Subgroups — C₅×#69643 #14308908

Order 1920.

564 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	C₅×#5500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅×#390	C₂	1	1	yes	maximal, characteristic
3	C₅×#5215	C₂	1	2	yes	maximal
4	C₅×#4530	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4523	C₂	1	2	yes	maximal, complemented
6	C₅×#2027	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₅×C₂²⋊C₄	—	2	2	no	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₅×#926	C₂²	1	2	yes	—
12	C₅×#924	—	2	2	no	—
13	C₂×C₅×#208	C₂²	1	2	yes	—
14	C₅×#890	—	2	2	no	—
15	C₅×#888	—	2	2	no	—
16	C₅×#832	—	2	2	no	—
17	C₅×#832	—	2	2	no	—
18	#69643	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₅×#544	—	3	1	no	—
20	C₅×#418	—	3	2	no	—
21	C₅×#390	D₃	1	1	yes	characteristic
22	C₅×#343	—	3	1	no	—
23	C₅×#339	—	3	2	no	—
24	C₂⁴×C₃×C₅	—	2	1	no	—
25	C₂×C₃×C₅×D₄	—	2	2	no	—
26	C₂×C₃×C₅×D₄	—	2	2	no	—
27	C₂×C₃×C₅×D₄	—	2	1	no	—
28	C₂×C₃×C₅×D₄	Q₈	1	2	yes	complemented
29	C₂²×C₃×C₄×C₅	—	2	1	no	—
30	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
31	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂×C₅×Dic₆	—	2	2	no	—
33	C₃×C₅×C₄⋊C₄	—	2	2	no	—
34	C₃×C₅×C₂²⋊C₄	—	4	4	no	—
35	C₃×C₄²×C₅	—	2	2	no	—
36	C₅×#217	—	2	2	no	—
37	C₅×C₄⋊Dic₃	—	2	2	no	—
38	C₅×C₄⋊Dic₃	D₄	1	2	yes	—
39	C₅×C₄⋊Dic₃	D₄	1	2	yes	—
40	C₅×Dic₃⋊C₄	—	2	4	no	—
41	C₄×C₅×Dic₃	—	2	2	no	—
42	C₅×#208	—	2	2	no	—
43	C₅×#208	—	2	2	no	—
44	C₂×C₅×C₃⋊C₈	—	2	2	no	—
45	#5500	C₂×C₅	1	1	yes	characteristic, complemented
46	C₃×#390	C₂×C₅	1	1	yes	characteristic
47	#5215	C₂×C₅	1	2	yes	—
48	#4530	C₂×C₅	1	1	yes	characteristic, complemented
49	#4523	C₂×C₅	1	2	yes	complemented
50	C₂²×C₅×D₄	C₂×D₃	1	1	yes	characteristic
51	C₂×C₅×M₁₆	—	3	2	no	—
52	C₅×C₄⋊Q₈	—	6	2	no	—
53	C₅×#125	—	6	2	no	—
54	C₅×C₂²⋊Q₈	—	6	2	no	—
55	C₅×#117	—	3	2	no	—
56	C₂×C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
57	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
58	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
59	C₅×#102	—	6	2	no	—
60	C₅×#102	—	6	2	no	—
61	#2027	—	3	1	no	Sylow
62	C₂³×C₃×C₅	—	2	2	no	—
63	C₂³×C₃×C₅	—	4	2	no	—
64	C₂³×C₃×C₅	—	4	2	no	—
65	C₂³×C₃×C₅	—	4	1	no	—
66	C₂³×C₃×C₅	—	4	1	no	—
67	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
68	C₃×C₅×D₄	—	2	4	no	—
69	C₃×C₅×D₄	—	4	2	no	—
70	C₂×C₃×C₄×C₅	—	2	2	no	—
71	C₂×C₃×C₄×C₅	—	2	2	no	—
72	C₂×C₃×C₄×C₅	—	4	1	no	—
73	C₂×C₃×C₄×C₅	—	4	1	no	—
74	C₂×C₃×C₄×C₅	—	4	2	no	—
75	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
76	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	#41	1	2	yes	—
78	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
79	C₂×C₅×Dic₃	—	2	2	no	—
80	C₂×C₅×Dic₃	—	2	2	no	—
