Subgroups — #14417798

Order 1920.

968 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#4050	C₂	1	1	yes	maximal, characteristic
2	#11754509	C₂	1	2	yes	maximal, complemented
3	C₂×#91101	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5101	C₂	1	1	yes	maximal, characteristic, complemented
5	#11748369	C₂	1	2	yes	maximal, complemented
6	#11747870	C₂	1	2	yes	maximal, complemented
7	#11747799	C₂	1	2	yes	maximal, complemented
8	Dic₃×#677	C₂	1	4	yes	maximal, complemented
9	#10610305	—	3	1	no	maximal, Hall
10	C₂×C₃×#709	C₂²	1	2	yes	—
11	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₁₅	C₂²	1	2	yes	—
13	C₅×#926	C₂²	1	1	yes	characteristic
14	C₂×C₅×#208	C₂²	1	1	yes	characteristic
15	C₂×C₃×Dic₂₀	C₂²	1	1	yes	characteristic
16	C₃×#4050	C₄	1	8	yes	complemented
17	C₂×C₃×D₄₀	C₂²	1	1	yes	characteristic
18	C₂×C₃×#677	C₂²	1	2	yes	—
19	#91559	—	2	4	no	—
20	Dic₃×D₂₀	C₂²	1	4	yes	—
21	D₅×C₄⋊Dic₃	—	2	2	no	—
22	#91358	—	2	2	no	—
23	Dic₃×Dic₁₀	C₂²	1	4	yes	—
24	#91101	C₂²	1	2	yes	—
25	#91101	C₂²	1	2	yes	—
26	C₅×#815	C₂²	1	2	yes	—
27	C₅×C₈×Dic₃	C₂²	1	2	yes	—
28	#90971	C₂²	1	4	yes	—
29	#90965	C₂²	1	4	yes	—
30	#83899	—	5	1	no	maximal, Hall
31	C₂×#4050	D₃	1	1	yes	characteristic
32	#65857	—	3	2	no	—
33	C₅×#405	—	3	1	no	—
34	#65422	—	3	2	no	—
35	C₂×#3949	—	3	1	no	—
36	#65005	—	3	2	no	—
37	#65000	—	3	2	no	—
38	C₄×#677	—	3	4	no	—
39	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₂²×Dic₁₅	—	2	2	no	—
41	C₂×C₃×C₂²⋊D₅	—	2	2	no	—
42	C₃×#709	—	2	4	no	—
43	C₃×#709	C₂×C₄	1	8	yes	complemented
44	C₂×C₃×D₂₀	C₂³	1	2	yes	—
45	C₂×C₃×C₄×D₅	—	2	2	no	—
46	C₂×C₃×Dic₁₀	C₂³	1	2	yes	—
47	C₂×D₅×Dic₃	—	2	4	no	—
48	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	complemented
49	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
51	C₄⋊Dic₁₅	—	2	2	no	—
52	C₄⋊Dic₁₅	C₂³	1	4	yes	—
53	C₅×#217	D₄	1	2	yes	—
54	C₅×C₄⋊Dic₃	D₄	1	2	yes	—
55	C₄×C₅×Dic₃	C₂³	1	2	yes	—
56	C₅×#208	—	2	2	no	—
57	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
58	C₃×Dic₂₀	C₂×C₄	1	4	yes	complemented
59	C₃×D₄₀	C₂×C₄	1	4	yes	complemented
60	C₃×#677	C₂×C₄	1	8	yes	complemented
61	#6668	—	2	4	no	—
62	#6665	—	2	4	no	—
63	Dic₃×Dic₅	—	2	4	no	—
64	C₂×C₃×#135	—	5	1	no	—
65	#5827	—	5	2	no	—
66	#5232	—	5	2	no	—
67	#5184	—	5	2	no	—
68	Dic₃×SD₁₆	—	5	4	no	—
69	#5152	—	5	2	no	—
70	C₂×#817	—	5	1	no	—
71	#5101	D₅	1	1	yes	characteristic, complemented
72	C₂×#709	C₂×D₃	1	2	yes	—
73	C₂×C₅×M₁₆	—	3	1	no	—
