Subgroups — C₂×C₅×#2237 #11956362

Order 1280.

586 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#467	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#427	C₂	1	1	yes	maximal, characteristic
3	C₅×#2237	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅×#367	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×#366	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×#358	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅×#358	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₅×C₂²⋊Q₈	—	2	1	no	—
12	C₂×C₅×#117	—	2	1	no	—
13	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
15	C₅×#467	—	2	2	no	—
16	C₅×#467	C₂²	1	4	yes	—
17	C₅×#427	—	2	2	no	—
18	C₅×#427	C₂²	1	4	yes	—
19	C₂×C₅×#107	C₂²	1	1	yes	characteristic
20	C₂×C₅×#107	C₂²	1	1	yes	characteristic
21	C₂×C₅×#107	C₂²	1	1	yes	characteristic
22	C₂×C₅×#106	—	2	1	no	—
23	C₂×C₅×#103	—	2	1	no	—
24	C₅×#367	C₂²	1	4	yes	complemented
25	C₅×#366	C₂²	1	4	yes	complemented
26	C₅×#358	C₂²	1	4	yes	complemented
27	C₅×#358	C₂²	1	4	yes	complemented
28	C₅×C₂²⋊C₁₆	C₂²	1	4	yes	—
29	C₂×#2237	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂²×C₅×Q₈	—	2	1	no	—
31	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
32	C₂×C₅×Q₁₆	—	2	1	no	—
33	C₂×C₅×Q₁₆	—	2	2	no	—
34	C₂×C₅×Q₁₆	—	2	2	no	—
35	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
36	C₂²×C₅×C₈	—	2	2	no	—
37	C₂²×C₅×C₈	—	2	2	no	—
38	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
39	C₂²×C₅×C₈	D₄	1	2	yes	—
40	C₂²×C₅×C₈	C₂³	1	2	yes	—
41	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
42	C₅×C₂²⋊Q₈	—	2	4	no	—
43	C₅×C₂²⋊Q₈	—	4	2	no	—
44	C₅×#117	—	2	4	no	—
45	C₅×#117	—	4	2	no	—
46	C₂×C₅×C₄⋊C₄	—	2	1	no	—
47	C₂×C₅×C₄⋊C₄	—	2	1	no	—
48	C₂×C₅×C₄⋊C₄	—	2	1	no	—
49	C₂×C₅×C₄⋊C₄	—	4	1	no	—
50	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
51	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
52	C₂×C₄²×C₅	—	4	1	no	—
53	C₂×C₅×C₁₆	—	2	2	no	—
54	C₂×C₅×C₁₆	—	2	2	no	—
55	C₂×C₅×C₁₆	C₂³	1	2	yes	—
56	C₂×C₅×C₁₆	C₂³	1	2	yes	—
57	C₅×#107	—	2	1	no	—
58	C₅×#107	—	2	1	no	—
59	C₅×#107	—	2	1	no	—
60	C₅×#107	C₂³	1	2	yes	—
61	C₅×#107	C₂³	1	2	yes	—
62	C₅×#107	C₂³	1	2	yes	—
63	C₅×#106	—	2	4	no	—
64	C₅×#103	—	2	4	no	—
65	C₂×#467	C₂×C₅	1	1	yes	characteristic
66	C₂×#427	C₂×C₅	1	1	yes	characteristic
67	#2237	C₂×C₅	1	8	yes	direct product factor
68	C₂×#367	C₂×C₅	1	1	yes	characteristic, complemented
69	C₂×#366	C₂×C₅	1	1	yes	characteristic, complemented
70	C₂×#358	C₂×C₅	1	1	yes	characteristic, complemented
