Subgroups — #14282155

Order 1920.

492 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#6817	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#432	C₂	1	1	yes	maximal, characteristic, complemented
3	#11751026	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746305	C₂	1	2	yes	maximal, complemented
5	#11746304	C₂	1	2	yes	maximal, complemented
6	#10601591	—	3	1	no	maximal, Hall
7	C₂×#6817	C₄	1	1	yes	characteristic
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₄×#1220	—	2	1	no	—
10	C₄×#1220	—	2	1	no	—
11	#91815	C₄	1	1	yes	characteristic
12	C₄×D₆₀	C₄	1	1	yes	characteristic
13	C₄×D₆₀	C₂²	1	1	yes	characteristic
14	#91761	—	2	1	no	—
15	C₄²×D₁₅	—	2	1	no	—
16	C₄×Dic₃₀	C₄	1	1	yes	characteristic
17	C₄×Dic₃₀	C₂²	1	1	yes	characteristic
18	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
19	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
20	C₃×C₅×#97	—	2	1	no	—
21	#91114	—	2	1	no	—
22	C₄⋊(C₃⋊#168)	—	2	1	no	—
23	C₄×C₃⋊#168	C₂²	1	2	yes	—
24	#70798	—	5	1	no	maximal, Hall
25	C₄×#709	—	3	1	no	—
26	C₅×#432	D₃	1	1	yes	characteristic, complemented
27	#65214	—	3	1	no	—
28	#64704	—	3	2	no	—
29	#64703	—	3	2	no	—
30	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
32	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
33	C₂×#1220	—	2	1	no	—
34	#6817	—	2	2	no	—
35	#6817	—	2	2	no	—
36	C₂×D₆₀	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×D₁₅	—	2	1	no	—
38	C₂×C₄×D₁₅	—	2	1	no	—
39	C₂×C₄×D₁₅	—	2	1	no	—
40	C₂×Dic₃₀	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃×C₅×C₈	—	2	2	no	—
42	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
43	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
44	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
45	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
46	#6719	—	2	1	no	—
47	#6714	—	2	1	no	—
48	#6714	—	2	1	no	—
49	C₄⋊Dic₁₅	C₂×C₄	1	1	yes	characteristic
50	C₄⋊Dic₁₅	C₂×C₄	1	1	yes	characteristic
51	#6712	—	2	1	no	—
52	#6712	—	2	1	no	—
53	C₄×Dic₁₅	—	2	1	no	—
54	C₄×Dic₁₅	—	2	1	no	—
55	C₄×Dic₁₅	—	2	1	no	—
56	C₂×C₃⋊#168	—	2	2	no	—
57	C₄×#235	—	5	1	no	—
58	C₃×#432	D₅	1	1	yes	characteristic, complemented
59	#4925	—	5	1	no	—
60	#4484	—	5	2	no	—
61	#4483	—	5	2	no	—
62	C₂×#709	—	3	1	no	—
63	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
64	C₄×C₂²⋊D₅	—	6	1	no	—
65	C₄×C₂²⋊D₅	—	6	1	no	—
66	#4072	—	3	1	no	—
67	C₄×D₂₀	—	3	1	no	—
68	C₄×D₂₀	—	3	1	no	—
69	#4018	—	6	1	no	—
70	C₄²×D₅	—	6	1	no	—
71	C₄×Dic₁₀	—	3	1	no	—
