Subgroups — C₃²×C₄×C₅×C₈×C₁₇ #72201747

Order 24480.

528 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	C₃²×C₄²×C₅×C₁₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×C₅×C₈×C₁₇	C₂	1	2	yes	maximal
3	C₃×C₄×C₅×C₈×C₁₇	C₃	1	4	yes	maximal, direct product factor
4	C₂×C₃²×C₄×C₅×C₁₇	C₄	1	2	yes	—
5	C₂×C₃²×C₄×C₅×C₁₇	C₂²	1	1	yes	characteristic
6	C₃²×C₅×C₈×C₁₇	C₄	1	4	yes	direct product factor
7	C₃²×C₄×C₈×C₁₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₃×C₄²×C₅×C₁₇	C₂×C₃	1	4	yes	—
9	C₂×C₃×C₅×C₈×C₁₇	C₂×C₃	1	8	yes	—
10	C₄×C₅×C₈×C₁₇	C₃²	1	1	yes	characteristic, direct product factor, Hall
11	C₃²×C₄²×C₁₇	C₂×C₅	1	1	yes	characteristic
12	C₂×C₃²×C₈×C₁₇	C₂×C₅	1	2	yes	—
13	C₂×C₃×C₄×C₅×C₁₇	C₃×C₄	1	8	yes	—
14	C₂×C₃×C₄×C₅×C₁₇	C₂²×C₃	1	4	yes	—
15	C₃×C₅×C₈×C₁₇	C₃×C₄	1	16	yes	direct product factor
16	C₂²×C₃²×C₅×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₃²×C₄×C₅×C₁₇	C₈	1	4	yes	direct product factor
18	C₃²×C₄×C₅×C₁₇	C₂×C₄	1	2	yes	—
19	C₃×C₄×C₈×C₁₇	C₃×C₅	1	4	yes	direct product factor
20	C₂×C₃²×C₅×C₁₇	C₄²	1	1	yes	characteristic
21	C₂×C₃²×C₅×C₁₇	C₂×C₈	1	2	yes	—
22	C₃²×C₄×C₅×C₈	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₂×C₅×C₈×C₁₇	C₂×C₃²	1	2	yes	—
24	C₄²×C₅×C₁₇	C₂×C₃²	1	1	yes	characteristic
25	C₂×C₃²×C₄×C₁₇	C₄×C₅	1	2	yes	—
26	C₂×C₃²×C₄×C₁₇	C₂²×C₅	1	1	yes	characteristic
27	C₃²×C₈×C₁₇	C₄×C₅	1	4	yes	direct product factor
28	C₂²×C₃×C₅×C₁₇	C₂×C₃×C₄	1	4	yes	—
29	C₃×C₄×C₅×C₁₇	C₃×C₈	1	16	yes	direct product factor
30	C₃×C₄×C₅×C₁₇	C₂×C₃×C₄	1	8	yes	—
31	C₂×C₃×C₈×C₁₇	C₂×C₃×C₅	1	8	yes	—
32	C₃×C₄²×C₁₇	C₂×C₃×C₅	1	4	yes	—
33	C₃²×C₅×C₁₇	C₄×C₈	1	1	yes	characteristic, direct product factor, Hall
34	C₂×C₃²×C₅×C₈	C₂×C₁₇	1	2	yes	—
35	C₃²×C₄²×C₅	C₂×C₁₇	1	1	yes	characteristic
36	C₂×C₄×C₅×C₁₇	C₃²×C₄	1	2	yes	—
37	C₂×C₄×C₅×C₁₇	C₂²×C₃²	1	1	yes	characteristic
38	C₅×C₈×C₁₇	C₃²×C₄	1	4	yes	direct product factor
39	C₂²×C₃²×C₁₇	C₂×C₄×C₅	1	1	yes	characteristic
40	C₃²×C₄×C₁₇	C₅×C₈	1	4	yes	direct product factor
41	C₃²×C₄×C₁₇	C₂×C₄×C₅	1	2	yes	—
42	C₄×C₈×C₁₇	C₃²×C₅	1	1	yes	characteristic, direct product factor, Hall
43	C₂×C₃×C₅×C₁₇	C₃×C₄²	1	4	yes	—
44	C₂×C₃×C₅×C₁₇	C₂×C₃×C₈	1	8	yes	—
45	C₃×C₄×C₅×C₈	C₃×C₁₇	1	4	yes	direct product factor
