Subgroups — #42122713

Order 27712.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28357280	C₂	1	1	yes	maximal, characteristic
2	#28357278	C₂	1	1	yes	maximal, characteristic
3	#28357203	C₂	1	1	yes	maximal, characteristic
4	#20294106	—	2	1	no	—
5	C₂²⋊#13181884	—	2	1	no	—
6	C₂×C₂²⋊D₄₃₃	C₂²	1	1	yes	characteristic
7	#20294076	—	2	1	no	—
8	C₂²⋊Dic₄₃₃	C₄	1	1	yes	characteristic, complemented
9	#20294074	—	2	1	no	—
10	C₂²×C₄×C₄₃₃	C₄	1	1	yes	characteristic, complemented
11	C₂×#13181884	—	4	1	no	—
12	#16812365	—	4	1	no	—
13	C₂²×D₄₃₃	D₄	1	1	yes	characteristic
14	C₂²⋊D₄₃₃	—	4	1	no	—
15	C₂×Dic₄₃₃	—	2	1	no	—
16	C₂×Dic₄₃₃	D₄	1	1	yes	characteristic
17	C₂³×C₄₃₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₄₃₃	—	2	1	no	—
19	C₂×C₄×C₄₃₃	—	4	1	no	—
20	C₂²×C₄₃₃	—	2	1	no	—
21	C₂²×C₄₃₃	—	4	1	no	—
22	C₂²×C₄₃₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×D₄₃₃	—	2	1	no	—
24	C₂×D₄₃₃	—	4	1	no	—
25	#13181884	—	8	1	no	—
26	C₄×C₄₃₃	—	4	1	no	—
27	Dic₄₃₃	—	2	1	no	—
28	Dic₄₃₃	—	4	1	no	—
29	C₂×C₄₃₃	—	2	1	no	—
30	C₂×C₄₃₃	—	4	1	no	—
31	C₂×C₄₃₃	#99	1	1	yes	characteristic
32	D₄₃₃	—	4	1	no	—
33	C₄₃₃	#352	1	1	yes	characteristic, complemented, Sylow
34	#352	—	433	1	no	maximal, Sylow
35	#123	—	433	1	no	—
36	#100	—	433	1	no	—
37	#99	—	433	1	no	—
38	C₂×D₄	—	433	1	no	—
39	C₂²×C₄	#13181884	1	1	yes	characteristic, complemented
40	M₁₆	—	866	1	no	—
41	C₄⋊C₄	—	866	1	no	—
42	C₂²⋊C₄	—	866	1	no	—
43	C₂²⋊C₄	—	866	1	no	—
44	C₂²⋊C₄	—	433	1	no	—
45	C₂³	—	433	1	no	—
46	C₂³	C₂×#13181884	1	1	yes	characteristic
47	D₄	—	1732	1	no	—
48	C₂×C₄	—	2	1	no	—
49	C₂×C₄	—	4	1	no	—
50	C₂×C₄	—	866	1	no	—
51	C₂×C₄	—	433	1	no	—
52	C₂×C₄	—	1732	1	no	—
53	C₈	—	1732	1	no	—
54	C₂²	—	2	1	no	—
55	C₂²	—	4	1	no	—
56	C₂²	—	866	1	no	—
57	C₂²	—	1732	1	no	—
58	C₂²	C₂²⋊#13181884	1	1	yes	characteristic
59	C₄	—	4	1	no	—
60	C₄	—	866	1	no	—
61	C₄	—	3464	1	no	—
62	C₄	—	1732	1	no	—
63	C₂	—	2	1	no	—
64	C₂	—	4	1	no	—
65	C₂	—	1732	1	no	—
66	C₂	#28357278	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#28357278
commutator	C₂³×C₄₃₃	C₂×C₄
Frattini	C₂³	C₂×#13181884
Fitting	C₂²×C₄×C₄₃₃	C₄
socle	C₂×C₄₃₃	#99
prime radical, cosocle	C₂×C₂²⋊D₄₃₃	C₂²
hypercenter	C₂²×C₄	#13181884
hypocenter	C₄₃₃	#352
LCS	G ▷ C₂³×C₄₃₃ ▷ C₂²×C₄₃₃ ▷ C₂×C₄₃₃ ▷ C₄₃₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
8. C₂²⋊Dic₄₃₃	C₄	semidirect
10. C₂²×C₄×C₄₃₃	C₄	semidirect
33. C₄₃₃	#352	semidirect
39. C₂²×C₄	#13181884	semidirect

Maximal subgroups

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

subgroup	index
#28357280	2
#28357278	2
#28357203	2
#352	433