Subgroups — C₅×#7876 #11930596

Order 1280.

662 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#432	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#423	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
4	C₅×#1277	C₂	1	4	yes	maximal, complemented
5	C₂²×C₅×M₁₆	—	2	1	no	—
6	C₂²×C₅×M₁₆	C₄	1	2	yes	—
7	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₅×#432	—	2	2	no	—
9	C₅×#432	C₄	1	4	yes	—
10	C₅×#423	—	2	2	no	—
11	C₅×#423	C₄	1	4	yes	—
12	C₂×C₅×C₄⋊C₈	—	2	1	no	—
13	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₅×C₂²⋊C₈	—	2	1	no	—
16	C₄×C₅×M₁₆	—	2	2	no	—
17	C₄×C₅×M₁₆	—	2	2	no	—
18	C₂×C₅×#97	—	2	1	no	—
19	C₂×C₅×#97	C₄	1	2	yes	—
20	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
21	#7876	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₅	Q₈	1	1	yes	characteristic
25	C₂×C₅×M₁₆	—	2	4	no	—
26	C₂×C₅×M₁₆	—	2	4	no	—
27	C₂×C₅×M₁₆	—	2	8	no	—
28	C₂×C₅×M₁₆	—	4	4	no	—
29	C₂²×C₅×C₈	—	2	2	no	—
30	C₂²×C₅×C₈	—	2	2	no	—
31	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₅	D₄	1	2	yes	—
37	C₂×C₄²×C₅	D₄	1	2	yes	—
38	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
39	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
40	C₅×C₄⋊C₈	—	2	4	no	—
41	C₅×C₄⋊C₈	—	2	2	no	—
42	C₅×C₄⋊C₈	—	2	2	no	—
43	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	—
44	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	—
45	C₅×C₂²⋊C₈	—	2	4	no	—
46	C₅×#97	—	2	4	no	—
47	C₅×#97	—	2	4	no	—
48	C₄×C₅×C₈	—	2	2	no	—
49	C₄×C₅×C₈	—	2	2	no	—
50	C₂×#432	C₂×C₅	1	1	yes	characteristic
51	C₂×#423	C₂×C₅	1	1	yes	characteristic
52	C₂×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic
53	#1277	C₂×C₅	1	4	yes	complemented
54	C₂⁴×C₅	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅	—	2	4	no	—
56	C₂²×C₄×C₅	—	2	2	no	—
57	C₂²×C₄×C₅	—	2	2	no	—
58	C₂²×C₄×C₅	—	4	2	no	—
59	C₂²×C₄×C₅	C₄²	1	2	yes	—
60	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
71	C₅×M₁₆	—	2	16	no	—
72	C₅×M₁₆	—	4	8	no	—
73	C₂×C₅×C₈	—	2	4	no	—
74	C₂×C₅×C₈	—	2	4	no	—
75	C₂×C₅×C₈	—	2	4	no	—
76	C₂×C₅×C₈	—	2	4	no	—
77	C₂×C₅×C₈	—	4	2	no	—
78	C₂×C₅×C₈	—	4	4	no	—
79	C₂×C₅×C₈	—	4	2	no	—
80	C₂×C₅×C₈	M₁₆	1	8	yes	complemented
81	C₄²×C₅	C₄²	1	2	yes	—
82	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
84	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
85	C₄²×C₅	C₄⋊C₄	1	2	yes	—
86	C₄²×C₅	C₄⋊C₄	1	2	yes	—
87	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
91	C₂²×M₁₆	—	2	1	no	—
92	C₂²×M₁₆	C₄×C₅	1	2	yes	—
93	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
94	#432	—	2	2	no	—
95	#432	C₄×C₅	1	4	yes	—
96	#423	—	2	2	no	—
97	#423	C₄×C₅	1	4	yes	—
98	C₂×C₄⋊C₈	—	2	1	no	—
99	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
100	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
101	C₂×C₂²⋊C₈	—	2	1	no	—
102	C₄×M₁₆	—	2	2	no	—
103	C₄×M₁₆	—	2	2	no	—
104	C₂×#97	—	2	1	no	—
105	C₂×#97	C₄×C₅	1	2	yes	—
106	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
107	C₂³×C₅	—	2	2	no	—
108	C₂³×C₅	—	2	2	no	—
109	C₂³×C₅	—	2	2	no	—
110	C₂³×C₅	#95	1	2	yes	—
111	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₅	—	2	2	no	—
113	C₂×C₄×C₅	—	2	2	no	—
114	C₂×C₄×C₅	—	2	2	no	—
115	C₂×C₄×C₅	—	2	2	no	—
116	C₂×C₄×C₅	—	2	4	no	—
117	C₂×C₄×C₅	—	2	4	no	—
118	C₂×C₄×C₅	—	2	4	no	—
119	C₂×C₄×C₅	—	2	2	no	—
120	C₂×C₄×C₅	—	2	2	no	—
121	C₂×C₄×C₅	—	4	2	no	—
122	C₂×C₄×C₅	—	4	2	no	—
123	C₂×C₄×C₅	#95	1	2	yes	—
124	C₂×C₄×C₅	#95	1	2	yes	—
125	C₂×C₄×C₅	#95	1	2	yes	—
126	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
133	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
134	C₂×C₄×C₅	#137	1	2	yes	—
135	C₅×C₈	—	2	8	no	—
136	C₅×C₈	—	4	4	no	—
137	C₅×C₈	—	4	4	no	—
138	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
139	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
140	C₂³×C₄	C₅×Q₈	1	1	yes	characteristic
141	C₂×M₁₆	—	2	4	no	—
142	C₂×M₁₆	—	2	4	no	—
143	C₂×M₁₆	—	2	8	no	—
144	C₂×M₁₆	—	4	4	no	—
145	C₂²×C₈	—	2	2	no	—
146	C₂²×C₈	—	2	2	no	—
147	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
148	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
149	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
150	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
