Subgroups — C₅×#7873 #11930593

Order 1280.

722 conjugacy classes of subgroups.

The subgroup lattice has 292 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	1	yes	maximal, characteristic
2	C₂²×C₅×C₄⋊C₈	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	C₄	1	2	yes	—
6	C₂³×C₅×C₈	—	2	1	no	—
7	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₅×#431	—	2	2	no	—
9	C₅×#431	C₄	1	4	yes	—
10	C₂×C₅×C₄⋊C₈	—	2	1	no	—
11	C₂×C₅×C₄⋊C₈	—	2	2	no	—
12	C₂×C₅×C₄⋊C₈	C₄	1	4	yes	—
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	C₄	1	2	yes	—
19	C₂×C₄×C₅×C₈	—	2	1	no	—
20	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
21	#7873	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	C₈	1	8	yes	complemented
28	C₂²×C₅×C₈	—	2	4	no	—
29	C₂²×C₅×C₈	—	2	2	no	—
30	C₂²×C₅×C₈	—	2	2	no	—
31	C₂²×C₅×C₈	—	4	4	no	—
32	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₅	D₄	1	2	yes	—
38	C₂×C₄²×C₅	D₄	1	2	yes	—
39	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
40	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
41	C₅×C₄⋊C₈	—	2	4	no	—
42	C₅×C₄⋊C₈	—	2	2	no	—
43	C₅×C₄⋊C₈	—	2	2	no	—
44	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	—
45	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	—
46	C₅×C₂²⋊C₈	—	2	4	no	—
47	C₅×#97	—	2	4	no	—
48	C₄×C₅×C₈	—	2	4	no	—
49	C₄×C₅×C₈	—	2	2	no	—
50	C₄×C₅×C₈	—	2	2	no	—
51	C₂×#431	C₂×C₅	1	1	yes	characteristic
52	C₂²×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic
53	C₂×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic
54	#1277	C₂×C₅	1	4	yes	complemented
55	C₂⁴×C₅	C₄⋊C₄	1	1	yes	characteristic
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	2	yes	—
66	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
69	C₂²×C₄×C₅	M₁₆	1	4	yes	—
70	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
73	C₅×M₁₆	—	2	16	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	—	2	4	no	—
78	C₂×C₅×C₈	—	4	8	no	—
79	C₂×C₅×C₈	—	4	2	no	—
80	C₂×C₅×C₈	—	4	4	no	—
81	C₂×C₅×C₈	—	4	2	no	—
82	C₂×C₅×C₈	C₂×C₈	1	8	yes	complemented
83	C₄²×C₅	C₄²	1	2	yes	—
84	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
85	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
86	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
87	C₄²×C₅	C₄⋊C₄	1	2	yes	—
88	C₄²×C₅	C₄⋊C₄	1	2	yes	—
89	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
93	C₂²×M₁₆	C₄×C₅	1	2	yes	—
94	C₂³×C₈	—	2	1	no	—
95	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
96	#431	—	2	2	no	—
97	#431	C₄×C₅	1	4	yes	—
98	C₂×C₄⋊C₈	—	2	1	no	—
99	C₂×C₄⋊C₈	—	2	2	no	—
100	C₂×C₄⋊C₈	C₄×C₅	1	4	yes	—
101	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
102	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
103	C₂×C₂²⋊C₈	—	2	1	no	—
104	C₄×M₁₆	—	2	2	no	—
105	C₄×M₁₆	—	2	2	no	—
106	C₂×#97	C₄×C₅	1	2	yes	—
107	C₂×C₄×C₈	—	2	1	no	—
108	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
109	C₂³×C₅	—	2	2	no	—
110	C₂³×C₅	—	2	2	no	—
111	C₂³×C₅	#95	1	2	yes	—
112	C₂³×C₅	C₄⋊C₈	1	4	yes	—
113	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅	—	2	2	no	—
115	C₂×C₄×C₅	—	2	2	no	—
116	C₂×C₄×C₅	—	2	2	no	—
117	C₂×C₄×C₅	—	2	4	no	—
118	C₂×C₄×C₅	—	2	2	no	—
119	C₂×C₄×C₅	—	2	2	no	—
120	C₂×C₄×C₅	—	4	2	no	—
121	C₂×C₄×C₅	—	4	2	no	—
122	C₂×C₄×C₅	#95	1	2	yes	—
123	C₂×C₄×C₅	#95	1	2	yes	—
124	C₂×C₄×C₅	#95	1	2	yes	—
125	C₂×C₄×C₅	C₄×C₈	1	4	yes	—
126	C₂×C₄×C₅	#97	1	4	yes	—
127	C₂×C₄×C₅	C₂²⋊C₈	1	8	yes	—
128	C₂×C₄×C₅	C₄⋊C₈	1	4	yes	—
129	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
136	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
137	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
138	C₂×C₄×C₅	#137	1	2	yes	—
139	C₅×C₈	—	2	8	no	—
140	C₅×C₈	—	4	4	no	—
141	C₅×C₈	—	4	4	no	—
142	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
143	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
144	C₂³×C₄	C₅×Q₈	1	1	yes	characteristic
145	C₂×M₁₆	—	2	4	no	—
146	C₂×M₁₆	—	2	4	no	—
147	C₂×M₁₆	C₅×C₈	1	8	yes	complemented
148	C₂²×C₈	—	2	4	no	—
149	C₂²×C₈	—	2	2	no	—
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	—	4	4	no	—
152	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
153	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
154	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
155	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
156	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
157	C₂×C₄²	C₅×D₄	1	2	yes	—
158	C₂×C₄²	C₅×D₄	1	2	yes	—
159	C₂×C₄²	C₅×Q₈	1	1	yes	characteristic
160	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
161	C₄⋊C₈	—	2	4	no	—
162	C₄⋊C₈	—	2	2	no	—
163	C₄⋊C₈	—	2	2	no	—
164	C₄⋊C₈	C₂×C₄×C₅	1	4	yes	—
