Subgroups — #14296027

Order 1920.

584 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#4053	C₂	1	1	yes	maximal, characteristic
2	C₂×#90976	C₂	1	1	yes	maximal, characteristic
3	#11748365	C₂	1	1	yes	maximal, characteristic
4	C₂×#90954	C₂	1	1	yes	maximal, characteristic, complemented
5	#11748324	C₂	1	1	yes	maximal, characteristic, complemented
6	#11746376	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4558	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599720	—	3	1	no	maximal, Hall
9	C₂×C₃×#709	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	#91815	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₁₅	C₂²	1	1	yes	characteristic
13	C₂×C₅×#208	C₂²	1	1	yes	characteristic
14	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
16	C₃×#4053	—	2	2	no	—
17	C₃×#4053	C₄	1	4	yes	complemented
18	C₂×C₃×#676	—	2	1	no	—
19	C₂×C₃×C₈×D₅	—	2	1	no	—
20	#90976	C₄	1	4	yes	complemented
21	#90971	C₄	1	2	yes	complemented
22	#90965	C₄	1	2	yes	complemented
23	#90954	—	2	1	no	—
24	#90954	—	2	2	no	—
25	#90951	—	2	1	no	—
26	#72009	—	5	1	no	maximal, Hall
27	C₂×#4053	D₃	1	1	yes	characteristic
28	C₂×#3957	—	3	1	no	—
29	#65411	—	3	1	no	—
30	#65402	—	3	1	no	—
31	C₂×#3947	—	3	1	no	—
32	C₅×#341	—	3	1	no	—
33	#64775	—	3	1	no	—
34	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₂²×Dic₁₅	—	2	1	no	—
36	C₂×C₃×C₂²⋊D₅	—	2	1	no	—
37	C₃×#709	—	2	1	no	—
38	C₃×#709	—	2	1	no	—
39	C₃×#709	C₂×C₄	1	2	yes	—
40	C₃×#709	C₂×C₄	1	2	yes	—
41	C₂×C₃×D₂₀	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃×C₄×D₅	—	2	1	no	—
43	C₂×C₃×Dic₁₀	C₂×C₄	1	1	yes	characteristic
44	C₃×C₅×M₁₆	—	2	1	no	—
45	C₃×C₅×M₁₆	C₂×C₄	1	2	yes	—
46	C₂×C₃×C₅×C₈	—	2	1	no	—
47	#6719	—	2	1	no	—
48	C₄⋊Dic₁₅	C₂×C₄	1	2	yes	—
49	C₄×Dic₁₅	—	2	1	no	—
50	C₄×Dic₁₅	—	2	1	no	—
51	C₄×Dic₁₅	C₂×C₄	1	2	yes	—
52	C₅×#208	—	2	1	no	—
53	C₅×#208	C₂×C₄	1	2	yes	—
54	C₂×C₅×C₃⋊C₈	—	2	1	no	—
55	C₂×C₅×C₃⋊C₈	—	2	1	no	—
56	C₂×C₅×C₃⋊C₈	—	2	1	no	—
57	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
58	C₂×C₃×#168	C₂×C₄	1	2	yes	—
59	C₂×C₃×#168	D₄	1	1	yes	characteristic
60	C₂×C₃×#168	D₄	1	1	yes	characteristic
61	C₂×C₃×#168	D₄	1	2	yes	—
62	C₃×#676	—	2	4	no	—
63	C₃×C₈×D₅	—	2	4	no	—
64	C₂×C₃×#131	—	5	1	no	—
65	C₂×#837	—	5	1	no	—
66	#5233	—	5	1	no	—
67	#5101	—	5	1	no	—
68	C₂×#815	—	5	1	no	—
69	#4558	D₅	1	1	yes	characteristic, complemented
70	#4555	—	5	1	no	—
71	C₂×#709	C₂×D₃	1	1	yes	characteristic
72	C₂×C₅×M₁₆	—	3	1	no	—
73	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
