Subgroups — C₅×#7855 #11930575

Order 1280.

706 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2920	C₂	1	2	yes	maximal
2	C₂×C₅×#376	C₂	1	1	yes	maximal, characteristic
3	C₅×#1274	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄²×C₅	C₄	1	2	yes	—
5	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
6	C₅×#432	—	2	2	no	—
7	C₅×#432	—	2	2	no	—
8	C₅×#431	C₄	1	4	yes	complemented
9	C₅×#431	C₂²	1	4	yes	—
10	C₅×#423	—	2	2	no	—
11	C₅×#407	—	2	2	no	—
12	C₄×C₅×M₁₆	C₄	1	4	yes	complemented
13	C₅×#376	—	2	2	no	—
14	C₅×#376	C₄	1	2	yes	—
15	C₅×#376	C₂²	1	2	yes	—
16	C₂×C₅×#95	—	2	2	no	—
17	#7855	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₄×C₅	—	2	2	no	—
19	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₅	D₄	1	2	yes	—
21	C₂×C₅×M₁₆	—	2	4	no	—
22	C₂×C₄²×C₅	—	2	2	no	—
23	C₂×C₄²×C₅	—	2	8	no	—
24	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₅	D₄	1	2	yes	—
29	C₂×C₄²×C₅	D₄	1	2	yes	—
30	C₂×C₄²×C₅	Q₈	1	2	yes	—
31	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
32	C₅×C₄⋊C₈	—	2	4	no	—
33	C₅×C₄⋊C₈	—	2	4	no	—
34	C₅×C₂²⋊C₈	—	2	4	no	—
35	C₅×C₂²⋊C₈	—	2	4	no	—
36	C₅×#97	—	2	4	no	—
37	C₄×C₅×C₈	C₂×C₄	1	8	yes	complemented
38	C₅×#95	—	2	4	no	—
39	C₅×#95	—	2	4	no	—
40	#2920	C₂×C₅	1	2	yes	—
41	C₂×#376	C₂×C₅	1	1	yes	characteristic
42	#1274	C₂×C₅	1	4	yes	complemented
43	C₂⁴×C₅	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅	—	2	4	no	—
45	C₂²×C₄×C₅	—	2	8	no	—
46	C₂²×C₄×C₅	—	2	4	no	—
47	C₂²×C₄×C₅	—	4	4	no	—
48	C₂²×C₄×C₅	—	4	8	no	—
49	C₂²×C₄×C₅	C₄²	1	2	yes	—
50	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
59	C₅×M₁₆	—	4	8	no	—
60	C₂×C₅×C₈	—	2	8	no	—
61	C₄²×C₅	—	2	8	no	—
62	C₄²×C₅	—	2	4	no	—
63	C₄²×C₅	C₄²	1	1	yes	characteristic
64	C₄²×C₅	C₄²	1	1	yes	characteristic
65	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
66	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
67	C₄²×C₅	C₄⋊C₄	1	2	yes	—
68	C₄²×C₅	C₄⋊C₄	1	2	yes	—
69	C₄²×C₅	C₄⋊C₄	1	2	yes	—
70	C₄²×C₅	M₁₆	1	8	yes	complemented
71	C₄²×C₅	C₂²×C₄	1	2	yes	—
72	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
74	C₂²×C₄²	C₄×C₅	1	2	yes	—
75	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
76	#432	—	2	2	no	—
77	#432	—	2	2	no	—
78	#431	C₄×C₅	1	4	yes	complemented
79	#431	C₂²×C₅	1	4	yes	—
80	#423	—	2	2	no	—
81	#407	—	2	2	no	—
82	C₄×M₁₆	C₄×C₅	1	4	yes	complemented
83	#376	—	2	2	no	—
84	#376	C₄×C₅	1	2	yes	—
85	#376	C₂²×C₅	1	2	yes	—
86	C₂×#95	—	2	2	no	—
87	C₂³×C₅	—	2	4	no	—
88	C₂³×C₅	—	2	2	no	—
89	C₂³×C₅	#95	1	2	yes	—
90	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₅	—	2	4	no	—
92	C₂×C₄×C₅	—	2	4	no	—
93	C₂×C₄×C₅	—	2	2	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	2	no	—
101	C₂×C₄×C₅	—	4	16	no	—
102	C₂×C₄×C₅	—	4	8	no	—
103	C₂×C₄×C₅	#95	1	2	yes	—
104	C₂×C₄×C₅	#95	1	2	yes	—
105	C₂×C₄×C₅	#95	1	2	yes	—
106	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
108	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
111	C₅×C₈	—	4	8	no	—
112	C₂³×C₄	—	2	2	no	—
113	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
114	C₂³×C₄	C₅×D₄	1	2	yes	—
115	C₂×M₁₆	—	2	4	no	—
116	C₂×C₄²	—	2	2	no	—
117	C₂×C₄²	—	2	8	no	—
118	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
119	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
120	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
121	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
122	C₂×C₄²	C₅×D₄	1	2	yes	—
123	C₂×C₄²	C₅×D₄	1	2	yes	—
124	C₂×C₄²	C₅×Q₈	1	2	yes	—
125	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
126	C₄⋊C₈	—	2	4	no	—
127	C₄⋊C₈	—	2	4	no	—
