Subgroups — C₅×#5608 #11755993

Order 960.

588 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#237	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#235	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#121	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#939	C₂	1	2	yes	maximal, complemented
5	C₅×#938	C₂	1	2	yes	maximal, complemented
6	C₅×#938	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#924	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#897	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×D₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#884	C₂	1	2	yes	maximal, complemented
11	C₅×#878	C₂	1	2	yes	maximal, complemented
12	C₅×#537	—	3	1	no	maximal, Hall
13	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₅×D₄	C₂²	1	2	yes	—
15	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
16	C₂×C₅×#67	C₂²	1	2	yes	—
17	C₂×C₅×#67	C₂²	1	1	yes	characteristic
18	C₂×C₅×#67	C₂²	1	1	yes	characteristic
19	C₂×C₅×#67	C₂²	1	2	yes	—
20	C₂²×C₅×Dic₃	C₂²	1	2	yes	—
21	C₅×#237	—	2	4	no	—
22	C₅×#235	—	2	4	no	—
23	C₂×C₅×D₁₂	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₅×D₃	C₂²	1	2	yes	—
26	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
27	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
28	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
29	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
30	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
31	C₅×#217	C₂²	1	2	yes	—
32	C₅×#217	C₂²	1	2	yes	—
33	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
34	C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
35	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
36	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
37	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
38	#5608	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
39	C₂×C₅×#41	—	3	1	no	—
40	C₂×C₅×#41	—	3	1	no	—
41	C₅×C₂²⋊Q₈	—	3	1	no	—
42	C₅×C₂²⋊Q₈	—	3	1	no	—
43	C₅×C₂²⋊Q₈	—	3	2	no	—
44	C₅×#121	—	3	1	no	—
45	C₅×#121	—	3	2	no	—
46	C₅×#121	—	3	2	no	—
47	C₅×#121	—	3	2	no	—
48	C₅×#121	D₃	1	1	yes	characteristic, complemented
49	C₂×C₅×C₄⋊C₄	—	3	1	no	—
50	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
51	C₂³×C₃×C₅	C₂³	1	2	yes	—
52	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
53	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
54	C₃×C₅×D₄	—	2	2	no	—
55	C₃×C₅×D₄	—	4	2	no	—
56	C₂×C₃×C₄×C₅	—	2	2	no	—
57	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
59	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
60	C₅×#67	—	2	2	no	—
61	C₅×#67	—	2	4	no	—
62	C₅×#67	—	2	2	no	—
63	C₅×#67	—	4	2	no	—
64	C₂×C₅×Dic₃	—	2	4	no	—
65	C₂×C₅×Dic₃	C₂³	1	2	yes	—
66	C₂×C₅×Dic₃	C₂³	1	2	yes	—
67	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
68	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
69	C₅×D₁₂	—	2	2	no	—
70	C₄×C₅×D₃	—	2	2	no	—
71	C₄×C₅×D₃	—	4	2	no	—
72	C₄×C₅×D₃	D₄	1	4	yes	complemented
73	C₅×Dic₆	—	2	2	no	—
74	C₅×Dic₆	—	2	2	no	—
75	C₂×#237	C₂×C₅	1	1	yes	characteristic, complemented
76	C₂×#235	C₂×C₅	1	1	yes	characteristic, complemented
77	C₃×#121	C₂×C₅	1	1	yes	characteristic, complemented
78	#939	C₂×C₅	1	2	yes	complemented
79	#938	C₂×C₅	1	2	yes	complemented
