Subgroups — C₂³×#41⋊C₃ #87529

Order 384.

1395 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41⋊C₃	C₂	1	14	yes	maximal, direct product factor
2	C₂³×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×#41	C₃	1	1	yes	maximal, characteristic, complemented, Sylow
4	C₂³×C₃×C₄	—	4	1	no	maximal
5	C₂×#41⋊C₃	C₂²	1	28	yes	direct product factor
6	C₂²×SL(2,𝔽₃)	C₂²	1	7	yes	complemented
7	C₂²×#41	—	3	14	no	—
8	C₂²×#41	C₂×C₃	1	14	yes	complemented
9	C₂³×Q₈	C₂×C₃	1	1	yes	characteristic
10	C₂³×D₄	—	3	1	no	—
11	C₂⁴×C₄	—	3	1	no	—
12	C₂⁴×C₃	—	4	1	no	—
13	C₂²×C₃×C₄	—	4	14	no	—
14	#41⋊C₃	C₂³	1	8	yes	direct product factor
15	C₂×SL(2,𝔽₃)	C₂³	1	7	yes	complemented
16	C₂⁵	—	3	1	no	—
17	C₂×#41	—	3	56	no	—
18	C₂×#41	—	3	84	no	—
19	C₂×#41	C₂²×C₃	1	28	yes	complemented
20	C₂²×Q₈	—	3	7	no	—
21	C₂²×Q₈	C₂²×C₃	1	7	yes	—
22	C₂²×D₄	—	3	14	no	—
23	C₂²×D₄	—	3	14	no	—
24	C₂³×C₄	—	3	1	no	—
25	C₂³×C₄	—	3	14	no	—
26	C₂³×C₄	—	3	14	no	—
27	C₂³×C₄	A₄	1	1	yes	characteristic
28	C₂³×C₃	—	4	8	no	—
29	C₂³×C₃	—	4	7	no	—
30	C₂×C₃×C₄	—	4	28	no	—
31	SL(2,𝔽₃)	C₂⁴	1	1	yes	characteristic, complemented
32	C₂⁴	—	3	14	no	—
33	C₂⁴	—	6	8	no	—
34	C₂⁴	C₂×A₄	1	1	yes	characteristic
35	#41	—	3	112	no	—
36	#41	—	3	56	no	—
37	#41	C₂³×C₃	1	8	yes	complemented
38	C₂×Q₈	—	3	14	no	—
39	C₂×Q₈	—	3	21	no	—
40	C₂×Q₈	C₂³×C₃	1	7	yes	—
41	C₂×D₄	—	3	28	no	—
42	C₂×D₄	—	3	84	no	—
43	C₂²×C₄	—	3	7	no	—
44	C₂²×C₄	—	3	7	no	—
45	C₂²×C₄	—	3	84	no	—
46	C₂²×C₄	—	3	28	no	—
47	C₂²×C₄	C₂×A₄	1	14	yes	—
48	C₂²×C₃	—	4	28	no	—
49	C₂²×C₃	—	4	7	no	—
50	C₃×C₄	—	4	8	no	—
51	C₂³	—	3	28	no	—
52	C₂³	—	6	56	no	—
53	C₂³	#41⋊C₃	1	8	yes	direct product factor
54	C₂³	C₂²×A₄	1	7	yes	—
55	Q₈	—	3	7	no	—
56	Q₈	—	3	14	no	—
57	Q₈	C₂⁴×C₃	1	1	yes	characteristic
58	D₄	—	3	8	no	—
59	D₄	—	3	56	no	—
60	C₂×C₄	—	3	21	no	—
61	C₂×C₄	—	3	7	no	—
62	C₂×C₄	—	3	56	no	—
63	C₂×C₄	—	3	8	no	—
64	C₂×C₄	C₂²×A₄	1	28	yes	—
65	C₂×C₃	—	4	14	no	—
66	C₂×C₃	—	4	1	no	—
67	C₂²	—	3	8	no	—
68	C₂²	—	6	56	no	—
69	C₂²	C₂×#41⋊C₃	1	28	yes	direct product factor
70	C₂²	C₂³×A₄	1	7	yes	—
71	C₄	—	3	7	no	—
72	C₄	—	3	1	no	—
73	C₄	C₂³×A₄	1	8	yes	—
74	C₃	—	4	1	no	Sylow
75	C₂	—	6	8	no	—
76	C₂	C₂²×#41⋊C₃	1	14	yes	direct product factor
77	C₂	C₂⁴×A₄	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₄	A₄
commutator, cosocle, hypocenter	Q₈	C₂⁴×C₃
Frattini	C₂	C₂⁴×A₄
Fitting	C₂³×#41	C₃
socle	C₂⁴	C₂×A₄
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#41⋊C₃	C₂	direct (N × H)
2. C₂³×SL(2,𝔽₃)	C₂	semidirect
3. C₂³×#41	C₃	semidirect
5. C₂×#41⋊C₃	C₂²	direct (N × H)
6. C₂²×SL(2,𝔽₃)	C₂²	semidirect
8. C₂²×#41	C₂×C₃	semidirect
14. #41⋊C₃	C₂³	direct (N × H)
15. C₂×SL(2,𝔽₃)	C₂³	semidirect
19. C₂×#41	C₂²×C₃	semidirect
31. SL(2,𝔽₃)	C₂⁴	semidirect
37. #41	C₂³×C₃	semidirect
53. C₂³	#41⋊C₃	direct (N × H)
69. C₂²	C₂×#41⋊C₃	direct (N × H)
76. C₂	C₂²×#41⋊C₃	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×#41⋊C₃	2
C₂³×SL(2,𝔽₃)	2
C₂³×#41	3
C₂³×C₃×C₄	4

Subgroups — C23×#41⋊C3 #87529

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#41⋊C₃ #87529