Subgroups — C₅×#7866 #11930586

Order 1280.

714 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#431	C₂	1	2	yes	maximal
2	C₄×C₅×#117	C₂	1	1	yes	maximal, characteristic
3	C₅×#1276	C₂	1	4	yes	maximal, complemented
4	C₂×C₅×#117	C₄	1	2	yes	—
5	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
6	C₅×#431	—	2	2	no	—
7	C₅×#431	—	2	2	no	—
8	C₅×#431	C₄	1	4	yes	complemented
9	C₅×#431	C₂²	1	4	yes	—
10	C₂×C₅×C₄⋊C₈	C₄	1	4	yes	complemented
11	C₂×C₅×C₂²⋊C₈	—	2	2	no	—
12	C₂×C₄×C₅×C₈	—	2	2	no	—
13	C₄×C₅×C₄⋊C₄	—	2	2	no	—
14	C₄×C₅×C₂²⋊C₄	—	2	2	no	—
15	C₅×#376	C₄	1	2	yes	—
16	C₄³×C₅	C₂²	1	2	yes	—
17	#7866	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₅	D₄	1	2	yes	—
20	C₂²×C₅×C₈	—	2	4	no	—
21	C₅×#117	—	2	4	no	—
22	C₅×#117	C₈	1	8	yes	complemented
23	C₂×C₅×C₄⋊C₄	—	2	2	no	—
24	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
25	C₂×C₄²×C₅	—	2	4	no	—
26	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₅	D₄	1	2	yes	—
31	C₂×C₄²×C₅	D₄	1	2	yes	—
32	C₂×C₄²×C₅	Q₈	1	2	yes	—
33	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
34	C₅×C₄⋊C₈	—	2	4	no	—
35	C₅×C₄⋊C₈	C₂×C₄	1	8	yes	—
36	C₅×C₂²⋊C₈	—	2	4	no	—
37	C₅×C₂²⋊C₈	—	2	4	no	—
38	C₄×C₅×C₈	—	2	4	no	—
39	C₄×C₅×C₈	—	2	4	no	—
40	C₅×#95	—	2	4	no	—
41	C₂×#431	C₂×C₅	1	2	yes	—
42	C₄×#117	C₂×C₅	1	1	yes	characteristic
43	#1276	C₂×C₅	1	4	yes	complemented
44	C₂⁴×C₅	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅	—	2	4	no	—
46	C₂²×C₄×C₅	—	4	4	no	—
47	C₂²×C₄×C₅	C₄²	1	2	yes	—
48	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
56	C₂²×C₄×C₅	M₁₆	1	4	yes	—
57	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
59	C₂×C₅×C₈	—	2	8	no	—
60	C₂×C₅×C₈	—	4	8	no	—
61	C₅×C₄⋊C₄	—	2	8	no	—
62	C₅×C₂²⋊C₄	—	2	8	no	—
63	C₄²×C₅	—	2	4	no	—
64	C₄²×C₅	—	4	8	no	—
65	C₄²×C₅	C₄²	1	1	yes	characteristic
66	C₄²×C₅	C₄²	1	1	yes	characteristic
67	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
69	C₄²×C₅	C₄⋊C₄	1	2	yes	—
70	C₄²×C₅	C₄⋊C₄	1	2	yes	—
71	C₄²×C₅	C₄⋊C₄	1	2	yes	—
72	C₄²×C₅	C₂×C₈	1	8	yes	complemented
73	C₄²×C₅	C₂²×C₄	1	2	yes	—
74	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
76	C₂×#117	C₄×C₅	1	2	yes	—
77	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
78	#431	—	2	2	no	—
79	#431	—	2	2	no	—
80	#431	C₄×C₅	1	4	yes	complemented
81	#431	C₂²×C₅	1	4	yes	—
82	C₂×C₄⋊C₈	C₄×C₅	1	4	yes	complemented
83	C₂×C₂²⋊C₈	—	2	2	no	—
84	C₂×C₄×C₈	—	2	2	no	—
85	C₄×C₄⋊C₄	—	2	2	no	—
86	C₄×C₂²⋊C₄	—	2	2	no	—
87	#376	C₄×C₅	1	2	yes	—
88	C₄³	C₂²×C₅	1	2	yes	—
89	C₂³×C₅	—	4	2	no	—
90	C₂³×C₅	#95	1	2	yes	—
91	C₂³×C₅	C₂²⋊C₈	1	4	yes	—
92	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₅	—	2	4	no	—
94	C₂×C₄×C₅	—	2	4	no	—
95	C₂×C₄×C₅	—	2	4	no	—
