Subgroups — #10611274

Order 640.

452 conjugacy classes of subgroups.

The subgroup lattice has 276 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	C₅×#436	C₂	1	1	yes	maximal, characteristic, complemented
3	#65417	C₂	1	2	yes	maximal, complemented
4	#65415	C₂	1	2	yes	maximal, complemented
5	#65301	C₂	1	1	yes	maximal, characteristic, complemented
6	#65127	C₂	1	1	yes	maximal, characteristic, complemented
7	#65122	C₂	1	1	yes	maximal, characteristic, complemented
8	#65018	C₂	1	2	yes	maximal, complemented
9	#65015	C₂	1	2	yes	maximal, complemented
10	#64977	C₂	1	1	yes	maximal, characteristic, complemented
11	#64974	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×D₅	C₄	1	1	yes	characteristic
13	C₂²×C₄×D₅	C₂²	1	2	yes	—
14	C₂×C₅×M₁₆	C₂²	1	2	yes	complemented
15	C₄×C₅×D₄	C₂²	1	1	yes	characteristic
16	D₄×Dic₅	C₄	1	1	yes	characteristic
17	C₄×C₂²⋊D₅	C₂²	1	2	yes	—
18	C₂×#681	C₂²	1	2	yes	complemented
19	C₂×#676	C₂²	1	2	yes	complemented
20	C₂×#676	C₂²	1	2	yes	complemented
21	#4039	C₄	1	1	yes	characteristic
22	D₅×C₄⋊C₄	C₄	1	1	yes	characteristic
23	#4027	C₄	1	2	yes	—
24	D₅×C₂²⋊C₄	C₄	1	2	yes	—
25	C₄×D₂₀	C₂²	1	1	yes	characteristic
26	C₄²×D₅	C₂²	1	1	yes	characteristic
27	C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
28	C₅×C₂²⋊C₈	C₂²	1	2	yes	complemented
29	C₄×C₅×C₈	C₂²	1	1	yes	characteristic, complemented
30	C₂²⋊#168	C₂²	1	2	yes	complemented
31	#3952	C₂²	1	2	yes	—
32	#3952	C₂²	1	1	yes	characteristic
33	#3952	C₂²	1	1	yes	characteristic
34	#3952	C₂²	1	2	yes	complemented
35	#3952	C₂²	1	2	yes	complemented
36	#3946	C₂²	1	1	yes	characteristic, complemented
37	#3946	C₂²	1	1	yes	characteristic, complemented
38	C₈×Dic₅	C₂²	1	2	yes	complemented
39	C₄⋊#168	C₂²	1	1	yes	characteristic, complemented
40	C₄×#168	C₂²	1	1	yes	characteristic, complemented
41	#2975	—	5	1	no	maximal, Sylow
42	C₂³×D₅	C₂×C₄	1	2	yes	—
43	C₂×C₅×D₄	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅	C₂³	1	2	yes	—
45	C₂×C₂²⋊D₅	C₂×C₄	1	2	yes	—
46	C₂²×Dic₅	C₂×C₄	1	2	yes	—
47	D₄×D₅	—	2	4	no	—
48	C₂×D₂₀	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₅	—	4	2	no	—
50	C₂×C₄×D₅	—	4	2	no	—
51	C₂×C₄×D₅	C₂×C₄	1	2	yes	—
52	C₂×C₄×D₅	C₂×C₄	1	2	yes	—
53	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
56	C₂×C₄×D₅	C₂³	1	2	yes	—
57	C₅×M₁₆	—	4	2	no	—
58	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
