Subgroups — #12069130

Order 1280.

710 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10611019	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₄₀	C₂	1	1	yes	maximal, characteristic
3	C₅×#1450	C₂	1	1	yes	maximal, characteristic, complemented
4	#10601550	C₂	1	2	yes	maximal, complemented
5	#10600522	C₂	1	1	yes	maximal, characteristic, complemented
6	#10598876	C₂	1	4	yes	maximal, complemented
7	#10598845	C₂	1	4	yes	maximal, complemented
8	C₂²×D₄₀	C₄	1	1	yes	characteristic
9	C₂×C₄×D₂₀	C₂²	1	2	yes	—
10	C₅×#431	C₂²	1	1	yes	characteristic
11	C₅×#431	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₂×#3953	C₄	1	2	yes	—
14	C₂×#3950	C₄	1	1	yes	characteristic
15	#65214	C₂²	1	2	yes	—
16	#65127	—	2	4	no	—
17	#65015	—	2	4	no	—
18	C₄×D₄₀	C₄	1	2	yes	—
19	C₄×D₄₀	C₄	1	2	yes	—
20	C₄×D₄₀	C₂²	1	2	yes	—
21	C₄×D₄₀	C₂²	1	2	yes	—
22	#64977	C₂²	1	8	yes	complemented
23	C₅×#322	C₂²	1	2	yes	complemented
24	C₅×C₈²	C₂²	1	2	yes	complemented
25	#64679	C₂²	1	8	yes	complemented
26	#64678	C₂²	1	2	yes	complemented
27	#64677	C₂²	1	2	yes	complemented
28	#17895	—	5	1	no	maximal, Sylow
29	C₂²×D₂₀	C₂×C₄	1	2	yes	—
30	C₂²×C₄×D₅	—	2	2	no	—
31	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
34	C₂×#685	—	2	2	no	—
35	C₂×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
36	C₂×D₄₀	—	2	4	no	—
37	C₂×D₄₀	C₂×C₄	1	2	yes	—
38	C₂×D₄₀	C₂×C₄	1	2	yes	—
39	C₂×#676	—	2	8	no	—
40	C₄×D₂₀	—	2	2	no	—
41	C₄×D₂₀	—	2	2	no	—
42	C₄×D₂₀	C₂×C₄	1	4	yes	—
43	C₄×D₂₀	C₂³	1	4	yes	—
44	C₅×C₄⋊C₈	—	2	1	no	—
45	C₅×C₄⋊C₈	D₄	1	2	yes	—
46	C₅×C₂²⋊C₈	—	2	1	no	—
47	C₅×C₂²⋊C₈	D₄	1	2	yes	—
48	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
49	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
50	C₄×C₅×C₈	C₂³	1	2	yes	—
51	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
52	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
53	C₄×C₅×C₈	C₂³	1	2	yes	—
54	C₂²⋊#168	—	2	2	no	—
55	#3953	C₂×C₄	1	4	yes	—
56	#3953	C₂×C₄	1	4	yes	—
57	#3952	—	2	8	no	—
58	#3950	C₂×C₄	1	2	yes	—
59	#3950	C₂×C₄	1	2	yes	—
60	C₄⋊#168	C₂³	1	4	yes	—
61	C₄×#168	—	2	2	no	—
62	#2975	—	5	2	no	—
63	C₂×C₄×D₈	—	5	1	no	—
64	#1594	—	5	4	no	—
65	#1589	—	5	4	no	—
66	#1564	—	5	1	no	—
67	#1478	—	5	2	no	—
68	#1450	D₅	1	1	yes	characteristic, complemented
69	C₂³×D₅	—	2	2	no	—
70	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	#41	1	1	yes	characteristic
73	C₂²×Dic₅	—	2	2	no	—
74	C₂×D₂₀	—	2	4	no	—
75	C₂×D₂₀	—	2	4	no	—
76	C₂×D₂₀	—	2	2	no	—
77	C₂×D₂₀	C₂²×C₄	1	4	yes	—
78	C₂×C₄×D₅	—	2	4	no	—
79	C₂×C₄×D₅	—	2	4	no	—
80	C₂×C₄×D₅	—	8	4	no	—
81	C₂×C₅×C₈	—	2	2	no	—
82	C₂×C₅×C₈	—	2	2	no	—
83	C₂×C₅×C₈	M₁₆	1	4	yes	complemented
84	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
89	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₅	#41	1	1	yes	characteristic
92	C₄²×C₅	C₂⁴	1	1	yes	characteristic
93	#685	—	2	4	no	—
94	#685	—	2	4	no	—
95	C₄⋊Dic₅	—	2	2	no	—
96	C₄⋊Dic₅	C₂²×C₄	1	4	yes	—
97	C₂×#168	—	2	4	no	—
98	D₄₀	—	2	4	no	—
99	D₄₀	—	2	4	no	—
100	#676	—	8	8	no	—
101	C₂²×D₈	—	5	1	no	—
102	C₂×C₄×D₄	—	5	2	no	—
103	C₄×D₈	—	5	2	no	—
104	C₄×D₈	—	5	2	no	—
105	C₄×D₈	—	5	2	no	—
