Subgroups — #12008519

Order 1280.

310 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2922	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	#10598209	C₂	1	2	yes	maximal, complemented
5	#10598180	C₂	1	2	yes	maximal
6	C₄×C₅×#41	C₄	1	1	yes	characteristic
7	C₅×#433	—	2	1	no	—
8	C₅×#433	—	2	1	no	—
9	C₅×#431	—	2	1	no	—
10	C₅×#431	—	2	1	no	—
11	C₅×#424	C₄	1	1	yes	characteristic
12	C₅×#424	C₂²	1	1	yes	characteristic
13	C₅×#408	C₄	1	1	yes	characteristic
14	C₅×#408	C₂²	1	1	yes	characteristic
15	C₄×C₅×M₁₆	C₄	1	1	yes	characteristic
16	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₂²⋊#672	C₂²	1	2	yes	—
18	C₂²⋊#672	C₂²	1	2	yes	—
19	C₄⋊#672	—	2	1	no	—
20	C₄⋊#672	—	2	1	no	—
21	#64681	—	2	1	no	—
22	C₄×#672	—	2	1	no	—
23	#11464	—	5	1	no	maximal, Sylow
24	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
25	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
29	C₄×C₅×Q₈	C₈	1	1	yes	characteristic
30	C₄×C₅×Q₈	C₈	1	1	yes	characteristic
31	C₄×C₅×D₄	—	2	1	no	—
32	C₄×C₅×D₄	—	2	1	no	—
33	C₄×C₅×D₄	C₈	1	1	yes	characteristic
34	C₄×C₅×D₄	C₈	1	1	yes	characteristic
35	C₅×#117	—	2	1	no	—
36	C₅×#117	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₅	—	2	1	no	—
38	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
39	C₅×C₄⋊C₈	—	2	1	no	—
40	C₅×C₄⋊C₈	—	2	1	no	—
41	C₅×C₄⋊C₈	—	2	1	no	—
42	C₅×C₄⋊C₈	—	2	1	no	—
43	C₅×C₂²⋊C₈	—	2	1	no	—
44	C₅×C₂²⋊C₈	—	2	1	no	—
45	C₅×C₂²⋊C₈	—	2	1	no	—
46	C₅×C₂²⋊C₈	—	2	1	no	—
47	C₅×#97	—	2	1	no	—
48	C₄×C₅×C₈	—	2	1	no	—
49	C₄×C₅×C₈	D₄	1	1	yes	characteristic
50	C₄×C₅×C₈	D₄	1	1	yes	characteristic
51	C₄×C₅×C₈	D₄	1	1	yes	characteristic
52	C₄×C₅×C₈	D₄	1	1	yes	characteristic
53	C₂×#672	—	2	2	no	—
54	C₂×#672	—	2	2	no	—
55	#2922	D₅	1	1	yes	characteristic
56	#2163	—	5	1	no	—
57	#2162	—	5	1	no	—
58	#1323	—	5	2	no	—
59	#1315	—	5	2	no	—
60	C₅×#41	—	4	1	no	—
61	C₅×#41	—	4	1	no	—
62	C₂×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
63	C₂×C₅×D₄	—	2	1	no	—
64	C₂×C₅×D₄	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₅	—	2	1	no	—
66	C₂²×C₄×C₅	—	2	1	no	—
67	C₂²×C₄×C₅	—	2	1	no	—
68	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
69	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₅×M₁₆	—	4	1	no	—
72	C₅×M₁₆	—	4	1	no	—
73	C₂×C₅×C₈	—	2	1	no	—
74	C₂×C₅×C₈	—	2	1	no	—
75	C₂×C₅×C₈	—	4	1	no	—
76	C₂×C₅×C₈	—	4	1	no	—
77	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
80	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
81	C₅×C₄⋊C₄	—	2	1	no	—
82	C₅×C₄⋊C₄	—	2	1	no	—
83	C₅×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
84	C₅×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
85	C₅×C₂²⋊C₄	—	2	1	no	—
86	C₅×C₂²⋊C₄	—	2	1	no	—
87	C₅×C₂²⋊C₄	—	2	1	no	—
88	C₄²×C₅	—	2	1	no	—
89	C₄²×C₅	—	2	1	no	—
90	C₄²×C₅	—	2	1	no	—
91	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