74	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
75	C₅×#117	—	3	1	no	—
76	C₂×C₄⋊Dic₅	—	3	2	no	—
77	C₂×Dic₂₀	C₂×D₃	1	1	yes	characteristic
78	#4050	Dic₃	1	8	yes	complemented
79	C₂×D₄₀	C₂×D₃	1	1	yes	characteristic
80	C₂×#677	C₂×D₃	1	2	yes	—
81	#4038	—	6	2	no	—
82	D₅×C₄⋊C₄	—	6	2	no	—
83	#4027	—	6	4	no	—
84	C₄×D₂₀	—	3	4	no	—
85	C₄×Dic₁₀	—	3	4	no	—
86	C₅×#97	—	3	2	no	—
87	C₄×C₅×C₈	—	3	2	no	—
88	#3953	—	3	4	no	—
89	#3949	—	3	2	no	—
90	#3949	—	3	2	no	—
91	#3948	—	3	4	no	—
92	#2952	—	15	1	no	Sylow
93	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₃×D₅	—	2	2	no	—
95	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
96	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
97	C₂×C₃×C₄×C₅	#41	1	2	yes	—
98	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
99	C₂×Dic₁₅	—	2	4	no	—
100	C₂×Dic₁₅	—	8	2	no	—
101	C₂×C₅×Dic₃	C₂×D₄	1	4	yes	—
102	C₃×C₂²⋊D₅	—	2	8	no	—
103	C₂×C₃×Dic₅	—	2	2	no	—
104	C₃×D₂₀	—	2	2	no	—
105	C₃×D₂₀	C₂²×C₄	1	4	yes	—
106	C₃×C₄×D₅	—	2	8	no	—
107	C₃×Dic₁₀	—	2	2	no	—
108	C₃×Dic₁₀	C₂²×C₄	1	4	yes	—
109	D₅×Dic₃	—	8	4	no	—
110	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
111	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
112	C₅×C₃⋊C₈	—	2	2	no	—
113	C₂×C₃×#41	—	5	2	no	—
114	C₃×#135	—	5	8	no	—
115	C₂×C₃×Q₁₆	—	5	1	no	—
116	C₂×C₃×SD₁₆	—	5	2	no	—
117	C₂×C₃×D₈	—	5	1	no	—
118	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
119	Q₈×Dic₃	—	5	4	no	—
120	D₄×Dic₃	—	5	4	no	—
121	D₄×Dic₃	—	10	4	no	—
122	#926	—	10	2	no	—
123	#926	C₂×D₅	1	1	yes	characteristic
124	C₂×C₄⋊Dic₃	—	5	2	no	—
125	C₂×C₄⋊Dic₃	—	10	2	no	—
126	C₂×#208	C₂×D₅	1	1	yes	characteristic
127	#835	—	5	4	no	—
128	#832	—	5	4	no	—
129	#817	—	5	2	no	—
130	#817	—	5	2	no	—
131	#815	C₂×D₅	1	2	yes	—
132	C₈×Dic₃	C₂×D₅	1	2	yes	—
133	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
134	C₂×C₂²⋊D₅	—	2	2	no	—
135	C₂²×Dic₅	—	6	2	no	—
136	#709	—	2	4	no	—
137	#709	C₂×Dic₃	1	8	yes	complemented
138	C₂×D₂₀	C₂²×D₃	1	2	yes	—
139	C₂×C₄×D₅	—	2	2	no	—
140	C₂×C₄×D₅	—	6	4	no	—
141	C₂×Dic₁₀	C₂²×D₃	1	2	yes	—
142	C₅×M₁₆	—	6	2	no	—
143	C₂×C₅×C₈	—	3	2	no	—
144	C₂×C₅×C₈	C₂×Dic₃	1	4	yes	complemented
145	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
146	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
147	C₅×C₄⋊C₄	—	3	2	no	—
148	C₅×C₂²⋊C₄	—	3	2	no	—
149	C₄²×C₅	—	3	2	no	—
150	#685	—	6	4	no	—
151	C₄⋊Dic₅	—	3	4	no	—
152	C₄⋊Dic₅	—	6	2	no	—
153	#683	—	6	4	no	—
154	C₄×Dic₅	—	6	4	no	—
