Subgroups — #14295012

Order 1920.

872 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄²×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂×#90955	C₂	1	2	yes	maximal
3	C₂×Dic₅×C₃⋊C₈	C₂	1	2	yes	maximal, complemented
4	#11746345	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4524	C₂	1	1	yes	maximal, characteristic, complemented
6	#10599392	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₄×D₅	C₄	1	2	yes	—
8	C₂²×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂²×C₃⋊#168	C₂²	1	2	yes	—
11	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
12	C₂×C₃×C₄×Dic₅	C₄	1	2	yes	—
13	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
14	C₃×C₄²×D₅	—	2	4	no	—
15	#90955	C₄	1	8	yes	complemented
16	Dic₅×C₃⋊C₈	—	2	4	no	—
17	C₄×#830	—	5	1	no	maximal, Hall
18	C₂×C₄²×D₅	D₃	1	1	yes	characteristic
19	C₂×#3952	—	3	2	no	—
20	C₂×C₈×Dic₅	—	3	2	no	—
21	C₅×#336	—	3	1	no	—
22	#64744	—	3	1	no	—
23	C₂³×C₃×D₅	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
25	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂²×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×Dic₅	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₄×D₅	—	2	2	no	—
29	C₂×C₃×C₄×D₅	—	2	1	no	—
30	C₂×C₃×C₄×D₅	—	2	1	no	—
31	C₂×C₃×C₄×D₅	—	2	4	no	—
32	C₂×C₃×C₄×D₅	—	2	4	no	—
33	C₂×C₃×C₄×D₅	C₈	1	8	yes	complemented
34	C₂×C₃×C₄×D₅	C₂×C₄	1	4	yes	—
35	C₃×C₄²×C₅	—	2	2	no	—
36	C₂×C₃⋊#168	—	2	2	no	—
37	C₂×C₃⋊#168	—	2	2	no	—
38	C₂×C₃⋊#168	C₂×C₄	1	4	yes	complemented
39	C₂×C₅×C₃⋊C₈	—	2	2	no	—
40	C₂×C₅×C₃⋊C₈	—	2	2	no	—
41	C₂×C₅×C₃⋊C₈	C₂×C₄	1	4	yes	complemented
42	C₃×C₄×Dic₅	—	2	4	no	—
43	C₃×C₄×Dic₅	D₄	1	4	yes	—
44	C₂²×C₃×C₄²	—	5	1	no	—
45	C₂×#830	—	5	2	no	—
46	C₂×C₄×C₃⋊C₈	—	5	2	no	—
47	#4524	—	5	1	no	—
48	#4524	D₅	1	1	yes	characteristic, complemented
49	C₂²×C₄×D₅	Dic₃	1	2	yes	—
50	C₂²×C₄×D₅	C₂×D₃	1	1	yes	characteristic
51	C₂²×C₅×C₈	—	3	2	no	—
52	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
53	C₂×C₄×Dic₅	Dic₃	1	2	yes	—
54	C₂×C₄×Dic₅	C₂×D₃	1	1	yes	characteristic
55	C₂²×#168	—	3	2	no	—
56	C₄²×D₅	—	2	4	no	—
57	#3952	—	3	8	no	—
58	C₈×Dic₅	—	6	4	no	—
59	C₄×C₂²⋊C₈	—	15	1	no	Sylow
60	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₃×D₅	—	2	1	no	—
62	C₂²×C₃×D₅	—	2	1	no	—
