Subgroups — #12069142

Order 1280.

614 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10611694	C₂	1	2	yes	maximal
2	C₄×D₅×Q₁₆	C₂	1	1	yes	maximal, characteristic
3	C₅×#1593	C₂	1	1	yes	maximal, characteristic, complemented
4	#10601744	C₂	1	2	yes	maximal, complemented
5	#10601699	C₂	1	1	yes	maximal, characteristic, complemented
6	#10599320	C₂	1	1	yes	maximal, characteristic
7	#10599317	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599024	C₂	1	2	yes	maximal
9	#10599018	C₂	1	2	yes	maximal, complemented
10	#10598882	C₂	1	1	yes	maximal, characteristic, complemented
11	#10598861	C₂	1	1	yes	maximal, characteristic
12	C₂×D₅×Q₁₆	C₄	1	1	yes	characteristic
13	C₄×Q₈×D₅	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×#689	C₂²	1	2	yes	—
18	#65313	C₂²	1	2	yes	—
19	D₅×#107	C₄	1	1	yes	characteristic
20	#65162	C₄	1	1	yes	characteristic
21	#65124	—	2	2	no	—
22	#65122	C₂²	1	2	yes	—
23	#65120	C₂²	1	2	yes	—
24	#65117	—	2	2	no	—
25	D₅×#103	C₄	1	2	yes	—
26	#65079	C₄	1	2	yes	—
27	#64994	—	2	2	no	—
28	#64990	—	2	2	no	—
29	C₄×Dic₂₀	C₂²	1	1	yes	characteristic
30	#64974	C₂²	1	1	yes	characteristic
31	#64974	C₂²	1	1	yes	characteristic
32	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
33	#64968	C₂²	1	2	yes	—
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	#64704	C₂²	1	2	yes	—
39	#64699	C₂²	1	2	yes	—
40	#64677	C₂²	1	1	yes	characteristic
41	#64676	C₂²	1	2	yes	—
42	#64675	C₂²	1	1	yes	characteristic
43	C₈×#168	C₂²	1	1	yes	characteristic
44	C₈×#168	C₂²	1	1	yes	characteristic
45	#17897	—	5	1	no	maximal, Sylow
46	C₂×Q₈×D₅	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×#689	C₂×C₄	1	2	yes	—
51	D₅×Q₁₆	—	2	4	no	—
52	C₂×Dic₂₀	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
54	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
55	D₅×C₄⋊C₄	—	2	2	no	—
56	D₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
57	#4033	—	2	2	no	—
58	#4033	C₂×C₄	1	2	yes	—
59	C₄²×D₅	—	2	2	no	—
60	C₄²×D₅	C₂³	1	1	yes	characteristic
61	C₄×Dic₁₀	—	2	2	no	—
62	C₄×Dic₁₀	C₂³	1	2	yes	—
63	C₅×#107	C₂×C₄	1	1	yes	characteristic
64	C₅×C₄⋊C₈	—	2	2	no	—
65	C₅×C₄⋊C₈	C₂³	1	2	yes	—
66	C₅×#103	C₂×C₄	1	2	yes	—
67	C₅×#97	—	2	2	no	—
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	#3967	C₂×C₄	1	2	yes	—
72	#3952	—	2	2	no	—
73	#3952	—	2	1	no	—
74	#3952	D₄	1	2	yes	—
75	#3950	C₂×C₄	1	1	yes	characteristic
76	#3948	C₂×C₄	1	2	yes	—
77	#3947	—	2	2	no	—
78	#3947	—	2	2	no	—
79	#3946	—	2	2	no	—
80	#3946	—	2	1	no	—
81	#3946	—	2	2	no	—
82	#3946	D₄	1	2	yes	—
