Subgroups — #11752709

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	C₄×D₆₀	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#872	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
4	#91461	C₂	1	2	yes	maximal, complemented
5	#91460	C₂	1	1	yes	maximal, characteristic, complemented
6	#91458	C₂	1	1	yes	maximal, characteristic, complemented
7	#91456	C₂	1	1	yes	maximal, characteristic, complemented
8	#91448	C₂	1	2	yes	maximal, complemented
9	#91381	C₂	1	1	yes	maximal, characteristic, complemented
10	#91354	C₂	1	2	yes	maximal, complemented
11	#91348	C₂	1	2	yes	maximal, complemented
12	#65869	—	3	1	no	maximal, Hall
13	C₂×D₆₀	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₁₅	C₂²	1	2	yes	—
15	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
16	C₂×C₃×D₂₀	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
18	C₂×#1203	C₂²	1	2	yes	—
19	C₂×#1202	C₂²	1	2	yes	—
20	C₂×#1202	C₂²	1	2	yes	—
21	C₂×#1201	C₂²	1	2	yes	—
22	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
23	#6714	C₂²	1	2	yes	—
24	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
25	C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
26	C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
27	C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
28	C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
29	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
30	C₃×#685	C₂²	1	2	yes	—
31	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
32	#6669	C₂²	1	2	yes	—
33	#6668	C₂²	1	2	yes	—
34	#6667	C₂²	1	2	yes	—
35	#6665	C₂²	1	2	yes	—
36	#5562	—	5	1	no	maximal, Hall
37	C₅×#117	—	3	1	no	—
38	#4076	—	3	1	no	—
39	#4076	—	3	1	no	—
40	#4040	—	3	2	no	—
41	#4040	—	3	2	no	—
42	#4030	—	3	2	no	—
43	#4030	—	3	2	no	—
44	#4020	—	3	1	no	—
45	C₄×D₂₀	—	3	1	no	—
46	C₄×D₂₀	D₃	1	1	yes	characteristic, complemented
47	#4016	—	3	1	no	—
48	C₂²×D₁₅	C₂³	1	2	yes	—
49	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
50	C₂²×C₃×D₅	C₂³	1	2	yes	—
51	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
54	C₂×Dic₁₅	C₂³	1	2	yes	—
55	D₆₀	—	2	2	no	—
56	C₄×D₁₅	—	4	2	no	—
57	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
58	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
59	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
60	C₄×C₅×D₃	—	2	2	no	—
61	C₂×C₃×Dic₅	C₂³	1	2	yes	—
62	C₃×D₂₀	—	2	2	no	—
63	C₃×C₄×D₅	—	4	2	no	—
64	#1203	—	4	2	no	—
65	#1202	—	4	2	no	—
66	#1202	—	4	2	no	—
67	#1201	—	4	2	no	—
