Subgroups — #12008623

Order 1280.

390 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#436	C₂	1	1	yes	maximal, characteristic
2	#10600538	C₂	1	1	yes	maximal, characteristic, complemented
3	#10600537	C₂	1	1	yes	maximal, characteristic, complemented
4	#10598164	C₂	1	4	yes	maximal, complemented
5	C₂²×C₅×M₁₆	C₄	1	2	yes	—
6	C₂×C₄×C₅×D₄	C₄	1	1	yes	characteristic
7	C₅×#436	—	2	1	no	—
8	C₅×#436	—	2	1	no	—
9	C₅×#436	—	2	1	no	—
10	C₅×#436	—	2	1	no	—
11	C₂×C₅×C₄⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₅×C₂²⋊C₈	C₂²	1	2	yes	—
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₂²⋊#672	C₂²	1	2	yes	—
15	C₂²⋊#672	C₂²	1	2	yes	—
16	C₄⋊#672	—	2	1	no	—
17	C₄⋊#672	—	2	1	no	—
18	#64681	—	2	1	no	—
19	C₄×#672	—	2	1	no	—
20	#11489	—	5	1	no	maximal, Sylow
21	C₂²×C₅×D₄	C₈	1	1	yes	characteristic
22	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
23	C₂×C₅×M₁₆	—	2	2	no	—
24	C₂×C₅×M₁₆	—	2	2	no	—
25	C₂×C₅×M₁₆	—	4	4	no	—
26	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
28	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
29	C₄×C₅×D₄	—	2	1	no	—
30	C₄×C₅×D₄	—	2	1	no	—
31	C₄×C₅×D₄	—	2	1	no	—
32	C₄×C₅×D₄	—	2	1	no	—
33	C₂×C₅×C₄⋊C₄	C₈	1	1	yes	characteristic
34	C₂×C₅×C₂²⋊C₄	C₈	1	2	yes	—
35	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₅×C₄⋊C₈	—	2	1	no	—
37	C₅×C₄⋊C₈	—	2	1	no	—
38	C₅×C₂²⋊C₈	—	2	2	no	—
39	C₅×C₂²⋊C₈	—	2	2	no	—
40	C₄×C₅×C₈	D₄	1	1	yes	characteristic
41	C₄×C₅×C₈	D₄	1	1	yes	characteristic
42	C₄×C₅×C₈	D₄	1	1	yes	characteristic
43	C₄×C₅×C₈	D₄	1	1	yes	characteristic
44	C₂×#672	—	2	2	no	—
45	C₂×#672	—	2	2	no	—
46	C₂×#436	D₅	1	1	yes	characteristic
47	#2163	—	5	1	no	—
48	#2162	—	5	1	no	—
49	#1314	—	5	4	no	—
50	C₂⁴×C₅	C₂×C₈	1	2	yes	—
51	C₂×C₅×D₄	—	2	1	no	—
52	C₂×C₅×D₄	—	2	1	no	—
53	C₂×C₅×D₄	—	4	2	no	—
54	C₂²×C₄×C₅	—	2	2	no	—
55	C₂²×C₄×C₅	—	2	2	no	—
56	C₂²×C₄×C₅	—	4	2	no	—
57	C₂²×C₄×C₅	—	4	2	no	—
58	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₅×M₁₆	—	4	4	no	—
63	C₅×M₁₆	—	4	4	no	—
64	C₂×C₅×C₈	—	2	2	no	—
65	C₂×C₅×C₈	—	2	2	no	—
66	C₂×C₅×C₈	—	4	1	no	—
67	C₂×C₅×C₈	—	4	2	no	—
68	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
71	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
72	C₅×C₄⋊C₄	—	2	1	no	—
73	C₅×C₄⋊C₄	—	2	1	no	—
74	C₅×C₂²⋊C₄	—	2	2	no	—
75	C₅×C₂²⋊C₄	—	2	2	no	—
76	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
77	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
78	C₄²×C₅	M₁₆	1	1	yes	characteristic
79	C₄²×C₅	M₁₆	1	1	yes	characteristic
80	#672	—	4	2	no	—
81	#672	—	4	2	no	—
82	C₂²×M₁₆	Dic₅	1	2	yes	—
83	C₂×C₄×D₄	Dic₅	1	1	yes	characteristic
84	#436	—	2	1	no	—
85	#436	—	2	1	no	—
86	#436	—	2	1	no	—
87	#436	—	2	1	no	—
88	C₂×C₄⋊C₈	Dic₅	1	1	yes	characteristic
89	C₂×C₂²⋊C₈	C₂×D₅	1	2	yes	—
90	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
91	C₄⋊C₁₆	—	10	1	no	—
92	C₄⋊C₁₆	—	10	1	no	—
93	C₂²⋊C₁₆	—	5	2	no	—
94	C₂²⋊C₁₆	—	5	2	no	—
95	#346	—	10	1	no	—
96	C₄×C₁₆	—	10	1	no	—
97	C₂³×C₅	—	2	2	no	—
98	C₂³×C₅	—	2	4	no	—
99	C₂³×C₅	—	4	2	no	—
100	C₂³×C₅	—	4	2	no	—
101	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
102	C₅×D₄	—	4	2	no	—
103	C₅×D₄	—	4	2	no	—
104	C₂×C₄×C₅	—	2	2	no	—
105	C₂×C₄×C₅	—	2	2	no	—
106	C₂×C₄×C₅	—	2	1	no	—
107	C₂×C₄×C₅	—	2	1	no	—
108	C₂×C₄×C₅	—	2	2	no	—
109	C₂×C₄×C₅	—	4	4	no	—
110	C₂×C₄×C₅	—	4	2	no	—
111	C₂×C₄×C₅	—	4	2	no	—
112	C₂×C₄×C₅	—	4	2	no	—
113	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
116	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
117	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
119	C₅×C₈	—	2	1	no	—
120	C₅×C₈	—	2	1	no	—
121	C₅×C₈	—	2	1	no	—
122	C₅×C₈	—	4	2	no	—
123	C₅×C₈	#100	1	2	yes	—
124	C₂²×D₄	#168	1	1	yes	characteristic
125	C₂³×C₄	C₂×Dic₅	1	2	yes	—
126	C₂×M₁₆	—	2	2	no	—
127	C₂×M₁₆	—	2	2	no	—
128	C₂×M₁₆	—	4	4	no	—
129	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
130	C₂²×C₈	C₂×Dic₅	1	2	yes	—
131	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
132	C₄×D₄	—	2	1	no	—
133	C₄×D₄	—	2	1	no	—
134	C₄×D₄	—	2	1	no	—
135	C₄×D₄	—	2	1	no	—
