Subgroups — C₂×C₅×#2228 #11956342

Order 1280.

606 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×C₅×D₈	C₂	1	1	yes	maximal, characteristic
3	C₅×#2228	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×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅×#357	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅×#357	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₅×#125	—	2	1	no	—
11	C₂×C₄×C₅×D₄	—	2	1	no	—
12	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
14	C₅×#499	—	2	2	no	—
15	C₅×#499	C₂²	1	4	yes	complemented
16	C₄×C₅×D₈	—	2	2	no	—
17	C₄×C₅×D₈	C₂²	1	4	yes	—
18	C₂×C₅×#107	C₂²	1	1	yes	characteristic
19	C₂×C₅×#107	C₂²	1	1	yes	characteristic
20	C₂×C₅×#107	C₂²	1	1	yes	characteristic
21	C₂×C₅×#106	—	2	1	no	—
22	C₂×C₅×#102	—	2	1	no	—
23	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
24	C₅×#367	C₂²	1	4	yes	complemented
25	C₅×#366	C₂²	1	4	yes	complemented
26	C₅×C₄⋊C₁₆	C₂²	1	4	yes	complemented
27	C₅×#357	C₂²	1	4	yes	—
28	C₅×#357	C₂²	1	4	yes	—
29	C₂×#2228	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂²×C₅×D₄	—	2	1	no	—
31	C₂³×C₄×C₅	—	4	1	no	—
32	C₂×C₅×D₈	—	2	1	no	—
33	C₂×C₅×D₈	—	2	2	no	—
34	C₂×C₅×D₈	Q₈	1	4	yes	—
35	C₂×C₅×D₈	C₂³	1	2	yes	—
36	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
37	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
38	C₅×#125	—	2	4	no	—
39	C₅×#125	—	4	2	no	—
40	C₄×C₅×D₄	—	2	4	no	—
41	C₄×C₅×D₄	—	4	2	no	—
42	C₂×C₅×C₄⋊C₄	—	2	1	no	—
43	C₂×C₅×C₄⋊C₄	—	2	1	no	—
44	C₂×C₅×C₄⋊C₄	—	2	1	no	—
45	C₂×C₅×C₄⋊C₄	—	4	1	no	—
46	C₂×C₅×C₄⋊C₄	—	4	1	no	—
47	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
48	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
49	C₂×C₅×C₁₆	—	2	2	no	—
50	C₂×C₅×C₁₆	—	2	2	no	—
51	C₂×C₅×C₁₆	C₂³	1	2	yes	—
52	C₂×C₅×C₁₆	C₂³	1	2	yes	—
53	C₅×#107	—	2	1	no	—
54	C₅×#107	—	2	1	no	—
55	C₅×#107	—	2	1	no	—
56	C₅×#107	C₂³	1	2	yes	—
57	C₅×#107	C₂³	1	2	yes	—
58	C₅×#107	C₂³	1	2	yes	—
59	C₅×#106	—	2	4	no	—
60	C₅×#102	—	2	4	no	—
61	C₄×C₅×C₈	D₄	1	2	yes	—
62	C₄×C₅×C₈	C₂³	1	2	yes	—
63	C₂×#499	C₂×C₅	1	1	yes	characteristic, complemented
64	C₂×C₄×D₈	C₂×C₅	1	1	yes	characteristic
65	#2228	C₂×C₅	1	8	yes	direct product factor
66	C₂×#367	C₂×C₅	1	1	yes	characteristic, complemented
67	C₂×#366	C₂×C₅	1	1	yes	characteristic, complemented
68	C₂×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
69	C₂×#357	C₂×C₅	1	1	yes	characteristic
70	C₂×#357	C₂×C₅	1	1	yes	characteristic
71	C₂⁴×C₅	—	4	1	no	—
72	C₂×C₅×D₄	—	2	4	no	—
73	C₂×C₅×D₄	—	2	2	no	—
74	C₂×C₅×D₄	—	4	1	no	—
75	C₂×C₅×D₄	—	4	2	no	—
76	C₂²×C₄×C₅	—	4	1	no	—
77	C₂²×C₄×C₅	—	4	1	no	—
78	C₂²×C₄×C₅	—	4	1	no	—
79	C₂²×C₄×C₅	—	4	4	no	—
80	C₂²×C₄×C₅	—	8	4	no	—
81	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
82	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₅×D₈	—	2	4	no	—
85	C₅×D₈	—	2	2	no	—
86	C₅×D₈	C₂×Q₈	1	4	yes	—
87	C₂×C₅×C₈	—	2	2	no	—
88	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
89	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
91	C₂×C₅×C₈	#41	1	2	yes	—
92	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
93	C₅×C₄⋊C₄	—	2	2	no	—
94	C₅×C₄⋊C₄	—	2	2	no	—
95	C₅×C₄⋊C₄	—	2	2	no	—
96	C₅×C₄⋊C₄	—	4	1	no	—
97	C₅×C₄⋊C₄	—	4	1	no	—
98	C₅×C₄⋊C₄	—	4	1	no	—
99	C₅×C₄⋊C₄	—	4	4	no	—
100	C₅×C₄⋊C₄	—	4	4	no	—
101	C₅×C₂²⋊C₄	—	4	4	no	—
