Subgroups — C₅×#8625 #11931345

Order 1280.

342 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#446	C₂	1	1	yes	maximal, characteristic
2	C₅×#1568	C₂	1	1	yes	maximal, characteristic
3	C₅×#1568	C₂	1	1	yes	maximal, characteristic
4	C₅×#1467	C₂	1	2	yes	maximal
5	C₅×#1290	C₂	1	2	yes	maximal
6	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
7	C₅×#446	—	2	1	no	—
8	C₅×#446	—	2	1	no	—
9	C₅×#446	—	2	1	no	—
10	C₅×#446	—	2	1	no	—
11	C₅×#432	C₂²	1	2	yes	—
12	C₅×#432	C₂²	1	2	yes	—
13	C₂×C₅×C₄⋊C₈	C₄	1	2	yes	—
14	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₅×#97	C₄	1	2	yes	—
16	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₅×#335	—	2	1	no	—
18	C₅×#335	—	2	1	no	—
19	C₅×#334	—	2	1	no	—
20	C₅×#334	—	2	1	no	—
21	C₅×#324	—	2	4	no	—
22	#8625	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂²×C₅×Q₈	Q₈	1	1	yes	characteristic
24	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
26	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₅×Q₈	—	2	1	no	—
28	C₄×C₅×Q₈	—	2	1	no	—
29	C₄×C₅×Q₈	—	2	1	no	—
30	C₄×C₅×Q₈	—	2	1	no	—
31	C₂×C₅×C₄⋊C₄	D₄	1	2	yes	—
32	C₂×C₅×C₄⋊C₄	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
35	C₅×C₄⋊C₈	—	2	2	no	—
36	C₅×C₄⋊C₈	—	2	2	no	—
37	C₅×C₄⋊C₈	—	2	1	no	—
38	C₅×C₄⋊C₈	—	2	1	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×#97	—	2	2	no	—
44	C₅×#97	—	2	2	no	—
45	C₅×#97	—	2	2	no	—
46	C₅×#97	—	2	2	no	—
47	C₄×C₅×C₈	D₄	1	1	yes	characteristic
48	C₄×C₅×C₈	D₄	1	1	yes	characteristic
49	C₄×C₅×C₈	D₄	1	1	yes	characteristic
50	C₄×C₅×C₈	D₄	1	1	yes	characteristic
51	C₂×#446	C₂×C₅	1	1	yes	characteristic
52	#1568	C₂×C₅	1	1	yes	characteristic
53	#1568	C₂×C₅	1	1	yes	characteristic
54	#1467	C₂×C₅	1	2	yes	—
55	#1290	C₂×C₅	1	2	yes	—
56	C₂×C₅×Q₈	—	2	1	no	—
57	C₂×C₅×Q₈	—	2	1	no	—
58	C₂×C₅×Q₈	—	4	2	no	—
59	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₅×C₈	—	2	2	no	—
65	C₂×C₅×C₈	—	2	2	no	—
66	C₂×C₅×C₈	—	4	1	no	—
67	C₂×C₅×C₈	—	4	2	no	—
68	C₂×C₅×C₈	—	4	2	no	—
69	C₂×C₅×C₈	—	4	2	no	—
70	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
74	C₅×C₄⋊C₄	—	2	2	no	—
75	C₅×C₄⋊C₄	—	2	2	no	—
76	C₅×C₄⋊C₄	—	2	1	no	—
77	C₅×C₄⋊C₄	—	2	1	no	—
78	C₄²×C₅	—	2	2	no	—
79	C₄²×C₅	—	2	2	no	—
80	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₅	#41	1	1	yes	characteristic
83	C₄²×C₅	#41	1	1	yes	characteristic
84	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
85	#446	—	2	1	no	—
86	#446	—	2	1	no	—
87	#446	—	2	1	no	—
88	#446	—	2	1	no	—
89	#432	C₂²×C₅	1	2	yes	—
90	#432	C₂²×C₅	1	2	yes	—
91	C₂×C₄⋊C₈	C₄×C₅	1	2	yes	—
92	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
93	C₂×#97	C₄×C₅	1	2	yes	—
94	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
95	#335	—	2	1	no	—
96	#335	—	2	1	no	—
97	#334	—	2	1	no	—
98	#334	—	2	1	no	—
99	#324	—	2	4	no	—
100	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₅×Q₈	—	4	2	no	—
102	C₅×Q₈	—	4	2	no	—
103	C₂×C₄×C₅	—	2	2	no	—
104	C₂×C₄×C₅	—	2	1	no	—
105	C₂×C₄×C₅	—	2	1	no	—
106	C₂×C₄×C₅	—	2	2	no	—
107	C₂×C₄×C₅	—	2	2	no	—
108	C₂×C₄×C₅	—	2	4	no	—
109	C₂×C₄×C₅	—	4	1	no	—
110	C₂×C₄×C₅	—	4	2	no	—
111	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₅	#117	1	1	yes	characteristic
113	C₂×C₄×C₅	#121	1	1	yes	characteristic
114	C₂×C₄×C₅	#121	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
117	C₅×C₈	—	2	1	no	—
118	C₅×C₈	—	2	1	no	—
119	C₅×C₈	—	2	1	no	—
120	C₅×C₈	—	4	2	no	—
121	C₅×C₈	—	8	2	no	—
122	C₅×C₈	—	8	2	no	—
123	C₅×C₈	#101	1	2	yes	—
124	C₂²×Q₈	C₅×Q₈	1	1	yes	characteristic
125	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
