Subgroups — C₅×#70749 #14285818

Order 1920.

336 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#431	C₂	1	1	yes	maximal, characteristic
2	C₅×#4924	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	C₅×#4457	C₂	1	4	yes	maximal, complemented
5	C₅×#1570	—	3	1	no	maximal, Hall
6	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₂×C₅×#208	—	2	1	no	—
8	C₂²×C₅×C₃⋊C₈	—	2	1	no	—
9	C₃×C₅×C₄⋊C₈	C₄	1	2	yes	—
10	C₃×C₅×C₂²⋊C₈	C₄	1	2	yes	—
11	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
12	C₅×C₄⋊(C₃⋊C₈)	—	2	1	no	—
13	C₅×C₄⋊(C₃⋊C₈)	—	2	1	no	—
14	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
15	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
16	#70749	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₅×#431	D₃	1	1	yes	characteristic
18	C₅×#423	—	3	1	no	—
19	C₂×C₅×C₄⋊C₈	—	3	1	no	—
20	C₅×#334	—	3	4	no	—
21	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
23	C₂²×C₃×C₄×C₅	Q₈	1	1	yes	characteristic
24	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	—
25	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
26	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
27	C₃×C₄²×C₅	Q₈	1	1	yes	characteristic
28	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
29	C₅×#208	—	4	2	no	—
30	C₂×C₅×C₃⋊C₈	—	2	2	no	—
31	C₂×C₅×C₃⋊C₈	—	2	2	no	—
32	C₂×C₅×C₃⋊C₈	—	4	2	no	—
33	C₃×#431	C₂×C₅	1	1	yes	characteristic
34	#4924	C₂×C₅	1	1	yes	characteristic
35	C₂×C₄⋊(C₃⋊C₈)	C₂×C₅	1	1	yes	characteristic
36	#4457	C₂×C₅	1	4	yes	complemented
37	C₂×C₅×M₁₆	—	6	1	no	—
38	C₂²×C₅×C₈	—	6	1	no	—
39	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
40	C₅×C₄⋊C₈	—	3	2	no	—
41	C₅×C₄⋊C₈	—	3	2	no	—
42	C₅×C₄⋊C₈	—	6	1	no	—
43	C₅×C₄⋊C₈	—	6	1	no	—
44	C₅×C₄⋊C₈	Dic₃	1	2	yes	—
45	C₅×C₂²⋊C₈	Dic₃	1	2	yes	—
46	C₄×C₅×C₈	C₂×D₃	1	2	yes	—
47	#1570	—	3	1	no	Sylow
48	C₂³×C₃×C₅	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	—	2	2	no	—
50	C₂×C₃×C₄×C₅	—	4	1	no	—
51	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
55	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
56	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
59	C₃×C₅×C₈	—	2	4	no	—
60	C₅×C₃⋊C₈	—	4	2	no	—
61	C₅×C₃⋊C₈	—	4	2	no	—
62	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
63	C₂×#208	—	2	1	no	—
64	C₂²×C₃⋊C₈	—	2	1	no	—
65	C₃×C₄⋊C₈	C₄×C₅	1	2	yes	—
66	C₃×C₂²⋊C₈	C₄×C₅	1	2	yes	—
67	C₃×C₄×C₈	C₂²×C₅	1	2	yes	—
68	C₄⋊(C₃⋊C₈)	—	2	1	no	—
69	C₄⋊(C₃⋊C₈)	—	2	1	no	—
70	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	2	yes	—
71	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	2	yes	—
72	C₂²×C₄×C₅	Dic₆	1	1	yes	characteristic
73	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
75	C₅×M₁₆	—	12	2	no	—
76	C₂×C₅×C₈	—	6	2	no	—
77	C₂×C₅×C₈	—	6	2	no	—
78	C₂×C₅×C₈	—	12	2	no	—
79	C₂×C₅×C₈	C₂×Dic₃	1	4	yes	—
80	C₄²×C₅	Dic₆	1	1	yes	characteristic