80	#938	C₂×C₅	1	1	yes	characteristic, complemented
81	#924	C₂×C₅	1	1	yes	characteristic, complemented
82	#897	C₂×C₅	1	1	yes	characteristic, complemented
83	D₃×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
84	#884	C₂×C₅	1	2	yes	complemented
85	#878	C₂×C₅	1	2	yes	complemented
86	C₅×#41	—	6	4	no	—
87	C₅×#41	—	6	4	no	—
88	C₂×C₅×Q₈	—	3	1	no	—
89	C₂×C₅×Q₈	—	3	1	no	—
90	C₂×C₅×D₄	—	3	1	no	—
91	C₂×C₅×D₄	—	3	1	no	—
92	C₂×C₅×D₄	—	3	2	no	—
93	C₂×C₅×D₄	—	3	2	no	—
94	C₂×C₅×D₄	—	3	1	no	—
95	C₂×C₅×D₄	C₂×D₃	1	1	yes	characteristic
96	C₂×C₅×D₄	C₂×D₃	1	2	yes	—
97	C₂²×C₄×C₅	—	3	1	no	—
98	C₂²×C₄×C₅	—	3	2	no	—
99	C₂²×C₄×C₅	—	3	1	no	—
100	C₂²×C₄×C₅	—	3	2	no	—
101	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
102	C₅×C₄⋊C₄	—	3	2	no	—
103	C₅×C₄⋊C₄	—	3	1	no	—
104	C₅×C₄⋊C₄	—	3	2	no	—
105	C₅×C₄⋊C₄	—	3	2	no	—
106	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
107	C₅×C₂²⋊C₄	—	3	2	no	—
108	C₅×C₂²⋊C₄	—	3	2	no	—
109	C₅×C₂²⋊C₄	—	3	2	no	—
110	C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
111	#537	—	3	1	no	Sylow
112	C₂²×C₃×C₅	—	2	2	no	—
113	C₂²×C₃×C₅	—	2	1	no	—
114	C₂²×C₃×C₅	—	2	2	no	—
115	C₂²×C₃×C₅	—	2	1	no	—
116	C₂²×C₃×C₅	—	2	1	no	—
117	C₂²×C₃×C₅	—	2	2	no	—
118	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
119	C₂×C₅×D₃	—	2	1	no	—
120	C₂×C₅×D₃	—	2	1	no	—
121	C₂×C₅×D₃	—	2	1	no	—
122	C₂×C₅×D₃	—	2	1	no	—
123	C₂×C₅×D₃	—	2	1	no	—
124	C₂×C₅×D₃	C₂×D₄	1	2	yes	—
125	C₃×C₄×C₅	—	2	1	no	—
126	C₃×C₄×C₅	—	2	2	no	—
127	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
128	C₅×Dic₃	—	2	1	no	—
129	C₅×Dic₃	—	2	2	no	—
130	C₅×Dic₃	—	2	2	no	—
131	C₅×Dic₃	C₂×D₄	1	2	yes	—
132	C₂×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
133	C₂×C₃×D₄	C₂²×C₅	1	2	yes	—
134	C₂²×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
135	C₂×#67	C₂²×C₅	1	2	yes	—
136	C₂×#67	C₂²×C₅	1	1	yes	characteristic
137	C₂×#67	C₂²×C₅	1	1	yes	characteristic
138	C₂×#67	C₂²×C₅	1	2	yes	—
139	C₂²×Dic₃	C₂²×C₅	1	2	yes	—
140	#237	—	2	4	no	—
141	#235	—	2	4	no	—
142	C₂×D₁₂	C₂²×C₅	1	1	yes	characteristic
143	C₂×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
144	C₂×C₄×D₃	C₂²×C₅	1	2	yes	—
145	C₂×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
146	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic
147	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic
148	C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
149	C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
150	#217	C₂²×C₅	1	2	yes	—
151	#217	C₂²×C₅	1	2	yes	—
152	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
153	C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
154	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
155	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
156	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
157	C₂³×C₅	—	3	1	no	—
158	C₂³×C₅	—	3	1	no	—
159	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
160	C₂³×C₅	C₂²×D₃	1	2	yes	—
161	C₅×Q₈	—	6	2	no	—
162	C₅×Q₈	—	6	2	no	—
163	C₅×D₄	—	2	2	no	—
164	C₅×D₄	—	4	2	no	—
165	C₅×D₄	—	6	2	no	—
166	C₅×D₄	—	6	2	no	—
167	C₅×D₄	—	6	2	no	—
168	C₅×D₄	—	6	4	no	—
169	C₅×D₄	—	12	2	no	—
170	C₂×C₄×C₅	—	2	2	no	—
171	C₂×C₄×C₅	—	3	1	no	—
172	C₂×C₄×C₅	—	3	4	no	—
173	C₂×C₄×C₅	—	3	2	no	—