155	Dic₂₀	C₂×Dic₃	1	4	yes	complemented
156	D₄₀	C₂×Dic₃	1	4	yes	complemented
157	#677	C₂×Dic₃	1	8	yes	complemented
158	C₂×#135	—	5	1	no	—
159	#520	—	15	2	no	—
160	#442	—	15	2	no	—
161	#441	—	15	2	no	—
162	C₄×SD₁₆	—	15	4	no	—
163	C₂×#106	—	15	1	no	—
164	#417	—	15	2	no	—
165	#405	—	3	1	no	—
166	C₂²×C₃×C₅	—	2	2	no	—
167	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
168	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
169	C₂×C₃×D₅	—	2	4	no	—
170	C₂×C₃×D₅	—	4	2	no	—
171	C₂×C₃×D₅	—	4	2	no	—
172	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
173	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
174	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
175	Dic₁₅	—	4	4	no	—
176	C₃×Dic₅	—	2	4	no	—
177	C₅×Dic₃	—	2	4	no	—
178	C₃×#41	—	5	8	no	—
179	C₃×#41	—	10	4	no	—
180	C₂×C₃×Q₈	—	5	2	no	—
181	C₂×C₃×D₄	—	5	2	no	—
182	C₂×C₃×D₄	—	10	2	no	—
183	C₂²×C₃×C₄	—	10	2	no	—
184	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
185	C₂²×Dic₃	—	10	2	no	—
186	C₂²×Dic₃	—	10	4	no	—
187	C₃×Q₁₆	—	5	4	no	—
188	C₃×SD₁₆	—	5	8	no	—
189	C₃×D₈	—	5	4	no	—
190	C₂×C₃×C₈	C₄×D₅	1	4	yes	complemented
191	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
192	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
193	#217	—	10	4	no	—
194	#217	D₂₀	1	2	yes	—
195	C₄⋊Dic₃	—	5	4	no	—
196	C₄⋊Dic₃	—	10	2	no	—
197	C₄⋊Dic₃	—	10	4	no	—
198	C₄⋊Dic₃	D₂₀	1	2	yes	—
199	C₄×Dic₃	—	10	4	no	—
200	C₄×Dic₃	C₂²×D₅	1	2	yes	—
201	#208	—	2	2	no	—
202	C₂×C₃⋊C₈	C₂²×D₅	1	2	yes	—
203	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
204	C₂²×D₅	—	2	2	no	—
205	C₂×C₄×C₅	—	3	4	no	—
206	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
207	C₂×C₄×C₅	#237	1	2	yes	—
208	C₂×C₄×C₅	C₂²×Dic₃	1	2	yes	—
209	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
210	C₂²⋊D₅	—	2	8	no	—
211	C₂×Dic₅	—	2	2	no	—
212	C₂×Dic₅	—	6	4	no	—
213	C₂×Dic₅	—	24	2	no	—
214	D₂₀	—	2	2	no	—
215	D₂₀	C₂²×Dic₃	1	4	yes	—
216	C₄×D₅	—	2	8	no	—
217	C₄×D₅	—	24	4	no	—
218	Dic₁₀	—	2	2	no	—
219	Dic₁₀	C₂²×Dic₃	1	4	yes	—
220	C₅×C₈	—	6	2	no	—
221	C₅×C₈	C₂²×Dic₃	1	2	yes	—
222	C₅×C₈	C₂²×Dic₃	1	2	yes	—
223	C₂×#41	—	5	2	no	—
224	#135	—	5	8	no	—
225	C₂×Q₁₆	—	5	1	no	—
226	C₂×SD₁₆	—	5	2	no	—
227	C₂×D₈	—	5	1	no	—
228	C₂×M₁₆	—	3	1	no	—
229	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
230	C₄×Q₈	—	15	4	no	—
231	C₄×D₄	—	15	4	no	—
232	C₄×D₄	—	30	4	no	—
233	#117	—	3	1	no	—
234	#117	—	30	2	no	—
235	C₂×C₄⋊C₄	—	15	2	no	—
236	C₂×C₄⋊C₄	—	30	2	no	—
