Subgroups — C₅×#69221 #14313891

Order 1920.

464 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#919	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#798	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×D₃×#107	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#358	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4564	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4509	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4491	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2139	—	3	1	no	maximal, Hall
9	C₂×C₅×#239	—	2	1	no	—
10	C₂×C₃×C₅×Q₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₅×#215	—	2	1	no	—
12	C₅×#919	—	2	2	no	—
13	C₅×#919	C₄	1	4	yes	complemented
14	C₂×C₅×D₂₄	C₂²	1	1	yes	characteristic
15	C₂×C₅×D₃×C₈	C₂²	1	1	yes	characteristic
16	C₅×D₃×C₄⋊C₄	—	2	1	no	—
17	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
18	C₃×C₅×#107	C₂²	1	1	yes	characteristic
19	C₅×#818	C₂²	1	1	yes	characteristic
20	C₂×C₅×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
21	C₅×#807	—	2	1	no	—
22	C₅×#798	—	2	2	no	—
23	#69221	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
24	C₂×C₅×#135	—	3	1	no	—
25	C₂×C₅×M₃₂	—	3	1	no	—
26	C₂×C₅×#107	—	3	1	no	—
27	C₅×#358	—	3	1	no	—
28	C₅×#358	D₃	1	1	yes	characteristic, complemented
29	C₅×#357	—	3	1	no	—
30	C₅×#357	—	3	1	no	—
31	C₂×C₃×C₅×Q₈	—	2	1	no	—
32	C₅×#239	—	2	4	no	—
33	C₅×#239	—	4	2	no	—
34	C₂×C₅×D₁₂	—	2	1	no	—
35	C₂×C₅×D₁₂	—	4	1	no	—
36	C₂×C₄×C₅×D₃	—	4	1	no	—
37	C₂×C₄×C₅×D₃	—	4	1	no	—
38	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
39	C₃×C₅×Q₁₆	—	2	1	no	—
40	C₃×C₅×Q₁₆	C₂×C₄	1	2	yes	complemented
41	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
42	C₃×C₅×C₄⋊C₄	—	2	1	no	—
43	C₅×#215	—	2	4	no	—
44	C₅×C₄⋊Dic₃	—	2	1	no	—
45	C₅×Dic₃⋊C₄	—	4	1	no	—
46	C₂×C₅×C₃⋊C₈	D₄	1	1	yes	characteristic
47	C₅×D₂₄	—	2	1	no	—
48	C₅×D₂₄	C₂×C₄	1	2	yes	complemented
49	C₅×D₃×C₈	C₂×C₄	1	2	yes	—
50	C₅×D₃×C₈	D₄	1	2	yes	—
51	C₃×C₅×C₁₆	—	2	1	no	—
52	C₅×C₃⋊C₁₆	—	2	1	no	—
53	C₂×#919	C₂×C₅	1	1	yes	characteristic
54	C₂×#798	C₂×C₅	1	1	yes	characteristic, complemented
55	D₃×#107	C₂×C₅	1	1	yes	characteristic, complemented
56	C₃×#358	C₂×C₅	1	1	yes	characteristic, complemented
57	#4564	C₂×C₅	1	1	yes	characteristic, complemented
