Subgroups — #12252413

Order 1280.

590 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10610280	C₂	1	1	yes	maximal, characteristic
2	C₅×#2108	C₂	1	1	yes	maximal, characteristic, complemented
3	#10602225	C₂	1	1	yes	maximal, characteristic
4	#10602213	C₂	1	1	yes	maximal, characteristic, complemented
5	#10602202	C₂	1	1	yes	maximal, characteristic
6	C₂×#64721	C₂	1	1	yes	maximal, characteristic, complemented
7	#10600674	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599867	C₂	1	2	yes	maximal, complemented
9	#10599866	C₂	1	2	yes	maximal, complemented
10	#10599846	C₂	1	2	yes	maximal, complemented
11	#10599845	C₂	1	2	yes	maximal, complemented
12	C₂×#4053	C₄	1	1	yes	characteristic
13	C₂×#4047	C₂²	1	1	yes	characteristic
14	#65856	C₄	1	1	yes	characteristic
15	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
16	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
17	C₅×#405	C₂²	1	1	yes	characteristic
18	M₁₆×Dic₅	C₄	1	1	yes	characteristic
19	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
20	C₂×#3944	C₂²	1	1	yes	characteristic
21	C₂²×#672	C₂²	1	1	yes	characteristic
22	#65286	C₄	1	1	yes	characteristic
23	#64997	C₄	1	2	yes	—
24	#64996	C₄	1	2	yes	—
25	#64978	C₂²	1	2	yes	—
26	C₄×C₈×D₅	C₂²	1	2	yes	—
27	C₅×#346	C₂²	1	2	yes	—
28	C₄×C₅×C₁₆	C₂²	1	2	yes	—
29	#64727	C₂²	1	2	yes	—
30	#64727	C₂²	1	2	yes	—
31	#64727	C₂²	1	2	yes	—
32	#64727	C₂²	1	2	yes	—
33	#64721	C₂²	1	1	yes	characteristic
34	#64721	C₂²	1	2	yes	—
35	#64721	C₂²	1	1	yes	characteristic
36	#64721	C₂²	1	2	yes	—
37	#64721	C₂²	1	1	yes	characteristic
38	#64721	C₂²	1	1	yes	characteristic
39	#64681	C₂²	1	2	yes	—
40	C₄×#672	C₂²	1	2	yes	—
41	#33423	—	5	1	no	maximal, Sylow
42	C₂×#709	C₂×C₄	1	1	yes	characteristic
43	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
45	C₅×#117	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
47	C₂×#681	C₂×C₄	1	1	yes	characteristic
48	C₂²×#168	C₂×C₄	1	1	yes	characteristic
49	#4053	—	2	4	no	—
50	#4047	—	2	2	no	—
51	#4047	—	2	2	no	—
52	C₂×#676	C₂×C₄	1	2	yes	—
53	C₂×#676	C₂×C₄	1	2	yes	—
54	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
55	C₂×C₈×D₅	C₂³	1	2	yes	—
56	#4039	C₈	1	1	yes	characteristic
57	#4039	C₈	1	1	yes	characteristic
58	#4033	C₈	1	1	yes	characteristic
59	#4033	C₈	1	1	yes	characteristic
60	#4027	C₈	1	2	yes	—
61	#4027	C₈	1	2	yes	—
62	#4018	C₂×C₄	1	2	yes	—
63	C₄²×D₅	C₂×C₄	1	2	yes	—
64	C₅×M₃₂	—	2	2	no	—
65	C₂×C₅×C₁₆	—	2	2	no	—
66	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
67	C₂×C₅×C₁₆	C₂³	1	2	yes	—
68	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
69	C₅×#97	C₂×C₄	1	2	yes	—
70	C₄×C₅×C₈	C₂³	1	2	yes	—
71	#3947	C₂×C₄	1	2	yes	—
72	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
73	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
74	C₈×Dic₅	C₂×C₄	1	2	yes	—
75	C₈×Dic₅	C₂³	1	1	yes	characteristic
76	C₈×Dic₅	C₂³	1	1	yes	characteristic
77	#3944	—	2	2	no	—
78	C₂×#672	—	2	2	no	—
79	C₂×#672	C₂³	1	1	yes	characteristic