174	C₂×C₄×C₅	—	3	1	no	—
175	C₂×C₄×C₅	—	3	2	no	—
176	C₂×C₄×C₅	—	6	4	no	—
177	C₂×C₄×C₅	—	6	2	no	—
178	C₂×C₄×C₅	—	12	2	no	—
179	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
180	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
182	C₂×#41	—	3	1	no	—
183	C₂×#41	—	3	1	no	—
184	C₂²⋊Q₈	—	3	1	no	—
185	C₂²⋊Q₈	—	3	1	no	—
186	C₂²⋊Q₈	—	3	2	no	—
187	#121	—	3	1	no	—
188	#121	—	3	2	no	—
189	#121	—	3	2	no	—
190	#121	—	3	2	no	—
191	#121	C₅×D₃	1	1	yes	characteristic, complemented
192	C₂×C₄⋊C₄	—	3	1	no	—
193	C₂×C₃×C₅	—	4	2	no	—
194	C₂×C₃×C₅	—	4	1	no	—
195	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
196	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
197	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
198	C₅×D₃	—	2	2	no	—
199	C₅×D₃	—	4	1	no	—
200	C₂³×C₃	C₂³×C₅	1	1	yes	characteristic
201	C₂³×C₃	C₂³×C₅	1	2	yes	—
202	C₂²×D₃	C₂³×C₅	1	1	yes	characteristic
203	C₂²×D₃	C₂³×C₅	1	1	yes	characteristic
204	C₃×D₄	—	2	2	no	—
205	C₃×D₄	—	4	2	no	—
206	C₂×C₃×C₄	—	2	2	no	—
207	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
209	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
210	#67	—	2	2	no	—
211	#67	—	2	4	no	—
212	#67	—	2	2	no	—
213	#67	—	4	2	no	—
214	C₂×Dic₃	—	2	4	no	—
215	C₂×Dic₃	C₂³×C₅	1	2	yes	—
216	C₂×Dic₃	C₂³×C₅	1	2	yes	—
217	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
218	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
219	D₁₂	—	2	2	no	—
220	C₄×D₃	—	2	2	no	—
221	C₄×D₃	—	4	2	no	—
222	C₄×D₃	C₅×D₄	1	4	yes	complemented
223	Dic₆	—	2	2	no	—
224	Dic₆	—	2	2	no	—
225	C₂²×C₅	—	2	1	no	—
226	C₂²×C₅	—	2	2	no	—
227	C₂²×C₅	—	2	1	no	—
228	C₂²×C₅	—	2	2	no	—
229	C₂²×C₅	—	2	1	no	—
230	C₂²×C₅	—	2	2	no	—
231	C₂²×C₅	—	3	2	no	—
232	C₂²×C₅	—	6	1	no	—
233	C₂²×C₅	—	6	1	no	—
234	C₂²×C₅	—	6	1	no	—
235	C₂²×C₅	—	6	1	no	—
236	C₂²×C₅	—	6	1	no	—
237	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
238	C₄×C₅	—	2	1	no	—
239	C₄×C₅	—	2	2	no	—
240	C₄×C₅	—	3	2	no	—
241	C₄×C₅	—	6	1	no	—
242	C₄×C₅	—	6	2	no	—
243	C₄×C₅	—	6	2	no	—
244	C₄×C₅	D₃×D₄	1	2	yes	—
245	#41	—	6	4	no	—
246	#41	—	6	4	no	—
247	C₂×Q₈	—	3	1	no	—
248	C₂×Q₈	—	3	1	no	—
249	C₂×D₄	—	3	1	no	—
250	C₂×D₄	—	3	1	no	—
251	C₂×D₄	—	3	2	no	—
252	C₂×D₄	—	3	2	no	—
253	C₂×D₄	—	3	1	no	—
254	C₂×D₄	C₂×C₅×D₃	1	1	yes	characteristic
255	C₂×D₄	C₂×C₅×D₃	1	2	yes	—
256	C₂²×C₄	—	3	1	no	—
257	C₂²×C₄	—	3	2	no	—
258	C₂²×C₄	—	3	1	no	—
259	C₂²×C₄	—	3	2	no	—
260	C₂²×C₄	C₂×C₅×D₃	1	1	yes	characteristic
261	C₄⋊C₄	—	3	2	no	—
262	C₄⋊C₄	—	3	1	no	—
263	C₄⋊C₄	—	3	2	no	—
264	C₄⋊C₄	—	3	2	no	—
265	C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
266	C₂²⋊C₄	—	3	2	no	—
267	C₂²⋊C₄	—	3	2	no	—
268	C₂²⋊C₄	—	3	2	no	—
269	C₂²⋊C₄	C₂×C₅×D₃	1	2	yes	—
270	C₃×C₅	#537	1	1	yes	characteristic, complemented, Hall
271	C₂²×C₃	—	2	2	no	—
272	C₂²×C₃	—	2	1	no	—
273	C₂²×C₃	—	2	2	no	—
274	C₂²×C₃	—	2	1	no	—
275	C₂²×C₃	—	2	1	no	—
276	C₂²×C₃	—	2	2	no	—
277	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
278	C₂×D₃	—	2	1	no	—
279	C₂×D₃	—	2	1	no	—
280	C₂×D₃	—	2	1	no	—
281	C₂×D₃	—	2	1	no	—
282	C₂×D₃	—	2	1	no	—
283	C₂×D₃	C₂×C₅×D₄	1	2	yes	—
284	C₃×C₄	—	2	1	no	—
285	C₃×C₄	—	2	2	no	—
286	C₃×C₄	C₂×C₅×D₄	1	2	yes	—
287	Dic₃	—	2	1	no	—
288	Dic₃	—	2	2	no	—
289	Dic₃	—	2	2	no	—
290	Dic₃	C₂×C₅×D₄	1	2	yes	—
