Subgroups — C₂²×C₅×#496 #11983353

Order 1280.

4038 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×SD₁₆	C₂	1	2	yes	maximal, complemented
2	C₂³×C₅×D₈	C₂	1	2	yes	maximal, complemented
3	C₂²×C₅×#127	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₅×#124	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×#496	C₂	1	24	yes	maximal, direct product factor
6	C₂²×C₅×#97	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₅×Q₈	C₂²	1	1	yes	characteristic
8	C₂³×C₅×D₄	—	2	2	no	—
9	C₂³×C₅×D₄	C₂²	1	1	yes	characteristic
10	C₂³×C₅×D₄	C₂²	1	2	yes	—
11	C₂²×C₅×SD₁₆	—	2	16	no	—
12	C₂²×C₅×SD₁₆	C₂²	1	24	yes	complemented
13	C₂²×C₅×D₈	—	2	16	no	—
14	C₂²×C₅×D₈	C₂²	1	24	yes	complemented
15	C₂³×C₅×C₈	C₂²	1	2	yes	—
16	C₂×C₅×#127	—	2	6	no	—
17	C₂×C₅×#127	C₂²	1	12	yes	complemented
18	C₂×C₅×#124	—	2	6	no	—
19	C₂×C₅×#124	C₂²	1	12	yes	complemented
20	C₂²×C₅×C₂²⋊C₄	—	2	2	no	—
21	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
22	C₅×#496	C₂²	1	64	yes	direct product factor
23	C₂×C₅×#97	C₂²	1	12	yes	complemented
24	C₂²×#496	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂⁵×C₅	—	2	1	no	—
26	C₂⁵×C₅	—	2	2	no	—
27	C₂²×C₅×Q₈	—	2	3	no	—
28	C₂²×C₅×Q₈	—	2	8	no	—
29	C₂²×C₅×Q₈	C₂³	1	6	yes	—
30	C₂²×C₅×D₄	—	2	8	no	—
31	C₂²×C₅×D₄	—	2	8	no	—
32	C₂²×C₅×D₄	—	2	8	no	—
33	C₂²×C₅×D₄	—	2	3	no	—
34	C₂²×C₅×D₄	—	2	6	no	—
35	C₂²×C₅×D₄	—	2	24	no	—
36	C₂²×C₅×D₄	—	4	16	no	—
37	C₂²×C₅×D₄	C₂³	1	6	yes	—
38	C₂²×C₅×D₄	C₂³	1	12	yes	—
39	C₂³×C₄×C₅	—	2	1	no	—
40	C₂³×C₄×C₅	D₄	1	2	yes	—
41	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
42	C₂×C₅×SD₁₆	—	2	96	no	—
43	C₂×C₅×SD₁₆	C₂³	1	32	yes	complemented
44	C₂×C₅×D₈	—	2	96	no	—
45	C₂×C₅×D₈	C₂³	1	32	yes	complemented
46	C₂²×C₅×C₈	D₄	1	16	yes	complemented
47	C₂²×C₅×C₈	C₂³	1	12	yes	—
48	C₅×#127	—	2	24	no	—
49	C₅×#127	C₂³	1	16	yes	complemented
50	C₅×#124	—	2	24	no	—
51	C₅×#124	C₂³	1	16	yes	complemented
52	C₂×C₅×C₂²⋊C₄	—	2	24	no	—
53	C₂×C₄²×C₅	D₄	1	6	yes	—
54	C₂×C₄²×C₅	C₂³	1	6	yes	—
55	C₅×#97	C₂³	1	16	yes	complemented