63	C₂²×C₃×D₅	—	2	2	no	—
64	C₂²×C₃×D₅	C₄²	1	2	yes	—
65	C₂²×C₃×D₅	C₂×C₈	1	4	yes	—
66	C₂×C₃×C₄×C₅	—	2	2	no	—
67	C₂×C₃×C₄×C₅	—	2	2	no	—
68	C₂×C₃×C₄×C₅	C₂×C₈	1	4	yes	—
69	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
70	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
71	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
73	C₂×C₃×Dic₅	—	2	1	no	—
74	C₂×C₃×Dic₅	—	2	1	no	—
75	C₂×C₃×Dic₅	C₄²	1	2	yes	—
76	C₂×C₃×Dic₅	C₂²⋊C₄	1	4	yes	—
77	C₂×C₃×Dic₅	C₂×C₈	1	4	yes	—
78	C₂×C₃×Dic₅	M₁₆	1	4	yes	—
79	C₃×C₄×D₅	—	2	4	no	—
80	C₃×C₄×D₅	—	2	4	no	—
81	C₃×C₄×D₅	—	2	16	no	—
82	C₃⋊#168	—	2	4	no	—
83	C₅×C₃⋊C₈	—	2	4	no	—
84	C₂³×C₃×C₄	—	5	2	no	—
85	C₂³×C₃×C₄	—	5	1	no	—
86	C₂×C₃×C₄²	—	5	1	no	—
87	C₂×C₃×C₄²	—	5	2	no	—
88	C₂×C₃×C₄²	—	10	4	no	—
89	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
90	C₂²×C₃⋊C₈	—	5	2	no	—
91	C₂²×C₃⋊C₈	C₂×D₅	1	2	yes	—
92	#830	—	5	8	no	—
93	C₄×C₃⋊C₈	—	10	4	no	—
94	C₂³×D₅	C₂×Dic₃	1	1	yes	characteristic
95	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
96	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
97	C₂²×Dic₅	C₂×Dic₃	1	1	yes	characteristic
98	C₂²×Dic₅	C₂×Dic₃	1	2	yes	—
99	C₂×C₄×D₅	—	2	1	no	—
100	C₂×C₄×D₅	—	2	1	no	—
101	C₂×C₄×D₅	—	2	2	no	—
102	C₂×C₄×D₅	—	2	4	no	—
103	C₂×C₄×D₅	—	2	4	no	—
104	C₂×C₄×D₅	C₃⋊C₈	1	8	yes	complemented
105	C₂×C₄×D₅	C₄×D₃	1	4	yes	—
106	C₂×C₅×C₈	—	3	4	no	—
107	C₂×C₅×C₈	—	6	2	no	—
108	C₂×C₅×C₈	—	6	2	no	—
109	C₄²×C₅	—	2	2	no	—
110	C₄×Dic₅	—	2	4	no	—
111	C₄×Dic₅	#67	1	4	yes	—
112	C₂×#168	—	3	4	no	—
113	C₂×#168	—	6	2	no	—
114	C₂×#168	—	6	2	no	—
115	C₂²×C₄²	—	5	1	no	—
116	C₂×C₂²⋊C₈	—	15	2	no	—
117	C₂×C₄×C₈	—	15	2	no	—
118	#336	—	3	1	no	—
119	#336	—	15	1	no	—
120	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
121	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂²×C₃×C₅	#117	1	1	yes	characteristic
123	C₂²×C₃×C₅	C₂²×C₈	1	2	yes	—
124	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
125	C₂×C₃×D₅	—	2	2	no	—
126	C₂×C₃×D₅	—	2	2	no	—
127	C₂×C₃×D₅	—	2	4	no	—
128	C₂×C₃×D₅	—	2	4	no	—
129	C₂×C₃×D₅	C₄×C₈	1	4	yes	complemented
130	C₃×C₄×C₅	—	2	4	no	—
131	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
132	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
133	C₃×Dic₅	—	2	2	no	—
