Subgroups — #10617160

Order 640.

456 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₄×D₅	C₂	1	1	yes	maximal, characteristic
2	#65773	C₂	1	2	yes	maximal, complemented
3	#65756	C₂	1	1	yes	maximal, characteristic, complemented
4	#65755	C₂	1	1	yes	maximal, characteristic, complemented
5	D₅⋊(C₂²⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#4003	C₂	1	1	yes	maximal, characteristic, complemented
7	#65704	C₂	1	2	yes	maximal, complemented
8	#65694	C₂	1	2	yes	maximal, complemented
9	#65693	C₂	1	2	yes	maximal, complemented
10	#65684	C₂	1	1	yes	maximal, characteristic, complemented
11	#65683	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×D₅	C₄	1	1	yes	characteristic
13	C₂²×C₄×D₅	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₅	C₂²	1	1	yes	characteristic
15	C₂×#704	C₂²	1	2	yes	complemented
16	C₂²×#170	C₂²	1	2	yes	complemented
17	C₂×#700	C₂²	1	2	yes	complemented
18	C₂×D₅⋊C₈	C₂²	1	2	yes	complemented
19	C₄×C₅×D₄	C₄	1	1	yes	characteristic
20	D₄×Dic₅	C₂²	1	1	yes	characteristic
21	C₄×C₂²⋊D₅	C₄	1	1	yes	characteristic
22	C₄×C₂²⋊D₅	C₄	1	1	yes	characteristic
23	#4039	C₂²	1	1	yes	characteristic
24	D₅×C₄⋊C₄	C₄	1	1	yes	characteristic
25	#4027	C₂²	1	1	yes	characteristic
26	#4027	C₂²	1	1	yes	characteristic
27	D₅×C₂²⋊C₄	C₄	1	1	yes	characteristic
28	D₅×C₂²⋊C₄	C₄	1	1	yes	characteristic
29	C₄×D₂₀	C₄	1	1	yes	characteristic
30	C₄²×D₅	C₂²	1	1	yes	characteristic
31	C₂²⋊#170	C₂²	1	2	yes	complemented
32	C₂²⋊#170	C₂²	1	2	yes	complemented
33	#4004	C₂²	1	2	yes	complemented
34	#4003	C₂²	1	1	yes	characteristic
35	#4003	C₂²	1	1	yes	characteristic
36	#4003	C₂²	1	1	yes	characteristic
37	#4003	C₂²	1	1	yes	characteristic
38	#4003	C₂²	1	2	yes	complemented
39	#4003	C₂²	1	2	yes	complemented
40	#4002	C₂²	1	2	yes	complemented
41	#4002	C₂²	1	2	yes	complemented
42	C₄⋊#170	C₂²	1	2	yes	complemented
43	#2973	—	5	1	no	maximal, Sylow
44	C₂³×D₅	C₂×C₄	1	1	yes	characteristic
45	C₂³×D₅	C₂×C₄	1	1	yes	characteristic
46	C₂×C₅×D₄	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
49	C₂×C₂²⋊D₅	C₂×C₄	1	1	yes	characteristic
50	C₂×C₂²⋊D₅	C₂×C₄	1	1	yes	characteristic
51	C₂²×Dic₅	C₂³	1	1	yes	characteristic
52	C₂²×Dic₅	C₂³	1	1	yes	characteristic
53	D₄×D₅	—	2	4	no	—
54	C₂×D₂₀	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×D₅	—	4	2	no	—
56	C₂×C₄×D₅	—	4	2	no	—
57	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
66	#704	—	4	2	no	—
67	C₂×#170	—	4	2	no	—
68	C₂×#170	C₂³	1	2	yes	—
69	C₂×#170	C₂³	1	2	yes	—
70	C₂×#170	C₂³	1	2	yes	—
71	C₂×#170	C₂³	1	2	yes	—
72	#700	—	4	2	no	—
73	D₅⋊C₈	—	4	2	no	—
74	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
