Subgroups — #14411422

Order 1920.

1128 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#4018	C₂	1	1	yes	maximal, characteristic
2	C₂×D₅×#208	C₂	1	2	yes	maximal
3	C₄×#6710	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₅×#208	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₅×#208	C₂	1	2	yes	maximal, complemented
6	#11746975	C₂	1	4	yes	maximal, complemented
7	#11746974	C₂	1	4	yes	maximal, complemented
8	#10610245	—	3	1	no	maximal, Hall
9	C₂²×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂×#6710	C₂²	1	2	yes	—
12	C₂×C₅×#208	C₂²	1	2	yes	—
13	C₂×C₃×#685	C₄	1	2	yes	—
14	C₂×C₃×#683	C₄	1	2	yes	—
15	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
16	C₃×#4018	C₄	1	4	yes	—
17	C₃×#4018	C₂²	1	2	yes	—
18	C₃×#4018	C₂²	1	2	yes	—
19	C₂×#6649	C₂²	1	4	yes	—
20	D₅×#208	C₄	1	16	yes	complemented
21	C₂×D₅×C₃⋊C₈	C₂²	1	4	yes	—
22	#91087	C₂²	1	2	yes	complemented
23	C₄×C₃⋊#168	C₂²	1	2	yes	complemented
24	C₅×#803	C₂²	1	2	yes	complemented
25	C₄×C₅×C₃⋊C₈	C₂²	1	2	yes	complemented
26	#90952	C₂²	1	4	yes	complemented
27	Dic₅×C₃⋊C₈	C₂²	1	4	yes	complemented
28	#82202	—	5	1	no	maximal, Hall
29	C₂×D₅×M₁₆	—	3	2	no	—
30	C₂×#4018	D₃	1	1	yes	characteristic
31	C₄×C₅×M₁₆	—	3	1	no	—
32	M₁₆×Dic₅	—	3	2	no	—
33	C₄×#681	—	3	1	no	—
34	#64997	—	3	4	no	—
35	#64978	—	3	4	no	—
36	C₂³×C₃×D₅	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
38	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
39	C₂²×C₃×Dic₅	C₂×C₄	1	2	yes	—
40	C₂²×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃×C₄×D₅	C₂×C₄	1	2	yes	—
42	C₂×C₃×C₄×D₅	C₂×C₄	1	8	yes	—
43	C₂×C₃×C₄×D₅	C₂³	1	2	yes	—
44	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
45	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
47	#6710	C₂×C₄	1	8	yes	—
48	C₂×C₃⋊#168	C₂³	1	4	yes	—
49	C₅×#208	C₂×C₄	1	8	yes	—
50	C₂×C₅×C₃⋊C₈	C₂³	1	4	yes	—
51	C₃×#685	C₂×C₄	1	4	yes	—
52	C₃×#685	C₂×C₄	1	4	yes	—
53	C₃×#683	C₂×C₄	1	4	yes	—
54	C₃×#683	C₂×C₄	1	4	yes	—
55	C₃×C₄×Dic₅	C₂×C₄	1	2	yes	—
56	C₃×C₄×Dic₅	C₂³	1	2	yes	—
57	#6649	C₂×C₄	1	16	yes	complemented
58	D₅×C₃⋊C₈	C₂×C₄	1	16	yes	complemented
59	C₂×C₃×#117	—	5	1	no	—
60	C₂²×#208	—	5	2	no	—
61	#4997	—	5	4	no	—
62	#4997	—	5	4	no	—
63	C₄×#208	—	5	1	no	—
64	C₄×#208	—	5	2	no	—
65	C₄×#208	D₅	1	1	yes	characteristic, complemented
66	C₂²×C₄×D₅	C₂×D₃	1	1	yes	characteristic
