Subgroups — C₂²×C₅×C₈×C₉×C₁₇ #72201749

Order 24480.

456 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅×C₉×C₁₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₈×C₉×C₁₇	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃×C₅×C₈×C₁₇	C₃	1	1	yes	maximal, characteristic
4	C₂³×C₅×C₉×C₁₇	C₄	1	1	yes	characteristic
5	C₂×C₄×C₅×C₉×C₁₇	C₄	1	3	yes	—
6	C₂×C₄×C₅×C₉×C₁₇	C₂²	1	3	yes	—
7	C₅×C₈×C₉×C₁₇	C₂²	1	4	yes	direct product factor
8	C₂²×C₈×C₉×C₁₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂²×C₃×C₄×C₅×C₁₇	C₂×C₃	1	1	yes	characteristic
10	C₂×C₃×C₅×C₈×C₁₇	C₂×C₃	1	6	yes	—
11	C₂²×C₅×C₈×C₁₇	C₉	1	1	yes	characteristic, direct product factor, Hall
12	C₂²×C₄×C₉×C₁₇	C₂×C₅	1	1	yes	characteristic
13	C₂×C₈×C₉×C₁₇	C₂×C₅	1	6	yes	direct product factor
14	C₂³×C₃×C₅×C₁₇	C₃×C₄	1	1	yes	characteristic
15	C₂×C₃×C₄×C₅×C₁₇	C₃×C₄	1	3	yes	—
16	C₂×C₃×C₄×C₅×C₁₇	C₂²×C₃	1	3	yes	—
17	C₃×C₅×C₈×C₁₇	C₂²×C₃	1	4	yes	—
18	C₂²×C₅×C₉×C₁₇	C₈	1	4	yes	direct product factor
19	C₂²×C₅×C₉×C₁₇	C₂×C₄	1	3	yes	—
20	C₄×C₅×C₉×C₁₇	C₂×C₄	1	3	yes	—
21	C₄×C₅×C₉×C₁₇	C₂³	1	1	yes	characteristic
22	C₂²×C₃×C₈×C₁₇	C₃×C₅	1	1	yes	characteristic
23	C₂×C₅×C₉×C₁₇	C₂×C₈	1	6	yes	direct product factor
24	C₂×C₅×C₉×C₁₇	C₂²×C₄	1	1	yes	characteristic
25	C₂²×C₅×C₈×C₉	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
26	C₂²×C₄×C₅×C₁₇	C₂×C₉	1	1	yes	characteristic
27	C₂×C₅×C₈×C₁₇	C₂×C₉	1	6	yes	direct product factor
28	C₂³×C₉×C₁₇	C₄×C₅	1	1	yes	characteristic
29	C₂×C₄×C₉×C₁₇	C₄×C₅	1	3	yes	—
30	C₂×C₄×C₉×C₁₇	C₂²×C₅	1	3	yes	—
31	C₈×C₉×C₁₇	C₂²×C₅	1	4	yes	direct product factor
32	C₂²×C₃×C₅×C₁₇	C₃×C₈	1	4	yes	—
33	C₂²×C₃×C₅×C₁₇	C₂×C₃×C₄	1	3	yes	—
34	C₃×C₄×C₅×C₁₇	C₂×C₃×C₄	1	3	yes	—
35	C₃×C₄×C₅×C₁₇	C₂³×C₃	1	1	yes	characteristic
36	C₂²×C₃×C₄×C₁₇	C₂×C₃×C₅	1	1	yes	characteristic
37	C₂×C₃×C₈×C₁₇	C₂×C₃×C₅	1	6	yes	—
38	C₅×C₉×C₁₇	C₂²×C₈	1	1	yes	characteristic, direct product factor, Hall
39	C₂²×C₄×C₅×C₉	C₂×C₁₇	1	1	yes	characteristic
40	C₂×C₅×C₈×C₉	C₂×C₁₇	1	6	yes	direct product factor
