Group #20241669 — subgroups

Subgroups — #20241669

Order 6016.

224 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 137 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	C₄×#10625329	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#15688845	C₂	1	2	yes	maximal, complemented
4	C₈×C₄₇⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₄₇×#431	C₂	1	1	yes	maximal, characteristic
6	C₂×#10625329	C₄	1	2	yes	—
7	C₂²×C₄₇⋊C₈	C₄	1	2	yes	—
8	#13093935	C₄	1	2	yes	—
9	C₄×C₄₇⋊C₈	C₄	1	2	yes	—
10	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
13	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₄₇	C₄	1	2	yes	—
15	C₂²⋊C₈×C₄₇	C₄	1	2	yes	—
16	C₄×C₈×C₄₇	C₂²	1	2	yes	—
17	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₄₇	C₂×C₄	1	4	yes	—
21	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₄₇	C₂³	1	1	yes	characteristic
25	#10625329	C₈	1	8	yes	complemented
26	C₂×C₄₇⋊C₈	—	2	4	no	—
27	C₂×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
28	C₂×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
29	C₂×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
30	C₂³×C₄₇	C₄²	1	1	yes	characteristic
31	C₂×C₄×C₄₇	—	2	2	no	—
32	C₂×C₄×C₄₇	—	2	2	no	—
33	C₂×C₄×C₄₇	C₄²	1	1	yes	characteristic
34	C₂×C₄×C₄₇	C₄²	1	1	yes	characteristic
35	C₂×C₄×C₄₇	C₄²	1	1	yes	characteristic
36	C₂×C₄×C₄₇	C₂×C₈	1	4	yes	—
37	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
40	C₈×C₄₇	—	2	4	no	—
41	C₄₇⋊C₈	—	2	2	no	—
42	C₄₇⋊C₈	C₂×C₈	1	8	yes	complemented
43	C₄₇⋊C₈	M₁₆	1	4	yes	complemented
44	C₂²×C₄₇	—	2	1	no	—
45	C₂²×C₄₇	—	2	1	no	—
46	C₂²×C₄₇	C₄×C₈	1	2	yes	—
47	C₂²×C₄₇	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₄₇	—	2	1	no	—
49	C₄×C₄₇	—	2	1	no	—
50	C₄×C₄₇	C₄×C₈	1	2	yes	—
51	C₄×C₄₇	C₂²×C₈	1	2	yes	—
52	C₄×C₄₇	C₂×M₁₆	1	2	yes	—
53	C₄×C₄₇	#131	1	2	yes	—
54	C₈×M₁₆	—	47	1	no	maximal, Sylow
55	C₂×C₄₇	—	2	2	no	—
56	C₂×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
57	C₂×C₄₇	C₄×M₁₆	1	1	yes	characteristic
58	C₂×C₄₇	#405	1	1	yes	characteristic
59	#431	D₄₇	1	1	yes	characteristic
60	C₄×M₁₆	—	47	1	no	—
61	C₂×C₄×C₈	—	47	1	no	—
62	#322	—	47	2	no	—
63	C₈²	—	47	2	no	—
64	C₄₇	C₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
65	C₂×M₁₆	—	47	2	no	—
66	C₂²×C₈	—	47	2	no	—
67	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
68	C₄⋊C₈	Dic₄₇	1	2	yes	—
69	C₂²⋊C₈	Dic₄₇	1	2	yes	—
70	#97	—	47	2	no	—
71	C₄×C₈	—	47	2	no	—
72	C₄×C₈	—	47	1	no	—
73	C₄×C₈	—	47	2	no	—
74	C₄×C₈	—	47	1	no	—
75	C₄×C₈	C₂×D₄₇	1	2	yes	—
76	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
77	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
78	C₂²×C₄	C₂×Dic₄₇	1	1	yes	characteristic
79	M₁₆	—	47	8	no	—
80	C₂×C₈	—	47	2	no	—
81	C₂×C₈	—	47	4	no	—
82	C₂×C₈	—	47	2	no	—
83	C₂×C₈	—	94	4	no	—
84	C₂×C₈	C₂×Dic₄₇	1	4	yes	—
85	C₄²	C₄×D₄₇	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	C₄×Dic₄₇	1	1	yes	characteristic
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	C₈×D₄₇	1	4	yes	—
93	C₂×C₄	C₄×Dic₄₇	1	1	yes	characteristic
94	C₂×C₄	C₄×Dic₄₇	1	1	yes	characteristic
95	C₂×C₄	C₄×Dic₄₇	1	1	yes	characteristic
96	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
97	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
98	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
99	C₈	—	2	4	no	—
100	C₈	—	47	8	no	—
101	C₈	—	47	4	no	—
102	C₈	—	94	2	no	—
103	C₂²	—	2	1	no	—
104	C₂²	—	2	1	no	—
105	C₂²	C₈×Dic₄₇	1	2	yes	—
106	C₂²	C₂×C₄×Dic₄₇	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	C₈×Dic₄₇	1	2	yes	—
110	C₄	C₂×C₈×D₄₇	1	2	yes	—
111	C₄	M₁₆×D₄₇	1	2	yes	—
112	C₄	#13094031	1	2	yes	—
113	C₂	—	2	2	no	—
114	C₂	C₂×C₈×Dic₄₇	1	1	yes	characteristic
115	C₂	M₁₆×Dic₄₇	1	1	yes	characteristic
116	C₂	#15689079	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#10625329

