Subgroups — C₅×#12600 #11935320

Order 1280.

514 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#405	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₂²⋊C₁₆	C₂	1	2	yes	maximal
3	C₂×C₅×#346	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1374	C₂	1	2	yes	maximal, complemented
6	C₂²×C₅×M₁₆	C₄	1	1	yes	characteristic
7	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₅×#117	C₄	1	1	yes	characteristic
9	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
10	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
11	C₅×#405	—	2	2	no	—
12	C₅×#405	—	2	2	no	—
13	C₂×C₅×#97	C₄	1	1	yes	characteristic
14	C₂×C₅×#97	C₄	1	1	yes	characteristic
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₅×C₂²⋊C₁₆	C₄	1	8	yes	complemented
18	C₅×#346	—	2	2	no	—
19	C₄×C₅×C₁₆	—	2	2	no	—
20	#12600	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂×C₅×M₁₆	—	2	2	no	—
23	C₂×C₅×M₁₆	—	2	2	no	—
24	C₂×C₅×M₁₆	C₈	1	4	yes	—
25	C₂²×C₅×C₈	—	2	1	no	—
26	C₂²×C₅×C₈	—	2	1	no	—
27	C₂²×C₅×C₈	—	2	2	no	—
28	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
30	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₅×#117	—	2	2	no	—
34	C₅×#117	—	2	2	no	—
35	C₂×C₅×C₄⋊C₄	C₈	1	2	yes	—
36	C₂×C₅×C₂²⋊C₄	C₈	1	2	yes	—
37	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
39	C₂×C₅×C₁₆	—	2	2	no	—
40	C₂×C₅×C₁₆	—	2	2	no	—
41	C₂×C₅×C₁₆	—	2	2	no	—
42	C₂×C₅×C₁₆	—	2	2	no	—
43	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
44	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
45	C₅×#97	—	2	2	no	—
46	C₅×#97	—	2	2	no	—
47	C₄×C₅×C₈	—	2	1	no	—
48	C₄×C₅×C₈	—	2	1	no	—
49	C₄×C₅×C₈	D₄	1	2	yes	—
50	C₄×C₅×C₈	D₄	1	2	yes	—
51	C₂×#405	C₂×C₅	1	1	yes	characteristic
52	C₂×C₂²⋊C₁₆	C₂×C₅	1	2	yes	—
53	C₂×#346	C₂×C₅	1	1	yes	characteristic, complemented
54	C₂×C₄×C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
55	#1374	C₂×C₅	1	2	yes	complemented
56	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₅	—	2	1	no	—
58	C₂²×C₄×C₅	—	2	1	no	—
59	C₂²×C₄×C₅	—	2	1	no	—
60	C₂²×C₄×C₅	—	2	1	no	—
61	C₂²×C₄×C₅	C₄²	1	2	yes	—
62	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
63	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
65	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
66	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
67	C₅×M₁₆	—	2	8	no	—
68	C₂×C₅×C₈	—	2	1	no	—
69	C₂×C₅×C₈	—	2	1	no	—
70	C₂×C₅×C₈	—	2	1	no	—
71	C₂×C₅×C₈	—	2	1	no	—
72	C₂×C₅×C₈	—	2	4	no	—
73	C₂×C₅×C₈	—	2	4	no	—
74	C₂×C₅×C₈	C₄²	1	2	yes	—
75	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
76	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
77	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
78	C₂×C₅×C₈	M₁₆	1	2	yes	—
79	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
80	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
81	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
82	C₂×C₅×C₈	#41	1	2	yes	—
83	C₅×C₄⋊C₄	—	2	4	no	—
84	C₅×C₂²⋊C₄	—	2	4	no	—
85	C₄²×C₅	—	2	1	no	—
86	C₄²×C₅	—	2	1	no	—
87	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
88	C₄²×C₅	M₁₆	1	2	yes	—
89	C₅×C₁₆	—	2	4	no	—
90	C₅×C₁₆	—	2	4	no	—
