Subgroups — C₂×C₅×#1591 #11940605

Order 1280.

670 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#446	C₂	1	2	yes	maximal
2	C₂×C₄×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic
3	C₅×#1591	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅×#335	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅×Q₈⋊C₈	C₂	1	2	yes	maximal
6	C₂×C₅×C₈²	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₅×Q₁₆	C₄	1	1	yes	characteristic
8	C₂×C₄×C₅×Q₈	C₂²	1	2	yes	—
9	C₅×#446	—	2	4	no	—
10	C₅×#446	C₂²	1	8	yes	—
11	C₄×C₅×Q₁₆	C₄	1	4	yes	—
12	C₄×C₅×Q₁₆	C₂²	1	4	yes	—
13	C₂×C₅×#107	C₄	1	1	yes	characteristic
14	C₂×C₅×C₄⋊C₈	—	2	2	no	—
15	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₅×#103	C₄	1	2	yes	—
17	C₂×C₅×#97	—	2	2	no	—
18	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	C₅×#335	C₂²	1	4	yes	—
22	C₅×Q₈⋊C₈	C₂²	1	8	yes	—
23	C₅×C₈²	C₂²	1	4	yes	—
24	C₂×#1591	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂²×C₅×Q₈	C₂×C₄	1	2	yes	—
26	C₂×C₅×Q₁₆	—	2	4	no	—
27	C₂×C₅×Q₁₆	C₂×C₄	1	4	yes	—
28	C₂²×C₅×C₈	—	2	1	no	—
29	C₂²×C₅×C₈	—	2	2	no	—
30	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
33	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
34	C₄×C₅×Q₈	—	2	2	no	—
35	C₄×C₅×Q₈	—	2	2	no	—
36	C₄×C₅×Q₈	C₂×C₄	1	4	yes	—
37	C₄×C₅×Q₈	C₂³	1	4	yes	—
38	C₂×C₅×C₄⋊C₄	—	2	2	no	—
39	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₅	—	2	2	no	—
41	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₅×#107	C₂×C₄	1	4	yes	—
43	C₅×C₄⋊C₈	—	2	8	no	—
44	C₅×C₄⋊C₈	—	2	2	no	—
45	C₅×C₄⋊C₈	C₂³	1	4	yes	—
46	C₅×#103	C₂×C₄	1	8	yes	—
47	C₅×#97	—	2	8	no	—
48	C₄×C₅×C₈	—	2	1	no	—
49	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
50	C₄×C₅×C₈	D₄	1	2	yes	—
51	C₄×C₅×C₈	C₂³	1	2	yes	—
52	C₄×C₅×C₈	C₂³	1	2	yes	—
53	C₄×C₅×C₈	C₂³	1	2	yes	—
54	C₂×#446	C₂×C₅	1	2	yes	—
55	C₂×C₄×Q₁₆	C₂×C₅	1	1	yes	characteristic
56	#1591	C₂×C₅	1	8	yes	direct product factor
57	C₂×#335	C₂×C₅	1	1	yes	characteristic
58	C₂×Q₈⋊C₈	C₂×C₅	1	2	yes	—
59	C₂×C₈²	C₂×C₅	1	1	yes	characteristic
60	C₂×C₅×Q₈	—	2	2	no	—
61	C₂×C₅×Q₈	—	2	4	no	—
62	C₂×C₅×Q₈	—	2	4	no	—
63	C₂×C₅×Q₈	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₅	—	2	2	no	—
65	C₂²×C₄×C₅	—	2	2	no	—
66	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	#41	1	1	yes	characteristic
69	C₅×Q₁₆	—	2	8	no	—
70	C₂×C₅×C₈	—	2	4	no	—
71	C₂×C₅×C₈	—	2	2	no	—
72	C₂×C₅×C₈	—	2	2	no	—
73	C₂×C₅×C₈	—	2	2	no	—
74	C₂×C₅×C₈	—	2	4	no	—
75	C₂×C₅×C₈	—	4	4	no	—
76	C₂×C₅×C₈	M₁₆	1	4	yes	—
77	C₂×C₅×C₈	Q₁₆	1	4	yes	complemented
78	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
79	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
80	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
81	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
82	C₅×C₄⋊C₄	—	2	4	no	—
83	C₅×C₄⋊C₄	—	2	2	no	—
84	C₅×C₄⋊C₄	—	2	4	no	—
85	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
86	C₄²×C₅	—	2	4	no	—
87	C₄²×C₅	—	2	4	no	—
