Group #35325400 — subgroups

Subgroups — #35325400

Order 21248.

230 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 67 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×Dic₁₃₂₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#19127541	C₂	1	2	yes	maximal
3	#24909714	C₂	1	2	yes	maximal
4	C₃₂×Dic₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₆₄×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₆₆₄	C₂²	1	2	yes	—
7	C₂×#14581738	—	2	2	no	—
8	#19128079	—	2	2	no	—
9	Dic₁₃₂₈	—	2	2	no	—
10	#19127541	—	2	4	no	—
11	C₂×C₈₃⋊C₃₂	C₂²	1	1	yes	characteristic
12	C₁₆×Dic₈₃	C₂²	1	1	yes	characteristic
13	C₂×Q₃₂×C₈₃	C₂²	1	2	yes	—
14	Q₆₄×C₈₃	—	2	2	no	—
15	C₂×C₃₂×C₈₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₃₃₂	—	2	2	no	—
17	C₂×#13050580	—	4	2	no	—
18	#14581790	—	4	2	no	—
19	Dic₆₆₄	—	2	2	no	—
20	Dic₆₆₄	—	2	2	no	—
21	#14581738	—	4	4	no	—
22	C₂×C₈₃⋊C₁₆	D₄	1	1	yes	characteristic
23	C₈×Dic₈₃	D₄	1	1	yes	characteristic
24	C₈₃⋊C₃₂	D₄	1	2	yes	—
25	C₂×Q₁₆×C₈₃	—	2	2	no	—
26	Q₃₂×C₈₃	—	2	2	no	—
27	Q₃₂×C₈₃	—	2	2	no	—
28	C₂×C₁₆×C₈₃	C₂³	1	1	yes	characteristic
29	C₃₂×C₈₃	D₄	1	2	yes	—
30	C₂×Dic₁₆₆	—	4	2	no	—
31	Dic₃₃₂	—	4	2	no	—
32	Dic₃₃₂	—	4	2	no	—
33	#13050580	—	8	4	no	—
34	C₂×C₈₃⋊C₈	D₈	1	1	yes	characteristic
35	Dic₈₃⋊C₄	—	8	4	no	—
36	C₄×Dic₈₃	D₈	1	1	yes	characteristic
37	C₈₃⋊C₁₆	D₈	1	2	yes	—
38	C₂×Q₈×C₈₃	—	4	2	no	—
39	Q₁₆×C₈₃	—	4	2	no	—
40	Q₁₆×C₈₃	—	4	2	no	—
41	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₈₃	C₂×D₄	1	1	yes	characteristic
43	C₁₆×C₈₃	C₂×D₄	1	1	yes	characteristic
44	Q₈×C₈₃	—	8	2	no	—
45	Q₈×C₈₃	—	8	2	no	—
46	C₂×C₄×C₈₃	—	8	2	no	—
47	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
48	C₂×Dic₈₃	—	8	2	no	—
49	C₂×Dic₈₃	D₁₆	1	2	yes	—
50	Dic₁₆₆	—	8	2	no	—
51	Dic₁₆₆	—	8	2	no	—
52	C₈×C₈₃	#127	1	1	yes	characteristic
53	C₈×C₈₃	C₂×D₈	1	1	yes	characteristic
54	C₈₃⋊C₈	D₁₆	1	2	yes	—
55	C₂²×C₈₃	C₂×D₁₆	1	1	yes	characteristic
56	C₄×C₈₃	—	16	2	no	—
