Subgroups — C₂×C₅×#2207 #11956296

Order 1280.

806 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×SD₃₂	C₂	1	2	yes	maximal, complemented
2	C₂×C₅×#493	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2207	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×#357	C₂	1	2	yes	maximal, complemented
7	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₅×D₈	—	2	1	no	—
9	C₂²×C₅×D₈	C₂²	1	2	yes	—
10	C₂×C₅×#127	—	2	1	no	—
11	C₂×C₄×C₅×Q₈	—	2	1	no	—
12	C₂×C₅×SD₃₂	—	2	8	no	—
13	C₂×C₅×SD₃₂	C₂²	1	8	yes	complemented
14	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
15	C₅×#493	—	2	2	no	—
16	C₅×#493	C₂²	1	4	yes	—
17	C₄×C₅×Q₁₆	—	2	2	no	—
18	C₄×C₅×Q₁₆	C₂²	1	4	yes	complemented
19	C₂×C₅×#107	C₂²	1	1	yes	characteristic
20	C₂×C₅×#103	—	2	1	no	—
21	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
22	C₅×C₄⋊C₁₆	C₂²	1	4	yes	complemented
23	C₅×#357	C₂²	1	8	yes	complemented
24	C₂×#2207	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂²×C₅×Q₈	—	2	1	no	—
26	C₂²×C₅×D₄	—	2	2	no	—
27	C₂²×C₅×D₄	—	4	1	no	—
28	C₂²×C₅×D₄	—	4	1	no	—
29	C₂×C₅×Q₁₆	—	2	1	no	—
30	C₂×C₅×Q₁₆	—	2	2	no	—
31	C₂×C₅×Q₁₆	D₄	1	4	yes	complemented
32	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
33	C₂×C₅×D₈	—	2	4	no	—
34	C₂×C₅×D₈	—	2	2	no	—
35	C₂×C₅×D₈	—	2	4	no	—
36	C₂×C₅×D₈	—	2	4	no	—
37	C₂×C₅×D₈	—	4	4	no	—
38	C₂×C₅×D₈	C₂³	1	4	yes	—
39	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
40	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
41	C₅×#127	—	2	4	no	—
42	C₅×#127	—	4	2	no	—
43	C₄×C₅×Q₈	—	2	4	no	—
44	C₄×C₅×Q₈	—	4	2	no	—
45	C₂×C₅×C₄⋊C₄	—	2	1	no	—
46	C₂×C₅×C₄⋊C₄	—	4	1	no	—
47	C₂×C₄²×C₅	—	4	1	no	—
48	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
49	C₅×SD₃₂	—	2	16	no	—
50	C₂×C₅×C₁₆	—	2	4	no	—
51	C₂×C₅×C₁₆	C₂³	1	4	yes	—
52	C₅×#107	—	2	1	no	—
53	C₅×#107	C₂³	1	2	yes	—
54	C₅×#103	—	2	4	no	—
55	C₄×C₅×C₈	D₄	1	2	yes	—
56	C₄×C₅×C₈	C₂³	1	2	yes	—
57	C₂²×SD₃₂	C₂×C₅	1	2	yes	complemented
58	C₂×#493	C₂×C₅	1	1	yes	characteristic
59	C₂×C₄×Q₁₆	C₂×C₅	1	1	yes	characteristic, complemented
60	#2207	C₂×C₅	1	8	yes	direct product factor
61	C₂×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
62	C₂×#357	C₂×C₅	1	2	yes	complemented
63	C₂⁴×C₅	—	4	2	no	—
64	C₂×C₅×Q₈	—	2	4	no	—
65	C₂×C₅×Q₈	—	2	2	no	—
66	C₂×C₅×Q₈	—	4	1	no	—
67	C₂×C₅×Q₈	—	4	2	no	—
68	C₂×C₅×D₄	—	2	4	no	—
69	C₂×C₅×D₄	—	4	4	no	—
70	C₂×C₅×D₄	—	4	4	no	—
71	C₂×C₅×D₄	—	4	4	no	—
72	C₂×C₅×D₄	—	4	2	no	—
73	C₂×C₅×D₄	—	4	4	no	—
74	C₂×C₅×D₄	—	8	4	no	—
75	C₂×C₅×D₄	—	8	4	no	—
76	C₂²×C₄×C₅	—	4	1	no	—
