Subgroups — #15750801

Order 3328.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₀₈	C₂	1	1	yes	maximal, characteristic
2	#13113036	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×C₆₄	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₁₀₄	—	2	1	no	—
5	C₂×C₁₃×C₃₂	C₂²	1	1	yes	characteristic
6	#11718096	—	2	1	no	—
7	Dic₂₀₈	—	2	1	no	—
8	Dic₂₀₈	C₄	1	2	yes	complemented
9	C₁₃×C₆₄	—	2	1	no	—
10	C₂×Dic₅₂	—	4	1	no	—
11	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
12	#88058	—	4	1	no	—
13	Dic₁₀₄	—	2	2	no	—
14	Dic₁₀₄	—	4	1	no	—
15	C₁₃×C₃₂	C₂×C₄	1	1	yes	characteristic
16	C₁₃×C₃₂	D₄	1	1	yes	characteristic
17	#7538	—	13	1	no	maximal, Sylow
18	C₂×Dic₂₆	—	8	1	no	—
19	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
20	C₄⋊Dic₁₃	—	8	1	no	—
21	Dic₅₂	—	4	2	no	—
22	Dic₅₂	—	8	1	no	—
23	C₁₃×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
24	C₁₃×C₁₆	SD₁₆	1	1	yes	characteristic
25	C₂×Q₆₄	—	13	1	no	—
26	C₂×C₆₄	D₁₃	1	1	yes	characteristic
27	#1423	—	13	1	no	—
28	C₂×C₄×C₁₃	D₁₆	1	1	yes	characteristic
29	C₂×Dic₁₃	—	16	1	no	—
30	C₂×Dic₁₃	—	16	1	no	—
31	Dic₂₆	—	8	2	no	—
32	Dic₂₆	—	16	1	no	—
33	C₈×C₁₃	#102	1	1	yes	characteristic
34	C₈×C₁₃	SD₃₂	1	1	yes	characteristic
35	C₂×Q₃₂	—	26	1	no	—
36	Q₆₄	—	13	2	no	—
37	Q₆₄	—	26	1	no	—
38	C₂×C₃₂	C₂×D₁₃	1	1	yes	characteristic
39	#366	—	26	1	no	—
40	C₆₄	—	2	1	no	—
41	C₂²×C₁₃	D₃₂	1	1	yes	characteristic
42	C₄×C₁₃	#357	1	1	yes	characteristic
43	C₄×C₁₃	SD₆₄	1	1	yes	characteristic
44	Dic₁₃	—	16	2	no	—
45	Dic₁₃	—	32	1	no	—
46	C₂×Q₁₆	—	52	1	no	—
47	Q₃₂	—	26	2	no	—
48	Q₃₂	—	52	1	no	—
49	C₂×C₁₆	D₅₂	1	1	yes	characteristic
50	#107	—	52	1	no	—
51	C₃₂	C₄×D₁₃	1	1	yes	characteristic
52	C₃₂	C₂²⋊D₁₃	1	1	yes	characteristic
53	C₂×C₁₃	#1415	1	1	yes	characteristic
54	C₂×C₁₃	SD₁₂₈	1	1	yes	characteristic
55	C₂×C₁₃	Q₁₂₈	1	1	yes	characteristic
56	C₂×Q₈	—	104	1	no	—
57	Q₁₆	—	52	2	no	—
58	Q₁₆	—	104	1	no	—
59	C₂×C₈	D₁₀₄	1	1	yes	characteristic
60	C₄⋊C₄	—	104	1	no	—
61	C₁₆	#6095	1	1	yes	characteristic
62	C₁₆	#6103	1	1	yes	characteristic
63	C₁₃	#7538	1	1	yes	characteristic, complemented, Sylow
64	Q₈	—	104	2	no	—
65	Q₈	—	208	1	no	—
66	C₂×C₄	—	208	1	no	—
67	C₂×C₄	—	208	1	no	—
68	C₂×C₄	D₂₀₈	1	1	yes	characteristic
69	C₈	#88040	1	1	yes	characteristic
70	C₈	#88061	1	1	yes	characteristic
71	C₂²	D₄₁₆	1	1	yes	characteristic
72	C₄	—	416	1	no	—
73	C₄	—	208	2	no	—
74	C₄	#11718041	1	1	yes	characteristic
75	C₄	#11718101	1	1	yes	characteristic
76	C₂	#13113015	1	1	yes	characteristic
77	C₂	#13113052	1	1	yes	characteristic
78	C₂	Dic₄₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄₁₆
commutator	C₁₃×C₃₂	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₁₃
Fitting	C₂×C₁₃×C₆₄	C₂
socle, prime radical	C₂²×C₁₃	D₃₂
cosocle	C₂×C₁₃×C₃₂	C₂²
hypercenter	C₂×C₆₄	D₁₃
hypocenter	C₁₃	#7538
LCS	G ▷ C₁₃×C₃₂ ▷ C₁₃×C₁₆ ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ 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
8. Dic₂₀₈	C₄	semidirect
63. C₁₃	#7538	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₂₀₈	2
#13113036	2
C₂×C₁₃×C₆₄	2
#7538	13