Subgroups — #12070009

Order 1280.

726 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	#10610201	C₂	1	1	yes	maximal, characteristic
2	#10610189	C₂	1	2	yes	maximal, complemented
3	C₂×C₅×#322	C₂	1	1	yes	maximal, characteristic, complemented
4	#10600528	C₂	1	2	yes	maximal, complemented
5	#10600524	C₂	1	1	yes	maximal, characteristic, complemented
6	#10598871	C₂	1	8	yes	maximal, complemented
7	C₂×#4047	—	2	2	no	—
8	#65811	C₂²	1	2	yes	—
9	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	#65404	C₄	1	4	yes	—
13	#65399	—	2	2	no	—
14	#65213	C₂²	1	2	yes	—
15	#64984	C₄	1	1	yes	characteristic
16	#64983	C₄	1	2	yes	—
17	C₄×D₄₀	C₂²	1	4	yes	—
18	#64977	—	2	2	no	—
19	#64977	C₂²	1	4	yes	—
20	C₈×D₂₀	—	2	2	no	—
21	C₈×D₂₀	C₂²	1	4	yes	—
22	#64971	C₄	1	1	yes	characteristic
23	C₅×#322	C₂²	1	4	yes	—
24	#64679	C₂²	1	8	yes	—
25	#64678	C₂²	1	4	yes	—
26	#17975	—	5	1	no	maximal, Sylow
27	C₂×#709	—	2	2	no	—
28	C₂²×C₅×C₈	—	2	1	no	—
29	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
30	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
31	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
33	#4076	—	2	2	no	—
34	#4076	C₂×C₄	1	4	yes	—
35	C₂×#681	—	2	2	no	—
36	C₂×D₄₀	C₂×C₄	1	4	yes	—
37	#4047	—	4	8	no	—
38	C₂×#676	—	2	4	no	—
39	C₂×C₈×D₅	—	2	4	no	—
40	#4021	—	2	2	no	—
41	#4020	C₂×C₄	1	2	yes	—
42	C₄×D₂₀	—	2	2	no	—
43	C₄×D₂₀	C₂³	1	4	yes	—
44	#4015	C₂×C₄	1	2	yes	—
45	C₄×C₅×C₈	—	2	1	no	—
46	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
47	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
48	C₄×C₅×C₈	D₄	1	2	yes	—
49	C₄×C₅×C₈	C₂³	1	2	yes	—
50	C₄×C₅×C₈	C₂³	1	2	yes	—
51	C₄×C₅×C₈	C₂³	1	2	yes	—
52	#3953	C₂×C₄	1	8	yes	—
53	#3952	—	2	4	no	—
54	#3952	—	2	4	no	—
55	#3950	C₂×C₄	1	4	yes	—
56	C₄⋊#168	—	2	2	no	—
57	C₄⋊#168	C₂³	1	4	yes	—
58	#3169	—	5	1	no	—
59	#2931	—	5	2	no	—
60	#1581	—	5	8	no	—
61	#1567	—	5	1	no	—
62	#1492	—	5	2	no	—
63	C₂×#322	D₅	1	1	yes	characteristic, complemented
64	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	#41	1	1	yes	characteristic
67	C₂×C₂²⋊D₅	—	2	4	no	—
68	#709	—	4	4	no	—
69	#709	—	4	4	no	—
70	C₂×D₂₀	—	2	2	no	—
71	C₂×D₂₀	—	2	2	no	—
72	C₂×D₂₀	C₂²×C₄	1	4	yes	—
73	C₂×C₄×D₅	—	2	4	no	—
74	C₂×Dic₁₀	—	2	2	no	—
75	C₂×C₅×C₈	—	2	2	no	—
76	C₂×C₅×C₈	—	2	2	no	—
77	C₂×C₅×C₈	—	4	2	no	—
78	C₂×C₅×C₈	—	4	2	no	—
79	C₂×C₅×C₈	D₈	1	4	yes	complemented
80	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
81	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
82	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
83	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
84	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₅	#41	1	1	yes	characteristic
87	C₄²×C₅	C₂⁴	1	1	yes	characteristic
88	#685	—	2	4	no	—
89	#685	—	2	4	no	—
90	C₄⋊Dic₅	—	2	2	no	—
91	C₄⋊Dic₅	C₂²×C₄	1	4	yes	—
92	#681	—	4	4	no	—
93	C₂×#168	—	2	4	no	—