83	C₈×Dic₅	—	2	2	no	—
84	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
85	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
86	#3942	C₂×C₄	1	2	yes	—
87	#3939	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	#2983	—	5	2	no	—
95	C₂×C₄×Q₁₆	—	5	1	no	—
96	#1593	—	5	1	no	—
97	#1593	—	5	1	no	—
98	#1593	—	5	1	no	—
99	#1593	D₅	1	1	yes	characteristic, complemented
100	#1591	—	5	2	no	—
101	#1591	—	5	2	no	—
102	#1564	—	5	1	no	—
103	#1479	—	5	2	no	—
104	#1450	—	5	1	no	—
105	C₂×C₅×Q₈	C₂²×C₄	1	2	yes	—
106	Q₈×D₅	—	2	4	no	—
107	Q₈×D₅	—	2	4	no	—
108	C₂×C₄×D₅	—	2	2	no	—
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	C₂×Dic₁₀	—	2	2	no	—
114	C₂×Dic₁₀	C₂²×C₄	1	2	yes	—
115	C₅×Q₁₆	—	2	2	no	—
116	C₂×C₅×C₈	—	2	2	no	—
117	C₂×C₅×C₈	—	2	1	no	—
118	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
121	C₅×C₄⋊C₄	—	2	2	no	—
122	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
123	C₄²×C₅	—	2	2	no	—
124	C₄²×C₅	C₂⁴	1	1	yes	characteristic
125	#689	—	2	4	no	—
126	C₄⋊Dic₅	—	2	2	no	—
127	C₄⋊Dic₅	C₂²×C₄	1	2	yes	—
128	#683	—	2	2	no	—
129	#683	—	2	2	no	—
130	#683	—	2	2	no	—
131	C₄×Dic₅	—	2	2	no	—
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	Dic₂₀	—	2	2	no	—
143	C₈×D₅	—	2	2	no	—
144	C₈×D₅	M₁₆	1	4	yes	—
145	C₂²×Q₁₆	—	5	1	no	—
146	C₂×C₄×Q₈	—	5	2	no	—
147	#446	—	5	2	no	—
148	#446	—	5	2	no	—
149	#446	—	5	2	no	—
150	#446	—	10	2	no	—
151	#446	—	10	2	no	—
152	#446	C₂×D₅	1	2	yes	—
153	C₄×Q₁₆	—	5	1	no	—
154	C₄×Q₁₆	—	5	2	no	—
155	C₄×Q₁₆	—	5	1	no	—
156	C₄×Q₁₆	—	5	2	no	—
157	C₄×Q₁₆	—	5	1	no	—
158	C₄×Q₁₆	C₂×D₅	1	1	yes	characteristic
159	#432	—	10	2	no	—
160	#432	—	10	2	no	—
161	#431	—	5	1	no	—
162	#431	—	5	1	no	—
163	#431	—	5	2	no	—
164	C₂×#107	—	5	1	no	—
165	C₂×#103	—	5	2	no	—
166	C₂×C₄×C₈	—	5	1	no	—
167	#335	—	5	2	no	—
168	#334	—	5	1	no	—
169	#334	C₂×D₅	1	1	yes	characteristic
170	Q₈⋊C₈	—	5	2	no	—
171	Q₈⋊C₈	—	5	2	no	—
172	Q₈⋊C₈	—	5	2	no	—
173	Q₈⋊C₈	C₂×D₅	1	2	yes	—
174	#322	—	5	1	no	—
175	#322	C₂×D₅	1	1	yes	characteristic
176	C₈²	—	5	1	no	—
177	C₈²	—	5	1	no	—
178	C₂²×D₅	C₄×D₄	1	1	yes	characteristic
179	C₅×Q₈	—	2	2	no	—
180	C₅×Q₈	—	2	2	no	—
181	C₂×C₄×C₅	—	2	2	no	—
182	C₂×C₄×C₅	—	2	2	no	—
183	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
184	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
186	C₂×Dic₅	—	2	2	no	—
187	C₂×Dic₅	—	2	2	no	—
188	C₂×Dic₅	C₄×D₄	1	1	yes	characteristic
189	C₂×Dic₅	C₄×D₄	1	1	yes	characteristic
190	C₂×Dic₅	C₄×D₄	1	1	yes	characteristic
191	C₄×D₅	—	2	2	no	—
192	C₄×D₅	—	4	1	no	—
193	C₄×D₅	—	4	4	no	—
