Subgroups — #11989595

Order 1280.

608 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#423	C₂	1	1	yes	maximal, characteristic
2	#10613791	C₂	1	1	yes	maximal, characteristic
3	#10613780	C₂	1	1	yes	maximal, characteristic
4	#10598206	C₂	1	4	yes	maximal, complemented
5	C₂²×C₅×M₁₆	—	2	1	no	—
6	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₅×#423	—	2	1	no	—
8	C₅×#423	—	2	1	no	—
9	C₅×#423	C₄	1	2	yes	complemented
10	C₅×#423	C₂²	1	2	yes	—
11	C₂×C₅×C₄⋊C₈	—	2	1	no	—
12	C₂×C₅×C₄⋊C₈	C₄	1	2	yes	complemented
13	#65501	—	2	1	no	—
14	#65500	C₄	1	2	yes	—
15	C₄×C₂²⋊Dic₅	—	2	1	no	—
16	#65496	—	2	1	no	—
17	#65226	C₄	1	2	yes	—
18	#65225	—	2	1	no	—
19	C₄×C₄⋊Dic₅	C₂²	1	2	yes	—
20	#65221	—	2	1	no	—
21	#65215	—	2	1	no	—
22	#65215	C₄	1	2	yes	complemented
23	#65214	—	2	1	no	—
24	#65214	C₂²	1	2	yes	—
25	#65213	C₄	1	2	yes	complemented
26	C₄×#681	—	2	1	no	—
27	#8093	—	5	1	no	maximal, Sylow
28	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
31	C₂×C₅×M₁₆	—	2	2	no	—
32	C₂×C₅×M₁₆	—	2	2	no	—
33	C₂×C₅×M₁₆	—	4	4	no	—
34	C₂²×C₅×C₈	—	2	2	no	—
35	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
45	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
46	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
47	C₂×C₂²⋊Dic₅	—	2	2	no	—
48	C₂×C₄⋊Dic₅	—	2	2	no	—
49	C₂×C₄×Dic₅	—	2	2	no	—
50	C₂×#681	—	2	2	no	—
51	C₅×C₄⋊C₈	—	2	2	no	—
52	C₅×C₄⋊C₈	—	2	2	no	—
53	C₅×C₄⋊C₈	—	2	2	no	—
54	C₅×C₄⋊C₈	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	C₂×C₄	1	4	yes	—
63	#3935	—	2	2	no	—
64	C₄×#168	—	2	2	no	—
65	#2920	—	5	1	no	—
66	C₂×#423	D₅	1	1	yes	characteristic
67	#2283	—	5	1	no	—
68	#1303	—	5	4	no	—
69	C₂⁴×C₅	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	—	2	2	no	—
71	C₂²×C₄×C₅	—	2	2	no	—
72	C₂²×C₄×C₅	—	4	2	no	—
73	C₂²×C₄×C₅	—	4	2	no	—
74	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₄²	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	C₂²⋊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	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
92	C₂²×Dic₅	—	2	4	no	—
93	C₅×M₁₆	—	4	4	no	—
94	C₅×M₁₆	—	4	4	no	—
95	C₂×C₅×C₈	—	2	4	no	—
96	C₂×C₅×C₈	—	4	4	no	—
97	C₄²×C₅	C₄²	1	1	yes	characteristic
98	C₄²×C₅	C₄²	1	1	yes	characteristic
99	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
100	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
101	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
102	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
103	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
104	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
105	C₄²×C₅	C₄⋊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	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
113	C₂²⋊Dic₅	—	4	4	no	—
114	C₄⋊Dic₅	—	4	4	no	—
115	C₄×Dic₅	—	4	2	no	—
116	C₄×Dic₅	—	4	2	no	—
117	#681	—	4	4	no	—
118	C₂×#168	—	2	4	no	—
119	C₂²×M₁₆	—	2	1	no	—
120	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
121	#432	—	5	2	no	—
122	#432	—	10	1	no	—
123	#431	—	5	2	no	—
124	#431	—	10	1	no	—
125	#423	—	2	1	no	—
126	#423	—	2	1	no	—
127	#423	—	5	2	no	—
128	#423	Dic₅	1	2	yes	complemented
129	#423	C₂×D₅	1	2	yes	—
130	C₂×C₄⋊C₈	—	2	1	no	—
131	C₂×C₄⋊C₈	Dic₅	1	2	yes	complemented
132	#407	—	10	1	no	—
133	C₄×M₁₆	—	10	1	no	—
134	#383	—	5	2	no	—
135	#382	—	10	1	no	—
136	#381	—	5	2	no	—
137	#380	—	10	1	no	—
138	C₄×C₄⋊C₄	—	5	2	no	—
139	C₄×C₂²⋊C₄	—	10	1	no	—
140	#376	—	10	1	no	—
