Group #20242758 — subgroups

Subgroups — #20242758

Order 6016.

156 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 61 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	#15689503	C₂	1	1	yes	maximal, characteristic
2	#15689408	C₂	1	2	yes	maximal
3	#15688876	C₂	1	1	yes	maximal, characteristic
4	#15688870	C₂	1	2	yes	maximal
5	C₈×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₈₈	—	2	1	no	—
7	C₂×Dic₁₈₈	—	2	1	no	—
8	#13094024	C₂²	1	2	yes	—
9	#13093955	—	2	2	no	—
10	C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
11	#13093948	—	2	2	no	—
12	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₄₇	—	2	1	no	—
14	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₄₇	—	2	1	no	—
16	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₄₇	D₄	1	1	yes	characteristic
18	C₂×Dic₉₄	—	2	2	no	—
19	C₂×C₈×C₄₇	—	2	1	no	—
20	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
21	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
22	C₄⋊C₄×C₄₇	—	2	1	no	—
23	C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
24	C₄²×C₄₇	—	2	1	no	—
25	C₄²×C₄₇	C₂³	1	1	yes	characteristic
26	C₄⋊Dic₄₇	—	2	2	no	—
27	C₄⋊Dic₄₇	—	2	2	no	—
28	C₂×C₄₇⋊C₈	—	2	2	no	—
29	Dic₁₈₈	—	4	2	no	—
30	Dic₁₈₈	—	4	2	no	—
31	Q₈×C₄₇	—	2	1	no	—
32	Q₈×C₄₇	Q₁₆	1	2	yes	complemented
33	C₂×C₄×C₄₇	—	2	1	no	—
34	C₂×C₄×C₄₇	—	2	1	no	—
35	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
38	C₂×Dic₄₇	—	4	2	no	—
39	Dic₉₄	—	4	2	no	—
40	Dic₉₄	—	4	2	no	—
41	C₈×C₄₇	—	2	1	no	—
42	C₈×C₄₇	—	4	1	no	—
43	C₈×C₄₇	SD₁₆	1	2	yes	—
44	C₄₇⋊C₈	—	4	2	no	—
45	C₂²×C₄₇	#121	1	1	yes	characteristic
46	C₄×C₄₇	—	2	2	no	—
47	C₄×C₄₇	—	2	1	no	—
48	C₄×C₄₇	—	4	1	no	—
49	C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
50	C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
51	C₄×C₄₇	#135	1	1	yes	characteristic
52	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
53	Dic₄₇	—	8	2	no	—
54	#1670	—	47	1	no	maximal, Sylow
55	C₂×C₄₇	#460	1	1	yes	characteristic
56	C₂×C₄₇	#467	1	1	yes	characteristic
57	C₂×C₄₇	#473	1	1	yes	characteristic
58	#494	—	47	1	no	—
59	#475	—	47	2	no	—
60	C₈×Q₈	D₄₇	1	1	yes	characteristic
61	#335	—	47	1	no	—
62	#332	—	47	2	no	—
63	C₄₇	#1670	1	1	yes	characteristic, complemented, Sylow
64	C₂×Q₁₆	—	94	1	no	—
65	C₂×Q₁₆	—	94	1	no	—
66	C₄⋊Q₈	—	47	2	no	—
67	C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
68	#106	—	94	2	no	—
69	C₄⋊C₈	—	2	1	no	—
70	C₄⋊C₈	—	47	2	no	—
71	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
72	#103	—	94	2	no	—
73	C₄×C₈	—	2	1	no	—
74	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
75	C₂×Q₈	—	94	2	no	—
76	C₂×Q₈	D₁₈₈	1	1	yes	characteristic
77	Q₁₆	—	188	2	no	—
78	Q₁₆	—	188	2	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	94	2	no	—
81	C₂×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
82	C₂×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
83	C₄⋊C₄	—	2	1	no	—
84	C₄⋊C₄	—	94	2	no	—
85	C₄⋊C₄	—	94	2	no	—
86	C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
87	C₄²	—	2	1	no	—
88	C₄²	C₂²×D₄₇	1	1	yes	characteristic
89	Q₈	—	2	1	no	—
90	Q₈	—	188	2	no	—
91	Q₈	—	188	2	no	—
92	Q₈	Dic₁₈₈	1	2	yes	complemented
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	188	2	no	—
96	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
97	C₂×C₄	#10625350	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
99	C₈	—	2	1	no	—
100	C₈	—	4	1	no	—
101	C₈	—	188	2	no	—
102	C₈	#10625336	1	2	yes	—
103	C₂²	#13093998	1	1	yes	characteristic
104	C₄	—	2	2	no	—
105	C₄	—	2	1	no	—
106	C₄	—	4	1	no	—
107	C₄	—	376	2	no	—
108	C₄	C₂×#10625336	1	1	yes	characteristic
109	C₄	C₂×Dic₁₈₈	1	1	yes	characteristic
110	C₄	#13094046	1	1	yes	characteristic
111	C₄	#13094049	1	1	yes	characteristic
112	C₂	#15689323	1	1	yes	characteristic
113	C₂	#15689364	1	1	yes	characteristic
114	C₂	#15689398	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689503

