Subgroups — C₅×#70850 #14291490

Order 1920.

344 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₈×D₄	C₂	1	1	yes	maximal, characteristic
2	C₅×#5007	C₂	1	2	yes	maximal
3	C₅×#4699	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4482	C₂	1	2	yes	maximal, complemented
5	C₅×#4458	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1669	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅×C₈	—	2	1	no	—
8	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
9	C₂×C₅×Dic₁₂	—	2	1	no	—
10	C₂×C₅×Dic₁₂	—	2	1	no	—
11	C₅×#869	C₂²	1	2	yes	—
12	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
14	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₅×#832	—	2	2	no	—
16	C₅×#807	—	2	2	no	—
17	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
18	#70850	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₅×#494	—	3	1	no	—
20	C₅×#474	—	3	2	no	—
21	C₅×C₈×D₄	D₃	1	1	yes	characteristic
22	C₅×#335	—	3	1	no	—
23	C₅×#332	—	3	2	no	—
24	C₂×C₃×C₅×D₄	D₄	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	—	2	1	no	—
26	C₂×C₅×Dic₆	—	2	2	no	—
27	C₂×C₃×C₅×C₈	—	2	1	no	—
28	C₂×C₃×C₅×C₈	—	4	2	no	—
29	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
30	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
31	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
32	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
33	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
34	C₅×C₄⋊Dic₃	—	2	2	no	—
35	C₅×C₄⋊Dic₃	—	2	2	no	—
36	C₂×C₅×C₃⋊C₈	—	2	2	no	—
37	C₅×Dic₁₂	—	4	2	no	—
38	C₅×Dic₁₂	—	4	2	no	—
39	C₃×C₈×D₄	C₂×C₅	1	1	yes	characteristic
40	#5007	C₂×C₅	1	2	yes	—
41	#4699	C₂×C₅	1	1	yes	characteristic, complemented
42	#4482	C₂×C₅	1	2	yes	complemented
43	#4458	C₂×C₅	1	1	yes	characteristic, complemented
44	C₂×C₅×Q₁₆	—	6	1	no	—
45	C₂×C₅×Q₁₆	—	6	1	no	—
46	C₂²×C₅×C₈	—	2	1	no	—
47	C₅×C₄⋊Q₈	—	3	2	no	—
48	C₄×C₅×D₄	C₂×D₃	1	1	yes	characteristic
49	C₅×#107	—	6	2	no	—
50	C₅×C₄⋊C₈	—	3	2	no	—
51	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
52	C₅×#102	—	6	2	no	—
53	C₅×C₂²⋊C₈	—	2	1	no	—
54	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
55	#1669	—	3	1	no	Sylow
56	C₂³×C₃×C₅	—	2	1	no	—
57	C₃×C₅×D₄	—	2	1	no	—
58	C₃×C₅×D₄	Q₁₆	1	2	yes	complemented
59	C₂×C₃×C₄×C₅	—	2	1	no	—
60	C₂×C₃×C₄×C₅	—	4	1	no	—
61	C₂×C₃×C₄×C₅	—	4	1	no	—
62	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
64	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
65	C₂×C₅×Dic₃	—	4	2	no	—
66	C₅×Dic₆	—	4	2	no	—
67	C₅×Dic₆	—	4	2	no	—
68	C₃×C₅×C₈	—	2	1	no	—
69	C₃×C₅×C₈	—	4	1	no	—
70	C₃×C₅×C₈	D₈	1	2	yes	—
71	C₅×C₃⋊C₈	—	4	2	no	—
72	C₂²×C₃×C₈	—	2	1	no	—
73	C₃×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
74	C₂×Dic₁₂	—	2	1	no	—
75	C₂×Dic₁₂	—	2	1	no	—
76	#869	C₂²×C₅	1	2	yes	—
77	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
78	C₃×C₂²⋊C₈	—	2	1	no	—
79	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
80	#832	—	2	2	no	—
81	#807	—	2	2	no	—
82	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	2	yes	—
83	C₂×C₅×Q₈	—	6	2	no	—
84	C₂×C₅×D₄	D₁₂	1	1	yes	characteristic
85	C₂²×C₄×C₅	—	2	1	no	—
86	C₅×Q₁₆	—	12	2	no	—
87	C₅×Q₁₆	—	12	2	no	—
88	C₂×C₅×C₈	—	2	1	no	—
89	C₂×C₅×C₈	—	4	2	no	—
