Subgroups — #14295657

Order 1920.

640 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#235	C₂	1	1	yes	maximal, characteristic
2	C₄×#6710	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65221	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#90979	C₂	1	2	yes	maximal
5	#11745977	C₂	1	2	yes	maximal, complemented
6	#10602436	—	3	1	no	maximal, Hall
7	C₂×C₅×#235	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂×#6710	C₂²	1	2	yes	—
10	C₄×C₅×#67	—	2	2	no	—
11	C₅×#926	C₂²	1	1	yes	characteristic
12	C₄×C₅×D₁₂	C₄	1	2	yes	complemented
13	C₅×#872	—	2	1	no	—
14	C₄²×C₅×D₃	—	2	1	no	—
15	C₄×C₅×Dic₆	C₄	1	2	yes	complemented
16	C₂×C₃×C₄×Dic₅	C₂²	1	2	yes	—
17	#91087	—	2	1	no	—
18	C₄×C₃⋊#168	—	2	1	no	—
19	C₃×#3963	—	2	2	no	—
20	#90979	C₄	1	8	yes	complemented
21	#69430	—	5	1	no	maximal, Hall
22	C₄×C₅×#41	—	3	1	no	—
23	C₂×#3969	—	3	2	no	—
24	#65221	D₃	1	1	yes	characteristic, complemented
25	C₄×#681	—	3	1	no	—
26	#64752	—	3	2	no	—
27	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
28	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂×C₅×#67	—	2	1	no	—
30	C₅×#235	—	2	2	no	—
31	C₅×#235	C₂×C₄	1	4	yes	—
32	C₂×C₅×D₁₂	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×C₅×D₃	—	2	1	no	—
34	C₂×C₄×C₅×D₃	—	2	2	no	—
35	C₂×C₅×Dic₆	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₃×Dic₅	—	2	2	no	—
37	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
38	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
39	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
40	#6710	—	2	2	no	—
41	#6710	C₂×C₄	1	4	yes	complemented
42	C₂×C₃⋊#168	—	2	2	no	—
43	C₅×#217	—	2	1	no	—
44	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
45	C₅×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
46	C₅×Dic₃⋊C₄	—	2	2	no	—
47	C₄×C₅×Dic₃	—	2	2	no	—
48	C₄×C₅×Dic₃	—	2	1	no	—
49	C₃×C₄×Dic₅	—	2	2	no	—
50	C₃×C₄×Dic₅	C₂×C₄	1	4	yes	complemented
51	C₄×#235	D₅	1	1	yes	characteristic
52	C₃×#376	—	5	1	no	—
53	C₂×#805	—	5	2	no	—
54	C₄×#208	—	5	1	no	—
55	#4523	—	5	2	no	—
56	C₂×C₅×#41	—	3	1	no	—
57	C₄×C₅×Q₈	—	3	2	no	—
58	C₄×C₅×D₄	—	3	2	no	—
59	C₄×C₅×D₄	—	6	2	no	—
60	C₅×#117	—	3	1	no	—
61	C₅×#117	—	6	1	no	—
62	C₂×C₄²×C₅	—	6	1	no	—
63	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
64	C₂×C₄×Dic₅	C₂×D₃	1	2	yes	—
65	C₂×#681	—	3	2	no	—
66	#3969	—	3	8	no	—
67	#3963	—	2	2	no	—
68	#3935	—	6	1	no	—
69	C₄×#168	—	6	1	no	—
70	#1759	—	15	1	no	Sylow
71	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₅×D₃	—	2	1	no	—
73	C₂×C₃×C₄×C₅	—	4	2	no	—
74	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
75	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
76	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