41	C₂³×C₅×C₁₇	C₄×C₉	1	1	yes	characteristic
42	C₂×C₄×C₅×C₁₇	C₄×C₉	1	3	yes	—
43	C₂×C₄×C₅×C₁₇	C₂²×C₉	1	3	yes	—
44	C₅×C₈×C₁₇	C₂²×C₉	1	4	yes	direct product factor
45	C₂²×C₉×C₁₇	C₅×C₈	1	4	yes	direct product factor
46	C₂²×C₉×C₁₇	C₂×C₄×C₅	1	3	yes	—
47	C₄×C₉×C₁₇	C₂×C₄×C₅	1	3	yes	—
48	C₄×C₉×C₁₇	C₂³×C₅	1	1	yes	characteristic
49	C₂²×C₈×C₁₇	C₅×C₉	1	1	yes	characteristic, direct product factor, Hall
50	C₂×C₃×C₅×C₁₇	C₂×C₃×C₈	1	6	yes	—
51	C₂×C₃×C₅×C₁₇	C₂²×C₃×C₄	1	1	yes	characteristic
52	C₂²×C₃×C₅×C₈	C₃×C₁₇	1	1	yes	characteristic
53	C₂³×C₃×C₁₇	C₃×C₄×C₅	1	1	yes	characteristic
54	C₂×C₃×C₄×C₁₇	C₃×C₄×C₅	1	3	yes	—
55	C₂×C₃×C₄×C₁₇	C₂²×C₃×C₅	1	3	yes	—
56	C₃×C₈×C₁₇	C₂²×C₃×C₅	1	4	yes	—
57	C₂³×C₅×C₉	C₄×C₁₇	1	1	yes	characteristic
58	C₂×C₄×C₅×C₉	C₄×C₁₇	1	3	yes	—
59	C₂×C₄×C₅×C₉	C₂²×C₁₇	1	3	yes	—
60	C₅×C₈×C₉	C₂²×C₁₇	1	4	yes	direct product factor
61	C₂²×C₅×C₁₇	C₈×C₉	1	4	yes	direct product factor
62	C₂²×C₅×C₁₇	C₂×C₄×C₉	1	3	yes	—
63	C₄×C₅×C₁₇	C₂×C₄×C₉	1	3	yes	—
64	C₄×C₅×C₁₇	C₂³×C₉	1	1	yes	characteristic
65	C₂×C₉×C₁₇	C₂×C₅×C₈	1	6	yes	direct product factor
66	C₂×C₉×C₁₇	C₂²×C₄×C₅	1	1	yes	characteristic
67	C₂²×C₈×C₉	C₅×C₁₇	1	1	yes	characteristic, direct product factor, Hall
68	C₂²×C₄×C₁₇	C₂×C₅×C₉	1	1	yes	characteristic
69	C₂×C₈×C₁₇	C₂×C₅×C₉	1	6	yes	direct product factor
70	C₃×C₅×C₁₇	C₂²×C₃×C₈	1	1	yes	characteristic
71	C₂²×C₃×C₄×C₅	C₂×C₃×C₁₇	1	1	yes	characteristic
72	C₂×C₃×C₅×C₈	C₂×C₃×C₁₇	1	6	yes	—
73	C₂²×C₃×C₁₇	C₃×C₅×C₈	1	4	yes	—
74	C₂²×C₃×C₁₇	C₂×C₃×C₄×C₅	1	3	yes	—
75	C₃×C₄×C₁₇	C₂×C₃×C₄×C₅	1	3	yes	—
76	C₃×C₄×C₁₇	C₂³×C₃×C₅	1	1	yes	characteristic
77	C₂²×C₅×C₉	C₈×C₁₇	1	4	yes	direct product factor
78	C₂²×C₅×C₉	C₂×C₄×C₁₇	1	3	yes	—
79	C₄×C₅×C₉	C₂×C₄×C₁₇	1	3	yes	—
80	C₄×C₅×C₉	C₂³×C₁₇	1	1	yes	characteristic
81	C₂×C₅×C₁₇	C₂×C₈×C₉	1	6	yes	direct product factor
82	C₂×C₅×C₁₇	C₂²×C₄×C₉	1	1	yes	characteristic
83	C₂²×C₅×C₈	C₉×C₁₇	1	1	yes	characteristic, direct product factor, Hall
84	C₉×C₁₇	C₂²×C₅×C₈	1	1	yes	characteristic, direct product factor, Hall
