Subgroups — C₂×C₃×C₄×C₇×C₉×C₁₇ #72202913

Order 25704.

320 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₇×C₉×C₁₇	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₇×C₉×C₁₇	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃²×C₄×C₇×C₁₇	C₃	1	1	yes	maximal, characteristic
4	C₂×C₄×C₇×C₉×C₁₇	C₃	1	3	yes	maximal, direct product factor
5	C₂×C₃×C₇×C₉×C₁₇	C₄	1	2	yes	direct product factor
6	C₂×C₃×C₇×C₉×C₁₇	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₄×C₉×C₁₇	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₃×C₇×C₉×C₁₇	C₂×C₄	1	1	yes	characteristic, direct product factor, Hall
9	C₂×C₃×C₄×C₇×C₁₇	C₉	1	3	yes	direct product factor
10	C₂×C₃×C₄×C₇×C₁₇	C₃²	1	1	yes	characteristic
11	C₂²×C₃²×C₇×C₁₇	C₂×C₃	1	1	yes	characteristic
12	C₃²×C₄×C₇×C₁₇	C₂×C₃	1	2	yes	—
13	C₂²×C₇×C₉×C₁₇	C₂×C₃	1	3	yes	—
14	C₄×C₇×C₉×C₁₇	C₂×C₃	1	6	yes	direct product factor
15	C₂×C₃²×C₇×C₁₇	C₃×C₄	1	2	yes	—
16	C₂×C₃²×C₇×C₁₇	C₂²×C₃	1	1	yes	characteristic
17	C₂×C₇×C₉×C₁₇	C₃×C₄	1	6	yes	direct product factor
18	C₂×C₇×C₉×C₁₇	C₂²×C₃	1	3	yes	—
19	C₂²×C₃×C₉×C₁₇	C₂×C₇	1	1	yes	characteristic
20	C₃×C₄×C₉×C₁₇	C₂×C₇	1	2	yes	direct product factor
21	C₂×C₃×C₄×C₇×C₉	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂²×C₃×C₇×C₁₇	C₂×C₉	1	3	yes	—
23	C₂²×C₃×C₇×C₁₇	C₂×C₃²	1	1	yes	characteristic
24	C₃×C₄×C₇×C₁₇	C₂×C₉	1	6	yes	direct product factor
25	C₃×C₄×C₇×C₁₇	C₂×C₃²	1	2	yes	—
26	C₂×C₃²×C₄×C₁₇	C₃×C₇	1	1	yes	characteristic
27	C₂×C₄×C₉×C₁₇	C₃×C₇	1	3	yes	direct product factor
28	C₃²×C₇×C₁₇	C₂×C₃×C₄	1	1	yes	characteristic
29	C₇×C₉×C₁₇	C₂×C₃×C₄	1	3	yes	direct product factor
30	C₂×C₄×C₇×C₁₇	C₃×C₉	1	1	yes	characteristic, direct product factor, Hall
31	C₂×C₃×C₉×C₁₇	C₄×C₇	1	2	yes	direct product factor
32	C₂×C₃×C₉×C₁₇	C₂²×C₇	1	1	yes	characteristic
33	C₂²×C₃×C₇×C₉	C₂×C₁₇	1	1	yes	characteristic
34	C₃×C₄×C₇×C₉	C₂×C₁₇	1	2	yes	direct product factor
35	C₂×C₃×C₇×C₁₇	C₄×C₉	1	6	yes	direct product factor
36	C₂×C₃×C₇×C₁₇	C₂²×C₉	1	3	yes	—
37	C₂×C₃×C₇×C₁₇	C₃²×C₄	1	2	yes	—
