Subgroups — #13109656

Order 1632.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11717860	C₂	1	1	yes	maximal, characteristic
2	#11717845	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#63242	C₂	1	1	yes	maximal, characteristic, complemented
4	#11717827	C₂	1	2	yes	maximal, complemented
5	#11717825	C₂	1	2	yes	maximal, complemented
6	#10587680	—	3	1	no	maximal, Hall
7	D₂₀₄	C₄	1	1	yes	characteristic
8	C₁₇×Dic₆	C₄	1	1	yes	characteristic
9	C₃×C₄×D₁₇	C₂²	1	1	yes	characteristic
10	#87974	C₄	1	2	yes	—
11	#87972	C₂²	1	2	yes	—
12	C₃⋊#3624	C₂²	1	2	yes	—
13	C₃×#3624	C₂²	1	1	yes	characteristic
14	C₃×#3624	C₂²	1	1	yes	characteristic
15	#63256	—	3	1	no	—
16	#63243	—	3	1	no	—
17	#63243	—	3	2	no	—
18	#63242	—	3	2	no	—
19	#63242	D₃	1	1	yes	characteristic, complemented
20	C₂×D₅₁	C₂×C₄	1	2	yes	—
21	C₂×C₃×D₁₇	C₂×C₄	1	1	yes	characteristic
22	C₃×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₃×Dic₁₇	C₂³	1	1	yes	characteristic
24	Dic₃×C₁₇	C₂×C₄	1	2	yes	—
25	Q₈×C₁₇	—	3	1	no	—
26	D₆₈	—	3	2	no	—
27	D₆₈	—	3	1	no	—
28	C₄×D₁₇	—	3	2	no	—
29	C₄×D₁₇	C₂×D₃	1	1	yes	characteristic
30	#3624	—	3	2	no	—
31	#3624	C₂×D₃	1	1	yes	characteristic
32	#3624	C₂×D₃	1	1	yes	characteristic
33	C₂×C₃×C₁₇	C₂²×C₄	1	1	yes	characteristic
34	D₅₁	—	2	2	no	—
35	C₃×D₁₇	—	2	1	no	—
36	#901	—	17	1	no	maximal, Hall
37	C₂×D₁₇	—	3	2	no	—
38	C₂×D₁₇	C₄×D₃	1	1	yes	characteristic
39	C₄×C₁₇	—	3	2	no	—
40	C₄×C₁₇	C₄×D₃	1	1	yes	characteristic
41	Dic₁₇	C₂²×D₃	1	1	yes	characteristic
42	C₃×C₁₇	#131	1	1	yes	characteristic, complemented, Hall
43	#235	—	17	1	no	—
44	C₂×C₃×C₈	—	17	1	no	—
45	#208	—	17	1	no	—
46	ΓL(1,𝔽₂₅)	—	17	2	no	—
47	D₃×C₈	—	17	2	no	—
48	C₂×C₁₇	C₂×C₄×D₃	1	1	yes	characteristic
49	D₁₇	—	2	1	no	—
50	D₁₇	—	6	2	no	—
51	#131	—	51	1	no	Sylow
52	C₂×C₃×C₄	—	17	1	no	—
53	#67	—	17	2	no	—
54	D₁₂	—	17	1	no	—
55	C₄×D₃	—	17	2	no	—
56	Dic₆	#595	1	1	yes	characteristic
57	C₃×C₈	—	17	1	no	—
58	C₃×C₈	—	17	1	no	—
59	C₃⋊C₈	—	17	2	no	—
60	C₁₇	#901	1	1	yes	characteristic, complemented, Sylow
61	#41	—	51	1	no	—
62	M₁₆	—	51	2	no	—
63	M₁₆	—	51	1	no	—
64	C₂×C₈	—	17	1	no	—
65	C₂×C₈	—	51	2	no	—
66	C₂²×C₃	—	17	1	no	—
67	C₂×D₃	—	17	2	no	—
68	C₃×C₄	—	17	1	no	—
69	C₃×C₄	C₂×#595	1	1	yes	characteristic
70	Dic₃	C₂×#595	1	2	yes	—
71	Q₈	—	3	1	no	—
72	D₄	—	51	2	no	—
73	D₄	—	51	1	no	—
74	C₂×C₄	—	17	1	no	—
75	C₂×C₄	—	51	2	no	—
76	C₈	—	17	1	no	—
77	C₈	—	17	1	no	—
78	C₈	—	51	2	no	—
79	C₂×C₃	—	34	1	no	—
80	C₂×C₃	C₂²×#595	1	1	yes	characteristic
81	D₃	—	34	2	no	—
82	C₂²	—	17	1	no	—
83	C₂²	—	51	2	no	—
84	C₄	—	3	2	no	—
85	C₄	—	17	1	no	—
86	C₄	D₃×#595	1	1	yes	characteristic
87	C₃	#10587680	1	1	yes	characteristic, complemented, Sylow
88	C₂	—	34	1	no	—
89	C₂	—	102	2	no	—
90	C₂	C₂×D₃×#595	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×D₃×#595
commutator, socle	C₂×C₃×C₁₇	C₂²×C₄
Fitting	C₃×C₄×C₁₇	C₂×C₄
prime radical	#11717860	C₂
cosocle	C₃×Dic₁₇	C₂³
hypercenter	C₄	D₃×#595
hypocenter	C₃×C₁₇	#131
LCS	G ▷ C₂×C₃×C₁₇ ▷ C₃×C₁₇
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. #11717845	C₂	semidirect
3. C₃×#63242	C₂	semidirect
4. #11717827	C₂	semidirect
5. #11717825	C₂	semidirect
19. #63242	D₃	semidirect
42. C₃×C₁₇	#131	semidirect
60. C₁₇	#901	semidirect
87. C₃	#10587680	semidirect

Maximal subgroups

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

subgroup	index
#11717860	2
#11717845	2
C₃×#63242	2
#11717827	2
#11717825	2
#10587680	3
#901	17