Subgroups — #10611269

Order 640.

500 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	C₂×#4053	C₂	1	2	yes	maximal, complemented
2	#65882	C₂	1	1	yes	maximal, characteristic
3	C₅×#436	C₂	1	1	yes	maximal, characteristic, complemented
4	#65417	C₂	1	2	yes	maximal, complemented
5	D₄×#168	C₂	1	1	yes	maximal, characteristic, complemented
6	#65126	C₂	1	1	yes	maximal, characteristic, complemented
7	#65123	C₂	1	1	yes	maximal, characteristic, complemented
8	#65016	C₂	1	2	yes	maximal, complemented
9	#65014	C₂	1	2	yes	maximal, complemented
10	#64977	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#709	C₂²	1	2	yes	—
13	C₂×C₅×M₁₆	C₂²	1	2	yes	—
14	C₄×C₅×D₄	C₂²	1	1	yes	characteristic
15	#4086	C₄	1	1	yes	characteristic
16	#4084	C₄	1	1	yes	characteristic
17	C₄×C₂²⋊D₅	C₂²	1	2	yes	—
18	C₂²×#168	C₂²	1	2	yes	—
19	#4053	—	2	8	no	—
20	C₂×#676	C₂²	1	2	yes	—
21	C₂×#676	C₂²	1	2	yes	—
22	C₂×C₈×D₅	C₂²	1	2	yes	—
23	C₂×C₈×D₅	C₂²	1	1	yes	characteristic
24	C₂×C₈×D₅	C₂²	1	1	yes	characteristic
25	#4041	C₄	1	1	yes	characteristic
26	#4034	C₄	1	1	yes	characteristic
27	#4031	C₄	1	2	yes	—
28	#4030	C₄	1	2	yes	—
29	C₄×D₂₀	C₂²	1	1	yes	characteristic
30	C₄²×D₅	C₂²	1	1	yes	characteristic
31	C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₅×C₂²⋊C₈	C₂²	1	2	yes	—
33	C₄×C₅×C₈	C₂²	1	1	yes	characteristic
34	C₂²⋊#168	C₂²	1	2	yes	—
35	#3952	C₂²	1	2	yes	—
36	#3952	C₂²	1	2	yes	—
37	#3946	C₂²	1	2	yes	—
38	C₈×Dic₅	C₂²	1	1	yes	characteristic
39	C₈×Dic₅	C₂²	1	1	yes	characteristic
40	C₄⋊#168	C₂²	1	1	yes	characteristic
41	C₄×#168	C₂²	1	1	yes	characteristic
42	#2931	—	5	1	no	maximal, Sylow
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	C₂×C₂²⋊D₅	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊D₅	C₂×C₄	1	2	yes	—
48	#709	—	2	4	no	—
49	#709	—	2	4	no	—
50	C₂×D₂₀	C₂×C₄	1	2	yes	—
51	C₂×D₂₀	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₅	C₂³	1	2	yes	—
55	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
56	C₂×Dic₁₀	C₂×C₄	1	2	yes	—
57	C₅×M₁₆	—	2	4	no	—
58	C₂×C₅×C₈	C₂³	1	2	yes	—
59	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
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	C₂×#168	—	2	4	no	—
73	C₂×#168	C₂³	1	1	yes	characteristic
74	C₂×#168	C₂³	1	2	yes	—
75	C₂×#168	C₂³	1	1	yes	characteristic
76	#676	—	2	4	no	—
77	#676	—	2	4	no	—
78	C₈×D₅	—	2	2	no	—
79	C₈×D₅	—	2	4	no	—
80	C₈×D₅	D₄	1	4	yes	complemented
81	C₂×#131	—	5	2	no	—
82	#532	—	5	1	no	—
83	#436	—	5	2	no	—
84	#436	—	5	1	no	—
85	#436	D₅	1	1	yes	characteristic, complemented
86	C₈×D₄	—	5	1	no	—
87	C₈×D₄	—	5	1	no	—
88	C₈×D₄	—	5	2	no	—
89	#423	—	5	1	no	—
