D₄

yes

characteristic

C₂×Dic₁₆₆

—

Dic₃₃₂

—

Dic₃₃₂

—

Dic₃₃₂

—

C₄⋊Dic₈₃

—

C₂×Q₈×C₈₃

—

Q₁₆×C₈₃

—

Q₁₆×C₈₃

—

Q₁₆×C₈₃

—

C₂×C₈×C₈₃

D₈

yes

characteristic

C₂×C₈×C₈₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

D₈

yes

characteristic

C₁₆×C₈₃

D₈

yes

—

C₁₆×C₈₃

C₂×D₄

yes

—

Q₈×C₈₃

—

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

D₁₆

yes

—

C₂×C₄×C₈₃

C₂×D₈

yes

characteristic

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

D₁₆

yes

—

C₈×C₈₃

#127

yes

characteristic

C₈×C₈₃

C₂×D₈

yes

characteristic

C₂²×C₈₃

C₂×D₁₆

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

Q₆₄

yes

complemented

C₄×C₈₃

#493

yes

characteristic

C₄×C₈₃

C₂×D₁₆

yes

characteristic

Dic₈₃

—

—

C₂×C₈₃

yes

C₂×C₈₃

C₂×Q₆₄

yes

—

C₂×Q₆₄

—

#2247

—

#2247

D₈₃

yes

characteristic

C₄×C₃₂

—

C₈₃

#13726

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

166

—

C₂×Q₃₂

C₂×D₈₃

yes

—

#494

—

#494

—

166

—

Q₆₄

—

166

—

C₂×C₃₂

—

C₄×C₁₆

C₂×D₈₃

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

332

—

C₂×Q₁₆

—

166

—

C₄⋊Q₈

—

C₄⋊Q₈

—

332

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

332

—

Q₃₂

—

166

—

C₂×C₁₆

C₂²⋊D₈₃

yes

characteristic

C₂×C₁₆

C₂²×D₈₃

yes

characteristic

C₄×C₈

C₂²⋊D₈₃

yes

characteristic

C₃₂

—

C₂×Q₈

—

C₂×Q₈

—

332

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

332

—

Q₁₆

—

332

—

Q₁₆

—

664

—

C₂×C₈

#13050575

yes

characteristic

100

C₂×C₈

C₂×C₂²⋊D₈₃

yes

characteristic

101

C₄⋊C₄

—

102

C₄⋊C₄

—

664

—

103

C₄²

#13050575

yes

characteristic

104

C₁₆

#13050575

yes

—

105

C₁₆

D₄×D₈₃

yes

—

106

Q₈

—

107

Q₈

—

108

Q₈

—

664

—

109

Q₈

—

664

—

110

C₂×C₄

—

111

C₂×C₄

—

664

—

112

C₂×C₄

#14581733

yes

—

113

C₂×C₄

C₂×#13050575

yes

characteristic

114

C₈

#14581733

yes

—

115

C₈

#14581788

yes

characteristic

116

C₈

C₂×#13050575

yes

characteristic

117

C₂²

C₂×#14581733

yes

characteristic

118

C₄

—

119

C₄

—

1328

—

120

C₄

#19127541

yes

complemented

121

C₄

#19128077

yes

characteristic

122

C₄

C₂×#14581733

yes

characteristic

123

C₂

#24909712

yes

characteristic

124

C₂

C₂×#19127541

yes

—

	subgroup	G/H
center	C₂²	C₂×#14581733
commutator, cosocle	C₂×C₁₆×C₈₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₈₃
Fitting	C₈₃×#2247	C₂
socle, prime radical	C₂²×C₈₃	C₂×D₁₆
hypercenter	#2247	D₈₃
hypocenter	C₈₃	#13726
LCS	G ▷ C₂×C₁₆×C₈₃ ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2247

subgroup

G/H

center

C₂²

C₂×#14581733

commutator, cosocle

C₂×C₁₆×C₈₃

C₂³

Frattini

C₂×C₁₆

C₂²×D₈₃

Fitting

C₈₃×#2247

C₂

socle, prime radical

C₂²×C₈₃

C₂×D₁₆

D₈₃

C₈₃

G ▷ C₂×C₁₆×C₈₃ ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2247

N	H ≅ G/N	type
56. C₄×C₈₃	Q₆₄	semidirect
67. C₈₃	#13726	semidirect
120. C₄	#19127541	semidirect

H ≅ G/N

type

56. C₄×C₈₃

Q₆₄

semidirect

67. C₈₃

#13726

semidirect

120. C₄

#19127541

semidirect

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

subgroup	index
C₂×#19127541	2
#24909716	2
C₄×C₈₃⋊C₃₂	2
C₈₃×#2247	2
#13726	83

subgroup

index

C₂×#19127541

#24909716

C₄×C₈₃⋊C₃₂

C₈₃×#2247

#13726

Subgroups — #35325401

Conjugacy classes of subgroups

Complements

Maximal subgroups