Subgroups — #12008616

Order 1280.

430 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₂₀	C₂	1	1	yes	maximal, characteristic
2	C₅×#2162	C₂	1	1	yes	maximal, characteristic, complemented
3	#10600537	C₂	1	1	yes	maximal, characteristic, complemented
4	#10598013	C₂	1	4	yes	maximal, complemented
5	C₂²×C₈×D₅	C₄	1	2	yes	—
6	C₂×C₄×D₂₀	C₄	1	1	yes	characteristic
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×#3952	C₂²	1	2	yes	—
9	C₂×C₄⋊#168	C₄	1	1	yes	characteristic
10	C₈×D₂₀	—	2	1	no	—
11	C₈×D₂₀	—	2	1	no	—
12	C₈×D₂₀	—	2	1	no	—
13	C₈×D₂₀	—	2	1	no	—
14	C₅×C₄⋊C₁₆	—	2	1	no	—
15	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
16	C₄×C₅×C₁₆	—	2	1	no	—
17	C₂²⋊#672	C₂²	1	2	yes	—
18	C₄⋊#672	—	2	1	no	—
19	C₄×#672	—	2	1	no	—
20	#11487	—	5	1	no	maximal, Sylow
21	C₂²×D₂₀	C₈	1	1	yes	characteristic
22	C₂²×C₄×D₅	C₂×C₄	1	2	yes	—
23	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂×#685	C₈	1	2	yes	—
27	C₂×C₄⋊Dic₅	C₈	1	1	yes	characteristic
28	C₂²×#168	C₂×C₄	1	2	yes	—
29	C₂×C₈×D₅	—	2	2	no	—
30	C₂×C₈×D₅	—	2	2	no	—
31	C₂×C₈×D₅	—	4	4	no	—
32	C₄×D₂₀	—	2	1	no	—
33	C₄×D₂₀	—	2	1	no	—
34	C₄×D₂₀	—	2	1	no	—
35	C₄×D₂₀	—	2	1	no	—
36	C₂×C₅×C₁₆	—	2	2	no	—
37	C₄×C₅×C₈	D₄	1	1	yes	characteristic
38	C₄×C₅×C₈	D₄	1	1	yes	characteristic
39	C₄×C₅×C₈	D₄	1	1	yes	characteristic
40	C₄×C₅×C₈	D₄	1	1	yes	characteristic
41	#3952	—	2	2	no	—
42	#3952	—	2	2	no	—
43	C₂×#672	—	2	2	no	—
44	C₄⋊#168	—	2	1	no	—
45	C₄⋊#168	—	2	1	no	—
46	C₂×C₈×D₄	—	5	1	no	—
47	#2162	—	5	1	no	—
48	#2162	D₅	1	1	yes	characteristic, complemented
49	#1314	—	5	4	no	—
50	C₂³×D₅	C₂×C₈	1	2	yes	—
51	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
54	C₂²×Dic₅	C₂×C₈	1	2	yes	—
55	C₂×D₂₀	—	2	1	no	—
56	C₂×D₂₀	—	2	1	no	—
57	C₂×D₂₀	—	2	2	no	—
58	C₂×D₂₀	C₁₆	1	4	yes	complemented
59	C₂×C₄×D₅	—	2	2	no	—
60	C₂×C₄×D₅	—	2	2	no	—
61	C₂×C₄×D₅	—	4	2	no	—
62	C₂×C₄×D₅	—	4	2	no	—
63	C₂×C₅×C₈	—	2	2	no	—
64	C₂×C₅×C₈	—	4	1	no	—
65	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
68	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₅	M₁₆	1	1	yes	characteristic
72	C₄²×C₅	M₁₆	1	1	yes	characteristic
73	#685	—	2	2	no	—
74	#685	—	2	2	no	—
75	C₄⋊Dic₅	—	2	1	no	—
76	C₄⋊Dic₅	—	2	1	no	—
77	C₂×#168	—	2	2	no	—
78	C₂×#168	—	4	2	no	—
79	C₈×D₅	—	4	4	no	—
80	C₈×D₅	—	4	4	no	—
81	C₅×C₁₆	—	4	2	no	—
82	#672	—	4	2	no	—
83	C₂³×C₈	—	5	2	no	—
84	C₂×C₄×D₄	—	5	1	no	—
85	C₈×D₄	—	10	1	no	—
86	C₈×D₄	—	10	1	no	—
87	C₈×D₄	—	10	1	no	—
88	C₈×D₄	—	10	1	no	—
89	C₂×C₄⋊C₈	—	5	1	no	—
90	C₂×C₂²⋊C₈	—	5	2	no	—
91	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
92	C₄⋊C₁₆	—	2	1	no	—
93	C₄⋊C₁₆	—	10	1	no	—
94	C₂²⋊C₁₆	—	5	2	no	—
95	C₂²⋊C₁₆	C₂×D₅	1	2	yes	—
96	C₄×C₁₆	—	2	1	no	—
97	C₄×C₁₆	—	10	1	no	—
