Subgroups — C₂×C₅×#1706 #11942313

Order 1280.

1046 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#493	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#492	C₂	1	2	yes	maximal, complemented
4	C₂×C₅×#486	C₂	1	2	yes	maximal, complemented
5	C₅×#1706	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₅×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₅×SD₁₆	—	2	4	no	—
8	C₂²×C₅×D₈	—	2	2	no	—
9	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₅×#127	C₂²	1	2	yes	—
11	C₅×#500	—	2	2	no	—
12	C₅×#500	C₂²	1	4	yes	complemented
13	C₅×#493	—	2	2	no	—
14	C₅×#493	C₂²	1	4	yes	—
15	C₅×#492	—	2	4	no	—
16	C₅×#492	C₂²	1	8	yes	complemented
17	C₅×#486	—	2	4	no	—
18	C₅×#486	C₂²	1	8	yes	complemented
19	C₂×C₅×#107	—	2	2	no	—
20	C₂×C₅×#102	—	2	4	no	—
21	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
24	C₅×C₈²	C₂²	1	4	yes	complemented
25	C₂×#1706	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂²×C₅×Q₈	—	2	2	no	—
27	C₂²×C₅×D₄	—	2	2	no	—
28	C₂²×C₅×D₄	—	2	2	no	—
29	C₂²×C₅×D₄	—	2	2	no	—
30	C₂×C₅×SD₁₆	—	2	16	no	—
31	C₂×C₅×SD₁₆	—	4	16	no	—
32	C₂×C₅×D₈	—	2	8	no	—
33	C₂×C₅×D₈	—	4	8	no	—
34	C₂²×C₅×C₈	D₄	1	2	yes	—
35	C₂²×C₅×C₈	D₄	1	2	yes	—
36	C₂²×C₅×C₈	D₄	1	2	yes	—
37	C₅×C₄⋊Q₈	—	2	2	no	—
38	C₅×C₄⋊Q₈	—	2	2	no	—
39	C₅×C₄⋊Q₈	—	2	2	no	—
40	C₅×C₄⋊Q₈	C₂³	1	4	yes	—
41	C₅×#127	—	2	2	no	—
42	C₅×#127	—	2	2	no	—
43	C₅×#127	—	2	2	no	—
44	C₅×#127	C₂³	1	4	yes	—
45	C₂×C₅×C₄⋊C₄	—	2	2	no	—
46	C₂×C₅×C₄⋊C₄	—	2	2	no	—
47	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
48	C₅×#107	—	2	8	no	—
49	C₅×#102	—	2	16	no	—
50	C₄×C₅×C₈	D₄	1	2	yes	—
51	C₄×C₅×C₈	D₄	1	2	yes	—
52	C₄×C₅×C₈	D₄	1	2	yes	—
53	C₄×C₅×C₈	C₂³	1	2	yes	—
54	C₄×C₅×C₈	C₂³	1	2	yes	—
55	C₄×C₅×C₈	C₂³	1	2	yes	—
56	C₂×#500	C₂×C₅	1	1	yes	characteristic, complemented
57	C₂×#493	C₂×C₅	1	1	yes	characteristic
58	C₂×#492	C₂×C₅	1	2	yes	complemented
59	C₂×#486	C₂×C₅	1	2	yes	complemented
60	#1706	C₂×C₅	1	8	yes	direct product factor
61	C₂×C₈²	C₂×C₅	1	1	yes	characteristic, complemented
62	C₂⁴×C₅	—	4	2	no	—
63	C₂×C₅×Q₈	—	2	4	no	—
64	C₂×C₅×Q₈	—	4	2	no	—
65	C₂×C₅×Q₈	—	4	8	no	—
66	C₂×C₅×D₄	—	2	4	no	—
67	C₂×C₅×D₄	—	2	4	no	—
68	C₂×C₅×D₄	—	2	4	no	—
69	C₂×C₅×D₄	—	4	2	no	—
70	C₂×C₅×D₄	—	4	2	no	—
71	C₂×C₅×D₄	—	4	2	no	—
72	C₂×C₅×D₄	—	4	8	no	—
73	C₂×C₅×D₄	—	4	8	no	—
74	C₂×C₅×D₄	—	4	8	no	—
75	C₂²×C₄×C₅	—	4	2	no	—
76	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₅×SD₁₆	—	4	32	no	—
80	C₅×D₈	—	4	16	no	—
81	C₂×C₅×C₈	—	2	4	no	—
82	C₂×C₅×C₈	D₈	1	8	yes	complemented
83	C₂×C₅×C₈	SD₁₆	1	8	yes	complemented
84	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
85	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
86	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
87	C₅×C₄⋊C₄	—	2	4	no	—
88	C₅×C₄⋊C₄	—	2	4	no	—
89	C₅×C₄⋊C₄	—	4	2	no	—
90	C₅×C₄⋊C₄	—	4	2	no	—
91	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₅	C₂⁴	1	1	yes	characteristic
95	C₂²×SD₁₆	—	2	4	no	—
96	C₂²×D₈	—	2	2	no	—
97	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
98	C₂×#127	C₂²×C₅	1	2	yes	—
99	#500	—	2	2	no	—
