Subgroups — C₅×#5562 #11755947

Order 960.

480 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₅×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#940	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#938	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#930	C₂	1	2	yes	maximal, complemented
5	C₅×#929	C₂	1	2	yes	maximal, complemented
6	C₅×#895	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#890	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#884	C₂	1	2	yes	maximal, complemented
9	C₅×#883	C₂	1	2	yes	maximal, complemented
10	C₄×C₅×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#872	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#540	—	3	1	no	maximal, Hall
13	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₄×C₅	C₂²	1	2	yes	—
15	C₂×C₅×#67	C₂²	1	2	yes	—
16	C₂×C₅×#67	C₂²	1	2	yes	—
17	C₂×C₅×#67	C₂²	1	2	yes	—
18	C₂×C₅×D₁₂	C₂²	1	1	yes	characteristic
19	C₂×C₅×D₁₂	C₂²	1	2	yes	—
20	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₅×D₃	C₂²	1	2	yes	—
22	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
25	C₅×#217	C₂²	1	2	yes	—
26	C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
27	C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
28	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
29	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
30	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
31	C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
32	C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
33	C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
34	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
35	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
36	#5562	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
37	C₅×#127	—	3	1	no	—
38	C₅×#125	—	3	1	no	—
39	C₅×#123	—	3	2	no	—
40	C₅×#123	—	3	2	no	—
41	C₅×#121	—	3	1	no	—
42	C₅×#121	—	3	2	no	—
43	C₅×#121	—	3	1	no	—
44	C₅×#121	—	3	2	no	—
45	C₄×C₅×D₄	—	3	1	no	—
46	C₄×C₅×D₄	D₃	1	1	yes	characteristic, complemented
47	C₅×#117	—	3	1	no	—
48	C₂³×C₃×C₅	C₂³	1	2	yes	—
49	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
50	C₂²×C₅×D₃	C₂³	1	2	yes	—
51	C₃×C₅×D₄	—	2	2	no	—
52	C₂×C₃×C₄×C₅	—	4	2	no	—
53	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
54	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
57	C₅×#67	—	4	2	no	—
58	C₅×#67	—	4	2	no	—
59	C₅×#67	—	4	2	no	—
60	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
61	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
62	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
63	C₂×C₅×Dic₃	C₂³	1	2	yes	—
64	C₅×D₁₂	—	2	2	no	—
65	C₅×D₁₂	—	4	2	no	—
66	C₄×C₅×D₃	—	2	2	no	—
67	C₄×C₅×D₃	—	4	2	no	—
68	C₃×C₄×D₄	C₂×C₅	1	1	yes	characteristic, complemented
69	#940	C₂×C₅	1	1	yes	characteristic, complemented
70	#938	C₂×C₅	1	1	yes	characteristic, complemented
71	#930	C₂×C₅	1	2	yes	complemented
72	#929	C₂×C₅	1	2	yes	complemented
73	#895	C₂×C₅	1	1	yes	characteristic, complemented
74	#890	C₂×C₅	1	1	yes	characteristic, complemented
75	#884	C₂×C₅	1	2	yes	complemented
76	#883	C₂×C₅	1	2	yes	complemented
77	C₄×D₁₂	C₂×C₅	1	1	yes	characteristic, complemented
78	#872	C₂×C₅	1	1	yes	characteristic, complemented
79	C₂×C₅×D₄	—	3	2	no	—
80	C₂×C₅×D₄	—	3	2	no	—
81	C₂×C₅×D₄	—	3	2	no	—
82	C₂×C₅×D₄	—	3	1	no	—
83	C₂×C₅×D₄	—	3	2	no	—
84	C₂×C₅×D₄	C₂×D₃	1	1	yes	characteristic
85	C₂²×C₄×C₅	—	3	1	no	—
86	C₂²×C₄×C₅	—	3	2	no	—
87	C₂²×C₄×C₅	C₂×D₃	1	2	yes	—
88	C₅×C₄⋊C₄	—	3	1	no	—
89	C₅×C₄⋊C₄	—	3	1	no	—
90	C₅×C₄⋊C₄	—	3	1	no	—
91	C₅×C₄⋊C₄	—	3	2	no	—
