Group #25013009 — subgroups

Subgroups — #25013009

Order 11008.

414 conjugacy classes of subgroups; only the 275 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 275 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×C₄⋊(C₄₃⋊C₁₆)	C₂	1	6	yes	maximal
2	C₄²×C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄₃⋊C₁₆	C₂²	1	4	yes	—
4	C₄⋊(C₄₃⋊C₁₆)	—	2	12	no	—
5	C₂×C₄×C₈×C₄₃	C₄	1	3	yes	—
6	C₂×C₄×C₈×C₄₃	C₂²	1	3	yes	—
7	C₄³×C₄₃	C₄	1	1	yes	characteristic
8	C₂×C₄₃⋊C₁₆	—	2	12	no	—
9	C₂²×C₈×C₄₃	C₂×C₄	1	3	yes	—
10	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
11	C₂×C₄²×C₄₃	C₈	1	3	yes	—
12	C₂×C₄²×C₄₃	C₈	1	1	yes	characteristic
13	C₂×C₄²×C₄₃	C₂×C₄	1	3	yes	—
14	C₄×C₈×C₄₃	—	2	6	no	—
15	C₄×C₈×C₄₃	D₄	1	6	yes	—
16	C₄×C₈×C₄₃	Q₈	1	6	yes	—
17	C₂²×C₄×C₄₃	C₂×C₈	1	3	yes	—
18	C₂²×C₄×C₄₃	C₂×C₈	1	3	yes	—
19	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
20	C₂×C₈×C₄₃	—	2	6	no	—
21	C₂×C₈×C₄₃	C₄⋊C₄	1	6	yes	—
22	C₂×C₈×C₄₃	C₂×D₄	1	3	yes	—
23	C₂×C₈×C₄₃	C₂×Q₈	1	3	yes	—
24	C₄²×C₄₃	—	2	6	no	—
25	C₄²×C₄₃	C₁₆	1	4	yes	complemented
26	C₄²×C₄₃	C₄⋊C₄	1	6	yes	—
27	C₄²×C₄₃	M₁₆	1	6	yes	—
28	C₄₃⋊C₁₆	—	4	4	no	—
29	C₂³×C₄₃	C₂²×C₈	1	1	yes	characteristic
30	C₂×C₄×C₄₃	—	2	6	no	—
31	C₂×C₄×C₄₃	C₄⋊C₈	1	6	yes	—
32	C₂×C₄×C₄₃	C₄⋊C₈	1	6	yes	—
33	C₂×C₄×C₄₃	C₂×C₁₆	1	6	yes	—
34	C₂×C₄×C₄₃	M₃₂	1	6	yes	—
35	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	3	yes	—
36	C₂×C₄×C₄₃	C₂×M₁₆	1	3	yes	—
37	C₈×C₄₃	—	2	6	no	—
38	C₈×C₄₃	#127	1	1	yes	characteristic
39	C₈×C₄₃	C₄⋊Q₈	1	3	yes	—
40	#12898	—	43	1	no	maximal, Sylow
41	C₂²×C₄₃	C₂×C₄⋊C₈	1	3	yes	—
42	C₂²×C₄₃	C₂²×C₁₆	1	1	yes	characteristic
43	C₂²×C₄₃	C₂×M₃₂	1	3	yes	—
44	C₄×C₄₃	—	2	6	no	—
45	C₄×C₄₃	C₄⋊C₁₆	1	12	yes	complemented
46	C₄×C₄₃	#384	1	1	yes	characteristic