90	#408	—	5	2	no	—
91	C₂×C₄×C₈	—	5	1	no	—
92	C₂³×C₅	C₂²×C₄	1	2	yes	—
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	C₂²⋊D₅	—	2	4	no	—
104	C₂²⋊D₅	—	2	4	no	—
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	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
109	D₂₀	—	2	4	no	—
110	D₂₀	—	2	2	no	—
111	C₄×D₅	—	2	2	no	—
112	C₄×D₅	—	2	2	no	—
113	C₄×D₅	—	2	2	no	—
114	C₄×D₅	—	2	2	no	—
115	C₄×D₅	C₂×D₄	1	2	yes	—
116	C₄×D₅	#41	1	2	yes	—
117	Dic₁₀	—	2	4	no	—
118	C₅×C₈	—	2	2	no	—
119	C₅×C₈	—	2	1	no	—
120	C₅×C₈	C₂×D₄	1	2	yes	—
121	#168	—	2	2	no	—
122	#168	—	2	1	no	—
123	#168	C₂×D₄	1	2	yes	—
124	C₂×#41	—	5	2	no	—
125	#131	—	10	8	no	—
126	C₂×M₁₆	—	5	2	no	—
127	C₂×M₁₆	—	5	2	no	—
128	C₂×M₁₆	C₂×D₅	1	2	yes	—
129	C₂²×C₈	—	5	1	no	—
130	C₂²×C₈	—	5	2	no	—
131	C₂²×C₈	—	5	2	no	—
132	C₂²×C₈	—	5	1	no	—
133	C₄⋊Q₈	—	5	1	no	—
134	#127	—	5	1	no	—
135	#124	—	5	2	no	—
136	#121	—	5	1	no	—
137	#121	—	5	1	no	—
138	#121	—	5	2	no	—
139	C₄×D₄	—	5	2	no	—
140	C₄×D₄	—	5	1	no	—
141	C₄×D₄	C₂×D₅	1	1	yes	characteristic
142	C₂×C₄²	—	5	1	no	—
143	C₄⋊C₈	—	5	2	no	—
144	C₄⋊C₈	—	5	1	no	—
145	C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
146	C₂²⋊C₈	—	5	2	no	—
147	C₂²⋊C₈	—	5	2	no	—
148	C₂²⋊C₈	—	5	2	no	—
149	C₂²⋊C₈	C₂×D₅	1	2	yes	—
150	C₄×C₈	—	5	1	no	—
151	C₄×C₈	—	5	1	no	—
152	C₄×C₈	—	5	1	no	—
153	C₄×C₈	C₂×D₅	1	1	yes	characteristic
154	C₂²×C₅	—	2	2	no	—
155	C₂²×C₅	—	2	2	no	—
156	C₂²×C₅	—	2	2	no	—
157	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
158	C₂×D₅	—	2	1	no	—
159	C₂×D₅	—	2	1	no	—
160	C₂×D₅	—	2	2	no	—
161	C₂×D₅	—	2	2	no	—
162	C₂×D₅	—	2	2	no	—
163	C₂×D₅	C₄×D₄	1	2	yes	—
164	C₄×C₅	—	2	2	no	—
165	C₄×C₅	—	2	1	no	—
166	C₄×C₅	#131	1	2	yes	complemented
167	C₄×C₅	C₂²×D₄	1	1	yes	characteristic
168	C₄×C₅	C₂×#41	1	1	yes	characteristic
169	Dic₅	—	2	2	no	—
170	Dic₅	—	2	1	no	—
171	Dic₅	C₄×D₄	1	2	yes	—
172	Dic₅	#131	1	2	yes	complemented
173	#41	—	10	4	no	—
174	#41	—	10	4	no	—
175	C₂×Q₈	—	5	2	no	—
176	C₂×D₄	—	5	2	no	—
177	C₂×D₄	—	5	2	no	—
178	C₂×D₄	—	5	2	no	—
179	C₂×D₄	—	5	1	no	—
180	C₂×D₄	—	5	2	no	—
181	C₂×D₄	C₄×D₅	1	1	yes	characteristic
182	C₂²×C₄	—	5	1	no	—
183	C₂²×C₄	—	5	1	no	—
184	C₂²×C₄	—	5	1	no	—
185	C₂²×C₄	—	5	2	no	—
186	C₂²×C₄	C₂²×D₅	1	2	yes	—
187	M₁₆	—	2	4	no	—
188	M₁₆	—	10	4	no	—
189	M₁₆	—	10	4	no	—
190	C₂×C₈	—	5	1	no	—
191	C₂×C₈	—	5	4	no	—
192	C₂×C₈	—	5	1	no	—
193	C₂×C₈	—	5	2	no	—
194	C₂×C₈	—	10	4	no	—
195	C₂×C₈	—	10	2	no	—
196	C₂×C₈	—	10	4	no	—
197	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
198	C₂×C₈	C₂²×D₅	1	2	yes	—
199	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
200	C₄⋊C₄	—	5	2	no	—
201	C₄⋊C₄	—	5	1	no	—