38	C₂×C₃×C₇×C₁₇	C₂²×C₃²	1	1	yes	characteristic
39	C₂²×C₃²×C₁₇	C₂×C₃×C₇	1	1	yes	characteristic
40	C₃²×C₄×C₁₇	C₂×C₃×C₇	1	2	yes	—
41	C₂²×C₉×C₁₇	C₂×C₃×C₇	1	3	yes	—
42	C₄×C₉×C₁₇	C₂×C₃×C₇	1	6	yes	direct product factor
43	C₂×C₃²×C₄×C₇	C₃×C₁₇	1	1	yes	characteristic
44	C₂×C₄×C₇×C₉	C₃×C₁₇	1	3	yes	direct product factor
45	C₂²×C₇×C₁₇	C₂×C₃×C₉	1	1	yes	characteristic
46	C₄×C₇×C₁₇	C₂×C₃×C₉	1	2	yes	direct product factor
47	C₃×C₉×C₁₇	C₂×C₄×C₇	1	1	yes	characteristic, direct product factor, Hall
48	C₂×C₃×C₄×C₁₇	C₇×C₉	1	3	yes	direct product factor
49	C₂×C₃×C₄×C₁₇	C₃²×C₇	1	1	yes	characteristic
50	C₂×C₃×C₇×C₉	C₄×C₁₇	1	2	yes	direct product factor
51	C₂×C₃×C₇×C₉	C₂²×C₁₇	1	1	yes	characteristic
52	C₃×C₇×C₁₇	C₂×C₄×C₉	1	3	yes	direct product factor
53	C₃×C₇×C₁₇	C₂×C₃²×C₄	1	1	yes	characteristic
54	C₂×C₃²×C₁₇	C₃×C₄×C₇	1	2	yes	—
55	C₂×C₃²×C₁₇	C₂²×C₃×C₇	1	1	yes	characteristic
56	C₂×C₉×C₁₇	C₃×C₄×C₇	1	6	yes	direct product factor
57	C₂×C₉×C₁₇	C₂²×C₃×C₇	1	3	yes	—
58	C₂²×C₃²×C₇	C₂×C₃×C₁₇	1	1	yes	characteristic
59	C₃²×C₄×C₇	C₂×C₃×C₁₇	1	2	yes	—
60	C₂²×C₇×C₉	C₂×C₃×C₁₇	1	3	yes	—
61	C₄×C₇×C₉	C₂×C₃×C₁₇	1	6	yes	direct product factor
62	C₂×C₇×C₁₇	C₃×C₄×C₉	1	2	yes	direct product factor
63	C₂×C₇×C₁₇	C₂²×C₃×C₉	1	1	yes	characteristic
64	C₂×C₃×C₄×C₉	C₇×C₁₇	1	1	yes	characteristic, direct product factor, Hall
65	C₂²×C₃×C₁₇	C₂×C₇×C₉	1	3	yes	—
66	C₂²×C₃×C₁₇	C₂×C₃²×C₇	1	1	yes	characteristic
67	C₃×C₄×C₁₇	C₂×C₇×C₉	1	6	yes	direct product factor
68	C₃×C₄×C₁₇	C₂×C₃²×C₇	1	2	yes	—
69	C₃×C₇×C₉	C₂×C₄×C₁₇	1	1	yes	characteristic, direct product factor, Hall
70	C₂×C₃×C₄×C₇	C₉×C₁₇	1	3	yes	direct product factor
71	C₂×C₃×C₄×C₇	C₃²×C₁₇	1	1	yes	characteristic
72	C₃²×C₁₇	C₂×C₃×C₄×C₇	1	1	yes	characteristic
73	C₉×C₁₇	C₂×C₃×C₄×C₇	1	3	yes	direct product factor
74	C₂×C₄×C₁₇	C₃×C₇×C₉	1	1	yes	characteristic, direct product factor, Hall
75	C₂×C₃²×C₇	C₃×C₄×C₁₇	1	2	yes	—
76	C₂×C₃²×C₇	C₂²×C₃×C₁₇	1	1	yes	characteristic
77	C₂×C₇×C₉	C₃×C₄×C₁₇	1	6	yes	direct product factor
78	C₂×C₇×C₉	C₂²×C₃×C₁₇	1	3	yes	—
