Subgroups — #12065025

Order 1280.

814 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10611008	C₂	1	1	yes	maximal, characteristic
2	C₄×#4053	C₂	1	1	yes	maximal, characteristic
3	#10610178	C₂	1	1	yes	maximal, characteristic
4	C₅×#1451	C₂	1	1	yes	maximal, characteristic, complemented
5	#10601533	C₂	1	1	yes	maximal, characteristic, complemented
6	#10600640	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₈×#168	C₂	1	1	yes	maximal, characteristic, complemented
8	#10598868	C₂	1	2	yes	maximal, complemented
9	#10598865	C₂	1	2	yes	maximal, complemented
10	#10598860	C₂	1	2	yes	maximal, complemented
11	#10598860	C₂	1	2	yes	maximal, complemented
12	C₂×#4053	C₄	1	2	yes	—
13	C₄×#709	C₂²	1	1	yes	characteristic
14	C₅×#431	C₂²	1	1	yes	characteristic
15	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	#65411	C₄	1	2	yes	—
18	#65400	C₄	1	4	yes	—
19	#65214	C₂²	1	1	yes	characteristic
20	C₂×C₄×#168	C₂²	1	1	yes	characteristic
21	C₂×C₄×#168	C₂²	1	1	yes	characteristic
22	#65127	C₄	1	2	yes	—
23	#65120	C₄	1	2	yes	—
24	#65019	C₄	1	4	yes	—
25	#64997	C₄	1	4	yes	—
26	#64977	C₄	1	2	yes	—
27	C₄×#676	C₂²	1	2	yes	—
28	C₄×#676	C₂²	1	2	yes	—
29	C₄×#676	C₂²	1	2	yes	—
30	#64974	C₂²	1	2	yes	—
31	#64974	C₂²	1	2	yes	—
32	C₄×C₈×D₅	C₂²	1	2	yes	—
33	#64968	C₄	1	2	yes	—
34	C₅×#322	C₂²	1	2	yes	—
35	C₅×#322	C₂²	1	2	yes	—
36	#64674	C₂²	1	2	yes	—
37	#64674	C₂²	1	2	yes	—
38	#64674	C₂²	1	2	yes	—
39	C₈×#168	C₂²	1	2	yes	—
40	C₈×#168	C₂²	1	1	yes	characteristic
41	C₈×#168	C₂²	1	1	yes	characteristic
42	C₈×#168	C₂²	1	1	yes	characteristic
43	C₈×#168	C₂²	1	1	yes	characteristic
44	#17335	—	5	1	no	maximal, Sylow
45	C₂×#709	C₂×C₄	1	1	yes	characteristic
46	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
47	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
49	C₄×C₂²⋊D₅	C₂×C₄	1	2	yes	—
50	C₄×C₂²⋊D₅	C₂×C₄	1	2	yes	—
51	#4072	C₂×C₄	1	1	yes	characteristic
52	C₂²×#168	C₂×C₄	1	2	yes	—
53	C₂²×#168	C₂×C₄	1	2	yes	—
54	#4053	—	2	8	no	—
55	C₂×#676	C₂×C₄	1	4	yes	—
56	C₂×#676	C₂×C₄	1	4	yes	—
57	C₂×#676	C₂×C₄	1	4	yes	—
58	C₂×C₈×D₅	C₂×C₄	1	4	yes	—
59	C₄×D₂₀	C₂×C₄	1	1	yes	characteristic
60	C₄×D₂₀	C₂×C₄	1	1	yes	characteristic
61	#4018	C₂×C₄	1	2	yes	—
62	C₄²×D₅	C₂³	1	2	yes	—
63	C₄×Dic₁₀	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₁₀	C₂×C₄	1	1	yes	characteristic
65	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
66	C₅×C₂²⋊C₈	C₂×C₄	1	2	yes	—
67	C₅×#97	C₂×C₄	1	2	yes	—
68	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
69	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
70	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
71	C₄×C₅×C₈	C₂³	1	2	yes	—
72	C₄×C₅×C₈	C₂³	1	2	yes	—
73	C₂²⋊#168	C₂×C₄	1	2	yes	—
74	#3952	C₂×C₄	1	4	yes	—
75	#3952	C₂×C₄	1	4	yes	—
76	#3947	C₂×C₄	1	4	yes	—
77	#3947	C₂×C₄	1	4	yes	—
78	#3947	C₂×C₄	1	4	yes	—
79	#3946	C₂×C₄	1	4	yes	—
80	#3946	C₂×C₄	1	4	yes	—
81	C₈×Dic₅	C₂×C₄	1	4	yes	—
82	C₄⋊#168	C₂×C₄	1	2	yes	—
