Subgroups — #42122815

Order 27712.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20294107	C₂	1	2	yes	maximal
2	C₄⋊#16812365	C₂	1	4	yes	maximal, complemented
3	C₄²×D₄₃₃	C₂	1	1	yes	maximal, characteristic
4	#20294107	—	2	4	no	—
5	C₂×#16812365	C₂²	1	4	yes	—
6	C₂×C₄×D₄₃₃	C₄	1	2	yes	—
7	C₂×C₄×D₄₃₃	C₂²	1	1	yes	characteristic
8	C₄×Dic₄₃₃	C₄	1	1	yes	characteristic
9	C₄×Dic₄₃₃	C₂²	1	2	yes	—
10	C₄²×C₄₃₃	C₄	1	1	yes	characteristic
11	#16812365	—	2	4	no	—
12	C₂²×D₄₃₃	C₂×C₄	1	1	yes	characteristic
13	C₂×Dic₄₃₃	C₂×C₄	1	2	yes	—
14	C₂×Dic₄₃₃	C₂³	1	1	yes	characteristic
15	C₄×D₄₃₃	—	2	4	no	—
16	C₄×D₄₃₃	D₄	1	2	yes	—
17	C₄×D₄₃₃	Q₈	1	2	yes	—
18	C₂×C₄×C₄₃₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₄₃₃	C₂×C₄	1	2	yes	—
20	C₂²×C₄₃₃	C₂²×C₄	1	1	yes	characteristic
21	C₂×D₄₃₃	—	2	2	no	—
22	C₂×D₄₃₃	C₄⋊C₄	1	2	yes	—
23	C₄×C₄₃₃	C₄⋊C₄	1	2	yes	—
24	C₄×C₄₃₃	M₁₆	1	4	yes	complemented
25	Dic₄₃₃	—	2	2	no	—
26	Dic₄₃₃	C₂×D₄	1	1	yes	characteristic
27	Dic₄₃₃	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₄₃₃	C₂×C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄₃₃	C₂×M₁₆	1	2	yes	—
30	D₄₃₃	—	2	2	no	—
31	C₄₃₃	#423	1	1	yes	characteristic, complemented, Sylow
32	#423	—	433	1	no	maximal, Sylow
33	C₂×M₁₆	—	433	2	no	—
34	C₂×C₄²	—	433	1	no	—
35	C₄⋊C₈	—	433	4	no	—
36	C₂²×C₄	—	433	2	no	—
37	C₂²×C₄	—	433	1	no	—
38	M₁₆	—	866	4	no	—
39	C₂×C₈	—	433	4	no	—
40	C₄²	—	433	2	no	—
41	C₄²	—	433	1	no	—
42	C₄²	#13181884	1	1	yes	characteristic
43	C₂³	—	433	1	no	—
44	C₂×C₄	—	866	4	no	—
45	C₂×C₄	—	433	2	no	—
46	C₂×C₄	—	433	1	no	—
47	C₂×C₄	—	433	2	no	—
48	C₂×C₄	—	433	2	no	—
49	C₂×C₄	C₂×#13181884	1	1	yes	characteristic
50	C₂×C₄	C₂×#13181884	1	2	yes	—
51	C₈	—	866	4	no	—
52	C₂²	—	866	2	no	—
53	C₂²	—	433	2	no	—
54	C₂²	C₂²×#13181884	1	1	yes	characteristic
55	C₄	—	866	2	no	—
56	C₄	—	433	1	no	—
57	C₄	—	433	1	no	—
58	C₄	C₄⋊#13181884	1	2	yes	—
59	C₄	#20294107	1	4	yes	complemented
60	C₂	—	866	2	no	—
61	C₂	C₂×C₄⋊#13181884	1	1	yes	characteristic
62	C₂	C₂×#20294107	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#13181884
commutator, socle	C₂²×C₄₃₃	C₂²×C₄
Fitting	C₄²×C₄₃₃	C₄
prime radical	C₂²×D₄₃₃	C₂×C₄
cosocle	C₂×Dic₄₃₃	C₂³
hypercenter	C₄²	#13181884
hypocenter	C₄₃₃	#423
LCS	G ▷ C₂²×C₄₃₃ ▷ C₄₃₃
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. C₄⋊#16812365	C₂	semidirect
24. C₄×C₄₃₃	M₁₆	semidirect
31. C₄₃₃	#423	semidirect
59. C₄	#20294107	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20294107	2
C₄⋊#16812365	2
C₄²×D₄₃₃	2
#423	433