Subgroups — C₅×#12992 #11935712

Order 1280.

582 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#466	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#427	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#357	C₂	1	2	yes	maximal, complemented
4	C₂×C₅×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1380	C₂	1	2	yes	maximal, complemented
6	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
7	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₅×#121	—	2	1	no	—
9	C₂×C₅×#117	—	2	1	no	—
10	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
11	C₅×#466	—	2	2	no	—
12	C₅×#466	C₄	1	4	yes	complemented
13	C₅×#427	—	2	4	no	—
14	C₂×C₅×#107	C₂²	1	1	yes	characteristic
15	C₂×C₅×#107	C₂²	1	2	yes	—
16	C₂×C₅×#106	—	2	1	no	—
17	C₂×C₅×#102	—	2	1	no	—
18	C₅×#357	—	2	4	no	—
19	C₅×C₂²⋊C₁₆	—	2	2	no	—
20	#12992	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₅×D₄	—	2	1	no	—
22	C₂²×C₅×D₄	—	4	1	no	—
23	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
24	C₂×C₅×D₈	—	2	2	no	—
25	C₂×C₅×D₈	—	2	1	no	—
26	C₂×C₅×D₈	—	2	2	no	—
27	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
28	C₂²×C₅×C₈	—	2	2	no	—
29	C₂²×C₅×C₈	—	2	2	no	—
30	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
31	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	D₄	1	2	yes	—
33	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
34	C₅×#121	—	2	4	no	—
35	C₅×#121	—	4	2	no	—
36	C₅×#117	—	2	4	no	—
37	C₅×#117	—	4	2	no	—
38	C₂×C₅×C₄⋊C₄	—	2	1	no	—
39	C₂×C₅×C₄⋊C₄	—	2	2	no	—
40	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
41	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
42	C₂×C₄²×C₅	—	4	1	no	—
43	C₂×C₅×C₁₆	—	2	2	no	—
44	C₂×C₅×C₁₆	—	2	2	no	—
45	C₂×C₅×C₁₆	—	2	2	no	—
46	C₅×#107	—	2	2	no	—
47	C₅×#107	—	2	1	no	—
48	C₅×#107	C₂×C₄	1	2	yes	—
49	C₅×#107	D₄	1	4	yes	complemented
50	C₅×#106	—	2	4	no	—
51	C₅×#102	—	2	4	no	—
52	C₂×#466	C₂×C₅	1	1	yes	characteristic
53	C₂×#427	C₂×C₅	1	1	yes	characteristic, complemented
54	C₂×#357	C₂×C₅	1	2	yes	complemented
55	C₂×C₂²⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
56	#1380	C₂×C₅	1	2	yes	complemented
57	C₂⁴×C₅	—	4	1	no	—
58	C₂⁴×C₅	D₈	1	1	yes	characteristic
59	C₂×C₅×D₄	—	2	2	no	—
60	C₂×C₅×D₄	—	4	4	no	—
61	C₂×C₅×D₄	—	4	1	no	—
62	C₂×C₅×D₄	—	4	1	no	—
63	C₂×C₅×D₄	—	4	1	no	—
64	C₂×C₅×D₄	—	4	1	no	—
65	C₂×C₅×D₄	—	4	1	no	—
66	C₂×C₅×D₄	—	8	4	no	—
67	C₂²×C₄×C₅	—	2	1	no	—
68	C₂²×C₄×C₅	—	2	1	no	—
69	C₂²×C₄×C₅	—	2	1	no	—
70	C₂²×C₄×C₅	—	2	1	no	—
71	C₂²×C₄×C₅	—	4	2	no	—
72	C₂²×C₄×C₅	—	4	1	no	—
73	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
74	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
76	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₅×D₈	—	2	4	no	—
78	C₅×D₈	—	4	2	no	—
79	C₂×C₅×C₈	—	2	1	no	—
80	C₂×C₅×C₈	—	2	1	no	—
81	C₂×C₅×C₈	—	2	4	no	—
82	C₂×C₅×C₈	—	2	4	no	—
83	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
84	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
86	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
87	C₂×C₅×C₈	#41	1	2	yes	—
88	C₅×C₄⋊C₄	—	2	2	no	—
89	C₅×C₄⋊C₄	—	2	4	no	—
90	C₅×C₄⋊C₄	—	4	1	no	—
91	C₅×C₄⋊C₄	—	4	2	no	—
92	C₅×C₂²⋊C₄	—	4	4	no	—
93	C₅×C₂²⋊C₄	—	4	4	no	—
94	C₄²×C₅	—	4	4	no	—
95	C₅×C₁₆	—	4	2	no	—
