Subgroups — #14411547

Order 1920.

1112 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	C₃×C₄×#709	C₂	1	1	yes	maximal, characteristic
2	C₂×#91283	C₂	1	2	yes	maximal
3	C₄×#6710	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#803	C₂	1	1	yes	maximal, characteristic, complemented
5	#11748318	C₂	1	2	yes	maximal, complemented
6	#11747003	C₂	1	2	yes	maximal, complemented
7	#11746975	C₂	1	4	yes	maximal, complemented
8	D₅×#803	C₂	1	2	yes	maximal, complemented
9	#10610217	—	3	1	no	maximal, Hall
10	C₂×C₃×#709	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₂×#6710	C₂²	1	2	yes	—
13	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
14	C₃×C₄×C₂²⋊D₅	C₄	1	4	yes	—
15	C₃×#4072	C₂²	1	1	yes	characteristic
16	C₃×C₄×D₂₀	C₄	1	2	yes	—
17	C₃×#4018	C₂²	1	2	yes	—
18	C₃×C₄²×D₅	C₂²	1	2	yes	—
19	C₃×C₄×Dic₁₀	C₄	1	2	yes	—
20	C₂×#6649	C₂²	1	4	yes	—
21	#91283	C₄	1	16	yes	complemented
22	C₂×D₅×C₃⋊C₈	C₂²	1	4	yes	—
23	#91087	C₂²	1	2	yes	—
24	C₄×C₃⋊#168	C₂²	1	2	yes	—
25	C₅×#803	C₂²	1	1	yes	characteristic
26	C₅×#803	C₂²	1	2	yes	—
27	C₅×#803	C₂²	1	1	yes	characteristic
28	#90952	C₂²	1	4	yes	—
29	Dic₅×C₃⋊C₈	C₂²	1	4	yes	—
30	#82511	—	5	1	no	maximal, Hall
31	C₂×#4047	—	3	2	no	—
32	C₄×#709	D₃	1	1	yes	characteristic
33	C₂×C₅×#97	—	3	1	no	—
34	#65390	—	3	2	no	—
35	C₄×#681	—	3	1	no	—
36	#64997	—	3	4	no	—
37	#64996	—	3	2	no	—
38	D₅×#97	—	3	2	no	—
39	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
40	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
41	C₂×C₃×C₂²⋊D₅	C₂×C₄	1	2	yes	—
42	C₃×#709	C₂×C₄	1	8	yes	—
43	C₂×C₃×D₂₀	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃×C₄×D₅	C₂×C₄	1	4	yes	—
45	C₂×C₃×C₄×D₅	C₂³	1	2	yes	—
46	C₂×C₃×Dic₁₀	C₂×C₄	1	1	yes	characteristic
47	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
48	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
49	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
50	#6710	C₂×C₄	1	8	yes	complemented
51	C₂×C₃⋊#168	C₂³	1	4	yes	—
52	C₂×C₅×C₃⋊C₈	C₂×C₄	1	8	yes	complemented
53	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
54	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
55	C₃×C₂²⋊Dic₅	C₂×C₄	1	2	yes	—
56	C₃×#685	C₂×C₄	1	4	yes	—
57	C₃×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
58	C₃×#683	C₂×C₄	1	4	yes	—
59	C₃×C₄×Dic₅	C₂×C₄	1	4	yes	—
