Subgroups — #14283994

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	C₃×#65811	C₂	1	1	yes	maximal, characteristic
2	C₄×#6710	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₅×#208	C₂	1	1	yes	maximal, characteristic, complemented
4	#11745846	C₂	1	4	yes	maximal, complemented
5	#10600652	—	3	1	no	maximal, Hall
6	C₂×C₃×#709	C₄	1	2	yes	—
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂×#6710	—	2	1	no	—
9	C₂×C₅×#208	—	2	1	no	—
10	C₃×#4076	—	2	1	no	—
11	C₃×#4076	—	2	1	no	—
12	C₃×#4021	C₂²	1	2	yes	—
13	C₃×#4020	C₄	1	1	yes	characteristic
14	C₃×C₄×D₂₀	—	2	1	no	—
15	C₃×C₄×D₂₀	—	2	1	no	—
16	C₃×#4015	C₄	1	1	yes	characteristic
17	#91087	C₂²	1	2	yes	—
18	C₄×C₃⋊#168	—	2	1	no	—
19	C₅×#803	C₂²	1	2	yes	—
20	C₄×C₅×C₃⋊C₈	—	2	1	no	—
21	#71797	—	5	1	no	maximal, Hall
22	#65811	D₃	1	1	yes	characteristic
23	C₄×C₅×M₁₆	—	3	1	no	—
24	C₄×#681	—	3	1	no	—
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×C₃×C₂²⋊D₅	—	2	2	no	—
30	C₃×#709	—	4	4	no	—
31	C₂×C₃×D₂₀	—	2	1	no	—
32	C₂×C₃×D₂₀	—	2	1	no	—
33	C₂×C₃×D₂₀	C₂×C₄	1	2	yes	—
34	C₂×C₃×C₄×D₅	—	2	2	no	—
35	C₂×C₃×Dic₁₀	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	#6710	—	2	4	no	—
41	C₂×C₃⋊#168	—	2	2	no	—
42	C₅×#208	—	2	4	no	—
43	C₂×C₅×C₃⋊C₈	—	2	2	no	—
44	C₃×#685	—	2	2	no	—
45	C₃×#685	—	2	2	no	—
46	C₃×C₄⋊Dic₅	—	2	1	no	—
47	C₃×C₄⋊Dic₅	—	2	1	no	—
48	C₃×#532	—	5	1	no	—
49	C₄×#208	—	5	1	no	—
50	C₄×#208	D₅	1	1	yes	characteristic, complemented
51	#4540	—	5	4	no	—
52	C₂×#709	Dic₃	1	2	yes	—
53	C₂×C₅×M₁₆	—	6	1	no	—
54	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
55	#4076	—	2	1	no	—
56	#4076	—	2	1	no	—
57	C₂×#681	—	6	1	no	—
58	#4021	C₂×D₃	1	2	yes	—
59	#4020	Dic₃	1	1	yes	characteristic
60	C₄×D₂₀	—	2	1	no	—
61	C₄×D₂₀	—	2	1	no	—
62	#4015	Dic₃	1	1	yes	characteristic
63	C₅×#97	—	3	2	no	—
64	C₄×C₅×C₈	—	6	1	no	—
65	#3935	—	3	2	no	—
66	C₄×#168	—	6	1	no	—
67	#1525	—	15	1	no	Sylow
68	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₃×D₅	—	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	C₃×C₂²⋊D₅	—	4	4	no	—
80	C₂×C₃×Dic₅	—	2	2	no	—
81	C₃×D₂₀	—	4	2	no	—
82	C₃×D₂₀	—	4	2	no	—
83	C₃×D₂₀	—	4	2	no	—
84	C₃×C₄×D₅	—	4	4	no	—
85	C₃×Dic₁₀	—	4	2	no	—
86	C₃⋊#168	—	2	4	no	—
87	C₅×C₃⋊C₈	—	2	4	no	—
88	C₂×C₃×#41	—	5	2	no	—
89	C₃×C₄⋊Q₈	—	5	1	no	—
90	C₃×#127	—	5	1	no	—
91	C₃×#124	—	5	2	no	—
92	C₃×#121	—	10	1	no	—
93	C₃×#121	—	10	1	no	—
94	C₃×C₄×D₄	—	10	1	no	—
95	C₃×C₄×D₄	—	10	1	no	—
96	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
97	C₂×#208	—	2	1	no	—
98	C₂×#208	—	10	1	no	—
99	#803	—	5	2	no	—
100	#803	C₂×D₅	1	2	yes	—
101	C₄×C₃⋊C₈	—	2	1	no	—
102	C₄×C₃⋊C₈	—	10	1	no	—
103	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
104	C₂²×C₄×C₅	#67	1	1	yes	characteristic
105	C₂²×C₄×C₅	#67	1	1	yes	characteristic
106	C₂×C₂²⋊D₅	—	2	2	no	—
107	#709	—	4	4	no	—
108	C₂×D₂₀	—	2	1	no	—
109	C₂×D₂₀	—	2	1	no	—
110	C₂×D₂₀	C₂×Dic₃	1	2	yes	—
111	C₂×C₄×D₅	—	2	2	no	—
112	C₂×Dic₁₀	C₂×Dic₃	1	2	yes	—
113	C₅×M₁₆	—	6	4	no	—
114	C₂×C₅×C₈	—	6	2	no	—
