Subgroups — #12005169

Order 1280.

802 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
2	#10613793	C₂	1	1	yes	maximal, characteristic
3	C₂×#65213	C₂	1	1	yes	maximal, characteristic
4	C₂×#64779	C₂	1	4	yes	maximal
5	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
6	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₂×#4072	C₂²	1	2	yes	—
8	C₂²×#681	C₂²	1	2	yes	—
9	C₄×C₅×M₁₆	—	2	1	no	—
10	C₄×C₅×M₁₆	—	2	1	no	—
11	C₄×C₅×M₁₆	C₄	1	2	yes	complemented
12	C₄×C₅×M₁₆	C₄	1	2	yes	complemented
13	C₂×C₅×#97	—	2	1	no	—
14	C₂×C₄×C₅×C₈	—	2	1	no	—
15	#65501	C₄	1	4	yes	—
16	#65225	—	2	1	no	—
17	#65224	—	2	1	no	—
18	C₄×C₄⋊Dic₅	—	2	2	no	—
19	#65213	—	2	2	no	—
20	#65213	—	2	1	no	—
21	#65213	—	2	1	no	—
22	C₂×C₄⋊#168	C₄	1	4	yes	complemented
23	#64779	—	2	8	no	—
24	#10855	—	5	1	no	maximal, Sylow
25	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
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	C₂×C₅×M₁₆	D₄	1	4	yes	—
32	C₂×C₅×M₁₆	Q₈	1	4	yes	—
33	C₂²×C₅×C₈	—	2	2	no	—
34	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₅	D₄	1	2	yes	—
38	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅	D₄	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	Q₈	1	2	yes	—
43	C₂×C₂²⋊Dic₅	C₂×C₄	1	4	yes	—
44	#4072	—	2	4	no	—
45	#4072	—	2	4	no	—
46	C₂×C₄⋊Dic₅	—	2	4	no	—
47	C₂×C₄⋊Dic₅	—	2	2	no	—
48	C₂×C₄×Dic₅	—	2	2	no	—
49	C₂×#681	—	2	4	no	—
50	C₂×#681	—	2	2	no	—
51	C₂×#681	—	2	2	no	—
52	C₂×#681	—	2	4	no	—
53	C₂²×#168	C₂×C₄	1	4	yes	—
54	C₅×#97	—	2	2	no	—
55	C₅×#97	—	2	2	no	—
56	C₄×C₅×C₈	—	2	2	no	—
57	C₄×C₅×C₈	—	2	2	no	—
58	#3963	—	2	4	no	—
59	C₄⋊#168	—	2	4	no	—
60	C₄⋊#168	—	2	4	no	—
61	C₂×#423	—	5	1	no	—
62	C₂×C₄×M₁₆	D₅	1	1	yes	characteristic
63	#2285	—	5	1	no	—
64	C₂×#344	—	5	4	no	—
65	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	—	4	4	no	—
67	C₂²×C₄×C₅	C₄²	1	2	yes	—
68	C₂²×C₄×C₅	C₄²	1	2	yes	—
69	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
81	C₂²×C₄×C₅	#41	1	2	yes	—
82	C₂²×C₄×C₅	#41	1	2	yes	—
83	C₂²×Dic₅	—	2	4	no	—
84	C₅×M₁₆	—	2	4	no	—
85	C₅×M₁₆	—	2	4	no	—
86	C₅×M₁₆	—	2	4	no	—
87	C₅×M₁₆	C₄⋊C₄	1	8	yes	complemented
88	C₂×C₅×C₈	—	2	4	no	—
89	C₂×C₅×C₈	—	2	4	no	—
90	C₂×C₅×C₈	—	2	4	no	—
91	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
92	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
93	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
94	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
95	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
96	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
97	C₄²×C₅	C₄⋊C₄	1	2	yes	—