102	C₄²×C₅	D₈	1	2	yes	—
103	C₄²×C₅	C₂×D₄	1	2	yes	—
104	C₅×C₁₆	—	2	2	no	—
105	C₅×C₁₆	—	2	2	no	—
106	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
107	C₂×#125	—	2	1	no	—
108	C₂×C₄×D₄	—	2	1	no	—
109	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
110	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
111	#499	—	2	2	no	—
112	#499	C₂²×C₅	1	4	yes	complemented
113	C₄×D₈	—	2	2	no	—
114	C₄×D₈	C₂²×C₅	1	4	yes	—
115	C₂×#107	C₂²×C₅	1	1	yes	characteristic
116	C₂×#107	C₂²×C₅	1	1	yes	characteristic
117	C₂×#107	C₂²×C₅	1	1	yes	characteristic
118	C₂×#106	—	2	1	no	—
119	C₂×#102	—	2	1	no	—
120	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
121	#367	C₂²×C₅	1	4	yes	complemented
122	#366	C₂²×C₅	1	4	yes	complemented
123	C₄⋊C₁₆	C₂²×C₅	1	4	yes	complemented
124	#357	C₂²×C₅	1	4	yes	—
125	#357	C₂²×C₅	1	4	yes	—
126	C₂³×C₅	—	4	2	no	—
127	C₂³×C₅	—	4	4	no	—
128	C₂³×C₅	—	8	2	no	—
129	C₂³×C₅	—	8	2	no	—
130	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
131	C₅×D₄	—	2	4	no	—
132	C₅×D₄	—	4	2	no	—
133	C₅×D₄	—	4	4	no	—
134	C₂×C₄×C₅	—	2	2	no	—
135	C₂×C₄×C₅	—	2	2	no	—
136	C₂×C₄×C₅	—	4	2	no	—
137	C₂×C₄×C₅	—	4	2	no	—
138	C₂×C₄×C₅	—	4	2	no	—
139	C₂×C₄×C₅	—	8	8	no	—
140	C₂×C₄×C₅	—	8	2	no	—
141	C₂×C₄×C₅	—	8	2	no	—
142	C₂×C₄×C₅	—	8	2	no	—
143	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
144	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
145	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
146	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₅	#137	1	2	yes	—
148	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
149	C₅×C₈	—	2	2	no	—
150	C₅×C₈	C₂²⋊Q₈	1	2	yes	—
151	C₅×C₈	C₂²×Q₈	1	1	yes	characteristic
152	C₅×C₈	C₂×#41	1	1	yes	characteristic
153	C₂²×D₄	—	2	1	no	—
154	C₂³×C₄	—	4	1	no	—
155	C₂×D₈	—	2	1	no	—
156	C₂×D₈	—	2	2	no	—
157	C₂×D₈	C₅×Q₈	1	4	yes	—
158	C₂×D₈	C₂³×C₅	1	2	yes	—
159	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
160	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
161	#125	—	2	4	no	—
162	#125	—	4	2	no	—
163	C₄×D₄	—	2	4	no	—
164	C₄×D₄	—	4	2	no	—
165	C₂×C₄⋊C₄	—	2	1	no	—
166	C₂×C₄⋊C₄	—	2	1	no	—
167	C₂×C₄⋊C₄	—	2	1	no	—
168	C₂×C₄⋊C₄	—	4	1	no	—
169	C₂×C₄⋊C₄	—	4	1	no	—
170	C₂×C₂²⋊C₄	—	4	1	no	—
171	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
172	C₂×C₁₆	—	2	2	no	—
173	C₂×C₁₆	—	2	2	no	—
174	C₂×C₁₆	C₂³×C₅	1	2	yes	—
175	C₂×C₁₆	C₂³×C₅	1	2	yes	—
176	#107	—	2	1	no	—
177	#107	—	2	1	no	—
178	#107	—	2	1	no	—
179	#107	C₂³×C₅	1	2	yes	—
180	#107	C₂³×C₅	1	2	yes	—
181	#107	C₂³×C₅	1	2	yes	—
182	#106	—	2	4	no	—
183	#102	—	2	4	no	—
184	C₄×C₈	C₅×D₄	1	2	yes	—
185	C₄×C₈	C₂³×C₅	1	2	yes	—
186	C₂²×C₅	—	4	4	no	—
187	C₂²×C₅	—	8	2	no	—
188	C₂²×C₅	—	8	2	no	—
189	C₂²×C₅	—	8	8	no	—
190	C₂²×C₅	#474	1	2	yes	—
191	C₂²×C₅	#508	1	2	yes	—
192	C₂²×C₅	#509	1	2	yes	—
193	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
194	C₄×C₅	—	2	2	no	—
195	C₄×C₅	—	2	2	no	—
196	C₄×C₅	—	8	2	no	—
197	C₄×C₅	—	8	2	no	—
198	C₄×C₅	—	8	2	no	—
199	C₄×C₅	#474	1	2	yes	—
200	C₄×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
201	C₄×C₅	C₂×#137	1	1	yes	characteristic
202	C₂⁴	—	4	1	no	—
203	C₂×D₄	—	2	4	no	—
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	—	4	1	no	—
206	C₂×D₄	—	4	2	no	—
207	C₂²×C₄	—	4	1	no	—
208	C₂²×C₄	—	4	1	no	—
209	C₂²×C₄	—	4	1	no	—
210	C₂²×C₄	—	4	4	no	—
