Subgroups — #14283995

Order 1920.

540 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11756236	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅×#208	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄×#681	C₂	1	1	yes	maximal, characteristic, complemented
4	#11745848	C₂	1	4	yes	maximal, complemented
5	#10600652	—	3	1	no	maximal, Hall
6	C₂×#6817	C₄	1	2	yes	—
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	#91819	—	2	1	no	—
9	#91819	—	2	1	no	—
10	#91764	C₂²	1	2	yes	—
11	#91763	C₄	1	1	yes	characteristic
12	C₄×D₆₀	—	2	1	no	—
13	C₄×D₆₀	—	2	1	no	—
14	#91758	C₄	1	1	yes	characteristic
15	C₂×C₅×#208	—	2	1	no	—
16	C₂×C₃×#681	—	2	1	no	—
17	C₅×#803	C₂²	1	2	yes	—
18	C₄×C₅×C₃⋊C₈	—	2	1	no	—
19	C₃×#3935	C₂²	1	2	yes	—
20	C₃×C₄×#168	—	2	1	no	—
21	#69859	—	5	1	no	maximal, Hall
22	#65811	—	3	1	no	—
23	C₄×C₅×M₁₆	—	3	1	no	—
24	C₄×#681	D₃	1	1	yes	characteristic, complemented
25	#64684	—	3	4	no	—
26	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
29	C₂×#1220	—	2	2	no	—
30	#6817	—	4	4	no	—
31	C₂×D₆₀	—	2	1	no	—
32	C₂×D₆₀	—	2	1	no	—
33	C₂×D₆₀	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₁₅	—	2	2	no	—
35	C₂×Dic₃₀	C₂×C₄	1	2	yes	—
36	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
38	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
39	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
40	#6714	—	2	2	no	—
41	#6714	—	2	2	no	—
42	C₄⋊Dic₁₅	—	2	1	no	—
43	C₄⋊Dic₁₅	—	2	1	no	—
44	C₅×#208	—	2	4	no	—
45	C₂×C₅×C₃⋊C₈	—	2	2	no	—
46	C₃×#681	—	2	4	no	—
47	C₂×C₃×#168	—	2	2	no	—
48	#5477	—	5	1	no	—
49	C₃×C₄×M₁₆	—	5	1	no	—
50	C₄×#208	D₅	1	1	yes	characteristic, complemented
51	#4464	—	5	4	no	—
52	C₂×#709	—	3	2	no	—
53	C₂×C₅×M₁₆	—	6	1	no	—
54	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
55	#4076	—	6	1	no	—
56	#4076	—	6	1	no	—
57	C₂×#681	—	2	1	no	—
58	#4021	—	3	2	no	—
59	#4020	—	3	1	no	—
60	C₄×D₂₀	—	6	1	no	—
61	C₄×D₂₀	—	6	1	no	—
62	#4015	—	3	1	no	—
63	C₅×#97	—	3	2	no	—
64	C₄×C₅×C₈	—	6	1	no	—
65	#3935	C₂×D₃	1	2	yes	—
66	C₄×#168	—	2	1	no	—
67	#1525	—	15	1	no	Sylow
68	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×D₁₅	—	2	2	no	—
70	C₂×C₃×C₄×C₅	—	2	2	no	—
71	C₂×C₃×C₄×C₅	—	2	2	no	—
72	C₂×C₃×C₄×C₅	—	2	2	no	—
73	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	#1220	—	4	4	no	—
80	C₂×Dic₁₅	—	2	2	no	—
81	D₆₀	—	4	2	no	—
82	D₆₀	—	4	2	no	—
83	D₆₀	—	4	2	no	—
84	C₄×D₁₅	—	4	4	no	—
85	Dic₃₀	—	4	2	no	—
86	C₃×#168	—	2	4	no	—
87	C₅×C₃⋊C₈	—	2	4	no	—
88	C₂×#235	—	5	2	no	—
