Subgroups — #44509698

Order 30848.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₉₂₈	C₂	1	4	yes	maximal
2	#29508097	C₂	1	2	yes	maximal
3	C₄×C₁₆×C₂₄₁	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₉₆₄	—	2	2	no	—
5	C₂×Dic₉₆₄	C₂²	1	4	yes	—
6	#21440642	—	2	2	no	—
7	Dic₁₉₂₈	—	2	8	no	—
8	C₂×C₁₆×C₂₄₁	C₂²	1	2	yes	—
9	C₄×C₈×C₂₄₁	C₂²	1	1	yes	characteristic
10	C₂×Dic₄₈₂	—	2	4	no	—
11	Dic₉₆₄	—	2	8	no	—
12	Dic₉₆₄	—	4	4	no	—
13	C₄⋊Dic₂₄₁	—	4	4	no	—
14	C₂×C₈×C₂₄₁	D₄	1	1	yes	characteristic
15	C₂×C₈×C₂₄₁	C₂³	1	1	yes	characteristic
16	C₄²×C₂₄₁	D₄	1	1	yes	characteristic
17	C₁₆×C₂₄₁	D₄	1	4	yes	—
18	Dic₄₈₂	—	4	4	no	—
19	Dic₄₈₂	—	4	4	no	—
20	C₂×Dic₂₄₁	—	4	4	no	—
21	C₂×C₄×C₂₄₁	D₈	1	2	yes	—
22	C₂×C₄×C₂₄₁	C₂×D₄	1	1	yes	characteristic
23	C₈×C₂₄₁	D₈	1	2	yes	—
24	C₈×C₂₄₁	C₂×D₄	1	2	yes	—
25	C₂²×C₂₄₁	C₂×D₈	1	1	yes	characteristic
26	C₄×C₂₄₁	Q₃₂	1	4	yes	complemented
27	C₄×C₂₄₁	#127	1	1	yes	characteristic
28	C₄×C₂₄₁	C₂×D₈	1	1	yes	characteristic
29	Dic₂₄₁	—	8	4	no	—
30	C₂×C₂₄₁	#493	1	1	yes	characteristic
31	C₂×C₂₄₁	C₂×Q₃₂	1	2	yes	—
32	C₂₄₁	#2247	1	1	yes	characteristic, complemented, Sylow
33	#2247	—	241	1	no	maximal, Sylow
34	C₂×Q₃₂	—	241	4	no	—
35	#494	—	241	2	no	—
36	C₄×C₁₆	D₂₄₁	1	1	yes	characteristic
37	C₂×Q₁₆	—	482	2	no	—
38	C₂×Q₁₆	—	241	4	no	—
39	C₄⋊Q₈	—	482	2	no	—
40	Q₃₂	—	482	8	no	—
41	C₂×C₁₆	C₂×D₂₄₁	1	2	yes	—
42	C₄×C₈	C₂×D₂₄₁	1	1	yes	characteristic
43	C₂×Q₈	—	482	4	no	—
44	Q₁₆	—	964	4	no	—
45	Q₁₆	—	482	8	no	—
46	C₂×C₈	D₉₆₄	1	1	yes	characteristic
47	C₂×C₈	C₂²×D₂₄₁	1	1	yes	characteristic
48	C₄⋊C₄	—	964	4	no	—
49	C₄²	D₉₆₄	1	1	yes	characteristic
50	C₁₆	D₉₆₄	1	4	yes	—
51	Q₈	—	964	4	no	—
52	Q₈	—	964	4	no	—
53	C₂×C₄	—	964	4	no	—
54	C₂×C₄	D₁₉₂₈	1	2	yes	—
55	C₂×C₄	C₂×D₉₆₄	1	1	yes	characteristic
56	C₈	D₁₉₂₈	1	2	yes	—
57	C₈	C₂×D₉₆₄	1	2	yes	—
58	C₂²	C₂×D₁₉₂₈	1	1	yes	characteristic
59	C₄	—	1928	4	no	—
60	C₄	Dic₁₉₂₈	1	4	yes	complemented
61	C₄	#21440639	1	1	yes	characteristic
62	C₄	C₂×D₁₉₂₈	1	1	yes	characteristic
63	C₂	#29508094	1	1	yes	characteristic
64	C₂	C₂×Dic₁₉₂₈	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₉₂₈
commutator, cosocle	C₂×C₈×C₂₄₁	C₂³
Frattini	C₂×C₈	C₂²×D₂₄₁
Fitting	C₄×C₁₆×C₂₄₁	C₂
socle, prime radical	C₂²×C₂₄₁	C₂×D₈
hypercenter	C₄×C₁₆	D₂₄₁
hypocenter	C₂₄₁	#2247
LCS	G ▷ C₂×C₈×C₂₄₁ ▷ C₄×C₂₄₁ ▷ C₂×C₂₄₁ ▷ C₂₄₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
26. C₄×C₂₄₁	Q₃₂	semidirect
32. C₂₄₁	#2247	semidirect
60. C₄	Dic₁₉₂₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₉₂₈	2
#29508097	2
C₄×C₁₆×C₂₄₁	2
#2247	241