194	C₄×D₅	—	8	2	no	—
195	C₄×D₅	C₂×M₁₆	1	2	yes	—
196	C₄×D₅	#131	1	2	yes	—
197	Dic₁₀	—	2	2	no	—
198	Dic₁₀	—	2	2	no	—
199	Dic₁₀	—	4	4	no	—
200	C₅×C₈	—	2	1	no	—
201	C₅×C₈	—	2	2	no	—
202	C₅×C₈	—	4	1	no	—
203	C₅×C₈	—	4	2	no	—
204	C₅×C₈	C₂×M₁₆	1	2	yes	—
205	#168	—	2	1	no	—
206	#168	—	2	2	no	—
207	#168	—	4	1	no	—
208	#168	—	4	2	no	—
209	#168	C₂×M₁₆	1	2	yes	—
210	C₂²×Q₈	—	5	2	no	—
211	C₂×Q₁₆	—	5	2	no	—
212	C₂×Q₁₆	—	5	1	no	—
213	C₂×Q₁₆	—	10	4	no	—
214	C₂×Q₁₆	C₄×D₅	1	1	yes	characteristic
215	C₂²×C₈	—	5	1	no	—
216	C₂²×C₈	—	5	1	no	—
217	C₄×Q₈	—	5	2	no	—
218	C₄×Q₈	—	5	2	no	—
219	C₄×Q₈	—	5	2	no	—
220	C₄×Q₈	—	10	2	no	—
221	C₄×Q₈	—	10	2	no	—
222	C₄×Q₈	C₂²×D₅	1	2	yes	—
223	C₂×C₄⋊C₄	—	5	2	no	—
224	C₂×C₄⋊C₄	—	10	2	no	—
225	C₂×C₄²	—	5	1	no	—
226	C₂×C₄²	—	10	2	no	—
227	#107	—	5	1	no	—
228	#107	—	5	2	no	—
229	#107	C₄×D₅	1	1	yes	characteristic
230	C₄⋊C₈	—	2	2	no	—
231	C₄⋊C₈	—	5	2	no	—
232	C₄⋊C₈	—	5	2	no	—
233	C₄⋊C₈	—	10	2	no	—
234	C₄⋊C₈	—	10	1	no	—
235	C₄⋊C₈	—	10	2	no	—
236	C₄⋊C₈	—	10	2	no	—
237	C₄⋊C₈	C₂²×D₅	1	2	yes	—
238	#103	—	5	2	no	—
239	#103	—	5	2	no	—
240	#103	—	5	2	no	—
241	#103	C₄×D₅	1	2	yes	—
242	C₂²⋊C₈	—	5	2	no	—
243	C₂²⋊C₈	—	10	1	no	—
244	C₂²⋊C₈	—	10	2	no	—
245	#97	—	2	2	no	—
246	#97	—	10	2	no	—
247	#97	—	10	2	no	—
248	#97	—	10	2	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	1	no	—
254	C₄×C₈	—	10	2	no	—
255	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
256	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
257	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
258	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
259	C₂×D₅	—	2	1	no	—
260	C₂×D₅	—	2	1	no	—
261	C₂×D₅	#436	1	2	yes	—
262	C₄×C₅	—	2	1	no	—
263	C₄×C₅	—	4	2	no	—
264	C₄×C₅	—	4	2	no	—
265	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
266	C₄×C₅	C₂×#131	1	1	yes	characteristic
267	C₄×C₅	#578	1	2	yes	—
268	Dic₅	—	2	1	no	—
269	Dic₅	—	2	1	no	—
270	Dic₅	—	4	2	no	—
271	Dic₅	—	4	2	no	—
272	Dic₅	#436	1	2	yes	—
273	C₂×Q₈	—	5	2	no	—
274	C₂×Q₈	—	10	2	no	—
275	C₂×Q₈	—	10	4	no	—
276	C₂×Q₈	—	10	4	no	—
277	C₂×Q₈	C₂×C₄×D₅	1	2	yes	—
278	C₂²×C₄	—	5	1	no	—
279	C₂²×C₄	—	5	1	no	—
280	C₂²×C₄	—	5	1	no	—
281	C₂²×C₄	—	10	2	no	—
282	C₂²×C₄	—	10	2	no	—
283	Q₁₆	—	2	2	no	—
284	Q₁₆	—	10	4	no	—
285	Q₁₆	—	10	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	5	1	no	—
289	C₂×C₈	—	5	4	no	—
290	C₂×C₈	—	5	1	no	—
291	C₂×C₈	—	5	2	no	—
292	C₂×C₈	—	10	2	no	—
293	C₂×C₈	—	10	2	no	—
294	C₂×C₈	—	10	1	no	—
295	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
