Subgroups — #10617453

Order 640.

458 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#66264	C₂	1	1	yes	maximal, characteristic, complemented
2	#66260	C₂	1	1	yes	maximal, characteristic, complemented
3	#66111	C₂	1	1	yes	maximal, characteristic
4	D₅⋊#104	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#4010	C₂	1	1	yes	maximal, characteristic, complemented
6	#65741	C₂	1	2	yes	maximal, complemented
7	#65740	C₂	1	2	yes	maximal, complemented
8	#65737	C₂	1	2	yes	maximal, complemented
9	#65736	C₂	1	2	yes	maximal, complemented
10	#65729	C₂	1	1	yes	maximal, characteristic, complemented
11	M₁₆×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic, complemented
12	D₅×#41	C₂²	1	1	yes	characteristic
13	C₂×#713	C₂²	1	1	yes	characteristic
14	#4133	—	2	1	no	—
15	#4133	—	2	1	no	—
16	#4133	C₂²	1	2	yes	complemented
17	#4133	C₂²	1	2	yes	complemented
18	#4133	C₂²	1	2	yes	complemented
19	#4131	—	2	2	no	—
20	#4131	C₂²	1	2	yes	complemented
21	C₂×C₄×AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
22	D₅⋊M₁₆	—	2	2	no	—
23	D₅⋊M₁₆	—	2	1	no	—
24	D₅⋊M₁₆	C₂²	1	1	yes	characteristic
25	D₅⋊M₁₆	C₂²	1	1	yes	characteristic
26	C₂×#700	—	2	1	no	—
27	C₂×#700	C₂²	1	1	yes	characteristic
28	C₅×#137	C₄	1	1	yes	characteristic, complemented
29	#4095	C₄	1	1	yes	characteristic, complemented
30	#4088	C₄	1	1	yes	characteristic, complemented
31	#4065	C₂²	1	2	yes	—
32	#4064	C₄	1	2	yes	complemented
33	#4062	C₂²	1	2	yes	—
34	#4060	C₄	1	2	yes	complemented
35	#4054	C₄	1	1	yes	characteristic, complemented
36	D₅×M₁₆	C₂²	1	1	yes	characteristic
37	#4010	C₂²	1	1	yes	characteristic, complemented
38	#4010	C₂²	1	1	yes	characteristic, complemented
39	#4010	C₂²	1	2	yes	complemented
40	#4010	C₂²	1	2	yes	complemented
41	#4008	C₂²	1	1	yes	characteristic, complemented
42	#4008	C₂²	1	1	yes	characteristic, complemented
43	#3996	C₂²	1	2	yes	complemented
44	#3995	C₂²	1	2	yes	complemented
45	#3992	C₂²	1	2	yes	complemented
46	C₈×AGL(1,𝔽₅)	C₂²	1	2	yes	complemented
47	#2956	—	5	1	no	maximal, Sylow
48	C₂²×AGL(1,𝔽₅)	—	2	1	no	—
49	C₅×#41	C₂×C₄	1	1	yes	characteristic
50	C₂×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
51	#713	—	2	1	no	—
52	#713	—	2	1	no	—
53	#713	C₂×C₄	1	2	yes	—
54	#713	C₂³	1	2	yes	—
