Subgroups — #11989250

Order 1280.

634 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×#65213	C₂	1	2	yes	maximal
3	#10598196	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
5	C₂²×#681	—	2	2	no	—
6	C₅×#431	—	2	2	no	—
7	C₅×#431	—	2	2	no	—
8	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₅×C₂²⋊C₈	C₄	1	2	yes	—
10	C₂×C₄×C₅×C₈	C₄	1	2	yes	—
11	#65213	—	2	2	no	—
12	#65213	—	2	2	no	—
13	#65213	C₄	1	4	yes	—
14	#65213	C₂²	1	4	yes	—
15	C₂×C₄⋊#168	—	2	2	no	—
16	C₂×C₄⋊#168	C₄	1	4	yes	—
17	#8001	—	5	1	no	maximal, Sylow
18	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
20	C₂³×C₄×C₅	Q₈	1	1	yes	characteristic
21	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
22	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₅	D₄	1	2	yes	—
27	C₂×C₄²×C₅	D₄	1	2	yes	—
28	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
29	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
30	C₂×#681	—	2	4	no	—
31	C₂×#681	—	2	4	no	—
32	C₂×#681	—	4	4	no	—
33	C₂×#681	—	4	4	no	—
34	C₂²×#168	—	2	4	no	—
35	C₅×C₄⋊C₈	—	2	4	no	—
36	C₅×C₂²⋊C₈	—	2	4	no	—
37	C₄×C₅×C₈	—	2	2	no	—
38	C₄×C₅×C₈	—	2	2	no	—
39	C₄⋊#168	—	2	4	no	—
40	C₄⋊#168	—	2	4	no	—
41	C₄⋊#168	—	2	4	no	—
42	C₄⋊#168	C₂×C₄	1	8	yes	—
43	C₂×#431	D₅	1	1	yes	characteristic
44	C₂×#423	—	5	2	no	—
45	#1292	—	5	4	no	—
46	C₂⁴×C₅	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅	—	4	2	no	—
48	C₂²×C₄×C₅	—	4	2	no	—
49	C₂²×C₄×C₅	C₄²	1	2	yes	—
50	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅	D₈	1	2	yes	—
59	C₂²×C₄×C₅	SD₁₆	1	4	yes	—
60	C₂²×C₄×C₅	Q₁₆	1	2	yes	—
61	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₅×C₈	—	2	4	no	—
65	C₂×C₅×C₈	—	4	4	no	—
66	C₄²×C₅	C₄²	1	1	yes	characteristic
67	C₄²×C₅	C₄²	1	1	yes	characteristic
68	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
69	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
70	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₅	C₄⋊C₄	1	2	yes	—
73	C₄²×C₅	C₄⋊C₄	1	2	yes	—
74	C₄²×C₅	C₂²×C₄	1	2	yes	—
75	C₄²×C₅	C₂×D₄	1	2	yes	—
76	#681	—	4	8	no	—
77	#681	—	4	4	no	—
78	#681	—	4	4	no	—
79	C₂×#168	—	2	8	no	—
80	C₂×#168	—	4	8	no	—
81	C₂²×M₁₆	—	10	2	no	—
82	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
83	#431	—	2	2	no	—
84	#431	—	2	2	no	—
85	#423	—	5	4	no	—
86	#423	—	5	4	no	—
87	#423	—	10	2	no	—
88	#423	—	10	2	no	—
89	C₂×C₄⋊C₈	—	5	4	no	—
90	C₂×C₄⋊C₈	—	10	2	no	—
91	C₂×C₄⋊C₈	C₂×D₅	1	2	yes	—
92	C₂×C₂²⋊C₈	Dic₅	1	2	yes	—
93	C₂×C₄×C₈	Dic₅	1	2	yes	—
94	C₂³×C₅	—	4	1	no	—
95	C₂³×C₅	—	4	1	no	—
96	C₂³×C₅	#95	1	2	yes	—
97	C₂³×C₅	#106	1	2	yes	—
98	C₂³×C₅	#107	1	2	yes	—
99	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₅	—	2	4	no	—
101	C₂×C₄×C₅	—	2	2	no	—
102	C₂×C₄×C₅	—	2	2	no	—
103	C₂×C₄×C₅	—	2	4	no	—
104	C₂×C₄×C₅	—	4	2	no	—