98	C₄²×C₅	C₄⋊C₄	1	2	yes	—
99	C₄²×C₅	C₄⋊C₄	1	2	yes	—
100	C₄²×C₅	C₄⋊C₄	1	2	yes	—
101	C₂²⋊Dic₅	—	2	8	no	—
102	C₄⋊Dic₅	—	4	8	no	—
103	C₄⋊Dic₅	—	4	4	no	—
104	C₄×Dic₅	—	4	2	no	—
105	C₄×Dic₅	—	4	2	no	—
106	#681	—	4	4	no	—
107	#681	—	4	2	no	—
108	#681	—	4	2	no	—
109	C₂×#168	—	2	4	no	—
110	C₂×#168	—	2	8	no	—
111	C₂²×M₁₆	—	5	2	no	—
112	C₂²×M₁₆	C₂×D₅	1	2	yes	—
113	C₂×#117	—	5	2	no	—
114	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
115	#423	—	10	2	no	—
116	#423	—	10	1	no	—
117	#423	—	10	1	no	—
118	C₂×C₄⋊C₈	—	5	4	no	—
119	C₄×M₁₆	—	2	1	no	—
120	C₄×M₁₆	—	2	1	no	—
121	C₄×M₁₆	Dic₅	1	2	yes	complemented
122	C₄×M₁₆	Dic₅	1	2	yes	complemented
123	C₂×#97	—	2	1	no	—
124	C₂×C₄×C₈	—	2	1	no	—
125	#384	—	10	1	no	—
126	#382	—	10	1	no	—
127	#380	—	5	4	no	—
128	C₄×C₄⋊C₄	—	10	2	no	—
129	#344	—	10	8	no	—
130	C₂³×C₅	—	4	2	no	—
131	C₂³×C₅	C₂×C₄²	1	1	yes	characteristic
132	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂³×C₅	C₄×D₄	1	2	yes	—
135	C₂³×C₅	C₄×Q₈	1	2	yes	—
136	C₂×C₄×C₅	—	2	4	no	—
137	C₂×C₄×C₅	—	2	4	no	—
138	C₂×C₄×C₅	—	2	4	no	—
139	C₂×C₄×C₅	—	4	2	no	—
140	C₂×C₄×C₅	—	4	2	no	—
141	C₂×C₄×C₅	—	4	2	no	—
142	C₂×C₄×C₅	#95	1	2	yes	—
143	C₂×C₄×C₅	#95	1	2	yes	—
144	C₂×C₄×C₅	#95	1	2	yes	—
145	C₂×C₄×C₅	#95	1	2	yes	—
146	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
150	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₅	#117	1	2	yes	—
152	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
153	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
154	C₂×C₄×C₅	#121	1	2	yes	—
155	C₂×C₄×C₅	#121	1	2	yes	—
156	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
157	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
158	C₂×C₄×C₅	#124	1	1	yes	characteristic
159	C₂×C₄×C₅	#124	1	1	yes	characteristic
160	C₂×C₄×C₅	#125	1	2	yes	—
161	C₂×C₄×C₅	#127	1	1	yes	characteristic
162	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
163	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
164	C₂×Dic₅	—	4	8	no	—
165	C₂×Dic₅	—	4	4	no	—
166	C₅×C₈	—	2	8	no	—
167	#168	—	4	4	no	—
168	C₂³×C₄	C₄×D₅	1	2	yes	—
169	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	10	4	no	—
174	C₂×M₁₆	—	10	2	no	—
175	C₂×M₁₆	—	10	4	no	—
176	C₂×M₁₆	—	10	2	no	—
177	C₂×M₁₆	Dic₁₀	1	4	yes	—
178	C₂×M₁₆	D₂₀	1	4	yes	—
179	C₂×M₁₆	C₂×Dic₅	1	4	yes	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	5	4	no	—
182	#117	—	10	4	no	—
183	#117	—	10	4	no	—
184	C₂×C₄⋊C₄	—	10	4	no	—
185	C₂×C₄⋊C₄	—	10	2	no	—
186	C₂×C₂²⋊C₄	—	5	4	no	—
187	C₂×C₄²	—	10	2	no	—
188	C₂×C₄²	Dic₁₀	1	2	yes	—
189	C₂×C₄²	C₄×D₅	1	2	yes	—