55	#713	C₂³	1	1	yes	characteristic
56	Q₈×D₅	C₂×C₄	1	1	yes	characteristic
57	#711	—	2	1	no	—
58	#711	C₂³	1	1	yes	characteristic
59	D₄×D₅	—	2	1	no	—
60	D₄×D₅	C₂×C₄	1	1	yes	characteristic
61	#709	—	2	1	no	—
62	#709	C₂×C₄	1	1	yes	characteristic
63	C₂×D₂₀	—	2	1	no	—
64	C₂×D₂₀	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×D₅	—	2	1	no	—
66	C₂×C₄×D₅	—	2	1	no	—
67	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
68	#704	—	2	1	no	—
69	#704	—	2	2	no	—
70	#704	D₄	1	2	yes	—
71	C₂×#170	—	2	1	no	—
72	C₂×#170	—	2	2	no	—
73	C₄×AGL(1,𝔽₅)	—	2	1	no	—
74	C₄×AGL(1,𝔽₅)	C₂³	1	1	yes	characteristic
75	C₄×AGL(1,𝔽₅)	C₂³	1	1	yes	characteristic
76	#700	—	2	1	no	—
77	#700	—	2	2	no	—
78	#700	—	2	2	no	—
79	#700	—	2	2	no	—
80	#700	D₄	1	1	yes	characteristic
81	#700	D₄	1	1	yes	characteristic
82	#700	C₂³	1	2	yes	—
83	#700	C₂³	1	1	yes	characteristic
84	#700	C₂³	1	1	yes	characteristic
85	D₅⋊C₈	—	2	1	no	—
86	D₅⋊C₈	—	2	2	no	—
87	D₅⋊C₈	—	2	2	no	—
88	D₅⋊C₈	—	2	2	no	—
89	D₅⋊C₈	C₂³	1	2	yes	—
90	C₅×Q₁₆	C₂×C₄	1	2	yes	complemented
91	C₅×SD₁₆	C₂×C₄	1	2	yes	complemented
92	C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
93	#689	C₂×C₄	1	2	yes	complemented
94	#688	C₂×C₄	1	2	yes	complemented
95	#688	C₂×C₄	1	2	yes	complemented
96	#686	C₂×C₄	1	2	yes	complemented
97	#681	C₂×C₄	1	1	yes	characteristic
98	D₄₀	C₂×C₄	1	2	yes	complemented
99	#677	C₂×C₄	1	2	yes	complemented
100	#676	C₂×C₄	1	2	yes	—
101	C₈×D₅	C₂³	1	2	yes	—
102	#575	—	5	1	no	—
103	#568	—	5	1	no	—
104	#568	—	5	1	no	—
105	#444	—	5	2	no	—
106	#444	—	5	2	no	—
107	#443	—	5	2	no	—
108	#443	—	5	2	no	—
109	#430	—	5	1	no	—
110	C₂×#104	—	5	1	no	—
111	C₂×#104	—	5	1	no	—
112	C₄×M₁₆	—	5	1	no	—
113	C₂²×D₅	—	2	1	no	—
114	C₂²×D₅	—	2	1	no	—
115	C₂²×D₅	#41	1	1	yes	characteristic
116	C₂×AGL(1,𝔽₅)	—	2	1	no	—
117	C₂×AGL(1,𝔽₅)	—	2	1	no	—
118	C₂×AGL(1,𝔽₅)	—	4	2	no	—
119	C₅×Q₈	—	2	1	no	—
120	C₅×Q₈	C₂²×C₄	1	1	yes	characteristic
121	C₅×Q₈	C₂²×C₄	1	2	yes	—
122	C₅×D₄	—	2	1	no	—
123	C₅×D₄	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₄×C₅	—	2	1	no	—
125	C₂×C₄×C₅	—	2	1	no	—
126	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
127	C₂²⋊D₅	—	2	1	no	—
128	C₂²⋊D₅	—	2	1	no	—
129	C₂²⋊D₅	—	2	1	no	—
130	C₂×Dic₅	—	2	1	no	—
131	C₂×Dic₅	C₂×D₄	1	1	yes	characteristic
