Group #26120787 — subgroups

Subgroups — #26120787

Order 12032.

350 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 97 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	#20243521	C₂	1	3	yes	maximal
2	#20241269	C₂	1	3	yes	maximal
3	C₄₇×#2845	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093935	C₂²	1	3	yes	—
5	#15688993	C₂²	1	1	yes	characteristic
6	#15688885	—	2	6	no	—
7	C₄₇×#398	—	2	6	no	—
8	C₄₇×#382	C₂²	1	3	yes	—
9	C₂²×C₄₇⋊C₈	—	2	3	no	—
10	C₂×C₄⋊Dic₄₇	—	2	3	no	—
11	#13093935	—	2	3	no	—
12	#13093935	Q₈	1	6	yes	—
13	C₂×C₄⋊C₄×C₄₇	—	2	6	no	—
14	C₂×C₄⋊C₄×C₄₇	D₄	1	6	yes	—
15	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
16	#95×C₄₇	—	2	2	no	—
17	#95×C₄₇	—	2	6	no	—
18	C₂²×C₄×C₄₇	—	2	6	no	—
19	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
20	C₂²×Dic₄₇	—	4	1	no	—
21	C₄⋊C₄×C₄₇	—	4	6	no	—
22	C₄⋊C₄×C₄₇	—	8	6	no	—
23	C₄²×C₄₇	C₂×Q₈	1	3	yes	—
24	C₄²×C₄₇	#41	1	1	yes	characteristic
25	C₄⋊Dic₄₇	—	4	6	no	—
26	C₂×C₄₇⋊C₈	—	2	6	no	—
27	C₂×C₄₇⋊C₈	—	4	3	no	—
28	C₂×C₄₇⋊C₈	—	4	3	no	—
29	C₂³×C₄₇	#120	1	1	yes	characteristic
30	C₂×C₄×C₄₇	—	2	3	no	—
31	C₂×C₄×C₄₇	—	2	3	no	—
32	C₂×C₄×C₄₇	—	4	6	no	—
33	C₂×C₄×C₄₇	—	4	6	no	—
34	C₂×C₄×C₄₇	—	4	6	no	—
35	C₂×C₄×C₄₇	C₂²⋊Q₈	1	3	yes	—
36	C₂×C₄×C₄₇	C₄⋊Q₈	1	3	yes	—
37	C₂×Dic₄₇	—	8	3	no	—
38	C₄₇⋊C₈	—	4	6	no	—
39	#10586	—	47	1	no	maximal, Sylow
40	C₂²×C₄₇	#395	1	1	yes	characteristic
41	C₂²×C₄₇	#454	1	3	yes	—
42	C₂²×C₄₇	#455	1	3	yes	—
43	C₄×C₄₇	—	2	3	no	—
44	C₄×C₄₇	—	2	3	no	—
45	C₄×C₄₇	—	8	6	no	—
46	Dic₄₇	—	16	1	no	—
47	#2845	D₄₇	1	1	yes	characteristic
48	#1836	—	47	3	no	—
49	#1295	—	47	3	no	—
50	C₂×C₄₇	#1658	1	1	yes	characteristic
51	C₂×C₄₇	#1662	1	3	yes	—
52	C₂×C₄₇	#2061	1	3	yes	—
53	C₂×#97	—	47	3	no	—
54	#398	—	2	6	no	—
55	#384	—	47	1	no	—
56	#382	C₂×D₄₇	1	3	yes	—
57	#340	—	94	6	no	—
58	C₄₇	#10586	1	1	yes	characteristic, complemented, Sylow
59	C₂²×C₈	—	94	3	no	—
60	C₂×C₄⋊C₄	—	2	6	no	—
61	C₂×C₄⋊C₄	—	94	3	no	—
62	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	6	yes	—
63	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
64	#97	—	47	6	no	—
65	#97	—	94	3	no	—
66	#95	—	2	2	no	—
67	#95	—	2	6	no	—
68	C₂²×C₄	—	2	6	no	—
69	C₂²×C₄	—	188	1	no	—
70	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
71	C₂×C₈	—	94	6	no	—
72	C₂×C₈	—	188	3	no	—
73	C₂×C₈	—	188	3	no	—
74	C₄⋊C₄	—	4	6	no	—
75	C₄⋊C₄	—	8	6	no	—
76	C₄⋊C₄	—	188	6	no	—
77	C₄²	Q₈×D₄₇	1	3	yes	—
78	C₄²	#10625354	1	1	yes	characteristic
79	C₂³	#13093993	1	1	yes	characteristic
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	2	3	no	—
82	C₂×C₄	—	4	6	no	—
83	C₂×C₄	—	4	6	no	—
84	C₂×C₄	—	4	6	no	—
85	C₂×C₄	—	376	3	no	—
86	C₂×C₄	#13094000	1	3	yes	—
87	C₂×C₄	#13094022	1	3	yes	—
88	C₈	—	188	6	no	—
89	C₂²	#15689042	1	1	yes	characteristic
90	C₂²	#15689296	1	3	yes	—
91	C₂²	#15689301	1	3	yes	—
92	C₄	—	2	3	no	—
93	C₄	—	2	3	no	—
94	C₄	—	8	6	no	—
95	C₄	—	752	1	no	—
96	C₂	#20242690	1	1	yes	characteristic
97	C₂	#20242711	1	3	yes	—
98	C₂	#20244735	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20243521

