Subgroups — #27213269

Order 13504.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20269999	C₂	1	1	yes	maximal, characteristic
2	#20269983	C₂	1	2	yes	maximal, complemented
3	#20269913	C₂	1	1	yes	maximal, characteristic, complemented
4	#20269905	C₂	1	2	yes	maximal, complemented
5	#97×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#16809962	C₂²	1	2	yes	—
7	Dic₈₄₄	C₄	1	1	yes	characteristic, complemented
8	#16809951	C₄	1	2	yes	complemented
9	D₁₆₈₈	C₄	1	1	yes	characteristic, complemented
10	#16809946	C₂²	1	2	yes	—
11	#16809945	—	2	2	no	—
12	C₈×D₂₁₁	—	2	2	no	—
13	C₂×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
15	C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
16	Dic₄₂₂	C₂×C₄	1	2	yes	—
17	D₈₄₄	C₂×C₄	1	2	yes	—
18	C₂²⋊D₂₁₁	—	2	2	no	—
19	C₄×D₂₁₁	—	2	2	no	—
20	C₂₁₁⋊C₈	—	2	2	no	—
21	C₂×C₄×C₂₁₁	—	2	1	no	—
22	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
23	C₈×C₂₁₁	C₂×C₄	1	1	yes	characteristic
24	C₈×C₂₁₁	C₂×C₄	1	1	yes	characteristic
25	C₈×C₂₁₁	D₄	1	2	yes	—
26	C₂²×C₂₁₁	#41	1	1	yes	characteristic
27	C₂×D₂₁₁	—	2	2	no	—
28	C₄×C₂₁₁	—	4	1	no	—
29	C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
31	Dic₂₁₁	—	2	2	no	—
32	C₂×C₂₁₁	—	2	1	no	—
33	C₂×C₂₁₁	C₄×D₄	1	1	yes	characteristic
34	D₂₁₁	—	4	2	no	—
35	C₂₁₁	#444	1	1	yes	characteristic, complemented, Sylow
36	#444	—	211	1	no	maximal, Sylow
37	#135	—	211	1	no	—
38	#131	—	211	2	no	—
39	#108	—	211	1	no	—
40	#104	—	211	2	no	—
41	#97	D₂₁₁	1	1	yes	characteristic, complemented
42	#41	—	211	2	no	—
43	Q₁₆	—	211	1	no	—
44	SD₁₆	—	211	2	no	—
45	D₈	—	211	1	no	—
46	M₁₆	—	422	2	no	—
47	M₁₆	—	211	2	no	—
48	C₂×C₈	—	422	2	no	—
49	C₂×C₈	C₂×D₂₁₁	1	1	yes	characteristic
50	C₂×C₈	C₂×D₂₁₁	1	1	yes	characteristic
51	C₄²	C₂×D₂₁₁	1	1	yes	characteristic
52	Q₈	—	211	2	no	—
53	D₄	—	422	2	no	—
54	D₄	—	211	2	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	422	2	no	—
57	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
58	C₈	—	422	2	no	—
59	C₈	C₄×D₂₁₁	1	1	yes	characteristic
60	C₈	C₄×D₂₁₁	1	1	yes	characteristic
61	C₈	D₈₄₄	1	2	yes	—
62	C₂²	—	422	2	no	—
63	C₂²	#16809962	1	1	yes	characteristic
64	C₄	—	4	1	no	—
65	C₄	—	422	2	no	—
66	C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
67	C₄	C₂×D₈₄₄	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	844	2	no	—
70	C₂	C₄×D₈₄₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₈₄₄
commutator	C₄×C₂₁₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₁₁
Fitting	#97×C₂₁₁	C₂
socle	C₂×C₂₁₁	C₄×D₄
prime radical	#20269999	C₂
cosocle	C₂×C₄×C₂₁₁	C₂³
hypercenter	#97	D₂₁₁
hypocenter	C₂₁₁	#444
LCS	G ▷ C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #97

Complements

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

N	H ≅ G/N	type
2. #20269983	C₂	semidirect
3. #20269913	C₂	semidirect
4. #20269905	C₂	semidirect
5. #97×C₂₁₁	C₂	semidirect
7. Dic₈₄₄	C₄	semidirect
8. #16809951	C₄	semidirect
9. D₁₆₈₈	C₄	semidirect
35. C₂₁₁	#444	semidirect
41. #97	D₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
#20269999	2
#20269983	2
#20269913	2
#20269905	2
#97×C₂₁₁	2
#444	211