Subgroups — C₅×#12994 #11935714

Order 1280.

530 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#467	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#427	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#358	C₂	1	2	yes	maximal, complemented
4	C₂×C₅×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₅×#1380	C₂	1	2	yes	maximal, complemented
6	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
7	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₅×C₂²⋊Q₈	—	2	1	no	—
9	C₂×C₅×#117	—	2	1	no	—
10	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
11	C₅×#467	—	2	2	no	—
12	C₅×#467	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×#103	—	2	1	no	—
18	C₅×#358	—	2	4	no	—
19	C₅×C₂²⋊C₁₆	—	2	2	no	—
20	#12994	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₅×Q₈	—	2	1	no	—
22	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
23	C₂×C₅×Q₁₆	—	2	2	no	—
24	C₂×C₅×Q₁₆	—	2	1	no	—
25	C₂×C₅×Q₁₆	—	2	2	no	—
26	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
27	C₂²×C₅×C₈	—	2	2	no	—
28	C₂²×C₅×C₈	—	2	2	no	—
29	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
30	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
31	C₂²×C₅×C₈	D₄	1	2	yes	—
32	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₅×C₂²⋊Q₈	—	2	4	no	—
34	C₅×C₂²⋊Q₈	—	4	2	no	—
35	C₅×#117	—	2	4	no	—
36	C₅×#117	—	4	2	no	—
37	C₂×C₅×C₄⋊C₄	—	2	1	no	—
38	C₂×C₅×C₄⋊C₄	—	2	2	no	—
39	C₂×C₅×C₄⋊C₄	—	4	1	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	—
50	C₅×#106	—	2	4	no	—
51	C₅×#103	—	2	4	no	—
52	C₂×#467	C₂×C₅	1	1	yes	characteristic
53	C₂×#427	C₂×C₅	1	1	yes	characteristic
54	C₂×#358	C₂×C₅	1	2	yes	complemented
55	C₂×C₂²⋊C₁₆	C₂×C₅	1	1	yes	characteristic
56	#1380	C₂×C₅	1	2	yes	complemented
57	C₂⁴×C₅	D₈	1	1	yes	characteristic
58	C₂×C₅×Q₈	—	2	2	no	—
59	C₂×C₅×Q₈	—	4	1	no	—
60	C₂×C₅×Q₈	—	4	1	no	—
61	C₂×C₅×Q₈	—	4	1	no	—
62	C₂×C₅×Q₈	—	4	1	no	—
63	C₂×C₅×Q₈	—	4	1	no	—
64	C₂²×C₄×C₅	—	2	1	no	—
65	C₂²×C₄×C₅	—	2	1	no	—
66	C₂²×C₄×C₅	—	2	1	no	—
67	C₂²×C₄×C₅	—	2	1	no	—
68	C₂²×C₄×C₅	—	4	2	no	—
69	C₂²×C₄×C₅	—	4	1	no	—
70	C₂²×C₄×C₅	—	4	1	no	—
71	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
72	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
73	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
74	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
75	C₅×Q₁₆	—	2	4	no	—
76	C₅×Q₁₆	—	4	2	no	—
77	C₂×C₅×C₈	—	2	1	no	—
78	C₂×C₅×C₈	—	2	1	no	—
79	C₂×C₅×C₈	—	2	4	no	—
80	C₂×C₅×C₈	—	2	4	no	—
81	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
82	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
84	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
85	C₂×C₅×C₈	#41	1	2	yes	—
86	C₅×C₄⋊C₄	—	2	2	no	—
87	C₅×C₄⋊C₄	—	2	4	no	—
88	C₅×C₄⋊C₄	—	4	4	no	—
89	C₅×C₄⋊C₄	—	4	1	no	—
90	C₅×C₄⋊C₄	—	4	2	no	—
91	C₅×C₂²⋊C₄	—	4	4	no	—
92	C₅×C₂²⋊C₄	—	4	4	no	—
93	C₄²×C₅	—	4	4	no	—
94	C₅×C₁₆	—	4	2	no	—
95	C₂²×Q₁₆	C₂²×C₅	1	1	yes	characteristic
96	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
