Subgroups — #12069819

Order 1280.

682 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10615074	C₂	1	1	yes	maximal, characteristic
2	#10615059	C₂	1	1	yes	maximal, characteristic, complemented
3	#10610772	C₂	1	1	yes	maximal, characteristic
4	C₅×#1493	C₂	1	1	yes	maximal, characteristic, complemented
5	#10601283	C₂	1	1	yes	maximal, characteristic, complemented
6	#10600528	C₂	1	1	yes	maximal, characteristic
7	C₂×C₈×#168	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599017	C₂	1	4	yes	maximal, complemented
9	#10599005	C₂	1	4	yes	maximal, complemented
10	C₅×#533	C₂²	1	1	yes	characteristic
11	C₂×#4053	—	2	1	no	—
12	#65811	C₂²	1	1	yes	characteristic
13	C₅×#423	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	#65418	—	2	1	no	—
16	#65377	C₄	1	1	yes	characteristic
17	#65374	C₄	1	1	yes	characteristic
18	#65370	C₄	1	1	yes	characteristic
19	#65368	C₄	1	1	yes	characteristic
20	#65337	C₄	1	4	yes	—
21	C₄×#688	C₂²	1	4	yes	—
22	#65313	—	2	1	no	—
23	#65313	C₂²	1	2	yes	—
24	Q₈×#168	—	2	1	no	—
25	Q₈×#168	C₂²	1	2	yes	—
26	#65288	—	2	1	no	—
27	#65286	—	2	1	no	—
28	#65213	C₂²	1	1	yes	characteristic
29	C₂×C₄×#168	C₂²	1	1	yes	characteristic
30	C₂×C₄×#168	C₂²	1	1	yes	characteristic
31	#65127	—	2	1	no	—
32	#65127	C₂²	1	2	yes	—
33	#65123	—	2	1	no	—
34	#65123	C₂²	1	2	yes	—
35	C₅×Q₈⋊C₈	C₂²	1	4	yes	—
36	#64700	C₂²	1	4	yes	—
37	#64679	C₂²	1	4	yes	—
38	C₈×#168	C₂²	1	2	yes	—
39	C₈×#168	C₂²	1	2	yes	—
40	#17957	—	5	1	no	maximal, Sylow
41	C₂×#709	—	2	1	no	—
42	C₂×C₅×M₁₆	—	2	1	no	—
43	C₂²×C₅×C₈	—	2	1	no	—
44	C₅×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
45	C₅×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
46	C₅×#125	—	2	1	no	—
47	C₅×C₂²⋊Q₈	—	2	1	no	—
48	C₅×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
49	C₄×C₅×Q₈	—	2	1	no	—
50	C₄×C₅×Q₈	C₂³	1	2	yes	—
51	C₅×#117	—	2	1	no	—
52	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
53	C₂×#688	C₂×C₄	1	4	yes	—
54	#4076	—	2	1	no	—
55	#4076	C₂×C₄	1	2	yes	—
56	C₂×#681	—	2	1	no	—
57	C₂²×#168	C₂×C₄	1	2	yes	—
58	C₂²×#168	D₄	1	1	yes	characteristic
59	C₂²×#168	D₄	1	1	yes	characteristic
60	#4053	—	4	4	no	—
61	C₂×#676	—	2	2	no	—
62	C₂×C₈×D₅	—	2	2	no	—
63	#4021	—	2	1	no	—
64	#4020	C₂×C₄	1	1	yes	characteristic
65	C₄×D₂₀	—	2	1	no	—
66	C₄×D₂₀	C₂³	1	2	yes	—