77	C₂²×C₄×C₅	—	4	1	no	—
78	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
79	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₅×Q₁₆	—	2	4	no	—
82	C₅×Q₁₆	—	2	2	no	—
83	C₅×Q₁₆	C₂×D₄	1	4	yes	complemented
84	C₅×D₈	—	2	4	no	—
85	C₅×D₈	—	2	4	no	—
86	C₅×D₈	—	4	8	no	—
87	C₅×D₈	—	4	4	no	—
88	C₂×C₅×C₈	—	2	2	no	—
89	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
90	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
91	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
92	C₂×C₅×C₈	#41	1	2	yes	—
93	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
94	C₅×C₄⋊C₄	—	2	2	no	—
95	C₅×C₄⋊C₄	—	4	1	no	—
96	C₅×C₄⋊C₄	—	4	4	no	—
97	C₄²×C₅	—	4	4	no	—
98	C₄²×C₅	D₈	1	2	yes	—
99	C₄²×C₅	C₂×D₄	1	2	yes	—
100	C₅×C₁₆	—	2	4	no	—
101	C₂²×Q₁₆	C₂²×C₅	1	1	yes	characteristic
102	C₂²×D₈	—	2	1	no	—
103	C₂²×D₈	C₂²×C₅	1	2	yes	—
104	C₂×#127	—	2	1	no	—
105	C₂×C₄×Q₈	—	2	1	no	—
106	C₂×SD₃₂	—	2	8	no	—
107	C₂×SD₃₂	C₂²×C₅	1	8	yes	complemented
108	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
109	#493	—	2	2	no	—
110	#493	C₂²×C₅	1	4	yes	—
111	C₄×Q₁₆	—	2	2	no	—
112	C₄×Q₁₆	C₂²×C₅	1	4	yes	complemented
113	C₂×#107	C₂²×C₅	1	1	yes	characteristic
114	C₂×#103	—	2	1	no	—
115	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
116	C₄⋊C₁₆	C₂²×C₅	1	4	yes	complemented
117	#357	C₂²×C₅	1	8	yes	complemented
118	C₂³×C₅	—	4	4	no	—
119	C₂³×C₅	—	8	4	no	—
120	C₂³×C₅	—	8	4	no	—
121	C₂³×C₅	—	8	4	no	—
122	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
123	C₅×Q₈	—	2	4	no	—
124	C₅×Q₈	—	4	2	no	—
125	C₅×Q₈	—	4	4	no	—
126	C₅×D₄	—	4	4	no	—
127	C₅×D₄	—	4	4	no	—
128	C₅×D₄	—	8	8	no	—
129	C₅×D₄	—	8	4	no	—
130	C₅×D₄	—	8	8	no	—
131	C₂×C₄×C₅	—	2	2	no	—
132	C₂×C₄×C₅	—	4	2	no	—
133	C₂×C₄×C₅	—	4	4	no	—
134	C₂×C₄×C₅	—	4	2	no	—
135	C₂×C₄×C₅	—	8	2	no	—
136	C₂×C₄×C₅	SD₃₂	1	4	yes	complemented
137	C₂×C₄×C₅	#121	1	2	yes	—
138	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
139	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
141	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
142	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
143	C₅×C₈	—	2	2	no	—
144	C₅×C₈	#121	1	2	yes	—
145	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
146	C₅×C₈	C₂×#41	1	1	yes	characteristic
147	C₂²×Q₈	—	2	1	no	—
148	C₂²×D₄	—	2	2	no	—
149	C₂²×D₄	—	4	1	no	—
150	C₂²×D₄	—	4	1	no	—
151	C₂×Q₁₆	—	2	1	no	—
152	C₂×Q₁₆	—	2	2	no	—
153	C₂×Q₁₆	C₅×D₄	1	4	yes	complemented
154	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
155	C₂×D₈	—	2	4	no	—
156	C₂×D₈	—	2	4	no	—
157	C₂×D₈	—	2	2	no	—
