Group #26106712 — subgroups

Subgroups — #26106712

Order 12032.

382 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 167 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	#20243133	C₂	1	3	yes	maximal
2	C₄₇⋊#1273	C₂	1	3	yes	maximal
3	C₄₇×#1273	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₄₇⋊C₈	C₂²	1	3	yes	—
5	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
6	#15688877	C₄	1	6	yes	—
7	#15688877	C₄	1	6	yes	—
8	C₂×C₄×C₈×C₄₇	C₂²	1	3	yes	—
9	C₂²×C₄₇⋊C₈	C₂×C₄	1	6	yes	—
10	C₂×C₄×Dic₄₇	C₂×C₄	1	3	yes	—
11	C₂×C₄×Dic₄₇	C₂×C₄	1	3	yes	—
12	C₄×C₄₇⋊C₈	—	2	3	no	—
13	C₄×C₄₇⋊C₈	—	2	3	no	—
14	C₂²×C₈×C₄₇	C₂×C₄	1	6	yes	—
15	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
16	C₄×C₈×C₄₇	—	2	6	no	—
17	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
18	C₂²×Dic₄₇	C₄²	1	1	yes	characteristic
19	C₂²×Dic₄₇	C₄²	1	3	yes	—
20	C₂×C₈×C₄₇	—	2	6	no	—
21	C₂×C₈×C₄₇	—	2	6	no	—
22	C₂×C₈×C₄₇	—	2	6	no	—
23	C₄²×C₄₇	#41	1	3	yes	—
24	C₄²×C₄₇	#41	1	1	yes	characteristic
25	C₄×Dic₄₇	—	2	6	no	—
26	C₄×Dic₄₇	—	2	6	no	—
27	C₂×C₄₇⋊C₈	—	2	6	no	—
28	C₂×C₄₇⋊C₈	—	2	6	no	—
29	C₂×C₄₇⋊C₈	—	2	6	no	—
30	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
31	C₂×C₄×C₄₇	#117	1	3	yes	—
32	C₂×C₄×C₄₇	#117	1	3	yes	—
33	C₂×C₄×C₄₇	#131	1	6	yes	—
34	C₂×C₄×C₄₇	#131	1	6	yes	—
35	C₂×Dic₄₇	—	2	3	no	—
36	C₂×Dic₄₇	—	2	3	no	—
37	C₂×Dic₄₇	—	2	6	no	—
38	C₈×C₄₇	—	4	6	no	—
39	C₄₇⋊C₈	—	4	6	no	—
40	#8135	—	47	1	no	maximal, Sylow
41	C₂²×C₄₇	#376	1	1	yes	characteristic
42	C₂²×C₄₇	#405	1	3	yes	—
43	C₂²×C₄₇	#405	1	3	yes	—
44	C₄×C₄₇	#433	1	6	yes	—
45	C₄×C₄₇	#433	1	6	yes	—
46	Dic₄₇	—	4	1	no	—
47	Dic₄₇	—	4	3	no	—
48	#1744	—	47	3	no	—
49	#1273	—	47	3	no	—
50	#1273	D₄₇	1	1	yes	characteristic
51	C₂×C₄₇	#1454	1	1	yes	characteristic
52	C₂×C₄₇	#1454	1	3	yes	—
53	C₂×C₄₇	#1746	1	3	yes	—
54	C₂×C₄×C₈	—	47	3	no	—
55	C₂×C₄×C₈	C₂×D₄₇	1	3	yes	—
56	C₄³	—	47	1	no	—
57	#336	—	47	6	no	—
58	#336	—	47	6	no	—
59	C₄₇	#8135	1	1	yes	characteristic, complemented, Sylow
60	C₂²×C₈	—	47	6	no	—
61	C₂²×C₈	C₄×D₄₇	1	6	yes	—
62	C₂×C₄²	—	47	3	no	—
63	C₂×C₄²	—	47	3	no	—
64	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
65	C₄×C₈	—	2	6	no	—
66	C₄×C₈	—	94	3	no	—
67	C₄×C₈	—	94	3	no	—
68	C₂²×C₄	—	47	1	no	—
69	C₂²×C₄	—	47	3	no	—
70	C₂²×C₄	C₂×C₄×D₄₇	1	3	yes	—
71	C₂×C₈	—	2	6	no	—
72	C₂×C₈	—	2	6	no	—
73	C₂×C₈	—	2	6	no	—
74	C₂×C₈	—	94	6	no	—
75	C₂×C₈	—	94	6	no	—
76	C₂×C₈	—	94	6	no	—
77	C₄²	—	94	6	no	—
78	C₄²	—	94	6	no	—
79	C₄²	#10625352	1	3	yes	—
80	C₄²	#10625354	1	1	yes	characteristic
81	C₂³	C₄²×D₄₇	1	1	yes	characteristic
82	C₂×C₄	—	94	3	no	—
83	C₂×C₄	—	94	3	no	—
84	C₂×C₄	—	94	6	no	—
85	C₂×C₄	#13093980	1	3	yes	—
86	C₂×C₄	#13093982	1	3	yes	—
87	C₂×C₄	#13094030	1	6	yes	—
88	C₂×C₄	#13094032	1	6	yes	—
89	C₈	—	4	6	no	—
90	C₈	—	188	6	no	—
91	C₂²	#15688968	1	1	yes	characteristic
92	C₂²	#15689077	1	3	yes	—
93	C₂²	#15689080	1	3	yes	—
94	C₄	—	188	1	no	—
95	C₄	—	188	3	no	—
96	C₄	#15689181	1	6	yes	—
97	C₄	#15689183	1	6	yes	—
98	C₂	#20241687	1	1	yes	characteristic
99	C₂	#20241689	1	3	yes	—
100	C₂	#20243138	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20243133