132	D₂₀	—	2	2	no	—
133	D₂₀	—	2	2	no	—
134	D₂₀	—	2	1	no	—
135	D₂₀	—	2	1	no	—
136	D₂₀	C₂²×C₄	1	2	yes	—
137	D₂₀	C₂²×C₄	1	1	yes	characteristic
138	C₄×D₅	—	2	1	no	—
139	C₄×D₅	—	2	1	no	—
140	C₄×D₅	—	2	2	no	—
141	C₄×D₅	—	2	1	no	—
142	C₄×D₅	—	2	2	no	—
143	C₄×D₅	C₂²×C₄	1	1	yes	characteristic
144	C₄×D₅	C₂×D₄	1	1	yes	characteristic
145	C₄×D₅	#41	1	1	yes	characteristic
146	C₄×D₅	C₂⁴	1	1	yes	characteristic
147	Dic₁₀	—	2	1	no	—
148	Dic₁₀	C₂²×C₄	1	1	yes	characteristic
149	#170	—	2	1	no	—
150	#170	—	2	1	no	—
151	#170	—	2	2	no	—
152	#170	C₂×D₄	1	2	yes	—
153	#170	C₂×D₄	1	2	yes	—
154	C₅×C₈	C₂²×C₄	1	2	yes	—
155	#168	C₂²×C₄	1	2	yes	—
156	C₂×#41	—	5	1	no	—
157	C₂×#41	—	5	1	no	—
158	#137	—	5	1	no	—
159	#137	—	5	2	no	—
160	#137	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
161	AGL(1,ℤ/8)	—	5	1	no	—
162	AGL(1,ℤ/8)	—	5	2	no	—
163	AGL(1,ℤ/8)	—	5	1	no	—
164	#135	—	5	2	no	—
165	#135	—	5	2	no	—
166	#131	—	5	2	no	—
167	#131	—	5	2	no	—
168	#131	—	5	2	no	—
169	#131	—	5	2	no	—
170	#131	—	10	2	no	—
171	#131	—	10	1	no	—
172	#131	—	10	1	no	—
173	C₂×M₁₆	—	5	1	no	—
174	C₂×M₁₆	—	5	1	no	—
175	C₂×M₁₆	—	5	1	no	—
176	C₂×M₁₆	—	5	1	no	—
177	C₂×M₁₆	—	10	1	no	—
178	C₂×M₁₆	—	10	1	no	—
179	C₂×M₁₆	—	10	2	no	—
180	C₂×C₄²	—	5	1	no	—
181	#108	—	5	2	no	—
182	#108	—	5	2	no	—
183	#104	—	5	1	no	—
184	#104	—	5	1	no	—
185	#104	—	5	1	no	—
186	#104	—	5	2	no	—
187	#104	—	5	1	no	—
188	#104	—	5	2	no	—
189	#97	—	5	2	no	—
190	C₄×C₈	—	5	2	no	—
191	C₂²×C₅	—	2	1	no	—
192	C₂²×C₅	C₄×D₄	1	1	yes	characteristic
193	C₂×D₅	—	2	2	no	—
194	C₂×D₅	—	2	1	no	—
195	C₂×D₅	—	4	1	no	—
196	C₂×D₅	—	8	1	no	—
197	C₂×D₅	—	8	1	no	—
198	C₂×D₅	C₄×D₄	1	1	yes	characteristic
199	C₂×D₅	C₂×#41	1	1	yes	characteristic
200	AGL(1,𝔽₅)	—	2	2	no	—
201	AGL(1,𝔽₅)	—	4	1	no	—
202	C₄×C₅	—	2	1	no	—
203	C₄×C₅	—	2	2	no	—
204	C₄×C₅	C₄×D₄	1	1	yes	characteristic
205	C₄×C₅	C₂³×C₄	1	1	yes	characteristic
206	Dic₅	—	2	1	no	—
207	Dic₅	C₄×D₄	1	1	yes	characteristic
208	Dic₅	C₂²×D₄	1	1	yes	characteristic
209	#41	—	5	1	no	—
210	#41	—	5	2	no	—
211	#41	—	5	2	no	—
212	#41	—	5	1	no	—
213	#41	—	5	1	no	—
214	#41	—	10	1	no	—
215	#41	—	10	1	no	—
216	#41	—	10	1	no	—
217	#41	—	10	1	no	—
218	#41	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
219	C₂×Q₈	—	5	1	no	—
