Group #20289753 — subgroups

Subgroups — #20289753

Order 6784.

160 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 90 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	C₂²×#11719910	C₂	1	2	yes	maximal
2	C₂×C₄×#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×#11719910	—	2	2	no	—
4	C₂×#11719910	—	2	2	no	—
5	C₂×#11719910	C₄	1	4	yes	complemented
6	C₂²×#88322	C₄	1	1	yes	characteristic
7	C₂²×#88322	C₂²	1	1	yes	characteristic
8	C₄×#88322	—	2	2	no	—
9	C₂×C₄²×C₅₃	C₄	1	1	yes	characteristic
10	C₂²×C₄×C₅₃	C₈	1	2	yes	—
11	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₄²×C₅₃	—	2	1	no	—
13	C₄²×C₅₃	—	2	1	no	—
14	C₂×#88322	—	2	1	no	—
15	C₂×#88322	—	2	1	no	—
16	C₂×#88322	C₈	1	2	yes	—
17	C₂×#88322	C₂×C₄	1	2	yes	—
18	C₂×#88322	D₄	1	1	yes	characteristic
19	C₂×#88322	D₄	1	2	yes	—
20	C₂×#88322	Q₈	1	1	yes	characteristic
21	#11719910	—	2	4	no	—
22	C₂³×C₅₃	C₂×C₈	1	1	yes	characteristic
23	C₂×C₄×C₅₃	—	2	2	no	—
24	C₂×C₄×C₅₃	—	2	2	no	—
25	C₂×C₄×C₅₃	C₁₆	1	4	yes	complemented
26	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
27	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₅₃	M₁₆	1	1	yes	characteristic
31	C₂×C₄×C₅₃	M₁₆	1	1	yes	characteristic
32	#88322	—	2	2	no	—
33	#88322	C₂²⋊C₄	1	2	yes	—
34	#88322	C₄⋊C₄	1	2	yes	—
35	C₂²×C₅₃	C₄×C₈	1	1	yes	characteristic
36	C₂²×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
37	C₂²×C₅₃	C₄⋊C₈	1	1	yes	characteristic
38	C₂²×C₅₃	C₂×C₁₆	1	2	yes	—
39	C₂²×C₅₃	M₃₂	1	2	yes	—
40	C₄×C₅₃	—	2	4	no	—
41	C₄×C₅₃	#95	1	1	yes	characteristic
42	C₄×C₅₃	#97	1	1	yes	characteristic
43	C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
44	C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
45	#1374	—	53	1	no	maximal, Sylow
46	C₂×C₅₃	#336	1	1	yes	characteristic
47	C₂×C₅₃	C₄×C₁₆	1	1	yes	characteristic
48	C₂×C₅₃	#346	1	1	yes	characteristic
49	C₂×C₅₃	C₂²⋊C₁₆	1	2	yes	—
50	C₂×C₅₃	C₄⋊C₁₆	1	2	yes	—
51	C₂²×C₁₆	—	53	2	no	—
52	C₂×C₄×C₈	—	53	1	no	—
53	C₅₃	#1374	1	1	yes	characteristic, complemented, Sylow
54	C₂²×C₈	—	53	1	no	—
55	C₂²×C₈	—	53	1	no	—
56	C₂×C₄²	#6157	1	1	yes	characteristic
57	C₂×C₁₆	—	53	4	no	—
58	C₂×C₁₆	—	106	2	no	—
59	C₂×C₁₆	—	106	2	no	—
60	C₄×C₈	—	106	2	no	—
61	C₂²×C₄	#88324	1	2	yes	—
62	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
63	C₂×C₈	—	53	2	no	—
64	C₂×C₈	—	53	1	no	—
65	C₂×C₈	—	53	1	no	—
66	C₂×C₈	—	53	2	no	—
67	C₂×C₈	—	53	2	no	—
68	C₂×C₈	—	106	1	no	—
69	C₂×C₈	—	106	1	no	—
70	C₄²	—	2	1	no	—
71	C₄²	—	2	1	no	—
72	C₁₆	—	106	4	no	—
73	C₂³	C₂×#88324	1	1	yes	characteristic
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	#11719910	1	4	yes	complemented
77	C₂×C₄	D₅₃⋊C₈	1	1	yes	characteristic
78	C₂×C₄	#11719936	1	1	yes	characteristic
79	C₂×C₄	C₄×#6157	1	1	yes	characteristic
80	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
81	C₂×C₄	#11719940	1	1	yes	characteristic
82	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
83	C₈	—	53	2	no	—
84	C₈	—	53	2	no	—
85	C₈	—	106	2	no	—
86	C₂²	C₄×#88324	1	1	yes	characteristic
87	C₂²	C₂²⋊#88324	1	1	yes	characteristic
88	C₂²	C₄⋊#88324	1	1	yes	characteristic
89	C₂²	C₂×#11719910	1	2	yes	—
90	C₂²	#13133291	1	2	yes	—
91	C₄	—	2	4	no	—
92	C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
93	C₄	#13133272	1	1	yes	characteristic
94	C₄	#13133274	1	1	yes	characteristic
95	C₄	#13133284	1	1	yes	characteristic
96	C₂	#16811489	1	1	yes	characteristic
97	C₂	C₄×#11719910	1	1	yes	characteristic
98	C₂	#16811511	1	1	yes	characteristic
99	C₂	C₂²⋊#11719910	1	2	yes	—
100	C₂	C₄⋊#11719910	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11719910