81	C₅×Dic₆	—	8	2	no	—
82	C₅×C₃⋊C₈	—	4	2	no	—
83	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
84	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
85	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
86	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
87	#926	C₂²×C₅	1	2	yes	—
88	#924	—	2	2	no	—
89	C₂×#208	C₂²×C₅	1	2	yes	—
90	#890	—	2	2	no	—
91	#888	—	2	2	no	—
92	#832	—	2	2	no	—
93	#832	—	2	2	no	—
94	C₂⁴×C₅	—	2	1	no	—
95	C₂×C₅×Q₈	—	6	2	no	—
96	C₂×C₅×D₄	—	2	1	no	—
97	C₂×C₅×D₄	—	2	2	no	—
98	C₂×C₅×D₄	—	2	2	no	—
99	C₂×C₅×D₄	Dic₆	1	2	yes	complemented
100	C₂²×C₄×C₅	—	2	1	no	—
101	C₂²×C₄×C₅	#67	1	2	yes	—
102	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
103	C₅×M₁₆	—	6	2	no	—
104	C₅×M₁₆	—	6	2	no	—
105	C₂×C₅×C₈	—	6	2	no	—
106	C₅×C₄⋊C₄	—	2	2	no	—
107	C₅×C₄⋊C₄	—	3	2	no	—
108	C₅×C₄⋊C₄	—	3	2	no	—
109	C₅×C₄⋊C₄	—	6	2	no	—
110	C₅×C₄⋊C₄	—	6	4	no	—
111	C₅×C₂²⋊C₄	—	4	4	no	—
112	C₅×C₂²⋊C₄	—	6	2	no	—
113	C₄²×C₅	—	2	2	no	—
114	C₄²×C₅	—	6	2	no	—
115	#544	—	3	1	no	—
116	#418	—	3	2	no	—
117	#390	C₅×D₃	1	1	yes	characteristic
118	#343	—	3	1	no	—
119	#339	—	3	2	no	—
120	C₂²×C₃×C₅	—	2	1	no	—
121	C₂²×C₃×C₅	—	2	1	no	—
122	C₂²×C₃×C₅	—	4	4	no	—
123	C₂²×C₃×C₅	—	4	2	no	—
124	C₂²×C₃×C₅	—	8	2	no	—
125	C₂²×C₃×C₅	—	8	1	no	—
126	C₂²×C₃×C₅	—	8	1	no	—
127	C₂²×C₃×C₅	#120	1	1	yes	characteristic
128	C₂²×C₃×C₅	C₂²⋊Q₈	1	1	yes	characteristic
129	C₂²×C₃×C₅	#124	1	1	yes	characteristic
130	C₃×C₄×C₅	—	4	2	no	—
131	C₃×C₄×C₅	—	4	2	no	—
132	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
133	C₃×C₄×C₅	#124	1	2	yes	—
134	C₅×Dic₃	—	4	2	no	—
135	C₅×Dic₃	—	4	2	no	—
136	C₂⁴×C₃	—	2	1	no	—
137	C₂×C₃×D₄	—	2	2	no	—
138	C₂×C₃×D₄	—	2	2	no	—
139	C₂×C₃×D₄	—	2	1	no	—
140	C₂×C₃×D₄	C₅×Q₈	1	2	yes	complemented
141	C₂²×C₃×C₄	—	2	1	no	—
142	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
143	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
144	C₂×Dic₆	—	2	2	no	—
145	C₃×C₄⋊C₄	—	2	2	no	—
146	C₃×C₂²⋊C₄	—	4	4	no	—
147	C₃×C₄²	—	2	2	no	—
148	#217	—	2	2	no	—
149	C₄⋊Dic₃	—	2	2	no	—
150	C₄⋊Dic₃	C₅×D₄	1	2	yes	—
151	C₄⋊Dic₃	C₅×D₄	1	2	yes	—
152	Dic₃⋊C₄	—	2	4	no	—
153	C₄×Dic₃	—	2	2	no	—
154	#208	—	2	2	no	—
155	#208	—	2	2	no	—
156	C₂×C₃⋊C₈	—	2	2	no	—
157	C₂³×C₅	—	2	2	no	—
158	C₂³×C₅	—	4	2	no	—
159	C₂³×C₅	—	4	2	no	—
160	C₂³×C₅	—	4	1	no	—
161	C₂³×C₅	—	4	1	no	—
162	C₂³×C₅	C₂×#67	1	1	yes	characteristic
163	C₅×Q₈	—	24	2	no	—
164	C₅×D₄	—	2	4	no	—
165	C₅×D₄	—	4	2	no	—
166	C₂×C₄×C₅	—	2	2	no	—
167	C₂×C₄×C₅	—	2	2	no	—
168	C₂×C₄×C₅	—	4	1	no	—
169	C₂×C₄×C₅	—	4	1	no	—
170	C₂×C₄×C₅	—	4	2	no	—
171	C₂×C₄×C₅	—	6	2	no	—
172	C₂×C₄×C₅	—	6	2	no	—
173	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
174	C₂×C₄×C₅	#235	1	1	yes	characteristic
