Subgroups — #16778210

Order 3328.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13130006	C₂	1	1	yes	maximal, characteristic
2	#13130000	C₂	1	1	yes	maximal, characteristic
3	D₁₃×Q₆₄	C₂	1	1	yes	maximal, characteristic
4	D₁₃×Q₃₂	—	2	1	no	—
5	#11718223	—	2	1	no	—
6	C₁₃×Q₆₄	C₄	1	1	yes	characteristic
7	#11718114	—	2	1	no	—
8	Dic₂₀₈	C₄	1	1	yes	characteristic
9	D₁₃×C₃₂	C₂²	1	1	yes	characteristic
10	#11718037	—	2	1	no	—
11	Q₁₆×D₁₃	—	4	1	no	—
12	#88104	—	4	1	no	—
13	C₁₃×Q₃₂	—	2	1	no	—
14	#88069	—	4	1	no	—
15	Dic₁₀₄	—	2	1	no	—
16	C₁₆×D₁₃	D₄	1	1	yes	characteristic
17	C₁₃×C₃₂	C₂×C₄	1	1	yes	characteristic
18	#88034	D₄	1	1	yes	characteristic
19	#7542	—	13	1	no	maximal, Sylow
20	Q₈×D₁₃	—	8	1	no	—
21	#6118	—	8	1	no	—
22	C₁₃×Q₁₆	—	4	1	no	—
23	#6107	—	8	1	no	—
24	Dic₅₂	—	4	1	no	—
25	C₈×D₁₃	D₈	1	1	yes	characteristic
26	C₁₃×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
27	#6090	SD₁₆	1	1	yes	characteristic
28	C₂×Q₆₄	—	13	1	no	—
29	M₁₂₈	—	13	1	no	—
30	#1425	—	13	1	no	—
31	Q₈×C₁₃	—	8	1	no	—
32	C₄×D₁₃	—	16	1	no	—
33	C₄×D₁₃	D₁₆	1	1	yes	characteristic
34	Dic₂₆	—	8	1	no	—
35	Dic₂₆	—	16	1	no	—
36	#1057	—	16	1	no	—
37	C₈×C₁₃	#102	1	1	yes	characteristic
38	#1055	SD₃₂	1	1	yes	characteristic
39	C₂×Q₃₂	—	26	1	no	—
40	Q₆₄	—	13	1	no	—
41	Q₆₄	—	26	1	no	—
42	Q₆₄	#261	1	1	yes	characteristic
43	C₂×C₃₂	—	13	1	no	—
44	#368	—	26	1	no	—
45	C₆₄	—	26	1	no	—
46	C₂×D₁₃	SD₆₄	1	1	yes	characteristic
47	C₄×C₁₃	—	16	1	no	—
48	C₄×C₁₃	#357	1	1	yes	characteristic
49	Dic₁₃	—	16	1	no	—
50	Dic₁₃	D₃₂	1	1	yes	characteristic
51	C₂×Q₁₆	—	52	1	no	—
52	Q₃₂	—	2	1	no	—
53	Q₃₂	—	26	1	no	—
54	Q₃₂	—	52	1	no	—
55	C₂×C₁₆	—	13	1	no	—
56	#108	—	52	1	no	—
57	C₃₂	—	13	1	no	—
58	C₃₂	C₂×#261	1	1	yes	characteristic
59	C₂×C₁₃	#1415	1	1	yes	characteristic
60	D₁₃	—	2	1	no	—
61	C₂×Q₈	—	104	1	no	—
62	Q₁₆	—	4	1	no	—
63	Q₁₆	—	52	1	no	—
64	Q₁₆	—	104	1	no	—
65	M₁₆	—	104	1	no	—
66	C₂×C₈	—	13	1	no	—
67	C₁₆	—	13	1	no	—
68	C₁₆	C₂²⋊#261	1	1	yes	characteristic
69	C₁₃	#7542	1	1	yes	characteristic, complemented, Sylow
70	Q₈	—	8	1	no	—
71	Q₈	—	104	1	no	—
72	Q₈	—	208	1	no	—
73	C₂×C₄	—	13	1	no	—
74	C₂×C₄	—	208	1	no	—
75	C₈	—	13	1	no	—
76	C₈	—	208	1	no	—
77	C₈	#88115	1	1	yes	characteristic
78	C₂²	—	13	1	no	—
79	C₄	—	13	1	no	—
80	C₄	—	16	1	no	—
81	C₄	—	208	1	no	—
82	C₄	#11718276	1	1	yes	characteristic
83	C₂	—	26	1	no	—
84	C₂	#13129985	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#13129985
commutator	C₁₃×C₃₂	C₂×C₄
Frattini	C₃₂	C₂×#261
Fitting	C₁₃×Q₆₄	C₄
socle	C₂×C₁₃	#1415
prime radical	C₂×D₁₃	SD₆₄
cosocle	D₁₃×C₃₂	C₂²
hypercenter	Q₆₄	#261
hypocenter	C₁₃	#7542
LCS	G ▷ C₁₃×C₃₂ ▷ C₁₃×C₁₆ ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ Q₆₄

Complements

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

N	H ≅ G/N	type
69. C₁₃	#7542	semidirect

Maximal subgroups

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

subgroup	index
#13130006	2
#13130000	2
D₁₃×Q₆₄	2
#7542	13