Subgroups — #11992009

Order 1280.

492 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10605031	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#64676	C₂	1	2	yes	maximal
4	#10600187	C₂	1	1	yes	maximal, characteristic
5	#10598146	C₂	1	2	yes	maximal
6	C₂×C₄×Dic₁₀	C₂²	1	1	yes	characteristic
7	C₂×C₅×C₄⋊C₈	—	2	1	no	—
8	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
9	C₄³×C₅	C₂²	1	1	yes	characteristic
10	C₄×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
11	#65222	—	2	1	no	—
12	#65222	—	2	1	no	—
13	#65221	—	2	1	no	—
14	C₄×#683	—	2	1	no	—
15	#65218	C₄	1	1	yes	characteristic
16	#65218	C₄	1	1	yes	characteristic
17	#65217	C₄	1	1	yes	characteristic
18	#65217	C₄	1	1	yes	characteristic
19	C₂×C₄⋊#168	—	2	1	no	—
20	C₂×C₄⋊#168	—	2	1	no	—
21	C₂×C₄⋊#168	C₂²	1	2	yes	—
22	C₅×#336	—	2	1	no	—
23	#64676	—	2	4	no	—
24	#8568	—	5	1	no	maximal, Sylow
25	C₂²×Dic₁₀	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₅×C₈	—	2	2	no	—
27	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
34	C₂×C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
37	C₂×#683	—	2	1	no	—
38	C₂×#683	—	2	1	no	—
39	C₂×#683	—	2	1	no	—
40	C₂×C₄×Dic₅	—	2	1	no	—
41	C₂×C₄×Dic₅	—	2	1	no	—
42	C₂²×#168	—	2	2	no	—
43	C₄×Dic₁₀	—	2	1	no	—
44	C₄×Dic₁₀	—	2	1	no	—
45	C₄×Dic₁₀	D₄	1	2	yes	—
46	C₄×Dic₁₀	Q₈	1	2	yes	—
47	C₅×C₄⋊C₈	—	2	4	no	—
48	C₄×C₅×C₈	—	2	2	no	—
49	C₄×C₅×C₈	—	2	2	no	—
50	#3963	—	2	1	no	—
51	#3963	—	2	1	no	—
52	#3963	—	2	1	no	—
53	#3963	—	2	1	no	—
54	#3963	—	2	1	no	—
55	#3963	—	2	1	no	—
56	#3963	—	2	1	no	—
57	C₄⋊#168	—	2	2	no	—
58	C₄⋊#168	—	2	2	no	—
59	C₄⋊#168	—	4	2	no	—
60	C₄⋊#168	—	4	2	no	—
61	#2295	—	5	1	no	—
62	#1842	—	5	1	no	—
63	C₄×C₄⋊C₈	D₅	1	1	yes	characteristic
64	C₂×Q₈⋊C₈	—	5	2	no	—
65	#1279	—	5	2	no	—
66	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
73	C₂²×Dic₅	—	2	1	no	—
74	C₂²×Dic₅	—	2	1	no	—
75	C₂²×Dic₅	—	2	1	no	—
76	C₂²×Dic₅	—	2	1	no	—
77	C₂×Dic₁₀	—	2	2	no	—
78	C₂×Dic₁₀	—	2	2	no	—
79	C₂×Dic₁₀	—	2	1	no	—
80	C₂×Dic₁₀	C₄⋊C₄	1	2	yes	—
81	C₂×C₅×C₈	—	2	4	no	—
82	C₂×C₅×C₈	—	4	2	no	—
83	C₂×C₅×C₈	—	4	2	no	—
84	C₄²×C₅	—	2	2	no	—
85	C₄²×C₅	—	2	2	no	—
86	C₄²×C₅	—	2	2	no	—
87	C₄²×C₅	—	2	1	no	—
88	C₄²×C₅	—	2	1	no	—
89	C₄²×C₅	—	4	1	no	—
90	C₄²×C₅	SD₁₆	1	2	yes	—
91	C₄²×C₅	Q₁₆	1	2	yes	—
92	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
94	C₄²×C₅	#41	1	1	yes	characteristic
95	C₄²×C₅	#41	1	1	yes	characteristic
96	C₄⋊Dic₅	—	2	1	no	—
97	C₄⋊Dic₅	—	2	2	no	—
98	C₄⋊Dic₅	—	2	2	no	—
