Subgroups — C₂×C₅×#2208 #11956294

Order 1280.

766 conjugacy classes of subgroups.

The subgroup lattice has 284 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
2	C₂×C₅×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₅×D₈	C₂	1	1	yes	maximal, characteristic
4	C₅×#2208	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×#358	C₂	1	2	yes	maximal, complemented
7	C₂²×C₅×Q₁₆	—	2	1	no	—
8	C₂²×C₅×Q₁₆	C₂²	1	2	yes	—
9	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₅×C₄⋊Q₈	—	2	1	no	—
11	C₂×C₄×C₅×D₄	—	2	1	no	—
12	C₂×C₅×SD₃₂	—	2	8	no	—
13	C₂×C₅×SD₃₂	C₂²	1	8	yes	—
14	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
15	C₅×#494	—	2	2	no	—
16	C₅×#494	C₂²	1	4	yes	complemented
17	C₄×C₅×D₈	—	2	2	no	—
18	C₄×C₅×D₈	C₂²	1	4	yes	—
19	C₂×C₅×#107	C₂²	1	1	yes	characteristic
20	C₂×C₅×#102	—	2	1	no	—
21	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
22	C₅×C₄⋊C₁₆	C₂²	1	4	yes	complemented
23	C₅×#358	C₂²	1	8	yes	complemented
24	C₂×#2208	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂²×C₅×Q₈	—	2	2	no	—
26	C₂²×C₅×D₄	—	2	1	no	—
27	C₂³×C₄×C₅	—	4	1	no	—
28	C₂×C₅×Q₁₆	—	2	4	no	—
29	C₂×C₅×Q₁₆	—	2	2	no	—
30	C₂×C₅×Q₁₆	—	2	4	no	—
31	C₂×C₅×Q₁₆	—	2	4	no	—
32	C₂×C₅×Q₁₆	—	4	4	no	—
33	C₂×C₅×Q₁₆	C₂³	1	4	yes	—
34	C₂×C₅×D₈	—	2	1	no	—
35	C₂×C₅×D₈	—	2	2	no	—
36	C₂×C₅×D₈	D₄	1	4	yes	—
37	C₂×C₅×D₈	C₂³	1	2	yes	—
38	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
39	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
40	C₅×C₄⋊Q₈	—	2	4	no	—
41	C₅×C₄⋊Q₈	—	4	2	no	—
42	C₄×C₅×D₄	—	2	4	no	—
43	C₄×C₅×D₄	—	4	2	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	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×#102	—	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	—
58	C₂×#494	C₂×C₅	1	1	yes	characteristic, complemented
59	C₂×C₄×D₈	C₂×C₅	1	1	yes	characteristic
60	#2208	C₂×C₅	1	8	yes	direct product factor
61	C₂×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
62	C₂×#358	C₂×C₅	1	2	yes	complemented
63	C₂⁴×C₅	—	4	1	no	—
64	C₂×C₅×Q₈	—	2	4	no	—
65	C₂×C₅×Q₈	—	4	2	no	—
66	C₂×C₅×Q₈	—	4	4	no	—
67	C₂×C₅×Q₈	—	4	4	no	—
68	C₂×C₅×D₄	—	2	4	no	—
69	C₂×C₅×D₄	—	2	2	no	—
70	C₂×C₅×D₄	—	4	1	no	—
71	C₂×C₅×D₄	—	4	2	no	—
72	C₂²×C₄×C₅	—	4	1	no	—
73	C₂²×C₄×C₅	—	4	2	no	—
74	C₂²×C₄×C₅	—	4	4	no	—
75	C₂²×C₄×C₅	—	8	4	no	—
76	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
77	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₅×Q₁₆	—	2	4	no	—
80	C₅×Q₁₆	—	2	4	no	—
81	C₅×Q₁₆	—	4	8	no	—
82	C₅×Q₁₆	—	4	4	no	—
83	C₅×D₈	—	2	4	no	—
84	C₅×D₈	—	2	2	no	—
85	C₅×D₈	C₂×D₄	1	4	yes	—
86	C₂×C₅×C₈	—	2	2	no	—
87	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
88	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
89	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
90	C₂×C₅×C₈	#41	1	2	yes	—
91	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
92	C₅×C₄⋊C₄	—	2	2	no	—
93	C₅×C₄⋊C₄	—	4	1	no	—
94	C₅×C₄⋊C₄	—	4	4	no	—
95	C₅×C₄⋊C₄	—	4	4	no	—
96	C₅×C₂²⋊C₄	—	4	4	no	—
97	C₄²×C₅	D₈	1	2	yes	—
98	C₄²×C₅	C₂×D₄	1	2	yes	—
99	C₅×C₁₆	—	2	4	no	—
100	C₂²×Q₁₆	—	2	1	no	—
101	C₂²×Q₁₆	C₂²×C₅	1	2	yes	—
102	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
103	C₂×C₄⋊Q₈	—	2	1	no	—