71	C₂×#358	C₂×C₅	1	1	yes	characteristic, complemented
72	C₂×C₂²⋊C₁₆	C₂×C₅	1	1	yes	characteristic
73	C₂⁴×C₅	D₈	1	1	yes	characteristic
74	C₂×C₅×Q₈	—	2	2	no	—
75	C₂×C₅×Q₈	—	4	1	no	—
76	C₂×C₅×Q₈	—	4	2	no	—
77	C₂×C₅×Q₈	—	4	2	no	—
78	C₂²×C₄×C₅	—	2	2	no	—
79	C₂²×C₄×C₅	—	2	2	no	—
80	C₂²×C₄×C₅	—	4	1	no	—
81	C₂²×C₄×C₅	—	4	1	no	—
82	C₂²×C₄×C₅	—	4	1	no	—
83	C₂²×C₄×C₅	—	4	1	no	—
84	C₂²×C₄×C₅	D₈	1	2	yes	—
85	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
86	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₅×Q₁₆	—	2	2	no	—
89	C₅×Q₁₆	—	2	2	no	—
90	C₅×Q₁₆	—	4	2	no	—
91	C₂×C₅×C₈	—	2	2	no	—
92	C₂×C₅×C₈	—	2	4	no	—
93	C₂×C₅×C₈	—	2	2	no	—
94	C₂×C₅×C₈	—	2	2	no	—
95	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
96	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂×C₅×C₈	#41	1	2	yes	—
98	C₂×C₅×C₈	#41	1	2	yes	—
99	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
100	C₅×C₄⋊C₄	—	2	2	no	—
101	C₅×C₄⋊C₄	—	2	2	no	—
102	C₅×C₄⋊C₄	—	2	2	no	—
103	C₅×C₄⋊C₄	—	4	1	no	—
104	C₅×C₄⋊C₄	—	4	1	no	—
105	C₅×C₄⋊C₄	—	4	1	no	—
106	C₅×C₄⋊C₄	—	4	4	no	—
107	C₅×C₂²⋊C₄	—	4	4	no	—
108	C₅×C₂²⋊C₄	—	4	4	no	—
109	C₄²×C₅	—	4	4	no	—
110	C₅×C₁₆	—	2	2	no	—
111	C₅×C₁₆	—	2	2	no	—
112	C₂²×Q₁₆	C₂²×C₅	1	1	yes	characteristic
113	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
114	C₂×C₂²⋊Q₈	—	2	1	no	—
115	C₂×#117	—	2	1	no	—
116	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
117	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
118	#467	—	2	2	no	—
119	#467	C₂²×C₅	1	4	yes	—
120	#427	—	2	2	no	—
121	#427	C₂²×C₅	1	4	yes	—
122	C₂×#107	C₂²×C₅	1	1	yes	characteristic
123	C₂×#107	C₂²×C₅	1	1	yes	characteristic
124	C₂×#107	C₂²×C₅	1	1	yes	characteristic
125	C₂×#106	—	2	1	no	—
126	C₂×#103	—	2	1	no	—
127	#367	C₂²×C₅	1	4	yes	complemented
128	#366	C₂²×C₅	1	4	yes	complemented
129	#358	C₂²×C₅	1	4	yes	complemented
130	#358	C₂²×C₅	1	4	yes	complemented
131	C₂²⋊C₁₆	C₂²×C₅	1	4	yes	—
132	C₂³×C₅	—	2	2	no	—
133	C₂³×C₅	—	2	2	no	—
134	C₂³×C₅	—	2	2	no	—
135	C₂³×C₅	C₂×D₈	1	2	yes	—
136	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
137	C₅×Q₈	—	4	2	no	—
138	C₅×Q₈	—	4	2	no	—
139	C₅×Q₈	—	8	2	no	—
140	C₂×C₄×C₅	—	2	2	no	—
141	C₂×C₄×C₅	—	2	2	no	—
142	C₂×C₄×C₅	—	2	2	no	—
143	C₂×C₄×C₅	—	2	4	no	—
144	C₂×C₄×C₅	—	4	2	no	—
145	C₂×C₄×C₅	—	4	2	no	—
146	C₂×C₄×C₅	—	4	2	no	—
147	C₂×C₄×C₅	—	4	2	no	—
148	C₂×C₄×C₅	—	8	2	no	—
149	C₂×C₄×C₅	—	8	2	no	—
150	C₂×C₄×C₅	—	8	2	no	—
151	C₂×C₄×C₅	—	8	2	no	—
152	C₂×C₄×C₅	#123	1	2	yes	—
153	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
