Subgroups — C₂²×C₅×#489 #11983326

Order 1280.

2486 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₅×#124	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#489	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₅×#103	C₂	1	2	yes	maximal, complemented
5	C₂²×C₅×#102	C₂	1	2	yes	maximal
6	C₂²×C₅×#97	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₅×Q₈	—	2	1	no	—
8	C₂³×C₅×Q₈	C₂²	1	1	yes	characteristic
9	C₂³×C₅×D₄	C₂²	1	1	yes	characteristic
10	C₂³×C₅×C₈	C₂²	1	2	yes	—
11	C₂×C₅×C₄⋊Q₈	—	2	6	no	—
12	C₂×C₅×C₄⋊Q₈	C₂²	1	12	yes	complemented
13	C₂×C₅×#124	—	2	6	no	—
14	C₂×C₅×#124	C₂²	1	12	yes	—
15	C₂²×C₅×C₄⋊C₄	—	2	1	no	—
16	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂²×C₅×C₂²⋊C₄	—	2	1	no	—
18	C₂²×C₅×C₂²⋊C₄	—	2	1	no	—
19	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
20	C₅×#489	C₂²	1	64	yes	direct product factor
21	C₂×C₅×#103	C₂²	1	24	yes	complemented
22	C₂×C₅×#102	C₂²	1	24	yes	—
23	C₂×C₅×#97	C₂²	1	12	yes	complemented
24	C₂²×#489	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂⁵×C₅	—	2	1	no	—
26	C₂²×C₅×Q₈	—	2	3	no	—
27	C₂²×C₅×Q₈	—	2	12	no	—
28	C₂²×C₅×Q₈	—	2	8	no	—
29	C₂²×C₅×Q₈	—	4	8	no	—
30	C₂²×C₅×Q₈	C₂³	1	6	yes	—
31	C₂²×C₅×D₄	—	2	3	no	—
32	C₂²×C₅×D₄	—	2	8	no	—
33	C₂²×C₅×D₄	C₂³	1	6	yes	—
34	C₂³×C₄×C₅	—	2	1	no	—
35	C₂³×C₄×C₅	—	2	2	no	—
36	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
39	C₂²×C₅×C₈	—	2	8	no	—
40	C₂²×C₅×C₈	C₂³	1	12	yes	—
41	C₅×C₄⋊Q₈	—	2	24	no	—
42	C₅×C₄⋊Q₈	C₂³	1	16	yes	complemented
43	C₅×#124	—	2	24	no	—
44	C₅×#124	C₂³	1	16	yes	—
45	C₂×C₅×C₄⋊C₄	—	2	12	no	—
46	C₂×C₅×C₄⋊C₄	—	2	6	no	—
47	C₂×C₅×C₄⋊C₄	C₂³	1	12	yes	—
48	C₂×C₅×C₂²⋊C₄	—	2	12	no	—
49	C₂×C₅×C₂²⋊C₄	—	2	12	no	—
50	C₂×C₄²×C₅	D₄	1	6	yes	—
51	C₂×C₄²×C₅	C₂³	1	6	yes	—
52	C₅×#103	C₂³	1	32	yes	complemented
53	C₅×#102	C₂³	1	32	yes	—
54	C₅×#97	C₂³	1	16	yes	complemented
55	C₂²×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic, complemented
56	C₂²×#124	C₂×C₅	1	1	yes	characteristic
57	C₂×#489	C₂×C₅	1	24	yes	direct product factor
58	C₂²×#103	C₂×C₅	1	2	yes	complemented
59	C₂²×#102	C₂×C₅	1	2	yes	—
60	C₂²×#97	C₂×C₅	1	1	yes	characteristic, complemented
61	C₂⁴×C₅	—	2	6	no	—
62	C₂⁴×C₅	—	4	4	no	—
63	C₂⁴×C₅	—	4	4	no	—
64	C₂⁴×C₅	—	4	4	no	—
65	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
66	C₂×C₅×Q₈	—	2	6	no	—
67	C₂×C₅×Q₈	—	2	16	no	—
68	C₂×C₅×Q₈	—	2	24	no	—
69	C₂×C₅×Q₈	—	4	48	no	—
