Group #20241417 — subgroups

Subgroups — #20241417

Order 6016.

140 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 35 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	#15689043	C₂	1	1	yes	maximal, characteristic
2	#15688849	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#343	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₉₄	C₄	1	1	yes	characteristic
5	C₂×Dic₄₇⋊C₄	—	2	1	no	—
6	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
7	C₄₇⋊#95	—	2	1	no	—
8	C₄₇⋊#95	C₄	1	1	yes	characteristic
9	C₂×M₁₆×C₄₇	—	2	1	no	—
10	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
13	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
14	C₂²×Dic₄₇	—	2	1	no	—
15	C₂²×Dic₄₇	—	2	1	no	—
16	C₂×Dic₉₄	—	4	1	no	—
17	C₂×Dic₉₄	—	4	2	no	—
18	M₁₆×C₄₇	—	2	2	no	—
19	M₁₆×C₄₇	—	4	1	no	—
20	C₂×C₈×C₄₇	—	4	1	no	—
21	C₄⋊C₄×C₄₇	—	4	1	no	—
22	Dic₄₇⋊C₄	—	4	2	no	—
23	C₂×C₄₇⋊C₈	—	4	1	no	—
24	C₂³×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₄₇	—	2	1	no	—
26	C₂×C₄×C₄₇	—	2	1	no	—
27	C₂×C₄×C₄₇	—	2	1	no	—
28	C₂×C₄×C₄₇	—	2	1	no	—
29	C₂×C₄×C₄₇	—	4	1	no	—
30	C₂×C₄×C₄₇	—	4	1	no	—
31	C₂×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
32	C₂×C₄×C₄₇	Q₁₆	1	1	yes	characteristic
33	C₂×Dic₄₇	—	2	2	no	—
34	C₂×Dic₄₇	—	2	2	no	—
35	C₂×Dic₄₇	—	4	1	no	—
36	C₂×Dic₄₇	—	4	1	no	—
37	C₂×Dic₄₇	—	4	1	no	—
38	C₂×Dic₄₇	—	4	1	no	—
39	Dic₉₄	—	4	2	no	—
40	Dic₉₄	—	4	2	no	—
41	C₈×C₄₇	—	4	2	no	—
42	C₄₇⋊C₈	—	8	1	no	—
43	C₂²×C₄₇	—	2	1	no	—
44	C₂²×C₄₇	—	2	1	no	—
45	C₂²×C₄₇	#99	1	1	yes	characteristic
46	C₂²×C₄₇	#103	1	1	yes	characteristic
47	C₂²×C₄₇	#104	1	1	yes	characteristic
48	C₄×C₄₇	—	2	1	no	—
49	C₄×C₄₇	—	2	1	no	—
50	C₄×C₄₇	—	4	1	no	—
51	C₄×C₄₇	—	4	1	no	—
52	Dic₄₇	—	4	2	no	—
53	Dic₄₇	—	4	2	no	—
54	#1354	—	47	1	no	maximal, Sylow
55	C₂×C₄₇	—	2	1	no	—
56	C₂×C₄₇	—	2	1	no	—
57	C₂×C₄₇	#328	1	1	yes	characteristic
58	C₂×C₄₇	AΣL(1,𝔽₁₆)	1	1	yes	characteristic
59	C₂×C₄₇	#356	1	1	yes	characteristic
60	#395	—	47	1	no	—
61	#343	D₄₇	1	1	yes	characteristic
62	#324	—	47	1	no	—
63	C₄₇	#1354	1	1	yes	characteristic, complemented, Sylow
64	C₂²×Q₈	—	47	1	no	—
65	C₂×M₁₆	—	2	1	no	—
66	C₂×C₄⋊C₄	—	94	1	no	—
67	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
68	C₂²⋊C₈	—	94	1	no	—
69	#95	—	47	1	no	—
70	#95	—	94	1	no	—
71	C₂×Q₈	—	188	1	no	—
72	C₂×Q₈	—	188	2	no	—
73	C₂²×C₄	—	94	1	no	—
74	C₂²×C₄	—	94	1	no	—
75	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
76	C₂²×C₄	D₁₈₈	1	1	yes	characteristic
77	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
78	M₁₆	—	2	2	no	—
79	M₁₆	—	4	1	no	—
80	C₂×C₈	—	4	1	no	—
81	C₂×C₈	—	188	1	no	—
82	C₄⋊C₄	—	4	1	no	—
83	C₄⋊C₄	—	188	2	no	—
84	C₂³	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
85	Q₈	—	188	2	no	—
86	Q₈	—	188	2	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	4	1	no	—
92	C₂×C₄	—	4	1	no	—
93	C₂×C₄	—	94	2	no	—
94	C₂×C₄	—	94	2	no	—
95	C₂×C₄	—	188	1	no	—
96	C₂×C₄	—	188	1	no	—
97	C₂×C₄	—	188	1	no	—
98	C₂×C₄	—	188	1	no	—
99	C₂×C₄	#10625335	1	1	yes	characteristic
100	C₂×C₄	#10625337	1	1	yes	characteristic
101	C₈	—	4	2	no	—
102	C₈	—	376	1	no	—
103	C₂²	—	2	1	no	—
104	C₂²	—	2	1	no	—
105	C₂²	#13093938	1	1	yes	characteristic
106	C₂²	#13093947	1	1	yes	characteristic
107	C₂²	#13093952	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	4	1	no	—
111	C₄	—	4	1	no	—
112	C₄	—	188	2	no	—
113	C₄	—	188	2	no	—
114	C₂	—	2	1	no	—
115	C₂	—	2	1	no	—
116	C₂	#15688859	1	1	yes	characteristic
117	C₂	#15688911	1	1	yes	characteristic
118	C₂	#15688921	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689043