88	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₅	#41	1	1	yes	characteristic
91	C₄²×C₅	C₂⁴	1	1	yes	characteristic
92	C₂²×Q₁₆	C₄×C₅	1	1	yes	characteristic
93	C₂×C₄×Q₈	C₂²×C₅	1	2	yes	—
94	#446	—	2	4	no	—
95	#446	C₂²×C₅	1	8	yes	—
96	C₄×Q₁₆	C₄×C₅	1	4	yes	—
97	C₄×Q₁₆	C₂²×C₅	1	4	yes	—
98	C₂×#107	C₄×C₅	1	1	yes	characteristic
99	C₂×C₄⋊C₈	—	2	2	no	—
100	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
101	C₂×#103	C₄×C₅	1	2	yes	—
102	C₂×#97	—	2	2	no	—
103	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
104	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
105	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
106	#335	C₂²×C₅	1	4	yes	—
107	Q₈⋊C₈	C₂²×C₅	1	8	yes	—
108	C₈²	C₂²×C₅	1	4	yes	—
109	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
110	C₅×Q₈	—	2	4	no	—
111	C₅×Q₈	—	2	4	no	—
112	C₅×Q₈	—	4	4	no	—
113	C₂×C₄×C₅	—	2	2	no	—
114	C₂×C₄×C₅	—	2	4	no	—
115	C₂×C₄×C₅	—	2	4	no	—
116	C₂×C₄×C₅	—	4	4	no	—
117	C₂×C₄×C₅	—	4	4	no	—
118	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
122	C₂×C₄×C₅	#131	1	2	yes	—
123	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
124	C₂×C₄×C₅	#135	1	2	yes	—
125	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
128	C₅×C₈	—	2	4	no	—
129	C₅×C₈	—	2	2	no	—
130	C₅×C₈	—	2	2	no	—
131	C₅×C₈	—	4	4	no	—
132	C₅×C₈	C₂×M₁₆	1	4	yes	—
133	C₅×C₈	C₂×Q₁₆	1	4	yes	complemented
134	C₂²×Q₈	C₂×C₄×C₅	1	2	yes	—
135	C₂×Q₁₆	—	2	4	no	—
136	C₂×Q₁₆	C₂×C₄×C₅	1	4	yes	—
137	C₂²×C₈	—	2	1	no	—
138	C₂²×C₈	—	2	2	no	—
139	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
140	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
141	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
142	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
143	C₄×Q₈	—	2	2	no	—
144	C₄×Q₈	—	2	2	no	—
145	C₄×Q₈	C₂×C₄×C₅	1	4	yes	—
146	C₄×Q₈	C₂³×C₅	1	4	yes	—
147	C₂×C₄⋊C₄	—	2	2	no	—
148	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
149	C₂×C₄²	—	2	2	no	—
150	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
151	#107	C₂×C₄×C₅	1	4	yes	—
152	C₄⋊C₈	—	2	8	no	—
153	C₄⋊C₈	—	2	2	no	—
154	C₄⋊C₈	C₂³×C₅	1	4	yes	—
155	#103	C₂×C₄×C₅	1	8	yes	—
156	#97	—	2	8	no	—
157	C₄×C₈	—	2	1	no	—
158	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
159	C₄×C₈	C₅×D₄	1	2	yes	—
160	C₄×C₈	C₂³×C₅	1	2	yes	—
161	C₄×C₈	C₂³×C₅	1	2	yes	—
162	C₄×C₈	C₂³×C₅	1	2	yes	—
163	C₂²×C₅	#436	1	2	yes	—
164	C₂²×C₅	C₄×Q₁₆	1	2	yes	—
165	C₂²×C₅	#443	1	2	yes	—
166	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
167	C₄×C₅	—	2	1	no	—
168	C₄×C₅	—	2	1	no	—
169	C₄×C₅	—	4	4	no	—
170	C₄×C₅	—	4	4	no	—
171	C₄×C₅	#436	1	2	yes	—
172	C₄×C₅	C₄×Q₁₆	1	2	yes	—
173	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
174	C₄×C₅	C₂×#131	1	1	yes	characteristic
175	C₄×C₅	C₂²×Q₁₆	1	1	yes	characteristic
176	C₄×C₅	C₂×#135	1	1	yes	characteristic
177	C₂×Q₈	—	2	2	no	—
178	C₂×Q₈	—	2	4	no	—
179	C₂×Q₈	—	2	4	no	—
180	C₂×Q₈	C₂²×C₄×C₅	1	4	yes	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