97	C₂×C₂²⋊Q₈	—	2	1	no	—
98	C₂×#117	—	2	1	no	—
99	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
100	#467	—	2	2	no	—
101	#467	C₄×C₅	1	4	yes	complemented
102	#427	—	2	4	no	—
103	C₂×#107	C₂²×C₅	1	1	yes	characteristic
104	C₂×#107	C₂²×C₅	1	2	yes	—
105	C₂×#106	—	2	1	no	—
106	C₂×#103	—	2	1	no	—
107	#358	—	2	4	no	—
108	C₂²⋊C₁₆	—	2	2	no	—
109	C₂³×C₅	—	2	1	no	—
110	C₂³×C₅	—	2	1	no	—
111	C₂³×C₅	—	2	2	no	—
112	C₂³×C₅	—	2	2	no	—
113	C₂³×C₅	#102	1	2	yes	—
114	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
115	C₅×Q₈	—	4	2	no	—
116	C₅×Q₈	—	4	2	no	—
117	C₅×Q₈	—	8	1	no	—
118	C₅×Q₈	—	8	1	no	—
119	C₂×C₄×C₅	—	2	1	no	—
120	C₂×C₄×C₅	—	2	1	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	—	4	2	no	—
126	C₂×C₄×C₅	—	4	2	no	—
127	C₂×C₄×C₅	—	4	4	no	—
128	C₂×C₄×C₅	—	8	1	no	—
129	C₂×C₄×C₅	—	8	1	no	—
130	C₂×C₄×C₅	—	8	2	no	—
131	C₂×C₄×C₅	—	8	2	no	—
132	C₂×C₄×C₅	—	8	2	no	—
133	C₂×C₄×C₅	#102	1	2	yes	—
134	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₅	#120	1	1	yes	characteristic
136	C₂×C₄×C₅	#123	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
138	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₂×C₄×C₅	#137	1	1	yes	characteristic
140	C₅×C₈	—	2	2	no	—
141	C₅×C₈	C₄×D₄	1	2	yes	—
142	C₅×C₈	#121	1	2	yes	—
143	C₂²×Q₈	—	2	1	no	—
144	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
145	C₂×Q₁₆	—	2	2	no	—
146	C₂×Q₁₆	—	2	1	no	—
147	C₂×Q₁₆	—	2	2	no	—
148	C₂×Q₁₆	C₂×C₄×C₅	1	2	yes	—
149	C₂²×C₈	—	2	2	no	—
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
152	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
153	C₂²×C₈	C₅×D₄	1	2	yes	—
154	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
155	C₂²⋊Q₈	—	2	4	no	—
156	C₂²⋊Q₈	—	4	2	no	—
157	#117	—	2	4	no	—
158	#117	—	4	2	no	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	—	2	1	no	—
161	C₂×C₄⋊C₄	—	4	1	no	—
162	C₂×C₂²⋊C₄	—	4	1	no	—
163	C₂×C₂²⋊C₄	—	4	1	no	—
164	C₂×C₄²	—	4	1	no	—
165	C₂×C₁₆	—	2	2	no	—
166	C₂×C₁₆	—	2	2	no	—
167	C₂×C₁₆	—	2	2	no	—
168	#107	—	2	2	no	—
169	#107	—	2	1	no	—
170	#107	C₂×C₄×C₅	1	2	yes	—
171	#107	C₅×D₄	1	4	yes	—
172	#106	—	2	4	no	—
173	#103	—	2	4	no	—
174	C₂²×C₅	—	2	2	no	—
175	C₂²×C₅	—	2	2	no	—
176	C₂²×C₅	—	2	2	no	—
177	C₂²×C₅	—	4	2	no	—
178	C₂²×C₅	—	4	2	no	—
179	C₂²×C₅	C₂×#102	1	1	yes	characteristic
180	C₂²×C₅	#447	1	1	yes	characteristic
181	C₂²×C₅	#480	1	1	yes	characteristic
182	C₂²×C₅	#508	1	2	yes	—
183	C₂²×C₅	#510	1	2	yes	—
184	C₄×C₅	—	2	2	no	—
185	C₄×C₅	—	8	2	no	—
186	C₄×C₅	—	8	2	no	—
187	C₄×C₅	—	8	4	no	—
188	C₄×C₅	#386	1	1	yes	characteristic
189	C₄×C₅	#417	1	1	yes	characteristic
190	C₄×C₅	#448	1	1	yes	characteristic
191	C₄×C₅	#481	1	1	yes	characteristic
192	C₂⁴	C₅×D₈	1	1	yes	characteristic
193	C₂×Q₈	—	2	2	no	—
194	C₂×Q₈	—	4	1	no	—
195	C₂×Q₈	—	4	1	no	—
196	C₂×Q₈	—	4	1	no	—
197	C₂×Q₈	—	4	1	no	—
198	C₂×Q₈	—	4	1	no	—
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	—	4	1	no	—
205	C₂²×C₄	—	4	1	no	—
206	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
