Subgroups — #10607921

Order 640.

472 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#4035	C₂	1	2	yes	maximal
2	C₅×#389	C₂	1	1	yes	maximal, characteristic
3	#65274	C₂	1	1	yes	maximal, characteristic
4	#65265	C₂	1	1	yes	maximal, characteristic
5	#65265	C₂	1	1	yes	maximal, characteristic
6	#65262	C₂	1	1	yes	maximal, characteristic
7	#65261	C₂	1	2	yes	maximal
8	C₄²×Dic₅	C₂	1	1	yes	maximal, characteristic
9	#65217	C₂	1	1	yes	maximal, characteristic
10	#64939	C₂	1	2	yes	maximal
11	#64936	C₂	1	2	yes	maximal
12	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₅	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
19	C₂×#683	C₂²	1	2	yes	—
20	C₂×#683	C₂²	1	2	yes	—
21	C₂×#683	C₂²	1	2	yes	—
22	C₂×#683	C₂²	1	2	yes	—
23	C₂×#683	C₂²	1	2	yes	—
24	C₂×#683	C₂²	1	2	yes	—
25	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
26	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
27	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
28	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
29	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
30	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
31	#4035	—	2	8	no	—
32	C₅×#95	C₂²	1	2	yes	—
33	#3963	C₂²	1	2	yes	—
34	#3963	C₂²	1	2	yes	—
35	#3963	C₂²	1	2	yes	—
36	#2859	—	5	1	no	maximal, Sylow
37	C₂²×C₄×C₅	C₂³	1	2	yes	—
38	C₂²×C₄×C₅	C₂³	1	2	yes	—
39	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
42	C₂²×Dic₅	C₂³	1	2	yes	—
43	C₂²×Dic₅	C₂³	1	2	yes	—
44	C₂²×Dic₅	C₂³	1	1	yes	characteristic
45	C₂²×Dic₅	C₂³	1	1	yes	characteristic
46	C₂²×Dic₅	C₂³	1	1	yes	characteristic
47	C₂²×Dic₅	C₂³	1	1	yes	characteristic
48	C₅×C₄⋊C₄	—	2	4	no	—
49	C₅×C₄⋊C₄	—	2	2	no	—
50	C₅×C₄⋊C₄	—	2	2	no	—
51	C₄²×C₅	—	2	2	no	—
52	C₄⋊Dic₅	—	2	2	no	—
53	C₄⋊Dic₅	—	2	2	no	—
54	C₄⋊Dic₅	—	2	4	no	—
55	#683	—	2	4	no	—
56	#683	—	2	4	no	—
57	#683	—	2	4	no	—
58	#683	—	2	4	no	—
59	#683	—	2	4	no	—
60	#683	—	2	4	no	—
61	C₄×Dic₅	—	2	2	no	—
62	C₄×Dic₅	—	2	2	no	—
63	C₄×Dic₅	—	2	2	no	—
64	C₄×Dic₅	D₄	1	4	yes	—
65	C₄×Dic₅	Q₈	1	4	yes	—
66	C₄×Dic₅	Q₈	1	4	yes	—
67	C₂×#125	—	5	2	no	—
68	#389	—	5	2	no	—
69	#389	—	5	1	no	—
70	#389	—	5	1	no	—
71	#389	—	5	1	no	—
72	#389	—	5	2	no	—
73	#389	D₅	1	1	yes	characteristic
74	#384	—	5	1	no	—
75	#384	—	5	1	no	—
76	#382	—	5	2	no	—
77	C₄³	—	5	1	no	—
78	C₂³×C₅	C₂⁴	1	1	yes	characteristic
79	C₂×C₄×C₅	—	2	1	no	—
80	C₂×C₄×C₅	—	2	1	no	—
81	C₂×C₄×C₅	—	2	1	no	—
82	C₂×C₄×C₅	—	2	1	no	—
83	C₂×C₄×C₅	—	2	2	no	—
84	C₂×C₄×C₅	—	2	2	no	—
85	C₂×C₄×C₅	—	2	2	no	—
86	C₂×C₄×C₅	—	2	2	no	—
87	C₂×C₄×C₅	—	2	2	no	—
