Subgroups — #12005207

Order 1280.

806 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	C₂²×C₅×#108	C₂	1	1	yes	maximal, characteristic
2	C₂²×#3951	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
4	C₂×#64778	C₂	1	2	yes	maximal
5	#10602772	C₂	1	2	yes	maximal
6	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
7	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄×Dic₅	C₂²	1	1	yes	characteristic
9	C₂²×#681	C₂²	1	2	yes	—
10	C₂³×#168	C₂²	1	1	yes	characteristic
11	C₂×C₅×#108	—	2	2	no	—
12	C₂×C₅×#108	—	2	2	no	—
13	C₂×C₅×#108	C₄	1	4	yes	complemented
14	C₂×#3951	—	2	2	no	—
15	C₂×#3951	—	2	2	no	—
16	C₂×#3951	C₄	1	4	yes	complemented
17	C₂×C₈×Dic₅	—	2	2	no	—
18	C₂×C₈×Dic₅	—	2	2	no	—
19	C₂×C₈×Dic₅	C₄	1	4	yes	—
20	#64778	—	2	4	no	—
21	#64770	—	2	4	no	—
22	#10864	—	5	1	no	maximal, Sylow
23	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₂³×Dic₅	Q₈	1	2	yes	—
25	C₂×C₅×M₁₆	—	2	2	no	—
26	C₂×C₅×M₁₆	—	2	2	no	—
27	C₂×C₅×M₁₆	—	2	2	no	—
28	C₂×C₅×M₁₆	—	2	2	no	—
29	C₂×C₅×M₁₆	—	2	2	no	—
30	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
31	C₂²×C₅×C₈	—	2	2	no	—
32	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
33	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
34	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
35	C₂²×C₅×C₈	D₄	1	2	yes	—
36	C₂²×C₅×C₈	D₄	1	2	yes	—
37	C₂²×C₅×C₈	D₄	1	2	yes	—
38	C₂²×C₅×C₈	Q₈	1	2	yes	—
39	C₂×C₄×Dic₅	—	2	1	no	—
40	C₂×C₄×Dic₅	—	2	1	no	—
41	C₂×C₄×Dic₅	—	2	1	no	—
42	C₂×C₄×Dic₅	—	2	1	no	—
43	C₂×C₄×Dic₅	C₂×C₄	1	2	yes	—
44	C₂×C₄×Dic₅	D₄	1	2	yes	—
45	C₂×#681	—	2	2	no	—
46	C₂×#681	—	2	2	no	—
47	C₂×#681	—	2	2	no	—
48	C₂×#681	—	2	2	no	—
49	C₂×#681	—	2	2	no	—
50	C₂×#681	C₂×C₄	1	4	yes	—
51	C₂²×#168	—	2	1	no	—
52	C₂²×#168	—	2	1	no	—
53	C₂²×#168	—	2	1	no	—
54	C₂²×#168	—	2	1	no	—
55	C₂²×#168	—	2	2	no	—
56	C₂²×#168	C₂×C₄	1	2	yes	—
57	C₂²×#168	D₄	1	2	yes	—
58	C₂²×#168	Q₈	1	2	yes	—
59	C₅×#108	—	2	8	no	—
60	#3951	—	2	8	no	—
61	C₈×Dic₅	—	2	8	no	—
62	C₂²×#108	—	5	1	no	—
63	C₂²×#108	D₅	1	1	yes	characteristic
64	C₂²×C₄×C₈	—	5	1	no	—
65	C₂×#341	—	5	2	no	—
66	#1728	—	5	2	no	—
67	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
78	C₂²×C₄×C₅	#41	1	1	yes	characteristic
79	C₂²×C₄×C₅	#41	1	1	yes	characteristic
80	C₂²×C₄×C₅	#41	1	1	yes	characteristic
81	C₂²×C₄×C₅	#41	1	1	yes	characteristic
82	C₂²×Dic₅	—	2	2	no	—
83	C₂²×Dic₅	—	2	2	no	—
84	C₂²×Dic₅	—	2	8	no	—
85	C₂²×Dic₅	C₄⋊C₄	1	4	yes	—
86	C₅×M₁₆	—	2	4	no	—
87	C₅×M₁₆	—	2	4	no	—
88	C₅×M₁₆	—	4	2	no	—
89	C₅×M₁₆	—	4	2	no	—
90	C₂×C₅×C₈	—	2	4	no	—
91	C₂×C₅×C₈	—	4	2	no	—
92	C₂×C₅×C₈	—	4	2	no	—
93	C₂×C₅×C₈	C₄²	1	4	yes	—
94	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
95	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
96	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
97	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
98	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
99	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
100	C₄×Dic₅	—	2	2	no	—
101	C₄×Dic₅	—	2	2	no	—