72	C₄×Dic₁₀	—	3	1	no	—
73	C₅×C₄⋊C₈	C₂×D₃	1	2	yes	—
74	C₅×C₂²⋊C₈	—	2	1	no	—
75	C₅×#97	—	2	1	no	—
76	C₂²⋊#168	—	6	1	no	—
77	C₄⋊#168	—	6	1	no	—
78	C₄×#168	—	3	2	no	—
79	#1488	—	15	1	no	Sylow
80	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×D₁₅	—	2	1	no	—
82	C₂×C₃×C₄×C₅	—	2	1	no	—
83	C₂×C₃×C₄×C₅	—	2	1	no	—
84	C₂×C₃×C₄×C₅	—	2	2	no	—
85	C₂×C₃×C₄×C₅	—	4	1	no	—
86	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
91	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
92	#1220	—	2	4	no	—
93	C₂×Dic₁₅	—	2	1	no	—
94	C₂×Dic₁₅	—	2	1	no	—
95	C₂×Dic₁₅	—	2	1	no	—
96	D₆₀	—	2	1	no	—
97	D₆₀	M₁₆	1	2	yes	—
98	C₄×D₁₅	—	2	4	no	—
99	C₄×D₁₅	—	4	2	no	—
100	C₄×D₁₅	—	4	2	no	—
101	Dic₃₀	—	2	1	no	—
102	Dic₃₀	M₁₆	1	2	yes	—
103	C₃×C₅×C₈	—	4	2	no	—
104	C₃⋊#168	—	4	2	no	—
105	C₂×#235	—	5	1	no	—
106	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
107	C₄×#67	—	10	1	no	—
108	C₄×#67	—	10	1	no	—
109	#926	—	5	1	no	—
110	C₄×D₁₂	—	5	1	no	—
111	C₄×D₁₂	—	5	1	no	—
112	#872	—	10	1	no	—
113	C₄²×D₃	—	10	1	no	—
114	C₄×Dic₆	—	5	1	no	—
115	C₄×Dic₆	—	5	1	no	—
116	C₃×C₄⋊C₈	C₂×D₅	1	2	yes	—
117	C₃×C₂²⋊C₈	—	2	1	no	—
118	C₃×#97	—	2	1	no	—
119	#830	—	10	1	no	—
120	C₄⋊(C₃⋊C₈)	—	10	1	no	—
121	C₄×C₃⋊C₈	—	5	2	no	—
122	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
123	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
124	C₂²×C₄×C₅	#67	1	1	yes	characteristic
125	C₂×C₂²⋊D₅	—	6	1	no	—
126	#709	—	6	2	no	—
127	#709	—	6	2	no	—
128	C₂×D₂₀	—	3	1	no	—
129	C₂×C₄×D₅	—	6	1	no	—
130	C₂×C₄×D₅	—	6	1	no	—
131	C₂×C₄×D₅	—	6	1	no	—
132	C₂×Dic₁₀	—	3	1	no	—
133	C₂×C₅×C₈	—	2	2	no	—
134	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
135	C₄²×C₅	D₁₂	1	1	yes	characteristic
136	C₄²×C₅	#67	1	1	yes	characteristic
137	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
138	C₂²⋊Dic₅	—	6	1	no	—
139	#685	—	6	1	no	—
140	#685	—	6	1	no	—
141	C₄⋊Dic₅	—	3	1	no	—
142	C₄⋊Dic₅	—	3	1	no	—
143	#683	—	6	1	no	—
144	#683	—	6	1	no	—
145	C₄×Dic₅	—	6	1	no	—
146	C₄×Dic₅	—	6	1	no	—
147	C₄×Dic₅	—	6	1	no	—
148	C₂×#168	—	6	2	no	—
149	C₄×#41	—	15	1	no	—
150	#432	D₁₅	1	1	yes	characteristic, complemented
151	#431	—	15	1	no	—
152	Q₈⋊C₈	—	15	2	no	—
153	D₄⋊C₈	—	15	2	no	—
154	C₂²×C₃×C₅	—	2	1	no	—
155	C₂²×C₃×C₅	—	2	1	no	—
156	C₂²×C₃×C₅	—	2	1	no	—
157	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
158	C₂×D₁₅	—	2	2	no	—
159	C₂×D₁₅	—	4	1	no	—
160	C₂×D₁₅	—	4	1	no	—
161	C₃×C₄×C₅	—	2	1	no	—
162	C₃×C₄×C₅	—	2	1	no	—
163	C₃×C₄×C₅	—	2	1	no	—
