Subgroups — C₃×C₄×C₄⋊C₄ #5252

Order 192.

194 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄⋊C₄	C₂	1	2	yes	maximal
2	C₂×C₃×C₄²	C₂	1	2	yes	maximal
3	C₂×C₃×C₄²	C₂	1	1	yes	maximal, characteristic
4	C₃×#95	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₄	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₄	C₂²	1	4	yes	—
8	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
10	C₃×C₄⋊C₄	C₄	1	8	yes	direct product factor
11	C₃×C₄²	—	2	4	no	—
12	C₃×C₄²	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄	C₂×C₃	1	2	yes	—
14	C₂×C₄²	C₂×C₃	1	2	yes	—
15	C₂×C₄²	C₂×C₃	1	1	yes	characteristic
16	#95	C₂×C₃	1	2	yes	—
17	C₂³×C₃	C₂³	1	1	yes	characteristic
18	C₂×C₃×C₄	—	2	4	no	—
19	C₂×C₃×C₄	—	2	2	no	—
20	C₂×C₃×C₄	—	2	4	no	—
21	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
22	C₂×C₃×C₄	C₂×C₄	1	8	yes	—
23	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
24	C₂×C₃×C₄	C₂×C₄	1	4	yes	—
25	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
26	C₂×C₃×C₄	D₄	1	2	yes	—
27	C₂×C₃×C₄	Q₈	1	2	yes	—
28	C₂²×C₄	C₂²×C₃	1	1	yes	characteristic
29	C₂²×C₄	C₂²×C₃	1	4	yes	—
30	C₂²×C₄	C₂²×C₃	1	1	yes	characteristic
31	C₂²×C₄	C₂²×C₃	1	1	yes	characteristic
32	C₄⋊C₄	C₃×C₄	1	8	yes	direct product factor
33	C₄²	—	2	4	no	—
34	C₄²	C₃×C₄	1	4	yes	complemented
35	C₂²×C₃	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₃	C₂²×C₄	1	2	yes	—
37	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₃	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₃	#41	1	2	yes	—
40	C₃×C₄	—	2	2	no	—
41	C₃×C₄	—	2	8	no	—
42	C₃×C₄	C₄²	1	4	yes	complemented
43	C₃×C₄	C₄⋊C₄	1	4	yes	direct product factor
44	C₂³	C₂³×C₃	1	1	yes	characteristic
45	C₂×C₄	—	2	4	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	2	4	no	—
48	C₂×C₄	C₂×C₃×C₄	1	2	yes	—
49	C₂×C₄	C₂×C₃×C₄	1	8	yes	—
50	C₂×C₄	C₂×C₃×C₄	1	2	yes	—
51	C₂×C₄	C₂×C₃×C₄	1	4	yes	—
52	C₂×C₄	C₂×C₃×C₄	1	2	yes	—
53	C₂×C₄	C₃×D₄	1	2	yes	—
54	C₂×C₄	C₃×Q₈	1	2	yes	—
55	C₂×C₃	C₂×C₄²	1	1	yes	characteristic
56	C₂×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₃	#117	1	1	yes	characteristic
58	C₂×C₃	C₄×D₄	1	2	yes	—
59	C₂×C₃	C₄×Q₈	1	2	yes	—
60	C₂²	C₂²×C₃×C₄	1	1	yes	characteristic
61	C₂²	C₂²×C₃×C₄	1	2	yes	—
62	C₂²	C₂×C₃×D₄	1	1	yes	characteristic
63	C₂²	C₂×C₃×Q₈	1	1	yes	characteristic
64	C₂²	C₃×#41	1	2	yes	—
65	C₄	—	2	2	no	—
66	C₄	—	2	8	no	—
67	C₄	C₃×C₄²	1	4	yes	complemented
68	C₄	C₃×C₄⋊C₄	1	4	yes	direct product factor
69	C₃	C₄×C₄⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
70	C₂	C₂×C₃×C₄²	1	1	yes	characteristic
71	C₂	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
72	C₂	C₃×#117	1	1	yes	characteristic
73	C₂	C₃×C₄×D₄	1	2	yes	—
74	C₂	C₃×C₄×Q₈	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃×C₄	C₂²
commutator	C₂	C₂×C₃×C₄²
Frattini, cosocle	C₂³	C₂³×C₃
socle, prime radical	C₂³×C₃	C₂³
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
5. C₄×C₄⋊C₄	C₃	direct (N × H)
10. C₃×C₄⋊C₄	C₄	direct (N × H)
12. C₃×C₄²	C₄	semidirect
32. C₄⋊C₄	C₃×C₄	direct (N × H)
34. C₄²	C₃×C₄	semidirect
42. C₃×C₄	C₄²	semidirect
43. C₃×C₄	C₄⋊C₄	direct (N × H)
67. C₄	C₃×C₄²	semidirect
68. C₄	C₃×C₄⋊C₄	direct (N × H)
69. C₃	C₄×C₄⋊C₄	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×C₄⋊C₄	2
C₂×C₃×C₄²	2
C₃×#95	2
C₄×C₄⋊C₄	3

Subgroups — C3×C4×C4⋊C4 #5252

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×C₄⋊C₄ #5252