296	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
297	C₂×C₈	D₄×D₅	1	2	yes	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	5	2	no	—
300	C₄⋊C₄	—	10	2	no	—
301	C₄⋊C₄	—	10	2	no	—
302	C₄⋊C₄	—	10	2	no	—
303	C₄⋊C₄	—	10	2	no	—
304	C₄⋊C₄	C₂×C₄×D₅	1	2	yes	—
305	C₄²	—	2	2	no	—
306	C₄²	—	5	1	no	—
307	C₄²	—	5	1	no	—
308	C₄²	—	5	1	no	—
309	C₄²	—	10	2	no	—
310	C₄²	—	10	2	no	—
311	C₄²	—	10	2	no	—
312	C₄²	C₂³×D₅	1	1	yes	characteristic
313	C₂×C₅	C₂×#436	1	1	yes	characteristic
314	C₂×C₅	#2950	1	1	yes	characteristic
315	C₂×C₅	#2957	1	1	yes	characteristic
316	D₅	—	2	2	no	—
317	C₂³	—	5	1	no	—
318	Q₈	—	2	2	no	—
319	Q₈	—	2	2	no	—
320	Q₈	—	10	2	no	—
321	Q₈	—	10	2	no	—
322	Q₈	—	20	4	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	5	1	no	—
326	C₂×C₄	—	5	2	no	—
327	C₂×C₄	—	5	2	no	—
328	C₂×C₄	—	5	1	no	—
329	C₂×C₄	—	5	1	no	—
330	C₂×C₄	—	10	2	no	—
331	C₂×C₄	—	10	2	no	—
332	C₂×C₄	—	10	2	no	—
333	C₂×C₄	—	20	1	no	—
334	C₂×C₄	—	20	4	no	—
335	C₂×C₄	—	40	2	no	—
336	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
337	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
338	C₂×C₄	D₅×#41	1	1	yes	characteristic
339	C₈	—	2	1	no	—
340	C₈	—	2	2	no	—
341	C₈	—	4	1	no	—
342	C₈	—	4	2	no	—
343	C₈	—	5	2	no	—
344	C₈	—	10	1	no	—
345	C₈	—	10	2	no	—
346	C₈	—	20	1	no	—
347	C₈	—	20	2	no	—
348	C₈	D₅×M₁₆	1	2	yes	—
349	C₅	#17897	1	1	yes	characteristic, complemented, Sylow
350	C₂²	—	5	2	no	—
351	C₂²	—	10	1	no	—
352	C₂²	—	10	1	no	—
353	C₂²	C₄×D₄×D₅	1	1	yes	characteristic
354	C₄	—	2	1	no	—
355	C₄	—	4	2	no	—
356	C₄	—	4	2	no	—
357	C₄	—	5	2	no	—
358	C₄	—	10	1	no	—
359	C₄	—	10	1	no	—
360	C₄	—	20	2	no	—
361	C₄	—	20	2	no	—
362	C₄	C₂×D₅×M₁₆	1	1	yes	characteristic
363	C₄	D₅×#131	1	1	yes	characteristic
364	C₄	#66113	1	2	yes	—
365	C₂	—	10	2	no	—
366	C₂	D₅×#436	1	1	yes	characteristic
367	C₂	#10611562	1	1	yes	characteristic
368	C₂	#10611630	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	C₂²×D₅	C₄×D₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	#1593	D₅
hypocenter	C₅	#17897
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
3. C₅×#1593	C₂	semidirect
4. #10601744	C₂	semidirect
5. #10601699	C₂	semidirect
7. #10599317	C₂	semidirect
9. #10599018	C₂	semidirect
10. #10598882	C₂	semidirect
99. #1593	D₅	semidirect
349. C₅	#17897	semidirect

Maximal subgroups

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

subgroup	index
#10611694	2
C₄×D₅×Q₁₆	2
C₅×#1593	2
#10601744	2
#10601699	2
#10599320	2
#10599317	2
#10599024	2
#10599018	2
#10598882	2
#10598861	2
#17897	5