75	C₅×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
76	C₅×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
77	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
78	C₂²⋊Dic₅	C₂×C₄	1	1	yes	characteristic
79	C₂²⋊Dic₅	C₂×C₄	1	1	yes	characteristic
80	#685	C₂×C₄	1	1	yes	characteristic
81	#685	C₂×C₄	1	1	yes	characteristic
82	#685	C₂×C₄	1	1	yes	characteristic
83	#685	C₂×C₄	1	1	yes	characteristic
84	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
85	#683	C₂×C₄	1	1	yes	characteristic
86	#683	C₂×C₄	1	1	yes	characteristic
87	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
88	C₄×Dic₅	C₂³	1	1	yes	characteristic
89	C₄×Dic₅	C₂³	1	1	yes	characteristic
90	C₂×C₄×D₄	—	5	1	no	—
91	#435	—	5	2	no	—
92	#435	—	5	2	no	—
93	#435	—	5	2	no	—
94	#435	—	5	2	no	—
95	#432	—	5	1	no	—
96	#432	—	5	1	no	—
97	#407	—	5	1	no	—
98	#407	—	5	1	no	—
99	C₂×C₂²⋊C₈	—	5	1	no	—
100	C₂×C₂²⋊C₈	—	5	1	no	—
101	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
102	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
103	C₂²×D₅	—	2	2	no	—
104	C₂²×D₅	—	4	1	no	—
105	C₂²×D₅	—	4	1	no	—
106	C₂²×D₅	—	4	1	no	—
107	C₂²×D₅	—	4	1	no	—
108	C₂²×D₅	M₁₆	1	4	yes	complemented
109	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
110	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
111	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
112	C₅×D₄	—	2	2	no	—
113	C₂×C₄×C₅	—	4	1	no	—
114	C₂×C₄×C₅	—	4	1	no	—
115	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
120	C₂²⋊D₅	—	2	2	no	—
121	C₂²⋊D₅	—	2	2	no	—
122	C₂×Dic₅	—	2	2	no	—
123	C₂×Dic₅	—	2	2	no	—
124	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
125	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
126	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
127	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
128	C₂×Dic₅	C₂⁴	1	1	yes	characteristic
129	D₂₀	—	2	2	no	—
130	C₄×D₅	—	2	2	no	—
131	C₄×D₅	—	2	2	no	—
132	C₄×D₅	—	2	2	no	—
133	C₄×D₅	—	2	2	no	—
134	C₄×D₅	—	4	1	no	—
135	C₄×D₅	—	4	1	no	—
136	C₄×D₅	—	4	1	no	—
137	C₄×D₅	—	4	1	no	—
138	C₄×D₅	—	4	1	no	—
139	#170	—	2	2	no	—
140	#170	—	2	2	no	—
141	#170	—	2	2	no	—
142	#170	—	2	2	no	—
143	C₂²×D₄	—	5	1	no	—
144	C₂³×C₄	—	5	1	no	—
145	C₂³×C₄	—	5	1	no	—
146	C₂×M₁₆	—	5	2	no	—
147	C₂×M₁₆	—	5	2	no	—
148	C₂²×C₈	—	5	2	no	—
149	C₂²×C₈	—	5	2	no	—
150	C₄×D₄	—	5	1	no	—
151	C₄×D₄	—	5	1	no	—
152	C₄×D₄	—	5	1	no	—
153	C₄×D₄	—	5	1	no	—
154	C₄×D₄	—	5	1	no	—
155	C₄×D₄	—	5	1	no	—
156	C₄×D₄	—	5	1	no	—
157	C₄×D₄	AGL(1,𝔽₅)	1	1	yes	characteristic
158	C₂×C₄⋊C₄	—	5	1	no	—
159	C₂×C₂²⋊C₄	—	5	1	no	—
160	C₂×C₂²⋊C₄	—	5	1	no	—
161	C₂×C₄²	—	5	1	no	—
162	C₄⋊C₈	—	5	2	no	—