80	C₂×#672	C₂³	1	1	yes	characteristic
81	C₂×#672	C₂³	1	2	yes	—
82	#3935	C₂×C₄	1	2	yes	—
83	C₄×#168	C₂×C₄	1	2	yes	—
84	C₄×#131	—	5	1	no	—
85	#2164	—	5	2	no	—
86	#2164	—	5	2	no	—
87	#2163	—	5	2	no	—
88	#2162	—	5	2	no	—
89	#2150	—	5	1	no	—
90	C₂×C₄⋊C₁₆	—	5	1	no	—
91	#2113	—	5	1	no	—
92	#2113	—	5	1	no	—
93	#2108	—	5	1	no	—
94	#2108	D₅	1	1	yes	characteristic, complemented
95	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₂²⋊D₅	C₂×C₈	1	2	yes	—
97	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
98	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
99	#709	—	2	2	no	—
100	#709	—	2	2	no	—
101	C₂×D₂₀	C₂×C₈	1	1	yes	characteristic
102	C₂×C₄×D₅	C₂×C₈	1	2	yes	—
103	C₂×C₄×D₅	C₂×C₈	1	2	yes	—
104	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
105	C₂×Dic₁₀	C₂×C₈	1	1	yes	characteristic
106	C₅×M₁₆	M₁₆	1	4	yes	—
107	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
109	C₂×C₅×C₈	#41	1	2	yes	—
110	C₂×C₅×C₈	#41	1	2	yes	—
111	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
112	C₅×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
113	C₅×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
114	C₅×C₂²⋊C₄	C₂×C₈	1	2	yes	—
115	C₄²×C₅	C₂²×C₄	1	2	yes	—
116	#685	C₂×C₈	1	2	yes	—
117	#685	C₂×C₈	1	2	yes	—
118	#683	C₂×C₈	1	2	yes	—
119	#683	C₂×C₈	1	2	yes	—
120	C₄×Dic₅	C₂×C₈	1	2	yes	—
121	C₄×Dic₅	C₂×C₈	1	2	yes	—
122	C₄×Dic₅	C₂×C₈	1	1	yes	characteristic
123	C₄×Dic₅	C₂×C₈	1	1	yes	characteristic
124	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
125	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
126	#681	—	2	2	no	—
127	C₂×#168	—	2	2	no	—
128	C₂×#168	C₂²×C₄	1	1	yes	characteristic
129	C₂×#168	C₂²×C₄	1	2	yes	—
130	C₂×#168	C₂²×C₄	1	1	yes	characteristic
131	#676	—	2	4	no	—
132	#676	—	2	4	no	—
133	C₈×D₅	—	2	4	no	—
134	C₈×D₅	—	2	2	no	—
135	C₈×D₅	—	2	2	no	—
136	C₅×C₁₆	—	2	1	no	—
137	C₅×C₁₆	—	2	1	no	—
138	C₅×C₁₆	—	2	2	no	—
139	#672	—	2	1	no	—
140	#672	—	2	1	no	—
141	#672	—	2	2	no	—
142	C₂×#131	—	5	1	no	—
143	C₂×#131	—	5	1	no	—
144	C₄×#41	—	5	1	no	—
145	C₂×M₃₂	—	5	1	no	—
146	C₂×M₃₂	C₂×D₅	1	1	yes	characteristic
147	C₂²×C₁₆	—	5	1	no	—
148	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
149	#405	—	5	1	no	—
150	#405	—	5	2	no	—
151	#405	—	5	2	no	—
152	#405	C₂×D₅	1	1	yes	characteristic
153	C₄×M₁₆	—	5	1	no	—
154	C₄×M₁₆	—	5	2	no	—
155	C₂×C₄×C₈	—	5	1	no	—
156	C₂×C₄×C₈	—	5	2	no	—
157	C₄⋊C₁₆	—	5	1	no	—
158	C₄⋊C₁₆	—	5	1	no	—
159	C₄⋊C₁₆	—	5	1	no	—
160	C₄⋊C₁₆	—	5	1	no	—
161	C₄⋊C₁₆	—	5	2	no	—
162	C₄⋊C₁₆	—	5	2	no	—
163	C₂²⋊C₁₆	—	5	2	no	—
164	C₂²⋊C₁₆	—	5	2	no	—
165	C₂²⋊C₁₆	—	5	2	no	—
166	C₂²⋊C₁₆	—	5	2	no	—
167	#346	—	5	2	no	—
168	#346	C₂×D₅	1	2	yes	—
169	C₄×C₁₆	—	5	2	no	—
170	C₄×C₁₆	C₂×D₅	1	2	yes	—
171	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
172	C₂²×D₅	C₂²×C₈	1	2	yes	—
173	C₂×C₄×C₅	#117	1	2	yes	—
174	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
175	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
