Subgroups — #12064997

Order 1280.

734 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10610186	C₂	1	1	yes	maximal, characteristic
2	#10610178	C₂	1	1	yes	maximal, characteristic
3	C₈×#709	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#322	C₂	1	1	yes	maximal, characteristic, complemented
5	#10600518	C₂	1	1	yes	maximal, characteristic, complemented
6	#10600517	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×#681	C₂	1	1	yes	maximal, characteristic, complemented
8	#10598867	C₂	1	2	yes	maximal, complemented
9	#10598866	C₂	1	2	yes	maximal, complemented
10	#10598864	C₂	1	2	yes	maximal, complemented
11	#10598860	C₂	1	2	yes	maximal, complemented
12	C₄×#709	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	#65400	C₄	1	4	yes	—
17	#65399	C₄	1	2	yes	—
18	C₈×C₂²⋊D₅	C₄	1	4	yes	—
19	#65391	C₄	1	2	yes	—
20	#65214	C₂²	1	1	yes	characteristic
21	#65214	C₂²	1	1	yes	characteristic
22	C₄×#681	C₂²	1	1	yes	characteristic
23	#64979	C₄	1	4	yes	—
24	#64977	C₄	1	2	yes	—
25	C₄×#676	C₂²	1	2	yes	—
26	C₈×D₂₀	C₄	1	2	yes	—
27	#64974	C₂²	1	2	yes	—
28	#64974	C₂²	1	2	yes	—
29	#64974	C₂²	1	2	yes	—
30	#64974	C₂²	1	2	yes	—
31	C₄×C₈×D₅	C₂²	1	2	yes	—
32	#64968	C₄	1	2	yes	—
33	C₈×Dic₁₀	C₄	1	2	yes	—
34	C₅×#322	C₂²	1	1	yes	characteristic
35	C₅×#322	C₂²	1	1	yes	characteristic
36	C₅×#322	C₂²	1	1	yes	characteristic
37	C₅×#322	C₂²	1	1	yes	characteristic
38	#64675	C₂²	1	2	yes	—
39	#64675	C₂²	1	2	yes	—
40	#64674	C₂²	1	2	yes	—
41	#64674	C₂²	1	2	yes	—
42	C₈×#168	C₂²	1	2	yes	—
43	C₈×#168	C₂²	1	2	yes	—
44	#17331	—	5	1	no	maximal, Sylow
45	C₂×#709	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
47	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
48	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
50	C₄×C₂²⋊D₅	C₂×C₄	1	2	yes	—
51	C₄×C₂²⋊D₅	C₂×C₄	1	2	yes	—
52	#4072	C₂×C₄	1	1	yes	characteristic
53	C₂×#681	C₂×C₄	1	2	yes	—
54	C₂×#676	C₂×C₄	1	4	yes	—
55	C₂×C₈×D₅	C₂×C₄	1	4	yes	—
56	C₄×D₂₀	C₂×C₄	1	1	yes	characteristic
57	C₄×D₂₀	C₂×C₄	1	1	yes	characteristic
58	#4018	C₂×C₄	1	2	yes	—
59	C₄²×D₅	C₂³	1	2	yes	—
60	C₄×Dic₁₀	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₁₀	C₂×C₄	1	1	yes	characteristic
62	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
63	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
64	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
65	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
66	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
67	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
68	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
69	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
70	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
71	C₂²⋊#168	C₂×C₄	1	2	yes	—
72	C₂²⋊#168	C₂×C₄	1	2	yes	—
73	#3952	C₂×C₄	1	4	yes	—
74	#3952	C₂×C₄	1	4	yes	—
75	#3952	C₂×C₄	1	4	yes	—
76	#3952	C₂×C₄	1	4	yes	—
77	#3947	C₂×C₄	1	4	yes	—
78	#3946	C₂×C₄	1	4	yes	—
79	#3946	C₂×C₄	1	4	yes	—
