Subgroups — C₅×#70854 #14291976

Order 1920.

332 conjugacy classes of subgroups.

The subgroup lattice has 312 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×#5019	C₂	1	1	yes	maximal, characteristic
3	C₅×#5007	C₂	1	1	yes	maximal, characteristic
4	C₅×#4681	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4486	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4482	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4457	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1679	—	3	1	no	maximal, Hall
9	C₂²×C₃×C₅×C₈	—	2	1	no	—
10	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
11	C₂×C₅×#214	—	2	1	no	—
12	C₅×#869	C₂²	1	1	yes	characteristic
13	C₅×#869	C₂²	1	1	yes	characteristic
14	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
16	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₅×#832	—	2	1	no	—
18	C₅×#816	—	2	1	no	—
19	C₅×#816	—	2	1	no	—
20	C₅×#810	—	2	1	no	—
21	C₅×#807	—	2	1	no	—
22	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
23	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
24	#70854	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₅×#487	—	3	1	no	—
26	C₅×#474	—	3	1	no	—
27	C₅×#461	—	3	1	no	—
28	C₅×C₈×D₄	D₃	1	1	yes	characteristic
29	C₅×#334	—	3	1	no	—
30	C₅×#332	—	3	1	no	—
31	C₅×#332	—	3	1	no	—
32	C₂×C₃×C₅×D₄	D₄	1	1	yes	characteristic
33	C₂²×C₃×C₄×C₅	—	2	1	no	—
34	C₂×C₅×Dic₆	—	2	1	no	—
35	C₂×C₅×Dic₆	—	2	1	no	—
36	C₂×C₃×C₅×C₈	—	2	1	no	—
37	C₂×C₃×C₅×C₈	—	4	2	no	—
38	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
39	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
40	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
41	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
42	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₅×#214	—	4	2	no	—
44	C₅×C₄⋊Dic₃	—	2	1	no	—
45	C₅×C₄⋊Dic₃	—	2	1	no	—
46	C₅×C₄⋊Dic₃	—	2	1	no	—
47	C₅×C₄⋊Dic₃	—	2	1	no	—
48	C₂×C₅×C₃⋊C₈	—	2	1	no	—
49	C₂×C₅×C₃⋊C₈	—	2	1	no	—
50	C₃×C₈×D₄	C₂×C₅	1	1	yes	characteristic
51	#5019	C₂×C₅	1	1	yes	characteristic
52	#5007	C₂×C₅	1	1	yes	characteristic
53	#4681	C₂×C₅	1	1	yes	characteristic, complemented
54	#4486	C₂×C₅	1	1	yes	characteristic, complemented
55	#4482	C₂×C₅	1	1	yes	characteristic, complemented
56	#4457	C₂×C₅	1	1	yes	characteristic, complemented
57	C₂×C₅×SD₁₆	—	6	1	no	—
58	C₂²×C₅×C₈	—	2	1	no	—
59	C₅×C₄⋊Q₈	—	3	1	no	—
60	C₅×C₄⋊Q₈	—	3	1	no	—
61	C₄×C₅×D₄	C₂×D₃	1	1	yes	characteristic
62	C₅×#107	—	6	1	no	—
63	C₅×C₄⋊C₈	—	3	1	no	—
64	C₅×C₄⋊C₈	—	3	1	no	—
65	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
66	C₅×#103	—	6	1	no	—
67	C₅×#103	—	6	1	no	—
68	C₅×#103	—	6	1	no	—