58	#4509	C₂×C₅	1	1	yes	characteristic, complemented
59	#4491	C₂×C₅	1	1	yes	characteristic, complemented
60	C₂×C₅×#41	—	6	1	no	—
61	C₅×#135	—	3	4	no	—
62	C₅×#135	—	6	2	no	—
63	C₂×C₅×Q₁₆	C₂×D₃	1	1	yes	characteristic
64	C₂×C₅×SD₁₆	—	6	1	no	—
65	C₂×C₅×D₈	—	3	1	no	—
66	C₂²×C₅×C₈	—	3	1	no	—
67	C₂×C₅×C₄⋊C₄	—	6	1	no	—
68	C₅×M₃₂	—	6	2	no	—
69	C₂×C₅×C₁₆	—	3	1	no	—
70	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
71	C₅×#107	—	3	1	no	—
72	C₅×#107	—	6	1	no	—
73	C₅×#107	C₂×D₃	1	1	yes	characteristic
74	#2139	—	3	1	no	Sylow
75	C₂²×C₅×D₃	—	4	1	no	—
76	C₂²×C₅×D₃	D₈	1	1	yes	characteristic
77	C₃×C₅×Q₈	—	2	2	no	—
78	C₃×C₅×Q₈	—	4	1	no	—
79	C₂×C₃×C₄×C₅	—	4	1	no	—
80	C₂×C₃×C₄×C₅	—	4	1	no	—
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂×C₅×Dic₃	—	4	1	no	—
83	C₂×C₅×Dic₃	D₈	1	1	yes	characteristic
84	C₅×D₁₂	—	2	2	no	—
85	C₅×D₁₂	—	4	4	no	—
86	C₅×D₁₂	—	4	1	no	—
87	C₄×C₅×D₃	—	4	4	no	—
88	C₄×C₅×D₃	—	8	2	no	—
89	C₄×C₅×D₃	C₂²⋊C₄	1	2	yes	—
90	C₄×C₅×D₃	SD₁₆	1	2	yes	—
91	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
92	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
93	C₅×C₃⋊C₈	C₂²⋊C₄	1	2	yes	—
94	C₂×#239	—	2	1	no	—
95	C₂×C₃×Q₁₆	C₂²×C₅	1	1	yes	characteristic
96	C₂×#215	—	2	1	no	—
97	#919	—	2	2	no	—
98	#919	C₄×C₅	1	4	yes	complemented
99	C₂×D₂₄	C₂²×C₅	1	1	yes	characteristic
100	C₂×D₃×C₈	C₂²×C₅	1	1	yes	characteristic
101	D₃×C₄⋊C₄	—	2	1	no	—
102	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
103	C₃×#107	C₂²×C₅	1	1	yes	characteristic
104	#818	C₂²×C₅	1	1	yes	characteristic
105	C₂×C₃⋊C₁₆	C₂²×C₅	1	1	yes	characteristic
106	#807	—	2	1	no	—
107	#798	—	2	2	no	—
108	C₅×#41	—	6	4	no	—
109	C₅×#41	—	12	2	no	—
110	C₂×C₅×Q₈	—	2	1	no	—
111	C₂×C₅×D₄	—	6	1	no	—
112	C₂×C₅×D₄	—	12	1	no	—
113	C₂²×C₄×C₅	—	3	1	no	—
114	C₂²×C₄×C₅	—	12	1	no	—
115	C₂²×C₄×C₅	—	12	1	no	—
116	C₅×Q₁₆	—	2	1	no	—
117	C₅×Q₁₆	C₄×D₃	1	2	yes	complemented
118	C₅×SD₁₆	—	6	4	no	—
119	C₅×D₈	—	3	2	no	—
120	C₅×D₈	—	6	1	no	—
121	C₂×C₅×C₈	—	3	1	no	—
122	C₂×C₅×C₈	—	3	2	no	—
123	C₂×C₅×C₈	—	3	2	no	—
124	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
125	C₅×C₄⋊C₄	—	2	1	no	—
126	C₅×C₄⋊C₄	—	6	1	no	—
127	C₅×C₄⋊C₄	—	12	1	no	—
128	C₅×C₁₆	—	2	1	no	—
129	C₅×C₁₆	—	6	1	no	—
130	C₂×#135	—	3	1	no	—
131	C₂×M₃₂	—	3	1	no	—
132	C₂×#107	—	3	1	no	—
133	#358	—	3	1	no	—
