Subgroups — C₂×C₅×#1274 #11930573

Order 1280.

758 conjugacy classes of subgroups.

The subgroup lattice has 264 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	2	yes	maximal
2	C₂×C₅×#376	C₂	1	1	yes	maximal, characteristic
3	C₅×#1274	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₄²×C₅	C₄	1	2	yes	—
5	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
6	C₅×#431	—	2	2	no	—
7	C₅×#431	—	2	2	no	—
8	C₅×#431	C₄	1	4	yes	complemented
9	C₅×#431	C₂²	1	4	yes	—
10	C₂×C₅×C₄⋊C₈	—	2	2	no	—
11	C₂×C₅×C₂²⋊C₈	—	2	2	no	—
12	C₂×C₄×C₅×C₈	C₄	1	4	yes	complemented
13	C₅×#376	—	2	2	no	—
14	C₅×#376	C₄	1	2	yes	—
15	C₅×#376	C₂²	1	2	yes	—
16	C₂×C₅×#95	—	2	2	no	—
17	C₂×#1274	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₄×C₅	—	2	2	no	—
19	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₅	D₄	1	2	yes	—
21	C₂²×C₅×C₈	—	2	4	no	—
22	C₂×C₄²×C₅	—	2	2	no	—
23	C₂×C₄²×C₅	—	2	4	no	—
24	C₂×C₄²×C₅	C₈	1	8	yes	complemented
25	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₅	D₄	1	2	yes	—
30	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
34	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
35	C₅×C₄⋊C₈	—	2	4	no	—
36	C₅×C₄⋊C₈	—	2	4	no	—
37	C₅×C₂²⋊C₈	—	2	4	no	—
38	C₅×C₂²⋊C₈	—	2	4	no	—
39	C₄×C₅×C₈	—	2	4	no	—
40	C₄×C₅×C₈	C₂×C₄	1	8	yes	complemented
41	C₅×#95	—	2	4	no	—
42	C₅×#95	—	2	4	no	—
43	C₂×#431	C₂×C₅	1	2	yes	—
44	C₂×#376	C₂×C₅	1	1	yes	characteristic
45	#1274	C₂×C₅	1	4	yes	direct product factor
46	C₂⁴×C₅	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅	—	2	8	no	—
48	C₂²×C₄×C₅	—	2	4	no	—
49	C₂²×C₄×C₅	—	4	4	no	—
50	C₂²×C₄×C₅	—	4	8	no	—
51	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
53	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
62	C₂²×C₄×C₅	M₁₆	1	4	yes	—
63	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
65	C₂×C₅×C₈	—	2	8	no	—
66	C₂×C₅×C₈	—	4	8	no	—
67	C₄²×C₅	—	2	8	no	—
68	C₄²×C₅	—	2	4	no	—
69	C₄²×C₅	C₄²	1	2	yes	—
70	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
71	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
72	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
73	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
74	C₄²×C₅	C₄⋊C₄	1	2	yes	—
75	C₄²×C₅	C₄⋊C₄	1	2	yes	—
76	C₄²×C₅	C₂×C₈	1	8	yes	complemented
77	C₄²×C₅	C₂²×C₄	1	2	yes	—
78	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
80	C₂²×C₄²	C₄×C₅	1	2	yes	—
81	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
82	#431	—	2	2	no	—
83	#431	—	2	2	no	—
84	#431	C₄×C₅	1	4	yes	complemented
85	#431	C₂²×C₅	1	4	yes	—
86	C₂×C₄⋊C₈	—	2	2	no	—
87	C₂×C₂²⋊C₈	—	2	2	no	—
88	C₂×C₄×C₈	C₄×C₅	1	4	yes	complemented
89	#376	—	2	2	no	—
90	#376	C₄×C₅	1	2	yes	—
91	#376	C₂²×C₅	1	2	yes	—
92	C₂×#95	—	2	2	no	—
93	C₂³×C₅	—	2	4	no	—
94	C₂³×C₅	#95	1	1	yes	characteristic
95	C₂³×C₅	#95	1	1	yes	characteristic
96	C₂³×C₅	C₂²⋊C₈	1	4	yes	—
97	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₅	—	2	4	no	—
99	C₂×C₄×C₅	—	2	4	no	—
100	C₂×C₄×C₅	—	2	8	no	—
101	C₂×C₄×C₅	—	4	2	no	—
102	C₂×C₄×C₅	—	4	2	no	—
103	C₂×C₄×C₅	—	4	2	no	—
