Subgroups — #12071050

Order 1280.

652 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10611676	C₂	1	2	yes	maximal, complemented
2	C₄×#4065	C₂	1	1	yes	maximal, characteristic
3	C₅×#1593	C₂	1	1	yes	maximal, characteristic, complemented
4	#10601746	C₂	1	2	yes	maximal, complemented
5	#10601699	C₂	1	1	yes	maximal, characteristic, complemented
6	#10599322	C₂	1	1	yes	maximal, characteristic, complemented
7	#10599311	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599031	C₂	1	2	yes	maximal, complemented
9	#10599009	C₂	1	2	yes	maximal, complemented
10	#10598870	C₂	1	1	yes	maximal, characteristic, complemented
11	#10598862	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#4065	C₄	1	1	yes	characteristic
13	C₄×#713	C₂²	1	2	yes	—
14	C₅×#446	C₂²	1	2	yes	—
15	C₄×C₅×Q₁₆	C₂²	1	1	yes	characteristic
16	Q₁₆×Dic₅	C₄	1	1	yes	characteristic
17	C₄×#688	C₂²	1	2	yes	—
18	#65313	C₂²	1	2	yes	—
19	#65169	C₄	1	1	yes	characteristic
20	#65161	C₄	1	1	yes	characteristic
21	#65126	C₂²	1	2	yes	—
22	#65125	—	2	2	no	—
23	#65123	C₂²	1	2	yes	—
24	D₅×C₄⋊C₈	—	2	2	no	—
25	#65093	C₄	1	2	yes	—
26	#65077	C₄	1	2	yes	—
27	#64995	—	2	2	no	—
28	#64994	—	2	2	no	—
29	C₄×D₄₀	C₂²	1	1	yes	characteristic
30	C₄×#676	C₂²	1	1	yes	characteristic
31	C₈×D₂₀	C₂²	1	2	yes	—
32	#64974	C₂²	1	1	yes	characteristic
33	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
34	C₅×#334	C₂²	1	1	yes	characteristic
35	C₅×Q₈⋊C₈	C₂²	1	2	yes	—
36	C₅×#322	C₂²	1	1	yes	characteristic
37	Q₈⋊#168	C₂²	1	2	yes	—
38	#64703	C₂²	1	2	yes	—
39	#64700	C₂²	1	2	yes	—
40	#64679	C₂²	1	2	yes	—
41	#64677	C₂²	1	1	yes	characteristic
42	#64675	C₂²	1	1	yes	characteristic
43	#64675	C₂²	1	1	yes	characteristic
44	#64674	C₂²	1	1	yes	characteristic
45	#18065	—	5	1	no	maximal, Sylow
46	C₂×#713	C₂×C₄	1	2	yes	—
47	C₂×C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic
48	C₄×C₅×Q₈	C₂³	1	2	yes	—
49	Q₈×Dic₅	C₂×C₄	1	2	yes	—
50	C₂×#688	C₂×C₄	1	2	yes	—
51	#4065	—	2	4	no	—
52	C₂×D₄₀	C₂×C₄	1	1	yes	characteristic
53	C₂×#676	—	2	1	no	—
54	C₂×#676	—	2	2	no	—
55	C₂×C₈×D₅	—	2	2	no	—
56	C₂×C₈×D₅	—	2	2	no	—
57	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
58	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
59	#4039	—	2	2	no	—
60	#4039	C₂×C₄	1	2	yes	—
61	#4038	—	2	2	no	—
62	#4038	C₂×C₄	1	2	yes	—
63	C₄×D₂₀	—	2	2	no	—
64	C₄×D₂₀	C₂³	1	2	yes	—
65	C₄²×D₅	—	2	2	no	—
