Subgroups — C₂×C₅×#1987 #11944028

Order 1280.

1510 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₅×#1987	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅×#340	C₂	1	4	yes	maximal
6	C₂³×C₅×Q₈	—	2	1	no	—
7	C₂³×C₅×C₈	C₂²	1	2	yes	—
8	C₂×C₅×C₄⋊Q₈	—	2	2	no	—
9	C₂×C₅×C₄⋊Q₈	—	2	4	no	—
10	C₂×C₅×C₄⋊Q₈	C₄	1	8	yes	complemented
11	C₂×C₅×C₄⋊Q₈	C₂²	1	4	yes	—
12	C₂²×C₅×C₄⋊C₄	—	2	1	no	—
13	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂²×C₄²×C₅	—	2	1	no	—
16	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
17	C₂×C₅×C₄⋊C₈	—	2	4	no	—
18	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
19	C₄×C₅×C₄⋊C₄	—	2	2	no	—
20	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
21	C₂×C₅×#95	—	2	1	no	—
22	C₅×#340	C₂²	1	16	yes	—
23	C₂×#1987	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂²×C₅×Q₈	—	2	8	no	—
25	C₂²×C₅×Q₈	—	2	4	no	—
26	C₂²×C₅×Q₈	—	4	8	no	—
27	C₂³×C₄×C₅	—	2	2	no	—
28	C₂³×C₄×C₅	—	2	2	no	—
29	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂²×C₅×C₈	—	2	4	no	—
33	C₂²×C₅×C₈	—	2	8	no	—
34	C₂²×C₅×C₈	C₂³	1	4	yes	—
35	C₅×C₄⋊Q₈	—	2	16	no	—
36	C₅×C₄⋊Q₈	—	2	8	no	—
37	C₅×C₄⋊Q₈	C₂×C₄	1	16	yes	complemented
38	C₂×C₅×C₄⋊C₄	—	2	8	no	—
39	C₂×C₅×C₄⋊C₄	—	2	2	no	—
40	C₂×C₅×C₄⋊C₄	—	2	4	no	—
41	C₂×C₅×C₄⋊C₄	—	2	2	no	—
42	C₂×C₅×C₄⋊C₄	—	2	4	no	—
43	C₂×C₅×C₄⋊C₄	—	2	4	no	—
44	C₂×C₅×C₄⋊C₄	C₂×C₄	1	8	yes	—
45	C₂×C₅×C₄⋊C₄	Q₈	1	8	yes	complemented
46	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
47	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
48	C₂×C₄²×C₅	—	2	4	no	—
49	C₂×C₄²×C₅	—	4	4	no	—
50	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₅	D₄	1	2	yes	—
52	C₂×C₄²×C₅	D₄	1	4	yes	—
53	C₂×C₄²×C₅	C₂³	1	2	yes	—
54	C₅×C₄⋊C₈	—	2	8	no	—
55	C₅×#95	—	2	4	no	—
56	C₂²×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
57	C₂²×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic
58	C₂×C₄×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
59	#1987	C₂×C₅	1	8	yes	direct product factor
60	C₂×#340	C₂×C₅	1	4	yes	—
61	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂×C₅×Q₈	—	2	16	no	—
63	C₂×C₅×Q₈	—	4	16	no	—
64	C₂×C₅×Q₈	—	4	32	no	—
65	C₂²×C₄×C₅	—	2	4	no	—
66	C₂²×C₄×C₅	—	2	8	no	—
67	C₂²×C₄×C₅	—	2	8	no	—
68	C₂²×C₄×C₅	—	2	4	no	—
69	C₂²×C₄×C₅	—	2	4	no	—
70	C₂²×C₄×C₅	—	4	4	no	—
71	C₂²×C₄×C₅	—	4	4	no	—
72	C₂²×C₄×C₅	—	4	8	no	—
73	C₂²×C₄×C₅	C₂²⋊C₄	1	4	yes	—
74	C₂²×C₄×C₅	C₂²⋊C₄	1	4	yes	—
75	C₂²×C₄×C₅	SD₁₆	1	4	yes	—
76	C₂²×C₄×C₅	Q₁₆	1	4	yes	—
77	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
83	C₂²×C₄×C₅	#41	1	4	yes	—
