Subgroups — #12071086

Order 1280.

600 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10611635	C₂	1	2	yes	maximal, complemented
2	#10611446	C₂	1	1	yes	maximal, characteristic
3	C₅×C₈×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#10601726	C₂	1	2	yes	maximal, complemented
5	#10601698	C₂	1	1	yes	maximal, characteristic, complemented
6	#10599360	C₂	1	1	yes	maximal, characteristic, complemented
7	#10599349	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599026	C₂	1	2	yes	maximal, complemented
9	#10599009	C₂	1	2	yes	maximal, complemented
10	#10598845	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×#676	C₂	1	1	yes	maximal, characteristic, complemented
12	#65914	C₂²	1	2	yes	—
13	C₅×C₈×Q₈	C₂²	1	2	yes	—
14	C₄×C₅×Q₁₆	C₂²	1	1	yes	characteristic
15	#65480	C₄	1	1	yes	characteristic
16	#65477	C₄	1	1	yes	characteristic
17	C₄×#688	C₂²	1	2	yes	—
18	#65313	C₂²	1	2	yes	—
19	#65172	C₄	1	1	yes	characteristic
20	#65167	C₄	1	1	yes	characteristic
21	#65129	—	2	2	no	—
22	#65125	—	2	2	no	—
23	#65123	C₂²	1	2	yes	—
24	#65122	C₂²	1	2	yes	—
25	#65096	C₄	1	2	yes	—
26	#65089	C₄	1	2	yes	—
27	C₄×D₄₀	C₂²	1	1	yes	characteristic
28	#64978	—	2	2	no	—
29	#64977	—	2	2	no	—
30	#64977	C₂²	1	2	yes	—
31	C₄×#676	C₂²	1	1	yes	characteristic
32	#64974	C₂²	1	1	yes	characteristic
33	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
34	C₅×#335	C₂²	1	1	yes	characteristic
35	C₅×Q₈⋊C₈	C₂²	1	2	yes	—
36	C₅×C₈²	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	#64678	C₂²	1	1	yes	characteristic
42	#64675	C₂²	1	1	yes	characteristic
43	#64674	C₂²	1	1	yes	characteristic
44	C₈×#168	C₂²	1	1	yes	characteristic
45	#18069	—	5	1	no	maximal, Sylow
46	C₂×C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic
47	C₄×C₅×Q₈	C₂³	1	2	yes	—
48	#4093	C₂×C₄	1	2	yes	—
49	#4092	C₂×C₄	1	2	yes	—
50	C₂×#688	C₂×C₄	1	2	yes	—
51	C₂×D₄₀	C₂×C₄	1	1	yes	characteristic
52	C₂×#676	—	2	2	no	—
53	C₂×#676	D₄	1	1	yes	characteristic
54	C₂×#676	D₄	1	1	yes	characteristic
55	C₂×C₈×D₅	—	2	2	no	—
56	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
57	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
58	#4044	—	2	2	no	—
59	#4041	C₂×C₄	1	2	yes	—
60	#4040	—	2	2	no	—
61	#4036	C₂×C₄	1	2	yes	—
62	C₄×D₂₀	—	2	2	no	—
63	C₄×D₂₀	C₂³	1	2	yes	—
64	#4018	—	2	2	no	—
65	C₄²×D₅	C₂³	1	1	yes	characteristic
66	C₅×#107	C₂×C₄	1	1	yes	characteristic
67	C₅×C₄⋊C₈	—	2	2	no	—
68	C₅×C₄⋊C₈	C₂³	1	2	yes	—
69	C₅×#103	C₂×C₄	1	2	yes	—
70	C₄×C₅×C₈	—	2	2	no	—
71	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
72	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
73	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
74	#3967	C₂×C₄	1	2	yes	—
75	#3953	C₂×C₄	1	2	yes	—
76	#3952	—	2	2	no	—
77	#3952	—	2	1	no	—
78	#3952	—	2	2	no	—
79	#3952	—	2	2	no	—
80	#3950	C₂×C₄	1	1	yes	characteristic
81	#3947	D₄	1	1	yes	characteristic
82	#3947	D₄	1	1	yes	characteristic