78	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	#41	1	2	yes	—
80	C₅×#67	—	2	4	no	—
81	C₂×C₅×Dic₃	—	2	1	no	—
82	C₂×C₅×Dic₃	—	2	2	no	—
83	C₅×D₁₂	—	2	1	no	—
84	C₅×D₁₂	C₄²	1	2	yes	—
85	C₄×C₅×D₃	—	2	4	no	—
86	C₄×C₅×D₃	—	4	4	no	—
87	C₅×Dic₆	—	2	1	no	—
88	C₅×Dic₆	C₄²	1	2	yes	—
89	C₂×C₃×Dic₅	—	2	4	no	—
90	C₂×C₃×Dic₅	—	4	4	no	—
91	C₃⋊#168	—	2	4	no	—
92	C₂×#235	C₂×D₅	1	1	yes	characteristic
93	C₂×C₃×C₄²	—	5	2	no	—
94	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
95	C₄×#67	—	2	2	no	—
96	#926	C₂×D₅	1	1	yes	characteristic
97	C₂×#208	—	5	2	no	—
98	C₄×D₁₂	Dic₅	1	2	yes	complemented
99	#872	—	2	1	no	—
100	C₄²×D₃	—	2	1	no	—
101	C₄×Dic₆	Dic₅	1	2	yes	complemented
102	C₃×#95	—	10	2	no	—
103	#805	—	5	8	no	—
104	#803	—	10	1	no	—
105	C₄×C₃⋊C₈	—	10	1	no	—
106	C₅×#41	—	3	4	no	—
107	C₅×#41	—	6	2	no	—
108	C₂×C₅×Q₈	—	3	1	no	—
109	C₂×C₅×D₄	—	3	1	no	—
110	C₂×C₅×D₄	—	6	1	no	—
111	C₂²×C₄×C₅	—	6	1	no	—
112	C₂²×C₄×C₅	—	6	2	no	—
113	C₂²×C₄×C₅	#67	1	2	yes	—
114	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
115	C₂²×Dic₅	—	2	2	no	—
116	C₅×C₄⋊C₄	—	3	2	no	—
117	C₅×C₄⋊C₄	—	6	2	no	—
118	C₅×C₂²⋊C₄	—	6	1	no	—
119	C₅×C₂²⋊C₄	—	6	2	no	—
120	C₄²×C₅	—	6	2	no	—
121	C₄²×C₅	—	6	1	no	—
122	C₄²×C₅	C₄×D₃	1	2	yes	—
123	C₄²×C₅	D₁₂	1	1	yes	characteristic
124	C₄²×C₅	D₁₂	1	1	yes	characteristic
125	C₄×Dic₅	—	2	2	no	—
126	C₄×Dic₅	C₄×D₃	1	4	yes	complemented
127	#681	—	3	4	no	—
128	#681	—	6	2	no	—
129	C₂×#168	—	6	2	no	—
130	C₄×#41	—	3	1	no	—
131	C₂×#104	—	15	2	no	—
132	C₄×M₁₆	—	15	1	no	—
133	#376	—	5	1	no	—
134	#339	—	15	2	no	—
135	C₂²×C₃×C₅	—	2	2	no	—
136	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
138	C₂×C₅×D₃	—	2	2	no	—
139	C₂×C₅×D₃	—	4	2	no	—
140	C₃×C₄×C₅	—	2	2	no	—
141	C₃×C₄×C₅	—	2	2	no	—
142	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
143	C₃×C₄×C₅	#117	1	1	yes	characteristic
144	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
145	C₃×Dic₅	—	4	4	no	—
146	C₅×Dic₃	—	2	2	no	—
147	C₅×Dic₃	—	4	2	no	—
148	C₂²×C₃×C₄	—	10	2	no	—
149	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
150	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
151	C₂×#67	—	2	1	no	—
152	#235	—	2	2	no	—
153	#235	C₄×D₅	1	4	yes	—
154	C₂×D₁₂	C₂×Dic₅	1	1	yes	characteristic
155	C₂×C₄×D₃	—	2	1	no	—
156	C₂×C₄×D₃	—	2	2	no	—
157	C₂×Dic₆	C₂×Dic₅	1	1	yes	characteristic
158	C₃×C₄²	—	5	4	no	—
159	C₃×C₄²	—	10	2	no	—
160	C₃×C₄²	C₂×Dic₅	1	2	yes	—
161	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
162	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
163	#217	—	2	1	no	—
164	(C₂×D₃)⋊C₄	—	2	2	no	—
165	C₄⋊Dic₃	C₂×Dic₅	1	2	yes	—
166	Dic₃⋊C₄	—	2	2	no	—
167	C₄×Dic₃	—	2	2	no	—
168	C₄×Dic₃	—	2	1	no	—
169	#208	—	5	4	no	—
170	#208	—	10	2	no	—
171	C₂×C₃⋊C₈	—	10	2	no	—
172	C₂³×C₅	—	6	1	no	—
