Subgroups — C₂×C₅×#2214 #11956313

Order 1280.

686 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#510	C₂	1	2	yes	maximal
2	C₂×C₅×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#443	C₂	1	1	yes	maximal, characteristic
4	C₅×#2214	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅×#364	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×#362	C₂	1	2	yes	maximal, complemented
7	C₂×C₅×#135	C₂²	1	1	yes	characteristic
8	C₂²×C₅×Q₁₆	—	2	1	no	—
9	C₂²×C₅×Q₁₆	C₂²	1	2	yes	—
10	C₂×C₅×#131	—	2	1	no	—
11	C₂×C₅×C₄⋊Q₈	—	2	1	no	—
12	C₅×#510	—	2	4	no	—
13	C₅×#510	C₂²	1	8	yes	—
14	C₂×C₅×Q₃₂	—	2	2	no	—
15	C₂×C₅×SD₃₂	—	2	2	no	—
16	C₂×C₅×M₃₂	C₂²	1	2	yes	—
17	C₅×#494	—	2	2	no	—
18	C₅×#494	C₂²	1	4	yes	complemented
19	C₅×#443	—	2	2	no	—
20	C₅×#443	C₂²	1	4	yes	—
21	C₂×C₅×#108	C₂²	1	1	yes	characteristic
22	C₂×C₅×#104	—	2	1	no	—
23	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
24	C₅×#364	C₂²	1	4	yes	complemented
25	C₅×#362	C₂²	1	8	yes	complemented
26	C₂×#2214	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂×C₅×#41	—	2	1	no	—
28	C₂²×C₅×Q₈	—	2	2	no	—
29	C₅×#135	—	2	1	no	—
30	C₅×#135	—	2	1	no	—
31	C₅×#135	D₄	1	2	yes	—
32	C₅×#135	C₂³	1	2	yes	—
33	C₂×C₅×Q₁₆	—	2	2	no	—
34	C₂×C₅×Q₁₆	—	2	2	no	—
35	C₂×C₅×Q₁₆	—	2	2	no	—
36	C₂×C₅×Q₁₆	—	2	4	no	—
37	C₂×C₅×Q₁₆	—	2	2	no	—
38	C₂×C₅×Q₁₆	—	2	2	no	—
39	C₂×C₅×Q₁₆	—	4	1	no	—
40	C₂×C₅×Q₁₆	—	4	1	no	—
41	C₂×C₅×Q₁₆	—	4	2	no	—
42	C₂×C₅×Q₁₆	D₄	1	1	yes	characteristic
43	C₂×C₅×Q₁₆	C₂³	1	4	yes	—
44	C₂×C₅×SD₁₆	—	2	1	no	—
45	C₂×C₅×D₈	D₄	1	1	yes	characteristic
46	C₅×#131	—	2	4	no	—
47	C₅×#131	—	4	2	no	—
48	C₂×C₅×M₁₆	—	2	1	no	—
49	C₂×C₅×M₁₆	—	4	1	no	—
50	C₂²×C₅×C₈	—	4	1	no	—
51	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
52	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
53	C₅×C₄⋊Q₈	—	2	4	no	—
54	C₅×C₄⋊Q₈	—	4	2	no	—
55	C₂×C₅×C₄⋊C₄	—	4	2	no	—
56	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
57	C₅×Q₃₂	—	2	8	no	—
58	C₅×SD₃₂	—	2	8	no	—
59	C₅×M₃₂	—	2	2	no	—
60	C₅×M₃₂	C₂³	1	4	yes	—
61	C₂×C₅×C₁₆	—	2	2	no	—
62	C₅×#108	—	2	1	no	—
63	C₅×#108	C₂³	1	2	yes	—
64	C₅×#104	—	2	4	no	—
65	C₄×C₅×C₈	D₄	1	2	yes	—
66	C₄×C₅×C₈	C₂³	1	2	yes	—
67	C₂×#510	C₂×C₅	1	2	yes	—
68	C₂×#494	C₂×C₅	1	1	yes	characteristic, complemented
69	C₂×#443	C₂×C₅	1	1	yes	characteristic
70	#2214	C₂×C₅	1	8	yes	direct product factor
71	C₂×#364	C₂×C₅	1	1	yes	characteristic, complemented
72	C₂×#362	C₂×C₅	1	2	yes	complemented
73	C₅×#41	—	2	2	no	—
74	C₅×#41	—	2	2	no	—
75	C₅×#41	—	4	1	no	—
76	C₅×#41	—	4	1	no	—
77	C₂×C₅×Q₈	—	2	1	no	—
78	C₂×C₅×Q₈	—	2	4	no	—
79	C₂×C₅×Q₈	—	4	2	no	—
80	C₂×C₅×Q₈	—	4	2	no	—
81	C₂×C₅×Q₈	—	4	2	no	—
82	C₂×C₅×Q₈	—	4	2	no	—
83	C₂×C₅×Q₈	—	4	2	no	—
84	C₂×C₅×D₄	—	2	1	no	—
85	C₂×C₅×D₄	—	4	1	no	—