47	C₄×C₄₃	#423	1	3	yes	—
48	C₂×C₄⋊C₁₆	—	43	6	no	—
49	C₄²×C₈	D₄₃	1	1	yes	characteristic
50	C₂×C₄₃	#1842	1	1	yes	characteristic
51	C₂×C₄₃	C₂×C₄⋊C₁₆	1	3	yes	—
52	C₂×C₄₃	#2150	1	3	yes	—
53	C₂²×C₁₆	—	43	4	no	—
54	C₂×C₄×C₈	Dic₄₃	1	3	yes	—
55	C₂×C₄×C₈	C₂×D₄₃	1	3	yes	—
56	C₄³	Dic₄₃	1	1	yes	characteristic
57	C₄⋊C₁₆	—	86	12	no	—
58	C₄₃	#12898	1	1	yes	characteristic, complemented, Sylow
59	C₂²×C₈	C₂×Dic₄₃	1	3	yes	—
60	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
61	C₂×C₄²	C₄₃⋊C₈	1	3	yes	—
62	C₂×C₄²	C₄₃⋊C₈	1	1	yes	characteristic
63	C₂×C₄²	C₂×Dic₄₃	1	3	yes	—
64	C₂×C₁₆	—	86	12	no	—
65	C₄×C₈	—	2	6	no	—
66	C₄×C₈	Dic₈₆	1	6	yes	—
67	C₄×C₈	D₁₇₂	1	6	yes	—
68	C₂²×C₄	C₂×C₄₃⋊C₈	1	3	yes	—
69	C₂²×C₄	C₂×C₄₃⋊C₈	1	3	yes	—
70	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
71	C₂×C₈	—	2	6	no	—
72	C₂×C₈	C₄⋊Dic₄₃	1	6	yes	—
73	C₂×C₈	C₂×Dic₈₆	1	3	yes	—
74	C₂×C₈	C₂×D₁₇₂	1	3	yes	—
75	C₄²	—	2	6	no	—
76	C₄²	C₄₃⋊C₁₆	1	4	yes	complemented
77	C₄²	#10621877	1	6	yes	—
78	C₄²	C₄⋊Dic₄₃	1	6	yes	—
79	C₁₆	—	172	4	no	—
80	C₂³	C₂²×C₄₃⋊C₈	1	1	yes	characteristic
81	C₂×C₄	—	2	6	no	—
82	C₂×C₄	C₄⋊(C₄₃⋊C₈)	1	6	yes	—
83	C₂×C₄	C₄⋊(C₄₃⋊C₈)	1	6	yes	—
84	C₂×C₄	C₂×C₄₃⋊C₁₆	1	6	yes	—
85	C₂×C₄	#13063265	1	6	yes	—
86	C₂×C₄	C₂×C₄⋊Dic₄₃	1	3	yes	—
87	C₂×C₄	C₂×#10621877	1	3	yes	—
88	C₈	—	2	6	no	—
89	C₈	#13063322	1	1	yes	characteristic
90	C₈	#13063325	1	3	yes	—
91	C₂²	C₂×C₄⋊(C₄₃⋊C₈)	1	3	yes	—
92	C₂²	C₂²×C₄₃⋊C₁₆	1	1	yes	characteristic
93	C₂²	C₂×#13063265	1	3	yes	—
94	C₄	—	2	6	no	—
95	C₄	C₄⋊(C₄₃⋊C₁₆)	1	12	yes	complemented
96	C₄	#14582769	1	1	yes	characteristic
97	C₄	#14582923	1	3	yes	—
98	C₂	#19134747	1	1	yes	characteristic
99	C₂	C₂×C₄⋊(C₄₃⋊C₁₆)	1	3	yes	—
100	C₂	#19136351	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₄₃⋊C₁₆)