60	C₃×C₄×Dic₅	C₂³	1	2	yes	—
61	#6649	C₂×C₄	1	16	yes	complemented
62	D₅×C₃⋊C₈	C₂×C₄	1	16	yes	complemented
63	C₃×C₄×#41	—	5	1	no	—
64	C₂×#948	—	5	2	no	—
65	#4997	—	5	2	no	—
66	#4997	—	5	4	no	—
67	C₄×#208	—	5	1	no	—
68	C₄×#208	—	5	2	no	—
69	C₂×#803	—	5	2	no	—
70	C₂×#803	D₅	1	1	yes	characteristic, complemented
71	C₂×#709	C₂×D₃	1	1	yes	characteristic
72	C₂²×C₅×C₈	—	3	2	no	—
73	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
74	C₄×C₂²⋊D₅	Dic₃	1	4	yes	—
75	#4072	C₂×D₃	1	1	yes	characteristic
76	C₂×#681	—	3	2	no	—
77	#4047	—	3	16	no	—
78	C₂×#676	—	3	4	no	—
79	C₂×C₈×D₅	—	3	4	no	—
80	C₄×D₂₀	Dic₃	1	2	yes	—
81	#4018	C₂×D₃	1	2	yes	—
82	C₄²×D₅	C₂×D₃	1	2	yes	—
83	C₄×Dic₁₀	Dic₃	1	2	yes	—
84	C₅×#97	—	3	1	no	—
85	C₅×#97	—	3	1	no	—
86	C₅×#97	—	3	2	no	—
87	#3947	—	3	4	no	—
88	C₈×Dic₅	—	3	4	no	—
89	#3935	—	3	2	no	—
90	C₄×#168	—	3	2	no	—
91	#2875	—	15	1	no	Sylow
92	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₃×D₅	C₂²×C₄	1	2	yes	—
94	C₂×C₃×C₄×C₅	—	4	2	no	—
95	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
96	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
97	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
99	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
100	C₃×C₂²⋊D₅	C₄²	1	8	yes	—
101	C₂×C₃×Dic₅	C₂²×C₄	1	4	yes	—
102	C₂×C₃×Dic₅	C₂²×C₄	1	2	yes	—
103	C₃×D₂₀	C₄²	1	4	yes	—
104	C₃×C₄×D₅	—	4	4	no	—
105	C₃×C₄×D₅	C₂²×C₄	1	8	yes	—
106	C₃×Dic₁₀	C₄²	1	4	yes	—
107	C₃⋊#168	C₂²×C₄	1	8	yes	—
108	C₅×C₃⋊C₈	C₂²×C₄	1	4	yes	—
109	C₅×C₃⋊C₈	C₂²×C₄	1	4	yes	—
110	C₂×C₃×#41	—	5	1	no	—
111	C₃×C₄×Q₈	—	5	2	no	—
112	C₃×C₄×D₄	—	5	4	no	—
113	C₃×C₄×D₄	—	5	2	no	—
114	C₃×#117	—	5	1	no	—
115	C₃×#117	—	5	2	no	—
116	C₂×C₃×C₄²	—	5	2	no	—
117	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
118	#948	—	5	16	no	—
119	C₂×#208	—	5	2	no	—
120	C₂×#208	—	5	4	no	—
121	C₂²×C₃⋊C₈	—	5	4	no	—
122	C₂²×C₃⋊C₈	C₂×D₅	1	2	yes	—
123	#803	—	5	2	no	—
124	#803	—	5	4	no	—
125	#803	C₂×D₅	1	1	yes	characteristic
126	#803	C₂×D₅	1	1	yes	characteristic
127	#803	C₂×D₅	1	2	yes	—
128	C₄×C₃⋊C₈	—	5	2	no	—
129	C₄×C₃⋊C₈	—	5	4	no	—
130	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
131	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
132	C₂×C₂²⋊D₅	C₂×Dic₃	1	2	yes	—
133	#709	C₄×D₃	1	8	yes	—
134	C₂×D₂₀	C₂×Dic₃	1	1	yes	characteristic
135	C₂×C₄×D₅	C₂×Dic₃	1	4	yes	—