158	C₂×D₈	—	2	4	no	—
159	C₂×D₈	—	4	4	no	—
160	C₂×D₈	C₂³×C₅	1	4	yes	—
161	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
162	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
163	#127	—	2	4	no	—
164	#127	—	4	2	no	—
165	C₄×Q₈	—	2	4	no	—
166	C₄×Q₈	—	4	2	no	—
167	C₂×C₄⋊C₄	—	2	1	no	—
168	C₂×C₄⋊C₄	—	4	1	no	—
169	C₂×C₄²	—	4	1	no	—
170	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
171	SD₃₂	—	2	16	no	—
172	C₂×C₁₆	—	2	4	no	—
173	C₂×C₁₆	C₂³×C₅	1	4	yes	—
174	#107	—	2	1	no	—
175	#107	C₂³×C₅	1	2	yes	—
176	#103	—	2	4	no	—
177	C₄×C₈	C₅×D₄	1	2	yes	—
178	C₄×C₈	C₂³×C₅	1	2	yes	—
179	C₂²×C₅	—	8	4	no	—
180	C₂²×C₅	—	8	4	no	—
181	C₂²×C₅	—	8	4	no	—
182	C₂²×C₅	—	16	8	no	—
183	C₂²×C₅	#459	1	2	yes	—
184	C₂²×C₅	C₂×SD₃₂	1	2	yes	—
185	C₂²×C₅	#509	1	2	yes	—
186	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
187	C₄×C₅	—	2	2	no	—
188	C₄×C₅	—	4	4	no	—
189	C₄×C₅	—	8	2	no	—
190	C₄×C₅	#459	1	2	yes	—
191	C₄×C₅	C₂×SD₃₂	1	4	yes	complemented
192	C₄×C₅	C₂×#121	1	1	yes	characteristic
193	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
194	C₂⁴	—	4	2	no	—
195	C₂×Q₈	—	2	4	no	—
196	C₂×Q₈	—	2	2	no	—
197	C₂×Q₈	—	4	1	no	—
198	C₂×Q₈	—	4	2	no	—
199	C₂×D₄	—	2	4	no	—
200	C₂×D₄	—	4	4	no	—
201	C₂×D₄	—	4	4	no	—
202	C₂×D₄	—	4	4	no	—
203	C₂×D₄	—	4	2	no	—
204	C₂×D₄	—	4	4	no	—
205	C₂×D₄	—	8	4	no	—
206	C₂×D₄	—	8	4	no	—
207	C₂²×C₄	—	4	1	no	—
208	C₂²×C₄	—	4	1	no	—
209	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
210	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
212	Q₁₆	—	2	4	no	—
213	Q₁₆	—	2	2	no	—
214	Q₁₆	C₂×C₅×D₄	1	4	yes	complemented
215	D₈	—	2	4	no	—
216	D₈	—	2	4	no	—
217	D₈	—	4	8	no	—
218	D₈	—	4	4	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
221	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
222	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
223	C₂×C₈	C₅×#41	1	2	yes	—
224	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	4	1	no	—
227	C₄⋊C₄	—	4	4	no	—
228	C₄²	—	4	4	no	—
229	C₄²	C₅×D₈	1	2	yes	—
230	C₄²	C₂×C₅×D₄	1	2	yes	—
231	C₁₆	—	2	4	no	—
232	C₂×C₅	—	16	4	no	—
233	C₂×C₅	#2207	1	4	yes	direct product factor
234	C₂×C₅	C₂×#459	1	1	yes	characteristic
235	C₂×C₅	C₂²×SD₃₂	1	1	yes	characteristic
236	C₂×C₅	C₂×#509	1	1	yes	characteristic
237	C₂³	—	4	4	no	—
238	C₂³	—	8	4	no	—
239	C₂³	—	8	4	no	—
240	C₂³	—	8	4	no	—
241	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
242	Q₈	—	2	4	no	—
243	Q₈	—	4	4	no	—
244	Q₈	—	4	2	no	—
245	D₄	—	4	4	no	—
246	D₄	—	4	4	no	—
247	D₄	—	8	8	no	—
248	D₄	—	8	4	no	—