84	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
85	C₂×C₅×C₈	—	2	4	no	—
86	C₂×C₅×C₈	—	2	8	no	—
87	C₂×C₅×C₈	—	4	8	no	—
88	C₅×C₄⋊C₄	—	2	16	no	—
89	C₅×C₄⋊C₄	—	2	8	no	—
90	C₅×C₄⋊C₄	—	2	8	no	—
91	C₅×C₄⋊C₄	—	2	4	no	—
92	C₅×C₄⋊C₄	—	2	4	no	—
93	C₅×C₄⋊C₄	C₂²×C₄	1	8	yes	—
94	C₅×C₄⋊C₄	C₂×Q₈	1	8	yes	complemented
95	C₄²×C₅	—	4	8	no	—
96	C₄²×C₅	C₂²⋊C₄	1	8	yes	—
97	C₄²×C₅	C₂²×C₄	1	4	yes	—
98	C₄²×C₅	C₂×D₄	1	4	yes	—
99	C₂³×Q₈	—	2	1	no	—
100	C₂³×C₈	C₂²×C₅	1	2	yes	—
101	C₂×C₄⋊Q₈	—	2	2	no	—
102	C₂×C₄⋊Q₈	—	2	4	no	—
103	C₂×C₄⋊Q₈	C₄×C₅	1	8	yes	complemented
104	C₂×C₄⋊Q₈	C₂²×C₅	1	4	yes	—
105	C₂²×C₄⋊C₄	—	2	1	no	—
106	C₂²×C₄⋊C₄	C₂²×C₅	1	2	yes	—
107	C₂²×C₄⋊C₄	C₂²×C₅	1	2	yes	—
108	C₂²×C₄²	—	2	1	no	—
109	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
110	C₂×C₄⋊C₈	—	2	4	no	—
111	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
112	C₄×C₄⋊C₄	—	2	2	no	—
113	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
114	C₂×#95	—	2	1	no	—
115	#340	C₂²×C₅	1	16	yes	—
116	C₂³×C₅	C₂×C₂²⋊C₄	1	2	yes	—
117	C₂³×C₅	C₂²⋊Q₈	1	4	yes	—
118	C₂³×C₅	C₂×SD₁₆	1	2	yes	—
119	C₂³×C₅	C₂×Q₁₆	1	2	yes	—
120	C₂³×C₅	AGL(1,ℤ/8)	1	4	yes	—
121	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
122	C₅×Q₈	—	4	32	no	—
123	C₂×C₄×C₅	—	2	4	no	—
124	C₂×C₄×C₅	—	2	8	no	—
125	C₂×C₄×C₅	—	2	8	no	—
126	C₂×C₄×C₅	—	2	8	no	—
127	C₂×C₄×C₅	—	4	4	no	—
128	C₂×C₄×C₅	—	4	4	no	—
129	C₂×C₄×C₅	—	4	8	no	—
130	C₂×C₄×C₅	—	4	16	no	—
131	C₂×C₄×C₅	—	4	16	no	—
132	C₂×C₄×C₅	#103	1	16	yes	—
133	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	4	yes	—
134	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
135	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	4	yes	—
136	C₂×C₄×C₅	C₄×Q₈	1	8	yes	—
137	C₂×C₄×C₅	C₂²⋊Q₈	1	4	yes	—
138	C₂×C₄×C₅	#124	1	4	yes	—
139	C₂×C₄×C₅	C₄⋊Q₈	1	4	yes	—
140	C₂×C₄×C₅	C₂×SD₁₆	1	4	yes	—
141	C₂×C₄×C₅	C₂×Q₁₆	1	4	yes	—
142	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂²×Q₈	1	2	yes	—
145	C₂×C₄×C₅	C₂×#41	1	2	yes	—
146	C₅×C₈	—	4	4	no	—
147	C₂²×Q₈	—	2	8	no	—
148	C₂²×Q₈	—	2	4	no	—
149	C₂²×Q₈	—	4	8	no	—
150	C₂³×C₄	—	2	2	no	—
151	C₂³×C₄	—	2	2	no	—
152	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
153	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
154	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
155	C₂²×C₈	—	2	4	no	—
156	C₂²×C₈	—	2	8	no	—
157	C₂²×C₈	C₂³×C₅	1	4	yes	—
158	C₄⋊Q₈	—	2	16	no	—
159	C₄⋊Q₈	—	2	8	no	—
160	C₄⋊Q₈	C₂×C₄×C₅	1	16	yes	complemented
161	C₂×C₄⋊C₄	—	2	8	no	—
162	C₂×C₄⋊C₄	—	2	4	no	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	2	4	no	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	—	2	4	no	—
