Subgroups — C₂×#28357214 #42122336

Order 27712.

246 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28357214	C₂	1	8	yes	maximal, direct product factor
2	C₂×#20294076	C₂	1	3	yes	maximal, complemented
3	C₂×#20294074	C₂	1	3	yes	maximal
4	C₂×C₄²×C₄₃₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₄₃₃	C₂²	1	1	yes	characteristic
6	C₂²×Dic₄₃₃	C₂²	1	3	yes	—
7	#20294076	C₂²	1	12	yes	complemented
8	#20294074	C₂²	1	12	yes	—
9	C₂²×C₄×C₄₃₃	C₂²	1	3	yes	—
10	C₄²×C₄₃₃	C₂²	1	4	yes	complemented
11	C₂²×D₄₃₃	—	2	6	no	—
12	C₂²×D₄₃₃	C₂³	1	2	yes	—
13	C₂×Dic₄₃₃	—	2	6	no	—
14	C₂×Dic₄₃₃	C₂³	1	6	yes	—
15	C₂³×C₄₃₃	C₂³	1	1	yes	characteristic
16	C₂×C₄×C₄₃₃	—	2	6	no	—
17	C₂×C₄×C₄₃₃	C₂³	1	6	yes	—
18	C₂²×C₄₃₃	#41	1	6	yes	—
19	C₂²×C₄₃₃	C₂⁴	1	1	yes	characteristic
20	C₂×D₄₃₃	—	2	6	no	—
21	C₂×D₄₃₃	—	4	4	no	—
22	C₄×C₄₃₃	—	2	6	no	—
23	Dic₄₃₃	—	2	6	no	—
24	C₂×C₄₃₃	#126	1	4	yes	complemented
25	C₂×C₄₃₃	C₂×#41	1	3	yes	—
26	D₄₃₃	—	4	2	no	—
27	C₄₃₃	C₂×#126	1	1	yes	characteristic, complemented, Sylow
28	C₂×#126	—	433	1	no	maximal, Sylow
29	#126	—	433	8	no	—
30	C₂×C₄⋊C₄	—	433	3	no	—
31	C₂×C₂²⋊C₄	—	433	3	no	—
32	C₂×C₄²	D₄₃₃	1	1	yes	characteristic, complemented
33	C₂⁴	—	433	1	no	—
34	C₂²×C₄	—	433	3	no	—
35	C₂²×C₄	C₂×D₄₃₃	1	3	yes	—
36	C₄⋊C₄	—	433	12	no	—
37	C₂²⋊C₄	—	433	12	no	—
38	C₄²	C₂×D₄₃₃	1	4	yes	complemented
39	C₂³	—	866	6	no	—
40	C₂³	—	433	2	no	—
41	C₂³	C₂²×D₄₃₃	1	1	yes	characteristic
42	C₂×C₄	—	2	6	no	—
43	C₂×C₄	—	866	6	no	—
44	C₂×C₄	—	433	6	no	—
45	C₂×C₄	C₂²×D₄₃₃	1	6	yes	—
46	C₂²	—	866	6	no	—
47	C₂²	—	1732	4	no	—
48	C₂²	#20294097	1	6	yes	—
49	C₂²	C₂³×D₄₃₃	1	1	yes	characteristic
50	C₄	—	2	6	no	—
51	C₄	—	866	6	no	—
52	C₂	—	1732	2	no	—
53	C₂	#28357214	1	4	yes	direct product factor
54	C₂	C₂×#20294097	1	3	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₄₃₃
commutator, cosocle	C₂²×C₄₃₃	C₂⁴
Frattini	C₂²	C₂³×D₄₃₃
Fitting	C₂×C₄²×C₄₃₃	C₂
socle	C₂³×C₄₃₃	C₂³
prime radical	C₂³×D₄₃₃	C₂²
hypercenter	C₂×C₄²	D₄₃₃
hypocenter	C₄₃₃	C₂×#126
LCS	G ▷ C₂²×C₄₃₃ ▷ C₄₃₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
1. #28357214	C₂	direct (N × H)
2. C₂×#20294076	C₂	semidirect
4. C₂×C₄²×C₄₃₃	C₂	semidirect
7. #20294076	C₂²	semidirect
10. C₄²×C₄₃₃	C₂²	semidirect
24. C₂×C₄₃₃	#126	semidirect
27. C₄₃₃	C₂×#126	semidirect
32. C₂×C₄²	D₄₃₃	semidirect
38. C₄²	C₂×D₄₃₃	semidirect
53. C₂	#28357214	direct (N × H)

Maximal subgroups

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

subgroup	index
#28357214	2
C₂×#20294076	2
C₂×#20294074	2
C₂×C₄²×C₄₃₃	2
C₂×#126	433

Subgroups — C2×#28357214 #42122336

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#28357214 #42122336