Subgroups — C₂×C₈×C₉²×C₁₇ #72199525

Order 22032.

506 conjugacy classes of subgroups.

The subgroup lattice has 108 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₉²×C₁₇	C₂	1	1	yes	maximal, characteristic
2	C₈×C₉²×C₁₇	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×C₈×C₉×C₁₇	C₃	1	4	yes	maximal
4	C₂²×C₉²×C₁₇	C₄	1	1	yes	characteristic
5	C₄×C₉²×C₁₇	C₄	1	1	yes	characteristic
6	C₄×C₉²×C₁₇	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₄×C₉×C₁₇	C₂×C₃	1	4	yes	—
8	C₃×C₈×C₉×C₁₇	C₂×C₃	1	8	yes	—
9	C₂×C₉²×C₁₇	C₈	1	2	yes	direct product factor
10	C₂×C₉²×C₁₇	C₂×C₄	1	1	yes	characteristic
11	C₂×C₃²×C₈×C₁₇	C₃²	1	1	yes	characteristic
12	C₂×C₈×C₉×C₁₇	C₉	1	12	yes	direct product factor
13	C₂²×C₃×C₉×C₁₇	C₃×C₄	1	4	yes	—
14	C₃×C₄×C₉×C₁₇	C₃×C₄	1	4	yes	—
15	C₃×C₄×C₉×C₁₇	C₂²×C₃	1	4	yes	—
16	C₉²×C₁₇	C₂×C₈	1	1	yes	characteristic, direct product factor, Hall
17	C₂×C₈×C₉²	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₃²×C₄×C₁₇	C₂×C₃²	1	1	yes	characteristic
19	C₃²×C₈×C₁₇	C₂×C₃²	1	2	yes	—
20	C₂×C₄×C₉×C₁₇	C₂×C₉	1	12	yes	—
21	C₈×C₉×C₁₇	C₂×C₉	1	24	yes	direct product factor
22	C₂×C₃×C₉×C₁₇	C₃×C₈	1	8	yes	—
23	C₂×C₃×C₉×C₁₇	C₂×C₃×C₄	1	4	yes	—
24	C₂×C₃×C₈×C₁₇	C₃×C₉	1	4	yes	—
25	C₂×C₄×C₉²	C₂×C₁₇	1	1	yes	characteristic
26	C₈×C₉²	C₂×C₁₇	1	2	yes	direct product factor
27	C₂²×C₃²×C₁₇	C₃²×C₄	1	1	yes	characteristic
28	C₃²×C₄×C₁₇	C₃²×C₄	1	1	yes	characteristic
29	C₃²×C₄×C₁₇	C₂²×C₃²	1	1	yes	characteristic
30	C₂²×C₉×C₁₇	C₄×C₉	1	12	yes	—
31	C₄×C₉×C₁₇	C₄×C₉	1	12	yes	—
32	C₄×C₉×C₁₇	C₂²×C₉	1	12	yes	—
33	C₃×C₉×C₁₇	C₂×C₃×C₈	1	4	yes	—
34	C₂×C₃×C₈×C₉	C₃×C₁₇	1	4	yes	—
35	C₂×C₃×C₄×C₁₇	C₂×C₃×C₉	1	4	yes	—
36	C₃×C₈×C₁₇	C₂×C₃×C₉	1	8	yes	—
37	C₂²×C₉²	C₄×C₁₇	1	1	yes	characteristic
38	C₄×C₉²	C₄×C₁₇	1	1	yes	characteristic
39	C₄×C₉²	C₂²×C₁₇	1	1	yes	characteristic
40	C₂×C₃²×C₁₇	C₃²×C₈	1	2	yes	—
41	C₂×C₃²×C₁₇	C₂×C₃²×C₄	1	1	yes	characteristic