167	C₂×C₄⋊C₄	C₂×C₄×C₅	1	8	yes	—
168	C₂×C₄⋊C₄	C₅×Q₈	1	8	yes	complemented
169	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
170	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
171	C₂×C₄²	—	2	4	no	—
172	C₂×C₄²	—	4	4	no	—
173	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
174	C₂×C₄²	C₅×D₄	1	2	yes	—
175	C₂×C₄²	C₅×D₄	1	4	yes	—
176	C₂×C₄²	C₂³×C₅	1	2	yes	—
177	C₄⋊C₈	—	2	8	no	—
178	#95	—	2	4	no	—
179	C₂²×C₅	#391	1	4	yes	—
180	C₂²×C₅	C₂×#103	1	4	yes	—
181	C₂²×C₅	#418	1	4	yes	—
182	C₂²×C₅	#476	1	8	yes	—
183	C₂²×C₅	#477	1	8	yes	—
184	C₂²×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
185	C₂²×C₅	C₂×C₂²⋊Q₈	1	2	yes	—
186	C₂²×C₅	C₂²×SD₁₆	1	1	yes	characteristic
187	C₂²×C₅	C₂²×Q₁₆	1	1	yes	characteristic
188	C₂²×C₅	C₂×AGL(1,ℤ/8)	1	2	yes	—
189	C₄×C₅	—	2	4	no	—
190	C₄×C₅	—	4	8	no	—
191	C₄×C₅	—	4	8	no	—
192	C₄×C₅	#391	1	4	yes	—
193	C₄×C₅	C₂×#103	1	8	yes	—
194	C₄×C₅	C₂×C₄×Q₈	1	2	yes	—
195	C₄×C₅	C₂×#124	1	1	yes	characteristic
196	C₄×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
197	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
198	C₂×Q₈	—	2	16	no	—
199	C₂×Q₈	—	4	32	no	—
200	C₂×Q₈	—	4	16	no	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	2	8	no	—
203	C₂²×C₄	—	2	8	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	4	4	no	—
207	C₂²×C₄	—	4	8	no	—
208	C₂²×C₄	—	4	4	no	—
209	C₂²×C₄	C₅×C₂²⋊C₄	1	4	yes	—
210	C₂²×C₄	C₅×C₂²⋊C₄	1	4	yes	—
211	C₂²×C₄	C₅×SD₁₆	1	4	yes	—
212	C₂²×C₄	C₅×Q₁₆	1	4	yes	—
213	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
214	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
215	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
217	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
218	C₂²×C₄	C₂×C₅×Q₈	1	4	yes	—
219	C₂²×C₄	C₅×#41	1	4	yes	—
220	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	8	no	—
223	C₂×C₈	—	4	8	no	—
224	C₄⋊C₄	—	2	16	no	—
225	C₄⋊C₄	—	2	8	no	—
226	C₄⋊C₄	—	2	8	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	C₂²×C₄×C₅	1	8	yes	—
230	C₄⋊C₄	C₂×C₅×Q₈	1	8	yes	complemented
231	C₄²	—	4	8	no	—
232	C₄²	C₅×C₂²⋊C₄	1	8	yes	—
233	C₄²	C₂²×C₄×C₅	1	4	yes	—
234	C₄²	C₂×C₅×D₄	1	4	yes	—
235	C₂×C₅	#1987	1	8	yes	direct product factor
236	C₂×C₅	C₂×#391	1	1	yes	characteristic
237	C₂×C₅	C₂²×#103	1	1	yes	characteristic
238	C₂×C₅	C₂×#418	1	1	yes	characteristic
239	C₂×C₅	C₂×#476	1	2	yes	—
240	C₂×C₅	C₂×#477	1	2	yes	—
241	C₂³	C₂×C₅×C₂²⋊C₄	1	2	yes	—
242	C₂³	C₅×C₂²⋊Q₈	1	4	yes	—
243	C₂³	C₂×C₅×SD₁₆	1	2	yes	—
244	C₂³	C₂×C₅×Q₁₆	1	2	yes	—
