Group #21436981 — subgroups

Subgroups — Q₈×Dic₂₃₆ #21436981

Order 7552.

336 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 149 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×Dic₁₁₈	C₂	1	2	yes	maximal
2	#16837807	C₂	1	3	yes	maximal
3	#16837698	C₂	1	6	yes	maximal
4	C₄×Dic₂₃₆	C₂	1	3	yes	maximal
5	C₈×Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
7	#14273727	C₂²	1	6	yes	—
8	#14273726	—	2	6	no	—
9	Q₈×Dic₅₉	—	2	2	no	—
10	C₄×Dic₁₁₈	C₂²	1	6	yes	—
11	#14273661	C₂²	1	6	yes	—
12	#14273661	C₂²	1	3	yes	—
13	#14273652	C₂²	1	6	yes	—
14	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₅₉	C₂²	1	3	yes	—
16	C₄×C₈×C₅₉	C₂²	1	3	yes	—
17	C₂×Q₈×C₅₉	D₄	1	1	yes	characteristic
18	C₂×Dic₁₁₈	—	2	6	no	—
19	C₂×Dic₁₁₈	C₂³	1	2	yes	—
20	C₂×C₈×C₅₉	C₂³	1	3	yes	—
21	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₅₉	D₄	1	3	yes	—
23	C₄²×C₅₉	C₂³	1	3	yes	—
24	C₄⋊Dic₅₉	—	2	6	no	—
25	C₄⋊Dic₅₉	C₂³	1	6	yes	—
26	Dic₅₉⋊C₄	—	2	6	no	—
27	C₄×Dic₅₉	—	2	6	no	—
28	Dic₂₃₆	Q₈	1	4	yes	direct product factor
29	Q₈×C₅₉	Q₁₆	1	4	yes	complemented
30	C₂×C₄×C₅₉	C₂×D₄	1	3	yes	—
31	C₂×C₄×C₅₉	C₂×D₄	1	3	yes	—
32	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
33	C₂×Dic₅₉	—	2	2	no	—
34	C₂×Dic₅₉	—	2	6	no	—
35	Dic₁₁₈	—	2	2	no	—
36	Dic₁₁₈	—	8	6	no	—
37	Dic₁₁₈	C₂×Q₈	1	4	yes	—
38	C₈×C₅₉	—	2	3	no	—
39	C₈×C₅₉	C₂×Q₈	1	2	yes	—
40	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
41	C₄×C₅₉	—	2	3	no	—
42	C₄×C₅₉	C₂×Q₁₆	1	6	yes	—
43	C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
44	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
45	Dic₅₉	—	2	4	no	—
46	Dic₅₉	—	4	6	no	—
47	Q₈×Q₁₆	—	59	1	no	maximal, Sylow
48	C₂×C₅₉	D₄×Q₈	1	1	yes	characteristic
49	C₂×C₅₉	C₂²×Q₁₆	1	1	yes	characteristic
50	C₂×C₅₉	#576	1	1	yes	characteristic
51	Q₈²	—	59	2	no	—
52	#500	—	59	3	no	—
53	#477	—	59	6	no	—
54	C₈×Q₈	D₅₉	1	1	yes	characteristic
55	C₄×Q₁₆	—	59	3	no	—
56	C₅₉	Q₈×Q₁₆	1	1	yes	characteristic, complemented, Sylow
57	C₂×Q₁₆	—	59	1	no	—
58	C₄⋊Q₈	—	59	6	no	—
59	C₄⋊Q₈	—	118	6	no	—
60	C₄×Q₈	—	59	6	no	—
61	C₄×Q₈	—	118	2	no	—
62	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
63	#107	—	59	3	no	—
64	#107	—	59	6	no	—
65	C₄⋊C₈	C₂×D₅₉	1	3	yes	—
66	#103	—	59	6	no	—
67	C₄×C₈	C₂×D₅₉	1	3	yes	—
68	C₂×Q₈	—	59	2	no	—
69	C₂×Q₈	—	118	6	no	—
70	C₂×Q₈	D₂₃₆	1	1	yes	characteristic
71	Q₁₆	—	59	4	no	—
72	C₂×C₈	C₂²×D₅₉	1	3	yes	—
73	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
74	C₄⋊C₄	—	59	6	no	—
75	C₄⋊C₄	—	118	6	no	—
76	C₄⋊C₄	—	118	6	no	—
77	C₄⋊C₄	D₂₃₆	1	3	yes	—
78	C₄²	—	118	6	no	—
79	C₄²	C₂²×D₅₉	1	3	yes	—
80	Q₈	—	59	4	no	—
81	Q₈	—	118	2	no	—
82	Q₈	—	472	6	no	—
83	Q₈	Dic₂₃₆	1	4	yes	direct product factor
84	C₂×C₄	—	118	2	no	—
85	C₂×C₄	—	118	6	no	—
86	C₂×C₄	C₂×D₂₃₆	1	3	yes	—
87	C₂×C₄	C₂×D₂₃₆	1	3	yes	—
88	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
89	C₈	—	2	3	no	—
90	C₈	Q₈×D₅₉	1	2	yes	—
91	C₂²	C₂²×D₂₃₆	1	1	yes	characteristic
92	C₄	—	2	3	no	—
93	C₄	—	118	4	no	—
94	C₄	—	236	6	no	—
95	C₄	C₂×Dic₂₃₆	1	6	yes	—
96	C₄	C₂×Q₈×D₅₉	1	1	yes	characteristic
97	C₄	#14273774	1	1	yes	characteristic
98	C₂	Q₈×D₂₃₆	1	1	yes	characteristic
99	C₂	C₂²×Dic₂₃₆	1	1	yes	characteristic
100	C₂	#16838207	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₁₈

