Subgroups — C₂×C₅×#1690 #11941865

Order 1280.

682 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#494	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#475	C₂	1	2	yes	maximal
3	C₂×C₅×#446	C₂	1	1	yes	maximal, characteristic
4	C₅×#1690	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅×#334	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×#332	C₂	1	2	yes	maximal
7	C₂²×C₅×Q₁₆	—	2	1	no	—
8	C₂²×C₅×Q₁₆	—	2	1	no	—
9	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
11	C₅×#494	—	2	2	no	—
12	C₅×#494	C₂²	1	4	yes	—
13	C₅×#475	—	2	4	no	—
14	C₅×#475	C₂²	1	8	yes	—
15	C₅×#446	—	2	2	no	—
16	C₅×#446	C₂²	1	4	yes	—
17	C₂×C₅×#106	—	2	2	no	—
18	C₂×C₅×C₄⋊C₈	—	2	1	no	—
19	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₅×#103	—	2	2	no	—
22	C₂×C₅×#97	—	2	1	no	—
23	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
24	C₅×#334	C₂²	1	4	yes	—
25	C₅×#332	C₂²	1	8	yes	—
26	C₂×#1690	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂²×C₅×Q₈	—	2	2	no	—
28	C₂²×C₅×Q₈	D₄	1	1	yes	characteristic
29	C₂×C₅×Q₁₆	—	2	4	no	—
30	C₂×C₅×Q₁₆	—	2	4	no	—
31	C₂×C₅×Q₁₆	—	4	4	no	—
32	C₂×C₅×Q₁₆	—	4	4	no	—
33	C₂²×C₅×C₈	—	2	1	no	—
34	C₂²×C₅×C₈	—	2	2	no	—
35	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
36	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
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	C₄×C₅×Q₈	—	2	1	no	—
42	C₄×C₅×Q₈	—	2	1	no	—
43	C₄×C₅×Q₈	D₄	1	2	yes	—
44	C₄×C₅×Q₈	C₂³	1	2	yes	—
45	C₂×C₅×C₄⋊C₄	—	2	1	no	—
46	C₂×C₅×C₄⋊C₄	—	2	2	no	—
47	C₂×C₅×C₄⋊C₄	—	2	2	no	—
48	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₅	—	2	1	no	—
50	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
51	C₅×#106	—	2	8	no	—
52	C₅×C₄⋊C₈	—	2	4	no	—
53	C₅×C₄⋊C₈	—	2	1	no	—
54	C₅×C₄⋊C₈	—	2	2	no	—
55	C₅×C₄⋊C₈	C₂³	1	2	yes	—
56	C₅×C₄⋊C₈	C₂³	1	4	yes	—
57	C₅×#103	—	2	8	no	—
58	C₅×#97	—	2	4	no	—
59	C₄×C₅×C₈	D₄	1	2	yes	—
60	C₄×C₅×C₈	C₂³	1	2	yes	—
61	C₂×#494	C₂×C₅	1	1	yes	characteristic
62	C₂×#475	C₂×C₅	1	2	yes	—
63	C₂×#446	C₂×C₅	1	1	yes	characteristic
64	#1690	C₂×C₅	1	8	yes	direct product factor
65	C₂×#334	C₂×C₅	1	1	yes	characteristic
66	C₂×#332	C₂×C₅	1	2	yes	—
67	C₂×C₅×Q₈	—	2	1	no	—
68	C₂×C₅×Q₈	—	2	4	no	—
69	C₂×C₅×Q₈	—	2	4	no	—
70	C₂×C₅×Q₈	—	4	2	no	—
71	C₂×C₅×Q₈	—	4	4	no	—
72	C₂×C₅×Q₈	—	4	4	no	—
73	C₂×C₅×Q₈	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₅	—	2	1	no	—
75	C₂²×C₄×C₅	—	2	1	no	—
76	C₂²×C₄×C₅	—	4	2	no	—
77	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	#41	1	1	yes	characteristic
80	C₅×Q₁₆	—	4	8	no	—
81	C₅×Q₁₆	—	4	8	no	—
82	C₂×C₅×C₈	—	2	2	no	—
83	C₂×C₅×C₈	—	2	2	no	—
84	C₂×C₅×C₈	—	2	4	no	—
85	C₂×C₅×C₈	—	4	2	no	—
86	C₂×C₅×C₈	—	4	4	no	—
87	C₂×C₅×C₈	SD₁₆	1	4	yes	—
88	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
89	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
90	C₅×C₄⋊C₄	—	2	2	no	—
91	C₅×C₄⋊C₄	—	2	1	no	—
92	C₅×C₄⋊C₄	—	2	2	no	—
