Subgroups — #13043557

Order 1280.

646 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	D₅⋊#504	C₂	1	2	yes	maximal, complemented
2	#10611628	C₂	1	1	yes	maximal, characteristic
3	#10604862	C₂	1	1	yes	maximal, characteristic, complemented
4	#10604861	C₂	1	1	yes	maximal, characteristic, complemented
5	#10604776	C₂	1	2	yes	maximal, complemented
6	#10604084	C₂	1	1	yes	maximal, characteristic, complemented
7	#10604083	C₂	1	1	yes	maximal, characteristic, complemented
8	#10604000	C₂	1	2	yes	maximal, complemented
9	#10603999	C₂	1	2	yes	maximal, complemented
10	#10603873	C₂	1	1	yes	maximal, characteristic, complemented
11	#10603868	C₂	1	1	yes	maximal, characteristic, complemented
12	#66102	C₄	1	1	yes	characteristic
13	D₅×#131	C₂²	1	2	yes	—
14	#65791	C₂²	1	2	yes	complemented
15	#65791	C₂²	1	2	yes	complemented
16	#65725	—	2	2	no	—
17	#65725	—	2	2	no	—
18	#65725	C₂²	1	2	yes	complemented
19	#65725	C₂²	1	2	yes	complemented
20	D₅⋊M₃₂	—	2	2	no	—
21	D₅⋊M₃₂	C₂²	1	2	yes	—
22	C₂×#3990	C₂²	1	1	yes	characteristic
23	C₂×D₅⋊C₁₆	—	2	2	no	—
24	C₂×D₅⋊C₁₆	C₂²	1	1	yes	characteristic
25	C₅×#444	C₄	1	1	yes	characteristic
26	#65485	C₂²	1	1	yes	characteristic
27	#65434	C₄	1	2	yes	—
28	#65184	C₂²	1	1	yes	characteristic
29	#65182	C₄	1	1	yes	characteristic
30	#65113	C₂²	1	2	yes	—
31	#65110	C₄	1	2	yes	—
32	#65006	C₄	1	1	yes	characteristic
33	#64997	C₂²	1	1	yes	characteristic
34	#64932	C₂²	1	2	yes	complemented
35	#64932	C₂²	1	2	yes	complemented
36	#64898	C₂²	1	2	yes	complemented
37	#64898	C₂²	1	2	yes	complemented
38	#64889	C₂²	1	1	yes	characteristic, complemented
39	#64889	C₂²	1	1	yes	characteristic, complemented
40	#64886	C₂²	1	1	yes	characteristic, complemented
41	#64885	C₂²	1	1	yes	characteristic, complemented
42	#64861	C₂²	1	2	yes	complemented
43	#64859	C₂²	1	1	yes	characteristic, complemented
44	C₄×#674	C₂²	1	1	yes	characteristic, complemented
45	#33441	—	5	1	no	maximal, Sylow
46	D₅×#41	C₂×C₄	1	2	yes	—
47	C₅×#135	C₂×C₄	1	1	yes	characteristic
48	C₅×#131	C₂×C₄	1	2	yes	—
49	#4096	C₂×C₄	1	2	yes	—
50	#4094	C₂³	1	2	yes	—
51	#4064	C₈	1	2	yes	—
52	#4061	C₈	1	2	yes	—
53	#4060	C₈	1	2	yes	—
54	#4057	C₈	1	2	yes	—
55	#4053	—	2	2	no	—
56	#4053	C₂³	1	2	yes	—
57	D₅×M₁₆	—	2	2	no	—
58	D₅×M₁₆	C₂×C₄	1	2	yes	—
59	#4050	C₂×C₄	1	1	yes	characteristic
60	#4047	—	2	2	no	—
61	#4047	C₂×C₄	1	2	yes	—
62	C₂×#676	C₂×C₄	1	1	yes	characteristic
63	C₂×C₈×D₅	—	2	2	no	—
64	C₂×C₈×D₅	C₂³	1	1	yes	characteristic
