Subgroups — C₂×C₃×#117 #5844

Order 192.

330 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂²×C₃×C₄	C₂²
commutator	C₂	C₂³×C₃×C₄
Frattini, cosocle	C₂²	C₂⁴×C₃
socle	C₂³×C₃	C₂³
prime radical	C₂⁴×C₃	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×#117	C₂	direct (N × H)
3. C₂×C₃×C₄⋊C₄	C₂	semidirect
5. C₂×C₃×C₄²	C₂	semidirect
6. C₂×#117	C₃	direct (N × H)
8. C₂²×C₃×C₄	C₄	semidirect
13. C₃×C₄⋊C₄	C₂²	semidirect
15. C₃×C₄²	C₂²	semidirect
17. #117	C₂×C₃	direct (N × H)
18. C₂×C₄⋊C₄	C₂×C₃	semidirect
20. C₂×C₄²	C₂×C₃	semidirect
28. C₂×C₃×C₄	C₂×C₄	semidirect
33. C₂²×C₄	C₃×C₄	semidirect
38. C₄⋊C₄	C₂²×C₃	semidirect
40. C₄²	C₂²×C₃	semidirect
57. C₂×C₄	C₂×C₃×C₄	semidirect
62. C₂×C₃	#117	direct (N × H)
74. C₃	C₂×#117	direct (N × H)
76. C₂	C₃×#117	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₄	2
C₃×#117	2
C₂×C₃×C₄⋊C₄	2
C₂×C₃×C₂²⋊C₄	2
C₂×C₃×C₄²	2
C₂×#117	3

Subgroups — C2×C3×#117 #5844

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#117 #5844