Subgroups — #14282880

Order 1920.

568 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	C₃×C₅×#532	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
3	#11751026	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746358	C₂	1	4	yes	maximal, complemented
5	#10602604	—	3	1	no	maximal, Hall
6	C₂×C₃×C₅×#41	—	2	1	no	—
7	C₃×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
8	C₃×C₅×#127	C₄	1	1	yes	characteristic, complemented
9	C₃×C₅×#124	—	2	1	no	—
10	C₃×C₅×#121	—	2	1	no	—
11	C₃×C₅×#121	C₂²	1	2	yes	—
12	C₃×C₄×C₅×D₄	—	2	1	no	—
13	C₃×C₄×C₅×D₄	C₄	1	2	yes	complemented
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₁₅	—	2	1	no	—
16	C₂×C₄×Dic₁₅	—	2	1	no	—
17	#91115	C₂²	1	2	yes	—
18	#91114	C₂²	1	2	yes	—
19	C₄⋊(C₃⋊#168)	—	2	1	no	—
20	C₄×C₃⋊#168	—	2	1	no	—
21	#71811	—	5	1	no	maximal, Hall
22	C₅×#532	D₃	1	1	yes	characteristic
23	C₄×C₄⋊Dic₅	—	3	1	no	—
24	#65214	—	3	1	no	—
25	#64757	—	3	4	no	—
26	C₃×C₅×#41	—	4	2	no	—
27	C₃×C₅×#41	—	4	2	no	—
28	C₂×C₃×C₅×Q₈	—	2	1	no	—
29	C₂×C₃×C₅×D₄	—	2	2	no	—
30	C₂×C₃×C₅×D₄	—	2	1	no	—
31	C₂×C₃×C₅×D₄	—	2	1	no	—
32	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
33	C₂²×C₃×C₄×C₅	—	2	2	no	—
34	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
35	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
36	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
37	C₂²×Dic₁₅	—	2	2	no	—
38	C₃×C₅×C₄⋊C₄	—	2	1	no	—
39	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
40	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
41	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
42	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
43	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
44	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
45	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
46	C₄⋊Dic₁₅	—	2	4	no	—
47	C₄×Dic₁₅	—	4	2	no	—
48	C₂×C₃⋊#168	—	2	2	no	—
49	C₃×#532	D₅	1	1	yes	characteristic
50	C₄×C₄⋊Dic₃	—	5	1	no	—
51	#4925	—	5	1	no	—
52	#4537	—	5	4	no	—
53	C₂×C₅×#41	—	2	1	no	—
54	C₅×C₄⋊Q₈	Dic₃	1	1	yes	characteristic, complemented
55	C₅×#127	Dic₃	1	1	yes	characteristic, complemented
56	C₅×#124	—	2	1	no	—
57	C₅×#121	—	2	1	no	—
58	C₅×#121	C₂×D₃	1	2	yes	—
59	C₄×C₅×D₄	—	2	1	no	—
60	C₄×C₅×D₄	Dic₃	1	2	yes	complemented
61	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
62	C₂×C₄⋊Dic₅	—	6	1	no	—
63	C₂×C₄×Dic₅	—	6	1	no	—
64	#3963	—	3	2	no	—
65	C₂²⋊#168	—	3	2	no	—
66	C₄⋊#168	—	6	1	no	—
67	C₄×#168	—	6	1	no	—
68	#1502	—	15	1	no	Sylow
69	C₂³×C₃×C₅	—	2	2	no	—
70	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
71	C₃×C₅×Q₈	—	4	2	no	—
72	C₃×C₅×D₄	—	4	2	no	—
73	C₃×C₅×D₄	—	4	2	no	—
74	C₃×C₅×D₄	—	4	2	no	—
75	C₃×C₅×D₄	—	4	2	no	—
76	C₃×C₅×D₄	—	4	2	no	—
77	C₂×C₃×C₄×C₅	—	2	2	no	—
78	C₂×C₃×C₄×C₅	—	2	2	no	—
79	C₂×C₃×C₄×C₅	—	4	1	no	—
80	C₂×C₃×C₄×C₅	—	4	4	no	—
81	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	D₈	1	2	yes	—
86	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
87	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₂×Dic₁₅	—	2	4	no	—
90	C₂×Dic₁₅	—	4	2	no	—
91	C₂×Dic₁₅	—	4	2	no	—