100	#500	C₂²×C₅	1	4	yes	complemented
101	#493	—	2	2	no	—
102	#493	C₂²×C₅	1	4	yes	—
103	#492	—	2	4	no	—
104	#492	C₂²×C₅	1	8	yes	complemented
105	#486	—	2	4	no	—
106	#486	C₂²×C₅	1	8	yes	complemented
107	C₂×#107	—	2	2	no	—
108	C₂×#102	—	2	4	no	—
109	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
110	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
111	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
112	C₈²	C₂²×C₅	1	4	yes	complemented
113	C₂³×C₅	—	4	4	no	—
114	C₂³×C₅	—	8	4	no	—
115	C₂³×C₅	—	8	4	no	—
116	C₂³×C₅	—	8	4	no	—
117	C₂³×C₅	#127	1	1	yes	characteristic
118	C₅×Q₈	—	4	8	no	—
119	C₅×Q₈	—	8	4	no	—
120	C₅×D₄	—	4	8	no	—
121	C₅×D₄	—	4	8	no	—
122	C₅×D₄	—	4	8	no	—
123	C₅×D₄	—	8	4	no	—
124	C₅×D₄	—	8	4	no	—
125	C₅×D₄	—	8	4	no	—
126	C₂×C₄×C₅	—	4	4	no	—
127	C₂×C₄×C₅	—	8	4	no	—
128	C₂×C₄×C₅	#127	1	1	yes	characteristic
129	C₂×C₄×C₅	#127	1	1	yes	characteristic
130	C₂×C₄×C₅	#127	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂×D₈	1	4	yes	—
132	C₂×C₄×C₅	C₂×SD₁₆	1	4	yes	—
133	C₂×C₄×C₅	#135	1	4	yes	—
134	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
137	C₅×C₈	—	2	4	no	—
138	C₅×C₈	C₂×D₈	1	8	yes	complemented
139	C₅×C₈	C₂×SD₁₆	1	8	yes	complemented
140	C₂²×Q₈	—	2	2	no	—
141	C₂²×D₄	—	2	2	no	—
142	C₂²×D₄	—	2	2	no	—
143	C₂²×D₄	—	2	2	no	—
144	C₂×SD₁₆	—	2	16	no	—
145	C₂×SD₁₆	—	4	16	no	—
146	C₂×D₈	—	2	8	no	—
147	C₂×D₈	—	4	8	no	—
148	C₂²×C₈	C₅×D₄	1	2	yes	—
149	C₂²×C₈	C₅×D₄	1	2	yes	—
150	C₂²×C₈	C₅×D₄	1	2	yes	—
151	C₄⋊Q₈	—	2	2	no	—
152	C₄⋊Q₈	—	2	2	no	—
153	C₄⋊Q₈	—	2	2	no	—
154	C₄⋊Q₈	C₂³×C₅	1	4	yes	—
155	#127	—	2	2	no	—
156	#127	—	2	2	no	—
157	#127	—	2	2	no	—
158	#127	C₂³×C₅	1	4	yes	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
162	#107	—	2	8	no	—
163	#102	—	2	16	no	—
164	C₄×C₈	C₅×D₄	1	2	yes	—
165	C₄×C₈	C₅×D₄	1	2	yes	—
166	C₄×C₈	C₅×D₄	1	2	yes	—
167	C₄×C₈	C₂³×C₅	1	2	yes	—
168	C₄×C₈	C₂³×C₅	1	2	yes	—
169	C₄×C₈	C₂³×C₅	1	2	yes	—
170	C₂²×C₅	—	8	4	no	—
171	C₂²×C₅	—	8	4	no	—
172	C₂²×C₅	—	8	4	no	—
173	C₂²×C₅	—	16	8	no	—
174	C₂²×C₅	#492	1	2	yes	—
175	C₂²×C₅	#493	1	2	yes	—
176	C₂²×C₅	#495	1	2	yes	—
177	C₂²×C₅	C₂×#127	1	1	yes	characteristic
178	C₄×C₅	—	8	4	no	—
179	C₄×C₅	#492	1	2	yes	—
180	C₄×C₅	#493	1	2	yes	—
181	C₄×C₅	#495	1	2	yes	—
182	C₄×C₅	C₂²×D₈	1	2	yes	—
183	C₄×C₅	C₂²×SD₁₆	1	2	yes	—
184	C₄×C₅	C₂×#135	1	2	yes	—
185	C₂⁴	—	4	2	no	—
186	C₂×Q₈	—	2	4	no	—
187	C₂×Q₈	—	4	2	no	—
188	C₂×Q₈	—	4	8	no	—
189	C₂×D₄	—	2	4	no	—
190	C₂×D₄	—	2	4	no	—
191	C₂×D₄	—	2	4	no	—
192	C₂×D₄	—	4	2	no	—
193	C₂×D₄	—	4	2	no	—
194	C₂×D₄	—	4	2	no	—
195	C₂×D₄	—	4	8	no	—
196	C₂×D₄	—	4	8	no	—
197	C₂×D₄	—	4	8	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
202	SD₁₆	—	4	32	no	—
203	D₈	—	4	16	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	C₅×D₈	1	8	yes	complemented
206	C₂×C₈	C₅×SD₁₆	1	8	yes	complemented
207	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
208	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
209	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₄⋊C₄	—	4	2	no	—
214	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
