Group #20251997 — subgroups

Subgroups — #20251997

Order 6016.

440 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 249 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	#15689890	C₂	1	1	yes	maximal, characteristic
2	C₂×#13094031	C₂	1	1	yes	maximal, characteristic
3	#15689171	C₂	1	6	yes	maximal, complemented
4	C₂×#13093959	C₂	1	6	yes	maximal, complemented
5	C₄₇×#568	C₂	1	1	yes	maximal, characteristic
6	#13094031	—	2	4	no	—
7	#13094031	C₄	1	8	yes	—
8	C₂×#10625329	C₂²	1	6	yes	—
9	C₂×#10625329	C₂²	1	3	yes	—
10	C₂²×C₄₇⋊C₈	C₂²	1	3	yes	—
11	#13093959	C₂²	1	4	yes	—
12	#13093959	C₂²	1	12	yes	—
13	C₂×#41×C₄₇	C₂²	1	1	yes	characteristic
14	#131×C₄₇	—	2	4	no	—
15	C₂×M₁₆×C₄₇	C₂²	1	6	yes	—
16	#41×C₄₇	C₂×C₄	1	4	yes	—
17	#41×C₄₇	D₄	1	4	yes	—
18	C₂×Q₈×C₄₇	Q₈	1	1	yes	characteristic
19	C₂×D₄×C₄₇	Q₈	1	3	yes	—
20	C₂²×C₄×C₄₇	C₂³	1	3	yes	—
21	M₁₆×C₄₇	—	2	6	no	—
22	M₁₆×C₄₇	C₂³	1	4	yes	—
23	C₂×C₈×C₄₇	—	2	6	no	—
24	#10625329	—	2	6	no	—
25	#10625329	C₂×C₄	1	12	yes	—
26	#10625329	C₂³	1	4	yes	—
27	C₂×C₄₇⋊C₈	—	2	6	no	—
28	C₂×C₄₇⋊C₈	C₂×C₄	1	12	yes	—
29	C₂×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
30	C₂×C₄₇⋊C₈	C₂³	1	3	yes	—
31	C₂³×C₄₇	C₂×Q₈	1	3	yes	—
32	Q₈×C₄₇	C₄⋊C₄	1	4	yes	—
33	D₄×C₄₇	C₄⋊C₄	1	12	yes	—
34	C₂×C₄×C₄₇	C₂²×C₄	1	6	yes	—
35	C₂×C₄×C₄₇	C₂×D₄	1	6	yes	—
36	C₂×C₄×C₄₇	C₂×Q₈	1	3	yes	—
37	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
38	C₈×C₄₇	—	2	4	no	—
39	C₄₇⋊C₈	—	2	4	no	—
40	C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
41	C₄₇⋊C₈	C₂²×C₄	1	6	yes	—
42	C₂²×C₄₇	—	4	3	no	—
43	C₂²×C₄₇	C₂×C₄⋊C₄	1	6	yes	—
44	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
45	C₄×C₄₇	C₂×C₄⋊C₄	1	6	yes	—
46	C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
47	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
48	#2916	—	47	1	no	maximal, Sylow
49	C₂×C₄₇	—	2	1	no	—
50	C₂×C₄₇	—	2	6	no	—
51	C₂×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
52	#568	—	47	1	no	—
53	#568	D₄₇	1	1	yes	characteristic
54	C₂×#131	—	47	1	no	—
55	#430	—	47	6	no	—
56	C₂×#108	—	47	6	no	—
57	C₄₇	#2916	1	1	yes	characteristic, complemented, Sylow
58	C₂×#41	C₂×D₄₇	1	1	yes	characteristic
59	#131	—	2	4	no	—
60	#131	—	47	8	no	—
61	#131	—	94	4	no	—
62	C₂×M₁₆	—	47	3	no	—
63	C₂×M₁₆	—	47	6	no	—
64	C₂×M₁₆	C₂×D₄₇	1	6	yes	—
65	C₂²×C₈	—	47	3	no	—
66	#108	—	47	4	no	—
67	#108	—	47	12	no	—
68	#41	C₄×D₄₇	1	4	yes	—
69	#41	C₂²⋊D₄₇	1	4	yes	—
70	C₂×Q₈	Dic₉₄	1	1	yes	characteristic
71	C₂×D₄	Dic₉₄	1	3	yes	—
72	C₂²×C₄	C₂²×D₄₇	1	3	yes	—
73	M₁₆	—	2	6	no	—
74	M₁₆	—	47	12	no	—
75	M₁₆	—	47	4	no	—
76	M₁₆	—	94	6	no	—
77	M₁₆	C₂²×D₄₇	1	4	yes	—
78	C₂×C₈	—	2	6	no	—
79	C₂×C₈	—	47	12	no	—
80	C₂×C₈	—	47	1	no	—
81	C₂×C₈	—	47	3	no	—
82	C₂×C₈	—	94	6	no	—
83	C₂³	C₂×Dic₉₄	1	3	yes	—
84	Q₈	Dic₄₇⋊C₄	1	4	yes	—
85	D₄	Dic₄₇⋊C₄	1	12	yes	—
86	C₂×C₄	C₂×Dic₉₄	1	3	yes	—
87	C₂×C₄	C₂×C₄×D₄₇	1	6	yes	—
88	C₂×C₄	C₂×C₂²⋊D₄₇	1	6	yes	—
89	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
90	C₈	—	2	4	no	—
91	C₈	—	47	2	no	—
92	C₈	—	47	6	no	—
93	C₈	—	94	4	no	—
94	C₂²	—	4	3	no	—
95	C₂²	C₂×Dic₄₇⋊C₄	1	6	yes	—
96	C₂²	C₂²×Dic₉₄	1	1	yes	characteristic
97	C₄	C₂×Dic₄₇⋊C₄	1	6	yes	—
98	C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
99	C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
100	C₂	—	2	1	no	—
101	C₂	—	2	6	no	—
102	C₂	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689890