83	C₄×#168	C₂×C₄	1	2	yes	—
84	C₄×#168	C₂×C₄	1	2	yes	—
85	C₄×#168	C₂³	1	1	yes	characteristic
86	C₄×#168	C₂³	1	1	yes	characteristic
87	C₄×#168	C₂³	1	1	yes	characteristic
88	C₄×#168	C₂³	1	1	yes	characteristic
89	C₄×#168	C₂³	1	2	yes	—
90	#2965	—	5	1	no	—
91	#2965	—	5	1	no	—
92	C₄×#131	—	5	1	no	—
93	#1455	—	5	2	no	—
94	#1453	—	5	2	no	—
95	#1451	—	5	1	no	—
96	#1451	D₅	1	1	yes	characteristic, complemented
97	#1450	—	5	1	no	—
98	C₈×M₁₆	—	5	2	no	—
99	C₈×M₁₆	—	5	2	no	—
100	C₂×C₈²	—	5	1	no	—
101	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₂²⋊D₅	C₄²	1	2	yes	—
105	#709	—	2	4	no	—
106	C₂×D₂₀	C₄²	1	1	yes	characteristic
107	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
108	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
109	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
110	C₂×Dic₁₀	C₄²	1	1	yes	characteristic
111	C₅×M₁₆	—	2	4	no	—
112	C₂×C₅×C₈	—	2	4	no	—
113	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
114	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
115	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
116	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
117	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
118	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
119	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
120	C₄²×C₅	C₂⁴	1	1	yes	characteristic
121	C₂²⋊Dic₅	C₄²	1	2	yes	—
122	#685	C₄²	1	2	yes	—
123	#685	C₄²	1	2	yes	—
124	C₄⋊Dic₅	C₄²	1	1	yes	characteristic
125	C₄⋊Dic₅	C₄²	1	1	yes	characteristic
126	#683	C₄²	1	2	yes	—
127	#683	C₄²	1	2	yes	—
128	C₄×Dic₅	C₂²×C₄	1	2	yes	—
129	C₄×Dic₅	C₂²×C₄	1	2	yes	—
130	C₄×Dic₅	C₂²×C₄	1	2	yes	—
131	C₂×#168	—	2	4	no	—
132	C₂×#168	M₁₆	1	8	yes	complemented
133	C₂×#168	C₂²×C₄	1	2	yes	—
134	C₂×#168	C₂²×C₄	1	4	yes	—
135	C₂×#168	C₂²×C₄	1	2	yes	—
136	C₂×#168	C₂²×C₄	1	2	yes	—
137	C₂×#168	C₂²×C₄	1	2	yes	—
138	#676	—	2	8	no	—
139	#676	—	2	8	no	—
140	#676	—	2	8	no	—
141	C₈×D₅	—	2	8	no	—
142	C₂×#131	—	5	2	no	—
143	C₄×#41	—	5	1	no	—
144	#446	—	5	2	no	—
145	#446	—	5	2	no	—
146	#436	—	5	2	no	—
147	#436	—	5	2	no	—
148	#435	—	5	4	no	—
149	#435	—	5	4	no	—
150	#431	—	5	1	no	—
151	#431	—	5	2	no	—
152	#431	—	5	2	no	—
153	#431	C₂×D₅	1	1	yes	characteristic
154	#405	—	5	2	no	—
155	#405	—	5	4	no	—
156	C₄×M₁₆	—	5	2	no	—
157	C₄×M₁₆	—	5	2	no	—
158	C₄×M₁₆	—	5	2	no	—
159	C₄×M₁₆	C₂×D₅	1	1	yes	characteristic
160	C₂×C₄×C₈	—	5	1	no	—
161	C₂×C₄×C₈	—	5	1	no	—
162	C₂×C₄×C₈	—	5	2	no	—
163	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
164	#322	—	5	2	no	—
165	#322	—	5	2	no	—
166	#322	—	5	2	no	—
167	#322	C₂×D₅	1	2	yes	—
168	#322	C₂×D₅	1	2	yes	—
169	C₈²	—	5	1	no	—
170	C₈²	—	5	1	no	—
171	C₈²	—	5	1	no	—
172	C₈²	—	5	1	no	—
173	C₈²	—	5	2	no	—
174	C₂³×C₅	C₂×C₄²	1	1	yes	characteristic
175	C₂²×D₅	C₂×C₄²	1	2	yes	—
176	C₂×C₄×C₅	—	2	2	no	—
177	C₂×C₄×C₅	—	2	2	no	—
178	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
179	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