151	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
152	C₂×C₄²	C₅×D₄	1	2	yes	—
153	C₂×C₄²	C₅×D₄	1	2	yes	—
154	C₂×C₄²	C₅×Q₈	1	1	yes	characteristic
155	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
156	C₄⋊C₈	—	2	4	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	C₂×C₄×C₅	1	4	yes	—
160	C₄⋊C₈	C₂×C₄×C₅	1	4	yes	—
161	C₂²⋊C₈	—	2	4	no	—
162	#97	—	2	4	no	—
163	#97	—	2	4	no	—
164	C₄×C₈	—	2	2	no	—
165	C₄×C₈	—	2	2	no	—
166	C₂²×C₅	—	2	2	no	—
167	C₂²×C₅	—	2	2	no	—
168	C₂²×C₅	—	2	2	no	—
169	C₂²×C₅	—	4	4	no	—
170	C₂²×C₅	C₂×#95	1	1	yes	characteristic
171	C₂²×C₅	#423	1	2	yes	—
172	C₂²×C₅	#428	1	2	yes	—
173	C₂²×C₅	#430	1	2	yes	—
174	C₄×C₅	—	2	2	no	—
175	C₄×C₅	—	2	2	no	—
176	C₄×C₅	—	2	2	no	—
177	C₄×C₅	—	2	1	no	—
178	C₄×C₅	—	2	1	no	—
179	C₄×C₅	C₄×M₁₆	1	2	yes	—
180	C₄×C₅	#407	1	2	yes	—
181	C₄×C₅	#417	1	2	yes	—
182	C₄×C₅	#418	1	1	yes	characteristic
183	C₄×C₅	#419	1	1	yes	characteristic
184	C₂⁴	C₅×C₄⋊C₄	1	1	yes	characteristic
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	C₄²×C₅	1	2	yes	—
190	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
191	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
192	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
193	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
194	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
195	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
196	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
197	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
198	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
201	M₁₆	—	2	16	no	—
202	M₁₆	—	4	8	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	4	4	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	—	4	2	no	—
210	C₂×C₈	C₅×M₁₆	1	8	yes	complemented
211	C₄²	C₄²×C₅	1	2	yes	—
212	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
213	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
214	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
215	C₄²	C₅×C₄⋊C₄	1	2	yes	—
216	C₄²	C₅×C₄⋊C₄	1	2	yes	—
217	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
218	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
219	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
220	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
221	C₂×C₅	—	4	2	no	—
222	C₂×C₅	#1569	1	4	yes	—
223	C₂×C₅	#1728	1	1	yes	characteristic
224	C₂×C₅	#1736	1	1	yes	characteristic
225	C₂×C₅	#1738	1	1	yes	characteristic
226	C₂³	—	2	2	no	—
227	C₂³	—	2	2	no	—
228	C₂³	—	2	2	no	—
229	C₂³	C₅×#95	1	2	yes	—
230	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	C₅×#95	1	2	yes	—
243	C₂×C₄	C₅×#95	1	2	yes	—
244	C₂×C₄	C₅×#95	1	2	yes	—
245	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
246	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
247	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
248	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
249	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
250	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
251	C₂×C₄	C₂×C₅×M₁₆	1	4	yes	—
252	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
253	C₂×C₄	C₅×#137	1	2	yes	—
254	C₈	—	2	8	no	—
255	C₈	—	4	4	no	—
256	C₈	—	4	4	no	—
257	C₅	#7876	1	1	yes	characteristic, direct product factor, Sylow
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	4	4	no	—
262	C₂²	C₂×C₅×#95	1	1	yes	characteristic
263	C₂²	C₅×#423	1	2	yes	—
264	C₂²	C₅×#428	1	2	yes	—
265	C₂²	C₅×#430	1	2	yes	—
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	2	no	—
271	C₄	C₄×C₅×M₁₆	1	2	yes	—
272	C₄	C₅×#407	1	2	yes	—
273	C₄	C₅×#417	1	2	yes	—
274	C₄	C₅×#418	1	1	yes	characteristic
275	C₄	C₅×#419	1	1	yes	characteristic
276	C₂	—	4	2	no	—
277	C₂	C₅×#1569	1	4	yes	—
278	C₂	C₅×#1728	1	1	yes	characteristic
279	C₂	C₅×#1736	1	1	yes	characteristic
280	C₂	C₅×#1738	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₅×#1277	C₂	semidirect
21. #7876	C₅	direct (N × H)
53. #1277	C₂×C₅	semidirect
80. C₂×C₅×C₈	M₁₆	semidirect
210. C₂×C₈	C₅×M₁₆	semidirect
257. C₅	#7876	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×#432	2
C₂×C₅×#423	2
C₂×C₄×C₅×M₁₆	2
C₅×#1277	2
#7876	5

Subgroups — C5×#7876 #11930596

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#7876 #11930596