Group #20283637 — subgroups

Subgroups — C₂×Q₈×#88322 #20283637

Order 6784.

440 conjugacy classes of subgroups; only the 333 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 333 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	Q₈×#88322	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊#88322	C₂	1	3	yes	maximal
3	C₂×C₄×#88322	C₂	1	3	yes	maximal
4	C₂×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#88322	C₂²	1	3	yes	—
6	C₂²×#88322	C₂²	1	1	yes	characteristic
7	C₄⋊#88322	C₂²	1	12	yes	—
8	C₄×#88322	C₂²	1	12	yes	—
9	C₂²×Q₈×C₅₃	C₄	1	1	yes	characteristic
10	C₄×Q₈×C₅₃	C₄	1	4	yes	—
11	C₄×Q₈×C₅₃	C₂²	1	4	yes	—
12	C₂×C₄⋊C₄×C₅₃	C₄	1	3	yes	—
13	C₂×C₄²×C₅₃	C₂²	1	3	yes	—
14	C₂×Q₈×C₅₃	C₈	1	8	yes	complemented
15	C₂×Q₈×C₅₃	C₂×C₄	1	4	yes	—
16	C₂²×C₄×C₅₃	C₂×C₄	1	3	yes	—
17	C₂²×C₄×C₅₃	C₂×C₄	1	3	yes	—
18	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
19	C₄⋊C₄×C₅₃	C₂×C₄	1	12	yes	—
20	C₄²×C₅₃	C₂×C₄	1	6	yes	—
21	C₄²×C₅₃	C₂³	1	6	yes	—
22	C₂×#88322	—	2	6	no	—
23	C₂×#88322	Q₈	1	4	yes	direct product factor
24	C₂×#88322	C₂³	1	6	yes	—
25	C₂×#88322	C₂³	1	2	yes	—
26	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
27	Q₈×C₅₃	C₂×C₈	1	16	yes	complemented
28	C₂×C₄×C₅₃	—	2	6	no	—
29	C₂×C₄×C₅₃	C₂×C₈	1	12	yes	—
30	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
31	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
33	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
34	C₂×C₄×C₅₃	#41	1	2	yes	—
35	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
36	#88322	—	2	6	no	—
37	#88322	C₂×Q₈	1	4	yes	direct product factor
38	C₂²×C₅₃	C₄×Q₈	1	2	yes	—
39	C₂²×C₅₃	C₂²×C₈	1	2	yes	—
40	C₂²×C₅₃	#131	1	2	yes	—
41	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
42	C₄×C₅₃	—	2	6	no	—
43	C₄×C₅₃	C₄×Q₈	1	2	yes	—
44	C₄×C₅₃	C₂²×C₈	1	12	yes	—
45	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
46	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
47	C₂×C₈×Q₈	—	53	1	no	maximal, Sylow
48	C₂×C₅₃	C₈×Q₈	1	4	yes	complemented
49	C₂×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
50	C₂×C₅₃	C₂³×C₈	1	1	yes	characteristic
51	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
52	C₂×C₄×Q₈	D₅₃	1	1	yes	characteristic
53	C₈×Q₈	—	53	8	no	—
54	C₂×C₄⋊C₈	—	53	3	no	—
55	C₂×C₄×C₈	—	53	3	no	—
56	C₅₃	C₂×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
57	C₂²×Q₈	Dic₅₃	1	1	yes	characteristic
58	C₂²×C₈	—	53	1	no	—
59	C₂²×C₈	—	53	3	no	—
60	C₄×Q₈	Dic₅₃	1	4	yes	—
61	C₄×Q₈	C₂×D₅₃	1	4	yes	—
62	C₂×C₄⋊C₄	Dic₅₃	1	3	yes	—
63	C₂×C₄²	C₂×D₅₃	1	3	yes	—
64	C₄⋊C₈	—	53	12	no	—
65	C₄×C₈	—	53	12	no	—
66	C₂×Q₈	#88322	1	8	yes	direct product factor
67	C₂×Q₈	C₂×Dic₅₃	1	4	yes	—
68	C₂²×C₄	C₂×Dic₅₃	1	3	yes	—
69	C₂²×C₄	C₂×Dic₅₃	1	3	yes	—
70	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
71	C₂×C₈	—	53	4	no	—
72	C₂×C₈	—	53	2	no	—
73	C₂×C₈	—	53	6	no	—
74	C₂×C₈	—	106	6	no	—
75	C₄⋊C₄	C₂×Dic₅₃	1	12	yes	—
76	C₄²	C₂×Dic₅₃	1	6	yes	—
77	C₄²	C₂²×D₅₃	1	6	yes	—
78	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
79	Q₈	C₂×#88322	1	16	yes	direct product factor
80	C₂×C₄	—	2	6	no	—
81	C₂×C₄	C₂×#88322	1	12	yes	—
82	C₂×C₄	Q₈×D₅₃	1	2	yes	—
83	C₂×C₄	#11719949	1	2	yes	—
84	C₂×C₄	C₂²×Dic₅₃	1	6	yes	—
85	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
86	C₂×C₄	C₂²×Dic₅₃	1	6	yes	—
87	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
88	C₈	—	53	4	no	—
89	C₈	—	106	6	no	—
90	C₂²	Q₈×Dic₅₃	1	2	yes	—
91	C₂²	C₂²×#88322	1	2	yes	—
92	C₂²	#13133224	1	2	yes	—
93	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
94	C₄	—	2	6	no	—
95	C₄	Q₈×Dic₅₃	1	2	yes	—
96	C₄	C₂²×#88322	1	12	yes	—
97	C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
98	C₄	C₂×#11719949	1	1	yes	characteristic
99	C₂	Q₈×#88322	1	4	yes	direct product factor
100	C₂	C₂×Q₈×Dic₅₃	1	1	yes	characteristic
101	C₂	C₂³×#88322	1	1	yes	characteristic
102	C₂	C₂×#13133224	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#88322