83	#3946	—	2	1	no	—
84	C₈×Dic₅	—	2	2	no	—
85	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
86	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
87	#3941	C₂×C₄	1	2	yes	—
88	#3940	C₂×C₄	1	2	yes	—
89	C₄⋊#168	—	2	2	no	—
90	C₄⋊#168	C₂³	1	2	yes	—
91	#3935	—	2	2	no	—
92	C₄×#168	C₂³	1	1	yes	characteristic
93	C₄×#168	C₂³	1	1	yes	characteristic
94	C₄×#168	C₂³	1	1	yes	characteristic
95	#3168	—	5	1	no	—
96	#2969	—	5	2	no	—
97	#1591	—	5	1	no	—
98	#1589	—	5	1	no	—
99	#1583	—	5	2	no	—
100	#1582	—	5	2	no	—
101	C₈×Q₁₆	D₅	1	1	yes	characteristic, complemented
102	C₈×D₈	—	5	1	no	—
103	#1564	—	5	1	no	—
104	#1482	—	5	2	no	—
105	C₈×M₁₆	—	5	1	no	—
106	C₂×C₅×Q₈	C₂²×C₄	1	2	yes	—
107	C₂×D₂₀	—	2	2	no	—
108	C₂×D₂₀	C₂²×C₄	1	2	yes	—
109	C₂×C₄×D₅	—	2	2	no	—
110	C₂×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×D₅	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×D₅	#41	1	1	yes	characteristic
113	C₅×Q₁₆	—	2	2	no	—
114	C₂×C₅×C₈	—	2	2	no	—
115	C₂×C₅×C₈	—	2	1	no	—
116	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
119	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
120	C₅×C₄⋊C₄	—	2	2	no	—
121	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
122	C₄²×C₅	—	2	2	no	—
123	C₄²×C₅	C₂⁴	1	1	yes	characteristic
124	#688	—	4	2	no	—
125	#685	—	2	2	no	—
126	#685	—	2	2	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	#683	—	2	2	no	—
133	#683	—	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	1	yes	characteristic
142	C₂×#168	C₂×D₄	1	1	yes	characteristic
143	D₄₀	—	2	2	no	—
144	#676	—	2	2	no	—
145	#676	—	2	2	no	—
146	#676	—	4	4	no	—
147	C₈×D₅	—	2	2	no	—
148	C₈×D₅	—	4	1	no	—
149	C₈×D₅	—	4	4	no	—
150	#529	—	5	2	no	—
151	#499	—	5	1	no	—
152	#495	—	5	1	no	—
153	#488	—	5	2	no	—
154	#467	—	5	1	no	—
155	#466	—	5	1	no	—
156	#465	—	5	2	no	—
157	#446	—	5	2	no	—
158	C₈×Q₈	C₂×D₅	1	2	yes	—
159	C₄×Q₁₆	C₂×D₅	1	1	yes	characteristic
160	C₄×SD₁₆	—	5	2	no	—
161	C₄×D₈	—	5	1	no	—
162	#436	—	5	2	no	—
163	#436	—	10	2	no	—
164	#435	—	10	2	no	—
165	C₈×D₄	—	5	2	no	—
166	#433	—	10	2	no	—
167	#431	—	5	1	no	—
168	#431	—	5	2	no	—
169	#405	—	10	2	no	—
170	C₄×M₁₆	—	5	1	no	—
171	C₂×C₄×C₈	—	5	1	no	—
172	#335	—	5	1	no	—
173	#335	C₂×D₅	1	1	yes	characteristic
174	#334	—	5	2	no	—
175	Q₈⋊C₈	—	5	2	no	—
176	Q₈⋊C₈	C₂×D₅	1	2	yes	—
177	D₄⋊C₈	—	5	2	no	—
178	D₄⋊C₈	—	5	2	no	—
179	#322	—	5	1	no	—
180	#322	—	5	1	no	—
181	C₈²	—	5	1	no	—
182	C₈²	C₂×D₅	1	1	yes	characteristic
183	C₂²×D₅	—	2	2	no	—
184	C₂²×D₅	C₄×D₄	1	1	yes	characteristic
185	C₅×Q₈	—	2	2	no	—
186	C₅×Q₈	—	2	2	no	—
187	C₂×C₄×C₅	—	2	2	no	—
188	C₂×C₄×C₅	—	2	2	no	—
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	C₂×Dic₅	—	2	2	no	—
193	C₂×Dic₅	C₄×D₄	1	1	yes	characteristic
194	C₂×Dic₅	C₄×D₄	1	1	yes	characteristic
195	C₂×Dic₅	C₄×D₄	1	1	yes	characteristic
196	D₂₀	—	2	2	no	—
197	D₂₀	—	2	2	no	—
198	D₂₀	—	8	2	no	—
199	C₄×D₅	—	2	1	no	—