42	C₂×C₉×C₁₇	C₈×C₉	1	24	yes	direct product factor
43	C₂×C₉×C₁₇	C₂×C₄×C₉	1	12	yes	—
44	C₂×C₈×C₁₇	C₉²	1	1	yes	characteristic, direct product factor, Hall
45	C₂×C₃×C₄×C₉	C₂×C₃×C₁₇	1	4	yes	—
46	C₃×C₈×C₉	C₂×C₃×C₁₇	1	8	yes	—
47	C₂²×C₃×C₁₇	C₃×C₄×C₉	1	4	yes	—
48	C₃×C₄×C₁₇	C₃×C₄×C₉	1	4	yes	—
49	C₃×C₄×C₁₇	C₂²×C₃×C₉	1	4	yes	—
50	C₂×C₉²	C₈×C₁₇	1	2	yes	direct product factor
51	C₂×C₉²	C₂×C₄×C₁₇	1	1	yes	characteristic
52	C₃²×C₁₇	C₂×C₃²×C₈	1	1	yes	characteristic
53	C₉×C₁₇	C₂×C₈×C₉	1	12	yes	direct product factor
54	C₂×C₃²×C₈	C₃²×C₁₇	1	1	yes	characteristic
55	C₂×C₈×C₉	C₉×C₁₇	1	12	yes	direct product factor
56	C₂×C₄×C₁₇	C₂×C₉²	1	1	yes	characteristic
57	C₈×C₁₇	C₂×C₉²	1	2	yes	direct product factor
58	C₂²×C₃×C₉	C₃×C₄×C₁₇	1	4	yes	—
59	C₃×C₄×C₉	C₃×C₄×C₁₇	1	4	yes	—
60	C₃×C₄×C₉	C₂²×C₃×C₁₇	1	4	yes	—
61	C₂×C₃×C₁₇	C₃×C₈×C₉	1	8	yes	—
62	C₂×C₃×C₁₇	C₂×C₃×C₄×C₉	1	4	yes	—
63	C₉²	C₂×C₈×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
64	C₂×C₃²×C₄	C₂×C₃²×C₁₇	1	1	yes	characteristic
65	C₃²×C₈	C₂×C₃²×C₁₇	1	2	yes	—
66	C₂×C₄×C₉	C₂×C₉×C₁₇	1	12	yes	—
67	C₈×C₉	C₂×C₉×C₁₇	1	24	yes	direct product factor
68	C₂²×C₁₇	C₄×C₉²	1	1	yes	characteristic
69	C₄×C₁₇	C₄×C₉²	1	1	yes	characteristic
70	C₄×C₁₇	C₂²×C₉²	1	1	yes	characteristic
71	C₂×C₃×C₉	C₃×C₈×C₁₇	1	8	yes	—
72	C₂×C₃×C₉	C₂×C₃×C₄×C₁₇	1	4	yes	—
73	C₃×C₁₇	C₂×C₃×C₈×C₉	1	4	yes	—
74	C₂×C₃×C₈	C₃×C₉×C₁₇	1	4	yes	—
75	C₂²×C₃²	C₃²×C₄×C₁₇	1	1	yes	characteristic
76	C₃²×C₄	C₃²×C₄×C₁₇	1	1	yes	characteristic
77	C₃²×C₄	C₂²×C₃²×C₁₇	1	1	yes	characteristic
78	C₂²×C₉	C₄×C₉×C₁₇	1	12	yes	—
79	C₄×C₉	C₄×C₉×C₁₇	1	12	yes	—
80	C₄×C₉	C₂²×C₉×C₁₇	1	12	yes	—
81	C₂×C₁₇	C₈×C₉²	1	2	yes	direct product factor
82	C₂×C₁₇	C₂×C₄×C₉²	1	1	yes	characteristic
83	C₃×C₉	C₂×C₃×C₈×C₁₇	1	4	yes	—
84	C₂×C₃×C₄	C₂×C₃×C₉×C₁₇	1	4	yes	—
85	C₃×C₈	C₂×C₃×C₉×C₁₇	1	8	yes	—
86	C₂×C₃²	C₃²×C₈×C₁₇	1	2	yes	—
87	C₂×C₃²	C₂×C₃²×C₄×C₁₇	1	1	yes	characteristic
88	C₂×C₉	C₈×C₉×C₁₇	1	24	yes	direct product factor
