Subgroups — C₂×C₅×#2209 #11956297

Order 1280.

702 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×Q₃₂	C₂	1	2	yes	maximal
2	C₂×C₅×#494	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic
4	C₅×#2209	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×#358	C₂	1	2	yes	maximal
7	C₂²×C₅×Q₁₆	—	2	1	no	—
8	C₂²×C₅×Q₁₆	C₂²	1	2	yes	—
9	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₅×C₄⋊Q₈	—	2	1	no	—
11	C₂×C₄×C₅×Q₈	—	2	1	no	—
12	C₂×C₅×Q₃₂	—	2	8	no	—
13	C₂×C₅×Q₃₂	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	—
17	C₄×C₅×Q₁₆	—	2	2	no	—
18	C₄×C₅×Q₁₆	C₂²	1	4	yes	—
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	—
23	C₅×#358	C₂²	1	8	yes	—
24	C₂×#2209	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂²×C₅×Q₈	—	2	1	no	—
26	C₂²×C₅×Q₈	—	2	2	no	—
27	C₂×C₅×Q₁₆	—	2	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	2	no	—
32	C₂×C₅×Q₁₆	—	2	4	no	—
33	C₂×C₅×Q₁₆	—	4	4	no	—
34	C₂×C₅×Q₁₆	D₄	1	4	yes	—
35	C₂×C₅×Q₁₆	C₂³	1	4	yes	—
36	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
37	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
38	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
39	C₅×C₄⋊Q₈	—	2	4	no	—
40	C₅×C₄⋊Q₈	—	4	2	no	—
41	C₄×C₅×Q₈	—	2	4	no	—
42	C₄×C₅×Q₈	—	4	2	no	—
43	C₂×C₅×C₄⋊C₄	—	2	1	no	—
44	C₂×C₅×C₄⋊C₄	—	4	1	no	—
45	C₂×C₅×C₄⋊C₄	—	4	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×Q₃₂	—	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×Q₃₂	C₂×C₅	1	2	yes	—
58	C₂×#494	C₂×C₅	1	1	yes	characteristic
59	C₂×C₄×Q₁₆	C₂×C₅	1	1	yes	characteristic
60	#2209	C₂×C₅	1	8	yes	direct product factor
61	C₂×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic
62	C₂×#358	C₂×C₅	1	2	yes	—
63	C₂×C₅×Q₈	—	2	4	no	—
64	C₂×C₅×Q₈	—	2	2	no	—
65	C₂×C₅×Q₈	—	2	4	no	—
66	C₂×C₅×Q₈	—	4	1	no	—
67	C₂×C₅×Q₈	—	4	2	no	—
68	C₂×C₅×Q₈	—	4	2	no	—
69	C₂×C₅×Q₈	—	4	4	no	—
70	C₂×C₅×Q₈	—	4	4	no	—
71	C₂²×C₄×C₅	—	4	1	no	—
72	C₂²×C₄×C₅	—	4	1	no	—
73	C₂²×C₄×C₅	—	4	2	no	—
74	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₅×Q₁₆	—	2	4	no	—
78	C₅×Q₁₆	—	2	4	no	—
79	C₅×Q₁₆	—	2	2	no	—
80	C₅×Q₁₆	—	2	4	no	—
81	C₅×Q₁₆	—	4	8	no	—
82	C₅×Q₁₆	—	4	4	no	—
83	C₅×Q₁₆	C₂×D₄	1	4	yes	—
84	C₂×C₅×C₈	—	2	2	no	—
85	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
86	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
87	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
88	C₂×C₅×C₈	#41	1	2	yes	—
89	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
90	C₅×C₄⋊C₄	—	2	2	no	—
91	C₅×C₄⋊C₄	—	4	1	no	—
92	C₅×C₄⋊C₄	—	4	4	no	—
93	C₅×C₄⋊C₄	—	4	4	no	—
94	C₅×C₄⋊C₄	—	4	4	no	—
95	C₄²×C₅	—	4	4	no	—
96	C₄²×C₅	D₈	1	2	yes	—
97	C₄²×C₅	C₂×D₄	1	2	yes	—
98	C₅×C₁₆	—	2	4	no	—
99	C₂²×Q₁₆	—	2	1	no	—
100	C₂²×Q₁₆	C₂²×C₅	1	2	yes	—
101	C₂²×Q₁₆	C₂²×C₅	1	1	yes	characteristic
102	C₂×C₄⋊Q₈	—	2	1	no	—