89	C₂×C₃×M₁₆	—	10	1	no	—
90	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
91	#930	—	10	1	no	—
92	#930	—	10	1	no	—
93	C₂×#208	—	2	1	no	—
94	#875	—	5	2	no	—
95	#874	—	5	1	no	—
96	C₄×D₁₂	—	10	1	no	—
97	C₄×D₁₂	—	10	1	no	—
98	#869	—	5	1	no	—
99	C₃×#97	—	5	2	no	—
100	C₃×C₄×C₈	—	10	1	no	—
101	#803	C₂×D₅	1	2	yes	—
102	C₄×C₃⋊C₈	—	2	1	no	—
103	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
104	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
105	C₂²×C₄×C₅	#67	1	1	yes	characteristic
106	C₂×C₂²⋊D₅	—	6	2	no	—
107	#709	—	12	4	no	—
108	C₂×D₂₀	—	3	2	no	—
109	C₂×D₂₀	—	6	1	no	—
110	C₂×D₂₀	—	6	1	no	—
111	C₂×C₄×D₅	—	6	2	no	—
112	C₂×Dic₁₀	—	3	2	no	—
113	C₅×M₁₆	—	6	4	no	—
114	C₂×C₅×C₈	—	6	2	no	—
115	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
116	C₄²×C₅	D₁₂	1	1	yes	characteristic
117	C₄²×C₅	#67	1	1	yes	characteristic
118	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
119	#685	—	6	2	no	—
120	#685	—	6	2	no	—
121	C₄⋊Dic₅	—	6	1	no	—
122	C₄⋊Dic₅	—	6	1	no	—
123	#681	—	2	4	no	—
124	C₂×#168	—	2	2	no	—
125	#532	—	15	1	no	—
126	C₄×M₁₆	—	3	1	no	—
127	C₄×M₁₆	—	5	1	no	—
128	#329	—	15	4	no	—
129	C₂²×C₃×C₅	—	2	1	no	—
130	C₂²×C₃×C₅	—	2	1	no	—
131	C₂²×C₃×C₅	—	2	1	no	—
132	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×D₁₅	—	4	2	no	—
134	C₂×D₁₅	—	4	2	no	—
135	C₂×D₁₅	—	4	2	no	—
136	C₃×C₄×C₅	—	2	1	no	—
137	C₃×C₄×C₅	—	2	1	no	—
138	C₃×C₄×C₅	—	2	1	no	—
139	C₃×C₄×C₅	#100	1	2	yes	—
140	C₃×C₄×C₅	#104	1	2	yes	—
141	C₃×C₄×C₅	#104	1	2	yes	—
142	Dic₁₅	—	4	2	no	—
143	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
144	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
145	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
146	C₂×#67	—	10	2	no	—
147	#235	—	20	4	no	—
148	C₂×D₁₂	—	5	2	no	—
149	C₂×D₁₂	—	10	1	no	—
150	C₂×D₁₂	—	10	1	no	—
151	C₂×C₄×D₃	—	10	2	no	—
152	C₂×Dic₆	—	5	2	no	—
153	C₃×M₁₆	—	10	4	no	—
154	C₂×C₃×C₈	—	10	2	no	—
155	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
156	C₃×C₄²	D₂₀	1	1	yes	characteristic
157	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
158	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
159	(C₂×D₃)⋊C₄	—	10	2	no	—
160	(C₂×D₃)⋊C₄	—	10	2	no	—
161	C₄⋊Dic₃	—	10	1	no	—
162	C₄⋊Dic₃	—	10	1	no	—
163	#208	—	2	4	no	—
164	C₂×C₃⋊C₈	—	2	2	no	—
165	C₂³×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
166	C₂²×D₅	—	6	2	no	—
167	C₂×C₄×C₅	—	2	2	no	—
168	C₂×C₄×C₅	—	2	2	no	—
169	C₂×C₄×C₅	—	2	2	no	—
170	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
