Group #20243017 — subgroups

Subgroups — #20243017

Order 6016.

220 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 65 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	#15689500	C₂	1	1	yes	maximal, characteristic
2	#15689497	C₂	1	2	yes	maximal, complemented
3	#15689471	C₂	1	2	yes	maximal, complemented
4	#15688844	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#498	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10625336	—	2	4	no	—
7	C₂×D₃₇₆	—	2	2	no	—
8	#13094021	C₂²	1	1	yes	characteristic
9	#13094021	C₂²	1	1	yes	characteristic
10	#13093944	—	2	4	no	—
11	C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
13	#106×C₄₇	—	2	2	no	—
14	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₄₇	—	2	2	no	—
16	C₂×D₁₈₈	—	2	2	no	—
17	C₂×D₁₈₈	—	2	1	no	—
18	C₂×D₁₈₈	—	2	1	no	—
19	C₂×D₁₈₈	—	2	2	no	—
20	C₂×C₈×C₄₇	D₄	1	2	yes	—
21	C₄⋊C₄×C₄₇	—	2	2	no	—
22	C₄⋊C₄×C₄₇	—	2	2	no	—
23	C₄²×C₄₇	C₂³	1	1	yes	characteristic
24	#10625336	—	4	8	no	—
25	C₂×C₄₇⋊C₈	D₄	1	4	yes	—
26	D₃₇₆	—	4	4	no	—
27	C₂²×D₄₇	—	4	1	no	—
28	C₂²×D₄₇	—	4	1	no	—
29	Q₈×C₄₇	—	4	4	no	—
30	C₂×C₄×C₄₇	—	4	2	no	—
31	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
32	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
33	D₁₈₈	—	4	2	no	—
34	D₁₈₈	—	4	2	no	—
35	D₁₈₈	—	4	4	no	—
36	D₁₈₈	—	4	4	no	—
37	C₈×C₄₇	SD₁₆	1	4	yes	complemented
38	C₄₇⋊C₈	—	2	4	no	—
39	C₂²×C₄₇	#127	1	1	yes	characteristic
40	C₂×D₄₇	—	8	1	no	—
41	C₂×D₄₇	—	8	1	no	—
42	C₂×D₄₇	—	8	2	no	—
43	C₂×D₄₇	—	8	2	no	—
44	C₄×C₄₇	—	8	2	no	—
45	C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
46	C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
47	#1715	—	47	1	no	maximal, Sylow
48	C₂×C₄₇	#492	1	1	yes	characteristic
49	C₂×C₄₇	#496	1	2	yes	—
50	D₄₇	—	16	1	no	—
51	D₄₇	—	16	1	no	—
52	#498	D₄₇	1	1	yes	characteristic, complemented
53	#493	—	47	1	no	—
54	#492	—	47	2	no	—
55	#486	—	47	2	no	—
56	#322	—	47	1	no	—
57	C₄₇	#1715	1	1	yes	characteristic, complemented, Sylow
58	C₂×SD₁₆	—	94	4	no	—
59	C₂×D₈	—	94	2	no	—
60	C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
61	#127	—	47	1	no	—
62	#127	—	47	1	no	—
63	#106	—	2	2	no	—
64	#102	—	94	4	no	—
65	C₄×C₈	—	47	2	no	—
66	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
67	C₂×Q₈	—	2	2	no	—
68	C₂×D₄	—	94	2	no	—
69	C₂×D₄	—	94	2	no	—
70	C₂×D₄	—	94	1	no	—
71	C₂×D₄	—	94	1	no	—
72	SD₁₆	—	188	8	no	—
73	D₈	—	188	4	no	—
74	C₂×C₈	—	47	4	no	—
75	C₂×C₈	C₂²⋊D₄₇	1	2	yes	—
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄²	C₂²×D₄₇	1	1	yes	characteristic
79	C₂³	—	188	1	no	—
80	C₂³	—	188	1	no	—
81	Q₈	—	4	4	no	—
82	D₄	—	188	2	no	—
83	D₄	—	188	2	no	—
84	D₄	—	188	4	no	—
85	D₄	—	188	4	no	—
86	C₂×C₄	—	4	2	no	—
87	C₂×C₄	D₄×D₄₇	1	2	yes	—
88	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
89	C₈	—	94	4	no	—
90	C₈	#10625336	1	4	yes	complemented
91	C₂²	—	376	1	no	—
92	C₂²	—	376	1	no	—
93	C₂²	—	376	2	no	—
94	C₂²	—	376	2	no	—
95	C₂²	#13094020	1	1	yes	characteristic
96	C₄	—	8	2	no	—
97	C₄	C₂×#10625336	1	2	yes	—
98	C₄	#13094051	1	4	yes	—
99	C₂	—	752	1	no	—
100	C₂	—	752	1	no	—
101	C₂	#15689497	1	1	yes	characteristic
102	C₂	#15689509	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#15689500