88	C₂×C₄×C₅	—	2	2	no	—
89	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
90	C₂×C₄×C₅	C₂×Q₈	1	2	yes	—
91	C₂×C₄×C₅	C₂×Q₈	1	2	yes	—
92	C₂×C₄×C₅	#41	1	2	yes	—
93	C₂×C₄×C₅	#41	1	2	yes	—
94	C₂×Dic₅	—	2	1	no	—
95	C₂×Dic₅	—	2	1	no	—
96	C₂×Dic₅	—	2	1	no	—
97	C₂×Dic₅	—	2	1	no	—
98	C₂×Dic₅	—	2	2	no	—
99	C₂×Dic₅	—	2	2	no	—
100	C₂×Dic₅	—	2	2	no	—
101	C₂×Dic₅	—	2	2	no	—
102	C₂×Dic₅	—	2	2	no	—
103	C₂×Dic₅	—	2	2	no	—
104	C₂×Dic₅	C₂×D₄	1	2	yes	—
105	C₂×Dic₅	C₂×D₄	1	2	yes	—
106	C₂×Dic₅	C₂×Q₈	1	2	yes	—
107	C₂×Dic₅	C₂×Q₈	1	2	yes	—
108	C₂×Dic₅	C₂×Q₈	1	2	yes	—
109	C₂×Dic₅	C₂×Q₈	1	2	yes	—
110	C₂×Dic₅	#41	1	2	yes	—
111	C₂×Dic₅	#41	1	2	yes	—
112	#125	—	10	8	no	—
113	C₂×C₄⋊C₄	—	5	2	no	—
114	C₂×C₄⋊C₄	—	5	2	no	—
115	C₂×C₄⋊C₄	—	5	2	no	—
116	C₂×C₄⋊C₄	—	5	2	no	—
117	C₂×C₄⋊C₄	—	5	2	no	—
118	C₂×C₄⋊C₄	—	5	2	no	—
119	C₂×C₄⋊C₄	—	5	2	no	—
120	C₂×C₄⋊C₄	—	5	1	no	—
121	C₂×C₄⋊C₄	—	5	1	no	—
122	C₂×C₄⋊C₄	C₂×D₅	1	2	yes	—
123	C₂×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
124	C₂×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
125	C₂×C₄²	—	5	1	no	—
126	C₂×C₄²	—	5	1	no	—
127	C₂×C₄²	—	5	1	no	—
128	C₂×C₄²	—	5	1	no	—
129	C₂×C₄²	—	5	1	no	—
130	C₂×C₄²	—	5	1	no	—
131	C₂×C₄²	C₂×D₅	1	1	yes	characteristic
132	#95	—	5	2	no	—
133	#95	—	5	2	no	—
134	#95	—	5	2	no	—
135	#95	C₂×D₅	1	2	yes	—
136	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
137	C₂²×C₅	C₂²×Q₈	1	1	yes	characteristic
138	C₂²×C₅	C₂²×Q₈	1	1	yes	characteristic
139	C₂²×C₅	C₂×#41	1	1	yes	characteristic
140	C₂²×C₅	C₂×#41	1	1	yes	characteristic
141	C₂²×C₅	C₂×#41	1	1	yes	characteristic
142	C₂²×C₅	C₂×#41	1	1	yes	characteristic
143	C₄×C₅	—	2	2	no	—
144	C₄×C₅	—	2	2	no	—
145	C₄×C₅	—	4	2	no	—
146	C₄×C₅	—	4	2	no	—
147	C₄×C₅	#125	1	4	yes	complemented
148	Dic₅	—	2	2	no	—
149	Dic₅	—	2	2	no	—
150	Dic₅	—	4	2	no	—
151	Dic₅	—	4	2	no	—
152	Dic₅	#125	1	4	yes	complemented
153	Dic₅	C₄⋊Q₈	1	4	yes	—
154	C₂²×C₄	—	5	1	no	—
155	C₂²×C₄	—	5	1	no	—
156	C₂²×C₄	—	5	1	no	—
157	C₂²×C₄	—	5	1	no	—
158	C₂²×C₄	—	5	2	no	—
159	C₂²×C₄	—	5	2	no	—
160	C₂²×C₄	C₂²×D₅	1	2	yes	—
161	C₂²×C₄	C₂²×D₅	1	2	yes	—
162	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
163	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
164	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	—	2	4	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	10	4	no	—
169	C₄⋊C₄	—	10	4	no	—
170	C₄⋊C₄	—	10	4	no	—
171	C₄⋊C₄	—	10	4	no	—
172	C₄⋊C₄	—	10	4	no	—
173	C₄⋊C₄	—	10	4	no	—
174	C₄⋊C₄	—	10	2	no	—
175	C₄⋊C₄	—	10	4	no	—
176	C₄⋊C₄	—	10	2	no	—
