Subgroups — C₃×#540 #5870

Order 192.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#127	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#123	C₂	1	4	yes	maximal, complemented
4	C₃×#121	C₂	1	4	yes	maximal, complemented
5	C₃×#121	C₂	1	2	yes	maximal, complemented
6	C₃×C₄×D₄	C₂	1	2	yes	maximal, complemented
7	C₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
8	#540	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂×C₃×D₄	C₂²	1	2	yes	—
10	C₂×C₃×D₄	C₂²	1	4	yes	—
11	C₂×C₃×D₄	C₂²	1	4	yes	—
12	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₄	C₂²	1	4	yes	—
14	C₃×C₄⋊C₄	C₂²	1	2	yes	—
15	C₃×C₄⋊C₄	C₂²	1	2	yes	—
16	C₃×C₄⋊C₄	C₂²	1	4	yes	—
17	C₃×C₂²⋊C₄	C₂²	1	2	yes	—
18	C₃×C₂²⋊C₄	C₂²	1	4	yes	—
19	C₃×C₂²⋊C₄	C₂²	1	4	yes	—
20	C₃×C₄²	C₂²	1	1	yes	characteristic
21	C₃×C₄²	C₂²	1	1	yes	characteristic
22	#127	C₂×C₃	1	1	yes	characteristic, complemented
23	#125	C₂×C₃	1	1	yes	characteristic, complemented
24	#123	C₂×C₃	1	4	yes	complemented
25	#121	C₂×C₃	1	4	yes	complemented
26	#121	C₂×C₃	1	2	yes	complemented
27	C₄×D₄	C₂×C₃	1	2	yes	complemented
28	#117	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂³×C₃	C₂³	1	1	yes	characteristic
30	C₂³×C₃	C₂³	1	4	yes	—
31	C₃×D₄	—	2	4	no	—
32	C₃×D₄	—	4	4	no	—
33	C₃×D₄	—	4	4	no	—
34	C₂×C₃×C₄	—	2	2	no	—
35	C₂×C₃×C₄	—	4	4	no	—
36	C₂×C₃×C₄	C₂³	1	2	yes	—
37	C₂×C₃×C₄	C₂³	1	2	yes	—
38	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
39	C₂×C₃×C₄	C₂³	1	4	yes	—
40	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
41	C₂×D₄	C₂²×C₃	1	2	yes	—
42	C₂×D₄	C₂²×C₃	1	4	yes	—
43	C₂×D₄	C₂²×C₃	1	4	yes	—
44	C₂²×C₄	C₂²×C₃	1	1	yes	characteristic
45	C₂²×C₄	C₂²×C₃	1	4	yes	—
46	C₄⋊C₄	C₂²×C₃	1	2	yes	—
47	C₄⋊C₄	C₂²×C₃	1	2	yes	—
48	C₄⋊C₄	C₂²×C₃	1	4	yes	—
49	C₂²⋊C₄	C₂²×C₃	1	2	yes	—
50	C₂²⋊C₄	C₂²×C₃	1	4	yes	—
51	C₂²⋊C₄	C₂²×C₃	1	4	yes	—
52	C₄²	C₂²×C₃	1	1	yes	characteristic
53	C₄²	C₂²×C₃	1	1	yes	characteristic
54	C₂²×C₃	—	2	1	no	—
55	C₂²×C₃	—	2	4	no	—
56	C₂²×C₃	—	2	4	no	—
57	C₂²×C₃	—	2	2	no	—
58	C₂²×C₃	—	2	4	no	—
59	C₂²×C₃	C₂⁴	1	1	yes	characteristic
60	C₃×C₄	—	2	1	no	—
61	C₃×C₄	—	2	2	no	—
62	C₃×C₄	—	2	4	no	—
63	C₃×C₄	—	2	2	no	—
64	C₃×C₄	#41	1	2	yes	—
65	C₂³	C₂³×C₃	1	1	yes	characteristic
66	C₂³	C₂³×C₃	1	4	yes	—
67	D₄	—	2	4	no	—
68	D₄	—	4	4	no	—
69	D₄	—	4	4	no	—
70	C₂×C₄	—	2	2	no	—
71	C₂×C₄	—	4	4	no	—
72	C₂×C₄	C₂³×C₃	1	2	yes	—
73	C₂×C₄	C₂³×C₃	1	2	yes	—
74	C₂×C₄	C₂³×C₃	1	1	yes	characteristic
75	C₂×C₄	C₂³×C₃	1	4	yes	—
76	C₂×C₄	C₂³×C₃	1	1	yes	characteristic
77	C₂×C₃	—	4	1	no	—
78	C₂×C₃	—	4	4	no	—
79	C₂×C₃	C₂×#41	1	1	yes	characteristic
80	C₂×C₃	H(5,𝔽₂)	1	2	yes	—
81	C₂²	—	2	1	no	—
82	C₂²	—	2	4	no	—
83	C₂²	—	2	4	no	—
84	C₂²	—	2	2	no	—
85	C₂²	—	2	4	no	—
86	C₂²	C₂⁴×C₃	1	1	yes	characteristic
87	C₄	—	2	1	no	—
88	C₄	—	2	2	no	—
89	C₄	—	2	4	no	—
90	C₄	—	2	2	no	—
91	C₄	C₃×#41	1	2	yes	—
92	C₃	#540	1	1	yes	characteristic, direct product factor, Sylow
93	C₂	—	4	1	no	—
94	C₂	—	4	4	no	—
95	C₂	C₂×C₃×#41	1	1	yes	characteristic
96	C₂	C₃×H(5,𝔽₂)	1	2	yes	—

	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. C₃×#127	C₂	semidirect
2. C₃×#125	C₂	semidirect
3. C₃×#123	C₂	semidirect
4. C₃×#121	C₂	semidirect
5. C₃×#121	C₂	semidirect
6. C₃×C₄×D₄	C₂	semidirect
7. C₃×#117	C₂	semidirect
8. #540	C₃	direct (N × H)
22. #127	C₂×C₃	semidirect
23. #125	C₂×C₃	semidirect
24. #123	C₂×C₃	semidirect
25. #121	C₂×C₃	semidirect
26. #121	C₂×C₃	semidirect
27. C₄×D₄	C₂×C₃	semidirect
28. #117	C₂×C₃	semidirect
92. C₃	#540	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#127	2
C₃×#125	2
C₃×#123	2
C₃×#121	2
C₃×C₄×D₄	2
C₃×#117	2
#540	3

Subgroups — C3×#540 #5870

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#540 #5870