C₂

yes

maximal, direct product factor

C₂×C₄⋊#88322

C₂

yes

maximal

C₂×C₄×#88322

C₂

yes

maximal

C₂×C₄×Q₈×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

—

C₂²×#88322

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

—

C₄×#88322

C₂²

yes

—

C₂²×Q₈×C₅₃

C₄

yes

characteristic

C₄×Q₈×C₅₃

C₄

yes

—

C₄×Q₈×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

C₄

yes

—

C₂×C₄²×C₅₃

C₂²

yes

—

C₂×Q₈×C₅₃

C₈

yes

complemented

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂³

yes

—

C₂×#88322

—

C₂×#88322

Q₈

yes

direct product factor

C₂×#88322

C₂³

yes

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

Q₈×C₅₃

C₂×C₈

yes

complemented

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

#88322

—

#88322

C₂×Q₈

yes

direct product factor

C₂²×C₅₃

C₄×Q₈

yes

—

C₂²×C₅₃

C₂²×C₈

yes

—

C₂²×C₅₃

#131

yes

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₄×Q₈

yes

—

C₄×C₅₃

C₂²×C₈

yes

—

C₄×C₅₃

C₂²×Q₈

yes

characteristic

C₄×C₅₃

C₂×#41

yes

characteristic

C₂×C₈×Q₈

—

maximal, Sylow

C₂×C₅₃

C₈×Q₈

yes

complemented

C₂×C₅₃

C₂×C₄×Q₈

yes

characteristic

C₂×C₅₃

C₂³×C₈

yes

characteristic

C₂×C₅₃

C₂×#131

yes

characteristic

C₂×C₄×Q₈

D₅₃

yes

characteristic

C₈×Q₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

C₅₃

C₂×C₈×Q₈

yes

characteristic, complemented, Sylow

C₂²×Q₈

Dic₅₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₄×Q₈

Dic₅₃

yes

—

C₄×Q₈

C₂×D₅₃

yes

—

C₂×C₄⋊C₄

Dic₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

—

C₄⋊C₈

—

C₄×C₈

—

C₂×Q₈

#88322

yes

direct product factor

C₂×Q₈

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₄⋊C₄

C₂×Dic₅₃

yes

—

C₄²

C₂×Dic₅₃

yes

—

C₄²

C₂²×D₅₃

yes

—

C₂³

C₂²×Dic₅₃

yes

characteristic

Q₈

C₂×#88322

yes

direct product factor

C₂×C₄

—

C₂×C₄

C₂×#88322

yes

—

C₂×C₄

Q₈×D₅₃

yes

—

C₂×C₄

#11719949

yes

—

C₂×C₄

C₂²×Dic₅₃

yes

—

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

C₂×C₄

C₂²×Dic₅₃

yes

—

C₂×C₄

C₂³×D₅₃

yes

characteristic

C₈

—

C₈

—

106

—

C₂²

Q₈×Dic₅₃

yes

—

C₂²

C₂²×#88322

yes

—

C₂²

#13133224

yes

—

C₂²

C₂³×Dic₅₃

yes

characteristic

C₄

—

C₄

Q₈×Dic₅₃

yes

—

C₄

C₂²×#88322

yes

—

C₄

C₂×Q₈×D₅₃

yes

C₄

yes

C₂

yes

direct product factor

100

C₂

C₂×Q₈×Dic₅₃

yes

characteristic

101

C₂

C₂³×#88322

yes

characteristic

102

C₂

C₂×#13133224

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₅₃
commutator	C₂×C₅₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂×C₄×Q₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×C₄
hypercenter	C₂×C₄×Q₈	D₅₃
hypocenter	C₅₃	C₂×C₈×Q₈
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂²×C₄

C₂²×D₅₃

commutator

C₂×C₅₃

C₂³×C₈

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₂×C₄×Q₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂²×C₄

hypercenter

C₂×C₄×Q₈

D₅₃

hypocenter

C₅₃

C₂×C₈×Q₈

LCS

G ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
1. Q₈×#88322	C₂	direct (N × H)
14. C₂×Q₈×C₅₃	C₈	semidirect
23. C₂×#88322	Q₈	direct (N × H)
27. Q₈×C₅₃	C₂×C₈	semidirect
37. #88322	C₂×Q₈	direct (N × H)
48. C₂×C₅₃	C₈×Q₈	semidirect
56. C₅₃	C₂×C₈×Q₈	semidirect
66. C₂×Q₈	#88322	direct (N × H)
79. Q₈	C₂×#88322	direct (N × H)
99. C₂	Q₈×#88322	direct (N × H)

H ≅ G/N

type

1. Q₈×#88322

C₂

direct (N × H)

14. C₂×Q₈×C₅₃

C₈

semidirect

23. C₂×#88322

Q₈

direct (N × H)

27. Q₈×C₅₃

C₂×C₈

semidirect

37. #88322

C₂×Q₈

direct (N × H)

48. C₂×C₅₃

C₈×Q₈

semidirect

56. C₅₃

C₂×C₈×Q₈

semidirect

66. C₂×Q₈

#88322

direct (N × H)

79. Q₈

C₂×#88322

direct (N × H)

99. C₂

Q₈×#88322

direct (N × H)

Subgroups — C₂×Q₈×#88322 #20283637

Conjugacy classes of subgroups

Complements

Subgroups — C2×Q8×#88322 #20283637

Conjugacy classes of subgroups

Complements

Subgroups — C₂×Q₈×#88322 #20283637