104	C₂×C₄×C₅	—	4	2	no	—
105	C₂×C₄×C₅	—	4	8	no	—
106	C₂×C₄×C₅	—	4	16	no	—
107	C₂×C₄×C₅	#95	1	2	yes	—
108	C₂×C₄×C₅	#95	1	2	yes	—
109	C₂×C₄×C₅	#95	1	1	yes	characteristic
110	C₂×C₄×C₅	#95	1	1	yes	characteristic
111	C₂×C₄×C₅	C₄×C₈	1	2	yes	—
112	C₂×C₄×C₅	C₄×C₈	1	2	yes	—
113	C₂×C₄×C₅	#97	1	2	yes	—
114	C₂×C₄×C₅	#97	1	2	yes	—
115	C₂×C₄×C₅	C₂²⋊C₈	1	4	yes	—
116	C₂×C₄×C₅	C₄⋊C₈	1	4	yes	—
117	C₂×C₄×C₅	C₄⋊C₈	1	4	yes	—
118	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
120	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
123	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
124	C₅×C₈	—	4	8	no	—
125	C₂³×C₄	—	2	2	no	—
126	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
127	C₂³×C₄	C₅×D₄	1	2	yes	—
128	C₂²×C₈	—	2	4	no	—
129	C₂×C₄²	—	2	2	no	—
130	C₂×C₄²	—	2	4	no	—
131	C₂×C₄²	C₅×C₈	1	8	yes	complemented
132	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
133	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
134	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
135	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
136	C₂×C₄²	C₅×D₄	1	2	yes	—
137	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
138	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
139	C₂×C₄²	C₅×Q₈	1	1	yes	characteristic
140	C₂×C₄²	C₅×Q₈	1	1	yes	characteristic
141	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
142	C₄⋊C₈	—	2	4	no	—
143	C₄⋊C₈	—	2	4	no	—
144	C₂²⋊C₈	—	2	4	no	—
145	C₂²⋊C₈	—	2	4	no	—
146	C₄×C₈	—	2	4	no	—
147	C₄×C₈	C₂×C₄×C₅	1	8	yes	complemented
148	#95	—	2	4	no	—
149	#95	—	2	4	no	—
150	C₂²×C₅	—	2	2	no	—
151	C₂²×C₅	—	2	2	no	—
152	C₂²×C₅	—	2	8	no	—
153	C₂²×C₅	#336	1	2	yes	—
154	C₂²×C₅	#336	1	2	yes	—
155	C₂²×C₅	H(3,ℤ/4)	1	4	yes	—
156	C₂²×C₅	C₂×#95	1	1	yes	characteristic
157	C₂²×C₅	C₂×C₂²⋊C₈	1	2	yes	—
158	C₄×C₅	—	2	2	no	—
159	C₄×C₅	—	2	2	no	—
160	C₄×C₅	—	2	2	no	—
161	C₄×C₅	—	2	2	no	—
162	C₄×C₅	—	4	8	no	—
163	C₄×C₅	#336	1	2	yes	—
164	C₄×C₅	#336	1	2	yes	—
165	C₄×C₅	C₂×C₄×C₈	1	1	yes	characteristic
166	C₄×C₅	C₂×#97	1	1	yes	characteristic
167	C₄×C₅	C₂×C₄⋊C₈	1	2	yes	—
168	C₂⁴	C₅×C₂²⋊C₄	1	1	yes	characteristic
169	C₂²×C₄	—	2	4	no	—
170	C₂²×C₄	—	2	8	no	—
171	C₂²×C₄	—	4	4	no	—
172	C₂²×C₄	—	4	8	no	—
173	C₂²×C₄	C₄²×C₅	1	1	yes	characteristic
174	C₂²×C₄	C₄²×C₅	1	1	yes	characteristic
175	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
176	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
177	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
178	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
179	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
180	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
181	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
182	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₅×C₈	1	4	yes	—
184	C₂²×C₄	C₅×M₁₆	1	4	yes	—
185	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
187	C₂×C₈	—	2	8	no	—
188	C₂×C₈	—	4	8	no	—
189	C₄²	—	2	4	no	—
190	C₄²	—	2	8	no	—
191	C₄²	C₄²×C₅	1	2	yes	—
192	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