59	C₂×C₅×C₈	C₂³	1	2	yes	—
60	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
61	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
62	C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
63	C₄²×C₅	C₂³	1	1	yes	characteristic
64	C₂²⋊Dic₅	C₂×C₄	1	2	yes	—
65	#685	C₂×C₄	1	2	yes	—
66	#685	C₂×C₄	1	2	yes	—
67	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
68	#683	C₂×C₄	1	2	yes	—
69	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₅	C₂³	1	1	yes	characteristic
72	#681	—	4	2	no	—
73	C₂×#168	C₂³	1	1	yes	characteristic
74	C₂×#168	C₂³	1	1	yes	characteristic
75	C₂×#168	C₂³	1	2	yes	—
76	#676	—	4	2	no	—
77	#676	—	4	2	no	—
78	C₂×C₄×D₄	—	5	1	no	—
79	#436	—	5	2	no	—
80	#436	—	5	1	no	—
81	#436	—	5	1	no	—
82	#436	—	5	2	no	—
83	#436	—	5	1	no	—
84	#436	D₅	1	1	yes	characteristic, complemented
85	#431	—	5	1	no	—
86	#431	—	5	1	no	—
87	#407	—	5	2	no	—
88	#407	—	5	2	no	—
89	C₂³×C₅	C₂²×C₄	1	2	yes	—
90	C₂²×D₅	—	2	4	no	—
91	C₂²×D₅	—	4	2	no	—
92	C₂²×D₅	—	4	2	no	—
93	C₂²×D₅	C₂²×C₄	1	2	yes	—
94	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
95	C₅×D₄	—	2	2	no	—
96	C₂×C₄×C₅	—	2	2	no	—
97	C₂×C₄×C₅	—	2	2	no	—
98	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
100	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
102	C₂²⋊D₅	—	2	4	no	—
103	C₂×Dic₅	—	4	2	no	—
104	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
105	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
106	C₂×Dic₅	C₂²×C₄	1	2	yes	—
107	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
108	D₂₀	—	2	2	no	—
109	C₄×D₅	—	2	2	no	—
110	C₄×D₅	—	2	2	no	—
111	C₄×D₅	—	2	1	no	—
112	C₄×D₅	—	2	1	no	—
113	C₄×D₅	—	4	2	no	—
114	C₄×D₅	—	4	2	no	—
115	C₄×D₅	—	4	1	no	—
116	C₄×D₅	M₁₆	1	4	yes	complemented
117	C₅×C₈	—	2	1	no	—
118	C₅×C₈	—	2	2	no	—
119	C₅×C₈	—	2	1	no	—
120	#168	—	2	1	no	—
121	#168	—	2	2	no	—
122	#168	—	2	1	no	—
123	C₂²×D₄	—	5	1	no	—
124	C₂³×C₄	—	5	2	no	—
125	C₂×M₁₆	—	5	2	no	—
126	C₂×M₁₆	—	5	2	no	—
127	C₂×M₁₆	—	5	2	no	—
128	C₂×M₁₆	C₂×D₅	1	2	yes	complemented
129	C₄×D₄	—	5	1	no	—
130	C₄×D₄	—	5	2	no	—
131	C₄×D₄	—	5	1	no	—
132	C₄×D₄	—	5	2	no	—
133	C₄×D₄	—	5	1	no	—
134	C₄×D₄	C₂×D₅	1	1	yes	characteristic
135	C₂×C₄⋊C₄	—	5	1	no	—
136	C₂×C₂²⋊C₄	—	5	2	no	—
137	C₂×C₄²	—	5	1	no	—
138	C₄⋊C₈	—	5	1	no	—
139	C₄⋊C₈	—	5	1	no	—
140	C₄⋊C₈	—	5	1	no	—
141	C₄⋊C₈	C₂×D₅	1	1	yes	characteristic, complemented
142	C₂²⋊C₈	—	5	2	no	—
