Group #26125333 — subgroups

Subgroups — #26125333

Order 12032.

228 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 123 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	#20245191	C₂	1	2	yes	maximal
2	#20241309	C₂	1	4	yes	maximal, complemented
3	C₄₇×#1450	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₄₇⋊C₁₆)	C₄	1	4	yes	—
5	C₂²⋊(C₄₇⋊C₁₆)	C₄	1	4	yes	—
6	#15688899	C₂²	1	4	yes	—
7	C₄₇×#431	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
9	C₄₇×#322	C₂²	1	2	yes	—
10	C₈²×C₄₇	C₄	1	2	yes	—
11	C₂×C₄₇⋊C₁₆	C₂×C₄	1	8	yes	—
12	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
13	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
14	C₄⋊C₈×C₄₇	—	2	2	no	—
15	C₂²⋊C₈×C₄₇	—	2	2	no	—
16	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
17	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
18	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
19	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
22	C₂×C₈×C₄₇	—	2	4	no	—
23	C₂×C₈×C₄₇	—	2	4	no	—
24	C₂×C₈×C₄₇	C₄²	1	2	yes	—
25	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
26	C₄²×C₄₇	C₄²	1	1	yes	characteristic
27	C₄²×C₄₇	M₁₆	1	2	yes	—
28	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
29	C₄₇⋊C₁₆	—	2	8	no	—
30	C₂³×C₄₇	#97	1	1	yes	characteristic
31	C₂×C₄×C₄₇	—	2	2	no	—
32	C₂×C₄×C₄₇	—	4	2	no	—
33	C₂×C₄×C₄₇	#97	1	2	yes	—
34	C₂×C₄×C₄₇	#97	1	1	yes	characteristic
35	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
36	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
37	C₈×C₄₇	—	2	2	no	—
38	C₈×C₄₇	—	4	4	no	—
39	C₈×C₄₇	#131	1	4	yes	—
40	#11416	—	47	1	no	maximal, Sylow
41	C₂²×C₄₇	—	2	2	no	—
42	C₂²×C₄₇	—	2	1	no	—
43	C₂²×C₄₇	C₂×#97	1	1	yes	characteristic
44	C₄×C₄₇	—	2	2	no	—
45	C₄×C₄₇	—	2	2	no	—
46	C₄×C₄₇	—	2	1	no	—
47	C₄×C₄₇	#405	1	2	yes	—
48	#2163	—	47	2	no	—
49	#1450	D₄₇	1	1	yes	characteristic
50	#1313	—	47	4	no	—
51	C₂×C₄₇	—	4	1	no	—
52	C₂×C₄₇	#1452	1	1	yes	characteristic
53	C₂×C₄₇	#2110	1	2	yes	—
54	#431	Dic₄₇	1	2	yes	—
55	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
56	C₄⋊C₁₆	—	47	4	no	—
57	C₂²⋊C₁₆	—	47	4	no	—
58	#346	—	47	4	no	—
59	#322	C₂×D₄₇	1	2	yes	—
60	C₈²	Dic₄₇	1	2	yes	—
61	C₄₇	#11416	1	1	yes	characteristic, complemented, Sylow
62	C₂²×C₈	C₄×D₄₇	1	2	yes	—
63	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
64	C₂×C₁₆	—	47	8	no	—
65	C₄⋊C₈	—	2	2	no	—
66	C₂²⋊C₈	—	2	2	no	—
67	C₄×C₈	C₄×D₄₇	1	2	yes	—
68	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
69	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
70	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
71	C₂²×C₄	#10625329	1	2	yes	—
72	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
73	C₂×C₈	—	2	4	no	—
74	C₂×C₈	—	2	4	no	—
75	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
76	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
77	C₄²	#10625329	1	2	yes	—
78	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
79	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
80	C₁₆	—	94	8	no	—
81	C₂³	#13093935	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	4	2	no	—
84	C₂×C₄	#13093935	1	1	yes	characteristic
85	C₂×C₄	#13093935	1	2	yes	—
86	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
87	C₂×C₄	C₂×#10625329	1	2	yes	—
88	C₈	—	2	2	no	—
89	C₈	—	4	4	no	—
90	C₈	#13094030	1	4	yes	—
91	C₂²	—	2	2	no	—
92	C₂²	—	2	1	no	—
93	C₂²	C₂×#13093935	1	1	yes	characteristic
94	C₄	—	2	2	no	—
95	C₄	—	2	2	no	—
96	C₄	—	2	1	no	—
97	C₄	#15689078	1	2	yes	—
98	C₂	—	4	1	no	—
99	C₂	#20241680	1	1	yes	characteristic
100	C₂	#20244993	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20245191