96	C₂×C₄×C₅	—	2	8	no	—
97	C₂×C₄×C₅	—	4	2	no	—
98	C₂×C₄×C₅	—	4	2	no	—
99	C₂×C₄×C₅	—	4	2	no	—
100	C₂×C₄×C₅	—	4	8	no	—
101	C₂×C₄×C₅	#95	1	2	yes	—
102	C₂×C₄×C₅	#95	1	2	yes	—
103	C₂×C₄×C₅	#95	1	2	yes	—
104	C₂×C₄×C₅	C₄×C₈	1	4	yes	—
105	C₂×C₄×C₅	#97	1	4	yes	—
106	C₂×C₄×C₅	C₂²⋊C₈	1	4	yes	—
107	C₂×C₄×C₅	C₄⋊C₈	1	8	yes	—
108	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
110	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
113	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
114	C₅×C₈	—	4	8	no	—
115	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
116	C₂³×C₄	C₅×D₄	1	2	yes	—
117	C₂²×C₈	—	2	4	no	—
118	#117	—	2	4	no	—
119	#117	C₅×C₈	1	8	yes	complemented
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₂²⋊C₄	—	2	2	no	—
122	C₂×C₄²	—	2	4	no	—
123	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
124	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
125	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
126	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
127	C₂×C₄²	C₅×D₄	1	2	yes	—
128	C₂×C₄²	C₅×D₄	1	2	yes	—
129	C₂×C₄²	C₅×Q₈	1	2	yes	—
130	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
131	C₄⋊C₈	—	2	4	no	—
132	C₄⋊C₈	C₂×C₄×C₅	1	8	yes	—
133	C₂²⋊C₈	—	2	4	no	—
134	C₂²⋊C₈	—	2	4	no	—
135	C₄×C₈	—	2	4	no	—
136	C₄×C₈	—	2	4	no	—
137	#95	—	2	4	no	—
138	C₂²×C₅	—	2	2	no	—
139	C₂²×C₅	—	2	2	no	—
140	C₂²×C₅	—	2	2	no	—
141	C₂²×C₅	—	4	4	no	—
142	C₂²×C₅	#336	1	4	yes	—
143	C₂²×C₅	C₂×#95	1	1	yes	characteristic
144	C₂²×C₅	C₂×C₂²⋊C₈	1	2	yes	—
145	C₄×C₅	—	2	2	no	—
146	C₄×C₅	—	2	2	no	—
147	C₄×C₅	—	2	2	no	—
148	C₄×C₅	—	4	8	no	—
149	C₄×C₅	#336	1	4	yes	—
150	C₄×C₅	C₂×C₄×C₈	1	1	yes	characteristic
151	C₄×C₅	C₂×#97	1	1	yes	characteristic
152	C₄×C₅	#421	1	4	yes	—
153	C₄×C₅	C₂×C₄⋊C₈	1	2	yes	—
154	C₂⁴	C₅×C₂²⋊C₄	1	1	yes	characteristic
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	—	4	4	no	—
157	C₂²×C₄	C₄²×C₅	1	2	yes	—
158	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
159	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
160	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
161	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
162	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
163	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
164	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
165	C₂²×C₄	C₂×C₅×C₈	1	4	yes	—
166	C₂²×C₄	C₅×M₁₆	1	4	yes	—
167	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
168	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
169	C₂×C₈	—	2	8	no	—
170	C₂×C₈	—	4	8	no	—
171	C₄⋊C₄	—	2	8	no	—
172	C₂²⋊C₄	—	2	8	no	—
173	C₄²	—	2	4	no	—
174	C₄²	—	4	8	no	—