98	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
99	C₂²×D₅	—	2	2	no	—
100	C₂²×D₅	—	2	2	no	—
101	C₂²×D₅	—	4	2	no	—
102	C₂²×D₅	—	4	2	no	—
103	C₂²×D₅	C₂×C₁₆	1	4	yes	—
104	C₂×C₄×C₅	—	2	1	no	—
105	C₂×C₄×C₅	—	2	1	no	—
106	C₂×C₄×C₅	—	4	1	no	—
107	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
108	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
109	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
110	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
111	C₂×C₄×C₅	C₂×C₁₆	1	2	yes	—
112	C₂×C₄×C₅	M₃₂	1	2	yes	—
113	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
115	C₂×Dic₅	—	2	2	no	—
116	C₂×Dic₅	—	4	2	no	—
117	D₂₀	—	2	4	no	—
118	D₂₀	—	2	4	no	—
119	C₄×D₅	—	4	4	no	—
120	C₄×D₅	—	4	2	no	—
121	C₄×D₅	—	4	2	no	—
122	C₅×C₈	—	2	1	no	—
123	C₅×C₈	—	2	1	no	—
124	C₅×C₈	—	2	1	no	—
125	C₅×C₈	#100	1	2	yes	—
126	#168	—	4	2	no	—
127	C₂²×D₄	—	5	1	no	—
128	C₂³×C₄	—	5	2	no	—
129	C₂²×C₈	—	5	2	no	—
130	C₂²×C₈	—	10	2	no	—
131	C₂²×C₈	—	10	2	no	—
132	C₂²×C₈	—	20	4	no	—
133	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
134	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
135	C₄×D₄	—	10	1	no	—
136	C₄×D₄	—	10	1	no	—
137	C₄×D₄	—	10	1	no	—
138	C₄×D₄	—	10	1	no	—
139	C₂×C₄⋊C₄	—	5	1	no	—
140	C₂×C₂²⋊C₄	—	5	2	no	—
141	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
142	C₂×C₁₆	—	2	2	no	—
143	C₂×C₁₆	—	10	2	no	—
144	C₄⋊C₈	—	10	1	no	—
145	C₄⋊C₈	—	10	1	no	—
146	C₂²⋊C₈	—	10	2	no	—
147	C₂²⋊C₈	—	10	2	no	—
148	C₄×C₈	D₂₀	1	1	yes	characteristic
149	C₄×C₈	D₂₀	1	1	yes	characteristic
150	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
151	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
152	C₂²×C₅	—	2	1	no	—
153	C₂²×C₅	—	2	1	no	—
154	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
155	C₂²×C₅	C₂²×C₁₆	1	1	yes	characteristic
156	C₂²×C₅	C₂×M₃₂	1	1	yes	characteristic
157	C₂×D₅	—	2	4	no	—
158	C₂×D₅	—	4	4	no	—
159	C₂×D₅	—	4	2	no	—
160	C₂×D₅	—	4	2	no	—
161	C₂×D₅	—	4	4	no	—
162	C₄×C₅	—	2	1	no	—
163	C₄×C₅	—	2	1	no	—
164	C₄×C₅	—	2	1	no	—
165	C₄×C₅	C₂²⋊C₁₆	1	2	yes	—
166	C₄×C₅	C₂²⋊C₁₆	1	2	yes	—
167	C₄×C₅	#410	1	1	yes	characteristic
168	C₄×C₅	C₂×#100	1	1	yes	characteristic
169	Dic₅	—	4	2	no	—
170	C₂⁴	—	5	2	no	—
171	C₂×D₄	—	5	4	no	—
172	C₂×D₄	—	10	2	no	—
173	C₂×D₄	—	10	1	no	—
174	C₂×D₄	—	10	1	no	—
175	C₂²×C₄	—	5	2	no	—
176	C₂²×C₄	—	10	2	no	—
177	C₂²×C₄	—	10	2	no	—
178	C₂²×C₄	—	20	2	no	—
179	C₂²×C₄	—	20	2	no	—
180	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
181	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	4	1	no	—
185	C₂×C₈	—	10	2	no	—
186	C₂×C₈	—	20	4	no	—
187	C₂×C₈	—	20	4	no	—
188	C₂×C₈	—	20	2	no	—
189	C₂×C₈	#685	1	1	yes	characteristic
190	C₂×C₈	#685	1	1	yes	characteristic
191	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
