Group #20241239 — subgroups

Subgroups — #20241239

Order 6016.

160 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 85 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	#15689176	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₄×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄₇⋊C₈	C₄	1	2	yes	—
5	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
6	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
7	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
8	C₂²⋊(C₄₇⋊C₈)	C₄	1	2	yes	—
9	C₄×C₄₇⋊C₈	—	2	1	no	—
10	C₂×M₁₆×C₄₇	C₄	1	2	yes	—
11	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
12	#97×C₄₇	—	2	1	no	—
13	C₄×C₈×C₄₇	—	2	1	no	—
14	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
17	M₁₆×C₄₇	—	2	2	no	—
18	M₁₆×C₄₇	C₈	1	4	yes	complemented
19	C₂×C₈×C₄₇	—	2	2	no	—
20	C₄²×C₄₇	D₄	1	1	yes	characteristic
21	C₄²×C₄₇	D₄	1	1	yes	characteristic
22	C₄²×C₄₇	D₄	1	1	yes	characteristic
23	C₄²×C₄₇	Q₈	1	1	yes	characteristic
24	C₂×C₄₇⋊C₈	—	2	2	no	—
25	C₂×C₄₇⋊C₈	—	2	2	no	—
26	C₂×C₄₇⋊C₈	—	4	2	no	—
27	C₂³×C₄₇	C₄²	1	1	yes	characteristic
28	C₂×C₄×C₄₇	—	2	1	no	—
29	C₂×C₄×C₄₇	—	2	1	no	—
30	C₂×C₄×C₄₇	—	4	1	no	—
31	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₄₇	C₂×C₈	1	2	yes	—
38	C₂×C₄×C₄₇	M₁₆	1	2	yes	—
39	C₈×C₄₇	—	2	4	no	—
40	C₄₇⋊C₈	—	4	2	no	—
41	C₄₇⋊C₈	—	4	2	no	—
42	C₂²×C₄₇	—	2	1	no	—
43	C₂²×C₄₇	—	2	1	no	—
44	C₂²×C₄₇	#95	1	1	yes	characteristic
45	C₂²×C₄₇	C₄×C₈	1	1	yes	characteristic
46	C₂²×C₄₇	#97	1	1	yes	characteristic
47	C₄×C₄₇	—	2	1	no	—
48	C₄×C₄₇	—	2	1	no	—
49	C₄×C₄₇	—	2	1	no	—
50	C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
51	C₄×C₄₇	#100	1	1	yes	characteristic
52	C₄×C₄₇	#101	1	1	yes	characteristic
53	C₄×C₄₇	C₄⋊C₈	1	2	yes	—
54	#1283	—	47	1	no	maximal, Sylow
55	C₂×C₄₇	—	2	1	no	—
56	C₂×C₄₇	—	2	1	no	—
57	C₂×C₄₇	#336	1	1	yes	characteristic
58	C₂×C₄₇	#343	1	1	yes	characteristic
59	C₂×C₄₇	#344	1	1	yes	characteristic
60	#431	—	47	1	no	—
61	C₂×C₄⋊C₈	—	47	1	no	—
62	C₄×M₁₆	D₄₇	1	1	yes	characteristic
63	C₄₇	#1283	1	1	yes	characteristic, complemented, Sylow
64	C₂×M₁₆	Dic₄₇	1	2	yes	—
65	C₂²×C₈	—	47	2	no	—
66	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
67	C₄⋊C₈	—	94	1	no	—
68	C₄⋊C₈	—	94	1	no	—
69	C₄⋊C₈	—	94	1	no	—
70	C₂²⋊C₈	—	47	2	no	—
71	#97	—	2	1	no	—
72	C₄×C₈	—	2	1	no	—
73	C₄×C₈	—	94	1	no	—
74	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
75	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
76	C₂²×C₄	C₂×Dic₄₇	1	1	yes	characteristic
77	M₁₆	—	2	2	no	—
78	M₁₆	C₄₇⋊C₈	1	4	yes	complemented
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	94	2	no	—
81	C₂×C₈	—	94	2	no	—
82	C₂×C₈	—	188	2	no	—
83	C₄²	Dic₉₄	1	1	yes	characteristic
84	C₄²	D₁₈₈	1	1	yes	characteristic
85	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
86	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
87	C₂³	C₄×Dic₄₇	1	1	yes	characteristic
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	4	1	no	—
91	C₂×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
92	C₂×C₄	#10625329	1	2	yes	—
93	C₂×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
94	C₂×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
95	C₂×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
96	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
97	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
98	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
99	C₈	—	2	4	no	—
100	C₈	—	188	2	no	—
101	C₈	—	188	2	no	—
102	C₂²	—	2	1	no	—
103	C₂²	—	2	1	no	—
104	C₂²	C₄₇⋊#95	1	1	yes	characteristic
105	C₂²	C₄×C₄₇⋊C₈	1	1	yes	characteristic
106	C₂²	#13093935	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
111	C₄	#13093940	1	1	yes	characteristic
112	C₄	#13093942	1	1	yes	characteristic
113	C₄	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
114	C₂	—	2	1	no	—
115	C₂	—	2	1	no	—
116	C₂	#15688878	1	1	yes	characteristic
117	C₂	#15688891	1	1	yes	characteristic
118	C₂	#15688895	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689176