C₂

yes

maximal, characteristic

#15689408

C₂

yes

maximal

#15688876

C₂

yes

maximal, characteristic

#15688870

C₂

yes

maximal

C₈×Q₈×C₄₇

C₂

yes

maximal, characteristic

C₂×Dic₁₈₈

—

C₂×Dic₁₈₈

—

#13094024

C₂²

yes

—

#13093955

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

#13093948

—

C₄×Q₈×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

C₂²

yes

characteristic

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

D₄

yes

characteristic

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

D₄

yes

characteristic

C₂×C₈×C₄₇

D₄

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

Dic₁₈₈

—

Dic₁₈₈

—

Q₈×C₄₇

—

Q₈×C₄₇

Q₁₆

yes

complemented

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

characteristic

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

SD₁₆

yes

—

C₄₇⋊C₈

—

C₂²×C₄₇

#121

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×SD₁₆

yes

characteristic

C₄×C₄₇

C₂×Q₁₆

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

—

C₈×Q₈

D₄₇

yes

characteristic

#335

—

#332

—

C₄₇

#1670

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

C₂×D₄₇

yes

characteristic

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₄₇

yes

characteristic

#103

—

C₄×C₈

—

C₄×C₈

C₂×D₄₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

D₁₈₈

yes

characteristic

Q₁₆

—

188

—

Q₁₆

—

188

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₄₇

yes

characteristic

C₂×C₈

C₂²⋊D₄₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₁₈₈

yes

characteristic

C₄²

—

C₄²

C₂²×D₄₇

yes

characteristic

Q₈

—

Q₈

—

188

—

Q₈

—

188

—

Q₈

Dic₁₈₈

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

188

—

C₂×C₄

C₂×D₁₈₈

yes

characteristic

C₂×C₄

#10625350

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

C₈

—

100

C₈

—

101

C₈

—

188

—

102

C₈

#10625336

yes

—

103

C₂²

#13093998

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

376

—

108

C₄

C₂×#10625336

yes

characteristic

109

C₄

C₂×Dic₁₈₈

yes

characteristic

110

C₄

#13094046

yes

characteristic

111

C₄

#13094049

yes

characteristic

112

C₂

#15689323

yes

characteristic

113

C₂

#15689364

yes

characteristic

114

C₂

#15689398

yes

characteristic

	subgroup	G/H
center	C₂²	#13093998
commutator	C₄²×C₄₇	C₂³
Frattini	C₄²	C₂²×D₄₇
Fitting	C₈×Q₈×C₄₇	C₂
socle	C₂²×C₄₇	#121
hypercenter	C₈×Q₈	D₄₇
hypocenter	C₄₇	#1670
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

subgroup

G/H

center

C₂²

#13093998

commutator

C₄²×C₄₇

C₂³

Frattini

C₄²

C₂²×D₄₇

Fitting

C₈×Q₈×C₄₇

C₂

socle

C₂²×C₄₇

#121

hypercenter

C₈×Q₈

D₄₇

hypocenter

C₄₇

#1670

LCS

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

N	H ≅ G/N	type
32. Q₈×C₄₇	Q₁₆	semidirect
63. C₄₇	#1670	semidirect
92. Q₈	Dic₁₈₈	semidirect

H ≅ G/N

type

32. Q₈×C₄₇

Q₁₆

semidirect

63. C₄₇

#1670

semidirect

92. Q₈

Dic₁₈₈

semidirect

Subgroups — #20242758

Conjugacy classes of subgroups

Complements