Group #20241271 — subgroups

Subgroups — #20241271

Order 6016.

188 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 105 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	C₂×C₄⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal
2	C₄₇×#384	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄₇⋊C₈	—	2	2	no	—
4	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
5	C₄⋊(C₄₇⋊C₈)	C₄	1	4	yes	complemented
6	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
7	C₂×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
8	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
9	C₂²×C₄×C₄₇	—	2	2	no	—
10	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
11	C₂²×C₄×C₄₇	D₄	1	2	yes	—
12	C₄⋊C₄×C₄₇	—	4	2	no	—
13	C₄⋊C₄×C₄₇	—	4	2	no	—
14	C₄⋊C₄×C₄₇	—	4	2	no	—
15	C₄²×C₄₇	C₂×C₄	1	2	yes	—
16	C₄²×C₄₇	D₄	1	1	yes	characteristic
17	C₄²×C₄₇	Q₈	1	1	yes	characteristic
18	C₂×C₄₇⋊C₈	—	2	4	no	—
19	C₂×C₄₇⋊C₈	—	4	2	no	—
20	C₂×C₄₇⋊C₈	—	4	2	no	—
21	C₂³×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₄₇	—	2	1	no	—
23	C₂×C₄×C₄₇	—	2	1	no	—
24	C₂×C₄×C₄₇	—	4	2	no	—
25	C₂×C₄×C₄₇	—	4	4	no	—
26	C₂×C₄×C₄₇	C₄²	1	1	yes	characteristic
27	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
28	C₂×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₄₇	D₈	1	2	yes	—
31	C₂×C₄×C₄₇	SD₁₆	1	2	yes	—
32	C₂×C₄×C₄₇	SD₁₆	1	2	yes	—
33	C₂×C₄×C₄₇	Q₁₆	1	2	yes	—
34	C₄₇⋊C₈	—	4	4	no	—
35	C₂²×C₄₇	#95	1	1	yes	characteristic
36	C₂²×C₄₇	#100	1	1	yes	characteristic
37	C₂²×C₄₇	#101	1	1	yes	characteristic
38	C₂²×C₄₇	#102	1	2	yes	—
39	C₂²×C₄₇	#103	1	2	yes	—
40	C₄×C₄₇	—	2	1	no	—
41	C₄×C₄₇	—	2	1	no	—
42	C₄×C₄₇	—	8	2	no	—
43	C₄×C₄₇	#102	1	2	yes	—
44	C₄×C₄₇	#103	1	2	yes	—
45	C₄×C₄₇	#106	1	2	yes	—
46	C₄×C₄₇	#107	1	2	yes	—
47	#1296	—	47	1	no	maximal, Sylow
48	C₂×C₄₇	#331	1	1	yes	characteristic
49	C₂×C₄₇	#332	1	2	yes	—
50	C₂×C₄₇	#333	1	1	yes	characteristic
51	C₂×C₄₇	#340	1	2	yes	—
52	C₂×C₄₇	#343	1	1	yes	characteristic
53	C₂×C₄⋊C₈	—	47	2	no	—
54	#384	D₄₇	1	1	yes	characteristic
55	C₄₇	#1296	1	1	yes	characteristic, complemented, Sylow
56	C₂²×C₈	—	94	2	no	—
57	C₂×C₄⋊C₄	—	2	2	no	—
58	C₂×C₄⋊C₄	Dic₄₇	1	2	yes	—
59	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
60	C₄⋊C₈	—	47	4	no	—
61	C₄⋊C₈	—	94	2	no	—
62	C₂²×C₄	—	2	2	no	—
63	C₂²×C₄	C₂×Dic₄₇	1	1	yes	characteristic
64	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
65	C₂×C₈	—	94	4	no	—
66	C₂×C₈	—	188	2	no	—
67	C₂×C₈	—	188	2	no	—
68	C₄⋊C₄	—	4	2	no	—
69	C₄⋊C₄	—	4	2	no	—
70	C₄⋊C₄	—	4	2	no	—
71	C₄²	Dic₉₄	1	1	yes	characteristic
72	C₄²	C₄×D₄₇	1	2	yes	—
73	C₄²	D₁₈₈	1	1	yes	characteristic
74	C₂³	C₂²⋊Dic₄₇	1	1	yes	characteristic
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	4	2	no	—
78	C₂×C₄	—	4	4	no	—
79	C₂×C₄	C₄×Dic₄₇	1	1	yes	characteristic
80	C₂×C₄	Dic₄₇⋊C₄	1	2	yes	—
81	C₂×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
82	C₂×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
83	C₂×C₄	#10625334	1	2	yes	—
84	C₂×C₄	#10625335	1	2	yes	—
85	C₂×C₄	#10625336	1	2	yes	—
86	C₂×C₄	#10625337	1	2	yes	—
87	C₈	—	188	4	no	—
88	C₂²	C₄₇⋊#95	1	1	yes	characteristic
89	C₂²	#13093941	1	1	yes	characteristic
90	C₂²	#13093943	1	1	yes	characteristic
91	C₂²	#13093945	1	2	yes	—
92	C₂²	#13093948	1	2	yes	—
93	C₄	—	2	1	no	—
94	C₄	—	2	1	no	—
95	C₄	—	8	2	no	—
96	C₄	#13093944	1	2	yes	—
97	C₄	#13093947	1	2	yes	—
98	C₄	#13093955	1	2	yes	—
99	C₄	#13093957	1	2	yes	—
100	C₂	#15688867	1	1	yes	characteristic
101	C₂	#15688869	1	2	yes	—
102	C₂	#15688872	1	1	yes	characteristic
103	C₂	#15688885	1	2	yes	—
104	C₂	#15688892	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₄₇⋊C₈)