177	C₄²	—	2	2	no	—
178	C₄²	—	5	4	no	—
179	C₄²	—	5	4	no	—
180	C₄²	—	5	4	no	—
181	C₄²	—	10	2	no	—
182	C₄²	—	10	2	no	—
183	C₄²	—	10	2	no	—
184	C₂×C₅	C₂×#125	1	1	yes	characteristic
185	C₂×C₅	C₂×#125	1	1	yes	characteristic
186	C₂×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
187	C₂×C₅	#532	1	1	yes	characteristic
188	C₂×C₅	#532	1	1	yes	characteristic
189	C₂×C₅	#533	1	1	yes	characteristic
190	C₂×C₅	#533	1	1	yes	characteristic
191	C₂³	C₂³×D₅	1	1	yes	characteristic
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	5	2	no	—
203	C₂×C₄	—	5	2	no	—
204	C₂×C₄	—	5	2	no	—
205	C₂×C₄	—	5	2	no	—
206	C₂×C₄	—	5	2	no	—
207	C₂×C₄	—	5	2	no	—
208	C₂×C₄	—	5	2	no	—
209	C₂×C₄	—	5	2	no	—
210	C₂×C₄	—	10	1	no	—
211	C₂×C₄	—	10	1	no	—
212	C₂×C₄	—	10	1	no	—
213	C₂×C₄	—	10	1	no	—
214	C₂×C₄	—	10	2	no	—
215	C₂×C₄	—	10	2	no	—
216	C₂×C₄	—	10	2	no	—
217	C₂×C₄	—	10	2	no	—
218	C₂×C₄	—	10	2	no	—
219	C₂×C₄	—	10	2	no	—
220	C₂×C₄	#709	1	2	yes	—
221	C₂×C₄	D₄×D₅	1	2	yes	—
222	C₂×C₄	#711	1	2	yes	—
223	C₂×C₄	Q₈×D₅	1	2	yes	—
224	C₂×C₄	Q₈×D₅	1	2	yes	—
225	C₅	#2859	1	1	yes	characteristic, complemented, Sylow
226	C₂²	C₂×#709	1	1	yes	characteristic
227	C₂²	C₂×D₄×D₅	1	1	yes	characteristic
228	C₂²	C₂×#711	1	1	yes	characteristic
229	C₂²	C₂×Q₈×D₅	1	1	yes	characteristic
230	C₂²	C₂×Q₈×D₅	1	1	yes	characteristic
231	C₂²	D₅×#41	1	1	yes	characteristic
232	C₂²	D₅×#41	1	1	yes	characteristic
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	—	4	2	no	—
236	C₄	—	4	2	no	—
237	C₄	—	5	4	no	—
238	C₄	—	5	4	no	—
239	C₄	—	10	2	no	—
240	C₄	—	10	2	no	—
241	C₄	—	20	2	no	—
242	C₄	—	20	2	no	—
243	C₄	#4035	1	4	yes	complemented
244	C₂	C₂×#4035	1	1	yes	characteristic
245	C₂	#65882	1	1	yes	characteristic
246	C₂	#65912	1	1	yes	characteristic
247	C₂	#65930	1	1	yes	characteristic
248	C₂	#65956	1	1	yes	characteristic
249	C₂	D₅×#125	1	1	yes	characteristic
250	C₂	D₅×C₄⋊Q₈	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅
commutator, socle, prime radical, cosocle	C₂³×C₅	C₂⁴
Fitting	C₅×#389	C₂
hypercenter	#389	D₅
hypocenter	C₅	#2859
LCS	G ▷ C₂³×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ #389

Complements

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

N	H ≅ G/N	type
147. C₄×C₅	#125	semidirect
152. Dic₅	#125	semidirect
225. C₅	#2859	semidirect
243. C₄	#4035	semidirect

Maximal subgroups

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

subgroup	index
C₂×#4035	2
C₅×#389	2
#65274	2
#65265	2
#65262	2
#65261	2
C₄²×Dic₅	2
#65217	2
#64939	2
#64936	2
#2859	5