Subgroups — #14294557

Order 1920.

848 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	C₂×C₃×C₅×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×#91815	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₃⋊#168	C₂	1	1	yes	maximal, characteristic
4	#11746350	C₂	1	4	yes	maximal, complemented
5	#10602373	—	3	1	no	maximal, Hall
6	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₃×C₅×#117	—	2	2	no	—
8	C₃×C₅×#117	C₄	1	4	yes	complemented
9	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₃×C₅×C₂²⋊C₄	—	2	1	no	—
11	C₂×C₃×C₄²×C₅	—	2	1	no	—
12	C₂×#6719	—	2	1	no	—
13	#91815	—	2	2	no	—
14	#91815	C₄	1	4	yes	complemented
15	C₂×C₄⋊Dic₁₅	C₂²	1	2	yes	—
16	C₂×C₄×Dic₁₅	—	2	1	no	—
17	C₂²×C₃⋊#168	—	2	2	no	—
18	C₂²×C₃⋊#168	—	2	2	no	—
19	C₂²×C₃⋊#168	C₄	1	4	yes	complemented
20	C₂²×C₃⋊#168	C₂²	1	2	yes	—
21	#71580	—	5	1	no	maximal, Hall
22	C₂×C₅×#117	D₃	1	1	yes	characteristic
23	C₂×#4072	—	3	1	no	—
24	C₂³×#168	—	3	1	no	—
25	#64749	—	3	4	no	—
26	C₂⁴×C₃×C₅	D₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	—	2	2	no	—
28	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
29	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
30	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
31	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
32	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
33	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
34	C₂²×C₃×C₄×C₅	Q₈	1	2	yes	—
35	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
36	C₂²×Dic₁₅	—	2	2	no	—
37	C₃×C₅×C₄⋊C₄	—	2	2	no	—
38	C₃×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
39	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
40	C₃×C₄²×C₅	—	2	4	no	—
41	#6719	—	2	4	no	—
42	C₄⋊Dic₁₅	—	2	2	no	—
43	C₄⋊Dic₁₅	C₂×C₄	1	4	yes	—
44	C₄×Dic₁₅	—	2	4	no	—
45	C₂×C₃⋊#168	—	2	8	no	—
46	C₂×C₃⋊#168	—	2	2	no	—
47	C₂×C₃⋊#168	—	2	2	no	—
48	C₂×C₃⋊#168	C₂×C₄	1	4	yes	—
49	C₂×C₃×#117	D₅	1	1	yes	characteristic
50	C₂×#926	—	5	1	no	—
51	C₂³×C₃⋊C₈	—	5	1	no	—
52	#4529	—	5	4	no	—
53	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
54	C₅×#117	—	2	2	no	—
55	C₅×#117	Dic₃	1	4	yes	complemented
56	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
57	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
58	C₂×C₄²×C₅	—	2	1	no	—
59	C₂×C₂²⋊Dic₅	—	6	1	no	—
60	#4072	—	3	4	no	—
61	#4072	—	6	2	no	—
62	C₂×C₄⋊Dic₅	—	3	2	no	—
63	C₂×C₄×Dic₅	—	6	1	no	—
64	C₂²×#168	—	3	4	no	—
65	C₂²×#168	—	3	2	no	—
66	C₂²×#168	—	6	2	no	—
67	C₂²×#168	—	6	2	no	—
68	#1735	—	15	1	no	Sylow
69	C₂³×C₃×C₅	—	2	4	no	—
70	C₂³×C₃×C₅	C₂²⋊C₄	1	2	yes	—
71	C₂³×C₃×C₅	C₂²⋊C₄	1	2	yes	—
72	C₂³×C₃×C₅	C₄⋊C₄	1	2	yes	—
73	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	—	2	4	no	—
75	C₂×C₃×C₄×C₅	—	4	4	no	—
76	C₂×C₃×C₄×C₅	C₄²	1	4	yes	—
77	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
78	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
79	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
80	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
81	C₂×C₃×C₄×C₅	C₄⋊C₄	1	4	yes	—
82	C₂×C₃×C₄×C₅	C₄⋊C₄	1	4	yes	—
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	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