85	C₂²×C₄×C₉	C₂×C₅×C₁₇	1	1	yes	characteristic
86	C₂×C₈×C₉	C₂×C₅×C₁₇	1	6	yes	direct product factor
87	C₂³×C₁₇	C₄×C₅×C₉	1	1	yes	characteristic
88	C₂×C₄×C₁₇	C₄×C₅×C₉	1	3	yes	—
89	C₂×C₄×C₁₇	C₂²×C₅×C₉	1	3	yes	—
90	C₈×C₁₇	C₂²×C₅×C₉	1	4	yes	direct product factor
91	C₂³×C₃×C₅	C₃×C₄×C₁₇	1	1	yes	characteristic
92	C₂×C₃×C₄×C₅	C₃×C₄×C₁₇	1	3	yes	—
93	C₂×C₃×C₄×C₅	C₂²×C₃×C₁₇	1	3	yes	—
94	C₃×C₅×C₈	C₂²×C₃×C₁₇	1	4	yes	—
95	C₂×C₃×C₁₇	C₂×C₃×C₅×C₈	1	6	yes	—
96	C₂×C₃×C₁₇	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
97	C₂²×C₃×C₈	C₃×C₅×C₁₇	1	1	yes	characteristic
98	C₂×C₅×C₉	C₂×C₈×C₁₇	1	6	yes	direct product factor
99	C₂×C₅×C₉	C₂²×C₄×C₁₇	1	1	yes	characteristic
100	C₅×C₁₇	C₂²×C₈×C₉	1	1	yes	characteristic, direct product factor, Hall
101	C₂²×C₄×C₅	C₂×C₉×C₁₇	1	1	yes	characteristic
102	C₂×C₅×C₈	C₂×C₉×C₁₇	1	6	yes	direct product factor
103	C₂³×C₉	C₄×C₅×C₁₇	1	1	yes	characteristic
104	C₂×C₄×C₉	C₄×C₅×C₁₇	1	3	yes	—
105	C₂×C₄×C₉	C₂²×C₅×C₁₇	1	3	yes	—
106	C₈×C₉	C₂²×C₅×C₁₇	1	4	yes	direct product factor
107	C₂²×C₁₇	C₅×C₈×C₉	1	4	yes	direct product factor
108	C₂²×C₁₇	C₂×C₄×C₅×C₉	1	3	yes	—
109	C₄×C₁₇	C₂×C₄×C₅×C₉	1	3	yes	—
110	C₄×C₁₇	C₂³×C₅×C₉	1	1	yes	characteristic
111	C₂²×C₃×C₅	C₃×C₈×C₁₇	1	4	yes	—
112	C₂²×C₃×C₅	C₂×C₃×C₄×C₁₇	1	3	yes	—
113	C₃×C₄×C₅	C₂×C₃×C₄×C₁₇	1	3	yes	—
114	C₃×C₄×C₅	C₂³×C₃×C₁₇	1	1	yes	characteristic
115	C₃×C₁₇	C₂²×C₃×C₅×C₈	1	1	yes	characteristic
116	C₂²×C₃×C₄	C₂×C₃×C₅×C₁₇	1	1	yes	characteristic
117	C₂×C₃×C₈	C₂×C₃×C₅×C₁₇	1	6	yes	—
118	C₅×C₉	C₂²×C₈×C₁₇	1	1	yes	characteristic, direct product factor, Hall
119	C₂³×C₅	C₄×C₉×C₁₇	1	1	yes	characteristic
120	C₂×C₄×C₅	C₄×C₉×C₁₇	1	3	yes	—
121	C₂×C₄×C₅	C₂²×C₉×C₁₇	1	3	yes	—
122	C₅×C₈	C₂²×C₉×C₁₇	1	4	yes	direct product factor
123	C₂²×C₉	C₅×C₈×C₁₇	1	4	yes	direct product factor
124	C₂²×C₉	C₂×C₄×C₅×C₁₇	1	3	yes	—
125	C₄×C₉	C₂×C₄×C₅×C₁₇	1	3	yes	—
126	C₄×C₉	C₂³×C₅×C₁₇	1	1	yes	characteristic
127	C₂×C₁₇	C₂×C₅×C₈×C₉	1	6	yes	direct product factor
