Group #26130996 — subgroups

Subgroups — #26130996

Order 12032.

354 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 115 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	#20245644	C₂	1	4	yes	maximal, complemented
2	#20243006	C₂	1	1	yes	maximal, characteristic, complemented
3	#20243003	C₂	1	1	yes	maximal, characteristic
4	C₈×C₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13093969	—	2	8	no	—
6	#15689528	—	2	2	no	—
7	#15689503	C₂²	1	2	yes	—
8	#15689500	—	2	2	no	—
9	#15689500	C₂²	1	2	yes	—
10	C₄×C₁₆×C₄₇	C₂²	1	2	yes	—
11	C₈²×C₄₇	C₂²	1	1	yes	characteristic
12	C₂×Dic₁₈₈	—	2	4	no	—
13	C₂×D₃₇₆	—	2	4	no	—
14	C₂×D₃₇₆	—	4	4	no	—
15	#13094024	—	2	2	no	—
16	#13094021	—	2	2	no	—
17	#13093969	—	4	16	no	—
18	#13093958	—	4	4	no	—
19	C₂×C₁₆×C₄₇	D₄	1	4	yes	—
20	C₄×C₈×C₄₇	D₄	1	2	yes	—
21	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
22	C₂×D₁₈₈	—	4	2	no	—
23	C₂×D₁₈₈	—	4	4	no	—
24	C₂×Dic₉₄	—	4	2	no	—
25	C₂×C₈×C₄₇	D₈	1	4	yes	—
26	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
27	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
28	C₄⋊Dic₄₇	—	4	4	no	—
29	Dic₁₈₈	—	4	8	no	—
30	D₃₇₆	—	4	8	no	—
31	D₃₇₆	—	8	8	no	—
32	C₁₆×C₄₇	D₈	1	8	yes	complemented
33	C₂²×D₄₇	—	8	2	no	—
34	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
35	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
36	C₂×Dic₄₇	—	8	2	no	—
37	D₁₈₈	—	8	4	no	—
38	D₁₈₈	—	8	8	no	—
39	Dic₉₄	—	8	4	no	—
40	C₈×C₄₇	SD₃₂	1	8	yes	complemented
41	C₈×C₄₇	C₂×D₈	1	4	yes	—
42	#12517	—	47	1	no	maximal, Sylow
43	C₂²×C₄₇	#493	1	1	yes	characteristic
44	C₂×D₄₇	—	16	2	no	—
45	C₂×D₄₇	—	16	4	no	—
46	C₄×C₄₇	#493	1	2	yes	—
47	C₄×C₄₇	C₂×SD₃₂	1	4	yes	—
48	Dic₄₇	—	16	2	no	—
49	#2248	—	47	4	no	—
50	#1713	—	47	1	no	—
51	#1712	—	47	1	no	—
52	C₈×C₁₆	D₄₇	1	1	yes	characteristic, complemented
53	C₂×C₄₇	#1712	1	1	yes	characteristic
54	C₂×C₄₇	#2248	1	2	yes	—
55	D₄₇	—	32	2	no	—
56	C₂×SD₃₂	—	94	8	no	—
57	#500	—	94	2	no	—
58	#494	—	47	2	no	—
59	#493	—	47	2	no	—
60	#493	—	94	2	no	—
61	C₄×C₁₆	C₂×D₄₇	1	2	yes	—
62	C₈²	C₂×D₄₇	1	1	yes	characteristic
63	C₄₇	#12517	1	1	yes	characteristic, complemented, Sylow
64	C₂×Q₁₆	—	94	4	no	—
65	C₂×D₈	—	94	4	no	—
66	C₂×D₈	—	188	4	no	—
67	C₄⋊Q₈	—	94	2	no	—
68	#127	—	94	2	no	—
69	SD₃₂	—	188	16	no	—
70	C₂×C₁₆	D₁₈₈	1	4	yes	—
71	#107	—	188	4	no	—
72	C₄×C₈	D₁₈₈	1	2	yes	—
73	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
74	C₂×Q₈	—	188	2	no	—
75	C₂×D₄	—	188	2	no	—
76	C₂×D₄	—	188	4	no	—
77	Q₁₆	—	188	8	no	—
78	D₈	—	376	8	no	—
79	D₈	—	188	8	no	—
80	C₂×C₈	D₃₇₆	1	4	yes	—
81	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
82	C₄⋊C₄	—	188	4	no	—
83	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
84	C₁₆	D₃₇₆	1	8	yes	complemented
85	C₂³	—	376	2	no	—
86	Q₈	—	376	4	no	—
87	D₄	—	376	4	no	—
88	D₄	—	376	8	no	—
89	C₂×C₄	—	376	2	no	—
90	C₂×C₄	#13094021	1	1	yes	characteristic
91	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
92	C₈	#13093969	1	8	yes	complemented
93	C₈	C₂×D₃₇₆	1	4	yes	—
94	C₂²	—	752	2	no	—
95	C₂²	—	752	4	no	—
96	C₂²	#15689500	1	1	yes	characteristic
97	C₄	—	752	2	no	—
98	C₄	#15689500	1	2	yes	—
99	C₄	C₂×#13093969	1	4	yes	—
100	C₂	—	1504	2	no	—
101	C₂	#20243003	1	1	yes	characteristic
102	C₂	#20245644	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20245644