57	C₄×C₈₃	#493	1	1	yes	characteristic
58	C₄×C₈₃	C₂×D₁₆	1	1	yes	characteristic
59	Dic₈₃	—	16	2	no	—
60	Dic₈₃	Q₆₄	1	4	yes	complemented
61	#13726	—	83	1	no	maximal, Sylow
62	C₂×C₈₃	#2246	1	1	yes	characteristic
63	C₂×C₈₃	C₂×Q₆₄	1	2	yes	—
64	C₂×Q₆₄	—	83	2	no	—
65	C₂×Q₆₄	—	83	1	no	—
66	C₂×Q₆₄	D₈₃	1	1	yes	characteristic
67	#2247	—	83	2	no	—
68	C₄×C₃₂	—	83	1	no	—
69	C₈₃	#13726	1	1	yes	characteristic, complemented, Sylow
70	C₂×Q₃₂	—	83	2	no	—
71	C₂×Q₃₂	—	166	2	no	—
72	C₂×Q₃₂	C₂×D₈₃	1	2	yes	—
73	#494	—	166	2	no	—
74	Q₆₄	—	2	2	no	—
75	Q₆₄	—	166	4	no	—
76	Q₆₄	—	166	2	no	—
77	C₂×C₃₂	—	83	1	no	—
78	C₂×C₃₂	C₂×D₈₃	1	1	yes	characteristic
79	C₄×C₁₆	—	83	1	no	—
80	C₂×Q₁₆	—	2	2	no	—
81	C₂×Q₁₆	—	332	2	no	—
82	C₂×Q₁₆	—	166	2	no	—
83	C₄⋊Q₈	—	332	2	no	—
84	Q₃₂	—	2	2	no	—
85	Q₃₂	—	2	2	no	—
86	Q₃₂	—	332	4	no	—
87	Q₃₂	—	166	2	no	—
88	Q₃₂	—	166	2	no	—
89	C₂×C₁₆	—	83	1	no	—
90	C₂×C₁₆	C₂²×D₈₃	1	1	yes	characteristic
91	C₄×C₈	—	83	1	no	—
92	C₃₂	—	83	2	no	—
93	C₃₂	C₂²⋊D₈₃	1	2	yes	—
94	C₂×Q₈	—	4	2	no	—
95	C₂×Q₈	—	332	2	no	—
96	Q₁₆	—	4	2	no	—
97	Q₁₆	—	4	2	no	—
98	Q₁₆	—	332	2	no	—
99	Q₁₆	—	332	2	no	—
100	Q₁₆	—	664	4	no	—
101	C₂×C₈	—	83	1	no	—
102	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
103	C₄⋊C₄	—	664	4	no	—
104	C₄²	—	83	1	no	—
105	C₁₆	—	83	2	no	—
106	C₁₆	D₄×D₈₃	1	1	yes	characteristic
107	C₁₆	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
108	Q₈	—	8	2	no	—
109	Q₈	—	8	2	no	—
110	Q₈	—	664	2	no	—
111	Q₈	—	664	2	no	—
112	C₂×C₄	—	8	2	no	—
113	C₂×C₄	—	83	2	no	—
114	C₂×C₄	—	664	2	no	—
115	C₂×C₄	D₈×D₈₃	1	1	yes	characteristic
116	C₈	—	83	2	no	—
117	C₈	#14581788	1	1	yes	characteristic
118	C₈	D₈×D₈₃	1	1	yes	characteristic
119	C₂²	D₁₆×D₈₃	1	1	yes	characteristic
120	C₄	—	16	2	no	—
121	C₄	—	1328	2	no	—
122	C₄	—	83	4	no	—
123	C₄	#19128076	1	1	yes	characteristic
124	C₄	D₁₆×D₈₃	1	1	yes	characteristic
125	C₂	#24909711	1	1	yes	characteristic
126	C₂	Q₆₄×D₈₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₁₃₂₈