C₂

yes

maximal, characteristic

#15688849

C₂

yes

maximal, characteristic

C₄₇×#343

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

C₄

yes

characteristic

C₂×Dic₄₇⋊C₄

—

C₂²⋊(C₄₇⋊C₈)

—

C₄₇⋊#95

—

C₄₇⋊#95

C₄

yes

characteristic

C₂×M₁₆×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

Dic₄₇⋊C₄

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

SD₁₆

yes

characteristic

C₂×C₄×C₄₇

Q₁₆

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

Dic₄₇

—

#1354

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#328

yes

characteristic

C₂×C₄₇

AΣL(1,𝔽₁₆)

yes

C₂×C₄₇

yes

—

D₄₇

yes

characteristic

#324

—

C₄₇

#1354

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×M₁₆

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

C₂²⋊C₈

—

#95

—

#95

—

C₂×Q₈

—

188

—

C₂×Q₈

—

188

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₄₇

yes

characteristic

C₂²×C₄

D₁₈₈

yes

characteristic

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

188

—

C₄⋊C₄

—

C₄⋊C₄

—

188

—

C₂³

(C₂×D₄₇)⋊C₄

yes

characteristic

Q₈

—

188

—

Q₈

—

188

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

188

—

C₂×C₄

—

188

—

C₂×C₄

—

188

—

C₂×C₄

—

188

—

C₂×C₄

#10625335

yes

characteristic

100

C₂×C₄

#10625337

yes

characteristic

101

C₈

—

102

C₈

—

376

—

103

C₂²

—

104

C₂²

—

105

C₂²

#13093938

yes

characteristic

106

C₂²

#13093947

yes

characteristic

107

C₂²

#13093952

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

188

—

113

C₄

—

188

—

114

C₂

—

115

C₂

—

116

C₂

#15688859

yes

characteristic

117

C₂

#15688911

yes

characteristic

118

C₂

#15688921

yes

characteristic

	subgroup	G/H
center	C₂²	#13093938
commutator	C₂²×C₄×C₄₇	C₂×C₄
Frattini	C₂×C₄⋊C₄	C₂×D₄₇
Fitting	C₄₇×#343	C₂
socle	C₂²×C₄₇	#99
hypercenter	#343	D₄₇
hypocenter	C₄₇	#1354
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #343

subgroup

G/H

center

C₂²

#13093938

commutator

C₂²×C₄×C₄₇

C₂×C₄

Frattini

C₂×C₄⋊C₄

C₂×D₄₇

Fitting

C₄₇×#343

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #343

N	H ≅ G/N	type
63. C₄₇	#1354	semidirect

H ≅ G/N

type

63. C₄₇

#1354

semidirect

Subgroups — #20241417

Conjugacy classes of subgroups

Complements