Subgroups — #11754440

Order 960.

468 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#91819	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#4070	C₂	1	1	yes	maximal, characteristic, complemented
4	#91556	C₂	1	2	yes	maximal, complemented
5	#91542	C₂	1	2	yes	maximal, complemented
6	#91428	C₂	1	1	yes	maximal, characteristic, complemented
7	#91403	C₂	1	2	yes	maximal, complemented
8	#91381	C₂	1	1	yes	maximal, characteristic, complemented
9	#91380	C₂	1	2	yes	maximal, complemented
10	#91374	C₂	1	1	yes	maximal, characteristic, complemented
11	Dic₃×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
12	#65813	—	3	1	no	maximal, Hall
13	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
14	C₂×#1220	C₂²	1	2	yes	—
15	C₂×D₆₀	C₂²	1	1	yes	characteristic
16	C₂²×C₅×Dic₃	C₂²	1	1	yes	characteristic
17	C₂²×C₅×Dic₃	C₂²	1	1	yes	characteristic
18	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
19	C₂×#1200	C₂²	1	2	yes	—
20	#6714	C₂²	1	2	yes	—
21	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
22	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
23	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
24	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
25	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
26	C₃×C₂²⋊Dic₅	C₂²	1	2	yes	—
27	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
28	C₃×#683	C₂²	1	2	yes	—
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	#5624	—	5	1	no	maximal, Hall
37	C₂×C₄²×C₅	—	3	1	no	—
38	#4076	—	3	1	no	—
39	#4075	—	3	2	no	—
40	C₄×C₂²⋊D₅	—	3	2	no	—
41	#4070	D₃	1	1	yes	characteristic, complemented
42	#4022	—	3	2	no	—
43	#4021	—	3	1	no	—
44	C₄×D₂₀	—	3	1	no	—
45	#4018	—	3	2	no	—
46	#4016	—	3	1	no	—
47	C₄×Dic₁₀	—	3	1	no	—
48	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
49	C₂²×D₁₅	C₂³	1	2	yes	—
50	C₂×C₃×C₄×C₅	—	2	2	no	—
51	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
53	#1220	—	2	4	no	—
54	C₂×Dic₁₅	C₂³	1	2	yes	—
55	D₆₀	—	2	2	no	—
56	C₂×C₅×Dic₃	—	2	2	no	—
57	C₂×C₅×Dic₃	—	2	2	no	—
58	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
59	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
60	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
61	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
62	C₂×C₃×Dic₅	C₂³	1	2	yes	—
63	C₂×C₃×Dic₅	C₂³	1	2	yes	—
64	C₃×Dic₁₀	—	2	2	no	—
65	#1200	—	2	4	no	—
66	C₃×C₂²⋊Q₈	—	5	1	no	—
67	#947	—	5	1	no	—
68	Q₈×Dic₃	—	5	1	no	—
69	#930	—	5	1	no	—
70	C₂×C₄×Dic₃	D₅	1	1	yes	characteristic, complemented
71	#898	—	5	2	no	—
72	#893	—	5	1	no	—
73	#892	—	5	2	no	—
74	#890	—	5	1	no	—
75	#886	—	5	2	no	—
76	#67⋊C₄	—	5	2	no	—
77	C₂²×C₄×C₅	—	3	1	no	—
78	C₂²×C₄×C₅	—	3	1	no	—
79	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
80	C₂×C₂²⋊D₅	—	3	2	no	—
81	C₂×D₂₀	—	3	1	no	—
82	C₂×C₄×D₅	—	3	2	no	—
83	C₂×Dic₁₀	C₂×D₃	1	1	yes	characteristic
84	C₄²×C₅	—	3	1	no	—
85	C₄²×C₅	—	3	1	no	—
86	C₄²×C₅	—	3	1	no	—
87	C₄²×C₅	—	3	1	no	—
88	C₂²⋊Dic₅	C₂×D₃	1	2	yes	—
89	#685	—	3	2	no	—
90	#685	—	3	2	no	—
91	#685	—	3	2	no	—
92	#685	—	3	2	no	—
93	C₄⋊Dic₅	—	3	1	no	—
94	C₄⋊Dic₅	C₂×D₃	1	1	yes	characteristic
95	#683	—	3	2	no	—
96	#683	—	3	2	no	—
97	#683	—	3	2	no	—
98	#683	C₂×D₃	1	2	yes	—
99	C₄×Dic₅	—	3	2	no	—