164	C₃×C₄×C₅	—	2	1	no	—
165	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
166	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
167	C₃×C₄×C₅	#135	1	2	yes	—
168	Dic₁₅	—	2	2	no	—
169	Dic₁₅	—	4	1	no	—
170	Dic₁₅	—	4	1	no	—
171	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
172	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
173	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
174	C₂×#67	—	10	1	no	—
175	#235	—	10	2	no	—
176	#235	—	10	2	no	—
177	C₂×D₁₂	—	5	1	no	—
178	C₂×C₄×D₃	—	10	1	no	—
179	C₂×C₄×D₃	—	10	1	no	—
180	C₂×C₄×D₃	—	10	1	no	—
181	C₂×Dic₆	—	5	1	no	—
182	C₂×C₃×C₈	—	2	2	no	—
183	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
184	C₃×C₄²	D₂₀	1	1	yes	characteristic
185	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
186	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
187	#217	—	10	1	no	—
188	(C₂×D₃)⋊C₄	—	10	1	no	—
189	(C₂×D₃)⋊C₄	—	10	1	no	—
190	C₄⋊Dic₃	—	5	1	no	—
191	C₄⋊Dic₃	—	5	1	no	—
192	Dic₃⋊C₄	—	10	1	no	—
193	Dic₃⋊C₄	—	10	1	no	—
194	C₄×Dic₃	—	10	1	no	—
195	C₄×Dic₃	—	10	1	no	—
196	C₄×Dic₃	—	10	1	no	—
197	C₂×C₃⋊C₈	—	10	2	no	—
198	C₂³×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
199	C₂²×D₅	—	6	1	no	—
200	C₂×C₄×C₅	—	2	1	no	—
201	C₂×C₄×C₅	—	2	1	no	—
202	C₂×C₄×C₅	—	2	2	no	—
203	C₂×C₄×C₅	—	4	1	no	—
204	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
205	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
206	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
207	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
208	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
209	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
210	C₂²⋊D₅	—	6	4	no	—
211	C₂×Dic₅	—	6	1	no	—
212	C₂×Dic₅	—	6	1	no	—
213	C₂×Dic₅	—	6	1	no	—
214	D₂₀	—	3	2	no	—
215	D₂₀	—	6	1	no	—
216	C₄×D₅	—	6	4	no	—
217	C₄×D₅	—	12	2	no	—
218	C₄×D₅	—	12	2	no	—
219	Dic₁₀	—	3	2	no	—
220	Dic₁₀	—	6	1	no	—
221	C₅×C₈	—	4	2	no	—
222	#168	—	12	2	no	—
223	C₂×#41	—	15	1	no	—
224	C₄×Q₈	—	15	1	no	—
225	C₄×Q₈	—	15	1	no	—
226	C₄×D₄	—	15	1	no	—
227	C₄×D₄	—	15	1	no	—
228	C₄×D₄	—	30	1	no	—
229	C₄×D₄	—	30	1	no	—
230	#117	—	15	1	no	—
231	#117	—	30	1	no	—
232	C₂×C₄²	—	30	1	no	—
233	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
234	C₄⋊C₈	—	30	1	no	—
235	C₄⋊C₈	C₂×D₁₅	1	2	yes	—
236	C₂²⋊C₈	—	2	1	no	—
237	C₂²⋊C₈	—	30	1	no	—
238	#97	—	2	1	no	—
239	C₄×C₈	—	15	2	no	—
240	C₂×C₃×C₅	—	4	1	no	—
241	C₂×C₃×C₅	#407	1	1	yes	characteristic
242	C₂×C₃×C₅	#416	1	1	yes	characteristic
