Group #26109994 — subgroups

Subgroups — #26109994

Order 12032.

220 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 99 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	#20242151	C₂	1	2	yes	maximal
2	#20241250	C₂	1	4	yes	maximal
3	C₄₇×#1452	C₂	1	1	yes	maximal, characteristic
4	#15689180	C₂²	1	4	yes	—
5	#15688876	—	2	2	no	—
6	#15688874	—	2	2	no	—
7	C₄₇×#431	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
9	C₄₇×#322	C₄	1	4	yes	—
10	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
11	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
12	#13093935	—	2	4	no	—
13	C₂²×C₈×C₄₇	Q₈	1	2	yes	—
14	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
15	C₄⋊C₈×C₄₇	—	2	2	no	—
16	C₂²⋊C₈×C₄₇	—	2	2	no	—
17	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
18	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₄₇	D₄	1	2	yes	—
20	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	#41	1	2	yes	—
22	C₂×C₈×C₄₇	—	2	4	no	—
23	C₂×C₈×C₄₇	—	4	2	no	—
24	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
25	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
26	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
27	C₄²×C₄₇	#41	1	2	yes	—
28	C₂×C₄₇⋊C₈	—	4	4	no	—
29	C₂³×C₄₇	#125	1	1	yes	characteristic
30	C₂×C₄×C₄₇	—	2	2	no	—
31	C₂×C₄×C₄₇	—	4	2	no	—
32	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₄₇	#117	1	2	yes	—
34	C₂×C₄×C₄₇	#124	1	2	yes	—
35	C₂×C₄×C₄₇	#125	1	1	yes	characteristic
36	C₈×C₄₇	—	2	4	no	—
37	C₈×C₄₇	—	4	4	no	—
38	C₄₇⋊C₈	—	8	4	no	—
39	#8799	—	47	1	no	maximal, Sylow
40	C₂²×C₄₇	—	2	2	no	—
41	C₂²×C₄₇	—	2	1	no	—
42	C₂²×C₄₇	#382	1	1	yes	characteristic
43	C₄×C₄₇	—	2	2	no	—
44	C₄×C₄₇	—	2	2	no	—
45	C₄×C₄₇	—	2	1	no	—
46	C₄×C₄₇	#430	1	2	yes	—
47	#1568	—	47	2	no	—
48	#1452	D₄₇	1	1	yes	characteristic
49	#1287	—	47	4	no	—
50	C₂×C₄₇	—	4	1	no	—
51	C₂×C₄₇	#1833	1	2	yes	—
52	C₂×C₄₇	#1838	1	1	yes	characteristic
53	#432	—	47	4	no	—
54	#431	C₂×D₄₇	1	2	yes	—
55	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
56	#335	—	94	2	no	—
57	#334	—	94	2	no	—
58	#322	Dic₄₇	1	4	yes	—
59	C₄₇	#8799	1	1	yes	characteristic, complemented, Sylow
60	C₂²×C₈	Dic₉₄	1	2	yes	—
61	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
62	C₄⋊C₈	—	2	2	no	—
63	C₄⋊C₈	—	94	4	no	—
64	C₂²⋊C₈	—	2	2	no	—
65	C₂²⋊C₈	—	94	4	no	—
66	#97	—	94	4	no	—
67	C₄×C₈	D₁₈₈	1	2	yes	—
68	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
69	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
70	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
71	C₂²×C₄	#10625350	1	2	yes	—
72	C₂×C₈	—	2	4	no	—
73	C₂×C₈	—	4	2	no	—
74	C₂×C₈	—	188	4	no	—
75	C₂×C₈	C₄⋊Dic₄₇	1	4	yes	—
76	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
77	C₄²	#10625350	1	2	yes	—
78	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
79	C₂³	#13094016	1	1	yes	characteristic
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
83	C₂×C₄	#13093983	1	2	yes	—
84	C₂×C₄	#13094012	1	2	yes	—
85	C₂×C₄	#13094016	1	1	yes	characteristic
86	C₈	—	2	4	no	—
87	C₈	—	4	4	no	—
88	C₈	—	376	4	no	—
89	C₂²	—	2	2	no	—
90	C₂²	—	2	1	no	—
91	C₂²	#15688989	1	1	yes	characteristic
92	C₄	—	2	2	no	—
93	C₄	—	2	2	no	—
94	C₄	—	2	1	no	—
95	C₄	#15689172	1	2	yes	—
96	C₂	—	4	1	no	—
97	C₂	#20243512	1	2	yes	—
98	C₂	#20243529	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20242151

C₂

yes

maximal

#20241250

C₂

yes

maximal

C₄₇×#1452

C₂

yes

maximal, characteristic

#15689180

C₂²

yes

—

#15688876

—

#15688874

—

C₄₇×#431

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#322

C₄

yes

—

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

#13093935

—

C₂²×C₈×C₄₇

Q₈

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₂²⋊C₈×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

#41

yes

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#125

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

#117

yes

—

C₂×C₄×C₄₇

#124

yes

—

C₂×C₄×C₄₇

#125

yes

characteristic

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

#8799

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#382

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#430

yes

—

#1568

—

#1452

D₄₇

yes

characteristic

#1287

—

C₂×C₄₇

—

C₂×C₄₇

#1833

yes

—

C₂×C₄₇

yes

—

C₂×D₄₇

yes

—

C₂×C₄×C₈

C₂×D₄₇

yes

—

—

Dic₄₇

yes

—

C₄₇

#8799

yes

characteristic, complemented, Sylow

C₂²×C₈

Dic₉₄

yes

—

C₂×C₄²

C₂²×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

D₁₈₈

yes

—

C₄×C₈

C₂×Dic₄₇

yes

—

C₄×C₈

C₂×Dic₄₇

yes

—

C₂²×C₄

C₂×Dic₉₄

yes

characteristic

C₂²×C₄

#10625350

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

188

—

C₂×C₈

C₄⋊Dic₄₇

yes

—

C₄²

C₂×D₁₈₈

yes

characteristic

C₄²

#10625350

yes

—

C₄²

C₂²×Dic₄₇

yes

characteristic

C₂³

#13094016

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄⋊Dic₄₇

yes

characteristic

C₂×C₄

#13093983

yes

—

C₂×C₄

#13094012

yes

—

C₂×C₄

#13094016

yes

characteristic

C₈

—

C₈

—

C₈

—

376

—

C₂²

—

C₂²

—

C₂²

#15688989

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#15689172

yes

—

C₂

—

C₂

#20243512

yes

—

C₂

#20243529

yes

characteristic

	subgroup	G/H
center	C₂²	#15688989
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1452	C₂
socle	C₂²×C₄₇	#382
prime radical	C₂³×C₄₇	#125
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1452	D₄₇
hypocenter	C₄₇	#8799
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

subgroup

G/H

center

C₂²

#15688989

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1452

C₂

C₂²×C₄₇

C₂³×C₄₇

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

N	H ≅ G/N	type
59. C₄₇	#8799	semidirect

H ≅ G/N

type

59. C₄₇

#8799

semidirect

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

subgroup	index
#20242151	2
#20241250	2
C₄₇×#1452	2
#8799	47

subgroup

index

#20242151

#20241250

C₄₇×#1452

#8799

Subgroups — #26109994

Conjugacy classes of subgroups

Complements

Maximal subgroups