100	#529	—	15	1	no	Sylow
101	C₂²×C₃×C₅	—	2	1	no	—
102	C₂²×C₃×C₅	—	2	1	no	—
103	C₂²×C₃×C₅	#41	1	2	yes	—
104	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
105	C₂×D₁₅	—	2	2	no	—
106	C₂×D₁₅	—	2	2	no	—
107	C₂×D₁₅	—	2	2	no	—
108	C₃×C₄×C₅	—	2	1	no	—
109	C₃×C₄×C₅	#41	1	2	yes	—
110	Dic₁₅	—	2	2	no	—
111	C₃×Dic₅	—	2	2	no	—
112	C₃×Dic₅	—	2	2	no	—
113	C₅×Dic₃	—	2	1	no	—
114	C₅×Dic₃	—	2	1	no	—
115	C₅×Dic₃	—	2	1	no	—
116	C₅×Dic₃	#41	1	2	yes	—
117	C₂×C₃×Q₈	—	5	1	no	—
118	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
119	C₂×#67	—	5	2	no	—
120	C₂²×Dic₃	C₂×D₅	1	1	yes	characteristic
121	C₂²×Dic₃	C₂×D₅	1	1	yes	characteristic
122	C₂×D₁₂	—	5	1	no	—
123	C₂×C₄×D₃	—	5	2	no	—
124	C₃×C₄⋊C₄	—	5	2	no	—
125	C₃×C₄⋊C₄	—	5	1	no	—
126	C₃×C₂²⋊C₄	—	5	2	no	—
127	(C₂×D₃)⋊C₄	—	5	2	no	—
128	(C₂×D₃)⋊C₄	—	5	2	no	—
129	(C₂×D₃)⋊C₄	—	5	2	no	—
130	(C₂×D₃)⋊C₄	—	5	2	no	—
131	C₄⋊Dic₃	—	5	1	no	—
132	C₄⋊Dic₃	—	5	2	no	—
133	C₄⋊Dic₃	—	5	2	no	—
134	Dic₃⋊C₄	—	5	2	no	—
135	C₄×Dic₃	—	5	2	no	—
136	C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
137	C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
138	C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
139	C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
140	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
141	C₂²×D₅	—	3	2	no	—
142	C₂×C₄×C₅	—	2	2	no	—
143	C₂×C₄×C₅	—	3	1	no	—
144	C₂×C₄×C₅	—	3	1	no	—
145	C₂×C₄×C₅	—	3	1	no	—
146	C₂×C₄×C₅	—	3	1	no	—
147	C₂×C₄×C₅	—	6	2	no	—
148	C₂×C₄×C₅	—	6	2	no	—
149	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
151	C₂²⋊D₅	—	6	4	no	—
152	C₂×Dic₅	—	3	2	no	—
153	C₂×Dic₅	C₂²×D₃	1	2	yes	—
154	C₂×Dic₅	C₂²×D₃	1	2	yes	—
155	D₂₀	—	6	2	no	—
156	C₄×D₅	—	6	4	no	—
157	Dic₁₀	—	2	2	no	—
158	#126	—	15	2	no	—
159	#125	—	15	1	no	—
160	#124	—	15	1	no	—
161	#123	—	15	2	no	—
162	C₂²⋊Q₈	—	5	1	no	—
163	#121	—	15	1	no	—
164	C₄×Q₈	—	15	1	no	—
165	C₄×D₄	—	15	1	no	—
166	C₄×D₄	—	15	2	no	—
167	#117	—	15	2	no	—
168	C₂×C₄²	—	3	1	no	—
169	C₂×C₃×C₅	—	2	2	no	—
170	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
171	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
172	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
173	D₁₅	—	4	2	no	—
174	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
175	C₂²×D₃	—	5	2	no	—
176	C₃×Q₈	—	10	2	no	—
177	C₂×C₃×C₄	—	2	2	no	—
178	C₂×C₃×C₄	—	5	2	no	—
179	C₂×C₃×C₄	—	5	2	no	—
180	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
182	#67	—	10	4	no	—
183	C₂×Dic₃	—	2	2	no	—
184	C₂×Dic₃	—	2	2	no	—
185	C₂×Dic₃	—	5	2	no	—
186	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
187	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
188	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
189	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
190	D₁₂	—	10	2	no	—
191	C₄×D₃	—	10	4	no	—
192	C₂²×C₅	—	2	1	no	—
193	C₂²×C₅	—	2	1	no	—
194	C₂²×C₅	#239	1	2	yes	—
195	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
196	C₂×D₅	—	6	2	no	—
197	C₂×D₅	—	6	2	no	—
198	C₂×D₅	—	6	2	no	—
199	C₄×C₅	—	2	1	no	—
200	C₄×C₅	—	3	2	no	—
201	C₄×C₅	—	6	1	no	—
202	C₄×C₅	—	6	1	no	—
203	C₄×C₅	—	6	1	no	—
204	C₄×C₅	#237	1	2	yes	—
205	Dic₅	—	2	2	no	—
206	Dic₅	—	2	2	no	—
207	Dic₅	—	6	2	no	—
208	C₂×Q₈	—	5	1	no	—
209	C₂×D₄	—	15	2	no	—
