Subgroups — #11752590

Order 960.

424 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#91764	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
4	#91418	C₂	1	1	yes	maximal, characteristic
5	#91411	C₂	1	2	yes	maximal, complemented
6	#91403	C₂	1	2	yes	maximal, complemented
7	#91380	C₂	1	2	yes	maximal, complemented
8	#91376	C₂	1	1	yes	maximal, characteristic, complemented
9	#91375	C₂	1	1	yes	maximal, characteristic
10	#91345	C₂	1	2	yes	maximal, complemented
11	#91333	C₂	1	1	yes	maximal, characteristic, complemented
12	#65902	—	3	1	no	maximal, Hall
13	C₂×D₆₀	C₂²	1	1	yes	characteristic
14	C₂×Dic₃₀	C₂²	1	1	yes	characteristic
15	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
16	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
17	C₂×#1200	C₂²	1	2	yes	—
18	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
19	#6714	C₂²	1	2	yes	—
20	#6714	C₂²	1	2	yes	—
21	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
22	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
23	C₄×C₅×Dic₃	C₂²	1	2	yes	—
24	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
25	C₃×#683	C₂²	1	1	yes	characteristic
26	C₃×#683	C₂²	1	1	yes	characteristic
27	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
28	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
29	#6669	C₂²	1	2	yes	—
30	#6668	C₂²	1	2	yes	—
31	#6668	C₂²	1	2	yes	—
32	#6667	C₂²	1	2	yes	—
33	#6667	C₂²	1	2	yes	—
34	#6667	C₂²	1	2	yes	—
35	Dic₃×Dic₅	C₂²	1	2	yes	—
36	#5584	—	5	1	no	maximal, Hall
37	C₄×C₅×Q₈	—	3	1	no	—
38	#4093	—	3	1	no	—
39	#4090	—	3	1	no	—
40	#4044	—	3	2	no	—
41	#4042	—	3	2	no	—
42	#4039	—	3	1	no	—
43	#4033	—	3	1	no	—
44	#4022	—	3	2	no	—
45	#4021	—	3	1	no	—
46	#4018	—	3	2	no	—
47	C₄×Dic₁₀	D₃	1	1	yes	characteristic, complemented
48	C₂²×D₁₅	C₂³	1	2	yes	—
49	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
52	C₂×Dic₁₅	C₂³	1	2	yes	—
53	D₆₀	—	2	2	no	—
54	Dic₃₀	—	2	2	no	—
55	C₂×C₅×Dic₃	C₂³	1	2	yes	—
56	C₂×C₅×Dic₃	C₂³	1	2	yes	—
57	C₅×Dic₆	—	2	2	no	—
58	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
59	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
60	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
61	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
62	C₃×Dic₁₀	—	2	2	no	—
63	#1200	—	2	4	no	—
64	C₃×C₄×Q₈	—	5	1	no	—
65	#946	—	5	2	no	—
66	#898	—	5	2	no	—
67	#898	—	5	2	no	—
68	#892	—	5	2	no	—
69	#875	—	5	1	no	—
70	#875	—	5	1	no	—
71	C₄×D₁₂	—	5	1	no	—
72	#869	—	5	1	no	—
73	C₄×Dic₆	—	5	1	no	—
74	C₄×Dic₆	D₅	1	1	yes	characteristic, complemented
75	C₂×C₅×Q₈	—	3	1	no	—
76	C₂×D₂₀	—	3	1	no	—
77	C₂×C₄×D₅	—	3	2	no	—
78	C₂×Dic₁₀	—	3	1	no	—
79	C₂×Dic₁₀	C₂×D₃	1	1	yes	characteristic
80	C₅×C₄⋊C₄	—	3	2	no	—
81	C₅×C₄⋊C₄	—	3	1	no	—
82	C₄²×C₅	—	3	2	no	—
83	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
84	#685	—	3	2	no	—
85	#685	—	3	2	no	—
86	#685	—	3	2	no	—
87	#685	—	3	2	no	—
88	#685	—	3	2	no	—
89	C₄⋊Dic₅	C₂×D₃	1	1	yes	characteristic
