Subgroups — #14413147

Order 1920.

968 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×#91815	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×#208	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×#681	C₂	1	1	yes	maximal, characteristic
4	#11748437	C₂	1	4	yes	maximal, complemented
5	#11748436	C₂	1	4	yes	maximal, complemented
6	#11748435	C₂	1	4	yes	maximal, complemented
7	#10615481	—	3	1	no	maximal, Hall
8	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
9	C₂×#6719	C₄	1	2	yes	—
10	#91815	C₄	1	2	yes	—
11	#91815	C₄	1	2	yes	—
12	#91815	C₂²	1	2	yes	—
13	#91815	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₁₅	C₄	1	2	yes	—
15	C₂×C₄×Dic₁₅	C₂²	1	2	yes	—
16	C₂×C₅×#208	—	2	4	no	—
17	C₂×C₅×#208	C₂²	1	2	yes	—
18	C₂×C₅×#208	C₂²	1	2	yes	—
19	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
20	C₂×C₃×#681	—	2	4	no	—
21	C₂×C₃×#681	C₂²	1	2	yes	—
22	C₂×C₃×#681	C₂²	1	2	yes	—
23	C₂²×C₃×#168	C₂²	1	2	yes	—
24	#90983	C₂²	1	16	yes	complemented
25	#83750	—	5	1	no	maximal, Hall
26	C₂²×C₅×M₁₆	—	3	1	no	—
27	C₂×#4072	—	3	1	no	—
28	C₂²×#681	D₃	1	1	yes	characteristic
29	#65408	—	3	4	no	—
30	#65407	—	3	4	no	—
31	#65391	—	3	4	no	—
32	C₂⁴×C₃×C₅	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₃×C₄×C₅	D₄	1	4	yes	—
37	C₂²×C₃×C₄×C₅	Q₈	1	4	yes	—
38	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
39	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
41	C₂²×Dic₁₅	C₂×C₄	1	4	yes	—
42	#6719	C₂×C₄	1	4	yes	—
43	#6719	C₂×C₄	1	4	yes	—
44	C₄⋊Dic₁₅	C₂×C₄	1	4	yes	—
45	C₄⋊Dic₁₅	C₂×C₄	1	4	yes	—
46	C₄×Dic₁₅	C₂×C₄	1	4	yes	—
47	C₄×Dic₁₅	C₂³	1	4	yes	—
48	C₅×#208	—	2	4	no	—
49	C₅×#208	—	2	4	no	—
50	C₂×C₅×C₃⋊C₈	—	2	4	no	—
51	C₂×C₅×C₃⋊C₈	C₂³	1	4	yes	—
52	C₃×#681	—	2	4	no	—
53	C₃×#681	—	2	4	no	—
54	C₂×C₃×#168	—	2	4	no	—
55	C₂×C₃×#168	C₂³	1	4	yes	—
56	C₂²×C₃×M₁₆	—	5	1	no	—
57	C₂×#926	—	5	1	no	—
58	C₂²×#208	D₅	1	1	yes	characteristic
59	#5119	—	5	4	no	—
60	#5118	—	5	4	no	—
61	#5102	—	5	4	no	—
62	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
63	C₂×C₅×M₁₆	—	3	2	no	—
64	C₂×C₅×M₁₆	—	3	2	no	—
65	C₂×C₅×M₁₆	—	6	4	no	—
66	C₂²×C₅×C₈	—	3	2	no	—
67	C₂×C₂²⋊Dic₅	—	3	2	no	—
68	#4072	—	3	2	no	—
69	#4072	—	3	2	no	—
70	#4072	—	3	2	no	—
71	#4072	—	3	2	no	—
72	C₂×C₄⋊Dic₅	—	3	2	no	—
73	C₂×C₄×Dic₅	—	3	2	no	—
74	C₂×#681	—	2	4	no	—
75	C₂×#681	C₂×D₃	1	2	yes	—
76	C₂×#681	C₂×D₃	1	2	yes	—
77	C₂²×#168	C₂×D₃	1	2	yes	—
78	#3946	—	3	16	no	—
79	#2905	—	15	1	no	Sylow
80	C₂³×C₃×C₅	—	4	2	no	—
81	C₂³×C₃×C₅	C₄⋊C₄	1	4	yes	—
82	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
83	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
84	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₄⋊C₄	1	4	yes	—
86	C₂×C₃×C₄×C₅	C₄⋊C₄	1	4	yes	—
87	C₂×C₃×C₄×C₅	C₄⋊C₄	1	4	yes	—
88	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
92	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
93	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
94	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
95	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
96	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