106	C₄×D₈	—	5	2	no	—
107	#436	—	5	8	no	—
108	#436	—	10	4	no	—
109	#431	—	5	2	no	—
110	#431	C₂×D₅	1	1	yes	characteristic
111	#431	C₂×D₅	1	1	yes	characteristic
112	C₂×#107	—	5	1	no	—
113	C₂×#102	—	5	2	no	—
114	#407	—	10	4	no	—
115	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
116	#335	—	5	2	no	—
117	#334	—	5	2	no	—
118	D₄⋊C₈	—	5	8	no	—
119	#322	C₂×D₅	1	2	yes	complemented
120	C₈²	C₂×D₅	1	2	yes	complemented
121	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
122	C₂²×D₅	—	2	4	no	—
123	C₂²×D₅	—	4	4	no	—
124	C₂²×D₅	—	8	2	no	—
125	C₂²×D₅	—	8	2	no	—
126	C₂×C₄×C₅	—	2	2	no	—
127	C₂×C₄×C₅	—	4	1	no	—
128	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
132	C₂×C₄×C₅	#131	1	2	yes	—
133	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
136	C₂×Dic₅	—	2	4	no	—
137	C₂×Dic₅	—	8	2	no	—
138	D₂₀	—	2	4	no	—
139	D₂₀	—	2	4	no	—
140	D₂₀	—	4	4	no	—
141	C₄×D₅	—	4	8	no	—
142	C₄×D₅	—	8	4	no	—
143	C₅×C₈	—	2	1	no	—
144	C₅×C₈	—	2	1	no	—
145	C₅×C₈	—	2	4	no	—
146	C₅×C₈	—	4	2	no	—
147	C₅×C₈	C₂×M₁₆	1	2	yes	—
148	C₅×C₈	C₂×M₁₆	1	2	yes	—
149	#168	—	4	4	no	—
150	C₂²×D₄	—	5	2	no	—
151	C₂³×C₄	—	10	2	no	—
152	C₂×D₈	—	5	2	no	—
153	C₂×D₈	—	5	2	no	—
154	C₂×D₈	—	10	4	no	—
155	C₂×M₁₆	—	10	8	no	—
156	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
157	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
158	C₄×D₄	—	5	4	no	—
159	C₄×D₄	—	5	4	no	—
160	C₄×D₄	—	10	2	no	—
161	C₄×D₄	—	10	2	no	—
162	C₂×C₄⋊C₄	—	5	2	no	—
163	C₂×C₂²⋊C₄	—	10	2	no	—
164	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
165	#107	—	5	2	no	—
166	#107	—	5	2	no	—
167	C₄⋊C₈	—	2	1	no	—
168	C₄⋊C₈	—	5	4	no	—
169	C₄⋊C₈	D₂₀	1	2	yes	—
170	#102	—	5	4	no	—
171	#102	—	5	4	no	—
172	C₂²⋊C₈	—	2	1	no	—
173	C₂²⋊C₈	—	10	2	no	—
174	C₂²⋊C₈	—	10	8	no	—
175	C₂²⋊C₈	D₂₀	1	2	yes	—
176	C₄×C₈	—	10	2	no	—
177	C₄×C₈	C₄×D₅	1	1	yes	characteristic
178	C₄×C₈	C₄×D₅	1	1	yes	characteristic
179	C₄×C₈	C₂²×D₅	1	2	yes	—
180	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
181	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
182	C₄×C₈	C₂²×D₅	1	2	yes	—
183	C₂²×C₅	—	2	1	no	—
184	C₂²×C₅	—	2	1	no	—
185	C₂²×C₅	#436	1	2	yes	—
186	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
187	C₂×D₅	—	4	4	no	—
188	C₂×D₅	—	4	4	no	—
189	C₂×D₅	—	4	4	no	—
190	C₂×D₅	—	8	8	no	—
191	C₄×C₅	—	2	1	no	—
192	C₄×C₅	—	2	1	no	—
193	C₄×C₅	—	2	1	no	—
194	C₄×C₅	#436	1	2	yes	—
195	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
196	C₄×C₅	C₂×#131	1	1	yes	characteristic
197	C₄×C₅	#576	1	2	yes	—
198	Dic₅	—	4	4	no	—
199	C₂⁴	—	10	2	no	—
200	C₂×D₄	—	5	4	no	—
201	C₂×D₄	—	10	2	no	—
202	C₂×D₄	—	10	4	no	—
203	C₂×D₄	—	10	4	no	—
204	C₂²×C₄	—	10	2	no	—
205	C₂²×C₄	—	10	4	no	—
206	C₂²×C₄	—	10	4	no	—
207	C₂²×C₄	—	40	4	no	—
208	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
210	C₂²×C₄	#709	1	1	yes	characteristic
211	D₈	—	10	4	no	—
212	D₈	—	10	4	no	—
213	M₁₆	—	40	8	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	10	4	no	—
217	C₂×C₈	#676	1	4	yes	complemented
218	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
219	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