243	C₂×C₃×C₅	#421	1	1	yes	characteristic
244	D₁₅	—	4	2	no	—
245	C₂³×C₃	#685	1	1	yes	characteristic
246	C₂²×D₃	—	10	1	no	—
247	C₂×C₃×C₄	—	2	2	no	—
248	C₂×C₃×C₄	—	2	1	no	—
249	C₂×C₃×C₄	—	2	1	no	—
250	C₂×C₃×C₄	—	4	1	no	—
251	C₂×C₃×C₄	#685	1	1	yes	characteristic
252	C₂×C₃×C₄	#685	1	1	yes	characteristic
253	C₂×C₃×C₄	#685	1	1	yes	characteristic
254	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
255	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
256	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
257	#67	—	10	4	no	—
258	C₂×Dic₃	—	10	1	no	—
259	C₂×Dic₃	—	10	1	no	—
260	C₂×Dic₃	—	10	1	no	—
261	D₁₂	—	5	2	no	—
262	D₁₂	—	10	1	no	—
263	C₄×D₃	—	10	4	no	—
264	C₄×D₃	—	20	2	no	—
265	C₄×D₃	—	20	2	no	—
266	Dic₆	—	5	2	no	—
267	Dic₆	—	10	1	no	—
268	C₃×C₈	—	4	2	no	—
269	C₃⋊C₈	—	20	2	no	—
270	C₂²×C₅	—	2	1	no	—
271	C₂²×C₅	—	2	1	no	—
272	C₂²×C₅	—	2	1	no	—
273	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
274	C₂×D₅	—	6	2	no	—
275	C₂×D₅	—	12	1	no	—
276	C₂×D₅	—	12	1	no	—
277	C₄×C₅	—	2	1	no	—
278	C₄×C₅	—	2	1	no	—
279	C₄×C₅	—	2	1	no	—
280	C₄×C₅	—	2	1	no	—
281	C₄×C₅	D₃×M₁₆	1	1	yes	characteristic
282	C₄×C₅	D₃×M₁₆	1	1	yes	characteristic
283	C₄×C₅	#950	1	2	yes	—
284	Dic₅	—	6	2	no	—
285	Dic₅	—	12	1	no	—
286	Dic₅	—	12	1	no	—
287	#41	—	30	2	no	—
288	#41	—	30	2	no	—
289	C₂×Q₈	—	15	1	no	—
290	C₂×D₄	—	15	1	no	—
291	C₂×D₄	—	30	1	no	—
292	C₂²×C₄	—	30	1	no	—
293	C₂²×C₄	—	30	1	no	—
294	C₂²×C₄	—	30	1	no	—
295	C₂²×C₄	C₄×D₁₅	1	1	yes	characteristic
296	C₂²×C₄	D₆₀	1	1	yes	characteristic
297	C₂²×C₄	#1220	1	1	yes	characteristic
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	30	2	no	—
300	C₄⋊C₄	—	15	1	no	—
301	C₄⋊C₄	—	15	1	no	—
302	C₄⋊C₄	—	30	1	no	—
303	C₄⋊C₄	—	30	1	no	—
304	C₂²⋊C₄	—	30	1	no	—
305	C₂²⋊C₄	—	30	1	no	—
306	C₂²⋊C₄	—	30	1	no	—
307	C₄²	—	30	1	no	—
308	C₄²	—	30	1	no	—
309	C₄²	—	30	1	no	—
310	C₄²	C₄×D₁₅	1	1	yes	characteristic
311	C₄²	D₆₀	1	1	yes	characteristic
312	C₄²	#1220	1	1	yes	characteristic
313	C₄²	C₂²×D₁₅	1	1	yes	characteristic
314	C₃×C₅	#1488	1	1	yes	characteristic, complemented, Hall
315	C₂²×C₃	—	2	1	no	—
316	C₂²×C₃	—	2	1	no	—
317	C₂²×C₃	—	2	1	no	—
318	C₂²×C₃	C₂×#685	1	1	yes	characteristic
319	C₂×D₃	—	10	2	no	—
320	C₂×D₃	—	20	1	no	—
321	C₂×D₃	—	20	1	no	—
322	C₃×C₄	—	2	1	no	—
323	C₃×C₄	—	2	1	no	—
324	C₃×C₄	—	2	1	no	—
325	C₃×C₄	—	2	1	no	—
326	C₃×C₄	D₅×M₁₆	1	1	yes	characteristic
327	C₃×C₄	D₅×M₁₆	1	1	yes	characteristic
328	C₃×C₄	#4096	1	2	yes	—
329	Dic₃	—	10	2	no	—