94	D₄₀	—	4	4	no	—
95	#676	—	4	8	no	—
96	C₈×D₅	—	4	8	no	—
97	C₂×#131	—	10	2	no	—
98	#532	—	5	2	no	—
99	#500	—	5	1	no	—
100	#493	—	5	1	no	—
101	#486	—	5	2	no	—
102	#466	—	5	4	no	—
103	C₄×D₈	—	5	4	no	—
104	#436	—	5	4	no	—
105	#436	—	10	2	no	—
106	C₈×D₄	—	5	4	no	—
107	C₈×D₄	—	10	2	no	—
108	#423	—	5	2	no	—
109	#408	—	10	2	no	—
110	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
111	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
112	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
113	#335	—	5	4	no	—
114	D₄⋊C₈	—	5	8	no	—
115	#322	C₂×D₅	1	4	yes	—
116	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
117	C₂²×D₅	—	2	4	no	—
118	C₂×C₄×C₅	—	2	2	no	—
119	C₂×C₄×C₅	—	2	2	no	—
120	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
124	C₂×C₄×C₅	#135	1	2	yes	—
125	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
128	C₂²⋊D₅	—	4	8	no	—
129	C₂×Dic₅	—	2	4	no	—
130	D₂₀	—	2	8	no	—
131	D₂₀	—	4	4	no	—
132	D₂₀	—	4	4	no	—
133	C₄×D₅	—	4	4	no	—
134	C₄×D₅	—	4	4	no	—
135	Dic₁₀	—	4	4	no	—
136	C₅×C₈	—	2	2	no	—
137	C₅×C₈	—	2	2	no	—
138	C₅×C₈	—	2	2	no	—
139	C₅×C₈	—	4	2	no	—
140	C₅×C₈	C₂×D₈	1	4	yes	—
141	#168	—	4	4	no	—
142	C₂×#41	—	10	2	no	—
143	C₂×D₈	—	5	4	no	—
144	#131	—	20	8	no	—
145	C₂×M₁₆	—	10	2	no	—
146	C₂×M₁₆	—	10	4	no	—
147	C₂²×C₈	—	2	1	no	—
148	C₂²×C₈	—	10	4	no	—
149	C₂²×C₈	C₄×D₅	1	2	yes	—
150	C₂²×C₈	D₂₀	1	1	yes	characteristic
151	C₂²×C₈	D₂₀	1	1	yes	characteristic
152	C₄⋊Q₈	—	5	2	no	—
153	#127	—	5	2	no	—
154	#124	—	10	2	no	—
155	#121	—	5	4	no	—
156	#121	—	10	2	no	—
157	C₄×D₄	—	5	4	no	—
158	C₄×D₄	—	10	2	no	—
159	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
160	#107	—	5	4	no	—
161	C₄⋊C₈	—	5	4	no	—
162	C₄⋊C₈	—	10	2	no	—
163	#102	—	5	8	no	—
164	C₂²⋊C₈	—	10	4	no	—
165	C₂²⋊C₈	—	10	4	no	—
166	C₄×C₈	—	2	1	no	—
167	C₄×C₈	C₄×D₅	1	1	yes	characteristic
168	C₄×C₈	C₄×D₅	1	1	yes	characteristic
169	C₄×C₈	D₂₀	1	2	yes	—
170	C₄×C₈	C₂²×D₅	1	2	yes	—
171	C₄×C₈	C₂²×D₅	1	2	yes	—
172	C₄×C₈	C₂²×D₅	1	2	yes	—
173	C₂²×C₅	—	2	1	no	—
174	C₂²×C₅	—	2	1	no	—
175	C₂²×C₅	C₄×D₈	1	2	yes	—
176	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
177	C₂×D₅	—	4	4	no	—
178	C₂×D₅	—	4	4	no	—
179	C₂×D₅	—	4	4	no	—
180	C₄×C₅	—	2	1	no	—
181	C₄×C₅	—	2	1	no	—
182	C₄×C₅	C₄×D₈	1	2	yes	—
183	C₄×C₅	#444	1	2	yes	—
184	C₄×C₅	#568	1	2	yes	—
185	C₄×C₅	C₂²×D₈	1	1	yes	characteristic
186	C₄×C₅	C₂×#135	1	1	yes	characteristic
187	Dic₅	—	4	4	no	—
188	#41	—	20	4	no	—
189	#41	—	20	4	no	—
190	C₂×Q₈	—	10	2	no	—
191	C₂×D₄	—	5	4	no	—
192	C₂×D₄	—	10	4	no	—
193	C₂×D₄	—	10	2	no	—
194	C₂×D₄	—	10	2	no	—
195	C₂²×C₄	—	10	4	no	—
196	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
197	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
198	C₂²×C₄	#709	1	1	yes	characteristic
199	D₈	—	20	4	no	—
200	M₁₆	—	20	4	no	—
201	M₁₆	—	20	8	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	10	4	no	—