99	C₄⋊Dic₅	C₄⋊C₄	1	2	yes	—
100	#683	—	2	2	no	—
101	#683	—	2	2	no	—
102	#683	—	4	2	no	—
103	#683	—	4	2	no	—
104	C₄×Dic₅	—	2	2	no	—
105	C₄×Dic₅	—	2	2	no	—
106	C₄×Dic₅	—	4	1	no	—
107	C₄×Dic₅	—	4	1	no	—
108	C₂×#168	—	4	2	no	—
109	C₂×#168	—	4	2	no	—
110	C₂×#168	—	4	2	no	—
111	C₂×C₄×Q₈	—	5	1	no	—
112	C₂×C₄⋊C₈	—	2	1	no	—
113	C₂×C₄⋊C₈	—	5	2	no	—
114	C₂×C₄⋊C₈	—	10	1	no	—
115	C₂×C₄⋊C₈	—	10	1	no	—
116	C₂×C₄×C₈	C₂×D₅	1	2	yes	—
117	#391	—	5	1	no	—
118	#391	—	5	1	no	—
119	#389	—	5	1	no	—
120	#389	—	5	1	no	—
121	#387	—	10	1	no	—
122	#387	—	10	1	no	—
123	C₄×C₄⋊C₄	—	5	1	no	—
124	C₄×C₄⋊C₄	—	10	1	no	—
125	#376	—	10	1	no	—
126	C₄³	C₂×D₅	1	1	yes	characteristic
127	#336	—	2	1	no	—
128	Q₈⋊C₈	—	10	4	no	—
129	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₅	—	2	1	no	—
131	C₂×C₄×C₅	—	2	1	no	—
132	C₂×C₄×C₅	—	2	1	no	—
133	C₂×C₄×C₅	—	2	1	no	—
134	C₂×C₄×C₅	—	2	1	no	—
135	C₂×C₄×C₅	—	2	1	no	—
136	C₂×C₄×C₅	—	2	1	no	—
137	C₂×C₄×C₅	—	2	1	no	—
138	C₂×C₄×C₅	—	2	1	no	—
139	C₂×C₄×C₅	—	2	1	no	—
140	C₂×C₄×C₅	—	2	1	no	—
141	C₂×C₄×C₅	#103	1	2	yes	—
142	C₂×C₄×C₅	#103	1	2	yes	—
143	C₂×C₄×C₅	#104	1	2	yes	—
144	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₅	#117	1	1	yes	characteristic
146	C₂×C₄×C₅	#121	1	1	yes	characteristic
147	C₂×C₄×C₅	#121	1	1	yes	characteristic
148	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
150	C₂×C₄×C₅	#131	1	2	yes	—
151	C₂×C₄×C₅	#131	1	2	yes	—
152	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
154	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
155	C₂×C₄×C₅	#137	1	1	yes	characteristic
156	C₂×Dic₅	—	2	2	no	—
157	C₂×Dic₅	—	2	2	no	—
158	C₂×Dic₅	—	4	1	no	—
159	C₂×Dic₅	—	4	1	no	—
160	C₂×Dic₅	—	4	1	no	—
161	C₂×Dic₅	—	4	1	no	—
162	C₂×Dic₅	—	4	1	no	—
163	C₂×Dic₅	—	4	1	no	—
164	C₂×Dic₅	—	4	1	no	—
165	C₂×Dic₅	—	4	1	no	—
166	C₂×Dic₅	—	4	1	no	—
167	C₂×Dic₅	—	4	1	no	—
168	Dic₁₀	—	2	4	no	—
169	Dic₁₀	—	4	2	no	—
170	C₅×C₈	—	4	4	no	—
171	#168	—	8	2	no	—
172	C₂²×Q₈	—	5	1	no	—
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	—	10	2	no	—
175	C₄×Q₈	—	5	2	no	—
176	C₄×Q₈	—	5	2	no	—
177	C₄×Q₈	—	10	1	no	—
178	C₄×Q₈	—	10	1	no	—
179	C₂×C₄⋊C₄	—	5	1	no	—
180	C₂×C₄⋊C₄	—	5	1	no	—
181	C₂×C₄⋊C₄	—	5	1	no	—
182	C₂×C₄⋊C₄	—	10	1	no	—
183	C₂×C₄⋊C₄	—	10	1	no	—
184	C₂×C₄⋊C₄	—	10	1	no	—
185	C₂×C₄²	—	10	1	no	—
186	C₂×C₄²	—	10	1	no	—
187	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
188	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
189	C₂×C₄²	D₂₀	1	1	yes	characteristic