104	C₂×C₄×D₄	—	2	1	no	—
105	C₂×SD₃₂	—	2	8	no	—
106	C₂×SD₃₂	C₂²×C₅	1	8	yes	—
107	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
108	#494	—	2	2	no	—
109	#494	C₂²×C₅	1	4	yes	complemented
110	C₄×D₈	—	2	2	no	—
111	C₄×D₈	C₂²×C₅	1	4	yes	—
112	C₂×#107	C₂²×C₅	1	1	yes	characteristic
113	C₂×#102	—	2	1	no	—
114	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
115	C₄⋊C₁₆	C₂²×C₅	1	4	yes	complemented
116	#358	C₂²×C₅	1	8	yes	complemented
117	C₂³×C₅	—	4	2	no	—
118	C₂³×C₅	—	4	4	no	—
119	C₂³×C₅	—	8	2	no	—
120	C₂³×C₅	—	8	2	no	—
121	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
122	C₅×Q₈	—	4	4	no	—
123	C₅×Q₈	—	4	4	no	—
124	C₅×Q₈	—	8	4	no	—
125	C₅×D₄	—	2	4	no	—
126	C₅×D₄	—	4	2	no	—
127	C₅×D₄	—	4	4	no	—
128	C₂×C₄×C₅	—	2	2	no	—
129	C₂×C₄×C₅	—	4	2	no	—
130	C₂×C₄×C₅	—	4	4	no	—
131	C₂×C₄×C₅	—	8	4	no	—
132	C₂×C₄×C₅	—	8	2	no	—
133	C₂×C₄×C₅	—	8	8	no	—
134	C₂×C₄×C₅	SD₃₂	1	4	yes	complemented
135	C₂×C₄×C₅	#121	1	2	yes	—
136	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
137	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
139	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
140	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
141	C₅×C₈	—	2	2	no	—
142	C₅×C₈	#121	1	2	yes	—
143	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
144	C₅×C₈	C₂×#41	1	1	yes	characteristic
145	C₂²×Q₈	—	2	2	no	—
146	C₂²×D₄	—	2	1	no	—
147	C₂³×C₄	—	4	1	no	—
148	C₂×Q₁₆	—	2	4	no	—
149	C₂×Q₁₆	—	2	4	no	—
150	C₂×Q₁₆	—	2	2	no	—
151	C₂×Q₁₆	—	2	4	no	—
152	C₂×Q₁₆	—	4	4	no	—
153	C₂×Q₁₆	C₂³×C₅	1	4	yes	—
154	C₂×D₈	—	2	1	no	—
155	C₂×D₈	—	2	2	no	—
156	C₂×D₈	C₅×D₄	1	4	yes	—
157	C₂×D₈	C₂³×C₅	1	2	yes	—
158	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
159	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
160	C₄⋊Q₈	—	2	4	no	—
161	C₄⋊Q₈	—	4	2	no	—
162	C₄×D₄	—	2	4	no	—
163	C₄×D₄	—	4	2	no	—
164	C₂×C₄⋊C₄	—	2	1	no	—
165	C₂×C₄⋊C₄	—	4	1	no	—
166	C₂×C₄⋊C₄	—	4	1	no	—
167	C₂×C₂²⋊C₄	—	4	1	no	—
168	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
169	SD₃₂	—	2	16	no	—
170	C₂×C₁₆	—	2	4	no	—
171	C₂×C₁₆	C₂³×C₅	1	4	yes	—
172	#107	—	2	1	no	—
173	#107	C₂³×C₅	1	2	yes	—
174	#102	—	2	4	no	—
175	C₄×C₈	C₅×D₄	1	2	yes	—
176	C₄×C₈	C₂³×C₅	1	2	yes	—
177	C₂²×C₅	—	4	4	no	—
178	C₂²×C₅	—	8	2	no	—
179	C₂²×C₅	—	8	2	no	—
180	C₂²×C₅	—	8	8	no	—
181	C₂²×C₅	#459	1	2	yes	—
182	C₂²×C₅	C₂×SD₃₂	1	2	yes	—
183	C₂²×C₅	#510	1	2	yes	—
184	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
185	C₄×C₅	—	2	2	no	—
186	C₄×C₅	—	8	2	no	—
187	C₄×C₅	—	8	4	no	—
188	C₄×C₅	#459	1	2	yes	—
189	C₄×C₅	C₂×SD₃₂	1	4	yes	complemented
190	C₄×C₅	C₂×#121	1	1	yes	characteristic
191	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
192	C₂⁴	—	4	1	no	—
193	C₂×Q₈	—	2	4	no	—
194	C₂×Q₈	—	4	2	no	—
195	C₂×Q₈	—	4	4	no	—
196	C₂×Q₈	—	4	4	no	—
197	C₂×D₄	—	2	4	no	—
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	—	4	1	no	—
200	C₂×D₄	—	4	2	no	—
201	C₂²×C₄	—	4	1	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	—	4	4	no	—
204	C₂²×C₄	—	8	4	no	—
205	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
206	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
208	Q₁₆	—	2	4	no	—
209	Q₁₆	—	2	4	no	—
210	Q₁₆	—	4	8	no	—
211	Q₁₆	—	4	4	no	—
