Group #20242528 — subgroups

Subgroups — #20242528

Order 6016.

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

normal subgroups only

This group has 206 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	#15689502	C₂	1	1	yes	maximal, characteristic
2	#15689418	C₂	1	2	yes	maximal
3	#15688872	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal
5	Q₈²×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#10625337	—	2	2	no	—
7	#13094024	C₂²	1	1	yes	characteristic
8	#13093957	—	2	2	no	—
9	C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
10	#13093948	—	2	2	no	—
11	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₄₇	—	2	2	no	—
13	C₄⋊Q₈×C₄₇	—	2	2	no	—
14	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₄×Q₈×C₄₇	—	2	2	no	—
16	C₄×Q₈×C₄₇	C₂²	1	2	yes	—
17	C₂×Q₈×C₄₇	—	2	2	no	—
18	C₂×Q₈×C₄₇	D₄	1	2	yes	—
19	C₂×Q₈×C₄₇	D₄	1	2	yes	—
20	C₂×Dic₉₄	—	2	1	no	—
21	C₄⋊C₄×C₄₇	—	2	4	no	—
22	C₄⋊C₄×C₄₇	—	2	2	no	—
23	C₄⋊C₄×C₄₇	—	2	2	no	—
24	C₄⋊C₄×C₄₇	D₄	1	2	yes	—
25	C₄²×C₄₇	—	2	2	no	—
26	C₄²×C₄₇	—	2	2	no	—
27	C₄²×C₄₇	C₂³	1	1	yes	characteristic
28	#10625337	—	4	4	no	—
29	C₄⋊Dic₄₇	—	2	2	no	—
30	C₂×C₄₇⋊C₈	—	2	1	no	—
31	C₂×C₄₇⋊C₈	—	2	2	no	—
32	Q₈×C₄₇	—	2	2	no	—
33	Q₈×C₄₇	—	4	2	no	—
34	Q₈×C₄₇	—	8	2	no	—
35	Q₈×C₄₇	Q₁₆	1	4	yes	complemented
36	C₂×C₄×C₄₇	—	2	2	no	—
37	C₂×C₄×C₄₇	—	2	2	no	—
38	C₂×C₄×C₄₇	—	2	2	no	—
39	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
41	C₂×Dic₄₇	—	4	1	no	—
42	Dic₉₄	—	4	2	no	—
43	C₄₇⋊C₈	—	2	2	no	—
44	C₄₇⋊C₈	—	4	2	no	—
45	C₂²×C₄₇	#120	1	1	yes	characteristic
46	C₄×C₄₇	—	2	4	no	—
47	C₄×C₄₇	—	2	1	no	—
48	C₄×C₄₇	—	4	2	no	—
49	C₄×C₄₇	—	4	2	no	—
50	C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
51	C₄×C₄₇	#137	1	2	yes	—
52	Dic₄₇	—	8	1	no	—
53	#1633	—	47	1	no	maximal, Sylow
54	C₂×C₄₇	#451	1	2	yes	—
55	C₂×C₄₇	#453	1	1	yes	characteristic
56	Q₈²	D₄₇	1	1	yes	characteristic
57	#494	—	47	1	no	—
58	#477	—	47	2	no	—
59	#333	—	47	1	no	—
60	Q₈⋊C₈	—	47	2	no	—
61	C₄₇	#1633	1	1	yes	characteristic, complemented, Sylow
62	C₂×Q₁₆	—	94	2	no	—
63	C₄⋊Q₈	—	2	2	no	—
64	C₄⋊Q₈	—	2	2	no	—
65	C₄⋊Q₈	—	47	1	no	—
66	C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
67	C₄×Q₈	—	2	2	no	—
68	C₄×Q₈	C₂×D₄₇	1	2	yes	—
69	#107	—	94	2	no	—
70	C₄⋊C₈	—	47	2	no	—
71	#103	—	94	2	no	—
72	C₄×C₈	—	47	1	no	—
73	C₂×Q₈	—	2	2	no	—
74	C₂×Q₈	—	94	1	no	—
75	C₂×Q₈	C₂²⋊D₄₇	1	2	yes	—
76	C₂×Q₈	C₂²⋊D₄₇	1	2	yes	—
77	Q₁₆	—	188	4	no	—
78	C₂×C₈	—	94	1	no	—
79	C₂×C₈	—	94	2	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	94	2	no	—
84	C₄⋊C₄	C₂²⋊D₄₇	1	2	yes	—
85	C₄²	—	2	2	no	—
86	C₄²	—	2	2	no	—
87	C₄²	C₂²×D₄₇	1	1	yes	characteristic
88	Q₈	—	2	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	8	2	no	—
91	Q₈	—	188	2	no	—
92	Q₈	#10625337	1	4	yes	complemented
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	188	1	no	—
97	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
99	C₈	—	94	2	no	—
100	C₈	—	188	2	no	—
101	C₂²	#13093993	1	1	yes	characteristic
102	C₄	—	2	4	no	—
103	C₄	—	2	1	no	—
104	C₄	—	4	2	no	—
105	C₄	—	4	2	no	—
106	C₄	—	376	1	no	—
107	C₄	C₂×#10625337	1	2	yes	—
108	C₄	#13094055	1	2	yes	—
109	C₂	#15689279	1	2	yes	—
110	C₂	#15689291	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689502