65	#4018	C₂×C₄	1	1	yes	characteristic
66	#3998	—	2	2	no	—
67	#3998	—	2	2	no	—
68	#3998	C₂³	1	2	yes	—
69	#3998	C₂³	1	2	yes	—
70	C₂×#674	—	2	2	no	—
71	C₂×#674	—	2	2	no	—
72	C₂×#674	C₂³	1	1	yes	characteristic
73	C₂×#674	C₂³	1	1	yes	characteristic
74	C₂×#674	C₂³	1	1	yes	characteristic
75	C₂×#674	C₂³	1	1	yes	characteristic
76	#3990	—	2	2	no	—
77	#3990	—	2	2	no	—
78	#3990	—	2	2	no	—
79	#3990	—	2	2	no	—
80	D₅⋊C₁₆	—	2	2	no	—
81	D₅⋊C₁₆	—	2	2	no	—
82	D₅⋊C₁₆	—	2	2	no	—
83	D₅⋊C₁₆	D₄	1	2	yes	—
84	D₅⋊C₁₆	D₄	1	2	yes	—
85	C₅×#108	C₂×C₄	1	1	yes	characteristic
86	C₅×#104	C₂×C₄	1	2	yes	—
87	C₅×#97	C₂×C₄	1	1	yes	characteristic
88	#3969	C₂×C₄	1	2	yes	—
89	#3957	C₂×C₄	1	2	yes	—
90	#3954	C₂×C₄	1	2	yes	—
91	#3951	C₂×C₄	1	1	yes	characteristic
92	#3947	C₂×C₄	1	1	yes	characteristic
93	C₈×Dic₅	C₂³	1	1	yes	characteristic
94	#3937	C₂×C₄	1	2	yes	—
95	C₄×#168	C₂³	1	1	yes	characteristic
96	C₂×#504	—	5	2	no	—
97	#2955	—	5	1	no	—
98	#2171	—	5	1	no	—
99	#2171	—	5	1	no	—
100	#2171	—	5	2	no	—
101	#2170	—	5	1	no	—
102	#2170	—	5	1	no	—
103	#2170	—	5	2	no	—
104	#2153	—	5	1	no	—
105	#2117	—	5	2	no	—
106	#2108	—	5	1	no	—
107	C₅×#41	C₂²×C₄	1	2	yes	—
108	#713	C₂×C₈	1	2	yes	—
109	Q₈×D₅	C₂×C₈	1	2	yes	—
110	#711	—	2	2	no	—
111	#711	C₂×C₈	1	2	yes	—
112	D₄×D₅	—	2	2	no	—
113	D₄×D₅	C₂×C₈	1	2	yes	—
114	#709	—	2	2	no	—
115	#709	C₂²×C₄	1	2	yes	—
116	C₂×C₄×D₅	—	2	2	no	—
117	C₂×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
118	C₅×Q₁₆	C₂×C₈	1	1	yes	characteristic
119	C₅×SD₁₆	C₂×C₈	1	2	yes	—
120	C₅×D₈	C₂×C₈	1	1	yes	characteristic
121	C₅×M₁₆	—	2	2	no	—
122	C₅×M₁₆	C₂²×C₄	1	2	yes	—
123	C₂×C₅×C₈	—	2	2	no	—
124	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
126	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
127	#689	C₂×C₈	1	2	yes	—
128	#688	C₂×C₈	1	2	yes	—
129	#687	C₂×C₈	1	2	yes	—
130	#686	C₂×C₈	1	2	yes	—
131	#685	—	2	1	no	—
132	#683	—	2	1	no	—
133	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
134	#681	—	2	2	no	—
135	#681	C₂²×C₄	1	2	yes	—
136	C₂×#168	—	2	2	no	—
137	C₂×#168	C₂²×C₄	1	1	yes	characteristic
138	C₂×#168	C₂⁴	1	1	yes	characteristic
139	Dic₂₀	C₂×C₈	1	1	yes	characteristic
140	D₄₀	C₂×C₈	1	1	yes	characteristic
141	#677	C₂×C₈	1	2	yes	—
142	#676	—	2	2	no	—
143	#676	—	2	2	no	—
144	#676	—	2	2	no	—
145	#676	C₂×C₈	1	2	yes	—
146	#676	C₂×C₈	1	2	yes	—
147	C₈×D₅	—	2	2	no	—
148	C₈×D₅	—	2	2	no	—
149	C₈×D₅	—	2	2	no	—
150	C₈×D₅	C₂×D₄	1	2	yes	—
151	C₈×D₅	#41	1	2	yes	—
152	#674	—	2	2	no	—