56	C₂³×SD₁₆	C₂×C₅	1	2	yes	complemented
57	C₂³×D₈	C₂×C₅	1	2	yes	complemented
58	C₂²×#127	C₂×C₅	1	1	yes	characteristic, complemented
59	C₂²×#124	C₂×C₅	1	1	yes	characteristic, complemented
60	C₂×#496	C₂×C₅	1	24	yes	direct product factor
61	C₂²×#97	C₂×C₅	1	1	yes	characteristic, complemented
62	C₂⁴×C₅	—	2	6	no	—
63	C₂⁴×C₅	—	2	12	no	—
64	C₂⁴×C₅	—	4	4	no	—
65	C₂⁴×C₅	—	4	8	no	—
66	C₂⁴×C₅	—	4	16	no	—
67	C₂⁴×C₅	—	4	8	no	—
68	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
69	C₂×C₅×Q₈	—	2	6	no	—
70	C₂×C₅×Q₈	—	2	24	no	—
71	C₂×C₅×Q₈	—	4	12	no	—
72	C₂×C₅×Q₈	C₂⁴	1	4	yes	—
73	C₂×C₅×D₄	—	2	24	no	—
74	C₂×C₅×D₄	—	2	24	no	—
75	C₂×C₅×D₄	—	2	24	no	—
76	C₂×C₅×D₄	—	2	6	no	—
77	C₂×C₅×D₄	—	2	12	no	—
78	C₂×C₅×D₄	—	2	32	no	—
79	C₂×C₅×D₄	—	4	96	no	—
80	C₂×C₅×D₄	—	4	12	no	—
81	C₂×C₅×D₄	—	4	24	no	—
82	C₂×C₅×D₄	C₂⁴	1	4	yes	—
83	C₂×C₅×D₄	C₂⁴	1	8	yes	—
84	C₂²×C₄×C₅	—	2	8	no	—
85	C₂²×C₄×C₅	—	2	6	no	—
86	C₂²×C₄×C₅	—	4	4	no	—
87	C₂²×C₄×C₅	C₂×D₄	1	12	yes	—
88	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
89	C₂²×C₄×C₅	C₂×D₄	1	8	yes	—
90	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
91	C₅×SD₁₆	—	2	64	no	—
92	C₅×D₈	—	2	64	no	—
93	C₂×C₅×C₈	C₂×D₄	1	48	yes	complemented
94	C₂×C₅×C₈	C₂⁴	1	8	yes	—
95	C₅×C₂²⋊C₄	—	2	32	no	—
96	C₄²×C₅	C₂×D₄	1	12	yes	—
97	C₄²×C₅	C₂⁴	1	4	yes	—
98	C₂³×Q₈	C₂²×C₅	1	1	yes	characteristic
99	C₂³×D₄	—	2	2	no	—
100	C₂³×D₄	C₂²×C₅	1	1	yes	characteristic
101	C₂³×D₄	C₂²×C₅	1	2	yes	—
102	C₂²×SD₁₆	—	2	16	no	—
103	C₂²×SD₁₆	C₂²×C₅	1	24	yes	complemented
104	C₂²×D₈	—	2	16	no	—
105	C₂²×D₈	C₂²×C₅	1	24	yes	complemented
106	C₂³×C₈	C₂²×C₅	1	2	yes	—
107	C₂×#127	—	2	6	no	—
108	C₂×#127	C₂²×C₅	1	12	yes	complemented
109	C₂×#124	—	2	6	no	—
110	C₂×#124	C₂²×C₅	1	12	yes	complemented
111	C₂²×C₂²⋊C₄	—	2	2	no	—
112	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
113	#496	C₂²×C₅	1	64	yes	direct product factor
114	C₂×#97	C₂²×C₅	1	12	yes	complemented
115	C₂³×C₅	—	2	4	no	—
116	C₂³×C₅	—	2	8	no	—
117	C₂³×C₅	—	4	12	no	—
118	C₂³×C₅	—	4	24	no	—
119	C₂³×C₅	—	4	48	no	—
