Subgroups — C₅×#7895 #11930615

Order 1280.

734 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₅×#2283	C₂	1	1	yes	maximal, characteristic
4	C₅×#1279	C₂	1	4	yes	maximal, complemented
5	C₂²×C₅×M₁₆	—	2	1	no	—
6	C₂³×C₅×C₈	—	2	1	no	—
7	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₅×C₄⋊C₈	—	2	1	no	—
9	C₂×C₅×C₄⋊C₈	—	2	2	no	—
10	C₂×C₅×C₄⋊C₈	C₄	1	4	yes	complemented
11	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
12	C₄×C₅×M₁₆	—	2	2	no	—
13	C₄×C₅×M₁₆	C₄	1	4	yes	complemented
14	C₂×C₅×#97	—	2	1	no	—
15	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
16	C₅×#383	—	2	1	no	—
17	C₅×#382	C₄	1	2	yes	—
18	C₅×#381	—	2	1	no	—
19	C₅×#380	C₄	1	2	yes	—
20	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
21	C₄×C₅×C₂²⋊C₄	—	2	1	no	—
22	C₅×#376	—	2	1	no	—
23	#7895	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
27	C₂×C₅×M₁₆	—	2	8	no	—
28	C₂×C₅×M₁₆	—	2	4	no	—
29	C₂²×C₅×C₈	—	2	2	no	—
30	C₂²×C₅×C₈	—	2	4	no	—
31	C₂²×C₅×C₈	—	2	2	no	—
32	C₂²×C₅×C₈	—	4	4	no	—
33	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
34	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
35	C₂×C₄²×C₅	—	2	2	no	—
36	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅	D₄	1	2	yes	—
42	C₂×C₄²×C₅	Q₈	1	2	yes	—
43	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
44	C₅×C₄⋊C₈	—	2	4	no	—
45	C₅×C₄⋊C₈	—	2	2	no	—
46	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	—
47	C₅×#97	—	2	4	no	—
48	C₄×C₅×C₈	—	2	2	no	—
49	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
50	C₅×#95	—	2	2	no	—
51	C₅×#95	—	2	2	no	—
52	C₅×#95	—	2	2	no	—
53	C₅×#95	—	2	2	no	—
54	C₂²×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic
55	C₂×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic
56	#2283	C₂×C₅	1	1	yes	characteristic
57	#1279	C₂×C₅	1	4	yes	complemented
58	C₂⁴×C₅	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅	—	2	4	no	—
60	C₂²×C₄×C₅	—	2	4	no	—
61	C₂²×C₄×C₅	—	2	2	no	—
62	C₂²×C₄×C₅	—	2	2	no	—
63	C₂²×C₄×C₅	—	2	4	no	—
64	C₂²×C₄×C₅	C₄²	1	2	yes	—
65	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₅×M₁₆	—	2	16	no	—
77	C₂×C₅×C₈	—	2	8	no	—
78	C₂×C₅×C₈	—	2	4	no	—
79	C₂×C₅×C₈	—	2	4	no	—
80	C₂×C₅×C₈	—	4	8	no	—
81	C₂×C₅×C₈	—	4	4	no	—
82	C₅×C₄⋊C₄	—	2	8	no	—
83	C₅×C₂²⋊C₄	—	4	4	no	—
84	C₄²×C₅	—	4	2	no	—
85	C₄²×C₅	—	4	2	no	—
86	C₄²×C₅	C₄²	1	2	yes	—
87	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
88	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
89	C₄²×C₅	C₄⋊C₄	1	2	yes	—
90	C₄²×C₅	C₄⋊C₄	1	2	yes	—
91	C₄²×C₅	C₄⋊C₄	1	2	yes	—
92	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
96	C₂²×M₁₆	—	2	1	no	—
97	C₂³×C₈	—	2	1	no	—
98	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
99	C₂×C₄⋊C₈	—	2	1	no	—
100	C₂×C₄⋊C₈	—	2	2	no	—
101	C₂×C₄⋊C₈	C₄×C₅	1	4	yes	complemented
102	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
103	C₄×M₁₆	—	2	2	no	—
104	C₄×M₁₆	C₄×C₅	1	4	yes	complemented
105	C₂×#97	—	2	1	no	—
106	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
107	#383	—	2	1	no	—
