Group #24984051 — subgroups

Subgroups — #24984051

Order 11008.

126 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 55 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	C₄×Dic₃₄₄	C₂	1	1	yes	maximal, characteristic
2	#19132667	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₂×C₄₃	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₃₄₄	C₄	1	1	yes	characteristic
5	C₄×Dic₁₇₂	—	2	1	no	—
6	#14582725	C₄	1	1	yes	characteristic
7	#14582704	—	2	1	no	—
8	#14582652	—	2	1	no	—
9	C₂×C₃₂×C₄₃	—	2	1	no	—
10	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₇₂	—	2	1	no	—
12	C₄×Dic₈₆	—	4	1	no	—
13	Dic₃₄₄	—	2	1	no	—
14	Dic₃₄₄	C₈	1	2	yes	complemented
15	#13063259	—	2	1	no	—
16	C₄⋊(C₄₃⋊C₈)	—	4	1	no	—
17	#13063250	—	4	1	no	—
18	C₂×C₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₆×C₄₃	D₄	1	1	yes	characteristic
20	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
21	C₃₂×C₄₃	—	2	2	no	—
22	C₂×Dic₈₆	—	4	1	no	—
23	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×C₈×C₄₃	SD₁₆	1	1	yes	characteristic
25	C₄²×C₄₃	D₈	1	1	yes	characteristic
26	C₄⋊Dic₄₃	—	4	1	no	—
27	Dic₄₃⋊C₄	—	8	1	no	—
28	C₄×Dic₄₃	—	8	1	no	—
29	C₂×C₄₃⋊C₈	—	8	1	no	—
30	Dic₁₇₂	—	2	2	no	—
31	Dic₁₇₂	—	4	1	no	—
32	C₁₆×C₄₃	—	2	1	no	—
33	C₁₆×C₄₃	C₂×C₈	1	1	yes	characteristic
34	C₁₆×C₄₃	M₁₆	1	1	yes	characteristic
35	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
36	C₂×C₄×C₄₃	D₁₆	1	1	yes	characteristic
37	C₂×C₄×C₄₃	SD₃₂	1	1	yes	characteristic
38	C₂×Dic₄₃	—	8	1	no	—
39	C₂×Dic₄₃	—	8	1	no	—
40	Dic₈₆	—	4	2	no	—
41	Dic₈₆	—	8	1	no	—
42	C₈×C₄₃	—	2	1	no	—
43	C₈×C₄₃	C₂²⋊C₈	1	1	yes	characteristic
44	C₈×C₄₃	#104	1	1	yes	characteristic
45	C₄₃⋊C₈	—	16	1	no	—
46	#7384	—	43	1	no	maximal, Sylow
47	C₂²×C₄₃	#357	1	1	yes	characteristic
48	C₄×C₄₃	—	2	1	no	—
49	C₄×C₄₃	D₄⋊C₈	1	1	yes	characteristic
50	C₄×C₄₃	#359	1	1	yes	characteristic
51	C₄×C₄₃	SD₆₄	1	1	yes	characteristic
52	C₄×C₄₃	Q₆₄	1	1	yes	characteristic
