Subgroups — #14286556

Order 1920.

364 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×D₄₀	C₂	1	1	yes	maximal, characteristic
2	#11747010	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃⋊C₈×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746279	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4454	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745835	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745835	C₂	1	1	yes	maximal, characteristic, complemented
8	#10598871	—	3	1	no	maximal, Hall
9	C₂×C₃×D₄₀	C₄	1	1	yes	characteristic
10	C₃×C₄×D₂₀	C₂²	1	1	yes	characteristic
11	C₃×C₄×D₂₀	C₂²	1	1	yes	characteristic
12	C₂×#6649	—	2	1	no	—
13	C₂×D₅×C₃⋊C₈	—	2	1	no	—
14	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
16	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
17	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₃×#3953	C₄	1	1	yes	characteristic
20	C₃×#3953	C₄	1	1	yes	characteristic
21	C₃×#3950	C₄	1	1	yes	characteristic
22	#90955	—	2	1	no	—
23	#90955	—	2	1	no	—
24	#70879	—	5	1	no	maximal, Hall
25	C₄×D₄₀	D₃	1	1	yes	characteristic
26	#64977	—	3	1	no	—
27	C₈×D₂₀	—	3	1	no	—
28	C₅×#322	—	3	1	no	—
29	#64679	—	3	1	no	—
30	#64679	—	3	1	no	—
31	#64678	—	3	1	no	—
32	C₂×C₃×D₂₀	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×D₂₀	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃×C₄×D₅	—	2	1	no	—
35	C₂×C₃×C₄×D₅	—	2	1	no	—
36	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
38	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
39	C₂×C₃⋊#168	—	2	1	no	—
40	C₂×C₃⋊#168	—	2	1	no	—
41	C₂×C₅×C₃⋊C₈	—	2	1	no	—
42	C₂×C₅×C₃⋊C₈	D₄	1	1	yes	characteristic
43	C₂×C₅×C₃⋊C₈	D₄	1	1	yes	characteristic
44	C₃×#685	—	2	1	no	—
45	C₃×#685	—	2	1	no	—
46	C₃×C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
47	C₃×C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
48	C₃×D₄₀	—	2	2	no	—
49	#6649	—	4	2	no	—
50	D₅×C₃⋊C₈	—	4	2	no	—
51	C₃×C₄×D₈	—	5	1	no	—
52	#5012	—	5	1	no	—
53	D₄×C₃⋊C₈	—	5	1	no	—
54	D₄⋊(C₃⋊C₈)	—	5	1	no	—
55	D₄⋊(C₃⋊C₈)	—	5	1	no	—
56	#4458	—	5	1	no	—
57	#4454	D₅	1	1	yes	characteristic, complemented
58	C₂×D₄₀	Dic₃	1	1	yes	characteristic
59	C₂×#676	—	6	1	no	—
60	C₂×C₈×D₅	—	6	1	no	—
61	C₄×D₂₀	C₂×D₃	1	1	yes	characteristic
62	C₄×D₂₀	C₂×D₃	1	1	yes	characteristic
63	C₄×C₅×C₈	—	3	1	no	—
64	C₄×C₅×C₈	—	3	1	no	—
65	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
66	#3953	Dic₃	1	1	yes	characteristic
67	#3953	Dic₃	1	1	yes	characteristic
68	#3952	—	6	1	no	—
69	#3952	—	6	1	no	—
70	#3950	Dic₃	1	1	yes	characteristic
71	C₄⋊#168	—	3	1	no	—
72	C₄⋊#168	—	3	1	no	—
73	#1581	—	15	1	no	Sylow
74	C₂²×C₃×D₅	—	2	1	no	—
75	C₂²×C₃×D₅	—	2	1	no	—
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	#41	1	1	yes	characteristic