330	Dic₃	—	20	1	no	—
331	Dic₃	—	20	1	no	—
332	C₂×C₅	—	4	1	no	—
333	C₂×C₅	#5002	1	1	yes	characteristic
334	C₂×C₅	#5126	1	1	yes	characteristic
335	C₂×C₅	#5139	1	1	yes	characteristic
336	D₅	—	12	2	no	—
337	C₂³	—	30	1	no	—
338	C₂³	#6714	1	1	yes	characteristic
339	Q₈	—	15	2	no	—
340	Q₈	—	30	1	no	—
341	D₄	—	15	2	no	—
342	D₄	—	30	4	no	—
343	D₄	—	30	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	30	4	no	—
349	C₂×C₄	—	30	1	no	—
350	C₂×C₄	—	30	1	no	—
351	C₂×C₄	—	30	1	no	—
352	C₂×C₄	—	60	2	no	—
353	C₂×C₄	—	60	2	no	—
354	C₂×C₄	#6714	1	1	yes	characteristic
355	C₂×C₄	#6714	1	1	yes	characteristic
356	C₂×C₄	#6714	1	1	yes	characteristic
357	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
358	C₂×C₄	C₂×D₆₀	1	1	yes	characteristic
359	C₂×C₄	C₂×#1220	1	1	yes	characteristic
360	C₈	—	4	2	no	—
361	C₈	—	60	2	no	—
362	C₂×C₃	—	4	1	no	—
363	C₂×C₃	#65291	1	1	yes	characteristic
364	C₂×C₃	#65415	1	1	yes	characteristic
365	C₂×C₃	#65428	1	1	yes	characteristic
366	D₃	—	20	2	no	—
367	C₅	#70798	1	1	yes	characteristic, complemented, Sylow
368	C₂²	—	2	1	no	—
369	C₂²	—	2	1	no	—
370	C₂²	—	2	1	no	—
371	C₂²	—	30	2	no	—
372	C₂²	—	60	1	no	—
373	C₂²	—	60	1	no	—
374	C₂²	C₂×#6714	1	1	yes	characteristic
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	30	2	no	—
380	C₄	—	60	1	no	—
381	C₄	—	60	1	no	—
382	C₄	M₁₆×D₁₅	1	1	yes	characteristic
383	C₄	M₁₆×D₁₅	1	1	yes	characteristic
384	C₄	#91839	1	2	yes	—
385	C₃	#10601591	1	1	yes	characteristic, complemented, Sylow
386	C₂	—	4	1	no	—
387	C₂	—	60	2	no	—
388	C₂	#11751103	1	1	yes	characteristic
389	C₂	#11751227	1	1	yes	characteristic
390	C₂	#11751240	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#1220
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂²×D₁₅
Fitting	C₃×C₅×#432	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂×#6817	C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	#432	D₁₅
hypocenter	C₃×C₅	#1488
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #432

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#432	C₂	semidirect
3. #11751026	C₂	semidirect
4. #11746305	C₂	semidirect
5. #11746304	C₂	semidirect
26. C₅×#432	D₃	semidirect
58. C₃×#432	D₅	semidirect
150. #432	D₁₅	semidirect
314. C₃×C₅	#1488	semidirect
367. C₅	#70798	semidirect
385. C₃	#10601591	semidirect

Maximal subgroups

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

subgroup	index
C₄×#6817	2
C₃×C₅×#432	2
#11751026	2
#11746305	2
#11746304	2
#10601591	3
#70798	5