105	C₂×C₄×C₅	—	4	2	no	—
106	C₂×C₄×C₅	—	4	1	no	—
107	C₂×C₄×C₅	—	4	1	no	—
108	C₂×C₄×C₅	#95	1	1	yes	characteristic
109	C₂×C₄×C₅	#95	1	2	yes	—
110	C₂×C₄×C₅	#95	1	1	yes	characteristic
111	C₂×C₄×C₅	#95	1	1	yes	characteristic
112	C₂×C₄×C₅	#95	1	1	yes	characteristic
113	C₂×C₄×C₅	#102	1	4	yes	—
114	C₂×C₄×C₅	#102	1	4	yes	—
115	C₂×C₄×C₅	#103	1	4	yes	—
116	C₂×C₄×C₅	#103	1	4	yes	—
117	C₂×C₄×C₅	#106	1	2	yes	—
118	C₂×C₄×C₅	#107	1	2	yes	—
119	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
122	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
125	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
126	C₅×C₈	—	4	4	no	—
127	#168	—	4	8	no	—
128	C₂³×C₄	Dic₁₀	1	1	yes	characteristic
129	C₂³×C₄	D₂₀	1	1	yes	characteristic
130	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
131	C₂×M₁₆	—	10	4	no	—
132	C₂×M₁₆	—	10	4	no	—
133	C₂×M₁₆	—	20	4	no	—
134	C₂×M₁₆	—	20	4	no	—
135	C₂²×C₈	—	10	4	no	—
136	C₂²×C₈	C₂×Dic₅	1	4	yes	—
137	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
138	C₂×C₄²	C₄×D₅	1	2	yes	—
139	C₂×C₄²	C₄×D₅	1	2	yes	—
140	C₂×C₄²	D₂₀	1	1	yes	characteristic
141	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
142	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
143	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
144	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
145	C₄⋊C₈	—	2	4	no	—
146	C₄⋊C₈	—	5	8	no	—
147	C₄⋊C₈	—	10	4	no	—
148	C₄⋊C₈	—	10	4	no	—
149	C₄⋊C₈	—	10	4	no	—
150	C₂²⋊C₈	—	2	4	no	—
151	C₄×C₈	—	2	2	no	—
152	C₄×C₈	—	2	2	no	—
153	C₂²×C₅	—	2	2	no	—
154	C₂²×C₅	—	2	1	no	—
155	C₂²×C₅	—	2	2	no	—
156	C₂²×C₅	—	2	1	no	—
157	C₂²×C₅	—	4	2	no	—
158	C₂²×C₅	—	4	2	no	—
159	C₂²×C₅	#340	1	4	yes	—
160	C₂²×C₅	C₂×#95	1	1	yes	characteristic
161	C₂²×C₅	C₂×#106	1	1	yes	characteristic
162	C₂²×C₅	C₂×#107	1	1	yes	characteristic
163	C₄×C₅	—	2	1	no	—
164	C₄×C₅	—	2	2	no	—
165	C₄×C₅	—	2	2	no	—
166	C₄×C₅	—	2	1	no	—
167	C₄×C₅	#338	1	2	yes	—
168	C₄×C₅	#340	1	2	yes	—
169	C₄×C₅	#340	1	2	yes	—
170	C₄×C₅	#413	1	2	yes	—
171	C₄×C₅	C₂×#102	1	2	yes	—
172	C₄×C₅	C₂×#103	1	2	yes	—
173	C₂⁴	C₄⋊Dic₅	1	1	yes	characteristic
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	—	4	2	no	—
176	C₂²×C₄	#677	1	4	yes	—
177	C₂²×C₄	D₄₀	1	2	yes	—
178	C₂²×C₄	Dic₂₀	1	2	yes	—
179	C₂²×C₄	C₄×Dic₅	1	2	yes	—
180	C₂²×C₄	#683	1	2	yes	—
181	C₂²×C₄	#683	1	2	yes	—
182	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
183	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
184	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
185	C₂²×C₄	#685	1	2	yes	—
186	C₂²×C₄	#685	1	2	yes	—
187	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
188	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
190	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