46	C₂×C₃×C₄×C₁₇	C₃×C₄×C₅	1	8	yes	—
47	C₂×C₃×C₄×C₁₇	C₂²×C₃×C₅	1	4	yes	—
48	C₃×C₈×C₁₇	C₃×C₄×C₅	1	16	yes	direct product factor
49	C₂×C₃²×C₄×C₅	C₄×C₁₇	1	2	yes	—
50	C₂×C₃²×C₄×C₅	C₂²×C₁₇	1	1	yes	characteristic
51	C₃²×C₅×C₈	C₄×C₁₇	1	4	yes	direct product factor
52	C₂²×C₅×C₁₇	C₂×C₃²×C₄	1	1	yes	characteristic
53	C₄×C₅×C₁₇	C₃²×C₈	1	4	yes	direct product factor
54	C₄×C₅×C₁₇	C₂×C₃²×C₄	1	2	yes	—
55	C₂×C₃²×C₁₇	C₄²×C₅	1	1	yes	characteristic
56	C₂×C₃²×C₁₇	C₂×C₅×C₈	1	2	yes	—
57	C₃²×C₄×C₈	C₅×C₁₇	1	1	yes	characteristic, direct product factor, Hall
58	C₂×C₈×C₁₇	C₂×C₃²×C₅	1	2	yes	—
59	C₄²×C₁₇	C₂×C₃²×C₅	1	1	yes	characteristic
60	C₃×C₅×C₁₇	C₃×C₄×C₈	1	4	yes	direct product factor
61	C₂×C₃×C₅×C₈	C₂×C₃×C₁₇	1	8	yes	—
62	C₃×C₄²×C₅	C₂×C₃×C₁₇	1	4	yes	—
63	C₂²×C₃×C₁₇	C₂×C₃×C₄×C₅	1	4	yes	—
64	C₃×C₄×C₁₇	C₃×C₅×C₈	1	16	yes	direct product factor
65	C₃×C₄×C₁₇	C₂×C₃×C₄×C₅	1	8	yes	—
66	C₂²×C₃²×C₅	C₂×C₄×C₁₇	1	1	yes	characteristic
67	C₃²×C₄×C₅	C₈×C₁₇	1	4	yes	direct product factor
68	C₃²×C₄×C₅	C₂×C₄×C₁₇	1	2	yes	—
69	C₂×C₅×C₁₇	C₃²×C₄²	1	1	yes	characteristic
70	C₂×C₅×C₁₇	C₂×C₃²×C₈	1	2	yes	—
71	C₄×C₅×C₈	C₃²×C₁₇	1	1	yes	characteristic, direct product factor, Hall
72	C₃²×C₁₇	C₄×C₅×C₈	1	1	yes	characteristic, direct product factor, Hall
73	C₂×C₃²×C₈	C₂×C₅×C₁₇	1	2	yes	—
74	C₃²×C₄²	C₂×C₅×C₁₇	1	1	yes	characteristic
75	C₂×C₄×C₁₇	C₃²×C₄×C₅	1	2	yes	—
76	C₂×C₄×C₁₇	C₂²×C₃²×C₅	1	1	yes	characteristic
77	C₈×C₁₇	C₃²×C₄×C₅	1	4	yes	direct product factor
78	C₂×C₃×C₄×C₅	C₃×C₄×C₁₇	1	8	yes	—
79	C₂×C₃×C₄×C₅	C₂²×C₃×C₁₇	1	4	yes	—
80	C₃×C₅×C₈	C₃×C₄×C₁₇	1	16	yes	direct product factor
81	C₂×C₃×C₁₇	C₃×C₄²×C₅	1	4	yes	—
82	C₂×C₃×C₁₇	C₂×C₃×C₅×C₈	1	8	yes	—
83	C₃×C₄×C₈	C₃×C₅×C₁₇	1	4	yes	direct product factor
84	C₂×C₃²×C₅	C₄²×C₁₇	1	1	yes	characteristic
85	C₂×C₃²×C₅	C₂×C₈×C₁₇	1	2	yes	—
86	C₅×C₁₇	C₃²×C₄×C₈	1	1	yes	characteristic, direct product factor, Hall
87	C₂×C₅×C₈	C₂×C₃²×C₁₇	1	2	yes	—
88	C₄²×C₅	C₂×C₃²×C₁₇	1	1	yes	characteristic
89	C₂×C₃²×C₄	C₄×C₅×C₁₇	1	2	yes	—
90	C₂×C₃²×C₄	C₂²×C₅×C₁₇	1	1	yes	characteristic
91	C₃²×C₈	C₄×C₅×C₁₇	1	4	yes	direct product factor
92	C₂²×C₁₇	C₂×C₃²×C₄×C₅	1	1	yes	characteristic
93	C₄×C₁₇	C₃²×C₅×C₈	1	4	yes	direct product factor
