Group #25017593 — subgroups

Subgroups — #25017593

Order 11008.

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×Dic₆₈₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582739	C₂	1	2	yes	maximal
3	#19136846	C₂	1	2	yes	maximal
4	C₃₂×Dic₄₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₃×Q₆₄	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₄₄	C₂²	1	2	yes	—
7	C₂×#13063271	—	2	2	no	—
8	#14583277	—	2	2	no	—
9	Dic₆₈₈	—	2	2	no	—
10	#14582739	—	2	4	no	—
11	C₂×C₄₃⋊C₃₂	C₂²	1	1	yes	characteristic
12	C₁₆×Dic₄₃	C₂²	1	1	yes	characteristic
13	C₂×Q₃₂×C₄₃	C₂²	1	2	yes	—
14	C₄₃×Q₆₄	—	2	2	no	—
15	C₂×C₃₂×C₄₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₇₂	—	2	2	no	—
17	C₂×#10621885	—	4	2	no	—
18	#13063323	—	4	2	no	—
19	Dic₃₄₄	—	2	2	no	—
20	Dic₃₄₄	—	2	2	no	—
21	#13063271	—	4	4	no	—
22	C₂×C₄₃⋊C₁₆	D₄	1	1	yes	characteristic
23	C₈×Dic₄₃	D₄	1	1	yes	characteristic
24	C₄₃⋊C₃₂	D₄	1	2	yes	—
25	C₂×Q₁₆×C₄₃	—	2	2	no	—
26	Q₃₂×C₄₃	—	2	2	no	—
27	Q₃₂×C₄₃	—	2	2	no	—
28	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
29	C₃₂×C₄₃	D₄	1	2	yes	—
30	C₂×Q₈×C₄₃	—	4	2	no	—
31	C₂×Dic₈₆	—	4	2	no	—
32	Q₁₆×C₄₃	—	4	2	no	—
33	Q₁₆×C₄₃	—	4	2	no	—
34	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
35	#10621885	—	8	4	no	—
36	Dic₄₃⋊C₄	—	8	4	no	—
37	C₄×Dic₄₃	D₈	1	1	yes	characteristic
38	C₂×C₄₃⋊C₈	D₈	1	1	yes	characteristic
39	Dic₁₇₂	—	4	2	no	—
40	Dic₁₇₂	—	4	2	no	—
41	C₁₆×C₄₃	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₄₃	C₂×D₄	1	1	yes	characteristic
43	C₄₃⋊C₁₆	D₈	1	2	yes	—
44	Q₈×C₄₃	—	8	2	no	—
45	Q₈×C₄₃	—	8	2	no	—
46	C₂×C₄×C₄₃	—	8	2	no	—
47	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
48	C₂×Dic₄₃	—	8	2	no	—
49	C₂×Dic₄₃	D₁₆	1	2	yes	—
50	Dic₈₆	—	8	2	no	—
51	Dic₈₆	—	8	2	no	—
52	C₈×C₄₃	#127	1	1	yes	characteristic
53	C₈×C₄₃	C₂×D₈	1	1	yes	characteristic
54	C₄₃⋊C₈	D₁₆	1	2	yes	—
55	#13726	—	43	1	no	maximal, Sylow
56	C₂²×C₄₃	C₂×D₁₆	1	1	yes	characteristic
57	C₄×C₄₃	—	16	2	no	—