69	C₅×#102	—	6	1	no	—
70	C₅×C₂²⋊C₈	—	2	1	no	—
71	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
72	#1679	—	3	1	no	Sylow
73	C₂³×C₃×C₅	—	2	1	no	—
74	C₃×C₅×D₄	—	2	1	no	—
75	C₃×C₅×D₄	SD₁₆	1	2	yes	—
76	C₂×C₃×C₄×C₅	—	2	1	no	—
77	C₂×C₃×C₄×C₅	—	4	1	no	—
78	C₂×C₃×C₄×C₅	—	4	1	no	—
79	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
82	C₂×C₅×Dic₃	—	4	1	no	—
83	C₂×C₅×Dic₃	—	4	1	no	—
84	C₅×Dic₆	—	4	1	no	—
85	C₅×Dic₆	—	4	1	no	—
86	C₅×Dic₆	—	4	1	no	—
87	C₅×Dic₆	—	4	1	no	—
88	C₃×C₅×C₈	—	2	1	no	—
89	C₃×C₅×C₈	—	2	1	no	—
90	C₃×C₅×C₈	—	4	1	no	—
91	C₅×C₃⋊C₈	—	4	1	no	—
92	C₅×C₃⋊C₈	—	4	1	no	—
93	C₂²×C₃×C₈	—	2	1	no	—
94	C₃×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
95	C₂×#214	—	2	1	no	—
96	#869	C₂²×C₅	1	1	yes	characteristic
97	#869	C₂²×C₅	1	1	yes	characteristic
98	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
99	C₃×C₂²⋊C₈	—	2	1	no	—
100	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
101	#832	—	2	1	no	—
102	#816	—	2	1	no	—
103	#816	—	2	1	no	—
104	#810	—	2	1	no	—
105	#807	—	2	1	no	—
106	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
107	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
108	C₂×C₅×Q₈	—	6	1	no	—
109	C₂×C₅×Q₈	—	6	1	no	—
110	C₂×C₅×D₄	D₁₂	1	1	yes	characteristic
111	C₂²×C₄×C₅	—	2	1	no	—
112	C₅×SD₁₆	—	12	2	no	—
113	C₂×C₅×C₈	—	2	1	no	—
114	C₂×C₅×C₈	—	4	2	no	—
115	C₂×C₅×C₈	—	6	1	no	—
116	C₂×C₅×C₈	—	6	1	no	—
117	C₂×C₅×C₈	#67	1	1	yes	characteristic
118	C₂×C₅×C₈	#67	1	1	yes	characteristic
119	C₅×C₄⋊C₄	—	6	1	no	—
120	C₅×C₄⋊C₄	—	6	1	no	—
121	C₅×C₄⋊C₄	—	6	1	no	—
122	C₅×C₄⋊C₄	—	6	1	no	—
123	C₅×C₄⋊C₄	D₁₂	1	1	yes	characteristic
124	C₅×C₂²⋊C₄	—	2	1	no	—
125	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
126	#487	—	3	1	no	—
127	#474	—	3	1	no	—
128	#461	—	3	1	no	—
129	C₈×D₄	C₅×D₃	1	1	yes	characteristic
130	#334	—	3	1	no	—
131	#332	—	3	1	no	—
132	#332	—	3	1	no	—
133	C₂²×C₃×C₅	—	2	2	no	—
134	C₂²×C₃×C₅	—	4	1	no	—
135	C₂²×C₃×C₅	—	4	1	no	—
136	C₂²×C₃×C₅	#121	1	1	yes	characteristic
137	C₃×C₄×C₅	—	2	1	no	—
138	C₃×C₄×C₅	—	4	1	no	—
139	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
140	C₃×C₄×C₅	#135	1	1	yes	characteristic
141	C₃×C₄×C₅	#135	1	1	yes	characteristic
142	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₅×Dic₃	—	8	1	no	—
144	C₅×Dic₃	—	8	1	no	—
145	C₂×C₃×D₄	C₅×D₄	1	1	yes	characteristic
146	C₂²×C₃×C₄	—	2	1	no	—
147	C₂×Dic₆	—	2	1	no	—
148	C₂×Dic₆	—	2	1	no	—
149	C₂×C₃×C₈	—	2	1	no	—
150	C₂×C₃×C₈	—	4	2	no	—
151	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
152	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
153	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