237	#106	—	15	2	no	—
238	#106	—	15	2	no	—
239	#103	—	15	4	no	—
240	#102	—	15	4	no	—
241	#97	—	3	2	no	—
242	C₄×C₈	—	3	2	no	—
243	C₂×C₃×C₅	—	2	2	no	—
244	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
245	C₂×C₃×C₅	#577	1	2	yes	—
246	C₃×D₅	—	4	4	no	—
247	C₂³×C₃	—	10	2	no	—
248	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
249	C₃×Q₈	—	5	4	no	—
250	C₃×Q₈	—	10	2	no	—
251	C₃×D₄	—	5	4	no	—
252	C₃×D₄	—	10	8	no	—
253	C₃×D₄	—	10	2	no	—
254	C₂×C₃×C₄	—	10	2	no	—
255	C₂×C₃×C₄	—	10	8	no	—
256	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
257	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
258	C₂×C₃×C₄	#709	1	2	yes	—
259	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
260	C₂×Dic₃	—	10	4	no	—
261	C₂×Dic₃	—	40	4	no	—
262	C₂×Dic₃	—	40	2	no	—
263	C₂×Dic₃	C₂×D₂₀	1	4	yes	—
264	C₃×C₈	C₂×C₄×D₅	1	2	yes	—
265	C₃×C₈	C₂×C₄×D₅	1	2	yes	—
266	C₃⋊C₈	—	2	2	no	—
267	C₂²×C₅	—	2	2	no	—
268	C₂²×C₅	D₄×Dic₃	1	2	yes	—
269	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
270	C₂×D₅	—	2	4	no	—
271	C₂×D₅	—	4	2	no	—
272	C₂×D₅	—	4	2	no	—
273	C₄×C₅	—	6	4	no	—
274	C₄×C₅	D₄×Dic₃	1	2	yes	—
275	C₄×C₅	C₂×#237	1	1	yes	characteristic
276	C₄×C₅	C₂³×Dic₃	1	1	yes	characteristic
277	Dic₅	—	2	4	no	—
278	Dic₅	—	12	4	no	—
279	#41	—	5	8	no	—
280	#41	—	10	4	no	—
281	C₂×Q₈	—	5	2	no	—
282	C₂×D₄	—	5	2	no	—
283	C₂×D₄	—	10	2	no	—
284	C₂²×C₄	—	10	2	no	—
285	C₂²×C₄	—	30	4	no	—
286	C₂²×C₄	—	30	2	no	—
287	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
288	Q₁₆	—	5	4	no	—
289	SD₁₆	—	5	8	no	—
290	D₈	—	5	4	no	—
291	M₁₆	—	6	2	no	—
292	C₂×C₈	—	3	2	no	—
293	C₂×C₈	D₅×Dic₃	1	4	yes	complemented
294	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
295	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
296	C₄⋊C₄	—	3	2	no	—
297	C₄⋊C₄	—	15	4	no	—
298	C₄⋊C₄	—	30	4	no	—
299	C₄⋊C₄	—	30	2	no	—
300	C₂²⋊C₄	—	3	2	no	—
301	C₂²⋊C₄	—	30	4	no	—
302	C₄²	—	3	2	no	—
303	C₄²	—	30	4	no	—
304	C₃×C₅	#2952	1	1	yes	characteristic, complemented, Hall
305	C₂²×C₃	—	2	2	no	—
306	C₂²×C₃	—	10	4	no	—
307	C₂²×C₃	—	20	2	no	—
308	C₂²×C₃	—	20	2	no	—
309	C₂²×C₃	C₄×D₂₀	1	2	yes	—
310	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
311	C₃×C₄	—	10	4	no	—
312	C₃×C₄	C₄×D₂₀	1	2	yes	—
313	C₃×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
314	C₃×C₄	C₂×#709	1	1	yes	characteristic
315	Dic₃	—	2	4	no	—
316	Dic₃	—	20	4	no	—
317	C₂×C₅	—	2	2	no	—
318	C₂×C₅	#5770	1	2	yes	—
319	C₂×C₅	C₂×D₄×Dic₃	1	1	yes	characteristic
