Subgroups — #10610803

Order 640.

402 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅×#117	C₂	1	1	yes	maximal, characteristic
2	C₅×#424	C₂	1	1	yes	maximal, characteristic, complemented
3	#65415	C₂	1	1	yes	maximal, characteristic
4	#65411	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#3952	C₂	1	1	yes	maximal, characteristic
6	#65390	C₂	1	1	yes	maximal, characteristic, complemented
7	#65288	C₂	1	1	yes	maximal, characteristic, complemented
8	#65129	C₂	1	2	yes	maximal, complemented
9	#65129	C₂	1	2	yes	maximal, complemented
10	#65124	C₂	1	2	yes	maximal, complemented
11	#65122	C₂	1	2	yes	maximal, complemented
12	C₂²×C₄×D₅	C₂²	1	1	yes	characteristic
13	C₂×C₅×M₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₅×#117	C₂²	1	1	yes	characteristic
16	#4072	C₂²	1	1	yes	characteristic
17	C₂×#681	C₂²	1	1	yes	characteristic
18	C₂²×#168	C₂²	1	1	yes	characteristic
19	#4038	C₄	1	1	yes	characteristic
20	#4038	C₄	1	1	yes	characteristic
21	D₅×C₄⋊C₄	C₄	1	1	yes	characteristic
22	D₅×C₄⋊C₄	C₄	1	1	yes	characteristic
23	D₅×C₂²⋊C₄	C₄	1	2	yes	—
24	#4023	C₄	1	2	yes	—
25	#4018	C₂²	1	2	yes	—
26	C₄²×D₅	C₂²	1	2	yes	—
27	C₅×C₄⋊C₈	C₂²	1	2	yes	complemented
28	C₅×C₄⋊C₈	C₂²	1	2	yes	complemented
29	#3952	C₂²	1	1	yes	characteristic
30	#3952	C₂²	1	2	yes	—
31	#3952	C₂²	1	1	yes	characteristic
32	#3952	C₂²	1	1	yes	characteristic
33	#3952	C₂²	1	1	yes	characteristic
34	#3952	C₂²	1	2	yes	—
35	#3947	C₂²	1	2	yes	complemented
36	#3947	C₂²	1	2	yes	complemented
37	C₈×Dic₅	C₂²	1	2	yes	complemented
38	C₈×Dic₅	C₂²	1	2	yes	complemented
39	C₄⋊#168	C₂²	1	2	yes	complemented
40	C₄⋊#168	C₂²	1	2	yes	complemented
41	#2924	—	5	1	no	maximal, Sylow
42	C₂³×D₅	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
44	C₂²×Dic₅	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₅	—	2	2	no	—
46	C₂×C₄×D₅	—	2	2	no	—
47	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₅	C₂×C₄	1	2	yes	—
50	C₂×C₄×D₅	C₂×C₄	1	2	yes	—
51	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
52	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
53	C₅×M₁₆	—	2	2	no	—
54	C₂×C₅×C₈	—	2	2	no	—
55	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
56	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
57	C₂×C₅×C₈	C₂³	1	2	yes	—
58	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
59	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
60	C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
61	C₄²×C₅	C₂³	1	2	yes	—
62	C₂²⋊Dic₅	C₂×C₄	1	2	yes	—
63	#685	C₂×C₄	1	2	yes	—
64	#685	C₂×C₄	1	2	yes	—
65	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
66	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
67	#683	C₂×C₄	1	2	yes	—
68	#683	C₂×C₄	1	2	yes	—
69	C₄×Dic₅	C₂×C₄	1	2	yes	—
70	C₄×Dic₅	C₂×C₄	1	2	yes	—
71	C₄×Dic₅	C₂³	1	2	yes	—
72	#681	—	2	2	no	—
73	C₂×#168	—	2	2	no	—
74	C₂×#168	C₂³	1	1	yes	characteristic
75	C₂×#168	C₂³	1	2	yes	—
76	C₂×#168	C₂³	1	1	yes	characteristic