173	C₂³×C₅	C₂×#67	1	1	yes	characteristic
174	C₅×Q₈	—	3	2	no	—
175	C₅×Q₈	—	6	1	no	—
176	C₅×D₄	—	3	2	no	—
177	C₅×D₄	—	6	4	no	—
178	C₅×D₄	—	6	1	no	—
179	C₂×C₄×C₅	—	4	2	no	—
180	C₂×C₄×C₅	—	6	1	no	—
181	C₂×C₄×C₅	—	6	4	no	—
182	C₂×C₄×C₅	—	6	2	no	—
183	C₂×C₄×C₅	—	12	4	no	—
184	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
185	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
186	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
187	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
188	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
189	C₂×C₄×C₅	#235	1	2	yes	—
190	C₂×Dic₅	—	2	4	no	—
191	C₂×Dic₅	—	4	4	no	—
192	#168	—	6	4	no	—
193	C₂×#41	—	3	1	no	—
194	C₂×M₁₆	—	15	2	no	—
195	C₄×Q₈	—	3	2	no	—
196	C₄×D₄	—	3	2	no	—
197	C₄×D₄	—	6	2	no	—
198	#117	—	3	1	no	—
199	#117	—	6	1	no	—
200	C₂×C₄²	—	5	2	no	—
201	C₂×C₄²	—	6	1	no	—
202	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
203	#104	—	15	8	no	—
204	#97	—	30	1	no	—
205	C₄×C₈	—	30	1	no	—
206	#95	—	10	2	no	—
207	C₂×C₃×C₅	—	2	2	no	—
208	C₂×C₃×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
209	C₂×C₃×C₅	#421	1	2	yes	—
210	C₅×D₃	—	4	2	no	—
211	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
212	C₂²×D₃	—	2	1	no	—
213	C₂×C₃×C₄	—	4	2	no	—
214	C₂×C₃×C₄	—	10	4	no	—
215	C₂×C₃×C₄	—	20	4	no	—
216	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
217	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
218	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
219	C₂×C₃×C₄	#709	1	2	yes	—
220	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
221	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
222	#67	—	2	4	no	—
223	C₂×Dic₃	—	2	1	no	—
224	C₂×Dic₃	—	2	2	no	—
225	D₁₂	—	2	1	no	—
226	D₁₂	C₄×Dic₅	1	2	yes	—
227	C₄×D₃	—	2	4	no	—
228	C₄×D₃	—	4	4	no	—
229	Dic₆	—	2	1	no	—
230	Dic₆	C₄×Dic₅	1	2	yes	—
231	C₃⋊C₈	—	10	4	no	—
232	C₂²×C₅	—	2	2	no	—
233	C₂²×C₅	—	6	2	no	—
234	C₂²×C₅	—	12	2	no	—
235	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
236	C₂²×C₅	C₄×#67	1	2	yes	—
237	C₄×C₅	—	2	2	no	—
238	C₄×C₅	—	2	2	no	—
239	C₄×C₅	—	6	2	no	—
240	C₄×C₅	—	12	2	no	—
241	C₄×C₅	C₄²×D₃	1	1	yes	characteristic
242	C₄×C₅	#872	1	1	yes	characteristic
243	C₄×C₅	C₄×D₁₂	1	2	yes	—
244	Dic₅	—	4	4	no	—
245	#41	—	3	4	no	—
246	#41	—	6	2	no	—
247	C₂×Q₈	—	3	1	no	—
248	C₂×D₄	—	3	1	no	—
249	C₂×D₄	—	6	1	no	—
250	C₂²×C₄	—	6	1	no	—
251	C₂²×C₄	—	6	2	no	—
252	C₂²×C₄	—	10	2	no	—
253	C₂²×C₄	#1201	1	2	yes	—
254	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
255	M₁₆	—	15	4	no	—
256	M₁₆	—	30	2	no	—
257	C₂×C₈	—	30	2	no	—
258	C₄⋊C₄	—	3	2	no	—
259	C₄⋊C₄	—	6	2	no	—
260	C₂²⋊C₄	—	6	1	no	—
261	C₂²⋊C₄	—	6	2	no	—
262	C₄²	—	5	4	no	—
263	C₄²	—	6	2	no	—
264	C₄²	—	6	1	no	—
265	C₄²	—	10	2	no	—
