Group #20241216 — subgroups

Subgroups — #20241216

Order 6016.

168 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 77 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	#15688969	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#431	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×Dic₄₇	C₄	1	2	yes	—
5	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
6	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
7	C₄×C₄₇⋊C₈	C₄	1	2	yes	complemented
8	C₄₇⋊#95	—	2	1	no	—
9	C₄₇⋊#95	—	2	1	no	—
10	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₄₇	—	2	1	no	—
12	C₂²⋊C₈×C₄₇	—	2	1	no	—
13	C₄×C₈×C₄₇	C₄	1	2	yes	complemented
14	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
17	C₂²×Dic₄₇	—	2	2	no	—
18	C₂×C₈×C₄₇	—	2	2	no	—
19	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₄²×C₄₇	D₄	1	1	yes	characteristic
22	C₄²×C₄₇	Q₈	1	1	yes	characteristic
23	C₄×Dic₄₇	—	2	2	no	—
24	C₄×Dic₄₇	C₈	1	4	yes	complemented
25	C₂×C₄₇⋊C₈	—	2	2	no	—
26	C₂³×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₄₇	—	4	1	no	—
28	C₂×C₄×C₄₇	—	4	1	no	—
29	C₂×C₄×C₄₇	C₄²	1	1	yes	characteristic
30	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₄₇	C₂×C₈	1	2	yes	—
36	C₂×C₄×C₄₇	M₁₆	1	2	yes	—
37	C₂×Dic₄₇	—	2	4	no	—
38	C₂×Dic₄₇	—	4	2	no	—
39	C₂×Dic₄₇	—	4	2	no	—
40	C₈×C₄₇	—	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	C₂²⋊C₈	1	2	yes	—
46	C₄×C₄₇	—	2	1	no	—
47	C₄×C₄₇	—	2	1	no	—
48	C₄×C₄₇	—	2	1	no	—
49	C₄×C₄₇	—	2	1	no	—
50	C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
51	C₄×C₄₇	#97	1	1	yes	characteristic
52	C₄×C₄₇	C₄⋊C₈	1	2	yes	—
53	Dic₄₇	—	4	4	no	—
54	#1274	—	47	1	no	maximal, Sylow
55	C₂×C₄₇	—	2	2	no	—
56	C₂×C₄₇	#336	1	1	yes	characteristic
57	C₂×C₄₇	H(3,ℤ/4)	1	1	yes	characteristic
58	C₂×C₄₇	H(3,ℤ/4)	1	1	yes	characteristic
59	#431	—	47	1	no	—
60	#431	D₄₇	1	1	yes	characteristic
61	#376	—	47	1	no	—
62	C₄₇	#1274	1	1	yes	characteristic, complemented, Sylow
63	C₂×C₄²	—	47	2	no	—
64	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
65	C₄⋊C₈	—	2	1	no	—
66	C₄⋊C₈	—	94	1	no	—
67	C₂²⋊C₈	—	2	1	no	—
68	C₂²⋊C₈	—	94	1	no	—
69	C₄×C₈	—	47	2	no	—
70	C₄×C₈	Dic₄₇	1	2	yes	complemented
71	#95	—	94	1	no	—
72	#95	—	94	1	no	—
73	C₂²×C₄	—	94	2	no	—
74	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
75	C₂²×C₄	D₁₈₈	1	1	yes	characteristic
76	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	94	2	no	—
79	C₄²	—	47	4	no	—
80	C₄²	—	94	2	no	—
81	C₄²	Dic₉₄	1	1	yes	characteristic
82	C₄²	C₄×D₄₇	1	1	yes	characteristic
83	C₄²	C₂×Dic₄₇	1	1	yes	characteristic
84	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
85	C₂³	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
86	C₂×C₄	—	4	1	no	—
87	C₂×C₄	—	4	1	no	—
88	C₂×C₄	—	94	4	no	—
89	C₂×C₄	—	188	2	no	—
90	C₂×C₄	—	188	2	no	—
91	C₂×C₄	C₈×D₄₇	1	2	yes	—
92	C₂×C₄	#10625324	1	2	yes	—
93	C₂×C₄	C₄×Dic₄₇	1	1	yes	characteristic
94	C₂×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
95	C₂×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
96	C₂×C₄	C₄⋊Dic₄₇	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	—	4	2	no	—
100	C₈	—	188	2	no	—
101	C₂²	—	2	1	no	—
102	C₂²	—	2	1	no	—
103	C₂²	C₄₇⋊#95	1	1	yes	characteristic
104	C₂²	(C₂×D₄₇)⋊C₈	1	2	yes	—
105	C₄	—	2	1	no	—
106	C₄	—	2	1	no	—
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	—	188	4	no	—
110	C₄	C₈×Dic₄₇	1	1	yes	characteristic
111	C₄	#13093934	1	1	yes	characteristic
112	C₄	Dic₄₇⋊C₈	1	2	yes	—
113	C₂	—	2	2	no	—
114	C₂	#15688877	1	1	yes	characteristic
115	C₂	#15688879	1	1	yes	characteristic
116	C₂	#15688880	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689176

