Subgroups — #12005221

Order 1280.

714 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#405	C₂	1	1	yes	maximal, characteristic
2	C₂²×#3951	C₂	1	2	yes	maximal
3	C₂×#64780	C₂	1	2	yes	maximal
4	#10602380	C₂	1	2	yes	maximal
5	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
6	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₅×#117	C₂²	1	1	yes	characteristic
8	C₂²×#681	C₂²	1	4	yes	—
9	C₅×#405	—	2	4	no	—
10	C₂×C₅×#97	C₄	1	2	yes	—
11	C₂×C₄×C₅×C₈	C₄	1	2	yes	—
12	C₂×#3951	—	2	4	no	—
13	C₂×#3951	—	2	4	no	—
14	C₂×#3951	C₄	1	8	yes	complemented
15	#64780	—	2	4	no	—
16	#64750	—	2	4	no	—
17	#10867	—	5	1	no	maximal, Sylow
18	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
19	C₂×C₅×M₁₆	—	2	2	no	—
20	C₂×C₅×M₁₆	—	2	2	no	—
21	C₂×C₅×M₁₆	D₄	1	2	yes	—
22	C₂×C₅×M₁₆	Q₈	1	2	yes	—
23	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
24	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	D₄	1	2	yes	—
28	C₂²×C₅×C₈	D₄	1	4	yes	—
29	C₂²×C₅×C₈	Q₈	1	2	yes	—
30	C₅×#117	—	2	2	no	—
31	C₅×#117	—	2	2	no	—
32	C₂×C₅×C₄⋊C₄	D₄	1	2	yes	—
33	C₂×C₅×C₂²⋊C₄	Q₈	1	2	yes	—
34	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
35	C₂×#681	—	2	4	no	—
36	C₂×#681	—	2	4	no	—
37	C₂×#681	—	2	4	no	—
38	C₂×#681	—	2	4	no	—
39	C₂×#681	—	2	4	no	—
40	C₂×#681	C₂×C₄	1	8	yes	—
41	C₂²×#168	—	2	4	no	—
42	C₅×#97	—	2	4	no	—
43	C₄×C₅×C₈	—	2	4	no	—
44	#3951	—	2	16	no	—
45	C₂²×#108	—	5	2	no	—
46	C₂×#405	D₅	1	1	yes	characteristic
47	#1742	—	5	2	no	—
48	C₂×#338	—	5	2	no	—
49	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
51	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₅	#41	1	2	yes	—
60	C₂²×C₄×C₅	#41	1	2	yes	—
61	C₅×M₁₆	—	2	4	no	—
62	C₅×M₁₆	—	2	4	no	—
63	C₂×C₅×C₈	—	2	2	no	—
64	C₂×C₅×C₈	—	2	2	no	—
65	C₂×C₅×C₈	C₄²	1	2	yes	—
66	C₂×C₅×C₈	C₄²	1	2	yes	—
67	C₂×C₅×C₈	C₂²⋊C₄	1	8	yes	—
68	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
69	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
70	C₂×C₅×C₈	C₄⋊C₄	1	8	yes	—
71	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
72	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
73	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
74	C₅×C₄⋊C₄	—	2	4	no	—
75	C₅×C₂²⋊C₄	—	2	4	no	—
76	C₄²×C₅	—	2	2	no	—
77	C₄²×C₅	—	2	2	no	—
78	#681	—	2	8	no	—
79	#681	—	2	8	no	—
80	#681	—	4	4	no	—
81	#681	—	4	4	no	—
82	C₂×#168	—	2	8	no	—
83	C₂×#168	—	4	4	no	—
84	C₂×#168	—	4	4	no	—
85	C₂²×M₁₆	—	5	4	no	—
86	C₂²×M₁₆	C₂×D₅	1	1	yes	characteristic
87	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
88	C₂×#117	C₂×D₅	1	1	yes	characteristic
89	C₂×#108	—	5	8	no	—
90	C₂×#108	—	10	4	no	—
91	C₂×#108	—	10	4	no	—
92	#405	—	2	4	no	—
93	C₂×#97	Dic₅	1	2	yes	—
94	C₂×C₄×C₈	Dic₅	1	2	yes	—
95	#343	—	10	4	no	—
96	#338	—	10	4	no	—
97	C₂³×C₅	—	4	2	no	—
98	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂³×C₅	C₄×Q₈	1	2	yes	—