128	C₂²⋊C₈	—	2	4	no	—
129	C₂²⋊C₈	—	2	4	no	—
130	#97	—	2	4	no	—
131	C₄×C₈	C₂×C₄×C₅	1	8	yes	complemented
132	#95	—	2	4	no	—
133	#95	—	2	4	no	—
134	C₂²×C₅	—	2	8	no	—
135	C₂²×C₅	—	2	2	no	—
136	C₂²×C₅	—	2	2	no	—
137	C₂²×C₅	—	2	2	no	—
138	C₂²×C₅	H(3,ℤ/4)	1	4	yes	—
139	C₂²×C₅	C₂×#95	1	1	yes	characteristic
140	C₂²×C₅	#407	1	2	yes	—
141	C₄×C₅	—	2	2	no	—
142	C₄×C₅	—	2	2	no	—
143	C₄×C₅	—	2	2	no	—
144	C₄×C₅	—	2	2	no	—
145	C₄×C₅	—	2	2	no	—
146	C₄×C₅	—	4	8	no	—
147	C₄×C₅	C₄×M₁₆	1	2	yes	—
148	C₄×C₅	#423	1	2	yes	—
149	C₂⁴	C₅×C₂²⋊C₄	1	1	yes	characteristic
150	C₂²×C₄	—	2	4	no	—
151	C₂²×C₄	—	2	8	no	—
152	C₂²×C₄	—	2	4	no	—
153	C₂²×C₄	—	4	4	no	—
154	C₂²×C₄	—	4	8	no	—
155	C₂²×C₄	C₄²×C₅	1	2	yes	—
156	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
157	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
158	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
159	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
160	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
161	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
162	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
163	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
164	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
165	M₁₆	—	4	8	no	—
166	C₂×C₈	—	2	8	no	—
167	C₄²	—	2	8	no	—
168	C₄²	—	2	4	no	—
169	C₄²	C₄²×C₅	1	1	yes	characteristic
170	C₄²	C₄²×C₅	1	1	yes	characteristic
171	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
172	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
173	C₄²	C₅×C₄⋊C₄	1	2	yes	—
174	C₄²	C₅×C₄⋊C₄	1	2	yes	—
175	C₄²	C₅×C₄⋊C₄	1	2	yes	—
176	C₄²	C₅×M₁₆	1	8	yes	complemented
177	C₄²	C₂²×C₄×C₅	1	2	yes	—
178	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
179	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
180	C₂×C₅	—	2	2	no	—
181	C₂×C₅	—	2	4	no	—
182	C₂×C₅	#1728	1	1	yes	characteristic
183	C₂×C₅	C₂×H(3,ℤ/4)	1	2	yes	—
184	C₂³	—	2	2	no	—
185	C₂³	—	2	4	no	—
186	C₂³	C₅×#95	1	2	yes	—
187	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	8	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	16	no	—
199	C₂×C₄	—	4	8	no	—
200	C₂×C₄	C₅×#95	1	2	yes	—
201	C₂×C₄	C₅×#95	1	2	yes	—
202	C₂×C₄	C₅×#95	1	2	yes	—
203	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
204	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
205	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
206	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
207	C₂×C₄	C₂×C₅×M₁₆	1	4	yes	—
208	C₈	—	4	8	no	—
209	C₅	#7855	1	1	yes	characteristic, direct product factor, Sylow
210	C₂²	—	2	8	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	C₅×H(3,ℤ/4)	1	4	yes	—
215	C₂²	C₂×C₅×#95	1	1	yes	characteristic
216	C₂²	C₅×#407	1	2	yes	—
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	2	no	—
221	C₄	—	2	2	no	—
222	C₄	—	4	8	no	—
223	C₄	C₄×C₅×M₁₆	1	2	yes	—
224	C₄	C₅×#423	1	2	yes	—
225	C₂	—	2	2	no	—
226	C₂	—	2	4	no	—
227	C₂	C₅×#1728	1	1	yes	characteristic
228	C₂	C₂×C₅×H(3,ℤ/4)	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₅×#1274	C₂	semidirect
8. C₅×#431	C₄	semidirect
12. C₄×C₅×M₁₆	C₄	semidirect
17. #7855	C₅	direct (N × H)
37. C₄×C₅×C₈	C₂×C₄	semidirect
42. #1274	C₂×C₅	semidirect
70. C₄²×C₅	M₁₆	semidirect
78. #431	C₄×C₅	semidirect
82. C₄×M₁₆	C₄×C₅	semidirect
131. C₄×C₈	C₂×C₄×C₅	semidirect
176. C₄²	C₅×M₁₆	semidirect
209. C₅	#7855	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×#2920	2
C₂×C₅×#376	2
C₅×#1274	2
#7855	5

Subgroups — C5×#7855 #11930575

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#7855 #11930575