249	D₄	—	8	8	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	8	2	no	—
255	C₂×C₄	C₅×SD₃₂	1	4	yes	complemented
256	C₂×C₄	C₅×#121	1	2	yes	—
257	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
258	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
259	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
260	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
261	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
262	C₈	—	2	2	no	—
263	C₈	C₅×#121	1	2	yes	—
264	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
265	C₈	C₂×C₅×#41	1	1	yes	characteristic
266	C₅	C₂×#2207	1	1	yes	characteristic, direct product factor, Sylow
267	C₂²	—	8	4	no	—
268	C₂²	—	8	4	no	—
269	C₂²	—	8	4	no	—
270	C₂²	—	16	8	no	—
271	C₂²	C₅×#459	1	2	yes	—
272	C₂²	C₂×C₅×SD₃₂	1	2	yes	—
273	C₂²	C₅×#509	1	2	yes	—
274	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
275	C₄	—	2	2	no	—
276	C₄	—	4	4	no	—
277	C₄	—	8	2	no	—
278	C₄	C₅×#459	1	2	yes	—
279	C₄	C₂×C₅×SD₃₂	1	4	yes	complemented
280	C₄	C₂×C₅×#121	1	1	yes	characteristic
281	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
282	C₂	—	16	4	no	—
283	C₂	C₅×#2207	1	4	yes	direct product factor
284	C₂	C₂×C₅×#459	1	1	yes	characteristic
285	C₂	C₂²×C₅×SD₃₂	1	1	yes	characteristic
286	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	C₂×C₅×#493	C₂
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. C₂²×C₅×SD₃₂	C₂	semidirect
3. C₂×C₄×C₅×Q₁₆	C₂	semidirect
4. C₅×#2207	C₂	direct (N × H)
5. C₂×C₅×C₄⋊C₁₆	C₂	semidirect
6. C₂×C₅×#357	C₂	semidirect
13. C₂×C₅×SD₃₂	C₂²	semidirect
18. C₄×C₅×Q₁₆	C₂²	semidirect
22. C₅×C₄⋊C₁₆	C₂²	semidirect
23. C₅×#357	C₂²	semidirect
24. C₂×#2207	C₅	direct (N × H)
31. C₂×C₅×Q₁₆	D₄	semidirect
57. C₂²×SD₃₂	C₂×C₅	semidirect
59. C₂×C₄×Q₁₆	C₂×C₅	semidirect
60. #2207	C₂×C₅	direct (N × H)
61. C₂×C₄⋊C₁₆	C₂×C₅	semidirect
62. C₂×#357	C₂×C₅	semidirect
83. C₅×Q₁₆	C₂×D₄	semidirect
107. C₂×SD₃₂	C₂²×C₅	semidirect
112. C₄×Q₁₆	C₂²×C₅	semidirect
116. C₄⋊C₁₆	C₂²×C₅	semidirect
117. #357	C₂²×C₅	semidirect
136. C₂×C₄×C₅	SD₃₂	semidirect
153. C₂×Q₁₆	C₅×D₄	semidirect
191. C₄×C₅	C₂×SD₃₂	semidirect
214. Q₁₆	C₂×C₅×D₄	semidirect
233. C₂×C₅	#2207	direct (N × H)
255. C₂×C₄	C₅×SD₃₂	semidirect
266. C₅	C₂×#2207	direct (N × H)
279. C₄	C₂×C₅×SD₃₂	semidirect
283. C₂	C₅×#2207	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×SD₃₂	2
C₂×C₅×#493	2
C₂×C₄×C₅×Q₁₆	2
C₅×#2207	2
C₂×C₅×C₄⋊C₁₆	2
C₂×C₅×#357	2
C₂×#2207	5

Subgroups — C2×C5×#2207 #11956296

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2207 #11956296