Subgroups — #14284675

Order 1920.

424 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5728	C₂	1	1	yes	maximal, characteristic
2	D₃×C₄⋊#168	C₂	1	1	yes	maximal, characteristic
3	#11746853	C₂	1	1	yes	maximal, characteristic
4	#11746367	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64766	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745858	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745857	C₂	1	1	yes	maximal, characteristic, complemented
8	#10602621	—	3	1	no	maximal, Hall
9	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₅×#946	C₄	1	2	yes	—
11	C₅×Q₈×Dic₃	C₄	1	2	yes	—
12	C₅×#897	—	2	1	no	—
13	C₅×#892	—	2	1	no	—
14	C₅×#890	—	2	1	no	—
15	C₅×#887	—	2	1	no	—
16	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
17	C₅×#869	C₂²	1	1	yes	characteristic
18	C₂×D₃×#168	—	2	1	no	—
19	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
20	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
21	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
22	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
23	Dic₃⋊#168	—	2	1	no	—
24	(C₂×D₃)⋊#168	—	2	1	no	—
25	Dic₃×#168	—	2	1	no	—
26	#68481	—	5	1	no	maximal, Hall
27	C₅×#533	—	3	1	no	—
28	#65214	—	3	1	no	—
29	C₂×C₄⋊#168	—	3	1	no	—
30	#64767	—	3	1	no	—
31	#64767	—	3	1	no	—
32	#64766	—	3	1	no	—
33	#64766	D₃	1	1	yes	characteristic, complemented
34	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
35	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
37	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
38	C₂×C₅×Dic₆	—	2	1	no	—
39	C₃×C₅×C₄⋊C₄	—	2	1	no	—
40	C₃×C₅×C₄⋊C₄	—	2	1	no	—
41	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₂×C₃⋊#168	—	2	1	no	—
43	C₂×C₃⋊#168	—	2	1	no	—
44	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
45	C₅×C₄⋊Dic₃	—	2	1	no	—
46	C₅×C₄⋊Dic₃	—	2	2	no	—
47	C₅×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
48	C₅×Dic₃⋊C₄	—	2	2	no	—
49	C₄×C₅×Dic₃	—	2	2	no	—
50	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
51	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
52	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
53	C₂×C₃×#168	—	2	1	no	—
54	C₂×C₃×#168	—	2	1	no	—
55	D₃×#168	—	4	2	no	—
56	#5728	D₅	1	1	yes	characteristic
57	#4833	—	5	1	no	—
58	D₃×C₄⋊C₈	—	5	1	no	—
59	C₃×#333	—	5	1	no	—
60	#4546	—	5	1	no	—
61	#4486	—	5	1	no	—
62	#4482	—	5	1	no	—
63	C₅×C₄⋊Q₈	—	3	1	no	—
64	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
65	C₅×#125	—	6	1	no	—
66	C₅×C₂²⋊Q₈	—	3	2	no	—
67	C₅×C₂²⋊Q₈	—	6	1	no	—
68	C₄×C₅×Q₈	—	3	2	no	—
69	C₄×C₅×Q₈	—	6	1	no	—
70	C₅×#117	—	6	1	no	—
71	C₂×C₄²×C₅	—	3	1	no	—
72	C₂²×#168	—	6	1	no	—
73	C₂²⋊#168	—	6	1	no	—
74	C₄⋊#168	—	3	1	no	—
75	C₄⋊#168	—	3	1	no	—
76	C₄⋊#168	—	6	1	no	—
77	C₄⋊#168	C₂×D₃	1	1	yes	characteristic
78	C₄⋊#168	C₂×D₃	1	1	yes	characteristic