171	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
172	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
175	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
176	C₂²⋊D₅	—	12	4	no	—
177	C₂×Dic₅	—	6	2	no	—
178	D₂₀	—	12	2	no	—
179	D₂₀	—	12	2	no	—
180	D₂₀	—	12	2	no	—
181	C₄×D₅	—	12	4	no	—
182	Dic₁₀	—	12	2	no	—
183	C₅×C₈	—	6	4	no	—
184	#168	—	2	4	no	—
185	C₂×#41	—	15	2	no	—
186	C₂×M₁₆	—	6	1	no	—
187	C₂×M₁₆	—	10	1	no	—
188	C₄⋊Q₈	—	15	1	no	—
189	#127	—	15	1	no	—
190	#124	—	15	2	no	—
191	#121	—	30	1	no	—
192	#121	—	30	1	no	—
193	C₄×D₄	—	30	1	no	—
194	C₄×D₄	—	30	1	no	—
195	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
196	#97	—	3	2	no	—
197	#97	—	5	2	no	—
198	C₄×C₈	—	6	1	no	—
199	C₄×C₈	—	10	1	no	—
200	C₂×C₃×C₅	—	4	1	no	—
201	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
202	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
203	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
204	D₁₅	—	8	2	no	—
205	C₂³×C₃	#685	1	1	yes	characteristic
206	C₂²×D₃	—	10	2	no	—
207	C₂×C₃×C₄	—	2	2	no	—
208	C₂×C₃×C₄	—	2	2	no	—
209	C₂×C₃×C₄	—	2	2	no	—
210	C₂×C₃×C₄	#685	1	1	yes	characteristic
211	C₂×C₃×C₄	#685	1	1	yes	characteristic
212	C₂×C₃×C₄	#685	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
214	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
216	#67	—	20	4	no	—
217	C₂×Dic₃	—	10	2	no	—
218	D₁₂	—	20	2	no	—
219	D₁₂	—	20	2	no	—
220	D₁₂	—	20	2	no	—
221	C₄×D₃	—	20	4	no	—
222	Dic₆	—	20	2	no	—
223	C₃×C₈	—	10	4	no	—
224	C₃⋊C₈	—	2	4	no	—
225	C₂²×C₅	—	2	1	no	—
226	C₂²×C₅	—	2	1	no	—
227	C₂²×C₅	—	2	1	no	—
228	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
229	C₂×D₅	—	12	2	no	—
230	C₂×D₅	—	12	2	no	—
231	C₂×D₅	—	12	2	no	—
232	C₄×C₅	—	2	1	no	—
233	C₄×C₅	—	2	1	no	—
234	C₄×C₅	—	2	1	no	—
235	C₄×C₅	#805	1	2	yes	—
236	C₄×C₅	#823	1	2	yes	—
237	C₄×C₅	#825	1	2	yes	—
238	Dic₅	—	12	2	no	—
239	#41	—	60	4	no	—
240	C₂×Q₈	—	15	2	no	—
241	C₂×D₄	—	15	2	no	—
242	C₂×D₄	—	30	2	no	—
243	C₂×D₄	—	30	1	no	—
244	C₂×D₄	—	30	1	no	—
245	C₂²×C₄	—	30	2	no	—
246	C₂²×C₄	#1200	1	1	yes	characteristic
247	C₂²×C₄	#1202	1	1	yes	characteristic
248	C₂²×C₄	#1203	1	1	yes	characteristic
249	M₁₆	—	6	4	no	—
250	M₁₆	—	10	4	no	—
251	C₂×C₈	—	6	2	no	—
252	C₂×C₈	—	10	2	no	—
253	C₄⋊C₄	—	30	1	no	—
254	C₄⋊C₄	—	30	1	no	—
255	C₂²⋊C₄	—	30	2	no	—
256	C₂²⋊C₄	—	30	2	no	—
257	C₄²	#1200	1	1	yes	characteristic
258	C₄²	#1202	1	1	yes	characteristic
259	C₄²	#1203	1	1	yes	characteristic