67	#4015	C₂×C₄	1	1	yes	characteristic
68	C₅×C₄⋊C₈	—	2	1	no	—
69	C₅×C₄⋊C₈	C₂³	1	2	yes	—
70	C₄×C₅×C₈	—	2	1	no	—
71	C₄×C₅×C₈	C₂³	1	2	yes	—
72	#3967	C₂×C₄	1	4	yes	—
73	#3952	—	2	2	no	—
74	#3952	—	2	2	no	—
75	#3941	C₂×C₄	1	4	yes	—
76	#3940	C₂×C₄	1	4	yes	—
77	C₄⋊#168	—	2	1	no	—
78	C₄⋊#168	—	2	2	no	—
79	C₄⋊#168	—	2	2	no	—
80	C₄⋊#168	C₂³	1	2	yes	—
81	#3935	—	2	2	no	—
82	C₄×#168	—	2	2	no	—
83	C₄×#168	C₂×C₄	1	1	yes	characteristic
84	C₄×#168	C₂×C₄	1	1	yes	characteristic
85	C₄×#168	D₄	1	1	yes	characteristic
86	C₄×#168	D₄	1	1	yes	characteristic
87	C₄×#168	C₂³	1	2	yes	—
88	C₄×#168	C₂³	1	1	yes	characteristic
89	C₄×#168	C₂³	1	1	yes	characteristic
90	#3167	—	5	1	no	—
91	#2960	—	5	1	no	—
92	#2931	—	5	1	no	—
93	#1590	—	5	4	no	—
94	C₈×SD₁₆	—	5	4	no	—
95	#1566	—	5	1	no	—
96	#1493	D₅	1	1	yes	characteristic, complemented
97	#1492	—	5	1	no	—
98	C₂×C₈²	—	5	1	no	—
99	C₂×C₅×Q₈	—	2	1	no	—
100	C₂×C₅×Q₈	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₅	#41	1	1	yes	characteristic
104	C₂×C₂²⋊D₅	—	2	2	no	—
105	#709	—	4	2	no	—
106	#709	—	4	2	no	—
107	C₂×D₂₀	—	2	1	no	—
108	C₂×D₂₀	—	2	1	no	—
109	C₂×D₂₀	C₂²×C₄	1	2	yes	—
110	C₂×C₄×D₅	—	2	2	no	—
111	C₂×Dic₁₀	—	2	1	no	—
112	C₅×M₁₆	—	4	2	no	—
113	C₂×C₅×C₈	—	2	4	no	—
114	C₂×C₅×C₈	—	2	2	no	—
115	C₂×C₅×C₈	—	2	2	no	—
116	C₅×C₄⋊C₄	—	2	1	no	—
117	C₅×C₄⋊C₄	—	2	1	no	—
118	C₅×C₄⋊C₄	—	2	1	no	—
119	C₅×C₄⋊C₄	—	2	2	no	—
120	C₅×C₄⋊C₄	—	2	2	no	—
121	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
122	C₅×C₂²⋊C₄	—	2	2	no	—
123	C₄²×C₅	—	2	2	no	—
124	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
125	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₅	#41	1	1	yes	characteristic
127	C₄²×C₅	C₂⁴	1	1	yes	characteristic
128	#688	—	2	8	no	—
129	#685	—	2	2	no	—
130	#685	—	2	2	no	—
131	C₄⋊Dic₅	—	2	1	no	—
132	C₄⋊Dic₅	C₂²×C₄	1	2	yes	—
133	#681	—	4	2	no	—
134	C₂×#168	—	2	2	no	—
135	C₂×#168	—	2	4	no	—
136	C₂×#168	—	2	2	no	—
137	C₂×#168	SD₁₆	1	4	yes	complemented
138	C₂×#168	C₂²×C₄	1	2	yes	—
139	C₂×#168	C₂²×C₄	1	2	yes	—
140	C₂×#168	C₂×D₄	1	1	yes	characteristic
141	C₂×#168	C₂×D₄	1	1	yes	characteristic
142	C₂×#168	C₂×D₄	1	1	yes	characteristic
143	C₂×#168	C₂×D₄	1	1	yes	characteristic
144	#676	—	4	4	no	—
145	C₈×D₅	—	4	4	no	—
146	C₂×#131	—	10	1	no	—
147	#533	C₂×D₅	1	1	yes	characteristic
148	#532	—	5	1	no	—
149	#498	—	5	1	no	—