134	#358	C₅×D₃	1	1	yes	characteristic, complemented
135	#357	—	3	1	no	—
136	#357	—	3	1	no	—
137	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
138	C₂×C₅×D₃	—	2	1	no	—
139	C₂×C₅×D₃	—	2	1	no	—
140	C₂×C₅×D₃	—	4	2	no	—
141	C₂×C₅×D₃	—	8	1	no	—
142	C₂×C₅×D₃	—	8	1	no	—
143	C₂×C₅×D₃	#102	1	2	yes	—
144	C₃×C₄×C₅	—	4	2	no	—
145	C₃×C₄×C₅	—	8	1	no	—
146	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
148	C₅×Dic₃	—	8	1	no	—
149	C₅×Dic₃	#102	1	2	yes	—
150	C₂×C₃×Q₈	—	2	1	no	—
151	#239	—	2	4	no	—
152	#239	—	4	2	no	—
153	C₂×D₁₂	—	2	1	no	—
154	C₂×D₁₂	—	4	1	no	—
155	C₂×C₄×D₃	—	4	1	no	—
156	C₂×C₄×D₃	—	4	1	no	—
157	C₂×C₄×D₃	C₅×D₄	1	1	yes	characteristic
158	C₃×Q₁₆	—	2	1	no	—
159	C₃×Q₁₆	C₂×C₄×C₅	1	2	yes	complemented
160	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
161	C₃×C₄⋊C₄	—	2	1	no	—
162	#215	—	2	4	no	—
163	C₄⋊Dic₃	—	2	1	no	—
164	Dic₃⋊C₄	—	4	1	no	—
165	C₂×C₃⋊C₈	C₅×D₄	1	1	yes	characteristic
166	D₂₄	—	2	1	no	—
167	D₂₄	C₂×C₄×C₅	1	2	yes	complemented
168	D₃×C₈	C₂×C₄×C₅	1	2	yes	—
169	D₃×C₈	C₅×D₄	1	2	yes	—
170	C₃×C₁₆	—	2	1	no	—
171	C₃⋊C₁₆	—	2	1	no	—
172	C₂³×C₅	—	3	1	no	—
173	C₂³×C₅	—	12	1	no	—
174	C₅×Q₈	—	2	2	no	—
175	C₅×Q₈	—	4	1	no	—
176	C₅×D₄	—	6	2	no	—
177	C₅×D₄	—	12	4	no	—
178	C₅×D₄	—	12	1	no	—
179	C₂×C₄×C₅	—	3	1	no	—
180	C₂×C₄×C₅	—	3	2	no	—
181	C₂×C₄×C₅	—	3	2	no	—
182	C₂×C₄×C₅	—	4	1	no	—
183	C₂×C₄×C₅	—	4	1	no	—
184	C₂×C₄×C₅	—	12	4	no	—
185	C₂×C₄×C₅	—	12	1	no	—
186	C₂×C₄×C₅	—	24	2	no	—
187	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
188	C₅×C₈	—	3	2	no	—
189	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
190	C₅×C₈	D₃×D₄	1	1	yes	characteristic
191	C₂×#41	—	6	1	no	—
192	#135	—	3	4	no	—
193	#135	—	6	2	no	—
194	C₂×Q₁₆	C₂×C₅×D₃	1	1	yes	characteristic
195	C₂×SD₁₆	—	6	1	no	—
196	C₂×D₈	—	3	1	no	—
197	C₂²×C₈	—	3	1	no	—
198	C₂×C₄⋊C₄	—	6	1	no	—
199	M₃₂	—	6	2	no	—
200	C₂×C₁₆	—	3	1	no	—
201	C₂×C₁₆	C₂×C₅×D₃	1	1	yes	characteristic
202	#107	—	3	1	no	—
203	#107	—	6	1	no	—
204	#107	C₂×C₅×D₃	1	1	yes	characteristic
205	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
206	C₂×C₃×C₅	#509	1	1	yes	characteristic
207	C₂×C₃×C₅	#510	1	1	yes	characteristic
208	C₅×D₃	—	2	2	no	—
209	C₅×D₃	—	8	2	no	—
210	C₂²×D₃	—	4	1	no	—
211	C₂²×D₃	C₅×D₈	1	1	yes	characteristic
212	C₃×Q₈	—	2	2	no	—
213	C₃×Q₈	—	4	1	no	—
214	C₂×C₃×C₄	—	4	1	no	—