207	C₂×C₈	—	20	8	no	—
208	C₂×C₈	D₄₀	1	4	yes	complemented
209	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
210	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
211	C₂×C₈	C₂×D₂₀	1	2	yes	—
212	C₂×C₈	C₂×D₂₀	1	2	yes	—
213	C₄⋊C₄	—	5	4	no	—
214	C₄⋊C₄	—	10	2	no	—
215	C₂²⋊C₄	—	10	4	no	—
216	C₂²⋊C₄	—	10	4	no	—
217	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
218	C₄²	C₂×D₂₀	1	1	yes	characteristic
219	C₄²	#709	1	1	yes	characteristic
220	C₄²	C₂³×D₅	1	1	yes	characteristic
221	C₂×C₅	—	2	2	no	—
222	C₂×C₅	#2932	1	1	yes	characteristic
223	C₂×C₅	C₂×C₄×D₈	1	1	yes	characteristic
224	C₂×C₅	C₂×#444	1	1	yes	characteristic
225	D₅	—	8	4	no	—
226	C₂³	—	10	4	no	—
227	C₂³	C₄×D₂₀	1	1	yes	characteristic
228	Q₈	—	20	4	no	—
229	D₄	—	10	8	no	—
230	D₄	—	20	8	no	—
231	D₄	—	20	4	no	—
232	D₄	—	20	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	10	4	no	—
236	C₂×C₄	—	20	4	no	—
237	C₂×C₄	—	20	4	no	—
238	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
239	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
240	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
241	C₂×C₄	C₂×D₄₀	1	2	yes	—
242	C₂×C₄	#4050	1	2	yes	—
243	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
244	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
245	C₂×C₄	C₂×#709	1	1	yes	characteristic
246	C₈	—	2	2	no	—
247	C₈	—	2	2	no	—
248	C₈	—	2	2	no	—
249	C₈	—	4	2	no	—
250	C₈	—	20	4	no	—
251	C₈	C₂×D₄₀	1	4	yes	—
252	C₅	#17975	1	1	yes	characteristic, complemented, Sylow
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	20	4	no	—
256	C₂²	—	20	4	no	—
257	C₂²	—	20	4	no	—
258	C₂²	C₄×D₄₀	1	2	yes	—
259	C₂²	C₂×C₄×D₂₀	1	1	yes	characteristic
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	20	4	no	—
263	C₄	C₄×D₄₀	1	2	yes	—
264	C₄	#65006	1	2	yes	—
265	C₄	C₂²×D₄₀	1	1	yes	characteristic
266	C₄	C₂×#4050	1	1	yes	characteristic
267	C₄	#66079	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	40	4	no	—
270	C₂	C₂×C₄×D₄₀	1	1	yes	characteristic
271	C₂	#10610222	1	1	yes	characteristic
272	C₂	C₂×#65006	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₀
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₄²	C₂³×D₅
Fitting	C₂×C₅×#322	C₂
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	#10610201	C₂
cosocle	C₄²×C₅	C₂⁴
hypercenter	C₂×#322	D₅
hypocenter	C₅	#17975
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
2. #10610189	C₂	semidirect
3. C₂×C₅×#322	C₂	semidirect
4. #10600528	C₂	semidirect
5. #10600524	C₂	semidirect
6. #10598871	C₂	semidirect
63. C₂×#322	D₅	semidirect
79. C₂×C₅×C₈	D₈	semidirect
208. C₂×C₈	D₄₀	semidirect
252. C₅	#17975	semidirect

Maximal subgroups

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

subgroup	index
#10610201	2
#10610189	2
C₂×C₅×#322	2
#10600528	2
#10600524	2
#10598871	2
#17975	5