190	C₂×C₄²	D₂₀	1	1	yes	characteristic
191	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
192	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
193	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
194	C₄⋊C₈	—	2	4	no	—
195	C₄⋊C₈	—	10	2	no	—
196	C₄⋊C₈	—	10	2	no	—
197	C₄⋊C₈	—	20	2	no	—
198	C₄⋊C₈	—	20	2	no	—
199	C₄×C₈	—	2	2	no	—
200	C₄×C₈	—	2	2	no	—
201	#95	—	10	1	no	—
202	#95	—	10	1	no	—
203	#95	—	10	1	no	—
204	#95	—	10	1	no	—
205	#95	—	10	1	no	—
206	#95	—	10	1	no	—
207	#95	—	10	1	no	—
208	C₂²×C₅	#380	1	1	yes	characteristic
209	C₂²×C₅	#408	1	1	yes	characteristic
210	C₂²×C₅	#408	1	1	yes	characteristic
211	C₂²×C₅	C₂×#103	1	1	yes	characteristic
212	C₂²×C₅	#417	1	1	yes	characteristic
213	C₂²×C₅	C₂×#104	1	1	yes	characteristic
214	C₂²×C₅	#421	1	1	yes	characteristic
215	C₄×C₅	—	2	1	no	—
216	C₄×C₅	—	2	1	no	—
217	C₄×C₅	—	2	2	no	—
218	C₄×C₅	—	2	2	no	—
219	C₄×C₅	—	2	2	no	—
220	C₄×C₅	—	4	1	no	—
221	C₄×C₅	#424	1	2	yes	—
222	C₄×C₅	#433	1	2	yes	—
223	C₄×C₅	#460	1	1	yes	characteristic
224	C₄×C₅	#462	1	1	yes	characteristic
225	C₄×C₅	#475	1	1	yes	characteristic
226	C₄×C₅	#477	1	1	yes	characteristic
227	Dic₅	—	4	2	no	—
228	Dic₅	—	4	2	no	—
229	Dic₅	—	8	1	no	—
230	Dic₅	—	8	1	no	—
231	C₂×Q₈	—	5	2	no	—
232	C₂×Q₈	—	10	1	no	—
233	C₂×Q₈	—	10	2	no	—
234	C₂×Q₈	—	10	2	no	—
235	C₂²×C₄	—	10	1	no	—
236	C₂²×C₄	—	10	1	no	—
237	C₂²×C₄	—	10	1	no	—
238	C₂²×C₄	—	10	1	no	—
239	C₂²×C₄	#685	1	1	yes	characteristic
240	C₂²×C₄	#685	1	1	yes	characteristic
241	C₂²×C₄	#685	1	1	yes	characteristic
242	C₂²×C₄	#685	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	4	2	no	—
248	C₂×C₈	—	4	2	no	—
249	C₂×C₈	—	20	2	no	—
250	C₂×C₈	—	20	2	no	—
251	C₂×C₈	—	20	2	no	—
252	C₄⋊C₄	—	5	2	no	—
253	C₄⋊C₄	—	10	1	no	—
254	C₄⋊C₄	—	10	2	no	—
255	C₄⋊C₄	—	10	2	no	—
256	C₄⋊C₄	—	10	2	no	—
257	C₄⋊C₄	—	10	2	no	—
258	C₄⋊C₄	—	20	2	no	—
259	C₄⋊C₄	—	20	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	2	1	no	—
265	C₄²	—	4	1	no	—
266	C₄²	—	10	2	no	—
267	C₄²	—	10	2	no	—
268	C₄²	—	20	1	no	—
269	C₄²	—	20	1	no	—
270	C₄²	#677	1	2	yes	—
271	C₄²	Dic₂₀	1	2	yes	—
272	C₄²	D₄×D₅	1	1	yes	characteristic
273	C₄²	#711	1	1	yes	characteristic
274	C₄²	Q₈×D₅	1	1	yes	characteristic
275	C₄²	#713	1	1	yes	characteristic
276	C₂×C₅	#1481	1	1	yes	characteristic
277	C₂×C₅	#1483	1	1	yes	characteristic
278	C₂×C₅	#1493	1	1	yes	characteristic
279	C₂×C₅	#1495	1	1	yes	characteristic
280	C₂×C₅	#1798	1	1	yes	characteristic
281	C₂×C₅	#1803	1	1	yes	characteristic