68	C₃×C₄×D₄	—	5	1	no	—
69	#940	—	5	1	no	—
70	#938	—	5	1	no	—
71	#930	—	5	2	no	—
72	#929	—	5	2	no	—
73	#895	—	5	1	no	—
74	#890	—	5	1	no	—
75	#884	—	5	2	no	—
76	#883	—	5	2	no	—
77	C₄×D₁₂	—	5	1	no	—
78	#872	D₅	1	1	yes	characteristic, complemented
79	C₂²×C₄×C₅	—	3	1	no	—
80	C₂×C₂²⋊D₅	—	3	2	no	—
81	C₂×C₂²⋊D₅	—	3	2	no	—
82	C₂×D₂₀	—	3	1	no	—
83	C₂×D₂₀	—	3	2	no	—
84	C₂×D₂₀	—	3	2	no	—
85	C₂×D₂₀	C₂×D₃	1	1	yes	characteristic
86	C₂×C₄×D₅	—	3	2	no	—
87	C₂×C₄×D₅	C₂×D₃	1	2	yes	—
88	C₅×C₄⋊C₄	—	3	1	no	—
89	C₅×C₄⋊C₄	—	3	1	no	—
90	C₅×C₂²⋊C₄	—	3	1	no	—
91	C₅×C₂²⋊C₄	—	3	1	no	—
92	C₄²×C₅	—	3	1	no	—
93	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
94	#685	—	3	2	no	—
95	#685	—	3	2	no	—
96	#685	—	3	2	no	—
97	#685	C₂×D₃	1	2	yes	—
98	C₄⋊Dic₅	—	3	1	no	—
99	C₄⋊Dic₅	C₂×D₃	1	1	yes	characteristic
100	#683	—	3	2	no	—
101	#683	—	3	2	no	—
102	#540	—	15	1	no	Sylow
103	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
104	C₂×D₁₅	—	2	2	no	—
105	C₂×D₁₅	—	2	2	no	—
106	C₂×D₁₅	—	2	2	no	—
107	C₂×C₅×D₃	—	2	1	no	—
108	C₂×C₅×D₃	—	2	1	no	—
109	C₂×C₅×D₃	—	2	1	no	—
110	C₂×C₃×D₅	—	2	2	no	—
111	C₂×C₃×D₅	—	2	2	no	—
112	C₂×C₃×D₅	—	2	2	no	—
113	C₃×C₄×C₅	—	2	1	no	—
114	C₃×C₄×C₅	—	2	1	no	—
115	C₃×C₄×C₅	#41	1	2	yes	—
116	Dic₁₅	—	2	2	no	—
117	C₃×Dic₅	—	2	2	no	—
118	C₅×Dic₃	—	2	1	no	—
119	C₅×Dic₃	—	2	1	no	—
120	C₅×Dic₃	—	2	1	no	—
121	C₂×C₃×D₄	—	5	1	no	—
122	C₂²×C₃×C₄	—	5	2	no	—
123	C₂×#67	—	5	2	no	—
124	C₂×#67	—	5	2	no	—
125	C₂×#67	—	5	2	no	—
126	C₂×D₁₂	—	5	1	no	—
127	C₂×D₁₂	—	5	2	no	—
128	C₂×C₄×D₃	—	5	2	no	—
129	C₂×C₄×D₃	C₂×D₅	1	1	yes	characteristic
130	C₃×C₄⋊C₄	—	5	1	no	—
131	C₃×C₂²⋊C₄	—	5	2	no	—
132	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
133	#217	—	5	2	no	—
134	(C₂×D₃)⋊C₄	—	5	2	no	—
135	(C₂×D₃)⋊C₄	—	5	2	no	—
136	(C₂×D₃)⋊C₄	C₂×D₅	1	1	yes	characteristic
137	(C₂×D₃)⋊C₄	C₂×D₅	1	1	yes	characteristic
138	C₄⋊Dic₃	—	5	1	no	—
139	C₄⋊Dic₃	—	5	2	no	—
140	Dic₃⋊C₄	—	5	2	no	—
141	Dic₃⋊C₄	C₂×D₅	1	1	yes	characteristic
142	Dic₃⋊C₄	C₂×D₅	1	1	yes	characteristic
143	C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
144	C₂³×C₅	—	3	1	no	—
145	C₂²×D₅	—	3	2	no	—
146	C₂²×D₅	C₂²×D₃	1	2	yes	—
147	C₂×C₄×C₅	—	3	1	no	—
148	C₂×C₄×C₅	—	3	1	no	—
149	C₂×C₄×C₅	—	3	1	no	—
150	C₂×C₄×C₅	—	6	2	no	—
151	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
154	C₂²⋊D₅	—	12	2	no	—
155	C₂²⋊D₅	—	12	2	no	—
156	C₂×Dic₅	—	3	2	no	—
157	C₂×Dic₅	C₂²×D₃	1	2	yes	—
158	D₂₀	—	2	2	no	—
159	D₂₀	—	6	2	no	—
160	D₂₀	—	12	2	no	—