97	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
98	C₂×Dic₁₅	—	4	4	no	—
99	C₂×Dic₁₅	C₂²×C₄	1	8	yes	—
100	C₃×#168	—	2	4	no	—
101	C₅×C₃⋊C₈	—	2	4	no	—
102	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
103	C₂×C₃×M₁₆	—	5	2	no	—
104	C₂×C₃×M₁₆	—	5	2	no	—
105	C₂×C₃×M₁₆	—	10	4	no	—
106	C₂²×C₃×C₈	—	5	2	no	—
107	C₂×#217	—	5	2	no	—
108	#926	—	5	2	no	—
109	#926	—	5	2	no	—
110	#926	—	5	2	no	—
111	#926	—	5	2	no	—
112	C₂×C₄⋊Dic₃	—	5	2	no	—
113	C₂×C₄×Dic₃	—	5	2	no	—
114	C₂×#208	—	2	4	no	—
115	C₂×#208	C₂×D₅	1	2	yes	—
116	C₂×#208	C₂×D₅	1	2	yes	—
117	C₂²×C₃⋊C₈	C₂×D₅	1	2	yes	—
118	Dic₃⋊C₈	—	5	16	no	—
119	C₂⁴×C₅	C₄×D₃	1	1	yes	characteristic
120	C₂²×C₄×C₅	Dic₆	1	4	yes	—
121	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
122	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
123	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
124	C₂²×C₄×C₅	#67	1	4	yes	—
125	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
126	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
127	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
128	C₂²×Dic₅	—	3	4	no	—
129	C₅×M₁₆	—	6	4	no	—
130	C₅×M₁₆	—	6	4	no	—
131	C₂×C₅×C₈	—	3	4	no	—
132	C₂×C₅×C₈	—	6	4	no	—
133	C₂²⋊Dic₅	—	3	4	no	—
134	C₂²⋊Dic₅	—	3	4	no	—
135	C₄⋊Dic₅	—	3	4	no	—
136	C₄⋊Dic₅	—	3	4	no	—
137	C₄×Dic₅	—	3	4	no	—
138	C₄×Dic₅	—	3	4	no	—
139	#681	—	2	4	no	—
140	#681	—	2	4	no	—
141	C₂×#168	—	2	4	no	—
142	C₂×#168	C₂²×D₃	1	4	yes	—
143	C₂²×M₁₆	—	3	1	no	—
144	C₂²×M₁₆	—	5	1	no	—
145	C₂×#117	—	15	1	no	—
146	#424	—	15	4	no	—
147	#424	—	15	4	no	—
148	#423	—	15	4	no	—
149	C₂²×C₃×C₅	—	2	2	no	—
150	C₂²×C₃×C₅	—	2	2	no	—
151	C₂²×C₃×C₅	—	2	2	no	—
152	C₂²×C₃×C₅	—	4	4	no	—
153	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	2	yes	—
154	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	2	yes	—
155	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	2	yes	—
156	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
157	C₃×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
158	C₃×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
159	C₃×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
160	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
161	C₃×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
162	Dic₁₅	—	2	8	no	—
163	C₂⁴×C₃	C₄×D₅	1	1	yes	characteristic
164	C₂²×C₃×C₄	Dic₁₀	1	4	yes	—
165	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
166	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
167	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
168	C₂²×C₃×C₄	C₂²⋊D₅	1	4	yes	—
169	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
170	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
171	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
172	C₂²×Dic₃	—	5	4	no	—
173	C₃×M₁₆	—	10	4	no	—
174	C₃×M₁₆	—	10	4	no	—
175	C₂×C₃×C₈	—	5	4	no	—
176	C₂×C₃×C₈	—	10	4	no	—
177	#217	—	5	4	no	—
178	#217	—	5	4	no	—
179	C₄⋊Dic₃	—	5	4	no	—
180	C₄⋊Dic₃	—	5	4	no	—
181	C₄×Dic₃	—	5	4	no	—
182	C₄×Dic₃	—	5	4	no	—
183	#208	—	2	4	no	—
184	#208	—	2	4	no	—
185	C₂×C₃⋊C₈	—	2	4	no	—
186	C₂×C₃⋊C₈	C₂²×D₅	1	4	yes	—
187	C₂³×C₅	—	4	2	no	—
