Subgroups — C₂×C₅×#2113 #11956052

Order 1280.

1190 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#131	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2113	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅×C₂²⋊C₁₆	C₂	1	4	yes	maximal, complemented
6	C₂²×C₅×#41	C₄	1	1	yes	characteristic
7	C₂×C₅×#131	—	2	4	no	—
8	C₂×C₅×#131	—	2	4	no	—
9	C₂×C₅×#131	C₄	1	4	yes	—
10	C₂×C₅×#131	C₂²	1	4	yes	—
11	C₂²×C₅×M₁₆	C₄	1	1	yes	characteristic
12	C₂²×C₅×M₁₆	C₄	1	2	yes	—
13	C₂³×C₅×C₈	C₂²	1	2	yes	—
14	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₅×M₃₂	—	2	4	no	—
16	C₂×C₅×M₃₂	C₂²	1	4	yes	complemented
17	C₂²×C₅×C₁₆	—	2	4	no	—
18	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
19	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
20	C₂²×C₅×C₁₆	C₂²	1	4	yes	complemented
21	C₅×C₂²⋊C₁₆	C₂²	1	16	yes	complemented
22	C₂×#2113	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂×C₅×#41	—	2	4	no	—
24	C₂×C₅×#41	—	2	4	no	—
25	C₂×C₅×#41	C₈	1	4	yes	—
26	C₂×C₅×#41	C₂×C₄	1	4	yes	—
27	C₂²×C₅×Q₈	C₈	1	1	yes	characteristic
28	C₂²×C₅×D₄	C₈	1	2	yes	—
29	C₂²×C₅×D₄	C₈	1	1	yes	characteristic
30	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
31	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
32	C₅×#131	—	2	16	no	—
33	C₅×#131	—	2	8	no	—
34	C₅×#131	—	2	8	no	—
35	C₂×C₅×M₁₆	—	2	4	no	—
36	C₂×C₅×M₁₆	—	2	8	no	—
37	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
38	C₂×C₅×M₁₆	C₂×C₄	1	8	yes	—
39	C₂²×C₅×C₈	—	2	4	no	—
40	C₂²×C₅×C₈	—	2	4	no	—
41	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
42	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
43	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
45	C₂²×C₅×C₈	D₄	1	4	yes	—
46	C₂²×C₅×C₈	D₄	1	4	yes	—
47	C₂²×C₅×C₈	C₂³	1	4	yes	—
48	C₂²×C₅×C₈	C₂³	1	2	yes	—
49	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
50	C₅×M₃₂	—	2	8	no	—
51	C₂×C₅×C₁₆	—	2	4	no	—
52	C₂×C₅×C₁₆	—	2	8	no	—
53	C₂×C₅×C₁₆	—	2	4	no	—
54	C₂×C₅×C₁₆	C₂³	1	4	yes	—
55	C₂×C₅×C₁₆	C₂³	1	4	yes	—
56	C₂²×#131	C₂×C₅	1	1	yes	characteristic
57	C₂²×M₃₂	C₂×C₅	1	1	yes	characteristic, complemented
58	C₂³×C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
59	#2113	C₂×C₅	1	8	yes	direct product factor
60	C₂×C₂²⋊C₁₆	C₂×C₅	1	4	yes	complemented
61	C₂⁴×C₅	C₂×C₈	1	2	yes	—
62	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
63	C₅×#41	—	2	16	no	—
64	C₅×#41	—	2	8	no	—
65	C₅×#41	—	2	8	no	—
66	C₂×C₅×Q₈	—	2	4	no	—
67	C₂×C₅×Q₈	C₂×C₈	1	4	yes	—
68	C₂×C₅×D₄	—	2	8	no	—
69	C₂×C₅×D₄	—	2	4	no	—
70	C₂×C₅×D₄	C₂×C₈	1	8	yes	—
71	C₂×C₅×D₄	C₂×C₈	1	4	yes	—
72	C₂²×C₄×C₅	—	2	4	no	—
73	C₂²×C₄×C₅	—	2	4	no	—
74	C₂²×C₄×C₅	C₂²⋊C₄	1	4	yes	—
75	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
76	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
77	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
79	C₂²×C₄×C₅	M₁₆	1	4	yes	—
80	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
81	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
83	C₅×M₁₆	—	2	8	no	—
84	C₅×M₁₆	—	2	16	no	—
85	C₂×C₅×C₈	—	2	4	no	—
86	C₂×C₅×C₈	—	2	8	no	—
87	C₂×C₅×C₈	—	2	4	no	—
88	C₂×C₅×C₈	—	2	4	no	—
89	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
90	C₂×C₅×C₈	C₂²⋊C₄	1	8	yes	—
91	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
92	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
94	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
95	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
96	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
97	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
98	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
99	C₅×C₁₆	—	2	4	no	—
100	C₅×C₁₆	—	2	4	no	—
101	C₂²×#41	C₄×C₅	1	1	yes	characteristic
102	C₂×#131	—	2	4	no	—