74	C₂²×C₅×C₈	—	3	1	no	—
75	#4072	—	3	1	no	—
76	C₂×C₄×Dic₅	—	3	1	no	—
77	C₂²×#168	C₂×D₃	1	1	yes	characteristic
78	#4053	—	2	2	no	—
79	#4053	Dic₃	1	4	yes	complemented
80	C₂×#676	—	2	1	no	—
81	C₂×C₈×D₅	—	2	1	no	—
82	#3957	—	3	4	no	—
83	#3953	—	3	2	no	—
84	#3948	—	3	2	no	—
85	#3947	—	6	2	no	—
86	#3947	—	6	1	no	—
87	C₈×Dic₅	—	6	1	no	—
88	#1765	—	15	1	no	Sylow
89	C₂³×C₃×C₅	#41	1	1	yes	characteristic
90	C₂²×C₃×D₅	—	2	1	no	—
91	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
95	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
96	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
97	C₂×Dic₁₅	—	2	2	no	—
98	C₂×Dic₁₅	—	2	2	no	—
99	C₂×Dic₁₅	—	4	2	no	—
100	C₃×C₂²⋊D₅	—	2	2	no	—
101	C₃×C₂²⋊D₅	—	2	2	no	—
102	C₂×C₃×Dic₅	—	2	1	no	—
103	C₃×D₂₀	—	2	1	no	—
104	C₃×D₂₀	C₄²	1	2	yes	—
105	C₃×C₄×D₅	—	2	2	no	—
106	C₃×C₄×D₅	—	2	2	no	—
107	C₃×Dic₁₀	—	2	1	no	—
108	C₃×Dic₁₀	C₄²	1	2	yes	—
109	C₃×C₅×C₈	—	2	2	no	—
110	C₃×#168	C₂²⋊C₄	1	2	yes	—
111	C₃×#168	C₂²⋊C₄	1	2	yes	—
112	C₅×C₃⋊C₈	—	2	2	no	—
113	C₅×C₃⋊C₈	—	2	2	no	—
114	C₂×C₃×#41	—	5	1	no	—
115	C₃×#131	—	5	4	no	—
116	C₃×#131	—	10	2	no	—
117	C₂×C₃×M₁₆	—	10	1	no	—
118	C₂×C₃×M₁₆	C₂×D₅	1	1	yes	characteristic
119	C₂²×C₃×C₈	—	5	1	no	—
120	C₂²×C₃×C₈	—	10	1	no	—
121	#926	—	5	1	no	—
122	C₂×C₄×Dic₃	—	5	1	no	—
123	C₂×#208	C₂×D₅	1	1	yes	characteristic
124	C₂²×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
125	#837	—	5	4	no	—
126	#835	—	5	2	no	—
127	#832	—	5	2	no	—
128	#815	—	10	2	no	—
129	#815	—	10	1	no	—
130	C₈×Dic₃	—	10	1	no	—
131	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
132	C₂×C₂²⋊D₅	—	2	1	no	—
133	C₂²×Dic₅	—	6	1	no	—
134	#709	—	2	1	no	—
135	#709	—	2	1	no	—
136	#709	C₄×D₃	1	2	yes	—
137	#709	C₂×Dic₃	1	2	yes	—
138	C₂×D₂₀	C₄×D₃	1	1	yes	characteristic
139	C₂×C₄×D₅	—	2	1	no	—
140	C₂×Dic₁₀	C₄×D₃	1	1	yes	characteristic
141	C₅×M₁₆	—	2	1	no	—
142	C₅×M₁₆	—	3	2	no	—
143	C₅×M₁₆	—	6	1	no	—
144	C₅×M₁₆	C₂×Dic₃	1	2	yes	—
145	C₂×C₅×C₈	—	2	1	no	—
146	C₂×C₅×C₈	—	3	2	no	—
147	C₂×C₅×C₈	—	6	1	no	—
148	C₂×C₅×C₈	—	6	1	no	—
149	C₂×C₅×C₈	—	6	1	no	—
150	C₂²⋊Dic₅	—	6	1	no	—
151	C₄⋊Dic₅	—	3	2	no	—
152	C₄×Dic₅	—	3	2	no	—
153	C₄×Dic₅	—	6	1	no	—
154	C₄×Dic₅	—	6	1	no	—
155	C₂×#168	C₂×Dic₃	1	2	yes	—
156	C₂×#168	#67	1	1	yes	characteristic
157	C₂×#168	#67	1	1	yes	characteristic
158	C₂×#168	#67	1	2	yes	—
159	#676	—	2	4	no	—
160	C₈×D₅	—	2	4	no	—
161	C₂×#131	—	5	1	no	—
162	C₂×#104	—	15	1	no	—
163	#416	—	15	1	no	—
164	#405	—	15	1	no	—
165	C₂×#97	—	15	1	no	—
166	#341	—	3	1	no	—
167	#339	—	15	1	no	—
