Subgroups — C₂²×C₄×#3622 #14537185

Order 2176.

760 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#3622	C₂	1	2	yes	maximal
2	C₂×C₄×#3622	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₄²×C₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₄×C₁₇	C₄	1	2	yes	—
5	C₂³×C₄×C₁₇	C₂²	1	1	yes	characteristic
6	C₂×C₄²×C₁₇	C₄	1	6	yes	—
7	C₂×C₄²×C₁₇	C₂²	1	6	yes	—
8	C₂²×#3622	C₄	1	16	yes	direct product factor
9	C₂²×#3622	C₂²	1	12	yes	—
10	C₄×#3622	C₂²	1	16	yes	direct product factor
11	C₂⁴×C₁₇	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₄×C₁₇	C₈	1	16	yes	complemented
13	C₂²×C₄×C₁₇	C₂×C₄	1	8	yes	—
14	C₂²×C₄×C₁₇	C₂×C₄	1	3	yes	—
15	C₂²×C₄×C₁₇	C₂×C₄	1	12	yes	—
16	C₂²×C₄×C₁₇	C₂³	1	3	yes	—
17	C₄²×C₁₇	C₂×C₄	1	12	yes	—
18	C₄²×C₁₇	C₂³	1	4	yes	—
19	C₂×#3622	C₂×C₄	1	48	yes	direct product factor
20	C₂×#3622	C₂³	1	8	yes	—
21	C₂³×C₁₇	C₄²	1	4	yes	—
22	C₂³×C₁₇	C₂×C₈	1	8	yes	—
23	C₂³×C₁₇	C₂²×C₄	1	3	yes	—
24	C₂×C₄×C₁₇	C₄²	1	12	yes	—
25	C₂×C₄×C₁₇	C₂×C₈	1	48	yes	complemented
26	C₂×C₄×C₁₇	C₂²×C₄	1	12	yes	—
27	C₂×C₄×C₁₇	C₂²×C₄	1	3	yes	—
28	C₂×C₄×C₁₇	C₂²×C₄	1	8	yes	—
29	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
30	#3622	C₂²×C₄	1	16	yes	direct product factor
31	C₂²×C₄×C₈	—	17	1	no	maximal, Sylow
32	C₂²×C₁₇	C₄×C₈	1	16	yes	complemented
33	C₂²×C₁₇	C₂×C₄²	1	6	yes	—
34	C₂²×C₁₇	C₂²×C₈	1	12	yes	—
35	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
36	C₄×C₁₇	C₂×C₄²	1	6	yes	—
37	C₄×C₁₇	C₂²×C₈	1	16	yes	complemented
38	C₄×C₁₇	C₂³×C₄	1	2	yes	—
39	C₂³×C₈	—	17	2	no	—
40	C₂²×C₄²	D₁₇	1	1	yes	characteristic
41	C₂×C₄×C₈	—	17	12	no	—
42	C₂×C₁₇	C₂×C₄×C₈	1	12	yes	complemented
43	C₂×C₁₇	C₂²×C₄²	1	1	yes	characteristic
44	C₂×C₁₇	C₂³×C₈	1	2	yes	—
45	C₂³×C₄	Dic₁₇	1	2	yes	—
46	C₂³×C₄	C₂×D₁₇	1	1	yes	characteristic
47	C₂²×C₈	—	17	16	no	—
48	C₂²×C₈	—	17	12	no	—
49	C₂×C₄²	Dic₁₇	1	6	yes	—
50	C₂×C₄²	C₂×D₁₇	1	6	yes	—
51	C₄×C₈	—	17	16	no	—
52	C₁₇	C₂²×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
53	C₂⁴	C₂×Dic₁₇	1	1	yes	characteristic
54	C₂²×C₄	#3622	1	16	yes	direct product factor
55	C₂²×C₄	C₄×D₁₇	1	8	yes	—
56	C₂²×C₄	C₂×Dic₁₇	1	12	yes	—
57	C₂²×C₄	C₂×Dic₁₇	1	3	yes	—
58	C₂²×C₄	C₂²×D₁₇	1	3	yes	—
59	C₂×C₈	—	17	48	no	—
60	C₂×C₈	—	17	8	no	—
61	C₄²	C₂×Dic₁₇	1	12	yes	—
62	C₄²	C₂²×D₁₇	1	4	yes	—
63	C₂³	C₂×#3622	1	8	yes	—
64	C₂³	C₄×Dic₁₇	1	4	yes	—
65	C₂³	C₂²×Dic₁₇	1	3	yes	—
66	C₂×C₄	C₂×#3622	1	48	yes	direct product factor
67	C₂×C₄	C₄×Dic₁₇	1	12	yes	—
68	C₂×C₄	C₂×C₄×D₁₇	1	12	yes	—
69	C₂×C₄	C₂²×Dic₁₇	1	8	yes	—
70	C₂×C₄	C₂²×Dic₁₇	1	3	yes	—
71	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
72	C₈	—	17	16	no	—
73	C₂²	C₄×#3622	1	16	yes	direct product factor
74	C₂²	C₂²×#3622	1	12	yes	—
75	C₂²	C₂×C₄×Dic₁₇	1	6	yes	—
76	C₂²	C₂³×Dic₁₇	1	1	yes	characteristic
77	C₄	C₂²×#3622	1	16	yes	direct product factor
78	C₄	C₂×C₄×Dic₁₇	1	6	yes	—
79	C₄	C₂²×C₄×D₁₇	1	2	yes	—
80	C₂	C₂×C₄×#3622	1	12	yes	direct product factor
81	C₂	C₂²×C₄×Dic₁₇	1	1	yes	characteristic
82	C₂	C₂³×#3622	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂²×C₄²	D₁₇
commutator, hypocenter	C₁₇	C₂²×C₄×C₈
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₂²×C₄²×C₁₇	C₂
socle, prime radical	C₂⁴×C₁₇	C₂×C₄
cosocle	C₂×C₄×C₁₇	C₂⁴
LCS	G ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×#3622	C₂	direct (N × H)
8. C₂²×#3622	C₄	direct (N × H)
10. C₄×#3622	C₂²	direct (N × H)
12. C₂²×C₄×C₁₇	C₈	semidirect
19. C₂×#3622	C₂×C₄	direct (N × H)
25. C₂×C₄×C₁₇	C₂×C₈	semidirect
30. #3622	C₂²×C₄	direct (N × H)
32. C₂²×C₁₇	C₄×C₈	semidirect
37. C₄×C₁₇	C₂²×C₈	semidirect
42. C₂×C₁₇	C₂×C₄×C₈	semidirect
52. C₁₇	C₂²×C₄×C₈	semidirect
54. C₂²×C₄	#3622	direct (N × H)
66. C₂×C₄	C₂×#3622	direct (N × H)
73. C₂²	C₄×#3622	direct (N × H)
77. C₄	C₂²×#3622	direct (N × H)
80. C₂	C₂×C₄×#3622	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×#3622	2
C₂×C₄×#3622	2
C₂²×C₄²×C₁₇	2
C₂²×C₄×C₈	17

Subgroups — C22×C4×#3622 #14537185

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×#3622 #14537185