128	C₂×C₁₇	C₂²×C₄×C₅×C₉	1	1	yes	characteristic
129	C₂²×C₈	C₅×C₉×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
130	C₂×C₃×C₅	C₂×C₃×C₈×C₁₇	1	6	yes	—
131	C₂×C₃×C₅	C₂²×C₃×C₄×C₁₇	1	1	yes	characteristic
132	C₂³×C₃	C₃×C₄×C₅×C₁₇	1	1	yes	characteristic
133	C₂×C₃×C₄	C₃×C₄×C₅×C₁₇	1	3	yes	—
134	C₂×C₃×C₄	C₂²×C₃×C₅×C₁₇	1	3	yes	—
135	C₃×C₈	C₂²×C₃×C₅×C₁₇	1	4	yes	—
136	C₂²×C₅	C₈×C₉×C₁₇	1	4	yes	direct product factor
137	C₂²×C₅	C₂×C₄×C₉×C₁₇	1	3	yes	—
138	C₄×C₅	C₂×C₄×C₉×C₁₇	1	3	yes	—
139	C₄×C₅	C₂³×C₉×C₁₇	1	1	yes	characteristic
140	C₂×C₉	C₂×C₅×C₈×C₁₇	1	6	yes	direct product factor
141	C₂×C₉	C₂²×C₄×C₅×C₁₇	1	1	yes	characteristic
142	C₁₇	C₂²×C₅×C₈×C₉	1	1	yes	characteristic, direct product factor, Sylow
143	C₂²×C₄	C₂×C₅×C₉×C₁₇	1	1	yes	characteristic
144	C₂×C₈	C₂×C₅×C₉×C₁₇	1	6	yes	direct product factor
145	C₃×C₅	C₂²×C₃×C₈×C₁₇	1	1	yes	characteristic
146	C₂²×C₃	C₃×C₅×C₈×C₁₇	1	4	yes	—
147	C₂²×C₃	C₂×C₃×C₄×C₅×C₁₇	1	3	yes	—
148	C₃×C₄	C₂×C₃×C₄×C₅×C₁₇	1	3	yes	—
149	C₃×C₄	C₂³×C₃×C₅×C₁₇	1	1	yes	characteristic
150	C₂×C₅	C₂×C₈×C₉×C₁₇	1	6	yes	direct product factor
151	C₂×C₅	C₂²×C₄×C₉×C₁₇	1	1	yes	characteristic
152	C₉	C₂²×C₅×C₈×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
153	C₂³	C₄×C₅×C₉×C₁₇	1	1	yes	characteristic
154	C₂×C₄	C₄×C₅×C₉×C₁₇	1	3	yes	—
155	C₂×C₄	C₂²×C₅×C₉×C₁₇	1	3	yes	—
156	C₈	C₂²×C₅×C₉×C₁₇	1	4	yes	direct product factor
157	C₂×C₃	C₂×C₃×C₅×C₈×C₁₇	1	6	yes	—
158	C₂×C₃	C₂²×C₃×C₄×C₅×C₁₇	1	1	yes	characteristic
159	C₅	C₂²×C₈×C₉×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
160	C₂²	C₅×C₈×C₉×C₁₇	1	4	yes	direct product factor
161	C₂²	C₂×C₄×C₅×C₉×C₁₇	1	3	yes	—
162	C₄	C₂×C₄×C₅×C₉×C₁₇	1	3	yes	—
163	C₄	C₂³×C₅×C₉×C₁₇	1	1	yes	characteristic
164	C₃	C₂²×C₃×C₅×C₈×C₁₇	1	1	yes	characteristic
165	C₂	C₂×C₅×C₈×C₉×C₁₇	1	6	yes	direct product factor
166	C₂	C₂²×C₄×C₅×C₉×C₁₇	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃×C₄	C₂³×C₃×C₅×C₁₇
socle, prime radical	C₂³×C₃×C₅×C₁₇	C₃×C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×C₈×C₉×C₁₇	C₂	direct (N × H)
