Subgroups — #16777005

Order 3328.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13129858	C₂	1	2	yes	maximal
2	#13112752	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#6092	—	2	2	no	—
4	C₄×C₁₃×C₁₆	C₄	1	1	yes	characteristic
5	C₄×#6090	C₄	1	1	yes	characteristic
6	C₈×#1055	C₂²	1	1	yes	characteristic
7	C₂×#6092	—	4	2	no	—
8	C₂×C₁₃×C₁₆	C₈	1	1	yes	characteristic
9	C₂×C₁₃×C₁₆	C₈	1	1	yes	characteristic
10	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
11	C₂×#6090	—	2	1	no	—
12	C₄×#1055	D₄	1	1	yes	characteristic
13	C₄×#1055	Q₈	1	1	yes	characteristic
14	#7099	—	13	1	no	maximal, Sylow
15	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
16	C₂×C₈×C₁₃	M₁₆	1	1	yes	characteristic
17	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
18	C₂×#1055	—	2	1	no	—
19	C₂×#1055	SD₁₆	1	2	yes	—
20	#6092	—	8	2	no	—
21	C₁₃×C₁₆	—	2	1	no	—
22	C₁₃×C₁₆	C₁₆	1	2	yes	complemented
23	#6090	—	2	2	no	—
24	#1367	—	13	2	no	—
25	#1311	—	13	1	no	—
26	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
27	C₂×C₄×C₁₃	#106	1	1	yes	characteristic
28	C₂×C₄×C₁₃	#108	1	1	yes	characteristic
29	C₈×C₁₃	—	2	1	no	—
30	C₈×C₁₃	C₂×C₁₆	1	1	yes	characteristic
31	C₈×C₁₃	M₃₂	1	1	yes	characteristic
32	#1055	—	2	2	no	—
33	#1055	—	4	1	no	—
34	C₄⋊C₁₆	—	26	2	no	—
35	C₄×C₁₆	—	13	1	no	—
36	C₄×C₁₆	#261	1	1	yes	characteristic
37	C₈²	—	13	1	no	—
38	C₂²×C₁₃	#334	1	1	yes	characteristic
39	C₄×C₁₃	—	2	1	no	—
40	C₄×C₁₃	C₄⋊C₁₆	1	1	yes	characteristic
41	C₄×C₁₃	#364	1	1	yes	characteristic
42	C₄×C₁₃	#365	1	2	yes	—
43	C₂×C₁₆	—	26	1	no	—
44	C₂×C₁₆	—	52	2	no	—
45	C₂×C₁₆	#1057	1	1	yes	characteristic
46	C₂×C₁₆	#1057	1	1	yes	characteristic
47	C₄×C₈	—	13	1	no	—
48	C₄×C₈	—	13	1	no	—
49	C₄×C₈	C₂×#261	1	1	yes	characteristic
50	C₂×C₁₃	#1367	1	1	yes	characteristic
51	C₂×C₁₃	#1368	1	1	yes	characteristic
52	C₂×C₁₃	#1369	1	1	yes	characteristic
53	C₂×C₈	—	13	2	no	—
54	C₂×C₈	—	26	1	no	—
55	C₂×C₈	#6118	1	1	yes	characteristic
56	C₂×C₈	C₂×#1057	1	1	yes	characteristic
57	C₄²	C₄⋊#261	1	1	yes	characteristic
58	C₁₆	—	2	1	no	—
59	C₁₆	—	26	2	no	—
60	C₁₆	—	104	2	no	—
61	C₁₆	#6092	1	2	yes	complemented
62	C₁₃	#7099	1	1	yes	characteristic, complemented, Sylow
63	C₂×C₄	#88101	1	1	yes	characteristic
64	C₂×C₄	#88103	1	1	yes	characteristic
65	C₂×C₄	C₄⋊#1057	1	1	yes	characteristic
66	C₈	—	2	1	no	—
67	C₈	—	26	2	no	—
68	C₈	—	52	1	no	—
69	C₈	C₂×#6092	1	1	yes	characteristic
70	C₈	#88106	1	1	yes	characteristic
71	C₂²	#11718253	1	1	yes	characteristic
72	C₄	—	2	1	no	—
73	C₄	#11718219	1	2	yes	—
74	C₄	C₄⋊#6092	1	1	yes	characteristic
75	C₄	#11718261	1	1	yes	characteristic
76	C₂	#13129858	1	1	yes	characteristic
77	C₂	#13129860	1	1	yes	characteristic
78	C₂	#13129863	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88101
commutator	C₈×C₁₃	C₂×C₁₆
Frattini	C₄×C₈	C₂×#261
Fitting	C₄×C₁₃×C₁₆	C₄
socle, prime radical	C₂²×C₁₃	#334
cosocle	C₈×#1055	C₂²
hypercenter	C₄×C₁₆	#261
hypocenter	C₁₃	#7099
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
22. C₁₃×C₁₆	C₁₆	semidirect
61. C₁₆	#6092	semidirect
62. C₁₃	#7099	semidirect

Maximal subgroups

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

subgroup	index
#13129858	2
#13112752	2
#7099	13