92	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
93	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
94	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
95	#672	—	4	2	no	—
96	#672	—	4	2	no	—
97	C₄×#41	Dic₅	1	1	yes	characteristic
98	#433	—	2	1	no	—
99	#433	—	2	1	no	—
100	#431	—	2	1	no	—
101	#431	—	2	1	no	—
102	#424	Dic₅	1	1	yes	characteristic
103	#424	C₂×D₅	1	1	yes	characteristic
104	#408	Dic₅	1	1	yes	characteristic
105	#408	C₂×D₅	1	1	yes	characteristic
106	C₄×M₁₆	Dic₅	1	1	yes	characteristic
107	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
108	C₄⋊C₁₆	—	10	1	no	—
109	C₄⋊C₁₆	—	10	1	no	—
110	C₂²⋊C₁₆	—	5	2	no	—
111	C₂²⋊C₁₆	—	5	2	no	—
112	#346	—	10	1	no	—
113	C₄×C₁₆	—	10	1	no	—
114	C₂³×C₅	—	2	1	no	—
115	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
116	C₅×Q₈	—	4	1	no	—
117	C₅×D₄	—	4	1	no	—
118	C₅×D₄	—	4	2	no	—
119	C₂×C₄×C₅	—	2	2	no	—
120	C₂×C₄×C₅	—	2	1	no	—
121	C₂×C₄×C₅	—	2	1	no	—
122	C₂×C₄×C₅	—	2	1	no	—
123	C₂×C₄×C₅	—	2	1	no	—
124	C₂×C₄×C₅	—	2	1	no	—
125	C₂×C₄×C₅	—	4	1	no	—
126	C₂×C₄×C₅	—	4	2	no	—
127	C₂×C₄×C₅	—	4	2	no	—
128	C₂×C₄×C₅	—	4	1	no	—
129	C₂×C₄×C₅	—	4	1	no	—
130	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
136	C₅×C₈	—	2	1	no	—
137	C₅×C₈	—	2	1	no	—
138	C₅×C₈	—	2	1	no	—
139	C₅×C₈	—	2	1	no	—
140	C₅×C₈	—	4	1	no	—
141	C₅×C₈	—	4	1	no	—
142	C₂×#41	C₂×Dic₅	1	1	yes	characteristic
143	C₂×M₁₆	C₂×Dic₅	1	1	yes	characteristic
144	C₂×M₁₆	C₂×Dic₅	1	1	yes	characteristic
145	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
146	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
147	C₄×Q₈	#168	1	1	yes	characteristic
148	C₄×Q₈	#168	1	1	yes	characteristic
149	C₄×D₄	—	2	1	no	—
150	C₄×D₄	—	2	1	no	—
151	C₄×D₄	#168	1	1	yes	characteristic
152	C₄×D₄	#168	1	1	yes	characteristic
153	#117	—	2	1	no	—
154	#117	C₂×Dic₅	1	1	yes	characteristic
155	C₂×C₄²	—	2	1	no	—
156	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
157	C₂×C₁₆	—	10	2	no	—
158	C₂×C₁₆	—	10	2	no	—
159	C₄⋊C₈	—	2	1	no	—
160	C₄⋊C₈	—	2	1	no	—
161	C₄⋊C₈	—	2	1	no	—
162	C₄⋊C₈	—	2	1	no	—
163	C₂²⋊C₈	—	2	1	no	—
164	C₂²⋊C₈	—	2	1	no	—
165	C₂²⋊C₈	—	2	1	no	—
166	C₂²⋊C₈	—	2	1	no	—
167	#97	—	2	1	no	—
168	C₄×C₈	—	2	1	no	—
169	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
170	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
171	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
172	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
173	C₂²×C₅	—	2	1	no	—
174	C₂²×C₅	—	2	1	no	—
175	C₂²×C₅	—	2	1	no	—
176	C₂²×C₅	—	4	1	no	—
177	C₂²×C₅	—	4	1	no	—
178	C₂²×C₅	—	4	1	no	—
179	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
180	C₄×C₅	—	2	1	no	—
181	C₄×C₅	—	2	1	no	—
182	C₄×C₅	—	2	1	no	—
183	C₄×C₅	—	2	1	no	—
184	C₄×C₅	—	4	1	no	—
185	C₄×C₅	—	4	1	no	—
186	C₄×C₅	—	4	1	no	—
187	C₄×C₅	#350	1	2	yes	—
188	C₄×C₅	#410	1	1	yes	characteristic
189	C₄×C₅	#413	1	1	yes	characteristic
