Subgroups — C₂²×#63223 #11763569

Order 1088.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×C₁₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#63223	C₂	1	12	yes	maximal, direct product factor
3	C₂²×#3622	C₂	1	2	yes	maximal, complemented
4	C₂⁴×C₁₇	C₄	1	1	yes	characteristic
5	C₂²×C₄×C₁₇	C₄	1	6	yes	—
6	C₂²×C₄×C₁₇	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₁₇	C₂²	1	6	yes	—
8	C₂²×C₄×C₁₇	C₂²	1	1	yes	characteristic
9	#63223	C₂²	1	16	yes	direct product factor
10	C₂×#3622	C₂²	1	12	yes	complemented
11	C₂³×C₁₇	—	2	4	no	—
12	C₂³×C₁₇	C₂×C₄	1	6	yes	—
13	C₂³×C₁₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₁₇	C₂×C₄	1	12	yes	—
15	C₂×C₄×C₁₇	C₂×C₄	1	6	yes	—
16	C₂×C₄×C₁₇	C₂×C₄	1	3	yes	—
17	C₂×C₄×C₁₇	C₂³	1	3	yes	—
18	C₂×C₄×C₁₇	C₂³	1	4	yes	—
19	#3622	C₂³	1	8	yes	complemented
20	C₂²×C₁₇	—	2	12	no	—
21	C₂²×C₁₇	M₁₆	1	4	yes	complemented
22	C₂²×C₁₇	C₂²×C₄	1	4	yes	—
23	C₂²×C₁₇	C₂²×C₄	1	3	yes	—
24	C₄×C₁₇	C₂²×C₄	1	3	yes	—
25	C₄×C₁₇	C₂²×C₄	1	4	yes	—
26	C₄×C₁₇	C₂⁴	1	1	yes	characteristic
27	C₂²×M₁₆	—	17	1	no	maximal, Sylow
28	C₂×C₁₇	—	2	4	no	—
29	C₂×C₁₇	C₂×M₁₆	1	6	yes	complemented
30	C₂×C₁₇	C₂³×C₄	1	1	yes	characteristic
31	C₂³×C₄	D₁₇	1	1	yes	characteristic
32	C₂×M₁₆	—	17	12	no	—
33	C₂²×C₈	—	17	2	no	—
34	C₁₇	C₂²×M₁₆	1	1	yes	characteristic, complemented, Sylow
35	C₂⁴	Dic₁₇	1	1	yes	characteristic
36	C₂²×C₄	Dic₁₇	1	1	yes	characteristic
37	C₂²×C₄	Dic₁₇	1	6	yes	—
38	C₂²×C₄	C₂×D₁₇	1	6	yes	—
39	C₂²×C₄	C₂×D₁₇	1	1	yes	characteristic
40	M₁₆	—	17	16	no	—
41	C₂×C₈	—	17	12	no	—
42	C₂³	—	2	4	no	—
43	C₂³	C₂×Dic₁₇	1	6	yes	—
44	C₂³	C₂×Dic₁₇	1	1	yes	characteristic
45	C₂×C₄	C₂×Dic₁₇	1	12	yes	—
46	C₂×C₄	C₂×Dic₁₇	1	6	yes	—
47	C₂×C₄	C₂×Dic₁₇	1	3	yes	—
48	C₂×C₄	C₂²×D₁₇	1	3	yes	—
49	C₂×C₄	C₂²×D₁₇	1	4	yes	—
50	C₈	—	17	8	no	—
51	C₂²	—	2	12	no	—
52	C₂²	#63223	1	4	yes	direct product factor
53	C₂²	C₂²×Dic₁₇	1	4	yes	—
54	C₂²	C₂²×Dic₁₇	1	3	yes	—
55	C₄	C₂²×Dic₁₇	1	4	yes	—
56	C₄	C₂²×Dic₁₇	1	3	yes	—
57	C₄	C₂³×D₁₇	1	1	yes	characteristic
58	C₂	—	2	4	no	—
59	C₂	C₂×#63223	1	6	yes	direct product factor
60	C₂	C₂³×Dic₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₇
commutator	C₂×C₁₇	C₂³×C₄
Frattini	C₄	C₂³×D₁₇
Fitting	C₂³×C₄×C₁₇	C₂
socle	C₂³×C₁₇	C₂×C₄
prime radical	C₂⁴×C₁₇	C₄
cosocle	C₄×C₁₇	C₂⁴
hypercenter	C₂³×C₄	D₁₇
hypocenter	C₁₇	C₂²×M₁₆
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×#63223	C₂	direct (N × H)
3. C₂²×#3622	C₂	semidirect
9. #63223	C₂²	direct (N × H)
10. C₂×#3622	C₂²	semidirect
19. #3622	C₂³	semidirect
21. C₂²×C₁₇	M₁₆	semidirect
29. C₂×C₁₇	C₂×M₁₆	semidirect
34. C₁₇	C₂²×M₁₆	semidirect
52. C₂²	#63223	direct (N × H)
59. C₂	C₂×#63223	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄×C₁₇	2
C₂×#63223	2
C₂²×#3622	2
C₂²×M₁₆	17

Subgroups — C22×#63223 #11763569

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#63223 #11763569