Subgroups — C₅×#71055 #14292171

Order 1920.

332 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#462	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#5088	C₂	1	1	yes	maximal, characteristic
3	C₅×#5027	C₂	1	1	yes	maximal, characteristic
4	C₅×Q₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4547	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4485	C₂	1	1	yes	maximal, characteristic
7	C₅×#4479	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1682	—	3	1	no	maximal, Hall
9	C₂×C₃×C₅×Q₁₆	—	2	1	no	—
10	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₅×#215	—	2	1	no	—
13	C₅×#874	C₂²	1	1	yes	characteristic
14	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₃×C₅×#103	—	2	1	no	—
16	C₅×#809	—	2	1	no	—
17	C₅×#809	—	2	1	no	—
18	C₅×#809	—	2	1	no	—
19	C₅×C₄⋊(C₃⋊C₈)	—	2	1	no	—
20	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
21	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
22	C₄×C₅×C₃⋊C₈	—	2	1	no	—
23	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
24	#71055	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₅×#486	—	3	1	no	—
26	C₅×#462	D₃	1	1	yes	characteristic, complemented
27	C₅×#460	—	3	1	no	—
28	C₅×C₈×Q₈	—	3	1	no	—
29	C₅×#335	—	3	1	no	—
30	C₅×#332	—	3	1	no	—
31	C₅×#331	—	3	1	no	—
32	C₂×C₃×C₅×Q₈	—	2	1	no	—
33	C₂×C₃×C₅×Q₈	D₄	1	1	yes	characteristic
34	C₂×C₅×D₁₂	—	2	1	no	—
35	C₂×C₅×D₁₂	—	2	1	no	—
36	C₂×C₅×D₁₂	—	2	1	no	—
37	C₃×C₅×Q₁₆	—	4	2	no	—
38	C₂×C₃×C₅×C₈	—	2	1	no	—
39	C₃×C₅×C₄⋊C₄	—	2	1	no	—
40	C₃×C₅×C₄⋊C₄	—	2	1	no	—
41	C₃×C₅×C₄⋊C₄	—	2	1	no	—
42	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
43	C₃×C₄²×C₅	—	2	1	no	—
44	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
45	C₅×#215	—	4	2	no	—
46	C₂×C₅×C₃⋊C₈	—	2	1	no	—
47	C₂×C₅×C₃⋊C₈	—	2	1	no	—
48	C₂×C₅×C₃⋊C₈	D₄	1	1	yes	characteristic
49	C₂×C₅×C₃⋊C₈	D₄	1	1	yes	characteristic
50	C₃×#462	C₂×C₅	1	1	yes	characteristic, complemented
51	#5088	C₂×C₅	1	1	yes	characteristic
52	#5027	C₂×C₅	1	1	yes	characteristic
53	Q₈×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
54	#4547	C₂×C₅	1	1	yes	characteristic, complemented
55	#4485	C₂×C₅	1	1	yes	characteristic
56	#4479	C₂×C₅	1	1	yes	characteristic, complemented
57	C₂×C₅×Q₁₆	—	2	1	no	—
58	C₂×C₅×SD₁₆	—	6	1	no	—
59	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
60	C₅×#127	—	3	1	no	—
61	C₄×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
62	C₅×C₄⋊C₈	—	3	1	no	—
63	C₅×C₄⋊C₈	—	3	1	no	—
64	C₅×C₄⋊C₈	—	6	1	no	—
65	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
66	C₅×#103	—	2	1	no	—
67	C₅×#102	—	6	1	no	—
68	C₅×#102	—	6	1	no	—
69	C₅×#102	—	6	1	no	—
70	C₄×C₅×C₈	—	3	1	no	—
71	C₄×C₅×C₈	—	6	1	no	—