86	C₂²×C₄×C₅	—	2	1	no	—
87	C₂²×C₄×C₅	—	4	1	no	—
88	C₂²×C₄×C₅	—	4	2	no	—
89	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
90	C₅×Q₁₆	—	2	1	no	—
91	C₅×Q₁₆	—	2	4	no	—
92	C₅×Q₁₆	—	2	4	no	—
93	C₅×Q₁₆	—	4	2	no	—
94	C₅×Q₁₆	—	4	2	no	—
95	C₅×Q₁₆	—	4	4	no	—
96	C₅×Q₁₆	—	4	4	no	—
97	C₅×Q₁₆	C₂×D₄	1	2	yes	—
98	C₅×SD₁₆	—	2	2	no	—
99	C₅×SD₁₆	—	2	2	no	—
100	C₅×D₈	—	2	1	no	—
101	C₅×D₈	C₂×D₄	1	2	yes	—
102	C₅×M₁₆	—	2	2	no	—
103	C₅×M₁₆	—	4	1	no	—
104	C₅×M₁₆	—	4	4	no	—
105	C₂×C₅×C₈	—	4	1	no	—
106	C₂×C₅×C₈	—	4	4	no	—
107	C₂×C₅×C₈	D₈	1	1	yes	characteristic
108	C₂×C₅×C₈	D₈	1	1	yes	characteristic
109	C₂×C₅×C₈	D₈	1	2	yes	—
110	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
111	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
112	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
113	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
114	C₂×C₅×C₈	#41	1	1	yes	characteristic
115	C₂×C₅×C₈	#41	1	1	yes	characteristic
116	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
117	C₅×C₄⋊C₄	—	4	8	no	—
118	C₄²×C₅	—	2	1	no	—
119	C₄²×C₅	C₂×D₄	1	2	yes	—
120	C₅×C₁₆	—	2	4	no	—
121	C₂×#135	C₂²×C₅	1	1	yes	characteristic
122	C₂²×Q₁₆	—	2	1	no	—
123	C₂²×Q₁₆	C₂²×C₅	1	2	yes	—
124	C₂×#131	—	2	1	no	—
125	C₂×C₄⋊Q₈	—	2	1	no	—
126	#510	—	2	4	no	—
127	#510	C₂²×C₅	1	8	yes	—
128	C₂×Q₃₂	—	2	2	no	—
129	C₂×SD₃₂	—	2	2	no	—
130	C₂×M₃₂	C₂²×C₅	1	2	yes	—
131	#494	—	2	2	no	—
132	#494	C₂²×C₅	1	4	yes	complemented
133	#443	—	2	2	no	—
134	#443	C₂²×C₅	1	4	yes	—
135	C₂×#108	C₂²×C₅	1	1	yes	characteristic
136	C₂×#104	—	2	1	no	—
137	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
138	#364	C₂²×C₅	1	4	yes	complemented
139	#362	C₂²×C₅	1	8	yes	complemented
140	C₂³×C₅	—	4	1	no	—
141	C₂³×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₅×Q₈	—	2	2	no	—
143	C₅×Q₈	—	4	1	no	—
144	C₅×Q₈	—	4	4	no	—
145	C₅×Q₈	—	4	4	no	—
146	C₅×Q₈	—	8	2	no	—
147	C₅×Q₈	—	8	2	no	—
148	C₅×D₄	—	2	2	no	—
149	C₅×D₄	—	4	2	no	—
150	C₅×D₄	—	4	2	no	—
151	C₅×D₄	—	4	1	no	—
152	C₂×C₄×C₅	—	2	4	no	—
153	C₂×C₄×C₅	—	2	2	no	—
154	C₂×C₄×C₅	—	4	1	no	—
155	C₂×C₄×C₅	—	4	2	no	—
156	C₂×C₄×C₅	—	4	2	no	—
157	C₂×C₄×C₅	—	4	4	no	—
158	C₂×C₄×C₅	—	8	2	no	—
159	C₂×C₄×C₅	—	8	2	no	—
160	C₂×C₄×C₅	#121	1	1	yes	characteristic
161	C₂×C₄×C₅	#121	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
164	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
165	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
166	C₅×C₈	—	4	2	no	—
167	C₅×C₈	#121	1	1	yes	characteristic
168	C₅×C₈	#121	1	1	yes	characteristic
169	C₅×C₈	C₂×D₈	1	2	yes	—
170	C₅×C₈	C₂×D₈	1	2	yes	—
171	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
172	C₅×C₈	C₂×#41	1	1	yes	characteristic
173	C₂×#41	—	2	1	no	—
174	C₂²×Q₈	—	2	2	no	—
