Subgroups — C₂×#3753 #64140

Order 288.

228 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	#3753	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#213	—	3	4	no	maximal
6	C₂³×C₃²	—	2	1	no	—
7	C₂²×C₃⋊D₃	—	2	1	no	—
8	C₃²×D₄	—	2	1	no	—
9	C₃²×D₄	C₂²	1	2	yes	complemented
10	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
11	C₄⋊(C₃⋊D₃)	—	2	1	no	—
12	C₄⋊(C₃⋊D₃)	C₂²	1	2	yes	complemented
13	C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	complemented
14	C₂×C₃×D₄	D₃	1	4	yes	complemented
15	C₂×D₁₂	—	3	4	no	—
16	#213	—	3	16	no	—
17	C₂×C₃⋊C₈	—	3	4	no	—
18	C₂²×C₃²	—	2	2	no	—
19	C₂²×C₃²	—	4	2	no	—
20	C₂²×C₃²	D₄	1	1	yes	characteristic
21	C₂×C₃⋊D₃	—	2	2	no	—
22	C₂×C₃⋊D₃	—	4	2	no	—
23	C₃²×C₄	D₄	1	1	yes	characteristic
24	C₃²×C₄	C₂³	1	1	yes	characteristic
25	C₂×D₈	—	9	1	no	Sylow
26	C₂³×C₃	—	2	4	no	—
27	C₂²×D₃	—	6	4	no	—
28	C₃×D₄	—	2	4	no	—
29	C₃×D₄	C₂×D₃	1	8	yes	complemented
30	C₂×C₃×C₄	C₂×D₃	1	4	yes	—
31	D₁₂	—	3	8	no	—
32	D₁₂	—	6	4	no	—
33	C₃⋊C₈	—	3	8	no	—
34	C₂×C₃²	—	4	2	no	—
35	C₂×C₃²	D₈	1	2	yes	complemented
36	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
37	C₃⋊D₃	—	4	2	no	—
38	C₂×D₄	—	9	1	no	—
39	C₂×D₄	C₃⋊D₃	1	1	yes	characteristic, complemented
40	D₈	—	9	4	no	—
41	C₂×C₈	—	9	1	no	—
42	C₂²×C₃	—	2	8	no	—
43	C₂²×C₃	—	4	8	no	—
44	C₂²×C₃	#67	1	4	yes	—
45	C₂×D₃	—	6	8	no	—
46	C₂×D₃	—	12	8	no	—
47	C₃×C₄	#67	1	4	yes	—
48	C₃×C₄	C₂²×D₃	1	4	yes	—
49	C₃²	C₂×D₈	1	1	yes	characteristic, complemented, Sylow
50	C₂³	—	2	1	no	—
51	C₂³	—	18	1	no	—
52	D₄	—	2	1	no	—
53	D₄	—	9	2	no	—
54	D₄	—	18	1	no	—
55	D₄	C₂×C₃⋊D₃	1	2	yes	complemented
56	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
57	C₈	—	9	2	no	—
58	C₂×C₃	—	4	8	no	—
59	C₂×C₃	#213	1	8	yes	complemented
60	C₂×C₃	C₂×#67	1	4	yes	—
61	D₃	—	12	8	no	—
62	C₂²	—	2	2	no	—
63	C₂²	—	4	2	no	—
64	C₂²	—	18	2	no	—
65	C₂²	—	36	2	no	—
66	C₂²	#638	1	1	yes	characteristic
67	C₄	#638	1	1	yes	characteristic
68	C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
69	C₃	C₂×#213	1	4	yes	complemented
70	C₂	—	4	2	no	—
71	C₂	—	36	2	no	—
72	C₂	#3753	1	2	yes	direct product factor
73	C₂	C₂×#638	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#638
commutator, cosocle	C₃²×C₄	C₂³
Frattini	C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₃²×D₄	C₂
socle	C₂²×C₃²	D₄
hypercenter	C₂×D₄	C₃⋊D₃
hypocenter	C₃²	C₂×D₈
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×D₄	C₂	semidirect
2. C₂×C₄⋊(C₃⋊D₃)	C₂	semidirect
3. #3753	C₂	direct (N × H)
4. C₂×C₃⋊(C₃⋊C₈)	C₂	semidirect
9. C₃²×D₄	C₂²	semidirect
12. C₄⋊(C₃⋊D₃)	C₂²	semidirect
13. C₃⋊(C₃⋊C₈)	C₂²	semidirect
14. C₂×C₃×D₄	D₃	semidirect
29. C₃×D₄	C₂×D₃	semidirect
35. C₂×C₃²	D₈	semidirect
39. C₂×D₄	C₃⋊D₃	semidirect
49. C₃²	C₂×D₈	semidirect
55. D₄	C₂×C₃⋊D₃	semidirect
59. C₂×C₃	#213	semidirect
69. C₃	C₂×#213	semidirect
72. C₂	#3753	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×D₄	2
C₂×C₄⋊(C₃⋊D₃)	2
#3753	2
C₂×C₃⋊(C₃⋊C₈)	2
C₂×#213	3

Subgroups — C2×#3753 #64140

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#3753 #64140