C₂

yes

maximal, characteristic

#15689418

C₂

yes

maximal

#15688872

C₂

yes

maximal, characteristic

Q₈⋊(C₄₇⋊C₈)

C₂

yes

maximal

Q₈²×C₄₇

C₂

yes

maximal, characteristic

C₂×#10625337

—

#13094024

C₂²

yes

characteristic

#13093957

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

#13093948

—

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

C₂²

yes

characteristic

C₄×Q₈×C₄₇

—

C₄×Q₈×C₄₇

C₂²

yes

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

D₄

yes

—

C₂×Q₈×C₄₇

D₄

yes

—

C₂×Dic₉₄

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

D₄

yes

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

#10625337

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

Q₁₆

yes

complemented

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×Dic₄₇

—

Dic₉₄

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

#120

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×Q₁₆

yes

—

C₄×C₄₇

#137

yes

—

Dic₄₇

—

#1633

—

maximal, Sylow

C₂×C₄₇

#451

yes

—

C₂×C₄₇

#453

yes

characteristic

Q₈²

D₄₇

yes

—

—

—

Q₈⋊C₈

—

C₄₇

#1633

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₄₇

yes

characteristic

C₄×Q₈

—

C₄×Q₈

C₂×D₄₇

yes

—

#107

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₄₇

yes

—

C₂×Q₈

C₂²⋊D₄₇

yes

—

Q₁₆

—

188

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊D₄₇

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×D₄₇

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

188

—

Q₈

#10625337

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

188

—

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

C₈

—

100

C₈

—

188

—

101

C₂²

#13093993

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

376

—

107

C₄

C₂×#10625337

yes

—

108

C₄

#13094055

yes

—

109

C₂

#15689279

yes

—

110

C₂

#15689291

yes

characteristic

	subgroup	G/H
center	C₂²	#13093993
commutator	C₄²×C₄₇	C₂³
Frattini	C₄²	C₂²×D₄₇
Fitting	Q₈²×C₄₇	C₂
socle	C₂²×C₄₇	#120
hypercenter	Q₈²	D₄₇
hypocenter	C₄₇	#1633
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

subgroup

G/H

center

C₂²

#13093993

commutator

C₄²×C₄₇

C₂³

Frattini

C₄²

C₂²×D₄₇

Fitting

Q₈²×C₄₇

C₂

socle

C₂²×C₄₇

#120

hypercenter

Q₈²

D₄₇

hypocenter

C₄₇

#1633

LCS

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

N	H ≅ G/N	type
35. Q₈×C₄₇	Q₁₆	semidirect
61. C₄₇	#1633	semidirect
92. Q₈	#10625337	semidirect

H ≅ G/N

type

35. Q₈×C₄₇

Q₁₆

semidirect

61. C₄₇

#1633

semidirect

92. Q₈

#10625337

semidirect

Subgroups — #20242528

Conjugacy classes of subgroups

Complements