Subgroups — C₂×C₄⋊#15688419 #37560422

Order 23872.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15688419	C₂	1	2	yes	maximal
2	C₄⋊#15688419	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×Dic₃₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#15688419	—	2	4	no	—
5	C₂×#15688419	C₂²	1	4	yes	—
6	C₂²×Dic₃₇₃	C₄	1	1	yes	characteristic
7	C₂²×Dic₃₇₃	C₂²	1	1	yes	characteristic
8	C₄×Dic₃₇₃	C₄	1	2	yes	—
9	C₄×Dic₃₇₃	C₂²	1	2	yes	—
10	C₂²×C₄×C₃₇₃	C₄	1	1	yes	characteristic
11	#15688419	—	2	4	no	—
12	C₂×Dic₃₇₃	—	2	2	no	—
13	C₂×Dic₃₇₃	C₂×C₄	1	2	yes	—
14	C₂×Dic₃₇₃	C₂×C₄	1	1	yes	characteristic
15	C₂×Dic₃₇₃	D₄	1	2	yes	—
16	C₂×Dic₃₇₃	Q₈	1	2	yes	—
17	C₂×Dic₃₇₃	C₂³	1	1	yes	characteristic
18	C₂³×C₃₇₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₃₇₃	C₈	1	4	yes	complemented
20	C₂×C₄×C₃₇₃	C₂×C₄	1	2	yes	—
21	C₂²×C₃₇₃	C₄⋊C₄	1	2	yes	—
22	C₂²×C₃₇₃	C₂×C₈	1	2	yes	—
23	C₂²×C₃₇₃	M₁₆	1	2	yes	—
24	C₂²×C₃₇₃	C₂²×C₄	1	1	yes	characteristic
25	C₄×C₃₇₃	C₂×C₈	1	4	yes	complemented
26	Dic₃₇₃	—	2	2	no	—
27	Dic₃₇₃	C₄⋊C₄	1	2	yes	—
28	Dic₃₇₃	C₂×D₄	1	1	yes	characteristic
29	Dic₃₇₃	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₃₇₃	C₄⋊C₈	1	4	yes	complemented
31	C₂×C₃₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₃₇₃	C₂²×C₈	1	1	yes	characteristic
33	C₂×C₃₇₃	C₂×M₁₆	1	1	yes	characteristic
34	C₃₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
35	C₂×C₄⋊C₈	—	373	1	no	maximal, Sylow
36	C₂²×C₈	—	373	2	no	—
37	C₂×C₄²	—	373	1	no	—
38	C₄⋊C₈	—	373	4	no	—
39	C₂²×C₄	—	373	1	no	—
40	C₂²×C₄	—	373	1	no	—
41	C₂²×C₄	#13093637	1	1	yes	characteristic
42	C₂×C₈	—	373	4	no	—
43	C₂×C₈	—	746	4	no	—
44	C₄²	—	373	2	no	—
45	C₄²	—	373	2	no	—
46	C₂³	C₂×#13093637	1	1	yes	characteristic
47	C₂×C₄	—	373	2	no	—
48	C₂×C₄	—	373	2	no	—
49	C₂×C₄	—	373	1	no	—
50	C₂×C₄	—	373	1	no	—
51	C₂×C₄	—	373	2	no	—
52	C₂×C₄	—	746	2	no	—
53	C₂×C₄	#15688419	1	4	yes	complemented
54	C₂×C₄	C₂×#13093637	1	2	yes	—
55	C₈	—	746	4	no	—
56	C₂²	C₄⋊#13093637	1	2	yes	—
57	C₂²	C₂×#15688419	1	2	yes	—
58	C₂²	#20238620	1	2	yes	—
59	C₂²	C₂²×#13093637	1	1	yes	characteristic
60	C₄	—	373	2	no	—
61	C₄	—	373	1	no	—
62	C₄	—	373	1	no	—
63	C₄	—	746	2	no	—
64	C₄	C₂×#15688419	1	4	yes	complemented
65	C₂	C₄⋊#15688419	1	4	yes	direct product factor
66	C₂	C₂×C₄⋊#13093637	1	1	yes	characteristic
67	C₂	C₂²×#15688419	1	1	yes	characteristic
68	C₂	C₂×#20238620	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#13093637
commutator	C₂×C₃₇₃	C₂²×C₈
Frattini	C₂²	C₂²×#13093637
Fitting	C₂²×C₄×C₃₇₃	C₄
socle, prime radical	C₂³×C₃₇₃	C₂×C₄
cosocle	C₂×Dic₃₇₃	C₂³
hypercenter	C₂²×C₄	#13093637
hypocenter	C₃₇₃	C₂×C₄⋊C₈
LCS	G ▷ C₂×C₃₇₃ ▷ C₃₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₄⋊#15688419	C₂	direct (N × H)
19. C₂×C₄×C₃₇₃	C₈	semidirect
25. C₄×C₃₇₃	C₂×C₈	semidirect
30. C₂×C₃₇₃	C₄⋊C₈	semidirect
34. C₃₇₃	C₂×C₄⋊C₈	semidirect
53. C₂×C₄	#15688419	semidirect
64. C₄	C₂×#15688419	semidirect
65. C₂	C₄⋊#15688419	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×#15688419	2
C₄⋊#15688419	2
C₂×C₄×Dic₃₇₃	2
C₂×C₄⋊C₈	373

Subgroups — C2×C4⋊#15688419 #37560422

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄⋊#15688419 #37560422