Subgroups — #27213268

Order 13504.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20269984	C₂	1	2	yes	maximal, complemented
2	#20269913	C₂	1	1	yes	maximal, characteristic, complemented
3	#20269904	C₂	1	2	yes	maximal, complemented
4	#20269887	C₂	1	1	yes	maximal, characteristic, complemented
5	#135×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	#16809946	—	2	2	no	—
7	#16809946	C₂²	1	2	yes	—
8	C₂×C₂₁₁⋊C₈	—	2	2	no	—
9	C₂×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
10	C₄×Dic₂₁₁	C₂²	1	1	yes	characteristic
11	#41×C₂₁₁	C₂²	1	2	yes	—
12	Q₁₆×C₂₁₁	C₄	1	1	yes	characteristic, complemented
13	SD₁₆×C₂₁₁	C₄	1	2	yes	complemented
14	D₈×C₂₁₁	C₄	1	1	yes	characteristic, complemented
15	C₂×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₁₁	—	2	1	no	—
17	C₂₁₁⋊C₈	—	2	2	no	—
18	C₂₁₁⋊C₈	D₄	1	2	yes	—
19	Q₈×C₂₁₁	C₂×C₄	1	2	yes	—
20	D₄×C₂₁₁	—	2	2	no	—
21	D₄×C₂₁₁	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₂₁₁	—	2	2	no	—
23	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
24	C₈×C₂₁₁	C₂×C₄	1	1	yes	characteristic
25	C₈×C₂₁₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₂₁₁	—	2	2	no	—
27	C₂²×C₂₁₁	#41	1	1	yes	characteristic
28	C₄×C₂₁₁	—	2	2	no	—
29	C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
31	Dic₂₁₁	—	4	1	no	—
32	C₂×C₂₁₁	—	2	1	no	—
33	C₂×C₂₁₁	—	4	2	no	—
34	C₂×C₂₁₁	C₄×D₄	1	1	yes	characteristic
35	C₂₁₁	#444	1	1	yes	characteristic, complemented, Sylow
36	#444	—	211	1	no	maximal, Sylow
37	#135	D₂₁₁	1	1	yes	characteristic
38	#131	—	211	2	no	—
39	#108	—	211	1	no	—
40	#104	—	211	2	no	—
41	#97	—	211	1	no	—
42	#41	C₂×D₂₁₁	1	2	yes	—
43	Q₁₆	Dic₂₁₁	1	1	yes	characteristic, complemented
44	SD₁₆	Dic₂₁₁	1	2	yes	complemented
45	D₈	Dic₂₁₁	1	1	yes	characteristic, complemented
46	M₁₆	—	422	2	no	—
47	M₁₆	—	211	2	no	—
48	C₂×C₈	—	422	2	no	—
49	C₂×C₈	—	211	1	no	—
50	C₂×C₈	C₂×D₂₁₁	1	1	yes	characteristic
51	C₄²	—	211	1	no	—
52	Q₈	C₂×Dic₂₁₁	1	2	yes	—
53	D₄	—	2	2	no	—
54	D₄	C₂×Dic₂₁₁	1	2	yes	—
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	422	1	no	—
57	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
58	C₈	—	422	2	no	—
59	C₈	—	211	2	no	—
60	C₈	C₂×Dic₂₁₁	1	1	yes	characteristic
61	C₈	C₂×Dic₂₁₁	1	1	yes	characteristic
62	C₂²	—	2	2	no	—
63	C₂²	#16809961	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	844	1	no	—
66	C₄	C₂²×Dic₂₁₁	1	1	yes	characteristic
67	C₄	D₄×D₂₁₁	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	4	2	no	—
70	C₂	D₄×Dic₂₁₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₂₁₁
commutator	C₄×C₂₁₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₁₁
Fitting, prime radical	#135×C₂₁₁	C₂
socle	C₂×C₂₁₁	C₄×D₄
cosocle	C₂×C₄×C₂₁₁	C₂³
hypercenter	#135	D₂₁₁
hypocenter	C₂₁₁	#444
LCS	G ▷ C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #20269984	C₂	semidirect
2. #20269913	C₂	semidirect
3. #20269904	C₂	semidirect
4. #20269887	C₂	semidirect
12. Q₁₆×C₂₁₁	C₄	semidirect
13. SD₁₆×C₂₁₁	C₄	semidirect
14. D₈×C₂₁₁	C₄	semidirect
35. C₂₁₁	#444	semidirect
43. Q₁₆	Dic₂₁₁	semidirect
44. SD₁₆	Dic₂₁₁	semidirect
45. D₈	Dic₂₁₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20269984	2
#20269913	2
#20269904	2
#20269887	2
#135×C₂₁₁	2
#444	211