77	C₂×#117	—	5	1	no	—
78	#433	—	5	2	no	—
79	#433	—	5	2	no	—
80	#432	—	5	2	no	—
81	#431	—	5	2	no	—
82	#424	—	5	1	no	—
83	#424	D₅	1	1	yes	characteristic, complemented
84	#407	—	5	1	no	—
85	C₂×C₂²⋊C₈	—	5	1	no	—
86	#405	—	5	1	no	—
87	#405	—	5	1	no	—
88	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
89	C₂²×D₅	—	2	2	no	—
90	C₂²×D₅	—	2	2	no	—
91	C₂²×D₅	—	4	1	no	—
92	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
93	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₅	#41	1	2	yes	—
98	C₂×C₄×C₅	#41	1	2	yes	—
99	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
100	C₂×Dic₅	—	2	2	no	—
101	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
102	C₂×Dic₅	C₂²×C₄	1	2	yes	—
103	C₂×Dic₅	C₂²×C₄	1	2	yes	—
104	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
105	C₄×D₅	—	2	4	no	—
106	C₄×D₅	—	2	2	no	—
107	C₄×D₅	—	2	2	no	—
108	C₄×D₅	—	2	1	no	—
109	C₄×D₅	—	2	1	no	—
110	C₄×D₅	—	2	1	no	—
111	C₄×D₅	—	2	1	no	—
112	C₄×D₅	—	4	2	no	—
113	C₅×C₈	—	2	1	no	—
114	C₅×C₈	—	2	1	no	—
115	C₅×C₈	—	2	2	no	—
116	#168	—	2	1	no	—
117	#168	—	2	1	no	—
118	#168	—	2	2	no	—
119	C₂³×C₄	—	5	1	no	—
120	C₂×M₁₆	—	5	1	no	—
121	C₂×M₁₆	C₂×D₅	1	1	yes	characteristic
122	C₂²×C₈	—	5	1	no	—
123	C₂²×C₈	C₂×D₅	1	1	yes	characteristic
124	#117	—	5	1	no	—
125	#117	—	5	1	no	—
126	#117	—	5	1	no	—
127	#117	—	5	2	no	—
128	#117	—	5	2	no	—
129	#117	C₂×D₅	1	1	yes	characteristic
130	C₂×C₄⋊C₄	—	5	1	no	—
131	C₂×C₄⋊C₄	—	5	1	no	—
132	C₂×C₂²⋊C₄	—	5	2	no	—
133	C₂×C₄²	—	5	2	no	—
134	C₄⋊C₈	—	5	2	no	—
135	C₄⋊C₈	—	5	2	no	—
136	C₄⋊C₈	C₂×D₅	1	2	yes	complemented
137	C₄⋊C₈	C₂×D₅	1	2	yes	complemented
138	C₂²⋊C₈	—	5	1	no	—
139	C₂²⋊C₈	—	5	1	no	—
140	C₂²⋊C₈	—	5	1	no	—
141	C₂²⋊C₈	—	5	1	no	—
142	C₂²⋊C₈	—	5	2	no	—
143	C₂²⋊C₈	—	5	2	no	—
144	#97	—	5	2	no	—
145	#97	—	5	2	no	—
146	C₄×C₈	—	5	2	no	—
147	C₄×C₈	—	5	2	no	—
148	C₂²×C₅	—	2	1	no	—
149	C₂²×C₅	—	2	1	no	—
150	C₂²×C₅	#117	1	2	yes	—
151	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
152	C₂×D₅	—	2	1	no	—
153	C₂×D₅	—	2	1	no	—
154	C₂×D₅	—	2	1	no	—
155	C₂×D₅	—	2	1	no	—
156	C₂×D₅	—	2	1	no	—
157	C₂×D₅	—	4	4	no	—
158	C₂×D₅	#131	1	2	yes	—
159	C₄×C₅	—	2	2	no	—
160	C₄×C₅	—	2	2	no	—
161	C₄×C₅	#117	1	2	yes	—
162	C₄×C₅	C₂×#41	1	1	yes	characteristic
163	C₄×C₅	C₂×#41	1	1	yes	characteristic
164	Dic₅	—	2	1	no	—
165	Dic₅	—	2	1	no	—
166	Dic₅	—	2	2	no	—
167	Dic₅	—	2	2	no	—
168	C₂⁴	—	5	1	no	—
169	C₂²×C₄	—	5	1	no	—
170	C₂²×C₄	—	5	2	no	—
171	C₂²×C₄	—	5	1	no	—
172	C₂²×C₄	—	5	1	no	—
173	C₂²×C₄	—	5	1	no	—
174	C₂²×C₄	—	5	1	no	—
175	C₂²×C₄	—	5	2	no	—
176	C₂²×C₄	—	10	2	no	—
177	C₂²×C₄	—	10	2	no	—
178	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
179	M₁₆	—	2	2	no	—
180	M₁₆	—	10	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	5	1	no	—
183	C₂×C₈	—	5	2	no	—
184	C₂×C₈	—	5	1	no	—
185	C₂×C₈	—	10	2	no	—
186	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
187	C₂×C₈	C₂²×D₅	1	2	yes	—
