Subgroups — C₄×C₅×#798 #14312575

Order 1920.

392 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#798	C₂	1	2	yes	maximal
2	C₄×C₅×D₃×C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4502	C₂	1	2	yes	maximal, complemented
5	C₄×C₅×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×C₅×M₃₂	—	3	1	no	maximal, Hall
7	C₂×C₅×D₃×C₈	C₄	1	1	yes	characteristic
8	C₂×C₅×D₃×C₈	C₂²	1	1	yes	characteristic
9	C₄²×C₅×D₃	C₄	1	1	yes	characteristic
10	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
11	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₅×C₈×Dic₃	C₄	1	1	yes	characteristic
13	C₅×C₈×Dic₃	C₂²	1	1	yes	characteristic
14	C₂×C₅×C₃⋊C₁₆	C₂²	1	2	yes	—
15	C₄×C₅×C₃⋊C₈	C₄	1	1	yes	characteristic
16	C₅×#798	C₄	1	8	yes	direct product factor
17	C₄×#798	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₅×M₃₂	—	3	2	no	—
19	C₂×C₄×C₅×C₈	—	3	1	no	—
20	C₅×#346	—	3	2	no	—
21	C₄×C₅×C₁₆	—	3	1	no	—
22	C₄×C₅×C₁₆	D₃	1	1	yes	characteristic, complemented
23	C₂×C₄×C₅×D₃	C₈	1	2	yes	—
24	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
27	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
28	C₄×C₅×Dic₃	C₈	1	2	yes	—
29	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
32	C₅×D₃×C₈	C₈	1	4	yes	—
33	C₅×D₃×C₈	C₂×C₄	1	4	yes	—
34	C₃×C₅×C₁₆	C₂×C₄	1	4	yes	complemented
35	C₅×C₃⋊C₁₆	C₂×C₄	1	4	yes	complemented
36	C₂×#798	C₂×C₅	1	2	yes	—
37	C₄×D₃×C₈	C₂×C₅	1	1	yes	characteristic
38	C₃×C₄×C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
39	#4502	C₂×C₅	1	2	yes	complemented
40	C₄×C₃⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
41	C₂²×C₅×C₈	—	3	1	no	—
42	C₂²×C₅×C₈	—	3	1	no	—
43	C₂×C₄²×C₅	—	3	1	no	—
44	C₅×M₃₂	—	3	8	no	—
45	C₂×C₅×C₁₆	—	3	2	no	—
46	C₂×C₅×C₁₆	C₂×D₃	1	2	yes	—
47	C₄×C₅×C₈	—	3	1	no	—
48	C₄×C₅×C₈	—	3	1	no	—
49	C₄×C₅×C₈	—	3	1	no	—
50	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
51	C₄×M₃₂	—	3	1	no	Sylow
52	C₂²×C₅×D₃	C₂×C₈	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
54	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₅×Dic₃	C₂×C₈	1	2	yes	—
56	C₂×C₅×Dic₃	C₂×C₈	1	1	yes	characteristic
57	C₄×C₅×D₃	—	2	4	no	—
58	C₄×C₅×D₃	C₄²	1	2	yes	—
59	C₄×C₅×D₃	C₂×C₈	1	2	yes	—
60	C₃×C₅×C₈	C₂×C₈	1	2	yes	—
61	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
62	C₅×C₃⋊C₈	C₄²	1	2	yes	—
63	C₅×C₃⋊C₈	C₂×C₈	1	2	yes	—
64	C₂×D₃×C₈	C₄×C₅	1	1	yes	characteristic
65	C₂×D₃×C₈	C₂²×C₅	1	1	yes	characteristic
66	C₄²×D₃	C₄×C₅	1	1	yes	characteristic
67	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	—
68	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
69	C₈×Dic₃	C₄×C₅	1	1	yes	characteristic
70	C₈×Dic₃	C₂²×C₅	1	1	yes	characteristic
71	C₂×C₃⋊C₁₆	C₂²×C₅	1	2	yes	—
72	C₄×C₃⋊C₈	C₄×C₅	1	1	yes	characteristic
73	#798	C₄×C₅	1	8	yes	direct product factor
74	C₂²×C₄×C₅	—	3	1	no	—
75	C₂²×C₄×C₅	—	3	2	no	—
76	C₂×C₅×C₈	—	3	1	no	—
77	C₂×C₅×C₈	—	3	1	no	—
78	C₂×C₅×C₈	—	3	4	no	—
79	C₂×C₅×C₈	—	3	4	no	—
80	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
81	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
82	C₄²×C₅	—	3	2	no	—
83	C₄²×C₅	—	3	1	no	—
84	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
85	C₅×C₁₆	—	3	4	no	—
86	C₅×C₁₆	C₄×D₃	1	4	yes	complemented
87	C₂×M₃₂	—	3	2	no	—
88	C₂×C₄×C₈	—	3	1	no	—
89	#346	—	3	2	no	—
90	C₄×C₁₆	—	3	1	no	—
91	C₄×C₁₆	C₅×D₃	1	1	yes	characteristic, complemented
92	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
93	C₂×C₅×D₃	—	2	2	no	—
94	C₂×C₅×D₃	C₄×C₈	1	2	yes	—
95	C₃×C₄×C₅	M₃₂	1	4	yes	complemented
96	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
97	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
98	C₅×Dic₃	—	2	2	no	—
99	C₅×Dic₃	C₄×C₈	1	2	yes	—
100	C₂×C₄×D₃	C₅×C₈	1	2	yes	—
101	C₂×C₄×D₃	C₂×C₄×C₅	1	1	yes	characteristic
