Subgroups — #16777009

Order 3328.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13129865	C₂	1	2	yes	maximal
2	C₁₆×#1055	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	—	2	1	no	—
9	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
10	C₂×#6090	C₈	1	1	yes	characteristic
11	C₂×#6090	C₈	1	1	yes	characteristic
12	C₄×#1055	D₄	1	1	yes	characteristic
13	C₄×#1055	Q₈	1	1	yes	characteristic
14	#7100	—	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	D₈	1	1	yes	characteristic
20	C₂×#1055	Q₁₆	1	1	yes	characteristic
21	#6092	—	8	2	no	—
22	C₁₃×C₁₆	—	2	2	no	—
23	#6090	—	2	1	no	—
24	#6090	C₁₆	1	2	yes	complemented
25	#1370	—	13	2	no	—
26	C₈×C₁₆	—	13	1	no	—
27	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
28	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
29	C₂×C₄×C₁₃	#108	1	1	yes	characteristic
30	C₈×C₁₃	—	2	1	no	—
31	C₈×C₁₃	C₂×C₁₆	1	1	yes	characteristic
32	C₈×C₁₃	M₃₂	1	1	yes	characteristic
33	#1055	—	2	2	no	—
34	#1055	—	2	1	no	—
35	#1055	D₁₆	1	1	yes	characteristic
36	#1055	Q₃₂	1	1	yes	characteristic
37	C₄⋊C₁₆	—	26	2	no	—
38	C₄×C₁₆	—	13	1	no	—
39	C₄×C₁₆	#261	1	1	yes	characteristic
40	C₈²	—	13	1	no	—
41	C₂²×C₁₃	#335	1	1	yes	characteristic
42	C₄×C₁₃	—	2	1	no	—
43	C₄×C₁₃	C₄⋊C₁₆	1	1	yes	characteristic
44	C₄×C₁₃	#364	1	1	yes	characteristic
45	C₄×C₁₃	#366	1	1	yes	characteristic
46	C₄×C₁₃	#368	1	1	yes	characteristic
47	C₂×C₁₆	—	2	1	no	—
48	C₂×C₁₆	—	13	1	no	—
49	C₂×C₁₆	—	13	1	no	—
50	C₂×C₁₆	—	52	2	no	—
51	C₄×C₈	—	13	1	no	—
52	C₄×C₈	—	13	1	no	—
53	C₄×C₈	C₂×#261	1	1	yes	characteristic
54	C₂×C₁₃	#1370	1	1	yes	characteristic
55	C₂×C₁₃	#1371	1	1	yes	characteristic
56	C₂×C₁₃	#1373	1	1	yes	characteristic
57	C₂×C₈	—	13	1	no	—
58	C₂×C₈	—	13	1	no	—
59	C₂×C₈	—	26	1	no	—
60	C₂×C₈	D₁₃⋊C₈	1	1	yes	characteristic
61	C₂×C₈	#6122	1	1	yes	characteristic
62	C₄²	C₄⋊#261	1	1	yes	characteristic
63	C₁₆	—	2	2	no	—
64	C₁₆	—	13	2	no	—
65	C₁₆	—	26	1	no	—
66	C₁₆	—	104	2	no	—
67	C₁₃	#7100	1	1	yes	characteristic, complemented, Sylow
68	C₂×C₄	#88102	1	1	yes	characteristic
69	C₂×C₄	#88103	1	1	yes	characteristic
70	C₂×C₄	#88112	1	1	yes	characteristic
71	C₈	—	2	1	no	—
72	C₈	—	13	1	no	—
73	C₈	—	13	1	no	—
74	C₈	—	26	2	no	—
75	C₈	—	26	1	no	—
76	C₈	D₁₃⋊C₁₆	1	1	yes	characteristic
77	C₈	#88098	1	1	yes	characteristic
78	C₂²	#11718256	1	1	yes	characteristic
79	C₄	—	2	1	no	—
80	C₄	#11718221	1	1	yes	characteristic
81	C₄	#11718224	1	1	yes	characteristic
82	C₄	#11718233	1	1	yes	characteristic
83	C₄	#11718260	1	1	yes	characteristic
84	C₂	#13129866	1	1	yes	characteristic
85	C₂	#13129868	1	1	yes	characteristic
86	C₂	#13129872	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88102
commutator	C₈×C₁₃	C₂×C₁₆
Frattini	C₄×C₈	C₂×#261
Fitting	C₄×C₁₃×C₁₆	C₄
socle, prime radical	C₂²×C₁₃	#335
cosocle	C₈×#1055	C₂²
hypercenter	C₄×C₁₆	#261
hypocenter	C₁₃	#7100
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
24. #6090	C₁₆	semidirect
67. C₁₃	#7100	semidirect

Maximal subgroups

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

subgroup	index
#13129865	2
C₁₆×#1055	2
#7100	13