153	#674	—	2	2	no	—
154	#674	—	2	1	no	—
155	#674	—	2	1	no	—
156	#674	C₂×D₄	1	1	yes	characteristic
157	#674	C₂×D₄	1	2	yes	—
158	#674	C₂×D₄	1	1	yes	characteristic
159	#575	—	5	1	no	—
160	C₂×#131	—	5	2	no	—
161	#504	—	5	2	no	—
162	#504	—	5	2	no	—
163	#504	—	5	2	no	—
164	#504	—	5	2	no	—
165	#504	—	10	2	no	—
166	#504	—	10	2	no	—
167	C₂×M₃₂	—	5	1	no	—
168	C₂×M₃₂	—	5	2	no	—
169	C₂×M₃₂	—	10	2	no	—
170	C₂²×C₁₆	—	5	1	no	—
171	C₂²×C₁₆	—	10	2	no	—
172	#444	—	5	2	no	—
173	#444	—	5	1	no	—
174	#444	AGL(1,𝔽₅)	1	1	yes	characteristic
175	#443	—	5	1	no	—
176	#443	—	5	1	no	—
177	#443	—	5	2	no	—
178	#430	—	5	1	no	—
179	#421	—	5	2	no	—
180	#405	—	5	1	no	—
181	#364	—	5	2	no	—
182	#364	—	5	1	no	—
183	#364	—	5	1	no	—
184	#350	—	5	2	no	—
185	#350	—	5	2	no	—
186	#350	—	5	2	no	—
187	#350	—	5	2	no	—
188	#346	—	5	1	no	—
189	#346	—	5	1	no	—
190	C₄×C₁₆	—	5	1	no	—
191	C₄×C₁₆	—	5	1	no	—
192	C₂²×D₅	—	2	2	no	—
193	C₂²×D₅	#131	1	1	yes	characteristic
194	C₅×Q₈	C₂²×C₈	1	2	yes	—
195	C₅×D₄	—	2	2	no	—
196	C₅×D₄	C₂²×C₈	1	2	yes	—
197	C₂×C₄×C₅	—	2	2	no	—
198	C₂×C₄×C₅	—	2	1	no	—
199	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
200	C₂²⋊D₅	—	2	2	no	—
201	C₂²⋊D₅	—	2	2	no	—
202	C₂²⋊D₅	—	2	2	no	—
203	C₂×Dic₅	—	2	2	no	—
204	C₂×Dic₅	—	2	1	no	—
205	C₂×Dic₅	#131	1	1	yes	characteristic
206	D₂₀	—	2	2	no	—
207	D₂₀	C₂²×C₈	1	2	yes	—
208	C₄×D₅	—	2	2	no	—
209	C₄×D₅	—	2	2	no	—
210	C₄×D₅	—	2	2	no	—
211	C₄×D₅	C₄×D₄	1	2	yes	—
212	C₄×D₅	C₂²×C₈	1	2	yes	—
213	Dic₁₀	—	2	2	no	—
214	Dic₁₀	C₂²×C₈	1	2	yes	—
215	C₅×C₈	—	2	2	no	—
216	C₅×C₈	C₄×D₄	1	2	yes	—
217	C₅×C₈	C₂²×C₈	1	1	yes	characteristic
218	C₅×C₈	C₂²×C₈	1	1	yes	characteristic
219	#168	—	2	2	no	—
220	#168	C₂²×C₈	1	2	yes	—
221	#168	C₂²×D₄	1	1	yes	characteristic
222	#168	C₂×#41	1	1	yes	characteristic
223	C₂×#41	—	5	2	no	—
224	#137	—	5	2	no	—
225	#137	—	5	2	no	—
226	AGL(1,ℤ/8)	—	5	2	no	—
227	AGL(1,ℤ/8)	—	5	2	no	—
228	#135	—	5	2	no	—
229	#135	—	5	1	no	—
230	#135	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
231	#131	—	5	2	no	—
232	#131	—	5	2	no	—
233	#131	—	5	2	no	—
234	#131	—	10	2	no	—
235	#131	—	10	2	no	—
236	#131	C₂×AGL(1,𝔽₅)	1	2	yes	—
237	C₂×M₁₆	—	5	1	no	—
238	C₂×M₁₆	—	5	2	no	—
239	C₂×M₁₆	—	10	2	no	—
240	C₂²×C₈	—	5	1	no	—
241	C₂²×C₈	—	10	2	no	—
242	#117	—	5	1	no	—
243	M₃₂	—	5	2	no	—
244	M₃₂	—	5	2	no	—
245	M₃₂	—	10	2	no	—
246	M₃₂	—	10	2	no	—
247	M₃₂	—	10	2	no	—
248	M₃₂	—	10	2	no	—
249	M₃₂	—	10	2	no	—
250	M₃₂	—	10	2	no	—
251	C₂×C₁₆	—	5	1	no	—