103	C₂×C₄×Q₈	—	2	1	no	—
104	C₂×Q₃₂	—	2	8	no	—
105	C₂×Q₃₂	C₂²×C₅	1	8	yes	—
106	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
107	#494	—	2	2	no	—
108	#494	C₂²×C₅	1	4	yes	—
109	C₄×Q₁₆	—	2	2	no	—
110	C₄×Q₁₆	C₂²×C₅	1	4	yes	—
111	C₂×#107	C₂²×C₅	1	1	yes	characteristic
112	C₂×#103	—	2	1	no	—
113	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
114	C₄⋊C₁₆	C₂²×C₅	1	4	yes	—
115	#358	C₂²×C₅	1	8	yes	—
116	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
117	C₅×Q₈	—	2	4	no	—
118	C₅×Q₈	—	4	2	no	—
119	C₅×Q₈	—	4	4	no	—
120	C₅×Q₈	—	4	4	no	—
121	C₅×Q₈	—	4	4	no	—
122	C₅×Q₈	—	8	4	no	—
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	—	4	2	no	—
125	C₂×C₄×C₅	—	4	4	no	—
126	C₂×C₄×C₅	—	4	2	no	—
127	C₂×C₄×C₅	—	4	4	no	—
128	C₂×C₄×C₅	—	8	4	no	—
129	C₂×C₄×C₅	—	8	2	no	—
130	C₂×C₄×C₅	Q₃₂	1	4	yes	complemented
131	C₂×C₄×C₅	#121	1	2	yes	—
132	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
133	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
136	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
137	C₅×C₈	—	2	2	no	—
138	C₅×C₈	#121	1	2	yes	—
139	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
140	C₅×C₈	C₂×#41	1	1	yes	characteristic
141	C₂²×Q₈	—	2	1	no	—
142	C₂²×Q₈	—	2	2	no	—
143	C₂×Q₁₆	—	2	1	no	—
144	C₂×Q₁₆	—	2	4	no	—
145	C₂×Q₁₆	—	2	4	no	—
146	C₂×Q₁₆	—	2	2	no	—
147	C₂×Q₁₆	—	2	2	no	—
148	C₂×Q₁₆	—	2	4	no	—
149	C₂×Q₁₆	—	4	4	no	—
150	C₂×Q₁₆	C₅×D₄	1	4	yes	—
151	C₂×Q₁₆	C₂³×C₅	1	4	yes	—
152	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
153	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
154	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
155	C₄⋊Q₈	—	2	4	no	—
156	C₄⋊Q₈	—	4	2	no	—
157	C₄×Q₈	—	2	4	no	—
158	C₄×Q₈	—	4	2	no	—
159	C₂×C₄⋊C₄	—	2	1	no	—
160	C₂×C₄⋊C₄	—	4	1	no	—
161	C₂×C₄⋊C₄	—	4	1	no	—
162	C₂×C₄⋊C₄	—	4	1	no	—
163	C₂×C₄²	—	4	1	no	—
164	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
165	Q₃₂	—	2	16	no	—
166	C₂×C₁₆	—	2	4	no	—
167	C₂×C₁₆	C₂³×C₅	1	4	yes	—
168	#107	—	2	1	no	—
169	#107	C₂³×C₅	1	2	yes	—
170	#103	—	2	4	no	—
171	C₄×C₈	C₅×D₄	1	2	yes	—
172	C₄×C₈	C₂³×C₅	1	2	yes	—
173	C₂²×C₅	#459	1	2	yes	—
174	C₂²×C₅	C₂×Q₃₂	1	2	yes	—
175	C₂²×C₅	#510	1	2	yes	—
176	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
177	C₄×C₅	—	2	2	no	—
178	C₄×C₅	—	4	4	no	—
179	C₄×C₅	—	8	2	no	—
180	C₄×C₅	—	8	4	no	—
181	C₄×C₅	#459	1	2	yes	—
182	C₄×C₅	C₂×Q₃₂	1	4	yes	complemented
183	C₄×C₅	C₂×#121	1	1	yes	characteristic
184	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
185	C₂×Q₈	—	2	4	no	—
186	C₂×Q₈	—	2	2	no	—
187	C₂×Q₈	—	2	4	no	—
188	C₂×Q₈	—	4	1	no	—
189	C₂×Q₈	—	4	2	no	—
190	C₂×Q₈	—	4	2	no	—
191	C₂×Q₈	—	4	4	no	—
192	C₂×Q₈	—	4	4	no	—
193	C₂²×C₄	—	4	1	no	—
194	C₂²×C₄	—	4	1	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
197	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