80	#3946	C₂×C₄	1	4	yes	—
81	#3946	C₂×C₄	1	4	yes	—
82	C₈×Dic₅	C₂×C₄	1	4	yes	—
83	C₄⋊#168	C₂×C₄	1	2	yes	—
84	C₄⋊#168	C₂×C₄	1	2	yes	—
85	#3935	C₂×C₄	1	2	yes	—
86	C₄×#168	C₂³	1	2	yes	—
87	C₄×#168	C₂³	1	2	yes	—
88	C₄×#168	C₂³	1	2	yes	—
89	#2965	—	5	1	no	—
90	C₈×#41	—	5	1	no	—
91	#2922	—	5	1	no	—
92	#1454	—	5	2	no	—
93	#1453	—	5	2	no	—
94	#1452	—	5	2	no	—
95	#1451	—	5	1	no	—
96	#1450	—	5	1	no	—
97	C₈×M₁₆	—	5	1	no	—
98	C₈×M₁₆	—	5	2	no	—
99	C₂×#322	D₅	1	1	yes	characteristic, complemented
100	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₂²⋊D₅	C₄²	1	2	yes	—
104	#709	—	2	4	no	—
105	C₂×D₂₀	C₄²	1	1	yes	characteristic
106	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
107	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
108	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
109	C₂×Dic₁₀	C₄²	1	1	yes	characteristic
110	C₂×C₅×C₈	—	2	4	no	—
111	C₂×C₅×C₈	—	2	4	no	—
112	C₂×C₅×C₈	—	4	2	no	—
113	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
114	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
115	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
116	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
117	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
118	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
119	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
120	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
121	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
122	C₄²×C₅	C₂⁴	1	1	yes	characteristic
123	C₂²⋊Dic₅	C₄²	1	2	yes	—
124	#685	C₄²	1	2	yes	—
125	#685	C₄²	1	2	yes	—
126	C₄⋊Dic₅	C₄²	1	1	yes	characteristic
127	C₄⋊Dic₅	C₄²	1	1	yes	characteristic
128	#683	C₄²	1	2	yes	—
129	#683	C₄²	1	2	yes	—
130	C₄×Dic₅	C₂²×C₄	1	2	yes	—
131	C₄×Dic₅	C₂²×C₄	1	2	yes	—
132	C₄×Dic₅	C₂²×C₄	1	2	yes	—
133	#681	—	2	4	no	—
134	C₂×#168	C₂²×C₄	1	4	yes	—
135	C₂×#168	C₂²×C₄	1	4	yes	—
136	C₂×#168	C₂²×C₄	1	4	yes	—
137	#676	—	2	8	no	—
138	C₈×D₅	—	4	4	no	—
139	C₄×#41	—	5	1	no	—
140	#446	—	5	2	no	—
141	C₈×Q₈	—	5	2	no	—
142	#436	—	5	2	no	—
143	#435	—	5	4	no	—
144	C₈×D₄	—	5	4	no	—
145	C₈×D₄	—	5	2	no	—
146	#433	—	5	4	no	—
147	#431	—	5	1	no	—
148	#431	—	5	1	no	—
149	#431	—	5	2	no	—
150	#431	—	5	2	no	—
151	#431	—	5	2	no	—
152	#431	—	5	2	no	—
153	#424	—	5	2	no	—
154	#408	—	5	2	no	—
155	C₄×M₁₆	—	5	1	no	—
156	C₄×M₁₆	—	5	2	no	—
157	C₂×C₄×C₈	—	5	2	no	—
158	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
159	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
160	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
161	#322	—	5	2	no	—
162	#322	—	5	2	no	—
163	#322	—	5	2	no	—
164	#322	—	5	2	no	—
165	#322	C₂×D₅	1	1	yes	characteristic
166	#322	C₂×D₅	1	1	yes	characteristic
167	#322	C₂×D₅	1	1	yes	characteristic
168	#322	C₂×D₅	1	1	yes	characteristic
169	C₈²	—	5	2	no	—
170	C₈²	—	5	2	no	—
171	C₂³×C₅	C₂×C₄²	1	1	yes	characteristic
172	C₂²×D₅	C₂×C₄²	1	2	yes	—
173	C₂×C₄×C₅	—	2	2	no	—
174	C₂×C₄×C₅	—	2	2	no	—
