Subgroups — C₂×C₄×#15688412 #37559368

Order 23872.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15688412	C₂	1	2	yes	maximal
2	C₄×#15688412	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄²×C₃₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#15688412	C₄	1	8	yes	direct product factor
5	C₂×#15688412	C₂²	1	4	yes	—
6	C₂²×C₄×C₃₇₃	C₄	1	2	yes	—
7	C₂²×C₄×C₃₇₃	C₂²	1	1	yes	characteristic
8	C₄²×C₃₇₃	C₄	1	2	yes	—
9	C₄²×C₃₇₃	C₂²	1	2	yes	—
10	#15688412	C₂×C₄	1	8	yes	direct product factor
11	C₂³×C₃₇₃	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄×C₃₇₃	C₈	1	8	yes	complemented
13	C₂×C₄×C₃₇₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₃₇₃	C₂×C₄	1	4	yes	—
15	C₂×C₄×C₃₇₃	C₂×C₄	1	4	yes	—
16	C₂×C₄×C₃₇₃	C₂³	1	1	yes	characteristic
17	C₂²×C₃₇₃	C₄²	1	2	yes	—
18	C₂²×C₃₇₃	C₂×C₈	1	4	yes	—
19	C₂²×C₃₇₃	C₂²×C₄	1	1	yes	characteristic
20	C₄×C₃₇₃	C₄²	1	2	yes	—
21	C₄×C₃₇₃	C₂×C₈	1	8	yes	complemented
22	C₄×C₃₇₃	C₂²×C₄	1	2	yes	—
23	C₂×C₃₇₃	C₄×C₈	1	4	yes	complemented
24	C₂×C₃₇₃	C₂×C₄²	1	1	yes	characteristic
25	C₂×C₃₇₃	C₂²×C₈	1	2	yes	—
26	C₃₇₃	C₂×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
27	C₂×C₄×C₈	—	373	1	no	maximal, Sylow
28	C₂²×C₈	—	373	2	no	—
29	C₂×C₄²	D₃₇₃	1	1	yes	characteristic
30	C₄×C₈	—	373	4	no	—
31	C₂²×C₄	Dic₃₇₃	1	2	yes	—
32	C₂²×C₄	C₂×D₃₇₃	1	1	yes	characteristic
33	C₂×C₈	—	373	8	no	—
34	C₂×C₈	—	373	4	no	—
35	C₄²	Dic₃₇₃	1	2	yes	—
36	C₄²	C₂×D₃₇₃	1	2	yes	—
37	C₂³	C₂×Dic₃₇₃	1	1	yes	characteristic
38	C₂×C₄	#15688412	1	8	yes	direct product factor
39	C₂×C₄	C₄×D₃₇₃	1	4	yes	—
40	C₂×C₄	C₂×Dic₃₇₃	1	4	yes	—
41	C₂×C₄	C₂×Dic₃₇₃	1	1	yes	characteristic
42	C₂×C₄	C₂²×D₃₇₃	1	1	yes	characteristic
43	C₈	—	373	8	no	—
44	C₂²	C₄×Dic₃₇₃	1	2	yes	—
45	C₂²	C₂×#15688412	1	4	yes	—
46	C₂²	C₂²×Dic₃₇₃	1	1	yes	characteristic
47	C₄	C₄×Dic₃₇₃	1	2	yes	—
48	C₄	C₂×#15688412	1	8	yes	direct product factor
49	C₄	C₂×C₄×D₃₇₃	1	2	yes	—
50	C₂	C₄×#15688412	1	4	yes	direct product factor
51	C₂	C₂×C₄×Dic₃₇₃	1	1	yes	characteristic
52	C₂	C₂²×#15688412	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂×C₄²	D₃₇₃
commutator, hypocenter	C₃₇₃	C₂×C₄×C₈
Frattini	C₂×C₄	C₂²×D₃₇₃
Fitting	C₂×C₄²×C₃₇₃	C₂
socle, prime radical	C₂³×C₃₇₃	C₂×C₄
cosocle	C₂×C₄×C₃₇₃	C₂³
LCS	G ▷ C₃₇₃
UCS	C₁ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. C₄×#15688412	C₂	direct (N × H)
4. C₂×#15688412	C₄	direct (N × H)
10. #15688412	C₂×C₄	direct (N × H)
12. C₂×C₄×C₃₇₃	C₈	semidirect
21. C₄×C₃₇₃	C₂×C₈	semidirect
23. C₂×C₃₇₃	C₄×C₈	semidirect
26. C₃₇₃	C₂×C₄×C₈	semidirect
38. C₂×C₄	#15688412	direct (N × H)
48. C₄	C₂×#15688412	direct (N × H)
50. C₂	C₄×#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
C₄×#15688412	2
C₂×C₄²×C₃₇₃	2
C₂×C₄×C₈	373

Subgroups — C2×C4×#15688412 #37559368

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#15688412 #37559368