161	D₂₀	—	12	2	no	—
162	C₄×D₅	—	4	2	no	—
163	C₄×D₅	—	12	2	no	—
164	#127	—	15	1	no	—
165	#125	—	15	1	no	—
166	#123	—	15	2	no	—
167	#123	—	15	2	no	—
168	#121	—	15	2	no	—
169	#121	—	15	2	no	—
170	#121	—	15	1	no	—
171	#121	—	15	1	no	—
172	C₄×D₄	—	5	1	no	—
173	C₄×D₄	—	15	1	no	—
174	#117	—	3	1	no	—
175	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
176	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
177	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
178	D₁₅	—	4	2	no	—
179	C₃×D₅	—	4	2	no	—
180	C₅×D₃	—	4	1	no	—
181	C₂³×C₃	—	5	2	no	—
182	C₂²×D₃	—	5	2	no	—
183	C₂²×D₃	C₂²×D₅	1	1	yes	characteristic
184	C₃×D₄	—	10	2	no	—
185	C₂×C₃×C₄	—	5	2	no	—
186	C₂×C₃×C₄	—	20	2	no	—
187	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
189	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
190	#67	—	20	2	no	—
191	#67	—	20	2	no	—
192	#67	—	20	2	no	—
193	C₂×Dic₃	—	5	2	no	—
194	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
195	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
196	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
197	D₁₂	—	10	2	no	—
198	D₁₂	—	20	2	no	—
199	C₄×D₃	—	2	2	no	—
200	C₄×D₃	—	20	2	no	—
201	C₂²×C₅	—	6	1	no	—
202	C₂²×C₅	—	6	1	no	—
203	C₂²×C₅	—	6	1	no	—
204	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
205	C₂×D₅	—	2	2	no	—
206	C₂×D₅	—	2	2	no	—
207	C₂×D₅	—	2	2	no	—
208	C₂×D₅	—	6	2	no	—
209	C₂×D₅	—	6	2	no	—
210	C₂×D₅	—	6	2	no	—
211	C₄×C₅	—	2	1	no	—
212	C₄×C₅	—	2	1	no	—
213	C₄×C₅	—	6	1	no	—
214	C₄×C₅	—	6	1	no	—
215	C₄×C₅	—	6	1	no	—
216	C₄×C₅	#235	1	2	yes	—
217	Dic₅	—	2	2	no	—
218	Dic₅	—	6	2	no	—
219	C₂×D₄	—	5	1	no	—
220	C₂×D₄	—	15	2	no	—
221	C₂×D₄	—	15	2	no	—
222	C₂×D₄	—	15	2	no	—
223	C₂×D₄	—	15	2	no	—
224	C₂×D₄	—	15	1	no	—
225	C₂²×C₄	—	3	1	no	—
226	C₂²×C₄	—	5	2	no	—
227	C₂²×C₄	—	15	2	no	—
228	C₄⋊C₄	—	3	1	no	—
229	C₄⋊C₄	—	3	1	no	—
230	C₄⋊C₄	—	5	1	no	—
231	C₄⋊C₄	—	15	2	no	—
232	C₄⋊C₄	—	15	2	no	—
233	C₄⋊C₄	—	15	1	no	—
234	C₂²⋊C₄	—	3	1	no	—
235	C₂²⋊C₄	—	3	1	no	—
236	C₂²⋊C₄	—	5	2	no	—
237	C₂²⋊C₄	—	15	2	no	—
238	C₂²⋊C₄	—	15	2	no	—
239	C₂²⋊C₄	—	15	2	no	—
240	C₄²	—	3	1	no	—
241	C₄²	D₃×D₅	1	1	yes	characteristic
242	C₃×C₅	#540	1	1	yes	characteristic, complemented, Hall
243	C₂²×C₃	—	10	2	no	—
244	C₂²×C₃	—	10	2	no	—
245	C₂²×C₃	—	10	2	no	—
246	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
247	C₂×D₃	—	2	1	no	—
248	C₂×D₃	—	2	1	no	—
249	C₂×D₃	—	2	1	no	—
250	C₂×D₃	—	10	2	no	—
251	C₂×D₃	—	10	2	no	—
252	C₂×D₃	—	10	2	no	—
253	C₃×C₄	—	2	1	no	—
