Subgroups — C₂³×#63247 #14545769

Order 2176.

1720 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#63247	C₂	1	28	yes	maximal, direct product factor
2	C₂³×#3624	C₂	1	2	yes	maximal, complemented
3	C₂⁴×Dic₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂⁵×C₁₇	C₄	1	1	yes	characteristic
5	C₂³×Dic₁₇	C₄	1	14	yes	—
6	C₂³×Dic₁₇	C₄	1	1	yes	characteristic
7	C₂³×Dic₁₇	C₂²	1	1	yes	characteristic
8	C₂³×Dic₁₇	C₂²	1	14	yes	—
9	C₂×#63247	C₂²	1	112	yes	direct product factor
10	C₂²×#3624	C₂²	1	28	yes	complemented
11	C₂⁴×C₁₇	—	2	8	no	—
12	C₂⁴×C₁₇	C₂×C₄	1	14	yes	—
13	C₂⁴×C₁₇	C₂×C₄	1	1	yes	characteristic
14	C₂²×Dic₁₇	C₂×C₄	1	84	yes	—
15	C₂²×Dic₁₇	C₂×C₄	1	7	yes	—
16	C₂²×Dic₁₇	C₂×C₄	1	14	yes	—
17	C₂²×Dic₁₇	C₂³	1	7	yes	—
18	C₂²×Dic₁₇	C₂³	1	28	yes	—
19	#63247	C₂³	1	64	yes	direct product factor
20	C₂×#3624	C₂³	1	56	yes	complemented
21	C₂³×C₁₇	—	2	56	no	—
22	C₂³×C₁₇	M₁₆	1	8	yes	complemented
23	C₂³×C₁₇	C₂²×C₄	1	28	yes	—
24	C₂³×C₁₇	C₂²×C₄	1	7	yes	—
25	C₂×Dic₁₇	C₂²×C₄	1	56	yes	—
26	C₂×Dic₁₇	C₂²×C₄	1	21	yes	—
27	C₂×Dic₁₇	C₂²×C₄	1	28	yes	—
28	C₂×Dic₁₇	C₂⁴	1	7	yes	—
29	C₂×Dic₁₇	C₂⁴	1	8	yes	—
30	#3624	C₂⁴	1	16	yes	complemented
31	C₂³×M₁₆	—	17	1	no	maximal, Sylow
32	C₂²×C₁₇	—	2	56	no	—
33	C₂²×C₁₇	C₂×M₁₆	1	28	yes	complemented
34	C₂²×C₁₇	C₂³×C₄	1	8	yes	—
35	C₂²×C₁₇	C₂³×C₄	1	7	yes	—
36	Dic₁₇	C₂³×C₄	1	7	yes	—
37	Dic₁₇	C₂³×C₄	1	8	yes	—
38	Dic₁₇	C₂⁵	1	1	yes	characteristic
39	C₂⁴×C₄	—	17	1	no	—
40	C₂²×M₁₆	—	17	28	no	—
41	C₂³×C₈	—	17	2	no	—
42	C₂×C₁₇	—	2	8	no	—
43	C₂×C₁₇	C₂²×M₁₆	1	14	yes	complemented
44	C₂×C₁₇	C₂⁴×C₄	1	1	yes	characteristic
45	C₂⁵	#595	1	1	yes	characteristic
46	C₂³×C₄	—	17	14	no	—
47	C₂³×C₄	—	17	14	no	—
48	C₂³×C₄	—	17	1	no	—
49	C₂³×C₄	—	17	1	no	—
50	C₂×M₁₆	—	17	112	no	—
51	C₂²×C₈	—	17	28	no	—
52	C₁₇	C₂³×M₁₆	1	1	yes	characteristic, complemented, Sylow
53	C₂⁴	—	2	8	no	—
54	C₂⁴	C₂×#595	1	14	yes	—
55	C₂⁴	C₂×#595	1	1	yes	characteristic
56	C₂²×C₄	—	17	84	no	—
57	C₂²×C₄	—	17	7	no	—
58	C₂²×C₄	—	17	14	no	—
59	C₂²×C₄	—	17	7	no	—
60	C₂²×C₄	—	17	28	no	—
61	M₁₆	—	17	64	no	—
62	C₂×C₈	—	17	56	no	—
63	C₂³	—	2	56	no	—
64	C₂³	#63247	1	8	yes	direct product factor
65	C₂³	C₂²×#595	1	28	yes	—
66	C₂³	C₂²×#595	1	7	yes	—
67	C₂×C₄	—	17	7	no	—
68	C₂×C₄	—	17	8	no	—
69	C₂×C₄	—	17	56	no	—
70	C₂×C₄	—	17	21	no	—
71	C₂×C₄	—	17	28	no	—
72	C₈	—	17	16	no	—
73	C₂²	—	2	56	no	—
74	C₂²	C₂×#63247	1	28	yes	direct product factor
75	C₂²	C₂³×#595	1	8	yes	—
76	C₂²	C₂³×#595	1	7	yes	—
77	C₄	—	17	1	no	—
78	C₄	—	17	7	no	—
79	C₄	—	17	8	no	—
80	C₂	—	2	8	no	—
81	C₂	C₂²×#63247	1	14	yes	direct product factor
82	C₂	C₂⁴×#595	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×#595
commutator	C₂×C₁₇	C₂⁴×C₄
Frattini	C₂	C₂⁴×#595
Fitting, prime radical	C₂⁵×C₁₇	C₄
socle	C₂⁴×C₁₇	C₂×C₄
cosocle	Dic₁₇	C₂⁵
hypercenter	C₂⁵	#595
hypocenter	C₁₇	C₂³×M₁₆
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂⁴ ◁ C₂⁵

Complements

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

N	H ≅ G/N	type
1. C₂²×#63247	C₂	direct (N × H)
2. C₂³×#3624	C₂	semidirect
9. C₂×#63247	C₂²	direct (N × H)
10. C₂²×#3624	C₂²	semidirect
19. #63247	C₂³	direct (N × H)
20. C₂×#3624	C₂³	semidirect
22. C₂³×C₁₇	M₁₆	semidirect
30. #3624	C₂⁴	semidirect
33. C₂²×C₁₇	C₂×M₁₆	semidirect
43. C₂×C₁₇	C₂²×M₁₆	semidirect
52. C₁₇	C₂³×M₁₆	semidirect
64. C₂³	#63247	direct (N × H)
74. C₂²	C₂×#63247	direct (N × H)
81. C₂	C₂²×#63247	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×#63247	2
C₂³×#3624	2
C₂⁴×Dic₁₇	2
C₂³×M₁₆	17

Subgroups — C23×#63247 #14545769

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#63247 #14545769