C₂

yes

maximal

C₄²×C₈×C₄₃

C₂

yes

maximal, characteristic

C₂²×C₄₃⋊C₁₆

C₂²

yes

—

C₄⋊(C₄₃⋊C₁₆)

—

C₂×C₄×C₈×C₄₃

C₄

yes

—

C₂×C₄×C₈×C₄₃

C₂²

yes

—

C₄³×C₄₃

C₄

yes

characteristic

C₂×C₄₃⋊C₁₆

—

C₂²×C₈×C₄₃

C₂×C₄

yes

—

C₂²×C₈×C₄₃

C₂³

yes

characteristic

C₂×C₄²×C₄₃

C₈

yes

—

C₂×C₄²×C₄₃

C₈

yes

characteristic

C₂×C₄²×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

—

C₄×C₈×C₄₃

D₄

yes

—

C₄×C₈×C₄₃

Q₈

yes

—

C₂²×C₄×C₄₃

C₂×C₈

yes

—

C₂²×C₄×C₄₃

C₂×C₈

yes

—

C₂²×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₄⋊C₄

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₂×C₈×C₄₃

C₂×Q₈

yes

—

C₄²×C₄₃

—

C₄²×C₄₃

C₁₆

yes

complemented

C₄²×C₄₃

C₄⋊C₄

yes

—

C₄²×C₄₃

M₁₆

yes

—

C₄₃⋊C₁₆

—

C₂³×C₄₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₄⋊C₈

yes

—

C₂×C₄×C₄₃

C₄⋊C₈

yes

—

C₂×C₄×C₄₃

C₂×C₁₆

yes

—

C₂×C₄×C₄₃

M₃₂

yes

—

C₂×C₄×C₄₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₄₃

C₂×M₁₆

yes

—

C₈×C₄₃

—

C₈×C₄₃

#127

yes

characteristic

C₈×C₄₃

C₄⋊Q₈

yes

—

#12898

—

maximal, Sylow

C₂²×C₄₃

C₂×C₄⋊C₈

yes

—

C₂²×C₄₃

C₂²×C₁₆

yes

characteristic

C₂²×C₄₃

C₂×M₃₂

yes

—

C₄×C₄₃

—

C₄×C₄₃

C₄⋊C₁₆

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

—

C₂×C₄⋊C₁₆

—

C₄²×C₈

D₄₃

yes

characteristic

C₂×C₄₃

#1842

yes

characteristic

C₂×C₄₃

C₂×C₄⋊C₁₆

yes

—

C₂×C₄₃

#2150

yes

—

C₂²×C₁₆

—

C₂×C₄×C₈

Dic₄₃

yes

—

C₂×C₄×C₈

C₂×D₄₃

yes

—

C₄³

Dic₄₃

yes

characteristic

C₄⋊C₁₆

—

C₄₃

#12898

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×Dic₄₃

yes

—

C₂²×C₈

C₂²×D₄₃

yes

characteristic

C₂×C₄²

C₄₃⋊C₈

yes

—

C₂×C₄²

C₄₃⋊C₈

yes

characteristic

C₂×C₄²

C₂×Dic₄₃

yes

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

Dic₈₆

yes

—

C₄×C₈

D₁₇₂

yes

—

C₂²×C₄

C₂×C₄₃⋊C₈

yes

—

C₂²×C₄

C₂×C₄₃⋊C₈

yes

—

C₂²×C₄

C₂²×Dic₄₃

yes

characteristic

C₂×C₈

—

C₂×C₈

C₄⋊Dic₄₃

yes

—

C₂×C₈

C₂×Dic₈₆

yes

—

C₂×C₈

C₂×D₁₇₂

yes

—

C₄²

—

C₄²

C₄₃⋊C₁₆

yes

complemented

C₄²

#10621877

yes

—

C₄²

C₄⋊Dic₄₃

yes

—

C₁₆

—

172

—

C₂³

C₂²×C₄₃⋊C₈

yes

characteristic

C₂×C₄

—

C₂×C₄

C₄⋊(C₄₃⋊C₈)

yes

—

C₂×C₄

C₄⋊(C₄₃⋊C₈)

yes

—

C₂×C₄

C₂×C₄₃⋊C₁₆

yes

—

C₂×C₄

#13063265

yes

—

C₂×C₄

C₂×C₄⋊Dic₄₃

yes

—

C₂×C₄

C₂×#10621877

yes

—

C₈

—

C₈

#13063322

yes

characteristic

C₈

#13063325

yes

—

C₂²

C₂×C₄⋊(C₄₃⋊C₈)

yes

—

C₂²

C₂²×C₄₃⋊C₁₆

yes

characteristic

C₂²

C₂×#13063265

yes

—

C₄

—

C₄

C₄⋊(C₄₃⋊C₁₆)

yes

C₄

yes

C₄

yes

—

C₂

#19134747

yes

characteristic

C₂

C₂×C₄⋊(C₄₃⋊C₁₆)

yes

—

100

C₂

#19136351

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₄₃
commutator	C₂²×C₄₃	C₂²×C₁₆
Fitting	C₄²×C₈×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	C₂²×C₈
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₄²×C₈	D₄₃
hypocenter	C₄₃	#12898
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₄₃

commutator

C₂²×C₄₃

C₂²×C₁₆

Fitting

C₄²×C₈×C₄₃

C₂

socle, prime radical

C₂³×C₄₃

C₂²×C₈

cosocle

C₂²×C₈×C₄₃

C₂³

hypercenter

C₄²×C₈

D₄₃

C₄₃

G ▷ C₂²×C₄₃ ▷ C₄₃

C₁ ◁ C₂²×C₈ ◁ C₄²×C₈

N	H ≅ G/N	type
25. C₄²×C₄₃	C₁₆	semidirect
45. C₄×C₄₃	C₄⋊C₁₆	semidirect
58. C₄₃	#12898	semidirect
76. C₄²	C₄₃⋊C₁₆	semidirect
95. C₄	C₄⋊(C₄₃⋊C₁₆)	semidirect

H ≅ G/N

type

25. C₄²×C₄₃

C₁₆

semidirect

45. C₄×C₄₃

C₄⋊C₁₆

semidirect

58. C₄₃

#12898

semidirect

76. C₄²

C₄₃⋊C₁₆

semidirect

95. C₄

C₄⋊(C₄₃⋊C₁₆)

semidirect

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

subgroup	index
C₂×C₄⋊(C₄₃⋊C₁₆)	2
C₄²×C₈×C₄₃	2
#12898	43

subgroup

index

C₂×C₄⋊(C₄₃⋊C₁₆)

C₄²×C₈×C₄₃

#12898

Subgroups — #25013009

Conjugacy classes of subgroups

Complements

Maximal subgroups