210	C₂×D₄	—	15	1	no	—
211	C₂²×C₄	—	3	1	no	—
212	C₂²×C₄	—	3	1	no	—
213	C₂²×C₄	—	15	2	no	—
214	C₂²×C₄	D₃×D₅	1	1	yes	characteristic
215	C₄⋊C₄	—	5	2	no	—
216	C₄⋊C₄	—	5	1	no	—
217	C₄⋊C₄	—	15	2	no	—
218	C₄⋊C₄	—	15	2	no	—
219	C₄⋊C₄	—	15	2	no	—
220	C₄⋊C₄	—	15	1	no	—
221	C₂²⋊C₄	—	5	2	no	—
222	C₂²⋊C₄	—	15	2	no	—
223	C₂²⋊C₄	—	15	2	no	—
224	C₂²⋊C₄	—	15	2	no	—
225	C₂²⋊C₄	—	15	2	no	—
226	C₄²	—	3	1	no	—
227	C₄²	—	3	1	no	—
228	C₄²	—	3	1	no	—
229	C₄²	—	3	1	no	—
230	C₄²	—	15	2	no	—
231	C₃×C₅	#529	1	1	yes	characteristic, complemented, Hall
232	C₂²×C₃	—	2	1	no	—
233	C₂²×C₃	—	2	1	no	—
234	C₂²×C₃	#709	1	2	yes	—
235	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
236	C₂×D₃	—	10	2	no	—
237	C₂×D₃	—	10	2	no	—
238	C₂×D₃	—	10	2	no	—
239	C₃×C₄	—	2	1	no	—
240	C₃×C₄	—	10	2	no	—
241	C₃×C₄	—	10	2	no	—
242	C₃×C₄	#709	1	2	yes	—
243	Dic₃	—	2	1	no	—
244	Dic₃	—	2	1	no	—
245	Dic₃	—	2	1	no	—
246	Dic₃	—	10	2	no	—
247	Dic₃	#709	1	2	yes	—
248	C₂×C₅	—	2	2	no	—
249	C₂×C₅	C₂×#237	1	1	yes	characteristic
250	C₂×C₅	C₂×#239	1	1	yes	characteristic
251	C₂×C₅	D₃×#41	1	1	yes	characteristic
252	D₅	—	12	2	no	—
253	C₂³	—	15	2	no	—
254	C₂³	C₂×D₃×D₅	1	1	yes	characteristic
255	Q₈	—	10	2	no	—
256	D₄	—	30	4	no	—
257	D₄	—	30	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	3	1	no	—
260	C₂×C₄	—	3	1	no	—
261	C₂×C₄	—	3	1	no	—
262	C₂×C₄	—	3	1	no	—
263	C₂×C₄	—	5	2	no	—
264	C₂×C₄	—	5	2	no	—
265	C₂×C₄	—	6	2	no	—
266	C₂×C₄	—	6	2	no	—
267	C₂×C₄	—	15	2	no	—
268	C₂×C₄	—	30	4	no	—
269	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
270	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
271	C₂×C₃	—	2	2	no	—
272	C₂×C₃	C₂×#709	1	1	yes	characteristic
273	C₂×C₃	C₂×#709	1	1	yes	characteristic
274	C₂×C₃	C₂×#709	1	1	yes	characteristic
275	D₃	—	20	2	no	—
276	C₅	#5624	1	1	yes	characteristic, complemented, Sylow
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	30	2	no	—
280	C₂²	—	30	2	no	—
281	C₂²	—	30	2	no	—
282	C₂²	#6769	1	2	yes	—
283	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	3	2	no	—
286	C₄	—	6	1	no	—
287	C₄	—	6	1	no	—
288	C₄	—	6	1	no	—
289	C₄	—	10	2	no	—
290	C₄	—	10	2	no	—
291	C₄	—	30	2	no	—
292	C₄	#6779	1	2	yes	—
293	C₃	#65813	1	1	yes	characteristic, complemented, Sylow
294	C₂	—	2	2	no	—
295	C₂	—	60	2	no	—
296	C₂	C₂×#6769	1	1	yes	characteristic
297	C₂	D₃×#709	1	1	yes	characteristic
298	C₂	C₂×#6779	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₅
commutator, socle, cosocle	C₂²×C₃×C₅	C₂⁴
Fitting	C₂²×C₃×C₄×C₅	C₂²
prime radical	#91819	C₂
hypercenter	C₂²×C₄	D₃×D₅
hypocenter	C₃×C₅	#529
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×C₅×Dic₃	C₂	semidirect
3. C₃×#4070	C₂	semidirect
4. #91556	C₂	semidirect
5. #91542	C₂	semidirect
6. #91428	C₂	semidirect
7. #91403	C₂	semidirect
8. #91381	C₂	semidirect
9. #91380	C₂	semidirect
10. #91374	C₂	semidirect
11. Dic₃×Dic₁₀	C₂	semidirect
41. #4070	D₃	semidirect
70. C₂×C₄×Dic₃	D₅	semidirect
231. C₃×C₅	#529	semidirect
276. C₅	#5624	semidirect
293. C₃	#65813	semidirect

Maximal subgroups

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

subgroup	index
#91819	2
C₂×C₄×C₅×Dic₃	2
C₃×#4070	2
#91556	2
#91542	2
#91428	2
#91403	2
#91381	2
#91380	2
#91374	2
Dic₃×Dic₁₀	2
#65813	3
#5624	5