175	#135	—	2	1	no	—
176	#135	—	2	1	no	—
177	#135	C₅×D₄	1	2	yes	—
178	#135	C₂³×C₅	1	2	yes	—
179	C₂×Q₁₆	—	2	2	no	—
180	C₂×Q₁₆	—	2	2	no	—
181	C₂×Q₁₆	—	2	4	no	—
182	C₂×Q₁₆	—	2	2	no	—
183	C₂×Q₁₆	—	2	2	no	—
184	C₂×Q₁₆	—	2	2	no	—
185	C₂×Q₁₆	—	4	1	no	—
186	C₂×Q₁₆	—	4	1	no	—
187	C₂×Q₁₆	—	4	2	no	—
188	C₂×Q₁₆	C₅×D₄	1	1	yes	characteristic
189	C₂×Q₁₆	C₂³×C₅	1	4	yes	—
190	C₂×SD₁₆	—	2	1	no	—
191	C₂×D₈	C₅×D₄	1	1	yes	characteristic
192	#131	—	2	4	no	—
193	#131	—	4	2	no	—
194	C₂×M₁₆	—	2	1	no	—
195	C₂×M₁₆	—	4	1	no	—
196	C₂²×C₈	—	4	1	no	—
197	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
198	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
199	C₄⋊Q₈	—	2	4	no	—
200	C₄⋊Q₈	—	4	2	no	—
201	C₂×C₄⋊C₄	—	4	2	no	—
202	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
203	Q₃₂	—	2	8	no	—
204	SD₃₂	—	2	8	no	—
205	M₃₂	—	2	2	no	—
206	M₃₂	C₂³×C₅	1	4	yes	—
207	C₂×C₁₆	—	2	2	no	—
208	#108	—	2	1	no	—
209	#108	C₂³×C₅	1	2	yes	—
210	#104	—	2	4	no	—
211	C₄×C₈	C₅×D₄	1	2	yes	—
212	C₄×C₈	C₂³×C₅	1	2	yes	—
213	C₂²×C₅	—	2	2	no	—
214	C₂²×C₅	—	4	2	no	—
215	C₂²×C₅	—	8	2	no	—
216	C₂²×C₅	#459	1	1	yes	characteristic
217	C₂²×C₅	#459	1	1	yes	characteristic
218	C₂²×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
219	C₄×C₅	—	2	4	no	—
220	C₄×C₅	—	4	2	no	—
221	C₄×C₅	—	8	4	no	—
222	C₄×C₅	#459	1	1	yes	characteristic
223	C₄×C₅	#459	1	1	yes	characteristic
224	C₄×C₅	C₂×#121	1	1	yes	characteristic
225	C₄×C₅	C₂²×D₈	1	1	yes	characteristic
226	#41	—	2	2	no	—
227	#41	—	2	2	no	—
228	#41	—	4	1	no	—
229	#41	—	4	1	no	—
230	C₂×Q₈	—	2	1	no	—
231	C₂×Q₈	—	2	4	no	—
232	C₂×Q₈	—	4	2	no	—
233	C₂×Q₈	—	4	2	no	—
234	C₂×Q₈	—	4	2	no	—
235	C₂×Q₈	—	4	2	no	—
236	C₂×Q₈	—	4	2	no	—
237	C₂×D₄	—	2	1	no	—
238	C₂×D₄	—	4	1	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	4	1	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
243	Q₁₆	—	2	1	no	—
244	Q₁₆	—	2	4	no	—
245	Q₁₆	—	2	4	no	—
246	Q₁₆	—	4	2	no	—
247	Q₁₆	—	4	2	no	—
248	Q₁₆	—	4	4	no	—
249	Q₁₆	—	4	4	no	—
250	Q₁₆	C₂×C₅×D₄	1	2	yes	—
251	SD₁₆	—	2	2	no	—
252	SD₁₆	—	2	2	no	—
253	D₈	—	2	1	no	—
254	D₈	C₂×C₅×D₄	1	2	yes	—
255	M₁₆	—	2	2	no	—
256	M₁₆	—	4	1	no	—
257	M₁₆	—	4	4	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	—	4	4	no	—
260	C₂×C₈	C₅×D₈	1	1	yes	characteristic
261	C₂×C₈	C₅×D₈	1	1	yes	characteristic
262	C₂×C₈	C₅×D₈	1	2	yes	—
263	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
264	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
265	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
266	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
267	C₂×C₈	C₅×#41	1	1	yes	characteristic
268	C₂×C₈	C₅×#41	1	1	yes	characteristic
269	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
270	C₄⋊C₄	—	4	8	no	—