70	C₂×C₅×Q₈	—	4	12	no	—
71	C₂×C₅×Q₈	C₂⁴	1	4	yes	—
72	C₂×C₅×D₄	—	2	6	no	—
73	C₂×C₅×D₄	—	2	24	no	—
74	C₂×C₅×D₄	—	4	12	no	—
75	C₂×C₅×D₄	C₂⁴	1	4	yes	—
76	C₂²×C₄×C₅	—	2	4	no	—
77	C₂²×C₄×C₅	—	2	4	no	—
78	C₂²×C₄×C₅	—	2	6	no	—
79	C₂²×C₄×C₅	—	2	12	no	—
80	C₂²×C₄×C₅	—	4	4	no	—
81	C₂²×C₄×C₅	—	4	8	no	—
82	C₂²×C₄×C₅	C₂×D₄	1	6	yes	—
83	C₂²×C₄×C₅	C₂×D₄	1	6	yes	—
84	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
85	C₂²×C₄×C₅	#41	1	8	yes	—
86	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
87	C₂×C₅×C₈	—	2	24	no	—
88	C₂×C₅×C₈	C₂⁴	1	8	yes	—
89	C₅×C₄⋊C₄	—	2	16	no	—
90	C₅×C₄⋊C₄	—	2	12	no	—
91	C₅×C₄⋊C₄	C₂⁴	1	8	yes	—
92	C₅×C₂²⋊C₄	—	2	16	no	—
93	C₅×C₂²⋊C₄	—	2	16	no	—
94	C₄²×C₅	C₂×D₄	1	12	yes	—
95	C₄²×C₅	C₂⁴	1	4	yes	—
96	C₂³×Q₈	—	2	1	no	—
97	C₂³×Q₈	C₂²×C₅	1	1	yes	characteristic
98	C₂³×D₄	C₂²×C₅	1	1	yes	characteristic
99	C₂³×C₈	C₂²×C₅	1	2	yes	—
100	C₂×C₄⋊Q₈	—	2	6	no	—
101	C₂×C₄⋊Q₈	C₂²×C₅	1	12	yes	complemented
102	C₂×#124	—	2	6	no	—
103	C₂×#124	C₂²×C₅	1	12	yes	—
104	C₂²×C₄⋊C₄	—	2	1	no	—
105	C₂²×C₄⋊C₄	C₂²×C₅	1	2	yes	—
106	C₂²×C₂²⋊C₄	—	2	1	no	—
107	C₂²×C₂²⋊C₄	—	2	1	no	—
108	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
109	#489	C₂²×C₅	1	64	yes	direct product factor
110	C₂×#103	C₂²×C₅	1	24	yes	complemented
111	C₂×#102	C₂²×C₅	1	24	yes	—
112	C₂×#97	C₂²×C₅	1	12	yes	complemented
113	C₂³×C₅	—	2	4	no	—
114	C₂³×C₅	—	4	12	no	—
115	C₂³×C₅	—	4	12	no	—
116	C₂³×C₅	—	4	12	no	—
117	C₂³×C₅	—	8	16	no	—
118	C₂³×C₅	#124	1	4	yes	—
119	C₂³×C₅	AGL(1,ℤ/8)	1	4	yes	—
120	C₂³×C₅	#137	1	4	yes	—
121	C₂³×C₅	C₂²×D₄	1	3	yes	—
122	C₅×Q₈	—	2	8	no	—
123	C₅×Q₈	—	4	32	no	—
124	C₅×Q₈	—	4	12	no	—
125	C₅×D₄	—	2	8	no	—
126	C₅×D₄	—	4	12	no	—
127	C₂×C₄×C₅	—	2	12	no	—
128	C₂×C₄×C₅	—	2	12	no	—
129	C₂×C₄×C₅	—	2	4	no	—
130	C₂×C₄×C₅	—	2	8	no	—
131	C₂×C₄×C₅	—	4	12	no	—
132	C₂×C₄×C₅	—	4	24	no	—
133	C₂×C₄×C₅	#124	1	12	yes	—
134	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
135	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
136	C₂×C₄×C₅	C₂²×D₄	1	3	yes	—
137	C₂×C₄×C₅	C₂×#41	1	12	yes	—
138	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
139	C₅×C₈	—	2	8	no	—
140	C₂⁵	—	2	1	no	—
141	C₂²×Q₈	—	2	3	no	—
142	C₂²×Q₈	—	2	12	no	—
143	C₂²×Q₈	—	2	8	no	—
144	C₂²×Q₈	—	4	8	no	—
145	C₂²×Q₈	C₂³×C₅	1	6	yes	—
146	C₂²×D₄	—	2	3	no	—
147	C₂²×D₄	—	2	8	no	—
148	C₂²×D₄	C₂³×C₅	1	6	yes	—
149	C₂³×C₄	—	2	1	no	—
150	C₂³×C₄	—	2	2	no	—