108	#382	C₄×C₅	1	2	yes	—
109	#381	—	2	1	no	—
110	#380	C₄×C₅	1	2	yes	—
111	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
112	C₄×C₂²⋊C₄	—	2	1	no	—
113	#376	—	2	1	no	—
114	C₂³×C₅	—	2	2	no	—
115	C₂³×C₅	—	2	2	no	—
116	C₂³×C₅	#95	1	2	yes	—
117	C₂³×C₅	#104	1	4	yes	—
118	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₅	—	2	4	no	—
120	C₂×C₄×C₅	—	2	4	no	—
121	C₂×C₄×C₅	—	2	2	no	—
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	—	2	2	no	—
125	C₂×C₄×C₅	—	2	4	no	—
126	C₂×C₄×C₅	—	2	2	no	—
127	C₂×C₄×C₅	—	2	2	no	—
128	C₂×C₄×C₅	—	2	4	no	—
129	C₂×C₄×C₅	—	2	4	no	—
130	C₂×C₄×C₅	—	4	4	no	—
131	C₂×C₄×C₅	—	4	4	no	—
132	C₂×C₄×C₅	—	4	4	no	—
133	C₂×C₄×C₅	#95	1	2	yes	—
134	C₂×C₄×C₅	#95	1	2	yes	—
135	C₂×C₄×C₅	#95	1	2	yes	—
136	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₅	#131	1	4	yes	—
143	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
144	C₂×C₄×C₅	#137	1	2	yes	—
145	C₅×C₈	—	2	8	no	—
146	C₅×C₈	—	4	4	no	—
147	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
148	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
149	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
150	C₂×M₁₆	—	2	8	no	—
151	C₂×M₁₆	—	2	4	no	—
152	C₂²×C₈	—	2	2	no	—
153	C₂²×C₈	—	2	4	no	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	4	4	no	—
156	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
157	C₂×C₂²⋊C₄	—	2	2	no	—
158	C₂×C₄²	—	2	2	no	—
159	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
160	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
161	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
162	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
163	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
164	C₂×C₄²	C₅×D₄	1	2	yes	—
165	C₂×C₄²	C₅×Q₈	1	2	yes	—
166	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
167	C₄⋊C₈	—	2	4	no	—
168	C₄⋊C₈	—	2	2	no	—
169	C₄⋊C₈	C₂×C₄×C₅	1	4	yes	—
170	#97	—	2	4	no	—
171	C₄×C₈	—	2	2	no	—
172	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
173	#95	—	2	2	no	—
174	#95	—	2	2	no	—
175	#95	—	2	2	no	—
176	#95	—	2	2	no	—
177	C₂²×C₅	—	2	2	no	—
178	C₂²×C₅	—	2	2	no	—
179	C₂²×C₅	—	2	8	no	—
180	C₂²×C₅	#339	1	4	yes	—
181	C₂²×C₅	C₂×#95	1	1	yes	characteristic
182	C₂²×C₅	#408	1	2	yes	—
183	C₂²×C₅	#417	1	2	yes	—
184	C₂²×C₅	C₂×#104	1	2	yes	—
185	C₄×C₅	—	2	2	no	—
186	C₄×C₅	—	2	1	no	—
187	C₄×C₅	—	2	2	no	—
188	C₄×C₅	—	2	1	no	—
189	C₄×C₅	—	2	2	no	—
190	C₄×C₅	—	8	4	no	—
191	C₄×C₅	#405	1	2	yes	—
192	C₄×C₅	#418	1	1	yes	characteristic
193	C₄×C₅	#419	1	1	yes	characteristic
194	C₄×C₅	#424	1	2	yes	—
195	C₄×C₅	#428	1	2	yes	—
196	C₂⁴	C₅×C₂²⋊C₄	1	1	yes	characteristic
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	C₄²×C₅	1	2	yes	—
203	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
204	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
205	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
206	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
207	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
208	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