115	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
116	C₄²×C₅	#67	1	1	yes	characteristic
117	C₄²×C₅	#67	1	1	yes	characteristic
118	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
119	#685	—	2	2	no	—
120	#685	—	2	2	no	—
121	C₄⋊Dic₅	—	2	1	no	—
122	C₄⋊Dic₅	—	2	1	no	—
123	#681	—	6	4	no	—
124	C₂×#168	—	6	2	no	—
125	#532	—	5	1	no	—
126	C₄×M₁₆	—	3	1	no	—
127	C₄×M₁₆	—	15	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×C₃×D₅	—	4	2	no	—
134	C₂×C₃×D₅	—	4	2	no	—
135	C₂×C₃×D₅	—	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	C₃×Dic₅	—	4	2	no	—
143	C₃×#41	—	20	4	no	—
144	C₂×C₃×Q₈	—	5	2	no	—
145	C₂×C₃×D₄	—	5	2	no	—
146	C₂×C₃×D₄	—	10	2	no	—
147	C₂×C₃×D₄	—	10	1	no	—
148	C₂×C₃×D₄	—	10	1	no	—
149	C₂²×C₃×C₄	—	10	2	no	—
150	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
151	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
152	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
153	C₃×C₄⋊C₄	—	10	1	no	—
154	C₃×C₄⋊C₄	—	10	1	no	—
155	C₃×C₂²⋊C₄	—	10	2	no	—
156	C₃×C₂²⋊C₄	—	10	2	no	—
157	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
158	C₃×C₄²	D₂₀	1	1	yes	characteristic
159	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
160	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
161	#208	—	2	4	no	—
162	#208	—	10	4	no	—
163	C₂×C₃⋊C₈	—	2	2	no	—
164	C₂×C₃⋊C₈	—	10	2	no	—
165	C₂³×C₅	#217	1	1	yes	characteristic
166	C₂²×D₅	—	2	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	#217	1	1	yes	characteristic
171	C₂×C₄×C₅	#217	1	1	yes	characteristic
172	C₂×C₄×C₅	#217	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
175	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
176	C₂²⋊D₅	—	4	4	no	—
177	C₂×Dic₅	—	2	2	no	—
178	D₂₀	—	4	2	no	—
179	D₂₀	—	4	2	no	—
180	D₂₀	—	4	2	no	—
181	C₄×D₅	—	4	4	no	—
182	Dic₁₀	—	4	2	no	—
183	C₅×C₈	—	6	4	no	—
184	#168	—	6	4	no	—
185	C₂×#41	—	5	2	no	—
186	C₂×M₁₆	—	6	1	no	—
187	C₂×M₁₆	—	30	1	no	—
188	C₄⋊Q₈	—	5	1	no	—
189	#127	—	5	1	no	—
190	#124	—	5	2	no	—
191	#121	—	10	1	no	—
192	#121	—	10	1	no	—
193	C₄×D₄	—	10	1	no	—
194	C₄×D₄	—	10	1	no	—
195	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
196	#97	—	3	2	no	—
197	#97	—	15	2	no	—
198	C₄×C₈	—	6	1	no	—
199	C₄×C₈	—	30	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	C₃×D₅	—	8	2	no	—
205	C₂³×C₃	—	10	2	no	—
206	C₂³×C₃	#685	1	1	yes	characteristic
207	C₃×Q₈	—	20	2	no	—
208	C₃×D₄	—	20	2	no	—
209	C₃×D₄	—	20	4	no	—
210	C₃×D₄	—	20	2	no	—
211	C₃×D₄	—	20	2	no	—
212	C₂×C₃×C₄	—	2	2	no	—
213	C₂×C₃×C₄	—	2	2	no	—
214	C₂×C₃×C₄	—	2	2	no	—
215	C₂×C₃×C₄	—	10	2	no	—
216	C₂×C₃×C₄	—	20	4	no	—
217	C₂×C₃×C₄	#685	1	1	yes	characteristic
218	C₂×C₃×C₄	#685	1	1	yes	characteristic
219	C₂×C₃×C₄	#685	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
221	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
222	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
223	C₃⋊C₈	—	2	4	no	—
224	C₃⋊C₈	—	10	4	no	—
225	C₂²×C₅	—	2	1	no	—
226	C₂²×C₅	—	2	1	no	—
227	C₂²×C₅	—	2	1	no	—