100	C₂³×C₅	C₂²⋊Q₈	1	2	yes	—
101	C₂³×C₅	C₄⋊Q₈	1	1	yes	characteristic
102	C₂³×C₅	C₄⋊Q₈	1	1	yes	characteristic
103	C₂×C₄×C₅	—	2	4	no	—
104	C₂×C₄×C₅	—	4	4	no	—
105	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
106	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
108	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₅	#117	1	1	yes	characteristic
112	C₂×C₄×C₅	#117	1	1	yes	characteristic
113	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
116	C₂×C₄×C₅	#121	1	2	yes	—
117	C₂×C₄×C₅	#121	1	2	yes	—
118	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₅	#124	1	2	yes	—
120	C₂×C₄×C₅	#125	1	2	yes	—
121	C₂×C₄×C₅	#127	1	1	yes	characteristic
122	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
123	C₅×C₈	—	2	4	no	—
124	C₅×C₈	#95	1	4	yes	—
125	C₅×C₈	#95	1	4	yes	—
126	#168	—	4	8	no	—
127	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
128	C₂×M₁₆	—	2	2	no	—
129	C₂×M₁₆	—	2	2	no	—
130	C₂×M₁₆	—	5	8	no	—
131	C₂×M₁₆	—	10	4	no	—
132	C₂×M₁₆	—	10	4	no	—
133	C₂×M₁₆	—	10	4	no	—
134	C₂×M₁₆	—	10	4	no	—
135	C₂×M₁₆	—	10	4	no	—
136	C₂×M₁₆	Dic₁₀	1	2	yes	—
137	C₂×M₁₆	D₂₀	1	2	yes	—
138	C₂²×C₈	—	10	4	no	—
139	C₂²×C₈	Dic₁₀	1	2	yes	—
140	C₂²×C₈	C₄×D₅	1	4	yes	—
141	C₂²×C₈	D₂₀	1	2	yes	—
142	C₂²×C₈	C₂×Dic₅	1	2	yes	—
143	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
144	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
145	C₂²×C₈	C₂²⋊D₅	1	4	yes	—
146	#117	—	2	2	no	—
147	#117	—	2	2	no	—
148	C₂×C₄⋊C₄	D₂₀	1	2	yes	—
149	C₂×C₂²⋊C₄	Dic₁₀	1	2	yes	—
150	C₂×C₄²	C₂×Dic₅	1	2	yes	—
151	#108	—	10	16	no	—
152	#97	—	2	4	no	—
153	C₄×C₈	—	2	4	no	—
154	C₂²×C₅	—	2	2	no	—
155	C₂²×C₅	—	2	4	no	—
156	C₂²×C₅	—	4	2	no	—
157	C₂²×C₅	—	4	1	no	—
158	C₂²×C₅	—	4	1	no	—
159	C₂²×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
160	C₂²×C₅	#380	1	1	yes	characteristic
161	C₂²×C₅	#384	1	1	yes	characteristic
162	C₂²×C₅	#389	1	2	yes	—
163	C₂²×C₅	#391	1	2	yes	—
164	C₄×C₅	—	4	4	no	—
165	C₄×C₅	C₂×#95	1	1	yes	characteristic
166	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
167	C₄×C₅	#380	1	1	yes	characteristic
168	C₄×C₅	#382	1	1	yes	characteristic
169	C₄×C₅	#389	1	2	yes	—
170	C₄×C₅	#390	1	2	yes	—
171	C₂⁴	C₂×Dic₁₀	1	1	yes	characteristic
172	C₂²×C₄	C₄⋊Dic₅	1	4	yes	—
173	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
174	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
175	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
176	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
177	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
178	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
179	C₂²×C₄	#709	1	2	yes	—
180	C₂²×C₄	#709	1	2	yes	—
181	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₂²⋊D₅	1	2	yes	—
183	M₁₆	—	2	4	no	—
184	M₁₆	—	2	4	no	—
185	M₁₆	—	10	8	no	—
186	M₁₆	—	10	8	no	—
187	M₁₆	—	20	4	no	—