220	C₂×Q₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
221	C₂×D₄	—	5	1	no	—
222	C₂×D₄	—	5	1	no	—
223	C₂×D₄	—	10	1	no	—
224	C₂×D₄	—	10	1	no	—
225	C₂²×C₄	—	5	1	no	—
226	C₂²×C₄	—	10	1	no	—
227	C₂²×C₄	—	10	1	no	—
228	C₂²×C₄	—	10	1	no	—
229	Q₁₆	—	5	2	no	—
230	Q₁₆	C₂×AGL(1,𝔽₅)	1	2	yes	complemented
231	SD₁₆	—	5	2	no	—
232	SD₁₆	—	5	2	no	—
233	SD₁₆	—	5	2	no	—
234	SD₁₆	C₂×AGL(1,𝔽₅)	1	2	yes	complemented
235	D₈	—	5	2	no	—
236	D₈	—	5	2	no	—
237	M₁₆	—	5	1	no	—
238	M₁₆	—	5	1	no	—
239	M₁₆	—	5	2	no	—
240	M₁₆	—	5	1	no	—
241	M₁₆	—	5	2	no	—
242	M₁₆	—	5	1	no	—
243	M₁₆	—	5	1	no	—
244	M₁₆	—	5	2	no	—
245	M₁₆	—	10	2	no	—
246	M₁₆	—	10	2	no	—
247	M₁₆	—	10	2	no	—
248	M₁₆	—	10	2	no	—
249	M₁₆	—	10	1	no	—
250	M₁₆	—	10	1	no	—
251	M₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
252	C₂×C₈	—	5	2	no	—
253	C₂×C₈	—	5	2	no	—
254	C₂×C₈	—	10	2	no	—
255	C₂×C₈	—	10	2	no	—
256	C₂×C₈	—	10	2	no	—
257	C₂×C₈	—	10	2	no	—
258	C₂×C₈	—	10	1	no	—
259	C₂×C₈	—	10	1	no	—
260	C₄²	—	5	1	no	—
261	C₄²	—	5	1	no	—
262	C₄²	—	10	1	no	—
263	C₂×C₅	—	2	1	no	—
264	C₂×C₅	—	4	1	no	—
265	C₂×C₅	C₂×C₄×D₄	1	1	yes	characteristic
266	D₅	—	2	1	no	—
267	D₅	—	2	1	no	—
268	D₅	—	4	2	no	—
269	D₅	—	4	1	no	—
270	C₂³	—	5	1	no	—
271	C₂³	—	10	1	no	—
272	C₂³	—	10	1	no	—
273	Q₈	—	2	1	no	—
274	Q₈	—	5	1	no	—
275	Q₈	—	10	1	no	—
276	Q₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
277	Q₈	C₂²×AGL(1,𝔽₅)	1	2	yes	—
278	D₄	—	2	1	no	—
279	D₄	—	5	2	no	—
280	D₄	—	5	1	no	—
281	D₄	—	10	1	no	—
282	D₄	—	10	2	no	—
283	D₄	—	10	2	no	—
284	D₄	—	10	1	no	—
285	D₄	—	10	1	no	—
286	D₄	—	10	1	no	—
287	D₄	—	10	1	no	—
288	D₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	5	1	no	—
292	C₂×C₄	—	5	1	no	—
293	C₂×C₄	—	5	1	no	—
294	C₂×C₄	—	5	1	no	—
295	C₂×C₄	—	5	1	no	—
296	C₂×C₄	—	10	1	no	—
297	C₂×C₄	—	10	2	no	—
298	C₂×C₄	—	10	2	no	—
299	C₂×C₄	—	10	1	no	—
300	C₂×C₄	—	10	1	no	—
301	C₂×C₄	—	10	1	no	—
302	C₂×C₄	—	10	1	no	—
303	C₂×C₄	—	10	1	no	—
304	C₂×C₄	—	20	2	no	—
305	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
306	C₈	—	5	2	no	—
307	C₈	—	5	2	no	—
308	C₈	—	5	2	no	—
309	C₈	—	10	1	no	—
310	C₈	—	10	1	no	—
311	C₈	—	10	2	no	—
312	C₈	C₂²×AGL(1,𝔽₅)	1	2	yes	—
313	C₅	#2956	1	1	yes	characteristic, complemented, Sylow
314	C₂²	—	2	1	no	—
315	C₂²	—	5	1	no	—
316	C₂²	—	5	1	no	—
