Subgroups — C₂³×#63242 #14545768

Order 2176.

1976 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#63242	C₂	1	28	yes	maximal, direct product factor
2	C₂³×#3624	C₂	1	2	yes	maximal, complemented
3	C₂³×C₄×D₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂⁴×D₁₇	C₄	1	1	yes	characteristic
5	C₂³×C₄×C₁₇	C₄	1	1	yes	characteristic
6	C₂³×Dic₁₇	C₂²	1	1	yes	characteristic
7	C₂²×C₄×D₁₇	C₄	1	14	yes	—
8	C₂²×C₄×D₁₇	C₂²	1	14	yes	—
9	C₂²×#3624	C₂²	1	28	yes	complemented
10	C₂×#63242	C₂²	1	112	yes	direct product factor
11	C₂⁴×C₁₇	C₂×C₄	1	1	yes	characteristic
12	C₂³×D₁₇	C₈	1	16	yes	complemented
13	C₂³×D₁₇	C₂×C₄	1	14	yes	—
14	C₂²×C₄×C₁₇	C₂×C₄	1	14	yes	—
15	C₂²×Dic₁₇	C₂×C₄	1	7	yes	—
16	C₂²×Dic₁₇	C₂³	1	7	yes	—
17	C₂×C₄×D₁₇	C₂×C₄	1	84	yes	—
18	C₂×C₄×D₁₇	C₂³	1	28	yes	—
19	C₂×#3624	C₂³	1	56	yes	complemented
20	#63242	C₂³	1	64	yes	direct product factor
21	C₂³×C₁₇	C₂×C₈	1	8	yes	complemented
22	C₂³×C₁₇	C₂²×C₄	1	7	yes	—
23	C₂²×D₁₇	C₂×C₈	1	112	yes	complemented
24	C₂²×D₁₇	C₂²×C₄	1	28	yes	—
25	C₂×C₄×C₁₇	C₂²×C₄	1	28	yes	—
26	C₂×Dic₁₇	C₂²×C₄	1	21	yes	—
27	C₂×Dic₁₇	C₂⁴	1	7	yes	—
28	C₄×D₁₇	C₂²×C₄	1	56	yes	—
29	C₄×D₁₇	C₂⁴	1	8	yes	—
30	#3624	C₂⁴	1	16	yes	complemented
31	C₂⁴×C₈	—	17	1	no	maximal, Sylow
32	C₂²×C₁₇	C₂²×C₈	1	28	yes	complemented
33	C₂²×C₁₇	C₂³×C₄	1	7	yes	—
34	C₂×D₁₇	C₂²×C₈	1	112	yes	complemented
35	C₂×D₁₇	C₂³×C₄	1	8	yes	—
36	C₄×C₁₇	C₂³×C₄	1	8	yes	—
37	Dic₁₇	C₂³×C₄	1	7	yes	—
38	Dic₁₇	C₂⁵	1	1	yes	characteristic
39	C₂⁴×C₄	—	17	1	no	—
40	C₂³×C₈	—	17	2	no	—
41	C₂³×C₈	—	17	28	no	—
42	C₂×C₁₇	C₂³×C₈	1	14	yes	complemented
43	C₂×C₁₇	C₂⁴×C₄	1	1	yes	characteristic
44	D₁₇	C₂³×C₈	1	16	yes	complemented
45	C₂⁵	—	17	1	no	—
46	C₂³×C₄	—	17	1	no	—
47	C₂³×C₄	—	17	14	no	—
48	C₂³×C₄	—	17	14	no	—
49	C₂³×C₄	#595	1	1	yes	characteristic
50	C₂²×C₈	—	17	112	no	—
51	C₂²×C₈	—	17	28	no	—
52	C₁₇	C₂⁴×C₈	1	1	yes	characteristic, complemented, Sylow
53	C₂⁴	—	17	16	no	—
54	C₂⁴	—	17	14	no	—
55	C₂⁴	C₂×#595	1	1	yes	characteristic
56	C₂²×C₄	—	17	7	no	—
57	C₂²×C₄	—	17	7	no	—
58	C₂²×C₄	—	17	84	no	—
59	C₂²×C₄	—	17	28	no	—
60	C₂²×C₄	C₂×#595	1	14	yes	—
61	C₂×C₈	—	17	64	no	—
62	C₂×C₈	—	17	56	no	—
63	C₂³	—	17	112	no	—
64	C₂³	—	17	28	no	—
65	C₂³	#63242	1	8	yes	direct product factor
66	C₂³	C₂²×#595	1	7	yes	—
67	C₂×C₄	—	17	7	no	—
68	C₂×C₄	—	17	21	no	—
69	C₂×C₄	—	17	8	no	—
70	C₂×C₄	—	17	56	no	—
71	C₂×C₄	C₂²×#595	1	28	yes	—
72	C₈	—	17	16	no	—
73	C₂²	—	17	112	no	—
74	C₂²	—	17	8	no	—
75	C₂²	C₂×#63242	1	28	yes	direct product factor
76	C₂²	C₂³×#595	1	7	yes	—
77	C₄	—	17	1	no	—
78	C₄	—	17	7	no	—
79	C₄	C₂³×#595	1	8	yes	—
80	C₂	—	17	16	no	—
81	C₂	C₂²×#63242	1	14	yes	direct product factor
82	C₂	C₂⁴×#595	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₄	#595
commutator, hypocenter	C₁₇	C₂⁴×C₈
Frattini	C₂	C₂⁴×#595
Fitting	C₂³×C₄×C₁₇	C₄
socle	C₂⁴×C₁₇	C₂×C₄
prime radical	C₂⁴×D₁₇	C₄
cosocle	Dic₁₇	C₂⁵
LCS	G ▷ C₁₇
UCS	C₁ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#63242	C₂	direct (N × H)
2. C₂³×#3624	C₂	semidirect
9. C₂²×#3624	C₂²	semidirect
10. C₂×#63242	C₂²	direct (N × H)
12. C₂³×D₁₇	C₈	semidirect
19. C₂×#3624	C₂³	semidirect
20. #63242	C₂³	direct (N × H)
21. C₂³×C₁₇	C₂×C₈	semidirect
23. C₂²×D₁₇	C₂×C₈	semidirect
30. #3624	C₂⁴	semidirect
32. C₂²×C₁₇	C₂²×C₈	semidirect
34. C₂×D₁₇	C₂²×C₈	semidirect
42. C₂×C₁₇	C₂³×C₈	semidirect
44. D₁₇	C₂³×C₈	semidirect
52. C₁₇	C₂⁴×C₈	semidirect
65. C₂³	#63242	direct (N × H)
75. C₂²	C₂×#63242	direct (N × H)
81. C₂	C₂²×#63242	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#63242	2
C₂³×#3624	2
C₂³×C₄×D₁₇	2
C₂⁴×C₈	17

Subgroups — C23×#63242 #14545768

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#63242 #14545768