Group #20241220 — subgroups

Subgroups — #20241220

Order 6016.

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

normal subgroups only

This group has 208 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	C₄²×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#431	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×Dic₄₇	—	2	1	no	—
5	C₂×C₄×Dic₄₇	—	2	1	no	—
6	C₂×C₄×Dic₄₇	C₄	1	2	yes	—
7	C₄⋊(C₄₇⋊C₈)	C₄	1	2	yes	complemented
8	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
9	C₄×C₄₇⋊C₈	—	2	1	no	—
10	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₄₇	C₄	1	2	yes	complemented
12	C₂²⋊C₈×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	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	4	no	—
24	C₄×Dic₄₇	—	2	4	no	—
25	C₄×Dic₄₇	—	2	2	no	—
26	C₄×Dic₄₇	C₈	1	4	yes	complemented
27	C₂×C₄₇⋊C₈	—	2	2	no	—
28	C₂³×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₄₇	—	2	2	no	—
30	C₂×C₄×C₄₇	—	2	2	no	—
31	C₂×C₄×C₄₇	C₄²	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	C₂×Dic₄₇	—	2	4	no	—
40	C₂×Dic₄₇	—	2	4	no	—
41	C₂×Dic₄₇	—	2	4	no	—
42	C₈×C₄₇	—	4	2	no	—
43	C₄₇⋊C₈	—	4	2	no	—
44	C₂²×C₄₇	—	2	1	no	—
45	C₂²×C₄₇	—	2	1	no	—
46	C₂²×C₄₇	#95	1	1	yes	characteristic
47	C₂²×C₄₇	C₂²⋊C₈	1	2	yes	—
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	#104	1	2	yes	—
53	C₄×C₄₇	#104	1	2	yes	—
54	C₄×C₄₇	C₄⋊C₈	1	2	yes	—
55	Dic₄₇	—	2	8	no	—
56	#1276	—	47	1	no	maximal, Sylow
57	C₂×C₄₇	—	2	2	no	—
58	C₂×C₄₇	#336	1	1	yes	characteristic
59	C₂×C₄₇	#339	1	1	yes	characteristic
60	C₂×C₄₇	#339	1	1	yes	characteristic
61	#431	—	47	1	no	—
62	#431	D₄₇	1	1	yes	characteristic
63	C₄³	—	47	1	no	—
64	C₄₇	#1276	1	1	yes	characteristic, complemented, Sylow
65	C₂×C₄²	—	47	2	no	—
66	C₂×C₄²	—	94	1	no	—
67	C₂×C₄²	—	94	1	no	—
68	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
69	C₄⋊C₈	—	47	2	no	—
70	C₄⋊C₈	Dic₄₇	1	2	yes	complemented
71	C₂²⋊C₈	—	2	1	no	—
72	C₂²⋊C₈	—	94	1	no	—
73	C₄×C₈	—	2	1	no	—
74	C₄×C₈	—	94	1	no	—
75	C₂²×C₄	—	94	2	no	—
76	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
77	C₂²×C₄	D₁₈₈	1	1	yes	characteristic
78	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	94	2	no	—
81	C₄²	—	47	4	no	—
82	C₄²	—	94	4	no	—
83	C₄²	—	94	4	no	—
84	C₄²	—	94	2	no	—
85	C₄²	Dic₉₄	1	1	yes	characteristic
86	C₄²	C₄×D₄₇	1	1	yes	characteristic
87	C₄²	C₂×Dic₄₇	1	1	yes	characteristic
88	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
89	C₂³	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	94	4	no	—
93	C₂×C₄	—	94	4	no	—
94	C₂×C₄	—	94	4	no	—
95	C₂×C₄	C₈×D₄₇	1	2	yes	—
96	C₂×C₄	#10625324	1	2	yes	—
97	C₂×C₄	C₄×Dic₄₇	1	1	yes	characteristic
98	C₂×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
99	C₂×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
100	C₂×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
101	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
102	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
103	C₈	—	4	2	no	—
104	C₈	—	188	2	no	—
105	C₂²	—	2	1	no	—
106	C₂²	—	2	1	no	—
107	C₂²	C₄₇⋊#95	1	1	yes	characteristic
108	C₂²	(C₂×D₄₇)⋊C₈	1	2	yes	—
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	94	8	no	—
112	C₄	C₈×Dic₄₇	1	1	yes	characteristic
113	C₄	#13093934	1	1	yes	characteristic
114	C₄	#13093950	1	2	yes	—
115	C₄	#13093951	1	2	yes	—
116	C₄	Dic₄₇⋊C₈	1	2	yes	—
117	C₂	—	2	2	no	—
118	C₂	#15688877	1	1	yes	characteristic
119	C₂	#15688882	1	1	yes	characteristic
120	C₂	#15688884	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689176