163	C₄⋊C₈	—	5	2	no	—
164	C₂²⋊C₈	—	5	1	no	—
165	C₂²⋊C₈	—	5	1	no	—
166	C₂²⋊C₈	—	5	1	no	—
167	C₂²⋊C₈	—	5	1	no	—
168	C₂²⋊C₈	—	5	2	no	—
169	C₂²⋊C₈	—	5	2	no	—
170	C₂²⋊C₈	—	5	2	no	—
171	C₂²⋊C₈	—	5	2	no	—
172	#97	—	5	2	no	—
173	#97	—	5	2	no	—
174	C₂²×C₅	—	2	1	no	—
175	C₂²×C₅	—	2	1	no	—
176	C₂²×C₅	—	2	1	no	—
177	C₂²×C₅	—	2	1	no	—
178	C₂²×C₅	—	2	1	no	—
179	C₂²×C₅	C₂×M₁₆	1	2	yes	—
180	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
181	C₂×D₅	—	2	1	no	—
182	C₂×D₅	—	2	1	no	—
183	C₂×D₅	—	2	1	no	—
184	C₂×D₅	—	2	1	no	—
185	C₂×D₅	—	2	1	no	—
186	C₂×D₅	—	2	1	no	—
187	C₂×D₅	—	2	1	no	—
188	C₂×D₅	—	4	4	no	—
189	C₂×D₅	—	4	4	no	—
190	C₂×D₅	C₂×M₁₆	1	2	yes	—
191	C₂×D₅	C₂×M₁₆	1	2	yes	—
192	C₄×C₅	—	2	1	no	—
193	C₄×C₅	—	2	1	no	—
194	C₄×C₅	—	2	1	no	—
195	C₄×C₅	—	2	1	no	—
196	C₄×C₅	—	2	1	no	—
197	Dic₅	—	2	1	no	—
198	Dic₅	—	2	1	no	—
199	Dic₅	—	2	1	no	—
200	Dic₅	—	2	1	no	—
201	Dic₅	H(5,𝔽₂)	1	2	yes	—
202	C₂⁴	—	5	1	no	—
203	C₂⁴	—	5	1	no	—
204	C₂×D₄	—	5	1	no	—
205	C₂×D₄	—	5	1	no	—
206	C₂×D₄	—	5	1	no	—
207	C₂×D₄	—	10	4	no	—
208	C₂×D₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
209	C₂²×C₄	—	5	1	no	—
210	C₂²×C₄	—	5	1	no	—
211	C₂²×C₄	—	5	1	no	—
212	C₂²×C₄	—	5	1	no	—
213	C₂²×C₄	—	5	1	no	—
214	C₂²×C₄	—	5	1	no	—
215	C₂²×C₄	—	5	1	no	—
216	C₂²×C₄	—	5	1	no	—
217	C₂²×C₄	—	5	1	no	—
218	C₂²×C₄	—	5	1	no	—
219	C₂²×C₄	—	5	1	no	—
220	C₂²×C₄	—	20	2	no	—
221	C₂²×C₄	—	20	2	no	—
222	C₂²×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
223	C₂²×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
224	M₁₆	—	20	2	no	—
225	M₁₆	—	20	2	no	—
226	C₂×C₈	—	5	2	no	—
227	C₂×C₈	—	5	2	no	—
228	C₂×C₈	—	5	2	no	—
229	C₂×C₈	—	5	2	no	—
230	C₂×C₈	—	20	2	no	—
231	C₂×C₈	—	20	2	no	—
232	C₄⋊C₄	—	5	1	no	—
233	C₄⋊C₄	—	5	1	no	—
234	C₄⋊C₄	—	5	1	no	—
235	C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
236	C₂²⋊C₄	—	5	1	no	—
237	C₂²⋊C₄	—	5	1	no	—
238	C₂²⋊C₄	—	5	1	no	—
239	C₂²⋊C₄	—	5	1	no	—
240	C₂²⋊C₄	—	5	1	no	—
241	C₂²⋊C₄	—	5	1	no	—
242	C₂²⋊C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
243	C₂²⋊C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
244	C₄²	—	5	1	no	—
245	C₄²	—	5	1	no	—
246	C₄²	—	5	1	no	—
247	C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
248	C₂×C₅	—	2	2	no	—
249	C₂×C₅	—	4	1	no	—
250	C₂×C₅	#518	1	1	yes	characteristic
251	C₂×C₅	C₂²×M₁₆	1	1	yes	characteristic
252	C₂×C₅	#568	1	1	yes	characteristic
253	D₅	—	2	2	no	—
254	D₅	—	2	2	no	—
255	D₅	—	4	1	no	—
256	C₂³	—	5	1	no	—
257	C₂³	—	5	1	no	—
258	C₂³	—	5	4	no	—
259	C₂³	—	5	1	no	—
260	C₂³	—	10	2	no	—
261	C₂³	—	20	1	no	—
262	C₂³	—	20	1	no	—
263	C₂³	—	20	1	no	—
264	C₂³	—	20	1	no	—
265	C₂³	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