81	C₄²×C₅	D₁₂	1	1	yes	characteristic
82	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
83	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
84	#431	C₅×D₃	1	1	yes	characteristic
85	#423	—	3	1	no	—
86	C₂×C₄⋊C₈	—	3	1	no	—
87	#334	—	3	4	no	—
88	C₂²×C₃×C₅	—	2	1	no	—
89	C₂²×C₃×C₅	—	2	1	no	—
90	C₂²×C₃×C₅	#106	1	2	yes	—
91	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₃×C₄×C₅	—	2	1	no	—
93	C₃×C₄×C₅	—	2	1	no	—
94	C₃×C₄×C₅	—	2	1	no	—
95	C₃×C₄×C₅	#106	1	2	yes	—
96	C₃×C₄×C₅	#131	1	2	yes	—
97	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
98	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
99	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
100	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
101	C₂²×C₃×C₄	C₅×Q₈	1	1	yes	characteristic
102	C₂×C₃×C₈	C₂×C₄×C₅	1	4	yes	—
103	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
104	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
105	C₃×C₄²	C₅×Q₈	1	1	yes	characteristic
106	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
107	#208	—	4	2	no	—
108	C₂×C₃⋊C₈	—	2	2	no	—
109	C₂×C₃⋊C₈	—	2	2	no	—
110	C₂×C₃⋊C₈	—	4	2	no	—
111	C₂³×C₅	C₄⋊Dic₃	1	1	yes	characteristic
112	C₂×C₄×C₅	—	2	2	no	—
113	C₂×C₄×C₅	—	4	1	no	—
114	C₂×C₄×C₅	#204	1	2	yes	—
115	C₂×C₄×C₅	#204	1	2	yes	—
116	C₂×C₄×C₅	C₄⋊Dic₃	1	1	yes	characteristic
117	C₂×C₄×C₅	C₄⋊Dic₃	1	1	yes	characteristic
118	C₂×C₄×C₅	C₄⋊Dic₃	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
122	C₅×C₈	—	2	4	no	—
123	C₅×C₈	—	12	2	no	—
124	C₅×C₈	—	12	2	no	—
125	C₂×M₁₆	—	6	1	no	—
126	C₂²×C₈	—	6	1	no	—
127	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
128	C₄⋊C₈	—	3	2	no	—
129	C₄⋊C₈	—	3	2	no	—
130	C₄⋊C₈	—	6	1	no	—
131	C₄⋊C₈	—	6	1	no	—
132	C₄⋊C₈	C₅×Dic₃	1	2	yes	—
133	C₂²⋊C₈	C₅×Dic₃	1	2	yes	—
134	C₄×C₈	C₂×C₅×D₃	1	2	yes	—
135	C₂×C₃×C₅	—	2	2	no	—
136	C₂×C₃×C₅	#424	1	1	yes	characteristic
137	C₂×C₃×C₅	C₂×#106	1	1	yes	characteristic
138	C₂×C₃×C₅	#430	1	1	yes	characteristic
139	C₂³×C₃	C₅×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₃×C₄	—	2	2	no	—
141	C₂×C₃×C₄	—	4	1	no	—
142	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₃×C₄	C₅×SD₁₆	1	2	yes	—
146	C₂×C₃×C₄	C₅×SD₁₆	1	2	yes	—
147	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
148	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
150	C₃×C₈	—	2	4	no	—
151	C₃⋊C₈	—	4	2	no	—
152	C₃⋊C₈	—	4	2	no	—
153	C₂²×C₅	—	2	1	no	—
154	C₂²×C₅	—	2	1	no	—
155	C₂²×C₅	#817	1	2	yes	—
156	C₂²×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
157	C₄×C₅	—	2	1	no	—
158	C₄×C₅	—	2	1	no	—
159	C₄×C₅	—	2	1	no	—
160	C₄×C₅	#817	1	2	yes	—
161	C₄×C₅	C₂×#204	1	1	yes	characteristic
162	C₄×C₅	C₂×#204	1	1	yes	characteristic
163	C₄×C₅	#948	1	2	yes	—
164	C₂²×C₄	C₅×Dic₆	1	1	yes	characteristic
165	C₂²×C₄	C₅×D₁₂	1	1	yes	characteristic
166	C₂²×C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