93	C₅×C₄⋊C₄	—	2	4	no	—
94	C₅×C₄⋊C₄	—	2	4	no	—
95	C₅×C₄⋊C₄	—	4	2	no	—
96	C₅×C₄⋊C₄	—	4	2	no	—
97	C₅×C₄⋊C₄	C₂×D₄	1	2	yes	—
98	C₄²×C₅	—	2	2	no	—
99	C₄²×C₅	—	2	2	no	—
100	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₅	#41	1	1	yes	characteristic
103	C₄²×C₅	C₂⁴	1	1	yes	characteristic
104	C₂²×Q₁₆	—	2	1	no	—
105	C₂²×Q₁₆	—	2	1	no	—
106	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
107	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
108	#494	—	2	2	no	—
109	#494	C₂²×C₅	1	4	yes	—
110	#475	—	2	4	no	—
111	#475	C₂²×C₅	1	8	yes	—
112	#446	—	2	2	no	—
113	#446	C₂²×C₅	1	4	yes	—
114	C₂×#106	—	2	2	no	—
115	C₂×C₄⋊C₈	—	2	1	no	—
116	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
117	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
118	C₂×#103	—	2	2	no	—
119	C₂×#97	—	2	1	no	—
120	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
121	#334	C₂²×C₅	1	4	yes	—
122	#332	C₂²×C₅	1	8	yes	—
123	C₂³×C₅	#121	1	1	yes	characteristic
124	C₅×Q₈	—	2	4	no	—
125	C₅×Q₈	—	4	2	no	—
126	C₅×Q₈	—	4	4	no	—
127	C₅×Q₈	—	4	4	no	—
128	C₅×Q₈	—	8	4	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	2	no	—
134	C₂×C₄×C₅	—	4	4	no	—
135	C₂×C₄×C₅	—	8	4	no	—
136	C₂×C₄×C₅	#121	1	1	yes	characteristic
137	C₂×C₄×C₅	#121	1	1	yes	characteristic
138	C₂×C₄×C₅	#121	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
140	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
141	C₂×C₄×C₅	#137	1	4	yes	—
142	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
145	C₅×C₈	—	2	2	no	—
146	C₅×C₈	—	4	2	no	—
147	C₅×C₈	—	4	4	no	—
148	C₅×C₈	C₂×SD₁₆	1	4	yes	—
149	C₂²×Q₈	—	2	2	no	—
150	C₂²×Q₈	C₅×D₄	1	1	yes	characteristic
151	C₂×Q₁₆	—	2	4	no	—
152	C₂×Q₁₆	—	2	4	no	—
153	C₂×Q₁₆	—	4	4	no	—
154	C₂×Q₁₆	—	4	4	no	—
155	C₂²×C₈	—	2	1	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
158	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
159	C₄⋊Q₈	—	2	2	no	—
160	C₄⋊Q₈	—	2	2	no	—
161	C₄⋊Q₈	—	2	2	no	—
162	C₄⋊Q₈	C₂³×C₅	1	4	yes	—
163	C₄×Q₈	—	2	1	no	—
164	C₄×Q₈	—	2	1	no	—
165	C₄×Q₈	C₅×D₄	1	2	yes	—
166	C₄×Q₈	C₂³×C₅	1	2	yes	—
167	C₂×C₄⋊C₄	—	2	1	no	—
168	C₂×C₄⋊C₄	—	2	2	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
171	C₂×C₄²	—	2	1	no	—
172	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
173	#106	—	2	8	no	—
174	C₄⋊C₈	—	2	4	no	—
175	C₄⋊C₈	—	2	1	no	—
176	C₄⋊C₈	—	2	2	no	—
177	C₄⋊C₈	C₂³×C₅	1	2	yes	—
178	C₄⋊C₈	C₂³×C₅	1	4	yes	—
179	#103	—	2	8	no	—
180	#97	—	2	4	no	—
181	C₄×C₈	C₅×D₄	1	2	yes	—
182	C₄×C₈	C₂³×C₅	1	2	yes	—
183	C₂²×C₅	#460	1	2	yes	—
184	C₂²×C₅	#470	1	2	yes	—
185	C₂²×C₅	#473	1	2	yes	—
186	C₂²×C₅	C₂×#121	1	1	yes	characteristic
187	C₄×C₅	—	2	1	no	—
188	C₄×C₅	—	2	1	no	—
189	C₄×C₅	—	4	2	no	—
190	C₄×C₅	—	4	2	no	—
191	C₄×C₅	—	8	4	no	—
192	C₄×C₅	#460	1	2	yes	—
193	C₄×C₅	#470	1	2	yes	—
194	C₄×C₅	C₂²×SD₁₆	1	1	yes	characteristic
195	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