282	C₂×C₅	#1806	1	1	yes	characteristic
283	C₂³	C₂×#685	1	1	yes	characteristic
284	Q₈	—	10	4	no	—
285	Q₈	—	20	2	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	10	2	no	—
298	C₂×C₄	—	10	2	no	—
299	C₂×C₄	—	20	1	no	—
300	C₂×C₄	—	20	1	no	—
301	C₂×C₄	—	20	1	no	—
302	C₂×C₄	—	20	1	no	—
303	C₂×C₄	—	20	1	no	—
304	C₂×C₄	—	20	1	no	—
305	C₂×C₄	—	20	1	no	—
306	C₂×C₄	—	20	1	no	—
307	C₂×C₄	—	20	1	no	—
308	C₂×C₄	—	20	1	no	—
309	C₂×C₄	#3937	1	2	yes	—
310	C₂×C₄	#3948	1	2	yes	—
311	C₂×C₄	#3948	1	2	yes	—
312	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
313	C₂×C₄	#4038	1	1	yes	characteristic
314	C₂×C₄	#4041	1	1	yes	characteristic
315	C₂×C₄	#4043	1	1	yes	characteristic
316	C₂×C₄	#4047	1	2	yes	—
317	C₂×C₄	C₂×#677	1	1	yes	characteristic
318	C₂×C₄	C₂×Dic₂₀	1	1	yes	characteristic
319	C₂×C₄	#4053	1	2	yes	—
320	C₂×C₄	#4054	1	1	yes	characteristic
321	C₂×C₄	#4055	1	1	yes	characteristic
322	C₂×C₄	#4084	1	1	yes	characteristic
323	C₂×C₄	#4092	1	1	yes	characteristic
324	C₈	—	4	4	no	—
325	C₈	—	40	2	no	—
326	C₅	#8568	1	1	yes	characteristic, complemented, Sylow
327	C₂²	C₂×#3937	1	1	yes	characteristic
328	C₂²	#65276	1	1	yes	characteristic
329	C₂²	#65294	1	1	yes	characteristic
330	C₂²	C₂×#3948	1	1	yes	characteristic
331	C₂²	#65399	1	1	yes	characteristic
332	C₂²	#65418	1	1	yes	characteristic
333	C₂²	#65422	1	1	yes	characteristic
334	C₄	—	2	2	no	—
335	C₄	—	2	2	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	2	no	—
339	C₄	—	4	1	no	—
340	C₄	—	20	2	no	—
341	C₄	—	20	2	no	—
342	C₄	—	40	1	no	—
343	C₄	—	40	1	no	—
344	C₄	#65128	1	2	yes	—
345	C₄	#65129	1	2	yes	—
346	C₄	#65137	1	1	yes	characteristic
347	C₄	#65138	1	1	yes	characteristic
348	C₄	#65139	1	1	yes	characteristic
349	C₄	#65140	1	1	yes	characteristic
350	C₂	#10600287	1	1	yes	characteristic
351	C₂	#10600531	1	1	yes	characteristic
352	C₂	#10600645	1	1	yes	characteristic
353	C₂	#10601269	1	1	yes	characteristic
354	C₂	#10601278	1	1	yes	characteristic
355	C₂	#10601557	1	1	yes	characteristic
356	C₂	#10601561	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₀
commutator	C₂²×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₄×C₅×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₄×C₄⋊C₈	D₅
hypocenter	C₅	#8568
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
326. C₅	#8568	semidirect

Maximal subgroups

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

subgroup	index
#10605031	2
C₄×C₅×C₄⋊C₈	2
C₂×#64676	2
#10600187	2
#10598146	2
#8568	5