C₂

yes

maximal, characteristic

C₄²×Dic₄₇

C₂

yes

maximal, characteristic

C₄₇×#431

C₂

yes

maximal, characteristic

C₂×C₄×Dic₄₇

—

C₂×C₄×Dic₄₇

—

C₂×C₄×Dic₄₇

C₄

yes

—

C₄⋊(C₄₇⋊C₈)

C₄

yes

complemented

C₂²⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₄

yes

complemented

C₂²⋊C₈×C₄₇

—

C₄×C₈×C₄₇

—

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

—

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×C₈

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

C₄×C₄₇

#104

yes

—

C₄×C₄₇

C₄⋊C₈

yes

—

Dic₄₇

—

#1276

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

D₄₇

yes

characteristic

C₄³

—

C₄₇

#1276

yes

characteristic, complemented, Sylow

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

Dic₄₇

yes

complemented

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

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

—

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

—

C₂×C₄

—

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

100

C₂×C₄

C₄⋊Dic₄₇

yes

characteristic

101

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

102

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

103

C₈

—

104

C₈

—

188

—

105

C₂²

—

106

C₂²

—

107

C₂²

C₄₇⋊#95

yes

characteristic

108

C₂²

(C₂×D₄₇)⋊C₈

yes

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

C₈×Dic₄₇

yes

characteristic

113

C₄

#13093934

yes

characteristic

114

C₄

#13093950

yes

—

115

C₄

#13093951

yes

—

116

C₄

Dic₄₇⋊C₈

yes

—

117

C₂

—

118

C₂

#15688877

yes

characteristic

119

C₂

#15688882

yes

characteristic

120

C₂

#15688884

yes

characteristic

	subgroup	G/H
center	C₄²	C₂²⋊D₄₇
commutator	C₄×C₄₇	C₄×C₈
Frattini	C₂×C₄²	C₂×D₄₇
Fitting	C₄₇×#431	C₂
socle	C₂²×C₄₇	#95
hypercenter	#431	D₄₇
hypocenter	C₄₇	#1276
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₄²

C₂²⋊D₄₇

commutator

C₄×C₄₇

C₄×C₈

Frattini

C₂×C₄²

C₂×D₄₇

Fitting

C₄₇×#431

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
7. C₄⋊(C₄₇⋊C₈)	C₄	semidirect
11. C₄⋊C₈×C₄₇	C₄	semidirect
26. C₄×Dic₄₇	C₈	semidirect
64. C₄₇	#1276	semidirect
70. C₄⋊C₈	Dic₄₇	semidirect

H ≅ G/N

type

7. C₄⋊(C₄₇⋊C₈)

C₄

semidirect

11. C₄⋊C₈×C₄₇

C₄

semidirect

26. C₄×Dic₄₇

C₈

semidirect

64. C₄₇

#1276

semidirect

70. C₄⋊C₈

Dic₄₇

semidirect

Subgroups — #20241220

Conjugacy classes of subgroups

Complements