58	C₄×C₄₃	#493	1	1	yes	characteristic
59	C₄×C₄₃	C₂×D₁₆	1	1	yes	characteristic
60	Dic₄₃	—	16	2	no	—
61	Dic₄₃	Q₆₄	1	4	yes	complemented
62	C₂×Q₆₄	—	43	2	no	—
63	C₂×Q₆₄	—	43	1	no	—
64	C₂×Q₆₄	D₄₃	1	1	yes	characteristic
65	#2247	—	43	2	no	—
66	C₄×C₃₂	—	43	1	no	—
67	C₂×C₄₃	#2246	1	1	yes	characteristic
68	C₂×C₄₃	C₂×Q₆₄	1	2	yes	—
69	C₂×Q₃₂	—	43	2	no	—
70	C₂×Q₃₂	—	86	2	no	—
71	C₂×Q₃₂	C₂×D₄₃	1	2	yes	—
72	#494	—	86	2	no	—
73	Q₆₄	—	2	2	no	—
74	Q₆₄	—	86	4	no	—
75	Q₆₄	—	86	2	no	—
76	C₂×C₃₂	—	43	1	no	—
77	C₂×C₃₂	C₂×D₄₃	1	1	yes	characteristic
78	C₄×C₁₆	—	43	1	no	—
79	C₄₃	#13726	1	1	yes	characteristic, complemented, Sylow
80	C₂×Q₁₆	—	2	2	no	—
81	C₂×Q₁₆	—	86	2	no	—
82	C₂×Q₁₆	—	172	2	no	—
83	C₄⋊Q₈	—	172	2	no	—
84	Q₃₂	—	2	2	no	—
85	Q₃₂	—	2	2	no	—
86	Q₃₂	—	86	2	no	—
87	Q₃₂	—	86	2	no	—
88	Q₃₂	—	172	4	no	—
89	C₂×C₁₆	—	43	1	no	—
90	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
91	C₄×C₈	—	43	1	no	—
92	C₃₂	—	43	2	no	—
93	C₃₂	C₂²⋊D₄₃	1	2	yes	—
94	C₂×Q₈	—	4	2	no	—
95	C₂×Q₈	—	172	2	no	—
96	Q₁₆	—	4	2	no	—
97	Q₁₆	—	4	2	no	—
98	Q₁₆	—	344	4	no	—
99	Q₁₆	—	172	2	no	—
100	Q₁₆	—	172	2	no	—
101	C₂×C₈	—	43	1	no	—
102	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
103	C₄⋊C₄	—	344	4	no	—
104	C₄²	—	43	1	no	—
105	C₁₆	—	43	2	no	—
106	C₁₆	D₄×D₄₃	1	1	yes	characteristic
107	C₁₆	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
108	Q₈	—	8	2	no	—
109	Q₈	—	8	2	no	—
110	Q₈	—	344	2	no	—
111	Q₈	—	344	2	no	—
112	C₂×C₄	—	8	2	no	—
113	C₂×C₄	—	43	2	no	—
114	C₂×C₄	—	344	2	no	—
115	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
116	C₈	—	43	2	no	—
117	C₈	#13063321	1	1	yes	characteristic
118	C₈	D₈×D₄₃	1	1	yes	characteristic
119	C₂²	D₁₆×D₄₃	1	1	yes	characteristic
120	C₄	—	16	2	no	—
121	C₄	—	43	4	no	—
122	C₄	—	688	2	no	—
123	C₄	#14583274	1	1	yes	characteristic
124	C₄	D₁₆×D₄₃	1	1	yes	characteristic
125	C₂	#19136843	1	1	yes	characteristic
126	C₂	Q₆₄×D₄₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₆₈₈