C₂

yes

maximal, characteristic

C₂×#19127541

C₂

yes

maximal

#24909714

C₂

yes

maximal

C₃₂×Dic₈₃

C₂

yes

maximal, characteristic

C₂×Q₆₄×C₈₃

C₂

yes

maximal, characteristic

C₂×Dic₆₆₄

C₂²

yes

—

C₂×#14581738

—

#19128079

—

Dic₁₃₂₈

—

#19127541

—

C₂×C₈₃⋊C₃₂

C₂²

yes

characteristic

C₁₆×Dic₈₃

C₂²

yes

characteristic

C₂×Q₃₂×C₈₃

C₂²

yes

—

Q₆₄×C₈₃

—

C₂×C₃₂×C₈₃

C₂²

yes

characteristic

C₂×Dic₃₃₂

—

C₂×#13050580

—

#14581790

—

Dic₆₆₄

—

Dic₆₆₄

—

#14581738

—

C₂×C₈₃⋊C₁₆

D₄

yes

characteristic

C₈×Dic₈₃

D₄

yes

characteristic

C₈₃⋊C₃₂

D₄

yes

—

C₂×Q₁₆×C₈₃

—

Q₃₂×C₈₃

—

Q₃₂×C₈₃

—

C₂×C₁₆×C₈₃

C₂³

yes

characteristic

C₃₂×C₈₃

D₄

yes

—

C₂×Dic₁₆₆

—

Dic₃₃₂

—

Dic₃₃₂

—

#13050580

—

C₂×C₈₃⋊C₈

D₈

yes

characteristic

Dic₈₃⋊C₄

—

C₄×Dic₈₃

D₈

yes

characteristic

C₈₃⋊C₁₆

D₈

yes

—

C₂×Q₈×C₈₃

—

Q₁₆×C₈₃

—

Q₁₆×C₈₃

—

C₂×C₈×C₈₃

C₂×D₄

yes

characteristic

C₁₆×C₈₃

C₂×D₄

yes

characteristic

C₁₆×C₈₃

C₂×D₄

yes

characteristic

Q₈×C₈₃

—

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×D₈

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

D₁₆

yes

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

#127

yes

characteristic

C₈×C₈₃

C₂×D₈

yes

characteristic

C₈₃⋊C₈

D₁₆

yes

—

C₂²×C₈₃

C₂×D₁₆

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

#493

yes

characteristic

C₄×C₈₃

C₂×D₁₆

yes

characteristic

Dic₈₃

—

Dic₈₃

Q₆₄

yes

—

C₂×C₈₃

yes

C₂×C₈₃

C₂×Q₆₄

yes

—

C₂×Q₆₄

—

C₂×Q₆₄

—

C₂×Q₆₄

D₈₃

yes

characteristic

#2247

—

C₄×C₃₂

—

C₈₃

#13726

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

—

166

—

C₂×Q₃₂

C₂×D₈₃

yes

—

#494

—

166

—

Q₆₄

—

Q₆₄

—

166

—

Q₆₄

—

166

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₈₃

yes

characteristic

C₄×C₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

332

—

C₂×Q₁₆

—

166

—

C₄⋊Q₈

—

332

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

332

—

Q₃₂

—

166

—

Q₃₂

—

166

—

C₂×C₁₆

—

C₂×C₁₆

C₂²×D₈₃

yes

characteristic

C₄×C₈

—

C₃₂

—

C₃₂

C₂²⋊D₈₃

yes

—

C₂×Q₈

—

C₂×Q₈

—

332

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

332

—

Q₁₆

—

332

—

100

Q₁₆

—

664

—

101

C₂×C₈

—

102

C₂×C₈

D₄×D₈₃

yes

characteristic

103

C₄⋊C₄

—

664

—

104

C₄²

—

105

C₁₆

—

106

C₁₆

D₄×D₈₃

yes

characteristic

107

C₁₆

C₂×C₂²⋊D₈₃

yes

characteristic

108

Q₈

—

109

Q₈

—

110

Q₈

—

664

—

111

Q₈

—

664

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

664

—

115

C₂×C₄

D₈×D₈₃

yes

characteristic

116

C₈

—

117

C₈

#14581788

yes

characteristic

118

C₈

D₈×D₈₃

yes

characteristic

119

C₂²

D₁₆×D₈₃

yes

characteristic

120

C₄

—

121

C₄

—

1328

—

122

C₄

—

123

C₄

#19128076

yes

characteristic

124

C₄

D₁₆×D₈₃

yes

characteristic

125

C₂

#24909711

yes

characteristic

126

C₂

Q₆₄×D₈₃

yes

—

	subgroup	G/H
center	C₂²	D₁₆×D₈₃
commutator, cosocle	C₂×C₁₆×C₈₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₈₃
Fitting	C₂×Q₆₄×C₈₃	C₂
socle, prime radical	C₂²×C₈₃	C₂×D₁₆
hypercenter	C₂×Q₆₄	D₈₃
hypocenter	C₈₃	#13726
LCS	G ▷ C₂×C₁₆×C₈₃ ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×Q₆₄

subgroup

G/H

center

C₂²

D₁₆×D₈₃

commutator, cosocle

C₂×C₁₆×C₈₃

C₂³

Frattini

C₂×C₁₆

C₂²×D₈₃

Fitting

C₂×Q₆₄×C₈₃

C₂

socle, prime radical

C₂²×C₈₃

C₂×D₁₆

hypercenter

C₂×Q₆₄

D₈₃

hypocenter

C₈₃

#13726

LCS

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

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×Q₆₄

N	H ≅ G/N	type
60. Dic₈₃	Q₆₄	semidirect
69. C₈₃	#13726	semidirect

H ≅ G/N

type

60. Dic₈₃

Q₆₄

semidirect

69. C₈₃

#13726

semidirect

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

subgroup	index
C₂×Dic₁₃₂₈	2
C₂×#19127541	2
#24909714	2
C₃₂×Dic₈₃	2
C₂×Q₆₄×C₈₃	2
#13726	83

subgroup

index

C₂×Dic₁₃₂₈

C₂×#19127541

#24909714

C₃₂×Dic₈₃

C₂×Q₆₄×C₈₃

#13726

Subgroups — #35325400

Conjugacy classes of subgroups

Complements

Maximal subgroups