Subgroups — #11989436

Order 1280.

696 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#117	C₂	1	1	yes	maximal, characteristic
2	#10613791	C₂	1	1	yes	maximal, characteristic
3	#10613780	C₂	1	1	yes	maximal, characteristic
4	#10598109	C₂	1	4	yes	maximal
5	C₂×C₅×#117	—	2	1	no	—
6	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₄×C₅×C₄⋊C₄	—	2	1	no	—
8	C₄×C₅×C₄⋊C₄	C₄	1	2	yes	complemented
9	C₄×C₅×C₂²⋊C₄	—	2	1	no	—
10	C₄×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
11	C₅×#376	C₄	1	2	yes	complemented
12	C₄³×C₅	—	2	1	no	—
13	#65501	—	2	1	no	—
14	#65500	—	2	1	no	—
15	C₄×C₂²⋊Dic₅	C₂²	1	2	yes	—
16	#65496	C₂²	1	2	yes	—
17	#65226	C₄	1	2	yes	complemented
18	#65225	C₄	1	2	yes	complemented
19	C₄×C₄⋊Dic₅	—	2	1	no	—
20	#65221	—	2	1	no	—
21	#65215	—	2	1	no	—
22	#65215	—	2	1	no	—
23	#65214	—	2	1	no	—
24	#65214	—	2	1	no	—
25	#65213	C₄	1	2	yes	complemented
26	C₄×#681	C₄	1	2	yes	complemented
27	#8052	—	5	1	no	maximal, Sylow
28	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₅×#117	—	2	8	no	—
32	C₂×C₅×C₄⋊C₄	—	2	2	no	—
33	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
34	C₂×C₄²×C₅	—	2	2	no	—
35	C₂×C₄²×C₅	—	2	2	no	—
36	C₂×C₄²×C₅	—	2	2	no	—
37	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
48	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
49	C₂×C₂²⋊Dic₅	—	2	2	no	—
50	C₂×C₄⋊Dic₅	—	2	2	no	—
51	C₂×C₄×Dic₅	—	2	2	no	—
52	C₂×#681	C₂×C₄	1	4	yes	—
53	C₅×#95	—	2	2	no	—
54	C₅×#95	C₂×C₄	1	4	yes	—
55	#3963	—	2	2	no	—
56	#3963	—	2	2	no	—
57	#3963	—	2	2	no	—
58	#3963	C₂×C₄	1	4	yes	—
59	C₂²⋊#168	—	2	2	no	—
60	C₂²⋊#168	—	2	2	no	—
61	C₄⋊#168	—	2	2	no	—
62	C₄⋊#168	—	2	2	no	—
63	#3935	—	2	2	no	—
64	C₄×#168	—	2	2	no	—
65	#2920	—	5	1	no	—
66	#2283	—	5	1	no	—
67	C₄×#117	D₅	1	1	yes	characteristic
68	#1297	—	5	4	no	—
69	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	—	2	4	no	—
71	C₂²×C₄×C₅	—	2	4	no	—
72	C₂²×C₄×C₅	—	4	2	no	—
73	C₂²×C₄×C₅	—	4	2	no	—
74	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
92	C₂²×Dic₅	—	2	4	no	—
93	C₅×C₄⋊C₄	—	2	8	no	—
94	C₅×C₂²⋊C₄	—	2	8	no	—
95	C₄²×C₅	—	2	8	no	—
96	C₄²×C₅	—	4	4	no	—
97	C₄²×C₅	—	4	4	no	—
98	C₄²×C₅	C₄²	1	1	yes	characteristic
99	C₄²×C₅	C₄²	1	1	yes	characteristic
100	C₄²×C₅	C₄²	1	1	yes	characteristic
101	C₄²×C₅	C₄²	1	1	yes	characteristic
102	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
103	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
104	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
105	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
106	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
107	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
108	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
109	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
