Subgroups — #14297489

Order 1920.

624 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#4072	C₂	1	1	yes	maximal, characteristic
2	C₂×#91114	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#830	C₂	1	1	yes	maximal, characteristic
4	C₂×#90982	C₂	1	2	yes	maximal, complemented
5	#11745966	C₂	1	2	yes	maximal, complemented
6	#10600702	—	3	1	no	maximal, Hall
7	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊#168	C₂²	1	2	yes	—
9	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
10	C₂×C₃×C₂²⋊Dic₅	C₄	1	2	yes	—
11	C₃×#4072	—	2	4	no	—
12	C₂×C₃×C₄⋊Dic₅	C₄	1	2	yes	—
13	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
15	#91114	—	2	2	no	—
16	C₅×#830	—	2	2	no	—
17	#90982	—	2	4	no	—
18	#71583	—	5	1	no	maximal, Hall
19	C₂×#4072	D₃	1	1	yes	characteristic
20	C₂×C₅×C₂²⋊C₈	—	3	1	no	—
21	C₂×C₂²⋊#168	—	3	1	no	—
22	C₂×#3946	—	3	2	no	—
23	#64770	—	3	2	no	—
24	C₂⁴×C₃×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	—	2	2	no	—
26	C₂²×C₃×C₄×C₅	—	2	2	no	—
27	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
29	C₂²×C₃×C₄×C₅	Q₈	1	2	yes	—
30	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂²×C₃×Dic₅	C₂×C₄	1	2	yes	—
32	C₂²×C₃×Dic₅	C₂×C₄	1	2	yes	—
33	C₂×C₃⋊#168	—	2	2	no	—
34	C₂×C₃⋊#168	—	2	2	no	—
35	C₂×C₃⋊#168	—	2	2	no	—
36	C₂×C₅×C₃⋊C₈	—	2	2	no	—
37	C₂×C₅×C₃⋊C₈	—	2	2	no	—
38	C₂×C₅×C₃⋊C₈	—	2	2	no	—
39	C₃×C₂²⋊Dic₅	—	2	4	no	—
40	C₃×C₄⋊Dic₅	—	2	4	no	—
41	C₃×C₄×Dic₅	—	2	2	no	—
42	C₃×C₄×Dic₅	D₄	1	4	yes	—
43	C₂×C₃×#117	—	5	1	no	—
44	C₂×#830	—	5	1	no	—
45	C₂×#830	D₅	1	1	yes	characteristic
46	C₂×C₄⋊(C₃⋊C₈)	—	5	2	no	—
47	#4524	—	5	2	no	—
48	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
49	C₂²×C₅×C₈	—	3	2	no	—
50	C₂×C₂²⋊Dic₅	Dic₃	1	2	yes	—
51	#4072	—	2	4	no	—
52	C₂×C₄⋊Dic₅	Dic₃	1	2	yes	—
53	C₂×C₄×Dic₅	C₂×D₃	1	1	yes	characteristic
54	C₂×C₄×Dic₅	C₂×D₃	1	1	yes	characteristic
55	C₂²×#168	—	3	2	no	—
56	C₅×C₂²⋊C₈	—	6	2	no	—
57	C₂²⋊#168	—	6	2	no	—
58	#3946	—	6	4	no	—
59	#1798	—	15	1	no	Sylow
60	C₂³×C₃×C₅	—	2	1	no	—
61	C₂³×C₃×C₅	—	2	1	no	—
62	C₂³×C₃×C₅	—	2	2	no	—
63	C₂³×C₃×C₅	—	2	2	no	—
64	C₂³×C₃×C₅	C₄⋊C₄	1	2	yes	—
65	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	—	2	1	no	—
67	C₂×C₃×C₄×C₅	—	2	1	no	—
68	C₂×C₃×C₄×C₅	—	2	4	no	—
69	C₂×C₃×C₄×C₅	—	2	4	no	—
70	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
71	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
73	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	#41	1	2	yes	—
75	C₂×C₃×Dic₅	—	2	2	no	—
76	C₂×C₃×Dic₅	—	2	2	no	—
77	C₂×C₃×Dic₅	—	2	2	no	—
78	C₂×C₃×Dic₅	—	2	2	no	—