215	C₂×C₃×C₄	—	4	1	no	—
216	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
217	C₂×Dic₃	—	4	1	no	—
218	C₂×Dic₃	C₅×D₈	1	1	yes	characteristic
219	D₁₂	—	2	2	no	—
220	D₁₂	—	4	4	no	—
221	D₁₂	—	4	1	no	—
222	C₄×D₃	—	4	4	no	—
223	C₄×D₃	—	8	2	no	—
224	C₄×D₃	C₅×C₂²⋊C₄	1	2	yes	—
225	C₄×D₃	C₅×SD₁₆	1	2	yes	—
226	C₃×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
227	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
228	C₃⋊C₈	C₅×C₂²⋊C₄	1	2	yes	—
229	C₂²×C₅	—	3	2	no	—
230	C₂²×C₅	—	6	1	no	—
231	C₂²×C₅	—	6	1	no	—
232	C₂²×C₅	—	12	2	no	—
233	C₂²×C₅	—	24	1	no	—
234	C₂²×C₅	—	24	1	no	—
235	C₂²×C₅	D₃×D₈	1	1	yes	characteristic
236	C₄×C₅	—	3	2	no	—
237	C₄×C₅	—	4	2	no	—
238	C₄×C₅	—	8	1	no	—
239	C₄×C₅	—	24	1	no	—
240	C₄×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
241	C₄×C₅	D₃×SD₁₆	1	1	yes	characteristic
242	#41	—	6	4	no	—
243	#41	—	12	2	no	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×D₄	—	6	1	no	—
246	C₂×D₄	—	12	1	no	—
247	C₂²×C₄	—	3	1	no	—
248	C₂²×C₄	—	12	1	no	—
249	C₂²×C₄	—	12	1	no	—
250	Q₁₆	—	2	1	no	—
251	Q₁₆	C₄×C₅×D₃	1	2	yes	complemented
252	SD₁₆	—	6	4	no	—
253	D₈	—	3	2	no	—
254	D₈	—	6	1	no	—
255	C₂×C₈	—	3	1	no	—
256	C₂×C₈	—	3	2	no	—
257	C₂×C₈	—	3	2	no	—
258	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	6	1	no	—
261	C₄⋊C₄	—	12	1	no	—
262	C₁₆	—	2	1	no	—
263	C₁₆	—	6	1	no	—
264	C₃×C₅	#2139	1	1	yes	characteristic, complemented, Hall
265	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
266	C₂×D₃	—	2	1	no	—
267	C₂×D₃	—	2	1	no	—
268	C₂×D₃	—	4	2	no	—
269	C₂×D₃	—	8	1	no	—
270	C₂×D₃	—	8	1	no	—
271	C₂×D₃	C₅×#102	1	2	yes	—
272	C₃×C₄	—	4	2	no	—
273	C₃×C₄	—	8	1	no	—
274	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
275	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
276	Dic₃	—	8	1	no	—
277	Dic₃	C₅×#102	1	2	yes	—
278	C₂×C₅	—	6	2	no	—
279	C₂×C₅	—	24	2	no	—
280	C₂×C₅	D₃×#102	1	1	yes	characteristic
281	C₂×C₅	#4914	1	1	yes	characteristic
282	C₂×C₅	#4918	1	1	yes	characteristic
283	C₂³	—	3	1	no	—
284	C₂³	—	12	1	no	—
285	Q₈	—	2	2	no	—
286	Q₈	—	4	1	no	—
287	D₄	—	6	2	no	—
288	D₄	—	12	4	no	—
289	D₄	—	12	1	no	—
290	C₂×C₄	—	3	1	no	—
291	C₂×C₄	—	3	2	no	—
292	C₂×C₄	—	3	2	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	12	4	no	—
296	C₂×C₄	—	12	1	no	—
297	C₂×C₄	—	24	2	no	—
