Group #26102999 — subgroups

Subgroups — #26102999

Order 12032.

152 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 51 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	#20241452	C₂	1	2	yes	maximal, complemented
2	C₄₇×#1709	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₄₇⋊C₁₆)	—	2	2	no	—
4	C₄₇×#494	C₄	1	1	yes	characteristic
5	C₄₇×#493	C₄	1	1	yes	characteristic
6	C₄₇×#486	—	2	2	no	—
7	C₈²×C₄₇	C₂²	1	1	yes	characteristic
8	C₂×C₄₇⋊C₁₆	—	4	2	no	—
9	C₂×Q₁₆×C₄₇	—	4	1	no	—
10	C₂×D₈×C₄₇	—	4	1	no	—
11	C₄⋊Q₈×C₄₇	—	2	1	no	—
12	#127×C₄₇	—	2	1	no	—
13	#102×C₄₇	—	4	2	no	—
14	#102×C₄₇	—	4	2	no	—
15	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
16	C₄×C₈×C₄₇	D₄	1	2	yes	—
17	C₂×Q₈×C₄₇	—	4	1	no	—
18	C₂×D₄×C₄₇	—	4	1	no	—
19	C₂×D₄×C₄₇	—	4	2	no	—
20	Q₁₆×C₄₇	—	8	2	no	—
21	D₈×C₄₇	—	8	2	no	—
22	C₂×C₈×C₄₇	—	2	1	no	—
23	C₂×C₈×C₄₇	D₈	1	2	yes	—
24	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
25	C₄⋊C₄×C₄₇	—	4	2	no	—
26	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₄₇⋊C₁₆	—	8	2	no	—
28	C₂³×C₄₇	—	8	1	no	—
29	Q₈×C₄₇	—	8	2	no	—
30	D₄×C₄₇	—	8	2	no	—
31	D₄×C₄₇	—	8	2	no	—
32	D₄×C₄₇	—	8	2	no	—
33	C₂×C₄×C₄₇	—	8	1	no	—
34	C₂×C₄×C₄₇	#100	1	1	yes	characteristic
35	C₂×C₄×C₄₇	#102	1	2	yes	—
36	C₈×C₄₇	—	2	2	no	—
37	C₈×C₄₇	—	2	2	no	—
38	C₈×C₄₇	—	2	2	no	—
39	#7092	—	47	1	no	maximal, Sylow
40	C₂²×C₄₇	—	16	1	no	—
41	C₂²×C₄₇	—	16	2	no	—
42	C₂²×C₄₇	#331	1	1	yes	characteristic
43	C₄×C₄₇	—	2	1	no	—
44	C₄×C₄₇	—	16	1	no	—
45	C₄×C₄₇	#359	1	2	yes	—
46	C₄×C₄₇	#360	1	2	yes	—
47	#1709	D₄₇	1	1	yes	characteristic
48	#1367	—	47	2	no	—
49	C₂×C₄₇	—	32	1	no	—
50	C₂×C₄₇	#1357	1	1	yes	characteristic
51	C₂×C₄₇	#1360	1	2	yes	—
52	#494	Dic₄₇	1	1	yes	characteristic
53	#493	Dic₄₇	1	1	yes	characteristic
54	#486	—	2	2	no	—
55	C₄⋊C₁₆	—	94	2	no	—
56	C₈²	C₂×D₄₇	1	1	yes	characteristic
57	C₄₇	#7092	1	1	yes	characteristic, complemented, Sylow
58	C₂×Q₁₆	—	4	1	no	—
59	C₂×D₈	—	4	1	no	—
60	C₄⋊Q₈	—	2	1	no	—
61	#127	—	2	1	no	—
62	C₂×C₁₆	—	188	2	no	—
63	#102	—	4	2	no	—
64	#102	—	4	2	no	—
65	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
66	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
67	C₂×Q₈	—	4	1	no	—
68	C₂×D₄	—	4	1	no	—
69	C₂×D₄	—	4	2	no	—
70	Q₁₆	—	8	2	no	—
71	D₈	—	8	2	no	—
72	C₂×C₈	—	2	1	no	—
73	C₂×C₈	#10625334	1	2	yes	—
74	C₂×C₈	#10625335	1	2	yes	—
75	C₄⋊C₄	—	4	2	no	—
76	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
77	C₁₆	—	376	2	no	—
78	C₂³	—	8	1	no	—
79	Q₈	—	8	2	no	—
80	D₄	—	8	2	no	—
81	D₄	—	8	2	no	—
82	D₄	—	8	2	no	—
83	C₂×C₄	—	8	1	no	—
84	C₂×C₄	#13093941	1	1	yes	characteristic
85	C₂×C₄	#13093945	1	2	yes	—
86	C₈	—	2	2	no	—
87	C₈	—	2	2	no	—
88	C₈	—	2	2	no	—
89	C₂²	—	16	1	no	—
90	C₂²	—	16	2	no	—
91	C₂²	#15688867	1	1	yes	characteristic
92	C₄	—	2	1	no	—
93	C₄	—	16	1	no	—
94	C₄	#15688930	1	2	yes	—
95	C₄	#15688933	1	2	yes	—
96	C₂	—	32	1	no	—
97	C₂	#20241425	1	1	yes	characteristic
98	C₂	#20241431	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20241452