66	C₄²×D₅	C₂³	1	1	yes	characteristic
67	C₅×#107	C₂×C₄	1	1	yes	characteristic
68	C₅×C₄⋊C₈	—	2	2	no	—
69	C₅×C₄⋊C₈	C₂³	1	2	yes	—
70	C₅×#103	C₂×C₄	1	2	yes	—
71	C₅×#97	—	2	2	no	—
72	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
73	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
74	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
75	#3967	C₂×C₄	1	2	yes	—
76	#3953	C₂×C₄	1	2	yes	—
77	#3952	—	2	2	no	—
78	#3952	—	2	2	no	—
79	#3952	D₄	1	2	yes	—
80	#3950	C₂×C₄	1	1	yes	characteristic
81	#3947	—	2	1	no	—
82	#3946	—	2	2	no	—
83	#3946	D₄	1	2	yes	—
84	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
85	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
86	#3941	C₂×C₄	1	2	yes	—
87	#3940	C₂×C₄	1	2	yes	—
88	C₄⋊#168	—	2	2	no	—
89	C₄⋊#168	C₂³	1	2	yes	—
90	#3935	—	2	2	no	—
91	C₄×#168	C₂³	1	1	yes	characteristic
92	C₄×#168	C₂³	1	1	yes	characteristic
93	C₄×#168	C₂³	1	1	yes	characteristic
94	#2990	—	5	2	no	—
95	C₄×#135	—	5	1	no	—
96	#1593	—	5	1	no	—
97	#1593	D₅	1	1	yes	characteristic, complemented
98	#1582	—	5	2	no	—
99	#1582	—	5	2	no	—
100	#1580	—	5	1	no	—
101	#1580	—	5	1	no	—
102	#1566	—	5	1	no	—
103	#1489	—	5	2	no	—
104	#1451	—	5	1	no	—
105	C₂×C₅×Q₈	C₂²×C₄	1	2	yes	—
106	#713	—	2	4	no	—
107	#713	—	4	2	no	—
108	C₂×D₂₀	—	2	2	no	—
109	C₂×D₂₀	C₂²×C₄	1	2	yes	—
110	C₂×C₄×D₅	—	2	2	no	—
111	C₂×C₄×D₅	—	2	2	no	—
112	C₂×C₄×D₅	—	2	2	no	—
113	C₂×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×D₅	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×D₅	#41	1	1	yes	characteristic
116	C₅×Q₁₆	—	2	2	no	—
117	C₂×C₅×C₈	—	2	2	no	—
118	C₂×C₅×C₈	—	2	1	no	—
119	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
122	C₅×C₄⋊C₄	—	2	2	no	—
123	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
124	C₄²×C₅	—	2	2	no	—
125	C₄²×C₅	C₂⁴	1	1	yes	characteristic
126	#688	—	2	4	no	—
127	#685	—	2	2	no	—
128	#685	—	2	2	no	—
129	#685	—	2	2	no	—
130	C₄⋊Dic₅	—	2	2	no	—
131	C₄⋊Dic₅	C₂²×C₄	1	2	yes	—
132	C₄×Dic₅	—	2	2	no	—
133	C₄×Dic₅	—	2	2	no	—
134	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
135	C₄×Dic₅	C₂×D₄	1	1	yes	characteristic
136	C₄×Dic₅	#41	1	1	yes	characteristic
137	C₂×#168	—	2	2	no	—
138	C₂×#168	—	2	1	no	—
139	C₂×#168	C₂²×C₄	1	1	yes	characteristic
140	C₂×#168	C₂²×C₄	1	1	yes	characteristic
141	C₂×#168	C₂×D₄	1	2	yes	—
142	D₄₀	M₁₆	1	4	yes	complemented
143	#676	—	4	2	no	—