94	C₄×C₁₇	C₂×C₃²×C₄×C₅	1	2	yes	—
95	C₂²×C₃×C₅	C₂×C₃×C₄×C₁₇	1	4	yes	—
96	C₃×C₄×C₅	C₃×C₈×C₁₇	1	16	yes	direct product factor
97	C₃×C₄×C₅	C₂×C₃×C₄×C₁₇	1	8	yes	—
98	C₃×C₁₇	C₃×C₄×C₅×C₈	1	4	yes	direct product factor
99	C₂×C₃×C₈	C₂×C₃×C₅×C₁₇	1	8	yes	—
100	C₃×C₄²	C₂×C₃×C₅×C₁₇	1	4	yes	—
101	C₃²×C₅	C₄×C₈×C₁₇	1	1	yes	characteristic, direct product factor, Hall
102	C₂×C₄×C₅	C₃²×C₄×C₁₇	1	2	yes	—
103	C₂×C₄×C₅	C₂²×C₃²×C₁₇	1	1	yes	characteristic
104	C₅×C₈	C₃²×C₄×C₁₇	1	4	yes	direct product factor
105	C₂²×C₃²	C₂×C₄×C₅×C₁₇	1	1	yes	characteristic
106	C₃²×C₄	C₅×C₈×C₁₇	1	4	yes	direct product factor
107	C₃²×C₄	C₂×C₄×C₅×C₁₇	1	2	yes	—
108	C₂×C₁₇	C₃²×C₄²×C₅	1	1	yes	characteristic
109	C₂×C₁₇	C₂×C₃²×C₅×C₈	1	2	yes	—
110	C₄×C₈	C₃²×C₅×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
111	C₂×C₃×C₅	C₃×C₄²×C₁₇	1	4	yes	—
112	C₂×C₃×C₅	C₂×C₃×C₈×C₁₇	1	8	yes	—
113	C₂×C₃×C₄	C₃×C₄×C₅×C₁₇	1	8	yes	—
114	C₂×C₃×C₄	C₂²×C₃×C₅×C₁₇	1	4	yes	—
115	C₃×C₈	C₃×C₄×C₅×C₁₇	1	16	yes	direct product factor
116	C₂²×C₅	C₂×C₃²×C₄×C₁₇	1	1	yes	characteristic
117	C₄×C₅	C₃²×C₈×C₁₇	1	4	yes	direct product factor
118	C₄×C₅	C₂×C₃²×C₄×C₁₇	1	2	yes	—
119	C₂×C₃²	C₄²×C₅×C₁₇	1	1	yes	characteristic
120	C₂×C₃²	C₂×C₅×C₈×C₁₇	1	2	yes	—
121	C₁₇	C₃²×C₄×C₅×C₈	1	1	yes	characteristic, direct product factor, Sylow
122	C₂×C₈	C₂×C₃²×C₅×C₁₇	1	2	yes	—
123	C₄²	C₂×C₃²×C₅×C₁₇	1	1	yes	characteristic
124	C₃×C₅	C₃×C₄×C₈×C₁₇	1	4	yes	direct product factor
125	C₂²×C₃	C₂×C₃×C₄×C₅×C₁₇	1	4	yes	—
126	C₃×C₄	C₃×C₅×C₈×C₁₇	1	16	yes	direct product factor
127	C₃×C₄	C₂×C₃×C₄×C₅×C₁₇	1	8	yes	—
128	C₂×C₅	C₂×C₃²×C₈×C₁₇	1	2	yes	—
129	C₂×C₅	C₃²×C₄²×C₁₇	1	1	yes	characteristic
130	C₃²	C₄×C₅×C₈×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
131	C₂×C₄	C₃²×C₄×C₅×C₁₇	1	2	yes	—
132	C₂×C₄	C₂²×C₃²×C₅×C₁₇	1	1	yes	characteristic
133	C₈	C₃²×C₄×C₅×C₁₇	1	4	yes	direct product factor
134	C₂×C₃	C₂×C₃×C₅×C₈×C₁₇	1	8	yes	—
135	C₂×C₃	C₃×C₄²×C₅×C₁₇	1	4	yes	—
136	C₅	C₃²×C₄×C₈×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
137	C₂²	C₂×C₃²×C₄×C₅×C₁₇	1	1	yes	characteristic
138	C₄	C₃²×C₅×C₈×C₁₇	1	4	yes	direct product factor
139	C₄	C₂×C₃²×C₄×C₅×C₁₇	1	2	yes	—
140	C₃	C₃×C₄×C₅×C₈×C₁₇	1	4	yes	direct product factor
141	C₂	C₂×C₃²×C₅×C₈×C₁₇	1	2	yes	—