67	C₂×C₅×M₁₆	—	3	2	no	—
68	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
69	C₂×#685	Dic₃	1	2	yes	—
70	C₂×#683	Dic₃	1	2	yes	—
71	C₂×C₄×Dic₅	C₂×D₃	1	1	yes	characteristic
72	C₂×#681	—	3	2	no	—
73	D₅×M₁₆	—	3	16	no	—
74	C₂×#676	—	3	4	no	—
75	C₂×C₈×D₅	—	3	4	no	—
76	#4018	Dic₃	1	4	yes	—
77	#4018	C₂×D₃	1	2	yes	—
78	#4018	C₂×D₃	1	2	yes	—
79	C₅×#97	—	3	2	no	—
80	C₄×C₅×C₈	—	3	2	no	—
81	#3947	—	3	4	no	—
82	C₈×Dic₅	—	3	4	no	—
83	#3935	—	3	2	no	—
84	C₄×#168	—	3	2	no	—
85	#2873	—	15	1	no	Sylow
86	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₃×D₅	—	4	2	no	—
88	C₂²×C₃×D₅	C₄²	1	4	yes	—
89	C₂²×C₃×D₅	C₂²×C₄	1	2	yes	—
90	C₂×C₃×C₄×C₅	—	4	2	no	—
91	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
92	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
93	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
95	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
96	C₂×C₃×Dic₅	—	4	2	no	—
97	C₂×C₃×Dic₅	C₄²	1	4	yes	—
98	C₂×C₃×Dic₅	C₂²×C₄	1	4	yes	—
99	C₂×C₃×Dic₅	C₂²×C₄	1	2	yes	—
100	C₃×C₄×D₅	C₄²	1	8	yes	—
101	C₃×C₄×D₅	C₂²×C₄	1	4	yes	—
102	C₃×C₄×D₅	C₂²×C₄	1	4	yes	—
103	C₃⋊#168	C₂²×C₄	1	8	yes	—
104	C₅×C₃⋊C₈	C₂²×C₄	1	8	yes	—
105	C₂³×C₃×C₄	—	5	1	no	—
106	C₃×#117	—	5	4	no	—
107	C₃×#117	—	5	2	no	—
108	C₃×#117	—	5	2	no	—
109	C₂×C₃×C₄⋊C₄	—	5	2	no	—
110	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
111	C₂×C₃×C₄²	—	5	1	no	—
112	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
113	C₂×#208	—	5	2	no	—
114	C₂×#208	—	5	16	no	—
115	C₂×#208	—	5	4	no	—
116	C₂×#208	C₂×D₅	1	2	yes	—
117	C₂²×C₃⋊C₈	—	5	4	no	—
118	#803	—	5	2	no	—
119	#803	—	5	4	no	—
120	#803	C₂×D₅	1	2	yes	complemented
121	C₄×C₃⋊C₈	—	5	2	no	—
122	C₄×C₃⋊C₈	—	5	4	no	—
123	C₄×C₃⋊C₈	C₂×D₅	1	2	yes	complemented
124	C₂³×D₅	C₂×Dic₃	1	1	yes	characteristic
125	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
126	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
127	C₂²×Dic₅	C₂×Dic₃	1	2	yes	—
128	C₂²×Dic₅	C₂×Dic₃	1	1	yes	characteristic
129	C₂×C₄×D₅	C₄×D₃	1	8	yes	—
130	C₂×C₄×D₅	C₂×Dic₃	1	2	yes	—
131	C₂×C₄×D₅	C₂²×D₃	1	2	yes	—
132	C₅×M₁₆	—	3	8	no	—
133	C₂×C₅×C₈	—	3	4	no	—
134	C₄²×C₅	C₂×Dic₃	1	2	yes	—
135	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
136	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
137	#685	C₂×Dic₃	1	4	yes	—