212	D₈	—	2	4	no	—
213	D₈	—	2	2	no	—
214	D₈	C₂×C₅×D₄	1	4	yes	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
217	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
218	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
219	C₂×C₈	C₅×#41	1	2	yes	—
220	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	4	1	no	—
223	C₄⋊C₄	—	4	4	no	—
224	C₄⋊C₄	—	4	4	no	—
225	C₂²⋊C₄	—	4	4	no	—
226	C₄²	C₅×D₈	1	2	yes	—
227	C₄²	C₂×C₅×D₄	1	2	yes	—
228	C₁₆	—	2	4	no	—
229	C₂×C₅	—	8	4	no	—
230	C₂×C₅	#2208	1	4	yes	direct product factor
231	C₂×C₅	C₂×#459	1	1	yes	characteristic
232	C₂×C₅	C₂²×SD₃₂	1	1	yes	characteristic
233	C₂×C₅	C₂×#510	1	1	yes	characteristic
234	C₂³	—	4	4	no	—
235	C₂³	—	4	2	no	—
236	C₂³	—	8	2	no	—
237	C₂³	—	8	2	no	—
238	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
239	Q₈	—	4	4	no	—
240	Q₈	—	4	4	no	—
241	Q₈	—	8	4	no	—
242	D₄	—	2	4	no	—
243	D₄	—	4	4	no	—
244	D₄	—	4	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	8	4	no	—
249	C₂×C₄	—	8	2	no	—
250	C₂×C₄	—	8	8	no	—
251	C₂×C₄	C₅×SD₃₂	1	4	yes	complemented
252	C₂×C₄	C₅×#121	1	2	yes	—
253	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
254	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
255	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
256	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
257	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
258	C₈	—	2	2	no	—
259	C₈	C₅×#121	1	2	yes	—
260	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
261	C₈	C₂×C₅×#41	1	1	yes	characteristic
262	C₅	C₂×#2208	1	1	yes	characteristic, direct product factor, Sylow
263	C₂²	—	4	4	no	—
264	C₂²	—	8	2	no	—
265	C₂²	—	8	2	no	—
266	C₂²	—	8	8	no	—
267	C₂²	C₅×#459	1	2	yes	—
268	C₂²	C₂×C₅×SD₃₂	1	2	yes	—
269	C₂²	C₅×#510	1	2	yes	—
270	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	8	4	no	—
273	C₄	—	8	2	no	—
274	C₄	C₅×#459	1	2	yes	—
275	C₄	C₂×C₅×SD₃₂	1	4	yes	complemented
276	C₄	C₂×C₅×#121	1	1	yes	characteristic
277	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
278	C₂	—	8	4	no	—
279	C₂	C₅×#2208	1	4	yes	direct product factor
280	C₂	C₂×C₅×#459	1	1	yes	characteristic
281	C₂	C₂²×C₅×SD₃₂	1	1	yes	characteristic
282	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	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
2. C₂×C₅×#494	C₂	semidirect
4. C₅×#2208	C₂	direct (N × H)
5. C₂×C₅×C₄⋊C₁₆	C₂	semidirect
6. C₂×C₅×#358	C₂	semidirect
16. C₅×#494	C₂²	semidirect
22. C₅×C₄⋊C₁₆	C₂²	semidirect
23. C₅×#358	C₂²	semidirect
24. C₂×#2208	C₅	direct (N × H)
58. C₂×#494	C₂×C₅	semidirect
60. #2208	C₂×C₅	direct (N × H)
61. C₂×C₄⋊C₁₆	C₂×C₅	semidirect
62. C₂×#358	C₂×C₅	semidirect
109. #494	C₂²×C₅	semidirect
115. C₄⋊C₁₆	C₂²×C₅	semidirect
116. #358	C₂²×C₅	semidirect
134. C₂×C₄×C₅	SD₃₂	semidirect
189. C₄×C₅	C₂×SD₃₂	semidirect
230. C₂×C₅	#2208	direct (N × H)
251. C₂×C₄	C₅×SD₃₂	semidirect
262. C₅	C₂×#2208	direct (N × H)
275. C₄	C₂×C₅×SD₃₂	semidirect
279. C₂	C₅×#2208	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×#494	2
C₂×C₄×C₅×D₈	2
C₅×#2208	2
C₂×C₅×C₄⋊C₁₆	2
C₂×C₅×#358	2
C₂×#2208	5

Subgroups — C2×C5×#2208 #11956294

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2208 #11956294