C₂

yes

maximal

C₂×C₄×#88322

C₂

yes

maximal, characteristic

C₂×#11719910

—

C₂×#11719910

—

C₂×#11719910

C₄

yes

complemented

C₂²×#88322

C₄

yes

characteristic

C₂²×#88322

C₂²

yes

characteristic

C₄×#88322

—

C₂×C₄²×C₅₃

C₄

yes

characteristic

C₂²×C₄×C₅₃

C₈

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

—

C₄²×C₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

C₈

yes

—

C₂×#88322

C₂×C₄

yes

—

C₂×#88322

D₄

yes

characteristic

C₂×#88322

D₄

yes

—

C₂×#88322

Q₈

yes

characteristic

#11719910

—

C₂³×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₁₆

yes

complemented

C₂×C₄×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

M₁₆

yes

characteristic

C₂×C₄×C₅₃

M₁₆

yes

characteristic

#88322

—

#88322

C₂²⋊C₄

yes

—

#88322

C₄⋊C₄

yes

—

C₂²×C₅₃

C₄×C₈

yes

characteristic

C₂²×C₅₃

C₂²⋊C₈

yes

characteristic

C₂²×C₅₃

C₄⋊C₈

yes

characteristic

C₂²×C₅₃

C₂×C₁₆

yes

—

C₂²×C₅₃

M₃₂

yes

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₂²⋊C₈

yes

characteristic

C₄×C₅₃

C₄⋊C₈

yes

—

C₂×C₅₃

yes

C₂×C₅₃

C₄×C₁₆

yes

characteristic

C₂×C₅₃

#346

yes

characteristic

C₂×C₅₃

C₂²⋊C₁₆

yes

—

C₂×C₅₃

C₄⋊C₁₆

yes

—

C₂²×C₁₆

—

C₂×C₄×C₈

—

C₅₃

#1374

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

#6157

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₂×C₁₆

—

106

—

C₄×C₈

—

106

—

C₂²×C₄

#88324

yes

—

C₂²×C₄

C₂×#6157

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄²

—

C₄²

—

C₁₆

—

106

—

C₂³

C₂×#88324

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#11719910

yes

complemented

C₂×C₄

D₅₃⋊C₈

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²⋊#6157

yes

characteristic

C₈

—

C₈

—

C₈

—

106

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#13133291

yes

—

C₄

—

C₄

yes

C₄

yes

C₄

yes

C₄

yes

C₂

yes

C₂

yes

C₂

yes

C₂

yes

—

100

C₂

C₄⋊#11719910

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂×#6157
commutator	C₂×C₅₃	C₄×C₁₆
Fitting	C₂×C₄²×C₅₃	C₄
socle	C₂³×C₅₃	C₂×C₈
hypercenter	C₂×C₄²	#6157
hypocenter	C₅₃	#1374
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

subgroup

G/H

center, Frattini

C₂²×C₄

C₂×#6157

commutator

C₂×C₅₃

C₄×C₁₆

Fitting

C₂×C₄²×C₅₃

C₄

socle

C₂³×C₅₃

C₂×C₈

C₂×C₄²

C₅₃

G ▷ C₂×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

N	H ≅ G/N	type
5. C₂×#11719910	C₄	semidirect
25. C₂×C₄×C₅₃	C₁₆	semidirect
53. C₅₃	#1374	semidirect
76. C₂×C₄	#11719910	semidirect

H ≅ G/N

type

5. C₂×#11719910

C₄

semidirect

25. C₂×C₄×C₅₃

C₁₆

semidirect

53. C₅₃

#1374

semidirect

76. C₂×C₄

#11719910

semidirect

Subgroups — #20289753

Conjugacy classes of subgroups

Complements