Subgroups — C₅×#7856 #11930576

Order 1280.

662 conjugacy classes of subgroups.

The subgroup lattice has 250 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	C₄×C₅×#117	C₂	1	1	yes	maximal, characteristic
3	C₅×#1274	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×#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	C₄×C₅×C₄⋊C₄	—	2	2	no	—
14	C₄×C₅×C₂²⋊C₄	—	2	2	no	—
15	C₅×#376	C₂²	1	2	yes	—
16	C₄³×C₅	C₄	1	2	yes	—
17	#7856	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	C₂×C₅×M₁₆	—	2	4	no	—
21	C₅×#117	—	2	8	no	—
22	C₂×C₅×C₄⋊C₄	—	2	2	no	—
23	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
24	C₂×C₄²×C₅	—	2	4	no	—
25	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₅	D₄	1	2	yes	—
30	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
34	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
35	C₅×C₄⋊C₈	—	2	4	no	—
36	C₅×C₄⋊C₈	—	2	4	no	—
37	C₅×C₂²⋊C₈	—	2	4	no	—
38	C₅×C₂²⋊C₈	—	2	4	no	—
39	C₅×#97	—	2	4	no	—
40	C₄×C₅×C₈	C₂×C₄	1	8	yes	—
41	C₅×#95	—	2	4	no	—
42	#2920	C₂×C₅	1	2	yes	—
43	C₄×#117	C₂×C₅	1	1	yes	characteristic
44	#1274	C₂×C₅	1	4	yes	complemented
45	C₂⁴×C₅	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅	—	2	4	no	—
47	C₂²×C₄×C₅	—	2	4	no	—
48	C₂²×C₄×C₅	—	4	4	no	—
49	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
50	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
51	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
61	C₅×M₁₆	—	4	8	no	—
62	C₂×C₅×C₈	—	2	8	no	—
63	C₅×C₄⋊C₄	—	2	8	no	—
64	C₅×C₂²⋊C₄	—	2	8	no	—
65	C₄²×C₅	—	2	4	no	—
66	C₄²×C₅	—	4	8	no	—
67	C₄²×C₅	C₄²	1	2	yes	—
68	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
69	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
70	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
71	C₄²×C₅	C₄⋊C₄	1	2	yes	—
72	C₄²×C₅	C₄⋊C₄	1	2	yes	—
73	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
74	C₄²×C₅	M₁₆	1	8	yes	complemented
75	C₄²×C₅	C₂²×C₄	1	2	yes	—
76	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
78	C₂×#117	C₄×C₅	1	2	yes	—
79	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
80	#432	—	2	2	no	—
81	#432	—	2	2	no	—
82	#431	C₄×C₅	1	4	yes	complemented
83	#431	C₂²×C₅	1	4	yes	—
84	#423	—	2	2	no	—
85	#407	—	2	2	no	—
86	C₄×M₁₆	C₄×C₅	1	4	yes	complemented
87	C₄×C₄⋊C₄	—	2	2	no	—
88	C₄×C₂²⋊C₄	—	2	2	no	—
89	#376	C₂²×C₅	1	2	yes	—
90	C₄³	C₄×C₅	1	2	yes	—
91	C₂³×C₅	—	2	2	no	—
92	C₂³×C₅	—	4	2	no	—
93	C₂³×C₅	#95	1	1	yes	characteristic
94	C₂³×C₅	#95	1	1	yes	characteristic
95	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅	—	2	2	no	—
97	C₂×C₄×C₅	—	2	4	no	—
98	C₂×C₄×C₅	—	2	4	no	—
99	C₂×C₄×C₅	—	2	4	no	—
100	C₂×C₄×C₅	—	2	2	no	—
101	C₂×C₄×C₅	—	2	2	no	—
102	C₂×C₄×C₅	—	2	2	no	—
103	C₂×C₄×C₅	—	2	2	no	—
104	C₂×C₄×C₅	—	2	8	no	—
105	C₂×C₄×C₅	—	4	2	no	—
106	C₂×C₄×C₅	—	4	2	no	—
107	C₂×C₄×C₅	—	4	2	no	—
108	C₂×C₄×C₅	—	4	8	no	—
109	C₂×C₄×C₅	#95	1	2	yes	—
110	C₂×C₄×C₅	#95	1	1	yes	characteristic
111	C₂×C₄×C₅	#95	1	1	yes	characteristic
112	C₂×C₄×C₅	#95	1	2	yes	—
113	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
115	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
118	C₅×C₈	—	4	8	no	—
119	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
120	C₂³×C₄	C₅×D₄	1	2	yes	—
121	C₂×M₁₆	—	2	4	no	—
122	#117	—	2	8	no	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₂²⋊C₄	—	2	2	no	—
125	C₂×C₄²	—	2	4	no	—
126	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
127	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
128	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
129	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
130	C₂×C₄²	C₅×D₄	1	2	yes	—
131	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
132	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
133	C₂×C₄²	C₅×Q₈	1	1	yes	characteristic