110	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
111	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
112	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
113	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
114	C₂²⋊Dic₅	—	4	4	no	—
115	C₄⋊Dic₅	—	4	4	no	—
116	C₄×Dic₅	—	4	4	no	—
117	#681	—	4	4	no	—
118	C₂×#168	—	2	4	no	—
119	C₂×#117	—	2	1	no	—
120	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
121	#432	—	10	1	no	—
122	#432	—	10	1	no	—
123	#431	—	10	1	no	—
124	#431	—	10	1	no	—
125	#423	—	5	2	no	—
126	#407	—	5	2	no	—
127	C₄×M₁₆	—	5	2	no	—
128	#383	—	5	2	no	—
129	#382	—	5	2	no	—
130	#381	—	10	1	no	—
131	#380	—	10	1	no	—
132	C₄×C₄⋊C₄	—	2	1	no	—
133	C₄×C₄⋊C₄	—	10	1	no	—
134	C₄×C₄⋊C₄	Dic₅	1	2	yes	complemented
135	C₄×C₂²⋊C₄	—	2	1	no	—
136	C₄×C₂²⋊C₄	—	5	2	no	—
137	C₄×C₂²⋊C₄	C₂×D₅	1	2	yes	—
138	#376	—	10	1	no	—
139	#376	Dic₅	1	2	yes	complemented
140	C₄³	—	2	1	no	—
141	C₂³×C₅	—	2	2	no	—
142	C₂³×C₅	—	2	2	no	—
143	C₂³×C₅	—	4	1	no	—
144	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₅	—	2	2	no	—
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	—	2	2	no	—
150	C₂×C₄×C₅	—	2	2	no	—
151	C₂×C₄×C₅	—	2	2	no	—
152	C₂×C₄×C₅	—	2	2	no	—
153	C₂×C₄×C₅	—	2	2	no	—
154	C₂×C₄×C₅	—	2	2	no	—
155	C₂×C₄×C₅	—	2	2	no	—
156	C₂×C₄×C₅	—	2	2	no	—
157	C₂×C₄×C₅	—	2	8	no	—
158	C₂×C₄×C₅	—	4	1	no	—
159	C₂×C₄×C₅	—	4	1	no	—
160	C₂×C₄×C₅	—	4	1	no	—
161	C₂×C₄×C₅	—	4	1	no	—
162	C₂×C₄×C₅	—	4	1	no	—
163	C₂×C₄×C₅	—	4	1	no	—
164	C₂×C₄×C₅	—	4	1	no	—
165	C₂×C₄×C₅	—	4	4	no	—
166	C₂×C₄×C₅	—	4	4	no	—
167	C₂×C₄×C₅	#95	1	1	yes	characteristic
168	C₂×C₄×C₅	#95	1	1	yes	characteristic
169	C₂×C₄×C₅	#95	1	1	yes	characteristic
170	C₂×C₄×C₅	#95	1	1	yes	characteristic
171	C₂×C₄×C₅	#95	1	1	yes	characteristic
172	C₂×C₄×C₅	#95	1	1	yes	characteristic
173	C₂×C₄×C₅	#95	1	1	yes	characteristic
174	C₂×C₄×C₅	#95	1	1	yes	characteristic
175	C₂×C₄×C₅	#104	1	4	yes	—
176	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
178	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
179	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
180	C₂×Dic₅	—	4	4	no	—
181	C₂×Dic₅	—	4	4	no	—
182	C₂×Dic₅	—	4	4	no	—
183	#168	—	4	4	no	—
184	C₂³×C₄	C₂²⋊D₅	1	1	yes	characteristic
185	C₂³×C₄	C₂²⋊D₅	1	1	yes	characteristic
186	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
187	C₂×M₁₆	—	5	4	no	—
188	#117	—	2	8	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	10	2	no	—
191	C₂×C₂²⋊C₄	—	2	2	no	—
192	C₂×C₂²⋊C₄	—	10	2	no	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	—	10	2	no	—
197	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
198	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
199	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
200	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