266	C₂³	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
267	D₄	—	2	2	no	—
268	D₄	—	10	2	no	—
269	D₄	—	10	2	no	—
270	D₄	—	10	2	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	5	1	no	—
274	C₂×C₄	—	5	1	no	—
275	C₂×C₄	—	5	1	no	—
276	C₂×C₄	—	5	1	no	—
277	C₂×C₄	—	5	1	no	—
278	C₂×C₄	—	10	2	no	—
279	C₂×C₄	—	10	2	no	—
280	C₂×C₄	—	10	2	no	—
281	C₂×C₄	—	10	2	no	—
282	C₂×C₄	—	10	2	no	—
283	C₂×C₄	—	10	2	no	—
284	C₂×C₄	—	20	1	no	—
285	C₂×C₄	—	20	1	no	—
286	C₂×C₄	—	20	1	no	—
287	C₂×C₄	—	20	1	no	—
288	C₂×C₄	—	20	1	no	—
289	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
290	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
291	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
292	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
293	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
294	C₈	—	10	2	no	—
295	C₈	—	10	2	no	—
296	C₈	—	10	2	no	—
297	C₈	—	10	2	no	—
298	C₅	#2973	1	1	yes	characteristic, complemented, Sylow
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	5	2	no	—
305	C₂²	—	5	2	no	—
306	C₂²	—	10	1	no	—
307	C₂²	—	10	1	no	—
308	C₂²	—	10	1	no	—
309	C₂²	—	10	1	no	—
310	C₂²	—	10	1	no	—
311	C₂²	—	10	1	no	—
312	C₂²	—	10	1	no	—
313	C₂²	—	20	4	no	—
314	C₂²	—	20	4	no	—
315	C₂²	D₅⋊M₁₆	1	2	yes	—
316	C₂²	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	5	2	no	—
323	C₄	—	10	1	no	—
324	C₄	—	10	1	no	—
325	C₄	—	10	1	no	—
326	C₄	—	10	1	no	—
327	C₂	—	2	2	no	—
328	C₂	—	4	1	no	—
329	C₂	—	10	2	no	—
330	C₂	—	10	2	no	—
331	C₂	—	20	1	no	—
332	C₂	C₂×D₅⋊M₁₆	1	1	yes	characteristic
333	C₂	#66258	1	1	yes	characteristic
334	C₂	#66264	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₄×C₅×D₄	C₄
prime radical	C₂×D₄×D₅	C₄
cosocle	C₂×Dic₅	C₂⁴
hypercenter	C₄×D₄	AGL(1,𝔽₅)
hypocenter	C₅	#2973
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. #65773	C₂	semidirect
3. #65756	C₂	semidirect
4. #65755	C₂	semidirect
5. D₅⋊(C₂²⋊C₈)	C₂	semidirect
6. C₂×#4003	C₂	semidirect
7. #65704	C₂	semidirect
8. #65694	C₂	semidirect
9. #65693	C₂	semidirect
10. #65684	C₂	semidirect
11. #65683	C₂	semidirect
15. C₂×#704	C₂²	semidirect
16. C₂²×#170	C₂²	semidirect
17. C₂×#700	C₂²	semidirect
18. C₂×D₅⋊C₈	C₂²	semidirect
31. C₂²⋊#170	C₂²	semidirect
32. C₂²⋊#170	C₂²	semidirect
33. #4004	C₂²	semidirect
38. #4003	C₂²	semidirect
39. #4003	C₂²	semidirect
40. #4002	C₂²	semidirect
41. #4002	C₂²	semidirect
42. C₄⋊#170	C₂²	semidirect
108. C₂²×D₅	M₁₆	semidirect
298. C₅	#2973	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₄×D₅	2
#65773	2
#65756	2
#65755	2
D₅⋊(C₂²⋊C₈)	2
C₂×#4003	2
#65704	2
#65694	2
#65693	2
#65684	2
#65683	2
#2973	5