154	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄×C₅	#137	1	2	yes	—
156	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
157	C₅×C₈	—	2	2	no	—
158	C₅×C₈	#123	1	2	yes	—
159	C₅×C₈	C₂×#41	1	1	yes	characteristic
160	C₅×C₈	C₂×#41	1	1	yes	characteristic
161	C₂²×Q₈	—	2	1	no	—
162	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
163	C₂×Q₁₆	—	2	1	no	—
164	C₂×Q₁₆	—	2	2	no	—
165	C₂×Q₁₆	—	2	2	no	—
166	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	—	2	2	no	—
169	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
170	C₂²×C₈	C₅×D₄	1	2	yes	—
171	C₂²×C₈	C₂³×C₅	1	2	yes	—
172	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
173	C₂²⋊Q₈	—	2	4	no	—
174	C₂²⋊Q₈	—	4	2	no	—
175	#117	—	2	4	no	—
176	#117	—	4	2	no	—
177	C₂×C₄⋊C₄	—	2	1	no	—
178	C₂×C₄⋊C₄	—	2	1	no	—
179	C₂×C₄⋊C₄	—	2	1	no	—
180	C₂×C₄⋊C₄	—	4	1	no	—
181	C₂×C₂²⋊C₄	—	4	1	no	—
182	C₂×C₂²⋊C₄	—	4	1	no	—
183	C₂×C₄²	—	4	1	no	—
184	C₂×C₁₆	—	2	2	no	—
185	C₂×C₁₆	—	2	2	no	—
186	C₂×C₁₆	C₂³×C₅	1	2	yes	—
187	C₂×C₁₆	C₂³×C₅	1	2	yes	—
188	#107	—	2	1	no	—
189	#107	—	2	1	no	—
190	#107	—	2	1	no	—
191	#107	C₂³×C₅	1	2	yes	—
192	#107	C₂³×C₅	1	2	yes	—
193	#107	C₂³×C₅	1	2	yes	—
194	#106	—	2	4	no	—
195	#103	—	2	4	no	—
196	C₂²×C₅	—	2	2	no	—
197	C₂²×C₅	—	2	2	no	—
198	C₂²×C₅	—	2	2	no	—
199	C₂²×C₅	—	4	4	no	—
200	C₂²×C₅	#480	1	2	yes	—
201	C₂²×C₅	#508	1	2	yes	—
202	C₂²×C₅	#510	1	2	yes	—
203	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
204	C₄×C₅	—	2	2	no	—
205	C₄×C₅	—	8	2	no	—
206	C₄×C₅	—	8	2	no	—
207	C₄×C₅	—	8	2	no	—
208	C₄×C₅	—	8	2	no	—
209	C₄×C₅	#480	1	2	yes	—
210	C₄×C₅	C₂×#123	1	1	yes	characteristic
211	C₄×C₅	C₂×#137	1	1	yes	characteristic
212	C₂⁴	C₅×D₈	1	1	yes	characteristic
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	—	4	1	no	—
215	C₂×Q₈	—	4	2	no	—
216	C₂×Q₈	—	4	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	4	1	no	—
220	C₂²×C₄	—	4	1	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
224	C₂²×C₄	C₅×D₈	1	2	yes	—
225	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
226	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
227	Q₁₆	—	2	2	no	—
228	Q₁₆	—	2	2	no	—
229	Q₁₆	—	4	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
235	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
236	C₂×C₈	C₅×#41	1	2	yes	—
237	C₂×C₈	C₅×#41	1	2	yes	—
238	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	4	1	no	—
243	C₄⋊C₄	—	4	1	no	—
244	C₄⋊C₄	—	4	1	no	—
245	C₄⋊C₄	—	4	4	no	—
246	C₂²⋊C₄	—	4	4	no	—
247	C₂²⋊C₄	—	4	4	no	—
248	C₄²	—	4	4	no	—
249	C₁₆	—	2	2	no	—
250	C₁₆	—	2	2	no	—