C₂

yes

maximal, complemented

#20243006

C₂

yes

maximal, characteristic, complemented

#20243003

C₂

yes

maximal, characteristic

C₈×C₁₆×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂×#13093969

—

#15689528

—

#15689503

C₂²

yes

—

#15689500

—

#15689500

C₂²

yes

—

C₄×C₁₆×C₄₇

C₂²

yes

—

C₈²×C₄₇

C₂²

yes

characteristic

C₂×Dic₁₈₈

—

C₂×D₃₇₆

—

C₂×D₃₇₆

—

—

—

—

—

C₂×C₁₆×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄⋊Dic₄₇

—

Dic₁₈₈

—

D₃₇₆

—

D₃₇₆

—

C₁₆×C₄₇

D₈

yes

complemented

C₂²×D₄₇

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

Dic₉₄

—

C₈×C₄₇

SD₃₂

yes

complemented

C₈×C₄₇

C₂×D₈

yes

—

—

C₂²×C₄₇

yes

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

#493

yes

—

C₄×C₄₇

C₂×SD₃₂

yes

—

Dic₄₇

—

#2248

—

#1713

—

#1712

—

C₈×C₁₆

D₄₇

yes

characteristic, complemented

C₂×C₄₇

#1712

yes

characteristic

C₂×C₄₇

#2248

yes

—

D₄₇

—

C₂×SD₃₂

—

—

—

—

—

C₄×C₁₆

C₂×D₄₇

yes

—

C₈²

C₂×D₄₇

yes

characteristic

C₄₇

#12517

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

188

—

C₄⋊Q₈

—

#127

—

SD₃₂

—

188

—

C₂×C₁₆

D₁₈₈

yes

—

#107

—

188

—

C₄×C₈

D₁₈₈

yes

—

C₄×C₈

C₂²×D₄₇

yes

characteristic

C₂×Q₈

—

188

—

C₂×D₄

—

188

—

C₂×D₄

—

188

—

Q₁₆

—

188

—

D₈

—

376

—

D₈

—

188

—

C₂×C₈

D₃₇₆

yes

—

C₂×C₈

C₂×D₁₈₈

yes

—

C₄⋊C₄

—

188

—

C₄²

C₂×D₁₈₈

yes

characteristic

C₁₆

D₃₇₆

yes

complemented

C₂³

—

376

—

Q₈

—

376

—

D₄

—

376

—

D₄

—

376

—

C₂×C₄

—

376

—

C₂×C₄

#13094021

yes

characteristic

C₂×C₄

C₂×D₃₇₆

yes

—

C₈

#13093969

yes

complemented

C₈

C₂×D₃₇₆

yes

—

C₂²

—

752

—

C₂²

—

752

—

C₂²

#15689500

yes

characteristic

C₄

—

752

—

C₄

#15689500

yes

—

C₄

C₂×#13093969

yes

—

100

C₂

—

1504

—

101

C₂

#20243003

yes

characteristic

102

C₂

#20245644

yes

—

	subgroup	G/H
center	C₂²	#15689500
commutator, cosocle	C₄×C₈×C₄₇	C₂³
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₈×C₁₆×C₄₇	C₂
socle	C₂²×C₄₇	#493
prime radical	#20243003	C₂
hypercenter	C₈×C₁₆	D₄₇
hypocenter	C₄₇	#12517
LCS	G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₂²

#15689500

commutator, cosocle

C₄×C₈×C₄₇

C₂³

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₈×C₁₆×C₄₇

C₂

C₂²×C₄₇

C₂

C₈×C₁₆

D₄₇

hypocenter

C₄₇

#12517

LCS

G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
1. #20245644	C₂	semidirect
2. #20243006	C₂	semidirect
4. C₈×C₁₆×C₄₇	C₂	semidirect
32. C₁₆×C₄₇	D₈	semidirect
40. C₈×C₄₇	SD₃₂	semidirect
52. C₈×C₁₆	D₄₇	semidirect
63. C₄₇	#12517	semidirect
84. C₁₆	D₃₇₆	semidirect
92. C₈	#13093969	semidirect

H ≅ G/N

type

1. #20245644

C₂

semidirect

2. #20243006

C₂

semidirect

4. C₈×C₁₆×C₄₇

C₂

semidirect

32. C₁₆×C₄₇

D₈

semidirect

40. C₈×C₄₇

SD₃₂

semidirect

52. C₈×C₁₆

D₄₇

semidirect

63. C₄₇

#12517

semidirect

84. C₁₆

D₃₇₆

semidirect

92. C₈

#13093969

semidirect

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

subgroup	index
#20245644	2
#20243006	2
#20243003	2
C₈×C₁₆×C₄₇	2
#12517	47

subgroup

index

C₈×C₁₆×C₄₇

Subgroups — #26130996

Conjugacy classes of subgroups

Complements

Maximal subgroups