Subgroups — C₅×#8664 #11931384

Order 1280.

434 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#436	C₂	1	1	yes	maximal, characteristic
2	C₅×#1837	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#1564	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1486	C₂	1	2	yes	maximal
5	C₅×#1301	C₂	1	2	yes	maximal, complemented
6	C₂²×C₅×M₁₆	—	2	1	no	—
7	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
8	C₅×#436	—	2	1	no	—
9	C₅×#436	—	2	1	no	—
10	C₅×#436	C₄	1	2	yes	—
11	C₅×#436	C₄	1	2	yes	—
12	C₅×#431	C₂²	1	2	yes	—
13	C₅×#423	C₂²	1	2	yes	—
14	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₅×C₂²⋊C₈	—	2	1	no	—
16	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₅×#341	—	2	1	no	—
18	C₅×#341	—	2	1	no	—
19	C₅×#335	—	2	1	no	—
20	C₅×#334	—	2	1	no	—
21	C₅×D₄⋊C₈	—	2	4	no	—
22	#8664	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂²×C₅×D₄	Q₈	1	1	yes	characteristic
24	C₂³×C₄×C₅	—	2	1	no	—
25	C₂×C₅×M₁₆	—	2	2	no	—
26	C₂×C₅×M₁₆	—	2	2	no	—
27	C₂×C₅×M₁₆	—	2	2	no	—
28	C₂×C₅×M₁₆	—	4	4	no	—
29	C₂²×C₅×C₈	—	2	1	no	—
30	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
31	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
32	C₄×C₅×D₄	—	2	1	no	—
33	C₄×C₅×D₄	—	2	1	no	—
34	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
35	C₄×C₅×D₄	D₄	1	2	yes	—
36	C₂×C₅×C₄⋊C₄	Q₈	1	1	yes	characteristic
37	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
38	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
39	C₅×C₄⋊C₈	—	2	1	no	—
40	C₅×C₄⋊C₈	—	2	2	no	—
41	C₅×C₄⋊C₈	—	2	2	no	—
42	C₅×C₄⋊C₈	—	2	2	no	—
43	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
44	C₅×C₂²⋊C₈	—	2	2	no	—
45	C₅×C₂²⋊C₈	—	2	2	no	—
46	C₅×C₂²⋊C₈	—	2	2	no	—
47	C₄×C₅×C₈	—	2	2	no	—
48	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
49	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
50	C₄×C₅×C₈	D₄	1	1	yes	characteristic
51	C₄×C₅×C₈	D₄	1	1	yes	characteristic
52	C₂×#436	C₂×C₅	1	1	yes	characteristic
53	#1837	C₂×C₅	1	1	yes	characteristic, complemented
54	#1564	C₂×C₅	1	1	yes	characteristic, complemented
55	#1486	C₂×C₅	1	2	yes	—
56	#1301	C₂×C₅	1	2	yes	complemented
57	C₂⁴×C₅	—	2	1	no	—
58	C₂×C₅×D₄	—	2	1	no	—
59	C₂×C₅×D₄	—	4	2	no	—
60	C₂×C₅×D₄	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₅	—	2	1	no	—
62	C₂²×C₄×C₅	—	2	2	no	—
63	C₂²×C₄×C₅	—	2	2	no	—
64	C₂²×C₄×C₅	—	4	2	no	—
65	C₂²×C₄×C₅	—	4	2	no	—
66	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₅	#41	1	1	yes	characteristic
69	C₅×M₁₆	—	4	4	no	—
70	C₅×M₁₆	—	4	4	no	—
71	C₅×M₁₆	—	4	2	no	—
72	C₅×M₁₆	—	4	2	no	—
73	C₂×C₅×C₈	—	2	2	no	—
74	C₂×C₅×C₈	—	2	2	no	—
75	C₂×C₅×C₈	—	2	2	no	—
76	C₂×C₅×C₈	—	4	2	no	—
77	C₂×C₅×C₈	—	4	2	no	—
78	C₂×C₅×C₈	—	4	2	no	—
79	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
83	C₅×C₄⋊C₄	—	2	1	no	—
84	C₅×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
85	C₅×C₂²⋊C₄	—	2	2	no	—
86	C₅×C₂²⋊C₄	—	2	2	no	—
