Subgroups — #12118186

Order 1280.

730 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#446	C₂	1	1	yes	maximal, characteristic
2	#10614193	C₂	1	2	yes	maximal
3	#10601248	C₂	1	2	yes	maximal
4	#10601248	C₂	1	2	yes	maximal
5	#10601246	C₂	1	2	yes	maximal
6	#10600609	C₂	1	2	yes	maximal
7	#10600607	C₂	1	2	yes	maximal
8	#10600559	C₂	1	1	yes	maximal, characteristic
9	C₄×#3950	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
11	C₅×#446	C₄	1	8	yes	complemented
12	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₅×#97	C₂²	1	2	yes	—
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	#65519	—	2	2	no	—
17	C₂×#3950	C₂²	1	1	yes	characteristic
18	C₂×#3950	C₂²	1	1	yes	characteristic
19	C₂×#3950	C₂²	1	2	yes	—
20	C₂×#3950	C₂²	1	1	yes	characteristic
21	C₂×#3950	C₂²	1	1	yes	characteristic
22	C₂×#3949	C₂²	1	2	yes	—
23	C₂×#3949	C₂²	1	2	yes	—
24	#65270	—	2	2	no	—
25	#65270	—	2	2	no	—
26	C₄⋊C₄×Dic₅	—	2	2	no	—
27	#65225	C₂²	1	2	yes	—
28	#65225	C₂²	1	2	yes	—
29	#65224	C₂²	1	2	yes	—
30	C₄×C₄⋊Dic₅	C₂²	1	2	yes	—
31	C₄×C₄⋊Dic₅	C₂²	1	2	yes	—
32	C₄×C₄⋊Dic₅	C₂²	1	2	yes	—
33	#64771	C₂²	1	2	yes	—
34	#64771	C₂²	1	2	yes	—
35	#64771	C₂²	1	2	yes	—
36	#22731	—	5	1	no	maximal, Sylow
37	C₂²×C₅×Q₈	D₄	1	1	yes	characteristic
38	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
39	C₂²×C₅×C₈	C₂³	1	2	yes	—
40	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
41	C₄×C₅×Q₈	C₂×C₄	1	4	yes	—
42	C₄×C₅×Q₈	Q₈	1	4	yes	—
43	C₂×C₅×C₄⋊C₄	D₄	1	2	yes	—
44	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₅	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₅	—	2	2	no	—
48	C₂×C₄⋊Dic₅	—	2	2	no	—
49	C₂×C₄⋊Dic₅	C₂³	1	2	yes	—
50	C₂×C₄⋊Dic₅	C₂³	1	2	yes	—
51	C₂×C₄⋊Dic₅	C₂³	1	2	yes	—
52	C₂×C₄⋊Dic₅	C₂³	1	2	yes	—
53	C₂×C₄×Dic₅	—	2	2	no	—
54	C₂×C₄×Dic₅	—	2	2	no	—
55	C₂×C₄×Dic₅	—	2	2	no	—
56	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	—
57	C₅×C₄⋊C₈	C₂×C₄	1	8	yes	—
58	C₅×#97	C₂×C₄	1	8	yes	—
59	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
60	#3963	—	2	2	no	—
61	#3963	—	2	2	no	—
62	#3963	—	2	2	no	—
63	#3963	—	2	2	no	—
64	#3963	—	2	2	no	—
65	#3963	—	2	2	no	—
66	#3963	—	2	2	no	—
67	#3950	—	2	2	no	—
68	#3950	—	4	1	no	—
69	#3950	—	4	2	no	—
70	#3950	—	4	1	no	—
71	#3950	Q₈	1	4	yes	complemented
72	#3949	—	2	4	no	—
73	#3949	—	4	2	no	—
74	C₂×#446	D₅	1	1	yes	characteristic
75	#2355	—	5	2	no	—
76	#1949	—	5	2	no	—
77	#1949	—	5	2	no	—
78	#1947	—	5	2	no	—
79	#1847	—	5	2	no	—
80	#1846	—	5	1	no	—
81	#1776	—	5	2	no	—
82	C₄×#107	—	5	1	no	—
83	C₂×C₅×Q₈	—	4	1	no	—
84	C₂×C₅×Q₈	—	4	1	no	—
85	C₂×C₅×Q₈	C₄⋊C₄	1	4	yes	—
86	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
91	C₂²×Dic₅	—	2	2	no	—
92	C₂²×Dic₅	—	2	2	no	—
