Subgroups — #14411424

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×C₅×#872	C₂	1	1	yes	maximal, characteristic
2	C₂×D₃×#681	C₂	1	2	yes	maximal
3	C₄×#6710	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄×#681	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₃×#681	C₂	1	2	yes	maximal, complemented
6	#11746979	C₂	1	4	yes	maximal, complemented
7	#11746977	C₂	1	4	yes	maximal, complemented
8	#10613933	—	3	1	no	maximal, Hall
9	C₂²×C₄×C₅×D₃	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×(C₂×D₃)⋊C₄	C₄	1	2	yes	—
13	C₂×C₅×Dic₃⋊C₄	C₄	1	2	yes	—
14	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
15	C₅×#872	C₄	1	4	yes	—
16	C₅×#872	C₂²	1	2	yes	—
17	C₅×#872	C₂²	1	2	yes	—
18	C₂×C₃×#681	C₂²	1	2	yes	—
19	C₂×#6650	C₂²	1	4	yes	—
20	D₃×#681	C₄	1	16	yes	complemented
21	C₂×D₃×#168	C₂²	1	4	yes	—
22	#91087	C₂²	1	2	yes	complemented
23	C₄×C₃⋊#168	C₂²	1	2	yes	complemented
24	C₃×#3935	C₂²	1	2	yes	complemented
25	C₃×C₄×#168	C₂²	1	2	yes	complemented
26	#90953	C₂²	1	4	yes	complemented
27	Dic₃×#168	C₂²	1	4	yes	complemented
28	#78514	—	5	1	no	maximal, Hall
29	C₂×C₅×#117	—	3	1	no	—
30	C₂²×#681	—	3	2	no	—
31	#65286	—	3	4	no	—
32	#65286	—	3	4	no	—
33	C₄×#681	—	3	2	no	—
34	C₄×#681	—	3	1	no	—
35	C₄×#681	D₃	1	1	yes	characteristic, complemented
36	C₂³×C₅×D₃	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×C₅×Dic₃	C₂×C₄	1	2	yes	—
40	C₂²×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×C₅×D₃	C₂×C₄	1	2	yes	—
42	C₂×C₄×C₅×D₃	C₂×C₄	1	8	yes	—
43	C₂×C₄×C₅×D₃	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×(C₂×D₃)⋊C₄	C₂×C₄	1	4	yes	—
50	C₅×(C₂×D₃)⋊C₄	C₂×C₄	1	4	yes	—
51	C₅×Dic₃⋊C₄	C₂×C₄	1	4	yes	—
52	C₅×Dic₃⋊C₄	C₂×C₄	1	4	yes	—
53	C₄×C₅×Dic₃	C₂×C₄	1	2	yes	—
54	C₄×C₅×Dic₃	C₂³	1	2	yes	—
55	C₃×#681	C₂×C₄	1	8	yes	—
56	C₂×C₃×#168	C₂³	1	4	yes	—
57	#6650	C₂×C₄	1	16	yes	complemented
58	D₃×#168	C₂×C₄	1	16	yes	complemented
59	C₂×D₃×M₁₆	—	5	2	no	—
60	C₂×#872	D₅	1	1	yes	characteristic
61	C₃×C₄×M₁₆	—	5	1	no	—
62	Dic₃×M₁₆	—	5	2	no	—
63	C₄×#208	—	5	1	no	—
64	#4708	—	5	4	no	—
65	#4689	—	5	4	no	—
66	C₂³×C₄×C₅	—	3	1	no	—
67	C₅×#117	—	3	4	no	—
68	C₅×#117	—	3	2	no	—
69	C₅×#117	—	3	2	no	—
70	C₂×C₅×C₄⋊C₄	—	3	2	no	—
71	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
72	C₂×C₄²×C₅	—	3	1	no	—
73	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
74	C₂×#681	—	3	2	no	—
75	C₂×#681	—	3	16	no	—
76	C₂×#681	—	3	4	no	—
77	C₂×#681	C₂×D₃	1	2	yes	—
78	C₂²×#168	—	3	4	no	—
79	#3935	—	3	2	no	—
80	#3935	—	3	4	no	—
81	#3935	C₂×D₃	1	2	yes	complemented
82	C₄×#168	—	3	2	no	—
83	C₄×#168	—	3	4	no	—
84	C₄×#168	C₂×D₃	1	2	yes	complemented