89	C₂×Dic₁₅	—	2	4	no	—
90	C₂×Dic₁₅	—	4	4	no	—
91	C₃⋊#168	—	2	4	no	—
92	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
93	C₃×#117	—	2	2	no	—
94	C₃×#117	Dic₅	1	4	yes	complemented
95	C₂×C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
96	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
97	C₂×C₃×C₄²	—	2	1	no	—
98	C₂×#217	—	10	1	no	—
99	#926	—	5	4	no	—
100	#926	—	10	2	no	—
101	C₂×C₄⋊Dic₃	—	5	2	no	—
102	C₂×C₄×Dic₃	—	10	1	no	—
103	C₂²×C₃⋊C₈	—	5	4	no	—
104	C₂²×C₃⋊C₈	—	5	2	no	—
105	C₂²×C₃⋊C₈	—	10	2	no	—
106	C₂²×C₃⋊C₈	—	10	2	no	—
107	C₂⁴×C₅	#67	1	1	yes	characteristic
108	C₂²×C₄×C₅	—	2	2	no	—
109	C₂²×C₄×C₅	Dic₆	1	2	yes	—
110	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
111	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
112	C₂²×C₄×C₅	D₁₂	1	2	yes	—
113	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
114	C₂²×C₄×C₅	#67	1	2	yes	—
115	C₂²×C₄×C₅	#67	1	1	yes	characteristic
116	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
117	C₂²×Dic₅	—	6	2	no	—
118	C₅×C₄⋊C₄	—	2	2	no	—
119	C₅×C₄⋊C₄	C₂×Dic₃	1	4	yes	—
120	C₅×C₂²⋊C₄	—	2	4	no	—
121	C₄²×C₅	—	2	4	no	—
122	C₂²⋊Dic₅	—	6	4	no	—
123	C₄⋊Dic₅	—	3	4	no	—
124	C₄⋊Dic₅	—	6	2	no	—
125	C₄×Dic₅	—	6	4	no	—
126	C₂×#168	—	3	4	no	—
127	C₂×#168	—	6	8	no	—
128	C₂×#168	—	6	2	no	—
129	C₂×#168	—	6	2	no	—
130	C₂³×C₈	—	15	1	no	—
131	C₂×#117	—	15	1	no	—
132	C₂×#117	D₁₅	1	1	yes	characteristic
133	#340	—	15	4	no	—
134	C₂²×C₃×C₅	—	2	4	no	—
135	C₂²×C₃×C₅	—	2	4	no	—
136	C₂²×C₃×C₅	—	2	4	no	—
137	C₂²×C₃×C₅	#95	1	4	yes	—
138	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂²×C₃×C₅	#135	1	4	yes	—
142	C₃×C₄×C₅	—	4	4	no	—
143	C₃×C₄×C₅	#95	1	4	yes	—
144	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
145	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
148	Dic₁₅	—	4	4	no	—
149	C₂⁴×C₃	C₂²⋊D₅	1	1	yes	characteristic
150	C₂²×C₃×C₄	—	2	2	no	—
151	C₂²×C₃×C₄	Dic₁₀	1	2	yes	—
152	C₂²×C₃×C₄	C₄×D₅	1	2	yes	—
153	C₂²×C₃×C₄	C₄×D₅	1	2	yes	—
154	C₂²×C₃×C₄	D₂₀	1	2	yes	—
155	C₂²×C₃×C₄	C₂×Dic₅	1	2	yes	—
156	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
157	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
158	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
159	C₂²×Dic₃	—	10	2	no	—
160	C₃×C₄⋊C₄	—	2	2	no	—
161	C₃×C₄⋊C₄	C₂×Dic₅	1	4	yes	—
162	C₃×C₂²⋊C₄	—	2	4	no	—
163	C₃×C₄²	—	2	4	no	—
164	#217	—	10	4	no	—
165	C₄⋊Dic₃	—	5	4	no	—
166	C₄⋊Dic₃	—	10	2	no	—
167	C₄×Dic₃	—	10	4	no	—
168	C₂×C₃⋊C₈	—	5	4	no	—
169	C₂×C₃⋊C₈	—	10	8	no	—
170	C₂×C₃⋊C₈	—	10	2	no	—
171	C₂×C₃⋊C₈	—	10	2	no	—
172	C₂³×C₅	—	2	4	no	—
173	C₂³×C₅	Dic₃⋊C₄	1	2	yes	—
174	C₂³×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
175	C₂³×C₅	#217	1	2	yes	—
176	C₂³×C₅	C₂×#67	1	1	yes	characteristic
177	C₂×C₄×C₅	—	2	4	no	—
178	C₂×C₄×C₅	—	4	4	no	—
179	C₂×C₄×C₅	C₄×Dic₃	1	4	yes	—
180	C₂×C₄×C₅	Dic₃⋊C₄	1	2	yes	—
181	C₂×C₄×C₅	Dic₃⋊C₄	1	4	yes	—
182	C₂×C₄×C₅	C₄⋊Dic₃	1	4	yes	—
183	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
184	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	4	yes	—
185	C₂×C₄×C₅	#217	1	2	yes	—