144	#676	—	8	2	no	—
145	C₈×D₅	—	2	2	no	—
146	C₈×D₅	—	4	1	no	—
147	C₈×D₅	—	8	2	no	—
148	C₈×D₅	—	8	2	no	—
149	C₂×#135	—	5	1	no	—
150	C₄×#41	—	5	2	no	—
151	#446	—	5	2	no	—
152	#446	C₂×D₅	1	2	yes	—
153	C₄×Q₁₆	—	5	1	no	—
154	C₄×Q₁₆	C₂×D₅	1	1	yes	characteristic
155	C₄×SD₁₆	—	5	2	no	—
156	C₄×SD₁₆	—	5	2	no	—
157	C₄×D₈	—	5	1	no	—
158	C₄×D₈	—	5	1	no	—
159	#435	—	10	2	no	—
160	#435	—	10	2	no	—
161	C₈×D₄	—	5	2	no	—
162	C₈×D₄	—	5	2	no	—
163	#432	—	10	2	no	—
164	#431	—	5	1	no	—
165	#427	—	5	1	no	—
166	#423	—	5	2	no	—
167	C₂×C₄⋊C₈	—	10	2	no	—
168	#416	—	5	2	no	—
169	C₄×M₁₆	—	5	1	no	—
170	C₂×C₄×C₈	—	5	1	no	—
171	#334	—	5	2	no	—
172	#334	—	5	1	no	—
173	#334	C₂×D₅	1	1	yes	characteristic
174	Q₈⋊C₈	—	5	2	no	—
175	Q₈⋊C₈	C₂×D₅	1	2	yes	—
176	D₄⋊C₈	—	5	2	no	—
177	D₄⋊C₈	—	5	2	no	—
178	#322	—	5	1	no	—
179	#322	—	5	1	no	—
180	#322	—	5	1	no	—
181	#322	C₂×D₅	1	1	yes	characteristic
182	C₂²×D₅	—	2	2	no	—
183	C₂²×D₅	C₄×D₄	1	1	yes	characteristic
184	C₅×Q₈	—	2	2	no	—
185	C₅×Q₈	—	2	2	no	—
186	C₂×C₄×C₅	—	2	2	no	—
187	C₂×C₄×C₅	—	2	2	no	—
188	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
189	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
190	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
191	C₂×Dic₅	—	2	2	no	—
192	C₂×Dic₅	C₄×D₄	1	1	yes	characteristic
193	C₂×Dic₅	C₄×D₄	1	1	yes	characteristic
194	C₂×Dic₅	C₄×D₄	1	1	yes	characteristic
195	D₂₀	—	2	2	no	—
196	D₂₀	—	4	4	no	—
197	D₂₀	C₂×M₁₆	1	4	yes	—
198	C₄×D₅	—	2	1	no	—
199	C₄×D₅	—	2	1	no	—
200	C₄×D₅	—	2	2	no	—
201	C₄×D₅	—	2	2	no	—
202	C₄×D₅	—	4	4	no	—
203	C₄×D₅	—	4	2	no	—
204	C₄×D₅	—	4	2	no	—
205	C₄×D₅	—	4	4	no	—
206	C₅×C₈	—	2	1	no	—
207	C₅×C₈	—	2	2	no	—
208	C₅×C₈	—	4	1	no	—
209	C₅×C₈	—	4	2	no	—
210	C₅×C₈	C₂×M₁₆	1	2	yes	—
211	#168	—	2	1	no	—
212	#168	—	2	1	no	—
213	#168	—	2	2	no	—
214	#168	—	4	1	no	—
215	#168	—	4	2	no	—
216	C₂×#41	—	5	2	no	—
217	#135	—	10	4	no	—
218	C₂×Q₁₆	C₄×D₅	1	1	yes	characteristic
219	C₂×SD₁₆	—	5	2	no	—
220	C₂×D₈	—	5	1	no	—
221	C₂×M₁₆	—	10	1	no	—
222	C₂×M₁₆	—	10	2	no	—
223	C₂²×C₈	—	5	1	no	—
224	C₂²×C₈	—	5	1	no	—
225	C₂²×C₈	—	10	2	no	—
226	C₂²×C₈	—	10	2	no	—
227	C₄×Q₈	—	5	2	no	—
228	C₄×Q₈	C₂²×D₅	1	2	yes	—
229	C₄×D₄	—	5	2	no	—
230	C₄×D₄	—	5	2	no	—
231	C₄×D₄	—	10	2	no	—
232	C₄×D₄	—	10	2	no	—
233	#117	—	5	2	no	—
234	#117	—	10	2	no	—
235	C₂×C₄²	—	5	1	no	—
236	C₂×C₄²	—	10	2	no	—
237	#107	—	5	1	no	—
238	#107	C₄×D₅	1	1	yes	characteristic
