Subgroups — C₅×#70751 #14285899

Order 1920.

300 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×#4926	C₂	1	1	yes	maximal, characteristic
3	C₅×#4925	C₂	1	1	yes	maximal, characteristic
4	C₅×#4458	C₂	1	2	yes	maximal, complemented
5	C₅×#4457	C₂	1	2	yes	maximal, complemented
6	C₅×#1572	—	3	1	no	maximal, Hall
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₃×C₅×C₄⋊C₈	C₄	1	2	yes	—
9	C₃×C₅×C₂²⋊C₈	C₄	1	2	yes	—
10	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₅×#830	—	2	1	no	—
13	C₅×#830	—	2	1	no	—
14	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
15	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
16	C₅×#803	—	2	1	no	—
17	C₄×C₅×C₃⋊C₈	—	2	1	no	—
18	#70751	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₅×#432	—	3	1	no	—
20	C₅×#431	—	3	1	no	—
21	C₅×#431	D₃	1	1	yes	characteristic
22	C₅×#335	—	3	2	no	—
23	C₅×#334	—	3	2	no	—
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	Q₈	1	1	yes	characteristic
27	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
29	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
30	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
31	C₃×C₄²×C₅	Q₈	1	1	yes	characteristic
32	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
33	C₂×C₅×C₃⋊C₈	—	2	2	no	—
34	C₂×C₅×C₃⋊C₈	—	2	2	no	—
35	C₃×#431	C₂×C₅	1	1	yes	characteristic
36	#4926	C₂×C₅	1	1	yes	characteristic
37	#4925	C₂×C₅	1	1	yes	characteristic
38	#4458	C₂×C₅	1	2	yes	complemented
39	#4457	C₂×C₅	1	2	yes	complemented
40	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
41	C₅×C₄⋊C₈	—	3	2	no	—
42	C₅×C₄⋊C₈	—	3	2	no	—
43	C₅×C₄⋊C₈	Dic₃	1	2	yes	—
44	C₅×C₂²⋊C₈	—	6	1	no	—
45	C₅×C₂²⋊C₈	—	6	1	no	—
46	C₅×C₂²⋊C₈	Dic₃	1	2	yes	—
47	C₅×#97	—	6	1	no	—
48	C₄×C₅×C₈	—	6	1	no	—
49	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
50	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
51	#1572	—	3	1	no	Sylow
52	C₂³×C₃×C₅	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	—	2	2	no	—
54	C₂×C₃×C₄×C₅	—	2	2	no	—
55	C₂×C₃×C₄×C₅	—	4	1	no	—
56	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
62	C₃×C₅×C₈	—	2	2	no	—
63	C₃×C₅×C₈	—	2	2	no	—
64	C₅×C₃⋊C₈	—	4	2	no	—
65	C₅×C₃⋊C₈	—	4	2	no	—
66	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
67	C₃×C₄⋊C₈	C₄×C₅	1	2	yes	—
68	C₃×C₂²⋊C₈	C₄×C₅	1	2	yes	—
69	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
70	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
71	#830	—	2	1	no	—
72	#830	—	2	1	no	—
73	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	2	yes	—
74	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	2	yes	—
75	#803	—	2	1	no	—
76	C₄×C₃⋊C₈	—	2	1	no	—
77	C₂²×C₄×C₅	Dic₆	1	1	yes	characteristic
78	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
80	C₂×C₅×C₈	—	6	2	no	—
81	C₂×C₅×C₈	—	6	2	no	—
82	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
83	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
84	C₄²×C₅	Dic₆	1	1	yes	characteristic
85	C₄²×C₅	D₁₂	1	1	yes	characteristic
86	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
87	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
88	#432	—	3	1	no	—
89	#431	—	3	1	no	—
90	#431	C₅×D₃	1	1	yes	characteristic
91	#335	—	3	2	no	—
92	#334	—	3	2	no	—
93	C₂²×C₃×C₅	—	2	1	no	—
94	C₂²×C₃×C₅	—	2	1	no	—
95	C₂²×C₃×C₅	—	2	1	no	—
96	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₃×C₄×C₅	—	2	1	no	—
98	C₃×C₄×C₅	—	2	1	no	—
99	C₃×C₄×C₅	—	2	1	no	—
100	C₃×C₄×C₅	—	2	1	no	—
101	C₃×C₄×C₅	#131	1	2	yes	—
102	C₃×C₄×C₅	#135	1	2	yes	—
103	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
104	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
105	C₂²×C₃×C₄	C₅×Q₈	1	1	yes	characteristic
106	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
107	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
108	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
109	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
110	C₃×C₄²	C₅×Q₈	1	1	yes	characteristic