201	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
202	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
203	C₂×C₄²	D₂₀	1	1	yes	characteristic
204	C₂×C₄²	D₂₀	1	1	yes	characteristic
205	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
206	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
207	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
208	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
209	C₄⋊C₈	—	10	2	no	—
210	C₄⋊C₈	—	10	2	no	—
211	C₂²⋊C₈	—	10	2	no	—
212	C₂²⋊C₈	—	10	2	no	—
213	#97	—	10	2	no	—
214	C₄×C₈	—	10	2	no	—
215	#95	—	2	2	no	—
216	#95	—	5	4	no	—
217	#95	—	10	2	no	—
218	#95	—	10	2	no	—
219	#95	—	10	2	no	—
220	#95	C₂×Dic₅	1	4	yes	—
221	C₂²×C₅	—	2	1	no	—
222	C₂²×C₅	—	2	1	no	—
223	C₂²×C₅	—	2	1	no	—
224	C₂²×C₅	—	2	1	no	—
225	C₂²×C₅	—	2	1	no	—
226	C₂²×C₅	—	2	1	no	—
227	C₂²×C₅	—	2	1	no	—
228	C₂²×C₅	—	4	4	no	—
229	C₂²×C₅	C₂×#95	1	1	yes	characteristic
230	C₂²×C₅	C₂×#104	1	2	yes	—
231	C₂²×C₅	#421	1	2	yes	—
232	C₄×C₅	—	2	1	no	—
233	C₄×C₅	—	2	1	no	—
234	C₄×C₅	—	2	1	no	—
235	C₄×C₅	—	2	1	no	—
236	C₄×C₅	—	2	1	no	—
237	C₄×C₅	—	2	1	no	—
238	C₄×C₅	—	2	1	no	—
239	C₄×C₅	—	2	1	no	—
240	C₄×C₅	—	2	1	no	—
241	C₄×C₅	—	4	8	no	—
242	C₄×C₅	#339	1	2	yes	—
243	C₄×C₅	#339	1	2	yes	—
244	C₄×C₅	#410	1	2	yes	—
245	Dic₅	—	8	4	no	—
246	C₂⁴	C₂×C₂²⋊D₅	1	1	yes	characteristic
247	C₂²×C₄	—	2	4	no	—
248	C₂²×C₄	—	2	4	no	—
249	C₂²×C₄	—	4	2	no	—
250	C₂²×C₄	—	4	2	no	—
251	C₂²×C₄	—	10	4	no	—
252	C₂²×C₄	#683	1	1	yes	characteristic
253	C₂²×C₄	#683	1	1	yes	characteristic
254	C₂²×C₄	#683	1	1	yes	characteristic
255	C₂²×C₄	#683	1	1	yes	characteristic
256	C₂²×C₄	#685	1	1	yes	characteristic
257	C₂²×C₄	#685	1	1	yes	characteristic
258	C₂²×C₄	#685	1	1	yes	characteristic
259	C₂²×C₄	#685	1	1	yes	characteristic
260	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
261	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
262	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
263	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
264	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
265	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
266	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
267	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
268	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
269	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
270	M₁₆	—	20	4	no	—
271	C₂×C₈	—	10	4	no	—
272	C₄⋊C₄	—	2	8	no	—
273	C₄⋊C₄	—	20	4	no	—
274	C₂²⋊C₄	—	2	8	no	—
275	C₂²⋊C₄	—	20	4	no	—
276	C₄²	—	2	8	no	—
277	C₄²	—	4	4	no	—
278	C₄²	—	4	4	no	—
279	C₄²	—	20	4	no	—
280	C₄²	C₄×Dic₅	1	1	yes	characteristic
281	C₄²	C₄×Dic₅	1	1	yes	characteristic
282	C₄²	C₄×Dic₅	1	1	yes	characteristic
283	C₄²	C₄×Dic₅	1	1	yes	characteristic