91	C₂²×M₁₆	C₄×C₅	1	1	yes	characteristic
92	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
93	C₂×#117	C₄×C₅	1	1	yes	characteristic
94	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
95	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
96	#405	—	2	2	no	—
97	#405	—	2	2	no	—
98	C₂×#97	C₄×C₅	1	1	yes	characteristic
99	C₂×#97	C₄×C₅	1	1	yes	characteristic
100	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
101	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
102	C₂²⋊C₁₆	C₄×C₅	1	8	yes	complemented
103	#346	—	2	2	no	—
104	C₄×C₁₆	—	2	2	no	—
105	C₂³×C₅	—	2	1	no	—
106	C₂³×C₅	—	2	1	no	—
107	C₂³×C₅	—	2	4	no	—
108	C₂³×C₅	C₄×C₈	1	2	yes	—
109	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₅	—	2	1	no	—
111	C₂×C₄×C₅	—	2	1	no	—
112	C₂×C₄×C₅	—	2	4	no	—
113	C₂×C₄×C₅	—	2	2	no	—
114	C₂×C₄×C₅	—	2	2	no	—
115	C₂×C₄×C₅	—	2	2	no	—
116	C₂×C₄×C₅	—	2	2	no	—
117	C₂×C₄×C₅	—	2	2	no	—
118	C₂×C₄×C₅	—	2	4	no	—
119	C₂×C₄×C₅	C₄×C₈	1	2	yes	—
120	C₂×C₄×C₅	C₂²⋊C₈	1	4	yes	—
121	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₅	#117	1	1	yes	characteristic
124	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
126	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
127	C₅×C₈	—	2	2	no	—
128	C₅×C₈	—	2	2	no	—
129	C₅×C₈	C₂²⋊C₈	1	4	yes	—
130	C₅×C₈	C₄×D₄	1	2	yes	—
131	C₅×C₈	C₄×D₄	1	2	yes	—
132	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
133	C₂×M₁₆	—	2	2	no	—
134	C₂×M₁₆	—	2	2	no	—
135	C₂×M₁₆	C₅×C₈	1	4	yes	—
136	C₂²×C₈	—	2	1	no	—
137	C₂²×C₈	—	2	1	no	—
138	C₂²×C₈	—	2	2	no	—
139	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
140	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
141	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
142	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
143	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
144	#117	—	2	2	no	—
145	#117	—	2	2	no	—
146	C₂×C₄⋊C₄	C₅×C₈	1	2	yes	—
147	C₂×C₂²⋊C₄	C₅×C₈	1	2	yes	—
148	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
149	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
150	C₂×C₁₆	—	2	2	no	—
151	C₂×C₁₆	—	2	2	no	—
152	C₂×C₁₆	—	2	2	no	—
153	C₂×C₁₆	—	2	2	no	—
154	C₂×C₁₆	C₂×C₄×C₅	1	4	yes	—
155	C₂×C₁₆	C₂×C₄×C₅	1	4	yes	—
156	#97	—	2	2	no	—
157	#97	—	2	2	no	—
158	C₄×C₈	—	2	1	no	—
159	C₄×C₈	—	2	1	no	—
160	C₄×C₈	C₅×D₄	1	2	yes	—
161	C₄×C₈	C₅×D₄	1	2	yes	—
162	C₂²×C₅	—	2	4	no	—
163	C₂²×C₅	—	2	2	no	—
164	C₂²×C₅	—	4	2	no	—
165	C₂²×C₅	—	4	2	no	—
166	C₂²×C₅	C₂×C₄×C₈	1	1	yes	characteristic
167	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
168	C₂²×C₅	#431	1	1	yes	characteristic
169	C₂²×C₅	#504	1	4	yes	—
170	C₄×C₅	—	2	4	no	—
171	C₄×C₅	—	2	2	no	—
172	C₄×C₅	—	4	2	no	—
173	C₄×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
174	C₄×C₅	C₄×M₁₆	1	1	yes	characteristic
175	C₄×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
176	C₄×C₅	#431	1	1	yes	characteristic
177	C₂⁴	C₂×C₅×C₈	1	1	yes	characteristic
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	C₄²×C₅	1	2	yes	—
183	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
184	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
