Subgroups — #27213193

Order 13504.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16809945	C₂	1	1	yes	maximal, characteristic
2	M₁₆×D₂₁₁	C₂	1	2	yes	maximal
3	#20269913	C₂	1	1	yes	maximal, characteristic, complemented
4	#20269912	C₂	1	2	yes	maximal, complemented
5	#108×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₂₁₁	C₂²	1	1	yes	characteristic
7	#16809946	C₂²	1	2	yes	—
8	#16809945	—	2	2	no	—
9	#16809945	C₄	1	4	yes	—
10	C₂×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₈×D₂₁₁	—	2	2	no	—
12	M₁₆×C₂₁₁	C₂²	1	2	yes	—
13	C₂×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₂²×D₂₁₁	Q₈	1	1	yes	characteristic
15	C₂×Dic₂₁₁	Q₈	1	1	yes	characteristic
16	C₄×D₂₁₁	C₂×C₄	1	2	yes	—
17	C₄×D₂₁₁	D₄	1	2	yes	—
18	C₂₁₁⋊C₈	—	2	2	no	—
19	C₂₁₁⋊C₈	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
21	C₈×C₂₁₁	—	2	2	no	—
22	C₈×C₂₁₁	C₂×C₄	1	2	yes	—
23	C₂²×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
24	C₂×D₂₁₁	—	4	1	no	—
25	C₂×D₂₁₁	C₄⋊C₄	1	2	yes	—
26	C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
27	C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
28	Dic₂₁₁	C₄⋊C₄	1	2	yes	—
29	C₂×C₂₁₁	—	2	1	no	—
30	C₂×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
31	D₂₁₁	—	2	2	no	—
32	C₂₁₁	#430	1	1	yes	characteristic, complemented, Sylow
33	#430	—	211	1	no	maximal, Sylow
34	C₂×M₁₆	—	211	2	no	—
35	C₂×M₁₆	—	211	1	no	—
36	#108	—	211	2	no	—
37	#108	—	211	1	no	—
38	#108	D₂₁₁	1	1	yes	characteristic, complemented
39	C₂²×C₄	—	211	1	no	—
40	M₁₆	—	422	2	no	—
41	M₁₆	—	211	2	no	—
42	M₁₆	—	211	4	no	—
43	M₁₆	C₂×D₂₁₁	1	2	yes	—
44	C₂×C₈	—	422	2	no	—
45	C₂×C₈	—	211	1	no	—
46	C₂×C₈	C₂×D₂₁₁	1	1	yes	characteristic
47	C₂³	—	211	1	no	—
48	C₂×C₄	—	211	1	no	—
49	C₂×C₄	—	211	2	no	—
50	C₂×C₄	—	211	2	no	—
51	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
52	C₈	—	2	2	no	—
53	C₈	—	422	2	no	—
54	C₈	—	211	2	no	—
55	C₈	C₄×D₂₁₁	1	2	yes	—
56	C₂²	—	844	1	no	—
57	C₂²	—	211	2	no	—
58	C₂²	Q₈×D₂₁₁	1	1	yes	characteristic
59	C₄	—	211	2	no	—
60	C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
61	C₄	D₄×D₂₁₁	1	1	yes	characteristic
62	C₂	—	2	1	no	—
63	C₂	—	422	2	no	—
64	C₂	C₄⋊C₄×D₂₁₁	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	C₂×C₄⋊C₄
prime radical	C₂²×D₂₁₁	Q₈
cosocle	C₂×C₄×C₂₁₁	C₂³
hypercenter	#108	D₂₁₁
hypocenter	C₂₁₁	#430
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
3. #20269913	C₂	semidirect
4. #20269912	C₂	semidirect
5. #108×C₂₁₁	C₂	semidirect
32. C₂₁₁	#430	semidirect
38. #108	D₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16809945	2
M₁₆×D₂₁₁	2
#20269913	2
#20269912	2
#108×C₂₁₁	2
#430	211