79	C₂×C₃×Dic₅	—	2	2	no	—
80	C₂×C₃×Dic₅	C₂²⋊C₄	1	4	yes	—
81	C₃⋊#168	—	4	2	no	—
82	C₅×C₃⋊C₈	—	4	2	no	—
83	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
84	C₃×#117	—	10	4	no	—
85	C₂×C₃×C₄⋊C₄	—	5	2	no	—
86	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
87	C₂×C₃×C₄²	—	5	1	no	—
88	C₂×C₃×C₄²	—	5	1	no	—
89	C₂²×C₃⋊C₈	—	5	2	no	—
90	C₂²×C₃⋊C₈	C₂×D₅	1	2	yes	—
91	#830	—	2	2	no	—
92	#830	—	10	2	no	—
93	C₄⋊(C₃⋊C₈)	—	10	4	no	—
94	C₂⁴×C₅	C₂×Dic₃	1	1	yes	characteristic
95	C₂²×C₄×C₅	—	2	2	no	—
96	C₂²×C₄×C₅	—	2	2	no	—
97	C₂²×C₄×C₅	Dic₆	1	2	yes	—
98	C₂²×C₄×C₅	D₁₂	1	2	yes	—
99	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
100	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
101	C₂²×Dic₅	C₂×Dic₃	1	2	yes	—
102	C₂²×Dic₅	C₂×Dic₃	1	2	yes	—
103	C₂×C₅×C₈	—	6	2	no	—
104	C₂×C₅×C₈	—	6	2	no	—
105	C₂×C₅×C₈	—	6	2	no	—
106	C₂²⋊Dic₅	—	2	4	no	—
107	C₄⋊Dic₅	—	2	4	no	—
108	C₄×Dic₅	—	2	2	no	—
109	C₄×Dic₅	#67	1	4	yes	—
110	C₂×#168	—	6	2	no	—
111	C₂×#168	—	6	2	no	—
112	C₂×#168	—	6	2	no	—
113	C₂×#117	—	5	1	no	—
114	C₂×C₄⋊C₈	—	15	2	no	—
115	C₂×C₂²⋊C₈	—	3	1	no	—
116	C₂×C₂²⋊C₈	—	15	1	no	—
117	#336	—	15	2	no	—
118	C₂²×C₃×C₅	—	2	2	no	—
119	C₂²×C₃×C₅	—	2	2	no	—
120	C₂²×C₃×C₅	—	2	2	no	—
121	C₂²×C₃×C₅	—	4	2	no	—
122	C₂²×C₃×C₅	—	4	2	no	—
123	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂²×C₃×C₅	#117	1	1	yes	characteristic
126	C₂²×C₃×C₅	#131	1	2	yes	—
127	C₂²×C₃×C₅	#131	1	2	yes	—
128	C₃×C₄×C₅	—	2	2	no	—
129	C₃×C₄×C₅	#121	1	2	yes	—
130	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
131	C₃×Dic₅	—	2	4	no	—
132	C₃×Dic₅	—	4	2	no	—
133	C₂⁴×C₃	C₄×D₅	1	1	yes	characteristic
134	C₂²×C₃×C₄	—	2	2	no	—
135	C₂²×C₃×C₄	—	2	2	no	—
136	C₂²×C₃×C₄	—	5	2	no	—
137	C₂²×C₃×C₄	—	5	2	no	—
138	C₂²×C₃×C₄	Dic₁₀	1	2	yes	—
139	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
140	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
141	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
142	C₃×C₄⋊C₄	—	10	4	no	—
143	C₃×C₂²⋊C₄	—	10	4	no	—
144	C₃×C₄²	—	5	4	no	—
145	C₃×C₄²	—	10	2	no	—
146	C₂×C₃⋊C₈	—	2	2	no	—
147	C₂×C₃⋊C₈	—	2	2	no	—
148	C₂×C₃⋊C₈	—	2	2	no	—
149	C₂×C₃⋊C₈	—	10	2	no	—
150	C₂×C₃⋊C₈	—	10	2	no	—
151	C₂×C₃⋊C₈	—	10	2	no	—
152	C₂³×C₅	—	2	1	no	—
153	C₂³×C₅	—	2	2	no	—
154	C₂³×C₅	—	2	2	no	—
155	C₂³×C₅	—	2	1	no	—
156	C₂³×C₅	C₄⋊Dic₃	1	2	yes	—
157	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
158	C₂×C₄×C₅	—	2	1	no	—
159	C₂×C₄×C₅	—	2	1	no	—
160	C₂×C₄×C₅	—	2	4	no	—
161	C₂×C₄×C₅	—	2	4	no	—
162	C₂×C₄×C₅	C₄⋊Dic₃	1	2	yes	—
163	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