182	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
183	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
184	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
185	C₂²⋊D₅	—	2	4	no	—
186	C₂×Dic₅	C₂×C₄²	1	2	yes	—
187	C₂×Dic₅	C₂×C₄²	1	2	yes	—
188	C₂×Dic₅	C₂×C₄²	1	2	yes	—
189	D₂₀	—	2	2	no	—
190	C₄×D₅	—	2	4	no	—
191	C₄×D₅	—	2	4	no	—
192	C₄×D₅	—	2	4	no	—
193	Dic₁₀	—	2	2	no	—
194	C₅×C₈	—	2	4	no	—
195	C₅×C₈	—	2	4	no	—
196	C₅×C₈	—	2	2	no	—
197	C₅×C₈	#131	1	4	yes	complemented
198	#168	—	2	4	no	—
199	#168	—	2	2	no	—
200	#168	C₂×M₁₆	1	4	yes	—
201	#168	C₂×M₁₆	1	4	yes	—
202	#168	#131	1	4	yes	complemented
203	C₂×#41	—	5	1	no	—
204	#131	—	10	8	no	—
205	C₂×M₁₆	—	5	4	no	—
206	C₂×M₁₆	—	5	4	no	—
207	C₂×M₁₆	—	5	4	no	—
208	C₂×M₁₆	C₄×D₅	1	2	yes	—
209	C₂²×C₈	—	5	2	no	—
210	C₂²×C₈	—	5	2	no	—
211	C₂²×C₈	—	5	4	no	—
212	C₂²×C₈	C₄×D₅	1	2	yes	—
213	C₄×Q₈	—	5	1	no	—
214	C₄×Q₈	—	5	1	no	—
215	C₄×D₄	—	5	2	no	—
216	C₄×D₄	—	5	2	no	—
217	C₄×D₄	—	5	1	no	—
218	C₄×D₄	—	5	1	no	—
219	#117	—	5	1	no	—
220	#117	—	5	2	no	—
221	C₂×C₄²	—	5	2	no	—
222	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
223	C₄⋊C₈	—	5	4	no	—
224	C₄⋊C₈	—	5	4	no	—
225	C₄⋊C₈	—	5	2	no	—
226	C₄⋊C₈	C₄×D₅	1	2	yes	—
227	C₂²⋊C₈	—	5	2	no	—
228	C₂²⋊C₈	—	5	4	no	—
229	C₂²⋊C₈	—	5	4	no	—
230	C₂²⋊C₈	C₄×D₅	1	2	yes	—
231	#97	—	5	4	no	—
232	#97	—	5	4	no	—
233	#97	—	5	4	no	—
234	#97	C₄×D₅	1	2	yes	—
235	C₄×C₈	—	5	4	no	—
236	C₄×C₈	—	5	2	no	—
237	C₄×C₈	—	5	2	no	—
238	C₄×C₈	—	5	1	no	—
239	C₄×C₈	—	5	1	no	—
240	C₄×C₈	—	5	1	no	—
241	C₄×C₈	—	5	1	no	—
242	C₄×C₈	—	5	2	no	—
243	C₄×C₈	C₄×D₅	1	2	yes	—
244	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
246	C₄×C₈	C₂²×D₅	1	2	yes	—
247	C₄×C₈	C₂²×D₅	1	2	yes	—
248	C₂²×C₅	—	2	1	no	—
249	C₂²×C₅	—	2	1	no	—
250	C₂²×C₅	C₄×M₁₆	1	2	yes	—
251	C₂²×C₅	C₂²×C₄²	1	1	yes	characteristic
252	C₂×D₅	—	2	2	no	—
253	C₂×D₅	—	2	2	no	—
254	C₂×D₅	—	2	2	no	—
255	C₄×C₅	—	2	1	no	—
256	C₄×C₅	—	2	1	no	—
257	C₄×C₅	C₄×M₁₆	1	2	yes	—
258	C₄×C₅	C₂²×M₁₆	1	2	yes	—
259	C₄×C₅	C₂×#131	1	2	yes	—
260	C₄×C₅	C₂×#131	1	2	yes	—
261	Dic₅	—	2	2	no	—
262	Dic₅	—	2	2	no	—
263	Dic₅	—	2	2	no	—
264	#41	—	10	4	no	—
265	C₂×Q₈	—	5	1	no	—
266	C₂×D₄	—	5	2	no	—
267	C₂×D₄	—	5	1	no	—
268	C₂²×C₄	—	5	2	no	—
269	C₂²×C₄	—	5	2	no	—
270	C₂²×C₄	—	5	2	no	—
271	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
272	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
273	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
274	M₁₆	—	2	4	no	—
275	M₁₆	—	10	8	no	—
276	M₁₆	—	10	8	no	—
277	M₁₆	—	10	8	no	—
278	C₂×C₈	—	2	4	no	—
279	C₂×C₈	—	5	2	no	—
280	C₂×C₈	—	5	2	no	—
281	C₂×C₈	—	5	2	no	—
282	C₂×C₈	—	5	8	no	—
283	C₂×C₈	—	5	2	no	—
284	C₂×C₈	—	5	4	no	—
285	C₂×C₈	—	10	8	no	—
286	C₂×C₈	—	10	4	no	—
287	C₂×C₈	C₂×C₄×D₅	1	4	yes	—
288	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
289	C₂×C₈	C₂×C₄×D₅	1	4	yes	—
290	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
291	C₄⋊C₄	—	5	2	no	—