266	C₄²	D₃×Dic₅	1	2	yes	—
267	C₄²	#1203	1	1	yes	characteristic
268	C₄²	#1203	1	1	yes	characteristic
269	C₃×C₅	#1759	1	1	yes	characteristic, complemented, Hall
270	C₂²×C₃	—	2	2	no	—
271	C₂²×C₃	C₄×C₂²⋊D₅	1	2	yes	—
272	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
273	C₂×D₃	—	2	2	no	—
274	C₂×D₃	—	4	2	no	—
275	C₃×C₄	—	2	2	no	—
276	C₃×C₄	—	2	2	no	—
277	C₃×C₄	—	20	4	no	—
278	C₃×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
279	C₃×C₄	#4072	1	1	yes	characteristic
280	C₃×C₄	C₄×C₂²⋊D₅	1	2	yes	—
281	Dic₃	—	2	2	no	—
282	Dic₃	—	4	2	no	—
283	C₂×C₅	—	2	2	no	—
284	C₂×C₅	—	12	2	no	—
285	C₂×C₅	C₄×(C₂×D₃)⋊C₄	1	1	yes	characteristic
286	C₂×C₅	#5005	1	2	yes	—
287	C₂³	—	6	1	no	—
288	C₂³	C₂×#1201	1	1	yes	characteristic
289	Q₈	—	3	2	no	—
290	Q₈	—	6	1	no	—
291	D₄	—	3	2	no	—
292	D₄	—	6	4	no	—
293	D₄	—	6	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	6	1	no	—
296	C₂×C₄	—	6	4	no	—
297	C₂×C₄	—	6	2	no	—
298	C₂×C₄	—	10	4	no	—
299	C₂×C₄	—	12	4	no	—
300	C₂×C₄	—	20	4	no	—
301	C₂×C₄	(C₂×D₃)⋊Dic₅	1	2	yes	—
302	C₂×C₄	(C₂×D₃)⋊Dic₅	1	2	yes	—
303	C₂×C₄	#6771	1	2	yes	—
304	C₂×C₄	C₄×D₃×D₅	1	2	yes	—
305	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
306	C₂×C₄	C₂×#1203	1	1	yes	characteristic
307	C₈	—	30	4	no	—
308	C₂×C₃	—	2	2	no	—
309	C₂×C₃	C₄×C₂²⋊Dic₅	1	1	yes	characteristic
310	C₂×C₃	#65525	1	2	yes	—
311	D₃	—	4	2	no	—
312	C₅	#69430	1	1	yes	characteristic, complemented, Sylow
313	C₂²	—	2	2	no	—
314	C₂²	—	6	2	no	—
315	C₂²	—	12	2	no	—
316	C₂²	C₄×#1201	1	2	yes	—
317	C₂²	C₂×(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	6	2	no	—
321	C₄	—	12	2	no	—
322	C₄	—	20	4	no	—
323	C₄	#91371	1	1	yes	characteristic
324	C₄	C₄×D₃×Dic₅	1	1	yes	characteristic
325	C₄	C₄×#1203	1	2	yes	—
326	C₃	#10602436	1	1	yes	characteristic, complemented, Sylow
327	C₂	—	2	2	no	—
328	C₂	—	12	2	no	—
329	C₂	#11748425	1	2	yes	—
330	C₂	C₄×(C₂×D₃)⋊Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#1203
commutator	C₃×C₄×C₅	C₂×C₄²
Frattini	C₂²×C₄	C₂×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂×C₅×#235	C₂²
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1759
LCS	G ▷ C₃×C₄×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. C₄×#6710	C₂	semidirect
3. C₃×#65221	C₂	semidirect
5. #11745977	C₂	semidirect
12. C₄×C₅×D₁₂	C₄	semidirect
15. C₄×C₅×Dic₆	C₄	semidirect
20. #90979	C₄	semidirect
24. #65221	D₃	semidirect
41. #6710	C₂×C₄	semidirect
50. C₃×C₄×Dic₅	C₂×C₄	semidirect
98. C₄×D₁₂	Dic₅	semidirect
101. C₄×Dic₆	Dic₅	semidirect
126. C₄×Dic₅	C₄×D₃	semidirect
269. C₃×C₅	#1759	semidirect
312. C₅	#69430	semidirect
326. C₃	#10602436	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×#235	2
C₄×#6710	2
C₃×#65221	2
C₂×#90979	2
#11745977	2
#10602436	3
#69430	5