79	C₇×C₁₇	C₂×C₃×C₄×C₉	1	1	yes	characteristic, direct product factor, Hall
80	C₂²×C₃×C₉	C₂×C₇×C₁₇	1	1	yes	characteristic
81	C₃×C₄×C₉	C₂×C₇×C₁₇	1	2	yes	direct product factor
82	C₂×C₃×C₁₇	C₄×C₇×C₉	1	6	yes	direct product factor
83	C₂×C₃×C₁₇	C₂²×C₇×C₉	1	3	yes	—
84	C₂×C₃×C₁₇	C₃²×C₄×C₇	1	2	yes	—
85	C₂×C₃×C₁₇	C₂²×C₃²×C₇	1	1	yes	characteristic
86	C₂²×C₃×C₇	C₂×C₉×C₁₇	1	3	yes	—
87	C₂²×C₃×C₇	C₂×C₃²×C₁₇	1	1	yes	characteristic
88	C₃×C₄×C₇	C₂×C₉×C₁₇	1	6	yes	direct product factor
89	C₃×C₄×C₇	C₂×C₃²×C₁₇	1	2	yes	—
90	C₂×C₃²×C₄	C₃×C₇×C₁₇	1	1	yes	characteristic
91	C₂×C₄×C₉	C₃×C₇×C₁₇	1	3	yes	direct product factor
92	C₂²×C₁₇	C₂×C₃×C₇×C₉	1	1	yes	characteristic
93	C₄×C₁₇	C₂×C₃×C₇×C₉	1	2	yes	direct product factor
94	C₃²×C₇	C₂×C₃×C₄×C₁₇	1	1	yes	characteristic
95	C₇×C₉	C₂×C₃×C₄×C₁₇	1	3	yes	direct product factor
96	C₂×C₄×C₇	C₃×C₉×C₁₇	1	1	yes	characteristic, direct product factor, Hall
97	C₂×C₃×C₉	C₄×C₇×C₁₇	1	2	yes	direct product factor
98	C₂×C₃×C₉	C₂²×C₇×C₁₇	1	1	yes	characteristic
99	C₃×C₁₇	C₂×C₄×C₇×C₉	1	3	yes	direct product factor
100	C₃×C₁₇	C₂×C₃²×C₄×C₇	1	1	yes	characteristic
101	C₂×C₃×C₇	C₄×C₉×C₁₇	1	6	yes	direct product factor
102	C₂×C₃×C₇	C₂²×C₉×C₁₇	1	3	yes	—
103	C₂×C₃×C₇	C₃²×C₄×C₁₇	1	2	yes	—
104	C₂×C₃×C₇	C₂²×C₃²×C₁₇	1	1	yes	characteristic
105	C₂²×C₃²	C₂×C₃×C₇×C₁₇	1	1	yes	characteristic
106	C₃²×C₄	C₂×C₃×C₇×C₁₇	1	2	yes	—
107	C₂²×C₉	C₂×C₃×C₇×C₁₇	1	3	yes	—
108	C₄×C₉	C₂×C₃×C₇×C₁₇	1	6	yes	direct product factor
109	C₂×C₁₇	C₃×C₄×C₇×C₉	1	2	yes	direct product factor
110	C₂×C₁₇	C₂²×C₃×C₇×C₉	1	1	yes	characteristic
111	C₂²×C₇	C₂×C₃×C₉×C₁₇	1	1	yes	characteristic
112	C₄×C₇	C₂×C₃×C₉×C₁₇	1	2	yes	direct product factor
113	C₃×C₉	C₂×C₄×C₇×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
114	C₂×C₃×C₄	C₇×C₉×C₁₇	1	3	yes	direct product factor
115	C₂×C₃×C₄	C₃²×C₇×C₁₇	1	1	yes	characteristic
116	C₃×C₇	C₂×C₄×C₉×C₁₇	1	3	yes	direct product factor
117	C₃×C₇	C₂×C₃²×C₄×C₁₇	1	1	yes	characteristic
118	C₂×C₃²	C₃×C₄×C₇×C₁₇	1	2	yes	—
119	C₂×C₃²	C₂²×C₃×C₇×C₁₇	1	1	yes	characteristic