92	C₃⋊#168	—	4	2	no	—
93	C₂×C₃×#41	—	2	1	no	—
94	C₃×C₄⋊Q₈	Dic₅	1	1	yes	characteristic, complemented
95	C₃×#127	Dic₅	1	1	yes	characteristic, complemented
96	C₃×#124	—	2	1	no	—
97	C₃×#121	—	2	1	no	—
98	C₃×#121	C₂×D₅	1	2	yes	—
99	C₃×C₄×D₄	—	2	1	no	—
100	C₃×C₄×D₄	Dic₅	1	2	yes	complemented
101	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
102	C₂×C₄⋊Dic₃	—	10	1	no	—
103	C₂×C₄×Dic₃	—	10	1	no	—
104	C₃⋊#95	—	5	2	no	—
105	#830	—	5	2	no	—
106	C₄⋊(C₃⋊C₈)	—	10	1	no	—
107	C₄×C₃⋊C₈	—	10	1	no	—
108	C₅×#41	—	4	2	no	—
109	C₅×#41	—	4	2	no	—
110	C₂×C₅×Q₈	—	2	1	no	—
111	C₂×C₅×D₄	—	2	2	no	—
112	C₂×C₅×D₄	—	2	1	no	—
113	C₂×C₅×D₄	—	2	1	no	—
114	C₂×C₅×D₄	C₂×Dic₃	1	2	yes	—
115	C₂²×C₄×C₅	—	2	2	no	—
116	C₂²×C₄×C₅	#67	1	1	yes	characteristic
117	C₂²×C₄×C₅	#67	1	1	yes	characteristic
118	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
119	C₂²×Dic₅	—	6	2	no	—
120	C₅×C₄⋊C₄	—	2	1	no	—
121	C₅×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
122	C₅×C₂²⋊C₄	—	2	2	no	—
123	C₅×C₂²⋊C₄	—	2	2	no	—
124	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
125	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
126	C₄²×C₅	#67	1	1	yes	characteristic
127	C₄²×C₅	#67	1	1	yes	characteristic
128	C₄⋊Dic₅	—	6	4	no	—
129	C₄×Dic₅	—	12	2	no	—
130	C₂×#168	—	6	2	no	—
131	#532	D₁₅	1	1	yes	characteristic
132	#431	—	15	1	no	—
133	C₄×C₄⋊C₄	—	15	1	no	—
134	#327	—	15	4	no	—
135	C₂²×C₃×C₅	—	2	1	no	—
136	C₂²×C₃×C₅	—	2	1	no	—
137	C₂²×C₃×C₅	—	4	2	no	—
138	C₂²×C₃×C₅	—	4	2	no	—
139	C₂²×C₃×C₅	—	4	2	no	—
140	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
142	C₂²×C₃×C₅	C₂×SD₁₆	1	1	yes	characteristic
143	C₃×C₄×C₅	—	2	1	no	—
144	C₃×C₄×C₅	—	2	1	no	—
145	C₃×C₄×C₅	—	2	1	no	—
146	C₃×C₄×C₅	—	4	2	no	—
147	C₃×C₄×C₅	#102	1	2	yes	—
148	C₃×C₄×C₅	#102	1	2	yes	—
149	C₃×C₄×C₅	#104	1	2	yes	—
150	Dic₁₅	—	4	4	no	—
151	C₃×#41	—	4	2	no	—
152	C₃×#41	—	4	2	no	—
153	C₂×C₃×Q₈	—	2	1	no	—
154	C₂×C₃×D₄	—	2	2	no	—
155	C₂×C₃×D₄	—	2	1	no	—
156	C₂×C₃×D₄	—	2	1	no	—
157	C₂×C₃×D₄	C₂×Dic₅	1	2	yes	—
158	C₂²×C₃×C₄	—	2	2	no	—
159	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
160	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
161	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
162	C₂²×Dic₃	—	10	2	no	—
163	C₃×C₄⋊C₄	—	2	1	no	—
164	C₃×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
165	C₃×C₂²⋊C₄	—	2	2	no	—
166	C₃×C₂²⋊C₄	—	2	2	no	—
167	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
168	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
169	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
170	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
171	C₄⋊Dic₃	—	10	4	no	—
172	C₄×Dic₃	—	20	2	no	—
173	C₂×C₃⋊C₈	—	10	2	no	—
174	C₂³×C₅	—	2	2	no	—
175	C₂³×C₅	C₂×#67	1	1	yes	characteristic
176	C₅×Q₈	—	4	2	no	—
177	C₅×D₄	—	4	2	no	—
178	C₅×D₄	—	4	2	no	—
179	C₅×D₄	—	4	2	no	—
180	C₅×D₄	—	4	2	no	—
181	C₅×D₄	—	4	2	no	—
182	C₂×C₄×C₅	—	2	2	no	—
183	C₂×C₄×C₅	—	2	2	no	—
184	C₂×C₄×C₅	—	4	1	no	—
185	C₂×C₄×C₅	—	4	4	no	—
186	C₂×C₄×C₅	#213	1	2	yes	—
187	C₂×C₄×C₅	#214	1	2	yes	—
188	C₂×C₄×C₅	#217	1	1	yes	characteristic
