Subgroups — #14280913

Order 1920.

624 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×D₁₂	C₂	1	1	yes	maximal, characteristic
2	#11751026	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65214	C₂	1	1	yes	maximal, characteristic, complemented
4	#11745949	C₂	1	4	yes	maximal, complemented
5	#10602394	—	3	1	no	maximal, Hall
6	C₂²×C₅×D₁₂	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₅×D₃	C₄	1	2	yes	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
10	C₂×C₅×C₄⋊Dic₃	C₄	1	1	yes	characteristic
11	C₄×C₅×D₁₂	—	2	1	no	—
12	C₄×C₅×D₁₂	—	2	1	no	—
13	C₄×C₅×D₁₂	—	2	1	no	—
14	C₄×C₅×D₁₂	—	2	1	no	—
15	#91114	C₂²	1	2	yes	—
16	C₄⋊(C₃⋊#168)	—	2	1	no	—
17	C₄×C₃⋊#168	—	2	1	no	—
18	C₃×C₂²⋊#168	C₂²	1	2	yes	—
19	C₃×C₄⋊#168	—	2	1	no	—
20	C₃×C₄×#168	—	2	1	no	—
21	#70783	—	5	1	no	maximal, Hall
22	C₂×C₄×C₅×D₄	—	3	1	no	—
23	#65214	—	3	1	no	—
24	#65214	D₃	1	1	yes	characteristic, complemented
25	#64745	—	3	4	no	—
26	C₂³×C₅×D₃	C₂×C₄	1	2	yes	—
27	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
30	C₂²×C₅×Dic₃	C₂×C₄	1	2	yes	—
31	C₂×C₅×D₁₂	—	2	1	no	—
32	C₂×C₅×D₁₂	—	2	1	no	—
33	C₂×C₅×D₁₂	—	2	2	no	—
34	C₂×C₅×D₁₂	C₈	1	4	yes	complemented
35	C₂×C₄×C₅×D₃	—	2	2	no	—
36	C₂×C₄×C₅×D₃	—	2	2	no	—
37	C₂×C₄×C₅×D₃	—	4	4	no	—
38	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
39	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
40	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₂×C₃⋊#168	—	2	2	no	—
43	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
44	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
45	C₅×C₄⋊Dic₃	—	2	1	no	—
46	C₅×C₄⋊Dic₃	—	2	1	no	—
47	C₂×C₃×#168	—	2	2	no	—
48	C₂×C₄×D₁₂	D₅	1	1	yes	characteristic
49	C₃×#431	—	5	1	no	—
50	#4925	—	5	1	no	—
51	#4468	—	5	4	no	—
52	C₂²×C₅×D₄	—	3	1	no	—
53	C₂³×C₄×C₅	—	3	2	no	—
54	C₄×C₅×D₄	—	6	1	no	—
55	C₄×C₅×D₄	—	6	1	no	—
56	C₄×C₅×D₄	—	6	1	no	—
57	C₄×C₅×D₄	—	6	1	no	—
58	C₂×C₅×C₄⋊C₄	—	3	1	no	—
59	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
60	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
61	C₂²⋊#168	—	3	2	no	—
62	C₂²⋊#168	C₂×D₃	1	2	yes	—
63	C₄⋊#168	—	2	1	no	—
64	C₄⋊#168	—	6	1	no	—
65	C₄×#168	—	2	1	no	—
66	C₄×#168	—	6	1	no	—
67	#1460	—	15	1	no	Sylow
68	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₅×D₃	—	2	2	no	—
70	C₂²×C₅×D₃	—	2	2	no	—
71	C₂²×C₅×D₃	—	4	2	no	—
72	C₂²×C₅×D₃	—	4	2	no	—
73	C₂²×C₅×D₃	C₂×C₈	1	4	yes	—
74	C₂×C₃×C₄×C₅	—	2	2	no	—
75	C₂×C₃×C₄×C₅	—	4	1	no	—
76	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
81	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
82	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂×C₅×Dic₃	—	2	2	no	—
85	C₂×C₅×Dic₃	—	4	2	no	—
86	C₅×D₁₂	—	2	4	no	—
87	C₅×D₁₂	—	2	4	no	—
88	C₄×C₅×D₃	—	4	4	no	—
89	C₄×C₅×D₃	—	4	4	no	—
90	C₃⋊#168	—	4	2	no	—
91	C₃×#168	—	4	2	no	—
92	C₂²×D₁₂	Dic₅	1	1	yes	characteristic
93	C₂²×C₄×D₃	Dic₅	1	2	yes	—
94	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
95	C₂×(C₂×D₃)⋊C₄	C₂×D₅	1	2	yes	—