185	C₂²×C₄	C₅×#41	1	1	yes	characteristic
186	Q₁₆	—	2	8	no	—
187	C₂×C₈	—	2	4	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	4	4	no	—
193	C₂×C₈	C₅×M₁₆	1	4	yes	—
194	C₂×C₈	C₅×Q₁₆	1	4	yes	complemented
195	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
196	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
197	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
198	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	2	4	no	—
202	C₄⋊C₄	C₂²×C₄×C₅	1	4	yes	—
203	C₄²	—	2	4	no	—
204	C₄²	—	2	4	no	—
205	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
206	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
207	C₄²	C₅×#41	1	1	yes	characteristic
208	C₄²	C₂⁴×C₅	1	1	yes	characteristic
209	C₂×C₅	#1591	1	4	yes	direct product factor
210	C₂×C₅	C₂×#436	1	1	yes	characteristic
211	C₂×C₅	C₂×C₄×Q₁₆	1	1	yes	characteristic
212	C₂×C₅	C₂×#443	1	1	yes	characteristic
213	C₂³	C₄×C₅×D₄	1	1	yes	characteristic
214	Q₈	—	2	4	no	—
215	Q₈	—	2	4	no	—
216	Q₈	—	4	4	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
223	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
224	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
225	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
226	C₂×C₄	C₅×#131	1	2	yes	—
227	C₂×C₄	C₂×C₅×Q₁₆	1	2	yes	—
228	C₂×C₄	C₅×#135	1	2	yes	—
229	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
230	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
231	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
232	C₈	—	2	4	no	—
233	C₈	—	2	2	no	—
234	C₈	—	2	2	no	—
235	C₈	—	4	4	no	—
236	C₈	C₂×C₅×M₁₆	1	4	yes	—
237	C₈	C₂×C₅×Q₁₆	1	4	yes	complemented
238	C₅	C₂×#1591	1	1	yes	characteristic, direct product factor, Sylow
239	C₂²	C₅×#436	1	2	yes	—
240	C₂²	C₄×C₅×Q₁₆	1	2	yes	—
241	C₂²	C₅×#443	1	2	yes	—
242	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	4	4	no	—
246	C₄	—	4	4	no	—
247	C₄	C₅×#436	1	2	yes	—
248	C₄	C₄×C₅×Q₁₆	1	2	yes	—
249	C₄	C₂²×C₅×M₁₆	1	1	yes	characteristic
250	C₄	C₂×C₅×#131	1	1	yes	characteristic
251	C₄	C₂²×C₅×Q₁₆	1	1	yes	characteristic
252	C₄	C₂×C₅×#135	1	1	yes	characteristic
253	C₂	C₅×#1591	1	4	yes	direct product factor
254	C₂	C₂×C₅×#436	1	1	yes	characteristic
255	C₂	C₂×C₄×C₅×Q₁₆	1	1	yes	characteristic
256	C₂	C₂×C₅×#443	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₄²	C₂⁴×C₅
socle, prime radical	C₂³×C₅	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1591	C₂	direct (N × H)
24. C₂×#1591	C₅	direct (N × H)
56. #1591	C₂×C₅	direct (N × H)
77. C₂×C₅×C₈	Q₁₆	semidirect
133. C₅×C₈	C₂×Q₁₆	semidirect
194. C₂×C₈	C₅×Q₁₆	semidirect
209. C₂×C₅	#1591	direct (N × H)
237. C₈	C₂×C₅×Q₁₆	semidirect
238. C₅	C₂×#1591	direct (N × H)
253. C₂	C₅×#1591	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×#446	2
C₂×C₄×C₅×Q₁₆	2
C₅×#1591	2
C₂×C₅×#335	2
C₂×C₅×Q₈⋊C₈	2
C₂×C₅×C₈²	2
C₂×#1591	5

Subgroups — C2×C5×#1591 #11940605

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1591 #11940605