72	#1682	—	3	1	no	Sylow
73	C₂²×C₅×D₃	—	4	1	no	—
74	C₃×C₅×Q₈	—	2	1	no	—
75	C₃×C₅×Q₈	—	4	1	no	—
76	C₃×C₅×Q₈	—	4	1	no	—
77	C₃×C₅×Q₈	D₈	1	2	yes	—
78	C₂×C₃×C₄×C₅	—	2	1	no	—
79	C₂×C₃×C₄×C₅	—	2	1	no	—
80	C₂×C₃×C₄×C₅	—	4	1	no	—
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
84	C₅×D₁₂	—	4	1	no	—
85	C₅×D₁₂	—	4	1	no	—
86	C₅×D₁₂	—	4	1	no	—
87	C₅×D₁₂	—	4	1	no	—
88	C₅×D₁₂	—	8	1	no	—
89	C₃×C₅×C₈	—	4	1	no	—
90	C₅×C₃⋊C₈	—	2	1	no	—
91	C₅×C₃⋊C₈	—	2	1	no	—
92	C₅×C₃⋊C₈	—	4	1	no	—
93	C₅×C₃⋊C₈	—	4	1	no	—
94	C₂×C₃×Q₁₆	—	2	1	no	—
95	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
96	C₃×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
97	C₂×#215	—	2	1	no	—
98	#874	C₂²×C₅	1	1	yes	characteristic
99	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
100	C₃×#103	—	2	1	no	—
101	#809	—	2	1	no	—
102	#809	—	2	1	no	—
103	#809	—	2	1	no	—
104	C₄⋊(C₃⋊C₈)	—	2	1	no	—
105	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
106	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
107	C₄×C₃⋊C₈	—	2	1	no	—
108	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
109	C₂×C₅×Q₈	—	2	1	no	—
110	C₂×C₅×Q₈	#67	1	1	yes	characteristic
111	C₂×C₅×D₄	—	6	1	no	—
112	C₂×C₅×D₄	—	6	1	no	—
113	C₂×C₅×D₄	—	6	1	no	—
114	C₅×Q₁₆	—	4	2	no	—
115	C₅×SD₁₆	—	12	2	no	—
116	C₂×C₅×C₈	—	2	1	no	—
117	C₂×C₅×C₈	—	3	1	no	—
118	C₂×C₅×C₈	—	3	1	no	—
119	C₂×C₅×C₈	—	6	1	no	—
120	C₂×C₅×C₈	—	6	1	no	—
121	C₅×C₄⋊C₄	—	2	1	no	—
122	C₅×C₄⋊C₄	—	2	1	no	—
123	C₅×C₄⋊C₄	—	2	1	no	—
124	C₅×C₄⋊C₄	#67	1	1	yes	characteristic
125	C₄²×C₅	—	2	1	no	—
126	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
127	#486	—	3	1	no	—
128	#462	C₅×D₃	1	1	yes	characteristic, complemented
129	#460	—	3	1	no	—
130	C₈×Q₈	—	3	1	no	—
131	#335	—	3	1	no	—
132	#332	—	3	1	no	—
133	#331	—	3	1	no	—
134	C₂²×C₃×C₅	#121	1	1	yes	characteristic
135	C₂×C₅×D₃	—	8	1	no	—
136	C₂×C₅×D₃	—	8	1	no	—
137	C₂×C₅×D₃	—	8	1	no	—
138	C₃×C₄×C₅	—	2	2	no	—
139	C₃×C₄×C₅	—	2	1	no	—
140	C₃×C₄×C₅	—	4	1	no	—
141	C₃×C₄×C₅	—	8	1	no	—
142	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
143	C₃×C₄×C₅	#135	1	1	yes	characteristic
144	C₃×C₄×C₅	#135	1	1	yes	characteristic
145	C₃×C₄×C₅	#137	1	1	yes	characteristic
146	C₂×C₃×Q₈	—	2	1	no	—
147	C₂×C₃×Q₈	C₅×D₄	1	1	yes	characteristic
148	C₂×D₁₂	—	2	1	no	—
149	C₂×D₁₂	—	2	1	no	—
150	C₂×D₁₂	—	2	1	no	—
151	C₃×Q₁₆	—	4	2	no	—
152	C₂×C₃×C₈	—	2	1	no	—
153	C₃×C₄⋊C₄	—	2	1	no	—
154	C₃×C₄⋊C₄	—	2	1	no	—
155	C₃×C₄⋊C₄	—	2	1	no	—
156	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
157	C₃×C₄²	—	2	1	no	—
158	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
159	#215	—	4	2	no	—
160	C₂×C₃⋊C₈	—	2	1	no	—
161	C₂×C₃⋊C₈	—	2	1	no	—