196	C₄×C₅	C₂×#137	1	2	yes	—
197	C₂×Q₈	—	2	1	no	—
198	C₂×Q₈	—	2	4	no	—
199	C₂×Q₈	—	2	4	no	—
200	C₂×Q₈	—	4	2	no	—
201	C₂×Q₈	—	4	4	no	—
202	C₂×Q₈	—	4	4	no	—
203	C₂×Q₈	C₂×C₅×D₄	1	2	yes	—
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	4	2	no	—
207	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
209	C₂²×C₄	C₅×#41	1	1	yes	characteristic
210	Q₁₆	—	4	8	no	—
211	Q₁₆	—	4	8	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	4	4	no	—
217	C₂×C₈	C₅×SD₁₆	1	4	yes	—
218	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
219	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	4	2	no	—
226	C₄⋊C₄	—	4	2	no	—
227	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
231	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
232	C₄²	C₅×#41	1	1	yes	characteristic
233	C₄²	C₂⁴×C₅	1	1	yes	characteristic
234	C₂×C₅	#1690	1	4	yes	direct product factor
235	C₂×C₅	C₂×#460	1	1	yes	characteristic
236	C₂×C₅	C₂×#470	1	1	yes	characteristic
237	C₂×C₅	C₂×#473	1	1	yes	characteristic
238	C₂³	C₅×#121	1	1	yes	characteristic
239	Q₈	—	2	4	no	—
240	Q₈	—	4	2	no	—
241	Q₈	—	4	4	no	—
242	Q₈	—	4	4	no	—
243	Q₈	—	8	4	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	8	4	no	—
251	C₂×C₄	C₅×#121	1	1	yes	characteristic
252	C₂×C₄	C₅×#121	1	1	yes	characteristic
253	C₂×C₄	C₅×#121	1	1	yes	characteristic
254	C₂×C₄	C₂×C₅×SD₁₆	1	2	yes	—
255	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
256	C₂×C₄	C₅×#137	1	4	yes	—
257	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
258	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
259	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
260	C₈	—	2	2	no	—
261	C₈	—	4	2	no	—
262	C₈	—	4	4	no	—
263	C₈	C₂×C₅×SD₁₆	1	4	yes	—
264	C₅	C₂×#1690	1	1	yes	characteristic, direct product factor, Sylow
265	C₂²	C₅×#460	1	2	yes	—
266	C₂²	C₅×#470	1	2	yes	—
267	C₂²	C₅×#473	1	2	yes	—
268	C₂²	C₂×C₅×#121	1	1	yes	characteristic
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	4	2	no	—
272	C₄	—	4	2	no	—
273	C₄	—	8	4	no	—
274	C₄	C₅×#460	1	2	yes	—
275	C₄	C₅×#470	1	2	yes	—
276	C₄	C₂²×C₅×SD₁₆	1	1	yes	characteristic
277	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
278	C₄	C₂×C₅×#137	1	2	yes	—
279	C₂	C₅×#1690	1	4	yes	direct product factor
280	C₂	C₂×C₅×#460	1	1	yes	characteristic
281	C₂	C₂×C₅×#470	1	1	yes	characteristic
282	C₂	C₂×C₅×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
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
4. C₅×#1690	C₂	direct (N × H)
26. C₂×#1690	C₅	direct (N × H)
64. #1690	C₂×C₅	direct (N × H)
234. C₂×C₅	#1690	direct (N × H)
264. C₅	C₂×#1690	direct (N × H)
279. C₂	C₅×#1690	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×#494	2
C₂×C₅×#475	2
C₂×C₅×#446	2
C₅×#1690	2
C₂×C₅×#334	2
C₂×C₅×#332	2
C₂×#1690	5

Subgroups — C2×C5×#1690 #11941865

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1690 #11941865