Subgroups — #16955378

Order 4352.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14531093	C₂	1	2	yes	maximal, complemented
2	Dic₅₄₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#14527000	C₂	1	2	yes	maximal, complemented
4	D₁₀₈₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₇×C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763241	—	2	2	no	—
7	Dic₂₇₂	C₂²	1	2	yes	—
8	#11762648	—	2	2	no	—
9	D₅₄₄	C₂²	1	2	yes	—
10	C₂×C₁₇×C₃₂	C₂²	1	1	yes	characteristic
11	C₁₇×C₆₄	C₂²	1	1	yes	characteristic
12	C₁₇×C₆₄	C₂²	1	1	yes	characteristic
13	#10587590	—	4	2	no	—
14	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
15	Dic₁₃₆	—	2	2	no	—
16	#10587471	—	4	2	no	—
17	D₂₇₂	—	2	2	no	—
18	C₁₇×C₃₂	D₄	1	1	yes	characteristic
19	C₁₇×C₃₂	C₂³	1	1	yes	characteristic
20	#63252	—	8	2	no	—
21	C₂×C₈×C₁₇	D₈	1	1	yes	characteristic
22	Dic₆₈	—	4	2	no	—
23	D₁₃₆	—	4	2	no	—
24	#63219	—	8	2	no	—
25	C₁₆×C₁₇	D₈	1	1	yes	characteristic
26	C₁₆×C₁₇	C₂×D₄	1	1	yes	characteristic
27	#13741	—	17	1	no	maximal, Sylow
28	C₂×C₄×C₁₇	D₁₆	1	1	yes	characteristic
29	C₂²⋊D₁₇	—	16	2	no	—
30	D₆₈	—	8	2	no	—
31	C₄×D₁₇	—	16	2	no	—
32	Dic₃₄	—	8	2	no	—
33	C₈×C₁₇	D₁₆	1	1	yes	characteristic
34	C₈×C₁₇	C₂×D₈	1	1	yes	characteristic
35	#2262	—	17	2	no	—
36	Q₁₂₈	—	17	1	no	—
37	SD₁₂₈	—	17	2	no	—
38	D₆₄	—	17	1	no	—
39	C₂×C₆₄	D₁₇	1	1	yes	characteristic, complemented
40	C₂²×C₁₇	D₃₂	1	1	yes	characteristic
41	C₂×D₁₇	—	16	2	no	—
42	C₄×C₁₇	D₃₂	1	1	yes	characteristic
43	C₄×C₁₇	C₂×D₁₆	1	1	yes	characteristic
44	Dic₁₇	—	16	2	no	—
45	#508	—	34	2	no	—
46	Q₆₄	—	17	2	no	—
47	SD₆₄	—	34	2	no	—
48	D₃₂	—	17	2	no	—
49	C₂×C₃₂	C₂×D₁₇	1	1	yes	characteristic
50	C₆₄	C₂×D₁₇	1	1	yes	characteristic
51	C₆₄	C₂×D₁₇	1	1	yes	characteristic
52	C₂×C₁₇	—	2	1	no	—
53	C₂×C₁₇	C₂×D₃₂	1	1	yes	characteristic
54	D₁₇	—	32	2	no	—
55	#135	—	68	2	no	—
56	Q₃₂	—	34	2	no	—
57	SD₃₂	—	68	2	no	—
58	D₁₆	—	34	2	no	—
59	C₂×C₁₆	D₆₈	1	1	yes	characteristic
60	C₃₂	D₆₈	1	1	yes	characteristic
61	C₃₂	C₂²×D₁₇	1	1	yes	characteristic
62	C₁₇	#13741	1	1	yes	characteristic, complemented, Sylow
63	#41	—	136	2	no	—
64	Q₁₆	—	68	2	no	—
65	SD₁₆	—	136	2	no	—
66	D₈	—	68	2	no	—
67	C₂×C₈	D₁₃₆	1	1	yes	characteristic
68	C₁₆	D₁₃₆	1	1	yes	characteristic
69	C₁₆	C₂×D₆₈	1	1	yes	characteristic
70	Q₈	—	136	2	no	—
71	D₄	—	272	2	no	—
72	D₄	—	136	2	no	—
73	C₂×C₄	—	272	2	no	—
74	C₂×C₄	D₂₇₂	1	1	yes	characteristic
75	C₈	D₂₇₂	1	1	yes	characteristic
76	C₈	C₂×D₁₃₆	1	1	yes	characteristic
77	C₂²	—	272	2	no	—
78	C₂²	D₅₄₄	1	1	yes	characteristic
79	C₄	—	272	2	no	—
80	C₄	D₅₄₄	1	1	yes	characteristic
81	C₄	C₂×D₂₇₂	1	1	yes	characteristic
82	C₂	—	2	1	no	—
83	C₂	—	544	2	no	—
84	C₂	C₂×D₅₄₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₅₄₄
commutator, cosocle	C₁₇×C₃₂	C₂³
Frattini	C₃₂	C₂²×D₁₇
Fitting	C₂×C₁₇×C₆₄	C₂
socle	C₂×C₁₇	C₂×D₃₂
hypercenter	C₂×C₆₄	D₁₇
hypocenter	C₁₇	#13741
LCS	G ▷ C₁₇×C₃₂ ▷ C₁₆×C₁₇ ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
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. #14531093	C₂	semidirect
2. Dic₅₄₄	C₂	semidirect
3. #14527000	C₂	semidirect
4. D₁₀₈₈	C₂	semidirect
5. C₂×C₁₇×C₆₄	C₂	semidirect
39. C₂×C₆₄	D₁₇	semidirect
62. C₁₇	#13741	semidirect

Maximal subgroups

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

subgroup	index
#14531093	2
Dic₅₄₄	2
#14527000	2
D₁₀₈₈	2
C₂×C₁₇×C₆₄	2
#13741	17