298	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
299	C₈	—	3	2	no	—
300	C₈	C₂×C₄×C₅×D₃	1	1	yes	characteristic
301	C₈	C₅×D₃×D₄	1	1	yes	characteristic
302	C₂×C₃	C₂×C₅×#102	1	1	yes	characteristic
303	C₂×C₃	C₅×#509	1	1	yes	characteristic
304	C₂×C₃	C₅×#510	1	1	yes	characteristic
305	D₃	—	2	2	no	—
306	D₃	—	8	2	no	—
307	C₅	#69221	1	1	yes	characteristic, direct product factor, Sylow
308	C₂²	—	3	2	no	—
309	C₂²	—	6	1	no	—
310	C₂²	—	6	1	no	—
311	C₂²	—	12	2	no	—
312	C₂²	—	24	1	no	—
313	C₂²	—	24	1	no	—
314	C₂²	C₅×D₃×D₈	1	1	yes	characteristic
315	C₄	—	3	2	no	—
316	C₄	—	4	2	no	—
317	C₄	—	8	1	no	—
318	C₄	—	24	1	no	—
319	C₄	C₅×D₃×C₂²⋊C₄	1	1	yes	characteristic
320	C₄	C₅×D₃×SD₁₆	1	1	yes	characteristic
321	C₃	C₅×#2139	1	1	yes	characteristic, complemented, Sylow
322	C₂	—	6	2	no	—
323	C₂	—	24	2	no	—
324	C₂	C₅×D₃×#102	1	1	yes	characteristic
325	C₂	C₅×#4914	1	1	yes	characteristic
326	C₂	C₅×#4918	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	D₃×D₈
commutator	C₃×C₈	C₂²×C₄×C₅
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₃×C₅×#358	C₂
socle	C₂²×C₃×C₅	C₂×D₈
prime radical	C₂×C₅×#919	C₂
cosocle	C₂×C₃×C₈	C₂³×C₅
hypercenter	C₅×#358	D₃
hypocenter	C₃	C₅×#2139
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₅×#358

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#798	C₂	semidirect
3. C₅×D₃×#107	C₂	semidirect
4. C₃×C₅×#358	C₂	semidirect
5. C₅×#4564	C₂	semidirect
6. C₅×#4509	C₂	semidirect
7. C₅×#4491	C₂	semidirect
13. C₅×#919	C₄	semidirect
23. #69221	C₅	direct (N × H)
28. C₅×#358	D₃	semidirect
40. C₃×C₅×Q₁₆	C₂×C₄	semidirect
48. C₅×D₂₄	C₂×C₄	semidirect
54. C₂×#798	C₂×C₅	semidirect
55. D₃×#107	C₂×C₅	semidirect
56. C₃×#358	C₂×C₅	semidirect
57. #4564	C₂×C₅	semidirect
58. #4509	C₂×C₅	semidirect
59. #4491	C₂×C₅	semidirect
98. #919	C₄×C₅	semidirect
117. C₅×Q₁₆	C₄×D₃	semidirect
134. #358	C₅×D₃	semidirect
159. C₃×Q₁₆	C₂×C₄×C₅	semidirect
167. D₂₄	C₂×C₄×C₅	semidirect
251. Q₁₆	C₄×C₅×D₃	semidirect
264. C₃×C₅	#2139	semidirect
307. C₅	#69221	direct (N × H)
321. C₃	C₅×#2139	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#919	2
C₂×C₅×#798	2
C₅×D₃×#107	2
C₃×C₅×#358	2
C₅×#4564	2
C₅×#4509	2
C₅×#4491	2
C₅×#2139	3
#69221	5

Subgroups — C5×#69221 #14313891

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69221 #14313891