102	C₄×Dic₅	—	2	2	no	—
103	C₄×Dic₅	—	2	2	no	—
104	#681	—	2	4	no	—
105	#681	—	2	4	no	—
106	#681	—	4	2	no	—
107	#681	—	4	2	no	—
108	C₂×#168	—	2	2	no	—
109	C₂×#168	—	2	2	no	—
110	C₂×#168	—	2	2	no	—
111	C₂×#168	—	2	2	no	—
112	C₂×#168	—	2	2	no	—
113	C₂×#168	—	2	2	no	—
114	C₂×#168	—	2	4	no	—
115	C₂×#168	—	4	2	no	—
116	C₂×#168	—	4	2	no	—
117	C₂×#168	C₄⋊C₄	1	4	yes	—
118	C₂²×M₁₆	—	5	2	no	—
119	C₂²×M₁₆	C₂×D₅	1	2	yes	—
120	C₂³×C₈	—	5	1	no	—
121	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
122	C₂²×C₄²	—	5	1	no	—
123	C₂×#108	—	2	2	no	—
124	C₂×#108	—	2	2	no	—
125	C₂×#108	—	5	4	no	—
126	C₂×#108	—	10	2	no	—
127	C₂×#108	—	10	2	no	—
128	C₂×#108	Dic₅	1	4	yes	complemented
129	C₂×C₄×C₈	—	5	4	no	—
130	C₂×C₄×C₈	—	10	2	no	—
131	C₂×C₄×C₈	—	10	2	no	—
132	#341	—	10	4	no	—
133	#336	—	10	4	no	—
134	C₂³×C₅	—	2	4	no	—
135	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂³×C₅	C₄×Q₈	1	1	yes	characteristic
137	C₂³×C₅	C₄×Q₈	1	1	yes	characteristic
138	C₂³×C₅	C₂²⋊Q₈	1	1	yes	characteristic
139	C₂³×C₅	C₂²⋊Q₈	1	1	yes	characteristic
140	C₂³×C₅	C₄⋊Q₈	1	1	yes	characteristic
141	C₂³×C₅	C₄⋊Q₈	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₅	#117	1	1	yes	characteristic
151	C₂×C₄×C₅	#117	1	1	yes	characteristic
152	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
154	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
155	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
156	C₂×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
157	C₂×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
158	C₂×C₄×C₅	#121	1	1	yes	characteristic
159	C₂×C₄×C₅	#121	1	1	yes	characteristic
160	C₂×C₄×C₅	#121	1	1	yes	characteristic
161	C₂×C₄×C₅	#121	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
164	C₂×C₄×C₅	#124	1	1	yes	characteristic
165	C₂×C₄×C₅	#124	1	1	yes	characteristic
166	C₂×C₄×C₅	#125	1	1	yes	characteristic
167	C₂×C₄×C₅	#125	1	1	yes	characteristic
168	C₂×C₄×C₅	#127	1	1	yes	characteristic
169	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
170	C₂×Dic₅	—	2	8	no	—
171	C₂×Dic₅	—	2	8	no	—
172	C₂×Dic₅	—	2	4	no	—
173	C₂×Dic₅	—	2	4	no	—
174	C₂×Dic₅	#108	1	8	yes	complemented
175	C₅×C₈	—	4	4	no	—
176	C₅×C₈	#95	1	8	yes	—
177	#168	—	2	4	no	—
178	#168	—	4	4	no	—
179	C₂³×C₄	—	5	2	no	—
180	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	5	4	no	—
187	C₂×M₁₆	—	10	2	no	—
188	C₂×M₁₆	—	10	2	no	—
189	C₂×M₁₆	—	10	2	no	—
190	C₂×M₁₆	—	10	2	no	—
191	C₂×M₁₆	—	10	2	no	—
192	C₂×M₁₆	C₂×Dic₅	1	4	yes	—
193	C₂²×C₈	—	2	2	no	—
194	C₂²×C₈	—	5	2	no	—
195	C₂²×C₈	—	5	2	no	—
196	C₂²×C₈	—	5	2	no	—
197	C₂²×C₈	—	10	1	no	—
198	C₂²×C₈	—	10	1	no	—
199	C₂²×C₈	—	10	1	no	—
200	C₂²×C₈	—	10	1	no	—
201	C₂²×C₈	—	10	2	no	—
202	C₂²×C₈	Dic₁₀	1	2	yes	—
203	C₂²×C₈	C₄×D₅	1	2	yes	—
204	C₂²×C₈	C₄×D₅	1	2	yes	—
205	C₂²×C₈	D₂₀	1	2	yes	—
206	C₂²×C₈	C₂×Dic₅	1	2	yes	—
207	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
208	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
209	C₂×C₄²	—	5	2	no	—
210	C₂×C₄²	—	5	2	no	—