120	C₂×C₉	C₃×C₄×C₇×C₁₇	1	6	yes	direct product factor
121	C₂×C₉	C₂²×C₃×C₇×C₁₇	1	3	yes	—
122	C₁₇	C₂×C₃×C₄×C₇×C₉	1	1	yes	characteristic, direct product factor, Sylow
123	C₂×C₇	C₃×C₄×C₉×C₁₇	1	2	yes	direct product factor
124	C₂×C₇	C₂²×C₃×C₉×C₁₇	1	1	yes	characteristic
125	C₂²×C₃	C₂×C₇×C₉×C₁₇	1	3	yes	—
126	C₂²×C₃	C₂×C₃²×C₇×C₁₇	1	1	yes	characteristic
127	C₃×C₄	C₂×C₇×C₉×C₁₇	1	6	yes	direct product factor
128	C₃×C₄	C₂×C₃²×C₇×C₁₇	1	2	yes	—
129	C₃²	C₂×C₃×C₄×C₇×C₁₇	1	1	yes	characteristic
130	C₉	C₂×C₃×C₄×C₇×C₁₇	1	3	yes	direct product factor
131	C₂×C₄	C₃×C₇×C₉×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
132	C₇	C₂×C₃×C₄×C₉×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
133	C₂×C₃	C₄×C₇×C₉×C₁₇	1	6	yes	direct product factor
134	C₂×C₃	C₂²×C₇×C₉×C₁₇	1	3	yes	—
135	C₂×C₃	C₃²×C₄×C₇×C₁₇	1	2	yes	—
136	C₂×C₃	C₂²×C₃²×C₇×C₁₇	1	1	yes	characteristic
137	C₂²	C₂×C₃×C₇×C₉×C₁₇	1	1	yes	characteristic
138	C₄	C₂×C₃×C₇×C₉×C₁₇	1	2	yes	direct product factor
139	C₃	C₂×C₄×C₇×C₉×C₁₇	1	3	yes	direct product factor
140	C₃	C₂×C₃²×C₄×C₇×C₁₇	1	1	yes	characteristic
141	C₂	C₃×C₄×C₇×C₉×C₁₇	1	2	yes	direct product factor
142	C₂	C₂²×C₃×C₇×C₉×C₁₇	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₃	C₂²×C₃²×C₇×C₁₇
socle, prime radical	C₂²×C₃²×C₇×C₁₇	C₂×C₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×C₇×C₉×C₁₇	C₂	direct (N × H)
4. C₂×C₄×C₇×C₉×C₁₇	C₃	direct (N × H)
5. C₂×C₃×C₇×C₉×C₁₇	C₄	direct (N × H)
7. C₂×C₃×C₄×C₉×C₁₇	C₇	direct (N × H)
8. C₃×C₇×C₉×C₁₇	C₂×C₄	direct (N × H)
9. C₂×C₃×C₄×C₇×C₁₇	C₉	direct (N × H)
14. C₄×C₇×C₉×C₁₇	C₂×C₃	direct (N × H)
17. C₂×C₇×C₉×C₁₇	C₃×C₄	direct (N × H)
20. C₃×C₄×C₉×C₁₇	C₂×C₇	direct (N × H)
21. C₂×C₃×C₄×C₇×C₉	C₁₇	direct (N × H)
24. C₃×C₄×C₇×C₁₇	C₂×C₉	direct (N × H)
27. C₂×C₄×C₉×C₁₇	C₃×C₇	direct (N × H)
29. C₇×C₉×C₁₇	C₂×C₃×C₄	direct (N × H)
30. C₂×C₄×C₇×C₁₇	C₃×C₉	direct (N × H)
31. C₂×C₃×C₉×C₁₇	C₄×C₇	direct (N × H)
34. C₃×C₄×C₇×C₉	C₂×C₁₇	direct (N × H)
35. C₂×C₃×C₇×C₁₇	C₄×C₉	direct (N × H)
42. C₄×C₉×C₁₇	C₂×C₃×C₇	direct (N × H)
