Subgroups — #11752573

Order 960.

428 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₃₀	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#870	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄×D₂₀	C₂	1	1	yes	maximal, characteristic
4	D₅×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#91407	C₂	1	2	yes	maximal
6	#91394	C₂	1	2	yes	maximal, complemented
7	#91366	C₂	1	2	yes	maximal, complemented
8	#91360	C₂	1	1	yes	maximal, characteristic, complemented
9	#91360	C₂	1	1	yes	maximal, characteristic, complemented
10	#91358	C₂	1	1	yes	maximal, characteristic, complemented
11	#91346	C₂	1	2	yes	maximal, complemented
12	#66028	—	3	1	no	maximal, Hall
13	C₂×Dic₃₀	C₂²	1	1	yes	characteristic
14	C₂×C₃×D₂₀	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
17	C₂×D₅×Dic₃	C₂²	1	2	yes	—
18	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
19	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
20	#6712	C₂²	1	2	yes	—
21	C₄×Dic₁₅	C₂²	1	2	yes	—
22	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
23	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
24	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
25	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
26	C₃×#685	C₂²	1	1	yes	characteristic
27	C₃×#685	C₂²	1	1	yes	characteristic
28	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
29	Dic₃⋊Dic₅	C₂²	1	2	yes	—
30	Dic₃⋊Dic₅	C₂²	1	2	yes	—
31	#6668	C₂²	1	2	yes	—
32	#6665	C₂²	1	2	yes	—
33	#6665	C₂²	1	2	yes	—
34	#6665	C₂²	1	2	yes	—
35	Dic₃×Dic₅	C₂²	1	2	yes	—
36	#5541	—	5	1	no	maximal, Hall
37	C₅×#125	—	3	1	no	—
38	#4044	—	3	2	no	—
39	#4043	—	3	1	no	—
40	#4043	—	3	1	no	—
41	#4040	—	3	2	no	—
42	#4038	—	3	1	no	—
43	#4038	—	3	2	no	—
44	D₅×C₄⋊C₄	—	3	1	no	—
45	#4036	—	3	2	no	—
46	C₄×D₂₀	D₃	1	1	yes	characteristic
47	C₄×Dic₁₀	—	3	1	no	—
48	C₂²×C₃×D₅	C₂³	1	1	yes	characteristic
49	C₂²×C₃×D₅	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×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
53	C₂×Dic₁₅	C₂³	1	2	yes	—
54	C₂×Dic₁₅	C₂³	1	2	yes	—
55	Dic₃₀	—	2	2	no	—
56	C₂×C₅×Dic₃	C₂³	1	2	yes	—
57	C₂×C₅×Dic₃	C₂³	1	2	yes	—
58	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
59	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
60	C₃×D₂₀	—	2	2	no	—
61	C₃×C₄×D₅	—	2	2	no	—
62	C₃×C₄×D₅	—	2	2	no	—
63	D₅×Dic₃	—	2	4	no	—
64	C₃×C₄×D₄	—	5	1	no	—
65	#935	—	5	2	no	—
66	#926	—	5	1	no	—
67	C₂×C₄⋊Dic₃	—	5	1	no	—
68	#924	—	5	1	no	—
69	#924	—	5	1	no	—
70	#889	—	5	2	no	—
71	#879	—	5	2	no	—
72	#877	—	5	2	no	—
73	#870	D₅	1	1	yes	characteristic, complemented
74	C₄×Dic₆	—	5	1	no	—
75	C₂×D₂₀	C₂×D₃	1	1	yes	characteristic
76	C₂×C₄×D₅	—	3	2	no	—
77	C₂×C₄×D₅	C₂×D₃	1	1	yes	characteristic
78	C₂×C₄×D₅	C₂×D₃	1	1	yes	characteristic
79	C₂×Dic₁₀	—	3	1	no	—
80	C₅×C₄⋊C₄	—	3	2	no	—
81	C₅×C₄⋊C₄	—	3	2	no	—
82	C₅×C₄⋊C₄	—	3	1	no	—
83	C₅×C₄⋊C₄	—	3	1	no	—
84	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
85	#685	—	3	2	no	—
86	#685	—	3	2	no	—
87	#685	—	3	2	no	—
88	#685	C₂×D₃	1	1	yes	characteristic
