Subgroups — C₂²×#65404 #12966392

Order 1280.

3694 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#4076	C₂	1	2	yes	maximal, complemented
3	C₂³×D₄₀	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#65404	C₂	1	24	yes	maximal, direct product factor
5	C₂²×#3953	C₂	1	2	yes	maximal, complemented
6	C₂²×#3950	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁴×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₂³×C₂²⋊D₅	—	2	2	no	—
9	C₂³×D₂₀	C₂²	1	2	yes	—
10	C₂³×C₅×C₈	—	2	8	no	—
11	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₂³×C₅×C₈	C₂²	1	12	yes	complemented
14	C₂×#4076	—	2	12	no	—
15	C₂×#4076	C₂²	1	24	yes	complemented
16	C₂²×#685	—	2	2	no	—
17	C₂²×C₄⋊Dic₅	C₂²	1	2	yes	—
18	C₂²×D₄₀	—	2	8	no	—
19	C₂²×D₄₀	C₂²	1	12	yes	complemented
20	#65404	C₂²	1	64	yes	direct product factor
21	C₂×#3953	C₂²	1	24	yes	complemented
22	C₂×#3950	C₂²	1	12	yes	complemented
23	C₂²×#466	—	5	1	no	maximal, Sylow
24	C₂⁵×C₅	D₄	1	1	yes	characteristic
25	C₂⁴×D₅	—	2	2	no	—
26	C₂³×C₄×C₅	—	2	4	no	—
27	C₂³×C₄×C₅	—	2	4	no	—
28	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₅	D₄	1	6	yes	—
30	C₂³×C₄×C₅	C₂³	1	6	yes	—
31	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂²×C₂²⋊D₅	—	2	24	no	—
33	C₂²×C₂²⋊D₅	—	4	16	no	—
34	C₂³×Dic₅	—	2	2	no	—
35	C₂²×D₂₀	—	2	8	no	—
36	C₂²×D₂₀	—	2	8	no	—
37	C₂²×D₂₀	—	2	6	no	—
38	C₂²×D₂₀	C₂³	1	12	yes	—
39	C₂²×C₅×C₈	—	2	4	no	—
40	C₂²×C₅×C₈	—	2	48	no	—
41	C₂²×C₅×C₈	D₄	1	8	yes	complemented
42	C₂²×C₅×C₈	C₂³	1	6	yes	—
43	C₂²×C₅×C₈	C₂³	1	6	yes	—
44	C₂²×C₅×C₈	C₂³	1	16	yes	complemented
45	#4076	—	2	48	no	—
46	#4076	C₂³	1	32	yes	complemented
47	C₂×#685	—	2	24	no	—
48	C₂×C₄⋊Dic₅	—	2	6	no	—
49	C₂×C₄⋊Dic₅	C₂³	1	12	yes	—
50	C₂×D₄₀	—	2	48	no	—
51	C₂×D₄₀	C₂³	1	16	yes	complemented
52	#3953	C₂³	1	32	yes	complemented
53	#3950	C₂³	1	16	yes	complemented
54	C₂³×D₈	—	5	1	no	—
55	C₂⁴×C₈	D₅	1	1	yes	characteristic, complemented
56	C₂²×#121	—	5	2	no	—
57	C₂×#466	—	5	24	no	—
58	C₂²×#107	—	5	1	no	—
59	C₂²×#102	—	5	2	no	—
60	C₂⁴×C₅	—	2	4	no	—
61	C₂⁴×C₅	—	2	4	no	—
62	C₂⁴×C₅	D₈	1	8	yes	complemented
63	C₂⁴×C₅	C₂×D₄	1	6	yes	—
64	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
65	C₂³×D₅	—	2	12	no	—
66	C₂³×D₅	—	4	8	no	—
67	C₂³×D₅	—	4	8	no	—
68	C₂³×D₅	—	4	8	no	—
69	C₂²×C₄×C₅	—	2	4	no	—
70	C₂²×C₄×C₅	—	2	24	no	—
71	C₂²×C₄×C₅	—	2	12	no	—
72	C₂²×C₄×C₅	—	2	12	no	—
73	C₂²×C₄×C₅	C₂×D₄	1	6	yes	—
74	C₂²×C₄×C₅	C₂×D₄	1	12	yes	—
75	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
76	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
77	C₂²×C₄×C₅	#41	1	4	yes	—