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₄₇⋊C₈)

C₂

yes

maximal, characteristic

C₄×M₁₆×C₄₇

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

C₄

yes

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

C₄

yes

—

C₄×C₄₇⋊C₈

—

C₂×M₁₆×C₄₇

C₄

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

#97×C₄₇

—

C₄×C₈×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

M₁₆×C₄₇

—

M₁₆×C₄₇

C₈

yes

complemented

C₂×C₈×C₄₇

—

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

Q₈

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₄²

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×C₈

yes

—

C₂×C₄×C₄₇

M₁₆

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#95

yes

characteristic

C₂²×C₄₇

C₄×C₈

yes

characteristic

C₂²×C₄₇

#97

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂²⋊C₈

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

C₄⋊C₈

yes

—

#1283

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄⋊C₈

—

C₄×M₁₆

D₄₇

yes

characteristic

C₄₇

#1283

yes

characteristic, complemented, Sylow

C₂×M₁₆

Dic₄₇

yes

—

C₂²×C₈

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

C₄×D₄₇

yes

characteristic

C₂²×C₄

C₄×D₄₇

yes

characteristic

C₂²×C₄

C₂×Dic₄₇

yes

characteristic

M₁₆

—

M₁₆

C₄₇⋊C₈

yes

complemented

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

188

—

C₄²

Dic₉₄

yes

characteristic

C₄²

D₁₈₈

yes

characteristic

C₄²

C₂²⋊D₄₇

yes

characteristic

C₄²

C₂²⋊D₄₇

yes

characteristic

C₂³

C₄×Dic₄₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄₇⋊C₈

yes

—

C₂×C₄

#10625329

yes

—

C₂×C₄

Dic₄₇⋊C₄

yes

characteristic

C₂×C₄

Dic₄₇⋊C₄

yes

characteristic

C₂×C₄

C₄⋊Dic₄₇

yes

characteristic

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

C₈

—

100

C₈

—

188

—

101

C₈

—

188

—

102

C₂²

—

103

C₂²

—

104

C₂²

C₄₇⋊#95

yes

characteristic

105

C₂²

C₄×C₄₇⋊C₈

yes

characteristic

106

C₂²

#13093935

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

C₂²⋊(C₄₇⋊C₈)

yes

—

111

C₄

#13093940

yes

characteristic

112

C₄

#13093942

yes

characteristic

113

C₄

C₄⋊(C₄₇⋊C₈)

yes

—

114

C₂

—

115

C₂

—

116

C₂

#15688878

yes

characteristic

117

C₂

#15688891

yes

characteristic

118

C₂

#15688895

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₄₇)⋊C₄
commutator	C₂²×C₄₇	C₄×C₈
Frattini	C₂×C₄²	C₂×D₄₇
Fitting	C₄×M₁₆×C₄₇	C₂
socle	C₂²×C₄₇	#95
hypercenter	C₄×M₁₆	D₄₇
hypocenter	C₄₇	#1283
LCS	G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂×C₄

(C₂×D₄₇)⋊C₄

commutator

C₂²×C₄₇

C₄×C₈

Frattini

C₂×C₄²

C₂×D₄₇

Fitting

C₄×M₁₆×C₄₇

C₂

socle

C₂²×C₄₇

#95

hypercenter

C₄×M₁₆

D₄₇

hypocenter

C₄₇

#1283

LCS

G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
18. M₁₆×C₄₇	C₈	semidirect
63. C₄₇	#1283	semidirect
78. M₁₆	C₄₇⋊C₈	semidirect

H ≅ G/N

type

18. M₁₆×C₄₇

C₈

semidirect

63. C₄₇

#1283

semidirect

78. M₁₆

C₄₇⋊C₈

semidirect

Subgroups — #20241239

Conjugacy classes of subgroups

Complements