188	M₁₆	—	20	4	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	10	8	no	—
192	C₂×C₈	—	20	4	no	—
193	C₂×C₈	—	20	4	no	—
194	C₂×C₈	C₄×Dic₅	1	2	yes	—
195	C₂×C₈	C₄×Dic₅	1	2	yes	—
196	C₂×C₈	#683	1	8	yes	—
197	C₂×C₈	C₄⋊Dic₅	1	4	yes	—
198	C₂×C₈	C₄⋊Dic₅	1	2	yes	—
199	C₂×C₈	C₄⋊Dic₅	1	2	yes	—
200	C₂×C₈	#685	1	8	yes	—
201	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
202	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
203	C₄⋊C₄	—	2	4	no	—
204	C₂²⋊C₄	—	2	4	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	2	no	—
207	C₂×C₅	—	2	1	no	—
208	C₂×C₅	—	2	1	no	—
209	C₂×C₅	—	2	2	no	—
210	C₂×C₅	—	2	2	no	—
211	C₂×C₅	C₄⋊#95	1	1	yes	characteristic
212	C₂×C₅	#1813	1	2	yes	—
213	C₂³	—	4	2	no	—
214	C₂³	C₄×Dic₁₀	1	2	yes	—
215	C₂³	#4015	1	1	yes	characteristic
216	C₂³	#4015	1	1	yes	characteristic
217	C₂³	#4070	1	2	yes	—
218	C₂³	C₂×C₄⋊Dic₅	1	1	yes	characteristic
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	C₄×Dic₁₀	1	2	yes	—
222	C₂×C₄	#4015	1	1	yes	characteristic
223	C₂×C₄	#4016	1	2	yes	—
224	C₂×C₄	C₄×D₂₀	1	2	yes	—
225	C₂×C₄	C₄×D₂₀	1	2	yes	—
226	C₂×C₄	#4020	1	1	yes	characteristic
227	C₂×C₄	#4021	1	2	yes	—
228	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
229	C₂×C₄	C₂×#683	1	2	yes	—
230	C₂×C₄	#4070	1	2	yes	—
231	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
232	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
233	C₂×C₄	#4072	1	1	yes	characteristic
234	C₂×C₄	#4072	1	1	yes	characteristic
235	C₂×C₄	C₂×#685	1	2	yes	—
236	C₂×C₄	#4076	1	2	yes	—
237	C₂×C₄	#4076	1	2	yes	—
238	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
239	C₈	—	2	4	no	—
240	C₈	—	20	8	no	—
241	C₈	#3963	1	4	yes	—
242	C₈	#3963	1	4	yes	—
243	C₅	#10867	1	1	yes	characteristic, complemented, Sylow
244	C₂²	—	2	2	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	4	1	no	—
248	C₂²	—	4	1	no	—
249	C₂²	#65217	1	2	yes	—
250	C₂²	#65218	1	2	yes	—
251	C₂²	C₄×C₄⋊Dic₅	1	1	yes	characteristic
252	C₂²	#65224	1	1	yes	characteristic
253	C₂²	#65501	1	1	yes	characteristic
254	C₄	—	4	4	no	—
255	C₄	#65217	1	2	yes	—
256	C₄	C₄×C₄⋊Dic₅	1	1	yes	characteristic
257	C₄	#65225	1	1	yes	characteristic
258	C₄	#65228	1	2	yes	—
259	C₄	C₂×#3963	1	1	yes	characteristic
260	C₄	#65501	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	2	2	no	—
263	C₂	—	2	1	no	—
264	C₂	—	2	1	no	—
265	C₂	#10602328	1	1	yes	characteristic
266	C₂	#10602807	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#4072
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅
Fitting	C₂×C₅×#405	C₂
socle	C₂²×C₅	#380
prime radical	C₂⁴×C₅	C₂×Q₈
cosocle	C₂³×C₄×C₅	C₂³
hypercenter	C₂×#405	D₅
hypocenter	C₅	#10867
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
14. C₂×#3951	C₄	semidirect
243. C₅	#10867	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#405	2
C₂²×#3951	2
C₂×#64780	2
#10602380	2
#10867	5