175	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
176	C₂×C₄×C₅	#237	1	2	yes	—
177	C₅×C₈	—	12	2	no	—
178	C₂²×D₄	C₂×C₅×D₃	1	1	yes	characteristic
179	C₂×M₁₆	—	3	2	no	—
180	C₄⋊Q₈	—	6	2	no	—
181	#125	—	6	2	no	—
182	C₂²⋊Q₈	—	6	2	no	—
183	#117	—	3	2	no	—
184	C₂×C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
185	C₂×C₂²⋊C₄	—	2	2	no	—
186	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
187	#102	—	6	2	no	—
188	#102	—	6	2	no	—
189	C₂×C₃×C₅	—	2	1	no	—
190	C₂×C₃×C₅	—	2	1	no	—
191	C₂×C₃×C₅	—	8	2	no	—
192	C₂×C₃×C₅	#393	1	1	yes	characteristic
193	C₂×C₃×C₅	#453	1	1	yes	characteristic
194	C₂×C₃×C₅	#454	1	1	yes	characteristic
195	C₂³×C₃	—	2	2	no	—
196	C₂³×C₃	—	4	2	no	—
197	C₂³×C₃	—	4	2	no	—
198	C₂³×C₃	—	4	1	no	—
199	C₂³×C₃	—	4	1	no	—
200	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
201	C₃×D₄	—	2	4	no	—
202	C₃×D₄	—	4	2	no	—
203	C₂×C₃×C₄	—	2	2	no	—
204	C₂×C₃×C₄	—	2	2	no	—
205	C₂×C₃×C₄	—	4	1	no	—
206	C₂×C₃×C₄	—	4	1	no	—
207	C₂×C₃×C₄	—	4	2	no	—
208	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
209	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
210	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
211	C₂×C₃×C₄	C₅×#41	1	2	yes	—
212	C₂×Dic₃	—	2	2	no	—
213	C₂×Dic₃	—	2	2	no	—
214	Dic₆	—	8	2	no	—
215	C₃⋊C₈	—	4	2	no	—
216	C₂²×C₅	—	2	1	no	—
217	C₂²×C₅	—	2	1	no	—
218	C₂²×C₅	—	4	4	no	—
219	C₂²×C₅	—	4	2	no	—
220	C₂²×C₅	—	8	2	no	—
221	C₂²×C₅	—	8	1	no	—
222	C₂²×C₅	—	8	1	no	—
223	C₂²×C₅	#924	1	1	yes	characteristic
224	C₂²×C₅	#936	1	1	yes	characteristic
225	C₂²×C₅	#937	1	1	yes	characteristic
226	C₄×C₅	—	4	2	no	—
227	C₄×C₅	—	4	2	no	—
228	C₄×C₅	—	12	2	no	—
229	C₄×C₅	—	12	2	no	—
230	C₄×C₅	#878	1	2	yes	—
231	C₄×C₅	#885	1	2	yes	—
232	C₂⁴	—	2	1	no	—
233	C₂×Q₈	—	6	2	no	—
234	C₂×D₄	—	2	1	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	C₅×Dic₆	1	2	yes	complemented
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	C₅×#67	1	2	yes	—
240	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
241	M₁₆	—	6	2	no	—
242	M₁₆	—	6	2	no	—
243	C₂×C₈	—	6	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	3	2	no	—
246	C₄⋊C₄	—	3	2	no	—
247	C₄⋊C₄	—	6	2	no	—
248	C₄⋊C₄	—	6	4	no	—
249	C₂²⋊C₄	—	4	4	no	—
250	C₂²⋊C₄	—	6	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	6	2	no	—
253	C₃×C₅	#2027	1	1	yes	characteristic, complemented, Hall
254	C₂²×C₃	—	2	1	no	—
255	C₂²×C₃	—	2	1	no	—
256	C₂²×C₃	—	4	4	no	—
257	C₂²×C₃	—	4	2	no	—
258	C₂²×C₃	—	8	2	no	—
259	C₂²×C₃	—	8	1	no	—
260	C₂²×C₃	—	8	1	no	—
261	C₂²×C₃	C₅×#120	1	1	yes	characteristic
262	C₂²×C₃	C₅×C₂²⋊Q₈	1	1	yes	characteristic
263	C₂²×C₃	C₅×#124	1	1	yes	characteristic
264	C₃×C₄	—	4	2	no	—
265	C₃×C₄	—	4	2	no	—
