Subgroups — C₂³×#15667725 #36378544

Order 22592.

338 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15667725	C₂	1	14	yes	maximal, direct product factor
2	C₂³×C₄×C₃₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#15667725	C₂²	1	28	yes	direct product factor
4	C₂⁴×C₃₅₃	C₄	1	1	yes	characteristic
5	C₂²×C₄×C₃₅₃	C₄	1	7	yes	—
6	C₂²×C₄×C₃₅₃	C₂²	1	7	yes	—
7	#15667725	C₂³	1	8	yes	direct product factor
8	C₂³×C₃₅₃	C₈	1	8	yes	complemented
9	C₂³×C₃₅₃	C₂×C₄	1	7	yes	—
10	C₂×C₄×C₃₅₃	C₂×C₄	1	21	yes	—
11	C₂×C₄×C₃₅₃	C₂³	1	7	yes	—
12	C₂²×C₃₅₃	C₂×C₈	1	28	yes	complemented
13	C₂²×C₃₅₃	C₂²×C₄	1	7	yes	—
14	C₄×C₃₅₃	C₂²×C₄	1	7	yes	—
15	C₄×C₃₅₃	C₂⁴	1	1	yes	characteristic
16	C₂×C₃₅₃	C₂²×C₈	1	14	yes	complemented
17	C₂×C₃₅₃	C₂³×C₄	1	1	yes	characteristic
18	C₃₅₃	C₂³×C₈	1	1	yes	characteristic, complemented, Sylow
19	C₂³×C₈	—	353	1	no	maximal, Sylow
20	C₂³×C₄	D₃₅₃	1	1	yes	characteristic
21	C₂²×C₈	—	353	14	no	—
22	C₂⁴	Dic₃₅₃	1	1	yes	characteristic
23	C₂²×C₄	Dic₃₅₃	1	7	yes	—
24	C₂²×C₄	C₂×D₃₅₃	1	7	yes	—
25	C₂×C₈	—	353	28	no	—
26	C₂³	#15667725	1	8	yes	direct product factor
27	C₂³	C₂×Dic₃₅₃	1	7	yes	—
28	C₂×C₄	C₂×Dic₃₅₃	1	21	yes	—
29	C₂×C₄	C₂²×D₃₅₃	1	7	yes	—
30	C₈	—	353	8	no	—
31	C₂²	C₂×#15667725	1	28	yes	direct product factor
32	C₂²	C₂²×Dic₃₅₃	1	7	yes	—
33	C₄	C₂²×Dic₃₅₃	1	7	yes	—
34	C₄	C₂³×D₃₅₃	1	1	yes	characteristic
35	C₂	C₂²×#15667725	1	14	yes	direct product factor
36	C₂	C₂³×Dic₃₅₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₄	D₃₅₃
commutator, hypocenter	C₃₅₃	C₂³×C₈
Frattini	C₄	C₂³×D₃₅₃
Fitting	C₂³×C₄×C₃₅₃	C₂
socle, prime radical	C₂⁴×C₃₅₃	C₄
cosocle	C₄×C₃₅₃	C₂⁴
LCS	G ▷ C₃₅₃
UCS	C₁ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#15667725	C₂	direct (N × H)
3. C₂×#15667725	C₂²	direct (N × H)
7. #15667725	C₂³	direct (N × H)
8. C₂³×C₃₅₃	C₈	semidirect
12. C₂²×C₃₅₃	C₂×C₈	semidirect
16. C₂×C₃₅₃	C₂²×C₈	semidirect
18. C₃₅₃	C₂³×C₈	semidirect
26. C₂³	#15667725	direct (N × H)
31. C₂²	C₂×#15667725	direct (N × H)
35. C₂	C₂²×#15667725	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#15667725	2
C₂³×C₄×C₃₅₃	2
C₂³×C₈	353

Subgroups — C23×#15667725 #36378544

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#15667725 #36378544