134	C₃×Dic₅	C₂²⋊C₈	1	8	yes	complemented
135	C₂⁴×C₃	—	5	1	no	—
136	C₂²×C₃×C₄	—	5	2	no	—
137	C₂²×C₃×C₄	—	5	1	no	—
138	C₂²×C₃×C₄	—	5	8	no	—
139	C₂²×C₃×C₄	—	5	4	no	—
140	C₂²×C₃×C₄	—	10	4	no	—
141	C₂²×C₃×C₄	—	10	4	no	—
142	C₂²×C₃×C₄	—	10	2	no	—
143	C₂²×C₃×C₄	—	10	1	no	—
144	C₂²×C₃×C₄	—	10	1	no	—
145	C₂²×C₃×C₄	C₄×D₅	1	2	yes	—
146	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
147	C₃×C₄²	—	2	2	no	—
148	C₃×C₄²	—	5	4	no	—
149	C₃×C₄²	—	10	4	no	—
150	C₂×C₃⋊C₈	—	2	2	no	—
151	C₂×C₃⋊C₈	—	2	2	no	—
152	C₂×C₃⋊C₈	—	5	4	no	—
153	C₂×C₃⋊C₈	—	10	2	no	—
154	C₂×C₃⋊C₈	—	10	2	no	—
155	C₂×C₃⋊C₈	C₄×D₅	1	4	yes	complemented
156	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
157	C₂²×D₅	—	2	2	no	—
158	C₂²×D₅	—	2	1	no	—
159	C₂²×D₅	—	2	1	no	—
160	C₂²×D₅	C₂×C₃⋊C₈	1	4	yes	—
161	C₂²×D₅	C₄×Dic₃	1	2	yes	—
162	C₂×C₄×C₅	—	2	2	no	—
163	C₂×C₄×C₅	—	2	2	no	—
164	C₂×C₄×C₅	C₂×C₃⋊C₈	1	4	yes	—
165	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
166	C₂×C₄×C₅	#235	1	1	yes	characteristic
167	C₂×C₄×C₅	#235	1	1	yes	characteristic
168	C₂×C₄×C₅	C₂×#67	1	2	yes	—
169	C₂×Dic₅	—	2	1	no	—
170	C₂×Dic₅	—	2	1	no	—
171	C₂×Dic₅	C₂×C₃⋊C₈	1	4	yes	—
172	C₂×Dic₅	#208	1	4	yes	—
173	C₂×Dic₅	C₄×Dic₃	1	2	yes	—
174	C₂×Dic₅	#217	1	4	yes	—
175	C₄×D₅	—	2	4	no	—
176	C₄×D₅	—	2	4	no	—
177	C₄×D₅	—	2	16	no	—
178	C₅×C₈	—	6	4	no	—
179	#168	—	6	4	no	—
180	C₂³×C₄	—	5	2	no	—
181	C₂³×C₄	—	5	1	no	—
182	C₂²×C₈	—	3	2	no	—
183	C₂²×C₈	—	15	2	no	—
184	C₂×C₄²	—	5	1	no	—
185	C₂×C₄²	—	5	2	no	—
186	C₂×C₄²	—	10	4	no	—
187	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
188	C₂²⋊C₈	—	15	8	no	—
189	C₄×C₈	—	30	4	no	—
190	C₂×C₃×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
191	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
192	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
193	C₂×C₃×C₅	C₂×C₂²⋊C₈	1	2	yes	—
194	C₂×C₃×C₅	#431	1	2	yes	—
195	C₃×D₅	—	2	4	no	—
196	C₂³×C₃	—	5	2	no	—
197	C₂³×C₃	—	5	4	no	—
198	C₂³×C₃	—	10	1	no	—
199	C₂³×C₃	—	10	1	no	—
200	C₂³×C₃	—	10	2	no	—
201	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	—	2	2	no	—
204	C₂×C₃×C₄	—	5	4	no	—
205	C₂×C₃×C₄	—	5	4	no	—
206	C₂×C₃×C₄	—	5	4	no	—
207	C₂×C₃×C₄	—	5	2	no	—
208	C₂×C₃×C₄	—	10	16	no	—
209	C₂×C₃×C₄	—	10	1	no	—
210	C₂×C₃×C₄	—	10	1	no	—