96	C₂×C₄⋊Dic₃	Dic₅	1	1	yes	characteristic
97	C₄×D₁₂	—	2	1	no	—
98	C₄×D₁₂	—	2	1	no	—
99	C₄×D₁₂	—	2	1	no	—
100	C₄×D₁₂	—	2	1	no	—
101	C₃×C₄⋊C₈	—	10	1	no	—
102	C₃×C₂²⋊C₈	—	5	2	no	—
103	C₃×C₄×C₈	—	10	1	no	—
104	#830	—	5	2	no	—
105	C₄⋊(C₃⋊C₈)	—	10	1	no	—
106	C₄×C₃⋊C₈	—	10	1	no	—
107	C₂⁴×C₅	—	3	2	no	—
108	C₂×C₅×D₄	—	3	4	no	—
109	C₂×C₅×D₄	—	6	2	no	—
110	C₂×C₅×D₄	—	6	1	no	—
111	C₂×C₅×D₄	—	6	1	no	—
112	C₂²×C₄×C₅	—	3	2	no	—
113	C₂²×C₄×C₅	—	6	2	no	—
114	C₂²×C₄×C₅	—	6	2	no	—
115	C₂²×C₄×C₅	—	12	4	no	—
116	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
117	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
118	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
119	C₅×C₄⋊C₄	—	6	1	no	—
120	C₅×C₄⋊C₄	—	6	1	no	—
121	C₅×C₂²⋊C₄	—	6	2	no	—
122	C₅×C₂²⋊C₄	—	6	2	no	—
123	C₄²×C₅	D₁₂	1	1	yes	characteristic
124	C₄²×C₅	D₁₂	1	1	yes	characteristic
125	C₄²×C₅	#67	1	1	yes	characteristic
126	C₄²×C₅	#67	1	1	yes	characteristic
127	C₂×#168	—	2	2	no	—
128	C₂×#168	—	6	2	no	—
129	C₂×C₄×D₄	—	3	1	no	—
130	#431	—	5	1	no	—
131	#431	—	15	1	no	—
132	#323	—	15	4	no	—
133	C₂²×C₃×C₅	—	2	1	no	—
134	C₂²×C₃×C₅	—	2	1	no	—
135	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
137	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
138	C₂×C₅×D₃	—	2	4	no	—
139	C₂×C₅×D₃	—	4	4	no	—
140	C₂×C₅×D₃	—	4	2	no	—
141	C₂×C₅×D₃	—	4	2	no	—
142	C₂×C₅×D₃	—	4	4	no	—
143	C₃×C₄×C₅	—	2	1	no	—
144	C₃×C₄×C₅	—	2	1	no	—
145	C₃×C₄×C₅	—	2	1	no	—
146	C₃×C₄×C₅	C₂²⋊C₈	1	2	yes	—
147	C₃×C₄×C₅	C₂²⋊C₈	1	2	yes	—
148	C₃×C₄×C₅	#100	1	2	yes	—
149	C₅×Dic₃	—	4	2	no	—
150	C₂³×D₃	C₂×Dic₅	1	2	yes	—
151	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
152	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
153	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
154	C₂²×Dic₃	C₂×Dic₅	1	2	yes	—
155	C₂×D₁₂	—	2	2	no	—
156	C₂×D₁₂	—	2	1	no	—
157	C₂×D₁₂	—	2	1	no	—
158	C₂×D₁₂	#168	1	4	yes	complemented
159	C₂×C₄×D₃	—	2	2	no	—
160	C₂×C₄×D₃	—	2	2	no	—
161	C₂×C₄×D₃	—	4	4	no	—
162	C₂×C₃×C₈	—	10	2	no	—
163	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
164	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
165	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
166	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
167	(C₂×D₃)⋊C₄	—	2	2	no	—
168	(C₂×D₃)⋊C₄	—	2	2	no	—
169	C₄⋊Dic₃	—	2	1	no	—
170	C₄⋊Dic₃	—	2	1	no	—
171	C₂×C₃⋊C₈	—	10	2	no	—
172	C₂³×C₅	—	3	4	no	—
173	C₂³×C₅	—	6	2	no	—
174	C₂³×C₅	—	6	2	no	—
175	C₂³×C₅	—	12	2	no	—
176	C₂³×C₅	—	12	2	no	—
177	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
178	C₅×D₄	—	6	4	no	—
179	C₅×D₄	—	6	4	no	—
180	C₂×C₄×C₅	—	2	2	no	—
181	C₂×C₄×C₅	—	4	1	no	—
182	C₂×C₄×C₅	—	6	2	no	—
183	C₂×C₄×C₅	—	12	4	no	—
184	C₂×C₄×C₅	—	12	4	no	—
185	C₂×C₄×C₅	—	12	2	no	—
186	C₂×C₄×C₅	D₃×C₈	1	2	yes	—
187	C₂×C₄×C₅	ΓL(1,𝔽₂₅)	1	2	yes	—
188	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
189	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
190	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