C₂

yes

maximal, complemented

C₄₇×#1709

C₂

yes

maximal, characteristic

C₄⋊(C₄₇⋊C₁₆)

—

C₄₇×#494

C₄

yes

characteristic

C₄₇×#493

C₄

yes

characteristic

C₄₇×#486

—

C₈²×C₄₇

C₂²

yes

characteristic

C₂×C₄₇⋊C₁₆

—

C₂×Q₁₆×C₄₇

—

C₂×D₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

#127×C₄₇

—

#102×C₄₇

—

#102×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

—

C₂×Q₈×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

Q₁₆×C₄₇

—

D₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

SD₁₆

yes

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂²⋊C₄

yes

characteristic

C₄₇⋊C₁₆

—

C₂³×C₄₇

—

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#100

yes

characteristic

C₂×C₄×C₄₇

#102

yes

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

#7092

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#331

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#359

yes

—

C₄×C₄₇

#360

yes

—

#1709

D₄₇

yes

characteristic

#1367

—

C₂×C₄₇

—

C₂×C₄₇

#1357

yes

characteristic

C₂×C₄₇

#1360

yes

—

#494

Dic₄₇

yes

characteristic

#493

Dic₄₇

yes

characteristic

#486

—

C₄⋊C₁₆

—

C₈²

C₂×D₄₇

yes

characteristic

C₄₇

#7092

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

—

188

—

#102

—

#102

—

C₄×C₈

C₂×Dic₄₇

yes

characteristic

C₄×C₈

C₂²⋊D₄₇

yes

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

D₈

—

C₂×C₈

—

C₂×C₈

#10625334

yes

—

C₂×C₈

#10625335

yes

—

C₄⋊C₄

—

C₄²

C₂²⋊Dic₄₇

yes

characteristic

C₁₆

—

376

—

C₂³

—

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

#13093941

yes

characteristic

C₂×C₄

#13093945

yes

—

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

#15688867

yes

characteristic

C₄

—

C₄

—

C₄

#15688930

yes

—

C₄

#15688933

yes

—

C₂

—

C₂

#20241425

yes

characteristic

C₂

#20241431

yes

—

	subgroup	G/H
center	C₂²	#15688867
commutator	C₄×C₈×C₄₇	C₂×C₄
Frattini	C₈²	C₂×D₄₇
Fitting	C₄₇×#1709	C₂
socle	C₂²×C₄₇	#331
prime radical	C₄₇×#493	C₄
cosocle	C₈²×C₄₇	C₂²
hypercenter	#1709	D₄₇
hypocenter	C₄₇	#7092
LCS	G ▷ C₄×C₈×C₄₇ ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

subgroup

G/H

center

C₂²

#15688867

commutator

C₄×C₈×C₄₇

C₂×C₄

Frattini

C₈²

C₂×D₄₇

Fitting

C₄₇×#1709

C₂

socle

C₂²×C₄₇

#331

prime radical

C₄₇×#493

C₄

cosocle

C₈²×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₄×C₈×C₄₇ ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

N	H ≅ G/N	type
1. #20241452	C₂	semidirect
57. C₄₇	#7092	semidirect

H ≅ G/N

type

1. #20241452

C₂

semidirect

57. C₄₇

#7092

semidirect

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

subgroup	index
#20241452	2
C₄₇×#1709	2
#7092	47

subgroup

index

#20241452

C₄₇×#1709

#7092

Subgroups — #26102999

Conjugacy classes of subgroups

Complements

Maximal subgroups