199	Q₁₆	—	2	4	no	—
200	Q₁₆	—	2	4	no	—
201	Q₁₆	—	2	2	no	—
202	Q₁₆	—	2	4	no	—
203	Q₁₆	—	4	8	no	—
204	Q₁₆	—	4	4	no	—
205	Q₁₆	C₂×C₅×D₄	1	4	yes	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
208	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
209	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
210	C₂×C₈	C₅×#41	1	2	yes	—
211	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	4	1	no	—
214	C₄⋊C₄	—	4	4	no	—
215	C₄⋊C₄	—	4	4	no	—
216	C₄⋊C₄	—	4	4	no	—
217	C₄²	—	4	4	no	—
218	C₄²	C₅×D₈	1	2	yes	—
219	C₄²	C₂×C₅×D₄	1	2	yes	—
220	C₁₆	—	2	4	no	—
221	C₂×C₅	#2209	1	4	yes	direct product factor
222	C₂×C₅	C₂×#459	1	1	yes	characteristic
223	C₂×C₅	C₂²×Q₃₂	1	1	yes	characteristic
224	C₂×C₅	C₂×#510	1	1	yes	characteristic
225	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
226	Q₈	—	2	4	no	—
227	Q₈	—	4	4	no	—
228	Q₈	—	4	2	no	—
229	Q₈	—	4	4	no	—
230	Q₈	—	4	4	no	—
231	Q₈	—	8	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	8	4	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	C₅×Q₃₂	1	4	yes	complemented
240	C₂×C₄	C₅×#121	1	2	yes	—
241	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
242	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
243	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
244	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
245	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
246	C₈	—	2	2	no	—
247	C₈	C₅×#121	1	2	yes	—
248	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
249	C₈	C₂×C₅×#41	1	1	yes	characteristic
250	C₅	C₂×#2209	1	1	yes	characteristic, direct product factor, Sylow
251	C₂²	C₅×#459	1	2	yes	—
252	C₂²	C₂×C₅×Q₃₂	1	2	yes	—
253	C₂²	C₅×#510	1	2	yes	—
254	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	4	4	no	—
257	C₄	—	8	4	no	—
258	C₄	—	8	2	no	—
259	C₄	C₅×#459	1	2	yes	—
260	C₄	C₂×C₅×Q₃₂	1	4	yes	complemented
261	C₄	C₂×C₅×#121	1	1	yes	characteristic
262	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
263	C₂	C₅×#2209	1	4	yes	direct product factor
264	C₂	C₂×C₅×#459	1	1	yes	characteristic
265	C₂	C₂²×C₅×Q₃₂	1	1	yes	characteristic
266	C₂	C₂×C₅×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
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
4. C₅×#2209	C₂	direct (N × H)
24. C₂×#2209	C₅	direct (N × H)
60. #2209	C₂×C₅	direct (N × H)
130. C₂×C₄×C₅	Q₃₂	semidirect
182. C₄×C₅	C₂×Q₃₂	semidirect
221. C₂×C₅	#2209	direct (N × H)
239. C₂×C₄	C₅×Q₃₂	semidirect
250. C₅	C₂×#2209	direct (N × H)
260. C₄	C₂×C₅×Q₃₂	semidirect
263. C₂	C₅×#2209	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×Q₃₂	2
C₂×C₅×#494	2
C₂×C₄×C₅×Q₁₆	2
C₅×#2209	2
C₂×C₅×C₄⋊C₁₆	2
C₂×C₅×#358	2
C₂×#2209	5

Subgroups — C2×C5×#2209 #11956297

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2209 #11956297