92	C₅×C₄⋊C₄	—	3	2	no	—
93	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
94	C₅×C₂²⋊C₄	—	3	2	no	—
95	C₅×C₂²⋊C₄	—	3	1	no	—
96	C₅×C₂²⋊C₄	—	3	1	no	—
97	C₅×C₂²⋊C₄	—	3	2	no	—
98	C₅×C₂²⋊C₄	—	3	2	no	—
99	C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
100	C₄²×C₅	—	3	1	no	—
101	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
102	#540	—	3	1	no	Sylow
103	C₂²×C₃×C₅	—	2	2	no	—
104	C₂²×C₃×C₅	—	2	2	no	—
105	C₂²×C₃×C₅	—	2	2	no	—
106	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
107	C₂×C₅×D₃	—	2	1	no	—
108	C₂×C₅×D₃	—	2	2	no	—
109	C₂×C₅×D₃	—	2	2	no	—
110	C₂×C₅×D₃	—	2	1	no	—
111	C₂×C₅×D₃	—	2	2	no	—
112	C₂×C₅×D₃	—	2	1	no	—
113	C₃×C₄×C₅	—	2	1	no	—
114	C₃×C₄×C₅	—	2	2	no	—
115	C₃×C₄×C₅	—	2	1	no	—
116	C₃×C₄×C₅	#41	1	2	yes	—
117	C₅×Dic₃	—	2	1	no	—
118	C₅×Dic₃	—	2	2	no	—
119	C₅×Dic₃	—	2	1	no	—
120	C₅×Dic₃	—	2	1	no	—
121	C₂×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
122	C₂²×C₃×C₄	C₂²×C₅	1	2	yes	—
123	C₂×#67	C₂²×C₅	1	2	yes	—
124	C₂×#67	C₂²×C₅	1	2	yes	—
125	C₂×#67	C₂²×C₅	1	2	yes	—
126	C₂×D₁₂	C₂²×C₅	1	1	yes	characteristic
127	C₂×D₁₂	C₂²×C₅	1	2	yes	—
128	C₂×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
129	C₂×C₄×D₃	C₂²×C₅	1	2	yes	—
130	C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
131	C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
132	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
133	#217	C₂²×C₅	1	2	yes	—
134	(C₂×D₃)⋊C₄	C₂²×C₅	1	1	yes	characteristic
135	(C₂×D₃)⋊C₄	C₂²×C₅	1	1	yes	characteristic
136	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
137	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
138	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
139	C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
140	Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
141	Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
142	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
143	C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
144	C₂³×C₅	—	3	1	no	—
145	C₂³×C₅	—	3	2	no	—
146	C₂³×C₅	C₂²×D₃	1	2	yes	—
147	C₅×D₄	—	2	2	no	—
148	C₅×D₄	—	6	2	no	—
149	C₅×D₄	—	12	2	no	—
150	C₅×D₄	—	12	2	no	—
151	C₅×D₄	—	12	2	no	—
152	C₅×D₄	—	12	2	no	—
153	C₂×C₄×C₅	—	3	1	no	—
154	C₂×C₄×C₅	—	3	1	no	—
155	C₂×C₄×C₅	—	3	1	no	—
156	C₂×C₄×C₅	—	3	2	no	—
157	C₂×C₄×C₅	—	4	2	no	—
158	C₂×C₄×C₅	—	6	2	no	—
159	C₂×C₄×C₅	—	12	2	no	—
160	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
161	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
163	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
164	#127	—	3	1	no	—
165	#125	—	3	1	no	—
166	#123	—	3	2	no	—
167	#123	—	3	2	no	—
168	#121	—	3	1	no	—
169	#121	—	3	2	no	—
170	#121	—	3	1	no	—
171	#121	—	3	2	no	—
172	C₄×D₄	—	3	1	no	—
173	C₄×D₄	C₅×D₃	1	1	yes	characteristic, complemented
174	#117	—	3	1	no	—
175	C₂×C₃×C₅	—	4	2	no	—
176	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
177	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
178	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
179	C₅×D₃	—	4	1	no	—
180	C₅×D₃	—	4	2	no	—
181	C₂³×C₃	C₂³×C₅	1	2	yes	—
182	C₂²×D₃	C₂³×C₅	1	1	yes	characteristic
183	C₂²×D₃	C₂³×C₅	1	2	yes	—
184	C₃×D₄	—	2	2	no	—
185	C₂×C₃×C₄	—	4	2	no	—
186	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
187	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
189	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
190	#67	—	4	2	no	—
191	#67	—	4	2	no	—
192	#67	—	4	2	no	—
193	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
194	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