190	C₂×C₄²	D₂₀	1	2	yes	—
191	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
192	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
193	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
194	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
195	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
196	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
197	C₄⋊C₈	—	10	4	no	—
198	C₄⋊C₈	—	10	4	no	—
199	#97	—	2	2	no	—
200	#97	—	2	2	no	—
201	C₄×C₈	—	2	2	no	—
202	C₄×C₈	—	2	2	no	—
203	#95	—	10	4	no	—
204	C₂²×C₅	—	2	2	no	—
205	C₂²×C₅	—	2	2	no	—
206	C₂²×C₅	—	2	2	no	—
207	C₂²×C₅	—	4	2	no	—
208	C₂²×C₅	—	4	2	no	—
209	C₂²×C₅	C₂×#95	1	1	yes	characteristic
210	C₂²×C₅	C₄×C₄⋊C₄	1	2	yes	—
211	C₂²×C₅	#389	1	2	yes	—
212	C₂²×C₅	#390	1	1	yes	characteristic
213	C₂²×C₅	#391	1	1	yes	characteristic
214	C₄×C₅	—	2	2	no	—
215	C₄×C₅	—	2	2	no	—
216	C₄×C₅	—	2	2	no	—
217	C₄×C₅	#344	1	4	yes	complemented
218	C₄×C₅	#380	1	2	yes	—
219	C₄×C₅	#382	1	1	yes	characteristic
220	C₄×C₅	#384	1	1	yes	characteristic
221	C₄×C₅	#389	1	2	yes	—
222	C₄×C₅	#390	1	1	yes	characteristic
223	C₄×C₅	#391	1	1	yes	characteristic
224	Dic₅	—	8	4	no	—
225	C₂⁴	C₂×C₄×D₅	1	1	yes	characteristic
226	C₂²×C₄	—	4	4	no	—
227	C₂²×C₄	—	10	4	no	—
228	C₂²×C₄	C₄×Dic₅	1	2	yes	—
229	C₂²×C₄	C₄×Dic₅	1	2	yes	—
230	C₂²×C₄	#683	1	2	yes	—
231	C₂²×C₄	#683	1	2	yes	—
232	C₂²×C₄	#685	1	2	yes	—
233	C₂²×C₄	#685	1	2	yes	—
234	C₂²×C₄	C₂×Dic₁₀	1	2	yes	—
235	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₂₀	1	2	yes	—
238	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
239	C₂²×C₄	#709	1	2	yes	—
240	C₂²×C₄	#709	1	2	yes	—
241	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
244	M₁₆	—	2	4	no	—
245	M₁₆	—	2	4	no	—
246	M₁₆	—	2	4	no	—
247	M₁₆	—	20	2	no	—
248	M₁₆	—	20	2	no	—
249	M₁₆	—	20	4	no	—
250	M₁₆	C₄⋊Dic₅	1	8	yes	complemented
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	—	10	8	no	—
255	C₂×C₈	—	10	4	no	—
256	C₄⋊C₄	—	20	8	no	—
257	C₄⋊C₄	—	20	4	no	—
258	C₂²⋊C₄	—	10	8	no	—
259	C₄²	—	20	2	no	—
260	C₄²	—	20	2	no	—
261	C₄²	#683	1	2	yes	—
262	C₄²	#683	1	2	yes	—
263	C₄²	C₄⋊Dic₅	1	2	yes	—
264	C₄²	C₄⋊Dic₅	1	2	yes	—
265	C₄²	#685	1	2	yes	—
266	C₄²	#685	1	2	yes	—
267	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
268	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
269	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
270	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
271	C₂×C₅	—	2	1	no	—
272	C₂×C₅	—	2	2	no	—
273	C₂×C₅	—	2	2	no	—
274	C₂×C₅	—	2	1	no	—
275	C₂×C₅	C₄⋊#95	1	1	yes	characteristic