103	C₂×#131	—	2	4	no	—
104	C₂×#131	C₄×C₅	1	4	yes	—
105	C₂×#131	C₂²×C₅	1	4	yes	—
106	C₂²×M₁₆	C₄×C₅	1	1	yes	characteristic
107	C₂²×M₁₆	C₄×C₅	1	2	yes	—
108	C₂³×C₈	C₂²×C₅	1	2	yes	—
109	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
110	C₂×M₃₂	—	2	4	no	—
111	C₂×M₃₂	C₂²×C₅	1	4	yes	complemented
112	C₂²×C₁₆	—	2	4	no	—
113	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
114	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
115	C₂²×C₁₆	C₂²×C₅	1	4	yes	complemented
116	C₂²⋊C₁₆	C₂²×C₅	1	16	yes	complemented
117	C₂³×C₅	—	2	4	no	—
118	C₂³×C₅	—	2	4	no	—
119	C₂³×C₅	—	2	8	no	—
120	C₂³×C₅	C₂²⋊C₈	1	4	yes	—
121	C₂³×C₅	C₂²×C₈	1	4	yes	—
122	C₂³×C₅	C₂²×C₈	1	2	yes	—
123	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
124	C₅×Q₈	—	2	8	no	—
125	C₅×D₄	—	2	16	no	—
126	C₅×D₄	—	2	8	no	—
127	C₂×C₄×C₅	—	2	4	no	—
128	C₂×C₄×C₅	—	2	4	no	—
129	C₂×C₄×C₅	—	2	4	no	—
130	C₂×C₄×C₅	—	2	8	no	—
131	C₂×C₄×C₅	C₂²⋊C₈	1	4	yes	—
132	C₂×C₄×C₅	C₂²⋊C₈	1	8	yes	—
133	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
134	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	4	yes	—
135	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
136	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
138	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
139	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
140	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
141	C₅×C₈	—	2	4	no	—
142	C₅×C₈	C₂×C₂²⋊C₄	1	2	yes	—
143	C₅×C₈	C₂×C₂²⋊C₄	1	4	yes	—
144	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
145	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
146	C₂×#41	—	2	4	no	—
147	C₂×#41	—	2	4	no	—
148	C₂×#41	C₅×C₈	1	4	yes	—
149	C₂×#41	C₂×C₄×C₅	1	4	yes	—
150	C₂²×Q₈	C₅×C₈	1	1	yes	characteristic
151	C₂²×D₄	C₅×C₈	1	2	yes	—
152	C₂²×D₄	C₅×C₈	1	1	yes	characteristic
153	C₂³×C₄	C₂×C₄×C₅	1	2	yes	—
154	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
155	#131	—	2	16	no	—
156	#131	—	2	8	no	—
157	#131	—	2	8	no	—
158	C₂×M₁₆	—	2	4	no	—
159	C₂×M₁₆	—	2	8	no	—
160	C₂×M₁₆	C₂×C₄×C₅	1	4	yes	—
161	C₂×M₁₆	C₂×C₄×C₅	1	8	yes	—
162	C₂²×C₈	—	2	4	no	—
163	C₂²×C₈	—	2	4	no	—
164	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
165	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
166	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
167	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
168	C₂²×C₈	C₅×D₄	1	4	yes	—
169	C₂²×C₈	C₅×D₄	1	4	yes	—
170	C₂²×C₈	C₂³×C₅	1	4	yes	—
171	C₂²×C₈	C₂³×C₅	1	2	yes	—
172	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
173	M₃₂	—	2	8	no	—
174	C₂×C₁₆	—	2	4	no	—
175	C₂×C₁₆	—	2	8	no	—
176	C₂×C₁₆	—	2	4	no	—
177	C₂×C₁₆	C₂³×C₅	1	4	yes	—
178	C₂×C₁₆	C₂³×C₅	1	4	yes	—
179	C₂²×C₅	—	2	8	no	—
180	C₂²×C₅	—	2	4	no	—
181	C₂²×C₅	—	2	4	no	—
182	C₂²×C₅	—	2	8	no	—
183	C₂²×C₅	—	4	8	no	—
184	C₂²×C₅	C₂×C₂²⋊C₈	1	4	yes	—
185	C₂²×C₅	C₂×C₂²⋊C₈	1	2	yes	—
186	C₂²×C₅	#504	1	4	yes	—
187	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
188	C₄×C₅	—	2	4	no	—
189	C₄×C₅	C₂×C₂²⋊C₈	1	2	yes	—
190	C₄×C₅	C₂×C₂²⋊C₈	1	4	yes	—
191	C₄×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
192	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
193	C₂⁴	C₂×C₅×C₈	1	2	yes	—
194	C₂⁴	C₂×C₅×C₈	1	1	yes	characteristic
195	#41	—	2	8	no	—
196	#41	—	2	8	no	—
197	#41	—	2	16	no	—
198	C₂×Q₈	—	2	4	no	—
199	C₂×Q₈	C₂×C₅×C₈	1	4	yes	—
200	C₂×D₄	—	2	8	no	—
201	C₂×D₄	—	2	4	no	—
202	C₂×D₄	C₂×C₅×C₈	1	8	yes	—
203	C₂×D₄	C₂×C₅×C₈	1	4	yes	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	C₅×C₂²⋊C₄	1	4	yes	—
207	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
208	C₂²×C₄	C₂×C₅×C₈	1	4	yes	—
209	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
211	C₂²×C₄	C₅×M₁₆	1	4	yes	—
