Subgroups — #42122851

Order 27712.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#13181884	C₂	1	2	yes	maximal
2	#28357288	C₂	1	1	yes	maximal, characteristic
3	#28357283	C₂	1	2	yes	maximal
4	#28357274	C₂	1	1	yes	maximal, characteristic
5	Q₁₆×D₄₃₃	C₂	1	1	yes	maximal, characteristic
6	#20294105	C₂²	1	1	yes	characteristic
7	C₄⋊#13181884	—	2	2	no	—
8	C₄⋊#13181884	C₂²	1	2	yes	—
9	C₄×#13181884	—	2	2	no	—
10	C₄×#13181884	C₂²	1	1	yes	characteristic
11	Q₈×D₄₃₃	C₂²	1	2	yes	—
12	Dic₁₇₃₂	C₄	1	1	yes	characteristic, complemented
13	#20294087	C₄	1	2	yes	complemented
14	C₈×D₄₃₃	C₂²	1	1	yes	characteristic
15	Q₁₆×C₄₃₃	C₄	1	1	yes	characteristic, complemented
16	C₂×#13181884	—	2	2	no	—
17	C₂×#13181884	D₄	1	2	yes	—
18	#16812365	—	2	1	no	—
19	Dic₈₆₆	—	2	2	no	—
20	Dic₈₆₆	C₂×C₄	1	2	yes	—
21	C₄×D₄₃₃	—	2	2	no	—
22	C₄×D₄₃₃	C₂³	1	1	yes	characteristic
23	#16812358	C₂×C₄	1	1	yes	characteristic
24	Q₈×C₄₃₃	C₂×C₄	1	2	yes	—
25	C₈×C₄₃₃	C₂×C₄	1	1	yes	characteristic
26	C₂×D₄₃₃	C₂×D₄	1	1	yes	characteristic
27	#13181884	—	2	2	no	—
28	#13181884	—	4	2	no	—
29	C₄×C₄₃₃	—	2	2	no	—
30	C₄×C₄₃₃	C₂²×C₄	1	1	yes	characteristic
31	Dic₄₃₃	—	2	2	no	—
32	Dic₄₃₃	#41	1	1	yes	characteristic
33	C₂×C₄₃₃	C₄×D₄	1	1	yes	characteristic
34	D₄₃₃	#137	1	2	yes	—
35	C₄₃₃	#441	1	1	yes	characteristic, complemented, Sylow
36	#441	—	433	1	no	maximal, Sylow
37	C₂×Q₁₆	—	433	1	no	—
38	C₄×Q₈	—	433	2	no	—
39	#106	—	433	1	no	—
40	#103	—	433	2	no	—
41	#97	—	433	1	no	—
42	C₂×Q₈	—	433	2	no	—
43	Q₁₆	—	433	2	no	—
44	Q₁₆	—	433	1	no	—
45	Q₁₆	#13181884	1	1	yes	characteristic, complemented
46	C₂×C₈	—	433	1	no	—
47	C₂×C₈	—	433	1	no	—
48	C₄⋊C₄	—	866	2	no	—
49	C₄⋊C₄	—	433	2	no	—
50	C₄²	—	866	2	no	—
51	C₄²	—	433	1	no	—
52	Q₈	—	866	2	no	—
53	Q₈	—	433	2	no	—
54	Q₈	C₂×#13181884	1	2	yes	—
55	C₂×C₄	—	866	2	no	—
56	C₂×C₄	—	866	2	no	—
57	C₂×C₄	—	433	2	no	—
58	C₂×C₄	—	433	1	no	—
59	C₈	—	866	1	no	—
60	C₈	—	433	1	no	—
61	C₈	C₂×#13181884	1	1	yes	characteristic
62	C₂²	—	433	1	no	—
63	C₄	—	2	2	no	—
64	C₄	—	866	2	no	—
65	C₄	—	866	2	no	—
66	C₄	—	433	1	no	—
67	C₄	—	1732	2	no	—
68	C₄	C₂²×#13181884	1	1	yes	characteristic
69	C₂	—	433	2	no	—
70	C₂	D₄×#13181884	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×#13181884
commutator	C₄×C₄₃₃	C₂²×C₄
Frattini	C₄	C₂²×#13181884
Fitting	Q₁₆×C₄₃₃	C₄
socle	C₂×C₄₃₃	C₄×D₄
prime radical	C₂×D₄₃₃	C₂×D₄
cosocle	C₄×D₄₃₃	C₂³
hypercenter	Q₁₆	#13181884
hypocenter	C₄₃₃	#441
LCS	G ▷ C₄×C₄₃₃ ▷ C₂×C₄₃₃ ▷ C₄₃₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
12. Dic₁₇₃₂	C₄	semidirect
13. #20294087	C₄	semidirect
15. Q₁₆×C₄₃₃	C₄	semidirect
35. C₄₃₃	#441	semidirect
45. Q₁₆	#13181884	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#13181884	2
#28357288	2
#28357283	2
#28357274	2
Q₁₆×D₄₃₃	2
#441	433