Subgroups — #14416549

Order 1920.

984 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91793	C₂	1	1	yes	maximal, characteristic
2	#11756281	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#427	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#91099	C₂	1	1	yes	maximal, characteristic, complemented
5	#11751103	C₂	1	2	yes	maximal, complemented
6	#11750990	C₂	1	4	yes	maximal, complemented
7	#11750970	C₂	1	2	yes	maximal, complemented
8	#11750954	C₂	1	2	yes	maximal, complemented
9	#10610897	—	3	1	no	maximal, Hall
10	C₂×#6817	C₂²	1	2	yes	—
11	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×#117	C₂²	1	2	yes	—
13	C₂×C₄×Dic₁₅	C₂²	1	1	yes	characteristic
14	C₂²×C₃⋊#168	C₂²	1	1	yes	characteristic
15	C₂×Dic₆₀	C₂²	1	1	yes	characteristic
16	#91793	C₄	1	8	yes	complemented
17	C₂×D₁₂₀	C₂²	1	1	yes	characteristic
18	C₂×#6706	C₂²	1	2	yes	—
19	#91782	C₂²	1	4	yes	—
20	#91776	C₂²	1	4	yes	—
21	#91770	—	2	4	no	—
22	#91761	—	2	2	no	—
23	C₄²×D₁₅	—	2	2	no	—
24	C₃×C₅×#107	C₂²	1	2	yes	—
25	C₃×C₅×#106	C₂²	1	2	yes	—
26	#91099	C₂²	1	2	yes	—
27	#91099	C₂²	1	2	yes	—
28	#91094	C₂²	1	4	yes	—
29	#91093	C₂²	1	4	yes	—
30	#79166	—	5	1	no	maximal, Hall
31	C₂×#4050	—	3	1	no	—
32	#65856	—	3	2	no	—
33	C₅×#427	D₃	1	1	yes	characteristic, complemented
34	C₂×#3947	—	3	1	no	—
35	#65291	—	3	2	no	—
36	#65178	—	3	4	no	—
37	#65158	—	3	2	no	—
38	#65142	—	3	2	no	—
39	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₂×#1220	—	2	2	no	—
41	C₂²×Dic₁₅	D₄	1	2	yes	—
42	#6817	—	2	4	no	—
43	#6817	C₂×C₄	1	8	yes	complemented
44	C₂×D₆₀	C₂³	1	2	yes	—
45	C₂×C₄×D₁₅	—	2	2	no	—
46	C₂×C₄×D₁₅	—	2	4	no	—
47	C₂×Dic₃₀	C₂³	1	2	yes	—
48	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	complemented
49	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
51	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
52	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
53	C₃×C₄²×C₅	—	2	2	no	—
54	#6714	—	2	4	no	—
55	#6712	—	2	4	no	—
56	C₄×Dic₁₅	—	2	4	no	—
57	C₄×Dic₁₅	D₄	1	2	yes	—
58	C₄×Dic₁₅	C₂³	1	2	yes	—
59	C₂×C₃⋊#168	—	2	2	no	—
60	C₂×C₃⋊#168	C₂³	1	2	yes	—
61	Dic₆₀	C₂×C₄	1	4	yes	complemented
62	D₁₂₀	C₂×C₄	1	4	yes	complemented
63	#6706	C₂×C₄	1	8	yes	complemented
64	C₂×#904	—	5	1	no	—
65	#5522	—	5	2	no	—
66	C₃×#427	D₅	1	1	yes	characteristic, complemented
67	C₂×#815	—	5	1	no	—
68	#5002	—	5	2	no	—
69	#4889	—	5	4	no	—
70	#4869	—	5	2	no	—
71	#4853	—	5	2	no	—
72	C₂×#709	—	3	2	no	—
73	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
74	C₅×#117	C₂×D₃	1	2	yes	—
75	C₂×C₄×Dic₅	—	3	1	no	—
76	C₂²×#168	—	3	1	no	—
77	C₂×Dic₂₀	—	3	1	no	—
78	#4050	—	3	8	no	—
79	C₂×D₄₀	—	3	1	no	—
80	C₂×#677	—	3	2	no	—
81	#4039	—	3	4	no	—
82	#4033	—	3	4	no	—
83	#4027	—	6	4	no	—
84	#4018	—	6	2	no	—
85	C₄²×D₅	—	6	2	no	—
86	C₅×#107	C₂×D₃	1	2	yes	—
87	C₅×#106	C₂×D₃	1	2	yes	—
88	#3947	—	3	2	no	—
89	#3947	—	3	2	no	—
90	#3942	—	3	4	no	—
91	#3941	—	3	4	no	—
92	#2943	—	15	1	no	Sylow
93	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
94	C₂²×D₁₅	—	2	2	no	—