162	C₂×C₃⋊C₈	C₅×D₄	1	1	yes	characteristic
163	C₂×C₃⋊C₈	C₅×D₄	1	1	yes	characteristic
164	C₂³×C₅	—	12	1	no	—
165	C₅×Q₈	—	2	1	no	—
166	C₅×Q₈	—	4	1	no	—
167	C₅×Q₈	—	4	1	no	—
168	C₅×Q₈	#213	1	2	yes	—
169	C₅×D₄	—	12	1	no	—
170	C₅×D₄	—	12	1	no	—
171	C₅×D₄	—	12	1	no	—
172	C₅×D₄	—	12	1	no	—
173	C₅×D₄	—	24	1	no	—
174	C₂×C₄×C₅	—	2	1	no	—
175	C₂×C₄×C₅	—	2	1	no	—
176	C₂×C₄×C₅	—	4	1	no	—
177	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
178	C₂×C₄×C₅	#237	1	1	yes	characteristic
179	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
180	C₅×C₈	—	4	1	no	—
181	C₅×C₈	—	6	1	no	—
182	C₅×C₈	—	6	1	no	—
183	C₅×C₈	—	12	1	no	—
184	C₅×C₈	—	12	1	no	—
185	C₂×Q₁₆	—	2	1	no	—
186	C₂×SD₁₆	—	6	1	no	—
187	C₄⋊Q₈	C₂×C₅×D₃	1	1	yes	characteristic
188	#127	—	3	1	no	—
189	C₄×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
190	C₄⋊C₈	—	3	1	no	—
191	C₄⋊C₈	—	3	1	no	—
192	C₄⋊C₈	—	6	1	no	—
193	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
194	#103	—	2	1	no	—
195	#102	—	6	1	no	—
196	#102	—	6	1	no	—
197	#102	—	6	1	no	—
198	C₄×C₈	—	3	1	no	—
199	C₄×C₈	—	6	1	no	—
200	C₂×C₃×C₅	#464	1	1	yes	characteristic
201	C₂×C₃×C₅	#466	1	1	yes	characteristic
202	C₂×C₃×C₅	#471	1	1	yes	characteristic
203	C₅×D₃	—	16	1	no	—
204	C₂²×D₃	—	4	1	no	—
205	C₃×Q₈	—	2	1	no	—
206	C₃×Q₈	—	4	1	no	—
207	C₃×Q₈	—	4	1	no	—
208	C₃×Q₈	C₅×D₈	1	2	yes	—
209	C₂×C₃×C₄	—	2	1	no	—
210	C₂×C₃×C₄	—	2	1	no	—
211	C₂×C₃×C₄	—	4	1	no	—
212	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
214	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
215	D₁₂	—	4	1	no	—
216	D₁₂	—	4	1	no	—
217	D₁₂	—	4	1	no	—
218	D₁₂	—	4	1	no	—
219	D₁₂	—	8	1	no	—
220	C₃×C₈	—	4	1	no	—
221	C₃⋊C₈	—	2	1	no	—
222	C₃⋊C₈	—	2	1	no	—
223	C₃⋊C₈	—	4	1	no	—
224	C₃⋊C₈	—	4	1	no	—
225	C₂²×C₅	—	24	1	no	—
226	C₂²×C₅	—	24	1	no	—
227	C₂²×C₅	—	24	1	no	—
228	C₂²×C₅	#939	1	1	yes	characteristic
229	C₄×C₅	—	2	1	no	—
230	C₄×C₅	—	2	2	no	—
231	C₄×C₅	—	4	1	no	—
232	C₄×C₅	—	8	1	no	—
233	C₄×C₅	#918	1	1	yes	characteristic
234	C₄×C₅	#919	1	1	yes	characteristic
235	C₄×C₅	C₂×#213	1	1	yes	characteristic
236	C₄×C₅	#950	1	1	yes	characteristic
237	C₂×Q₈	—	2	1	no	—
238	C₂×Q₈	C₅×#67	1	1	yes	characteristic
239	C₂×D₄	—	6	1	no	—
240	C₂×D₄	—	6	1	no	—
241	C₂×D₄	—	6	1	no	—
242	Q₁₆	—	4	2	no	—
243	SD₁₆	—	12	2	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	3	1	no	—
246	C₂×C₈	—	3	1	no	—
247	C₂×C₈	—	6	1	no	—
248	C₂×C₈	—	6	1	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	C₅×#67	1	1	yes	characteristic
253	C₄²	—	2	1	no	—
254	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
255	C₃×C₅	#1682	1	1	yes	characteristic, complemented, Hall
256	C₂²×C₃	C₅×#121	1	1	yes	characteristic