175	C₄²	C₄²×C₅	1	1	yes	characteristic
176	C₄²	C₄²×C₅	1	1	yes	characteristic
177	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
178	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
179	C₄²	C₅×C₄⋊C₄	1	2	yes	—
180	C₄²	C₅×C₄⋊C₄	1	2	yes	—
181	C₄²	C₅×C₄⋊C₄	1	2	yes	—
182	C₄²	C₂×C₅×C₈	1	8	yes	complemented
183	C₄²	C₂²×C₄×C₅	1	2	yes	—
184	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
185	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
186	C₂×C₅	—	2	2	no	—
187	C₂×C₅	—	2	4	no	—
188	C₂×C₅	C₂×#336	1	1	yes	characteristic
189	C₂×C₅	#1733	1	2	yes	—
190	C₂³	—	4	2	no	—
191	C₂³	C₅×#95	1	2	yes	—
192	C₂³	C₅×C₂²⋊C₈	1	4	yes	—
193	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	8	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	4	8	no	—
202	C₂×C₄	C₅×#95	1	2	yes	—
203	C₂×C₄	C₅×#95	1	2	yes	—
204	C₂×C₄	C₅×#95	1	2	yes	—
205	C₂×C₄	C₄×C₅×C₈	1	4	yes	—
206	C₂×C₄	C₅×#97	1	4	yes	—
207	C₂×C₄	C₅×C₂²⋊C₈	1	4	yes	—
208	C₂×C₄	C₅×C₄⋊C₈	1	8	yes	—
209	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
210	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
211	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
212	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
213	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
214	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
215	C₈	—	4	8	no	—
216	C₅	#7866	1	1	yes	characteristic, direct product factor, Sylow
217	C₂²	—	2	2	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	4	4	no	—
221	C₂²	C₅×#336	1	4	yes	—
222	C₂²	C₂×C₅×#95	1	1	yes	characteristic
223	C₂²	C₂×C₅×C₂²⋊C₈	1	2	yes	—
224	C₄	—	2	2	no	—
225	C₄	—	2	2	no	—
226	C₄	—	2	2	no	—
227	C₄	—	4	8	no	—
228	C₄	C₅×#336	1	4	yes	—
229	C₄	C₂×C₄×C₅×C₈	1	1	yes	characteristic
230	C₄	C₂×C₅×#97	1	1	yes	characteristic
231	C₄	C₅×#421	1	4	yes	—
232	C₄	C₂×C₅×C₄⋊C₈	1	2	yes	—
233	C₂	—	2	2	no	—
234	C₂	—	2	4	no	—
235	C₂	C₂×C₅×#336	1	1	yes	characteristic
236	C₂	C₅×#1733	1	2	yes	—

	subgroup	G/H
center	C₄²×C₅	C₂×D₄
commutator	C₄	C₂×C₄×C₅×C₈
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	C₂×#95
prime radical	C₂⁴×C₅	C₂²⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1276	C₂	semidirect
8. C₅×#431	C₄	semidirect
10. C₂×C₅×C₄⋊C₈	C₄	semidirect
17. #7866	C₅	direct (N × H)
22. C₅×#117	C₈	semidirect
43. #1276	C₂×C₅	semidirect
72. C₄²×C₅	C₂×C₈	semidirect
80. #431	C₄×C₅	semidirect
82. C₂×C₄⋊C₈	C₄×C₅	semidirect
119. #117	C₅×C₈	semidirect
182. C₄²	C₂×C₅×C₈	semidirect
216. C₅	#7866	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×#431	2
C₄×C₅×#117	2
C₅×#1276	2
#7866	5

Subgroups — C5×#7866 #11930586

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#7866 #11930586