85	#2873	—	15	1	no	Sylow
86	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₅×D₃	—	4	2	no	—
88	C₂²×C₅×D₃	C₄²	1	4	yes	—
89	C₂²×C₅×D₃	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×C₅×Dic₃	—	4	2	no	—
97	C₂×C₅×Dic₃	C₄²	1	4	yes	—
98	C₂×C₅×Dic₃	C₂²×C₄	1	4	yes	—
99	C₂×C₅×Dic₃	C₂²×C₄	1	2	yes	—
100	C₄×C₅×D₃	C₄²	1	8	yes	—
101	C₄×C₅×D₃	C₂²×C₄	1	4	yes	—
102	C₄×C₅×D₃	C₂²×C₄	1	4	yes	—
103	C₃⋊#168	C₂²×C₄	1	8	yes	—
104	C₃×#168	C₂²×C₄	1	8	yes	—
105	C₂²×C₄×D₃	C₂×D₅	1	1	yes	characteristic
106	C₂×C₃×M₁₆	—	5	2	no	—
107	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
108	C₂×(C₂×D₃)⋊C₄	Dic₅	1	2	yes	—
109	C₂×Dic₃⋊C₄	Dic₅	1	2	yes	—
110	C₂×C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
111	C₂×#208	—	5	2	no	—
112	D₃×M₁₆	—	5	16	no	—
113	C₂×ΓL(1,𝔽₂₅)	—	5	4	no	—
114	C₂×D₃×C₈	—	5	4	no	—
115	#872	Dic₅	1	4	yes	—
116	#872	C₂×D₅	1	2	yes	—
117	#872	C₂×D₅	1	2	yes	—
118	C₃×#97	—	5	2	no	—
119	C₃×C₄×C₈	—	5	2	no	—
120	#815	—	5	4	no	—
121	C₈×Dic₃	—	5	4	no	—
122	#803	—	5	2	no	—
123	C₄×C₃⋊C₈	—	5	2	no	—
124	C₂⁴×C₅	—	3	1	no	—
125	C₂²×C₄×C₅	—	3	1	no	—
126	C₂²×C₄×C₅	—	3	2	no	—
127	C₂²×C₄×C₅	—	3	8	no	—
128	C₂²×C₄×C₅	—	3	2	no	—
129	C₂²×C₄×C₅	—	3	2	no	—
130	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
131	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
132	C₅×C₄⋊C₄	—	3	4	no	—
133	C₅×C₄⋊C₄	—	3	4	no	—
134	C₅×C₂²⋊C₄	—	3	4	no	—
135	C₅×C₂²⋊C₄	—	3	4	no	—
136	C₄²×C₅	—	3	2	no	—
137	C₄²×C₅	—	3	2	no	—
138	C₄²×C₅	C₄×D₃	1	2	yes	—
139	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
140	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
141	#681	—	3	8	no	—
142	#681	—	3	16	no	—
143	#681	C₄×D₃	1	8	yes	—
144	C₂×#168	—	3	4	no	—
145	C₂×#168	—	3	16	no	—
146	C₂×#168	C₂²×D₃	1	4	yes	—
147	C₂²×M₁₆	—	15	2	no	—
148	C₂×#117	—	3	1	no	—
149	#405	—	15	4	no	—
150	#405	—	15	4	no	—
151	C₄×M₁₆	—	5	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×C₅×D₃	—	2	4	no	—
158	C₂×C₅×D₃	—	4	4	no	—
159	C₂×C₅×D₃	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	C₅×Dic₃	—	2	4	no	—
165	C₅×Dic₃	C₂×C₄²	1	4	yes	—
166	C₂³×D₃	C₂×Dic₅	1	1	yes	characteristic
167	C₂²×C₃×C₄	C₂×Dic₅	1	2	yes	—
168	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
169	C₂²×Dic₃	C₂×Dic₅	1	2	yes	—
170	C₂²×Dic₃	C₂×Dic₅	1	1	yes	characteristic
171	C₂×C₄×D₃	C₄×D₅	1	8	yes	—
172	C₂×C₄×D₃	C₂×Dic₅	1	2	yes	—
173	C₂×C₄×D₃	C₂²×D₅	1	2	yes	—
174	C₃×M₁₆	—	5	8	no	—
175	C₂×C₃×C₈	—	5	4	no	—
176	C₃×C₄²	C₂×Dic₅	1	2	yes	—
177	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
178	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
179	(C₂×D₃)⋊C₄	C₂×Dic₅	1	4	yes	—