195	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
196	C₂×Dic₃	C₂³×C₅	1	2	yes	—
197	D₁₂	—	2	2	no	—
198	D₁₂	—	4	2	no	—
199	C₄×D₃	—	2	2	no	—
200	C₄×D₃	—	4	2	no	—
201	C₂²×C₅	—	2	2	no	—
202	C₂²×C₅	—	2	2	no	—
203	C₂²×C₅	—	2	2	no	—
204	C₂²×C₅	—	6	1	no	—
205	C₂²×C₅	—	6	2	no	—
206	C₂²×C₅	—	6	1	no	—
207	C₂²×C₅	—	6	2	no	—
208	C₂²×C₅	—	6	2	no	—
209	C₂²×C₅	—	6	1	no	—
210	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
211	C₄×C₅	—	2	1	no	—
212	C₄×C₅	—	2	2	no	—
213	C₄×C₅	—	2	1	no	—
214	C₄×C₅	—	6	1	no	—
215	C₄×C₅	—	6	1	no	—
216	C₄×C₅	—	6	2	no	—
217	C₄×C₅	—	6	1	no	—
218	C₄×C₅	#235	1	2	yes	—
219	C₂×D₄	—	3	2	no	—
220	C₂×D₄	—	3	2	no	—
221	C₂×D₄	—	3	2	no	—
222	C₂×D₄	—	3	1	no	—
223	C₂×D₄	—	3	2	no	—
224	C₂×D₄	C₂×C₅×D₃	1	1	yes	characteristic
225	C₂²×C₄	—	3	1	no	—
226	C₂²×C₄	—	3	2	no	—
227	C₂²×C₄	C₂×C₅×D₃	1	2	yes	—
228	C₄⋊C₄	—	3	1	no	—
229	C₄⋊C₄	—	3	1	no	—
230	C₄⋊C₄	—	3	1	no	—
231	C₄⋊C₄	—	3	2	no	—
232	C₄⋊C₄	—	3	2	no	—
233	C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
234	C₂²⋊C₄	—	3	2	no	—
235	C₂²⋊C₄	—	3	1	no	—
236	C₂²⋊C₄	—	3	1	no	—
237	C₂²⋊C₄	—	3	2	no	—
238	C₂²⋊C₄	—	3	2	no	—
239	C₂²⋊C₄	C₂×C₅×D₃	1	2	yes	—
240	C₄²	—	3	1	no	—
241	C₄²	C₂×C₅×D₃	1	1	yes	characteristic
242	C₃×C₅	#540	1	1	yes	characteristic, complemented, Hall
243	C₂²×C₃	—	2	2	no	—
244	C₂²×C₃	—	2	2	no	—
245	C₂²×C₃	—	2	2	no	—
246	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
247	C₂×D₃	—	2	1	no	—
248	C₂×D₃	—	2	2	no	—
249	C₂×D₃	—	2	2	no	—
250	C₂×D₃	—	2	1	no	—
251	C₂×D₃	—	2	2	no	—
252	C₂×D₃	—	2	1	no	—
253	C₃×C₄	—	2	1	no	—
254	C₃×C₄	—	2	2	no	—
255	C₃×C₄	—	2	1	no	—
256	C₃×C₄	C₅×#41	1	2	yes	—
257	Dic₃	—	2	1	no	—
258	Dic₃	—	2	2	no	—
259	Dic₃	—	2	1	no	—
260	Dic₃	—	2	1	no	—
261	C₂×C₅	—	4	2	no	—
262	C₂×C₅	—	12	1	no	—
263	C₂×C₅	—	12	2	no	—
264	C₂×C₅	C₂×#235	1	1	yes	characteristic
265	C₂×C₅	#1004	1	1	yes	characteristic
266	C₂×C₅	#1009	1	1	yes	characteristic
267	C₂³	—	3	1	no	—
268	C₂³	—	3	2	no	—
269	C₂³	C₂²×C₅×D₃	1	2	yes	—
270	D₄	—	2	2	no	—
271	D₄	—	6	2	no	—
272	D₄	—	12	2	no	—
273	D₄	—	12	2	no	—
274	D₄	—	12	2	no	—
275	D₄	—	12	2	no	—
276	C₂×C₄	—	3	1	no	—
277	C₂×C₄	—	3	1	no	—
278	C₂×C₄	—	3	1	no	—
279	C₂×C₄	—	3	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	6	2	no	—
282	C₂×C₄	—	12	2	no	—
283	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
284	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
285	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
286	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
287	C₂×C₃	—	4	2	no	—
288	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
289	C₂×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
290	C₂×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
291	D₃	—	4	1	no	—
292	D₃	—	4	2	no	—
293	C₅	#5562	1	1	yes	characteristic, direct product factor, Sylow
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	6	1	no	—
298	C₂²	—	6	2	no	—
299	C₂²	—	6	1	no	—
300	C₂²	—	6	2	no	—
301	C₂²	—	6	2	no	—
302	C₂²	—	6	1	no	—
303	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	1	no	—
307	C₄	—	6	1	no	—
308	C₄	—	6	1	no	—
309	C₄	—	6	2	no	—
310	C₄	—	6	1	no	—
311	C₄	C₅×#235	1	2	yes	—
312	C₃	C₅×#540	1	1	yes	characteristic, complemented, Sylow
313	C₂	—	4	2	no	—
314	C₂	—	12	1	no	—
315	C₂	—	12	2	no	—
316	C₂	C₂×C₅×#235	1	1	yes	characteristic
317	C₂	C₅×#1004	1	1	yes	characteristic
318	C₂	C₅×#1009	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×C₄×C₅×D₄	C₂
socle	C₂²×C₃×C₅	C₂⁴
hypercenter	C₄×C₅×D₄	D₃
hypocenter	C₃	C₅×#540
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄×C₅×D₄