120	C₂³×C₅	—	4	24	no	—
121	C₂³×C₅	—	8	32	no	—
122	C₂³×C₅	—	8	16	no	—
123	C₂³×C₅	#127	1	4	yes	—
124	C₂³×C₅	AGL(1,ℤ/8)	1	8	yes	—
125	C₂³×C₅	C₂²×D₄	1	3	yes	—
126	C₅×Q₈	—	2	8	no	—
127	C₅×Q₈	—	4	12	no	—
128	C₅×D₄	—	2	8	no	—
129	C₅×D₄	—	2	8	no	—
130	C₅×D₄	—	2	8	no	—
131	C₅×D₄	—	4	64	no	—
132	C₅×D₄	—	4	12	no	—
133	C₅×D₄	—	4	24	no	—
134	C₂×C₄×C₅	—	2	24	no	—
135	C₂×C₄×C₅	—	2	4	no	—
136	C₂×C₄×C₅	—	4	12	no	—
137	C₂×C₄×C₅	#127	1	12	yes	—
138	C₂×C₄×C₅	C₂²×D₄	1	8	yes	—
139	C₂×C₄×C₅	C₂²×D₄	1	3	yes	—
140	C₂×C₄×C₅	C₂²×D₄	1	12	yes	—
141	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
142	C₅×C₈	C₂²×D₄	1	16	yes	complemented
143	C₂⁵	—	2	1	no	—
144	C₂⁵	—	2	2	no	—
145	C₂²×Q₈	—	2	3	no	—
146	C₂²×Q₈	—	2	8	no	—
147	C₂²×Q₈	C₂³×C₅	1	6	yes	—
148	C₂²×D₄	—	2	8	no	—
149	C₂²×D₄	—	2	8	no	—
150	C₂²×D₄	—	2	8	no	—
151	C₂²×D₄	—	2	3	no	—
152	C₂²×D₄	—	2	6	no	—
153	C₂²×D₄	—	2	24	no	—
154	C₂²×D₄	—	4	16	no	—
155	C₂²×D₄	C₂³×C₅	1	6	yes	—
156	C₂²×D₄	C₂³×C₅	1	12	yes	—
157	C₂³×C₄	—	2	1	no	—
158	C₂³×C₄	C₅×D₄	1	2	yes	—
159	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
160	C₂×SD₁₆	—	2	96	no	—
161	C₂×SD₁₆	C₂³×C₅	1	32	yes	complemented
162	C₂×D₈	—	2	96	no	—
163	C₂×D₈	C₂³×C₅	1	32	yes	complemented
164	C₂²×C₈	C₅×D₄	1	16	yes	complemented
165	C₂²×C₈	C₂³×C₅	1	12	yes	—
166	#127	—	2	24	no	—
167	#127	C₂³×C₅	1	16	yes	complemented
168	#124	—	2	24	no	—
169	#124	C₂³×C₅	1	16	yes	complemented
170	C₂×C₂²⋊C₄	—	2	24	no	—
171	C₂×C₄²	C₅×D₄	1	6	yes	—
172	C₂×C₄²	C₂³×C₅	1	6	yes	—
173	#97	C₂³×C₅	1	16	yes	complemented
174	C₂²×C₅	—	4	4	no	—
175	C₂²×C₅	—	4	8	no	—
176	C₂²×C₅	—	4	16	no	—
177	C₂²×C₅	—	4	8	no	—
178	C₂²×C₅	—	8	48	no	—
179	C₂²×C₅	—	8	24	no	—
180	C₂²×C₅	#496	1	16	yes	direct product factor
181	C₂²×C₅	C₂×#127	1	6	yes	—
182	C₂²×C₅	C₂×AGL(1,ℤ/8)	1	12	yes	—
183	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
184	C₄×C₅	—	2	8	no	—
185	C₄×C₅	—	4	4	no	—
186	C₄×C₅	C₂×#127	1	6	yes	—
187	C₄×C₅	C₂³×D₄	1	2	yes	—
188	C₂⁴	—	2	6	no	—
189	C₂⁴	—	2	12	no	—
190	C₂⁴	—	4	4	no	—
191	C₂⁴	—	4	8	no	—
192	C₂⁴	—	4	16	no	—