189	C₂×C₄×C₅	#217	1	1	yes	characteristic
190	C₂×C₄×C₅	#217	1	1	yes	characteristic
191	C₂×C₄×C₅	#217	1	1	yes	characteristic
192	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
193	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
194	C₂×Dic₅	—	6	4	no	—
195	C₂×Dic₅	—	12	2	no	—
196	C₂×Dic₅	—	12	2	no	—
197	#168	—	12	2	no	—
198	C₂×#41	—	2	1	no	—
199	C₄⋊Q₈	Dic₁₅	1	1	yes	characteristic, complemented
200	#127	Dic₁₅	1	1	yes	characteristic, complemented
201	#124	—	2	1	no	—
202	#121	—	2	1	no	—
203	#121	C₂×D₁₅	1	2	yes	—
204	C₄×D₄	—	2	1	no	—
205	C₄×D₄	Dic₁₅	1	2	yes	complemented
206	C₂×C₄⋊C₄	—	30	1	no	—
207	C₂×C₄²	—	30	1	no	—
208	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
209	C₄⋊C₈	—	30	1	no	—
210	C₂²⋊C₈	—	15	2	no	—
211	C₄×C₈	—	30	1	no	—
212	#95	—	15	2	no	—
213	C₂×C₃×C₅	—	2	1	no	—
214	C₂×C₃×C₅	—	2	1	no	—
215	C₂×C₃×C₅	—	8	2	no	—
216	C₂×C₃×C₅	#410	1	1	yes	characteristic
217	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
218	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
219	C₂³×C₃	—	2	2	no	—
220	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
221	C₃×Q₈	—	4	2	no	—
222	C₃×D₄	—	4	2	no	—
223	C₃×D₄	—	4	2	no	—
224	C₃×D₄	—	4	2	no	—
225	C₃×D₄	—	4	2	no	—
226	C₃×D₄	—	4	2	no	—
227	C₂×C₃×C₄	—	2	2	no	—
228	C₂×C₃×C₄	—	2	2	no	—
229	C₂×C₃×C₄	—	4	1	no	—
230	C₂×C₃×C₄	—	4	4	no	—
231	C₂×C₃×C₄	#686	1	2	yes	—
232	C₂×C₃×C₄	#687	1	2	yes	—
233	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
234	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
235	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
236	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
237	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
238	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
239	C₂×Dic₃	—	10	4	no	—
240	C₂×Dic₃	—	20	2	no	—
241	C₂×Dic₃	—	20	2	no	—
242	C₃⋊C₈	—	20	2	no	—
243	C₂²×C₅	—	2	1	no	—
244	C₂²×C₅	—	2	1	no	—
245	C₂²×C₅	—	4	2	no	—
246	C₂²×C₅	—	4	2	no	—
247	C₂²×C₅	—	4	2	no	—
248	C₂²×C₅	C₂×#213	1	1	yes	characteristic
249	C₂²×C₅	C₂×#214	1	1	yes	characteristic
250	C₂²×C₅	C₂×#217	1	1	yes	characteristic
251	C₄×C₅	—	2	1	no	—
252	C₄×C₅	—	2	1	no	—
253	C₄×C₅	—	2	1	no	—
254	C₄×C₅	—	4	2	no	—
255	C₄×C₅	#832	1	2	yes	—
256	C₄×C₅	#832	1	2	yes	—
257	C₄×C₅	#837	1	2	yes	—
258	Dic₅	—	12	4	no	—
259	#41	—	4	2	no	—
260	#41	—	4	2	no	—
261	C₂×Q₈	—	2	1	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	—	2	1	no	—
264	C₂×D₄	—	2	1	no	—
265	C₂×D₄	C₂×Dic₁₅	1	2	yes	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	30	2	no	—
268	C₂²×C₄	#1220	1	1	yes	characteristic
269	C₂²×C₄	#1220	1	1	yes	characteristic
270	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
271	C₂×C₈	—	30	2	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	30	4	no	—
274	C₄⋊C₄	C₂×Dic₁₅	1	2	yes	—
275	C₂²⋊C₄	—	2	2	no	—
276	C₂²⋊C₄	—	2	2	no	—
277	C₄²	—	60	2	no	—
278	C₄²	C₂×Dic₁₅	1	1	yes	characteristic
279	C₄²	C₂×Dic₁₅	1	1	yes	characteristic
280	C₄²	#1220	1	1	yes	characteristic
281	C₄²	#1220	1	1	yes	characteristic
282	C₃×C₅	#1502	1	1	yes	characteristic, complemented, Hall
283	C₂²×C₃	—	2	1	no	—
284	C₂²×C₃	—	2	1	no	—