C₂

yes

maximal

#16837807

C₂

yes

maximal

#16837698

C₂

yes

maximal

C₄×Dic₂₃₆

C₂

yes

maximal

C₈×Q₈×C₅₉

C₂

yes

maximal, characteristic

C₂×Dic₂₃₆

C₂²

yes

characteristic

#14273727

C₂²

yes

—

#14273726

—

Q₈×Dic₅₉

—

C₄×Dic₁₁₈

C₂²

yes

—

#14273661

C₂²

yes

—

#14273661

C₂²

yes

—

#14273652

C₂²

yes

—

C₄×Q₈×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

—

C₄×C₈×C₅₉

C₂²

yes

—

C₂×Q₈×C₅₉

D₄

yes

characteristic

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

C₂³

yes

—

C₂×C₈×C₅₉

C₂³

yes

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

D₄

yes

—

C₄²×C₅₉

C₂³

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂³

yes

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

Dic₂₃₆

Q₈

yes

direct product factor

Q₈×C₅₉

Q₁₆

yes

complemented

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

C₂×Q₈

yes

—

C₈×C₅₉

—

C₈×C₅₉

C₂×Q₈

yes

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

C₂×Q₁₆

yes

—

C₄×C₅₉

C₂²×Q₈

yes

characteristic

C₄×C₅₉

2₋¹⁺⁴

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Q₈×Q₁₆

—

maximal, Sylow

C₂×C₅₉

D₄×Q₈

yes

characteristic

C₂×C₅₉

C₂²×Q₁₆

yes

characteristic

C₂×C₅₉

#576

yes

characteristic

Q₈²

—

#500

—

#477

—

C₈×Q₈

D₅₉

yes

characteristic

C₄×Q₁₆

—

C₅₉

Q₈×Q₁₆

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

118

—

C₄×Q₈

—

C₄×Q₈

—

118

—

C₄×Q₈

C₂×D₅₉

yes

characteristic

#107

—

#107

—

C₄⋊C₈

C₂×D₅₉

yes

—

#103

—

C₄×C₈

C₂×D₅₉

yes

—

C₂×Q₈

—

C₂×Q₈

—

118

—

C₂×Q₈

D₂₃₆

yes

characteristic

Q₁₆

—

C₂×C₈

C₂²×D₅₉

yes

—

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

D₂₃₆

yes

—

C₄²

—

118

—

C₄²

C₂²×D₅₉

yes

—

Q₈

—

Q₈

—

118

—

Q₈

—

472

—

Q₈

Dic₂₃₆

yes

direct product factor

C₂×C₄

—

118

—

C₂×C₄

—

118

—

C₂×C₄

C₂×D₂₃₆

yes

—

C₂×C₄

C₂×D₂₃₆

yes

—

C₂×C₄

C₂³×D₅₉

yes

characteristic

C₈

—

C₈

Q₈×D₅₉

yes

—

C₂²

C₂²×D₂₃₆

yes

characteristic

C₄

—

C₄

—

118

—

C₄

—

236

—

C₄

C₂×Dic₂₃₆

yes

—

C₄

C₂×Q₈×D₅₉

yes

characteristic

C₄

#14273774

yes

characteristic

C₂

Q₈×D₂₃₆

yes

characteristic

C₂

C₂²×Dic₂₃₆

yes

characteristic

100

C₂

#16838207

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₈×Q₈×C₅₉	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	C₈×Q₈	D₅₉
hypocenter	C₅₉	Q₈×Q₁₆
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×D₂₃₆

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₈×Q₈×C₅₉

C₂

socle

C₂²×C₅₉

C₂²×D₄

hypercenter

C₈×Q₈

D₅₉

hypocenter

C₅₉

Q₈×Q₁₆

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
28. Dic₂₃₆	Q₈	direct (N × H)
29. Q₈×C₅₉	Q₁₆	semidirect
56. C₅₉	Q₈×Q₁₆	semidirect
83. Q₈	Dic₂₃₆	direct (N × H)

H ≅ G/N

type

28. Dic₂₃₆

Q₈

direct (N × H)

29. Q₈×C₅₉

Q₁₆

semidirect

56. C₅₉

Q₈×Q₁₆

semidirect

83. Q₈

Dic₂₃₆

direct (N × H)

Subgroups — Q₈×Dic₂₃₆ #21436981

Conjugacy classes of subgroups

Complements

Subgroups — Q8×Dic236 #21436981

Conjugacy classes of subgroups

Complements

Subgroups — Q₈×Dic₂₃₆ #21436981