Subgroups — C₅×#8607 #11931327

Order 1280.

372 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2922	C₂	1	1	yes	maximal, characteristic
2	C₅×#1564	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#334	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1477	C₂	1	2	yes	maximal
5	C₅×#1283	C₂	1	2	yes	maximal, complemented
6	C₄×C₅×#41	C₂²	1	1	yes	characteristic
7	C₅×#433	—	2	1	no	—
8	C₅×#433	—	2	1	no	—
9	C₅×#431	—	2	2	no	—
10	C₅×#431	C₂²	1	2	yes	—
11	C₅×#424	C₄	1	2	yes	—
12	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
13	C₅×#408	C₄	1	2	yes	—
14	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₅×#335	—	2	1	no	—
17	C₅×#334	—	2	2	no	—
18	C₅×#334	—	2	1	no	—
19	C₅×Q₈⋊C₈	—	2	2	no	—
20	C₅×D₄⋊C₈	—	2	2	no	—
21	#8607	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
23	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
24	C₂²×C₅×C₈	—	2	2	no	—
25	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
26	C₄×C₅×Q₈	D₄	1	1	yes	characteristic
27	C₄×C₅×Q₈	Q₈	1	1	yes	characteristic
28	C₄×C₅×D₄	—	2	1	no	—
29	C₄×C₅×D₄	—	2	1	no	—
30	C₄×C₅×D₄	D₄	1	1	yes	characteristic
31	C₄×C₅×D₄	Q₈	1	1	yes	characteristic
32	C₅×#117	—	2	1	no	—
33	C₅×#117	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅	—	2	1	no	—
35	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
36	C₅×C₄⋊C₈	—	2	2	no	—
37	C₅×C₄⋊C₈	—	2	2	no	—
38	C₅×C₄⋊C₈	—	2	2	no	—
39	C₅×C₄⋊C₈	—	2	2	no	—
40	C₅×C₄⋊C₈	—	2	2	no	—
41	C₅×C₂²⋊C₈	—	2	2	no	—
42	C₅×C₂²⋊C₈	—	2	2	no	—
43	C₅×C₂²⋊C₈	—	2	2	no	—
44	C₅×#97	—	2	1	no	—
45	C₄×C₅×C₈	—	2	1	no	—
46	C₄×C₅×C₈	—	2	2	no	—
47	C₄×C₅×C₈	D₄	1	2	yes	—
48	C₄×C₅×C₈	D₄	1	1	yes	characteristic
49	C₄×C₅×C₈	D₄	1	1	yes	characteristic
50	#2922	C₂×C₅	1	1	yes	characteristic
51	#1564	C₂×C₅	1	1	yes	characteristic, complemented
52	C₂×#334	C₂×C₅	1	1	yes	characteristic, complemented
53	#1477	C₂×C₅	1	2	yes	—
54	#1283	C₂×C₅	1	2	yes	complemented
55	C₅×#41	—	2	4	no	—
56	C₂×C₅×Q₈	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₅×D₄	—	2	1	no	—
58	C₂×C₅×D₄	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅	—	2	1	no	—
60	C₂²×C₄×C₅	—	2	1	no	—
61	C₂²×C₄×C₅	—	2	1	no	—
62	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅	#41	1	1	yes	characteristic
64	C₂²×C₄×C₅	#41	1	1	yes	characteristic
65	C₅×M₁₆	—	2	4	no	—
66	C₂×C₅×C₈	—	2	4	no	—
67	C₂×C₅×C₈	—	2	2	no	—
68	C₂×C₅×C₈	—	4	2	no	—
69	C₂×C₅×C₈	—	4	2	no	—
70	C₂×C₅×C₈	—	8	2	no	—
71	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
72	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
73	C₅×C₄⋊C₄	—	2	1	no	—
74	C₅×C₄⋊C₄	—	2	1	no	—
75	C₅×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
76	C₅×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