168	C₂²×C₃×C₅	—	2	2	no	—
169	C₂²×C₃×C₅	#117	1	1	yes	characteristic
170	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
171	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
172	C₂×C₃×D₅	—	2	2	no	—
173	C₂×C₃×D₅	—	4	2	no	—
174	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
175	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
176	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
177	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
178	Dic₁₅	—	4	2	no	—
179	Dic₁₅	—	4	2	no	—
180	C₃×Dic₅	—	2	2	no	—
181	C₃×#41	—	5	2	no	—
182	C₃×#41	—	5	2	no	—
183	C₃×#41	—	10	1	no	—
184	C₃×#41	—	10	1	no	—
185	C₂×C₃×Q₈	—	5	1	no	—
186	C₂×C₃×D₄	—	5	1	no	—
187	C₂×C₃×D₄	—	10	1	no	—
188	C₂²×C₃×C₄	—	10	1	no	—
189	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
190	C₂²×Dic₃	—	10	1	no	—
191	C₃×M₁₆	—	2	1	no	—
192	C₃×M₁₆	—	10	4	no	—
193	C₃×M₁₆	C₄×D₅	1	2	yes	—
194	C₂×C₃×C₈	—	2	1	no	—
195	C₂×C₃×C₈	—	5	2	no	—
196	C₂×C₃×C₈	—	5	1	no	—
197	C₂×C₃×C₈	—	5	1	no	—
198	C₂×C₃×C₈	—	5	2	no	—
199	C₂×C₃×C₈	—	10	4	no	—
200	#217	—	10	1	no	—
201	C₄⋊Dic₃	—	5	2	no	—
202	C₄×Dic₃	—	5	2	no	—
203	C₄×Dic₃	—	10	1	no	—
204	C₄×Dic₃	—	10	1	no	—
205	#208	—	2	1	no	—
206	#208	C₄×D₅	1	2	yes	—
207	C₂×C₃⋊C₈	—	2	1	no	—
208	C₂×C₃⋊C₈	—	2	1	no	—
209	C₂×C₃⋊C₈	—	2	1	no	—
210	C₂×C₃⋊C₈	C₄×D₅	1	2	yes	—
211	C₂³×C₅	#235	1	1	yes	characteristic
212	C₂²×D₅	—	2	1	no	—
213	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
214	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
215	C₂×C₄×C₅	#235	1	1	yes	characteristic
216	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
217	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
218	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
219	C₂²⋊D₅	—	2	2	no	—
220	C₂²⋊D₅	—	2	2	no	—
221	C₂×Dic₅	—	2	1	no	—
222	C₂×Dic₅	—	6	2	no	—
223	C₂×Dic₅	—	6	2	no	—
224	C₂×Dic₅	—	12	2	no	—
225	D₂₀	—	2	1	no	—
226	D₂₀	C₄×Dic₃	1	2	yes	—
227	C₄×D₅	—	2	2	no	—
228	C₄×D₅	—	2	2	no	—
229	Dic₁₀	—	2	1	no	—
230	Dic₁₀	C₄×Dic₃	1	2	yes	—
231	C₅×C₈	—	2	2	no	—
232	C₅×C₈	—	6	2	no	—
233	C₅×C₈	—	6	2	no	—
234	#168	#217	1	2	yes	—
235	#168	#217	1	2	yes	—
236	C₂×#41	—	5	1	no	—
237	#131	—	5	4	no	—
238	#131	—	10	2	no	—
239	C₂×M₁₆	—	3	1	no	—
240	C₂×M₁₆	—	10	1	no	—
241	C₂×M₁₆	D₃×D₅	1	1	yes	characteristic
242	C₂²×C₈	—	3	1	no	—
243	C₂²×C₈	—	5	1	no	—
244	C₂²×C₈	—	10	1	no	—
245	#117	—	15	1	no	—
246	C₂×C₄²	—	15	1	no	—
247	#104	—	15	4	no	—
248	#103	—	15	2	no	—
249	#102	—	15	2	no	—
250	#97	—	30	2	no	—
251	#97	—	30	1	no	—
252	C₄×C₈	—	30	1	no	—