245	C₂³	C₅×AGL(1,ℤ/8)	1	4	yes	—
246	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
247	Q₈	—	4	32	no	—
248	C₂×C₄	—	2	8	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	2	8	no	—
252	C₂×C₄	—	4	16	no	—
253	C₂×C₄	—	4	16	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	8	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	C₅×#103	1	16	yes	—
258	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	4	yes	—
259	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
260	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	4	yes	—
261	C₂×C₄	C₄×C₅×Q₈	1	8	yes	—
262	C₂×C₄	C₅×C₂²⋊Q₈	1	4	yes	—
263	C₂×C₄	C₅×#124	1	4	yes	—
264	C₂×C₄	C₅×C₄⋊Q₈	1	4	yes	—
265	C₂×C₄	C₂×C₅×SD₁₆	1	4	yes	—
266	C₂×C₄	C₂×C₅×Q₁₆	1	4	yes	—
267	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
268	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
269	C₂×C₄	C₂²×C₅×Q₈	1	2	yes	—
270	C₂×C₄	C₂×C₅×#41	1	2	yes	—
271	C₈	—	4	4	no	—
272	C₅	C₂×#1987	1	1	yes	characteristic, direct product factor, Sylow
273	C₂²	C₅×#391	1	4	yes	—
274	C₂²	C₂×C₅×#103	1	4	yes	—
275	C₂²	C₅×#418	1	4	yes	—
276	C₂²	C₅×#476	1	8	yes	—
277	C₂²	C₅×#477	1	8	yes	—
278	C₂²	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
279	C₂²	C₂×C₅×C₂²⋊Q₈	1	2	yes	—
280	C₂²	C₂²×C₅×SD₁₆	1	1	yes	characteristic
281	C₂²	C₂²×C₅×Q₁₆	1	1	yes	characteristic
282	C₂²	C₂×C₅×AGL(1,ℤ/8)	1	2	yes	—
283	C₄	—	2	4	no	—
284	C₄	—	4	8	no	—
285	C₄	—	4	8	no	—
286	C₄	C₅×#391	1	4	yes	—
287	C₄	C₂×C₅×#103	1	8	yes	—
288	C₄	C₂×C₄×C₅×Q₈	1	2	yes	—
289	C₄	C₂×C₅×#124	1	1	yes	characteristic
290	C₄	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
291	C₂	C₅×#1987	1	8	yes	direct product factor
292	C₂	C₂×C₅×#391	1	1	yes	characteristic
293	C₂	C₂²×C₅×#103	1	1	yes	characteristic
294	C₂	C₂×C₅×#418	1	1	yes	characteristic
295	C₂	C₂×C₅×#476	1	2	yes	—
296	C₂	C₂×C₅×#477	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₅×#1987	C₂	direct (N × H)
10. C₂×C₅×C₄⋊Q₈	C₄	semidirect
23. C₂×#1987	C₅	direct (N × H)
37. C₅×C₄⋊Q₈	C₂×C₄	semidirect
45. C₂×C₅×C₄⋊C₄	Q₈	semidirect
59. #1987	C₂×C₅	direct (N × H)
94. C₅×C₄⋊C₄	C₂×Q₈	semidirect
103. C₂×C₄⋊Q₈	C₄×C₅	semidirect
160. C₄⋊Q₈	C₂×C₄×C₅	semidirect
168. C₂×C₄⋊C₄	C₅×Q₈	semidirect
230. C₄⋊C₄	C₂×C₅×Q₈	semidirect
235. C₂×C₅	#1987	direct (N × H)
272. C₅	C₂×#1987	direct (N × H)
291. C₂	C₅×#1987	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₅×C₄⋊Q₈	2
C₂²×C₅×C₄⋊C₈	2
C₂×C₄×C₅×C₄⋊C₄	2
C₅×#1987	2
C₂×C₅×#340	2
C₂×#1987	5

Subgroups — C2×C5×#1987 #11944028

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1987 #11944028