228	C₂²×C₅	C₂×#217	1	1	yes	characteristic
229	C₂×D₅	—	4	2	no	—
230	C₂×D₅	—	4	2	no	—
231	C₂×D₅	—	4	2	no	—
232	C₄×C₅	—	2	1	no	—
233	C₄×C₅	—	2	1	no	—
234	C₄×C₅	—	2	1	no	—
235	C₄×C₅	#833	1	2	yes	—
236	C₄×C₅	#837	1	2	yes	—
237	C₄×C₅	#837	1	2	yes	—
238	Dic₅	—	4	2	no	—
239	#41	—	20	4	no	—
240	C₂×Q₈	—	5	2	no	—
241	C₂×D₄	—	5	2	no	—
242	C₂×D₄	—	10	2	no	—
243	C₂×D₄	—	10	1	no	—
244	C₂×D₄	—	10	1	no	—
245	C₂²×C₄	—	10	2	no	—
246	C₂²×C₄	D₅×Dic₃	1	1	yes	characteristic
247	C₂²×C₄	#1201	1	1	yes	characteristic
248	C₂²×C₄	#1202	1	1	yes	characteristic
249	M₁₆	—	6	4	no	—
250	M₁₆	—	30	4	no	—
251	C₂×C₈	—	6	2	no	—
252	C₂×C₈	—	30	2	no	—
253	C₄⋊C₄	—	10	1	no	—
254	C₄⋊C₄	—	10	1	no	—
255	C₂²⋊C₄	—	10	2	no	—
256	C₂²⋊C₄	—	10	2	no	—
257	C₄²	D₅×Dic₃	1	1	yes	characteristic
258	C₄²	#1201	1	1	yes	characteristic
259	C₄²	#1202	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	—	20	2	no	—
266	C₂²×C₃	—	20	2	no	—
267	C₂²×C₃	—	20	2	no	—
268	C₂²×C₃	C₂×#685	1	1	yes	characteristic
269	C₃×C₄	—	2	1	no	—
270	C₃×C₄	—	2	1	no	—
271	C₃×C₄	—	2	1	no	—
272	C₃×C₄	—	20	2	no	—
273	C₃×C₄	#3937	1	2	yes	—
274	C₃×C₄	#3955	1	2	yes	—
275	C₃×C₄	#3957	1	2	yes	—
276	C₂×C₅	—	4	1	no	—
277	C₂×C₅	C₂×#833	1	1	yes	characteristic
278	C₂×C₅	C₂×#837	1	1	yes	characteristic
279	C₂×C₅	C₂×#837	1	1	yes	characteristic
280	D₅	—	8	2	no	—
281	C₂³	—	10	2	no	—
282	C₂³	#6665	1	1	yes	characteristic
283	Q₈	—	20	2	no	—
284	D₄	—	20	2	no	—
285	D₄	—	20	4	no	—
286	D₄	—	20	2	no	—
287	D₄	—	20	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	10	2	no	—
292	C₂×C₄	—	20	4	no	—
293	C₂×C₄	#6665	1	1	yes	characteristic
294	C₂×C₄	#6665	1	1	yes	characteristic
295	C₂×C₄	#6665	1	1	yes	characteristic
296	C₂×C₄	C₂×D₅×Dic₃	1	1	yes	characteristic
297	C₂×C₄	C₂×#1201	1	1	yes	characteristic
298	C₂×C₄	C₂×#1202	1	1	yes	characteristic
299	C₈	—	6	4	no	—
300	C₈	—	30	4	no	—
301	C₂×C₃	—	4	1	no	—
302	C₂×C₃	—	40	2	no	—
303	C₂×C₃	C₂×#3937	1	1	yes	characteristic
304	C₂×C₃	C₂×#3955	1	1	yes	characteristic
305	C₂×C₃	C₂×#3957	1	1	yes	characteristic
306	C₅	#71797	1	1	yes	characteristic, complemented, Sylow
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	20	2	no	—
311	C₂²	—	20	2	no	—
312	C₂²	—	20	2	no	—
313	C₂²	C₂×#6665	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	20	2	no	—
318	C₄	#90958	1	2	yes	—
319	C₄	#90976	1	2	yes	—
320	C₄	#90977	1	2	yes	—
321	C₃	#10600652	1	1	yes	characteristic, complemented, Sylow
322	C₂	—	4	1	no	—
323	C₂	—	40	2	no	—
324	C₂	C₂×#90958	1	1	yes	characteristic
325	C₂	C₂×#90976	1	1	yes	characteristic
326	C₂	C₂×#90977	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6665
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	C₃×#65811	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×#6710	C₂	semidirect
3. C₄×C₅×#208	C₂	semidirect
4. #11745846	C₂	semidirect
50. C₄×#208	D₅	semidirect
261. C₃×C₅	#1525	semidirect
306. C₅	#71797	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
C₃×#65811	2
C₄×#6710	2
C₄×C₅×#208	2
#11745846	2
#10600652	3
#71797	5