253	C₂×C₃×C₅	—	2	1	no	—
254	C₂×C₃×C₅	—	2	1	no	—
255	C₂×C₃×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
256	C₂×C₃×C₅	#444	1	2	yes	—
257	C₃×D₅	—	4	2	no	—
258	C₂³×C₃	—	10	1	no	—
259	C₂³×C₃	#713	1	1	yes	characteristic
260	C₃×Q₈	—	5	2	no	—
261	C₃×Q₈	—	10	1	no	—
262	C₃×D₄	—	5	2	no	—
263	C₃×D₄	—	10	2	no	—
264	C₃×D₄	—	10	2	no	—
265	C₃×D₄	—	10	1	no	—
266	C₂×C₃×C₄	—	10	1	no	—
267	C₂×C₃×C₄	—	10	2	no	—
268	C₂×C₃×C₄	—	10	2	no	—
269	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
270	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
271	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
272	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
273	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
274	C₂×C₃×C₄	#711	1	1	yes	characteristic
275	C₂×Dic₃	—	10	2	no	—
276	C₂×Dic₃	—	10	2	no	—
277	C₂×Dic₃	—	20	2	no	—
278	C₃×C₈	—	2	2	no	—
279	C₃×C₈	—	5	2	no	—
280	C₃×C₈	—	5	2	no	—
281	C₃⋊C₈	—	2	2	no	—
282	C₃⋊C₈	—	2	2	no	—
283	C₂²×C₅	—	2	2	no	—
284	C₂²×C₅	#926	1	1	yes	characteristic
285	C₂²×C₅	C₄×#67	1	1	yes	characteristic
286	C₂²×C₅	C₄×#67	1	1	yes	characteristic
287	C₂×D₅	—	2	2	no	—
288	C₂×D₅	—	4	2	no	—
289	C₄×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
290	C₄×C₅	C₄×#67	1	1	yes	characteristic
291	C₄×C₅	C₄×#67	1	1	yes	characteristic
292	C₄×C₅	C₂×#217	1	1	yes	characteristic
293	Dic₅	—	2	2	no	—
294	Dic₅	—	12	2	no	—
295	Dic₅	—	12	2	no	—
296	#41	—	5	2	no	—
297	#41	—	5	2	no	—
298	#41	—	10	1	no	—
299	#41	—	10	1	no	—
300	C₂×Q₈	—	5	1	no	—
301	C₂×D₄	—	5	1	no	—
302	C₂×D₄	—	10	1	no	—
303	C₂²×C₄	—	10	1	no	—
304	C₂²×C₄	—	30	1	no	—
305	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
306	M₁₆	—	2	1	no	—
307	M₁₆	—	3	2	no	—
308	M₁₆	—	6	1	no	—
309	M₁₆	—	10	4	no	—
310	M₁₆	D₅×Dic₃	1	2	yes	—
311	C₂×C₈	—	2	1	no	—
312	C₂×C₈	—	3	2	no	—
313	C₂×C₈	—	5	2	no	—
314	C₂×C₈	—	5	2	no	—
315	C₂×C₈	—	5	1	no	—
316	C₂×C₈	—	5	1	no	—
317	C₂×C₈	—	6	1	no	—
318	C₂×C₈	—	6	1	no	—
319	C₂×C₈	—	6	1	no	—
320	C₂×C₈	—	10	4	no	—
321	C₄⋊C₄	—	15	2	no	—
322	C₂²⋊C₄	—	30	1	no	—
323	C₄²	—	15	2	no	—
324	C₄²	—	30	1	no	—
325	C₄²	—	30	1	no	—
326	C₃×C₅	#1765	1	1	yes	characteristic, complemented, Hall
327	C₂²×C₃	—	2	2	no	—
328	C₂²×C₃	—	10	2	no	—
329	C₂²×C₃	—	20	2	no	—
330	C₂²×C₃	#4038	1	1	yes	characteristic
331	C₂²×C₃	#4039	1	1	yes	characteristic
332	C₂²×C₃	#4039	1	1	yes	characteristic
333	C₃×C₄	—	10	2	no	—
334	C₃×C₄	C₄²×D₅	1	1	yes	characteristic
335	C₃×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
336	C₃×C₄	#4027	1	1	yes	characteristic