212	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
213	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
214	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
215	M₁₆	—	2	8	no	—
216	M₁₆	—	2	16	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	8	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	C₅×C₂²⋊C₄	1	4	yes	—
222	C₂×C₈	C₅×C₂²⋊C₄	1	8	yes	—
223	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
224	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
225	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
226	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
227	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
228	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
229	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
230	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
231	C₁₆	—	2	4	no	—
232	C₁₆	—	2	4	no	—
233	C₂×C₅	—	2	4	no	—
234	C₂×C₅	—	4	4	no	—
235	C₂×C₅	#2113	1	4	yes	direct product factor
236	C₂×C₅	C₂²×C₂²⋊C₈	1	1	yes	characteristic
237	C₂×C₅	C₂×#504	1	2	yes	—
238	C₂³	—	2	4	no	—
239	C₂³	—	2	4	no	—
240	C₂³	—	2	8	no	—
241	C₂³	C₅×C₂²⋊C₈	1	4	yes	—
242	C₂³	C₂²×C₅×C₈	1	4	yes	—
243	C₂³	C₂²×C₅×C₈	1	2	yes	—
244	C₂³	C₂²×C₅×C₈	1	1	yes	characteristic
245	Q₈	—	2	8	no	—
246	D₄	—	2	16	no	—
247	D₄	—	2	8	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	8	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	C₅×C₂²⋊C₈	1	4	yes	—
253	C₂×C₄	C₅×C₂²⋊C₈	1	8	yes	—
254	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
255	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	4	yes	—
256	C₂×C₄	C₂²×C₅×C₈	1	4	yes	—
257	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
258	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
259	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
260	C₂×C₄	C₂×C₅×M₁₆	1	4	yes	—
261	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
262	C₈	—	2	4	no	—
263	C₈	C₂×C₅×C₂²⋊C₄	1	2	yes	—
264	C₈	C₂×C₅×C₂²⋊C₄	1	4	yes	—
265	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
266	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
267	C₅	C₂×#2113	1	1	yes	characteristic, direct product factor, Sylow
268	C₂²	—	2	8	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	2	8	no	—
272	C₂²	—	4	8	no	—
273	C₂²	C₂×C₅×C₂²⋊C₈	1	4	yes	—
274	C₂²	C₂×C₅×C₂²⋊C₈	1	2	yes	—
275	C₂²	C₅×#504	1	4	yes	—
276	C₂²	C₂³×C₅×C₈	1	1	yes	characteristic
277	C₄	—	2	4	no	—
278	C₄	C₂×C₅×C₂²⋊C₈	1	2	yes	—
279	C₄	C₂×C₅×C₂²⋊C₈	1	4	yes	—
280	C₄	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
281	C₄	C₂²×C₅×M₁₆	1	1	yes	characteristic
282	C₂	—	2	4	no	—
283	C₂	—	4	4	no	—
284	C₂	C₅×#2113	1	4	yes	direct product factor
285	C₂	C₂²×C₅×C₂²⋊C₈	1	1	yes	characteristic
286	C₂	C₂×C₅×#504	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅×C₈	C₂³
commutator	C₂²	C₂³×C₅×C₈
Frattini, cosocle	C₂×C₈	C₂⁴×C₅
socle	C₂³×C₅	C₂²×C₈
prime radical	C₂²×C₅×#41	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₅×M₃₂	C₂	semidirect
3. C₂³×C₅×C₁₆	C₂	semidirect
4. C₅×#2113	C₂	direct (N × H)
5. C₂×C₅×C₂²⋊C₁₆	C₂	semidirect
16. C₂×C₅×M₃₂	C₂²	semidirect
20. C₂²×C₅×C₁₆	C₂²	semidirect
21. C₅×C₂²⋊C₁₆	C₂²	semidirect
22. C₂×#2113	C₅	direct (N × H)
57. C₂²×M₃₂	C₂×C₅	semidirect
58. C₂³×C₁₆	C₂×C₅	semidirect
59. #2113	C₂×C₅	direct (N × H)
60. C₂×C₂²⋊C₁₆	C₂×C₅	semidirect
111. C₂×M₃₂	C₂²×C₅	semidirect
115. C₂²×C₁₆	C₂²×C₅	semidirect
116. C₂²⋊C₁₆	C₂²×C₅	semidirect
235. C₂×C₅	#2113	direct (N × H)
267. C₅	C₂×#2113	direct (N × H)
284. C₂	C₅×#2113	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×#131	2
C₂²×C₅×M₃₂	2
C₂³×C₅×C₁₆	2
C₅×#2113	2
C₂×C₅×C₂²⋊C₁₆	2
C₂×#2113	5

Subgroups — C2×C5×#2113 #11956052

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2113 #11956052