Subgroups — #42122685

Order 27712.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#13181884	C₂	1	2	yes	maximal
2	C₂×C₈×D₄₃₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×#13181884	—	2	2	no	—
4	C₄⋊#13181884	—	2	2	no	—
5	C₄⋊#13181884	C₄	1	4	yes	complemented
6	C₂×C₄×D₄₃₃	C₂²	1	1	yes	characteristic
7	C₂×#16812358	C₄	1	1	yes	characteristic, complemented
8	C₈×D₄₃₃	—	2	2	no	—
9	C₂×C₈×C₄₃₃	C₄	1	1	yes	characteristic, complemented
10	C₂×#13181884	—	2	4	no	—
11	C₂×#13181884	—	4	2	no	—
12	C₂×#13181884	—	4	2	no	—
13	C₂²×D₄₃₃	D₄	1	1	yes	characteristic
14	C₂×Dic₄₃₃	Q₈	1	1	yes	characteristic
15	C₄×D₄₃₃	C₂×C₄	1	2	yes	—
16	C₄×D₄₃₃	D₄	1	2	yes	—
17	#16812358	—	2	1	no	—
18	C₂×C₄×C₄₃₃	C₂×C₄	1	1	yes	characteristic
19	C₈×C₄₃₃	—	2	1	no	—
20	C₂²×C₄₃₃	C₄⋊C₄	1	1	yes	characteristic
21	C₂×D₄₃₃	C₂²⋊C₄	1	2	yes	—
22	C₂×D₄₃₃	D₈	1	1	yes	characteristic
23	C₂×D₄₃₃	SD₁₆	1	2	yes	—
24	C₂×D₄₃₃	Q₁₆	1	1	yes	characteristic
25	#13181884	—	4	4	no	—
26	C₄×C₄₃₃	C₄²	1	1	yes	characteristic
27	C₄×C₄₃₃	C₂²⋊C₄	1	1	yes	characteristic
28	Dic₄₃₃	C₄⋊C₄	1	2	yes	—
29	C₂×C₄₃₃	#95	1	1	yes	characteristic
30	C₂×C₄₃₃	#106	1	1	yes	characteristic
31	C₂×C₄₃₃	#107	1	1	yes	characteristic
32	D₄₃₃	#102	1	2	yes	—
33	D₄₃₃	#103	1	2	yes	—
34	C₄₃₃	#340	1	1	yes	characteristic, complemented, Sylow
35	#340	—	433	1	no	maximal, Sylow
36	C₂²×C₈	—	433	1	no	—
37	C₂×C₄⋊C₄	—	433	2	no	—
38	C₂²×C₄	—	866	2	no	—
39	C₂²×C₄	—	433	1	no	—
40	C₂×C₈	—	866	2	no	—
41	C₂×C₈	—	433	1	no	—
42	C₂×C₈	#13181884	1	1	yes	characteristic, complemented
43	C₄⋊C₄	—	866	2	no	—
44	C₄⋊C₄	—	433	4	no	—
45	C₂³	—	433	1	no	—
46	C₂×C₄	—	866	4	no	—
47	C₂×C₄	—	433	1	no	—
48	C₂×C₄	—	433	2	no	—
49	C₂×C₄	—	433	2	no	—
50	C₂×C₄	—	1732	2	no	—
51	C₂×C₄	—	1732	2	no	—
52	C₂×C₄	C₂×#13181884	1	1	yes	characteristic
53	C₈	—	2	1	no	—
54	C₈	—	866	1	no	—
55	C₂²	—	433	2	no	—
56	C₂²	—	433	2	no	—
57	C₂²	—	433	1	no	—
58	C₂²	—	433	1	no	—
59	C₂²	C₄⋊#13181884	1	1	yes	characteristic
60	C₄	—	433	2	no	—
61	C₄	—	1732	4	no	—
62	C₄	C₄×#13181884	1	1	yes	characteristic
63	C₄	C₂²⋊#13181884	1	1	yes	characteristic
64	C₂	—	433	2	no	—
65	C₂	—	433	2	no	—
66	C₂	(C₂×#13181884)⋊C₄	1	1	yes	characteristic
67	C₂	#28357288	1	1	yes	characteristic
68	C₂	#28357289	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₄⋊#13181884
commutator	C₄×C₄₃₃	C₄²
Frattini	C₂×C₄	C₂×#13181884
Fitting	C₂×C₈×C₄₃₃	C₄
socle	C₂²×C₄₃₃	C₄⋊C₄
prime radical	C₂²×D₄₃₃	D₄
cosocle	C₂×C₄×D₄₃₃	C₂²
hypercenter	C₂×C₈	#13181884
hypocenter	C₄₃₃	#340
LCS	G ▷ C₄×C₄₃₃ ▷ C₂×C₄₃₃ ▷ C₄₃₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
5. C₄⋊#13181884	C₄	semidirect
7. C₂×#16812358	C₄	semidirect
9. C₂×C₈×C₄₃₃	C₄	semidirect
34. C₄₃₃	#340	semidirect
42. C₂×C₈	#13181884	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#13181884	2
C₂×C₈×D₄₃₃	2
#340	433