87	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₅	#41	1	1	yes	characteristic
91	C₂²×M₁₆	—	2	1	no	—
92	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
93	#436	—	2	1	no	—
94	#436	—	2	1	no	—
95	#436	C₄×C₅	1	2	yes	—
96	#436	C₄×C₅	1	2	yes	—
97	#431	C₂²×C₅	1	2	yes	—
98	#423	C₂²×C₅	1	2	yes	—
99	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
100	C₂×C₂²⋊C₈	—	2	1	no	—
101	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
102	#341	—	2	1	no	—
103	#341	—	2	1	no	—
104	#335	—	2	1	no	—
105	#334	—	2	1	no	—
106	D₄⋊C₈	—	2	4	no	—
107	C₂³×C₅	—	2	2	no	—
108	C₂³×C₅	—	4	2	no	—
109	C₂³×C₅	—	4	2	no	—
110	C₂³×C₅	—	4	2	no	—
111	C₂³×C₅	C₂²⋊Q₈	1	1	yes	characteristic
112	C₅×D₄	—	2	4	no	—
113	C₅×D₄	—	4	2	no	—
114	C₂×C₄×C₅	—	2	2	no	—
115	C₂×C₄×C₅	—	2	2	no	—
116	C₂×C₄×C₅	—	4	1	no	—
117	C₂×C₄×C₅	—	4	2	no	—
118	C₂×C₄×C₅	—	4	2	no	—
119	C₂×C₄×C₅	—	4	2	no	—
120	C₂×C₄×C₅	—	4	4	no	—
121	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₅	#117	1	1	yes	characteristic
124	C₂×C₄×C₅	#121	1	1	yes	characteristic
125	C₂×C₄×C₅	#121	1	1	yes	characteristic
126	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₅	#135	1	2	yes	—
128	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
129	C₅×C₈	—	2	1	no	—
130	C₅×C₈	—	2	1	no	—
131	C₅×C₈	—	2	1	no	—
132	C₅×C₈	—	2	1	no	—
133	C₅×C₈	—	4	2	no	—
134	C₅×C₈	—	8	2	no	—
135	C₅×C₈	—	8	2	no	—
136	C₂²×D₄	C₅×Q₈	1	1	yes	characteristic
137	C₂³×C₄	—	2	1	no	—
138	C₂×M₁₆	—	2	2	no	—
139	C₂×M₁₆	—	2	2	no	—
140	C₂×M₁₆	—	2	2	no	—
141	C₂×M₁₆	—	4	4	no	—
142	C₂²×C₈	—	2	1	no	—
143	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
144	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
145	C₄×D₄	—	2	1	no	—
146	C₄×D₄	—	2	1	no	—
147	C₄×D₄	C₂×C₄×C₅	1	2	yes	—
148	C₄×D₄	C₅×D₄	1	2	yes	—
149	C₂×C₄⋊C₄	C₅×Q₈	1	1	yes	characteristic
150	C₂×C₂²⋊C₄	—	2	1	no	—
151	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
152	C₄⋊C₈	—	2	1	no	—
153	C₄⋊C₈	—	2	2	no	—
154	C₄⋊C₈	—	2	2	no	—
155	C₄⋊C₈	—	2	2	no	—
156	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
157	C₂²⋊C₈	—	2	2	no	—
158	C₂²⋊C₈	—	2	2	no	—
159	C₂²⋊C₈	—	2	2	no	—
160	C₄×C₈	—	2	2	no	—
161	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
162	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
163	C₄×C₈	C₅×D₄	1	1	yes	characteristic
164	C₄×C₈	C₅×D₄	1	1	yes	characteristic
165	C₂²×C₅	—	2	1	no	—
166	C₂²×C₅	—	2	1	no	—
167	C₂²×C₅	—	4	2	no	—
168	C₂²×C₅	—	4	2	no	—
169	C₂²×C₅	—	4	2	no	—
170	C₂²×C₅	—	8	4	no	—
171	C₂²×C₅	#380	1	1	yes	characteristic
172	C₂²×C₅	#478	1	2	yes	—
173	C₄×C₅	—	2	1	no	—
174	C₄×C₅	—	2	1	no	—
175	C₄×C₅	—	2	1	no	—
176	C₄×C₅	—	4	2	no	—
177	C₄×C₅	#416	1	1	yes	characteristic
178	C₄×C₅	#418	1	1	yes	characteristic
179	C₄×C₅	#430	1	2	yes	—
180	C₄×C₅	#463	1	2	yes	—
181	C₂⁴	—	2	1	no	—
182	C₂×D₄	—	2	1	no	—
183	C₂×D₄	—	4	2	no	—
184	C₂×D₄	C₅×C₄⋊C₄	1	2	yes	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