77	C₅×C₂²⋊C₄	—	2	1	no	—
78	C₅×C₂²⋊C₄	—	2	1	no	—
79	C₅×C₂²⋊C₄	—	2	1	no	—
80	C₄²×C₅	—	2	1	no	—
81	C₄²×C₅	—	2	1	no	—
82	C₄²×C₅	—	2	1	no	—
83	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
87	C₄×#41	C₂²×C₅	1	1	yes	characteristic
88	#433	—	2	1	no	—
89	#433	—	2	1	no	—
90	#431	—	2	2	no	—
91	#431	C₂²×C₅	1	2	yes	—
92	#424	C₄×C₅	1	2	yes	—
93	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
94	#408	C₄×C₅	1	2	yes	—
95	C₄×M₁₆	C₂²×C₅	1	1	yes	characteristic
96	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
97	#335	—	2	1	no	—
98	#334	—	2	2	no	—
99	#334	—	2	1	no	—
100	Q₈⋊C₈	—	2	2	no	—
101	D₄⋊C₈	—	2	2	no	—
102	C₂³×C₅	—	2	1	no	—
103	C₂³×C₅	#117	1	1	yes	characteristic
104	C₅×Q₈	—	2	2	no	—
105	C₅×D₄	—	2	2	no	—
106	C₅×D₄	—	4	2	no	—
107	C₂×C₄×C₅	—	2	2	no	—
108	C₂×C₄×C₅	—	2	1	no	—
109	C₂×C₄×C₅	—	2	1	no	—
110	C₂×C₄×C₅	—	2	1	no	—
111	C₂×C₄×C₅	—	4	1	no	—
112	C₂×C₄×C₅	—	4	2	no	—
113	C₂×C₄×C₅	—	4	2	no	—
114	C₂×C₄×C₅	—	4	2	no	—
115	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅	#121	1	1	yes	characteristic
118	C₂×C₄×C₅	#121	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×C₄×C₅	#131	1	4	yes	—
122	C₅×C₈	—	2	2	no	—
123	C₅×C₈	—	2	2	no	—
124	C₅×C₈	—	4	2	no	—
125	C₅×C₈	—	8	2	no	—
126	C₅×C₈	—	8	2	no	—
127	C₂×#41	C₂×C₄×C₅	1	1	yes	characteristic
128	C₂×M₁₆	C₂×C₄×C₅	1	2	yes	—
129	C₂²×C₈	—	2	2	no	—
130	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
131	C₄×Q₈	C₅×D₄	1	1	yes	characteristic
132	C₄×Q₈	C₅×Q₈	1	1	yes	characteristic
133	C₄×D₄	—	2	1	no	—
134	C₄×D₄	—	2	1	no	—
135	C₄×D₄	C₅×D₄	1	1	yes	characteristic
136	C₄×D₄	C₅×Q₈	1	1	yes	characteristic
137	#117	—	2	1	no	—
138	#117	C₂×C₄×C₅	1	1	yes	characteristic
139	C₂×C₄²	—	2	1	no	—
140	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
141	C₄⋊C₈	—	2	2	no	—
142	C₄⋊C₈	—	2	2	no	—
143	C₄⋊C₈	—	2	2	no	—
144	C₄⋊C₈	—	2	2	no	—
145	C₄⋊C₈	—	2	2	no	—
146	C₂²⋊C₈	—	2	2	no	—
147	C₂²⋊C₈	—	2	2	no	—
148	C₂²⋊C₈	—	2	2	no	—
149	#97	—	2	1	no	—
150	C₄×C₈	—	2	1	no	—
151	C₄×C₈	—	2	2	no	—
152	C₄×C₈	C₅×D₄	1	2	yes	—
153	C₄×C₈	C₅×D₄	1	1	yes	characteristic
154	C₄×C₈	C₅×D₄	1	1	yes	characteristic
155	C₂²×C₅	—	2	1	no	—
156	C₂²×C₅	—	2	1	no	—
157	C₂²×C₅	—	4	1	no	—
158	C₂²×C₅	—	4	1	no	—
159	C₂²×C₅	—	4	1	no	—
160	C₂²×C₅	#380	1	1	yes	characteristic
161	C₂²×C₅	#433	1	2	yes	—
162	C₄×C₅	—	2	1	no	—
163	C₄×C₅	—	2	1	no	—
164	C₄×C₅	—	2	1	no	—
165	C₄×C₅	—	4	1	no	—
166	C₄×C₅	—	4	1	no	—
167	C₄×C₅	—	4	1	no	—
168	C₄×C₅	#408	1	2	yes	—
169	C₄×C₅	#424	1	2	yes	—
170	C₄×C₅	#471	1	2	yes	—
171	#41	—	2	4	no	—
172	C₂×Q₈	C₅×C₄⋊C₄	1	1	yes	characteristic
173	C₂×D₄	—	2	1	no	—
174	C₂×D₄	C₅×C₄⋊C₄	1	1	yes	characteristic
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