136	C₂×C₄×D₅	C₂²×D₃	1	2	yes	—
137	C₂×Dic₁₀	C₂×Dic₃	1	1	yes	characteristic
138	C₂×C₅×C₈	—	3	8	no	—
139	C₂×C₅×C₈	—	3	2	no	—
140	C₂×C₅×C₈	—	3	2	no	—
141	C₄²×C₅	C₂×Dic₃	1	2	yes	—
142	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
143	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
144	C₂²⋊Dic₅	C₂×Dic₃	1	2	yes	—
145	#685	C₂×Dic₃	1	4	yes	—
146	C₄⋊Dic₅	C₂×Dic₃	1	2	yes	—
147	#683	C₂×Dic₃	1	4	yes	—
148	C₄×Dic₅	C₂×Dic₃	1	4	yes	—
149	C₄×Dic₅	C₂²×D₃	1	2	yes	—
150	#681	—	3	8	no	—
151	C₂×#168	—	3	4	no	—
152	#676	—	3	16	no	—
153	C₈×D₅	—	3	16	no	—
154	C₂×#131	—	15	2	no	—
155	C₄×#41	—	5	1	no	—
156	#405	—	15	4	no	—
157	#405	—	15	2	no	—
158	C₄×M₁₆	—	15	2	no	—
159	C₄×M₁₆	—	15	1	no	—
160	C₂×#97	—	3	1	no	—
161	C₂×#97	—	15	2	no	—
162	C₂²×C₃×C₅	—	2	2	no	—
163	C₂²×C₃×C₅	C₂×C₄²	1	2	yes	—
164	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
165	C₂×C₃×D₅	—	2	4	no	—
166	C₂×C₃×D₅	C₂×C₄²	1	4	yes	—
167	C₃×C₄×C₅	—	2	2	no	—
168	C₃×C₄×C₅	—	2	2	no	—
169	C₃×C₄×C₅	C₂×C₄²	1	2	yes	—
170	C₃×C₄×C₅	C₂³×C₄	1	2	yes	—
171	C₃×Dic₅	—	2	4	no	—
172	C₃×Dic₅	C₂×C₄²	1	4	yes	—
173	C₃×#41	—	5	8	no	—
174	C₂×C₃×Q₈	—	5	1	no	—
175	C₂×C₃×D₄	—	5	2	no	—
176	C₂×C₃×D₄	—	5	1	no	—
177	C₂²×C₃×C₄	—	5	2	no	—
178	C₂²×C₃×C₄	—	5	4	no	—
179	C₂²×C₃×C₄	C₄×D₅	1	2	yes	—
180	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
181	C₃×C₄⋊C₄	—	5	4	no	—
182	C₃×C₄⋊C₄	—	5	2	no	—
183	C₃×C₂²⋊C₄	—	5	2	no	—
184	C₃×C₂²⋊C₄	—	5	4	no	—
185	C₃×C₄²	—	5	4	no	—
186	C₃×C₄²	—	5	2	no	—
187	C₃×C₄²	C₄×D₅	1	2	yes	—
188	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
189	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
190	#208	—	5	8	no	—
191	#208	—	5	16	no	—
192	C₂×C₃⋊C₈	—	5	4	no	—
193	C₂×C₃⋊C₈	—	5	16	no	—
194	C₂×C₃⋊C₈	C₄×D₅	1	8	yes	complemented
195	C₂×C₃⋊C₈	C₂²×D₅	1	2	yes	—
196	C₂×C₃⋊C₈	C₂²×D₅	1	2	yes	—
197	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
198	C₂²×D₅	C₂²×Dic₃	1	2	yes	—
199	C₂×C₄×C₅	—	4	2	no	—
200	C₂×C₄×C₅	C₂×C₄×D₃	1	4	yes	—
201	C₂×C₄×C₅	C₂²×Dic₃	1	2	yes	—
202	C₂×C₄×C₅	C₂²×Dic₃	1	2	yes	—
203	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
204	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
205	C₂²⋊D₅	C₄×Dic₃	1	8	yes	—
206	C₂×Dic₅	C₂²×Dic₃	1	4	yes	—
207	C₂×Dic₅	C₂²×Dic₃	1	2	yes	—
208	D₂₀	C₄×Dic₃	1	4	yes	—
209	C₄×D₅	—	4	4	no	—