Complements

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

N	H ≅ G/N	type
1. C₃×C₄×C₅×D₄	C₂	semidirect
2. C₅×#940	C₂	semidirect
3. C₅×#938	C₂	semidirect
4. C₅×#930	C₂	semidirect
5. C₅×#929	C₂	semidirect
6. C₅×#895	C₂	semidirect
7. C₅×#890	C₂	semidirect
8. C₅×#884	C₂	semidirect
9. C₅×#883	C₂	semidirect
10. C₄×C₅×D₁₂	C₂	semidirect
11. C₅×#872	C₂	semidirect
36. #5562	C₅	direct (N × H)
46. C₄×C₅×D₄	D₃	semidirect
68. C₃×C₄×D₄	C₂×C₅	semidirect
69. #940	C₂×C₅	semidirect
70. #938	C₂×C₅	semidirect
71. #930	C₂×C₅	semidirect
72. #929	C₂×C₅	semidirect
73. #895	C₂×C₅	semidirect
74. #890	C₂×C₅	semidirect
75. #884	C₂×C₅	semidirect
76. #883	C₂×C₅	semidirect
77. C₄×D₁₂	C₂×C₅	semidirect
78. #872	C₂×C₅	semidirect
173. C₄×D₄	C₅×D₃	semidirect
242. C₃×C₅	#540	semidirect
293. C₅	#5562	direct (N × H)
312. C₃	C₅×#540	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₅×D₄	2
C₅×#940	2
C₅×#938	2
C₅×#930	2
C₅×#929	2
C₅×#895	2
C₅×#890	2
C₅×#884	2
C₅×#883	2
C₄×C₅×D₁₂	2
C₅×#872	2
C₅×#540	3
#5562	5

Subgroups — C5×#5562 #11755947

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5562 #11755947