188	C₂³×C₅	Dic₃⋊C₄	1	4	yes	—
189	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
190	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
191	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
192	C₂×C₄×C₅	Dic₃⋊C₄	1	4	yes	—
193	C₂×C₄×C₅	Dic₃⋊C₄	1	4	yes	—
194	C₂×C₄×C₅	Dic₃⋊C₄	1	4	yes	—
195	C₂×C₄×C₅	C₂×Dic₆	1	2	yes	—
196	C₂×C₄×C₅	C₂×Dic₆	1	2	yes	—
197	C₂×C₄×C₅	C₂×Dic₆	1	2	yes	—
198	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
199	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
200	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
201	C₂×C₄×C₅	C₂×#67	1	2	yes	—
202	C₂×C₄×C₅	C₂×#67	1	2	yes	—
203	C₂×C₄×C₅	C₂×#67	1	2	yes	—
204	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
205	C₂×Dic₅	—	3	8	no	—
206	C₂×Dic₅	—	12	4	no	—
207	C₅×C₈	—	6	4	no	—
208	#168	—	2	4	no	—
209	C₂³×C₄	D₃×D₅	1	1	yes	characteristic
210	C₂×M₁₆	—	3	2	no	—
211	C₂×M₁₆	—	3	2	no	—
212	C₂×M₁₆	—	5	2	no	—
213	C₂×M₁₆	—	5	2	no	—
214	C₂×M₁₆	—	6	4	no	—
215	C₂×M₁₆	—	10	4	no	—
216	C₂²×C₈	—	3	2	no	—
217	C₂²×C₈	—	5	2	no	—
218	#117	—	15	2	no	—
219	#117	—	15	2	no	—
220	#117	—	15	2	no	—
221	#117	—	15	2	no	—
222	C₂×C₄⋊C₄	—	15	2	no	—
223	C₂×C₂²⋊C₄	—	15	2	no	—
224	C₂×C₄²	—	15	2	no	—
225	C₄⋊C₈	—	15	16	no	—
226	C₂×C₃×C₅	—	2	2	no	—
227	C₂×C₃×C₅	—	2	2	no	—
228	C₂×C₃×C₅	—	2	2	no	—
229	C₂×C₃×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
230	C₂×C₃×C₅	#568	1	2	yes	—
231	C₂³×C₃	—	4	2	no	—
232	C₂³×C₃	#683	1	4	yes	—
233	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
234	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
235	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
236	C₂×C₃×C₄	#683	1	4	yes	—
237	C₂×C₃×C₄	#683	1	4	yes	—
238	C₂×C₃×C₄	#683	1	4	yes	—
239	C₂×C₃×C₄	C₂×Dic₁₀	1	2	yes	—
240	C₂×C₃×C₄	C₂×Dic₁₀	1	2	yes	—
241	C₂×C₃×C₄	C₂×Dic₁₀	1	2	yes	—
242	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
243	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
244	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
245	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
246	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
247	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
248	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
249	C₂×Dic₃	—	5	8	no	—
250	C₂×Dic₃	—	20	4	no	—
251	C₃×C₈	—	10	4	no	—
252	C₃⋊C₈	—	2	4	no	—
253	C₂²×C₅	—	2	2	no	—
254	C₂²×C₅	—	2	2	no	—
255	C₂²×C₅	—	2	2	no	—
256	C₂²×C₅	—	4	4	no	—
257	C₂²×C₅	C₂×Dic₃⋊C₄	1	2	yes	—
258	C₂²×C₅	C₂×Dic₃⋊C₄	1	2	yes	—
259	C₂²×C₅	C₂×Dic₃⋊C₄	1	2	yes	—
260	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
261	C₄×C₅	C₂×Dic₃⋊C₄	1	2	yes	—
262	C₄×C₅	C₂×Dic₃⋊C₄	1	2	yes	—
263	C₄×C₅	C₂×Dic₃⋊C₄	1	2	yes	—
264	C₄×C₅	C₂²×Dic₆	1	1	yes	characteristic
265	C₄×C₅	C₂²×#67	1	1	yes	characteristic
266	Dic₅	—	6	8	no	—
267	C₂⁴	#1200	1	1	yes	characteristic
268	C₂²×C₄	—	15	4	no	—
269	C₂²×C₄	#1200	1	1	yes	characteristic
270	C₂²×C₄	#1200	1	1	yes	characteristic
271	C₂²×C₄	#1200	1	1	yes	characteristic
272	C₂²×C₄	#1201	1	4	yes	—
273	C₂²×C₄	#1204	1	4	yes	—
274	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
275	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