151	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
152	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
153	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
154	C₂²×C₈	—	2	8	no	—
155	C₂²×C₈	C₂³×C₅	1	12	yes	—
156	C₄⋊Q₈	—	2	24	no	—
157	C₄⋊Q₈	C₂³×C₅	1	16	yes	complemented
158	#124	—	2	24	no	—
159	#124	C₂³×C₅	1	16	yes	—
160	C₂×C₄⋊C₄	—	2	12	no	—
161	C₂×C₄⋊C₄	—	2	6	no	—
162	C₂×C₄⋊C₄	C₂³×C₅	1	12	yes	—
163	C₂×C₂²⋊C₄	—	2	12	no	—
164	C₂×C₂²⋊C₄	—	2	12	no	—
165	C₂×C₄²	C₅×D₄	1	6	yes	—
166	C₂×C₄²	C₂³×C₅	1	6	yes	—
167	#103	C₂³×C₅	1	32	yes	complemented
168	#102	C₂³×C₅	1	32	yes	—
169	#97	C₂³×C₅	1	16	yes	complemented
170	C₂²×C₅	—	4	4	no	—
171	C₂²×C₅	—	4	4	no	—
172	C₂²×C₅	—	4	4	no	—
173	C₂²×C₅	—	8	24	no	—
174	C₂²×C₅	#489	1	16	yes	direct product factor
175	C₂²×C₅	C₂×#124	1	6	yes	—
176	C₂²×C₅	C₂×AGL(1,ℤ/8)	1	6	yes	—
177	C₂²×C₅	C₂×#137	1	6	yes	—
178	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
179	C₄×C₅	—	2	4	no	—
180	C₄×C₅	—	2	4	no	—
181	C₄×C₅	—	4	4	no	—
182	C₄×C₅	—	4	8	no	—
183	C₄×C₅	C₂×#124	1	6	yes	—
184	C₄×C₅	C₂²×#41	1	2	yes	—
185	C₂⁴	—	2	6	no	—
186	C₂⁴	—	4	4	no	—
187	C₂⁴	—	4	4	no	—
188	C₂⁴	—	4	4	no	—
189	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
190	C₂×Q₈	—	2	6	no	—
191	C₂×Q₈	—	2	16	no	—
192	C₂×Q₈	—	2	24	no	—
193	C₂×Q₈	—	4	48	no	—
194	C₂×Q₈	—	4	12	no	—
195	C₂×Q₈	C₂⁴×C₅	1	4	yes	—
196	C₂×D₄	—	2	6	no	—
197	C₂×D₄	—	2	24	no	—
198	C₂×D₄	—	4	12	no	—
199	C₂×D₄	C₂⁴×C₅	1	4	yes	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	2	6	no	—
203	C₂²×C₄	—	2	12	no	—
204	C₂²×C₄	—	4	4	no	—
205	C₂²×C₄	—	4	8	no	—
206	C₂²×C₄	C₂×C₅×D₄	1	6	yes	—
207	C₂²×C₄	C₂×C₅×D₄	1	6	yes	—
208	C₂²×C₄	C₂×C₅×D₄	1	3	yes	—
209	C₂²×C₄	C₅×#41	1	8	yes	—
210	C₂²×C₄	C₂⁴×C₅	1	3	yes	—
211	C₂×C₈	—	2	24	no	—
212	C₂×C₈	C₂⁴×C₅	1	8	yes	—
213	C₄⋊C₄	—	2	16	no	—
214	C₄⋊C₄	—	2	12	no	—
215	C₄⋊C₄	C₂⁴×C₅	1	8	yes	—
216	C₂²⋊C₄	—	2	16	no	—
217	C₂²⋊C₄	—	2	16	no	—
218	C₄²	C₂×C₅×D₄	1	12	yes	—
219	C₄²	C₂⁴×C₅	1	4	yes	—
220	C₂×C₅	—	8	4	no	—
221	C₂×C₅	C₂×#489	1	12	yes	direct product factor
222	C₂×C₅	C₂²×#124	1	1	yes	characteristic
223	C₂×C₅	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
224	C₂×C₅	C₂²×#137	1	1	yes	characteristic
225	C₂³	—	2	4	no	—
226	C₂³	—	4	12	no	—
227	C₂³	—	4	12	no	—
228	C₂³	—	4	12	no	—
229	C₂³	—	8	16	no	—
230	C₂³	C₅×#124	1	4	yes	—
231	C₂³	C₅×AGL(1,ℤ/8)	1	4	yes	—
232	C₂³	C₅×#137	1	4	yes	—