90	#683	—	3	2	no	—
91	#683	—	3	2	no	—
92	#683	—	3	2	no	—
93	#683	C₂×D₃	1	1	yes	characteristic
94	#683	C₂×D₃	1	1	yes	characteristic
95	C₄×Dic₅	—	3	2	no	—
96	C₄×Dic₅	C₂×D₃	1	1	yes	characteristic
97	C₄×Dic₅	C₂×D₃	1	1	yes	characteristic
98	#556	—	15	1	no	Sylow
99	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
100	C₂×D₁₅	—	2	2	no	—
101	C₂×D₁₅	—	2	2	no	—
102	C₂×D₁₅	—	2	2	no	—
103	C₃×C₄×C₅	—	2	1	no	—
104	C₃×C₄×C₅	—	2	1	no	—
105	C₃×C₄×C₅	#41	1	2	yes	—
106	Dic₁₅	—	2	2	no	—
107	C₃×Dic₅	—	2	1	no	—
108	C₃×Dic₅	—	2	1	no	—
109	C₃×Dic₅	—	2	1	no	—
110	C₃×Dic₅	#41	1	2	yes	—
111	C₅×Dic₃	—	2	2	no	—
112	C₅×Dic₃	—	2	2	no	—
113	C₂×C₃×Q₈	—	5	1	no	—
114	C₂×D₁₂	—	5	1	no	—
115	C₂×C₄×D₃	—	5	2	no	—
116	C₂×Dic₆	—	5	1	no	—
117	C₂×Dic₆	C₂×D₅	1	1	yes	characteristic
118	C₃×C₄⋊C₄	—	5	1	no	—
119	C₃×C₄⋊C₄	—	5	1	no	—
120	C₃×C₄⋊C₄	—	5	1	no	—
121	C₃×C₄²	—	5	1	no	—
122	C₃×C₄²	—	5	1	no	—
123	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
124	(C₂×D₃)⋊C₄	—	5	2	no	—
125	(C₂×D₃)⋊C₄	—	5	2	no	—
126	(C₂×D₃)⋊C₄	—	5	2	no	—
127	(C₂×D₃)⋊C₄	—	5	2	no	—
128	(C₂×D₃)⋊C₄	—	5	2	no	—
129	C₄⋊Dic₃	—	5	2	no	—
130	C₄⋊Dic₃	—	5	2	no	—
131	C₄⋊Dic₃	C₂×D₅	1	1	yes	characteristic
132	Dic₃⋊C₄	—	5	2	no	—
133	Dic₃⋊C₄	C₂×D₅	1	2	yes	—
134	C₄×Dic₃	—	5	2	no	—
135	C₄×Dic₃	C₂×D₅	1	2	yes	—
136	C₂²×D₅	—	3	2	no	—
137	C₅×Q₈	—	6	2	no	—
138	C₂×C₄×C₅	—	3	2	no	—
139	C₂×C₄×C₅	—	3	2	no	—
140	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
143	C₂×Dic₅	—	3	2	no	—
144	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
145	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
146	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
147	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
148	D₂₀	—	6	2	no	—
149	C₄×D₅	—	6	4	no	—
150	Dic₁₀	—	2	2	no	—
151	Dic₁₀	—	6	2	no	—
152	C₄⋊Q₈	—	15	1	no	—
153	#126	—	15	2	no	—
154	#126	—	15	2	no	—
155	#124	—	15	1	no	—
156	#124	—	15	1	no	—
157	C₂²⋊Q₈	—	15	2	no	—
158	C₄×Q₈	—	3	1	no	—
159	C₄×Q₈	—	5	1	no	—
160	C₄×Q₈	—	15	1	no	—
161	C₄×D₄	—	15	1	no	—
162	#117	—	15	2	no	—
163	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
164	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
165	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
166	D₁₅	—	4	2	no	—
167	C₂²×D₃	—	5	2	no	—
168	C₃×Q₈	—	10	2	no	—
169	C₂×C₃×C₄	—	5	1	no	—
170	C₂×C₃×C₄	—	5	1	no	—
171	C₂×C₃×C₄	—	5	1	no	—
172	C₂×C₃×C₄	—	5	1	no	—
173	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
176	C₂×Dic₃	—	5	2	no	—
177	C₂×Dic₃	C₂²×D₅	1	2	yes	—
178	C₂×Dic₃	C₂²×D₅	1	2	yes	—
179	D₁₂	—	10	2	no	—
180	C₄×D₃	—	10	4	no	—
181	Dic₆	—	2	2	no	—
182	Dic₆	—	10	2	no	—
183	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
184	C₂×D₅	—	6	2	no	—
185	C₂×D₅	—	6	2	no	—
186	C₂×D₅	—	6	2	no	—
187	C₄×C₅	—	2	1	no	—
188	C₄×C₅	—	2	1	no	—
189	C₄×C₅	—	6	2	no	—
190	C₄×C₅	—	6	2	no	—
191	C₄×C₅	#239	1	2	yes	—
192	Dic₅	—	2	1	no	—
193	Dic₅	—	2	1	no	—
194	Dic₅	—	2	1	no	—
195	Dic₅	—	6	2	no	—
