Subgroups — #14286543

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	#11751113	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₅×#213	C₂	1	1	yes	maximal, characteristic
3	#11746998	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×D₄⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
5	#11745869	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745837	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745833	C₂	1	1	yes	maximal, characteristic, complemented
8	#10601672	—	3	1	no	maximal, Hall
9	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
10	C₂×#6710	—	2	1	no	—
11	C₂×C₅×#213	C₄	1	1	yes	characteristic
12	C₄×C₅×D₁₂	C₂²	1	1	yes	characteristic
13	C₂×D₃×#168	—	2	1	no	—
14	#91114	—	2	1	no	—
15	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
16	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
17	C₅×#832	C₄	1	1	yes	characteristic
18	C₅×#809	C₄	1	1	yes	characteristic
19	C₅×#807	C₄	1	1	yes	characteristic
20	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
22	C₃×C₄×#168	C₂²	1	1	yes	characteristic
23	(C₂×D₃)⋊#168	—	2	1	no	—
24	#68153	—	5	1	no	maximal, Hall
25	C₄×C₅×D₈	—	3	1	no	—
26	#65301	—	3	1	no	—
27	D₄×#168	—	3	1	no	—
28	D₄⋊#168	—	3	1	no	—
29	D₄⋊#168	D₃	1	1	yes	characteristic, complemented
30	#64678	—	3	1	no	—
31	#64674	—	3	1	no	—
32	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₃×C₄×C₅	—	2	1	no	—
34	C₂×C₅×D₁₂	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×C₅×D₃	—	2	1	no	—
36	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
37	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
38	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
39	#6710	—	4	2	no	—
40	C₂×C₃⋊#168	—	2	1	no	—
41	C₂×C₃⋊#168	D₄	1	1	yes	characteristic
42	C₂×C₃⋊#168	D₄	1	1	yes	characteristic
43	C₅×#213	—	2	2	no	—
44	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
45	C₅×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
48	C₂×C₃×#168	—	2	1	no	—
49	C₂×C₃×#168	—	2	1	no	—
50	D₃×#168	—	4	2	no	—
51	C₄×#213	D₅	1	1	yes	characteristic
52	#5012	—	5	1	no	—
53	#4834	—	5	1	no	—
54	C₃×D₄⋊C₈	—	5	1	no	—
55	#4479	—	5	1	no	—
56	#4459	—	5	1	no	—
57	#4454	—	5	1	no	—
58	C₂×C₅×D₈	—	3	1	no	—
59	C₄×C₅×D₄	—	3	1	no	—
60	C₄×C₅×D₄	C₂×D₃	1	1	yes	characteristic
61	C₂×#681	—	6	1	no	—
62	C₂²×#168	—	6	1	no	—
63	C₅×#107	—	3	1	no	—
64	C₅×#102	—	3	1	no	—
65	C₅×#102	—	3	1	no	—
66	C₄×C₅×C₈	—	3	1	no	—
67	C₂²⋊#168	—	6	1	no	—
68	C₂²⋊#168	—	6	1	no	—
69	C₄⋊#168	—	3	1	no	—
70	C₄⋊#168	C₂×D₃	1	1	yes	characteristic
71	C₄×#168	—	3	1	no	—
72	C₄×#168	C₂×D₃	1	1	yes	characteristic
73	#1581	—	15	1	no	Sylow
74	C₂³×C₃×C₅	—	2	1	no	—
75	C₂²×C₅×D₃	—	2	1	no	—
76	C₃×C₅×D₄	—	2	1	no	—
77	C₃×C₅×D₄	—	2	1	no	—
78	C₂×C₃×C₄×C₅	—	2	1	no	—