53	Dic₄₃	—	8	2	no	—
54	Dic₄₃	—	16	1	no	—
55	C₄×Q₃₂	—	43	1	no	—
56	C₄×C₃₂	D₄₃	1	1	yes	characteristic
57	#1371	—	43	1	no	—
58	C₂×C₄₃	#1331	1	1	yes	characteristic
59	C₂×C₄₃	#1416	1	1	yes	characteristic
60	C₂×C₄₃	#1417	1	1	yes	characteristic
61	C₂×Q₃₂	—	43	1	no	—
62	C₄×Q₁₆	—	86	1	no	—
63	C₂×C₃₂	—	2	1	no	—
64	#366	—	43	1	no	—
65	#358	—	86	1	no	—
66	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
67	#335	—	86	1	no	—
68	C₄₃	#7384	1	1	yes	characteristic, complemented, Sylow
69	C₂×Q₁₆	—	86	1	no	—
70	C₄×Q₈	—	172	1	no	—
71	Q₃₂	—	43	2	no	—
72	Q₃₂	—	86	1	no	—
73	C₂×C₁₆	C₄×D₄₃	1	1	yes	characteristic
74	C₂×C₁₆	C₂²⋊D₄₃	1	1	yes	characteristic
75	#107	—	86	1	no	—
76	C₄⋊C₈	—	172	1	no	—
77	#103	—	172	1	no	—
78	C₄×C₈	D₁₇₂	1	1	yes	characteristic
79	C₃₂	—	2	2	no	—
80	C₂×Q₈	—	172	1	no	—
81	Q₁₆	—	86	2	no	—
82	Q₁₆	—	172	1	no	—
83	C₂×C₈	—	344	1	no	—
84	C₂×C₈	#10621873	1	1	yes	characteristic
85	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
86	C₄⋊C₄	—	344	1	no	—
87	C₄⋊C₄	—	172	1	no	—
88	C₄²	—	344	1	no	—
89	C₄²	D₃₄₄	1	1	yes	characteristic
90	C₁₆	—	2	1	no	—
91	C₁₆	C₈×D₄₃	1	1	yes	characteristic
92	C₁₆	#10621872	1	1	yes	characteristic
93	Q₈	—	344	1	no	—
94	Q₈	—	172	2	no	—
95	C₂×C₄	—	344	1	no	—
96	C₂×C₄	—	344	1	no	—
97	C₂×C₄	#13063247	1	1	yes	characteristic
98	C₂×C₄	D₆₈₈	1	1	yes	characteristic
99	C₂×C₄	#13063270	1	1	yes	characteristic
100	C₈	—	2	1	no	—
101	C₈	—	688	1	no	—
102	C₈	(C₂×D₄₃)⋊C₈	1	1	yes	characteristic
103	C₈	#13063253	1	1	yes	characteristic
104	C₂²	#14582701	1	1	yes	characteristic
105	C₄	—	2	1	no	—
106	C₄	—	344	2	no	—
107	C₄	—	688	1	no	—
108	C₄	#14582628	1	1	yes	characteristic
109	C₄	#14582707	1	1	yes	characteristic
110	C₄	#14582738	1	1	yes	characteristic
111	C₄	Dic₆₈₈	1	1	yes	characteristic
112	C₂	#19132557	1	1	yes	characteristic
113	C₂	#19132777	1	1	yes	characteristic
114	C₂	#19132780	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Dic₃₄₄

