Subgroups — #13117151

Order 1664.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11718565	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₂₀₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718041	C₂	1	2	yes	maximal, complemented
5	D₄₁₆	C₂	1	2	yes	maximal, complemented
6	#88158	—	2	1	no	—
7	C₂×D₁₀₄	—	2	1	no	—
8	C₂×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
9	Dic₁₀₄	C₂²	1	1	yes	characteristic, complemented
10	#88040	—	2	1	no	—
11	D₂₀₈	—	2	1	no	—
12	D₂₀₈	C₂²	1	2	yes	—
13	D₂₀₈	C₂²	1	1	yes	characteristic, complemented
14	C₁₃×C₃₂	C₂²	1	2	yes	complemented
15	#6127	—	4	1	no	—
16	C₂×D₅₂	—	4	1	no	—
17	C₂×C₈×C₁₃	D₄	1	1	yes	characteristic
18	Dic₅₂	—	2	1	no	—
19	D₁₀₄	—	2	2	no	—
20	D₁₀₄	—	2	1	no	—
21	D₁₀₄	—	4	1	no	—
22	#6095	—	4	1	no	—
23	C₁₃×C₁₆	D₄	1	1	yes	characteristic
24	C₁₃×C₁₆	C₂³	1	1	yes	characteristic
25	#2263	—	13	1	no	maximal, Sylow
26	C₂²×D₁₃	—	8	1	no	—
27	C₂×C₄×C₁₃	D₈	1	1	yes	characteristic
28	C₂²⋊D₁₃	—	8	1	no	—
29	D₅₂	—	4	1	no	—
30	D₅₂	—	4	2	no	—
31	D₅₂	—	8	1	no	—
32	C₄×D₁₃	—	8	1	no	—
33	Dic₂₆	—	4	1	no	—
34	C₈×C₁₃	D₈	1	1	yes	characteristic
35	C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
36	#508	—	13	1	no	—
37	C₂×D₁₆	—	13	1	no	—
38	SD₆₄	—	13	2	no	—
39	D₃₂	—	13	2	no	—
40	M₆₄	D₁₃	1	1	yes	characteristic, complemented
41	C₂²×C₁₃	D₁₆	1	1	yes	characteristic
42	C₂×D₁₃	—	8	1	no	—
43	C₂×D₁₃	—	8	2	no	—
44	C₂×D₁₃	—	16	2	no	—
45	C₄×C₁₃	D₁₆	1	1	yes	characteristic
46	C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
47	Dic₁₃	—	8	1	no	—
48	#135	—	26	1	no	—
49	C₂×D₈	—	26	1	no	—
50	Q₃₂	—	13	1	no	—
51	SD₃₂	—	26	1	no	—
52	D₁₆	—	13	1	no	—
53	D₁₆	—	13	2	no	—
54	D₁₆	—	26	1	no	—
55	C₂×C₁₆	C₂×D₁₃	1	1	yes	characteristic
56	C₃₂	C₂×D₁₃	1	2	yes	complemented
57	C₂×C₁₃	—	2	1	no	—
58	C₂×C₁₃	C₂×D₁₆	1	1	yes	characteristic
59	D₁₃	—	16	1	no	—
60	D₁₃	—	16	2	no	—
61	#41	—	52	1	no	—
62	C₂×D₄	—	52	1	no	—
63	Q₁₆	—	26	1	no	—
64	SD₁₆	—	52	1	no	—
65	D₈	—	26	2	no	—
66	D₈	—	26	1	no	—
67	D₈	—	52	1	no	—
68	C₂×C₈	D₅₂	1	1	yes	characteristic
69	C₁₆	D₅₂	1	1	yes	characteristic
70	C₁₆	C₂²×D₁₃	1	1	yes	characteristic
71	C₁₃	#2263	1	1	yes	characteristic, complemented, Sylow
72	C₂³	—	104	1	no	—
73	Q₈	—	52	1	no	—
74	D₄	—	52	2	no	—
75	D₄	—	52	1	no	—
76	D₄	—	104	1	no	—
77	D₄	—	104	1	no	—
78	C₂×C₄	—	104	1	no	—
79	C₂×C₄	D₁₀₄	1	1	yes	characteristic
80	C₈	D₁₀₄	1	1	yes	characteristic
81	C₈	C₂×D₅₂	1	1	yes	characteristic
82	C₂²	—	104	1	no	—
83	C₂²	—	104	2	no	—
84	C₂²	—	208	2	no	—
85	C₂²	D₂₀₈	1	1	yes	characteristic
86	C₄	—	104	1	no	—
87	C₄	D₂₀₈	1	1	yes	characteristic
88	C₄	C₂×D₁₀₄	1	1	yes	characteristic
89	C₂	—	2	1	no	—
90	C₂	—	208	2	no	—
91	C₂	—	208	1	no	—
92	C₂	C₂×D₂₀₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₂₀₈
commutator, cosocle	C₁₃×C₁₆	C₂³
Frattini	C₁₆	C₂²×D₁₃
Fitting	C₁₃×M₆₄	C₂
socle	C₂×C₁₃	C₂×D₁₆
hypercenter	M₆₄	D₁₃
hypocenter	C₁₃	#2263
LCS	G ▷ C₁₃×C₁₆ ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
1. #11718565	C₂	semidirect
2. C₂×D₂₀₈	C₂	semidirect
3. C₁₃×M₆₄	C₂	semidirect
4. #11718041	C₂	semidirect
5. D₄₁₆	C₂	semidirect
9. Dic₁₀₄	C₂²	semidirect
13. D₂₀₈	C₂²	semidirect
14. C₁₃×C₃₂	C₂²	semidirect
40. M₆₄	D₁₃	semidirect
56. C₃₂	C₂×D₁₃	semidirect
71. C₁₃	#2263	semidirect

Maximal subgroups

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

subgroup	index
#11718565	2
C₂×D₂₀₈	2
C₁₃×M₆₄	2
#11718041	2
D₄₁₆	2
#2263	13