89	C₂×C₉	C₂×C₄×C₉×C₁₇	1	12	yes	—
90	C₁₇	C₂×C₈×C₉²	1	1	yes	characteristic, direct product factor, Sylow
91	C₂×C₈	C₉²×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
92	C₂²×C₃	C₃×C₄×C₉×C₁₇	1	4	yes	—
93	C₃×C₄	C₃×C₄×C₉×C₁₇	1	4	yes	—
94	C₃×C₄	C₂²×C₃×C₉×C₁₇	1	4	yes	—
95	C₃²	C₂×C₃²×C₈×C₁₇	1	1	yes	characteristic
96	C₉	C₂×C₈×C₉×C₁₇	1	12	yes	direct product factor
97	C₂×C₄	C₂×C₉²×C₁₇	1	1	yes	characteristic
98	C₈	C₂×C₉²×C₁₇	1	2	yes	direct product factor
99	C₂×C₃	C₃×C₈×C₉×C₁₇	1	8	yes	—
100	C₂×C₃	C₂×C₃×C₄×C₉×C₁₇	1	4	yes	—
101	C₂²	C₄×C₉²×C₁₇	1	1	yes	characteristic
102	C₄	C₄×C₉²×C₁₇	1	1	yes	characteristic
103	C₄	C₂²×C₉²×C₁₇	1	1	yes	characteristic
104	C₃	C₂×C₃×C₈×C₉×C₁₇	1	4	yes	—
105	C₂	C₈×C₉²×C₁₇	1	2	yes	direct product factor
106	C₂	C₂×C₄×C₉²×C₁₇	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃²×C₄	C₂²×C₃²×C₁₇
socle, prime radical	C₂²×C₃²×C₁₇	C₃²×C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈×C₉²×C₁₇	C₂	direct (N × H)
9. C₂×C₉²×C₁₇	C₈	direct (N × H)
12. C₂×C₈×C₉×C₁₇	C₉	direct (N × H)
16. C₉²×C₁₇	C₂×C₈	direct (N × H)
17. C₂×C₈×C₉²	C₁₇	direct (N × H)
21. C₈×C₉×C₁₇	C₂×C₉	direct (N × H)
26. C₈×C₉²	C₂×C₁₇	direct (N × H)
42. C₂×C₉×C₁₇	C₈×C₉	direct (N × H)
44. C₂×C₈×C₁₇	C₉²	direct (N × H)
50. C₂×C₉²	C₈×C₁₇	direct (N × H)
53. C₉×C₁₇	C₂×C₈×C₉	direct (N × H)
55. C₂×C₈×C₉	C₉×C₁₇	direct (N × H)
57. C₈×C₁₇	C₂×C₉²	direct (N × H)
63. C₉²	C₂×C₈×C₁₇	direct (N × H)
67. C₈×C₉	C₂×C₉×C₁₇	direct (N × H)
81. C₂×C₁₇	C₈×C₉²	direct (N × H)
88. C₂×C₉	C₈×C₉×C₁₇	direct (N × H)
90. C₁₇	C₂×C₈×C₉²	direct (N × H)
91. C₂×C₈	C₉²×C₁₇	direct (N × H)
96. C₉	C₂×C₈×C₉×C₁₇	direct (N × H)
98. C₈	C₂×C₉²×C₁₇	direct (N × H)
105. C₂	C₈×C₉²×C₁₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₉²×C₁₇	2
C₈×C₉²×C₁₇	2
C₂×C₃×C₈×C₉×C₁₇	3
C₂×C₈×C₉²	17

Subgroups — C2×C8×C92×C17 #72199525

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×C₉²×C₁₇ #72199525