210	C₄×D₅	C₂×C₄×D₃	1	8	yes	—
211	Dic₁₀	C₄×Dic₃	1	4	yes	—
212	C₅×C₈	—	3	4	no	—
213	C₅×C₈	—	3	4	no	—
214	#168	—	3	8	no	—
215	C₂×#41	—	5	1	no	—
216	#131	—	15	16	no	—
217	C₂×M₁₆	—	15	4	no	—
218	C₂×M₁₆	—	15	2	no	—
219	C₂²×C₈	—	3	2	no	—
220	C₂²×C₈	—	15	4	no	—
221	C₄×Q₈	—	5	2	no	—
222	C₄×D₄	—	5	4	no	—
223	C₄×D₄	—	5	2	no	—
224	#117	—	5	1	no	—
225	#117	—	5	2	no	—
226	C₂×C₄²	—	5	2	no	—
227	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
228	#97	—	3	1	no	—
229	#97	—	3	1	no	—
230	#97	—	3	2	no	—
231	#97	—	15	4	no	—
232	#97	—	15	2	no	—
233	C₄×C₈	—	15	4	no	—
234	C₄×C₈	—	15	2	no	—
235	C₂×C₃×C₅	—	2	2	no	—
236	C₂×C₃×C₅	C₂²×C₄²	1	1	yes	characteristic
237	C₂×C₃×C₅	#568	1	2	yes	—
238	C₃×D₅	—	2	4	no	—
239	C₂³×C₃	—	5	2	no	—
240	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
241	C₃×Q₈	—	5	4	no	—
242	C₃×D₄	—	5	8	no	—
243	C₃×D₄	—	5	4	no	—
244	C₂×C₃×C₄	—	4	2	no	—
245	C₂×C₃×C₄	—	5	2	no	—
246	C₂×C₃×C₄	—	5	8	no	—
247	C₂×C₃×C₄	—	5	4	no	—
248	C₂×C₃×C₄	—	20	4	no	—
249	C₂×C₃×C₄	C₂×C₄×D₅	1	4	yes	—
250	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
251	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
252	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
253	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
254	C₃⋊C₈	—	5	8	no	—
255	C₃⋊C₈	C₂×C₄×D₅	1	4	yes	—
256	C₃⋊C₈	C₂×C₄×D₅	1	4	yes	—
257	C₂²×C₅	—	2	2	no	—
258	C₂²×C₅	C₂×C₄×Dic₃	1	2	yes	—
259	C₂²×C₅	C₂³×Dic₃	1	1	yes	characteristic
260	C₂×D₅	—	2	4	no	—
261	C₂×D₅	C₂×C₄×Dic₃	1	4	yes	—
262	C₄×C₅	—	2	2	no	—
263	C₄×C₅	—	2	2	no	—
264	C₄×C₅	C₂×C₄×Dic₃	1	2	yes	—
265	C₄×C₅	C₂²×C₄×D₃	1	2	yes	—
266	Dic₅	—	2	4	no	—
267	Dic₅	C₂×C₄×Dic₃	1	4	yes	—
268	#41	—	5	8	no	—
269	C₂×Q₈	—	5	1	no	—
270	C₂×D₄	—	5	2	no	—
271	C₂×D₄	—	5	1	no	—
272	C₂²×C₄	—	5	2	no	—
273	C₂²×C₄	—	5	4	no	—
274	C₂²×C₄	D₅×Dic₃	1	2	yes	—
275	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
276	M₁₆	—	15	16	no	—
277	M₁₆	—	15	8	no	—
278	C₂×C₈	—	3	8	no	—
279	C₂×C₈	—	3	2	no	—
280	C₂×C₈	—	3	2	no	—
281	C₂×C₈	—	15	16	no	—
282	C₂×C₈	—	15	4	no	—
283	C₄⋊C₄	—	5	4	no	—
284	C₄⋊C₄	—	5	2	no	—
285	C₂²⋊C₄	—	5	2	no	—
286	C₂²⋊C₄	—	5	4	no	—
287	C₄²	—	5	4	no	—
288	C₄²	—	5	2	no	—
289	C₄²	D₅×Dic₃	1	2	yes	—
290	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
291	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