180	(C₂×D₃)⋊C₄	C₂×Dic₅	1	4	yes	—
181	Dic₃⋊C₄	C₂×Dic₅	1	4	yes	—
182	Dic₃⋊C₄	C₂×Dic₅	1	4	yes	—
183	C₄×Dic₃	C₂×Dic₅	1	2	yes	—
184	C₄×Dic₃	C₂²×D₅	1	2	yes	—
185	#208	—	5	8	no	—
186	C₂×C₃⋊C₈	—	5	4	no	—
187	ΓL(1,𝔽₂₅)	—	5	16	no	—
188	D₃×C₈	—	5	16	no	—
189	C₂³×C₅	—	3	2	no	—
190	C₂³×C₅	—	3	4	no	—
191	C₂³×C₅	—	12	2	no	—
192	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
193	C₂×C₄×C₅	—	3	4	no	—
194	C₂×C₄×C₅	—	3	2	no	—
195	C₂×C₄×C₅	—	3	8	no	—
196	C₂×C₄×C₅	—	3	4	no	—
197	C₂×C₄×C₅	—	3	4	no	—
198	C₂×C₄×C₅	—	3	4	no	—
199	C₂×C₄×C₅	—	4	2	no	—
200	C₂×C₄×C₅	—	12	2	no	—
201	C₂×C₄×C₅	C₂×C₄×D₃	1	4	yes	—
202	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
203	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
204	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
205	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
206	#168	—	3	8	no	—
207	#168	C₂×C₄×D₃	1	8	yes	—
208	C₂³×C₄	—	3	1	no	—
209	C₂×M₁₆	—	5	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	—	3	4	no	—
215	#117	—	3	2	no	—
216	#117	—	3	2	no	—
217	C₂×C₄⋊C₄	—	3	2	no	—
218	C₂×C₂²⋊C₄	—	3	2	no	—
219	C₂×C₄²	—	3	1	no	—
220	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
221	#97	—	5	2	no	—
222	#97	—	15	4	no	—
223	#97	—	15	2	no	—
224	C₄×C₈	—	5	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	C₅×D₃	—	2	4	no	—
231	C₂³×C₃	C₂²×Dic₅	1	1	yes	characteristic
232	C₂²×D₃	—	4	2	no	—
233	C₂²×D₃	C₄×Dic₅	1	4	yes	—
234	C₂²×D₃	C₂²×Dic₅	1	2	yes	—
235	C₂×C₃×C₄	—	4	2	no	—
236	C₂×C₃×C₄	C₂×C₄×D₅	1	4	yes	—
237	C₂×C₃×C₄	C₂²×Dic₅	1	2	yes	—
238	C₂×C₃×C₄	C₂²×Dic₅	1	2	yes	—
239	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
240	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
241	C₂×Dic₃	—	4	2	no	—
242	C₂×Dic₃	C₄×Dic₅	1	4	yes	—
243	C₂×Dic₃	C₂²×Dic₅	1	4	yes	—
244	C₂×Dic₃	C₂²×Dic₅	1	2	yes	—
245	C₄×D₃	C₄×Dic₅	1	8	yes	—
246	C₄×D₃	C₂×C₄×D₅	1	4	yes	—
247	C₄×D₃	C₂×C₄×D₅	1	4	yes	—
248	C₃×C₈	—	5	8	no	—
249	C₃⋊C₈	—	5	8	no	—
250	C₂²×C₅	—	2	2	no	—
251	C₂²×C₅	—	3	4	no	—
252	C₂²×C₅	—	6	4	no	—
253	C₂²×C₅	—	12	4	no	—
254	C₂²×C₅	C₄²×D₃	1	2	yes	—
255	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
256	C₄×C₅	—	2	2	no	—
257	C₄×C₅	—	2	2	no	—
258	C₄×C₅	—	3	4	no	—
259	C₄×C₅	—	6	4	no	—
260	C₄×C₅	C₄²×D₃	1	2	yes	—
261	C₄×C₅	C₂²×C₄×D₃	1	2	yes	—
262	C₂⁴	—	3	1	no	—
263	C₂²×C₄	—	3	1	no	—
264	C₂²×C₄	—	3	8	no	—
265	C₂²×C₄	—	3	2	no	—
266	C₂²×C₄	—	3	2	no	—
267	C₂²×C₄	—	3	2	no	—
268	C₂²×C₄	D₃×Dic₅	1	2	yes	—
269	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
270	M₁₆	—	5	8	no	—
271	M₁₆	—	15	16	no	—
272	M₁₆	—	15	8	no	—
273	C₂×C₈	—	5	4	no	—