193	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
194	C₄²	C₅×C₄⋊C₄	1	2	yes	—
195	C₄²	C₅×C₄⋊C₄	1	2	yes	—
196	C₄²	C₅×C₄⋊C₄	1	1	yes	characteristic
197	C₄²	C₅×C₄⋊C₄	1	1	yes	characteristic
198	C₄²	C₂×C₅×C₈	1	8	yes	complemented
199	C₄²	C₂²×C₄×C₅	1	2	yes	—
200	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
201	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
202	C₂×C₅	—	2	2	no	—
203	C₂×C₅	—	2	2	no	—
204	C₂×C₅	#1274	1	4	yes	direct product factor
205	C₂×C₅	C₂×#336	1	1	yes	characteristic
206	C₂×C₅	C₂×H(3,ℤ/4)	1	2	yes	—
207	C₂³	—	2	4	no	—
208	C₂³	C₅×#95	1	1	yes	characteristic
209	C₂³	C₅×#95	1	1	yes	characteristic
210	C₂³	C₅×C₂²⋊C₈	1	4	yes	—
211	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	16	no	—
220	C₂×C₄	—	4	8	no	—
221	C₂×C₄	C₅×#95	1	2	yes	—
222	C₂×C₄	C₅×#95	1	2	yes	—
223	C₂×C₄	C₅×#95	1	1	yes	characteristic
224	C₂×C₄	C₅×#95	1	1	yes	characteristic
225	C₂×C₄	C₄×C₅×C₈	1	2	yes	—
226	C₂×C₄	C₄×C₅×C₈	1	2	yes	—
227	C₂×C₄	C₅×#97	1	2	yes	—
228	C₂×C₄	C₅×#97	1	2	yes	—
229	C₂×C₄	C₅×C₂²⋊C₈	1	4	yes	—
230	C₂×C₄	C₅×C₄⋊C₈	1	4	yes	—
231	C₂×C₄	C₅×C₄⋊C₈	1	4	yes	—
232	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
233	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
234	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
235	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
236	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
237	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
238	C₈	—	4	8	no	—
239	C₅	C₂×#1274	1	1	yes	characteristic, direct product factor, Sylow
240	C₂²	—	2	8	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	C₅×#336	1	2	yes	—
244	C₂²	C₅×#336	1	2	yes	—
245	C₂²	C₅×H(3,ℤ/4)	1	4	yes	—
246	C₂²	C₂×C₅×#95	1	1	yes	characteristic
247	C₂²	C₂×C₅×C₂²⋊C₈	1	2	yes	—
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	4	8	no	—
253	C₄	C₅×#336	1	2	yes	—
254	C₄	C₅×#336	1	2	yes	—
255	C₄	C₂×C₄×C₅×C₈	1	1	yes	characteristic
256	C₄	C₂×C₅×#97	1	1	yes	characteristic
257	C₄	C₂×C₅×C₄⋊C₈	1	2	yes	—
258	C₂	—	2	2	no	—
259	C₂	—	2	2	no	—
260	C₂	C₅×#1274	1	4	yes	direct product factor
261	C₂	C₂×C₅×#336	1	1	yes	characteristic
262	C₂	C₂×C₅×H(3,ℤ/4)	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₄²
commutator	C₄	C₂×C₄×C₅×C₈
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂³×C₅	#95
prime radical	C₂⁴×C₅	C₂²⋊C₄
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1274	C₂	direct (N × H)
8. C₅×#431	C₄	semidirect
12. C₂×C₄×C₅×C₈	C₄	semidirect
17. C₂×#1274	C₅	direct (N × H)
24. C₂×C₄²×C₅	C₈	semidirect
40. C₄×C₅×C₈	C₂×C₄	semidirect
45. #1274	C₂×C₅	direct (N × H)
76. C₄²×C₅	C₂×C₈	semidirect
84. #431	C₄×C₅	semidirect
88. C₂×C₄×C₈	C₄×C₅	semidirect
131. C₂×C₄²	C₅×C₈	semidirect
147. C₄×C₈	C₂×C₄×C₅	semidirect
198. C₄²	C₂×C₅×C₈	semidirect
204. C₂×C₅	#1274	direct (N × H)
239. C₅	C₂×#1274	direct (N × H)
260. C₂	C₅×#1274	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
C₂×C₅×#376	2
C₅×#1274	2
C₂×#1274	5

Subgroups — C2×C5×#1274 #11930573

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1274 #11930573