186	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
187	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
188	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
189	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
190	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
191	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
192	C₂×Dic₅	—	6	4	no	—
193	C₂×Dic₅	—	12	4	no	—
194	#168	—	6	4	no	—
195	C₂³×C₄	C₂×D₁₅	1	1	yes	characteristic
196	C₂²×C₈	—	15	4	no	—
197	C₂²×C₈	—	15	2	no	—
198	C₂²×C₈	—	30	2	no	—
199	C₂²×C₈	—	30	2	no	—
200	#117	—	2	2	no	—
201	#117	—	15	4	no	—
202	#117	—	30	2	no	—
203	#117	Dic₁₅	1	4	yes	complemented
204	C₂×C₄⋊C₄	—	15	2	no	—
205	C₂×C₄⋊C₄	C₂×D₁₅	1	2	yes	—
206	C₂×C₂²⋊C₄	—	2	1	no	—
207	C₂×C₂²⋊C₄	—	30	1	no	—
208	C₂×C₄²	—	2	1	no	—
209	C₂×C₄²	—	30	1	no	—
210	C₂×C₃×C₅	—	2	4	no	—
211	C₂×C₃×C₅	C₂×#95	1	1	yes	characteristic
212	C₂×C₃×C₅	#416	1	2	yes	—
213	C₂×C₃×C₅	#416	1	2	yes	—
214	C₂×C₃×C₅	#427	1	2	yes	—
215	C₂³×C₃	—	2	4	no	—
216	C₂³×C₃	#683	1	2	yes	—
217	C₂³×C₃	#685	1	2	yes	—
218	C₂³×C₃	C₂²⋊Dic₅	1	2	yes	—
219	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
220	C₂×C₃×C₄	—	2	4	no	—
221	C₂×C₃×C₄	—	4	4	no	—
222	C₂×C₃×C₄	C₄×Dic₅	1	4	yes	—
223	C₂×C₃×C₄	#683	1	2	yes	—
224	C₂×C₃×C₄	#683	1	4	yes	—
225	C₂×C₃×C₄	C₄⋊Dic₅	1	4	yes	—
226	C₂×C₃×C₄	#685	1	2	yes	—
227	C₂×C₃×C₄	#685	1	4	yes	—
228	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
229	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
230	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
231	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
232	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
233	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
234	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
235	C₂×Dic₃	—	10	4	no	—
236	C₂×Dic₃	—	20	4	no	—
237	C₃⋊C₈	—	10	4	no	—
238	C₂²×C₅	—	2	4	no	—
239	C₂²×C₅	—	2	4	no	—
240	C₂²×C₅	—	2	4	no	—
241	C₂²×C₅	C₃⋊#95	1	4	yes	—
242	C₂²×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
243	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
244	C₂²×C₅	#950	1	4	yes	—
245	C₂²×C₅	C₂×#217	1	1	yes	characteristic
246	C₄×C₅	—	4	4	no	—
247	C₄×C₅	C₃⋊#95	1	4	yes	—
248	C₄×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
249	C₄×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
250	C₄×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
251	C₄×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
252	Dic₅	—	12	4	no	—
253	C₂⁴	#1220	1	1	yes	characteristic
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	30	2	no	—
256	C₂²×C₄	Dic₃₀	1	2	yes	—
257	C₂²×C₄	C₄×D₁₅	1	2	yes	—
258	C₂²×C₄	C₄×D₁₅	1	2	yes	—
259	C₂²×C₄	D₆₀	1	2	yes	—
260	C₂²×C₄	C₂×Dic₁₅	1	2	yes	—
261	C₂²×C₄	#1220	1	2	yes	—
262	C₂²×C₄	#1220	1	1	yes	characteristic
263	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
264	C₂×C₈	—	15	4	no	—
265	C₂×C₈	—	30	8	no	—
266	C₂×C₈	—	30	2	no	—
267	C₂×C₈	—	30	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	15	4	no	—
270	C₄⋊C₄	—	30	2	no	—
271	C₄⋊C₄	C₂×Dic₁₅	1	4	yes	—
272	C₂²⋊C₄	—	2	4	no	—
273	C₂²⋊C₄	—	30	4	no	—
274	C₄²	—	2	4	no	—
275	C₄²	—	30	4	no	—