284	C₄²	#683	1	1	yes	characteristic
285	C₄²	#683	1	1	yes	characteristic
286	C₄²	#683	1	1	yes	characteristic
287	C₄²	#683	1	1	yes	characteristic
288	C₄²	C₄⋊Dic₅	1	1	yes	characteristic
289	C₄²	C₄⋊Dic₅	1	1	yes	characteristic
290	C₄²	C₄⋊Dic₅	1	1	yes	characteristic
291	C₄²	C₄⋊Dic₅	1	1	yes	characteristic
292	C₄²	#685	1	1	yes	characteristic
293	C₄²	#685	1	1	yes	characteristic
294	C₄²	#685	1	1	yes	characteristic
295	C₄²	#685	1	1	yes	characteristic
296	C₂×C₅	—	2	2	no	—
297	C₂×C₅	—	2	2	no	—
298	C₂×C₅	—	4	1	no	—
299	C₂×C₅	C₂×#339	1	1	yes	characteristic
300	C₂×C₅	#1733	1	1	yes	characteristic
301	C₂×C₅	#1740	1	1	yes	characteristic
302	C₂³	—	2	2	no	—
303	C₂³	—	2	2	no	—
304	C₂³	—	4	1	no	—
305	C₂³	C₂×#683	1	1	yes	characteristic
306	C₂³	C₂×#685	1	1	yes	characteristic
307	C₂³	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	8	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	4	4	no	—
327	C₂×C₄	—	4	4	no	—
328	C₂×C₄	—	20	4	no	—
329	C₂×C₄	—	20	4	no	—
330	C₂×C₄	—	20	4	no	—
331	C₂×C₄	#3937	1	4	yes	—
332	C₂×C₄	#3963	1	1	yes	characteristic
333	C₂×C₄	#3963	1	1	yes	characteristic
334	C₂×C₄	#3963	1	1	yes	characteristic
335	C₂×C₄	#3963	1	1	yes	characteristic
336	C₂×C₄	#3963	1	1	yes	characteristic
337	C₂×C₄	#3963	1	1	yes	characteristic
338	C₂×C₄	#3963	1	1	yes	characteristic
339	C₂×C₄	#3963	1	1	yes	characteristic
340	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
341	C₂×C₄	C₂×#683	1	1	yes	characteristic
342	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
343	C₂×C₄	C₂×#685	1	1	yes	characteristic
344	C₈	—	20	4	no	—
345	C₅	#8052	1	1	yes	characteristic, complemented, Sylow
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	4	4	no	—
354	C₂²	C₂×#3937	1	2	yes	—
355	C₂²	C₂×#3963	1	1	yes	characteristic
356	C₂²	#65525	1	2	yes	—
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	4	8	no	—
367	C₄	—	40	4	no	—
368	C₄	#64743	1	2	yes	—
369	C₄	#64743	1	2	yes	—
370	C₄	#65295	1	2	yes	—
371	C₂	—	2	2	no	—
372	C₂	—	2	2	no	—
373	C₂	—	4	1	no	—
374	C₂	C₂×#64743	1	1	yes	characteristic
375	C₂	#10602367	1	1	yes	characteristic
376	C₂	#10602386	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#685
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅
Fitting	C₄×C₅×#117	C₂
socle	C₂²×C₅	C₂×#95
prime radical	C₂⁴×C₅	C₂×D₄
cosocle	C₂³×C₄×C₅	C₂³
hypercenter	C₄×#117	D₅
hypocenter	C₅	#8052
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
8. C₄×C₅×C₄⋊C₄	C₄	semidirect
11. C₅×#376	C₄	semidirect
17. #65226	C₄	semidirect
18. #65225	C₄	semidirect
25. #65213	C₄	semidirect
26. C₄×#681	C₄	semidirect
134. C₄×C₄⋊C₄	Dic₅	semidirect
139. #376	Dic₅	semidirect
345. C₅	#8052	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×#117	2
#10613791	2
#10613780	2
#10598109	2
#8052	5