200	C₄×D₅	—	2	1	no	—
201	C₄×D₅	—	2	1	no	—
202	C₄×D₅	—	2	1	no	—
203	C₄×D₅	—	4	1	no	—
204	C₄×D₅	—	4	2	no	—
205	C₄×D₅	—	4	2	no	—
206	C₅×C₈	—	2	1	no	—
207	C₅×C₈	—	2	1	no	—
208	C₅×C₈	—	4	1	no	—
209	C₅×C₈	—	4	2	no	—
210	C₅×C₈	#131	1	2	yes	—
211	C₅×C₈	#135	1	2	yes	—
212	#168	—	2	1	no	—
213	#168	—	2	1	no	—
214	#168	—	2	1	no	—
215	#168	—	4	1	no	—
216	#168	—	4	2	no	—
217	#168	#135	1	2	yes	—
218	C₂×Q₁₆	C₄×D₅	1	1	yes	characteristic
219	C₂×SD₁₆	—	5	2	no	—
220	C₂×D₈	—	5	1	no	—
221	C₂×M₁₆	—	5	1	no	—
222	C₂×M₁₆	—	5	1	no	—
223	C₂×M₁₆	—	10	2	no	—
224	C₂²×C₈	—	5	1	no	—
225	C₂²×C₈	—	5	1	no	—
226	C₂²×C₈	—	10	2	no	—
227	#126	—	10	2	no	—
228	#125	—	5	2	no	—
229	#124	—	5	2	no	—
230	#123	—	10	2	no	—
231	C₂²⋊Q₈	—	5	2	no	—
232	#121	—	5	2	no	—
233	C₄×Q₈	C₂²×D₅	1	2	yes	—
234	C₄×D₄	—	5	2	no	—
235	C₄×D₄	—	10	2	no	—
236	#117	—	10	2	no	—
237	C₂×C₄²	—	5	1	no	—
238	#107	—	5	1	no	—
239	#107	C₄×D₅	1	1	yes	characteristic
240	#106	—	5	2	no	—
241	C₄⋊C₈	—	2	2	no	—
242	C₄⋊C₈	—	5	2	no	—
243	C₄⋊C₈	—	10	2	no	—
244	C₄⋊C₈	—	10	1	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	—	10	2	no	—
251	C₂²⋊C₈	—	10	1	no	—
252	C₂²⋊C₈	—	10	2	no	—
253	C₂²⋊C₈	—	10	2	no	—
254	#97	—	5	1	no	—
255	#97	—	5	1	no	—
256	#97	—	10	2	no	—
257	C₄×C₈	—	2	2	no	—
258	C₄×C₈	—	5	1	no	—
259	C₄×C₈	—	5	1	no	—
260	C₄×C₈	—	5	1	no	—
261	C₄×C₈	—	5	1	no	—
262	C₄×C₈	—	5	1	no	—
263	C₄×C₈	—	10	2	no	—
264	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
265	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
266	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
267	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
268	C₂×D₅	—	2	1	no	—
269	C₂×D₅	—	2	1	no	—
270	C₂×D₅	—	2	1	no	—
271	C₂×D₅	—	4	2	no	—
272	C₂×D₅	—	4	2	no	—
273	C₂×D₅	—	4	2	no	—
274	C₄×C₅	—	2	1	no	—
275	C₄×C₅	—	4	2	no	—
276	C₄×C₅	—	4	2	no	—
277	C₄×C₅	C₂×#131	1	1	yes	characteristic
278	C₄×C₅	#568	1	1	yes	characteristic
279	C₄×C₅	C₂×#135	1	1	yes	characteristic
280	C₄×C₅	C₂×#135	1	1	yes	characteristic
281	Dic₅	—	2	1	no	—
282	Dic₅	—	2	1	no	—
283	Dic₅	—	2	1	no	—
284	Dic₅	—	4	2	no	—
285	C₂×Q₈	C₂×C₄×D₅	1	2	yes	—
286	C₂×D₄	—	5	2	no	—
287	C₂×D₄	—	10	2	no	—
288	C₂²×C₄	—	5	1	no	—
289	C₂²×C₄	—	5	1	no	—
290	C₂²×C₄	—	5	1	no	—
291	C₂²×C₄	—	10	2	no	—
292	Q₁₆	—	2	2	no	—
293	SD₁₆	—	20	2	no	—
294	D₈	—	10	2	no	—
295	M₁₆	—	10	2	no	—
296	M₁₆	—	10	2	no	—
297	M₁₆	—	20	4	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	1	no	—
300	C₂×C₈	—	5	1	no	—
301	C₂×C₈	—	5	1	no	—
302	C₂×C₈	—	5	1	no	—
303	C₂×C₈	—	5	1	no	—
304	C₂×C₈	—	10	2	no	—
305	C₂×C₈	—	10	2	no	—
306	C₂×C₈	—	10	1	no	—
307	C₂×C₈	—	20	1	no	—
308	C₂×C₈	—	20	4	no	—
309	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
310	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
311	C₂×C₈	D₄×D₅	1	1	yes	characteristic
312	C₂×C₈	D₄×D₅	1	1	yes	characteristic
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	5	2	no	—