150	#492	—	5	1	no	—
151	#487	—	5	1	no	—
152	#486	—	5	1	no	—
153	#465	—	5	4	no	—
154	#446	—	5	2	no	—
155	#446	—	10	1	no	—
156	C₈×Q₈	—	5	2	no	—
157	C₈×Q₈	—	10	1	no	—
158	C₄×SD₁₆	—	5	4	no	—
159	#436	—	5	2	no	—
160	#436	—	10	1	no	—
161	C₈×D₄	—	5	2	no	—
162	C₈×D₄	—	10	1	no	—
163	#424	—	10	1	no	—
164	#423	—	5	1	no	—
165	#423	C₂×D₅	1	1	yes	characteristic
166	#408	—	10	1	no	—
167	#405	—	10	1	no	—
168	C₂×C₄×C₈	—	5	1	no	—
169	C₂×C₄×C₈	—	5	1	no	—
170	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
171	#334	—	5	4	no	—
172	Q₈⋊C₈	C₂×D₅	1	4	yes	—
173	D₄⋊C₈	—	5	4	no	—
174	C₈²	—	5	2	no	—
175	C₈²	—	5	2	no	—
176	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
177	C₂²×D₅	—	2	2	no	—
178	C₅×Q₈	—	2	4	no	—
179	C₅×Q₈	—	4	2	no	—
180	C₂×C₄×C₅	—	2	2	no	—
181	C₂×C₄×C₅	—	2	2	no	—
182	C₂×C₄×C₅	—	2	2	no	—
183	C₂×C₄×C₅	—	2	2	no	—
184	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
185	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
186	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
187	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
188	C₂×C₄×C₅	#135	1	2	yes	—
189	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
190	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
192	C₂²⋊D₅	—	4	4	no	—
193	C₂×Dic₅	—	2	2	no	—
194	D₂₀	—	2	4	no	—
195	D₂₀	—	4	2	no	—
196	D₂₀	—	4	2	no	—
197	C₄×D₅	—	4	2	no	—
198	C₄×D₅	—	4	2	no	—
199	Dic₁₀	—	4	2	no	—
200	C₅×C₈	—	2	4	no	—
201	C₅×C₈	—	4	2	no	—
202	#168	—	2	2	no	—
203	#168	—	2	1	no	—
204	#168	—	2	1	no	—
205	#168	—	4	2	no	—
206	#168	#131	1	4	yes	complemented
207	#168	C₂×SD₁₆	1	2	yes	—
208	#168	C₂×SD₁₆	1	2	yes	—
209	C₂×#41	—	10	1	no	—
210	C₂×SD₁₆	—	5	4	no	—
211	#131	—	20	4	no	—
212	C₂×M₁₆	—	2	1	no	—
213	C₂×M₁₆	—	10	2	no	—
214	C₂×M₁₆	—	10	1	no	—
215	C₂²×C₈	—	2	1	no	—
216	C₂²×C₈	—	5	1	no	—
217	C₂²×C₈	—	5	2	no	—
218	C₂²×C₈	—	5	1	no	—
219	C₂²×C₈	—	10	2	no	—
220	C₄⋊Q₈	—	5	1	no	—
221	C₄⋊Q₈	C₄×D₅	1	1	yes	characteristic
222	C₄⋊Q₈	C₄×D₅	1	1	yes	characteristic
223	#127	—	5	1	no	—
224	#125	—	2	1	no	—
225	#124	—	10	1	no	—
226	C₂²⋊Q₈	—	2	1	no	—
227	C₂²⋊Q₈	C₄×D₅	1	2	yes	—
228	#121	—	5	2	no	—
229	#121	—	10	1	no	—
230	C₄×Q₈	—	2	1	no	—
231	C₄×Q₈	C₂²×D₅	1	2	yes	—
232	C₄×D₄	—	5	2	no	—
233	C₄×D₄	—	10	1	no	—
234	#117	—	2	1	no	—
235	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
236	#106	—	5	4	no	—
