Subgroups — #15783973

Order 3328.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13117145	C₂	1	2	yes	maximal, complemented
2	Dic₄₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#13113052	C₂	1	2	yes	maximal, complemented
4	D₈₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₃×C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718565	—	2	2	no	—
7	C₂×C₁₃×C₃₂	C₂²	1	1	yes	characteristic
8	Dic₂₀₈	C₂²	1	2	yes	—
9	#11718041	—	2	2	no	—
10	D₄₁₆	C₂²	1	2	yes	—
11	C₁₃×C₆₄	C₂²	1	1	yes	characteristic
12	C₁₃×C₆₄	C₂²	1	1	yes	characteristic
13	#88158	—	4	2	no	—
14	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
15	Dic₁₀₄	—	2	2	no	—
16	#88040	—	4	2	no	—
17	D₂₀₈	—	2	2	no	—
18	C₁₃×C₃₂	D₄	1	1	yes	characteristic
19	C₁₃×C₃₂	C₂³	1	1	yes	characteristic
20	#13741	—	13	1	no	maximal, Sylow
21	#6127	—	8	2	no	—
22	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
23	Dic₅₂	—	4	2	no	—
24	D₁₀₄	—	4	2	no	—
25	#6095	—	8	2	no	—
26	C₁₃×C₁₆	D₈	1	1	yes	characteristic
27	C₁₃×C₁₆	C₂×D₄	1	1	yes	characteristic
28	#2262	—	13	2	no	—
29	Q₁₂₈	—	13	1	no	—
30	SD₁₂₈	—	13	2	no	—
31	D₆₄	—	13	1	no	—
32	C₂×C₆₄	D₁₃	1	1	yes	characteristic, complemented
33	C₂×C₄×C₁₃	D₁₆	1	1	yes	characteristic
34	C₂²⋊D₁₃	—	16	2	no	—
35	D₅₂	—	8	2	no	—
36	C₄×D₁₃	—	16	2	no	—
37	Dic₂₆	—	8	2	no	—
38	C₈×C₁₃	D₁₆	1	1	yes	characteristic
39	C₈×C₁₃	C₂×D₈	1	1	yes	characteristic
40	#508	—	26	2	no	—
41	Q₆₄	—	13	2	no	—
42	SD₆₄	—	26	2	no	—
43	D₃₂	—	13	2	no	—
44	C₂×C₃₂	C₂×D₁₃	1	1	yes	characteristic
45	C₆₄	C₂×D₁₃	1	1	yes	characteristic
46	C₆₄	C₂×D₁₃	1	1	yes	characteristic
47	C₂²×C₁₃	D₃₂	1	1	yes	characteristic
48	C₂×D₁₃	—	16	2	no	—
49	C₄×C₁₃	D₃₂	1	1	yes	characteristic
50	C₄×C₁₃	C₂×D₁₆	1	1	yes	characteristic
51	Dic₁₃	—	16	2	no	—
52	#135	—	52	2	no	—
53	Q₃₂	—	26	2	no	—
54	SD₃₂	—	52	2	no	—
55	D₁₆	—	26	2	no	—
56	C₂×C₁₆	D₅₂	1	1	yes	characteristic
57	C₃₂	D₅₂	1	1	yes	characteristic
58	C₃₂	C₂²×D₁₃	1	1	yes	characteristic
59	C₂×C₁₃	—	2	1	no	—
60	C₂×C₁₃	C₂×D₃₂	1	1	yes	characteristic
61	D₁₃	—	32	2	no	—
62	#41	—	104	2	no	—
63	Q₁₆	—	52	2	no	—
64	SD₁₆	—	104	2	no	—
65	D₈	—	52	2	no	—
66	C₂×C₈	D₁₀₄	1	1	yes	characteristic
67	C₁₆	D₁₀₄	1	1	yes	characteristic
68	C₁₆	C₂×D₅₂	1	1	yes	characteristic
69	C₁₃	#13741	1	1	yes	characteristic, complemented, Sylow
70	Q₈	—	104	2	no	—
71	D₄	—	104	2	no	—
72	D₄	—	208	2	no	—
73	C₂×C₄	—	208	2	no	—
74	C₂×C₄	D₂₀₈	1	1	yes	characteristic
75	C₈	D₂₀₈	1	1	yes	characteristic
76	C₈	C₂×D₁₀₄	1	1	yes	characteristic
77	C₂²	—	208	2	no	—
78	C₂²	D₄₁₆	1	1	yes	characteristic
79	C₄	—	208	2	no	—
80	C₄	D₄₁₆	1	1	yes	characteristic
81	C₄	C₂×D₂₀₈	1	1	yes	characteristic
82	C₂	—	2	1	no	—
83	C₂	—	416	2	no	—
84	C₂	C₂×D₄₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄₁₆
commutator, cosocle	C₁₃×C₃₂	C₂³
Frattini	C₃₂	C₂²×D₁₃
Fitting	C₂×C₁₃×C₆₄	C₂
socle	C₂×C₁₃	C₂×D₃₂
hypercenter	C₂×C₆₄	D₁₃
hypocenter	C₁₃	#13741
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
1. #13117145	C₂	semidirect
2. Dic₄₁₆	C₂	semidirect
3. #13113052	C₂	semidirect
4. D₈₃₂	C₂	semidirect
5. C₂×C₁₃×C₆₄	C₂	semidirect
32. C₂×C₆₄	D₁₃	semidirect
69. C₁₃	#13741	semidirect

Maximal subgroups

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

subgroup	index
#13117145	2
Dic₄₁₆	2
#13113052	2
D₈₃₂	2
C₂×C₁₃×C₆₄	2
#13741	13