207	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
208	C₂²×C₄	C₅×SD₁₆	1	2	yes	—
209	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
210	Q₁₆	—	2	4	no	—
211	Q₁₆	—	4	2	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
217	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
218	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
219	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
220	C₂×C₈	C₅×#41	1	2	yes	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	4	1	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₄⋊C₄	—	4	4	no	—
226	C₂²⋊C₄	—	4	4	no	—
227	C₂²⋊C₄	—	4	4	no	—
228	C₄²	—	4	4	no	—
229	C₁₆	—	4	2	no	—
230	C₂×C₅	—	4	2	no	—
231	C₂×C₅	#1874	1	1	yes	characteristic
232	C₂×C₅	#2138	1	1	yes	characteristic
233	C₂×C₅	#2141	1	1	yes	characteristic
234	C₂×C₅	#2189	1	2	yes	—
235	C₂×C₅	#2237	1	2	yes	—
236	C₂³	—	2	1	no	—
237	C₂³	—	2	1	no	—
238	C₂³	—	2	2	no	—
239	C₂³	—	2	2	no	—
240	C₂³	C₅×#102	1	2	yes	—
241	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
242	Q₈	—	4	2	no	—
243	Q₈	—	4	2	no	—
244	Q₈	—	8	1	no	—
245	Q₈	—	8	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	8	1	no	—
256	C₂×C₄	—	8	1	no	—
257	C₂×C₄	—	8	2	no	—
258	C₂×C₄	—	8	2	no	—
259	C₂×C₄	—	8	2	no	—
260	C₂×C₄	C₅×#102	1	2	yes	—
261	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
262	C₂×C₄	C₅×#120	1	1	yes	characteristic
263	C₂×C₄	C₅×#123	1	1	yes	characteristic
264	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
265	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
266	C₂×C₄	C₅×#137	1	1	yes	characteristic
267	C₈	—	2	2	no	—
268	C₈	C₄×C₅×D₄	1	2	yes	—
269	C₈	C₅×#121	1	2	yes	—
270	C₅	#12994	1	1	yes	characteristic, direct product factor, Sylow
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	4	2	no	—
275	C₂²	—	4	2	no	—
276	C₂²	C₂×C₅×#102	1	1	yes	characteristic
277	C₂²	C₅×#447	1	1	yes	characteristic
278	C₂²	C₅×#480	1	1	yes	characteristic
279	C₂²	C₅×#508	1	2	yes	—
280	C₂²	C₅×#510	1	2	yes	—
281	C₄	—	2	2	no	—
282	C₄	—	8	2	no	—
283	C₄	—	8	2	no	—
284	C₄	—	8	4	no	—
285	C₄	C₅×#386	1	1	yes	characteristic
286	C₄	C₅×#417	1	1	yes	characteristic
287	C₄	C₅×#448	1	1	yes	characteristic
288	C₄	C₅×#481	1	1	yes	characteristic
289	C₂	—	4	2	no	—
290	C₂	C₅×#1874	1	1	yes	characteristic
291	C₂	C₅×#2138	1	1	yes	characteristic
292	C₂	C₅×#2141	1	1	yes	characteristic
293	C₂	C₅×#2189	1	2	yes	—
294	C₂	C₅×#2237	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	C₂⁴×C₅	D₈
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
3. C₂×C₅×#358	C₂	semidirect
5. C₅×#1380	C₂	semidirect
12. C₅×#467	C₄	semidirect
20. #12994	C₅	direct (N × H)
54. C₂×#358	C₂×C₅	semidirect
56. #1380	C₂×C₅	semidirect
101. #467	C₄×C₅	semidirect
270. C₅	#12994	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×#467	2
C₂×C₅×#427	2
C₂×C₅×#358	2
C₂×C₅×C₂²⋊C₁₆	2
C₅×#1380	2
#12994	5

Subgroups — C5×#12994 #11935714

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12994 #11935714