257	C₂×D₃	—	8	1	no	—
258	C₂×D₃	—	8	1	no	—
259	C₂×D₃	—	8	1	no	—
260	C₃×C₄	—	2	2	no	—
261	C₃×C₄	—	2	1	no	—
262	C₃×C₄	—	4	1	no	—
263	C₃×C₄	—	8	1	no	—
264	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
265	C₃×C₄	C₅×#135	1	1	yes	characteristic
266	C₃×C₄	C₅×#135	1	1	yes	characteristic
267	C₃×C₄	C₅×#137	1	1	yes	characteristic
268	C₂×C₅	—	48	1	no	—
269	C₂×C₅	#5038	1	1	yes	characteristic
270	C₂×C₅	#5185	1	1	yes	characteristic
271	C₂×C₅	#5203	1	1	yes	characteristic
272	C₂³	—	12	1	no	—
273	Q₈	—	2	1	no	—
274	Q₈	—	4	1	no	—
275	Q₈	—	4	1	no	—
276	Q₈	C₅×#213	1	2	yes	—
277	D₄	—	12	1	no	—
278	D₄	—	12	1	no	—
279	D₄	—	12	1	no	—
280	D₄	—	12	1	no	—
281	D₄	—	24	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
286	C₂×C₄	C₅×#237	1	1	yes	characteristic
287	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
288	C₈	—	4	1	no	—
289	C₈	—	6	1	no	—
290	C₈	—	6	1	no	—
291	C₈	—	12	1	no	—
292	C₈	—	12	1	no	—
293	C₂×C₃	C₅×#464	1	1	yes	characteristic
294	C₂×C₃	C₅×#466	1	1	yes	characteristic
295	C₂×C₃	C₅×#471	1	1	yes	characteristic
296	D₃	—	16	1	no	—
297	C₅	#71055	1	1	yes	characteristic, direct product factor, Sylow
298	C₂²	—	24	1	no	—
299	C₂²	—	24	1	no	—
300	C₂²	—	24	1	no	—
301	C₂²	C₅×#939	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	2	no	—
304	C₄	—	4	1	no	—
305	C₄	—	8	1	no	—
306	C₄	C₅×#918	1	1	yes	characteristic
307	C₄	C₅×#919	1	1	yes	characteristic
308	C₄	C₂×C₅×#213	1	1	yes	characteristic
309	C₄	C₅×#950	1	1	yes	characteristic
310	C₃	C₅×#1682	1	1	yes	characteristic, complemented, Sylow
311	C₂	—	48	1	no	—
312	C₂	C₅×#5038	1	1	yes	characteristic
313	C₂	C₅×#5185	1	1	yes	characteristic
314	C₂	C₅×#5203	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	#939
commutator, cosocle	C₃×C₄²	C₂³×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#462	C₂
socle	C₂²×C₃×C₅	#121
prime radical	C₅×#874	C₂²
hypercenter	C₅×#462	D₃
hypocenter	C₃	C₅×#1682
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#462

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#462	C₂	semidirect
4. C₅×Q₈×C₃⋊C₈	C₂	semidirect
5. C₅×#4547	C₂	semidirect
7. C₅×#4479	C₂	semidirect
24. #71055	C₅	direct (N × H)
26. C₅×#462	D₃	semidirect
50. C₃×#462	C₂×C₅	semidirect
53. Q₈×C₃⋊C₈	C₂×C₅	semidirect
54. #4547	C₂×C₅	semidirect
56. #4479	C₂×C₅	semidirect
128. #462	C₅×D₃	semidirect
255. C₃×C₅	#1682	semidirect
297. C₅	#71055	direct (N × H)
310. C₃	C₅×#1682	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#462	2
C₅×#5088	2
C₅×#5027	2
C₅×Q₈×C₃⋊C₈	2
C₅×#4547	2
C₅×#4485	2
C₅×#4479	2
C₅×#1682	3
#71055	5

Subgroups — C5×#71055 #14292171

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71055 #14292171