95	C₂×C₃×C₄×C₅	—	2	4	no	—
96	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
97	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
98	C₂×C₃×C₄×C₅	#41	1	2	yes	—
99	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
100	#1220	—	2	8	no	—
101	C₂×Dic₁₅	—	2	2	no	—
102	C₂×Dic₁₅	—	4	2	no	—
103	C₂×Dic₁₅	C₂×D₄	1	4	yes	—
104	D₆₀	—	2	2	no	—
105	D₆₀	C₂²×C₄	1	4	yes	—
106	C₄×D₁₅	—	2	8	no	—
107	C₄×D₁₅	—	4	8	no	—
108	Dic₃₀	—	2	2	no	—
109	Dic₃₀	C₂²×C₄	1	4	yes	—
110	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
111	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
112	C₃⋊#168	—	2	2	no	—
113	C₂×#235	—	5	2	no	—
114	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
115	C₃×#117	C₂×D₅	1	2	yes	—
116	C₂×C₄×Dic₃	—	5	1	no	—
117	C₂²×C₃⋊C₈	—	5	1	no	—
118	C₂×Dic₁₂	—	5	1	no	—
119	#904	—	5	8	no	—
120	C₂×D₂₄	—	5	1	no	—
121	C₂×#204	—	5	2	no	—
122	#893	—	5	4	no	—
123	#887	—	5	4	no	—
124	#67⋊C₄	—	10	4	no	—
125	#872	—	10	2	no	—
126	C₄²×D₃	—	10	2	no	—
127	C₃×#107	C₂×D₅	1	2	yes	—
128	C₃×#106	C₂×D₅	1	2	yes	—
129	#815	—	5	2	no	—
130	#815	—	5	2	no	—
131	#810	—	5	4	no	—
132	#809	—	5	4	no	—
133	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
134	C₂×C₂²⋊D₅	—	6	2	no	—
135	C₂²×Dic₅	—	3	2	no	—
136	#709	—	3	8	no	—
137	#709	—	6	4	no	—
138	C₂×D₂₀	—	3	2	no	—
139	C₂×C₄×D₅	—	6	2	no	—
140	C₂×C₄×D₅	—	6	4	no	—
141	C₂×Dic₁₀	—	3	2	no	—
142	C₂×C₅×C₈	C₄×D₃	1	4	yes	complemented
143	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
144	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
145	C₅×C₄⋊C₄	C₂²×D₃	1	4	yes	—
146	C₅×C₂²⋊C₄	—	2	2	no	—
147	C₄²×C₅	—	2	2	no	—
148	#685	—	6	4	no	—
149	#683	—	6	4	no	—
150	C₄×Dic₅	—	3	2	no	—
151	C₄×Dic₅	—	3	2	no	—
152	C₄×Dic₅	—	6	4	no	—
153	C₂×#168	—	3	2	no	—
154	C₂×#168	—	6	2	no	—
155	Dic₂₀	—	3	4	no	—
156	D₄₀	—	3	4	no	—
157	#677	—	3	8	no	—
158	C₂×#135	—	15	1	no	—
159	C₄×#41	—	15	2	no	—
160	#442	—	15	2	no	—
161	#441	—	15	2	no	—
162	#440	—	15	4	no	—
163	#427	D₁₅	1	1	yes	characteristic, complemented
164	#416	—	15	2	no	—
165	C₂×#97	—	15	1	no	—
166	C₂²×C₃×C₅	—	2	2	no	—
167	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
168	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
169	C₂×D₁₅	—	2	4	no	—
170	C₂×D₁₅	—	4	4	no	—
171	C₃×C₄×C₅	—	4	4	no	—
172	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
173	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
174	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
175	Dic₁₅	—	2	4	no	—
176	Dic₁₅	—	2	4	no	—
177	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
178	C₂×#67	—	10	2	no	—
179	C₂²×Dic₃	—	5	2	no	—
180	#235	—	5	8	no	—
181	#235	—	10	4	no	—
182	C₂×D₁₂	—	5	2	no	—
183	C₂×C₄×D₃	—	10	2	no	—
184	C₂×C₄×D₃	—	10	4	no	—
185	C₂×Dic₆	—	5	2	no	—
186	C₂×C₃×C₈	C₄×D₅	1	4	yes	complemented
187	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
188	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
189	C₃×C₄⋊C₄	C₂²×D₅	1	4	yes	—
190	C₃×C₂²⋊C₄	—	2	2	no	—
191	C₃×C₄²	—	2	2	no	—
192	(C₂×D₃)⋊C₄	—	10	4	no	—
