Subgroups — C₅×#8606 #11931326

Order 1280.

372 conjugacy classes of subgroups.

The subgroup lattice has 262 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×#335	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	2	no	—
17	C₅×#335	—	2	1	no	—
18	C₅×#334	—	2	1	no	—
19	C₅×Q₈⋊C₈	—	2	2	no	—
20	C₅×D₄⋊C₈	—	2	2	no	—
21	#8606	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	1	yes	characteristic
48	C₄×C₅×C₈	D₄	1	2	yes	—
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×#335	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	2	no	—
98	#335	—	2	1	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	1	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	—	2	1	no	—
112	C₂×C₄×C₅	—	4	1	no	—
113	C₂×C₄×C₅	—	4	2	no	—
114	C₂×C₄×C₅	—	4	2	no	—
115	C₂×C₄×C₅	—	4	1	no	—
116	C₂×C₄×C₅	—	4	1	no	—
117	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₅	#121	1	1	yes	characteristic
120	C₂×C₄×C₅	#121	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₅	#131	1	4	yes	—
124	C₅×C₈	—	2	2	no	—
125	C₅×C₈	—	2	2	no	—
126	C₅×C₈	—	4	2	no	—
127	C₅×C₈	—	8	2	no	—
128	C₅×C₈	—	8	2	no	—
129	C₂×#41	C₂×C₄×C₅	1	1	yes	characteristic
130	C₂×M₁₆	C₂×C₄×C₅	1	2	yes	—
131	C₂²×C₈	—	2	2	no	—
132	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
133	C₄×Q₈	C₅×D₄	1	1	yes	characteristic
134	C₄×Q₈	C₅×Q₈	1	1	yes	characteristic
135	C₄×D₄	—	2	1	no	—
136	C₄×D₄	—	2	1	no	—
137	C₄×D₄	C₅×D₄	1	1	yes	characteristic
138	C₄×D₄	C₅×Q₈	1	1	yes	characteristic
139	#117	—	2	1	no	—
140	#117	C₂×C₄×C₅	1	1	yes	characteristic
141	C₂×C₄²	—	2	1	no	—
142	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
143	C₄⋊C₈	—	2	2	no	—
144	C₄⋊C₈	—	2	2	no	—
145	C₄⋊C₈	—	2	2	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	2	2	no	—
148	C₂²⋊C₈	—	2	2	no	—
149	C₂²⋊C₈	—	2	2	no	—
150	C₂²⋊C₈	—	2	2	no	—
151	#97	—	2	1	no	—
152	C₄×C₈	—	2	1	no	—
153	C₄×C₈	—	2	2	no	—
154	C₄×C₈	C₅×D₄	1	1	yes	characteristic
155	C₄×C₈	C₅×D₄	1	2	yes	—
156	C₄×C₈	C₅×D₄	1	1	yes	characteristic
157	C₂²×C₅	—	2	1	no	—
158	C₂²×C₅	—	2	1	no	—
159	C₂²×C₅	—	4	1	no	—
160	C₂²×C₅	—	4	1	no	—
161	C₂²×C₅	—	4	1	no	—
162	C₂²×C₅	#380	1	1	yes	characteristic
163	C₂²×C₅	#433	1	2	yes	—
164	C₄×C₅	—	2	1	no	—
165	C₄×C₅	—	2	1	no	—
166	C₄×C₅	—	2	1	no	—
167	C₄×C₅	—	4	1	no	—
168	C₄×C₅	—	4	1	no	—
169	C₄×C₅	—	4	1	no	—
170	C₄×C₅	#408	1	2	yes	—
171	C₄×C₅	#424	1	2	yes	—
172	C₄×C₅	#472	1	1	yes	characteristic
173	C₄×C₅	#473	1	1	yes	characteristic
174	#41	—	2	4	no	—
175	C₂×Q₈	C₅×C₄⋊C₄	1	1	yes	characteristic
176	C₂×D₄	—	2	1	no	—
177	C₂×D₄	C₅×C₄⋊C₄	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	C₂²×C₄×C₅	1	1	yes	characteristic
182	C₂²×C₄	C₅×#41	1	1	yes	characteristic
183	C₂²×C₄	C₅×#41	1	1	yes	characteristic
184	M₁₆	—	2	4	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	8	2	no	—
190	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
191	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
192	C₄⋊C₄	—	2	1	no	—
193	C₄⋊C₄	—	2	1	no	—
194	C₄⋊C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
195	C₄⋊C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
196	C₂²⋊C₄	—	2	1	no	—
197	C₂²⋊C₄	—	2	1	no	—
198	C₂²⋊C₄	—	2	1	no	—
199	C₄²	—	2	1	no	—
200	C₄²	—	2	1	no	—
201	C₄²	—	2	1	no	—
202	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
203	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
204	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
205	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
206	C₂×C₅	—	2	2	no	—
207	C₂×C₅	—	8	1	no	—
208	C₂×C₅	#1798	1	1	yes	characteristic
209	C₂×C₅	#1814	1	1	yes	characteristic
210	C₂×C₅	#1816	1	1	yes	characteristic
211	C₂³	—	2	1	no	—
212	C₂³	C₅×#117	1	1	yes	characteristic
213	Q₈	—	2	2	no	—
214	D₄	—	2	2	no	—
215	D₄	—	4	2	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
227	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
228	C₂×C₄	C₅×#121	1	1	yes	characteristic
229	C₂×C₄	C₅×#121	1	1	yes	characteristic
230	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
231	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
232	C₂×C₄	C₅×#131	1	4	yes	—
233	C₈	—	2	2	no	—
234	C₈	—	2	2	no	—
235	C₈	—	4	2	no	—
236	C₈	—	8	2	no	—
237	C₈	—	8	2	no	—
238	C₅	#8606	1	1	yes	characteristic, direct product factor, Sylow
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	4	1	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	C₅×#380	1	1	yes	characteristic
245	C₂²	C₅×#433	1	2	yes	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	4	1	no	—
250	C₄	—	4	1	no	—
251	C₄	—	4	1	no	—
252	C₄	C₅×#408	1	2	yes	—
253	C₄	C₅×#424	1	2	yes	—
254	C₄	C₅×#472	1	1	yes	characteristic
255	C₄	C₅×#473	1	1	yes	characteristic
256	C₂	—	2	2	no	—
257	C₂	—	8	1	no	—
258	C₂	C₅×#1798	1	1	yes	characteristic
259	C₂	C₅×#1814	1	1	yes	characteristic
260	C₂	C₅×#1816	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×#335	C₂	semidirect
5. C₅×#1283	C₂	semidirect
21. #8606	C₅	direct (N × H)
51. #1564	C₂×C₅	semidirect
52. C₂×#335	C₂×C₅	semidirect
54. #1283	C₂×C₅	semidirect
238. C₅	#8606	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×#335	2
C₅×#1477	2
C₅×#1283	2
#8606	5

Subgroups — C5×#8606 #11931326

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8606 #11931326