320	D₅	—	4	4	no	—
321	C₂³	—	10	2	no	—
322	C₂³	D₃×D₂₀	1	1	yes	characteristic
323	Q₈	—	5	4	no	—
324	Q₈	—	10	2	no	—
325	D₄	—	5	4	no	—
326	D₄	—	10	8	no	—
327	D₄	—	10	2	no	—
328	C₂×C₄	—	3	4	no	—
329	C₂×C₄	—	10	2	no	—
330	C₂×C₄	—	10	8	no	—
331	C₂×C₄	—	30	4	no	—
332	C₂×C₄	—	120	4	no	—
333	C₂×C₄	—	120	2	no	—
334	C₂×C₄	#6765	1	2	yes	—
335	C₂×C₄	D₃×D₂₀	1	1	yes	characteristic
336	C₂×C₄	C₂×D₅×Dic₃	1	2	yes	—
337	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
338	C₈	—	6	2	no	—
339	C₈	C₂×D₅×Dic₃	1	2	yes	—
340	C₈	C₂×D₅×Dic₃	1	2	yes	—
341	C₂×C₃	—	2	2	no	—
342	C₂×C₃	—	20	4	no	—
343	C₂×C₃	C₂×C₄×D₂₀	1	1	yes	characteristic
344	C₂×C₃	#66085	1	2	yes	—
345	C₅	#83899	1	1	yes	characteristic, complemented, Sylow
346	C₂²	—	2	2	no	—
347	C₂²	—	10	4	no	—
348	C₂²	—	20	2	no	—
349	C₂²	—	20	2	no	—
350	C₂²	Dic₃×D₂₀	1	2	yes	—
351	C₂²	C₂×D₃×D₂₀	1	1	yes	characteristic
352	C₄	—	6	4	no	—
353	C₄	—	10	4	no	—
354	C₄	—	60	4	no	—
355	C₄	Dic₃×D₂₀	1	2	yes	—
356	C₄	C₂×#6765	1	1	yes	characteristic
357	C₄	C₂²×D₅×Dic₃	1	1	yes	characteristic
358	C₃	#10610305	1	1	yes	characteristic, complemented, Sylow
359	C₂	—	2	2	no	—
360	C₂	—	20	4	no	—
361	C₂	#11753972	1	2	yes	—
362	C₂	C₂×Dic₃×D₂₀	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₃×D₂₀
commutator	C₃×C₄×C₅	C₂³×C₄
Frattini	C₂×C₄	C₂²×D₃×D₅
Fitting	C₂²×C₃×C₅×C₈	C₂²
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂×C₃×#4050	C₂
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₂²×C₈	D₃×D₅
hypocenter	C₃×C₅	#2952
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. #11754509	C₂	semidirect
3. C₂×#91101	C₂	semidirect
4. C₅×#5101	C₂	semidirect
5. #11748369	C₂	semidirect
6. #11747870	C₂	semidirect
7. #11747799	C₂	semidirect
8. Dic₃×#677	C₂	semidirect
16. C₃×#4050	C₄	semidirect
43. C₃×#709	C₂×C₄	semidirect
48. C₂×C₃×C₅×C₈	C₂×C₄	semidirect
58. C₃×Dic₂₀	C₂×C₄	semidirect
59. C₃×D₄₀	C₂×C₄	semidirect
60. C₃×#677	C₂×C₄	semidirect
71. #5101	D₅	semidirect
78. #4050	Dic₃	semidirect
137. #709	C₂×Dic₃	semidirect
144. C₂×C₅×C₈	C₂×Dic₃	semidirect
155. Dic₂₀	C₂×Dic₃	semidirect
156. D₄₀	C₂×Dic₃	semidirect
157. #677	C₂×Dic₃	semidirect
190. C₂×C₃×C₈	C₄×D₅	semidirect
293. C₂×C₈	D₅×Dic₃	semidirect
304. C₃×C₅	#2952	semidirect
345. C₅	#83899	semidirect
358. C₃	#10610305	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#4050	2
#11754509	2
C₂×#91101	2
C₅×#5101	2
#11748369	2
#11747870	2
#11747799	2
Dic₃×#677	2
#10610305	3
#83899	5