193	Dic₃⋊C₄	—	10	4	no	—
194	C₄×Dic₃	—	5	2	no	—
195	C₄×Dic₃	—	5	2	no	—
196	C₄×Dic₃	—	10	4	no	—
197	C₂×C₃⋊C₈	—	5	2	no	—
198	C₂×C₃⋊C₈	—	10	2	no	—
199	Dic₁₂	—	5	4	no	—
200	D₂₄	—	5	4	no	—
201	#204	—	5	8	no	—
202	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
203	C₂²×D₅	—	6	2	no	—
204	C₂×C₄×C₅	—	2	4	no	—
205	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
206	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
207	C₂×C₄×C₅	#239	1	2	yes	—
208	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
209	C₂²⋊D₅	—	6	8	no	—
210	C₂×Dic₅	—	3	4	no	—
211	C₂×Dic₅	—	6	2	no	—
212	C₂×Dic₅	—	12	2	no	—
213	D₂₀	—	3	4	no	—
214	D₂₀	—	6	2	no	—
215	C₄×D₅	—	6	8	no	—
216	C₄×D₅	—	12	8	no	—
217	Dic₁₀	—	3	4	no	—
218	Dic₁₀	—	6	2	no	—
219	C₅×C₈	C₂×C₄×D₃	1	2	yes	—
220	C₅×C₈	C₂×C₄×D₃	1	2	yes	—
221	#168	—	6	2	no	—
222	C₂×#41	—	15	2	no	—
223	#135	—	15	8	no	—
224	C₂×Q₁₆	—	15	1	no	—
225	C₂×SD₁₆	—	15	2	no	—
226	C₂×D₈	—	15	1	no	—
227	C₂²×C₈	—	15	1	no	—
228	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
229	C₄×Q₈	—	15	4	no	—
230	C₄×D₄	—	15	4	no	—
231	C₄×D₄	—	30	4	no	—
232	#117	—	30	2	no	—
233	#117	C₂×D₁₅	1	2	yes	—
234	C₂×C₄²	—	15	1	no	—
235	C₂×C₄²	—	30	2	no	—
236	#107	C₂×D₁₅	1	2	yes	—
237	#106	C₂×D₁₅	1	2	yes	—
238	#103	—	15	4	no	—
239	#102	—	15	4	no	—
240	#97	—	15	2	no	—
241	#97	—	15	2	no	—
242	C₂×C₃×C₅	—	2	2	no	—
243	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
244	C₂×C₃×C₅	#575	1	2	yes	—
245	D₁₅	—	4	4	no	—
246	C₂³×C₃	D₄×D₅	1	1	yes	characteristic
247	C₂²×D₃	—	10	2	no	—
248	C₂×C₃×C₄	—	2	4	no	—
249	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
250	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
251	C₂×C₃×C₄	#713	1	2	yes	—
252	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
253	#67	—	10	8	no	—
254	C₂×Dic₃	—	5	4	no	—
255	C₂×Dic₃	—	10	2	no	—
256	C₂×Dic₃	—	20	2	no	—
257	D₁₂	—	5	4	no	—
258	D₁₂	—	10	2	no	—
259	C₄×D₃	—	10	8	no	—
260	C₄×D₃	—	20	8	no	—
261	Dic₆	—	5	4	no	—
262	Dic₆	—	10	2	no	—
263	C₃×C₈	C₂×C₄×D₅	1	2	yes	—
264	C₃×C₈	C₂×C₄×D₅	1	2	yes	—
265	C₃⋊C₈	—	10	2	no	—
266	C₂²×C₅	—	2	2	no	—
267	C₂²×C₅	#893	1	2	yes	—
268	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
269	C₂×D₅	—	6	4	no	—
270	C₂×D₅	—	12	4	no	—
271	C₄×C₅	—	4	4	no	—
272	C₄×C₅	#893	1	2	yes	—
273	C₄×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
274	C₄×C₅	C₂×#239	1	1	yes	characteristic
275	Dic₅	—	6	4	no	—
276	Dic₅	—	6	4	no	—
277	#41	—	15	8	no	—
278	#41	—	30	4	no	—
279	C₂×Q₈	—	15	2	no	—
280	C₂×D₄	—	15	2	no	—
281	C₂×D₄	—	30	2	no	—
282	C₂²×C₄	—	15	2	no	—
283	C₂²×C₄	—	30	4	no	—
284	C₂²×C₄	—	30	2	no	—
285	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
286	Q₁₆	—	15	4	no	—
287	SD₁₆	—	15	8	no	—
288	D₈	—	15	4	no	—
289	C₂×C₈	—	15	2	no	—
290	C₂×C₈	—	30	2	no	—
291	C₂×C₈	C₄×D₁₅	1	4	yes	complemented
292	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
293	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
294	C₄⋊C₄	—	30	4	no	—
295	C₄⋊C₄	C₂²×D₁₅	1	4	yes	—
296	C₂²⋊C₄	—	2	2	no	—