79	C₂×C₃×C₄×C₅	—	4	1	no	—
80	C₂×C₃×C₄×C₅	—	4	1	no	—
81	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
84	C₂×C₅×Dic₃	—	2	1	no	—
85	C₅×D₁₂	—	2	1	no	—
86	C₅×D₁₂	M₁₆	1	2	yes	complemented
87	C₄×C₅×D₃	—	4	1	no	—
88	C₄×C₅×D₃	—	4	1	no	—
89	C₃⋊#168	—	2	1	no	—
90	C₃⋊#168	—	4	1	no	—
91	C₃⋊#168	D₈	1	2	yes	complemented
92	C₃×#168	—	4	1	no	—
93	C₃×#168	—	4	1	no	—
94	C₅×C₃⋊C₈	—	2	1	no	—
95	C₅×C₃⋊C₈	—	2	1	no	—
96	C₃×C₄×D₄	C₂×D₅	1	1	yes	characteristic
97	C₂×#213	Dic₅	1	1	yes	characteristic
98	C₂×#208	—	10	1	no	—
99	C₂×D₃×C₈	—	10	1	no	—
100	C₄×D₁₂	C₂×D₅	1	1	yes	characteristic
101	C₃×C₄⋊C₈	—	5	1	no	—
102	C₃×C₄×C₈	—	5	1	no	—
103	#832	Dic₅	1	1	yes	characteristic
104	#830	—	10	1	no	—
105	(C₂×D₃)⋊C₈	—	10	1	no	—
106	#809	Dic₅	1	1	yes	characteristic
107	#807	Dic₅	1	1	yes	characteristic
108	C₄⋊(C₃⋊C₈)	—	5	1	no	—
109	C₄×C₃⋊C₈	—	5	1	no	—
110	C₄×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
111	C₂×C₅×D₄	—	3	1	no	—
112	C₂×C₅×D₄	C₄×D₃	1	1	yes	characteristic
113	C₂²×C₄×C₅	—	2	1	no	—
114	C₂²×C₄×C₅	—	6	1	no	—
115	C₅×D₈	—	6	2	no	—
116	C₂×C₅×C₈	—	3	1	no	—
117	C₂×C₅×C₈	—	3	1	no	—
118	C₅×C₄⋊C₄	—	3	1	no	—
119	C₅×C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
120	C₅×C₂²⋊C₄	—	2	1	no	—
121	C₅×C₂²⋊C₄	—	6	1	no	—
122	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
123	#681	—	12	2	no	—
124	C₂×#168	—	2	1	no	—
125	C₂×#168	—	2	1	no	—
126	C₂×#168	—	3	1	no	—
127	C₂×#168	—	3	1	no	—
128	C₂×#168	—	6	1	no	—
129	C₂×#168	—	12	2	no	—
130	C₄×D₈	—	3	1	no	—
131	#436	—	15	1	no	—
132	C₈×D₄	—	15	1	no	—
133	#335	—	15	1	no	—
134	D₄⋊C₈	—	5	1	no	—
135	D₄⋊C₈	—	15	1	no	—
136	#322	—	15	1	no	—
137	C₂²×C₃×C₅	—	4	1	no	—
138	C₂²×C₃×C₅	—	4	1	no	—
139	C₂²×C₃×C₅	—	4	1	no	—
140	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
141	C₂×C₅×D₃	—	2	2	no	—
142	C₂×C₅×D₃	—	4	1	no	—
143	C₂×C₅×D₃	—	4	1	no	—
144	C₃×C₄×C₅	—	2	1	no	—
145	C₃×C₄×C₅	—	4	1	no	—
146	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
147	C₃×C₄×C₅	#131	1	1	yes	characteristic
148	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
149	C₃×C₄×C₅	#135	1	1	yes	characteristic
150	C₅×Dic₃	—	4	1	no	—
151	C₂×C₃×D₄	C₂×Dic₅	1	1	yes	characteristic
152	C₂²×C₃×C₄	—	2	1	no	—
153	C₂×D₁₂	C₂×Dic₅	1	1	yes	characteristic
154	C₂×C₄×D₃	—	2	1	no	—
155	C₂×C₃×C₈	—	10	1	no	—
156	C₂×C₃×C₈	—	10	1	no	—
157	C₃×C₄⋊C₄	C₂×Dic₅	1	1	yes	characteristic
158	C₃×C₂²⋊C₄	—	2	1	no	—
159	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
160	#213	—	2	2	no	—
161	(C₂×D₃)⋊C₄	—	2	1	no	—
162	C₄⋊Dic₃	C₂×Dic₅	1	1	yes	characteristic
163	#208	—	20	2	no	—
164	C₂×C₃⋊C₈	—	5	1	no	—
165	C₂×C₃⋊C₈	—	5	1	no	—
166	C₂×C₃⋊C₈	—	10	1	no	—
167	C₂×C₃⋊C₈	C₂×Dic₅	1	1	yes	characteristic
168	C₂×C₃⋊C₈	C₂×Dic₅	1	1	yes	characteristic
169	D₃×C₈	—	20	2	no	—