211	C₂×C₄²	—	10	1	no	—
212	C₂×C₄²	—	10	1	no	—
213	C₂×C₄²	—	10	1	no	—
214	C₂×C₄²	—	10	1	no	—
215	#108	—	2	8	no	—
216	#108	—	10	8	no	—
217	C₄×C₈	—	10	8	no	—
218	C₂²×C₅	—	2	2	no	—
219	C₂²×C₅	—	2	2	no	—
220	C₂²×C₅	—	2	2	no	—
221	C₂²×C₅	—	2	2	no	—
222	C₂²×C₅	—	2	2	no	—
223	C₂²×C₅	—	2	2	no	—
224	C₂²×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
225	C₂²×C₅	#380	1	1	yes	characteristic
226	C₂²×C₅	#384	1	1	yes	characteristic
227	C₂²×C₅	#389	1	1	yes	characteristic
228	C₂²×C₅	#389	1	1	yes	characteristic
229	C₂²×C₅	#391	1	1	yes	characteristic
230	C₂²×C₅	#391	1	1	yes	characteristic
231	C₂²×C₅	C₂×#108	1	4	yes	—
232	C₄×C₅	C₂×#95	1	1	yes	characteristic
233	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
234	C₄×C₅	#380	1	1	yes	characteristic
235	C₄×C₅	#382	1	1	yes	characteristic
236	C₄×C₅	#389	1	1	yes	characteristic
237	C₄×C₅	#389	1	1	yes	characteristic
238	C₄×C₅	#390	1	1	yes	characteristic
239	C₄×C₅	#390	1	1	yes	characteristic
240	Dic₅	—	2	8	no	—
241	C₂⁴	Q₈×D₅	1	1	yes	characteristic
242	C₂²×C₄	—	5	4	no	—
243	C₂²×C₄	—	10	8	no	—
244	C₂²×C₄	—	10	2	no	—
245	C₂²×C₄	—	10	2	no	—
246	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
248	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
249	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
250	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
251	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
252	C₂²×C₄	#711	1	1	yes	characteristic
253	C₂²×C₄	#711	1	1	yes	characteristic
254	C₂²×C₄	Q₈×D₅	1	1	yes	characteristic
255	C₂²×C₄	#713	1	1	yes	characteristic
256	C₂²×C₄	#713	1	1	yes	characteristic
257	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
258	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
259	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
260	M₁₆	—	2	4	no	—
261	M₁₆	—	2	4	no	—
262	M₁₆	—	4	2	no	—
263	M₁₆	—	4	2	no	—
264	M₁₆	—	10	4	no	—
265	M₁₆	—	10	4	no	—
266	M₁₆	—	20	2	no	—
267	M₁₆	—	20	2	no	—
268	C₂×C₈	—	2	4	no	—
269	C₂×C₈	—	4	2	no	—
270	C₂×C₈	—	4	2	no	—
271	C₂×C₈	—	5	4	no	—
272	C₂×C₈	—	10	2	no	—
273	C₂×C₈	—	10	2	no	—
274	C₂×C₈	—	10	2	no	—
275	C₂×C₈	—	10	2	no	—
276	C₂×C₈	—	10	2	no	—
277	C₂×C₈	—	10	2	no	—
278	C₂×C₈	—	10	4	no	—
279	C₂×C₈	—	20	2	no	—
280	C₂×C₈	—	20	2	no	—
281	C₂×C₈	C₄×Dic₅	1	4	yes	—
282	C₂×C₈	#683	1	4	yes	—
283	C₂×C₈	#683	1	4	yes	—
284	C₂×C₈	C₄⋊Dic₅	1	4	yes	—
285	C₂×C₈	#685	1	4	yes	—
286	C₂×C₈	#685	1	4	yes	—
287	C₂×C₈	C₂²⋊Dic₅	1	4	yes	—
288	C₄²	—	10	2	no	—
289	C₄²	—	10	2	no	—
290	C₄²	—	10	2	no	—
291	C₄²	—	10	2	no	—
292	C₂×C₅	—	2	1	no	—
293	C₂×C₅	—	2	1	no	—
294	C₂×C₅	—	2	1	no	—
295	C₂×C₅	—	2	1	no	—
296	C₂×C₅	C₄⋊#95	1	1	yes	characteristic
297	C₂×C₅	C₄×#108	1	2	yes	—
298	C₂×C₅	#1810	1	2	yes	—
299	C₂×C₅	#1848	1	2	yes	—
300	C₂³	—	2	4	no	—
301	C₂³	#4033	1	1	yes	characteristic
302	C₂³	#4034	1	1	yes	characteristic
303	C₂³	D₅×C₄⋊C₄	1	1	yes	characteristic
304	C₂³	#4043	1	1	yes	characteristic
305	C₂³	#4090	1	1	yes	characteristic
306	C₂³	Q₈×Dic₅	1	1	yes	characteristic