297	C₂²⋊C₄	—	30	4	no	—
298	C₄²	—	2	2	no	—
299	C₄²	—	15	2	no	—
300	C₄²	—	15	2	no	—
301	C₄²	—	30	4	no	—
302	C₃×C₅	#2943	1	1	yes	characteristic, complemented, Hall
303	C₂²×C₃	—	2	2	no	—
304	C₂²×C₃	#4039	1	2	yes	—
305	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
306	C₂×D₃	—	10	4	no	—
307	C₂×D₃	—	20	4	no	—
308	C₃×C₄	—	4	4	no	—
309	C₃×C₄	#4039	1	2	yes	—
310	C₃×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
311	C₃×C₄	C₂×#713	1	1	yes	characteristic
312	Dic₃	—	10	4	no	—
313	Dic₃	—	10	4	no	—
314	C₂×C₅	—	2	2	no	—
315	C₂×C₅	C₂×#893	1	1	yes	characteristic
316	C₂×C₅	#5768	1	2	yes	—
317	D₅	—	12	4	no	—
318	C₂³	—	30	2	no	—
319	C₂³	D₄×D₁₅	1	1	yes	characteristic
320	Q₈	—	15	4	no	—
321	Q₈	—	30	2	no	—
322	D₄	—	15	4	no	—
323	D₄	—	30	8	no	—
324	D₄	—	30	2	no	—
325	C₂×C₄	—	2	4	no	—
326	C₂×C₄	—	15	4	no	—
327	C₂×C₄	—	30	2	no	—
328	C₂×C₄	—	30	8	no	—
329	C₂×C₄	—	60	8	no	—
330	C₂×C₄	—	60	2	no	—
331	C₂×C₄	C₂×C₄×D₁₅	1	2	yes	—
332	C₂×C₄	D₄×D₁₅	1	1	yes	characteristic
333	C₂×C₄	#6821	1	2	yes	—
334	C₂×C₄	C₂³×D₁₅	1	1	yes	characteristic
335	C₈	—	30	2	no	—
336	C₈	C₂×C₄×D₁₅	1	2	yes	—
337	C₈	C₂×C₄×D₁₅	1	2	yes	—
338	C₂×C₃	—	2	2	no	—
339	C₂×C₃	C₂×#4039	1	1	yes	characteristic
340	C₂×C₃	#66102	1	2	yes	—
341	D₃	—	20	4	no	—
342	C₅	#79166	1	1	yes	characteristic, complemented, Sylow
343	C₂²	—	2	2	no	—
344	C₂²	—	30	4	no	—
345	C₂²	—	60	4	no	—
346	C₂²	#91782	1	2	yes	—
347	C₂²	C₂×D₄×D₁₅	1	1	yes	characteristic
348	C₄	—	4	4	no	—
349	C₄	—	30	4	no	—
350	C₄	—	30	4	no	—
351	C₄	#91782	1	2	yes	—
352	C₄	C₂²×C₄×D₁₅	1	1	yes	characteristic
353	C₄	C₂×#6821	1	1	yes	characteristic
354	C₃	#10610897	1	1	yes	characteristic, complemented, Sylow
355	C₂	—	2	2	no	—
356	C₂	—	60	4	no	—
357	C₂	C₂×#91782	1	1	yes	characteristic
358	C₂	#11756527	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₅
commutator	C₃×C₄×C₅	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₅
Fitting	C₃×C₅×#427	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂×#91793	C₂
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	#427	D₁₅
hypocenter	C₃×C₅	#2943
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

Complements

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

N	H ≅ G/N	type
2. #11756281	C₂	semidirect
3. C₃×C₅×#427	C₂	semidirect
4. C₂×#91099	C₂	semidirect
5. #11751103	C₂	semidirect
6. #11750990	C₂	semidirect
7. #11750970	C₂	semidirect
8. #11750954	C₂	semidirect
16. #91793	C₄	semidirect
33. C₅×#427	D₃	semidirect
43. #6817	C₂×C₄	semidirect
48. C₂×C₃×C₅×C₈	C₂×C₄	semidirect
61. Dic₆₀	C₂×C₄	semidirect
62. D₁₂₀	C₂×C₄	semidirect
63. #6706	C₂×C₄	semidirect
66. C₃×#427	D₅	semidirect
142. C₂×C₅×C₈	C₄×D₃	semidirect
163. #427	D₁₅	semidirect
186. C₂×C₃×C₈	C₄×D₅	semidirect
291. C₂×C₈	C₄×D₁₅	semidirect
302. C₃×C₅	#2943	semidirect
342. C₅	#79166	semidirect
354. C₃	#10610897	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91793	2
#11756281	2
C₃×C₅×#427	2
C₂×#91099	2
#11751103	2
#11750990	2
#11750970	2
#11750954	2
#10610897	3
#79166	5