276	C₂×C₅	C₂×#344	1	2	yes	—
277	C₂³	—	4	2	no	—
278	C₂³	C₄×Dic₁₀	1	2	yes	—
279	C₂³	C₄×D₂₀	1	2	yes	—
280	C₂³	C₂×C₄×Dic₅	1	1	yes	characteristic
281	C₂³	C₂×#683	1	1	yes	characteristic
282	C₂³	C₂×#685	1	1	yes	characteristic
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	20	8	no	—
290	C₂×C₄	—	20	4	no	—
291	C₂×C₄	#3963	1	2	yes	—
292	C₂×C₄	#3963	1	2	yes	—
293	C₂×C₄	#3963	1	2	yes	—
294	C₂×C₄	#3963	1	2	yes	—
295	C₂×C₄	C₄×Dic₁₀	1	2	yes	—
296	C₂×C₄	#4015	1	2	yes	—
297	C₂×C₄	#4015	1	1	yes	characteristic
298	C₂×C₄	#4016	1	2	yes	—
299	C₂×C₄	C₄×D₂₀	1	2	yes	—
300	C₂×C₄	#4020	1	1	yes	characteristic
301	C₂×C₄	#4021	1	1	yes	characteristic
302	C₂×C₄	#4021	1	1	yes	characteristic
303	C₂×C₄	C₂×#683	1	1	yes	characteristic
304	C₂×C₄	#4070	1	2	yes	—
305	C₂×C₄	#4070	1	2	yes	—
306	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
307	C₂×C₄	C₂×C₄⋊Dic₅	1	2	yes	—
308	C₂×C₄	#4072	1	2	yes	—
309	C₂×C₄	C₂×#685	1	1	yes	characteristic
310	C₂×C₄	#4076	1	2	yes	—
311	C₂×C₄	#4076	1	2	yes	—
312	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
313	C₈	—	2	8	no	—
314	C₈	—	20	4	no	—
315	C₅	#10855	1	1	yes	characteristic, complemented, Sylow
316	C₂²	—	2	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	#65217	1	2	yes	—
322	C₂²	#65218	1	1	yes	characteristic
323	C₂²	C₄×C₄⋊Dic₅	1	2	yes	—
324	C₂²	#65228	1	1	yes	characteristic
325	C₂²	C₂×#3963	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	2	no	—
329	C₄	—	40	4	no	—
330	C₄	#64779	1	4	yes	complemented
331	C₄	#65217	1	2	yes	—
332	C₄	#65218	1	1	yes	characteristic
333	C₄	#65224	1	1	yes	characteristic
334	C₄	#65225	1	1	yes	characteristic
335	C₄	#65228	1	1	yes	characteristic
336	C₄	#65501	1	2	yes	—
337	C₂	—	2	1	no	—
338	C₂	—	2	2	no	—
339	C₂	—	2	1	no	—
340	C₂	—	2	2	no	—
341	C₂	#10602328	1	1	yes	characteristic
342	C₂	C₂×#64779	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#685
commutator	C₂³×C₅	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅
Fitting	C₂×C₄×C₅×M₁₆	C₂
socle	C₂²×C₅	C₂×#95
prime radical	C₂⁴×C₅	C₂²×C₄
cosocle	C₂³×C₄×C₅	C₂³
hypercenter	C₂×C₄×M₁₆	D₅
hypocenter	C₅	#10855
LCS	G ▷ C₂³×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
11. C₄×C₅×M₁₆	C₄	semidirect
12. C₄×C₅×M₁₆	C₄	semidirect
22. C₂×C₄⋊#168	C₄	semidirect
87. C₅×M₁₆	C₄⋊C₄	semidirect
121. C₄×M₁₆	Dic₅	semidirect
122. C₄×M₁₆	Dic₅	semidirect
217. C₄×C₅	#344	semidirect
250. M₁₆	C₄⋊Dic₅	semidirect
315. C₅	#10855	semidirect
330. C₄	#64779	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅×M₁₆	2
#10613793	2
C₂×#65213	2
C₂×#64779	2
#10855	5