260	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
261	C₃×C₅	#1525	1	1	yes	characteristic, complemented, Hall
262	C₂²×C₃	—	2	1	no	—
263	C₂²×C₃	—	2	1	no	—
264	C₂²×C₃	—	2	1	no	—
265	C₂²×C₃	C₂×#685	1	1	yes	characteristic
266	C₂×D₃	—	20	2	no	—
267	C₂×D₃	—	20	2	no	—
268	C₂×D₃	—	20	2	no	—
269	C₃×C₄	—	2	1	no	—
270	C₃×C₄	—	2	1	no	—
271	C₃×C₄	—	2	1	no	—
272	C₃×C₄	#3937	1	2	yes	—
273	C₃×C₄	#3955	1	2	yes	—
274	C₃×C₄	#3957	1	2	yes	—
275	Dic₃	—	20	2	no	—
276	C₂×C₅	—	4	1	no	—
277	C₂×C₅	C₂×#805	1	1	yes	characteristic
278	C₂×C₅	C₂×#823	1	1	yes	characteristic
279	C₂×C₅	C₂×#825	1	1	yes	characteristic
280	D₅	—	24	2	no	—
281	C₂³	—	30	2	no	—
282	C₂³	#6667	1	1	yes	characteristic
283	Q₈	—	60	2	no	—
284	D₄	—	60	2	no	—
285	D₄	—	60	4	no	—
286	D₄	—	60	2	no	—
287	D₄	—	60	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	30	2	no	—
292	C₂×C₄	—	60	4	no	—
293	C₂×C₄	#6667	1	1	yes	characteristic
294	C₂×C₄	#6667	1	1	yes	characteristic
295	C₂×C₄	#6667	1	1	yes	characteristic
296	C₂×C₄	C₂×#1200	1	1	yes	characteristic
297	C₂×C₄	C₂×#1202	1	1	yes	characteristic
298	C₂×C₄	C₂×#1203	1	1	yes	characteristic
299	C₈	—	6	4	no	—
300	C₈	—	10	4	no	—
301	C₂×C₃	—	4	1	no	—
302	C₂×C₃	C₂×#3937	1	1	yes	characteristic
303	C₂×C₃	C₂×#3955	1	1	yes	characteristic
304	C₂×C₃	C₂×#3957	1	1	yes	characteristic
305	D₃	—	40	2	no	—
306	C₅	#69859	1	1	yes	characteristic, complemented, Sylow
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	60	2	no	—
311	C₂²	—	60	2	no	—
312	C₂²	—	60	2	no	—
313	C₂²	C₂×#6667	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	60	2	no	—
318	C₄	#90960	1	2	yes	—
319	C₄	#90980	1	2	yes	—
320	C₄	#90981	1	2	yes	—
321	C₃	#10600652	1	1	yes	characteristic, complemented, Sylow
322	C₂	—	4	1	no	—
323	C₂	—	120	2	no	—
324	C₂	C₂×#90960	1	1	yes	characteristic
325	C₂	C₂×#90980	1	1	yes	characteristic
326	C₂	C₂×#90981	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6667
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	#11756236	C₂
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1525
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. C₄×C₅×#208	C₂	semidirect
3. C₃×C₄×#681	C₂	semidirect
4. #11745848	C₂	semidirect
24. C₄×#681	D₃	semidirect
50. C₄×#208	D₅	semidirect
261. C₃×C₅	#1525	semidirect
306. C₅	#69859	semidirect
321. C₃	#10600652	semidirect

Maximal subgroups

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

subgroup	index
#11756236	2
C₄×C₅×#208	2
C₃×C₄×#681	2
#11745848	2
#10600652	3
#69859	5