185	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
187	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
188	M₁₆	—	2	8	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	4	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	C₄²×C₅	1	2	yes	—
196	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
197	C₂×C₈	C₂×C₅×C₈	1	2	yes	—
198	C₂×C₈	C₂×C₅×C₈	1	2	yes	—
199	C₂×C₈	C₅×M₁₆	1	2	yes	—
200	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
201	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
202	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
203	C₂×C₈	C₅×#41	1	2	yes	—
204	C₄⋊C₄	—	2	4	no	—
205	C₂²⋊C₄	—	2	4	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	2	1	no	—
208	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
209	C₄²	C₅×M₁₆	1	2	yes	—
210	C₁₆	—	2	4	no	—
211	C₁₆	—	2	4	no	—
212	C₂×C₅	—	4	2	no	—
213	C₂×C₅	C₄×C₂²⋊C₈	1	1	yes	characteristic
214	C₂×C₅	#2108	1	2	yes	—
215	C₂×C₅	#2113	1	2	yes	—
216	C₂×C₅	#2164	1	2	yes	—
217	C₂³	—	2	1	no	—
218	C₂³	—	2	1	no	—
219	C₂³	—	2	4	no	—
220	C₂³	C₄×C₅×C₈	1	2	yes	—
221	C₂³	C₂²×C₅×C₈	1	1	yes	characteristic
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	C₄×C₅×C₈	1	2	yes	—
232	C₂×C₄	C₅×C₂²⋊C₈	1	4	yes	—
233	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
234	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
235	C₂×C₄	C₅×#117	1	1	yes	characteristic
236	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
237	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
238	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
239	C₈	—	2	2	no	—
240	C₈	—	2	2	no	—
241	C₈	C₅×C₂²⋊C₈	1	4	yes	—
242	C₈	C₄×C₅×D₄	1	2	yes	—
243	C₈	C₄×C₅×D₄	1	2	yes	—
244	C₅	#12600	1	1	yes	characteristic, direct product factor, Sylow
245	C₂²	—	2	4	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	4	2	no	—
248	C₂²	—	4	2	no	—
249	C₂²	C₂×C₄×C₅×C₈	1	1	yes	characteristic
250	C₂²	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
251	C₂²	C₅×#431	1	1	yes	characteristic
252	C₂²	C₅×#504	1	4	yes	—
253	C₄	—	2	4	no	—
254	C₄	—	2	2	no	—
255	C₄	—	4	2	no	—
256	C₄	C₄×C₅×C₂²⋊C₄	1	1	yes	characteristic
257	C₄	C₄×C₅×M₁₆	1	1	yes	characteristic
258	C₄	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
259	C₄	C₅×#431	1	1	yes	characteristic
260	C₂	—	4	2	no	—
261	C₂	C₄×C₅×C₂²⋊C₈	1	1	yes	characteristic
262	C₂	C₅×#2108	1	2	yes	—
263	C₂	C₅×#2113	1	2	yes	—
264	C₂	C₅×#2164	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅×C₈	C₂³
commutator	C₂²	C₂×C₄×C₅×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂³×C₅	C₂²×C₈
prime radical	C₂⁴×C₅	C₂×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₅×#346	C₂	semidirect
4. C₂×C₄×C₅×C₁₆	C₂	semidirect
5. C₅×#1374	C₂	semidirect
17. C₅×C₂²⋊C₁₆	C₄	semidirect
20. #12600	C₅	direct (N × H)
53. C₂×#346	C₂×C₅	semidirect
54. C₂×C₄×C₁₆	C₂×C₅	semidirect
55. #1374	C₂×C₅	semidirect
102. C₂²⋊C₁₆	C₄×C₅	semidirect
244. C₅	#12600	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#405	2
C₂×C₅×C₂²⋊C₁₆	2
C₂×C₅×#346	2
C₂×C₄×C₅×C₁₆	2
C₅×#1374	2
#12600	5

Subgroups — C5×#12600 #11935320

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12600 #11935320