Subgroups — C₂³×#15688412 #37560179

Order 23872.

338 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15688412	C₂	1	14	yes	maximal, direct product factor
2	C₂³×C₄×C₃₇₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688412	C₂²	1	28	yes	direct product factor
4	C₂⁴×C₃₇₃	C₄	1	1	yes	characteristic
5	C₂²×C₄×C₃₇₃	C₄	1	7	yes	—
6	C₂²×C₄×C₃₇₃	C₂²	1	7	yes	—
7	#15688412	C₂³	1	8	yes	direct product factor
8	C₂³×C₃₇₃	C₈	1	8	yes	complemented
9	C₂³×C₃₇₃	C₂×C₄	1	7	yes	—
10	C₂×C₄×C₃₇₃	C₂×C₄	1	21	yes	—
11	C₂×C₄×C₃₇₃	C₂³	1	7	yes	—
12	C₂²×C₃₇₃	C₂×C₈	1	28	yes	complemented
13	C₂²×C₃₇₃	C₂²×C₄	1	7	yes	—
14	C₄×C₃₇₃	C₂²×C₄	1	7	yes	—
15	C₄×C₃₇₃	C₂⁴	1	1	yes	characteristic
16	C₂×C₃₇₃	C₂²×C₈	1	14	yes	complemented
17	C₂×C₃₇₃	C₂³×C₄	1	1	yes	characteristic
18	C₃₇₃	C₂³×C₈	1	1	yes	characteristic, complemented, Sylow
19	C₂³×C₈	—	373	1	no	maximal, Sylow
20	C₂³×C₄	D₃₇₃	1	1	yes	characteristic
21	C₂²×C₈	—	373	14	no	—
22	C₂⁴	Dic₃₇₃	1	1	yes	characteristic
23	C₂²×C₄	Dic₃₇₃	1	7	yes	—
24	C₂²×C₄	C₂×D₃₇₃	1	7	yes	—
25	C₂×C₈	—	373	28	no	—
26	C₂³	#15688412	1	8	yes	direct product factor
27	C₂³	C₂×Dic₃₇₃	1	7	yes	—
28	C₂×C₄	C₂×Dic₃₇₃	1	21	yes	—
29	C₂×C₄	C₂²×D₃₇₃	1	7	yes	—
30	C₈	—	373	8	no	—
31	C₂²	C₂×#15688412	1	28	yes	direct product factor
32	C₂²	C₂²×Dic₃₇₃	1	7	yes	—
33	C₄	C₂²×Dic₃₇₃	1	7	yes	—
34	C₄	C₂³×D₃₇₃	1	1	yes	characteristic
35	C₂	C₂²×#15688412	1	14	yes	direct product factor
36	C₂	C₂³×Dic₃₇₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₄	D₃₇₃
commutator, hypocenter	C₃₇₃	C₂³×C₈
Frattini	C₄	C₂³×D₃₇₃
Fitting	C₂³×C₄×C₃₇₃	C₂
socle, prime radical	C₂⁴×C₃₇₃	C₄
cosocle	C₄×C₃₇₃	C₂⁴
LCS	G ▷ C₃₇₃
UCS	C₁ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#15688412	C₂	direct (N × H)
3. C₂×#15688412	C₂²	direct (N × H)
7. #15688412	C₂³	direct (N × H)
8. C₂³×C₃₇₃	C₈	semidirect
12. C₂²×C₃₇₃	C₂×C₈	semidirect
16. C₂×C₃₇₃	C₂²×C₈	semidirect
18. C₃₇₃	C₂³×C₈	semidirect
26. C₂³	#15688412	direct (N × H)
31. C₂²	C₂×#15688412	direct (N × H)
35. C₂	C₂²×#15688412	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×#15688412	2
C₂³×C₄×C₃₇₃	2
C₂³×C₈	373

Subgroups — C23×#15688412 #37560179

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#15688412 #37560179