Subgroups — #16955319

Order 4352.

270 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₅₄₄	C₂	1	4	yes	maximal, complemented
2	#14531034	C₂	1	2	yes	maximal, complemented
3	C₄×C₁₇×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₂₇₂	—	2	2	no	—
5	C₂×D₂₇₂	C₂²	1	4	yes	—
6	#11763186	—	2	2	no	—
7	D₅₄₄	—	2	8	no	—
8	C₂×C₁₇×C₃₂	C₂²	1	2	yes	—
9	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
10	C₂×D₁₃₆	—	2	4	no	—
11	C₂×D₁₃₆	—	4	2	no	—
12	#10587560	—	4	2	no	—
13	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
14	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
15	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
16	D₂₇₂	—	2	8	no	—
17	D₂₇₂	—	4	4	no	—
18	C₁₇×C₃₂	D₄	1	4	yes	complemented
19	C₂×D₆₈	—	4	4	no	—
20	C₂×D₆₈	—	8	4	no	—
21	C₂×C₈×C₁₇	D₈	1	1	yes	characteristic
22	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
23	C₄²×C₁₇	D₈	1	1	yes	characteristic
24	D₁₃₆	—	4	4	no	—
25	D₁₃₆	—	4	4	no	—
26	D₁₃₆	—	8	4	no	—
27	C₁₆×C₁₇	D₈	1	2	yes	—
28	C₁₆×C₁₇	C₂×D₄	1	2	yes	—
29	#13725	—	17	1	no	maximal, Sylow
30	C₂²×D₁₇	—	8	4	no	—
31	C₂×C₄×C₁₇	D₁₆	1	2	yes	—
32	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
33	D₆₈	—	8	4	no	—
34	D₆₈	—	8	4	no	—
35	D₆₈	—	16	8	no	—
36	C₈×C₁₇	D₁₆	1	2	yes	—
37	C₈×C₁₇	#127	1	1	yes	characteristic
38	C₈×C₁₇	C₂×D₈	1	1	yes	characteristic
39	C₂×D₃₂	—	17	4	no	—
40	#2246	—	17	2	no	—
41	C₄×C₃₂	D₁₇	1	1	yes	characteristic, complemented
42	C₂²×C₁₇	C₂×D₁₆	1	1	yes	characteristic
43	C₂×D₁₇	—	16	4	no	—
44	C₂×D₁₇	—	16	8	no	—
45	C₄×C₁₇	D₃₂	1	4	yes	complemented
46	C₄×C₁₇	#493	1	1	yes	characteristic
47	C₄×C₁₇	C₂×D₁₆	1	1	yes	characteristic
48	C₂×D₁₆	—	17	4	no	—
49	C₂×D₁₆	—	34	2	no	—
50	#493	—	34	2	no	—
51	D₃₂	—	34	8	no	—
52	C₂×C₃₂	C₂×D₁₇	1	2	yes	—
53	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
54	C₂×C₁₇	#2246	1	1	yes	characteristic
55	C₂×C₁₇	C₂×D₃₂	1	2	yes	—
56	D₁₇	—	32	4	no	—
57	C₂×D₈	—	34	4	no	—
58	C₂×D₈	—	68	2	no	—
59	#127	—	68	2	no	—
60	D₁₆	—	34	8	no	—
61	D₁₆	—	68	4	no	—
62	C₂×C₁₆	D₆₈	1	1	yes	characteristic
63	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
64	C₄×C₈	D₆₈	1	1	yes	characteristic
65	C₃₂	D₆₈	1	4	yes	complemented
66	C₁₇	#13725	1	1	yes	characteristic, complemented, Sylow
67	C₂×D₄	—	68	4	no	—
68	C₂×D₄	—	136	4	no	—
69	D₈	—	68	4	no	—
70	D₈	—	68	4	no	—
71	D₈	—	136	4	no	—
72	C₂×C₈	D₁₃₆	1	1	yes	characteristic
73	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
74	C₄²	D₁₃₆	1	1	yes	characteristic
75	C₁₆	D₁₃₆	1	2	yes	—
76	C₁₆	C₂×D₆₈	1	2	yes	—
77	C₂³	—	136	4	no	—
78	D₄	—	272	8	no	—
79	D₄	—	136	4	no	—
80	D₄	—	136	4	no	—
81	C₂×C₄	D₂₇₂	1	2	yes	—
82	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
83	C₈	D₂₇₂	1	2	yes	—
84	C₈	#10587560	1	1	yes	characteristic
85	C₈	C₂×D₁₃₆	1	1	yes	characteristic
86	C₂²	—	272	8	no	—
87	C₂²	—	272	4	no	—
88	C₂²	C₂×D₂₇₂	1	1	yes	characteristic
89	C₄	D₅₄₄	1	4	yes	complemented
90	C₄	#11763186	1	1	yes	characteristic
91	C₄	C₂×D₂₇₂	1	1	yes	characteristic
92	C₂	—	544	4	no	—
93	C₂	#14531034	1	1	yes	characteristic
94	C₂	C₂×D₅₄₄	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₂₇₂
commutator, cosocle	C₂×C₁₆×C₁₇	C₂³
Frattini	C₂×C₁₆	C₂²×D₁₇
Fitting	C₄×C₁₇×C₃₂	C₂
socle	C₂²×C₁₇	C₂×D₁₆
hypercenter	C₄×C₃₂	D₁₇
hypocenter	C₁₇	#13725
LCS	G ▷ C₂×C₁₆×C₁₇ ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×D₅₄₄	C₂	semidirect
2. #14531034	C₂	semidirect
3. C₄×C₁₇×C₃₂	C₂	semidirect
18. C₁₇×C₃₂	D₄	semidirect
41. C₄×C₃₂	D₁₇	semidirect
45. C₄×C₁₇	D₃₂	semidirect
65. C₃₂	D₆₈	semidirect
66. C₁₇	#13725	semidirect
89. C₄	D₅₄₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₅₄₄	2
#14531034	2
C₄×C₁₇×C₃₂	2
#13725	17