190	#41	—	4	1	no	—
191	#41	—	4	1	no	—
192	C₂×Q₈	C₂×#168	1	1	yes	characteristic
193	C₂×D₄	—	2	1	no	—
194	C₂×D₄	C₂×#168	1	1	yes	characteristic
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	#681	1	1	yes	characteristic
199	C₂²×C₄	#681	1	1	yes	characteristic
200	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
201	M₁₆	—	4	1	no	—
202	M₁₆	—	4	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	—	4	1	no	—
206	C₂×C₈	—	4	1	no	—
207	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
208	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
209	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
210	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
211	C₄⋊C₄	—	2	1	no	—
212	C₄⋊C₄	—	2	1	no	—
213	C₄⋊C₄	C₂×#168	1	1	yes	characteristic
214	C₄⋊C₄	C₂×#168	1	1	yes	characteristic
215	C₂²⋊C₄	—	2	1	no	—
216	C₂²⋊C₄	—	2	1	no	—
217	C₂²⋊C₄	—	2	1	no	—
218	C₄²	—	2	1	no	—
219	C₄²	—	2	1	no	—
220	C₄²	—	2	1	no	—
221	C₄²	C₂×#168	1	1	yes	characteristic
222	C₄²	C₂×#168	1	1	yes	characteristic
223	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
224	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
225	C₁₆	—	20	2	no	—
226	C₁₆	—	20	2	no	—
227	C₂×C₅	—	4	1	no	—
228	C₂×C₅	—	8	1	no	—
229	C₂×C₅	#1458	1	1	yes	characteristic
230	C₂×C₅	C₂×#350	1	1	yes	characteristic
231	C₂×C₅	#2117	1	1	yes	characteristic
232	C₂³	—	2	1	no	—
233	C₂³	C₂×#681	1	1	yes	characteristic
234	Q₈	—	4	1	no	—
235	D₄	—	4	1	no	—
236	D₄	—	4	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
249	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
250	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
251	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
252	C₂×C₄	C₂²×#168	1	1	yes	characteristic
253	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	4	1	no	—
259	C₈	—	4	1	no	—
260	C₅	#11464	1	1	yes	characteristic, complemented, Sylow
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	C₂×C₂²⋊#168	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	4	1	no	—
273	C₄	—	4	1	no	—
274	C₄	—	4	1	no	—
275	C₄	#64781	1	2	yes	—
276	C₄	#65526	1	1	yes	characteristic
277	C₄	#65529	1	1	yes	characteristic
278	C₂	—	4	1	no	—
279	C₂	—	8	1	no	—
280	C₂	#10602364	1	1	yes	characteristic
281	C₂	C₂×#64781	1	1	yes	characteristic
282	C₂	#10602799	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₅
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#2922	C₂
socle	C₂²×C₅	C₂×C₂²⋊C₈
prime radical	C₂×C₅×#41	C₂×C₄
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#2922	D₅
hypocenter	C₅	#11464
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2922

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. #10598209	C₂	semidirect
260. C₅	#11464	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2922	2
#10600538	2
#10600537	2
#10598209	2
#10598180	2
#11464	5