254	C₃×C₄	—	2	1	no	—
255	C₃×C₄	—	10	2	no	—
256	C₃×C₄	#709	1	2	yes	—
257	Dic₃	—	2	1	no	—
258	Dic₃	—	2	1	no	—
259	Dic₃	—	2	1	no	—
260	Dic₃	—	10	2	no	—
261	C₂×C₅	—	12	1	no	—
262	C₂×C₅	C₂×#235	1	1	yes	characteristic
263	C₂×C₅	#1004	1	1	yes	characteristic
264	C₂×C₅	#1009	1	1	yes	characteristic
265	D₅	—	4	2	no	—
266	D₅	—	12	2	no	—
267	C₂³	—	3	1	no	—
268	C₂³	—	5	2	no	—
269	C₂³	—	15	2	no	—
270	D₄	—	10	2	no	—
271	D₄	—	30	2	no	—
272	D₄	—	60	2	no	—
273	D₄	—	60	2	no	—
274	D₄	—	60	2	no	—
275	D₄	—	60	2	no	—
276	C₂×C₄	—	3	1	no	—
277	C₂×C₄	—	3	1	no	—
278	C₂×C₄	—	3	1	no	—
279	C₂×C₄	—	5	2	no	—
280	C₂×C₄	—	6	2	no	—
281	C₂×C₄	—	15	2	no	—
282	C₂×C₄	—	20	2	no	—
283	C₂×C₄	—	60	2	no	—
284	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
285	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
286	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
287	C₂×C₃	—	20	2	no	—
288	C₂×C₃	C₂×#709	1	1	yes	characteristic
289	C₂×C₃	#4149	1	1	yes	characteristic
290	C₂×C₃	#4149	1	1	yes	characteristic
291	D₃	—	4	1	no	—
292	D₃	—	20	2	no	—
293	C₅	#5562	1	1	yes	characteristic, complemented, Sylow
294	C₂²	—	6	1	no	—
295	C₂²	—	6	1	no	—
296	C₂²	—	6	1	no	—
297	C₂²	—	10	2	no	—
298	C₂²	—	10	2	no	—
299	C₂²	—	10	2	no	—
300	C₂²	—	30	2	no	—
301	C₂²	—	30	2	no	—
302	C₂²	—	30	2	no	—
303	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	6	1	no	—
307	C₄	—	6	1	no	—
308	C₄	—	6	1	no	—
309	C₄	—	10	2	no	—
310	C₄	—	30	2	no	—
311	C₄	#6771	1	2	yes	—
312	C₃	#65869	1	1	yes	characteristic, complemented, Sylow
313	C₂	—	12	1	no	—
314	C₂	—	20	2	no	—
315	C₂	—	60	2	no	—
316	C₂	C₂×#6771	1	1	yes	characteristic
317	C₂	#92017	1	1	yes	characteristic
318	C₂	#92019	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₅
commutator, socle, cosocle	C₂²×C₃×C₅	C₂⁴
Fitting	C₃×C₄²×C₅	C₂²
hypercenter	C₄²	D₃×D₅
hypocenter	C₃×C₅	#540
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
1. C₄×D₆₀	C₂	semidirect
2. C₅×#872	C₂	semidirect
3. C₃×C₄×D₂₀	C₂	semidirect
4. #91461	C₂	semidirect
5. #91460	C₂	semidirect
6. #91458	C₂	semidirect
7. #91456	C₂	semidirect
8. #91448	C₂	semidirect
9. #91381	C₂	semidirect
10. #91354	C₂	semidirect
11. #91348	C₂	semidirect
46. C₄×D₂₀	D₃	semidirect
78. #872	D₅	semidirect
242. C₃×C₅	#540	semidirect
293. C₅	#5562	semidirect
312. C₃	#65869	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₆₀	2
C₅×#872	2
C₃×C₄×D₂₀	2
#91461	2
#91460	2
#91458	2
#91456	2
#91448	2
#91381	2
#91354	2
#91348	2
#65869	3
#5562	5