215	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
216	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
217	C₄²	C₂⁴×C₅	1	1	yes	characteristic
218	C₂×C₅	—	16	4	no	—
219	C₂×C₅	#1706	1	4	yes	direct product factor
220	C₂×C₅	C₂×#492	1	1	yes	characteristic
221	C₂×C₅	C₂×#493	1	1	yes	characteristic
222	C₂×C₅	C₂×#495	1	1	yes	characteristic
223	C₂³	—	4	4	no	—
224	C₂³	—	8	4	no	—
225	C₂³	—	8	4	no	—
226	C₂³	—	8	4	no	—
227	C₂³	C₅×#127	1	1	yes	characteristic
228	Q₈	—	4	8	no	—
229	Q₈	—	8	4	no	—
230	D₄	—	4	8	no	—
231	D₄	—	4	8	no	—
232	D₄	—	4	8	no	—
233	D₄	—	8	4	no	—
234	D₄	—	8	4	no	—
235	D₄	—	8	4	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	8	4	no	—
238	C₂×C₄	C₅×#127	1	1	yes	characteristic
239	C₂×C₄	C₅×#127	1	1	yes	characteristic
240	C₂×C₄	C₅×#127	1	1	yes	characteristic
241	C₂×C₄	C₂×C₅×D₈	1	4	yes	—
242	C₂×C₄	C₂×C₅×SD₁₆	1	4	yes	—
243	C₂×C₄	C₅×#135	1	4	yes	—
244	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
245	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
246	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
247	C₈	—	2	4	no	—
248	C₈	C₂×C₅×D₈	1	8	yes	complemented
249	C₈	C₂×C₅×SD₁₆	1	8	yes	complemented
250	C₅	C₂×#1706	1	1	yes	characteristic, direct product factor, Sylow
251	C₂²	—	8	4	no	—
252	C₂²	—	8	4	no	—
253	C₂²	—	8	4	no	—
254	C₂²	—	16	8	no	—
255	C₂²	C₅×#492	1	2	yes	—
256	C₂²	C₅×#493	1	2	yes	—
257	C₂²	C₅×#495	1	2	yes	—
258	C₂²	C₂×C₅×#127	1	1	yes	characteristic
259	C₄	—	8	4	no	—
260	C₄	C₅×#492	1	2	yes	—
261	C₄	C₅×#493	1	2	yes	—
262	C₄	C₅×#495	1	2	yes	—
263	C₄	C₂²×C₅×D₈	1	2	yes	—
264	C₄	C₂²×C₅×SD₁₆	1	2	yes	—
265	C₄	C₂×C₅×#135	1	2	yes	—
266	C₂	—	16	4	no	—
267	C₂	C₅×#1706	1	4	yes	direct product factor
268	C₂	C₂×C₅×#492	1	1	yes	characteristic
269	C₂	C₂×C₅×#493	1	1	yes	characteristic
270	C₂	C₂×C₅×#495	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
prime radical	C₂×C₅×#493	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#500	C₂	semidirect
3. C₂×C₅×#492	C₂	semidirect
4. C₂×C₅×#486	C₂	semidirect
5. C₅×#1706	C₂	direct (N × H)
6. C₂×C₅×C₈²	C₂	semidirect
12. C₅×#500	C₂²	semidirect
16. C₅×#492	C₂²	semidirect
18. C₅×#486	C₂²	semidirect
24. C₅×C₈²	C₂²	semidirect
25. C₂×#1706	C₅	direct (N × H)
56. C₂×#500	C₂×C₅	semidirect
58. C₂×#492	C₂×C₅	semidirect
59. C₂×#486	C₂×C₅	semidirect
60. #1706	C₂×C₅	direct (N × H)
61. C₂×C₈²	C₂×C₅	semidirect
82. C₂×C₅×C₈	D₈	semidirect
83. C₂×C₅×C₈	SD₁₆	semidirect
100. #500	C₂²×C₅	semidirect
104. #492	C₂²×C₅	semidirect
106. #486	C₂²×C₅	semidirect
112. C₈²	C₂²×C₅	semidirect
138. C₅×C₈	C₂×D₈	semidirect
139. C₅×C₈	C₂×SD₁₆	semidirect
205. C₂×C₈	C₅×D₈	semidirect
206. C₂×C₈	C₅×SD₁₆	semidirect
219. C₂×C₅	#1706	direct (N × H)
248. C₈	C₂×C₅×D₈	semidirect
249. C₈	C₂×C₅×SD₁₆	semidirect
250. C₅	C₂×#1706	direct (N × H)
267. C₂	C₅×#1706	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×#500	2
C₂×C₅×#493	2
C₂×C₅×#492	2
C₂×C₅×#486	2
C₅×#1706	2
C₂×C₅×C₈²	2
C₂×#1706	5

Subgroups — C2×C5×#1706 #11942313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1706 #11942313