134	C₂×C₄²	C₅×Q₈	1	1	yes	characteristic
135	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
136	C₄⋊C₈	—	2	4	no	—
137	C₄⋊C₈	—	2	4	no	—
138	C₂²⋊C₈	—	2	4	no	—
139	C₂²⋊C₈	—	2	4	no	—
140	#97	—	2	4	no	—
141	C₄×C₈	C₂×C₄×C₅	1	8	yes	—
142	#95	—	2	4	no	—
143	C₂²×C₅	—	2	2	no	—
144	C₂²×C₅	—	2	1	no	—
145	C₂²×C₅	—	2	1	no	—
146	C₂²×C₅	—	2	2	no	—
147	C₂²×C₅	—	2	2	no	—
148	C₂²×C₅	—	4	4	no	—
149	C₂²×C₅	C₂×#95	1	1	yes	characteristic
150	C₂²×C₅	#407	1	2	yes	—
151	C₄×C₅	—	2	1	no	—
152	C₄×C₅	—	2	1	no	—
153	C₄×C₅	—	2	2	no	—
154	C₄×C₅	—	2	2	no	—
155	C₄×C₅	—	2	2	no	—
156	C₄×C₅	—	4	8	no	—
157	C₄×C₅	H(3,ℤ/4)	1	4	yes	—
158	C₄×C₅	C₄×M₁₆	1	2	yes	—
159	C₄×C₅	#423	1	2	yes	—
160	C₂⁴	C₅×C₂²⋊C₄	1	1	yes	characteristic
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	4	4	no	—
164	C₂²×C₄	C₄²×C₅	1	1	yes	characteristic
165	C₂²×C₄	C₄²×C₅	1	1	yes	characteristic
166	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
167	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
168	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
169	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
170	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
171	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
172	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
173	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
174	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
175	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
176	M₁₆	—	4	8	no	—
177	C₂×C₈	—	2	8	no	—
178	C₄⋊C₄	—	2	8	no	—
179	C₂²⋊C₄	—	2	8	no	—
180	C₄²	—	2	4	no	—
181	C₄²	—	4	8	no	—
182	C₄²	C₄²×C₅	1	2	yes	—
183	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
184	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
185	C₄²	C₅×C₄⋊C₄	1	2	yes	—
186	C₄²	C₅×C₄⋊C₄	1	2	yes	—
187	C₄²	C₅×C₄⋊C₄	1	1	yes	characteristic
188	C₄²	C₅×C₄⋊C₄	1	1	yes	characteristic
189	C₄²	C₅×M₁₆	1	8	yes	complemented
190	C₄²	C₂²×C₄×C₅	1	2	yes	—
191	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
192	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
193	C₂×C₅	—	2	2	no	—
194	C₂×C₅	—	4	1	no	—
195	C₂×C₅	—	4	1	no	—
196	C₂×C₅	#1728	1	1	yes	characteristic
197	C₂×C₅	C₂×H(3,ℤ/4)	1	2	yes	—
198	C₂³	—	2	2	no	—
199	C₂³	—	4	2	no	—
200	C₂³	C₅×#95	1	1	yes	characteristic
201	C₂³	C₅×#95	1	1	yes	characteristic
202	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	8	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	8	no	—
216	C₂×C₄	C₅×#95	1	2	yes	—
217	C₂×C₄	C₅×#95	1	1	yes	characteristic
218	C₂×C₄	C₅×#95	1	1	yes	characteristic
219	C₂×C₄	C₅×#95	1	2	yes	—
220	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
221	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
222	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
223	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
224	C₂×C₄	C₂×C₅×M₁₆	1	4	yes	—
225	C₈	—	4	8	no	—
226	C₅	#7856	1	1	yes	characteristic, direct product factor, Sylow
227	C₂²	—	2	2	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	4	4	no	—
233	C₂²	C₂×C₅×#95	1	1	yes	characteristic
234	C₂²	C₅×#407	1	2	yes	—
235	C₄	—	2	1	no	—
236	C₄	—	2	1	no	—
237	C₄	—	2	2	no	—
238	C₄	—	2	2	no	—
239	C₄	—	2	2	no	—
240	C₄	—	4	8	no	—
241	C₄	C₅×H(3,ℤ/4)	1	4	yes	—
242	C₄	C₄×C₅×M₁₆	1	2	yes	—
243	C₄	C₅×#423	1	2	yes	—
244	C₂	—	2	2	no	—
245	C₂	—	4	1	no	—
246	C₂	—	4	1	no	—
247	C₂	C₅×#1728	1	1	yes	characteristic
248	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. #7856	C₅	direct (N × H)
44. #1274	C₂×C₅	semidirect
74. C₄²×C₅	M₁₆	semidirect
82. #431	C₄×C₅	semidirect
86. C₄×M₁₆	C₄×C₅	semidirect
189. C₄²	C₅×M₁₆	semidirect
226. C₅	#7856	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
C₄×C₅×#117	2
C₅×#1274	2
#7856	5

Subgroups — C5×#7856 #11930576

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#7856 #11930576