239	#106	—	5	2	no	—
240	C₄⋊C₈	—	2	2	no	—
241	C₄⋊C₈	—	5	2	no	—
242	C₄⋊C₈	—	5	2	no	—
243	C₄⋊C₈	—	10	2	no	—
244	C₄⋊C₈	—	10	2	no	—
245	C₄⋊C₈	C₂²×D₅	1	2	yes	—
246	#103	—	5	2	no	—
247	#103	C₄×D₅	1	2	yes	—
248	#102	—	5	2	no	—
249	#102	—	5	2	no	—
250	C₂²⋊C₈	—	5	2	no	—
251	C₂²⋊C₈	—	10	2	no	—
252	C₂²⋊C₈	—	10	2	no	—
253	#97	—	2	2	no	—
254	#97	—	10	2	no	—
255	#97	—	10	1	no	—
256	C₄×C₈	—	5	1	no	—
257	C₄×C₈	—	5	1	no	—
258	C₄×C₈	—	5	1	no	—
259	C₄×C₈	—	5	1	no	—
260	C₄×C₈	—	5	1	no	—
261	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
262	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
263	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
264	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
265	C₂×D₅	—	2	1	no	—
266	C₂×D₅	—	2	4	no	—
267	C₂×D₅	—	2	1	no	—
268	C₂×D₅	—	2	1	no	—
269	C₂×D₅	—	4	2	no	—
270	C₂×D₅	—	4	2	no	—
271	C₄×C₅	—	2	1	no	—
272	C₄×C₅	—	4	2	no	—
273	C₄×C₅	—	4	2	no	—
274	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
275	C₄×C₅	#568	1	1	yes	characteristic
276	C₄×C₅	#577	1	2	yes	—
277	Dic₅	—	2	1	no	—
278	Dic₅	—	2	1	no	—
279	Dic₅	—	4	2	no	—
280	Dic₅	#444	1	2	yes	—
281	#41	—	10	4	no	—
282	#41	—	20	2	no	—
283	C₂×Q₈	C₂×C₄×D₅	1	2	yes	—
284	C₂×D₄	—	5	2	no	—
285	C₂×D₄	—	10	2	no	—
286	C₂²×C₄	—	5	1	no	—
287	C₂²×C₄	—	5	1	no	—
288	C₂²×C₄	—	5	1	no	—
289	C₂²×C₄	—	10	2	no	—
290	C₂²×C₄	—	10	2	no	—
291	C₂²×C₄	—	10	2	no	—
292	Q₁₆	—	2	2	no	—
293	SD₁₆	—	10	4	no	—
294	D₈	—	5	4	no	—
295	M₁₆	—	20	2	no	—
296	M₁₆	—	40	2	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	2	1	no	—
299	C₂×C₈	—	5	1	no	—
300	C₂×C₈	—	5	1	no	—
301	C₂×C₈	—	5	2	no	—
302	C₂×C₈	—	10	2	no	—
303	C₂×C₈	—	10	2	no	—
304	C₂×C₈	—	10	1	no	—
305	C₂×C₈	—	20	1	no	—
306	C₂×C₈	—	40	2	no	—
307	C₂×C₈	—	40	2	no	—
308	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
309	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
310	C₂×C₈	D₄×D₅	1	2	yes	—
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	5	2	no	—
313	C₄⋊C₄	—	10	2	no	—
314	C₄⋊C₄	C₂×C₄×D₅	1	2	yes	—
315	C₂²⋊C₄	—	10	2	no	—
316	C₂²⋊C₄	—	10	2	no	—
317	C₂²⋊C₄	—	10	2	no	—
318	C₄²	—	2	2	no	—
319	C₄²	—	5	1	no	—
320	C₄²	—	5	1	no	—
321	C₄²	—	5	1	no	—
322	C₄²	—	10	2	no	—
323	C₄²	—	10	2	no	—
324	C₄²	C₂³×D₅	1	1	yes	characteristic
325	C₂×C₅	D₄×M₁₆	1	1	yes	characteristic
326	C₂×C₅	#2952	1	1	yes	characteristic
327	C₂×C₅	C₂×#444	1	1	yes	characteristic
328	D₅	—	4	1	no	—