271	C₄²	—	2	1	no	—
272	C₄²	C₂×C₅×D₄	1	2	yes	—
273	C₁₆	—	2	4	no	—
274	C₂×C₅	—	2	1	no	—
275	C₂×C₅	—	2	1	no	—
276	C₂×C₅	—	8	2	no	—
277	C₂×C₅	#2214	1	2	yes	direct product factor
278	C₂×C₅	C₂×#459	1	1	yes	characteristic
279	C₂³	—	4	1	no	—
280	C₂³	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
281	Q₈	—	2	2	no	—
282	Q₈	—	4	1	no	—
283	Q₈	—	4	4	no	—
284	Q₈	—	4	4	no	—
285	Q₈	—	8	2	no	—
286	Q₈	—	8	2	no	—
287	D₄	—	2	2	no	—
288	D₄	—	4	2	no	—
289	D₄	—	4	2	no	—
290	D₄	—	4	1	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	8	2	no	—
298	C₂×C₄	—	8	2	no	—
299	C₂×C₄	C₅×#121	1	1	yes	characteristic
300	C₂×C₄	C₅×#121	1	1	yes	characteristic
301	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
302	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
303	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
304	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
305	C₈	—	4	2	no	—
306	C₈	C₅×#121	1	1	yes	characteristic
307	C₈	C₅×#121	1	1	yes	characteristic
308	C₈	C₂×C₅×D₈	1	2	yes	—
309	C₈	C₂×C₅×D₈	1	2	yes	—
310	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
311	C₈	C₂×C₅×#41	1	1	yes	characteristic
312	C₅	C₂×#2214	1	1	yes	characteristic, direct product factor, Sylow
313	C₂²	—	2	2	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	8	2	no	—
316	C₂²	C₅×#459	1	1	yes	characteristic
317	C₂²	C₅×#459	1	1	yes	characteristic
318	C₂²	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
319	C₄	—	2	4	no	—
320	C₄	—	4	2	no	—
321	C₄	—	8	4	no	—
322	C₄	C₅×#459	1	1	yes	characteristic
323	C₄	C₅×#459	1	1	yes	characteristic
324	C₄	C₂×C₅×#121	1	1	yes	characteristic
325	C₄	C₂²×C₅×D₈	1	1	yes	characteristic
326	C₂	—	2	1	no	—
327	C₂	—	2	1	no	—
328	C₂	—	8	2	no	—
329	C₂	C₅×#2214	1	2	yes	direct product factor
330	C₂	C₂×C₅×#459	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	#459
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂×C₅×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#494	C₂	semidirect
4. C₅×#2214	C₂	direct (N × H)
5. C₂×C₅×#364	C₂	semidirect
6. C₂×C₅×#362	C₂	semidirect
18. C₅×#494	C₂²	semidirect
24. C₅×#364	C₂²	semidirect
25. C₅×#362	C₂²	semidirect
26. C₂×#2214	C₅	direct (N × H)
68. C₂×#494	C₂×C₅	semidirect
70. #2214	C₂×C₅	direct (N × H)
71. C₂×#364	C₂×C₅	semidirect
72. C₂×#362	C₂×C₅	semidirect
132. #494	C₂²×C₅	semidirect
138. #364	C₂²×C₅	semidirect
139. #362	C₂²×C₅	semidirect
277. C₂×C₅	#2214	direct (N × H)
312. C₅	C₂×#2214	direct (N × H)
329. C₂	C₅×#2214	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×#510	2
C₂×C₅×#494	2
C₂×C₅×#443	2
C₅×#2214	2
C₂×C₅×#364	2
C₂×C₅×#362	2
C₂×#2214	5

Subgroups — C2×C5×#2214 #11956313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2214 #11956313