136	C₂×C₄⋊C₄	#168	1	1	yes	characteristic
137	C₂×C₂²⋊C₄	#168	1	2	yes	—
138	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
139	C₂×C₁₆	—	10	2	no	—
140	C₂×C₁₆	—	10	2	no	—
141	C₄⋊C₈	—	2	1	no	—
142	C₄⋊C₈	—	2	1	no	—
143	C₂²⋊C₈	—	2	2	no	—
144	C₂²⋊C₈	—	2	2	no	—
145	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
146	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
147	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
148	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
149	C₂²×C₅	—	2	1	no	—
150	C₂²×C₅	—	2	1	no	—
151	C₂²×C₅	—	4	2	no	—
152	C₂²×C₅	—	4	2	no	—
153	C₂²×C₅	—	4	2	no	—
154	C₂²×C₅	—	8	4	no	—
155	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
156	C₂²×C₅	#504	1	2	yes	—
157	C₄×C₅	—	2	1	no	—
158	C₄×C₅	—	2	1	no	—
159	C₄×C₅	—	2	1	no	—
160	C₄×C₅	—	2	1	no	—
161	C₄×C₅	—	4	2	no	—
162	C₄×C₅	#349	1	2	yes	—
163	C₄×C₅	#410	1	1	yes	characteristic
164	C₄×C₅	C₂×#100	1	1	yes	characteristic
165	C₂⁴	C₂×#168	1	2	yes	—
166	C₂×D₄	—	2	1	no	—
167	C₂×D₄	—	2	1	no	—
168	C₂×D₄	—	4	2	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	—	4	2	no	—
172	C₂²×C₄	—	4	2	no	—
173	C₂²×C₄	C₂×#168	1	1	yes	characteristic
174	C₂²×C₄	C₂×#168	1	1	yes	characteristic
175	C₂²×C₄	C₂×#168	1	2	yes	—
176	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
177	M₁₆	—	4	4	no	—
178	M₁₆	—	4	4	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	4	1	no	—
182	C₂×C₈	—	4	2	no	—
183	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
184	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
185	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
186	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
187	C₄⋊C₄	—	2	1	no	—
188	C₄⋊C₄	—	2	1	no	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₄²	#681	1	1	yes	characteristic
192	C₄²	#681	1	1	yes	characteristic
193	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
194	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
195	C₁₆	—	20	2	no	—
196	C₁₆	—	20	2	no	—
197	C₂×C₅	—	2	2	no	—
198	C₂×C₅	—	8	2	no	—
199	C₂×C₅	#1460	1	1	yes	characteristic
200	C₂×C₅	#2113	1	1	yes	characteristic
201	C₂×C₅	C₂×#349	1	1	yes	characteristic
202	C₂³	—	2	2	no	—
203	C₂³	—	2	4	no	—
204	C₂³	—	4	2	no	—
205	C₂³	—	4	2	no	—
206	C₂³	C₂²×#168	1	1	yes	characteristic
207	D₄	—	4	2	no	—
208	D₄	—	4	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
219	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
220	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
221	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
222	C₂×C₄	C₂×#681	1	1	yes	characteristic
223	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
224	C₈	—	2	1	no	—
225	C₈	—	2	1	no	—
226	C₈	—	2	1	no	—
227	C₈	—	4	2	no	—
228	C₈	#3965	1	2	yes	—
229	C₅	#11489	1	1	yes	characteristic, complemented, Sylow
230	C₂²	—	2	1	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	4	2	no	—
233	C₂²	—	4	2	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	8	4	no	—
236	C₂²	#65429	1	2	yes	—
237	C₂²	C₂×C₂²⋊#168	1	1	yes	characteristic
238	C₄	—	2	1	no	—
239	C₄	—	2	1	no	—
240	C₄	—	2	1	no	—
241	C₄	—	2	1	no	—
242	C₄	—	4	2	no	—
243	C₄	#64773	1	2	yes	—
244	C₄	C₂×#3965	1	1	yes	characteristic
245	C₄	#65529	1	1	yes	characteristic
246	C₂	—	2	2	no	—
247	C₂	—	8	2	no	—
248	C₂	#10602394	1	1	yes	characteristic
249	C₂	C₂×#64773	1	1	yes	characteristic
250	C₂	#10602797	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₅
commutator	C₂³×C₅	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂×C₅×#436	C₂
socle	C₂²×C₅	C₂×C₂²⋊C₈
prime radical	C₂²×C₅×D₄	C₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×#436	D₅
hypocenter	C₅	#11489
LCS	G ▷ C₂³×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#436

Complements

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

N	H ≅ G/N	type
2. #10600538	C₂	semidirect
3. #10600537	C₂	semidirect
4. #10598164	C₂	semidirect
229. C₅	#11489	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#436	2
#10600538	2
#10600537	2
#10598164	2
#11489	5