164	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
165	C₂×C₄×C₅	#235	1	2	yes	—
166	C₂×C₄×C₅	C₂×#67	1	2	yes	—
167	C₂×Dic₅	—	2	2	no	—
168	C₂×Dic₅	—	2	2	no	—
169	C₂×Dic₅	—	2	2	no	—
170	C₂×Dic₅	—	2	2	no	—
171	C₂×Dic₅	—	2	2	no	—
172	C₂×Dic₅	#217	1	4	yes	—
173	C₅×C₈	—	12	2	no	—
174	#168	—	12	2	no	—
175	C₂³×C₄	D₃×D₅	1	1	yes	characteristic
176	C₂²×C₈	—	3	2	no	—
177	C₂²×C₈	—	15	2	no	—
178	#117	—	10	4	no	—
179	C₂×C₄⋊C₄	—	5	2	no	—
180	C₂×C₂²⋊C₄	—	5	2	no	—
181	C₂×C₄²	—	5	1	no	—
182	C₂×C₄²	—	5	1	no	—
183	C₄⋊C₈	—	30	4	no	—
184	C₂²⋊C₈	—	6	2	no	—
185	C₂²⋊C₈	—	30	2	no	—
186	C₂×C₃×C₅	—	4	2	no	—
187	C₂×C₃×C₅	#380	1	1	yes	characteristic
188	C₂×C₃×C₅	#408	1	2	yes	—
189	C₂×C₃×C₅	#424	1	1	yes	characteristic
190	C₂×C₃×C₅	#424	1	1	yes	characteristic
191	C₂×C₃×C₅	#433	1	2	yes	—
192	C₂³×C₃	—	2	1	no	—
193	C₂³×C₃	—	2	1	no	—
194	C₂³×C₃	—	2	2	no	—
195	C₂³×C₃	—	2	2	no	—
196	C₂³×C₃	#683	1	2	yes	—
197	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
198	C₂×C₃×C₄	—	2	1	no	—
199	C₂×C₃×C₄	—	2	1	no	—
200	C₂×C₃×C₄	—	2	4	no	—
201	C₂×C₃×C₄	—	2	4	no	—
202	C₂×C₃×C₄	—	5	4	no	—
203	C₂×C₃×C₄	—	10	2	no	—
204	C₂×C₃×C₄	—	10	2	no	—
205	C₂×C₃×C₄	—	10	2	no	—
206	C₂×C₃×C₄	—	10	2	no	—
207	C₂×C₃×C₄	—	10	2	no	—
208	C₂×C₃×C₄	#683	1	2	yes	—
209	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
210	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
211	C₂×C₃×C₄	#711	1	2	yes	—
212	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
213	C₃⋊C₈	—	4	2	no	—
214	C₃⋊C₈	—	20	2	no	—
215	C₂²×C₅	—	2	2	no	—
216	C₂²×C₅	—	2	2	no	—
217	C₂²×C₅	—	2	2	no	—
218	C₂²×C₅	—	4	2	no	—
219	C₂²×C₅	—	4	2	no	—
220	C₂²×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
221	C₂²×C₅	#926	1	1	yes	characteristic
222	C₂²×C₅	#948	1	2	yes	—
223	C₂²×C₅	#948	1	2	yes	—
224	C₂²×C₅	C₂×#217	1	1	yes	characteristic
225	C₄×C₅	—	2	2	no	—
226	C₄×C₅	#924	1	2	yes	—
227	C₄×C₅	#930	1	2	yes	—
228	Dic₅	—	2	4	no	—
229	Dic₅	—	4	2	no	—
230	C₂⁴	D₅×Dic₃	1	1	yes	characteristic
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	5	2	no	—
234	C₂²×C₄	—	5	2	no	—
235	C₂²×C₄	D₅×Dic₃	1	1	yes	characteristic
236	C₂²×C₄	#1203	1	2	yes	—
237	C₂²×C₄	#1204	1	2	yes	—
238	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
239	C₂×C₈	—	6	2	no	—
240	C₂×C₈	—	6	2	no	—
241	C₂×C₈	—	6	2	no	—
242	C₂×C₈	—	30	2	no	—
243	C₂×C₈	—	30	2	no	—
244	C₂×C₈	—	30	2	no	—
245	C₄⋊C₄	—	10	4	no	—
246	C₂²⋊C₄	—	10	4	no	—
247	C₄²	—	5	4	no	—
248	C₄²	—	10	2	no	—
249	C₃×C₅	#1798	1	1	yes	characteristic, complemented, Hall
250	C₂²×C₃	—	2	2	no	—
251	C₂²×C₃	—	2	2	no	—
