Subgroups — #27170801

Order 12928.

198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265580	C₂	1	2	yes	maximal, complemented
2	#20265557	C₂	1	1	yes	maximal, characteristic, complemented
3	#20265523	C₂	1	2	yes	maximal
4	#20264898	C₂	1	1	yes	maximal, characteristic, complemented
5	#492×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₀₄	—	2	2	no	—
7	#15692255	C₂²	1	2	yes	—
8	#15692188	—	2	4	no	—
9	#15692176	—	2	4	no	—
10	C₄×#11717579	C₂²	1	2	yes	—
11	C₂×SD₁₆×C₁₀₁	—	2	2	no	—
12	C₄⋊Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
13	#127×C₁₀₁	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₀₂	—	2	2	no	—
16	Dic₄₀₄	—	4	4	no	—
17	C₂×#11717579	D₄	1	4	yes	—
18	C₄⋊Dic₁₀₁	—	2	2	no	—
19	C₄⋊Dic₁₀₁	—	2	2	no	—
20	C₂×Q₈×C₁₀₁	—	2	1	no	—
21	C₂×D₄×C₁₀₁	—	2	1	no	—
22	C₂×D₄×C₁₀₁	—	2	2	no	—
23	SD₁₆×C₁₀₁	—	4	4	no	—
24	C₂×C₈×C₁₀₁	D₄	1	2	yes	—
25	C₄⋊C₄×C₁₀₁	—	2	2	no	—
26	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
27	C₂³×C₁₀₁	—	4	1	no	—
28	Q₈×C₁₀₁	—	4	2	no	—
29	D₄×C₁₀₁	—	4	2	no	—
30	D₄×C₁₀₁	—	4	4	no	—
31	C₂×C₄×C₁₀₁	—	4	1	no	—
32	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
33	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
34	C₂×Dic₁₀₁	—	4	2	no	—
35	Dic₂₀₂	—	4	4	no	—
36	C₈×C₁₀₁	D₈	1	4	yes	—
37	#11717579	—	2	4	no	—
38	C₂²×C₁₀₁	—	8	1	no	—
39	C₂²×C₁₀₁	—	8	2	no	—
40	C₂²×C₁₀₁	#127	1	1	yes	characteristic
41	C₄×C₁₀₁	—	8	1	no	—
42	C₄×C₁₀₁	C₂×D₈	1	2	yes	—
43	C₄×C₁₀₁	#137	1	4	yes	—
44	Dic₁₀₁	—	8	2	no	—
45	C₂×C₁₀₁	—	16	1	no	—
46	C₂×C₁₀₁	#493	1	1	yes	characteristic
47	C₂×C₁₀₁	#497	1	2	yes	—
48	#1722	—	101	1	no	maximal, Sylow
49	C₁₀₁	#1722	1	1	yes	characteristic, complemented, Sylow
50	#500	—	101	2	no	—
51	#494	—	101	1	no	—
52	#492	D₁₀₁	1	1	yes	characteristic
53	#486	—	101	2	no	—
54	#322	—	101	1	no	—
55	C₂×Q₁₆	—	202	2	no	—
56	C₂×SD₁₆	—	2	2	no	—
57	C₄⋊Q₈	—	101	2	no	—
58	C₄⋊Q₈	C₂×D₁₀₁	1	1	yes	characteristic
59	#127	C₂×D₁₀₁	1	1	yes	characteristic
60	#107	—	202	4	no	—
61	#102	—	202	4	no	—
62	C₄×C₈	—	101	2	no	—
63	C₄×C₈	C₂×D₁₀₁	1	1	yes	characteristic
64	C₂×Q₈	—	2	1	no	—
65	C₂×Q₈	—	202	2	no	—
66	C₂×D₄	—	2	2	no	—
67	C₂×D₄	—	2	1	no	—
68	Q₁₆	—	404	4	no	—
69	SD₁₆	—	4	4	no	—
70	C₂×C₈	—	101	4	no	—
71	C₂×C₈	C₂²⋊D₁₀₁	1	2	yes	—
72	C₄⋊C₄	—	2	2	no	—
73	C₄⋊C₄	—	202	2	no	—
74	C₄⋊C₄	—	202	2	no	—
75	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
76	C₂³	—	4	1	no	—
77	Q₈	—	4	2	no	—
78	Q₈	—	404	4	no	—
79	D₄	—	4	2	no	—
80	D₄	—	4	4	no	—
81	C₂×C₄	—	4	1	no	—
82	C₂×C₄	—	404	2	no	—
83	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
84	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
85	C₈	—	202	4	no	—
86	C₈	#13108059	1	4	yes	—
87	C₂²	—	8	1	no	—
88	C₂²	—	8	2	no	—
89	C₂²	#15692251	1	1	yes	characteristic
90	C₄	—	8	1	no	—
91	C₄	—	808	2	no	—
92	C₄	C₂×#13108059	1	2	yes	—
93	C₄	#15692284	1	4	yes	—
94	C₂	—	16	1	no	—
95	C₂	#20265553	1	1	yes	characteristic
96	C₂	#20265568	1	2	yes	—

	subgroup	G/H
center	C₂²	#15692251
commutator, cosocle	C₄²×C₁₀₁	C₂³
Frattini	C₄²	C₂²×D₁₀₁
Fitting	#492×C₁₀₁	C₂
socle	C₂²×C₁₀₁	#127
prime radical	#127×C₁₀₁	C₂²
hypercenter	#492	D₁₀₁
hypocenter	C₁₀₁	#1722
LCS	G ▷ C₄²×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

Complements

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

N	H ≅ G/N	type
1. #20265580	C₂	semidirect
2. #20265557	C₂	semidirect
4. #20264898	C₂	semidirect
49. C₁₀₁	#1722	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
#20265523	2
#20264898	2
#492×C₁₀₁	2
#1722	101