102	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
103	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
104	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
105	C₄×Dic₃	C₅×C₈	1	2	yes	—
106	C₄×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
107	C₂×C₃⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
108	C₂×C₃⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
109	D₃×C₈	C₅×C₈	1	4	yes	—
110	D₃×C₈	C₂×C₄×C₅	1	4	yes	—
111	C₃×C₁₆	C₂×C₄×C₅	1	4	yes	complemented
112	C₃⋊C₁₆	C₂×C₄×C₅	1	4	yes	complemented
113	C₂³×C₅	—	3	1	no	—
114	C₂×C₄×C₅	—	3	1	no	—
115	C₂×C₄×C₅	—	3	2	no	—
116	C₂×C₄×C₅	—	3	2	no	—
117	C₂×C₄×C₅	—	3	2	no	—
118	C₂×C₄×C₅	—	6	4	no	—
119	C₂×C₄×C₅	D₃×C₈	1	2	yes	—
120	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
121	C₅×C₈	—	3	2	no	—
122	C₅×C₈	—	3	2	no	—
123	C₅×C₈	D₃×C₈	1	2	yes	—
124	C₅×C₈	C₂×C₄×D₃	1	2	yes	—
125	C₂²×C₈	—	3	1	no	—
126	C₂²×C₈	—	3	1	no	—
127	C₂×C₄²	—	3	1	no	—
128	M₃₂	—	3	8	no	—
129	C₂×C₁₆	—	3	2	no	—
130	C₂×C₁₆	C₂×C₅×D₃	1	2	yes	—
131	C₄×C₈	—	3	1	no	—
132	C₄×C₈	—	3	1	no	—
133	C₄×C₈	—	3	1	no	—
134	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
135	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
136	C₂×C₃×C₅	C₂×M₃₂	1	2	yes	—
137	C₅×D₃	—	2	2	no	—
138	C₂²×D₃	C₂×C₅×C₈	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂×C₅×C₈	1	2	yes	—
140	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
141	C₂×Dic₃	C₂×C₅×C₈	1	2	yes	—
142	C₂×Dic₃	C₂×C₅×C₈	1	1	yes	characteristic
143	C₄×D₃	—	2	4	no	—
144	C₄×D₃	C₄²×C₅	1	2	yes	—
145	C₄×D₃	C₂×C₅×C₈	1	2	yes	—
146	C₃×C₈	C₂×C₅×C₈	1	2	yes	—
147	C₃×C₈	C₂²×C₄×C₅	1	2	yes	—
148	C₃⋊C₈	C₄²×C₅	1	2	yes	—
149	C₃⋊C₈	C₂×C₅×C₈	1	2	yes	—
150	C₂²×C₅	—	3	2	no	—
151	C₂²×C₅	—	6	2	no	—
152	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
153	C₄×C₅	—	3	2	no	—
154	C₄×C₅	—	6	2	no	—
155	C₄×C₅	#798	1	4	yes	direct product factor
156	C₄×C₅	C₄²×D₃	1	1	yes	characteristic
157	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
158	C₂²×C₄	—	3	1	no	—
159	C₂²×C₄	—	3	2	no	—
160	C₂×C₈	—	3	1	no	—
161	C₂×C₈	—	3	1	no	—
162	C₂×C₈	—	3	4	no	—
163	C₂×C₈	—	3	4	no	—
164	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
165	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
166	C₄²	—	3	2	no	—
167	C₄²	—	3	1	no	—
168	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
169	C₁₆	—	3	4	no	—
170	C₁₆	C₄×C₅×D₃	1	4	yes	complemented
171	C₃×C₅	C₄×M₃₂	1	1	yes	characteristic, complemented, Hall
172	C₂²×C₃	C₂²×C₅×C₈	1	1	yes	characteristic
173	C₂×D₃	—	2	2	no	—
174	C₂×D₃	C₄×C₅×C₈	1	2	yes	—
175	C₃×C₄	C₅×M₃₂	1	4	yes	complemented
176	C₃×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
177	C₃×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
178	Dic₃	—	2	2	no	—
179	Dic₃	C₄×C₅×C₈	1	2	yes	—
180	C₂×C₅	—	6	2	no	—
181	C₂×C₅	C₄×D₃×C₈	1	1	yes	characteristic
182	C₂×C₅	C₂×#798	1	2	yes	—
183	C₂³	—	3	1	no	—
184	C₂×C₄	—	3	1	no	—
185	C₂×C₄	—	3	2	no	—
186	C₂×C₄	—	3	2	no	—
187	C₂×C₄	—	3	2	no	—
188	C₂×C₄	—	6	4	no	—
189	C₂×C₄	C₅×D₃×C₈	1	2	yes	—
190	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
191	C₈	—	3	2	no	—
192	C₈	—	3	2	no	—
193	C₈	C₅×D₃×C₈	1	2	yes	—
194	C₈	C₂×C₄×C₅×D₃	1	2	yes	—
195	C₂×C₃	C₂×C₄×C₅×C₈	1	1	yes	characteristic
196	C₂×C₃	C₂×C₅×M₃₂	1	2	yes	—
197	D₃	—	2	2	no	—
198	C₅	C₄×#798	1	1	yes	characteristic, direct product factor, Sylow
199	C₂²	—	3	2	no	—
200	C₂²	—	6	2	no	—
201	C₂²	C₂×C₅×D₃×C₈	1	1	yes	characteristic
202	C₄	—	3	2	no	—
203	C₄	—	6	2	no	—
204	C₄	C₅×#798	1	4	yes	direct product factor
205	C₄	C₄²×C₅×D₃	1	1	yes	characteristic
206	C₄	C₂×C₅×D₃×C₈	1	1	yes	characteristic
207	C₃	C₄×C₅×M₃₂	1	1	yes	characteristic, complemented, Sylow
208	C₂	—	6	2	no	—
209	C₂	C₄×C₅×D₃×C₈	1	1	yes	characteristic
210	C₂	C₂×C₅×#798	1	2	yes	—