154	C₃×C₂²⋊C₄	—	2	1	no	—
155	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
156	#214	—	4	2	no	—
157	C₄⋊Dic₃	—	2	1	no	—
158	C₄⋊Dic₃	—	2	1	no	—
159	C₄⋊Dic₃	—	2	1	no	—
160	C₄⋊Dic₃	—	2	1	no	—
161	C₂×C₃⋊C₈	—	2	1	no	—
162	C₂×C₃⋊C₈	—	2	1	no	—
163	C₂³×C₅	—	2	1	no	—
164	C₅×Q₈	—	12	1	no	—
165	C₅×Q₈	—	12	1	no	—
166	C₅×Q₈	—	12	1	no	—
167	C₅×Q₈	—	12	1	no	—
168	C₅×D₄	—	2	1	no	—
169	C₅×D₄	#204	1	2	yes	—
170	C₂×C₄×C₅	—	2	1	no	—
171	C₂×C₄×C₅	—	4	1	no	—
172	C₂×C₄×C₅	—	4	1	no	—
173	C₂×C₄×C₅	—	12	1	no	—
174	C₂×C₄×C₅	—	12	1	no	—
175	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
176	C₂×C₄×C₅	#235	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
178	C₅×C₈	—	2	1	no	—
179	C₅×C₈	—	2	1	no	—
180	C₅×C₈	—	4	1	no	—
181	C₅×C₈	—	12	1	no	—
182	C₅×C₈	—	12	1	no	—
183	C₂×SD₁₆	—	6	1	no	—
184	C₂²×C₈	—	2	1	no	—
185	C₄⋊Q₈	—	3	1	no	—
186	C₄⋊Q₈	—	3	1	no	—
187	C₄×D₄	C₂×C₅×D₃	1	1	yes	characteristic
188	#107	—	6	1	no	—
189	C₄⋊C₈	—	3	1	no	—
190	C₄⋊C₈	—	3	1	no	—
191	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
192	#103	—	6	1	no	—
193	#103	—	6	1	no	—
194	#103	—	6	1	no	—
195	#102	—	6	1	no	—
196	C₂²⋊C₈	—	2	1	no	—
197	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
198	C₂×C₃×C₅	—	4	2	no	—
199	C₂×C₃×C₅	#463	1	1	yes	characteristic
200	C₂×C₃×C₅	#465	1	1	yes	characteristic
201	C₂×C₃×C₅	#473	1	1	yes	characteristic
202	C₂³×C₃	—	2	1	no	—
203	C₃×D₄	—	2	1	no	—
204	C₃×D₄	C₅×SD₁₆	1	2	yes	—
205	C₂×C₃×C₄	—	2	1	no	—
206	C₂×C₃×C₄	—	4	1	no	—
207	C₂×C₃×C₄	—	4	1	no	—
208	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
210	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
211	C₂×Dic₃	—	4	1	no	—
212	C₂×Dic₃	—	4	1	no	—
213	Dic₆	—	4	1	no	—
214	Dic₆	—	4	1	no	—
215	Dic₆	—	4	1	no	—
216	Dic₆	—	4	1	no	—
217	C₃×C₈	—	2	1	no	—
218	C₃×C₈	—	2	1	no	—
219	C₃×C₈	—	4	1	no	—
220	C₃⋊C₈	—	4	1	no	—
221	C₃⋊C₈	—	4	1	no	—
222	C₂²×C₅	—	2	2	no	—
223	C₂²×C₅	—	4	1	no	—
224	C₂²×C₅	—	4	1	no	—
225	C₂²×C₅	#930	1	1	yes	characteristic
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	—	4	1	no	—
228	C₄×C₅	—	24	1	no	—
229	C₄×C₅	—	24	1	no	—
230	C₄×C₅	C₂×#204	1	1	yes	characteristic
231	C₄×C₅	#904	1	1	yes	characteristic
232	C₄×C₅	#932	1	1	yes	characteristic
233	C₄×C₅	#950	1	1	yes	characteristic
234	C₂×Q₈	—	6	1	no	—
235	C₂×Q₈	—	6	1	no	—
236	C₂×D₄	C₅×D₁₂	1	1	yes	characteristic
237	C₂²×C₄	—	2	1	no	—
238	SD₁₆	—	12	2	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	—	6	1	no	—
242	C₂×C₈	—	6	1	no	—
243	C₂×C₈	C₅×#67	1	1	yes	characteristic
244	C₂×C₈	C₅×#67	1	1	yes	characteristic
245	C₄⋊C₄	—	6	1	no	—
246	C₄⋊C₄	—	6	1	no	—