7. C₅×C₈×C₉×C₁₇	C₂²	direct (N × H)
8. C₂²×C₈×C₉×C₁₇	C₅	direct (N × H)
11. C₂²×C₅×C₈×C₁₇	C₉	direct (N × H)
13. C₂×C₈×C₉×C₁₇	C₂×C₅	direct (N × H)
18. C₂²×C₅×C₉×C₁₇	C₈	direct (N × H)
23. C₂×C₅×C₉×C₁₇	C₂×C₈	direct (N × H)
25. C₂²×C₅×C₈×C₉	C₁₇	direct (N × H)
27. C₂×C₅×C₈×C₁₇	C₂×C₉	direct (N × H)
31. C₈×C₉×C₁₇	C₂²×C₅	direct (N × H)
38. C₅×C₉×C₁₇	C₂²×C₈	direct (N × H)
40. C₂×C₅×C₈×C₉	C₂×C₁₇	direct (N × H)
44. C₅×C₈×C₁₇	C₂²×C₉	direct (N × H)
45. C₂²×C₉×C₁₇	C₅×C₈	direct (N × H)
49. C₂²×C₈×C₁₇	C₅×C₉	direct (N × H)
60. C₅×C₈×C₉	C₂²×C₁₇	direct (N × H)
61. C₂²×C₅×C₁₇	C₈×C₉	direct (N × H)
65. C₂×C₉×C₁₇	C₂×C₅×C₈	direct (N × H)
67. C₂²×C₈×C₉	C₅×C₁₇	direct (N × H)
69. C₂×C₈×C₁₇	C₂×C₅×C₉	direct (N × H)
77. C₂²×C₅×C₉	C₈×C₁₇	direct (N × H)
81. C₂×C₅×C₁₇	C₂×C₈×C₉	direct (N × H)
83. C₂²×C₅×C₈	C₉×C₁₇	direct (N × H)
84. C₉×C₁₇	C₂²×C₅×C₈	direct (N × H)
86. C₂×C₈×C₉	C₂×C₅×C₁₇	direct (N × H)
90. C₈×C₁₇	C₂²×C₅×C₉	direct (N × H)
98. C₂×C₅×C₉	C₂×C₈×C₁₇	direct (N × H)
100. C₅×C₁₇	C₂²×C₈×C₉	direct (N × H)
102. C₂×C₅×C₈	C₂×C₉×C₁₇	direct (N × H)
106. C₈×C₉	C₂²×C₅×C₁₇	direct (N × H)
107. C₂²×C₁₇	C₅×C₈×C₉	direct (N × H)
118. C₅×C₉	C₂²×C₈×C₁₇	direct (N × H)
122. C₅×C₈	C₂²×C₉×C₁₇	direct (N × H)
123. C₂²×C₉	C₅×C₈×C₁₇	direct (N × H)
127. C₂×C₁₇	C₂×C₅×C₈×C₉	direct (N × H)
129. C₂²×C₈	C₅×C₉×C₁₇	direct (N × H)
136. C₂²×C₅	C₈×C₉×C₁₇	direct (N × H)
140. C₂×C₉	C₂×C₅×C₈×C₁₇	direct (N × H)
142. C₁₇	C₂²×C₅×C₈×C₉	direct (N × H)
144. C₂×C₈	C₂×C₅×C₉×C₁₇	direct (N × H)
150. C₂×C₅	C₂×C₈×C₉×C₁₇	direct (N × H)
152. C₉	C₂²×C₅×C₈×C₁₇	direct (N × H)
156. C₈	C₂²×C₅×C₉×C₁₇	direct (N × H)
159. C₅	C₂²×C₈×C₉×C₁₇	direct (N × H)
160. C₂²	C₅×C₈×C₉×C₁₇	direct (N × H)
165. C₂	C₂×C₅×C₈×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×C₄×C₅×C₉×C₁₇	2
C₂×C₅×C₈×C₉×C₁₇	2
C₂²×C₃×C₅×C₈×C₁₇	3
C₂²×C₈×C₉×C₁₇	5
C₂²×C₅×C₈×C₉	17

Subgroups — C22×C5×C8×C9×C17 #72201749

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×C₈×C₉×C₁₇ #72201749