317	C₂²	—	10	2	no	—
318	C₂²	—	10	1	no	—
319	C₂²	—	20	1	no	—
320	C₂²	—	40	1	no	—
321	C₂²	—	40	1	no	—
322	C₂²	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	2	2	no	—
325	C₄	—	5	1	no	—
326	C₄	—	5	1	no	—
327	C₄	—	10	2	no	—
328	C₄	—	10	1	no	—
329	C₄	—	20	1	no	—
330	C₄	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
331	C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
332	C₂	—	2	1	no	—
333	C₂	—	4	1	no	—
334	C₂	—	10	1	no	—
335	C₂	—	10	1	no	—
336	C₂	—	20	2	no	—
337	C₂	—	20	1	no	—
338	C₂	C₂×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×AGL(1,𝔽₅)
commutator	C₄×C₅	C₂³×C₄
Frattini	C₄	C₂³×AGL(1,𝔽₅)
Fitting	C₅×#137	C₄
socle	C₂×C₅	C₂×C₄×D₄
prime radical	#66111	C₂
cosocle	C₄×D₅	C₂⁴
hypercenter	#137	AGL(1,𝔽₅)
hypocenter	C₅	#2956
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

Complements

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

N	H ≅ G/N	type
1. #66264	C₂	semidirect
2. #66260	C₂	semidirect
4. D₅⋊#104	C₂	semidirect
5. C₂×#4010	C₂	semidirect
6. #65741	C₂	semidirect
7. #65740	C₂	semidirect
8. #65737	C₂	semidirect
9. #65736	C₂	semidirect
10. #65729	C₂	semidirect
11. M₁₆×AGL(1,𝔽₅)	C₂	semidirect
16. #4133	C₂²	semidirect
17. #4133	C₂²	semidirect
18. #4133	C₂²	semidirect
20. #4131	C₂²	semidirect
28. C₅×#137	C₄	semidirect
29. #4095	C₄	semidirect
30. #4088	C₄	semidirect
32. #4064	C₄	semidirect
34. #4060	C₄	semidirect
35. #4054	C₄	semidirect
37. #4010	C₂²	semidirect
38. #4010	C₂²	semidirect
39. #4010	C₂²	semidirect
40. #4010	C₂²	semidirect
41. #4008	C₂²	semidirect
42. #4008	C₂²	semidirect
43. #3996	C₂²	semidirect
44. #3995	C₂²	semidirect
45. #3992	C₂²	semidirect
46. C₈×AGL(1,𝔽₅)	C₂²	semidirect
90. C₅×Q₁₆	C₂×C₄	semidirect
91. C₅×SD₁₆	C₂×C₄	semidirect
93. #689	C₂×C₄	semidirect
94. #688	C₂×C₄	semidirect
95. #688	C₂×C₄	semidirect
96. #686	C₂×C₄	semidirect
98. D₄₀	C₂×C₄	semidirect
99. #677	C₂×C₄	semidirect
160. #137	AGL(1,𝔽₅)	semidirect
230. Q₁₆	C₂×AGL(1,𝔽₅)	semidirect
234. SD₁₆	C₂×AGL(1,𝔽₅)	semidirect
313. C₅	#2956	semidirect

Maximal subgroups

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

subgroup	index
#66264	2
#66260	2
#66111	2
D₅⋊#104	2
C₂×#4010	2
#65741	2
#65740	2
#65737	2
#65736	2
#65729	2
M₁₆×AGL(1,𝔽₅)	2
#2956	5