292	C₄⋊C₄	—	5	2	no	—
293	C₄⋊C₄	—	5	1	no	—
294	C₄⋊C₄	—	5	1	no	—
295	C₂²⋊C₄	—	5	2	no	—
296	C₂²⋊C₄	—	5	2	no	—
297	C₂²⋊C₄	—	5	2	no	—
298	C₄²	—	5	2	no	—
299	C₄²	—	5	2	no	—
300	C₄²	—	5	2	no	—
301	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
302	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
303	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
304	C₄²	C₂³×D₅	1	1	yes	characteristic
305	C₂×C₅	—	2	2	no	—
306	C₂×C₅	C₂×C₄×M₁₆	1	1	yes	characteristic
307	C₂×C₅	C₄×#131	1	1	yes	characteristic
308	C₂×C₅	C₄×#131	1	1	yes	characteristic
309	D₅	—	4	2	no	—
310	C₂³	—	5	2	no	—
311	C₂³	C₄²×D₅	1	1	yes	characteristic
312	Q₈	—	10	2	no	—
313	D₄	—	10	4	no	—
314	D₄	—	10	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	5	2	no	—
318	C₂×C₄	—	5	2	no	—
319	C₂×C₄	—	5	2	no	—
320	C₂×C₄	—	10	4	no	—
321	C₂×C₄	—	10	4	no	—
322	C₂×C₄	—	10	4	no	—
323	C₂×C₄	C₄²×D₅	1	1	yes	characteristic
324	C₂×C₄	C₄²×D₅	1	1	yes	characteristic
325	C₂×C₄	C₄²×D₅	1	1	yes	characteristic
326	C₂×C₄	D₅×M₁₆	1	4	yes	—
327	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
328	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
329	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
330	C₈	—	2	4	no	—
331	C₈	—	2	2	no	—
332	C₈	—	2	4	no	—
333	C₈	—	5	4	no	—
334	C₈	—	5	4	no	—
335	C₈	—	5	4	no	—
336	C₈	—	10	4	no	—
337	C₈	—	10	2	no	—
338	C₈	#4053	1	4	yes	complemented
339	C₅	#17335	1	1	yes	characteristic, complemented, Sylow
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	10	2	no	—
343	C₂²	—	10	2	no	—
344	C₂²	—	10	2	no	—
345	C₂²	#64996	1	2	yes	—
346	C₂²	C₂×C₄²×D₅	1	1	yes	characteristic
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	10	2	no	—
350	C₄	—	10	2	no	—
351	C₄	—	10	2	no	—
352	C₄	#64996	1	2	yes	—
353	C₄	C₂×D₅×M₁₆	1	2	yes	—
354	C₄	C₂×#4053	1	2	yes	—
355	C₄	D₅×#131	1	2	yes	—
356	C₂	—	2	2	no	—
357	C₂	—	20	2	no	—
358	C₂	C₂×#64996	1	1	yes	characteristic
359	C₂	C₄×#4053	1	1	yes	characteristic
360	C₂	#10610280	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₅
commutator, socle	C₂²×C₅	C₂²×C₄²
Fitting	C₅×#1451	C₂
prime radical	C₂×#709	C₂×C₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	#1451	D₅
hypocenter	C₅	#17335
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₄² ◁ #1451

Complements

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

N	H ≅ G/N	type
4. C₅×#1451	C₂	semidirect
5. #10601533	C₂	semidirect
6. #10600640	C₂	semidirect
7. C₂×C₈×#168	C₂	semidirect
8. #10598868	C₂	semidirect
9. #10598865	C₂	semidirect
10. #10598860	C₂	semidirect
11. #10598860	C₂	semidirect
96. #1451	D₅	semidirect
132. C₂×#168	M₁₆	semidirect
197. C₅×C₈	#131	semidirect
202. #168	#131	semidirect
338. C₈	#4053	semidirect
339. C₅	#17335	semidirect

Maximal subgroups

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

subgroup	index
#10611008	2
C₄×#4053	2
#10610178	2
C₅×#1451	2
#10601533	2
#10600640	2
C₂×C₈×#168	2
#10598868	2
#10598865	2
#10598860	2
#17335	5