111	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
112	C₂×C₃⋊C₈	—	2	2	no	—
113	C₂×C₃⋊C₈	—	2	2	no	—
114	C₂³×C₅	C₄⋊Dic₃	1	1	yes	characteristic
115	C₂×C₄×C₅	—	2	2	no	—
116	C₂×C₄×C₅	—	2	2	no	—
117	C₂×C₄×C₅	—	4	1	no	—
118	C₂×C₄×C₅	C₄⋊Dic₃	1	1	yes	characteristic
119	C₂×C₄×C₅	C₄⋊Dic₃	1	1	yes	characteristic
120	C₂×C₄×C₅	C₄⋊Dic₃	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
123	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
124	C₅×C₈	—	2	2	no	—
125	C₅×C₈	—	2	2	no	—
126	C₅×C₈	—	12	2	no	—
127	C₅×C₈	—	12	2	no	—
128	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
129	C₄⋊C₈	—	3	2	no	—
130	C₄⋊C₈	—	3	2	no	—
131	C₄⋊C₈	C₅×Dic₃	1	2	yes	—
132	C₂²⋊C₈	—	6	1	no	—
133	C₂²⋊C₈	—	6	1	no	—
134	C₂²⋊C₈	C₅×Dic₃	1	2	yes	—
135	#97	—	6	1	no	—
136	C₄×C₈	—	6	1	no	—
137	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
138	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
139	C₂×C₃×C₅	—	4	1	no	—
140	C₂×C₃×C₅	#424	1	1	yes	characteristic
141	C₂×C₃×C₅	#427	1	1	yes	characteristic
142	C₂×C₃×C₅	#430	1	1	yes	characteristic
143	C₂³×C₃	C₅×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	2	no	—
146	C₂×C₃×C₄	—	4	1	no	—
147	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
152	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
153	C₃×C₈	—	2	2	no	—
154	C₃×C₈	—	2	2	no	—
155	C₃⋊C₈	—	4	2	no	—
156	C₃⋊C₈	—	4	2	no	—
157	C₂²×C₅	—	2	1	no	—
158	C₂²×C₅	—	2	1	no	—
159	C₂²×C₅	—	2	1	no	—
160	C₂²×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
161	C₄×C₅	—	2	1	no	—
162	C₄×C₅	—	2	1	no	—
163	C₄×C₅	—	2	1	no	—
164	C₄×C₅	—	2	1	no	—
165	C₄×C₅	#904	1	2	yes	—
166	C₄×C₅	#948	1	2	yes	—
167	C₂²×C₄	C₅×Dic₆	1	1	yes	characteristic
168	C₂²×C₄	C₅×D₁₂	1	1	yes	characteristic
169	C₂²×C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
170	C₂×C₈	—	6	2	no	—
171	C₂×C₈	—	6	2	no	—
172	C₂×C₈	C₂×C₅×Dic₃	1	2	yes	—
173	C₂×C₈	C₂×C₅×Dic₃	1	2	yes	—
174	C₄²	C₅×Dic₆	1	1	yes	characteristic
175	C₄²	C₅×D₁₂	1	1	yes	characteristic
176	C₄²	C₂×C₅×Dic₃	1	1	yes	characteristic
177	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
178	C₃×C₅	#1572	1	1	yes	characteristic, complemented, Hall
179	C₂²×C₃	—	2	1	no	—
180	C₂²×C₃	—	2	1	no	—
181	C₂²×C₃	—	2	1	no	—
182	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
183	C₃×C₄	—	2	1	no	—
184	C₃×C₄	—	2	1	no	—
185	C₃×C₄	—	2	1	no	—
186	C₃×C₄	—	2	1	no	—
187	C₃×C₄	C₅×#131	1	2	yes	—
188	C₃×C₄	C₅×#135	1	2	yes	—
189	C₂×C₅	—	4	1	no	—
190	C₂×C₅	#4999	1	1	yes	characteristic
191	C₂×C₅	#5106	1	1	yes	characteristic
192	C₂×C₅	#5125	1	1	yes	characteristic
193	C₂³	C₅×C₄⋊Dic₃	1	1	yes	characteristic
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	4	1	no	—
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	2	no	—
204	C₈	—	2	2	no	—
205	C₈	—	12	2	no	—
206	C₈	—	12	2	no	—
207	C₂×C₃	—	4	1	no	—
208	C₂×C₃	C₅×#424	1	1	yes	characteristic
209	C₂×C₃	C₅×#427	1	1	yes	characteristic
210	C₂×C₃	C₅×#430	1	1	yes	characteristic
211	C₅	#70751	1	1	yes	characteristic, direct product factor, Sylow
212	C₂²	—	2	1	no	—
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	C₂×C₅×C₄⋊Dic₃	1	1	yes	characteristic
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	C₅×#904	1	2	yes	—
221	C₄	C₅×#948	1	2	yes	—
222	C₃	C₅×#1572	1	1	yes	characteristic, complemented, Sylow
223	C₂	—	4	1	no	—
224	C₂	C₅×#4999	1	1	yes	characteristic
225	C₂	C₅×#5106	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×#1572
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×#4458	C₂	semidirect
5. C₅×#4457	C₂	semidirect
18. #70751	C₅	direct (N × H)
38. #4458	C₂×C₅	semidirect
39. #4457	C₂×C₅	semidirect
178. C₃×C₅	#1572	semidirect
211. C₅	#70751	direct (N × H)
222. C₃	C₅×#1572	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×#4926	2
C₅×#4925	2
C₅×#4458	2
C₅×#4457	2
C₅×#1572	3
#70751	5

Subgroups — C5×#70751 #14285899

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70751 #14285899