179	C₂²×C₄	C₅×#41	1	1	yes	characteristic
180	C₂²×C₄	C₅×#41	1	1	yes	characteristic
181	M₁₆	—	2	4	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	—	8	2	no	—
187	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
188	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
192	C₄⋊C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
193	C₂²⋊C₄	—	2	1	no	—
194	C₂²⋊C₄	—	2	1	no	—
195	C₂²⋊C₄	—	2	1	no	—
196	C₄²	—	2	1	no	—
197	C₄²	—	2	1	no	—
198	C₄²	—	2	1	no	—
199	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
200	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
201	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
202	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
203	C₂×C₅	—	2	2	no	—
204	C₂×C₅	—	8	1	no	—
205	C₂×C₅	#1798	1	1	yes	characteristic
206	C₂×C₅	#1814	1	1	yes	characteristic
207	C₂×C₅	#1815	1	1	yes	characteristic
208	C₂³	—	2	1	no	—
209	C₂³	C₅×#117	1	1	yes	characteristic
210	Q₈	—	2	2	no	—
211	D₄	—	2	2	no	—
212	D₄	—	4	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	4	1	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
222	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
223	C₂×C₄	C₅×#121	1	1	yes	characteristic
224	C₂×C₄	C₅×#121	1	1	yes	characteristic
225	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
226	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
227	C₂×C₄	C₅×#131	1	4	yes	—
228	C₈	—	2	2	no	—
229	C₈	—	2	2	no	—
230	C₈	—	4	2	no	—
231	C₈	—	8	2	no	—
232	C₈	—	8	2	no	—
233	C₅	#8607	1	1	yes	characteristic, direct product factor, Sylow
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	4	1	no	—
237	C₂²	—	4	1	no	—
238	C₂²	—	4	1	no	—
239	C₂²	C₅×#380	1	1	yes	characteristic
240	C₂²	C₅×#433	1	2	yes	—
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	4	1	no	—
245	C₄	—	4	1	no	—
246	C₄	—	4	1	no	—
247	C₄	C₅×#408	1	2	yes	—
248	C₄	C₅×#424	1	2	yes	—
249	C₄	C₅×#471	1	2	yes	—
250	C₂	—	2	2	no	—
251	C₂	—	8	1	no	—
252	C₂	C₅×#1798	1	1	yes	characteristic
253	C₂	C₅×#1814	1	1	yes	characteristic
254	C₂	C₅×#1815	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	#121
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	#380
prime radical	C₂×C₅×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#1564	C₂	semidirect
3. C₂×C₅×#334	C₂	semidirect
5. C₅×#1283	C₂	semidirect
21. #8607	C₅	direct (N × H)
51. #1564	C₂×C₅	semidirect
52. C₂×#334	C₂×C₅	semidirect
54. #1283	C₂×C₅	semidirect
233. C₅	#8607	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#2922	2
C₅×#1564	2
C₂×C₅×#334	2
C₅×#1477	2
C₅×#1283	2
#8607	5

Subgroups — C5×#8607 #11931327

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8607 #11931327