170	C₂³×C₅	—	2	1	no	—
171	C₂³×C₅	—	6	1	no	—
172	C₅×D₄	—	2	1	no	—
173	C₅×D₄	—	2	1	no	—
174	C₅×D₄	—	3	2	no	—
175	C₅×D₄	—	6	1	no	—
176	C₂×C₄×C₅	—	2	1	no	—
177	C₂×C₄×C₅	—	4	1	no	—
178	C₂×C₄×C₅	—	4	1	no	—
179	C₂×C₄×C₅	—	6	1	no	—
180	C₂×C₄×C₅	—	12	1	no	—
181	C₂×C₄×C₅	—	12	1	no	—
182	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
183	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
184	C₂×C₄×C₅	#237	1	1	yes	characteristic
185	C₅×C₈	—	6	1	no	—
186	C₅×C₈	—	6	1	no	—
187	#168	—	3	2	no	—
188	#168	—	4	1	no	—
189	#168	—	4	1	no	—
190	#168	—	6	1	no	—
191	#168	—	12	1	no	—
192	C₂×D₈	—	3	1	no	—
193	C₂×M₁₆	—	30	1	no	—
194	C₂²×C₈	—	30	1	no	—
195	C₄×D₄	—	3	1	no	—
196	C₄×D₄	D₃×D₅	1	1	yes	characteristic
197	#107	—	3	1	no	—
198	C₄⋊C₈	—	5	1	no	—
199	C₄⋊C₈	—	15	1	no	—
200	#102	—	3	1	no	—
201	#102	—	3	1	no	—
202	C₂²⋊C₈	—	30	1	no	—
203	C₂²⋊C₈	—	30	1	no	—
204	C₄×C₈	—	3	1	no	—
205	C₄×C₈	—	5	1	no	—
206	C₄×C₈	—	15	1	no	—
207	C₂×C₃×C₅	—	8	1	no	—
208	C₂×C₃×C₅	#435	1	1	yes	characteristic
209	C₂×C₃×C₅	C₄×D₈	1	1	yes	characteristic
210	C₂×C₃×C₅	#444	1	1	yes	characteristic
211	C₅×D₃	—	4	2	no	—
212	C₂³×C₃	—	2	1	no	—
213	C₂²×D₃	—	2	1	no	—
214	C₃×D₄	—	2	1	no	—
215	C₃×D₄	—	2	1	no	—
216	C₂×C₃×C₄	—	2	1	no	—
217	C₂×C₃×C₄	—	4	1	no	—
218	C₂×C₃×C₄	—	4	1	no	—
219	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
220	C₂×C₃×C₄	#711	1	1	yes	characteristic
221	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
222	C₂×Dic₃	—	2	1	no	—
223	D₁₂	—	2	1	no	—
224	D₁₂	#681	1	2	yes	complemented
225	C₄×D₃	—	4	1	no	—
226	C₄×D₃	—	4	1	no	—
227	C₃×C₈	—	20	1	no	—
228	C₃×C₈	—	20	1	no	—
229	C₃⋊C₈	—	2	1	no	—
230	C₃⋊C₈	—	2	1	no	—
231	C₃⋊C₈	—	5	2	no	—
232	C₃⋊C₈	—	10	1	no	—
233	C₃⋊C₈	—	20	1	no	—
234	C₂²×C₅	—	4	1	no	—
235	C₂²×C₅	—	4	1	no	—
236	C₂²×C₅	—	4	1	no	—
237	C₂²×C₅	—	6	2	no	—
238	C₂²×C₅	—	12	1	no	—
239	C₂²×C₅	—	12	1	no	—
240	C₂²×C₅	#67⋊C₄	1	1	yes	characteristic
241	C₄×C₅	—	2	1	no	—
242	C₄×C₅	—	4	1	no	—
243	C₄×C₅	—	12	1	no	—
244	C₄×C₅	#901	1	1	yes	characteristic
245	C₄×C₅	D₃×M₁₆	1	1	yes	characteristic
246	C₄×C₅	D₃×D₈	1	1	yes	characteristic
247	C₄×C₅	#916	1	1	yes	characteristic
248	C₂×D₄	—	3	1	no	—
249	C₂×D₄	D₃×Dic₅	1	1	yes	characteristic
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	—	6	1	no	—
252	D₈	—	6	2	no	—
253	M₁₆	—	60	2	no	—
254	C₂×C₈	—	3	1	no	—
255	C₂×C₈	—	3	1	no	—
256	C₂×C₈	—	10	1	no	—
257	C₂×C₈	—	10	1	no	—
258	C₂×C₈	—	15	1	no	—
259	C₂×C₈	—	15	1	no	—
260	C₂×C₈	—	30	1	no	—
261	C₂×C₈	—	60	2	no	—
262	C₄⋊C₄	—	3	1	no	—
263	C₄⋊C₄	D₃×Dic₅	1	1	yes	characteristic
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	6	1	no	—
266	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
267	C₃×C₅	#1581	1	1	yes	characteristic, complemented, Hall
268	C₂²×C₃	—	4	1	no	—