188	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
189	C₄⋊C₄	—	5	2	no	—
190	C₄⋊C₄	—	5	2	no	—
191	C₄⋊C₄	—	5	1	no	—
192	C₄⋊C₄	—	5	1	no	—
193	C₄⋊C₄	C₄×D₅	1	1	yes	characteristic
194	C₄⋊C₄	C₄×D₅	1	1	yes	characteristic
195	C₂²⋊C₄	—	5	2	no	—
196	C₂²⋊C₄	—	5	2	no	—
197	C₂²⋊C₄	—	5	2	no	—
198	C₂²⋊C₄	C₄×D₅	1	2	yes	—
199	C₄²	—	5	2	no	—
200	C₄²	—	5	2	no	—
201	C₄²	—	5	2	no	—
202	C₄²	C₂²×D₅	1	2	yes	—
203	C₂×C₅	—	2	2	no	—
204	C₂×C₅	C₂×#117	1	1	yes	characteristic
205	C₂×C₅	C₂×#131	1	1	yes	characteristic
206	C₂×C₅	#568	1	1	yes	characteristic
207	D₅	—	2	2	no	—
208	D₅	—	4	1	no	—
209	C₂³	—	5	1	no	—
210	C₂³	—	5	1	no	—
211	C₂³	—	10	2	no	—
212	C₂³	—	10	2	no	—
213	C₂³	—	20	1	no	—
214	C₂³	C₂×C₄×D₅	1	1	yes	characteristic
215	C₂×C₄	—	5	1	no	—
216	C₂×C₄	—	5	2	no	—
217	C₂×C₄	—	5	2	no	—
218	C₂×C₄	—	5	1	no	—
219	C₂×C₄	—	10	4	no	—
220	C₂×C₄	—	10	2	no	—
221	C₂×C₄	—	10	1	no	—
222	C₂×C₄	—	10	1	no	—
223	C₂×C₄	—	10	1	no	—
224	C₂×C₄	—	10	1	no	—
225	C₂×C₄	—	10	2	no	—
226	C₂×C₄	—	10	2	no	—
227	C₂×C₄	—	20	2	no	—
228	C₂×C₄	C₂×C₄×D₅	1	2	yes	—
229	C₂×C₄	C₂×C₄×D₅	1	2	yes	—
230	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
231	C₂×C₄	#711	1	2	yes	—
232	C₂×C₄	#713	1	2	yes	—
233	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	2	2	no	—
237	C₈	—	10	1	no	—
238	C₈	—	10	1	no	—
239	C₈	—	10	2	no	—
240	C₅	#2924	1	1	yes	characteristic, complemented, Sylow
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	5	2	no	—
244	C₂²	—	10	1	no	—
245	C₂²	—	10	1	no	—
246	C₂²	—	10	1	no	—
247	C₂²	—	10	1	no	—
248	C₂²	—	10	1	no	—
249	C₂²	—	20	4	no	—
250	C₂²	#4038	1	2	yes	—
251	C₂²	C₂²×C₄×D₅	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	10	2	no	—
255	C₄	—	10	2	no	—
256	C₄	—	10	1	no	—
257	C₄	—	10	1	no	—
258	C₄	#4038	1	2	yes	—
259	C₄	C₂×#711	1	1	yes	characteristic
260	C₄	C₂×#713	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	10	2	no	—
263	C₂	—	20	1	no	—
264	C₂	C₂×#4038	1	1	yes	characteristic
265	C₂	D₅×#131	1	1	yes	characteristic
266	C₂	#66084	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₅×#424	C₂
prime radical	C₂³×D₅	C₂×C₄
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	#424	D₅
hypocenter	C₅	#2924
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
2. C₅×#424	C₂	semidirect
4. #65411	C₂	semidirect
6. #65390	C₂	semidirect
7. #65288	C₂	semidirect
8. #65129	C₂	semidirect
9. #65129	C₂	semidirect
10. #65124	C₂	semidirect
11. #65122	C₂	semidirect
27. C₅×C₄⋊C₈	C₂²	semidirect
28. C₅×C₄⋊C₈	C₂²	semidirect
35. #3947	C₂²	semidirect
36. #3947	C₂²	semidirect
37. C₈×Dic₅	C₂²	semidirect
38. C₈×Dic₅	C₂²	semidirect
39. C₄⋊#168	C₂²	semidirect
40. C₄⋊#168	C₂²	semidirect
83. #424	D₅	semidirect
136. C₄⋊C₈	C₂×D₅	semidirect
137. C₄⋊C₈	C₂×D₅	semidirect
240. C₅	#2924	semidirect

Maximal subgroups

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

subgroup	index
D₅×#117	2
C₅×#424	2
#65415	2
#65411	2
C₂×#3952	2
#65390	2
#65288	2
#65129	2
#65124	2
#65122	2
#2924	5