93	C₂²×Dic₅	—	2	2	no	—
94	C₂²×Dic₅	—	2	2	no	—
95	C₂×C₅×C₈	—	2	2	no	—
96	C₂×C₅×C₈	—	2	1	no	—
97	C₂×C₅×C₈	—	2	1	no	—
98	C₂×C₅×C₈	—	2	2	no	—
99	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
100	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
101	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
102	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
103	C₂×C₅×C₈	#41	1	2	yes	—
104	C₅×C₄⋊C₄	C₄⋊C₄	1	8	yes	—
105	C₅×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
106	C₄²×C₅	C₂²×C₄	1	2	yes	—
107	C₄²×C₅	C₂²×C₄	1	4	yes	—
108	C₄²×C₅	C₂×Q₈	1	2	yes	—
109	C₄²×C₅	C₂×Q₈	1	4	yes	—
110	C₄⋊Dic₅	—	2	2	no	—
111	C₄⋊Dic₅	—	2	2	no	—
112	C₄⋊Dic₅	—	2	2	no	—
113	C₄⋊Dic₅	—	2	2	no	—
114	C₄⋊Dic₅	—	2	2	no	—
115	C₄⋊Dic₅	—	2	2	no	—
116	C₄⋊Dic₅	—	2	2	no	—
117	C₄⋊Dic₅	—	4	4	no	—
118	C₄⋊Dic₅	—	8	2	no	—
119	C₄⋊Dic₅	C₂×Q₈	1	4	yes	—
120	C₄×Dic₅	—	4	4	no	—
121	C₄×Dic₅	—	8	2	no	—
122	C₄×Dic₅	—	8	2	no	—
123	C₂×C₄×Q₈	C₂×D₅	1	1	yes	characteristic
124	#446	Dic₅	1	8	yes	complemented
125	C₂×#107	—	5	1	no	—
126	C₂×#107	—	5	1	no	—
127	C₂×#107	—	5	1	no	—
128	C₂×#107	—	5	1	no	—
129	C₂×#107	—	5	2	no	—
130	C₂×#106	—	5	2	no	—
131	C₂×#106	—	5	2	no	—
132	C₂×C₄⋊C₈	C₂×D₅	1	2	yes	—
133	C₂×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
134	C₂×#97	C₂×D₅	1	2	yes	—
135	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
136	#391	—	10	2	no	—
137	#387	—	10	2	no	—
138	#387	—	10	2	no	—
139	#384	—	5	2	no	—
140	#382	—	5	2	no	—
141	#382	—	5	2	no	—
142	C₄×C₄⋊C₄	—	5	2	no	—
143	C₄×C₄⋊C₄	—	5	2	no	—
144	C₄×C₄⋊C₄	—	5	2	no	—
145	C₄×C₄⋊C₄	—	10	2	no	—
146	#340	—	5	2	no	—
147	#340	—	5	2	no	—
148	#340	—	5	2	no	—
149	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
150	C₅×Q₈	—	4	4	no	—
151	C₂×C₄×C₅	—	2	2	no	—
152	C₂×C₄×C₅	—	2	2	no	—
153	C₂×C₄×C₅	—	2	1	no	—
154	C₂×C₄×C₅	—	2	1	no	—
155	C₂×C₄×C₅	—	2	1	no	—
156	C₂×C₄×C₅	—	2	2	no	—
157	C₂×C₄×C₅	—	2	1	no	—
158	C₂×C₄×C₅	—	2	2	no	—
159	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
160	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
161	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
162	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
163	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
164	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
165	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
166	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
167	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
168	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
169	C₂×Dic₅	—	2	4	no	—
170	C₂×Dic₅	—	4	2	no	—
171	C₂×Dic₅	—	4	2	no	—
172	C₂×Dic₅	—	4	2	no	—
173	C₂×Dic₅	—	4	2	no	—
174	C₂×Dic₅	—	4	2	no	—
175	C₂×Dic₅	—	4	2	no	—
176	C₂×Dic₅	—	4	2	no	—