252	C₂²×C₃	—	2	2	no	—
253	C₂²×C₃	—	4	2	no	—
254	C₂²×C₃	—	4	2	no	—
255	C₂²×C₃	#4023	1	1	yes	characteristic
256	C₂²×C₃	D₅×C₂²⋊C₄	1	1	yes	characteristic
257	C₂²×C₃	#4047	1	2	yes	—
258	C₂²×C₃	#4053	1	2	yes	—
259	C₂²×C₃	C₂×#683	1	1	yes	characteristic
260	C₃×C₄	—	2	2	no	—
261	C₃×C₄	—	10	4	no	—
262	C₃×C₄	—	20	2	no	—
263	C₃×C₄	#4024	1	2	yes	—
264	C₃×C₄	#4085	1	2	yes	—
265	C₂×C₅	—	4	2	no	—
266	C₂×C₅	#4999	1	1	yes	characteristic
267	C₂×C₅	#4999	1	1	yes	characteristic
268	C₂×C₅	#5000	1	2	yes	—
269	C₂×C₅	#5212	1	1	yes	characteristic
270	C₂×C₅	#5234	1	2	yes	—
271	C₂³	—	2	1	no	—
272	C₂³	—	2	2	no	—
273	C₂³	—	2	2	no	—
274	C₂³	—	2	1	no	—
275	C₂³	#6668	1	2	yes	—
276	C₂³	C₂×D₅×Dic₃	1	1	yes	characteristic
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	5	4	no	—
282	C₂×C₄	—	10	2	no	—
283	C₂×C₄	—	10	2	no	—
284	C₂×C₄	—	10	2	no	—
285	C₂×C₄	—	10	2	no	—
286	C₂×C₄	—	10	2	no	—
287	C₂×C₄	#6668	1	2	yes	—
288	C₂×C₄	#6782	1	2	yes	—
289	C₂×C₄	C₂×#1203	1	1	yes	characteristic
290	C₂×C₄	C₂×#1204	1	1	yes	characteristic
291	C₂×C₄	D₅×#67	1	2	yes	—
292	C₈	—	12	2	no	—
293	C₈	—	60	2	no	—
294	C₂×C₃	—	4	2	no	—
295	C₂×C₃	#65009	1	2	yes	—
296	C₂×C₃	#65016	1	2	yes	—
297	C₂×C₃	#65231	1	1	yes	characteristic
298	C₂×C₃	#65391	1	1	yes	characteristic
299	C₂×C₃	#65408	1	1	yes	characteristic
300	C₅	#71583	1	1	yes	characteristic, complemented, Sylow
301	C₂²	—	2	2	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	4	2	no	—
305	C₂²	—	4	2	no	—
306	C₂²	#91283	1	2	yes	—
307	C₂²	#91284	1	2	yes	—
308	C₂²	#91528	1	1	yes	characteristic
309	C₂²	C₂×#6668	1	1	yes	characteristic
310	C₂²	D₅×#217	1	1	yes	characteristic
311	C₄	—	2	2	no	—
312	C₄	—	10	4	no	—
313	C₄	—	20	2	no	—
314	C₄	#91566	1	2	yes	—
315	C₄	#91574	1	2	yes	—
316	C₃	#10600702	1	1	yes	characteristic, complemented, Sylow
317	C₂	—	4	2	no	—
318	C₂	#11748319	1	2	yes	—
319	C₂	#11748432	1	1	yes	characteristic
320	C₂	#11748433	1	1	yes	characteristic
321	C₂	#11748482	1	2	yes	—
322	C₂	#11749506	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂×D₃×D₅
commutator, socle	C₂³×C₃×C₅	C₂²×C₄
Fitting	C₂³×C₃×C₄×C₅	C₂²
prime radical	C₂⁴×C₃×C₅	C₂×C₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂³×C₄	D₃×D₅
hypocenter	C₃×C₅	#1798
LCS	G ▷ C₂³×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
4. C₂×#90982	C₂	semidirect
5. #11745966	C₂	semidirect
249. C₃×C₅	#1798	semidirect
300. C₅	#71583	semidirect
316. C₃	#10600702	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#4072	2
C₂×#91114	2
C₂×C₅×#830	2
C₂×#90982	2
#11745966	2
#10600702	3
#71583	5