44. C₂×C₄×C₇×C₉	C₃×C₁₇	direct (N × H)
46. C₄×C₇×C₁₇	C₂×C₃×C₉	direct (N × H)
47. C₃×C₉×C₁₇	C₂×C₄×C₇	direct (N × H)
48. C₂×C₃×C₄×C₁₇	C₇×C₉	direct (N × H)
50. C₂×C₃×C₇×C₉	C₄×C₁₇	direct (N × H)
52. C₃×C₇×C₁₇	C₂×C₄×C₉	direct (N × H)
56. C₂×C₉×C₁₇	C₃×C₄×C₇	direct (N × H)
61. C₄×C₇×C₉	C₂×C₃×C₁₇	direct (N × H)
62. C₂×C₇×C₁₇	C₃×C₄×C₉	direct (N × H)
64. C₂×C₃×C₄×C₉	C₇×C₁₇	direct (N × H)
67. C₃×C₄×C₁₇	C₂×C₇×C₉	direct (N × H)
69. C₃×C₇×C₉	C₂×C₄×C₁₇	direct (N × H)
70. C₂×C₃×C₄×C₇	C₉×C₁₇	direct (N × H)
73. C₉×C₁₇	C₂×C₃×C₄×C₇	direct (N × H)
74. C₂×C₄×C₁₇	C₃×C₇×C₉	direct (N × H)
77. C₂×C₇×C₉	C₃×C₄×C₁₇	direct (N × H)
79. C₇×C₁₇	C₂×C₃×C₄×C₉	direct (N × H)
81. C₃×C₄×C₉	C₂×C₇×C₁₇	direct (N × H)
82. C₂×C₃×C₁₇	C₄×C₇×C₉	direct (N × H)
88. C₃×C₄×C₇	C₂×C₉×C₁₇	direct (N × H)
91. C₂×C₄×C₉	C₃×C₇×C₁₇	direct (N × H)
93. C₄×C₁₇	C₂×C₃×C₇×C₉	direct (N × H)
95. C₇×C₉	C₂×C₃×C₄×C₁₇	direct (N × H)
96. C₂×C₄×C₇	C₃×C₉×C₁₇	direct (N × H)
97. C₂×C₃×C₉	C₄×C₇×C₁₇	direct (N × H)
99. C₃×C₁₇	C₂×C₄×C₇×C₉	direct (N × H)
101. C₂×C₃×C₇	C₄×C₉×C₁₇	direct (N × H)
108. C₄×C₉	C₂×C₃×C₇×C₁₇	direct (N × H)
109. C₂×C₁₇	C₃×C₄×C₇×C₉	direct (N × H)
112. C₄×C₇	C₂×C₃×C₉×C₁₇	direct (N × H)
113. C₃×C₉	C₂×C₄×C₇×C₁₇	direct (N × H)
114. C₂×C₃×C₄	C₇×C₉×C₁₇	direct (N × H)
116. C₃×C₇	C₂×C₄×C₉×C₁₇	direct (N × H)
120. C₂×C₉	C₃×C₄×C₇×C₁₇	direct (N × H)
122. C₁₇	C₂×C₃×C₄×C₇×C₉	direct (N × H)
123. C₂×C₇	C₃×C₄×C₉×C₁₇	direct (N × H)
127. C₃×C₄	C₂×C₇×C₉×C₁₇	direct (N × H)
130. C₉	C₂×C₃×C₄×C₇×C₁₇	direct (N × H)
131. C₂×C₄	C₃×C₇×C₉×C₁₇	direct (N × H)
132. C₇	C₂×C₃×C₄×C₉×C₁₇	direct (N × H)
133. C₂×C₃	C₄×C₇×C₉×C₁₇	direct (N × H)
138. C₄	C₂×C₃×C₇×C₉×C₁₇	direct (N × H)
139. C₃	C₂×C₄×C₇×C₉×C₁₇	direct (N × H)
141. C₂	C₃×C₄×C₇×C₉×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
C₂²×C₃×C₇×C₉×C₁₇	2
C₃×C₄×C₇×C₉×C₁₇	2
C₂×C₃²×C₄×C₇×C₁₇	3
C₂×C₄×C₇×C₉×C₁₇	3
C₂×C₃×C₄×C₉×C₁₇	7
C₂×C₃×C₄×C₇×C₉	17

Subgroups — C2×C3×C4×C7×C9×C17 #72202913

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₄×C₇×C₉×C₁₇ #72202913