C₂

yes

maximal

C₄₇×#384

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

C₄

yes

complemented

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₄

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

Q₈

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄²

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₄⋊C₄

yes

—

C₂×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

D₈

yes

—

C₂×C₄×C₄₇

SD₁₆

yes

—

C₂×C₄×C₄₇

SD₁₆

yes

—

C₂×C₄×C₄₇

Q₁₆

yes

—

C₄₇⋊C₈

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

#103

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#102

yes

—

C₄×C₄₇

#103

yes

—

C₄×C₄₇

#106

yes

—

C₄×C₄₇

#107

yes

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#333

yes

characteristic

C₂×C₄₇

#340

yes

—

C₂×C₄₇

#343

yes

characteristic

C₂×C₄⋊C₈

—

#384

D₄₇

yes

characteristic

C₄₇

#1296

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₄₇

yes

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₄₇

yes

characteristic

C₂²×C₄

C₂²⋊D₄₇

yes

—

C₂×C₈

—

C₂×C₈

—

188

—

C₂×C₈

—

188

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Dic₉₄

yes

characteristic

C₄²

C₄×D₄₇

yes

—

C₄²

D₁₈₈

yes

characteristic

C₂³

C₂²⋊Dic₄₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₄₇

yes

characteristic

C₂×C₄

Dic₄₇⋊C₄

yes

—

C₂×C₄

C₄⋊Dic₄₇

yes

characteristic

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

—

C₂×C₄

#10625334

yes

—

C₂×C₄

#10625335

yes

—

C₂×C₄

#10625336

yes

—

C₂×C₄

#10625337

yes

—

C₈

—

188

—

C₂²

C₄₇⋊#95

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#13093948

yes

—

C₄

—

C₄

—

C₄

—

C₄

#13093944

yes

—

C₄

#13093947

yes

—

C₄

#13093955

yes

—

C₄

#13093957

yes

—

100

C₂

#15688867

yes

characteristic

101

C₂

#15688869

yes

—

102

C₂

#15688872

yes

characteristic

103

C₂

#15688885

yes

—

104

C₂

#15688892

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₄²
Frattini	C₂×C₄²	C₂×D₄₇
Fitting	C₄₇×#384	C₂
socle	C₂³×C₄₇	C₂²⋊C₄
hypercenter	#384	D₄₇
hypocenter	C₄₇	#1296
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂²⋊Dic₄₇

commutator

C₂×C₄×C₄₇

C₄²

Frattini

C₂×C₄²

C₂×D₄₇

Fitting

C₄₇×#384

C₂

socle

C₂³×C₄₇

C₂²⋊C₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
5. C₄⋊(C₄₇⋊C₈)	C₄	semidirect
55. C₄₇	#1296	semidirect

H ≅ G/N

type

5. C₄⋊(C₄₇⋊C₈)

C₄

semidirect

55. C₄₇

#1296

semidirect

Subgroups — #20241271

Conjugacy classes of subgroups

Complements