90	C₂×C₅×C₈	—	6	2	no	—
91	C₂×C₅×C₈	#67	1	1	yes	characteristic
92	C₂×C₅×C₈	#67	1	1	yes	characteristic
93	C₅×C₄⋊C₄	—	6	2	no	—
94	C₅×C₄⋊C₄	—	6	2	no	—
95	C₅×C₄⋊C₄	D₁₂	1	1	yes	characteristic
96	C₅×C₂²⋊C₄	—	2	1	no	—
97	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
98	#494	—	3	1	no	—
99	#474	—	3	2	no	—
100	C₈×D₄	C₅×D₃	1	1	yes	characteristic
101	#335	—	3	1	no	—
102	#332	—	3	2	no	—
103	C₂²×C₃×C₅	—	2	2	no	—
104	C₂²×C₃×C₅	—	4	1	no	—
105	C₂²×C₃×C₅	—	4	1	no	—
106	C₂²×C₃×C₅	#121	1	1	yes	characteristic
107	C₃×C₄×C₅	—	2	1	no	—
108	C₃×C₄×C₅	—	4	1	no	—
109	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
110	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
111	C₃×C₄×C₅	#135	1	1	yes	characteristic
112	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
113	C₅×Dic₃	—	8	2	no	—
114	C₂×C₃×D₄	C₅×D₄	1	1	yes	characteristic
115	C₂²×C₃×C₄	—	2	1	no	—
116	C₂×Dic₆	—	2	2	no	—
117	C₂×C₃×C₈	—	2	1	no	—
118	C₂×C₃×C₈	—	4	2	no	—
119	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
120	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
121	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
122	C₃×C₂²⋊C₄	—	2	1	no	—
123	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
124	C₄⋊Dic₃	—	2	2	no	—
125	C₄⋊Dic₃	—	2	2	no	—
126	C₂×C₃⋊C₈	—	2	2	no	—
127	Dic₁₂	—	4	2	no	—
128	Dic₁₂	—	4	2	no	—
129	C₂³×C₅	—	2	1	no	—
130	C₅×Q₈	—	12	2	no	—
131	C₅×Q₈	—	12	2	no	—
132	C₅×D₄	—	2	1	no	—
133	C₅×D₄	Dic₁₂	1	2	yes	complemented
134	C₂×C₄×C₅	—	2	1	no	—
135	C₂×C₄×C₅	—	4	1	no	—
136	C₂×C₄×C₅	—	4	1	no	—
137	C₂×C₄×C₅	—	12	2	no	—
138	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
139	C₂×C₄×C₅	#235	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
141	C₅×C₈	—	2	1	no	—
142	C₅×C₈	—	4	1	no	—
143	C₅×C₈	—	12	2	no	—
144	C₅×C₈	#213	1	2	yes	—
145	C₂×Q₁₆	—	6	1	no	—
146	C₂×Q₁₆	—	6	1	no	—
147	C₂²×C₈	—	2	1	no	—
148	C₄⋊Q₈	—	3	2	no	—
149	C₄×D₄	C₂×C₅×D₃	1	1	yes	characteristic
150	#107	—	6	2	no	—
151	C₄⋊C₈	—	3	2	no	—
152	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
153	#102	—	6	2	no	—
154	C₂²⋊C₈	—	2	1	no	—
155	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
156	C₂×C₃×C₅	—	4	2	no	—
157	C₂×C₃×C₅	#459	1	1	yes	characteristic
158	C₂×C₃×C₅	#467	1	1	yes	characteristic
159	C₂×C₃×C₅	#473	1	1	yes	characteristic
160	C₂³×C₃	—	2	1	no	—
161	C₃×D₄	—	2	1	no	—
162	C₃×D₄	C₅×Q₁₆	1	2	yes	complemented
163	C₂×C₃×C₄	—	2	1	no	—
164	C₂×C₃×C₄	—	4	1	no	—
165	C₂×C₃×C₄	—	4	1	no	—
166	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
167	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
168	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
169	C₂×Dic₃	—	4	2	no	—
170	Dic₆	—	4	2	no	—
171	Dic₆	—	4	2	no	—
172	C₃×C₈	—	2	1	no	—
173	C₃×C₈	—	4	1	no	—
174	C₃×C₈	C₅×D₈	1	2	yes	—
175	C₃⋊C₈	—	4	2	no	—
176	C₂²×C₅	—	2	2	no	—
177	C₂²×C₅	—	4	1	no	—
178	C₂²×C₅	—	4	1	no	—
179	C₂²×C₅	#930	1	1	yes	characteristic
180	C₄×C₅	—	2	1	no	—
181	C₄×C₅	—	4	1	no	—
182	C₄×C₅	—	24	2	no	—
183	C₄×C₅	#904	1	1	yes	characteristic
184	C₄×C₅	C₂×Dic₁₂	1	1	yes	characteristic
185	C₄×C₅	C₂×#213	1	1	yes	characteristic
186	C₄×C₅	#949	1	1	yes	characteristic
187	C₂×Q₈	—	6	2	no	—
188	C₂×D₄	C₅×D₁₂	1	1	yes	characteristic