307	C₂³	#4092	1	1	yes	characteristic
308	C₂×C₄	—	5	8	no	—
309	C₂×C₄	—	10	8	no	—
310	C₂×C₄	—	10	8	no	—
311	C₂×C₄	—	10	4	no	—
312	C₂×C₄	—	10	4	no	—
313	C₂×C₄	#4033	1	1	yes	characteristic
314	C₂×C₄	#4034	1	1	yes	characteristic
315	C₂×C₄	#4036	1	1	yes	characteristic
316	C₂×C₄	#4036	1	1	yes	characteristic
317	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
318	C₂×C₄	#4038	1	1	yes	characteristic
319	C₂×C₄	#4038	1	1	yes	characteristic
320	C₂×C₄	#4039	1	1	yes	characteristic
321	C₂×C₄	#4039	1	1	yes	characteristic
322	C₂×C₄	#4041	1	1	yes	characteristic
323	C₂×C₄	#4041	1	1	yes	characteristic
324	C₂×C₄	#4043	1	1	yes	characteristic
325	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
326	C₂×C₄	C₂×#683	1	1	yes	characteristic
327	C₂×C₄	C₂×#683	1	1	yes	characteristic
328	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
329	C₂×C₄	C₂×#685	1	1	yes	characteristic
330	C₂×C₄	C₂×#685	1	1	yes	characteristic
331	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
332	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
333	C₂×C₄	#4082	1	1	yes	characteristic
334	C₂×C₄	#4084	1	1	yes	characteristic
335	C₂×C₄	#4084	1	1	yes	characteristic
336	C₂×C₄	#4086	1	1	yes	characteristic
337	C₂×C₄	Q₈×Dic₅	1	1	yes	characteristic
338	C₂×C₄	#4092	1	1	yes	characteristic
339	C₂×C₄	#4093	1	1	yes	characteristic
340	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
341	C₈	—	4	4	no	—
342	C₈	—	10	4	no	—
343	C₈	—	20	4	no	—
344	C₈	#3963	1	8	yes	—
345	C₅	#10864	1	1	yes	characteristic, complemented, Sylow
346	C₂²	—	2	2	no	—
347	C₂²	—	2	2	no	—
348	C₂²	—	2	2	no	—
349	C₂²	—	2	2	no	—
350	C₂²	—	2	2	no	—
351	C₂²	—	2	2	no	—
352	C₂²	D₅×#108	1	4	yes	—
353	C₂²	#65262	1	1	yes	characteristic
354	C₂²	#65263	1	1	yes	characteristic
355	C₂²	C₄⋊C₄×Dic₅	1	1	yes	characteristic
356	C₂²	#65265	1	1	yes	characteristic
357	C₂²	#65274	1	1	yes	characteristic
358	C₂²	#65276	1	1	yes	characteristic
359	C₂²	#65519	1	1	yes	characteristic
360	C₄	—	10	8	no	—
361	C₄	#65262	1	1	yes	characteristic
362	C₄	C₄⋊C₄×Dic₅	1	1	yes	characteristic
363	C₄	#65266	1	1	yes	characteristic
364	C₄	#65274	1	1	yes	characteristic
365	C₄	#65276	1	1	yes	characteristic
366	C₄	#65277	1	1	yes	characteristic
367	C₄	C₂×#3963	1	1	yes	characteristic
368	C₄	#65510	1	1	yes	characteristic
369	C₂	—	2	1	no	—
370	C₂	—	2	1	no	—
371	C₂	—	2	1	no	—
372	C₂	—	2	1	no	—
373	C₂	Dic₅×#108	1	2	yes	—
374	C₂	#10601486	1	2	yes	—
375	C₂	#10601504	1	2	yes	—
376	C₂	#10602356	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅
Fitting	C₂²×C₅×#108	C₂
socle	C₂³×C₅	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅	C₂×Q₈
cosocle	C₂³×C₄×C₅	C₂³
hypercenter	C₂²×#108	D₅
hypocenter	C₅	#10864
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
13. C₂×C₅×#108	C₄	semidirect
16. C₂×#3951	C₄	semidirect
128. C₂×#108	Dic₅	semidirect
174. C₂×Dic₅	#108	semidirect
345. C₅	#10864	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅×#108	2
C₂²×#3951	2
C₂²×C₈×Dic₅	2
C₂×#64778	2
#10602772	2
#10864	5