211	C₂×C₃×C₄	—	10	4	no	—
212	C₂×C₃×C₄	—	10	4	no	—
213	C₂×C₃×C₄	C₈×D₅	1	4	yes	—
214	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
215	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
216	C₂×C₃×C₄	#711	1	1	yes	characteristic
217	C₂×C₃×C₄	#713	1	1	yes	characteristic
218	C₃⋊C₈	—	2	4	no	—
219	C₃⋊C₈	—	10	4	no	—
220	C₂²×C₅	C₂²×C₃⋊C₈	1	2	yes	—
221	C₂²×C₅	C₂×#208	1	2	yes	—
222	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
223	C₂²×C₅	#926	1	1	yes	characteristic
224	C₂²×C₅	C₂×#217	1	1	yes	characteristic
225	C₂×D₅	—	2	2	no	—
226	C₂×D₅	—	2	4	no	—
227	C₂×D₅	—	2	4	no	—
228	C₂×D₅	—	2	2	no	—
229	C₂×D₅	C₄×C₃⋊C₈	1	4	yes	complemented
230	C₄×C₅	—	2	4	no	—
231	C₄×C₅	C₄×#67	1	2	yes	—
232	C₄×C₅	C₄×#67	1	2	yes	—
233	Dic₅	—	2	2	no	—
234	Dic₅	#830	1	8	yes	complemented
235	C₂⁴	—	5	1	no	—
236	C₂²×C₄	—	5	2	no	—
237	C₂²×C₄	—	5	1	no	—
238	C₂²×C₄	—	5	4	no	—
239	C₂²×C₄	—	5	8	no	—
240	C₂²×C₄	—	10	4	no	—
241	C₂²×C₄	—	10	4	no	—
242	C₂²×C₄	—	10	1	no	—
243	C₂²×C₄	—	10	1	no	—
244	C₂²×C₄	—	10	2	no	—
245	C₂²×C₄	D₅×Dic₃	1	2	yes	—
246	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
247	C₂×C₈	—	3	4	no	—
248	C₂×C₈	—	6	2	no	—
249	C₂×C₈	—	6	2	no	—
250	C₂×C₈	—	15	4	no	—
251	C₂×C₈	—	30	2	no	—
252	C₂×C₈	—	30	2	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	5	4	no	—
255	C₄²	—	10	4	no	—
256	C₃×C₅	C₄×C₂²⋊C₈	1	1	yes	characteristic, complemented, Hall
257	C₂²×C₃	—	5	4	no	—
258	C₂²×C₃	—	10	4	no	—
259	C₂²×C₃	—	10	2	no	—
260	C₂²×C₃	—	10	2	no	—
261	C₂²×C₃	—	10	4	no	—
262	C₂²×C₃	C₄²×D₅	1	1	yes	characteristic
263	C₂²×C₃	D₅×C₂²⋊C₄	1	1	yes	characteristic
264	C₂²×C₃	#4038	1	1	yes	characteristic
265	C₂²×C₃	C₂×C₈×D₅	1	2	yes	—
266	C₂²×C₃	D₅×M₁₆	1	2	yes	—
267	C₃×C₄	—	2	4	no	—
268	C₃×C₄	—	5	8	no	—
269	C₃×C₄	—	10	2	no	—
270	C₃×C₄	#4027	1	2	yes	—
271	C₃×C₄	#4039	1	2	yes	—
272	C₂×C₅	C₂×C₄×C₃⋊C₈	1	1	yes	characteristic
273	C₂×C₅	C₄×#208	1	1	yes	characteristic
274	C₂×C₅	#4925	1	2	yes	—
275	C₂×C₅	C₂×#830	1	2	yes	—
276	C₂×C₅	C₄×#217	1	1	yes	characteristic
277	D₅	—	2	4	no	—
278	C₂³	—	5	4	no	—
279	C₂³	—	5	2	no	—
280	C₂³	—	10	2	no	—
281	C₂³	—	10	1	no	—
282	C₂³	—	10	1	no	—
283	C₂³	C₂×D₅×Dic₃	1	1	yes	characteristic
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	5	4	no	—