126	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
127	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
128	C₄×Q₈	—	2	1	no	—
129	C₄×Q₈	—	2	1	no	—
130	C₄×Q₈	—	2	1	no	—
131	C₄×Q₈	—	2	1	no	—
132	C₂×C₄⋊C₄	C₅×D₄	1	2	yes	—
133	C₂×C₄⋊C₄	C₅×Q₈	1	1	yes	characteristic
134	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
135	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
136	C₄⋊C₈	—	2	2	no	—
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	—	2	1	no	—
139	C₄⋊C₈	—	2	1	no	—
140	C₄⋊C₈	—	2	2	no	—
141	C₄⋊C₈	—	2	2	no	—
142	C₂²⋊C₈	—	2	2	no	—
143	C₂²⋊C₈	—	2	2	no	—
144	#97	—	2	2	no	—
145	#97	—	2	2	no	—
146	#97	—	2	2	no	—
147	#97	—	2	2	no	—
148	C₄×C₈	C₅×D₄	1	1	yes	characteristic
149	C₄×C₈	C₅×D₄	1	1	yes	characteristic
150	C₄×C₈	C₅×D₄	1	1	yes	characteristic
151	C₄×C₈	C₅×D₄	1	1	yes	characteristic
152	C₂²×C₅	—	2	1	no	—
153	C₂²×C₅	—	2	1	no	—
154	C₂²×C₅	#380	1	1	yes	characteristic
155	C₂²×C₅	#430	1	2	yes	—
156	C₄×C₅	—	2	1	no	—
157	C₄×C₅	—	2	1	no	—
158	C₄×C₅	—	2	1	no	—
159	C₄×C₅	—	2	1	no	—
160	C₄×C₅	—	2	1	no	—
161	C₄×C₅	—	4	2	no	—
162	C₄×C₅	—	4	2	no	—
163	C₄×C₅	C₂×#101	1	1	yes	characteristic
164	C₄×C₅	#413	1	1	yes	characteristic
165	C₂×Q₈	—	2	1	no	—
166	C₂×Q₈	—	2	1	no	—
167	C₂×Q₈	—	4	2	no	—
168	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
169	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
170	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
171	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	4	1	no	—
176	C₂×C₈	—	4	2	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	—	4	2	no	—
179	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
180	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
181	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
182	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	2	1	no	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄²	—	2	2	no	—
188	C₄²	—	2	2	no	—
189	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
190	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
191	C₄²	C₅×#41	1	1	yes	characteristic
192	C₄²	C₅×#41	1	1	yes	characteristic
193	C₂×C₅	—	2	2	no	—
194	C₂×C₅	#1801	1	1	yes	characteristic
195	C₂×C₅	#1811	1	1	yes	characteristic
196	C₂×C₅	#1812	1	1	yes	characteristic
197	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
198	Q₈	—	4	2	no	—
199	Q₈	—	4	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
209	C₂×C₄	C₅×#117	1	1	yes	characteristic
210	C₂×C₄	C₅×#121	1	1	yes	characteristic
211	C₂×C₄	C₅×#121	1	1	yes	characteristic
212	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
213	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
214	C₈	—	2	1	no	—
215	C₈	—	2	1	no	—
216	C₈	—	2	1	no	—
217	C₈	—	4	2	no	—
218	C₈	—	8	2	no	—
219	C₈	—	8	2	no	—
220	C₈	C₅×#101	1	2	yes	—
221	C₅	#8625	1	1	yes	characteristic, direct product factor, Sylow
222	C₂²	—	2	1	no	—
223	C₂²	—	2	1	no	—
224	C₂²	C₅×#380	1	1	yes	characteristic
225	C₂²	C₅×#430	1	2	yes	—
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	4	2	no	—
232	C₄	—	4	2	no	—
233	C₄	C₂×C₅×#101	1	1	yes	characteristic
234	C₄	C₅×#413	1	1	yes	characteristic
235	C₂	—	2	2	no	—
236	C₂	C₅×#1801	1	1	yes	characteristic
237	C₂	C₅×#1811	1	1	yes	characteristic
238	C₂	C₅×#1812	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
22. #8625	C₅	direct (N × H)
221. C₅	#8625	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×#446	2
C₅×#1568	2
C₅×#1467	2
C₅×#1290	2
#8625	5

Subgroups — C5×#8625 #11931345

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8625 #11931345