Subgroups — C₂×C₃×#135 #5902

Order 192.

266 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×#135	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×SD₁₆	C₂	1	2	yes	maximal, complemented
5	C₂×C₃×D₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₃×C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#135	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₃×#41	—	2	4	no	—
9	C₃×#41	C₂²	1	8	yes	complemented
10	C₂×C₃×Q₈	C₂²	1	2	yes	—
11	C₂×C₃×D₄	—	2	2	no	—
12	C₂×C₃×D₄	C₂²	1	2	yes	—
13	C₂²×C₃×C₄	—	2	2	no	—
14	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
15	C₃×Q₁₆	C₂²	1	4	yes	complemented
16	C₃×SD₁₆	C₂²	1	8	yes	complemented
17	C₃×D₈	C₂²	1	4	yes	complemented
18	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₈	C₂²	1	4	yes	complemented
21	C₂×#41	C₂×C₃	1	2	yes	complemented
22	#135	C₂×C₃	1	8	yes	direct product factor
23	C₂×Q₁₆	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂×SD₁₆	C₂×C₃	1	2	yes	complemented
25	C₂×D₈	C₂×C₃	1	1	yes	characteristic, complemented
26	C₂²×C₈	C₂×C₃	1	1	yes	characteristic, complemented
27	C₂³×C₃	—	2	2	no	—
28	C₂³×C₃	D₄	1	1	yes	characteristic
29	C₃×Q₈	—	2	2	no	—
30	C₃×Q₈	C₂³	1	4	yes	—
31	C₃×D₄	—	2	8	no	—
32	C₃×D₄	—	2	2	no	—
33	C₃×D₄	C₂³	1	4	yes	—
34	C₂×C₃×C₄	—	2	2	no	—
35	C₂×C₃×C₄	—	2	8	no	—
36	C₂×C₃×C₄	D₄	1	2	yes	—
37	C₂×C₃×C₄	D₄	1	1	yes	characteristic
38	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
39	C₂×C₃×C₄	C₂³	1	2	yes	—
40	C₃×C₈	C₂³	1	2	yes	—
41	C₃×C₈	C₂³	1	2	yes	—
42	#41	—	2	4	no	—
43	#41	C₂²×C₃	1	8	yes	complemented
44	C₂×Q₈	C₂²×C₃	1	2	yes	—
45	C₂×D₄	—	2	2	no	—
46	C₂×D₄	C₂²×C₃	1	2	yes	—
47	C₂²×C₄	—	2	2	no	—
48	C₂²×C₄	C₂²×C₃	1	1	yes	characteristic
49	Q₁₆	C₂²×C₃	1	4	yes	complemented
50	SD₁₆	C₂²×C₃	1	8	yes	complemented
51	D₈	C₂²×C₃	1	4	yes	complemented
52	C₂×C₈	C₂²×C₃	1	1	yes	characteristic
53	C₂×C₈	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₈	C₂²×C₃	1	4	yes	complemented
55	C₂²×C₃	—	2	2	no	—
56	C₂²×C₃	—	2	4	no	—
57	C₂²×C₃	—	4	4	no	—
58	C₂²×C₃	C₂×D₄	1	2	yes	—
59	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
60	C₃×C₄	—	2	4	no	—
61	C₃×C₄	C₂×D₄	1	2	yes	—
62	C₃×C₄	C₂×D₄	1	1	yes	characteristic
63	C₃×C₄	C₂⁴	1	1	yes	characteristic
64	C₂³	—	2	2	no	—
65	C₂³	C₃×D₄	1	1	yes	characteristic
66	Q₈	—	2	2	no	—
67	Q₈	C₂³×C₃	1	4	yes	—
68	D₄	—	2	8	no	—
69	D₄	—	2	2	no	—
70	D₄	C₂³×C₃	1	4	yes	—
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	—	2	8	no	—
73	C₂×C₄	C₃×D₄	1	1	yes	characteristic
74	C₂×C₄	C₃×D₄	1	2	yes	—
75	C₂×C₄	C₂³×C₃	1	1	yes	characteristic
76	C₂×C₄	C₂³×C₃	1	2	yes	—
77	C₈	C₂³×C₃	1	2	yes	—
78	C₈	C₂³×C₃	1	2	yes	—
79	C₂×C₃	—	2	2	no	—
80	C₂×C₃	—	4	4	no	—
81	C₂×C₃	#135	1	2	yes	direct product factor
82	C₂×C₃	C₂²×D₄	1	1	yes	characteristic
83	C₂²	—	2	2	no	—
84	C₂²	—	2	4	no	—
85	C₂²	—	4	4	no	—
86	C₂²	C₂×C₃×D₄	1	2	yes	—
87	C₂²	C₂×C₃×D₄	1	1	yes	characteristic
88	C₄	—	2	4	no	—
89	C₄	C₂×C₃×D₄	1	2	yes	—
90	C₄	C₂×C₃×D₄	1	1	yes	characteristic
91	C₄	C₂⁴×C₃	1	1	yes	characteristic
92	C₃	C₂×#135	1	1	yes	characteristic, direct product factor, Sylow
93	C₂	—	2	2	no	—
94	C₂	—	4	4	no	—
95	C₂	C₃×#135	1	2	yes	direct product factor
96	C₂	C₂²×C₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	D₄
commutator, Frattini, cosocle	C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄ ◁ 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×#135	C₂	direct (N × H)
3. C₂×C₃×Q₁₆	C₂	semidirect
4. C₂×C₃×SD₁₆	C₂	semidirect
5. C₂×C₃×D₈	C₂	semidirect
6. C₂²×C₃×C₈	C₂	semidirect
7. C₂×#135	C₃	direct (N × H)
9. C₃×#41	C₂²	semidirect
15. C₃×Q₁₆	C₂²	semidirect
16. C₃×SD₁₆	C₂²	semidirect
17. C₃×D₈	C₂²	semidirect
20. C₂×C₃×C₈	C₂²	semidirect
21. C₂×#41	C₂×C₃	semidirect
22. #135	C₂×C₃	direct (N × H)
23. C₂×Q₁₆	C₂×C₃	semidirect
24. C₂×SD₁₆	C₂×C₃	semidirect
25. C₂×D₈	C₂×C₃	semidirect
26. C₂²×C₈	C₂×C₃	semidirect
43. #41	C₂²×C₃	semidirect
49. Q₁₆	C₂²×C₃	semidirect
50. SD₁₆	C₂²×C₃	semidirect
51. D₈	C₂²×C₃	semidirect
54. C₂×C₈	C₂²×C₃	semidirect
81. C₂×C₃	#135	direct (N × H)
92. C₃	C₂×#135	direct (N × H)
95. C₂	C₃×#135	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×#135	2
C₂×C₃×Q₁₆	2
C₂×C₃×SD₁₆	2
C₂×C₃×D₈	2
C₂²×C₃×C₈	2
C₂×#135	3

Subgroups — C2×C3×#135 #5902

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#135 #5902