165	C₄⋊C₈	C₂×C₄×C₅	1	4	yes	—
166	C₂²⋊C₈	—	2	4	no	—
167	#97	—	2	4	no	—
168	C₄×C₈	—	2	4	no	—
169	C₄×C₈	—	2	2	no	—
170	C₄×C₈	—	2	2	no	—
171	C₂²×C₅	—	2	8	no	—
172	C₂²×C₅	—	2	2	no	—
173	C₂²×C₅	—	2	2	no	—
174	C₂²×C₅	#336	1	4	yes	—
175	C₂²×C₅	C₂×#95	1	1	yes	characteristic
176	C₂²×C₅	C₂×C₄⋊C₈	1	2	yes	—
177	C₂²×C₅	#428	1	2	yes	—
178	C₂²×C₅	#430	1	2	yes	—
179	C₄×C₅	—	2	2	no	—
180	C₄×C₅	—	2	2	no	—
181	C₄×C₅	—	2	1	no	—
182	C₄×C₅	—	2	1	no	—
183	C₄×C₅	#336	1	4	yes	—
184	C₄×C₅	C₂×C₄×C₈	1	1	yes	characteristic
185	C₄×C₅	C₂×#97	1	1	yes	characteristic
186	C₄×C₅	C₂×C₂²⋊C₈	1	2	yes	—
187	C₄×C₅	#417	1	2	yes	—
188	C₄×C₅	#418	1	1	yes	characteristic
189	C₄×C₅	#419	1	1	yes	characteristic
190	C₂⁴	C₅×C₄⋊C₄	1	1	yes	characteristic
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	C₄²×C₅	1	2	yes	—
195	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
196	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
197	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
198	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
199	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
200	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
201	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
202	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₅×C₈	1	4	yes	—
204	C₂²×C₄	C₅×M₁₆	1	4	yes	—
205	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
208	M₁₆	—	2	16	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	4	8	no	—
214	C₂×C₈	—	4	4	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	C₂×C₅×C₈	1	8	yes	complemented
218	C₄²	C₄²×C₅	1	2	yes	—
219	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
220	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
221	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
222	C₄²	C₅×C₄⋊C₄	1	2	yes	—
223	C₄²	C₅×C₄⋊C₄	1	2	yes	—
224	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
225	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
226	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
227	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
228	C₂×C₅	—	2	4	no	—
229	C₂×C₅	#1565	1	4	yes	—
230	C₂×C₅	C₂×#336	1	1	yes	characteristic
231	C₂×C₅	#1736	1	1	yes	characteristic
232	C₂×C₅	#1738	1	1	yes	characteristic
233	C₂³	—	2	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	C₅×#95	1	2	yes	—
236	C₂³	C₅×C₄⋊C₈	1	4	yes	—
237	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	C₅×#95	1	2	yes	—
247	C₂×C₄	C₅×#95	1	2	yes	—
248	C₂×C₄	C₅×#95	1	2	yes	—
249	C₂×C₄	C₄×C₅×C₈	1	4	yes	—
250	C₂×C₄	C₅×#97	1	4	yes	—
251	C₂×C₄	C₅×C₂²⋊C₈	1	8	yes	—
252	C₂×C₄	C₅×C₄⋊C₈	1	4	yes	—
253	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
254	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
255	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
256	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
257	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
258	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
260	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
261	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
262	C₂×C₄	C₅×#137	1	2	yes	—
263	C₈	—	2	8	no	—
264	C₈	—	4	4	no	—
265	C₈	—	4	4	no	—
266	C₅	#7873	1	1	yes	characteristic, direct product factor, Sylow
267	C₂²	—	2	8	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	2	no	—
270	C₂²	C₅×#336	1	4	yes	—
271	C₂²	C₂×C₅×#95	1	1	yes	characteristic
272	C₂²	C₂×C₅×C₄⋊C₈	1	2	yes	—
273	C₂²	C₅×#428	1	2	yes	—
274	C₂²	C₅×#430	1	2	yes	—
275	C₄	—	2	2	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	2	no	—
279	C₄	C₅×#336	1	4	yes	—
280	C₄	C₂×C₄×C₅×C₈	1	1	yes	characteristic
281	C₄	C₂×C₅×#97	1	1	yes	characteristic
282	C₄	C₂×C₅×C₂²⋊C₈	1	2	yes	—
283	C₄	C₅×#417	1	2	yes	—
284	C₄	C₅×#418	1	1	yes	characteristic
285	C₄	C₅×#419	1	1	yes	characteristic
286	C₂	—	2	4	no	—
287	C₂	C₅×#1565	1	4	yes	—
288	C₂	C₂×C₅×#336	1	1	yes	characteristic
289	C₂	C₅×#1736	1	1	yes	characteristic
290	C₂	C₅×#1738	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₄	C₂×C₄×C₅×C₈
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂³×C₅	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅	C₄⋊C₄
LCS	G ▷ 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. #7873	C₅	direct (N × H)
27. C₂×C₅×M₁₆	C₈	semidirect
54. #1277	C₂×C₅	semidirect
82. C₂×C₅×C₈	C₂×C₈	semidirect
147. C₂×M₁₆	C₅×C₈	semidirect
217. C₂×C₈	C₂×C₅×C₈	semidirect
266. C₅	#7873	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
C₂²×C₅×C₄⋊C₈	2
C₂×C₄×C₅×M₁₆	2
C₅×#1277	2
#7873	5

Subgroups — C5×#7873 #11930593

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#7873 #11930593