138	#685	C₂×Dic₃	1	4	yes	—
139	#683	C₂×Dic₃	1	4	yes	—
140	#683	C₂×Dic₃	1	4	yes	—
141	C₄×Dic₅	C₂×Dic₃	1	2	yes	—
142	C₄×Dic₅	C₂²×D₃	1	2	yes	—
143	#681	—	3	8	no	—
144	C₂×#168	—	3	4	no	—
145	#676	—	3	16	no	—
146	C₈×D₅	—	3	16	no	—
147	C₂²×M₁₆	—	15	2	no	—
148	C₂×#117	—	5	1	no	—
149	#405	—	15	4	no	—
150	#405	—	15	4	no	—
151	C₄×M₁₆	—	3	1	no	—
152	C₄×M₁₆	—	15	2	no	—
153	C₄×M₁₆	—	15	1	no	—
154	C₂²×C₃×C₅	—	2	2	no	—
155	C₂²×C₃×C₅	C₂×C₄²	1	2	yes	—
156	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
157	C₂×C₃×D₅	—	2	4	no	—
158	C₂×C₃×D₅	—	4	4	no	—
159	C₂×C₃×D₅	C₂×C₄²	1	4	yes	—
160	C₃×C₄×C₅	—	2	2	no	—
161	C₃×C₄×C₅	—	2	2	no	—
162	C₃×C₄×C₅	C₂×C₄²	1	2	yes	—
163	C₃×C₄×C₅	C₂³×C₄	1	2	yes	—
164	C₃×Dic₅	—	2	4	no	—
165	C₃×Dic₅	C₂×C₄²	1	4	yes	—
166	C₂⁴×C₃	—	5	1	no	—
167	C₂²×C₃×C₄	—	5	1	no	—
168	C₂²×C₃×C₄	—	5	8	no	—
169	C₂²×C₃×C₄	—	5	2	no	—
170	C₂²×C₃×C₄	—	5	2	no	—
171	C₂²×C₃×C₄	—	5	2	no	—
172	C₂²×C₃×C₄	C₄×D₅	1	2	yes	—
173	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
174	C₃×C₄⋊C₄	—	5	4	no	—
175	C₃×C₄⋊C₄	—	5	4	no	—
176	C₃×C₂²⋊C₄	—	5	4	no	—
177	C₃×C₂²⋊C₄	—	5	4	no	—
178	C₃×C₄²	—	5	2	no	—
179	C₃×C₄²	—	5	2	no	—
180	C₃×C₄²	C₄×D₅	1	2	yes	—
181	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
182	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
183	#208	—	5	8	no	—
184	#208	—	5	16	no	—
185	#208	C₄×D₅	1	8	yes	—
186	C₂×C₃⋊C₈	—	5	4	no	—
187	C₂×C₃⋊C₈	—	5	16	no	—
188	C₂×C₃⋊C₈	C₂²×D₅	1	4	yes	—
189	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
190	C₂²×D₅	—	4	2	no	—
191	C₂²×D₅	C₄×Dic₃	1	4	yes	—
192	C₂²×D₅	C₂²×Dic₃	1	2	yes	—
193	C₂×C₄×C₅	—	4	2	no	—
194	C₂×C₄×C₅	C₂×C₄×D₃	1	4	yes	—
195	C₂×C₄×C₅	C₂²×Dic₃	1	2	yes	—
196	C₂×C₄×C₅	C₂²×Dic₃	1	2	yes	—
197	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
198	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
199	C₂×Dic₅	—	4	2	no	—
200	C₂×Dic₅	C₄×Dic₃	1	4	yes	—
201	C₂×Dic₅	C₂²×Dic₃	1	4	yes	—
202	C₂×Dic₅	C₂²×Dic₃	1	2	yes	—
203	C₄×D₅	C₄×Dic₃	1	8	yes	—
204	C₄×D₅	C₂×C₄×D₃	1	4	yes	—
205	C₄×D₅	C₂×C₄×D₃	1	4	yes	—
206	C₅×C₈	—	3	8	no	—
207	#168	—	3	8	no	—
208	C₂³×C₄	—	5	1	no	—
209	C₂×M₁₆	—	3	2	no	—
210	C₂×M₁₆	—	15	4	no	—
211	C₂×M₁₆	—	15	16	no	—
212	C₂×M₁₆	—	15	2	no	—
213	C₂²×C₈	—	15	4	no	—
214	#117	—	5	4	no	—
215	#117	—	5	2	no	—
216	#117	—	5	2	no	—
217	C₂×C₄⋊C₄	—	5	2	no	—