C₂

yes

maximal, characteristic

C₂×C₄×C₄₇⋊C₈

C₂

yes

maximal, characteristic

#15688845

C₂

yes

maximal, complemented

C₈×C₄₇⋊C₈

C₂

yes

maximal, complemented

C₄₇×#431

C₂

yes

maximal, characteristic

C₂×#10625329

C₄

yes

—

C₂²×C₄₇⋊C₈

C₄

yes

—

#13093935

C₄

yes

—

C₄×C₄₇⋊C₈

C₄

yes

—

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₄×C₄₇⋊C₈

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₄

yes

—

C₂²⋊C₈×C₄₇

C₄

yes

—

C₄×C₈×C₄₇

C₂²

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

C₂³

yes

characteristic

#10625329

C₈

yes

complemented

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂×C₄

yes

—

C₂×C₄₇⋊C₈

C₂×C₄

yes

—

C₂×C₄₇⋊C₈

C₂×C₄

yes

—

C₂³×C₄₇

C₄²

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄²

yes

characteristic

C₂×C₄×C₄₇

C₄²

yes

characteristic

C₂×C₄×C₄₇

C₄²

yes

characteristic

C₂×C₄×C₄₇

C₂×C₈

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

C₂×C₈

yes

complemented

C₄₇⋊C₈

M₁₆

yes

complemented

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₄×C₈

yes

—

C₂²×C₄₇

C₂×C₄²

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄×C₈

yes

—

C₄×C₄₇

C₂²×C₈

yes

—

C₄×C₄₇

C₂×M₁₆

yes

—

C₄×C₄₇

#131

yes

—

C₈×M₁₆

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₂×C₄×C₈

yes

characteristic

C₂×C₄₇

C₄×M₁₆

yes

C₂×C₄₇

yes

D₄₇

yes

characteristic

C₄×M₁₆

—

C₂×C₄×C₈

—

#322

—

C₈²

—

C₄₇

C₈×M₁₆

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

Dic₄₇

yes

—

C₂²⋊C₈

Dic₄₇

yes

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄₇

yes

—

C₂²×C₄

C₄×D₄₇

yes

characteristic

C₂²×C₄

C₄×D₄₇

yes

characteristic

C₂²×C₄

C₂×Dic₄₇

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₄₇

yes

—

C₄²

C₄×D₄₇

yes

characteristic

C₄²

C₄×D₄₇

yes

characteristic

C₄²

C₂×Dic₄₇

yes

characteristic

C₄²

C₂²×D₄₇

yes

characteristic

C₂³

C₄×Dic₄₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₈×D₄₇

yes

—

C₂×C₄

C₄×Dic₄₇

yes

characteristic

C₂×C₄

C₄×Dic₄₇

yes

characteristic

C₂×C₄

C₄×Dic₄₇

yes

characteristic

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

C₈

—

100

C₈

—

101

C₈

—

102

C₈

—

103

C₂²

—

104

C₂²

—

105

C₂²

C₈×Dic₄₇

yes

—

106

C₂²

C₂×C₄×Dic₄₇

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

C₈×Dic₄₇

yes

—

110

C₄

C₂×C₈×D₄₇

yes

—

111

C₄

M₁₆×D₄₇

yes

—

112

C₄

#13094031

yes

—

113

C₂

—

114

C₂

C₂×C₈×Dic₄₇

yes

characteristic

115

C₂

M₁₆×Dic₄₇

yes

characteristic

116

C₂

#15689079

yes

characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₄₇
commutator	C₂×C₄₇	C₂×C₄×C₈
Fitting	C₄₇×#431	C₂
socle	C₂²×C₄₇	C₂×C₄²
hypercenter	#431	D₄₇
hypocenter	C₄₇	C₈×M₁₆
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ #431

subgroup

G/H

center, Frattini

C₄²

C₂²×D₄₇

commutator

C₂×C₄₇

C₂×C₄×C₈

Fitting

C₄₇×#431

C₂

socle

C₂²×C₄₇

C₂×C₄²

hypercenter

#431

D₄₇

hypocenter

C₄₇

C₈×M₁₆

LCS

G ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₄² ◁ #431

N	H ≅ G/N	type
3. #15688845	C₂	semidirect
4. C₈×C₄₇⋊C₈	C₂	semidirect
25. #10625329	C₈	semidirect
42. C₄₇⋊C₈	C₂×C₈	semidirect
43. C₄₇⋊C₈	M₁₆	semidirect
64. C₄₇	C₈×M₁₆	semidirect

H ≅ G/N

type

3. #15688845

C₂

semidirect

4. C₈×C₄₇⋊C₈

C₂

semidirect

25. #10625329

C₈

semidirect

42. C₄₇⋊C₈

C₂×C₈

semidirect

43. C₄₇⋊C₈

M₁₆

semidirect

64. C₄₇

C₈×M₁₆

semidirect

Subgroups — #20241669

Conjugacy classes of subgroups

Complements