191	M₁₆	—	20	8	no	—
192	M₁₆	—	20	4	no	—
193	M₁₆	—	20	4	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	4	4	no	—
196	C₂×C₈	—	10	8	no	—
197	C₂×C₈	—	20	8	no	—
198	C₄²	C₄×Dic₅	1	1	yes	characteristic
199	C₄²	C₄×Dic₅	1	1	yes	characteristic
200	C₄²	#683	1	2	yes	—
201	C₄²	#683	1	2	yes	—
202	C₄²	#685	1	2	yes	—
203	C₄²	#685	1	2	yes	—
204	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
205	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
206	C₄²	C₂×C₄×D₅	1	2	yes	—
207	C₄²	C₂×C₂²⋊D₅	1	2	yes	—
208	C₂×C₅	—	2	4	no	—
209	C₂×C₅	—	2	1	no	—
210	C₂×C₅	—	2	1	no	—
211	C₂×C₅	C₂×#338	1	1	yes	characteristic
212	C₂×C₅	C₂×#340	1	1	yes	characteristic
213	C₂×C₅	#1742	1	1	yes	characteristic
214	C₂³	—	4	1	no	—
215	C₂³	—	4	1	no	—
216	C₂³	#3949	1	2	yes	—
217	C₂³	#3950	1	2	yes	—
218	C₂³	#3963	1	2	yes	—
219	C₂³	C₂×C₄⋊Dic₅	1	1	yes	characteristic
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	#3948	1	4	yes	—
229	C₂×C₄	#3948	1	4	yes	—
230	C₂×C₄	#3949	1	2	yes	—
231	C₂×C₄	#3950	1	2	yes	—
232	C₂×C₄	#3953	1	4	yes	—
233	C₂×C₄	#3953	1	4	yes	—
234	C₂×C₄	#3963	1	1	yes	characteristic
235	C₂×C₄	#3963	1	2	yes	—
236	C₂×C₄	#3963	1	1	yes	characteristic
237	C₂×C₄	#3963	1	1	yes	characteristic
238	C₂×C₄	#3963	1	1	yes	characteristic
239	C₂×C₄	C₂×#677	1	2	yes	—
240	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
241	C₂×C₄	C₂×Dic₂₀	1	1	yes	characteristic
242	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
243	C₂×C₄	C₂×#683	1	2	yes	—
244	C₂×C₄	C₂×#685	1	2	yes	—
245	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
246	C₈	—	4	4	no	—
247	C₈	—	20	8	no	—
248	C₅	#8001	1	1	yes	characteristic, complemented, Sylow
249	C₂²	—	2	2	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	4	2	no	—
254	C₂²	—	4	2	no	—
255	C₂²	#64771	1	4	yes	—
256	C₂²	C₂×#3949	1	1	yes	characteristic
257	C₂²	C₂×#3950	1	1	yes	characteristic
258	C₂²	C₂×#3963	1	1	yes	characteristic
259	C₄	—	2	1	no	—
260	C₄	—	2	2	no	—
261	C₄	—	2	2	no	—
262	C₄	—	2	1	no	—
263	C₄	#64771	1	2	yes	—
264	C₄	#64771	1	2	yes	—
265	C₄	#64780	1	2	yes	—
266	C₄	C₂×#3948	1	2	yes	—
267	C₄	C₂×#3953	1	2	yes	—
268	C₄	#65526	1	2	yes	—
269	C₂	—	2	4	no	—
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	#10602380	1	1	yes	characteristic
273	C₂	C₂×#64771	1	1	yes	characteristic
274	C₂	C₂×#64780	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#3963
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₂×C₅×#431	C₂
socle	C₂²×C₅	C₂×#95
prime radical	C₂⁴×C₅	C₄⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₂×#431	D₅
hypocenter	C₅	#8001
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #10598196	C₂	semidirect
248. C₅	#8001	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#431	2
C₂×#65213	2
#10598196	2
#8001	5