191	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
192	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
193	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
194	#168	—	4	2	no	—
195	#168	—	12	2	no	—
196	C₂²×D₄	—	3	1	no	—
197	C₂³×C₄	—	3	2	no	—
198	C₄×D₄	—	6	1	no	—
199	C₄×D₄	—	6	1	no	—
200	C₄×D₄	—	6	1	no	—
201	C₄×D₄	—	6	1	no	—
202	C₂×C₄⋊C₄	—	3	1	no	—
203	C₂×C₂²⋊C₄	—	3	2	no	—
204	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
205	C₄⋊C₈	—	10	1	no	—
206	C₄⋊C₈	—	30	1	no	—
207	C₂²⋊C₈	—	5	2	no	—
208	C₂²⋊C₈	—	15	2	no	—
209	C₄×C₈	—	10	1	no	—
210	C₄×C₈	—	30	1	no	—
211	C₂×C₃×C₅	—	2	1	no	—
212	C₂×C₃×C₅	—	2	1	no	—
213	C₂×C₃×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
214	C₂×C₃×C₅	#410	1	1	yes	characteristic
215	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
216	C₅×D₃	—	4	4	no	—
217	C₂³×C₃	C₂²×Dic₅	1	1	yes	characteristic
218	C₂²×D₃	—	2	2	no	—
219	C₂²×D₃	—	2	2	no	—
220	C₂²×D₃	—	4	2	no	—
221	C₂²×D₃	—	4	2	no	—
222	C₂²×D₃	C₂×#168	1	4	yes	—
223	C₂×C₃×C₄	—	2	2	no	—
224	C₂×C₃×C₄	—	4	1	no	—
225	C₂×C₃×C₄	C₂×#168	1	2	yes	—
226	C₂×C₃×C₄	#681	1	2	yes	—
227	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
228	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
229	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
230	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
231	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
232	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
233	C₂×Dic₃	—	2	2	no	—
234	C₂×Dic₃	—	4	2	no	—
235	D₁₂	—	2	4	no	—
236	D₁₂	—	2	4	no	—
237	C₄×D₃	—	4	4	no	—
238	C₄×D₃	—	4	4	no	—
239	C₃×C₈	—	20	2	no	—
240	C₃⋊C₈	—	20	2	no	—
241	C₂²×C₅	—	2	1	no	—
242	C₂²×C₅	—	2	1	no	—
243	C₂²×C₅	—	6	4	no	—
244	C₂²×C₅	—	12	4	no	—
245	C₂²×C₅	—	12	4	no	—
246	C₂²×C₅	—	12	2	no	—
247	C₂²×C₅	—	12	2	no	—
248	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
249	C₂²×C₅	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
250	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
251	C₄×C₅	—	2	1	no	—
252	C₄×C₅	—	2	1	no	—
253	C₄×C₅	—	2	1	no	—
254	C₄×C₅	—	12	2	no	—
255	C₄×C₅	(C₂×D₃)⋊C₈	1	2	yes	—
256	C₄×C₅	(C₂×D₃)⋊C₈	1	2	yes	—
257	C₄×C₅	#823	1	2	yes	—
258	C₂⁴	—	3	2	no	—
259	C₂×D₄	—	3	4	no	—
260	C₂×D₄	—	6	1	no	—
261	C₂×D₄	—	6	1	no	—
262	C₂×D₄	—	6	2	no	—
263	C₂²×C₄	—	3	2	no	—
264	C₂²×C₄	—	6	2	no	—
265	C₂²×C₄	—	6	2	no	—
266	C₂²×C₄	—	12	4	no	—
267	C₂²×C₄	D₃×Dic₅	1	1	yes	characteristic
268	C₂²×C₄	D₃×Dic₅	1	1	yes	characteristic
269	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
270	C₂×C₈	—	10	2	no	—
271	C₂×C₈	—	30	2	no	—
272	C₄⋊C₄	—	6	1	no	—
273	C₄⋊C₄	—	6	1	no	—
274	C₂²⋊C₄	—	6	2	no	—
275	C₂²⋊C₄	—	6	2	no	—
276	C₄²	#1201	1	1	yes	characteristic
277	C₄²	#1201	1	1	yes	characteristic
278	C₄²	#1203	1	1	yes	characteristic
279	C₄²	#1203	1	1	yes	characteristic
280	C₃×C₅	#1460	1	1	yes	characteristic, complemented, Hall
281	C₂²×C₃	—	2	1	no	—
282	C₂²×C₃	—	2	1	no	—
283	C₂²×C₃	C₂²×#168	1	1	yes	characteristic
