Subgroups — #16955364

Order 4352.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14531077	C₂	1	1	yes	maximal, characteristic
2	#14526994	C₂	1	2	yes	maximal, complemented
3	C₂×#11762528	C₂	1	1	yes	maximal, characteristic, complemented
4	D₁₇×C₆₄	C₂	1	2	yes	maximal, complemented
5	C₁₇×M₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763225	C₄	1	1	yes	characteristic
7	#11762643	C₄	1	1	yes	characteristic
8	#11762642	C₄	1	2	yes	—
9	C₃₂×D₁₇	C₂²	1	2	yes	—
10	#11762528	C₂²	1	1	yes	characteristic
11	#11762528	C₂²	1	1	yes	characteristic
12	C₂×C₁₇×C₃₂	C₂²	1	1	yes	characteristic
13	C₁₇×C₆₄	C₂²	1	2	yes	—
14	#10587587	C₈	1	1	yes	characteristic
15	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
16	#10587483	C₈	1	1	yes	characteristic
17	#10587469	C₈	1	2	yes	—
18	C₁₆×D₁₇	C₂×C₄	1	2	yes	—
19	C₁₇×C₃₂	C₂×C₄	1	1	yes	characteristic
20	C₁₇×C₃₂	C₂³	1	1	yes	characteristic
21	#10587465	C₂×C₄	1	2	yes	—
22	#63252	C₁₆	1	1	yes	characteristic
23	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
24	#63223	C₁₆	1	1	yes	characteristic
25	#63218	C₁₆	1	2	yes	—
26	C₈×D₁₇	C₂×C₈	1	2	yes	—
27	C₁₆×C₁₇	C₂×C₈	1	1	yes	characteristic
28	C₁₆×C₁₇	C₂²×C₄	1	1	yes	characteristic
29	#63214	C₂×C₈	1	2	yes	—
30	#13737	—	17	1	no	maximal, Sylow
31	C₂×C₄×C₁₇	C₂×C₁₆	1	1	yes	characteristic
32	C₂²⋊D₁₇	C₃₂	1	2	yes	—
33	D₆₈	C₃₂	1	1	yes	characteristic
34	C₄×D₁₇	C₂×C₁₆	1	2	yes	—
35	Dic₃₄	C₃₂	1	1	yes	characteristic
36	C₈×C₁₇	C₂×C₁₆	1	1	yes	characteristic
37	C₈×C₁₇	C₂²×C₈	1	1	yes	characteristic
38	#3622	C₂×C₁₆	1	2	yes	—
39	#2258	—	17	1	no	—
40	M₁₂₈	—	17	2	no	—
41	M₁₂₈	D₁₇	1	1	yes	characteristic, complemented
42	C₂×C₆₄	—	17	1	no	—
43	C₂×C₆₄	—	17	2	no	—
44	C₂²×C₁₇	C₂×C₃₂	1	1	yes	characteristic
45	C₂×D₁₇	C₂×C₃₂	1	2	yes	—
46	C₄×C₁₇	C₂×C₃₂	1	1	yes	characteristic
47	C₄×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
48	Dic₁₇	C₂×C₃₂	1	2	yes	—
49	#504	—	17	1	no	—
50	M₆₄	—	17	1	no	—
51	M₆₄	—	17	2	no	—
52	C₂×C₃₂	—	17	2	no	—
53	C₂×C₃₂	C₂×D₁₇	1	1	yes	characteristic
54	C₆₄	—	17	1	no	—
55	C₆₄	—	17	1	no	—
56	C₆₄	C₂×D₁₇	1	2	yes	—
57	C₂×C₁₇	—	2	1	no	—
58	C₂×C₁₇	C₂²×C₃₂	1	1	yes	characteristic
59	D₁₇	—	2	2	no	—
60	#131	—	17	1	no	—
61	M₃₂	—	17	1	no	—
62	M₃₂	—	17	2	no	—
63	C₂×C₁₆	—	17	2	no	—
64	C₂×C₁₆	C₄×D₁₇	1	1	yes	characteristic
65	C₃₂	—	17	2	no	—
66	C₃₂	C₄×D₁₇	1	1	yes	characteristic
67	C₃₂	C₂²×D₁₇	1	1	yes	characteristic
68	C₁₇	#13737	1	1	yes	characteristic, complemented, Sylow
69	#41	—	17	1	no	—
70	M₁₆	—	17	1	no	—
71	M₁₆	—	17	2	no	—
72	C₂×C₈	—	17	2	no	—
73	C₂×C₈	C₈×D₁₇	1	1	yes	characteristic
74	C₁₆	—	17	2	no	—
75	C₁₆	C₈×D₁₇	1	1	yes	characteristic
76	C₁₆	C₂×C₄×D₁₇	1	1	yes	characteristic
77	Q₈	—	17	1	no	—
78	D₄	—	17	2	no	—
79	D₄	—	17	1	no	—
80	C₂×C₄	—	17	2	no	—
81	C₂×C₄	C₁₆×D₁₇	1	1	yes	characteristic
82	C₈	—	17	2	no	—
83	C₈	C₁₆×D₁₇	1	1	yes	characteristic
84	C₈	C₂×C₈×D₁₇	1	1	yes	characteristic
85	C₂²	—	17	2	no	—
86	C₂²	C₃₂×D₁₇	1	1	yes	characteristic
87	C₄	—	17	2	no	—
88	C₄	C₃₂×D₁₇	1	1	yes	characteristic
89	C₄	C₂×C₁₆×D₁₇	1	1	yes	characteristic
90	C₂	—	2	1	no	—
91	C₂	—	34	2	no	—
92	C₂	C₂×C₃₂×D₁₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃₂	C₂²×D₁₇
commutator, socle	C₂×C₁₇	C₂²×C₃₂
Fitting	C₁₇×M₁₂₈	C₂
prime radical	#63252	C₁₆
cosocle	C₁₇×C₃₂	C₂³
hypercenter	M₁₂₈	D₁₇
hypocenter	C₁₇	#13737
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₃₂ ◁ M₁₂₈

Complements

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

N	H ≅ G/N	type
2. #14526994	C₂	semidirect
3. C₂×#11762528	C₂	semidirect
4. D₁₇×C₆₄	C₂	semidirect
5. C₁₇×M₁₂₈	C₂	semidirect
41. M₁₂₈	D₁₇	semidirect
68. C₁₇	#13737	semidirect

Maximal subgroups

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

subgroup	index
#14531077	2
#14526994	2
C₂×#11762528	2
D₁₇×C₆₄	2
C₁₇×M₁₂₈	2
#13737	17