C₂

yes

maximal, characteristic

#15688969

C₂

yes

maximal, characteristic

C₄₇×#431

C₂

yes

maximal, characteristic

C₂×C₄×Dic₄₇

C₄

yes

—

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

C₄

yes

complemented

C₄₇⋊#95

—

C₄₇⋊#95

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

—

C₂²⋊C₈×C₄₇

—

C₄×C₈×C₄₇

C₄

yes

complemented

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×Dic₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

Q₈

yes

characteristic

C₄×Dic₄₇

—

C₄×Dic₄₇

C₈

yes

complemented

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄²

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

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#95

yes

characteristic

C₂²×C₄₇

C₂²⋊C₈

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄×C₈

yes

characteristic

C₄×C₄₇

#97

yes

characteristic

C₄×C₄₇

C₄⋊C₈

yes

—

Dic₄₇

—

#1274

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

D₄₇

yes

characteristic

#376

—

C₄₇

#1274

yes

characteristic, complemented, Sylow

C₂×C₄²

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

Dic₄₇

yes

complemented

#95

—

#95

—

C₂²×C₄

—

C₂²×C₄

C₄×D₄₇

yes

characteristic

C₂²×C₄

D₁₈₈

yes

characteristic

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₄²

Dic₉₄

yes

characteristic

C₄²

C₄×D₄₇

yes

characteristic

C₄²

C₂×Dic₄₇

yes

characteristic

C₄²

C₂²⋊D₄₇

yes

characteristic

C₂³

(C₂×D₄₇)⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

188

—

C₂×C₄

—

188

—

C₂×C₄

C₈×D₄₇

yes

—

C₂×C₄

#10625324

yes

—

C₂×C₄

C₄×Dic₄₇

yes

characteristic

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

C₂²⋊Dic₄₇

yes

characteristic

C₈

—

100

C₈

—

188

—

101

C₂²

—

102

C₂²

—

103

C₂²

C₄₇⋊#95

yes

characteristic

104

C₂²

(C₂×D₄₇)⋊C₈

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

188

—

110

C₄

C₈×Dic₄₇

yes

characteristic

111

C₄

#13093934

yes

characteristic

112

C₄

Dic₄₇⋊C₈

yes

—

113

C₂

—

114

C₂

#15688877

yes

characteristic

115

C₂

#15688879

yes

characteristic

116

C₂

#15688880

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₄×Dic₄₇
commutator	C₄×C₄₇	C₄×C₈
Frattini	C₂×C₄²	C₂×D₄₇
Fitting	C₄₇×#431	C₂
socle	C₂²×C₄₇	#95
hypercenter	#431	D₄₇
hypocenter	C₄₇	#1274
LCS	G ▷ C₄×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #431

subgroup

G/H

center

C₂×C₄

C₄×Dic₄₇

commutator

C₄×C₄₇

C₄×C₈

Frattini

C₂×C₄²

C₂×D₄₇

Fitting

C₄₇×#431

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₄×C₄₇ ▷ C₄₇

C₁ ◁ C₂×C₄ ◁ #431

N	H ≅ G/N	type
7. C₄×C₄₇⋊C₈	C₄	semidirect
13. C₄×C₈×C₄₇	C₄	semidirect
24. C₄×Dic₄₇	C₈	semidirect
62. C₄₇	#1274	semidirect
70. C₄×C₈	Dic₄₇	semidirect

H ≅ G/N

type

7. C₄×C₄₇⋊C₈

C₄

semidirect

13. C₄×C₈×C₄₇

C₄

semidirect

24. C₄×Dic₄₇

C₈

semidirect

62. C₄₇

#1274

semidirect

70. C₄×C₈

Dic₄₇

semidirect

Subgroups — #20241216

Conjugacy classes of subgroups

Complements