285	C₂²×C₃	—	4	2	no	—
286	C₂²×C₃	—	4	2	no	—
287	C₂²×C₃	—	4	2	no	—
288	C₂²×C₃	C₂×#686	1	1	yes	characteristic
289	C₂²×C₃	C₂×#687	1	1	yes	characteristic
290	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
291	C₃×C₄	—	2	1	no	—
292	C₃×C₄	—	2	1	no	—
293	C₃×C₄	—	2	1	no	—
294	C₃×C₄	—	4	2	no	—
295	C₃×C₄	#3964	1	2	yes	—
296	C₃×C₄	#3964	1	2	yes	—
297	C₃×C₄	#3969	1	2	yes	—
298	Dic₃	—	20	4	no	—
299	C₂×C₅	—	2	1	no	—
300	C₂×C₅	—	2	1	no	—
301	C₂×C₅	—	8	2	no	—
302	C₂×C₅	C₂×#832	1	1	yes	characteristic
303	C₂×C₅	C₂×#837	1	1	yes	characteristic
304	C₂×C₅	#5240	1	1	yes	characteristic
305	C₂³	—	2	2	no	—
306	C₂³	C₂×#1220	1	1	yes	characteristic
307	Q₈	—	4	2	no	—
308	D₄	—	4	2	no	—
309	D₄	—	4	2	no	—
310	D₄	—	4	2	no	—
311	D₄	—	4	2	no	—
312	D₄	—	4	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	4	no	—
317	C₂×C₄	—	30	4	no	—
318	C₂×C₄	—	60	2	no	—
319	C₂×C₄	—	60	2	no	—
320	C₂×C₄	#6715	1	2	yes	—
321	C₂×C₄	#6716	1	2	yes	—
322	C₂×C₄	#6719	1	1	yes	characteristic
323	C₂×C₄	#6719	1	1	yes	characteristic
324	C₂×C₄	#6719	1	1	yes	characteristic
325	C₂×C₄	#6719	1	1	yes	characteristic
326	C₂×C₄	C₂²×Dic₁₅	1	1	yes	characteristic
327	C₂×C₄	C₂×#1220	1	1	yes	characteristic
328	C₈	—	60	2	no	—
329	C₂×C₃	—	2	1	no	—
330	C₂×C₃	—	2	1	no	—
331	C₂×C₃	—	8	2	no	—
332	C₂×C₃	C₂×#3964	1	1	yes	characteristic
333	C₂×C₃	C₂×#3969	1	1	yes	characteristic
334	C₂×C₃	#65529	1	1	yes	characteristic
335	C₅	#71811	1	1	yes	characteristic, complemented, Sylow
336	C₂²	—	2	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	4	2	no	—
339	C₂²	—	4	2	no	—
340	C₂²	—	4	2	no	—
341	C₂²	C₂×#6715	1	1	yes	characteristic
342	C₂²	C₂×#6716	1	1	yes	characteristic
343	C₂²	C₂×#6719	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	2	1	no	—
347	C₄	—	4	2	no	—
348	C₄	—	60	4	no	—
349	C₄	#91116	1	2	yes	—
350	C₄	#91116	1	2	yes	—
351	C₄	#91121	1	2	yes	—
352	C₃	#10602604	1	1	yes	characteristic, complemented, Sylow
353	C₂	—	2	1	no	—
354	C₂	—	2	1	no	—
355	C₂	—	8	2	no	—
356	C₂	C₂×#91116	1	1	yes	characteristic
357	C₂	C₂×#91121	1	1	yes	characteristic
358	C₂	#11751341	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6719
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting, prime radical	C₃×C₅×#532	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	#532	D₁₅
hypocenter	C₃×C₅	#1502
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

Complements

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

N	H ≅ G/N	type
2. C₄×C₄⋊Dic₁₅	C₂	semidirect
3. #11751026	C₂	semidirect
4. #11746358	C₂	semidirect
7. C₃×C₅×C₄⋊Q₈	C₄	semidirect
8. C₃×C₅×#127	C₄	semidirect
13. C₃×C₄×C₅×D₄	C₄	semidirect
54. C₅×C₄⋊Q₈	Dic₃	semidirect
55. C₅×#127	Dic₃	semidirect
60. C₄×C₅×D₄	Dic₃	semidirect
94. C₃×C₄⋊Q₈	Dic₅	semidirect
95. C₃×#127	Dic₅	semidirect
100. C₃×C₄×D₄	Dic₅	semidirect
199. C₄⋊Q₈	Dic₁₅	semidirect
200. #127	Dic₁₅	semidirect
205. C₄×D₄	Dic₁₅	semidirect
282. C₃×C₅	#1502	semidirect
335. C₅	#71811	semidirect
352. C₃	#10602604	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#532	2
C₄×C₄⋊Dic₁₅	2
#11751026	2
#11746358	2
#10602604	3
#71811	5