143	C₂²⋊C₈	—	5	2	no	—
144	C₂²⋊C₈	—	5	1	no	—
145	C₂²⋊C₈	—	5	1	no	—
146	C₂²⋊C₈	—	5	2	no	—
147	C₂²⋊C₈	—	5	2	no	—
148	C₂²⋊C₈	C₂×D₅	1	2	yes	complemented
149	C₄×C₈	—	5	2	no	—
150	C₄×C₈	—	5	1	no	—
151	C₄×C₈	C₂×D₅	1	1	yes	characteristic, complemented
152	C₂²×C₅	—	2	2	no	—
153	C₂²×C₅	—	2	2	no	—
154	C₂²×C₅	—	2	2	no	—
155	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
156	C₂×D₅	—	2	1	no	—
157	C₂×D₅	—	2	1	no	—
158	C₂×D₅	—	2	2	no	—
159	C₂×D₅	—	2	2	no	—
160	C₂×D₅	—	2	2	no	—
161	C₂×D₅	—	4	8	no	—
162	C₂×D₅	C₂×M₁₆	1	2	yes	—
163	C₄×C₅	—	2	2	no	—
164	C₄×C₅	—	2	1	no	—
165	C₄×C₅	C₂×M₁₆	1	2	yes	—
166	C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
167	C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
168	Dic₅	—	2	1	no	—
169	Dic₅	—	2	2	no	—
170	Dic₅	—	2	1	no	—
171	Dic₅	C₂×M₁₆	1	2	yes	—
172	C₂⁴	—	5	2	no	—
173	C₂×D₄	—	5	2	no	—
174	C₂×D₄	—	5	1	no	—
175	C₂×D₄	—	10	4	no	—
176	C₂×D₄	C₄×D₅	1	1	yes	characteristic
177	C₂²×C₄	—	5	1	no	—
178	C₂²×C₄	—	5	2	no	—
179	C₂²×C₄	—	5	1	no	—
180	C₂²×C₄	—	5	2	no	—
181	C₂²×C₄	—	5	1	no	—
182	C₂²×C₄	—	5	2	no	—
183	C₂²×C₄	—	5	2	no	—
184	C₂²×C₄	—	20	2	no	—
185	C₂²×C₄	—	20	2	no	—
186	C₂²×C₄	C₂²×D₅	1	2	yes	—
187	M₁₆	—	4	2	no	—
188	M₁₆	—	20	2	no	—
189	M₁₆	—	20	2	no	—
190	M₁₆	—	20	2	no	—
191	C₂×C₈	—	5	1	no	—
192	C₂×C₈	—	5	2	no	—
193	C₂×C₈	—	5	1	no	—
194	C₂×C₈	C₂²×D₅	1	2	yes	—
195	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
196	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
197	C₄⋊C₄	—	5	2	no	—
198	C₄⋊C₄	—	5	1	no	—
199	C₄⋊C₄	C₄×D₅	1	1	yes	characteristic
200	C₂²⋊C₄	—	5	2	no	—
201	C₂²⋊C₄	—	5	2	no	—
202	C₂²⋊C₄	—	5	2	no	—
203	C₂²⋊C₄	C₄×D₅	1	2	yes	—
204	C₄²	—	5	1	no	—
205	C₄²	—	5	1	no	—
206	C₄²	—	5	1	no	—
207	C₄²	C₂²×D₅	1	1	yes	characteristic
208	C₂×C₅	—	4	2	no	—
209	C₂×C₅	#518	1	1	yes	characteristic
210	C₂×C₅	C₂²×M₁₆	1	1	yes	characteristic
211	C₂×C₅	#568	1	1	yes	characteristic
212	D₅	—	2	2	no	—
213	D₅	—	4	2	no	—
214	C₂³	—	5	1	no	—
215	C₂³	—	5	2	no	—
216	C₂³	—	10	4	no	—
217	C₂³	—	20	2	no	—
218	C₂³	—	20	2	no	—
219	C₂³	C₂×C₄×D₅	1	2	yes	—
220	D₄	—	2	2	no	—
221	D₄	—	10	4	no	—
222	D₄	—	10	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	5	1	no	—
226	C₂×C₄	—	5	4	no	—
227	C₂×C₄	—	5	1	no	—
228	C₂×C₄	—	5	1	no	—
229	C₂×C₄	—	5	2	no	—
230	C₂×C₄	—	10	2	no	—
231	C₂×C₄	—	10	1	no	—
232	C₂×C₄	—	10	2	no	—
233	C₂×C₄	—	10	1	no	—