211	C₂²×C₄	—	8	4	no	—
212	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
213	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
215	D₈	—	2	4	no	—
216	D₈	—	2	2	no	—
217	D₈	C₂×C₅×Q₈	1	4	yes	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
220	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
221	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
222	C₂×C₈	C₅×#41	1	2	yes	—
223	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	4	1	no	—
228	C₄⋊C₄	—	4	1	no	—
229	C₄⋊C₄	—	4	1	no	—
230	C₄⋊C₄	—	4	4	no	—
231	C₄⋊C₄	—	4	4	no	—
232	C₂²⋊C₄	—	4	4	no	—
233	C₄²	C₅×D₈	1	2	yes	—
234	C₄²	C₂×C₅×D₄	1	2	yes	—
235	C₁₆	—	2	2	no	—
236	C₁₆	—	2	2	no	—
237	C₂×C₅	—	8	4	no	—
238	C₂×C₅	#2228	1	4	yes	direct product factor
239	C₂×C₅	C₂×#474	1	1	yes	characteristic
240	C₂×C₅	C₂×#508	1	1	yes	characteristic
241	C₂×C₅	C₂×#509	1	1	yes	characteristic
242	C₂³	—	4	4	no	—
243	C₂³	—	4	2	no	—
244	C₂³	—	8	2	no	—
245	C₂³	—	8	2	no	—
246	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
247	D₄	—	2	4	no	—
248	D₄	—	4	4	no	—
249	D₄	—	4	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	8	8	no	—
256	C₂×C₄	—	8	2	no	—
257	C₂×C₄	—	8	2	no	—
258	C₂×C₄	—	8	2	no	—
259	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
260	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
261	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
262	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
263	C₂×C₄	C₅×#137	1	2	yes	—
264	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
265	C₈	—	2	2	no	—
266	C₈	C₅×C₂²⋊Q₈	1	2	yes	—
267	C₈	C₂²×C₅×Q₈	1	1	yes	characteristic
268	C₈	C₂×C₅×#41	1	1	yes	characteristic
269	C₅	C₂×#2228	1	1	yes	characteristic, direct product factor, Sylow
270	C₂²	—	4	4	no	—
271	C₂²	—	8	2	no	—
272	C₂²	—	8	2	no	—
273	C₂²	—	8	8	no	—
274	C₂²	C₅×#474	1	2	yes	—
275	C₂²	C₅×#508	1	2	yes	—
276	C₂²	C₅×#509	1	2	yes	—
277	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	8	2	no	—
281	C₄	—	8	2	no	—
282	C₄	—	8	2	no	—
283	C₄	C₅×#474	1	2	yes	—
284	C₄	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic
285	C₄	C₂×C₅×#137	1	1	yes	characteristic
286	C₂	—	8	4	no	—
287	C₂	C₅×#2228	1	4	yes	direct product factor
288	C₂	C₂×C₅×#474	1	1	yes	characteristic
289	C₂	C₂×C₅×#508	1	1	yes	characteristic
290	C₂	C₂×C₅×#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂²×C₅×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#499	C₂	semidirect
3. C₅×#2228	C₂	direct (N × H)
4. C₂×C₅×#367	C₂	semidirect
5. C₂×C₅×#366	C₂	semidirect
6. C₂×C₅×C₄⋊C₁₆	C₂	semidirect
15. C₅×#499	C₂²	semidirect
24. C₅×#367	C₂²	semidirect
25. C₅×#366	C₂²	semidirect
26. C₅×C₄⋊C₁₆	C₂²	semidirect
29. C₂×#2228	C₅	direct (N × H)
63. C₂×#499	C₂×C₅	semidirect
65. #2228	C₂×C₅	direct (N × H)
66. C₂×#367	C₂×C₅	semidirect
67. C₂×#366	C₂×C₅	semidirect
68. C₂×C₄⋊C₁₆	C₂×C₅	semidirect
112. #499	C₂²×C₅	semidirect
121. #367	C₂²×C₅	semidirect
122. #366	C₂²×C₅	semidirect
123. C₄⋊C₁₆	C₂²×C₅	semidirect
238. C₂×C₅	#2228	direct (N × H)
269. C₅	C₂×#2228	direct (N × H)
287. C₂	C₅×#2228	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×#499	2
C₂×C₄×C₅×D₈	2
C₅×#2228	2
C₂×C₅×#367	2
C₂×C₅×#366	2
C₂×C₅×C₄⋊C₁₆	2
C₂×C₅×#357	2
C₂×#2228	5

Subgroups — C2×C5×#2228 #11956342

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2228 #11956342