284	C₂²×C₃	C₂×#681	1	1	yes	characteristic
285	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
286	C₂×D₃	—	2	4	no	—
287	C₂×D₃	—	4	4	no	—
288	C₂×D₃	—	4	2	no	—
289	C₂×D₃	—	4	2	no	—
290	C₂×D₃	—	4	4	no	—
291	C₃×C₄	—	2	1	no	—
292	C₃×C₄	—	2	1	no	—
293	C₃×C₄	—	2	1	no	—
294	C₃×C₄	C₂²⋊#168	1	2	yes	—
295	C₃×C₄	C₂²⋊#168	1	2	yes	—
296	C₃×C₄	#3965	1	2	yes	—
297	Dic₃	—	4	2	no	—
298	C₂×C₅	—	2	1	no	—
299	C₂×C₅	—	2	1	no	—
300	C₂×C₅	—	12	4	no	—
301	C₂×C₅	#5006	1	1	yes	characteristic
302	C₂×C₅	C₂×(C₂×D₃)⋊C₈	1	1	yes	characteristic
303	C₂×C₅	C₂×#823	1	1	yes	characteristic
304	C₂³	—	3	4	no	—
305	C₂³	—	6	2	no	—
306	C₂³	—	6	2	no	—
307	C₂³	—	12	2	no	—
308	C₂³	—	12	2	no	—
309	C₂³	C₂×D₃×Dic₅	1	1	yes	characteristic
310	D₄	—	6	4	no	—
311	D₄	—	6	4	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	6	2	no	—
315	C₂×C₄	—	12	4	no	—
316	C₂×C₄	—	12	4	no	—
317	C₂×C₄	—	12	2	no	—
318	C₂×C₄	D₃×#168	1	2	yes	—
319	C₂×C₄	#6650	1	2	yes	—
320	C₂×C₄	(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
321	C₂×C₄	(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
322	C₂×C₄	(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
323	C₂×C₄	(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
324	C₂×C₄	C₂×#1201	1	1	yes	characteristic
325	C₂×C₄	C₂×#1203	1	1	yes	characteristic
326	C₈	—	20	2	no	—
327	C₈	—	60	2	no	—
328	C₂×C₃	—	2	1	no	—
329	C₂×C₃	—	2	1	no	—
330	C₂×C₃	C₂×C₂²⋊#168	1	1	yes	characteristic
331	C₂×C₃	C₂×#3965	1	1	yes	characteristic
332	C₂×C₃	#65529	1	1	yes	characteristic
333	D₃	—	4	4	no	—
334	C₅	#70783	1	1	yes	characteristic, complemented, Sylow
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	6	4	no	—
338	C₂²	—	12	4	no	—
339	C₂²	—	12	4	no	—
340	C₂²	—	12	2	no	—
341	C₂²	—	12	2	no	—
342	C₂²	C₂×D₃×#168	1	1	yes	characteristic
343	C₂²	C₂×#6650	1	1	yes	characteristic
344	C₂²	C₂×(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
345	C₄	—	2	1	no	—
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	12	2	no	—
349	C₄	(C₂×D₃)⋊#168	1	2	yes	—
350	C₄	(C₂×D₃)⋊#168	1	2	yes	—
351	C₄	#90959	1	2	yes	—
352	C₃	#10602394	1	1	yes	characteristic, complemented, Sylow
353	C₂	—	2	1	no	—
354	C₂	—	2	1	no	—
355	C₂	—	12	4	no	—
356	C₂	C₂×(C₂×D₃)⋊#168	1	1	yes	characteristic
357	C₂	C₂×#90959	1	1	yes	characteristic
358	C₂	#11748625	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₃)⋊Dic₅
commutator	C₂²×C₃×C₅	C₂²×C₈
Frattini	C₂²×C₄	C₂×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×C₅×D₁₂	C₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1460
LCS	G ▷ C₂²×C₃×C₅ ▷ 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
2. #11751026	C₂	semidirect
3. C₃×#65214	C₂	semidirect
4. #11745949	C₂	semidirect
24. #65214	D₃	semidirect
34. C₂×C₅×D₁₂	C₈	semidirect
158. C₂×D₁₂	#168	semidirect
280. C₃×C₅	#1460	semidirect
334. C₅	#70783	semidirect
352. C₃	#10602394	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅×D₁₂	2
#11751026	2
C₃×#65214	2
#11745949	2
#10602394	3
#70783	5