78	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
79	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
80	C₂×C₂²⋊D₅	—	2	32	no	—
81	C₂×C₂²⋊D₅	—	4	96	no	—
82	C₂²×Dic₅	—	2	12	no	—
83	C₂²×Dic₅	—	4	8	no	—
84	C₂×D₂₀	—	2	24	no	—
85	C₂×D₂₀	—	2	24	no	—
86	C₂×D₂₀	—	2	12	no	—
87	C₂×D₂₀	—	4	24	no	—
88	C₂×D₂₀	C₂⁴	1	8	yes	—
89	C₂×C₅×C₈	—	2	12	no	—
90	C₂×C₅×C₈	—	2	32	no	—
91	C₂×C₅×C₈	C₂×D₄	1	24	yes	complemented
92	C₂×C₅×C₈	C₂⁴	1	4	yes	—
93	C₂×C₅×C₈	C₂⁴	1	4	yes	—
94	#685	—	2	32	no	—
95	C₄⋊Dic₅	—	2	12	no	—
96	C₄⋊Dic₅	C₂⁴	1	8	yes	—
97	D₄₀	—	2	32	no	—
98	C₂³×D₄	—	5	2	no	—
99	C₂³×D₄	—	10	2	no	—
100	C₂⁴×C₄	C₂×D₅	1	1	yes	characteristic
101	C₂²×D₈	—	5	12	no	—
102	C₂²×D₈	—	10	8	no	—
103	C₂³×C₈	—	2	8	no	—
104	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
105	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
106	C₂³×C₈	C₂×D₅	1	12	yes	complemented
107	C₂×#121	—	5	24	no	—
108	C₂×#121	—	10	12	no	—
109	C₂²×C₄⋊C₄	—	5	2	no	—
110	C₂²×C₂²⋊C₄	—	10	2	no	—
111	#466	—	5	64	no	—
112	C₂×#107	—	5	12	no	—
113	C₂×#102	—	5	24	no	—
114	C₂³×C₅	—	2	12	no	—
115	C₂³×C₅	—	2	12	no	—
116	C₂³×C₅	—	2	32	no	—
117	C₂³×C₅	#121	1	4	yes	—
118	C₂³×C₅	C₂×D₈	1	4	yes	—
119	C₂³×C₅	C₂×D₈	1	24	yes	complemented
120	C₂³×C₅	#135	1	4	yes	—
121	C₂³×C₅	C₂²×D₄	1	4	yes	—
122	C₂³×C₅	C₂²×D₄	1	3	yes	—
123	C₂²×D₅	—	2	8	no	—
124	C₂²×D₅	—	4	24	no	—
125	C₂²×D₅	—	4	24	no	—
126	C₂²×D₅	—	4	24	no	—
127	C₂²×D₅	—	8	32	no	—
128	C₂×C₄×C₅	—	2	12	no	—
129	C₂×C₄×C₅	—	2	24	no	—
130	C₂×C₄×C₅	—	2	4	no	—
131	C₂×C₄×C₅	—	2	4	no	—
132	C₂×C₄×C₅	#121	1	12	yes	—
133	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
134	C₂×C₄×C₅	C₂²×D₄	1	3	yes	—
135	C₂×C₄×C₅	C₂²×D₄	1	6	yes	—
136	C₂×C₄×C₅	C₂×#41	1	6	yes	—
137	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
138	C₂²⋊D₅	—	4	64	no	—
139	C₂×Dic₅	—	2	8	no	—
140	C₂×Dic₅	—	4	24	no	—
141	D₂₀	—	2	8	no	—
142	D₂₀	—	2	8	no	—
143	D₂₀	—	4	24	no	—
144	C₅×C₈	—	2	4	no	—
145	C₅×C₈	C₂²×D₄	1	8	yes	complemented
146	C₂⁵	—	10	2	no	—
147	C₂⁵	D₂₀	1	1	yes	characteristic
148	C₂²×D₄	—	5	12	no	—
149	C₂²×D₄	—	10	24	no	—
150	C₂²×D₄	—	10	6	no	—
151	C₂²×D₄	—	10	8	no	—
152	C₂²×D₄	—	10	8	no	—
153	C₂²×D₄	—	20	16	no	—
154	C₂³×C₄	—	2	4	no	—
155	C₂³×C₄	—	2	4	no	—
156	C₂³×C₄	—	10	2	no	—
157	C₂³×C₄	D₂₀	1	6	yes	—
158	C₂³×C₄	D₂₀	1	1	yes	characteristic
159	C₂³×C₄	C₂²×D₅	1	6	yes	—
160	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
161	C₂×D₈	—	5	16	no	—
162	C₂×D₈	—	10	48	no	—
163	C₂²×C₈	—	2	4	no	—
164	C₂²×C₈	—	2	48	no	—
165	C₂²×C₈	C₂²⋊D₅	1	8	yes	complemented
166	C₂²×C₈	C₂²×D₅	1	6	yes	—