C₂

yes

maximal, characteristic

#19132667

C₂

yes

maximal, characteristic

C₄×C₃₂×C₄₃

C₂

yes

maximal, characteristic

C₂×Dic₃₄₄

C₄

yes

characteristic

C₄×Dic₁₇₂

—

#14582725

C₄

yes

characteristic

#14582704

—

#14582652

—

C₂×C₃₂×C₄₃

—

C₄×C₁₆×C₄₃

C₂²

yes

characteristic

C₂×Dic₁₇₂

—

C₄×Dic₈₆

—

Dic₃₄₄

—

Dic₃₄₄

C₈

yes

complemented

#13063259

—

C₄⋊(C₄₃⋊C₈)

—

#13063250

—

C₂×C₁₆×C₄₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

characteristic

C₃₂×C₄₃

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₄₃

SD₁₆

yes

characteristic

C₄²×C₄₃

D₈

yes

characteristic

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

Dic₁₇₂

—

Dic₁₇₂

—

C₁₆×C₄₃

—

C₁₆×C₄₃

C₂×C₈

yes

characteristic

C₁₆×C₄₃

M₁₆

yes

characteristic

C₂×C₄×C₄₃

#102

yes

characteristic

C₂×C₄×C₄₃

D₁₆

yes

characteristic

C₂×C₄×C₄₃

SD₃₂

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

C₂²⋊C₈

yes

characteristic

C₈×C₄₃

#104

yes

characteristic

C₄₃⋊C₈

—

—

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

D₄⋊C₈

yes

characteristic

C₄×C₄₃

#359

yes

characteristic

C₄×C₄₃

SD₆₄

yes

characteristic

C₄×C₄₃

Q₆₄

yes

characteristic

Dic₄₃

—

Dic₄₃

—

C₄×Q₃₂

—

C₄×C₃₂

D₄₃

yes

characteristic

#1371

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×Q₃₂

—

C₄×Q₁₆

—

C₂×C₃₂

—

#366

—

#358

—

C₄×C₁₆

C₂×D₄₃

yes

characteristic

#335

—

C₄₃

#7384

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

172

—

Q₃₂

—

Q₃₂

—

C₂×C₁₆

C₄×D₄₃

yes

characteristic

C₂×C₁₆

C₂²⋊D₄₃

yes

characteristic

#107

—

C₄⋊C₈

—

172

—

#103

—

172

—

C₄×C₈

D₁₇₂

yes

characteristic

C₃₂

—

C₂×Q₈

—

172

—

Q₁₆

—

Q₁₆

—

172

—

C₂×C₈

—

344

—

C₂×C₈

#10621873

yes

characteristic

C₂×C₈

(C₂×D₄₃)⋊C₄

yes

characteristic

C₄⋊C₄

—

344

—

C₄⋊C₄

—

172

—

C₄²

—

344

—

C₄²

D₃₄₄

yes

characteristic

C₁₆

—

C₁₆

C₈×D₄₃

yes

characteristic

C₁₆

#10621872

yes

characteristic

Q₈

—

344

—

Q₈

—

172

—

C₂×C₄

—

344

—

C₂×C₄

—

344

—

C₂×C₄

#13063247

yes

characteristic

C₂×C₄

D₆₈₈

yes

characteristic

C₂×C₄

#13063270

yes

characteristic

100

C₈

—

101

C₈

—

688

—

102

C₈

(C₂×D₄₃)⋊C₈

yes

characteristic

103

C₈

#13063253

yes

characteristic

104

C₂²

#14582701

yes

characteristic

105

C₄

—

106

C₄

—

344

—

107

C₄

—

688

—

108

C₄

#14582628

yes

characteristic

109

C₄

#14582707

yes

characteristic

110

C₄

#14582738

yes

characteristic

111

C₄

Dic₆₈₈

yes

characteristic

112

C₂

#19132557

yes

characteristic

113

C₂

#19132777

yes

characteristic

114

C₂

#19132780

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₆₈₈
commutator	C₁₆×C₄₃	C₂×C₈
Frattini	C₄×C₁₆	C₂×D₄₃
Fitting	C₄×C₃₂×C₄₃	C₂
socle, prime radical	C₂²×C₄₃	#357
cosocle	C₄×C₁₆×C₄₃	C₂²
hypercenter	C₄×C₃₂	D₄₃
hypocenter	C₄₃	#7384
LCS	G ▷ C₁₆×C₄₃ ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

subgroup

G/H

center

C₂×C₄

D₆₈₈

commutator

C₁₆×C₄₃

C₂×C₈

Frattini

C₄×C₁₆

C₂×D₄₃

Fitting

C₄×C₃₂×C₄₃

C₂

socle, prime radical

C₂²×C₄₃

#357

cosocle

C₄×C₁₆×C₄₃

C₂²

hypercenter

C₄×C₃₂

D₄₃

hypocenter

C₄₃

#7384

LCS

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

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

N	H ≅ G/N	type
14. Dic₃₄₄	C₈	semidirect
68. C₄₃	#7384	semidirect

H ≅ G/N

type

14. Dic₃₄₄

C₈

semidirect

68. C₄₃

#7384

semidirect

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

subgroup	index
C₄×Dic₃₄₄	2
#19132667	2
C₄×C₃₂×C₄₃	2
#7384	43

subgroup

index

C₄×Dic₃₄₄

#19132667

C₄×C₃₂×C₄₃

#7384

Subgroups — #24984051

Conjugacy classes of subgroups

Complements

Maximal subgroups