Subgroups — #16949036

Order 4352.

398 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14531034	C₂	1	4	yes	maximal, complemented
2	#14528399	C₂	1	2	yes	maximal, complemented
3	C₈×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₂₇₂	—	2	8	no	—
5	#11763186	—	2	4	no	—
6	#11763186	C₂²	1	4	yes	—
7	C₄×C₁₆×C₁₇	C₂²	1	2	yes	—
8	C₈²×C₁₇	C₂²	1	1	yes	characteristic
9	C₂×D₁₃₆	—	2	8	no	—
10	C₂×D₁₃₆	—	4	8	no	—
11	#10587560	—	2	4	no	—
12	C₂×C₁₆×C₁₇	D₄	1	4	yes	—
13	C₄×C₈×C₁₇	D₄	1	2	yes	—
14	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
15	D₂₇₂	—	4	16	no	—
16	C₂×D₆₈	—	4	4	no	—
17	C₂×D₆₈	—	4	8	no	—
18	C₂×C₈×C₁₇	D₈	1	4	yes	—
19	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
20	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
21	D₁₃₆	—	4	16	no	—
22	D₁₃₆	—	8	16	no	—
23	C₁₆×C₁₇	D₈	1	8	yes	complemented
24	#12515	—	17	1	no	maximal, Sylow
25	C₂²×D₁₇	—	8	4	no	—
26	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
27	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
28	D₆₈	—	8	8	no	—
29	D₆₈	—	8	16	no	—
30	C₈×C₁₇	D₁₆	1	8	yes	complemented
31	C₈×C₁₇	C₂×D₈	1	4	yes	—
32	#2246	—	17	4	no	—
33	#1712	—	17	2	no	—
34	C₈×C₁₆	D₁₇	1	1	yes	characteristic, complemented
35	C₂²×C₁₇	#493	1	1	yes	characteristic
36	C₂×D₁₇	—	16	4	no	—
37	C₂×D₁₇	—	16	8	no	—
38	C₄×C₁₇	#493	1	2	yes	—
39	C₄×C₁₇	C₂×D₁₆	1	4	yes	—
40	C₂×D₁₆	—	34	8	no	—
41	#493	—	17	4	no	—
42	#493	—	34	4	no	—
43	C₄×C₁₆	C₂×D₁₇	1	2	yes	—
44	C₈²	C₂×D₁₇	1	1	yes	characteristic
45	C₂×C₁₇	#1712	1	1	yes	characteristic
46	C₂×C₁₇	#2246	1	2	yes	—
47	D₁₇	—	32	4	no	—
48	C₂×D₈	—	34	8	no	—
49	C₂×D₈	—	68	8	no	—
50	#127	—	34	4	no	—
51	D₁₆	—	68	16	no	—
52	C₂×C₁₆	D₆₈	1	4	yes	—
53	C₄×C₈	D₆₈	1	2	yes	—
54	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
55	C₁₇	#12515	1	1	yes	characteristic, complemented, Sylow
56	C₂×D₄	—	68	4	no	—
57	C₂×D₄	—	68	8	no	—
58	D₈	—	68	16	no	—
59	D₈	—	136	16	no	—
60	C₂×C₈	D₁₃₆	1	4	yes	—
61	C₂×C₈	C₂×D₆₈	1	2	yes	—
62	C₄²	C₂×D₆₈	1	1	yes	characteristic
63	C₁₆	D₁₃₆	1	8	yes	complemented
64	C₂³	—	136	4	no	—
65	D₄	—	136	16	no	—
66	D₄	—	136	8	no	—
67	C₂×C₄	#10587560	1	1	yes	characteristic
68	C₂×C₄	C₂×D₁₃₆	1	2	yes	—
69	C₈	D₂₇₂	1	8	yes	complemented
70	C₈	C₂×D₁₃₆	1	4	yes	—
71	C₂²	—	272	4	no	—
72	C₂²	—	272	8	no	—
73	C₂²	#11763186	1	1	yes	characteristic
74	C₄	#11763186	1	2	yes	—
75	C₄	C₂×D₂₇₂	1	4	yes	—
76	C₂	—	544	4	no	—
77	C₂	#14528399	1	1	yes	characteristic
78	C₂	#14531034	1	2	yes	—

	subgroup	G/H
center	C₂²	#11763186
commutator, cosocle	C₄×C₈×C₁₇	C₂³
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₈×C₁₆×C₁₇	C₂
socle	C₂²×C₁₇	#493
hypercenter	C₈×C₁₆	D₁₇
hypocenter	C₁₇	#12515
LCS	G ▷ C₄×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
1. #14531034	C₂	semidirect
2. #14528399	C₂	semidirect
3. C₈×C₁₆×C₁₇	C₂	semidirect
23. C₁₆×C₁₇	D₈	semidirect
30. C₈×C₁₇	D₁₆	semidirect
34. C₈×C₁₆	D₁₇	semidirect
55. C₁₇	#12515	semidirect
63. C₁₆	D₁₃₆	semidirect
69. C₈	D₂₇₂	semidirect

Maximal subgroups

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

subgroup	index
#14531034	2
#14528399	2
C₈×C₁₆×C₁₇	2
#12515	17