276	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
277	M₁₆	—	6	4	no	—
278	M₁₆	—	6	4	no	—
279	M₁₆	—	10	4	no	—
280	M₁₆	—	10	4	no	—
281	C₂×C₈	—	3	4	no	—
282	C₂×C₈	—	5	4	no	—
283	C₂×C₈	—	6	4	no	—
284	C₂×C₈	—	10	4	no	—
285	C₄⋊C₄	—	15	4	no	—
286	C₄⋊C₄	—	15	4	no	—
287	C₂²⋊C₄	—	15	4	no	—
288	C₂²⋊C₄	—	15	4	no	—
289	C₄²	—	15	4	no	—
290	C₄²	—	15	4	no	—
291	C₃×C₅	#2905	1	1	yes	characteristic, complemented, Hall
292	C₂²×C₃	—	2	2	no	—
293	C₂²×C₃	—	2	2	no	—
294	C₂²×C₃	—	2	2	no	—
295	C₂²×C₃	—	4	4	no	—
296	C₂²×C₃	C₂×#683	1	2	yes	—
297	C₂²×C₃	C₂×#683	1	2	yes	—
298	C₂²×C₃	C₂×#683	1	2	yes	—
299	C₂²×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
300	C₃×C₄	C₂×#683	1	2	yes	—
301	C₃×C₄	C₂×#683	1	2	yes	—
302	C₃×C₄	C₂×#683	1	2	yes	—
303	C₃×C₄	C₂²×Dic₁₀	1	1	yes	characteristic
304	C₃×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
305	Dic₃	—	10	8	no	—
306	C₂×C₅	—	2	2	no	—
307	C₂×C₅	—	2	2	no	—
308	C₂×C₅	—	2	2	no	—
309	C₂×C₅	#5745	1	2	yes	—
310	C₂×C₅	C₂²×Dic₃⋊C₄	1	1	yes	characteristic
311	C₂³	—	4	2	no	—
312	C₂³	#6669	1	4	yes	—
313	C₂³	C₂×#1200	1	1	yes	characteristic
314	C₂³	C₂×#1200	1	1	yes	characteristic
315	C₂³	C₂×#1200	1	1	yes	characteristic
316	C₂×C₄	—	15	8	no	—
317	C₂×C₄	—	60	4	no	—
318	C₂×C₄	#6669	1	4	yes	—
319	C₂×C₄	#6669	1	4	yes	—
320	C₂×C₄	#6669	1	4	yes	—
321	C₂×C₄	C₂×#1200	1	1	yes	characteristic
322	C₂×C₄	C₂×#1200	1	1	yes	characteristic
323	C₂×C₄	C₂×#1200	1	1	yes	characteristic
324	C₂×C₄	C₂×#1201	1	2	yes	—
325	C₂×C₄	C₂×#1201	1	2	yes	—
326	C₂×C₄	C₂×#1201	1	2	yes	—
327	C₂×C₄	C₂×#1204	1	2	yes	—
328	C₂×C₄	C₂×#1204	1	2	yes	—
329	C₂×C₄	C₂×#1204	1	2	yes	—
330	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
331	C₈	—	6	4	no	—
332	C₈	—	10	4	no	—
333	C₂×C₃	—	2	2	no	—
334	C₂×C₃	—	2	2	no	—
335	C₂×C₃	—	2	2	no	—
336	C₂×C₃	#66079	1	2	yes	—
337	C₂×C₃	C₂²×#683	1	1	yes	characteristic
338	C₅	#83750	1	1	yes	characteristic, complemented, Sylow
339	C₂²	—	2	2	no	—
340	C₂²	—	2	2	no	—
341	C₂²	—	2	2	no	—
342	C₂²	—	4	4	no	—
343	C₂²	C₂×#6669	1	2	yes	—
344	C₂²	C₂×#6669	1	2	yes	—
345	C₂²	C₂×#6669	1	2	yes	—
346	C₂²	C₂²×#1200	1	1	yes	characteristic
347	C₄	—	30	8	no	—
348	C₄	C₂×#6669	1	2	yes	—
349	C₄	C₂×#6669	1	2	yes	—
350	C₄	C₂×#6669	1	2	yes	—
351	C₄	C₂²×#1201	1	1	yes	characteristic
352	C₄	C₂²×#1204	1	1	yes	characteristic
353	C₃	#10615481	1	1	yes	characteristic, complemented, Sylow
354	C₂	—	2	2	no	—
355	C₂	—	2	2	no	—
356	C₂	—	2	2	no	—
357	C₂	#11754130	1	2	yes	—
358	C₂	C₂²×#6669	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃×D₅
commutator, socle	C₂²×C₃×C₅	C₂³×C₄
Fitting	C₂³×C₃×C₄×C₅	C₂²
prime radical	C₂⁴×C₃×C₅	C₂×C₄
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₂³×C₄	D₃×D₅
hypocenter	C₃×C₅	#2905
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
4. #11748437	C₂	semidirect
5. #11748436	C₂	semidirect
6. #11748435	C₂	semidirect
24. #90983	C₂²	semidirect
291. C₃×C₅	#2905	semidirect
338. C₅	#83750	semidirect
353. C₃	#10615481	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91815	2
C₂²×C₅×#208	2
C₂²×C₃×#681	2
#11748437	2
#11748436	2
#11748435	2
#10615481	3
#83750	5