Subgroups — #11763652

Order 1088.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊Dic₁₇	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊#3624	C₂	1	2	yes	maximal, complemented
3	C₂⁴×C₁₇	C₄	1	1	yes	characteristic
4	C₂²×Dic₁₇	C₄	1	1	yes	characteristic
5	C₂²×Dic₁₇	C₂²	1	1	yes	characteristic
6	C₂×#3624	—	2	2	no	—
7	C₂²⋊Dic₁₇	—	2	2	no	—
8	C₂³×C₁₇	—	2	1	no	—
9	C₂³×C₁₇	—	2	1	no	—
10	C₂³×C₁₇	—	4	1	no	—
11	C₂³×C₁₇	—	4	1	no	—
12	C₂³×C₁₇	C₈	1	2	yes	complemented
13	C₂³×C₁₇	C₂×C₄	1	1	yes	characteristic
14	C₂×Dic₁₇	—	2	1	no	—
15	C₂×Dic₁₇	—	4	1	no	—
16	C₂×Dic₁₇	—	4	1	no	—
17	C₂×Dic₁₇	D₄	1	2	yes	—
18	#3624	—	4	2	no	—
19	C₂²×C₁₇	—	2	1	no	—
20	C₂²×C₁₇	—	2	1	no	—
21	C₂²×C₁₇	—	2	2	no	—
22	C₂²×C₁₇	—	4	2	no	—
23	C₂²×C₁₇	—	4	2	no	—
24	C₂²×C₁₇	—	4	1	no	—
25	C₂²×C₁₇	—	4	1	no	—
26	C₂²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₁₇	C₂×C₈	1	1	yes	characteristic
28	C₂²×C₁₇	M₁₆	1	1	yes	characteristic
29	Dic₁₇	—	2	2	no	—
30	Dic₁₇	—	4	1	no	—
31	#323	—	17	1	no	maximal, Sylow
32	C₂×C₁₇	—	2	1	no	—
33	C₂×C₁₇	—	2	1	no	—
34	C₂×C₁₇	—	4	2	no	—
35	C₂×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
36	C₂×C₁₇	#99	1	1	yes	characteristic
37	C₂×C₁₇	#100	1	1	yes	characteristic
38	C₂×C₂²⋊C₄	—	17	1	no	—
39	C₂²⋊C₈	—	17	2	no	—
40	C₁₇	#323	1	1	yes	characteristic, complemented, Sylow
41	C₂⁴	#595	1	1	yes	characteristic
42	C₂²×C₄	—	17	1	no	—
43	C₂²×C₄	—	17	1	no	—
44	C₂×C₈	—	34	2	no	—
45	C₂²⋊C₄	—	34	2	no	—
46	C₂³	—	2	1	no	—
47	C₂³	—	2	1	no	—
48	C₂³	—	4	1	no	—
49	C₂³	—	4	1	no	—
50	C₂³	#3624	1	2	yes	complemented
51	C₂³	C₂×#595	1	1	yes	characteristic
52	C₂×C₄	—	17	2	no	—
53	C₂×C₄	—	34	1	no	—
54	C₂×C₄	—	68	1	no	—
55	C₂×C₄	—	68	1	no	—
56	C₈	—	68	2	no	—
57	C₂²	—	2	1	no	—
58	C₂²	—	2	1	no	—
59	C₂²	—	2	2	no	—
60	C₂²	—	4	2	no	—
61	C₂²	—	4	2	no	—
62	C₂²	—	4	1	no	—
63	C₂²	—	4	1	no	—
64	C₂²	C₂×#3624	1	1	yes	characteristic
65	C₂²	#63247	1	1	yes	characteristic
66	C₂²	C₂²⋊#595	1	1	yes	characteristic
67	C₄	—	34	2	no	—
68	C₄	—	68	1	no	—
69	C₂	—	2	1	no	—
70	C₂	—	2	1	no	—
71	C₂	—	4	2	no	—
72	C₂	#10587551	1	1	yes	characteristic
73	C₂	C₂²⋊#3624	1	1	yes	characteristic
74	C₂	#10587553	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#595
commutator	C₂²×C₁₇	C₂×C₈
Frattini	C₂³	C₂×#595
Fitting, prime radical	C₂⁴×C₁₇	C₄
socle	C₂²×C₁₇	C₂²⋊C₄
cosocle	C₂²×Dic₁₇	C₂²
hypercenter	C₂⁴	#595
hypocenter	C₁₇	#323
LCS	G ▷ C₂²×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
2. C₂²⋊#3624	C₂	semidirect
12. C₂³×C₁₇	C₈	semidirect
40. C₁₇	#323	semidirect
50. C₂³	#3624	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊Dic₁₇	2
C₂²⋊#3624	2
#323	17