C₂

yes

maximal, characteristic

C₂×#13094031

C₂

yes

maximal, characteristic

#15689171

C₂

yes

maximal, complemented

C₂×#13093959

C₂

yes

maximal, complemented

C₄₇×#568

C₂

yes

maximal, characteristic

#13094031

—

#13094031

C₄

yes

—

C₂×#10625329

C₂²

yes

—

C₂×#10625329

C₂²

yes

—

C₂²×C₄₇⋊C₈

C₂²

yes

—

#13093959

C₂²

yes

—

#13093959

C₂²

yes

—

C₂×#41×C₄₇

C₂²

yes

characteristic

#131×C₄₇

—

C₂×M₁₆×C₄₇

C₂²

yes

—

#41×C₄₇

C₂×C₄

yes

—

#41×C₄₇

D₄

yes

—

C₂×Q₈×C₄₇

Q₈

yes

characteristic

C₂×D₄×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

M₁₆×C₄₇

—

M₁₆×C₄₇

C₂³

yes

—

C₂×C₈×C₄₇

—

#10625329

—

#10625329

C₂×C₄

yes

—

#10625329

C₂³

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂×C₄

yes

—

C₂×C₄₇⋊C₈

C₂³

yes

characteristic

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂³×C₄₇

C₂×Q₈

yes

—

Q₈×C₄₇

C₄⋊C₄

yes

—

D₄×C₄₇

C₄⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

C₂²×C₄

yes

—

C₄₇⋊C₈

C₂²×C₄

yes

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₄⋊C₄

yes

—

C₂²×C₄₇

C₂²×Q₈

yes

characteristic

C₄×C₄₇

C₂×C₄⋊C₄

yes

—

C₄×C₄₇

C₂³×C₄

yes

characteristic

C₄×C₄₇

C₂²×D₄

yes

characteristic

#2916

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

C₂²×C₄⋊C₄

yes

characteristic

#568

—

#568

D₄₇

yes

—

—

—

C₄₇

#2916

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₄₇

yes

—

—

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₄₇

yes

—

C₂²×C₈

—

#108

—

#108

—

#41

C₄×D₄₇

yes

—

#41

C₂²⋊D₄₇

yes

—

C₂×Q₈

Dic₉₄

yes

characteristic

C₂×D₄

Dic₉₄

yes

—

C₂²×C₄

C₂²×D₄₇

yes

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

C₂²×D₄₇

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂³

C₂×Dic₉₄

yes

—

Q₈

Dic₄₇⋊C₄

yes

—

D₄

Dic₄₇⋊C₄

yes

—

C₂×C₄

C₂×Dic₉₄

yes

—

C₂×C₄

C₂×C₄×D₄₇

yes

—

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

C₂×C₄

C₂³×D₄₇

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

C₂×Dic₄₇⋊C₄

yes

—

C₂²

C₂²×Dic₉₄

yes

characteristic

C₄

C₂×Dic₄₇⋊C₄

yes

—

C₄

C₂²×C₄×D₄₇

yes

characteristic

C₄

C₂²×C₂²⋊D₄₇

yes

characteristic

100

C₂

—

101

C₂

—

102

C₂

C₂²×Dic₄₇⋊C₄

yes

characteristic

	subgroup	G/H
center	C₄	C₂²×C₂²⋊D₄₇
commutator	C₄×C₄₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄₇×#568	C₂
socle	C₂×C₄₇	C₂²×C₄⋊C₄
hypercenter	#568	D₄₇
hypocenter	C₄₇	#2916
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂×#41 ◁ #568

subgroup

G/H

center

C₄

C₂²×C₂²⋊D₄₇

commutator

C₄×C₄₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₄₇×#568

C₂

socle

C₂×C₄₇

C₂²×C₄⋊C₄

D₄₇

C₄₇

G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₄ ◁ C₂×#41 ◁ #568

N	H ≅ G/N	type
3. #15689171	C₂	semidirect
4. C₂×#13093959	C₂	semidirect
57. C₄₇	#2916	semidirect

H ≅ G/N

type

3. #15689171

C₂

semidirect

4. C₂×#13093959

C₂

semidirect

57. C₄₇

#2916

semidirect

Subgroups — #20251997

Conjugacy classes of subgroups

Complements