189	C₂²×C₄	—	2	1	no	—
190	Q₁₆	—	12	2	no	—
191	Q₁₆	—	12	2	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	4	2	no	—
194	C₂×C₈	—	6	2	no	—
195	C₂×C₈	C₅×#67	1	1	yes	characteristic
196	C₂×C₈	C₅×#67	1	1	yes	characteristic
197	C₄⋊C₄	—	6	2	no	—
198	C₄⋊C₄	—	6	2	no	—
199	C₄⋊C₄	C₅×D₁₂	1	1	yes	characteristic
200	C₂²⋊C₄	—	2	1	no	—
201	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
202	C₃×C₅	#1669	1	1	yes	characteristic, complemented, Hall
203	C₂²×C₃	—	2	2	no	—
204	C₂²×C₃	—	4	1	no	—
205	C₂²×C₃	—	4	1	no	—
206	C₂²×C₃	C₅×#121	1	1	yes	characteristic
207	C₃×C₄	—	2	1	no	—
208	C₃×C₄	—	4	1	no	—
209	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
210	C₃×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
211	C₃×C₄	C₅×#135	1	1	yes	characteristic
212	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
213	Dic₃	—	8	2	no	—
214	C₂×C₅	—	4	2	no	—
215	C₂×C₅	#5015	1	1	yes	characteristic
216	C₂×C₅	#5116	1	1	yes	characteristic
217	C₂×C₅	#5143	1	1	yes	characteristic
218	C₂³	—	2	1	no	—
219	Q₈	—	12	2	no	—
220	Q₈	—	12	2	no	—
221	D₄	—	2	1	no	—
222	D₄	C₅×Dic₁₂	1	2	yes	complemented
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	12	2	no	—
227	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
228	C₂×C₄	C₅×#235	1	1	yes	characteristic
229	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
230	C₈	—	2	1	no	—
231	C₈	—	4	1	no	—
232	C₈	—	12	2	no	—
233	C₈	C₅×#213	1	2	yes	—
234	C₂×C₃	—	4	2	no	—
235	C₂×C₃	C₅×#459	1	1	yes	characteristic
236	C₂×C₃	C₅×#467	1	1	yes	characteristic
237	C₂×C₃	C₅×#473	1	1	yes	characteristic
238	C₅	#70850	1	1	yes	characteristic, direct product factor, Sylow
239	C₂²	—	2	2	no	—
240	C₂²	—	4	1	no	—
241	C₂²	—	4	1	no	—
242	C₂²	C₅×#930	1	1	yes	characteristic
243	C₄	—	2	1	no	—
244	C₄	—	4	1	no	—
245	C₄	—	24	2	no	—
246	C₄	C₅×#904	1	1	yes	characteristic
247	C₄	C₂×C₅×Dic₁₂	1	1	yes	characteristic
248	C₄	C₂×C₅×#213	1	1	yes	characteristic
249	C₄	C₅×#949	1	1	yes	characteristic
250	C₃	C₅×#1669	1	1	yes	characteristic, complemented, Sylow
251	C₂	—	4	2	no	—
252	C₂	C₅×#5015	1	1	yes	characteristic
253	C₂	C₅×#5116	1	1	yes	characteristic
254	C₂	C₅×#5143	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	#930
commutator, cosocle	C₃×C₄²	C₂³×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×C₈×D₄	C₂
socle	C₂²×C₃×C₅	#121
prime radical	C₂×C₃×C₅×D₄	D₄
hypercenter	C₅×C₈×D₄	D₃
hypocenter	C₃	C₅×#1669
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×C₈×D₄

Complements

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

N	H ≅ G/N	type
3. C₅×#4699	C₂	semidirect
4. C₅×#4482	C₂	semidirect
5. C₅×#4458	C₂	semidirect
18. #70850	C₅	direct (N × H)
41. #4699	C₂×C₅	semidirect
42. #4482	C₂×C₅	semidirect
43. #4458	C₂×C₅	semidirect
58. C₃×C₅×D₄	Q₁₆	semidirect
133. C₅×D₄	Dic₁₂	semidirect
162. C₃×D₄	C₅×Q₁₆	semidirect
202. C₃×C₅	#1669	semidirect
222. D₄	C₅×Dic₁₂	semidirect
238. C₅	#70850	direct (N × H)
250. C₃	C₅×#1669	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×C₈×D₄	2
C₅×#5007	2
C₅×#4699	2
C₅×#4482	2
C₅×#4458	2
C₅×#1669	3
#70850	5

Subgroups — C5×#70850 #14291490

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70850 #14291490