167	M₁₆	—	12	2	no	—
168	C₂×C₈	—	6	2	no	—
169	C₂×C₈	—	6	2	no	—
170	C₂×C₈	—	12	2	no	—
171	C₂×C₈	C₂×C₅×Dic₃	1	4	yes	—
172	C₄²	C₅×Dic₆	1	1	yes	characteristic
173	C₄²	C₅×D₁₂	1	1	yes	characteristic
174	C₄²	C₂×C₅×Dic₃	1	1	yes	characteristic
175	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
176	C₃×C₅	#1570	1	1	yes	characteristic, complemented, Hall
177	C₂²×C₃	—	2	1	no	—
178	C₂²×C₃	—	2	1	no	—
179	C₂²×C₃	C₅×#106	1	2	yes	—
180	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
181	C₃×C₄	—	2	1	no	—
182	C₃×C₄	—	2	1	no	—
183	C₃×C₄	—	2	1	no	—
184	C₃×C₄	C₅×#106	1	2	yes	—
185	C₃×C₄	C₅×#131	1	2	yes	—
186	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
187	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
188	C₂×C₅	—	2	2	no	—
189	C₂×C₅	#4999	1	1	yes	characteristic
190	C₂×C₅	C₂×#817	1	1	yes	characteristic
191	C₂×C₅	#5125	1	1	yes	characteristic
192	C₂³	C₅×C₄⋊Dic₃	1	1	yes	characteristic
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	C₅×#204	1	2	yes	—
196	C₂×C₄	C₅×#204	1	2	yes	—
197	C₂×C₄	C₅×C₄⋊Dic₃	1	1	yes	characteristic
198	C₂×C₄	C₅×C₄⋊Dic₃	1	1	yes	characteristic
199	C₂×C₄	C₅×C₄⋊Dic₃	1	1	yes	characteristic
200	C₂×C₄	C₂×C₅×Dic₆	1	1	yes	characteristic
201	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
202	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
203	C₈	—	2	4	no	—
204	C₈	—	12	2	no	—
205	C₈	—	12	2	no	—
206	C₂×C₃	—	2	2	no	—
207	C₂×C₃	C₅×#424	1	1	yes	characteristic
208	C₂×C₃	C₂×C₅×#106	1	1	yes	characteristic
209	C₂×C₃	C₅×#430	1	1	yes	characteristic
210	C₅	#70749	1	1	yes	characteristic, direct product factor, Sylow
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	C₅×#817	1	2	yes	—
214	C₂²	C₂×C₅×C₄⋊Dic₃	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	C₅×#817	1	2	yes	—
219	C₄	C₂×C₅×#204	1	1	yes	characteristic
220	C₄	C₂×C₅×#204	1	1	yes	characteristic
221	C₄	C₅×#948	1	2	yes	—
222	C₃	C₅×#1570	1	1	yes	characteristic, complemented, Sylow
223	C₂	—	2	2	no	—
224	C₂	C₅×#4999	1	1	yes	characteristic
225	C₂	C₂×C₅×#817	1	1	yes	characteristic
226	C₂	C₅×#5125	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#431	C₂
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂³×C₃×C₅	C₄⋊C₄
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×#431	D₃
hypocenter	C₃	C₅×#1570
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₅×#431

Complements

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

N	H ≅ G/N	type
4. C₅×#4457	C₂	semidirect
16. #70749	C₅	direct (N × H)
36. #4457	C₂×C₅	semidirect
176. C₃×C₅	#1570	semidirect
210. C₅	#70749	direct (N × H)
222. C₃	C₅×#1570	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#431	2
C₅×#4924	2
C₂×C₅×C₄⋊(C₃⋊C₈)	2
C₅×#4457	2
C₅×#1570	3
#70749	5

Subgroups — C5×#70749 #14285818

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70749 #14285818