252	C₂×C₁₆	—	5	1	no	—
253	C₂×C₁₆	—	5	2	no	—
254	C₂×C₁₆	—	5	2	no	—
255	C₂×C₁₆	—	5	1	no	—
256	C₂×C₁₆	—	5	1	no	—
257	C₂×C₁₆	—	10	2	no	—
258	C₂×C₁₆	—	10	2	no	—
259	C₂×C₁₆	—	10	2	no	—
260	C₂×C₁₆	—	10	2	no	—
261	C₂×C₁₆	—	10	2	no	—
262	#108	—	5	2	no	—
263	#108	—	5	1	no	—
264	#108	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
265	#104	—	5	2	no	—
266	#104	—	5	2	no	—
267	#104	—	5	2	no	—
268	#104	C₂×AGL(1,𝔽₅)	1	2	yes	—
269	#97	—	5	1	no	—
270	#97	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
271	C₄×C₈	—	5	1	no	—
272	C₄×C₈	—	5	1	no	—
273	C₂²×C₅	—	2	2	no	—
274	C₂²×C₅	C₂×#131	1	1	yes	characteristic
275	C₂×D₅	—	2	2	no	—
276	C₂×D₅	—	4	1	no	—
277	C₂×D₅	—	4	4	no	—
278	C₂×D₅	C₈×D₄	1	2	yes	—
279	C₄×C₅	—	2	2	no	—
280	C₄×C₅	—	4	1	no	—
281	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
282	C₄×C₅	C₂³×C₈	1	1	yes	characteristic
283	Dic₅	—	2	2	no	—
284	Dic₅	—	4	1	no	—
285	Dic₅	C₈×D₄	1	2	yes	—
286	#41	—	5	2	no	—
287	#41	—	5	2	no	—
288	#41	—	5	2	no	—
289	#41	—	10	2	no	—
290	#41	—	10	2	no	—
291	#41	C₂²×AGL(1,𝔽₅)	1	2	yes	—
292	C₂×Q₈	—	5	2	no	—
293	C₂×D₄	—	5	2	no	—
294	C₂×D₄	—	10	2	no	—
295	C₂²×C₄	—	5	1	no	—
296	C₂²×C₄	—	10	2	no	—
297	Q₁₆	—	5	2	no	—
298	Q₁₆	—	5	1	no	—
299	Q₁₆	D₅⋊C₈	1	1	yes	characteristic
300	SD₁₆	—	5	2	no	—
301	SD₁₆	—	5	2	no	—
302	SD₁₆	—	5	2	no	—
303	SD₁₆	D₅⋊C₈	1	2	yes	—
304	D₈	—	5	2	no	—
305	D₈	—	5	1	no	—
306	D₈	D₅⋊C₈	1	1	yes	characteristic
307	M₁₆	—	2	2	no	—
308	M₁₆	—	5	2	no	—
309	M₁₆	—	5	2	no	—
310	M₁₆	—	5	2	no	—
311	M₁₆	—	10	2	no	—
312	M₁₆	—	10	2	no	—
313	M₁₆	—	10	2	no	—
314	M₁₆	—	10	2	no	—
315	M₁₆	C₂²×AGL(1,𝔽₅)	1	2	yes	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	5	1	no	—
318	C₂×C₈	—	5	1	no	—
319	C₂×C₈	—	5	2	no	—
320	C₂×C₈	—	5	2	no	—
321	C₂×C₈	—	10	2	no	—
322	C₂×C₈	—	10	2	no	—
323	C₂×C₈	—	10	2	no	—
324	C₂×C₈	—	10	2	no	—
325	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
326	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
327	C₄⋊C₄	—	10	1	no	—
328	C₂²⋊C₄	—	10	1	no	—
329	C₄²	—	5	1	no	—
330	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
331	C₁₆	—	5	1	no	—
332	C₁₆	—	5	2	no	—
333	C₁₆	—	5	1	no	—
334	C₁₆	—	10	1	no	—
335	C₁₆	—	10	1	no	—
336	C₁₆	—	10	2	no	—
337	C₁₆	—	10	2	no	—
338	C₂×C₅	—	2	1	no	—
339	C₂×C₅	—	4	2	no	—
340	C₂×C₅	C₂×C₈×D₄	1	1	yes	characteristic
341	D₅	—	2	2	no	—
342	D₅	—	4	2	no	—
343	C₂³	—	5	1	no	—
344	C₂³	—	10	2	no	—
345	Q₈	—	5	2	no	—
346	Q₈	—	10	2	no	—
347	Q₈	C₂×D₅⋊C₈	1	2	yes	—
348	D₄	—	2	2	no	—
