Subgroups — C₃×#537 #5867

Order 192.

294 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#41	C₂	1	2	yes	maximal, complemented
2	C₃×C₂²⋊Q₈	C₂	1	4	yes	maximal, complemented
3	C₃×#121	C₂	1	4	yes	maximal, complemented
4	C₃×#121	C₂	1	4	yes	maximal, complemented
5	C₂×C₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#537	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₃×#41	—	2	8	no	—
8	C₂×C₃×Q₈	C₂²	1	2	yes	—
9	C₂×C₃×D₄	C₂²	1	4	yes	—
10	C₂×C₃×D₄	C₂²	1	2	yes	—
11	C₂×C₃×D₄	C₂²	1	4	yes	—
12	C₂²×C₃×C₄	C₂²	1	4	yes	—
13	C₂²×C₃×C₄	C₂²	1	2	yes	—
14	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
15	C₃×C₄⋊C₄	C₂²	1	4	yes	—
16	C₃×C₄⋊C₄	C₂²	1	4	yes	—
17	C₃×C₂²⋊C₄	C₂²	1	8	yes	—
18	C₂×#41	C₂×C₃	1	2	yes	complemented
19	C₂²⋊Q₈	C₂×C₃	1	4	yes	complemented
20	#121	C₂×C₃	1	4	yes	complemented
21	#121	C₂×C₃	1	4	yes	complemented
22	C₂×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂³×C₃	C₂³	1	4	yes	—
24	C₂³×C₃	C₂³	1	1	yes	characteristic
25	C₃×Q₈	—	2	4	no	—
26	C₃×D₄	—	2	8	no	—
27	C₃×D₄	—	2	4	no	—
28	C₃×D₄	—	4	4	no	—
29	C₂×C₃×C₄	—	2	8	no	—
30	C₂×C₃×C₄	—	4	2	no	—
31	C₂×C₃×C₄	D₄	1	4	yes	complemented
32	C₂×C₃×C₄	C₂³	1	4	yes	—
33	C₂×C₃×C₄	C₂³	1	4	yes	—
34	C₂×C₃×C₄	C₂³	1	2	yes	—
35	#41	—	2	8	no	—
36	C₂×Q₈	C₂²×C₃	1	2	yes	—
37	C₂×D₄	C₂²×C₃	1	4	yes	—
38	C₂×D₄	C₂²×C₃	1	2	yes	—
39	C₂×D₄	C₂²×C₃	1	4	yes	—
40	C₂²×C₄	C₂²×C₃	1	4	yes	—
41	C₂²×C₄	C₂²×C₃	1	2	yes	—
42	C₂²×C₄	C₂²×C₃	1	1	yes	characteristic
43	C₄⋊C₄	C₂²×C₃	1	4	yes	—
44	C₄⋊C₄	C₂²×C₃	1	4	yes	—
45	C₂²⋊C₄	C₂²×C₃	1	8	yes	—
46	C₂²×C₃	—	2	1	no	—
47	C₂²×C₃	—	2	1	no	—
48	C₂²×C₃	—	2	4	no	—
49	C₂²×C₃	—	2	4	no	—
50	C₂²×C₃	—	2	4	no	—
51	C₂²×C₃	C₂×D₄	1	2	yes	—
52	C₂²×C₃	C₂⁴	1	1	yes	characteristic
53	C₃×C₄	—	2	4	no	—
54	C₃×C₄	—	2	4	no	—
55	C₃×C₄	C₂×D₄	1	4	yes	—
56	C₂³	C₂³×C₃	1	4	yes	—
57	C₂³	C₂³×C₃	1	1	yes	characteristic
58	Q₈	—	2	4	no	—
59	D₄	—	2	8	no	—
60	D₄	—	2	4	no	—
61	D₄	—	4	4	no	—
62	C₂×C₄	—	2	8	no	—
63	C₂×C₄	—	4	2	no	—
64	C₂×C₄	C₃×D₄	1	4	yes	complemented
65	C₂×C₄	C₂³×C₃	1	4	yes	—
66	C₂×C₄	C₂³×C₃	1	4	yes	—
67	C₂×C₄	C₂³×C₃	1	2	yes	—
68	C₂×C₃	—	2	2	no	—
69	C₂×C₃	—	4	4	no	—
70	C₂×C₃	C₂²×D₄	1	1	yes	characteristic
71	C₂×C₃	H(5,𝔽₂)	1	1	yes	characteristic
72	C₂×C₃	2₋¹⁺⁴	1	1	yes	characteristic
73	C₂²	—	2	1	no	—
74	C₂²	—	2	1	no	—
75	C₂²	—	2	4	no	—
76	C₂²	—	2	4	no	—
77	C₂²	—	2	4	no	—
78	C₂²	C₂×C₃×D₄	1	2	yes	—
79	C₂²	C₂⁴×C₃	1	1	yes	characteristic
80	C₄	—	2	4	no	—
81	C₄	—	2	4	no	—
82	C₄	C₂×C₃×D₄	1	4	yes	—
83	C₃	#537	1	1	yes	characteristic, direct product factor, Sylow
84	C₂	—	2	2	no	—
85	C₂	—	4	4	no	—
86	C₂	C₂²×C₃×D₄	1	1	yes	characteristic
87	C₂	C₃×H(5,𝔽₂)	1	1	yes	characteristic
88	C₂	C₃×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#41	C₂	semidirect
2. C₃×C₂²⋊Q₈	C₂	semidirect
3. C₃×#121	C₂	semidirect
4. C₃×#121	C₂	semidirect
5. C₂×C₃×C₄⋊C₄	C₂	semidirect
6. #537	C₃	direct (N × H)
18. C₂×#41	C₂×C₃	semidirect
19. C₂²⋊Q₈	C₂×C₃	semidirect
20. #121	C₂×C₃	semidirect
21. #121	C₂×C₃	semidirect
22. C₂×C₄⋊C₄	C₂×C₃	semidirect
31. C₂×C₃×C₄	D₄	semidirect
64. C₂×C₄	C₃×D₄	semidirect
83. C₃	#537	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×C₃×#41	2
C₃×C₂²⋊Q₈	2
C₃×#121	2
C₂×C₃×C₄⋊C₄	2
#537	3

Subgroups — C3×#537 #5867

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#537 #5867