237	C₄⋊C₈	—	2	1	no	—
238	C₄⋊C₈	—	5	2	no	—
239	C₄⋊C₈	—	10	2	no	—
240	C₄⋊C₈	—	10	2	no	—
241	C₄⋊C₈	—	10	1	no	—
242	C₄⋊C₈	C₂²×D₅	1	2	yes	—
243	#103	—	5	4	no	—
244	#102	—	5	4	no	—
245	C₂²⋊C₈	—	10	2	no	—
246	C₂²⋊C₈	—	10	2	no	—
247	#97	—	10	2	no	—
248	C₄×C₈	—	2	1	no	—
249	C₄×C₈	—	5	1	no	—
250	C₄×C₈	—	5	1	no	—
251	C₄×C₈	—	5	1	no	—
252	C₄×C₈	—	5	1	no	—
253	C₄×C₈	—	5	2	no	—
254	C₄×C₈	—	5	1	no	—
255	C₄×C₈	—	5	1	no	—
256	C₄×C₈	—	10	2	no	—
257	C₄×C₈	C₂²×D₅	1	2	yes	—
258	C₂²×C₅	—	2	1	no	—
259	C₂²×C₅	—	2	1	no	—
260	C₂²×C₅	C₄×SD₁₆	1	2	yes	—
261	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
262	C₂×D₅	—	4	2	no	—
263	C₂×D₅	—	4	2	no	—
264	C₂×D₅	—	4	2	no	—
265	C₄×C₅	—	2	1	no	—
266	C₄×C₅	—	2	1	no	—
267	C₄×C₅	—	4	2	no	—
268	C₄×C₅	—	4	2	no	—
269	C₄×C₅	C₄×SD₁₆	1	2	yes	—
270	C₄×C₅	#443	1	2	yes	—
271	C₄×C₅	C₂×#131	1	2	yes	—
272	C₄×C₅	C₂²×SD₁₆	1	1	yes	characteristic
273	C₄×C₅	C₂×#135	1	1	yes	characteristic
274	Dic₅	—	4	2	no	—
275	#41	—	20	2	no	—
276	#41	—	20	2	no	—
277	C₂×Q₈	—	2	1	no	—
278	C₂×Q₈	—	10	1	no	—
279	C₂×Q₈	C₂×C₄×D₅	1	2	yes	—
280	C₂×D₄	—	5	2	no	—
281	C₂×D₄	—	10	2	no	—
282	C₂×D₄	—	10	1	no	—
283	C₂×D₄	—	10	1	no	—
284	C₂²×C₄	—	10	2	no	—
285	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
286	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
287	C₂²×C₄	#711	1	1	yes	characteristic
288	SD₁₆	—	10	8	no	—
289	M₁₆	—	4	2	no	—
290	M₁₆	—	20	4	no	—
291	M₁₆	—	20	2	no	—
292	C₂×C₈	—	2	4	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	5	2	no	—
296	C₂×C₈	—	5	1	no	—
297	C₂×C₈	—	5	1	no	—
298	C₂×C₈	—	5	4	no	—
299	C₂×C₈	—	5	2	no	—
300	C₂×C₈	—	5	1	no	—
301	C₂×C₈	—	5	1	no	—
302	C₂×C₈	—	10	4	no	—
303	C₂×C₈	—	10	2	no	—
304	C₂×C₈	—	10	2	no	—
305	C₂×C₈	—	20	4	no	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	2	1	no	—
308	C₄⋊C₄	—	2	1	no	—
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	5	2	no	—
312	C₄⋊C₄	—	10	1	no	—
313	C₄⋊C₄	C₂×C₄×D₅	1	2	yes	—
314	C₂²⋊C₄	—	2	2	no	—
315	C₂²⋊C₄	—	10	2	no	—
316	C₂²⋊C₄	—	10	2	no	—
317	C₄²	—	2	2	no	—
318	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
319	C₄²	D₄×D₅	1	1	yes	characteristic
320	C₄²	#711	1	1	yes	characteristic
321	C₄²	C₂³×D₅	1	1	yes	characteristic
322	C₂×C₅	—	2	2	no	—
323	C₂×C₅	#2931	1	1	yes	characteristic
324	C₂×C₅	C₂×C₄×SD₁₆	1	1	yes	characteristic