177	C₂×Dic₅	—	4	2	no	—
178	C₂×Dic₅	—	4	2	no	—
179	C₂×Dic₅	—	4	2	no	—
180	C₂×Dic₅	—	4	2	no	—
181	C₅×C₈	—	2	4	no	—
182	C₅×C₈	—	2	2	no	—
183	C₅×C₈	C₄×Q₈	1	4	yes	—
184	C₂²×Q₈	D₂₀	1	1	yes	characteristic
185	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
186	C₂²×C₈	C₂²×D₅	1	2	yes	—
187	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
188	C₄×Q₈	Dic₁₀	1	4	yes	—
189	C₄×Q₈	C₂×Dic₅	1	4	yes	—
190	C₂×C₄⋊C₄	—	5	2	no	—
191	C₂×C₄⋊C₄	—	5	2	no	—
192	C₂×C₄⋊C₄	—	5	2	no	—
193	C₂×C₄⋊C₄	—	5	2	no	—
194	C₂×C₄⋊C₄	—	10	2	no	—
195	C₂×C₄⋊C₄	—	10	2	no	—
196	C₂×C₄⋊C₄	D₂₀	1	2	yes	—
197	C₂×C₄⋊C₄	D₂₀	1	1	yes	characteristic
198	C₂×C₄²	—	10	2	no	—
199	C₂×C₄²	—	10	2	no	—
200	C₂×C₄²	—	10	2	no	—
201	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
202	C₂×C₄²	C₂²×D₅	1	2	yes	—
203	#107	—	5	4	no	—
204	#107	—	10	2	no	—
205	#107	—	20	1	no	—
206	#107	—	20	2	no	—
207	#107	—	20	1	no	—
208	#106	—	10	4	no	—
209	#106	—	20	2	no	—
210	C₄⋊C₈	C₂×Dic₅	1	4	yes	—
211	C₄⋊C₈	C₂×Dic₅	1	8	yes	—
212	#97	C₂×Dic₅	1	8	yes	—
213	C₄×C₈	C₂×Dic₅	1	4	yes	—
214	#95	—	10	2	no	—
215	#95	—	10	2	no	—
216	#95	—	10	2	no	—
217	#95	—	10	2	no	—
218	#95	—	10	2	no	—
219	#95	—	10	2	no	—
220	#95	—	10	2	no	—
221	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
222	C₂²×C₅	D₄×Q₈	1	1	yes	characteristic
223	C₂²×C₅	#550	1	1	yes	characteristic
224	C₂²×C₅	#575	1	1	yes	characteristic
225	C₂²×C₅	#575	1	1	yes	characteristic
226	C₂²×C₅	#576	1	1	yes	characteristic
227	C₂²×C₅	#578	1	1	yes	characteristic
228	C₄×C₅	—	2	4	no	—
229	C₄×C₅	—	2	2	no	—
230	C₄×C₅	—	2	2	no	—
231	C₄×C₅	—	2	4	no	—
232	C₄×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
233	C₄×C₅	#520	1	1	yes	characteristic
234	C₄×C₅	#557	1	1	yes	characteristic
235	C₄×C₅	Q₈²	1	1	yes	characteristic
236	Dic₅	—	4	4	no	—
237	Dic₅	—	8	2	no	—
238	Dic₅	—	8	2	no	—
239	Dic₅	—	8	2	no	—
240	C₂×Q₈	—	4	1	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×Q₈	C₄⋊Dic₅	1	4	yes	—
243	C₂²×C₄	—	10	2	no	—
244	C₂²×C₄	—	10	2	no	—
245	C₂²×C₄	—	10	2	no	—
246	C₂²×C₄	—	10	2	no	—
247	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
249	C₂²×C₄	C₂×D₂₀	1	2	yes	—
250	C₂²×C₄	C₂×D₂₀	1	2	yes	—
251	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	Q₈×D₅	1	2	yes	—
257	C₂×C₈	#713	1	2	yes	—
258	C₂×C₈	C₂²×Dic₅	1	2	yes	—
259	C₂×C₈	C₂²×Dic₅	1	4	yes	—
260	C₂×C₈	C₂²×Dic₅	1	2	yes	—
261	C₄⋊C₄	—	5	4	no	—
262	C₄⋊C₄	—	10	2	no	—
263	C₄⋊C₄	—	10	2	no	—
264	C₄⋊C₄	—	10	2	no	—
265	C₄⋊C₄	—	10	2	no	—
266	C₄⋊C₄	—	10	2	no	—
267	C₄⋊C₄	—	10	2	no	—
268	C₄⋊C₄	—	10	2	no	—
269	C₄⋊C₄	—	20	4	no	—
270	C₄⋊C₄	—	40	2	no	—
271	C₄⋊C₄	C₄⋊Dic₅	1	8	yes	—
272	C₄⋊C₄	C₄⋊Dic₅	1	4	yes	—
273	C₄²	—	20	4	no	—
274	C₄²	—	40	2	no	—
275	C₄²	—	40	2	no	—