287	C₂×C₄	—	5	4	no	—
288	C₂×C₄	—	5	4	no	—
289	C₂×C₄	—	5	2	no	—
290	C₂×C₄	—	10	16	no	—
291	C₂×C₄	—	10	1	no	—
292	C₂×C₄	—	10	1	no	—
293	C₂×C₄	—	10	4	no	—
294	C₂×C₄	—	10	4	no	—
295	C₂×C₄	D₅×C₃⋊C₈	1	4	yes	—
296	C₂×C₄	#6772	1	1	yes	characteristic
297	C₂×C₄	#6773	1	1	yes	characteristic
298	C₂×C₄	C₄×D₃×D₅	1	2	yes	—
299	C₂×C₄	D₅×#67	1	2	yes	—
300	C₈	—	6	4	no	—
301	C₈	—	30	4	no	—
302	C₂×C₃	—	10	4	no	—
303	C₂×C₃	#64955	1	1	yes	characteristic
304	C₂×C₃	C₄×C₈×D₅	1	1	yes	characteristic
305	C₂×C₃	#64996	1	1	yes	characteristic
306	C₂×C₃	D₅×C₂²⋊C₈	1	2	yes	—
307	C₂×C₃	#65122	1	2	yes	—
308	C₅	C₄×#830	1	1	yes	characteristic, complemented, Sylow
309	C₂²	—	5	4	no	—
310	C₂²	—	10	4	no	—
311	C₂²	—	10	2	no	—
312	C₂²	—	10	4	no	—
313	C₂²	—	10	2	no	—
314	C₂²	C₂×D₅×C₃⋊C₈	1	2	yes	—
315	C₂²	D₅×#208	1	2	yes	—
316	C₂²	#91358	1	1	yes	characteristic
317	C₂²	C₄×D₅×Dic₃	1	1	yes	characteristic
318	C₂²	D₅×#217	1	1	yes	characteristic
319	C₄	—	2	4	no	—
320	C₄	—	5	8	no	—
321	C₄	—	10	2	no	—
322	C₄	#91440	1	2	yes	—
323	C₄	#91444	1	2	yes	—
324	C₃	#10599392	1	1	yes	characteristic, complemented, Sylow
325	C₂	—	10	4	no	—
326	C₂	#11746869	1	2	yes	—
327	C₂	C₄×D₅×C₃⋊C₈	1	1	yes	characteristic
328	C₂	#11747003	1	1	yes	characteristic
329	C₂	D₅×#830	1	2	yes	—
330	C₂	#11748702	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂×D₃×D₅
commutator	C₂×C₃×C₅	C₂×C₄×C₈
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂³×C₃×C₅	C₂²×C₄
prime radical	C₂³×C₃×D₅	C₂×C₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	C₄×C₂²⋊C₈
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
3. C₂×Dic₅×C₃⋊C₈	C₂	semidirect
4. #11746345	C₂	semidirect
5. C₅×#4524	C₂	semidirect
15. #90955	C₄	semidirect
33. C₂×C₃×C₄×D₅	C₈	semidirect
38. C₂×C₃⋊#168	C₂×C₄	semidirect
41. C₂×C₅×C₃⋊C₈	C₂×C₄	semidirect
48. #4524	D₅	semidirect
104. C₂×C₄×D₅	C₃⋊C₈	semidirect
129. C₂×C₃×D₅	C₄×C₈	semidirect
134. C₃×Dic₅	C₂²⋊C₈	semidirect
155. C₂×C₃⋊C₈	C₄×D₅	semidirect
229. C₂×D₅	C₄×C₃⋊C₈	semidirect
234. Dic₅	#830	semidirect
256. C₃×C₅	C₄×C₂²⋊C₈	semidirect
308. C₅	C₄×#830	semidirect
324. C₃	#10599392	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄²×D₅	2
C₂×#90955	2
C₂×Dic₅×C₃⋊C₈	2
#11746345	2
C₅×#4524	2
#10599392	3
C₄×#830	5