Subgroups — C₃×C₄×C₂²⋊C₄ #5251

Order 192.

258 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	C₂×C₃×C₂²⋊C₄	C₂	1	2	yes	maximal
3	C₂×C₃×C₄²	C₂	1	2	yes	maximal, complemented
4	C₃×#95	C₂	1	2	yes	maximal, complemented
5	C₄×C₂²⋊C₄	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂⁴×C₃	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₄	—	2	4	no	—
8	C₂²×C₃×C₄	C₄	1	4	yes	complemented
9	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₄	C₂²	1	4	yes	—
11	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
12	C₃×C₂²⋊C₄	C₄	1	8	yes	direct product factor
13	C₃×C₄²	—	2	4	no	—
14	C₂³×C₄	C₂×C₃	1	1	yes	characteristic
15	C₂×C₂²⋊C₄	C₂×C₃	1	2	yes	—
16	C₂×C₄²	C₂×C₃	1	2	yes	complemented
17	#95	C₂×C₃	1	2	yes	complemented
18	C₂³×C₃	—	2	2	no	—
19	C₂³×C₃	—	2	2	no	—
20	C₂³×C₃	C₂×C₄	1	4	yes	—
21	C₂³×C₃	C₂×C₄	1	2	yes	—
22	C₂³×C₃	C₂³	1	1	yes	characteristic
23	C₂×C₃×C₄	—	2	4	no	—
24	C₂×C₃×C₄	—	2	2	no	—
25	C₂×C₃×C₄	—	2	4	no	—
26	C₂×C₃×C₄	—	2	8	no	—
27	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₄	C₂×C₄	1	8	yes	complemented
30	C₂×C₃×C₄	D₄	1	4	yes	—
31	C₂⁴	C₂²×C₃	1	1	yes	characteristic
32	C₂²×C₄	—	2	4	no	—
33	C₂²×C₄	C₃×C₄	1	4	yes	complemented
34	C₂²×C₄	C₂²×C₃	1	1	yes	characteristic
35	C₂²×C₄	C₂²×C₃	1	4	yes	—
36	C₂²×C₄	C₂²×C₃	1	1	yes	characteristic
37	C₂²⋊C₄	C₃×C₄	1	8	yes	direct product factor
38	C₄²	—	2	4	no	—
39	C₂²×C₃	—	2	2	no	—
40	C₂²×C₃	—	2	2	no	—
41	C₂²×C₃	—	2	8	no	—
42	C₂²×C₃	C₄²	1	4	yes	complemented
43	C₂²×C₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₃	C₂²×C₄	1	2	yes	—
45	C₂²×C₃	C₂×D₄	1	2	yes	—
46	C₂²×C₃	#41	1	2	yes	—
47	C₃×C₄	—	2	2	no	—
48	C₃×C₄	—	2	8	no	—
49	C₃×C₄	C₂²⋊C₄	1	4	yes	direct product factor
50	C₂³	—	2	2	no	—
51	C₂³	—	2	2	no	—
52	C₂³	C₂×C₃×C₄	1	4	yes	—
53	C₂³	C₂×C₃×C₄	1	2	yes	—
54	C₂³	C₂³×C₃	1	1	yes	characteristic
55	C₂×C₄	—	2	4	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	2	4	no	—
58	C₂×C₄	—	2	8	no	—
59	C₂×C₄	C₂×C₃×C₄	1	2	yes	—
60	C₂×C₄	C₂×C₃×C₄	1	2	yes	—
61	C₂×C₄	C₂×C₃×C₄	1	8	yes	complemented
62	C₂×C₄	C₃×D₄	1	4	yes	—
63	C₂×C₃	—	2	4	no	—
64	C₂×C₃	C₂×C₄²	1	1	yes	characteristic
65	C₂×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₃	#117	1	1	yes	characteristic
67	C₂×C₃	C₄×D₄	1	4	yes	—
68	C₂²	—	2	2	no	—
69	C₂²	—	2	2	no	—
70	C₂²	—	2	8	no	—
71	C₂²	C₃×C₄²	1	4	yes	complemented
72	C₂²	C₂²×C₃×C₄	1	1	yes	characteristic
73	C₂²	C₂²×C₃×C₄	1	2	yes	—
74	C₂²	C₂×C₃×D₄	1	2	yes	—
75	C₂²	C₃×#41	1	2	yes	—
76	C₄	—	2	2	no	—
77	C₄	—	2	8	no	—
78	C₄	C₃×C₂²⋊C₄	1	4	yes	direct product factor
79	C₃	C₄×C₂²⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
80	C₂	—	2	4	no	—
81	C₂	C₂×C₃×C₄²	1	1	yes	characteristic
82	C₂	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
83	C₂	C₃×#117	1	1	yes	characteristic
84	C₂	C₃×C₄×D₄	1	4	yes	—

	subgroup	G/H
center	C₂²×C₃×C₄	C₂²
commutator	C₂	C₂×C₃×C₄²
Frattini, cosocle	C₂³	C₂³×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
3. C₂×C₃×C₄²	C₂	semidirect
4. C₃×#95	C₂	semidirect
5. C₄×C₂²⋊C₄	C₃	direct (N × H)
8. C₂²×C₃×C₄	C₄	semidirect
12. C₃×C₂²⋊C₄	C₄	direct (N × H)
16. C₂×C₄²	C₂×C₃	semidirect
17. #95	C₂×C₃	semidirect
29. C₂×C₃×C₄	C₂×C₄	semidirect
33. C₂²×C₄	C₃×C₄	semidirect
37. C₂²⋊C₄	C₃×C₄	direct (N × H)
42. C₂²×C₃	C₄²	semidirect
49. C₃×C₄	C₂²⋊C₄	direct (N × H)
61. C₂×C₄	C₂×C₃×C₄	semidirect
71. C₂²	C₃×C₄²	semidirect
78. C₄	C₃×C₂²⋊C₄	direct (N × H)
79. C₃	C₄×C₂²⋊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
C₂³×C₃×C₄	2
C₂×C₃×C₂²⋊C₄	2
C₂×C₃×C₄²	2
C₃×#95	2
C₄×C₂²⋊C₄	3

Subgroups — C3×C4×C22⋊C4 #5251

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×C₂²⋊C₄ #5251