Subgroups — #27213262

Order 13504.

106 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	#20269985	C₂	1	2	yes	maximal, complemented
3	#20269903	C₂	1	2	yes	maximal, complemented
4	C₈×Dic₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#108×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	#16809945	—	2	2	no	—
11	C₂×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₈×D₂₁₁	—	2	2	no	—
13	C₄×Dic₂₁₁	C₂²	1	1	yes	characteristic
14	M₁₆×C₂₁₁	C₂²	1	2	yes	—
15	C₂×C₈×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	C₂×Dic₂₁₁	—	2	1	no	—
20	C₄×D₂₁₁	—	2	2	no	—
21	C₂₁₁⋊C₈	D₄	1	1	yes	characteristic
22	C₂₁₁⋊C₈	D₄	1	1	yes	characteristic
23	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
24	C₈×C₂₁₁	—	2	2	no	—
25	C₈×C₂₁₁	C₂×C₄	1	1	yes	characteristic
26	C₈×C₂₁₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₂₁₁	#41	1	1	yes	characteristic
28	C₂×D₂₁₁	—	2	2	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	Dic₂₁₁	—	2	2	no	—
33	C₂×C₂₁₁	—	2	1	no	—
34	C₂×C₂₁₁	C₄×D₄	1	1	yes	characteristic
35	D₂₁₁	—	4	2	no	—
36	C₂₁₁	#443	1	1	yes	characteristic, complemented, Sylow
37	#443	—	211	1	no	maximal, Sylow
38	#135	—	211	1	no	—
39	#131	—	211	2	no	—
40	#108	D₂₁₁	1	1	yes	characteristic, complemented
41	#104	—	211	2	no	—
42	C₄×C₈	—	211	1	no	—
43	#41	—	211	2	no	—
44	Q₁₆	—	211	1	no	—
45	SD₁₆	—	211	2	no	—
46	D₈	—	211	1	no	—
47	M₁₆	—	422	2	no	—
48	M₁₆	C₂×D₂₁₁	1	2	yes	—
49	C₂×C₈	—	422	2	no	—
50	C₂×C₈	—	211	1	no	—
51	C₂×C₈	C₂×D₂₁₁	1	1	yes	characteristic
52	C₄²	—	211	1	no	—
53	Q₈	—	211	2	no	—
54	D₄	—	422	2	no	—
55	D₄	—	211	2	no	—
56	C₂×C₄	—	422	1	no	—
57	C₂×C₄	—	422	2	no	—
58	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
59	C₈	—	2	2	no	—
60	C₈	—	211	1	no	—
61	C₈	—	211	1	no	—
62	C₈	C₄×D₂₁₁	1	1	yes	characteristic
63	C₈	C₄×D₂₁₁	1	1	yes	characteristic
64	C₂²	—	422	2	no	—
65	C₂²	#16809963	1	1	yes	characteristic
66	C₄	—	422	2	no	—
67	C₄	—	422	2	no	—
68	C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
69	C₄	D₄×D₂₁₁	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	844	2	no	—
72	C₂	#20269939	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₂₁₁
commutator	C₄×C₂₁₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₁₁
Fitting	#108×C₂₁₁	C₂
socle	C₂×C₂₁₁	C₄×D₄
prime radical	#20269999	C₂
cosocle	C₂×C₄×C₂₁₁	C₂³
hypercenter	#108	D₂₁₁
hypocenter	C₂₁₁	#443
LCS	G ▷ C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

Complements

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

N	H ≅ G/N	type
2. #20269985	C₂	semidirect
3. #20269903	C₂	semidirect
4. C₈×Dic₂₁₁	C₂	semidirect
5. #108×C₂₁₁	C₂	semidirect
7. Dic₈₄₄	C₄	semidirect
8. #16809951	C₄	semidirect
9. D₁₆₈₈	C₄	semidirect
36. C₂₁₁	#443	semidirect
40. #108	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
#20269985	2
#20269903	2
C₈×Dic₂₁₁	2
#108×C₂₁₁	2
#443	211