C₂

yes

maximal

C₄₇⋊#1273

C₂

yes

maximal

C₄₇×#1273

C₂

yes

maximal, characteristic

C₂×C₄×C₄₇⋊C₈

C₂²

yes

—

C₄²×Dic₄₇

C₂²

yes

characteristic

#15688877

C₄

yes

—

#15688877

C₄

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

—

C₂²×C₄₇⋊C₈

C₂×C₄

yes

—

C₂×C₄×Dic₄₇

C₂×C₄

yes

—

C₂×C₄×Dic₄₇

C₂×C₄

yes

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

C₂²×C₈×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄×C₈×C₄₇

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

—

C₂²×Dic₄₇

C₄²

yes

characteristic

C₂²×Dic₄₇

C₄²

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

#41

yes

—

C₄²×C₄₇

#41

yes

characteristic

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₄₇

#117

yes

—

C₂×C₄×C₄₇

#117

yes

—

C₂×C₄×C₄₇

#131

yes

—

C₂×C₄×C₄₇

#131

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

#405

yes

—

C₄×C₄₇

#433

yes

—

C₄×C₄₇

#433

yes

—

Dic₄₇

—

Dic₄₇

—

#1744

—

#1273

—

#1273

D₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#1746

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₄₇

yes

—

C₄³

—

#336

—

#336

—

C₄₇

#8135

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₄×D₄₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₄₇

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×D₄₇

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₄²

#10625352

yes

—

C₄²

#10625354

yes

characteristic

C₂³

C₄²×D₄₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#13093980

yes

—

C₂×C₄

#13093982

yes

—

C₂×C₄

#13094030

yes

—

C₂×C₄

#13094032

yes

—

C₈

—

C₈

—

188

—

C₂²

#15688968

yes

characteristic

C₂²

#15689077

yes

—

C₂²

#15689080

yes

—

C₄

—

188

—

C₄

—

188

—

C₄

#15689181

yes

—

C₄

#15689183

yes

—

C₂

#20241687

yes

characteristic

C₂

#20241689

yes

—

100

C₂

#20243138

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₄₇
commutator, socle, prime radical	C₂³×C₄₇	C₂×C₄²
Fitting	C₄₇×#1273	C₂
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1273	D₄₇
hypocenter	C₄₇	#8135
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center, Frattini

C₂×C₄²

C₂²×D₄₇

commutator, socle, prime radical

C₂³×C₄₇

C₂×C₄²

Fitting

C₄₇×#1273

C₂

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
59. C₄₇	#8135	semidirect

H ≅ G/N

type

59. C₄₇

#8135

semidirect

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

subgroup	index
#20243133	2
C₄₇⋊#1273	2
C₄₇×#1273	2
#8135	47

subgroup

index

#20243133

C₄₇⋊#1273

C₄₇×#1273

#8135

Subgroups — #26106712

Conjugacy classes of subgroups

Complements

Maximal subgroups