C₂

yes

maximal

#20241269

C₂

yes

maximal

C₄₇×#2845

C₂

yes

maximal, characteristic

C₂×#13093935

C₂²

yes

—

#15688993

C₂²

yes

characteristic

#15688885

—

C₄₇×#398

—

C₄₇×#382

C₂²

yes

—

C₂²×C₄₇⋊C₈

—

C₂×C₄⋊Dic₄₇

—

#13093935

—

#13093935

Q₈

yes

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#95×C₄₇

—

#95×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×Dic₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×Q₈

yes

—

C₄²×C₄₇

#41

yes

characteristic

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#120

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₇

C₄⋊Q₈

yes

—

C₂×Dic₄₇

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

#455

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

#2845

D₄₇

yes

characteristic

#1836

—

#1295

—

C₂×C₄₇

#1658

yes

characteristic

C₂×C₄₇

#1662

yes

—

C₂×C₄₇

#2061

yes

—

C₂×#97

—

#398

—

#384

—

#382

C₂×D₄₇

yes

—

#340

—

C₄₇

#10586

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²⋊D₄₇

yes

—

C₂×C₄²

C₂²×D₄₇

yes

—

—

—

—

C₂²×C₄

—

C₂²×C₄

—

188

—

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

C₂×C₈

—

C₂×C₈

—

188

—

C₂×C₈

—

188

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

188

—

C₄²

Q₈×D₄₇

yes

—

C₄²

yes

C₂³

yes

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

376

—

C₂×C₄

#13094000

yes

—

C₂×C₄

#13094022

yes

—

C₈

—

188

—

C₂²

#15689042

yes

characteristic

C₂²

#15689296

yes

—

C₂²

#15689301

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

752

—

C₂

#20242690

yes

characteristic

C₂

#20242711

yes

—

C₂

#20244735

yes

—

	subgroup	G/H
center	C₂³	#13093993
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#2845	C₂
socle, prime radical	C₂³×C₄₇	#120
hypercenter	#2845	D₄₇
hypocenter	C₄₇	#10586
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2845

subgroup

G/H

center

C₂³

#13093993

commutator, cosocle

C₂×C₄²×C₄₇

C₂³

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#2845

C₂

C₂³×C₄₇

D₄₇

C₄₇

G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2845

N	H ≅ G/N	type
58. C₄₇	#10586	semidirect

H ≅ G/N

type

58. C₄₇

#10586

semidirect

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

subgroup	index
#20243521	2
#20241269	2
C₄₇×#2845	2
#10586	47

subgroup

index

#20243521

#20241269

C₄₇×#2845

#10586

Subgroups — #26120787

Conjugacy classes of subgroups

Complements

Maximal subgroups