325	C₂×C₅	C₂×#443	1	1	yes	characteristic
326	D₅	—	8	2	no	—
327	C₂³	—	10	2	no	—
328	C₂³	#4027	1	1	yes	characteristic
329	Q₈	—	2	4	no	—
330	Q₈	—	4	2	no	—
331	Q₈	—	20	2	no	—
332	D₄	—	10	4	no	—
333	D₄	—	20	4	no	—
334	D₄	—	20	2	no	—
335	D₄	—	20	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	10	2	no	—
341	C₂×C₄	—	20	2	no	—
342	C₂×C₄	—	20	2	no	—
343	C₂×C₄	#4027	1	1	yes	characteristic
344	C₂×C₄	#4027	1	1	yes	characteristic
345	C₂×C₄	#4027	1	1	yes	characteristic
346	C₂×C₄	D₅×SD₁₆	1	2	yes	—
347	C₂×C₄	#4065	1	2	yes	—
348	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
349	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
350	C₂×C₄	C₂×#711	1	1	yes	characteristic
351	C₈	—	2	4	no	—
352	C₈	—	4	2	no	—
353	C₈	—	5	4	no	—
354	C₈	—	5	2	no	—
355	C₈	—	5	2	no	—
356	C₈	—	10	2	no	—
357	C₈	—	10	1	no	—
358	C₈	—	10	1	no	—
359	C₈	—	20	2	no	—
360	C₅	#17957	1	1	yes	characteristic, complemented, Sylow
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	20	2	no	—
364	C₂²	—	20	2	no	—
365	C₂²	—	20	2	no	—
366	C₂²	#65077	1	2	yes	—
367	C₂²	C₂×#4027	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	4	2	no	—
371	C₄	—	4	2	no	—
372	C₄	—	20	2	no	—
373	C₄	#65077	1	2	yes	—
374	C₄	#65113	1	2	yes	—
375	C₄	D₅×#131	1	2	yes	—
376	C₄	C₂×D₅×SD₁₆	1	1	yes	characteristic
377	C₄	C₂×#4065	1	1	yes	characteristic
378	C₂	—	2	2	no	—
379	C₂	—	40	2	no	—
380	C₂	#10610388	1	1	yes	characteristic
381	C₂	C₂×#65077	1	1	yes	characteristic
382	C₂	C₂×#65113	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₄²	C₂³×D₅
Fitting	C₅×#1493	C₂
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	#65811	C₂²
cosocle	C₄²×C₅	C₂⁴
hypercenter	#1493	D₅
hypocenter	C₅	#17957
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1493

Complements

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

N	H ≅ G/N	type
2. #10615059	C₂	semidirect
4. C₅×#1493	C₂	semidirect
5. #10601283	C₂	semidirect
7. C₂×C₈×#168	C₂	semidirect
8. #10599017	C₂	semidirect
9. #10599005	C₂	semidirect
96. #1493	D₅	semidirect
137. C₂×#168	SD₁₆	semidirect
206. #168	#131	semidirect
360. C₅	#17957	semidirect

Maximal subgroups

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

subgroup	index
#10615074	2
#10615059	2
#10610772	2
C₅×#1493	2
#10601283	2
#10600528	2
C₂×C₈×#168	2
#10599017	2
#10599005	2
#17957	5