79	C₄×#168	—	6	1	no	—
80	#1542	—	15	1	no	Sylow
81	C₂²×C₅×D₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₃×C₅×Q₈	—	4	2	no	—
83	C₂×C₃×C₄×C₅	—	2	2	no	—
84	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₂×C₅×Dic₃	—	2	2	no	—
88	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
91	C₄×C₅×D₃	—	2	1	no	—
92	C₄×C₅×D₃	—	2	1	no	—
93	C₄×C₅×D₃	—	4	1	no	—
94	C₄×C₅×D₃	SD₁₆	1	2	yes	—
95	C₄×C₅×D₃	Q₁₆	1	2	yes	—
96	C₅×Dic₆	—	4	1	no	—
97	C₅×Dic₆	—	4	1	no	—
98	C₃⋊#168	—	4	1	no	—
99	C₃⋊#168	—	4	1	no	—
100	C₃×#168	—	4	1	no	—
101	C₃×#168	—	4	1	no	—
102	C₃×C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
103	#946	Dic₅	1	2	yes	—
104	Q₈×Dic₃	Dic₅	1	2	yes	—
105	C₂×D₃×C₈	—	10	1	no	—
106	#897	—	2	1	no	—
107	#892	—	2	1	no	—
108	#890	—	2	1	no	—
109	#887	—	2	1	no	—
110	C₄²×D₃	C₂×D₅	1	1	yes	characteristic
111	#869	C₂×D₅	1	1	yes	characteristic
112	C₃×C₄⋊C₈	—	5	1	no	—
113	C₃×C₄⋊C₈	—	5	1	no	—
114	(C₂×D₃)⋊C₈	—	10	1	no	—
115	Dic₃⋊C₈	—	10	1	no	—
116	C₈×Dic₃	—	10	1	no	—
117	C₄⋊(C₃⋊C₈)	—	5	1	no	—
118	C₄⋊(C₃⋊C₈)	—	5	1	no	—
119	C₂×C₅×Q₈	—	6	1	no	—
120	C₂×C₅×Q₈	C₄×D₃	1	2	yes	—
121	C₂²×C₄×C₅	—	3	1	no	—
122	C₂²×C₄×C₅	—	3	1	no	—
123	C₂²×C₄×C₅	—	3	1	no	—
124	C₅×C₄⋊C₄	—	2	1	no	—
125	C₅×C₄⋊C₄	—	2	1	no	—
126	C₅×C₄⋊C₄	—	3	2	no	—
127	C₅×C₄⋊C₄	—	6	2	no	—
128	C₅×C₄⋊C₄	—	6	2	no	—
129	C₅×C₄⋊C₄	—	6	1	no	—
130	C₅×C₂²⋊C₄	—	6	2	no	—
131	C₄²×C₅	—	3	1	no	—
132	C₄²×C₅	—	3	1	no	—
133	C₄²×C₅	—	3	1	no	—
134	C₄²×C₅	—	6	2	no	—
135	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
136	C₂×#168	—	2	1	no	—
137	C₂×#168	—	2	1	no	—
138	C₂×#168	—	6	1	no	—
139	C₂×#168	—	6	1	no	—
140	C₂×#168	—	12	2	no	—
141	#533	—	3	1	no	—
142	#431	—	15	1	no	—
143	C₂×C₄⋊C₈	—	15	1	no	—
144	#333	—	5	1	no	—
145	#333	—	15	1	no	—
146	#332	—	15	1	no	—
147	#332	—	15	1	no	—
148	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₅×D₃	—	2	1	no	—
150	C₂×C₅×D₃	—	2	1	no	—
151	C₂×C₅×D₃	#103	1	2	yes	—
152	C₃×C₄×C₅	—	2	1	no	—
153	C₃×C₄×C₅	—	4	2	no	—
154	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
155	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
156	C₃×C₄×C₅	#135	1	1	yes	characteristic
157	C₃×C₄×C₅	#135	1	1	yes	characteristic
158	C₅×Dic₃	—	2	1	no	—
159	C₅×Dic₃	—	2	1	no	—
160	C₅×Dic₃	—	4	2	no	—
161	C₅×Dic₃	#103	1	2	yes	—
162	C₂×C₃×Q₈	C₂×Dic₅	1	2	yes	—
163	C₂×C₄×D₃	C₂×Dic₅	1	1	yes	characteristic
164	C₂×C₄×D₃	C₂²⋊D₅	1	1	yes	characteristic
165	C₂×C₄×D₃	C₂²⋊D₅	1	1	yes	characteristic
166	C₂×Dic₆	—	2	1	no	—
167	C₂×C₃×C₈	—	10	1	no	—
168	C₂×C₃×C₈	—	10	1	no	—
169	C₃×C₄⋊C₄	—	2	1	no	—
170	C₃×C₄⋊C₄	—	2	1	no	—
171	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
172	(C₂×D₃)⋊C₄	—	2	2	no	—
173	C₄⋊Dic₃	—	2	1	no	—
174	C₄⋊Dic₃	—	2	2	no	—
175	C₄⋊Dic₃	C₂×Dic₅	1	2	yes	—
176	Dic₃⋊C₄	—	2	2	no	—
177	C₄×Dic₃	—	2	2	no	—
