Subgroups — #15750804

Order 3328.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13117145	C₂	1	1	yes	maximal, characteristic
2	#13113040	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718565	—	2	1	no	—
5	C₂×C₁₃×C₃₂	C₂²	1	1	yes	characteristic
6	#11718098	—	2	1	no	—
7	Dic₂₀₈	C₄	1	1	yes	characteristic
8	#11718041	—	2	1	no	—
9	D₄₁₆	C₄	1	1	yes	characteristic
10	C₁₃×C₆₄	—	2	1	no	—
11	#88158	—	4	1	no	—
12	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
13	#88059	—	4	1	no	—
14	Dic₁₀₄	—	2	1	no	—
15	#88040	—	4	1	no	—
16	D₂₀₈	—	2	1	no	—
17	C₁₃×C₃₂	C₂×C₄	1	1	yes	characteristic
18	C₁₃×C₃₂	D₄	1	1	yes	characteristic
19	#7539	—	13	1	no	maximal, Sylow
20	#6127	—	8	1	no	—
21	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
22	#6099	—	8	1	no	—
23	Dic₅₂	—	4	1	no	—
24	D₁₀₄	—	4	1	no	—
25	#6095	—	8	1	no	—
26	C₁₃×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
27	C₁₃×C₁₆	SD₁₆	1	1	yes	characteristic
28	#2262	—	13	1	no	—
29	C₂×C₆₄	D₁₃	1	1	yes	characteristic, complemented
30	#1425	—	13	1	no	—
31	C₂×C₄×C₁₃	D₁₆	1	1	yes	characteristic
32	C₂²⋊D₁₃	—	16	1	no	—
33	D₅₂	—	8	1	no	—
34	C₄×D₁₃	—	16	1	no	—
35	Dic₂₆	—	8	1	no	—
36	C₈×C₁₃	#102	1	1	yes	characteristic
37	C₈×C₁₃	SD₃₂	1	1	yes	characteristic
38	#1055	—	16	1	no	—
39	#508	—	26	1	no	—
40	Q₆₄	—	13	1	no	—
41	SD₆₄	—	26	1	no	—
42	D₃₂	—	13	1	no	—
43	C₂×C₃₂	C₂×D₁₃	1	1	yes	characteristic
44	#368	—	26	1	no	—
45	C₆₄	—	2	1	no	—
46	C₂²×C₁₃	SD₆₄	1	1	yes	characteristic
47	C₂×D₁₃	—	16	1	no	—
48	C₄×C₁₃	#357	1	1	yes	characteristic
49	C₄×C₁₃	D₃₂	1	1	yes	characteristic
50	Dic₁₃	—	16	1	no	—
51	#135	—	52	1	no	—
52	Q₃₂	—	26	1	no	—
53	SD₃₂	—	52	1	no	—
54	D₁₆	—	26	1	no	—
55	C₂×C₁₆	D₅₂	1	1	yes	characteristic
56	#108	—	52	1	no	—
57	C₃₂	C₄×D₁₃	1	1	yes	characteristic
58	C₃₂	C₂²⋊D₁₃	1	1	yes	characteristic
59	C₂×C₁₃	—	2	1	no	—
60	C₂×C₁₃	#1415	1	1	yes	characteristic
61	D₁₃	—	32	1	no	—
62	#41	—	104	1	no	—
63	Q₁₆	—	52	1	no	—
64	SD₁₆	—	104	1	no	—
65	D₈	—	52	1	no	—
66	M₁₆	—	104	1	no	—
67	C₂×C₈	D₁₀₄	1	1	yes	characteristic
68	C₁₆	#6095	1	1	yes	characteristic
69	C₁₆	#6103	1	1	yes	characteristic
70	C₁₃	#7539	1	1	yes	characteristic, complemented, Sylow
71	Q₈	—	104	1	no	—
72	D₄	—	104	1	no	—
73	D₄	—	208	1	no	—
74	C₂×C₄	—	208	1	no	—
75	C₂×C₄	D₂₀₈	1	1	yes	characteristic
76	C₈	—	208	1	no	—
77	C₈	#88040	1	1	yes	characteristic
78	C₈	#88061	1	1	yes	characteristic
79	C₂²	—	208	1	no	—
80	C₂²	#11718041	1	1	yes	characteristic
81	C₄	—	208	1	no	—
82	C₄	D₄₁₆	1	1	yes	characteristic
83	C₄	#11718101	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	416	1	no	—
86	C₂	#13113015	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄₁₆
commutator	C₁₃×C₃₂	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₁₃
Fitting	C₂×C₁₃×C₆₄	C₂
socle	C₂×C₁₃	#1415
prime radical	#13117145	C₂
cosocle	C₂×C₁₃×C₃₂	C₂²
hypercenter	C₂×C₆₄	D₁₃
hypocenter	C₁₃	#7539
LCS	G ▷ C₁₃×C₃₂ ▷ C₁₃×C₁₆ ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ C₂×C₆₄

Complements

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

N	H ≅ G/N	type
2. #13113040	C₂	semidirect
3. C₂×C₁₃×C₆₄	C₂	semidirect
29. C₂×C₆₄	D₁₃	semidirect
70. C₁₃	#7539	semidirect

Maximal subgroups

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

subgroup	index
#13117145	2
#13113040	2
C₂×C₁₃×C₆₄	2
#7539	13