291	C₂×C₅	—	4	1	no	—
292	C₂×C₅	—	4	2	no	—
293	C₂×C₅	—	6	2	no	—
294	C₂×C₅	—	12	1	no	—
295	C₂×C₅	C₂×D₃×D₄	1	1	yes	characteristic
296	C₂×C₅	#1004	1	1	yes	characteristic
297	C₂×C₅	#1010	1	1	yes	characteristic
298	C₂³	—	3	1	no	—
299	C₂³	—	3	1	no	—
300	C₂³	C₂²×C₅×D₃	1	1	yes	characteristic
301	C₂³	C₂²×C₅×D₃	1	2	yes	—
302	Q₈	—	6	2	no	—
303	Q₈	—	6	2	no	—
304	D₄	—	2	2	no	—
305	D₄	—	4	2	no	—
306	D₄	—	6	2	no	—
307	D₄	—	6	2	no	—
308	D₄	—	6	2	no	—
309	D₄	—	6	4	no	—
310	D₄	—	12	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	3	1	no	—
313	C₂×C₄	—	3	4	no	—
314	C₂×C₄	—	3	2	no	—
315	C₂×C₄	—	3	1	no	—
316	C₂×C₄	—	3	2	no	—
317	C₂×C₄	—	6	4	no	—
318	C₂×C₄	—	6	2	no	—
319	C₂×C₄	—	12	2	no	—
320	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
321	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
322	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
323	C₂×C₃	—	4	2	no	—
324	C₂×C₃	—	4	1	no	—
325	C₂×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
326	C₂×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
327	C₂×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
328	D₃	—	2	2	no	—
329	D₃	—	4	1	no	—
330	C₅	#5608	1	1	yes	characteristic, direct product factor, Sylow
331	C₂²	—	2	1	no	—
332	C₂²	—	2	2	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	2	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	3	2	no	—
338	C₂²	—	6	1	no	—
339	C₂²	—	6	1	no	—
340	C₂²	—	6	1	no	—
341	C₂²	—	6	1	no	—
342	C₂²	—	6	1	no	—
343	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	2	2	no	—
346	C₄	—	3	2	no	—
347	C₄	—	6	2	no	—
348	C₄	—	6	1	no	—
349	C₄	—	6	2	no	—
350	C₄	C₅×D₃×D₄	1	2	yes	—
351	C₃	C₅×#537	1	1	yes	characteristic, complemented, Sylow
352	C₂	—	4	1	no	—
353	C₂	—	4	2	no	—
354	C₂	—	6	2	no	—
355	C₂	—	12	1	no	—
356	C₂	C₂×C₅×D₃×D₄	1	1	yes	characteristic
357	C₂	C₅×#1004	1	1	yes	characteristic
358	C₂	C₅×#1010	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₃
Fitting	C₃×C₅×#121	C₂
socle	C₂²×C₃×C₅	C₂⁴
hypercenter	C₅×#121	D₃
hypocenter	C₃	C₅×#537
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×#121

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#237	C₂	semidirect
2. C₂×C₅×#235	C₂	semidirect
3. C₃×C₅×#121	C₂	semidirect
4. C₅×#939	C₂	semidirect
5. C₅×#938	C₂	semidirect
6. C₅×#938	C₂	semidirect
7. C₅×#924	C₂	semidirect
8. C₅×#897	C₂	semidirect
9. C₅×D₃×C₄⋊C₄	C₂	semidirect
10. C₅×#884	C₂	semidirect
11. C₅×#878	C₂	semidirect
38. #5608	C₅	direct (N × H)
48. C₅×#121	D₃	semidirect
72. C₄×C₅×D₃	D₄	semidirect
75. C₂×#237	C₂×C₅	semidirect
76. C₂×#235	C₂×C₅	semidirect
77. C₃×#121	C₂×C₅	semidirect
78. #939	C₂×C₅	semidirect
79. #938	C₂×C₅	semidirect
80. #938	C₂×C₅	semidirect
81. #924	C₂×C₅	semidirect
82. #897	C₂×C₅	semidirect
83. D₃×C₄⋊C₄	C₂×C₅	semidirect
84. #884	C₂×C₅	semidirect
85. #878	C₂×C₅	semidirect
191. #121	C₅×D₃	semidirect
222. C₄×D₃	C₅×D₄	semidirect
270. C₃×C₅	#537	semidirect
330. C₅	#5608	direct (N × H)
351. C₃	C₅×#537	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#237	2
C₂×C₅×#235	2
C₃×C₅×#121	2
C₅×#939	2
C₅×#938	2
C₅×#924	2
C₅×#897	2
C₅×D₃×C₄⋊C₄	2
C₅×#884	2
C₅×#878	2
C₅×#537	3
#5608	5

Subgroups — C5×#5608 #11755993

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5608 #11755993