337	C₃×C₄	#4027	1	1	yes	characteristic
338	Dic₃	—	20	2	no	—
339	Dic₃	—	20	2	no	—
340	C₂×C₅	—	2	1	no	—
341	C₂×C₅	—	2	1	no	—
342	C₂×C₅	#5145	1	2	yes	—
343	C₂×C₅	C₄×#217	1	1	yes	characteristic
344	D₅	—	4	2	no	—
345	C₂³	—	10	1	no	—
346	C₂³	#6773	1	1	yes	characteristic
347	Q₈	—	5	2	no	—
348	Q₈	—	10	1	no	—
349	D₄	—	5	2	no	—
350	D₄	—	10	2	no	—
351	D₄	—	10	2	no	—
352	D₄	—	10	1	no	—
353	C₂×C₄	—	10	1	no	—
354	C₂×C₄	—	10	2	no	—
355	C₂×C₄	—	10	2	no	—
356	C₂×C₄	—	30	2	no	—
357	C₂×C₄	—	30	2	no	—
358	C₂×C₄	—	60	2	no	—
359	C₂×C₄	#6772	1	1	yes	characteristic
360	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
361	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
362	C₂×C₄	C₂×D₅×Dic₃	1	1	yes	characteristic
363	C₂×C₄	D₅×#67	1	1	yes	characteristic
364	C₂×C₄	D₅×#67	1	1	yes	characteristic
365	C₈	—	2	2	no	—
366	C₈	—	5	2	no	—
367	C₈	—	5	2	no	—
368	C₈	—	6	2	no	—
369	C₈	—	6	2	no	—
370	C₂×C₃	—	2	1	no	—
371	C₂×C₃	—	2	1	no	—
372	C₂×C₃	—	20	2	no	—
373	C₂×C₃	#64955	1	1	yes	characteristic
374	C₂×C₃	#65183	1	2	yes	—
375	C₅	#72009	1	1	yes	characteristic, complemented, Sylow
376	C₂²	—	2	2	no	—
377	C₂²	—	10	2	no	—
378	C₂²	—	20	2	no	—
379	C₂²	#91358	1	1	yes	characteristic
380	C₂²	#91444	1	1	yes	characteristic
381	C₂²	#91444	1	1	yes	characteristic
382	C₄	—	10	2	no	—
383	C₄	—	60	2	no	—
384	C₄	—	60	2	no	—
385	C₄	C₄×D₅×Dic₃	1	1	yes	characteristic
386	C₄	#91440	1	1	yes	characteristic
387	C₄	#91440	1	1	yes	characteristic
388	C₄	D₅×#217	1	1	yes	characteristic
389	C₃	#10599720	1	1	yes	characteristic, complemented, Sylow
390	C₂	—	2	1	no	—
391	C₂	—	2	1	no	—
392	C₂	—	20	2	no	—
393	C₂	#11748206	1	2	yes	—
394	C₂	#11748702	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₅×#67
commutator	C₃×C₄×C₅	C₂×C₄²
Frattini	C₂²×C₄	C₂×D₃×D₅
Fitting	C₂×C₃×C₅×M₁₆	C₂²
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂×C₃×#709	C₂²
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×M₁₆	D₃×D₅
hypocenter	C₃×C₅	#1765
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
4. C₂×#90954	C₂	semidirect
5. #11748324	C₂	semidirect
6. #11746376	C₂	semidirect
7. C₅×#4558	C₂	semidirect
17. C₃×#4053	C₄	semidirect
20. #90976	C₄	semidirect
21. #90971	C₄	semidirect
22. #90965	C₄	semidirect
69. #4558	D₅	semidirect
79. #4053	Dic₃	semidirect
326. C₃×C₅	#1765	semidirect
375. C₅	#72009	semidirect
389. C₃	#10599720	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#4053	2
C₂×#90976	2
#11748365	2
C₂×#90954	2
#11748324	2
#11746376	2
C₅×#4558	2
#10599720	3
#72009	5