175	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
176	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
178	C₂×C₄×C₅	#131	1	4	yes	—
179	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
182	C₂²⋊D₅	—	2	4	no	—
183	C₂×Dic₅	C₂×C₄²	1	2	yes	—
184	C₂×Dic₅	C₂×C₄²	1	2	yes	—
185	C₂×Dic₅	C₂×C₄²	1	2	yes	—
186	D₂₀	—	2	2	no	—
187	C₄×D₅	—	2	4	no	—
188	C₄×D₅	—	2	4	no	—
189	C₄×D₅	—	2	4	no	—
190	Dic₁₀	—	2	2	no	—
191	C₅×C₈	—	2	2	no	—
192	C₅×C₈	—	2	2	no	—
193	C₅×C₈	—	2	2	no	—
194	C₅×C₈	—	2	2	no	—
195	C₅×C₈	—	2	2	no	—
196	C₅×C₈	#131	1	4	yes	—
197	#168	—	2	4	no	—
198	#168	—	2	4	no	—
199	#168	—	2	4	no	—
200	C₂×#41	—	5	1	no	—
201	C₂×M₁₆	—	5	2	no	—
202	C₂×M₁₆	—	5	4	no	—
203	C₂²×C₈	—	5	4	no	—
204	C₂²×C₈	C₄×D₅	1	2	yes	—
205	C₂²×C₈	C₄×D₅	1	2	yes	—
206	C₂²×C₈	C₄×D₅	1	2	yes	—
207	C₄×Q₈	—	5	1	no	—
208	C₄×Q₈	—	5	1	no	—
209	C₄×D₄	—	5	2	no	—
210	C₄×D₄	—	5	2	no	—
211	C₄×D₄	—	5	1	no	—
212	C₄×D₄	—	5	1	no	—
213	#117	—	5	1	no	—
214	#117	—	5	2	no	—
215	C₂×C₄²	—	5	2	no	—
216	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
217	C₄⋊C₈	—	5	4	no	—
218	C₄⋊C₈	—	5	4	no	—
219	C₄⋊C₈	—	5	4	no	—
220	C₄⋊C₈	—	5	4	no	—
221	C₄⋊C₈	—	5	2	no	—
222	C₄⋊C₈	—	5	2	no	—
223	C₂²⋊C₈	—	5	2	no	—
224	C₂²⋊C₈	—	5	2	no	—
225	C₂²⋊C₈	—	5	4	no	—
226	C₂²⋊C₈	—	5	4	no	—
227	C₂²⋊C₈	—	5	4	no	—
228	C₂²⋊C₈	—	5	4	no	—
229	#97	—	5	4	no	—
230	#97	—	5	2	no	—
231	C₄×C₈	—	5	4	no	—
232	C₄×C₈	—	5	2	no	—
233	C₄×C₈	—	5	2	no	—
234	C₄×C₈	—	5	2	no	—
235	C₄×C₈	C₄×D₅	1	2	yes	—
236	C₄×C₈	C₄×D₅	1	2	yes	—
237	C₄×C₈	C₄×D₅	1	2	yes	—
238	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
239	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
240	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
241	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
242	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
244	C₂²×C₅	—	2	1	no	—
245	C₂²×C₅	—	2	1	no	—
246	C₂²×C₅	#405	1	2	yes	—
247	C₂²×C₅	C₂²×C₄²	1	1	yes	characteristic
248	C₂×D₅	—	2	2	no	—
249	C₂×D₅	—	2	2	no	—
250	C₂×D₅	—	2	2	no	—
251	C₄×C₅	—	2	1	no	—
252	C₄×C₅	—	2	1	no	—
253	C₄×C₅	#405	1	2	yes	—
254	C₄×C₅	C₂×#131	1	2	yes	—
255	C₄×C₅	C₂×#131	1	2	yes	—
256	C₄×C₅	#568	1	2	yes	—
257	Dic₅	—	2	2	no	—
258	Dic₅	—	2	2	no	—
259	Dic₅	—	2	2	no	—
260	#41	—	10	4	no	—
261	C₂×Q₈	—	5	1	no	—
262	C₂×D₄	—	5	2	no	—
263	C₂×D₄	—	5	1	no	—
264	C₂²×C₄	—	5	2	no	—
265	C₂²×C₄	—	5	2	no	—
266	C₂²×C₄	—	5	2	no	—
267	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
268	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
269	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
270	M₁₆	—	10	4	no	—
271	M₁₆	—	10	8	no	—
272	C₂×C₈	—	2	4	no	—
273	C₂×C₈	—	2	4	no	—
274	C₂×C₈	—	4	2	no	—
275	C₂×C₈	—	5	4	no	—
276	C₂×C₈	—	5	4	no	—
277	C₂×C₈	—	5	4	no	—
278	C₂×C₈	—	20	4	no	—
279	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
280	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
281	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