193	C₂⁴	—	4	8	no	—
194	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
195	C₂×Q₈	—	2	6	no	—
196	C₂×Q₈	—	2	24	no	—
197	C₂×Q₈	—	4	12	no	—
198	C₂×Q₈	C₂⁴×C₅	1	4	yes	—
199	C₂×D₄	—	2	24	no	—
200	C₂×D₄	—	2	24	no	—
201	C₂×D₄	—	2	24	no	—
202	C₂×D₄	—	2	6	no	—
203	C₂×D₄	—	2	12	no	—
204	C₂×D₄	—	2	32	no	—
205	C₂×D₄	—	4	96	no	—
206	C₂×D₄	—	4	12	no	—
207	C₂×D₄	—	4	24	no	—
208	C₂×D₄	C₂⁴×C₅	1	4	yes	—
209	C₂×D₄	C₂⁴×C₅	1	8	yes	—
210	C₂²×C₄	—	2	8	no	—
211	C₂²×C₄	—	2	6	no	—
212	C₂²×C₄	—	4	4	no	—
213	C₂²×C₄	C₂×C₅×D₄	1	12	yes	—
214	C₂²×C₄	C₂×C₅×D₄	1	3	yes	—
215	C₂²×C₄	C₂×C₅×D₄	1	8	yes	—
216	C₂²×C₄	C₂⁴×C₅	1	3	yes	—
217	SD₁₆	—	2	64	no	—
218	D₈	—	2	64	no	—
219	C₂×C₈	C₂×C₅×D₄	1	48	yes	complemented
220	C₂×C₈	C₂⁴×C₅	1	8	yes	—
221	C₂²⋊C₄	—	2	32	no	—
222	C₄²	C₂×C₅×D₄	1	12	yes	—
223	C₄²	C₂⁴×C₅	1	4	yes	—
224	C₂×C₅	—	8	8	no	—
225	C₂×C₅	—	8	4	no	—
226	C₂×C₅	C₂×#496	1	12	yes	direct product factor
227	C₂×C₅	C₂²×#127	1	1	yes	characteristic
228	C₂×C₅	C₂²×AGL(1,ℤ/8)	1	2	yes	—
229	C₂³	—	2	4	no	—
230	C₂³	—	2	8	no	—
231	C₂³	—	4	12	no	—
232	C₂³	—	4	24	no	—
233	C₂³	—	4	48	no	—
234	C₂³	—	4	24	no	—
235	C₂³	—	8	32	no	—
236	C₂³	—	8	16	no	—
237	C₂³	C₅×#127	1	4	yes	—
238	C₂³	C₅×AGL(1,ℤ/8)	1	8	yes	—
239	C₂³	C₂²×C₅×D₄	1	3	yes	—
240	Q₈	—	2	8	no	—
241	Q₈	—	4	12	no	—
242	D₄	—	2	8	no	—
243	D₄	—	2	8	no	—
244	D₄	—	2	8	no	—
245	D₄	—	4	64	no	—
246	D₄	—	4	12	no	—
247	D₄	—	4	24	no	—
248	C₂×C₄	—	2	24	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	4	12	no	—
251	C₂×C₄	C₅×#127	1	12	yes	—
252	C₂×C₄	C₂²×C₅×D₄	1	8	yes	—
253	C₂×C₄	C₂²×C₅×D₄	1	3	yes	—
254	C₂×C₄	C₂²×C₅×D₄	1	12	yes	—
255	C₂×C₄	C₂⁵×C₅	1	1	yes	characteristic
256	C₈	C₂²×C₅×D₄	1	16	yes	complemented
257	C₅	C₂²×#496	1	1	yes	characteristic, direct product factor, Sylow
258	C₂²	—	4	4	no	—
259	C₂²	—	4	8	no	—
260	C₂²	—	4	16	no	—
261	C₂²	—	4	8	no	—
262	C₂²	—	8	48	no	—
263	C₂²	—	8	24	no	—
264	C₂²	C₅×#496	1	16	yes	direct product factor
265	C₂²	C₂×C₅×#127	1	6	yes	—
266	C₂²	C₂×C₅×AGL(1,ℤ/8)	1	12	yes	—
267	C₂²	C₂³×C₅×D₄	1	1	yes	characteristic
