Subgroups — #27169172

Order 12928.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265550	C₂	1	1	yes	maximal, characteristic
2	#20264998	C₂	1	1	yes	maximal, characteristic, complemented
3	#364×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13108063	—	2	1	no	—
5	C₂×#13108063	—	2	1	no	—
6	#15692255	C₄	1	1	yes	characteristic
7	#15692252	C₄	1	1	yes	characteristic
8	#15692195	—	2	1	no	—
9	M₃₂×C₁₀₁	—	2	1	no	—
10	C₄×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
11	C₂×Dic₂₀₂	—	2	1	no	—
12	C₂×D₄₀₄	—	2	1	no	—
13	C₂×D₄₀₄	—	4	1	no	—
14	#13108063	—	4	1	no	—
15	#13108063	—	4	1	no	—
16	#13108063	—	4	1	no	—
17	#13108063	—	4	1	no	—
18	C₄⋊Dic₁₀₁	—	4	1	no	—
19	#13108049	—	4	1	no	—
20	C₂×C₈×C₁₀₁	D₄	1	1	yes	characteristic
21	C₂×C₈×C₁₀₁	D₄	1	1	yes	characteristic
22	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
23	C₁₆×C₁₀₁	—	4	1	no	—
24	C₂²×D₁₀₁	—	4	1	no	—
25	C₂×C₄×C₁₀₁	—	2	1	no	—
26	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×Dic₁₀₁	—	4	1	no	—
28	D₄₀₄	—	4	1	no	—
29	D₄₀₄	—	4	1	no	—
30	D₄₀₄	—	8	2	no	—
31	Dic₂₀₂	—	4	1	no	—
32	Dic₂₀₂	—	4	1	no	—
33	C₈×C₁₀₁	D₈	1	1	yes	characteristic
34	C₈×C₁₀₁	D₈	1	1	yes	characteristic
35	C₈×C₁₀₁	SD₁₆	1	1	yes	characteristic
36	C₈×C₁₀₁	SD₁₆	1	1	yes	characteristic
37	C₂²×C₁₀₁	#100	1	1	yes	characteristic
38	C₂×D₁₀₁	—	8	2	no	—
39	C₂×D₁₀₁	—	8	1	no	—
40	C₄×C₁₀₁	—	2	2	no	—
41	C₄×C₁₀₁	#102	1	1	yes	characteristic
42	C₄×C₁₀₁	#102	1	1	yes	characteristic
43	Dic₁₀₁	—	8	1	no	—
44	C₂×C₁₀₁	—	2	1	no	—
45	C₂×C₁₀₁	#331	1	1	yes	characteristic
46	D₁₀₁	—	16	1	no	—
47	#1359	—	101	1	no	maximal, Sylow
48	C₁₀₁	#1359	1	1	yes	characteristic, complemented, Sylow
49	#492	—	101	1	no	—
50	#364	—	101	1	no	—
51	#364	D₁₀₁	1	1	yes	characteristic, complemented
52	C₂×SD₁₆	—	202	1	no	—
53	C₂×SD₁₆	—	202	1	no	—
54	C₄⋊Q₈	—	101	1	no	—
55	#127	—	101	1	no	—
56	M₃₂	—	2	1	no	—
57	M₃₂	—	202	1	no	—
58	C₄×C₈	C₂×D₁₀₁	1	1	yes	characteristic
59	C₂×Q₈	—	202	1	no	—
60	C₂×D₄	—	404	1	no	—
61	C₂×D₄	—	202	1	no	—
62	SD₁₆	—	404	1	no	—
63	SD₁₆	—	404	1	no	—
64	SD₁₆	—	404	1	no	—
65	SD₁₆	—	404	1	no	—
66	C₂×C₈	D₄₀₄	1	1	yes	characteristic
67	C₂×C₈	C₂²⋊D₁₀₁	1	1	yes	characteristic
68	C₄⋊C₄	—	404	1	no	—
69	C₄²	C₄×D₁₀₁	1	1	yes	characteristic
70	C₁₆	—	4	1	no	—
71	C₁₆	—	404	1	no	—
72	C₂³	—	404	1	no	—
73	Q₈	—	404	1	no	—
74	Q₈	—	404	1	no	—
75	D₄	—	808	2	no	—
76	D₄	—	404	1	no	—
77	D₄	—	404	1	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	404	1	no	—
80	C₂×C₄	#13108051	1	1	yes	characteristic
81	C₈	#13108059	1	1	yes	characteristic
82	C₈	D₈₀₈	1	1	yes	characteristic
83	C₈	#13108061	1	1	yes	characteristic
84	C₈	#13108063	1	1	yes	characteristic
85	C₂²	—	808	2	no	—
86	C₂²	—	808	1	no	—
87	C₂²	#15692171	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	—	808	1	no	—
90	C₄	#15692175	1	1	yes	characteristic
91	C₄	#15692177	1	1	yes	characteristic
92	C₂	—	2	1	no	—
93	C₂	—	1616	1	no	—
94	C₂	#20264920	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20264920
commutator	C₄²×C₁₀₁	C₂×C₄
Frattini	C₄×C₈	C₂×D₁₀₁
Fitting	#364×C₁₀₁	C₂
socle	C₂×C₁₀₁	#331
prime radical	#15692252	C₄
cosocle	C₄×C₈×C₁₀₁	C₂²
hypercenter	#364	D₁₀₁
hypocenter	C₁₀₁	#1359
LCS	G ▷ C₄²×C₁₀₁ ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
2. #20264998	C₂	semidirect
3. #364×C₁₀₁	C₂	semidirect
48. C₁₀₁	#1359	semidirect
51. #364	D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
#20265550	2
#20264998	2
#364×C₁₀₁	2
#1359	101