233	C₂³	C₂²×C₅×D₄	1	3	yes	—
234	Q₈	—	2	8	no	—
235	Q₈	—	4	32	no	—
236	Q₈	—	4	12	no	—
237	D₄	—	2	8	no	—
238	D₄	—	4	12	no	—
239	C₂×C₄	—	2	12	no	—
240	C₂×C₄	—	2	12	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	8	no	—
243	C₂×C₄	—	4	12	no	—
244	C₂×C₄	—	4	24	no	—
245	C₂×C₄	C₅×#124	1	12	yes	—
246	C₂×C₄	C₂²×C₅×D₄	1	4	yes	—
247	C₂×C₄	C₂²×C₅×D₄	1	4	yes	—
248	C₂×C₄	C₂²×C₅×D₄	1	3	yes	—
249	C₂×C₄	C₂×C₅×#41	1	12	yes	—
250	C₂×C₄	C₂⁵×C₅	1	1	yes	characteristic
251	C₈	—	2	8	no	—
252	C₅	C₂²×#489	1	1	yes	characteristic, direct product factor, Sylow
253	C₂²	—	4	4	no	—
254	C₂²	—	4	4	no	—
255	C₂²	—	4	4	no	—
256	C₂²	—	8	24	no	—
257	C₂²	C₅×#489	1	16	yes	direct product factor
258	C₂²	C₂×C₅×#124	1	6	yes	—
259	C₂²	C₂×C₅×AGL(1,ℤ/8)	1	6	yes	—
260	C₂²	C₂×C₅×#137	1	6	yes	—
261	C₂²	C₂³×C₅×D₄	1	1	yes	characteristic
262	C₄	—	2	4	no	—
263	C₄	—	2	4	no	—
264	C₄	—	4	4	no	—
265	C₄	—	4	8	no	—
266	C₄	C₂×C₅×#124	1	6	yes	—
267	C₄	C₂²×C₅×#41	1	2	yes	—
268	C₂	—	8	4	no	—
269	C₂	C₂×C₅×#489	1	12	yes	direct product factor
270	C₂	C₂²×C₅×#124	1	1	yes	characteristic
271	C₂	C₂²×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
272	C₂	C₂²×C₅×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅
prime radical	C₂³×C₅×D₄	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×C₄⋊Q₈	C₂	semidirect
3. C₂×C₅×#489	C₂	direct (N × H)
4. C₂²×C₅×#103	C₂	semidirect
6. C₂²×C₅×#97	C₂	semidirect
12. C₂×C₅×C₄⋊Q₈	C₂²	semidirect
20. C₅×#489	C₂²	direct (N × H)
21. C₂×C₅×#103	C₂²	semidirect
23. C₂×C₅×#97	C₂²	semidirect
24. C₂²×#489	C₅	direct (N × H)
42. C₅×C₄⋊Q₈	C₂³	semidirect
52. C₅×#103	C₂³	semidirect
54. C₅×#97	C₂³	semidirect
55. C₂²×C₄⋊Q₈	C₂×C₅	semidirect
57. C₂×#489	C₂×C₅	direct (N × H)
58. C₂²×#103	C₂×C₅	semidirect
60. C₂²×#97	C₂×C₅	semidirect
101. C₂×C₄⋊Q₈	C₂²×C₅	semidirect
109. #489	C₂²×C₅	direct (N × H)
110. C₂×#103	C₂²×C₅	semidirect
112. C₂×#97	C₂²×C₅	semidirect
157. C₄⋊Q₈	C₂³×C₅	semidirect
167. #103	C₂³×C₅	semidirect
169. #97	C₂³×C₅	semidirect
174. C₂²×C₅	#489	direct (N × H)
221. C₂×C₅	C₂×#489	direct (N × H)
252. C₅	C₂²×#489	direct (N × H)
257. C₂²	C₅×#489	direct (N × H)
269. C₂	C₂×C₅×#489	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×C₄⋊Q₈	2
C₂²×C₅×#124	2
C₂×C₅×#489	2
C₂²×C₅×#103	2
C₂²×C₅×#102	2
C₂²×C₅×#97	2
C₂²×#489	5

Subgroups — C22×C5×#489 #11983326

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#489 #11983326