Subgroups — C₂²×Dic₂₄₈ #16858824

Order 3968.

360 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₁₂₄	C₂	1	2	yes	maximal
2	C₂×Dic₂₄₈	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₆₂	—	2	2	no	—
5	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
6	C₂×Dic₁₂₄	—	2	6	no	—
7	C₂×Dic₁₂₄	C₂²	1	12	yes	—
8	C₂×C₁₆×C₃₁	C₂²	1	6	yes	—
9	Dic₂₄₈	C₂²	1	16	yes	direct product factor
10	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
11	C₂²×Dic₃₁	—	4	2	no	—
12	C₂×Dic₆₂	—	2	12	no	—
13	C₂×Dic₆₂	—	4	6	no	—
14	C₂×C₈×C₃₁	D₄	1	3	yes	—
15	C₂×C₈×C₃₁	C₂³	1	3	yes	—
16	Dic₁₂₄	—	2	12	no	—
17	Dic₁₂₄	C₂³	1	8	yes	—
18	C₁₆×C₃₁	C₂³	1	4	yes	—
19	C₂³×C₃₁	D₈	1	1	yes	characteristic
20	C₂×C₄×C₃₁	D₈	1	3	yes	—
21	C₂×C₄×C₃₁	C₂×D₄	1	3	yes	—
22	C₂×Dic₃₁	—	4	12	no	—
23	Dic₆₂	—	2	8	no	—
24	Dic₆₂	—	4	12	no	—
25	C₈×C₃₁	C₂×D₄	1	3	yes	—
26	C₈×C₃₁	C₂⁴	1	1	yes	characteristic
27	C₂²×Q₃₂	—	31	1	no	maximal, Sylow
28	C₂²×C₃₁	Q₃₂	1	4	yes	complemented
29	C₂²×C₃₁	C₂×D₈	1	3	yes	—
30	C₄×C₃₁	C₂×D₈	1	3	yes	—
31	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
32	Dic₃₁	—	4	8	no	—
33	C₂²×Q₁₆	—	31	2	no	—
34	C₂×Q₃₂	—	31	12	no	—
35	C₂²×C₁₆	D₃₁	1	1	yes	characteristic
36	C₂×C₃₁	C₂×Q₃₂	1	6	yes	complemented
37	C₂×C₃₁	C₂²×D₈	1	1	yes	characteristic
38	C₂²×Q₈	—	62	2	no	—
39	C₂×Q₁₆	—	31	12	no	—
40	C₂×Q₁₆	—	62	6	no	—
41	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
42	Q₃₂	—	31	16	no	—
43	C₂×C₁₆	C₂×D₃₁	1	6	yes	—
44	C₃₁	C₂²×Q₃₂	1	1	yes	characteristic, complemented, Sylow
45	C₂×Q₈	—	62	12	no	—
46	C₂×Q₈	—	124	6	no	—
47	C₂²×C₄	—	124	2	no	—
48	C₂²×C₄	D₁₂₄	1	1	yes	characteristic
49	Q₁₆	—	31	8	no	—
50	Q₁₆	—	62	12	no	—
51	C₂×C₈	D₁₂₄	1	3	yes	—
52	C₂×C₈	C₂²×D₃₁	1	3	yes	—
53	C₁₆	C₂²×D₃₁	1	4	yes	—
54	C₂³	D₂₄₈	1	1	yes	characteristic
55	Q₈	—	62	8	no	—
56	Q₈	—	124	12	no	—
57	C₂×C₄	—	124	12	no	—
58	C₂×C₄	D₂₄₈	1	3	yes	—
59	C₂×C₄	C₂×D₁₂₄	1	3	yes	—
60	C₈	C₂×D₁₂₄	1	3	yes	—
61	C₈	C₂³×D₃₁	1	1	yes	characteristic
62	C₂²	Dic₂₄₈	1	4	yes	direct product factor
63	C₂²	C₂×D₂₄₈	1	3	yes	—
64	C₄	—	124	8	no	—
65	C₄	C₂×D₂₄₈	1	3	yes	—
66	C₄	C₂²×D₁₂₄	1	1	yes	characteristic
67	C₂	C₂×Dic₂₄₈	1	6	yes	direct product factor
68	C₂	C₂²×D₂₄₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₂₄₈
commutator, cosocle	C₈×C₃₁	C₂⁴
Frattini	C₈	C₂³×D₃₁
Fitting	C₂²×C₁₆×C₃₁	C₂
socle, prime radical	C₂³×C₃₁	D₈
hypercenter	C₂²×C₁₆	D₃₁
hypocenter	C₃₁	C₂²×Q₃₂
LCS	G ▷ C₈×C₃₁ ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₂₄₈	C₂	direct (N × H)
9. Dic₂₄₈	C₂²	direct (N × H)
28. C₂²×C₃₁	Q₃₂	semidirect
36. C₂×C₃₁	C₂×Q₃₂	semidirect
44. C₃₁	C₂²×Q₃₂	semidirect
62. C₂²	Dic₂₄₈	direct (N × H)
67. C₂	C₂×Dic₂₄₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Dic₁₂₄	2
C₂×Dic₂₄₈	2
C₂²×C₁₆×C₃₁	2
C₂²×Q₃₂	31

Subgroups — C22×Dic248 #16858824

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Dic₂₄₈ #16858824