Subgroups — #27173385

Order 12928.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₈₀₈	C₂	1	4	yes	maximal
2	#20265557	C₂	1	2	yes	maximal
3	C₄×C₁₆×C₁₀₁	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₄₀₄	—	2	2	no	—
5	C₂×Dic₄₀₄	C₂²	1	4	yes	—
6	#15692255	—	2	2	no	—
7	Dic₈₀₈	—	2	8	no	—
8	C₂×C₁₆×C₁₀₁	C₂²	1	2	yes	—
9	C₄×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
10	C₂×Dic₂₀₂	—	2	4	no	—
11	Dic₄₀₄	—	2	8	no	—
12	Dic₄₀₄	—	4	4	no	—
13	C₄⋊Dic₁₀₁	—	4	4	no	—
14	C₂×C₈×C₁₀₁	D₄	1	1	yes	characteristic
15	C₂×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
16	C₄²×C₁₀₁	D₄	1	1	yes	characteristic
17	C₁₆×C₁₀₁	D₄	1	4	yes	—
18	C₂×C₄×C₁₀₁	D₈	1	2	yes	—
19	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
20	C₂×Dic₁₀₁	—	4	4	no	—
21	Dic₂₀₂	—	4	4	no	—
22	Dic₂₀₂	—	4	4	no	—
23	C₈×C₁₀₁	D₈	1	2	yes	—
24	C₈×C₁₀₁	C₂×D₄	1	2	yes	—
25	C₂²×C₁₀₁	C₂×D₈	1	1	yes	characteristic
26	C₄×C₁₀₁	Q₃₂	1	4	yes	complemented
27	C₄×C₁₀₁	#127	1	1	yes	characteristic
28	C₄×C₁₀₁	C₂×D₈	1	1	yes	characteristic
29	Dic₁₀₁	—	8	4	no	—
30	C₂×C₁₀₁	#493	1	1	yes	characteristic
31	C₂×C₁₀₁	C₂×Q₃₂	1	2	yes	—
32	#2247	—	101	1	no	maximal, Sylow
33	C₁₀₁	#2247	1	1	yes	characteristic, complemented, Sylow
34	C₂×Q₃₂	—	101	4	no	—
35	#494	—	101	2	no	—
36	C₄×C₁₆	D₁₀₁	1	1	yes	characteristic
37	C₂×Q₁₆	—	101	4	no	—
38	C₂×Q₁₆	—	202	2	no	—
39	C₄⋊Q₈	—	202	2	no	—
40	Q₃₂	—	202	8	no	—
41	C₂×C₁₆	C₂×D₁₀₁	1	2	yes	—
42	C₄×C₈	C₂×D₁₀₁	1	1	yes	characteristic
43	C₂×Q₈	—	202	4	no	—
44	Q₁₆	—	404	4	no	—
45	Q₁₆	—	202	8	no	—
46	C₂×C₈	D₄₀₄	1	1	yes	characteristic
47	C₂×C₈	C₂²×D₁₀₁	1	1	yes	characteristic
48	C₄⋊C₄	—	404	4	no	—
49	C₄²	D₄₀₄	1	1	yes	characteristic
50	C₁₆	D₄₀₄	1	4	yes	—
51	Q₈	—	404	4	no	—
52	Q₈	—	404	4	no	—
53	C₂×C₄	—	404	4	no	—
54	C₂×C₄	D₈₀₈	1	2	yes	—
55	C₂×C₄	C₂×D₄₀₄	1	1	yes	characteristic
56	C₈	D₈₀₈	1	2	yes	—
57	C₈	C₂×D₄₀₄	1	2	yes	—
58	C₂²	C₂×D₈₀₈	1	1	yes	characteristic
59	C₄	—	808	4	no	—
60	C₄	Dic₈₀₈	1	4	yes	complemented
61	C₄	#15692252	1	1	yes	characteristic
62	C₄	C₂×D₈₀₈	1	1	yes	characteristic
63	C₂	#20265554	1	1	yes	characteristic
64	C₂	C₂×Dic₈₀₈	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈₀₈
commutator, cosocle	C₂×C₈×C₁₀₁	C₂³
Frattini	C₂×C₈	C₂²×D₁₀₁
Fitting	C₄×C₁₆×C₁₀₁	C₂
socle, prime radical	C₂²×C₁₀₁	C₂×D₈
hypercenter	C₄×C₁₆	D₁₀₁
hypocenter	C₁₀₁	#2247
LCS	G ▷ C₂×C₈×C₁₀₁ ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
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×C₁₀₁	Q₃₂	semidirect
33. C₁₀₁	#2247	semidirect
60. C₄	Dic₈₀₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₈₀₈	2
#20265557	2
C₄×C₁₆×C₁₀₁	2
#2247	101