79	C₂×C₃×Dic₅	—	2	1	no	—
80	C₂×C₃×Dic₅	—	2	1	no	—
81	C₃×D₂₀	—	2	1	no	—
82	C₃×D₂₀	—	2	1	no	—
83	C₃×D₂₀	—	2	1	no	—
84	C₃×D₂₀	M₁₆	1	2	yes	complemented
85	C₃×C₄×D₅	—	4	1	no	—
86	C₃×C₄×D₅	—	4	1	no	—
87	C₃×C₄×D₅	—	4	1	no	—
88	C₃×C₄×D₅	—	4	1	no	—
89	C₃×C₅×C₈	—	2	1	no	—
90	C₃×C₅×C₈	—	2	1	no	—
91	C₃⋊#168	—	4	1	no	—
92	C₃⋊#168	—	4	1	no	—
93	C₅×C₃⋊C₈	—	2	1	no	—
94	C₅×C₃⋊C₈	—	4	1	no	—
95	C₅×C₃⋊C₈	D₈	1	2	yes	complemented
96	C₂×C₃×D₈	—	5	1	no	—
97	C₃×C₄×D₄	—	5	1	no	—
98	C₃×C₄×D₄	—	5	1	no	—
99	C₂×#208	—	10	1	no	—
100	C₂²×C₃⋊C₈	—	10	1	no	—
101	C₃×#107	—	5	1	no	—
102	C₃×#102	—	5	1	no	—
103	C₃×#102	—	5	1	no	—
104	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
105	#830	—	10	1	no	—
106	#830	—	10	1	no	—
107	C₄⋊(C₃⋊C₈)	—	5	1	no	—
108	C₄⋊(C₃⋊C₈)	—	5	1	no	—
109	C₄×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
110	C₄×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
111	C₂×D₂₀	C₂×Dic₃	1	1	yes	characteristic
112	C₂×D₂₀	C₂×Dic₃	1	1	yes	characteristic
113	C₂×C₄×D₅	—	2	1	no	—
114	C₂×C₄×D₅	—	2	1	no	—
115	C₂×C₅×C₈	—	3	1	no	—
116	C₂×C₅×C₈	—	3	1	no	—
117	C₂×C₅×C₈	—	6	1	no	—
118	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
119	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
120	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
121	#685	—	2	1	no	—
122	#685	—	2	1	no	—
123	C₄⋊Dic₅	C₂×Dic₃	1	1	yes	characteristic
124	C₄⋊Dic₅	C₂×Dic₃	1	1	yes	characteristic
125	C₂×#168	—	6	1	no	—
126	C₂×#168	—	6	1	no	—
127	D₄₀	—	2	2	no	—
128	#676	—	12	2	no	—
129	C₈×D₅	—	12	2	no	—
130	C₄×D₈	—	5	1	no	—
131	#436	—	15	1	no	—
132	C₈×D₄	—	15	1	no	—
133	#335	—	15	1	no	—
134	D₄⋊C₈	—	15	1	no	—
135	D₄⋊C₈	—	15	1	no	—
136	#322	—	3	1	no	—
137	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
138	C₂×C₃×D₅	—	2	2	no	—
139	C₂×C₃×D₅	—	4	1	no	—
140	C₂×C₃×D₅	—	4	1	no	—
141	C₂×C₃×D₅	—	4	1	no	—
142	C₂×C₃×D₅	—	4	1	no	—
143	C₂×C₃×D₅	—	4	1	no	—
144	C₃×C₄×C₅	—	2	1	no	—
145	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
146	C₃×C₄×C₅	#131	1	1	yes	characteristic
147	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
148	C₃×C₄×C₅	#135	1	1	yes	characteristic
149	C₃×Dic₅	—	4	1	no	—
150	C₃×Dic₅	—	4	1	no	—
151	C₂×C₃×D₄	—	5	1	no	—
152	C₂×C₃×D₄	—	5	1	no	—
153	C₂²×C₃×C₄	—	10	1	no	—
154	C₂²×C₃×C₄	—	10	1	no	—
155	C₃×D₈	—	10	2	no	—
156	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
157	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
158	C₃×C₄⋊C₄	—	5	1	no	—
159	C₃×C₄⋊C₄	—	5	1	no	—
160	C₃×C₂²⋊C₄	—	10	1	no	—
161	C₃×C₂²⋊C₄	—	10	1	no	—
162	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
163	#208	—	20	2	no	—
164	C₂×C₃⋊C₈	—	2	1	no	—
165	C₂×C₃⋊C₈	—	10	1	no	—
166	C₂×C₃⋊C₈	—	10	1	no	—
167	C₂×C₃⋊C₈	—	20	2	no	—
168	C₂×C₃⋊C₈	D₂₀	1	1	yes	characteristic