176	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
178	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
179	C₂²⋊D₅	—	2	4	no	—
180	C₂×Dic₅	—	2	2	no	—
181	C₂×Dic₅	—	4	1	no	—
182	C₂×Dic₅	C₂²×C₈	1	1	yes	characteristic
183	C₂×Dic₅	C₂²×C₈	1	2	yes	—
184	C₂×Dic₅	C₂²×C₈	1	1	yes	characteristic
185	C₂×Dic₅	C₂²×C₈	1	1	yes	characteristic
186	C₂×Dic₅	C₂²×C₈	1	1	yes	characteristic
187	D₂₀	—	2	2	no	—
188	C₄×D₅	—	2	2	no	—
189	C₄×D₅	—	2	2	no	—
190	C₄×D₅	—	2	4	no	—
191	C₄×D₅	—	4	2	no	—
192	Dic₁₀	—	2	2	no	—
193	C₅×C₈	#117	1	2	yes	—
194	C₅×C₈	C₂×M₁₆	1	4	yes	—
195	C₅×C₈	C₂×#41	1	1	yes	characteristic
196	C₅×C₈	C₂×#41	1	1	yes	characteristic
197	#168	—	2	2	no	—
198	#168	—	2	1	no	—
199	#168	—	2	1	no	—
200	C₂×#41	—	5	1	no	—
201	#131	—	10	4	no	—
202	#131	—	10	2	no	—
203	#131	—	10	2	no	—
204	C₂×M₁₆	—	5	1	no	—
205	C₂×M₁₆	—	5	2	no	—
206	C₂×M₁₆	—	5	2	no	—
207	C₂×M₁₆	C₄×D₅	1	1	yes	characteristic
208	C₂²×C₈	—	5	1	no	—
209	C₂²×C₈	—	5	2	no	—
210	C₂²×C₈	—	5	2	no	—
211	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
212	C₄×Q₈	—	5	1	no	—
213	C₄×Q₈	—	5	1	no	—
214	C₄×D₄	—	5	1	no	—
215	C₄×D₄	—	5	1	no	—
216	C₄×D₄	—	5	2	no	—
217	C₄×D₄	—	5	2	no	—
218	#117	—	5	2	no	—
219	#117	C₄×D₅	1	1	yes	characteristic
220	C₂×C₄²	—	5	1	no	—
221	C₂×C₄²	—	5	2	no	—
222	M₃₂	—	2	2	no	—
223	M₃₂	—	10	2	no	—
224	C₂×C₁₆	—	2	2	no	—
225	C₂×C₁₆	—	5	1	no	—
226	C₂×C₁₆	—	5	2	no	—
227	C₂×C₁₆	—	5	1	no	—
228	C₂×C₁₆	—	10	2	no	—
229	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
230	C₂×C₁₆	C₂²×D₅	1	2	yes	—
231	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
232	#97	—	5	2	no	—
233	#97	—	5	2	no	—
234	#97	C₄×D₅	1	2	yes	—
235	C₄×C₈	—	5	1	no	—
236	C₄×C₈	—	5	2	no	—
237	C₄×C₈	—	5	1	no	—
238	C₄×C₈	—	5	1	no	—
239	C₄×C₈	—	5	1	no	—
240	C₄×C₈	—	5	2	no	—
241	C₄×C₈	C₂²×D₅	1	2	yes	—
242	C₂²×C₅	—	2	1	no	—
243	C₂²×C₅	—	2	1	no	—
244	C₂²×C₅	#431	1	2	yes	—
245	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
246	C₂×D₅	—	2	2	no	—
247	C₂×D₅	—	2	2	no	—
248	C₂×D₅	—	2	2	no	—
249	C₄×C₅	—	2	2	no	—
250	C₄×C₅	—	2	2	no	—
251	C₄×C₅	#431	1	2	yes	—
252	C₄×C₅	C₂×#117	1	1	yes	characteristic
253	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
254	Dic₅	—	2	1	no	—
255	Dic₅	—	2	1	no	—
256	Dic₅	—	2	1	no	—
257	Dic₅	—	2	2	no	—
258	Dic₅	#504	1	2	yes	—
259	#41	—	10	2	no	—
260	#41	—	10	2	no	—
261	C₂×Q₈	—	5	1	no	—
262	C₂×D₄	—	5	2	no	—
263	C₂×D₄	—	5	1	no	—
264	C₂²×C₄	—	5	1	no	—
265	C₂²×C₄	—	5	1	no	—
266	C₂²×C₄	—	5	2	no	—
267	C₂²×C₄	—	5	2	no	—
268	C₂²×C₄	—	5	2	no	—
269	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
270	M₁₆	—	10	2	no	—
271	M₁₆	—	10	4	no	—
272	M₁₆	—	10	4	no	—
273	M₁₆	#676	1	4	yes	—
274	C₂×C₈	—	5	1	no	—
275	C₂×C₈	—	5	2	no	—
276	C₂×C₈	—	5	1	no	—
277	C₂×C₈	—	10	2	no	—
278	C₂×C₈	—	10	2	no	—
279	C₂×C₈	—	10	2	no	—
280	C₂×C₈	—	10	4	no	—
281	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
282	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
283	C₂×C₈	#709	1	2	yes	—
284	C₂×C₈	#709	1	2	yes	—