276	C₄²	C₂×Dic₁₀	1	2	yes	—
277	C₄²	C₂×Dic₁₀	1	4	yes	—
278	C₄²	C₂²×Dic₅	1	2	yes	—
279	C₄²	C₂²×Dic₅	1	4	yes	—
280	C₂×C₅	Q₈×C₄⋊C₄	1	1	yes	characteristic
281	C₂×C₅	#2911	1	1	yes	characteristic
282	C₂×C₅	#2913	1	1	yes	characteristic
283	C₂×C₅	#3389	1	1	yes	characteristic
284	C₂×C₅	#3390	1	1	yes	characteristic
285	C₂×C₅	#3402	1	1	yes	characteristic
286	C₂×C₅	#3403	1	1	yes	characteristic
287	C₂³	C₂²×D₂₀	1	1	yes	characteristic
288	Q₈	—	4	4	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	10	4	no	—
298	C₂×C₄	—	20	2	no	—
299	C₂×C₄	—	20	2	no	—
300	C₂×C₄	—	20	2	no	—
301	C₂×C₄	—	20	2	no	—
302	C₂×C₄	—	20	2	no	—
303	C₂×C₄	—	20	2	no	—
304	C₂×C₄	—	20	2	no	—
305	C₂×C₄	—	20	2	no	—
306	C₂×C₄	—	20	2	no	—
307	C₂×C₄	—	20	2	no	—
308	C₂×C₄	—	20	2	no	—
309	C₂×C₄	C₂×C₄⋊Dic₅	1	2	yes	—
310	C₂×C₄	C₂×C₄⋊Dic₅	1	4	yes	—
311	C₂×C₄	C₂×C₄⋊Dic₅	1	2	yes	—
312	C₂×C₄	C₂×C₄⋊Dic₅	1	4	yes	—
313	C₂×C₄	C₂²×Dic₁₀	1	1	yes	characteristic
314	C₂×C₄	C₂×Q₈×D₅	1	1	yes	characteristic
315	C₂×C₄	C₂×#713	1	1	yes	characteristic
316	C₂×C₄	#4149	1	1	yes	characteristic
317	C₂×C₄	#4150	1	1	yes	characteristic
318	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
319	C₈	—	2	4	no	—
320	C₈	—	2	2	no	—
321	C₈	Q₈×Dic₅	1	4	yes	—
322	C₅	#22731	1	1	yes	characteristic, complemented, Sylow
323	C₂²	Q₈×D₂₀	1	1	yes	characteristic
324	C₂²	#65911	1	1	yes	characteristic
325	C₂²	#66082	1	1	yes	characteristic
326	C₂²	#66082	1	1	yes	characteristic
327	C₂²	#66086	1	1	yes	characteristic
328	C₂²	#66087	1	1	yes	characteristic
329	C₂²	C₂²×C₄⋊Dic₅	1	1	yes	characteristic
330	C₄	—	2	4	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	2	no	—
333	C₄	—	2	4	no	—
334	C₄	—	20	4	no	—
335	C₄	—	40	2	no	—
336	C₄	—	40	2	no	—
337	C₄	—	40	2	no	—
338	C₄	Q₈×Dic₁₀	1	1	yes	characteristic
339	C₄	#65904	1	1	yes	characteristic
340	C₄	C₂×Q₈×Dic₅	1	1	yes	characteristic
341	C₄	#66161	1	1	yes	characteristic
342	C₂	#10611686	1	1	yes	characteristic
343	C₂	#10611687	1	1	yes	characteristic
344	C₂	#10611710	1	1	yes	characteristic
345	C₂	#10611718	1	1	yes	characteristic
346	C₂	Q₈×C₄⋊Dic₅	1	1	yes	characteristic
347	C₂	#10615483	1	1	yes	characteristic
348	C₂	#10615516	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₂₀
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅
Fitting	C₂×C₅×#446	C₂
socle, prime radical	C₂³×C₅	C₂²×D₄
cosocle	C₂²×C₄×C₅	C₂⁴
hypercenter	C₂×#446	D₅
hypocenter	C₅	#22731
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
11. C₅×#446	C₄	semidirect
71. #3950	Q₈	semidirect
124. #446	Dic₅	semidirect
322. C₅	#22731	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#446	2
#10614193	2
#10601248	2
#10601246	2
#10600609	2
#10600607	2
#10600559	2
C₄×#3950	2
#22731	5