169	C₂×C₃⋊C₈	D₂₀	1	1	yes	characteristic
170	C₂²×D₅	—	2	1	no	—
171	C₂²×D₅	—	2	1	no	—
172	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
173	C₂×C₄×C₅	#237	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
175	C₂×Dic₅	—	2	1	no	—
176	C₂×Dic₅	—	2	1	no	—
177	D₂₀	—	2	1	no	—
178	D₂₀	—	2	1	no	—
179	D₂₀	—	2	1	no	—
180	D₂₀	#208	1	2	yes	complemented
181	C₄×D₅	—	4	1	no	—
182	C₄×D₅	—	4	1	no	—
183	C₄×D₅	—	4	1	no	—
184	C₄×D₅	—	4	1	no	—
185	C₅×C₈	—	2	1	no	—
186	C₅×C₈	—	2	1	no	—
187	C₅×C₈	—	3	2	no	—
188	C₅×C₈	—	6	1	no	—
189	C₅×C₈	—	12	1	no	—
190	#168	—	12	1	no	—
191	#168	—	12	1	no	—
192	C₂×D₈	—	5	1	no	—
193	C₂×M₁₆	—	30	1	no	—
194	C₂²×C₈	—	30	1	no	—
195	C₄×D₄	—	5	1	no	—
196	C₄×D₄	—	5	1	no	—
197	#107	—	5	1	no	—
198	C₄⋊C₈	—	15	1	no	—
199	C₄⋊C₈	—	15	1	no	—
200	#102	—	5	1	no	—
201	#102	—	5	1	no	—
202	C₂²⋊C₈	—	30	1	no	—
203	C₂²⋊C₈	—	30	1	no	—
204	C₄×C₈	—	3	1	no	—
205	C₄×C₈	—	3	1	no	—
206	C₄×C₈	D₃×D₅	1	1	yes	characteristic
207	C₂×C₃×C₅	#435	1	1	yes	characteristic
208	C₂×C₃×C₅	C₄×D₈	1	1	yes	characteristic
209	C₂×C₃×C₅	#444	1	1	yes	characteristic
210	C₃×D₅	—	4	2	no	—
211	C₃×D₅	—	8	1	no	—
212	C₂³×C₃	—	10	1	no	—
213	C₂³×C₃	—	10	1	no	—
214	C₃×D₄	—	5	2	no	—
215	C₃×D₄	—	10	1	no	—
216	C₃×D₄	—	10	1	no	—
217	C₃×D₄	—	10	1	no	—
218	C₂×C₃×C₄	—	10	1	no	—
219	C₂×C₃×C₄	—	10	1	no	—
220	C₂×C₃×C₄	—	20	1	no	—
221	C₂×C₃×C₄	—	20	1	no	—
222	C₂×C₃×C₄	—	20	1	no	—
223	C₂×C₃×C₄	—	20	1	no	—
224	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
225	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
226	C₂×C₃×C₄	#709	1	1	yes	characteristic
227	C₃×C₈	—	2	1	no	—
228	C₃×C₈	—	2	1	no	—
229	C₃⋊C₈	—	2	1	no	—
230	C₃⋊C₈	—	4	1	no	—
231	C₃⋊C₈	—	20	1	no	—
232	C₃⋊C₈	—	20	1	no	—
233	C₃⋊C₈	D₄₀	1	2	yes	complemented
234	C₂²×C₅	D₄×Dic₃	1	1	yes	characteristic
235	C₂×D₅	—	2	2	no	—
236	C₂×D₅	—	4	1	no	—
237	C₂×D₅	—	4	1	no	—
238	C₂×D₅	—	4	1	no	—
239	C₂×D₅	—	4	1	no	—
240	C₂×D₅	—	4	1	no	—
241	C₄×C₅	—	2	1	no	—
242	C₄×C₅	D₃×D₈	1	1	yes	characteristic
243	C₄×C₅	#912	1	1	yes	characteristic
244	C₄×C₅	C₂×#208	1	1	yes	characteristic
245	C₄×C₅	#948	1	1	yes	characteristic
246	Dic₅	—	4	1	no	—
247	Dic₅	—	4	1	no	—
248	C₂×D₄	—	5	1	no	—
249	C₂×D₄	—	5	1	no	—
250	C₂²×C₄	—	10	1	no	—
251	C₂²×C₄	—	10	1	no	—
252	D₈	—	10	2	no	—
253	M₁₆	—	60	2	no	—
254	C₂×C₈	—	3	1	no	—
255	C₂×C₈	—	3	1	no	—
256	C₂×C₈	—	6	1	no	—
257	C₂×C₈	—	30	1	no	—
258	C₂×C₈	—	30	1	no	—
259	C₂×C₈	—	60	2	no	—
260	C₂×C₈	D₅×Dic₃	1	1	yes	characteristic
261	C₂×C₈	D₅×Dic₃	1	1	yes	characteristic
262	C₄⋊C₄	—	5	1	no	—
263	C₄⋊C₄	—	5	1	no	—
264	C₂²⋊C₄	—	10	1	no	—
265	C₂²⋊C₄	—	10	1	no	—
266	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
267	C₃×C₅	#1581	1	1	yes	characteristic, complemented, Hall
268	C₂²×C₃	—	10	2	no	—