178	C₄×Dic₃	C₂×Dic₅	1	1	yes	characteristic
179	C₄×Dic₃	C₂²⋊D₅	1	1	yes	characteristic
180	C₄×Dic₃	C₂²⋊D₅	1	1	yes	characteristic
181	C₂×C₃⋊C₈	—	10	1	no	—
182	C₂×C₃⋊C₈	—	10	1	no	—
183	D₃×C₈	—	20	2	no	—
184	C₂³×C₅	—	3	1	no	—
185	C₅×Q₈	—	4	2	no	—
186	C₅×Q₈	—	12	1	no	—
187	C₅×Q₈	—	12	1	no	—
188	C₂×C₄×C₅	—	2	2	no	—
189	C₂×C₄×C₅	—	3	1	no	—
190	C₂×C₄×C₅	—	3	2	no	—
191	C₂×C₄×C₅	—	3	2	no	—
192	C₂×C₄×C₅	—	3	1	no	—
193	C₂×C₄×C₅	—	3	1	no	—
194	C₂×C₄×C₅	—	6	1	no	—
195	C₂×C₄×C₅	—	6	1	no	—
196	C₂×C₄×C₅	—	6	2	no	—
197	C₂×C₄×C₅	—	12	1	no	—
198	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
199	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
200	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
201	#168	—	4	1	no	—
202	#168	—	4	1	no	—
203	#168	—	12	1	no	—
204	#168	—	12	1	no	—
205	C₂²×C₈	—	30	1	no	—
206	C₄⋊Q₈	—	3	1	no	—
207	C₄⋊Q₈	D₃×D₅	1	1	yes	characteristic
208	#125	—	6	1	no	—
209	C₂²⋊Q₈	—	3	2	no	—
210	C₂²⋊Q₈	—	6	1	no	—
211	C₄×Q₈	—	3	2	no	—
212	C₄×Q₈	—	6	1	no	—
213	#117	—	6	1	no	—
214	C₂×C₄²	—	3	1	no	—
215	C₄⋊C₈	—	5	1	no	—
216	C₄⋊C₈	—	5	1	no	—
217	C₄⋊C₈	—	15	1	no	—
218	C₄⋊C₈	—	15	1	no	—
219	C₄⋊C₈	—	30	1	no	—
220	C₂²⋊C₈	—	30	1	no	—
221	C₄×C₈	—	30	1	no	—
222	C₂×C₃×C₅	#413	1	1	yes	characteristic
223	C₂×C₃×C₅	C₂×#103	1	1	yes	characteristic
224	C₂×C₃×C₅	#416	1	1	yes	characteristic
225	C₅×D₃	—	2	2	no	—
226	C₂²×D₃	C₂²⋊Dic₅	1	1	yes	characteristic
227	C₃×Q₈	—	4	2	no	—
228	C₂×C₃×C₄	—	2	2	no	—
229	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
230	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
231	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
232	C₂×Dic₃	—	2	2	no	—
233	C₂×Dic₃	C₂²⋊Dic₅	1	1	yes	characteristic
234	C₂×Dic₃	C₂²⋊Dic₅	1	1	yes	characteristic
235	C₂×Dic₃	C₂²⋊Dic₅	1	1	yes	characteristic
236	C₄×D₃	—	2	1	no	—
237	C₄×D₃	—	2	1	no	—
238	C₄×D₃	—	4	1	no	—
239	C₄×D₃	#688	1	2	yes	—
240	C₄×D₃	#689	1	2	yes	—
241	Dic₆	—	4	1	no	—
242	Dic₆	—	4	1	no	—
243	C₃×C₈	—	20	1	no	—
244	C₃×C₈	—	20	1	no	—
245	C₃⋊C₈	—	20	1	no	—
246	C₃⋊C₈	—	20	1	no	—
247	C₂²×C₅	—	3	2	no	—
248	C₂²×C₅	—	6	1	no	—
249	C₂²×C₅	—	6	1	no	—
250	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
251	C₄×C₅	—	2	1	no	—
252	C₄×C₅	—	3	2	no	—
253	C₄×C₅	—	4	2	no	—
254	C₄×C₅	—	6	1	no	—
255	C₄×C₅	—	6	1	no	—
256	C₄×C₅	—	12	2	no	—
257	C₄×C₅	D₃×SD₁₆	1	1	yes	characteristic
258	C₄×C₅	#916	1	1	yes	characteristic
259	C₄×C₅	D₃×Q₁₆	1	1	yes	characteristic
260	C₄×C₅	#919	1	1	yes	characteristic
261	C₂×Q₈	—	6	1	no	—
262	C₂×Q₈	D₃×Dic₅	1	2	yes	—
263	C₂²×C₄	—	3	1	no	—
264	C₂²×C₄	—	3	1	no	—
265	C₂²×C₄	—	3	1	no	—
266	C₂×C₈	—	10	1	no	—
267	C₂×C₈	—	10	1	no	—
268	C₂×C₈	—	30	1	no	—
269	C₂×C₈	—	30	1	no	—
270	C₂×C₈	—	60	2	no	—
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	3	2	no	—
274	C₄⋊C₄	—	6	2	no	—
275	C₄⋊C₄	—	6	2	no	—
276	C₄⋊C₄	—	6	1	no	—