269	C₂²×C₃	—	4	1	no	—
270	C₂²×C₃	—	4	1	no	—
271	C₂²×C₃	D₄×Dic₅	1	1	yes	characteristic
272	C₂×D₃	—	2	2	no	—
273	C₂×D₃	—	4	1	no	—
274	C₂×D₃	—	4	1	no	—
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	4	1	no	—
277	C₃×C₄	D₅×D₈	1	1	yes	characteristic
278	C₃×C₄	#4058	1	1	yes	characteristic
279	C₃×C₄	C₂×#681	1	1	yes	characteristic
280	C₃×C₄	#4094	1	1	yes	characteristic
281	Dic₃	—	4	1	no	—
282	C₂×C₅	—	8	1	no	—
283	C₂×C₅	—	12	2	no	—
284	C₂×C₅	#4728	1	1	yes	characteristic
285	C₂×C₅	#4756	1	1	yes	characteristic
286	C₂×C₅	#4823	1	1	yes	characteristic
287	C₂³	—	2	1	no	—
288	C₂³	—	6	1	no	—
289	D₄	—	2	1	no	—
290	D₄	—	2	1	no	—
291	D₄	—	3	2	no	—
292	D₄	—	6	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	6	1	no	—
297	C₂×C₄	—	12	1	no	—
298	C₂×C₄	—	12	1	no	—
299	C₂×C₄	#6780	1	1	yes	characteristic
300	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
301	C₂×C₄	#6790	1	1	yes	characteristic
302	C₈	—	6	1	no	—
303	C₈	—	6	1	no	—
304	C₈	—	15	2	no	—
305	C₈	—	20	1	no	—
306	C₈	—	20	1	no	—
307	C₈	—	30	1	no	—
308	C₈	—	60	1	no	—
309	C₂×C₃	—	8	1	no	—
310	C₂×C₃	#65300	1	1	yes	characteristic
311	C₂×C₃	D₈×Dic₅	1	1	yes	characteristic
312	C₂×C₃	#65486	1	1	yes	characteristic
313	D₃	—	4	2	no	—
314	C₅	#68153	1	1	yes	characteristic, complemented, Sylow
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	6	2	no	—
319	C₂²	—	12	1	no	—
320	C₂²	—	12	1	no	—
321	C₂²	#67⋊Dic₅	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	12	1	no	—
325	C₄	#91286	1	1	yes	characteristic
326	C₄	D₃×#681	1	1	yes	characteristic
327	C₄	#91481	1	1	yes	characteristic
328	C₄	#91499	1	1	yes	characteristic
329	C₃	#10601672	1	1	yes	characteristic, complemented, Sylow
330	C₂	—	8	1	no	—
331	C₂	—	12	2	no	—
332	C₂	#11748496	1	1	yes	characteristic
333	C₂	#11748994	1	1	yes	characteristic
334	C₂	#11749443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6790
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₃×C₄×C₅×D₄	C₂²
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂×C₅×#213	C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄×D₄	D₃×D₅
hypocenter	C₃×C₅	#1581
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #11751113	C₂	semidirect
3. #11746998	C₂	semidirect
4. C₃×D₄⋊#168	C₂	semidirect
5. #11745869	C₂	semidirect
6. #11745837	C₂	semidirect
7. #11745833	C₂	semidirect
29. D₄⋊#168	D₃	semidirect
86. C₅×D₁₂	M₁₆	semidirect
91. C₃⋊#168	D₈	semidirect
224. D₁₂	#681	semidirect
267. C₃×C₅	#1581	semidirect
314. C₅	#68153	semidirect
329. C₃	#10601672	semidirect

Maximal subgroups

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

subgroup	index
#11751113	2
C₄×C₅×#213	2
#11746998	2
C₃×D₄⋊#168	2
#11745869	2
#11745837	2
#11745833	2
#10601672	3
#68153	5