315	C₄⋊C₄	—	10	2	no	—
316	C₄⋊C₄	—	10	2	no	—
317	C₄⋊C₄	—	10	2	no	—
318	C₄⋊C₄	C₂×C₄×D₅	1	2	yes	—
319	C₂²⋊C₄	—	10	2	no	—
320	C₂²⋊C₄	—	10	2	no	—
321	C₂²⋊C₄	—	10	2	no	—
322	C₂²⋊C₄	—	10	2	no	—
323	C₂²⋊C₄	—	10	2	no	—
324	C₄²	—	2	2	no	—
325	C₄²	—	5	1	no	—
326	C₄²	—	5	1	no	—
327	C₄²	—	5	1	no	—
328	C₄²	C₂³×D₅	1	1	yes	characteristic
329	C₂×C₅	#2935	1	1	yes	characteristic
330	C₂×C₅	C₄×#135	1	1	yes	characteristic
331	C₂×C₅	#2957	1	1	yes	characteristic
332	D₅	—	4	1	no	—
333	D₅	—	8	2	no	—
334	C₂³	—	5	1	no	—
335	C₂³	—	10	2	no	—
336	Q₈	—	2	2	no	—
337	Q₈	—	2	2	no	—
338	D₄	—	10	2	no	—
339	D₄	—	10	2	no	—
340	D₄	—	40	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	5	1	no	—
344	C₂×C₄	—	5	1	no	—
345	C₂×C₄	—	5	1	no	—
346	C₂×C₄	—	10	1	no	—
347	C₂×C₄	—	10	1	no	—
348	C₂×C₄	—	10	1	no	—
349	C₂×C₄	—	10	1	no	—
350	C₂×C₄	—	10	2	no	—
351	C₂×C₄	—	20	1	no	—
352	C₂×C₄	—	20	2	no	—
353	C₂×C₄	—	20	2	no	—
354	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
355	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
356	C₂×C₄	D₅×#41	1	1	yes	characteristic
357	C₈	—	2	1	no	—
358	C₈	—	2	1	no	—
359	C₈	—	4	1	no	—
360	C₈	—	4	2	no	—
361	C₈	—	5	2	no	—
362	C₈	—	10	1	no	—
363	C₈	—	10	1	no	—
364	C₈	—	10	1	no	—
365	C₈	—	20	1	no	—
366	C₈	—	20	2	no	—
367	C₈	#4047	1	2	yes	—
368	C₈	#4065	1	2	yes	—
369	C₅	#18069	1	1	yes	characteristic, complemented, Sylow
370	C₂²	—	10	1	no	—
371	C₂²	—	10	1	no	—
372	C₂²	—	10	1	no	—
373	C₂²	—	20	2	no	—
374	C₂²	—	20	2	no	—
375	C₂²	—	20	2	no	—
376	C₂²	C₄×D₄×D₅	1	1	yes	characteristic
377	C₄	—	2	1	no	—
378	C₄	—	4	2	no	—
379	C₄	—	4	2	no	—
380	C₄	—	10	1	no	—
381	C₄	—	10	1	no	—
382	C₄	—	10	1	no	—
383	C₄	—	20	2	no	—
384	C₄	C₂×#4047	1	1	yes	characteristic
385	C₄	#66084	1	1	yes	characteristic
386	C₄	C₂×#4065	1	1	yes	characteristic
387	C₄	D₅×#135	1	1	yes	characteristic
388	C₂	—	20	1	no	—
389	C₂	—	40	2	no	—
390	C₂	#10611156	1	1	yes	characteristic
391	C₂	C₄×#4065	1	1	yes	characteristic
392	C₂	#10611621	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×C₈×Q₁₆	C₂
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	#65172	C₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	C₈×Q₁₆	D₅
hypocenter	C₅	#18069
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈×Q₁₆

Complements

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

N	H ≅ G/N	type
1. #10611635	C₂	semidirect
3. C₅×C₈×Q₁₆	C₂	semidirect
4. #10601726	C₂	semidirect
5. #10601698	C₂	semidirect
6. #10599360	C₂	semidirect
7. #10599349	C₂	semidirect
8. #10599026	C₂	semidirect
9. #10599009	C₂	semidirect
10. #10598845	C₂	semidirect
11. C₈×#676	C₂	semidirect
101. C₈×Q₁₆	D₅	semidirect
369. C₅	#18069	semidirect

Maximal subgroups

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

subgroup	index
#10611635	2
#10611446	2
C₅×C₈×Q₁₆	2
#10601726	2
#10601698	2
#10599360	2
#10599349	2
#10599026	2
#10599009	2
#10598845	2
C₈×#676	2
#18069	5