C₂

yes

maximal

#20241309

C₂

yes

maximal, complemented

C₄₇×#1450

C₂

yes

maximal, characteristic

C₄⋊(C₄₇⋊C₁₆)

C₄

yes

—

C₂²⋊(C₄₇⋊C₁₆)

C₄

yes

—

#15688899

C₂²

yes

—

C₄₇×#431

C₄

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#322

C₂²

yes

—

C₈²×C₄₇

C₄

yes

—

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

—

C₂²×C₈×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄⋊C₈×C₄₇

—

C₂²⋊C₈×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₄²

yes

characteristic

C₂²×C₄×C₄₇

M₁₆

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄²

yes

—

C₂×C₈×C₄₇

C₂²×C₄

yes

—

C₄²×C₄₇

C₄²

yes

characteristic

C₄²×C₄₇

M₁₆

yes

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄₇⋊C₁₆

—

C₂³×C₄₇

#97

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#97

yes

—

C₂×C₄×C₄₇

#97

yes

characteristic

C₂×C₄×C₄₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

#131

yes

—

#11416

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×#97

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#405

yes

—

#2163

—

#1450

D₄₇

yes

characteristic

#1313

—

C₂×C₄₇

—

C₂×C₄₇

#1452

yes

characteristic

C₂×C₄₇

#2110

yes

—

#431

Dic₄₇

yes

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

#322

C₂×D₄₇

yes

—

C₈²

Dic₄₇

yes

—

C₄₇

#11416

yes

characteristic, complemented, Sylow

C₂²×C₈

C₄×D₄₇

yes

—

C₂×C₄²

C₂×Dic₄₇

yes

characteristic

C₂×C₁₆

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

C₄×D₄₇

yes

—

C₄×C₈

C₂×Dic₄₇

yes

—

C₄×C₈

C₂×Dic₄₇

yes

characteristic

C₄×C₈

C₂²×D₄₇

yes

characteristic

C₂²×C₄

#10625329

yes

—

C₂²×C₄

C₄×Dic₄₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×Dic₄₇

yes

—

C₂×C₈

C₂×C₄×D₄₇

yes

—

C₄²

#10625329

yes

—

C₄²

C₄×Dic₄₇

yes

characteristic

C₄²

C₂²×Dic₄₇

yes

characteristic

C₁₆

—

C₂³

#13093935

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#13093935

yes

characteristic

C₂×C₄

#13093935

yes

—

C₂×C₄

C₂×C₄×Dic₄₇

yes

characteristic

C₂×C₄

C₂×#10625329

yes

—

C₈

—

C₈

—

C₈

#13094030

yes

—

C₂²

—

C₂²

—

C₂²

C₂×#13093935

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#15689078

yes

—

C₂

—

C₂

#20241680

yes

characteristic

100

C₂

#20244993

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#1450	C₂
socle	C₂²×C₄₇	C₂×#97
prime radical	C₂³×C₄₇	#97
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	#1450	D₄₇
hypocenter	C₄₇	#11416
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #1450

subgroup

G/H

center

C₂×C₄

C₂×C₄×Dic₄₇

commutator

C₂×C₄×C₄₇

C₂×C₄²

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₄₇×#1450

C₂

socle

C₂²×C₄₇

C₂×#97

prime radical

C₂³×C₄₇

#97

cosocle

C₄×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₄₇

C₁ ◁ C₂×C₄ ◁ #1450

N	H ≅ G/N	type
2. #20241309	C₂	semidirect
61. C₄₇	#11416	semidirect

H ≅ G/N

type

2. #20241309

C₂

semidirect

61. C₄₇

#11416

semidirect

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

subgroup	index
#20245191	2
#20241309	2
C₄₇×#1450	2
#11416	47

subgroup

index

#20245191

#20241309

C₄₇×#1450

#11416

Subgroups — #26125333

Conjugacy classes of subgroups

Complements

Maximal subgroups