Subgroups — #27170748

Order 12928.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265580	C₂	1	4	yes	maximal
2	#20265557	C₂	1	1	yes	maximal, characteristic
3	C₈×#11717579	C₂	1	1	yes	maximal, characteristic
4	#494×C₁₀₁	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₄₀₄	—	2	2	no	—
6	#15692255	C₂²	1	2	yes	—
7	#15692188	—	2	8	no	—
8	C₄×#11717579	C₂²	1	2	yes	—
9	C₂×Q₁₆×C₁₀₁	—	2	2	no	—
10	C₄⋊Q₈×C₁₀₁	C₂²	1	2	yes	—
11	C₄×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
12	C₂×Dic₂₀₂	—	2	2	no	—
13	Dic₄₀₄	—	4	4	no	—
14	C₂×#11717579	D₄	1	4	yes	—
15	C₄⋊Dic₁₀₁	—	2	4	no	—
16	C₂×Q₈×C₁₀₁	—	2	2	no	—
17	Q₁₆×C₁₀₁	—	4	4	no	—
18	C₂×C₈×C₁₀₁	D₄	1	2	yes	—
19	C₄⋊C₄×C₁₀₁	—	2	4	no	—
20	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
21	Q₈×C₁₀₁	—	4	4	no	—
22	C₂×C₄×C₁₀₁	—	4	2	no	—
23	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
24	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
25	C₂×Dic₁₀₁	—	4	2	no	—
26	Dic₂₀₂	—	4	4	no	—
27	C₈×C₁₀₁	D₈	1	4	yes	—
28	#11717579	Q₁₆	1	8	yes	complemented
29	C₂²×C₁₀₁	#127	1	1	yes	characteristic
30	C₄×C₁₀₁	—	8	2	no	—
31	C₄×C₁₀₁	C₂×D₈	1	2	yes	—
32	C₄×C₁₀₁	C₂×Q₁₆	1	4	yes	—
33	Dic₁₀₁	—	8	2	no	—
34	C₂×C₁₀₁	#493	1	1	yes	characteristic
35	C₂×C₁₀₁	#494	1	2	yes	—
36	#1713	—	101	1	no	maximal, Sylow
37	C₁₀₁	#1713	1	1	yes	characteristic, complemented, Sylow
38	#500	—	101	4	no	—
39	#494	—	101	1	no	—
40	#494	D₁₀₁	1	1	yes	characteristic
41	C₈²	—	101	1	no	—
42	C₂×Q₁₆	—	2	2	no	—
43	C₂×Q₁₆	—	202	2	no	—
44	C₄⋊Q₈	—	101	2	no	—
45	C₄⋊Q₈	C₂×D₁₀₁	1	2	yes	—
46	#107	—	202	8	no	—
47	C₄×C₈	—	101	2	no	—
48	C₄×C₈	C₂×D₁₀₁	1	1	yes	characteristic
49	C₂×Q₈	—	2	2	no	—
50	C₂×Q₈	—	202	2	no	—
51	Q₁₆	—	4	4	no	—
52	Q₁₆	—	404	4	no	—
53	C₂×C₈	—	101	4	no	—
54	C₂×C₈	C₂²⋊D₁₀₁	1	2	yes	—
55	C₄⋊C₄	—	2	4	no	—
56	C₄⋊C₄	—	202	4	no	—
57	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
58	Q₈	—	4	4	no	—
59	Q₈	—	404	4	no	—
60	C₂×C₄	—	4	2	no	—
61	C₂×C₄	—	404	2	no	—
62	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
63	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
64	C₈	—	101	8	no	—
65	C₈	#13108059	1	4	yes	—
66	C₂²	#15692251	1	1	yes	characteristic
67	C₄	—	8	2	no	—
68	C₄	—	808	2	no	—
69	C₄	C₂×#13108059	1	2	yes	—
70	C₄	Q₁₆×D₁₀₁	1	4	yes	—
71	C₂	#20265553	1	1	yes	characteristic
72	C₂	#20265555	1	2	yes	—

	subgroup	G/H
center	C₂²	#15692251
commutator, cosocle	C₄²×C₁₀₁	C₂³
Frattini	C₄²	C₂²×D₁₀₁
Fitting	#494×C₁₀₁	C₂
socle, prime radical	C₂²×C₁₀₁	#127
hypercenter	#494	D₁₀₁
hypocenter	C₁₀₁	#1713
LCS	G ▷ C₄²×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

Complements

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

N	H ≅ G/N	type
28. #11717579	Q₁₆	semidirect
37. C₁₀₁	#1713	semidirect

Maximal subgroups

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

subgroup	index
#20265580	2
#20265557	2
C₈×#11717579	2
#494×C₁₀₁	2
#1713	101