277	C₂²⋊C₄	—	6	2	no	—
278	C₄²	—	3	1	no	—
279	C₄²	—	3	1	no	—
280	C₄²	—	3	1	no	—
281	C₄²	—	6	2	no	—
282	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
283	C₃×C₅	#1542	1	1	yes	characteristic, complemented, Hall
284	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
285	C₂×D₃	—	2	1	no	—
286	C₂×D₃	—	2	1	no	—
287	C₂×D₃	#3967	1	2	yes	—
288	C₃×C₄	—	2	1	no	—
289	C₃×C₄	—	4	2	no	—
290	C₃×C₄	C₂×#688	1	1	yes	characteristic
291	C₃×C₄	C₂×#689	1	1	yes	characteristic
292	C₃×C₄	#4096	1	1	yes	characteristic
293	C₃×C₄	#4096	1	1	yes	characteristic
294	Dic₃	—	2	1	no	—
295	Dic₃	—	2	1	no	—
296	Dic₃	—	4	2	no	—
297	Dic₃	#3967	1	2	yes	—
298	C₂×C₅	—	6	2	no	—
299	C₂×C₅	#4745	1	1	yes	characteristic
300	C₂×C₅	D₃×#103	1	1	yes	characteristic
301	C₂×C₅	#4804	1	1	yes	characteristic
302	C₂³	—	3	1	no	—
303	Q₈	—	4	2	no	—
304	Q₈	—	12	1	no	—
305	Q₈	—	12	1	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	3	1	no	—
308	C₂×C₄	—	3	2	no	—
309	C₂×C₄	—	3	2	no	—
310	C₂×C₄	—	3	1	no	—
311	C₂×C₄	—	3	1	no	—
312	C₂×C₄	—	6	1	no	—
313	C₂×C₄	—	6	1	no	—
314	C₂×C₄	—	6	2	no	—
315	C₂×C₄	—	12	1	no	—
316	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
317	C₂×C₄	D₃×C₂²⋊D₅	1	1	yes	characteristic
318	C₂×C₄	D₃×C₂²⋊D₅	1	1	yes	characteristic
319	C₈	—	20	1	no	—
320	C₈	—	20	1	no	—
321	C₈	—	60	1	no	—
322	C₈	—	60	1	no	—
323	C₂×C₃	C₂×#3967	1	1	yes	characteristic
324	C₂×C₃	#65522	1	1	yes	characteristic
325	C₂×C₃	#65526	1	1	yes	characteristic
326	D₃	—	2	2	no	—
327	C₅	#68481	1	1	yes	characteristic, complemented, Sylow
328	C₂²	—	3	2	no	—
329	C₂²	—	6	1	no	—
330	C₂²	—	6	1	no	—
331	C₂²	D₃×C₂²⋊Dic₅	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	3	2	no	—
334	C₄	—	4	2	no	—
335	C₄	—	6	1	no	—
336	C₄	—	6	1	no	—
337	C₄	—	12	2	no	—
338	C₄	D₃×#688	1	1	yes	characteristic
339	C₄	D₃×#689	1	1	yes	characteristic
340	C₄	#91517	1	1	yes	characteristic
341	C₄	#91519	1	1	yes	characteristic
342	C₃	#10602621	1	1	yes	characteristic, complemented, Sylow
343	C₂	—	6	2	no	—
344	C₂	#11749126	1	1	yes	characteristic
345	C₂	D₃×#3967	1	1	yes	characteristic
346	C₂	#11749211	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃×C₂²⋊Dic₅
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₃×C₅×C₄⋊Q₈	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×C₅×D₃	C₂²⋊C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄⋊Q₈	D₃×D₅
hypocenter	C₃×C₅	#1542
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
4. #11746367	C₂	semidirect
5. C₃×#64766	C₂	semidirect
6. #11745858	C₂	semidirect
7. #11745857	C₂	semidirect
33. #64766	D₃	semidirect
283. C₃×C₅	#1542	semidirect
327. C₅	#68481	semidirect
342. C₃	#10602621	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5728	2
D₃×C₄⋊#168	2
#11746853	2
#11746367	2
C₃×#64766	2
#11745858	2
#11745857	2
#10602621	3
#68481	5