292	C₃×C₅	#2875	1	1	yes	characteristic, complemented, Hall
293	C₂²×C₃	—	2	2	no	—
294	C₂²×C₃	—	5	4	no	—
295	C₂²×C₃	—	10	4	no	—
296	C₂²×C₃	C₄²×D₅	1	2	yes	—
297	C₂²×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
298	C₃×C₄	—	2	2	no	—
299	C₃×C₄	—	2	2	no	—
300	C₃×C₄	—	5	4	no	—
301	C₃×C₄	—	10	4	no	—
302	C₃×C₄	C₄²×D₅	1	2	yes	—
303	C₃×C₄	C₂²×C₄×D₅	1	2	yes	—
304	C₂×C₅	—	2	2	no	—
305	C₂×C₅	C₂²×C₄×Dic₃	1	1	yes	characteristic
306	C₂×C₅	#5819	1	2	yes	—
307	D₅	—	2	4	no	—
308	C₂³	—	5	2	no	—
309	C₂³	C₂×D₅×Dic₃	1	1	yes	characteristic
310	Q₈	—	5	4	no	—
311	D₄	—	5	8	no	—
312	D₄	—	5	4	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	5	2	no	—
315	C₂×C₄	—	5	8	no	—
316	C₂×C₄	—	5	4	no	—
317	C₂×C₄	—	20	4	no	—
318	C₂×C₄	C₄×D₃×D₅	1	4	yes	—
319	C₂×C₄	C₂×D₅×Dic₃	1	2	yes	—
320	C₂×C₄	C₂×D₅×Dic₃	1	2	yes	—
321	C₂×C₄	C₂×D₅×Dic₃	1	1	yes	characteristic
322	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
323	C₈	—	3	4	no	—
324	C₈	—	3	4	no	—
325	C₈	—	15	8	no	—
326	C₂×C₃	—	2	2	no	—
327	C₂×C₃	—	10	4	no	—
328	C₂×C₃	C₂×C₄²×D₅	1	1	yes	characteristic
329	C₂×C₃	#66079	1	2	yes	—
330	C₅	#82511	1	1	yes	characteristic, complemented, Sylow
331	C₂²	—	2	2	no	—
332	C₂²	—	5	4	no	—
333	C₂²	—	10	4	no	—
334	C₂²	C₄×D₅×Dic₃	1	2	yes	—
335	C₂²	C₂²×D₅×Dic₃	1	1	yes	characteristic
336	C₄	—	2	2	no	—
337	C₄	—	2	2	no	—
338	C₄	—	5	4	no	—
339	C₄	—	10	4	no	—
340	C₄	C₄×D₅×Dic₃	1	2	yes	—
341	C₄	C₂×C₄×D₃×D₅	1	2	yes	—
342	C₃	#10610217	1	1	yes	characteristic, complemented, Sylow
343	C₂	—	2	2	no	—
344	C₂	—	10	4	no	—
345	C₂	#11754120	1	2	yes	—
346	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	C₂×C₃×#709	C₂²
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#2875
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. C₂×C₅×#803	C₂	semidirect
5. #11748318	C₂	semidirect
6. #11747003	C₂	semidirect
7. #11746975	C₂	semidirect
8. D₅×#803	C₂	semidirect
21. #91283	C₄	semidirect
50. #6710	C₂×C₄	semidirect
52. C₂×C₅×C₃⋊C₈	C₂×C₄	semidirect
61. #6649	C₂×C₄	semidirect
62. D₅×C₃⋊C₈	C₂×C₄	semidirect
70. C₂×#803	D₅	semidirect
194. C₂×C₃⋊C₈	C₄×D₅	semidirect
292. C₃×C₅	#2875	semidirect
330. C₅	#82511	semidirect
342. C₃	#10610217	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×#709	2
C₂×#91283	2
C₄×#6710	2
C₂×C₅×#803	2
#11748318	2
#11747003	2
#11746975	2
D₅×#803	2
#10610217	3
#82511	5