251	C₂×C₅	—	4	2	no	—
252	C₂×C₅	#2237	1	4	yes	direct product factor
253	C₂×C₅	C₂×#480	1	1	yes	characteristic
254	C₂×C₅	C₂×#508	1	1	yes	characteristic
255	C₂×C₅	C₂×#510	1	1	yes	characteristic
256	C₂³	—	2	2	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	C₂×C₅×D₈	1	2	yes	—
260	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
261	Q₈	—	4	2	no	—
262	Q₈	—	4	2	no	—
263	Q₈	—	8	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	8	2	no	—
273	C₂×C₄	—	8	2	no	—
274	C₂×C₄	—	8	2	no	—
275	C₂×C₄	—	8	2	no	—
276	C₂×C₄	C₅×#123	1	2	yes	—
277	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
278	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
279	C₂×C₄	C₅×#137	1	2	yes	—
280	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
281	C₈	—	2	2	no	—
282	C₈	C₅×#123	1	2	yes	—
283	C₈	C₂×C₅×#41	1	1	yes	characteristic
284	C₈	C₂×C₅×#41	1	1	yes	characteristic
285	C₅	C₂×#2237	1	1	yes	characteristic, direct product factor, Sylow
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	4	no	—
290	C₂²	C₅×#480	1	2	yes	—
291	C₂²	C₅×#508	1	2	yes	—
292	C₂²	C₅×#510	1	2	yes	—
293	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
294	C₄	—	2	2	no	—
295	C₄	—	8	2	no	—
296	C₄	—	8	2	no	—
297	C₄	—	8	2	no	—
298	C₄	—	8	2	no	—
299	C₄	C₅×#480	1	2	yes	—
300	C₄	C₂×C₅×#123	1	1	yes	characteristic
301	C₄	C₂×C₅×#137	1	1	yes	characteristic
302	C₂	—	4	2	no	—
303	C₂	C₅×#2237	1	4	yes	direct product factor
304	C₂	C₂×C₅×#480	1	1	yes	characteristic
305	C₂	C₂×C₅×#508	1	1	yes	characteristic
306	C₂	C₂×C₅×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂⁴×C₅	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#2237	C₂	direct (N × H)
4. C₂×C₅×#367	C₂	semidirect
5. C₂×C₅×#366	C₂	semidirect
6. C₂×C₅×#358	C₂	semidirect
7. C₂×C₅×#358	C₂	semidirect
24. C₅×#367	C₂²	semidirect
25. C₅×#366	C₂²	semidirect
26. C₅×#358	C₂²	semidirect
27. C₅×#358	C₂²	semidirect
29. C₂×#2237	C₅	direct (N × H)
67. #2237	C₂×C₅	direct (N × H)
68. C₂×#367	C₂×C₅	semidirect
69. C₂×#366	C₂×C₅	semidirect
70. C₂×#358	C₂×C₅	semidirect
71. C₂×#358	C₂×C₅	semidirect
127. #367	C₂²×C₅	semidirect
128. #366	C₂²×C₅	semidirect
129. #358	C₂²×C₅	semidirect
130. #358	C₂²×C₅	semidirect
252. C₂×C₅	#2237	direct (N × H)
285. C₅	C₂×#2237	direct (N × H)
303. C₂	C₅×#2237	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×#467	2
C₂×C₅×#427	2
C₅×#2237	2
C₂×C₅×#367	2
C₂×C₅×#366	2
C₂×C₅×#358	2
C₂×C₅×C₂²⋊C₁₆	2
C₂×#2237	5

Subgroups — C2×C5×#2237 #11956362

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2237 #11956362