89	#685	C₂×D₃	1	1	yes	characteristic
90	C₄⋊Dic₅	—	3	1	no	—
91	C₄⋊Dic₅	—	3	2	no	—
92	C₄⋊Dic₅	—	3	2	no	—
93	C₄⋊Dic₅	C₂×D₃	1	1	yes	characteristic
94	#683	—	3	2	no	—
95	#683	—	3	2	no	—
96	C₄×Dic₅	—	3	2	no	—
97	C₄×Dic₅	—	3	2	no	—
98	#552	—	15	1	no	Sylow
99	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
100	C₂×C₃×D₅	—	2	1	no	—
101	C₂×C₃×D₅	—	2	1	no	—
102	C₂×C₃×D₅	—	2	1	no	—
103	C₂×C₃×D₅	—	2	1	no	—
104	C₂×C₃×D₅	—	2	1	no	—
105	C₂×C₃×D₅	#41	1	2	yes	—
106	C₃×C₄×C₅	—	2	1	no	—
107	C₃×C₄×C₅	—	2	1	no	—
108	C₃×C₄×C₅	#41	1	2	yes	—
109	Dic₁₅	—	2	2	no	—
110	Dic₁₅	—	2	2	no	—
111	C₃×Dic₅	—	2	1	no	—
112	C₃×Dic₅	—	2	1	no	—
113	C₅×Dic₃	—	2	2	no	—
114	C₅×Dic₃	—	2	2	no	—
115	C₂×C₃×D₄	—	5	1	no	—
116	C₂²×C₃×C₄	—	5	1	no	—
117	C₂²×C₃×C₄	—	5	1	no	—
118	C₂²×Dic₃	—	5	2	no	—
119	C₂×Dic₆	—	5	1	no	—
120	C₃×C₄⋊C₄	—	5	1	no	—
121	C₃×C₂²⋊C₄	—	5	1	no	—
122	C₃×C₂²⋊C₄	—	5	1	no	—
123	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
124	#217	—	5	2	no	—
125	#217	—	5	2	no	—
126	#217	—	5	2	no	—
127	C₄⋊Dic₃	—	5	1	no	—
128	C₄⋊Dic₃	—	5	2	no	—
129	C₄⋊Dic₃	C₂×D₅	1	1	yes	characteristic
130	C₄⋊Dic₃	C₂×D₅	1	1	yes	characteristic
131	Dic₃⋊C₄	—	5	2	no	—
132	Dic₃⋊C₄	—	5	2	no	—
133	Dic₃⋊C₄	—	5	2	no	—
134	Dic₃⋊C₄	C₂×D₅	1	2	yes	—
135	Dic₃⋊C₄	C₂×D₅	1	2	yes	—
136	C₄×Dic₃	—	5	2	no	—
137	C₄×Dic₃	—	5	2	no	—
138	C₂²×D₅	C₂²×D₃	1	1	yes	characteristic
139	C₂²×D₅	C₂²×D₃	1	1	yes	characteristic
140	C₂×C₄×C₅	—	3	2	no	—
141	C₂×C₄×C₅	—	3	2	no	—
142	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
145	C₂×Dic₅	—	3	2	no	—
146	C₂×Dic₅	—	3	2	no	—
147	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
148	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
149	D₂₀	—	2	2	no	—
150	C₄×D₅	—	2	2	no	—
151	C₄×D₅	—	2	2	no	—
152	C₄×D₅	—	6	4	no	—
153	Dic₁₀	—	6	2	no	—
154	#126	—	15	2	no	—
155	#125	—	3	1	no	—
156	#125	—	15	2	no	—
157	#123	—	15	2	no	—
158	C₂²⋊Q₈	—	15	1	no	—
159	C₂²⋊Q₈	—	15	1	no	—
160	C₄×Q₈	—	15	1	no	—
161	C₄×D₄	—	5	1	no	—
162	#117	—	15	2	no	—
163	#117	—	15	1	no	—
164	C₂×C₄⋊C₄	—	15	1	no	—
165	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
166	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
167	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
168	C₃×D₅	—	2	2	no	—
169	C₃×D₅	—	4	1	no	—
170	C₂³×C₃	—	5	1	no	—
171	C₂³×C₃	—	5	1	no	—
172	C₃×D₄	—	10	2	no	—
173	C₂×C₃×C₄	—	5	1	no	—
174	C₂×C₃×C₄	—	5	1	no	—
175	C₂×C₃×C₄	—	10	2	no	—
176	C₂×C₃×C₄	—	10	2	no	—
177	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
180	C₂×Dic₃	—	5	2	no	—
181	C₂×Dic₃	—	5	2	no	—
182	C₂×Dic₃	—	10	4	no	—
183	C₂×Dic₃	C₂²×D₅	1	2	yes	—
184	C₂×Dic₃	C₂²×D₅	1	2	yes	—
185	Dic₆	—	10	2	no	—
186	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
187	C₂×D₅	—	2	1	no	—
188	C₂×D₅	—	2	1	no	—
189	C₂×D₅	—	2	1	no	—
190	C₂×D₅	—	2	1	no	—
191	C₂×D₅	—	2	1	no	—
192	C₂×D₅	#235	1	2	yes	—
193	C₄×C₅	—	2	1	no	—
194	C₄×C₅	—	2	1	no	—
195	C₄×C₅	—	6	2	no	—
196	C₄×C₅	—	6	2	no	—
197	C₄×C₅	#237	1	2	yes	—
198	Dic₅	—	2	1	no	—
199	Dic₅	—	2	1	no	—
200	Dic₅	—	6	2	no	—