282	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
283	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
284	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
285	C₄⋊C₄	—	5	2	no	—
286	C₄⋊C₄	—	5	2	no	—
287	C₄⋊C₄	—	5	1	no	—
288	C₄⋊C₄	—	5	1	no	—
289	C₂²⋊C₄	—	5	2	no	—
290	C₂²⋊C₄	—	5	2	no	—
291	C₂²⋊C₄	—	5	2	no	—
292	C₄²	—	5	2	no	—
293	C₄²	—	5	2	no	—
294	C₄²	—	5	2	no	—
295	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
296	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
297	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
298	C₄²	C₂³×D₅	1	1	yes	characteristic
299	C₂×C₅	—	2	2	no	—
300	C₂×C₅	C₂×#405	1	1	yes	characteristic
301	C₂×C₅	C₄×#131	1	1	yes	characteristic
302	C₂×C₅	#2875	1	1	yes	characteristic
303	D₅	—	4	2	no	—
304	C₂³	—	5	2	no	—
305	C₂³	C₄²×D₅	1	1	yes	characteristic
306	Q₈	—	10	2	no	—
307	D₄	—	10	4	no	—
308	D₄	—	10	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	5	2	no	—
312	C₂×C₄	—	5	2	no	—
313	C₂×C₄	—	5	2	no	—
314	C₂×C₄	—	10	4	no	—
315	C₂×C₄	—	10	4	no	—
316	C₂×C₄	—	10	4	no	—
317	C₂×C₄	C₄²×D₅	1	1	yes	characteristic
318	C₂×C₄	C₄²×D₅	1	1	yes	characteristic
319	C₂×C₄	C₄²×D₅	1	1	yes	characteristic
320	C₂×C₄	#4053	1	4	yes	—
321	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
322	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
323	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
324	C₈	—	2	2	no	—
325	C₈	—	2	2	no	—
326	C₈	—	2	2	no	—
327	C₈	—	2	2	no	—
328	C₈	—	2	2	no	—
329	C₈	—	10	4	no	—
330	C₈	—	10	4	no	—
331	C₈	—	10	4	no	—
332	C₈	#4047	1	4	yes	—
333	C₅	#17331	1	1	yes	characteristic, complemented, Sylow
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	10	2	no	—
337	C₂²	—	10	2	no	—
338	C₂²	—	10	2	no	—
339	C₂²	#64997	1	2	yes	—
340	C₂²	C₂×C₄²×D₅	1	1	yes	characteristic
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	10	2	no	—
344	C₄	—	10	2	no	—
345	C₄	—	10	2	no	—
346	C₄	#64997	1	2	yes	—
347	C₄	C₂×#4047	1	2	yes	—
348	C₄	C₂×#4053	1	2	yes	—
349	C₄	#66079	1	2	yes	—
350	C₂	—	2	2	no	—
351	C₂	—	20	2	no	—
352	C₂	C₄×#4047	1	1	yes	characteristic
353	C₂	C₂×#64997	1	1	yes	characteristic
354	C₂	#10610217	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₅
commutator, socle	C₂²×C₅	C₂²×C₄²
Fitting	C₂×C₅×#322	C₂
prime radical	C₂×#709	C₂×C₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	C₂×#322	D₅
hypocenter	C₅	#17331
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
4. C₂×C₅×#322	C₂	semidirect
5. #10600518	C₂	semidirect
6. #10600517	C₂	semidirect
7. C₈×#681	C₂	semidirect
8. #10598867	C₂	semidirect
9. #10598866	C₂	semidirect
10. #10598864	C₂	semidirect
11. #10598860	C₂	semidirect
99. C₂×#322	D₅	semidirect
333. C₅	#17331	semidirect

Maximal subgroups

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

subgroup	index
#10610186	2
#10610178	2
C₈×#709	2
C₂×C₅×#322	2
#10600518	2
#10600517	2
C₈×#681	2
#10598867	2
#10598866	2
#10598864	2
#10598860	2
#17331	5