C₂

yes

maximal, characteristic

C₂×#14582739

C₂

yes

maximal

#19136846

C₂

yes

maximal

C₃₂×Dic₄₃

C₂

yes

maximal, characteristic

C₂×C₄₃×Q₆₄

C₂

yes

maximal, characteristic

C₂×Dic₃₄₄

C₂²

yes

—

C₂×#13063271

—

#14583277

—

Dic₆₈₈

—

#14582739

—

C₂×C₄₃⋊C₃₂

C₂²

yes

characteristic

C₁₆×Dic₄₃

C₂²

yes

characteristic

C₂×Q₃₂×C₄₃

C₂²

yes

—

C₄₃×Q₆₄

—

C₂×C₃₂×C₄₃

C₂²

yes

characteristic

C₂×Dic₁₇₂

—

C₂×#10621885

—

#13063323

—

Dic₃₄₄

—

Dic₃₄₄

—

#13063271

—

C₂×C₄₃⋊C₁₆

D₄

yes

characteristic

C₈×Dic₄₃

D₄

yes

characteristic

C₄₃⋊C₃₂

D₄

yes

—

C₂×Q₁₆×C₄₃

—

Q₃₂×C₄₃

—

Q₃₂×C₄₃

—

C₂×C₁₆×C₄₃

C₂³

yes

characteristic

C₃₂×C₄₃

D₄

yes

—

C₂×Q₈×C₄₃

—

C₂×Dic₈₆

—

Q₁₆×C₄₃

—

Q₁₆×C₄₃

—

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

#10621885

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

D₈

yes

characteristic

C₂×C₄₃⋊C₈

D₈

yes

characteristic

Dic₁₇₂

—

Dic₁₇₂

—

C₁₆×C₄₃

C₂×D₄

yes

characteristic

C₁₆×C₄₃

C₂×D₄

yes

characteristic

C₄₃⋊C₁₆

D₈

yes

—

Q₈×C₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

D₁₆

yes

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

#127

yes

characteristic

C₈×C₄₃

C₂×D₈

yes

characteristic

C₄₃⋊C₈

D₁₆

yes

—

#13726

—

maximal, Sylow

C₂²×C₄₃

C₂×D₁₆

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

#493

yes

characteristic

C₄×C₄₃

C₂×D₁₆

yes

characteristic

Dic₄₃

—

Dic₄₃

Q₆₄

yes

complemented

C₂×Q₆₄

—

C₂×Q₆₄

—

C₂×Q₆₄

D₄₃

yes

characteristic

#2247

—

C₄×C₃₂

—

C₂×C₄₃

#2246

yes

characteristic

C₂×C₄₃

C₂×Q₆₄

yes

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

C₂×D₄₃

yes

—

#494

—

Q₆₄

—

Q₆₄

—

Q₆₄

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₄₃

yes

characteristic

C₄×C₁₆

—

C₄₃

#13726

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

172

—

C₄⋊Q₈

—

172

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

172

—

C₂×C₁₆

—

C₂×C₁₆

C₂²×D₄₃

yes

characteristic

C₄×C₈

—

C₃₂

—

C₃₂

C₂²⋊D₄₃

yes

—

C₂×Q₈

—

C₂×Q₈

—

172

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

344

—

Q₁₆

—

172

—

100

Q₁₆

—

172

—

101

C₂×C₈

—

102

C₂×C₈

D₄×D₄₃

yes

characteristic

103

C₄⋊C₄

—

344

—

104

C₄²

—

105

C₁₆

—

106

C₁₆

D₄×D₄₃

yes

characteristic

107

C₁₆

C₂×C₂²⋊D₄₃

yes

characteristic

108

Q₈

—

109

Q₈

—

110

Q₈

—

344

—

111

Q₈

—

344

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

344

—

115

C₂×C₄

D₈×D₄₃

yes

characteristic

116

C₈

—

117

C₈

#13063321

yes

characteristic

118

C₈

D₈×D₄₃

yes

characteristic

119

C₂²

D₁₆×D₄₃

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

688

—

123

C₄

#14583274

yes

characteristic

124

C₄

D₁₆×D₄₃

yes

characteristic

125

C₂

#19136843

yes

characteristic

126

C₂

Q₆₄×D₄₃

yes

—

	subgroup	G/H
center	C₂²	D₁₆×D₄₃
commutator, cosocle	C₂×C₁₆×C₄₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₄₃
Fitting	C₂×C₄₃×Q₆₄	C₂
socle, prime radical	C₂²×C₄₃	C₂×D₁₆
hypercenter	C₂×Q₆₄	D₄₃
hypocenter	C₄₃	#13726
LCS	G ▷ C₂×C₁₆×C₄₃ ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×Q₆₄

subgroup

G/H

center

C₂²

D₁₆×D₄₃

commutator, cosocle

C₂×C₁₆×C₄₃

C₂³

Frattini

C₂×C₁₆

C₂²×D₄₃

Fitting

C₂×C₄₃×Q₆₄

C₂

socle, prime radical

C₂²×C₄₃

C₂×D₁₆

hypercenter

C₂×Q₆₄

D₄₃

hypocenter

C₄₃

#13726

LCS

G ▷ C₂×C₁₆×C₄₃ ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

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

N	H ≅ G/N	type
61. Dic₄₃	Q₆₄	semidirect
79. C₄₃	#13726	semidirect

H ≅ G/N

type

61. Dic₄₃

Q₆₄

semidirect

79. C₄₃

#13726

semidirect

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

subgroup	index
C₂×Dic₆₈₈	2
C₂×#14582739	2
#19136846	2
C₃₂×Dic₄₃	2
C₂×C₄₃×Q₆₄	2
#13726	43

subgroup

index

C₂×Dic₆₈₈

C₂×#14582739

#19136846

C₃₂×Dic₄₃

C₂×C₄₃×Q₆₄

#13726

Subgroups — #25017593

Conjugacy classes of subgroups

Complements

Maximal subgroups