Subgroups — #27169179

Order 12928.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265554	C₂	1	1	yes	maximal, characteristic
2	#20264930	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊C₁₆×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₈₀₈	—	2	1	no	—
5	C₂×D₈₀₈	C₄	1	2	yes	—
6	#15692252	—	2	1	no	—
7	C₄⋊#11717579	—	2	1	no	—
8	C₂×C₁₆×C₁₀₁	—	2	1	no	—
9	C₄×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
10	C₂×D₄₀₄	—	2	2	no	—
11	C₂×D₄₀₄	—	4	1	no	—
12	C₂×D₄₀₄	—	4	1	no	—
13	D₈₀₈	—	4	2	no	—
14	D₈₀₈	—	4	1	no	—
15	D₈₀₈	—	4	1	no	—
16	C₂×#11717579	—	4	1	no	—
17	C₂×C₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₁₀₁	D₄	1	1	yes	characteristic
19	C₄²×C₁₀₁	D₄	1	1	yes	characteristic
20	C₁₆×C₁₀₁	—	4	1	no	—
21	C₂²×D₁₀₁	—	4	2	no	—
22	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₄×C₁₀₁	D₈	1	1	yes	characteristic
24	C₂×C₄×C₁₀₁	SD₁₆	1	1	yes	characteristic
25	D₄₀₄	—	4	2	no	—
26	D₄₀₄	—	4	2	no	—
27	D₄₀₄	—	8	2	no	—
28	D₄₀₄	—	8	1	no	—
29	D₄₀₄	—	8	1	no	—
30	C₈×C₁₀₁	—	2	1	no	—
31	C₈×C₁₀₁	D₈	1	1	yes	characteristic
32	C₈×C₁₀₁	SD₁₆	1	1	yes	characteristic
33	#11717579	—	8	1	no	—
34	C₂²×C₁₀₁	#102	1	1	yes	characteristic
35	C₂×D₁₀₁	—	8	2	no	—
36	C₂×D₁₀₁	—	8	2	no	—
37	C₂×D₁₀₁	—	8	2	no	—
38	C₄×C₁₀₁	—	2	1	no	—
39	C₄×C₁₀₁	#100	1	1	yes	characteristic
40	C₄×C₁₀₁	#102	1	1	yes	characteristic
41	C₄×C₁₀₁	D₁₆	1	1	yes	characteristic
42	C₄×C₁₀₁	SD₃₂	1	1	yes	characteristic
43	C₂×C₁₀₁	#331	1	1	yes	characteristic
44	C₂×C₁₀₁	#357	1	1	yes	characteristic
45	C₂×C₁₀₁	#361	1	1	yes	characteristic
46	D₁₀₁	—	16	2	no	—
47	#1361	—	101	1	no	maximal, Sylow
48	C₁₀₁	#1361	1	1	yes	characteristic, complemented, Sylow
49	#493	—	101	1	no	—
50	C₄⋊C₁₆	D₁₀₁	1	1	yes	characteristic, complemented
51	#335	—	101	1	no	—
52	C₂×D₈	—	101	2	no	—
53	C₂×D₈	—	202	1	no	—
54	#127	—	202	1	no	—
55	C₂×C₁₆	—	2	1	no	—
56	C₄⋊C₈	—	202	1	no	—
57	C₄×C₈	C₂×D₁₀₁	1	1	yes	characteristic
58	C₂×D₄	—	404	1	no	—
59	C₂×D₄	—	404	1	no	—
60	C₂×D₄	—	202	2	no	—
61	D₈	—	404	2	no	—
62	D₈	—	404	1	no	—
63	D₈	—	404	1	no	—
64	C₂×C₈	—	404	1	no	—
65	C₂×C₈	C₄×D₁₀₁	1	1	yes	characteristic
66	C₂×C₈	C₂²⋊D₁₀₁	1	1	yes	characteristic
67	C₄²	D₄₀₄	1	1	yes	characteristic
68	C₁₆	—	4	1	no	—
69	C₂³	—	404	2	no	—
70	D₄	—	808	2	no	—
71	D₄	—	808	1	no	—
72	D₄	—	808	1	no	—
73	D₄	—	404	2	no	—
74	D₄	—	404	2	no	—
75	C₂×C₄	#13108051	1	1	yes	characteristic
76	C₂×C₄	D₈₀₈	1	1	yes	characteristic
77	C₂×C₄	#13108063	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	808	1	no	—
80	C₈	#13108059	1	1	yes	characteristic
81	C₈	#13108061	1	1	yes	characteristic
82	C₂²	—	808	2	no	—
83	C₂²	—	808	2	no	—
84	C₂²	—	808	2	no	—
85	C₂²	#15692177	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	#15692171	1	1	yes	characteristic
88	C₄	#15692175	1	1	yes	characteristic
89	C₄	D₁₆₁₆	1	1	yes	characteristic
90	C₄	#15692200	1	1	yes	characteristic
91	C₂	—	1616	2	no	—
92	C₂	#20264920	1	1	yes	characteristic
93	C₂	#20264979	1	1	yes	characteristic
94	C₂	#20264991	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15692177
commutator	C₂×C₈×C₁₀₁	C₂×C₄
Frattini	C₄×C₈	C₂×D₁₀₁
Fitting	C₄⋊C₁₆×C₁₀₁	C₂
socle	C₂²×C₁₀₁	#102
prime radical	#20265554	C₂
cosocle	C₄×C₈×C₁₀₁	C₂²
hypercenter	C₄⋊C₁₆	D₁₀₁
hypocenter	C₁₀₁	#1361
LCS	G ▷ C₂×C₈×C₁₀₁ ▷ C₂×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
2. #20264930	C₂	semidirect
3. C₄⋊C₁₆×C₁₀₁	C₂	semidirect
48. C₁₀₁	#1361	semidirect
50. C₄⋊C₁₆	D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
#20265554	2
#20264930	2
C₄⋊C₁₆×C₁₀₁	2
#1361	101