	subgroup	G/H
center	C₄×C₅×C₈	C₂×D₃
commutator	C₂×C₃	C₂×C₄×C₅×C₈
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₃×C₄×C₅×C₁₆	C₂
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂²×C₅×D₃	C₂×C₈
cosocle	C₂×C₃×C₈	C₂³×C₅
hypercenter	C₄×C₅×C₁₆	D₃
hypocenter	C₃	C₄×C₅×M₃₂
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅×C₈ ◁ C₄×C₅×C₁₆

Complements

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

N	H ≅ G/N	type
3. C₃×C₄×C₅×C₁₆	C₂	semidirect
4. C₅×#4502	C₂	semidirect
5. C₄×C₅×C₃⋊C₁₆	C₂	semidirect
16. C₅×#798	C₄	direct (N × H)
17. C₄×#798	C₅	direct (N × H)
22. C₄×C₅×C₁₆	D₃	semidirect
34. C₃×C₅×C₁₆	C₂×C₄	semidirect
35. C₅×C₃⋊C₁₆	C₂×C₄	semidirect
38. C₃×C₄×C₁₆	C₂×C₅	semidirect
39. #4502	C₂×C₅	semidirect
40. C₄×C₃⋊C₁₆	C₂×C₅	semidirect
73. #798	C₄×C₅	direct (N × H)
86. C₅×C₁₆	C₄×D₃	semidirect
91. C₄×C₁₆	C₅×D₃	semidirect
95. C₃×C₄×C₅	M₃₂	semidirect
111. C₃×C₁₆	C₂×C₄×C₅	semidirect
112. C₃⋊C₁₆	C₂×C₄×C₅	semidirect
155. C₄×C₅	#798	direct (N × H)
170. C₁₆	C₄×C₅×D₃	semidirect
171. C₃×C₅	C₄×M₃₂	semidirect
175. C₃×C₄	C₅×M₃₂	semidirect
198. C₅	C₄×#798	direct (N × H)
204. C₄	C₅×#798	direct (N × H)
207. C₃	C₄×C₅×M₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#798	2
C₄×C₅×D₃×C₈	2
C₃×C₄×C₅×C₁₆	2
C₅×#4502	2
C₄×C₅×C₃⋊C₁₆	2
C₄×C₅×M₃₂	3
C₄×#798	5

Subgroups — C4×C5×#798 #14312575

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅×#798 #14312575