167	C₂²×C₈	C₂²×D₅	1	6	yes	—
168	C₂²×C₈	C₂²×D₅	1	16	yes	complemented
169	#121	—	5	32	no	—
170	#121	—	10	48	no	—
171	C₂×C₄⋊C₄	—	5	12	no	—
172	C₂×C₄⋊C₄	—	10	6	no	—
173	C₂×C₂²⋊C₄	—	10	24	no	—
174	#107	—	5	16	no	—
175	#102	—	5	32	no	—
176	C₂²×C₅	—	2	4	no	—
177	C₂²×C₅	—	2	48	no	—
178	C₂²×C₅	—	2	4	no	—
179	C₂²×C₅	#466	1	16	yes	complemented
180	C₂²×C₅	C₂×#121	1	6	yes	—
181	C₂²×C₅	C₂²×D₈	1	6	yes	—
182	C₂²×C₅	C₂²×D₈	1	8	yes	complemented
183	C₂²×C₅	C₂×#135	1	6	yes	—
184	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
185	C₂×D₅	—	4	8	no	—
186	C₂×D₅	—	4	8	no	—
187	C₂×D₅	—	4	8	no	—
188	C₂×D₅	—	8	48	no	—
189	C₄×C₅	—	2	4	no	—
190	C₄×C₅	C₂×#121	1	6	yes	—
191	C₄×C₅	C₂³×D₄	1	1	yes	characteristic
192	C₄×C₅	C₂²×#41	1	1	yes	characteristic
193	Dic₅	—	4	8	no	—
194	C₂⁴	—	2	4	no	—
195	C₂⁴	—	2	4	no	—
196	C₂⁴	—	10	12	no	—
197	C₂⁴	—	20	8	no	—
198	C₂⁴	—	20	8	no	—
199	C₂⁴	—	20	8	no	—
200	C₂⁴	D₄₀	1	8	yes	complemented
201	C₂⁴	C₂×D₂₀	1	6	yes	—
202	C₂⁴	C₂×D₂₀	1	1	yes	characteristic
203	C₂×D₄	—	5	8	no	—
204	C₂×D₄	—	10	32	no	—
205	C₂×D₄	—	10	12	no	—
206	C₂×D₄	—	10	24	no	—
207	C₂×D₄	—	10	24	no	—
208	C₂×D₄	—	20	96	no	—
209	C₂×D₄	—	20	24	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	12	no	—
212	C₂²×C₄	—	2	12	no	—
213	C₂²×C₄	—	2	24	no	—
214	C₂²×C₄	—	10	12	no	—
215	C₂²×C₄	—	20	8	no	—
216	C₂²×C₄	C₂×D₂₀	1	6	yes	—
217	C₂²×C₄	C₂×D₂₀	1	12	yes	—
218	C₂²×C₄	C₂×D₂₀	1	3	yes	—
219	C₂²×C₄	#709	1	4	yes	—
220	C₂²×C₄	C₂×C₂²⋊D₅	1	4	yes	—
221	C₂²×C₄	C₂³×D₅	1	4	yes	—
222	C₂²×C₄	C₂³×D₅	1	3	yes	—
223	D₈	—	10	32	no	—
224	C₂×C₈	—	2	12	no	—
225	C₂×C₈	—	2	32	no	—
226	C₂×C₈	C₂×C₂²⋊D₅	1	24	yes	complemented
227	C₂×C₈	C₂³×D₅	1	4	yes	—
228	C₂×C₈	C₂³×D₅	1	4	yes	—
229	C₄⋊C₄	—	5	8	no	—
230	C₄⋊C₄	—	10	12	no	—
231	C₂²⋊C₄	—	10	32	no	—
232	C₂×C₅	—	2	8	no	—
233	C₂×C₅	C₂×#466	1	12	yes	complemented
234	C₂×C₅	C₂²×#121	1	1	yes	characteristic
235	C₂×C₅	C₂³×D₈	1	1	yes	characteristic
236	C₂×C₅	C₂²×#135	1	1	yes	characteristic
237	D₅	—	8	8	no	—
238	C₂³	—	2	12	no	—
239	C₂³	—	2	12	no	—
240	C₂³	—	2	32	no	—
241	C₂³	—	10	8	no	—
242	C₂³	—	20	24	no	—
243	C₂³	—	20	24	no	—
244	C₂³	—	20	24	no	—
245	C₂³	—	40	32	no	—
246	C₂³	C₂×D₄₀	1	4	yes	—
247	C₂³	C₂×D₄₀	1	24	yes	complemented
248	C₂³	#4050	1	4	yes	—
249	C₂³	#4076	1	4	yes	—
250	C₂³	C₂²×D₂₀	1	4	yes	—
251	C₂³	C₂²×D₂₀	1	3	yes	—
252	D₄	—	10	8	no	—
253	D₄	—	10	8	no	—
254	D₄	—	20	64	no	—
255	D₄	—	20	24	no	—
256	C₂×C₄	—	2	12	no	—
257	C₂×C₄	—	2	24	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	10	8	no	—
261	C₂×C₄	—	20	24	no	—
262	C₂×C₄	#4076	1	12	yes	—