141	C₂³×C₅	—	2	2	no	—
142	C₂³×C₅	—	2	2	no	—
143	C₂³×C₅	—	4	1	no	—
144	C₂³×C₅	#95	1	1	yes	characteristic
145	C₂³×C₅	#95	1	1	yes	characteristic
146	C₂³×C₅	C₂×C₂²⋊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	1	no	—
151	C₂×C₄×C₅	—	2	1	no	—
152	C₂×C₄×C₅	—	2	1	no	—
153	C₂×C₄×C₅	—	2	1	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	2	no	—
158	C₂×C₄×C₅	—	2	2	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	1	no	—
166	C₂×C₄×C₅	#95	1	1	yes	characteristic
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	C₂×C₄²	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
175	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
176	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
178	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
179	C₂×C₄×C₅	#137	1	2	yes	—
180	C₂×Dic₅	—	4	4	no	—
181	C₂×Dic₅	—	4	4	no	—
182	C₂×Dic₅	—	4	4	no	—
183	C₅×C₈	—	4	4	no	—
184	#168	—	4	4	no	—
185	C₂³×C₄	C₄×D₅	1	1	yes	characteristic
186	C₂³×C₄	D₂₀	1	1	yes	characteristic
187	C₂³×C₄	C₂²⋊D₅	1	1	yes	characteristic
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	4	4	no	—
191	C₂×M₁₆	—	10	2	no	—
192	C₂²×C₈	—	2	2	no	—
193	C₂×C₄⋊C₄	—	10	2	no	—
194	C₂×C₂²⋊C₄	—	10	2	no	—
195	C₂×C₄²	—	10	2	no	—
196	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
197	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
198	C₂×C₄²	C₄×D₅	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	D₂₀	1	1	yes	characteristic
202	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
203	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
204	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
205	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
206	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
207	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
208	C₄⋊C₈	—	2	2	no	—
209	C₄⋊C₈	—	2	2	no	—
210	C₄⋊C₈	—	2	2	no	—
211	C₄⋊C₈	—	5	4	no	—
212	C₄⋊C₈	—	10	2	no	—
213	C₄⋊C₈	C₂×Dic₅	1	4	yes	—
214	C₂²⋊C₈	—	10	2	no	—
215	C₂²⋊C₈	—	10	2	no	—
216	#97	—	10	2	no	—
217	C₄×C₈	—	10	2	no	—
218	#95	—	5	4	no	—
219	#95	—	10	2	no	—
220	#95	—	10	2	no	—
221	#95	—	10	2	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	—	2	1	no	—
229	C₂²×C₅	—	4	4	no	—
230	C₂²×C₅	#344	1	2	yes	—
231	C₂²×C₅	C₂×#95	1	1	yes	characteristic
232	C₂²×C₅	#417	1	2	yes	—
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	—	2	1	no	—
242	C₄×C₅	#418	1	1	yes	characteristic
243	C₄×C₅	#419	1	1	yes	characteristic
244	C₄×C₅	#421	1	2	yes	—
245	C₄×C₅	#428	1	2	yes	—
246	Dic₅	—	8	4	no	—
247	C₂⁴	#685	1	1	yes	characteristic
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	4	2	no	—
251	C₂²×C₄	—	4	2	no	—
252	C₂²×C₄	—	10	4	no	—
253	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
254	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
255	C₂²×C₄	#683	1	1	yes	characteristic
256	C₂²×C₄	#683	1	1	yes	characteristic
257	C₂²×C₄	#683	1	1	yes	characteristic
258	C₂²×C₄	#683	1	1	yes	characteristic
259	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
260	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
261	C₂²×C₄	#685	1	1	yes	characteristic
262	C₂²×C₄	#685	1	1	yes	characteristic
263	C₂²×C₄	#685	1	1	yes	characteristic
264	C₂²×C₄	#685	1	1	yes	characteristic
265	C₂²×C₄	#685	1	1	yes	characteristic
266	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
267	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