218	C₂×C₂²⋊C₄	—	5	2	no	—
219	C₂×C₄²	—	5	1	no	—
220	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
221	#97	—	3	2	no	—
222	#97	—	15	4	no	—
223	#97	—	15	2	no	—
224	C₄×C₈	—	3	2	no	—
225	C₄×C₈	—	15	4	no	—
226	C₄×C₈	—	15	2	no	—
227	C₂×C₃×C₅	—	2	2	no	—
228	C₂×C₃×C₅	C₂²×C₄²	1	1	yes	characteristic
229	C₂×C₃×C₅	#568	1	2	yes	—
230	C₃×D₅	—	2	4	no	—
231	C₂³×C₃	—	5	2	no	—
232	C₂³×C₃	—	5	4	no	—
233	C₂³×C₃	—	20	2	no	—
234	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
235	C₂×C₃×C₄	—	4	2	no	—
236	C₂×C₃×C₄	—	5	2	no	—
237	C₂×C₃×C₄	—	5	4	no	—
238	C₂×C₃×C₄	—	5	4	no	—
239	C₂×C₃×C₄	—	5	4	no	—
240	C₂×C₃×C₄	—	5	8	no	—
241	C₂×C₃×C₄	—	5	4	no	—
242	C₂×C₃×C₄	—	20	2	no	—
243	C₂×C₃×C₄	C₂×C₄×D₅	1	4	yes	—
244	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
245	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
246	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
247	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
248	C₃⋊C₈	—	5	8	no	—
249	C₃⋊C₈	C₂×C₄×D₅	1	8	yes	—
250	C₂²×C₅	—	2	2	no	—
251	C₂²×C₅	C₂×C₄×Dic₃	1	2	yes	—
252	C₂²×C₅	C₂³×Dic₃	1	1	yes	characteristic
253	C₂×D₅	—	2	4	no	—
254	C₂×D₅	—	4	4	no	—
255	C₂×D₅	C₂×C₄×Dic₃	1	4	yes	—
256	C₄×C₅	—	2	2	no	—
257	C₄×C₅	—	2	2	no	—
258	C₄×C₅	C₂×C₄×Dic₃	1	2	yes	—
259	C₄×C₅	C₂²×C₄×D₃	1	2	yes	—
260	Dic₅	—	2	4	no	—
261	Dic₅	C₂×C₄×Dic₃	1	4	yes	—
262	C₂⁴	—	5	1	no	—
263	C₂²×C₄	—	5	1	no	—
264	C₂²×C₄	—	5	8	no	—
265	C₂²×C₄	—	5	2	no	—
266	C₂²×C₄	—	5	2	no	—
267	C₂²×C₄	—	5	2	no	—
268	C₂²×C₄	D₅×Dic₃	1	2	yes	—
269	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
270	M₁₆	—	3	8	no	—
271	M₁₆	—	15	16	no	—
272	M₁₆	—	15	8	no	—
273	C₂×C₈	—	3	4	no	—
274	C₂×C₈	—	15	16	no	—
275	C₂×C₈	—	15	4	no	—
276	C₄⋊C₄	—	5	4	no	—
277	C₄⋊C₄	—	5	4	no	—
278	C₂²⋊C₄	—	5	4	no	—
279	C₂²⋊C₄	—	5	4	no	—
280	C₄²	—	5	2	no	—
281	C₄²	—	5	2	no	—
282	C₄²	D₅×Dic₃	1	2	yes	—
283	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
284	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
285	C₃×C₅	#2873	1	1	yes	characteristic, complemented, Hall
286	C₂²×C₃	—	2	2	no	—
287	C₂²×C₃	—	5	4	no	—
288	C₂²×C₃	—	10	4	no	—
289	C₂²×C₃	—	20	4	no	—
290	C₂²×C₃	C₄²×D₅	1	2	yes	—
291	C₂²×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
292	C₃×C₄	—	2	2	no	—
293	C₃×C₄	—	2	2	no	—
294	C₃×C₄	—	5	4	no	—
295	C₃×C₄	—	10	4	no	—
296	C₃×C₄	C₄²×D₅	1	2	yes	—