285	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
286	C₄⋊C₄	—	5	2	no	—
287	C₄⋊C₄	—	5	2	no	—
288	C₄⋊C₄	C₈×D₅	1	1	yes	characteristic
289	C₄⋊C₄	C₈×D₅	1	1	yes	characteristic
290	C₂²⋊C₄	—	5	2	no	—
291	C₂²⋊C₄	—	5	2	no	—
292	C₂²⋊C₄	C₈×D₅	1	2	yes	—
293	C₄²	—	5	1	no	—
294	C₄²	—	5	1	no	—
295	C₄²	—	5	1	no	—
296	C₄²	—	5	2	no	—
297	C₄²	—	5	1	no	—
298	C₄²	—	5	2	no	—
299	C₄²	C₂×C₄×D₅	1	2	yes	—
300	C₁₆	—	2	1	no	—
301	C₁₆	—	2	1	no	—
302	C₁₆	—	2	2	no	—
303	C₁₆	—	10	1	no	—
304	C₁₆	—	10	1	no	—
305	C₁₆	—	10	2	no	—
306	C₂×C₅	—	2	2	no	—
307	C₂×C₅	C₂×#431	1	1	yes	characteristic
308	C₂×C₅	C₂×#504	1	1	yes	characteristic
309	C₂×C₅	#3407	1	1	yes	characteristic
310	D₅	—	4	2	no	—
311	C₂³	—	5	2	no	—
312	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
313	Q₈	—	10	2	no	—
314	D₄	—	10	4	no	—
315	D₄	—	10	2	no	—
316	C₂×C₄	—	5	1	no	—
317	C₂×C₄	—	5	1	no	—
318	C₂×C₄	—	5	1	no	—
319	C₂×C₄	—	5	1	no	—
320	C₂×C₄	—	5	2	no	—
321	C₂×C₄	—	10	4	no	—
322	C₂×C₄	—	10	2	no	—
323	C₂×C₄	—	10	2	no	—
324	C₂×C₄	—	10	2	no	—
325	C₂×C₄	—	20	2	no	—
326	C₂×C₄	—	20	1	no	—
327	C₂×C₄	#4018	1	2	yes	—
328	C₂×C₄	C₂×C₈×D₅	1	2	yes	—
329	C₂×C₄	C₂×C₈×D₅	1	2	yes	—
330	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
331	C₂×C₄	C₂×#676	1	2	yes	—
332	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
333	C₈	—	10	2	no	—
334	C₈	—	10	1	no	—
335	C₈	—	10	1	no	—
336	C₈	#4018	1	2	yes	—
337	C₈	C₂×#676	1	4	yes	—
338	C₈	C₂×#709	1	1	yes	characteristic
339	C₈	C₂×#709	1	1	yes	characteristic
340	C₅	#33423	1	1	yes	characteristic, complemented, Sylow
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	10	2	no	—
344	C₂²	—	10	2	no	—
345	C₂²	—	10	2	no	—
346	C₂²	#64974	1	2	yes	—
347	C₂²	C₂²×C₈×D₅	1	1	yes	characteristic
348	C₄	—	2	2	no	—
349	C₄	—	2	2	no	—
350	C₄	—	5	2	no	—
351	C₄	—	10	2	no	—
352	C₄	—	10	1	no	—
353	C₄	—	10	1	no	—
354	C₄	—	10	1	no	—
355	C₄	#64974	1	2	yes	—
356	C₄	C₂×#4018	1	1	yes	characteristic
357	C₄	C₂²×#676	1	1	yes	characteristic
358	C₂	—	2	2	no	—
359	C₂	—	20	2	no	—
360	C₂	C₂×#64974	1	1	yes	characteristic
361	C₂	D₅×#504	1	1	yes	characteristic
362	C₂	#10613743	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₈
Fitting	C₅×#2108	C₂
prime radical	C₂×#709	C₂×C₄
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	#2108	D₅
hypocenter	C₅	#33423
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ #2108

Complements

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

N	H ≅ G/N	type
2. C₅×#2108	C₂	semidirect
4. #10602213	C₂	semidirect
6. C₂×#64721	C₂	semidirect
7. #10600674	C₂	semidirect
8. #10599867	C₂	semidirect
9. #10599866	C₂	semidirect
10. #10599846	C₂	semidirect
11. #10599845	C₂	semidirect
94. #2108	D₅	semidirect
340. C₅	#33423	semidirect

Maximal subgroups

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

subgroup	index
#10610280	2
C₅×#2108	2
#10602225	2
#10602213	2
#10602202	2
C₂×#64721	2
#10600674	2
#10599867	2
#10599866	2
#10599846	2
#10599845	2
#33423	5