142	C₂	C₃²×C₄²×C₅×C₁₇	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₄	C₂²×C₃²×C₅×C₁₇
socle, prime radical	C₂²×C₃²×C₅×C₁₇	C₂×C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₄×C₅×C₈×C₁₇	C₃	direct (N × H)
6. C₃²×C₅×C₈×C₁₇	C₄	direct (N × H)
7. C₃²×C₄×C₈×C₁₇	C₅	direct (N × H)
10. C₄×C₅×C₈×C₁₇	C₃²	direct (N × H)
15. C₃×C₅×C₈×C₁₇	C₃×C₄	direct (N × H)
17. C₃²×C₄×C₅×C₁₇	C₈	direct (N × H)
19. C₃×C₄×C₈×C₁₇	C₃×C₅	direct (N × H)
22. C₃²×C₄×C₅×C₈	C₁₇	direct (N × H)
27. C₃²×C₈×C₁₇	C₄×C₅	direct (N × H)
29. C₃×C₄×C₅×C₁₇	C₃×C₈	direct (N × H)
33. C₃²×C₅×C₁₇	C₄×C₈	direct (N × H)
38. C₅×C₈×C₁₇	C₃²×C₄	direct (N × H)
40. C₃²×C₄×C₁₇	C₅×C₈	direct (N × H)
42. C₄×C₈×C₁₇	C₃²×C₅	direct (N × H)
45. C₃×C₄×C₅×C₈	C₃×C₁₇	direct (N × H)
48. C₃×C₈×C₁₇	C₃×C₄×C₅	direct (N × H)
51. C₃²×C₅×C₈	C₄×C₁₇	direct (N × H)
53. C₄×C₅×C₁₇	C₃²×C₈	direct (N × H)
57. C₃²×C₄×C₈	C₅×C₁₇	direct (N × H)
60. C₃×C₅×C₁₇	C₃×C₄×C₈	direct (N × H)
64. C₃×C₄×C₁₇	C₃×C₅×C₈	direct (N × H)
67. C₃²×C₄×C₅	C₈×C₁₇	direct (N × H)
71. C₄×C₅×C₈	C₃²×C₁₇	direct (N × H)
72. C₃²×C₁₇	C₄×C₅×C₈	direct (N × H)
77. C₈×C₁₇	C₃²×C₄×C₅	direct (N × H)
80. C₃×C₅×C₈	C₃×C₄×C₁₇	direct (N × H)
83. C₃×C₄×C₈	C₃×C₅×C₁₇	direct (N × H)
86. C₅×C₁₇	C₃²×C₄×C₈	direct (N × H)
91. C₃²×C₈	C₄×C₅×C₁₇	direct (N × H)
93. C₄×C₁₇	C₃²×C₅×C₈	direct (N × H)
96. C₃×C₄×C₅	C₃×C₈×C₁₇	direct (N × H)
98. C₃×C₁₇	C₃×C₄×C₅×C₈	direct (N × H)
101. C₃²×C₅	C₄×C₈×C₁₇	direct (N × H)
104. C₅×C₈	C₃²×C₄×C₁₇	direct (N × H)
106. C₃²×C₄	C₅×C₈×C₁₇	direct (N × H)
110. C₄×C₈	C₃²×C₅×C₁₇	direct (N × H)
115. C₃×C₈	C₃×C₄×C₅×C₁₇	direct (N × H)
117. C₄×C₅	C₃²×C₈×C₁₇	direct (N × H)
121. C₁₇	C₃²×C₄×C₅×C₈	direct (N × H)
124. C₃×C₅	C₃×C₄×C₈×C₁₇	direct (N × H)
126. C₃×C₄	C₃×C₅×C₈×C₁₇	direct (N × H)
130. C₃²	C₄×C₅×C₈×C₁₇	direct (N × H)
133. C₈	C₃²×C₄×C₅×C₁₇	direct (N × H)
136. C₅	C₃²×C₄×C₈×C₁₇	direct (N × H)
138. C₄	C₃²×C₅×C₈×C₁₇	direct (N × H)
140. C₃	C₃×C₄×C₅×C₈×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
C₃²×C₄²×C₅×C₁₇	2
C₂×C₃²×C₅×C₈×C₁₇	2
C₃×C₄×C₅×C₈×C₁₇	3
C₃²×C₄×C₈×C₁₇	5
C₃²×C₄×C₅×C₈	17

Subgroups — C32×C4×C5×C8×C17 #72201747

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₄×C₅×C₈×C₁₇ #72201747