263	C₂×C₄	C₂²×D₂₀	1	4	yes	—
264	C₂×C₄	C₂²×D₂₀	1	3	yes	—
265	C₂×C₄	C₂×#709	1	6	yes	—
266	C₂×C₄	C₂²×C₂²⋊D₅	1	6	yes	—
267	C₂×C₄	C₂⁴×D₅	1	1	yes	characteristic
268	C₈	—	2	4	no	—
269	C₈	C₂²×C₂²⋊D₅	1	8	yes	complemented
270	C₅	C₂²×#466	1	1	yes	characteristic, complemented, Sylow
271	C₂²	—	2	4	no	—
272	C₂²	—	2	48	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	20	8	no	—
275	C₂²	—	20	8	no	—
276	C₂²	—	20	8	no	—
277	C₂²	—	40	48	no	—
278	C₂²	#65404	1	16	yes	direct product factor
279	C₂²	C₂²×D₄₀	1	6	yes	—
280	C₂²	C₂²×D₄₀	1	8	yes	complemented
281	C₂²	C₂×#4050	1	6	yes	—
282	C₂²	C₂×#4076	1	6	yes	—
283	C₂²	C₂³×D₂₀	1	1	yes	characteristic
284	C₄	—	2	4	no	—
285	C₄	—	20	8	no	—
286	C₄	C₂×#4076	1	6	yes	—
287	C₄	C₂²×#709	1	1	yes	characteristic
288	C₄	C₂³×C₂²⋊D₅	1	1	yes	characteristic
289	C₂	—	2	8	no	—
290	C₂	—	40	8	no	—
291	C₂	C₂×#65404	1	12	yes	direct product factor
292	C₂	C₂³×D₄₀	1	1	yes	characteristic
293	C₂	C₂²×#4050	1	1	yes	characteristic
294	C₂	C₂²×#4076	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×D₂₀
commutator, cosocle	C₂×C₄×C₅	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₅
Fitting	C₂⁴×C₅×C₈	C₂
socle	C₂⁴×C₅	C₂×D₄
hypercenter	C₂⁴×C₈	D₅
hypocenter	C₅	C₂²×#466
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₅×C₈	C₂	semidirect
2. C₂²×#4076	C₂	semidirect
3. C₂³×D₄₀	C₂	semidirect
4. C₂×#65404	C₂	direct (N × H)
5. C₂²×#3953	C₂	semidirect
6. C₂²×#3950	C₂	semidirect
13. C₂³×C₅×C₈	C₂²	semidirect
15. C₂×#4076	C₂²	semidirect
19. C₂²×D₄₀	C₂²	semidirect
20. #65404	C₂²	direct (N × H)
21. C₂×#3953	C₂²	semidirect
22. C₂×#3950	C₂²	semidirect
41. C₂²×C₅×C₈	D₄	semidirect
44. C₂²×C₅×C₈	C₂³	semidirect
46. #4076	C₂³	semidirect
51. C₂×D₄₀	C₂³	semidirect
52. #3953	C₂³	semidirect
53. #3950	C₂³	semidirect
55. C₂⁴×C₈	D₅	semidirect
62. C₂⁴×C₅	D₈	semidirect
91. C₂×C₅×C₈	C₂×D₄	semidirect
106. C₂³×C₈	C₂×D₅	semidirect
119. C₂³×C₅	C₂×D₈	semidirect
145. C₅×C₈	C₂²×D₄	semidirect
165. C₂²×C₈	C₂²⋊D₅	semidirect
168. C₂²×C₈	C₂²×D₅	semidirect
179. C₂²×C₅	#466	semidirect
182. C₂²×C₅	C₂²×D₈	semidirect
200. C₂⁴	D₄₀	semidirect
226. C₂×C₈	C₂×C₂²⋊D₅	semidirect
233. C₂×C₅	C₂×#466	semidirect
247. C₂³	C₂×D₄₀	semidirect
269. C₈	C₂²×C₂²⋊D₅	semidirect
270. C₅	C₂²×#466	semidirect
278. C₂²	#65404	direct (N × H)
280. C₂²	C₂²×D₄₀	semidirect
291. C₂	C₂×#65404	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₅×C₈	2
C₂²×#4076	2
C₂³×D₄₀	2
C₂×#65404	2
C₂²×#3953	2
C₂²×#3950	2
C₂²×#466	5

Subgroups — C22×#65404 #12966392

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#65404 #12966392