220	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
221	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
222	C₂×C₈	C₂×D₂₀	1	4	yes	—
223	C₄⋊C₄	—	5	4	no	—
224	C₄⋊C₄	—	10	2	no	—
225	C₂²⋊C₄	—	10	4	no	—
226	C₂²⋊C₄	—	10	4	no	—
227	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
228	C₄²	C₂×D₂₀	1	1	yes	characteristic
229	C₄²	#709	1	1	yes	characteristic
230	C₄²	C₂³×D₅	1	1	yes	characteristic
231	C₂×C₅	—	2	2	no	—
232	C₂×C₅	C₂×#436	1	1	yes	characteristic
233	C₂×C₅	#2948	1	1	yes	characteristic
234	C₂×C₅	#2957	1	1	yes	characteristic
235	D₅	—	8	4	no	—
236	C₂³	—	10	4	no	—
237	C₂³	—	20	4	no	—
238	C₂³	—	40	2	no	—
239	C₂³	—	40	2	no	—
240	C₂³	C₄×D₂₀	1	1	yes	characteristic
241	D₄	—	10	4	no	—
242	D₄	—	10	4	no	—
243	D₄	—	20	4	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	10	4	no	—
247	C₂×C₄	—	20	8	no	—
248	C₂×C₄	—	40	4	no	—
249	C₂×C₄	—	40	2	no	—
250	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
251	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
252	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
253	C₂×C₄	C₂×#676	1	2	yes	—
254	C₂×C₄	#4047	1	2	yes	—
255	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
256	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
257	C₂×C₄	C₂×#709	1	1	yes	characteristic
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₈	—	2	4	no	—
261	C₈	—	4	2	no	—
262	C₈	—	20	4	no	—
263	C₈	C₂×#676	1	2	yes	—
264	C₈	C₂×#676	1	2	yes	—
265	C₅	#17895	1	1	yes	characteristic, complemented, Sylow
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	20	4	no	—
269	C₂²	—	20	4	no	—
270	C₂²	—	20	4	no	—
271	C₂²	—	40	8	no	—
272	C₂²	#64977	1	2	yes	—
273	C₂²	C₂×C₄×D₂₀	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	20	4	no	—
278	C₄	#64977	1	2	yes	—
279	C₄	C₂²×#676	1	1	yes	characteristic
280	C₄	C₂×#4047	1	1	yes	characteristic
281	C₄	#66086	1	2	yes	—
282	C₂	—	2	2	no	—
283	C₂	—	40	4	no	—
284	C₂	C₂×#64977	1	1	yes	characteristic
285	C₂	#10610292	1	1	yes	characteristic
286	C₂	#10610311	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₀
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₄²	C₂³×D₅
Fitting	C₅×#1450	C₂
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₂²×D₄₀	C₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	#1450	D₅
hypocenter	C₅	#17895
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1450

Complements

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

N	H ≅ G/N	type
1. #10611019	C₂	semidirect
3. C₅×#1450	C₂	semidirect
4. #10601550	C₂	semidirect
5. #10600522	C₂	semidirect
6. #10598876	C₂	semidirect
7. #10598845	C₂	semidirect
22. #64977	C₂²	semidirect
23. C₅×#322	C₂²	semidirect
24. C₅×C₈²	C₂²	semidirect
25. #64679	C₂²	semidirect
26. #64678	C₂²	semidirect
27. #64677	C₂²	semidirect
68. #1450	D₅	semidirect
83. C₂×C₅×C₈	M₁₆	semidirect
119. #322	C₂×D₅	semidirect
120. C₈²	C₂×D₅	semidirect
217. C₂×C₈	#676	semidirect
265. C₅	#17895	semidirect

Maximal subgroups

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

subgroup	index
#10611019	2
C₂×C₄×D₄₀	2
C₅×#1450	2
#10601550	2
#10600522	2
#10598876	2
#10598845	2
#17895	5