269	C₂²×C₃	—	20	1	no	—
270	C₂²×C₃	—	20	1	no	—
271	C₂²×C₃	—	20	1	no	—
272	C₂²×C₃	—	20	1	no	—
273	C₂²×C₃	—	20	1	no	—
274	C₂²×C₃	C₄×D₂₀	1	1	yes	characteristic
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	20	1	no	—
277	C₃×C₄	—	20	1	no	—
278	C₃×C₄	C₂×D₄₀	1	1	yes	characteristic
279	C₃×C₄	#4050	1	1	yes	characteristic
280	C₃×C₄	D₅×M₁₆	1	1	yes	characteristic
281	C₃×C₄	#4053	1	1	yes	characteristic
282	C₂×C₅	#5011	1	1	yes	characteristic
283	C₂×C₅	Dic₃×D₈	1	1	yes	characteristic
284	C₂×C₅	#5197	1	1	yes	characteristic
285	D₅	—	4	2	no	—
286	D₅	—	8	1	no	—
287	C₂³	—	10	1	no	—
288	C₂³	—	10	1	no	—
289	D₄	—	5	2	no	—
290	D₄	—	10	1	no	—
291	D₄	—	10	1	no	—
292	D₄	—	10	1	no	—
293	C₂×C₄	—	10	1	no	—
294	C₂×C₄	—	10	1	no	—
295	C₂×C₄	—	20	1	no	—
296	C₂×C₄	—	20	1	no	—
297	C₂×C₄	—	20	1	no	—
298	C₂×C₄	—	20	1	no	—
299	C₂×C₄	#6765	1	1	yes	characteristic
300	C₂×C₄	D₃×D₂₀	1	1	yes	characteristic
301	C₂×C₄	C₂×D₅×Dic₃	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	2	1	no	—
304	C₈	—	3	2	no	—
305	C₈	—	6	1	no	—
306	C₈	—	12	1	no	—
307	C₈	—	60	1	no	—
308	C₈	—	60	1	no	—
309	C₂×C₃	—	20	2	no	—
310	C₂×C₃	—	40	1	no	—
311	C₂×C₃	C₄×D₄₀	1	1	yes	characteristic
312	C₂×C₃	#64995	1	1	yes	characteristic
313	C₂×C₃	#65006	1	1	yes	characteristic
314	C₅	#70879	1	1	yes	characteristic, complemented, Sylow
315	C₂²	—	10	2	no	—
316	C₂²	—	20	1	no	—
317	C₂²	—	20	1	no	—
318	C₂²	—	20	1	no	—
319	C₂²	—	20	1	no	—
320	C₂²	—	20	1	no	—
321	C₂²	Dic₃×D₂₀	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	20	1	no	—
324	C₄	—	20	1	no	—
325	C₄	D₃×D₄₀	1	1	yes	characteristic
326	C₄	#91273	1	1	yes	characteristic
327	C₄	D₅×#208	1	1	yes	characteristic
328	C₄	#91284	1	1	yes	characteristic
329	C₃	#10598871	1	1	yes	characteristic, complemented, Sylow
330	C₂	—	20	2	no	—
331	C₂	—	40	1	no	—
332	C₂	#11746991	1	1	yes	characteristic
333	C₂	Dic₃×D₄₀	1	1	yes	characteristic
334	C₂	#11748067	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₃×D₂₀
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₃×C₄×C₅×C₈	C₂²
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂×C₃×D₄₀	C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄×C₈	D₃×D₅
hypocenter	C₃×C₅	#1581
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #11747010	C₂	semidirect
3. C₃⋊C₈×D₂₀	C₂	semidirect
4. #11746279	C₂	semidirect
5. C₅×#4454	C₂	semidirect
6. #11745835	C₂	semidirect
7. #11745835	C₂	semidirect
57. #4454	D₅	semidirect
84. C₃×D₂₀	M₁₆	semidirect
95. C₅×C₃⋊C₈	D₈	semidirect
180. D₂₀	#208	semidirect
233. C₃⋊C₈	D₄₀	semidirect
267. C₃×C₅	#1581	semidirect
314. C₅	#70879	semidirect
329. C₃	#10598871	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×D₄₀	2
#11747010	2
C₃⋊C₈×D₂₀	2
#11746279	2
C₅×#4454	2
#11745835	2
#10598871	3
#70879	5