268	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
269	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
270	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
271	M₁₆	—	4	4	no	—
272	M₁₆	—	4	4	no	—
273	M₁₆	—	20	4	no	—
274	C₂×C₈	—	2	4	no	—
275	C₂×C₈	—	4	4	no	—
276	C₂×C₈	—	10	4	no	—
277	C₄⋊C₄	—	20	4	no	—
278	C₂²⋊C₄	—	20	4	no	—
279	C₄²	—	20	2	no	—
280	C₄²	—	20	2	no	—
281	C₄²	C₄×Dic₅	1	1	yes	characteristic
282	C₄²	C₄×Dic₅	1	1	yes	characteristic
283	C₄²	#683	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	C₄⋊Dic₅	1	1	yes	characteristic
288	C₄²	C₄⋊Dic₅	1	1	yes	characteristic
289	C₄²	#685	1	1	yes	characteristic
290	C₄²	#685	1	1	yes	characteristic
291	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
292	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
293	C₄²	C₂×Dic₁₀	1	1	yes	characteristic
294	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
295	C₄²	C₂²×Dic₅	1	1	yes	characteristic
296	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
297	C₂×C₅	—	2	2	no	—
298	C₂×C₅	—	2	2	no	—
299	C₂×C₅	—	4	1	no	—
300	C₂×C₅	#1733	1	1	yes	characteristic
301	C₂×C₅	#1736	1	1	yes	characteristic
302	C₂×C₅	C₂×#344	1	1	yes	characteristic
303	C₂³	—	2	2	no	—
304	C₂³	—	2	2	no	—
305	C₂³	—	4	1	no	—
306	C₂³	#3963	1	1	yes	characteristic
307	C₂³	#3963	1	1	yes	characteristic
308	C₂³	C₂×#685	1	1	yes	characteristic
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	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	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	1	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	20	4	no	—
329	C₂×C₄	—	20	4	no	—
330	C₂×C₄	—	20	4	no	—
331	C₂×C₄	#3963	1	1	yes	characteristic
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	C₂×C₄×Dic₅	1	1	yes	characteristic
338	C₂×C₄	C₂×#683	1	1	yes	characteristic
339	C₂×C₄	C₂×#683	1	1	yes	characteristic
340	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
341	C₂×C₄	C₂×#685	1	1	yes	characteristic
342	C₂×C₄	#4078	1	2	yes	—
343	C₂×C₄	#4088	1	2	yes	—
344	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
345	C₈	—	4	4	no	—
346	C₈	—	20	4	no	—
347	C₅	#8093	1	1	yes	characteristic, complemented, Sylow
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	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	4	4	no	—
356	C₂²	#64776	1	2	yes	—
357	C₂²	#65255	1	2	yes	—
358	C₂²	C₂×#3963	1	1	yes	characteristic
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	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	40	4	no	—
369	C₄	#65253	1	2	yes	—
370	C₄	#65428	1	2	yes	—
371	C₄	#65504	1	1	yes	characteristic
372	C₄	#65514	1	1	yes	characteristic
373	C₂	—	2	2	no	—
374	C₂	—	2	2	no	—
375	C₂	—	4	1	no	—
376	C₂	#10602351	1	1	yes	characteristic
377	C₂	#10602775	1	1	yes	characteristic
378	C₂	C₂×#64776	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₅
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₂×C₅×#423	C₂
socle	C₂²×C₅	C₂×#95
prime radical	C₂⁴×C₅	C₂²⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₂×#423	D₅
hypocenter	C₅	#8093
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
4. #10598206	C₂	semidirect
9. C₅×#423	C₄	semidirect
12. C₂×C₅×C₄⋊C₈	C₄	semidirect
22. #65215	C₄	semidirect
25. #65213	C₄	semidirect
128. #423	Dic₅	semidirect
131. C₂×C₄⋊C₈	Dic₅	semidirect
347. C₅	#8093	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#423	2
#10613791	2
#10613780	2
#10598206	2
#8093	5