202	C₄⋊C₄	C₄×D₅	1	1	yes	characteristic
203	C₂²⋊C₄	—	5	2	no	—
204	C₂²⋊C₄	—	5	2	no	—
205	C₂²⋊C₄	—	5	2	no	—
206	C₂²⋊C₄	C₄×D₅	1	2	yes	—
207	C₄²	—	5	1	no	—
208	C₄²	—	5	1	no	—
209	C₄²	—	5	1	no	—
210	C₄²	C₂²×D₅	1	1	yes	characteristic
211	C₂×C₅	—	4	2	no	—
212	C₂×C₅	C₂×C₄×D₄	1	1	yes	characteristic
213	C₂×C₅	C₂×#131	1	1	yes	characteristic
214	C₂×C₅	C₂×#131	1	1	yes	characteristic
215	D₅	—	2	2	no	—
216	D₅	—	4	2	no	—
217	C₂³	—	5	1	no	—
218	C₂³	—	5	2	no	—
219	C₂³	C₂×C₄×D₅	1	2	yes	—
220	Q₈	—	10	4	no	—
221	D₄	—	2	2	no	—
222	D₄	—	10	4	no	—
223	D₄	—	10	4	no	—
224	D₄	—	10	4	no	—
225	D₄	—	10	4	no	—
226	D₄	—	10	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	5	1	no	—
230	C₂×C₄	—	5	1	no	—
231	C₂×C₄	—	5	2	no	—
232	C₂×C₄	—	5	2	no	—
233	C₂×C₄	—	5	1	no	—
234	C₂×C₄	—	5	2	no	—
235	C₂×C₄	—	10	2	no	—
236	C₂×C₄	—	10	2	no	—
237	C₂×C₄	—	10	2	no	—
238	C₂×C₄	—	10	2	no	—
239	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
240	C₂×C₄	C₂×C₄×D₅	1	2	yes	—
241	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
242	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	—	2	1	no	—
245	C₈	—	5	2	no	—
246	C₈	—	10	2	no	—
247	C₈	—	10	1	no	—
248	C₈	D₄×D₅	1	2	yes	—
249	C₅	#2931	1	1	yes	characteristic, complemented, Sylow
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	5	2	no	—
254	C₂²	—	10	1	no	—
255	C₂²	—	10	1	no	—
256	C₂²	—	10	2	no	—
257	C₂²	—	10	2	no	—
258	C₂²	—	10	2	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	—	5	2	no	—
264	C₄	—	10	1	no	—
265	C₄	—	10	2	no	—
266	C₄	#4053	1	2	yes	complemented
267	C₄	C₂×D₄×D₅	1	1	yes	characteristic
268	C₄	D₅×#41	1	1	yes	characteristic
269	C₂	—	4	2	no	—
270	C₂	—	10	2	no	—
271	C₂	—	20	2	no	—
272	C₂	C₄×D₄×D₅	1	1	yes	characteristic
273	C₂	C₂×#4053	1	1	yes	characteristic
274	C₂	D₅×#131	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	#65882	C₂
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	#436	D₅
hypocenter	C₅	#2931
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
1. C₂×#4053	C₂	semidirect
3. C₅×#436	C₂	semidirect
4. #65417	C₂	semidirect
5. D₄×#168	C₂	semidirect
6. #65126	C₂	semidirect
7. #65123	C₂	semidirect
8. #65016	C₂	semidirect
9. #65014	C₂	semidirect
10. #64977	C₂	semidirect
11. C₄×C₈×D₅	C₂	semidirect
80. C₈×D₅	D₄	semidirect
85. #436	D₅	semidirect
166. C₄×C₅	#131	semidirect
172. Dic₅	#131	semidirect
249. C₅	#2931	semidirect
266. C₄	#4053	semidirect

Maximal subgroups

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

subgroup	index
C₂×#4053	2
#65882	2
C₅×#436	2
#65417	2
D₄×#168	2
#65126	2
#65123	2
#65016	2
#65014	2
#64977	2
C₄×C₈×D₅	2
#2931	5