201	Dic₅	—	6	2	no	—
202	C₂×Q₈	—	15	1	no	—
203	C₂×D₄	—	5	1	no	—
204	C₂²×C₄	—	5	1	no	—
205	C₂²×C₄	—	5	1	no	—
206	C₂²×C₄	—	15	2	no	—
207	C₄⋊C₄	—	3	1	no	—
208	C₄⋊C₄	—	3	1	no	—
209	C₄⋊C₄	—	3	2	no	—
210	C₄⋊C₄	—	3	2	no	—
211	C₄⋊C₄	—	5	1	no	—
212	C₄⋊C₄	—	15	2	no	—
213	C₄⋊C₄	—	15	2	no	—
214	C₄⋊C₄	—	15	2	no	—
215	C₄⋊C₄	—	15	1	no	—
216	C₄⋊C₄	—	15	2	no	—
217	C₂²⋊C₄	—	5	1	no	—
218	C₂²⋊C₄	—	5	1	no	—
219	C₂²⋊C₄	—	15	2	no	—
220	C₂²⋊C₄	—	15	2	no	—
221	C₂²⋊C₄	—	15	2	no	—
222	C₄²	—	15	2	no	—
223	C₄²	—	15	2	no	—
224	C₄²	D₃×D₅	1	1	yes	characteristic
225	C₃×C₅	#552	1	1	yes	characteristic, complemented, Hall
226	C₂²×C₃	—	5	2	no	—
227	C₂²×C₃	—	10	1	no	—
228	C₂²×C₃	—	10	1	no	—
229	C₂²×C₃	—	10	1	no	—
230	C₂²×C₃	—	10	1	no	—
231	C₂²×C₃	—	10	1	no	—
232	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
233	C₃×C₄	—	2	1	no	—
234	C₃×C₄	—	2	1	no	—
235	C₃×C₄	—	10	1	no	—
236	C₃×C₄	—	10	1	no	—
237	C₃×C₄	#713	1	2	yes	—
238	Dic₃	—	2	2	no	—
239	Dic₃	—	2	2	no	—
240	Dic₃	—	10	2	no	—
241	Dic₃	—	10	2	no	—
242	C₂×C₅	C₂×#235	1	1	yes	characteristic
243	C₂×C₅	C₂×#237	1	1	yes	characteristic
244	C₂×C₅	#1010	1	1	yes	characteristic
245	D₅	—	2	2	no	—
246	D₅	—	4	1	no	—
247	C₂³	—	5	1	no	—
248	C₂³	—	5	1	no	—
249	Q₈	—	30	2	no	—
250	D₄	—	10	2	no	—
251	C₂×C₄	—	3	2	no	—
252	C₂×C₄	—	3	2	no	—
253	C₂×C₄	—	5	1	no	—
254	C₂×C₄	—	5	1	no	—
255	C₂×C₄	—	10	2	no	—
256	C₂×C₄	—	10	2	no	—
257	C₂×C₄	—	15	2	no	—
258	C₂×C₄	—	15	2	no	—
259	C₂×C₄	—	30	4	no	—
260	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
261	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
262	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
263	C₂×C₃	—	10	2	no	—
264	C₂×C₃	—	20	1	no	—
265	C₂×C₃	C₂×#713	1	1	yes	characteristic
266	C₂×C₃	D₅×#41	1	1	yes	characteristic
267	C₂×C₃	#4150	1	1	yes	characteristic
268	C₅	#5541	1	1	yes	characteristic, complemented, Sylow
269	C₂²	—	5	2	no	—
270	C₂²	—	10	1	no	—
271	C₂²	—	10	1	no	—
272	C₂²	—	10	1	no	—
273	C₂²	—	10	1	no	—
274	C₂²	—	10	1	no	—
275	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	6	2	no	—
279	C₄	—	6	2	no	—
280	C₄	—	10	1	no	—
281	C₄	—	10	1	no	—
282	C₄	—	30	2	no	—
283	C₄	—	30	2	no	—
284	C₄	#6766	1	2	yes	—
285	C₃	#66028	1	1	yes	characteristic, complemented, Sylow
286	C₂	—	10	2	no	—
287	C₂	—	20	1	no	—
288	C₂	C₂×#6766	1	1	yes	characteristic
289	C₂	#92001	1	1	yes	characteristic
290	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×C₃×D₂₀	C₂²
hypercenter	C₄²	D₃×D₅
hypocenter	C₃×C₅	#552
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×Dic₃₀	C₂	semidirect
2. C₅×#870	C₂	semidirect
4. D₅×C₄⋊Dic₃	C₂	semidirect
6. #91394	C₂	semidirect
7. #91366	C₂	semidirect
8. #91360	C₂	semidirect
9. #91360	C₂	semidirect
10. #91358	C₂	semidirect
11. #91346	C₂	semidirect
73. #870	D₅	semidirect
225. C₃×C₅	#552	semidirect
268. C₅	#5541	semidirect
285. C₃	#66028	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₃₀	2
C₅×#870	2
C₃×C₄×D₂₀	2
D₅×C₄⋊Dic₃	2
#91407	2
#91394	2
#91366	2
#91360	2
#91358	2
#91346	2
#66028	3
#5541	5