96	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
97	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
98	C₂×#121	—	2	1	no	—
99	C₂×#117	—	2	1	no	—
100	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
101	#466	—	2	2	no	—
102	#466	C₄×C₅	1	4	yes	complemented
103	#427	—	2	4	no	—
104	C₂×#107	C₂²×C₅	1	1	yes	characteristic
105	C₂×#107	C₂²×C₅	1	2	yes	—
106	C₂×#106	—	2	1	no	—
107	C₂×#102	—	2	1	no	—
108	#357	—	2	4	no	—
109	C₂²⋊C₁₆	—	2	2	no	—
110	C₂³×C₅	—	2	1	no	—
111	C₂³×C₅	—	2	1	no	—
112	C₂³×C₅	—	2	2	no	—
113	C₂³×C₅	—	2	2	no	—
114	C₂³×C₅	—	4	2	no	—
115	C₂³×C₅	—	8	1	no	—
116	C₂³×C₅	—	8	1	no	—
117	C₂³×C₅	—	8	2	no	—
118	C₂³×C₅	—	8	2	no	—
119	C₂³×C₅	#102	1	2	yes	—
120	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
121	C₅×D₄	—	4	2	no	—
122	C₅×D₄	—	4	2	no	—
123	C₅×D₄	—	8	8	no	—
124	C₅×D₄	—	8	1	no	—
125	C₅×D₄	—	8	1	no	—
126	C₂×C₄×C₅	—	2	1	no	—
127	C₂×C₄×C₅	—	2	1	no	—
128	C₂×C₄×C₅	—	2	2	no	—
129	C₂×C₄×C₅	—	2	2	no	—
130	C₂×C₄×C₅	—	2	2	no	—
131	C₂×C₄×C₅	—	2	2	no	—
132	C₂×C₄×C₅	—	4	2	no	—
133	C₂×C₄×C₅	—	4	4	no	—
134	C₂×C₄×C₅	—	8	2	no	—
135	C₂×C₄×C₅	—	8	2	no	—
136	C₂×C₄×C₅	—	8	2	no	—
137	C₂×C₄×C₅	#102	1	2	yes	—
138	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₅	#120	1	1	yes	characteristic
140	C₂×C₄×C₅	#123	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
142	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₄×C₅	#137	1	1	yes	characteristic
144	C₅×C₈	—	2	2	no	—
145	C₅×C₈	C₄×D₄	1	2	yes	—
146	C₅×C₈	#121	1	2	yes	—
147	C₂²×D₄	—	2	1	no	—
148	C₂²×D₄	—	4	1	no	—
149	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
150	C₂×D₈	—	2	2	no	—
151	C₂×D₈	—	2	1	no	—
152	C₂×D₈	—	2	2	no	—
153	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
157	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
158	C₂²×C₈	C₅×D₄	1	2	yes	—
159	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
160	#121	—	2	4	no	—
161	#121	—	4	2	no	—
162	#117	—	2	4	no	—
163	#117	—	4	2	no	—
164	C₂×C₄⋊C₄	—	2	2	no	—
165	C₂×C₄⋊C₄	—	2	1	no	—
166	C₂×C₂²⋊C₄	—	4	1	no	—
167	C₂×C₂²⋊C₄	—	4	1	no	—
168	C₂×C₄²	—	4	1	no	—
169	C₂×C₁₆	—	2	2	no	—
170	C₂×C₁₆	—	2	2	no	—
171	C₂×C₁₆	—	2	2	no	—
172	#107	—	2	2	no	—
173	#107	—	2	1	no	—
174	#107	C₂×C₄×C₅	1	2	yes	—
175	#107	C₅×D₄	1	4	yes	complemented
176	#106	—	2	4	no	—
177	#102	—	2	4	no	—
178	C₂²×C₅	—	2	2	no	—
179	C₂²×C₅	—	2	2	no	—
180	C₂²×C₅	—	2	2	no	—
181	C₂²×C₅	—	4	2	no	—
182	C₂²×C₅	—	4	2	no	—
183	C₂²×C₅	—	8	2	no	—
184	C₂²×C₅	—	8	2	no	—
185	C₂²×C₅	—	8	2	no	—
186	C₂²×C₅	—	16	2	no	—
187	C₂²×C₅	—	16	2	no	—
188	C₂²×C₅	C₂×#102	1	1	yes	characteristic
189	C₂²×C₅	#447	1	1	yes	characteristic
190	C₂²×C₅	#480	1	1	yes	characteristic
191	C₂²×C₅	#508	1	2	yes	—
192	C₂²×C₅	#509	1	2	yes	—
193	C₄×C₅	—	2	2	no	—
194	C₄×C₅	—	8	2	no	—
195	C₄×C₅	—	8	4	no	—
196	C₄×C₅	#386	1	1	yes	characteristic
197	C₄×C₅	#417	1	1	yes	characteristic
198	C₄×C₅	#448	1	1	yes	characteristic
199	C₄×C₅	#481	1	1	yes	characteristic
200	C₂⁴	—	4	1	no	—
201	C₂⁴	C₅×D₈	1	1	yes	characteristic
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	—	4	4	no	—
204	C₂×D₄	—	4	1	no	—
205	C₂×D₄	—	4	1	no	—
206	C₂×D₄	—	4	1	no	—
207	C₂×D₄	—	4	1	no	—
208	C₂×D₄	—	4	1	no	—
209	C₂×D₄	—	8	4	no	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
217	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