234	C₂×C₄	—	20	2	no	—
235	C₂×C₄	—	20	2	no	—
236	C₂×C₄	—	20	1	no	—
237	C₂×C₄	—	20	2	no	—
238	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
239	C₂×C₄	C₂×C₄×D₅	1	2	yes	—
240	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
241	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
242	C₈	—	2	1	no	—
243	C₈	—	2	2	no	—
244	C₈	—	2	1	no	—
245	C₈	—	10	1	no	—
246	C₈	—	10	2	no	—
247	C₈	—	10	1	no	—
248	C₅	#2975	1	1	yes	characteristic, complemented, Sylow
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	5	2	no	—
253	C₂²	—	10	1	no	—
254	C₂²	—	10	1	no	—
255	C₂²	—	10	2	no	—
256	C₂²	—	10	2	no	—
257	C₂²	—	10	2	no	—
258	C₂²	—	20	8	no	—
259	C₂²	C₂²×C₄×D₅	1	1	yes	characteristic
260	C₄	—	2	1	no	—
261	C₄	—	2	2	no	—
262	C₄	—	5	2	no	—
263	C₄	—	10	1	no	—
264	C₄	—	10	1	no	—
265	C₄	—	10	2	no	—
266	C₄	D₅×M₁₆	1	2	yes	—
267	C₄	#4144	1	1	yes	characteristic
268	C₄	#4149	1	1	yes	characteristic
269	C₂	—	4	2	no	—
270	C₂	—	10	2	no	—
271	C₂	—	20	2	no	—
272	C₂	#65879	1	1	yes	characteristic
273	C₂	C₂×D₅×M₁₆	1	1	yes	characteristic
274	C₂	#66084	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₅×#436	C₂
prime radical	C₂×D₄×D₅	C₄
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	#436	D₅
hypocenter	C₅	#2975
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

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

N	H ≅ G/N	type
2. C₅×#436	C₂	semidirect
3. #65417	C₂	semidirect
4. #65415	C₂	semidirect
5. #65301	C₂	semidirect
6. #65127	C₂	semidirect
7. #65122	C₂	semidirect
8. #65018	C₂	semidirect
9. #65015	C₂	semidirect
10. #64977	C₂	semidirect
11. #64974	C₂	semidirect
14. C₂×C₅×M₁₆	C₂²	semidirect
18. C₂×#681	C₂²	semidirect
19. C₂×#676	C₂²	semidirect
20. C₂×#676	C₂²	semidirect
27. C₅×C₄⋊C₈	C₂²	semidirect
28. C₅×C₂²⋊C₈	C₂²	semidirect
29. C₄×C₅×C₈	C₂²	semidirect
30. C₂²⋊#168	C₂²	semidirect
34. #3952	C₂²	semidirect
35. #3952	C₂²	semidirect
36. #3946	C₂²	semidirect
37. #3946	C₂²	semidirect
38. C₈×Dic₅	C₂²	semidirect
39. C₄⋊#168	C₂²	semidirect
40. C₄×#168	C₂²	semidirect
84. #436	D₅	semidirect
116. C₄×D₅	M₁₆	semidirect
128. C₂×M₁₆	C₂×D₅	semidirect
141. C₄⋊C₈	C₂×D₅	semidirect
148. C₂²⋊C₈	C₂×D₅	semidirect
151. C₄×C₈	C₂×D₅	semidirect
248. C₅	#2975	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
C₅×#436	2
#65417	2
#65415	2
#65301	2
#65127	2
#65122	2
#65018	2
#65015	2
#64977	2
#64974	2
#2975	5