192	C₂²×C₄	C₅×#41	1	1	yes	characteristic
193	M₁₆	—	4	4	no	—
194	M₁₆	—	4	4	no	—
195	M₁₆	—	4	2	no	—
196	M₁₆	—	4	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	4	2	no	—
202	C₂×C₈	—	4	2	no	—
203	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
204	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
205	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
206	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
207	C₄⋊C₄	—	2	1	no	—
208	C₄⋊C₄	C₅×C₄⋊C₄	1	2	yes	—
209	C₂²⋊C₄	—	2	2	no	—
210	C₂²⋊C₄	—	2	2	no	—
211	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
212	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
213	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
214	C₄²	C₅×#41	1	1	yes	characteristic
215	C₂×C₅	—	2	2	no	—
216	C₂×C₅	—	8	2	no	—
217	C₂×C₅	#1804	1	1	yes	characteristic
218	C₂×C₅	#1811	1	1	yes	characteristic
219	C₂×C₅	#1814	1	1	yes	characteristic
220	C₂³	—	2	2	no	—
221	C₂³	—	4	2	no	—
222	C₂³	—	4	2	no	—
223	C₂³	—	4	2	no	—
224	C₂³	C₅×C₂²⋊Q₈	1	1	yes	characteristic
225	D₄	—	2	4	no	—
226	D₄	—	4	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	4	1	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
235	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
236	C₂×C₄	C₅×#117	1	1	yes	characteristic
237	C₂×C₄	C₅×#121	1	1	yes	characteristic
238	C₂×C₄	C₅×#121	1	1	yes	characteristic
239	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
240	C₂×C₄	C₅×#135	1	2	yes	—
241	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
242	C₈	—	2	1	no	—
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	4	2	no	—
247	C₈	—	8	2	no	—
248	C₈	—	8	2	no	—
249	C₅	#8664	1	1	yes	characteristic, direct product factor, Sylow
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	4	2	no	—
253	C₂²	—	4	2	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	8	4	no	—
256	C₂²	C₅×#380	1	1	yes	characteristic
257	C₂²	C₅×#478	1	2	yes	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	4	2	no	—
262	C₄	C₅×#416	1	1	yes	characteristic
263	C₄	C₅×#418	1	1	yes	characteristic
264	C₄	C₅×#430	1	2	yes	—
265	C₄	C₅×#463	1	2	yes	—
266	C₂	—	2	2	no	—
267	C₂	—	8	2	no	—
268	C₂	C₅×#1804	1	1	yes	characteristic
269	C₂	C₅×#1811	1	1	yes	characteristic
270	C₂	C₅×#1814	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	#121
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	#380
prime radical	C₂²×C₅×D₄	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ 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×#1837	C₂	semidirect
3. C₅×#1564	C₂	semidirect
5. C₅×#1301	C₂	semidirect
22. #8664	C₅	direct (N × H)
53. #1837	C₂×C₅	semidirect
54. #1564	C₂×C₅	semidirect
56. #1301	C₂×C₅	semidirect
249. C₅	#8664	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×#436	2
C₅×#1837	2
C₅×#1564	2
C₅×#1486	2
C₅×#1301	2
#8664	5

Subgroups — C5×#8664 #11931384

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8664 #11931384