218	C₂²×C₄	C₅×SD₁₆	1	2	yes	—
219	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
220	D₈	—	2	4	no	—
221	D₈	—	4	2	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
227	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
228	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
229	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
230	C₂×C₈	C₅×#41	1	2	yes	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	4	1	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₂²⋊C₄	—	4	4	no	—
236	C₂²⋊C₄	—	4	4	no	—
237	C₄²	—	4	4	no	—
238	C₁₆	—	4	2	no	—
239	C₂×C₅	—	4	2	no	—
240	C₂×C₅	—	16	2	no	—
241	C₂×C₅	#1874	1	1	yes	characteristic
242	C₂×C₅	#2138	1	1	yes	characteristic
243	C₂×C₅	#2140	1	1	yes	characteristic
244	C₂×C₅	#2186	1	2	yes	—
245	C₂×C₅	#2234	1	2	yes	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	2	2	no	—
249	C₂³	—	2	2	no	—
250	C₂³	—	4	2	no	—
251	C₂³	—	8	1	no	—
252	C₂³	—	8	1	no	—
253	C₂³	—	8	2	no	—
254	C₂³	—	8	2	no	—
255	C₂³	C₅×#102	1	2	yes	—
256	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
257	D₄	—	4	2	no	—
258	D₄	—	4	2	no	—
259	D₄	—	8	8	no	—
260	D₄	—	8	1	no	—
261	D₄	—	8	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	8	2	no	—
271	C₂×C₄	—	8	2	no	—
272	C₂×C₄	—	8	2	no	—
273	C₂×C₄	C₅×#102	1	2	yes	—
274	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
275	C₂×C₄	C₅×#120	1	1	yes	characteristic
276	C₂×C₄	C₅×#123	1	1	yes	characteristic
277	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
278	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
279	C₂×C₄	C₅×#137	1	1	yes	characteristic
280	C₈	—	2	2	no	—
281	C₈	C₄×C₅×D₄	1	2	yes	—
282	C₈	C₅×#121	1	2	yes	—
283	C₅	#12992	1	1	yes	characteristic, direct product factor, Sylow
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	2	no	—
289	C₂²	—	8	2	no	—
290	C₂²	—	8	2	no	—
291	C₂²	—	8	2	no	—
292	C₂²	—	16	2	no	—
293	C₂²	—	16	2	no	—
294	C₂²	C₂×C₅×#102	1	1	yes	characteristic
295	C₂²	C₅×#447	1	1	yes	characteristic
296	C₂²	C₅×#480	1	1	yes	characteristic
297	C₂²	C₅×#508	1	2	yes	—
298	C₂²	C₅×#509	1	2	yes	—
299	C₄	—	2	2	no	—
300	C₄	—	8	2	no	—
301	C₄	—	8	4	no	—
302	C₄	C₅×#386	1	1	yes	characteristic
303	C₄	C₅×#417	1	1	yes	characteristic
304	C₄	C₅×#448	1	1	yes	characteristic
305	C₄	C₅×#481	1	1	yes	characteristic
306	C₂	—	4	2	no	—
307	C₂	—	16	2	no	—
308	C₂	C₅×#1874	1	1	yes	characteristic
309	C₂	C₅×#2138	1	1	yes	characteristic
310	C₂	C₅×#2140	1	1	yes	characteristic
311	C₂	C₅×#2186	1	2	yes	—
312	C₂	C₅×#2234	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂×C₅×#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#427	C₂	semidirect
3. C₂×C₅×#357	C₂	semidirect
4. C₂×C₅×C₂²⋊C₁₆	C₂	semidirect
5. C₅×#1380	C₂	semidirect
12. C₅×#466	C₄	semidirect
20. #12992	C₅	direct (N × H)
49. C₅×#107	D₄	semidirect
53. C₂×#427	C₂×C₅	semidirect
54. C₂×#357	C₂×C₅	semidirect
55. C₂×C₂²⋊C₁₆	C₂×C₅	semidirect
56. #1380	C₂×C₅	semidirect
102. #466	C₄×C₅	semidirect
175. #107	C₅×D₄	semidirect
283. C₅	#12992	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×#466	2
C₂×C₅×#427	2
C₂×C₅×#357	2
C₂×C₅×C₂²⋊C₁₆	2
C₅×#1380	2
#12992	5

Subgroups — C5×#12992 #11935712

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12992 #11935712