196	Dic₅	#235	1	2	yes	—
197	C₂×Q₈	—	3	1	no	—
198	C₂×Q₈	—	5	1	no	—
199	C₂×Q₈	—	15	1	no	—
200	C₂×D₄	—	15	1	no	—
201	C₂²×C₄	—	15	2	no	—
202	C₄⋊C₄	—	3	1	no	—
203	C₄⋊C₄	—	3	2	no	—
204	C₄⋊C₄	—	5	1	no	—
205	C₄⋊C₄	—	5	1	no	—
206	C₄⋊C₄	—	5	1	no	—
207	C₄⋊C₄	—	15	2	no	—
208	C₄⋊C₄	—	15	2	no	—
209	C₄⋊C₄	—	15	2	no	—
210	C₂²⋊C₄	—	15	2	no	—
211	C₂²⋊C₄	—	15	2	no	—
212	C₂²⋊C₄	—	15	2	no	—
213	C₂²⋊C₄	—	15	2	no	—
214	C₂²⋊C₄	—	15	2	no	—
215	C₄²	—	3	2	no	—
216	C₄²	—	5	1	no	—
217	C₄²	—	5	1	no	—
218	C₄²	—	15	2	no	—
219	C₄²	D₃×D₅	1	1	yes	characteristic
220	C₃×C₅	#556	1	1	yes	characteristic, complemented, Hall
221	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
222	C₂×D₃	—	10	2	no	—
223	C₂×D₃	—	10	2	no	—
224	C₂×D₃	—	10	2	no	—
225	C₃×C₄	—	2	1	no	—
226	C₃×C₄	—	2	1	no	—
227	C₃×C₄	—	5	2	no	—
228	C₃×C₄	—	10	1	no	—
229	C₃×C₄	—	10	1	no	—
230	C₃×C₄	—	10	1	no	—
231	C₃×C₄	#709	1	2	yes	—
232	Dic₃	—	2	2	no	—
233	Dic₃	—	2	2	no	—
234	Dic₃	—	10	2	no	—
235	C₂×C₅	C₂×#235	1	1	yes	characteristic
236	C₂×C₅	C₂×#239	1	1	yes	characteristic
237	C₂×C₅	#1010	1	1	yes	characteristic
238	D₅	—	12	2	no	—
239	C₂³	—	15	2	no	—
240	Q₈	—	6	2	no	—
241	Q₈	—	10	2	no	—
242	Q₈	—	30	2	no	—
243	D₄	—	30	2	no	—
244	C₂×C₄	—	3	2	no	—
245	C₂×C₄	—	3	2	no	—
246	C₂×C₄	—	5	1	no	—
247	C₂×C₄	—	5	1	no	—
248	C₂×C₄	—	5	1	no	—
249	C₂×C₄	—	5	1	no	—
250	C₂×C₄	—	15	2	no	—
251	C₂×C₄	—	30	4	no	—
252	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
253	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
254	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
255	C₂×C₃	C₂×#709	1	1	yes	characteristic
256	C₂×C₃	#4147	1	1	yes	characteristic
257	C₂×C₃	D₅×#41	1	1	yes	characteristic
258	D₃	—	20	2	no	—
259	C₅	#5584	1	1	yes	characteristic, complemented, Sylow
260	C₂²	—	30	2	no	—
261	C₂²	—	30	2	no	—
262	C₂²	—	30	2	no	—
263	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	5	2	no	—
267	C₄	—	6	2	no	—
268	C₄	—	6	2	no	—
269	C₄	—	10	1	no	—
270	C₄	—	10	1	no	—
271	C₄	—	10	1	no	—
272	C₄	—	30	2	no	—
273	C₄	#6769	1	2	yes	—
274	C₃	#65902	1	1	yes	characteristic, complemented, Sylow
275	C₂	—	60	2	no	—
276	C₂	#92000	1	1	yes	characteristic
277	C₂	C₂×#6769	1	1	yes	characteristic
278	C₂	D₅×#235	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
prime radical	C₂×D₆₀	C₂²
hypercenter	C₄²	D₃×D₅
hypocenter	C₃×C₅	#556
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
2. C₄×C₅×Dic₆	C₂	semidirect
3. C₃×C₄×Dic₁₀	C₂	semidirect
5. #91411	C₂	semidirect
6. #91403	C₂	semidirect
7. #91380	C₂	semidirect
8. #91376	C₂	semidirect
10. #91345	C₂	semidirect
11. #91333	C₂	semidirect
47. C₄×Dic₁₀	D₃	semidirect
74. C₄×Dic₆	D₅	semidirect
220. C₃×C₅	#556	semidirect
259. C₅	#5584	semidirect
274. C₃	#65902	semidirect

Maximal subgroups

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

subgroup	index
#91764	2
C₄×C₅×Dic₆	2
C₃×C₄×Dic₁₀	2
#91418	2
#91411	2
#91403	2
#91380	2
#91376	2
#91375	2
#91345	2
#91333	2
#65902	3
#5584	5