268	C₄	—	2	8	no	—
269	C₄	—	4	4	no	—
270	C₄	C₂×C₅×#127	1	6	yes	—
271	C₄	C₂³×C₅×D₄	1	2	yes	—
272	C₂	—	8	8	no	—
273	C₂	—	8	4	no	—
274	C₂	C₂×C₅×#496	1	12	yes	direct product factor
275	C₂	C₂²×C₅×#127	1	1	yes	characteristic
276	C₂	C₂²×C₅×AGL(1,ℤ/8)	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅
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. C₂³×C₅×SD₁₆	C₂	semidirect
2. C₂³×C₅×D₈	C₂	semidirect
3. C₂²×C₅×#127	C₂	semidirect
4. C₂²×C₅×#124	C₂	semidirect
5. C₂×C₅×#496	C₂	direct (N × H)
6. C₂²×C₅×#97	C₂	semidirect
12. C₂²×C₅×SD₁₆	C₂²	semidirect
14. C₂²×C₅×D₈	C₂²	semidirect
17. C₂×C₅×#127	C₂²	semidirect
19. C₂×C₅×#124	C₂²	semidirect
22. C₅×#496	C₂²	direct (N × H)
23. C₂×C₅×#97	C₂²	semidirect
24. C₂²×#496	C₅	direct (N × H)
43. C₂×C₅×SD₁₆	C₂³	semidirect
45. C₂×C₅×D₈	C₂³	semidirect
46. C₂²×C₅×C₈	D₄	semidirect
49. C₅×#127	C₂³	semidirect
51. C₅×#124	C₂³	semidirect
55. C₅×#97	C₂³	semidirect
56. C₂³×SD₁₆	C₂×C₅	semidirect
57. C₂³×D₈	C₂×C₅	semidirect
58. C₂²×#127	C₂×C₅	semidirect
59. C₂²×#124	C₂×C₅	semidirect
60. C₂×#496	C₂×C₅	direct (N × H)
61. C₂²×#97	C₂×C₅	semidirect
93. C₂×C₅×C₈	C₂×D₄	semidirect
103. C₂²×SD₁₆	C₂²×C₅	semidirect
105. C₂²×D₈	C₂²×C₅	semidirect
108. C₂×#127	C₂²×C₅	semidirect
110. C₂×#124	C₂²×C₅	semidirect
113. #496	C₂²×C₅	direct (N × H)
114. C₂×#97	C₂²×C₅	semidirect
142. C₅×C₈	C₂²×D₄	semidirect
161. C₂×SD₁₆	C₂³×C₅	semidirect
163. C₂×D₈	C₂³×C₅	semidirect
164. C₂²×C₈	C₅×D₄	semidirect
167. #127	C₂³×C₅	semidirect
169. #124	C₂³×C₅	semidirect
173. #97	C₂³×C₅	semidirect
180. C₂²×C₅	#496	direct (N × H)
219. C₂×C₈	C₂×C₅×D₄	semidirect
226. C₂×C₅	C₂×#496	direct (N × H)
256. C₈	C₂²×C₅×D₄	semidirect
257. C₅	C₂²×#496	direct (N × H)
264. C₂²	C₅×#496	direct (N × H)
274. C₂	C₂×C₅×#496	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₅×SD₁₆	2
C₂³×C₅×D₈	2
C₂²×C₅×#127	2
C₂²×C₅×#124	2
C₂×C₅×#496	2
C₂²×C₅×#97	2
C₂²×#496	5

Subgroups — C22×C5×#496 #11983353

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#496 #11983353