266	C₃×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
267	C₃×C₄	C₅×#124	1	2	yes	—
268	Dic₃	—	4	2	no	—
269	Dic₃	—	4	2	no	—
270	C₂×C₅	—	2	1	no	—
271	C₂×C₅	—	2	1	no	—
272	C₂×C₅	—	8	2	no	—
273	C₂×C₅	#4947	1	1	yes	characteristic
274	C₂×C₅	#5062	1	1	yes	characteristic
275	C₂×C₅	#5077	1	1	yes	characteristic
276	C₂³	—	2	2	no	—
277	C₂³	—	4	2	no	—
278	C₂³	—	4	2	no	—
279	C₂³	—	4	1	no	—
280	C₂³	—	4	1	no	—
281	C₂³	C₂×C₅×#67	1	1	yes	characteristic
282	Q₈	—	24	2	no	—
283	D₄	—	2	4	no	—
284	D₄	—	4	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	6	2	no	—
291	C₂×C₄	—	6	2	no	—
292	C₂×C₄	C₂×C₅×Dic₆	1	1	yes	characteristic
293	C₂×C₄	C₅×#235	1	1	yes	characteristic
294	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
295	C₂×C₄	C₅×#237	1	2	yes	—
296	C₈	—	12	2	no	—
297	C₂×C₃	—	2	1	no	—
298	C₂×C₃	—	2	1	no	—
299	C₂×C₃	—	8	2	no	—
300	C₂×C₃	C₅×#393	1	1	yes	characteristic
301	C₂×C₃	C₅×#453	1	1	yes	characteristic
302	C₂×C₃	C₅×#454	1	1	yes	characteristic
303	C₅	#69643	1	1	yes	characteristic, direct product factor, Sylow
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	4	4	no	—
307	C₂²	—	4	2	no	—
308	C₂²	—	8	2	no	—
309	C₂²	—	8	1	no	—
310	C₂²	—	8	1	no	—
311	C₂²	C₅×#924	1	1	yes	characteristic
312	C₂²	C₅×#936	1	1	yes	characteristic
313	C₂²	C₅×#937	1	1	yes	characteristic
314	C₄	—	4	2	no	—
315	C₄	—	4	2	no	—
316	C₄	—	12	2	no	—
317	C₄	—	12	2	no	—
318	C₄	C₅×#878	1	2	yes	—
319	C₄	C₅×#885	1	2	yes	—
320	C₃	C₅×#2027	1	1	yes	characteristic, complemented, Sylow
321	C₂	—	2	1	no	—
322	C₂	—	2	1	no	—
323	C₂	—	8	2	no	—
324	C₂	C₅×#4947	1	1	yes	characteristic
325	C₂	C₅×#5062	1	1	yes	characteristic
326	C₂	C₅×#5077	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	#937
commutator, cosocle	C₂²×C₃×C₄	C₂³×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#390	C₂
socle	C₂²×C₃×C₅	#120
prime radical	C₂²×C₃×C₅×D₄	C₂²
hypercenter	C₅×#390	D₃
hypocenter	C₃	C₅×#2027
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#390

Complements

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

N	H ≅ G/N	type
1. C₅×#5500	C₂	semidirect
4. C₅×#4530	C₂	semidirect
5. C₅×#4523	C₂	semidirect
18. #69643	C₅	direct (N × H)
28. C₂×C₃×C₅×D₄	Q₈	semidirect
45. #5500	C₂×C₅	semidirect
48. #4530	C₂×C₅	semidirect
49. #4523	C₂×C₅	semidirect
99. C₂×C₅×D₄	Dic₆	semidirect
140. C₂×C₃×D₄	C₅×Q₈	semidirect
237. C₂×D₄	C₅×Dic₆	semidirect
253. C₃×C₅	#2027	semidirect
303. C₅	#69643	direct (N × H)
320. C₃	C₅×#2027	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5500	2
C₃×C₅×#390	2
C₅×#5215	2
C₅×#4530	2
C₅×#4523	2
C₅×#2027	3
#69643	5

Subgroups — C5×#69643 #14308908

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69643 #14308908