209	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
210	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
211	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
214	M₁₆	—	2	16	no	—
215	C₂×C₈	—	2	8	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	4	8	no	—
219	C₂×C₈	—	4	4	no	—
220	C₄⋊C₄	—	2	8	no	—
221	C₂²⋊C₄	—	4	4	no	—
222	C₄²	—	4	2	no	—
223	C₄²	—	4	2	no	—
224	C₄²	C₄²×C₅	1	2	yes	—
225	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
226	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
227	C₄²	C₅×C₄⋊C₄	1	2	yes	—
228	C₄²	C₅×C₄⋊C₄	1	2	yes	—
229	C₄²	C₅×C₄⋊C₄	1	2	yes	—
230	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
231	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
232	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
233	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
234	C₂×C₅	—	2	4	no	—
235	C₂×C₅	#1483	1	4	yes	—
236	C₂×C₅	#1729	1	1	yes	characteristic
237	C₂×C₅	C₂×#339	1	1	yes	characteristic
238	C₂×C₅	#1736	1	1	yes	characteristic
239	C₂³	—	2	2	no	—
240	C₂³	—	2	2	no	—
241	C₂³	C₅×#95	1	2	yes	—
242	C₂³	C₅×#104	1	4	yes	—
243	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	C₅×#95	1	2	yes	—
259	C₂×C₄	C₅×#95	1	2	yes	—
260	C₂×C₄	C₅×#95	1	2	yes	—
261	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
262	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
263	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
264	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
265	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
266	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
267	C₂×C₄	C₅×#131	1	4	yes	—
268	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
269	C₂×C₄	C₅×#137	1	2	yes	—
270	C₈	—	2	8	no	—
271	C₈	—	4	4	no	—
272	C₅	#7895	1	1	yes	characteristic, direct product factor, Sylow
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	8	no	—
276	C₂²	C₅×#339	1	4	yes	—
277	C₂²	C₂×C₅×#95	1	1	yes	characteristic
278	C₂²	C₅×#408	1	2	yes	—
279	C₂²	C₅×#417	1	2	yes	—
280	C₂²	C₂×C₅×#104	1	2	yes	—
281	C₄	—	2	2	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	8	4	no	—
287	C₄	C₅×#405	1	2	yes	—
288	C₄	C₅×#418	1	1	yes	characteristic
289	C₄	C₅×#419	1	1	yes	characteristic
290	C₄	C₅×#424	1	2	yes	—
291	C₄	C₅×#428	1	2	yes	—
292	C₂	—	2	4	no	—
293	C₂	C₅×#1483	1	4	yes	—
294	C₂	C₅×#1729	1	1	yes	characteristic
295	C₂	C₂×C₅×#339	1	1	yes	characteristic
296	C₂	C₅×#1736	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×C₂²⋊C₄
prime radical	C₂⁴×C₅	C₂²⋊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×#1279	C₂	semidirect
10. C₂×C₅×C₄⋊C₈	C₄	semidirect
13. C₄×C₅×M₁₆	C₄	semidirect
23. #7895	C₅	direct (N × H)
57. #1279	C₂×C₅	semidirect
101. C₂×C₄⋊C₈	C₄×C₅	semidirect
104. C₄×M₁₆	C₄×C₅	semidirect
272. C₅	#7895	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₅×C₄⋊C₈	2
C₂×C₄×C₅×M₁₆	2
C₅×#2283	2
C₅×#1279	2
#7895	5

Subgroups — C5×#7895 #11930615

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#7895 #11930615