247	C₄⋊C₄	—	6	1	no	—
248	C₄⋊C₄	—	6	1	no	—
249	C₄⋊C₄	C₅×D₁₂	1	1	yes	characteristic
250	C₂²⋊C₄	—	2	1	no	—
251	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
252	C₃×C₅	#1679	1	1	yes	characteristic, complemented, Hall
253	C₂²×C₃	—	2	2	no	—
254	C₂²×C₃	—	4	1	no	—
255	C₂²×C₃	—	4	1	no	—
256	C₂²×C₃	C₅×#121	1	1	yes	characteristic
257	C₃×C₄	—	2	1	no	—
258	C₃×C₄	—	4	1	no	—
259	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
260	C₃×C₄	C₅×#135	1	1	yes	characteristic
261	C₃×C₄	C₅×#135	1	1	yes	characteristic
262	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
263	Dic₃	—	8	1	no	—
264	Dic₃	—	8	1	no	—
265	C₂×C₅	—	4	2	no	—
266	C₂×C₅	#5016	1	1	yes	characteristic
267	C₂×C₅	#5114	1	1	yes	characteristic
268	C₂×C₅	#5143	1	1	yes	characteristic
269	C₂³	—	2	1	no	—
270	Q₈	—	12	1	no	—
271	Q₈	—	12	1	no	—
272	Q₈	—	12	1	no	—
273	Q₈	—	12	1	no	—
274	D₄	—	2	1	no	—
275	D₄	C₅×#204	1	2	yes	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	12	1	no	—
280	C₂×C₄	—	12	1	no	—
281	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
282	C₂×C₄	C₅×#235	1	1	yes	characteristic
283	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	4	1	no	—
287	C₈	—	12	1	no	—
288	C₈	—	12	1	no	—
289	C₂×C₃	—	4	2	no	—
290	C₂×C₃	C₅×#463	1	1	yes	characteristic
291	C₂×C₃	C₅×#465	1	1	yes	characteristic
292	C₂×C₃	C₅×#473	1	1	yes	characteristic
293	C₅	#70854	1	1	yes	characteristic, direct product factor, Sylow
294	C₂²	—	2	2	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	C₅×#930	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	4	1	no	—
300	C₄	—	24	1	no	—
301	C₄	—	24	1	no	—
302	C₄	C₂×C₅×#204	1	1	yes	characteristic
303	C₄	C₅×#904	1	1	yes	characteristic
304	C₄	C₅×#932	1	1	yes	characteristic
305	C₄	C₅×#950	1	1	yes	characteristic
306	C₃	C₅×#1679	1	1	yes	characteristic, complemented, Sylow
307	C₂	—	4	2	no	—
308	C₂	C₅×#5016	1	1	yes	characteristic
309	C₂	C₅×#5114	1	1	yes	characteristic
310	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×#1679
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
4. C₅×#4681	C₂	semidirect
5. C₅×#4486	C₂	semidirect
6. C₅×#4482	C₂	semidirect
7. C₅×#4457	C₂	semidirect
24. #70854	C₅	direct (N × H)
53. #4681	C₂×C₅	semidirect
54. #4486	C₂×C₅	semidirect
55. #4482	C₂×C₅	semidirect
56. #4457	C₂×C₅	semidirect
252. C₃×C₅	#1679	semidirect
293. C₅	#70854	direct (N × H)
306. C₃	C₅×#1679	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×#5019	2
C₅×#5007	2
C₅×#4681	2
C₅×#4486	2
C₅×#4482	2
C₅×#4457	2
C₅×#1679	3
#70854	5

Subgroups — C5×#70854 #14291976

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70854 #14291976