C₂

yes

maximal, characteristic

#15689497

C₂

yes

maximal, complemented

#15689471

C₂

yes

maximal, complemented

#15688844

C₂

yes

maximal, characteristic, complemented

C₄₇×#498

C₂

yes

maximal, characteristic, complemented

C₂×#10625336

—

C₂×D₃₇₆

—

#13094021

C₂²

yes

characteristic

#13094021

C₂²

yes

characteristic

#13093944

—

C₄×C₄₇⋊C₈

C₂²

yes

—

C₄⋊Q₈×C₄₇

C₂²

yes

—

#106×C₄₇

—

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×C₈×C₄₇

D₄

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

#10625336

—

C₂×C₄₇⋊C₈

D₄

yes

—

D₃₇₆

—

C₂²×D₄₇

—

C₂²×D₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₈×C₄₇

SD₁₆

yes

complemented

C₄₇⋊C₈

—

C₂²×C₄₇

#127

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×SD₁₆

yes

—

C₄×C₄₇

AGL(1,ℤ/8)

yes

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

—

D₄₇

—

D₄₇

—

#498

D₄₇

yes

characteristic, complemented

—

—

—

—

C₄₇

#1715

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

C₂×D₄₇

yes

—

—

—

—

—

C₄×C₈

—

C₄×C₈

C₂×D₄₇

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

188

—

D₈

—

188

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₄₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×D₄₇

yes

characteristic

C₂³

—

188

—

C₂³

—

188

—

Q₈

—

D₄

—

188

—

D₄

—

188

—

D₄

—

188

—

D₄

—

188

—

C₂×C₄

—

C₂×C₄

D₄×D₄₇

yes

—

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

C₈

—

C₈

#10625336

yes

complemented

C₂²

—

376

—

C₂²

—

376

—

C₂²

—

376

—

C₂²

—

376

—

C₂²

#13094020

yes

characteristic

C₄

—

C₄

C₂×#10625336

yes

—

C₄

#13094051

yes

—

C₂

—

752

—

100

C₂

—

752

—

101

C₂

#15689497

yes

characteristic

102

C₂

#15689509

yes

—

	subgroup	G/H
center	C₂²	#13094020
commutator	C₄²×C₄₇	C₂³
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#498	C₂
socle	C₂²×C₄₇	#127
hypercenter	#498	D₄₇
hypocenter	C₄₇	#1715
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#13094020

commutator

C₄²×C₄₇

C₂³

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#498

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
2. #15689497	C₂	semidirect
3. #15689471	C₂	semidirect
4. #15688844	C₂	semidirect
5. C₄₇×#498	C₂	semidirect
37. C₈×C₄₇	SD₁₆	semidirect
52. #498	D₄₇	semidirect
57. C₄₇	#1715	semidirect
90. C₈	#10625336	semidirect

H ≅ G/N

type

2. #15689497

C₂

semidirect

3. #15689471

C₂

semidirect

4. #15688844

C₂

semidirect

5. C₄₇×#498

C₂

semidirect

37. C₈×C₄₇

SD₁₆

semidirect

52. #498

D₄₇

semidirect

57. C₄₇

#1715

semidirect

90. C₈

#10625336

semidirect

Subgroups — #20243017

Conjugacy classes of subgroups

Complements