274	C₂×C₈	—	15	16	no	—
275	C₂×C₈	—	15	4	no	—
276	C₄⋊C₄	—	3	4	no	—
277	C₄⋊C₄	—	3	4	no	—
278	C₂²⋊C₄	—	3	4	no	—
279	C₂²⋊C₄	—	3	4	no	—
280	C₄²	—	3	2	no	—
281	C₄²	—	3	2	no	—
282	C₄²	D₃×Dic₅	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	C₂×C₄×Dic₅	1	2	yes	—
288	C₂²×C₃	C₂³×Dic₅	1	1	yes	characteristic
289	C₂×D₃	—	2	4	no	—
290	C₂×D₃	—	4	4	no	—
291	C₂×D₃	C₂×C₄×Dic₅	1	4	yes	—
292	C₃×C₄	—	2	2	no	—
293	C₃×C₄	—	2	2	no	—
294	C₃×C₄	C₂×C₄×Dic₅	1	2	yes	—
295	C₃×C₄	C₂²×C₄×D₅	1	2	yes	—
296	Dic₃	—	2	4	no	—
297	Dic₃	C₂×C₄×Dic₅	1	4	yes	—
298	C₂×C₅	—	2	2	no	—
299	C₂×C₅	—	6	4	no	—
300	C₂×C₅	C₂×C₄²×D₃	1	1	yes	characteristic
301	C₂×C₅	#5745	1	2	yes	—
302	C₂³	—	3	2	no	—
303	C₂³	—	3	4	no	—
304	C₂³	—	12	2	no	—
305	C₂³	C₂×D₃×Dic₅	1	1	yes	characteristic
306	C₂×C₄	—	3	4	no	—
307	C₂×C₄	—	3	2	no	—
308	C₂×C₄	—	3	8	no	—
309	C₂×C₄	—	3	4	no	—
310	C₂×C₄	—	3	4	no	—
311	C₂×C₄	—	3	4	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	12	2	no	—
314	C₂×C₄	C₄×D₃×D₅	1	4	yes	—
315	C₂×C₄	C₂×D₃×Dic₅	1	2	yes	—
316	C₂×C₄	C₂×D₃×Dic₅	1	2	yes	—
317	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
318	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
319	C₈	—	5	8	no	—
320	C₈	—	15	8	no	—
321	C₂×C₃	—	2	2	no	—
322	C₂×C₃	C₂²×C₄×Dic₅	1	1	yes	characteristic
323	C₂×C₃	#66153	1	2	yes	—
324	D₃	—	2	4	no	—
325	C₅	#78514	1	1	yes	characteristic, complemented, Sylow
326	C₂²	—	2	2	no	—
327	C₂²	—	3	4	no	—
328	C₂²	—	6	4	no	—
329	C₂²	—	12	4	no	—
330	C₂²	C₄×D₃×Dic₅	1	2	yes	—
331	C₂²	C₂²×D₃×Dic₅	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	3	4	no	—
335	C₄	—	6	4	no	—
336	C₄	C₄×D₃×Dic₅	1	2	yes	—
337	C₄	C₂×C₄×D₃×D₅	1	2	yes	—
338	C₃	#10613933	1	1	yes	characteristic, complemented, Sylow
339	C₂	—	2	2	no	—
340	C₂	—	6	4	no	—
341	C₂	#11754125	1	2	yes	—
342	C₂	C₂×C₄×D₃×Dic₅	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×C₅×D₃	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. C₃×C₄×#681	C₂	semidirect
5. Dic₃×#681	C₂	semidirect
6. #11746979	C₂	semidirect
7. #11746977	C₂	semidirect
20. D₃×#681	C₄	semidirect
22. #91087	C₂²	semidirect
23. C₄×C₃⋊#168	C₂²	semidirect
24. C₃×#3935	C₂²	semidirect
25. C₃×C₄×#168	C₂²	semidirect
26. #90953	C₂²	semidirect
27. Dic₃×#168	C₂²	semidirect
35. C₄×#681	D₃	semidirect
57. #6650	C₂×C₄	semidirect
58. D₃×#168	C₂×C₄	semidirect
81. #3935	C₂×D₃	semidirect
84. C₄×#168	C₂×D₃	semidirect
285. C₃×C₅	#2873	semidirect
325. C₅	#78514	semidirect
338. C₃	#10613933	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#872	2
C₂×D₃×#681	2
C₄×#6710	2
C₃×C₄×#681	2
Dic₃×#681	2
#11746979	2
#11746977	2
#10613933	3
#78514	5