349	D₄	—	5	2	no	—
350	D₄	—	10	2	no	—
351	D₄	—	10	2	no	—
352	D₄	—	10	2	no	—
353	D₄	—	10	2	no	—
354	D₄	C₂×D₅⋊C₈	1	2	yes	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	5	1	no	—
358	C₂×C₄	—	5	2	no	—
359	C₂×C₄	—	5	2	no	—
360	C₂×C₄	—	10	2	no	—
361	C₂×C₄	—	10	2	no	—
362	C₂×C₄	—	10	2	no	—
363	C₂×C₄	—	10	2	no	—
364	C₂×C₄	—	10	1	no	—
365	C₂×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
366	C₈	—	2	2	no	—
367	C₈	—	5	2	no	—
368	C₈	—	5	1	no	—
369	C₈	—	5	1	no	—
370	C₈	—	10	2	no	—
371	C₈	C₂×D₅⋊C₈	1	1	yes	characteristic
372	C₈	C₂×D₅⋊C₈	1	1	yes	characteristic
373	C₈	D₄×AGL(1,𝔽₅)	1	2	yes	—
374	C₅	#33441	1	1	yes	characteristic, complemented, Sylow
375	C₂²	—	2	2	no	—
376	C₂²	—	5	2	no	—
377	C₂²	—	10	2	no	—
378	C₂²	—	20	1	no	—
379	C₂²	—	20	4	no	—
380	C₂²	D₅⋊#131	1	1	yes	characteristic
381	C₄	—	2	2	no	—
382	C₄	—	4	1	no	—
383	C₄	—	5	2	no	—
384	C₄	—	10	2	no	—
385	C₄	—	20	1	no	—
386	C₄	C₂²×D₅⋊C₈	1	1	yes	characteristic
387	C₄	C₂×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
388	C₂	—	2	1	no	—
389	C₂	—	4	2	no	—
390	C₂	—	10	2	no	—
391	C₂	—	20	2	no	—
392	C₂	D₄×D₅⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×AGL(1,𝔽₅)
commutator	C₄×C₅	C₂³×C₈
Frattini	C₂×C₄	C₂³×AGL(1,𝔽₅)
Fitting	C₅×#444	C₄
socle	C₂×C₅	C₂×C₈×D₄
prime radical	#66102	C₄
cosocle	C₂×#168	C₂⁴
hypercenter	#444	AGL(1,𝔽₅)
hypocenter	C₅	#33441
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #444

Complements

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

N	H ≅ G/N	type
1. D₅⋊#504	C₂	semidirect
3. #10604862	C₂	semidirect
4. #10604861	C₂	semidirect
5. #10604776	C₂	semidirect
6. #10604084	C₂	semidirect
7. #10604083	C₂	semidirect
8. #10604000	C₂	semidirect
9. #10603999	C₂	semidirect
10. #10603873	C₂	semidirect
11. #10603868	C₂	semidirect
14. #65791	C₂²	semidirect
15. #65791	C₂²	semidirect
18. #65725	C₂²	semidirect
19. #65725	C₂²	semidirect
34. #64932	C₂²	semidirect
35. #64932	C₂²	semidirect
36. #64898	C₂²	semidirect
37. #64898	C₂²	semidirect
38. #64889	C₂²	semidirect
39. #64889	C₂²	semidirect
40. #64886	C₂²	semidirect
41. #64885	C₂²	semidirect
42. #64861	C₂²	semidirect
43. #64859	C₂²	semidirect
44. C₄×#674	C₂²	semidirect
374. C₅	#33441	semidirect

Maximal subgroups

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

subgroup	index
D₅⋊#504	2
#10611628	2
#10604862	2
#10604861	2
#10604776	2
#10604084	2
#10604083	2
#10604000	2
#10603999	2
#10603873	2
#10603868	2
#33441	5