276	C₃×C₅	#1735	1	1	yes	characteristic, complemented, Hall
277	C₂²×C₃	—	2	4	no	—
278	C₂²×C₃	—	2	4	no	—
279	C₂²×C₃	—	2	4	no	—
280	C₂²×C₃	#3963	1	4	yes	—
281	C₂²×C₃	C₂×#683	1	1	yes	characteristic
282	C₂²×C₃	C₂×#685	1	1	yes	characteristic
283	C₂²×C₃	#4096	1	4	yes	—
284	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
285	C₃×C₄	—	4	4	no	—
286	C₃×C₄	#3963	1	4	yes	—
287	C₃×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
288	C₃×C₄	C₂×#683	1	1	yes	characteristic
289	C₃×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
290	C₃×C₄	C₂×#685	1	1	yes	characteristic
291	Dic₃	—	20	4	no	—
292	C₂×C₅	—	2	4	no	—
293	C₂×C₅	#4998	1	2	yes	—
294	C₂×C₅	#5002	1	2	yes	—
295	C₂×C₅	C₂×C₃⋊#95	1	1	yes	characteristic
296	C₂×C₅	#5233	1	2	yes	—
297	C₂³	—	2	4	no	—
298	C₂³	#6712	1	2	yes	—
299	C₂³	#6714	1	2	yes	—
300	C₂³	#6719	1	2	yes	—
301	C₂³	C₂×#1220	1	1	yes	characteristic
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	—	30	4	no	—
305	C₂×C₄	—	60	4	no	—
306	C₂×C₄	C₄×Dic₁₅	1	4	yes	—
307	C₂×C₄	#6712	1	4	yes	—
308	C₂×C₄	#6712	1	2	yes	—
309	C₂×C₄	C₄⋊Dic₁₅	1	4	yes	—
310	C₂×C₄	#6714	1	4	yes	—
311	C₂×C₄	#6714	1	2	yes	—
312	C₂×C₄	#6719	1	2	yes	—
313	C₂×C₄	C₂×Dic₃₀	1	1	yes	characteristic
314	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
315	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
316	C₂×C₄	C₂×D₆₀	1	1	yes	characteristic
317	C₂×C₄	C₂²×Dic₁₅	1	1	yes	characteristic
318	C₂×C₄	C₂×#1220	1	1	yes	characteristic
319	C₈	—	30	4	no	—
320	C₂×C₃	—	2	4	no	—
321	C₂×C₃	#65287	1	2	yes	—
322	C₂×C₃	#65291	1	2	yes	—
323	C₂×C₃	C₂×#3963	1	1	yes	characteristic
324	C₂×C₃	#65522	1	2	yes	—
325	C₅	#71580	1	1	yes	characteristic, complemented, Sylow
326	C₂²	—	2	4	no	—
327	C₂²	—	2	4	no	—
328	C₂²	—	2	4	no	—
329	C₂²	#91115	1	4	yes	—
330	C₂²	C₂×#6712	1	1	yes	characteristic
331	C₂²	C₂×#6714	1	1	yes	characteristic
332	C₂²	#91839	1	4	yes	—
333	C₂²	C₂×#6719	1	1	yes	characteristic
334	C₄	—	4	4	no	—
335	C₄	—	60	4	no	—
336	C₄	#91115	1	4	yes	—
337	C₄	C₂×C₄×Dic₁₅	1	1	yes	characteristic
338	C₄	C₂×#6712	1	1	yes	characteristic
339	C₄	C₂×C₄⋊Dic₁₅	1	1	yes	characteristic
340	C₄	C₂×#6714	1	1	yes	characteristic
341	C₃	#10602373	1	1	yes	characteristic, complemented, Sylow
342	C₂	—	2	4	no	—
343	C₂	#11751099	1	2	yes	—
344	C₂	#11751103	1	2	yes	—
345	C₂	C₂×#91115	1	1	yes	characteristic
346	C₂	#11751334	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	#1220
commutator	C₃×C₄×C₅	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₂×C₃×C₅×#117	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂⁴×C₃×C₅	D₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×#117	D₁₅
hypocenter	C₃×C₅	#1735
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
4. #11746350	C₂	semidirect
8. C₃×C₅×#117	C₄	semidirect
14. #91815	C₄	semidirect
19. C₂²×C₃⋊#168	C₄	semidirect
55. C₅×#117	Dic₃	semidirect
94. C₃×#117	Dic₅	semidirect
203. #117	Dic₁₅	semidirect
276. C₃×C₅	#1735	semidirect
325. C₅	#71580	semidirect
341. C₃	#10602373	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×#117	2
C₂×#91815	2
C₂³×C₃⋊#168	2
#11746350	2
#10602373	3
#71580	5