297	C₃×C₄	C₂²×C₄×D₅	1	2	yes	—
298	C₂×C₅	—	2	2	no	—
299	C₂×C₅	C₂²×C₄×Dic₃	1	1	yes	characteristic
300	C₂×C₅	#5819	1	2	yes	—
301	D₅	—	2	4	no	—
302	C₂³	—	5	2	no	—
303	C₂³	—	5	4	no	—
304	C₂³	—	20	2	no	—
305	C₂³	C₂×D₅×Dic₃	1	1	yes	characteristic
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	5	2	no	—
308	C₂×C₄	—	5	4	no	—
309	C₂×C₄	—	5	4	no	—
310	C₂×C₄	—	5	4	no	—
311	C₂×C₄	—	5	8	no	—
312	C₂×C₄	—	5	4	no	—
313	C₂×C₄	—	20	2	no	—
314	C₂×C₄	C₄×D₃×D₅	1	4	yes	—
315	C₂×C₄	C₂×D₅×Dic₃	1	2	yes	—
316	C₂×C₄	C₂×D₅×Dic₃	1	2	yes	—
317	C₂×C₄	C₂×D₅×Dic₃	1	1	yes	characteristic
318	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
319	C₈	—	3	8	no	—
320	C₈	—	15	8	no	—
321	C₂×C₃	—	2	2	no	—
322	C₂×C₃	—	10	4	no	—
323	C₂×C₃	C₂×C₄²×D₅	1	1	yes	characteristic
324	C₂×C₃	#66079	1	2	yes	—
325	C₅	#82202	1	1	yes	characteristic, complemented, Sylow
326	C₂²	—	2	2	no	—
327	C₂²	—	5	4	no	—
328	C₂²	—	10	4	no	—
329	C₂²	—	20	4	no	—
330	C₂²	C₄×D₅×Dic₃	1	2	yes	—
331	C₂²	C₂²×D₅×Dic₃	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	5	4	no	—
335	C₄	—	10	4	no	—
336	C₄	C₄×D₅×Dic₃	1	2	yes	—
337	C₄	C₂×C₄×D₃×D₅	1	2	yes	—
338	C₃	#10610245	1	1	yes	characteristic, complemented, Sylow
339	C₂	—	2	2	no	—
340	C₂	—	10	4	no	—
341	C₂	#11754120	1	2	yes	—
342	C₂	C₂×C₄×D₅×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃×D₅
commutator	C₂×C₃×C₅	C₂²×C₄²
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂²×C₃×C₅	C₂³×C₄
prime radical	C₂³×C₃×D₅	C₂×C₄
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#2873
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
3. C₄×#6710	C₂	semidirect
4. C₄×C₅×#208	C₂	semidirect
5. Dic₅×#208	C₂	semidirect
6. #11746975	C₂	semidirect
7. #11746974	C₂	semidirect
20. D₅×#208	C₄	semidirect
22. #91087	C₂²	semidirect
23. C₄×C₃⋊#168	C₂²	semidirect
24. C₅×#803	C₂²	semidirect
25. C₄×C₅×C₃⋊C₈	C₂²	semidirect
26. #90952	C₂²	semidirect
27. Dic₅×C₃⋊C₈	C₂²	semidirect
57. #6649	C₂×C₄	semidirect
58. D₅×C₃⋊C₈	C₂×C₄	semidirect
65. C₄×#208	D₅	semidirect
120. #803	C₂×D₅	semidirect
123. C₄×C₃⋊C₈	C₂×D₅	semidirect
285. C₃×C₅	#2873	semidirect
325. C₅	#82202	semidirect
338. C₃	#10610245	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#4018	2
C₂×D₅×#208	2
C₄×#6710	2
C₄×C₅×#208	2
Dic₅×#208	2
#11746975	2
#11746974	2
#10610245	3
#82202	5