329	D₅	—	4	4	no	—
330	C₂³	—	5	1	no	—
331	C₂³	—	10	2	no	—
332	Q₈	—	2	2	no	—
333	Q₈	—	2	2	no	—
334	D₄	—	5	4	no	—
335	D₄	—	10	2	no	—
336	D₄	—	20	4	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	5	1	no	—
340	C₂×C₄	—	5	1	no	—
341	C₂×C₄	—	5	1	no	—
342	C₂×C₄	—	10	2	no	—
343	C₂×C₄	—	10	1	no	—
344	C₂×C₄	—	10	1	no	—
345	C₂×C₄	—	10	2	no	—
346	C₂×C₄	—	10	2	no	—
347	C₂×C₄	—	20	4	no	—
348	C₂×C₄	—	20	4	no	—
349	C₂×C₄	—	20	2	no	—
350	C₂×C₄	—	20	2	no	—
351	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
352	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
353	C₂×C₄	D₅×#41	1	1	yes	characteristic
354	C₈	—	2	1	no	—
355	C₈	—	2	2	no	—
356	C₈	—	4	1	no	—
357	C₈	—	4	2	no	—
358	C₈	—	10	1	no	—
359	C₈	—	10	1	no	—
360	C₈	—	10	2	no	—
361	C₈	—	20	1	no	—
362	C₈	—	20	2	no	—
363	C₈	D₅×M₁₆	1	2	yes	—
364	C₅	#18065	1	1	yes	characteristic, complemented, Sylow
365	C₂²	—	10	1	no	—
366	C₂²	—	10	4	no	—
367	C₂²	—	10	1	no	—
368	C₂²	—	10	1	no	—
369	C₂²	—	20	2	no	—
370	C₂²	—	20	2	no	—
371	C₂²	C₄×D₄×D₅	1	1	yes	characteristic
372	C₄	—	2	1	no	—
373	C₄	—	4	2	no	—
374	C₄	—	4	2	no	—
375	C₄	—	5	2	no	—
376	C₄	—	10	1	no	—
377	C₄	—	10	1	no	—
378	C₄	—	20	2	no	—
379	C₄	C₂×D₅×M₁₆	1	1	yes	characteristic
380	C₄	#66084	1	1	yes	characteristic
381	C₄	#66112	1	2	yes	—
382	C₂	—	20	1	no	—
383	C₂	—	20	4	no	—
384	C₂	#10611271	1	1	yes	characteristic
385	C₂	#10611563	1	1	yes	characteristic
386	C₂	D₅×#444	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×#1593	C₂
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₂×#4065	C₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	#1593	D₅
hypocenter	C₅	#18065
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1593

Complements

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

N	H ≅ G/N	type
1. #10611676	C₂	semidirect
3. C₅×#1593	C₂	semidirect
4. #10601746	C₂	semidirect
5. #10601699	C₂	semidirect
6. #10599322	C₂	semidirect
7. #10599311	C₂	semidirect
8. #10599031	C₂	semidirect
9. #10599009	C₂	semidirect
10. #10598870	C₂	semidirect
11. #10598862	C₂	semidirect
97. #1593	D₅	semidirect
142. D₄₀	M₁₆	semidirect
364. C₅	#18065	semidirect

Maximal subgroups

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

subgroup	index
#10611676	2
C₄×#4065	2
C₅×#1593	2
#10601746	2
#10601699	2
#10599322	2
#10599311	2
#10599031	2
#10599009	2
#10598870	2
#10598862	2
#18065	5