Subgroups — C₃×#1453 #72168

Order 384.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#431	C₂	1	2	yes	maximal
2	C₃×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#322	C₂	1	2	yes	maximal, complemented
5	C₃×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
6	#1453	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂×C₃×M₁₆	C₄	1	2	yes	—
8	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
9	C₃×C₄⋊C₈	C₄	1	4	yes	—
10	C₃×C₂²⋊C₈	C₄	1	4	yes	—
11	C₃×#97	C₄	1	2	yes	—
12	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₃×C₄×C₈	C₂²	1	2	yes	—
15	C₃×C₄×C₈	C₂²	1	2	yes	—
16	#431	C₂×C₃	1	2	yes	—
17	C₄×M₁₆	C₂×C₃	1	1	yes	characteristic
18	#322	C₂×C₃	1	1	yes	characteristic, complemented
19	#322	C₂×C₃	1	2	yes	complemented
20	C₈²	C₂×C₃	1	1	yes	characteristic, complemented
21	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
22	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
23	C₃×M₁₆	—	2	4	no	—
24	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
25	C₂×C₃×C₈	C₂×C₄	1	4	yes	—
26	C₂×C₃×C₈	C₂×C₄	1	4	yes	—
27	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
28	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
29	C₃×C₄²	C₂×C₄	1	2	yes	—
30	C₃×C₄²	C₂³	1	1	yes	characteristic
31	C₂×M₁₆	C₃×C₄	1	2	yes	—
32	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
33	C₄⋊C₈	C₃×C₄	1	4	yes	—
34	C₂²⋊C₈	C₃×C₄	1	4	yes	—
35	#97	C₃×C₄	1	2	yes	—
36	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
37	C₄×C₈	C₂²×C₃	1	2	yes	—
38	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
39	C₄×C₈	C₂²×C₃	1	2	yes	—
40	C₂³×C₃	C₄²	1	1	yes	characteristic
41	C₂×C₃×C₄	—	2	2	no	—
42	C₂×C₃×C₄	—	2	4	no	—
43	C₂×C₃×C₄	C₄²	1	1	yes	characteristic
44	C₂×C₃×C₄	C₄²	1	2	yes	—
45	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
47	C₃×C₈	—	2	2	no	—
48	C₃×C₈	—	2	4	no	—
49	C₃×C₈	—	2	4	no	—
50	C₃×C₈	M₁₆	1	4	yes	—
51	C₂²×C₄	C₂×C₃×C₄	1	2	yes	—
52	C₂²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
53	M₁₆	—	2	4	no	—
54	C₂×C₈	C₂×C₃×C₄	1	2	yes	—
55	C₂×C₈	C₂×C₃×C₄	1	4	yes	—
56	C₂×C₈	C₂×C₃×C₄	1	4	yes	—
57	C₂×C₈	C₂×C₃×C₄	1	2	yes	—
58	C₄²	C₂×C₃×C₄	1	1	yes	characteristic
59	C₄²	C₂×C₃×C₄	1	2	yes	—
60	C₄²	C₂³×C₃	1	1	yes	characteristic
61	C₂²×C₃	—	2	1	no	—
62	C₂²×C₃	—	2	2	no	—
63	C₂²×C₃	C₂×C₄²	1	1	yes	characteristic
64	C₃×C₄	—	2	1	no	—
65	C₃×C₄	—	2	2	no	—
66	C₃×C₄	C₂×M₁₆	1	2	yes	—
67	C₃×C₄	#131	1	4	yes	—
68	C₂³	C₃×C₄²	1	1	yes	characteristic
69	C₂×C₄	—	2	2	no	—
70	C₂×C₄	—	2	4	no	—
71	C₂×C₄	C₃×C₄²	1	1	yes	characteristic
72	C₂×C₄	C₃×C₄²	1	2	yes	—
73	C₂×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
74	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
75	C₈	—	2	2	no	—
76	C₈	—	2	4	no	—
77	C₈	—	2	4	no	—
78	C₈	C₃×M₁₆	1	4	yes	—
79	C₂×C₃	—	4	1	no	—
80	C₂×C₃	C₄×M₁₆	1	1	yes	characteristic
81	C₂×C₃	#405	1	2	yes	—
82	C₂²	—	2	1	no	—
83	C₂²	—	2	2	no	—
84	C₂²	C₂×C₃×C₄²	1	1	yes	characteristic
85	C₄	—	2	1	no	—
86	C₄	—	2	2	no	—
87	C₄	C₂×C₃×M₁₆	1	2	yes	—
88	C₄	C₃×#131	1	4	yes	—
89	C₃	#1453	1	1	yes	characteristic, direct product factor, Sylow
90	C₂	—	4	1	no	—
91	C₂	C₃×C₄×M₁₆	1	1	yes	characteristic
92	C₂	C₃×#405	1	2	yes	—

	subgroup	G/H
center	C₃×C₄²	C₂³
commutator	C₂²	C₂×C₃×C₄²
Frattini, cosocle	C₄²	C₂³×C₃
socle	C₂²×C₃	C₂×C₄²
prime radical	C₂³×C₃	C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×#322	C₂	semidirect
4. C₃×#322	C₂	semidirect
5. C₃×C₈²	C₂	semidirect
6. #1453	C₃	direct (N × H)
18. #322	C₂×C₃	semidirect
19. #322	C₂×C₃	semidirect
20. C₈²	C₂×C₃	semidirect
89. C₃	#1453	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×#431	2
C₃×C₄×M₁₆	2
C₃×#322	2
C₃×C₈²	2
#1453	3

Subgroups — C3×#1453 #72168

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#1453 #72168