192	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
193	C₄⋊C₄	—	10	1	no	—
194	C₄⋊C₄	—	10	1	no	—
195	C₂²⋊C₄	—	10	2	no	—
196	C₂²⋊C₄	—	10	2	no	—
197	C₄²	#676	1	1	yes	characteristic
198	C₄²	#676	1	1	yes	characteristic
199	C₄²	#685	1	1	yes	characteristic
200	C₄²	#685	1	1	yes	characteristic
201	C₁₆	—	4	2	no	—
202	C₁₆	—	20	2	no	—
203	C₂×C₅	—	2	1	no	—
204	C₂×C₅	—	2	1	no	—
205	C₂×C₅	#1460	1	1	yes	characteristic
206	C₂×C₅	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
207	C₂×C₅	#2115	1	1	yes	characteristic
208	D₅	—	4	4	no	—
209	C₂³	—	5	4	no	—
210	C₂³	—	10	2	no	—
211	C₂³	—	10	2	no	—
212	C₂³	—	20	2	no	—
213	C₂³	—	20	2	no	—
214	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
215	D₄	—	10	4	no	—
216	D₄	—	10	4	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	4	1	no	—
220	C₂×C₄	—	10	2	no	—
221	C₂×C₄	—	20	4	no	—
222	C₂×C₄	—	20	2	no	—
223	C₂×C₄	—	20	2	no	—
224	C₂×C₄	—	20	2	no	—
225	C₂×C₄	D₅×C₁₆	1	2	yes	—
226	C₂×C₄	#3930	1	2	yes	—
227	C₂×C₄	#3952	1	1	yes	characteristic
228	C₂×C₄	#3952	1	1	yes	characteristic
229	C₂×C₄	#3952	1	1	yes	characteristic
230	C₂×C₄	#3952	1	1	yes	characteristic
231	C₂×C₄	C₂×#676	1	1	yes	characteristic
232	C₂×C₄	C₂×#685	1	1	yes	characteristic
233	C₈	—	2	1	no	—
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	20	2	no	—
237	C₈	#3955	1	2	yes	—
238	C₅	#11487	1	1	yes	characteristic, complemented, Sylow
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	10	4	no	—
242	C₂²	—	20	4	no	—
243	C₂²	—	20	4	no	—
244	C₂²	—	20	2	no	—
245	C₂²	—	20	2	no	—
246	C₂²	C₂×D₅×C₁₆	1	1	yes	characteristic
247	C₂²	C₂×#3930	1	1	yes	characteristic
248	C₂²	C₂×#3952	1	1	yes	characteristic
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	20	2	no	—
253	C₄	#64727	1	2	yes	—
254	C₄	#64727	1	2	yes	—
255	C₄	#65295	1	1	yes	characteristic
256	C₄	C₂×#3955	1	1	yes	characteristic
257	C₂	—	2	1	no	—
258	C₂	—	2	1	no	—
259	C₂	—	20	4	no	—
260	C₂	#10601576	1	1	yes	characteristic
261	C₂	C₂×#64727	1	1	yes	characteristic
262	C₂	#10602228	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	#685
commutator	C₂²×C₅	C₂²×C₁₆
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#2162	C₂
socle	C₂²×C₅	C₂×C₂²⋊C₈
prime radical	C₂²×D₂₀	C₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#2162	D₅
hypocenter	C₅	#11487
LCS	G ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2162

Complements

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

N	H ≅ G/N	type
2. C₅×#2162	C₂	semidirect
3. #10600537	C₂	semidirect
4. #10598013	C₂	semidirect
48. #2162	D₅	semidirect
58. C₂×D₂₀	C₁₆	semidirect
238. C₅	#11487	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₂₀	2
C₅×#2162	2
#10600537	2
#10598013	2
#11487	5