Subgroups — #12014374

Order 1280.

444 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2875	C₂	1	1	yes	maximal, characteristic
2	#10602799	C₂	1	2	yes	maximal
3	C₂×#64686	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#64686	C₂	1	1	yes	maximal, characteristic, complemented
5	#10598163	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#131	C₄	1	1	yes	characteristic
7	C₂×C₅×#131	C₂²	1	1	yes	characteristic
8	C₄×C₅×#41	C₄	1	1	yes	characteristic
9	C₅×#405	—	2	1	no	—
10	C₅×#405	—	2	1	no	—
11	C₅×#405	C₄	1	1	yes	characteristic
12	C₅×#405	C₂²	1	1	yes	characteristic
13	C₄×C₅×M₁₆	—	2	1	no	—
14	C₄×C₅×M₁₆	C₄	1	1	yes	characteristic
15	C₂×C₅×#97	—	2	1	no	—
16	C₂×C₅×#97	C₂²	1	1	yes	characteristic
17	C₂×#3944	C₂²	1	2	yes	—
18	C₂×#3944	C₂²	1	2	yes	—
19	#64781	C₄	1	8	yes	complemented
20	#64686	—	2	1	no	—
21	#64686	—	2	1	no	—
22	#64686	—	2	1	no	—
23	#64686	—	2	1	no	—
24	#12609	—	5	1	no	maximal, Sylow
25	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
26	C₅×#131	—	2	2	no	—
27	C₅×#131	—	2	2	no	—
28	C₅×#131	—	2	2	no	—
29	C₅×#131	C₂×C₄	1	4	yes	—
30	C₂×C₅×M₁₆	—	2	1	no	—
31	C₂×C₅×M₁₆	—	2	1	no	—
32	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
33	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₅×C₈	—	2	1	no	—
35	C₂²×C₅×C₈	—	2	1	no	—
36	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
38	C₄×C₅×Q₈	—	2	1	no	—
39	C₄×C₅×D₄	—	2	1	no	—
40	C₄×C₅×D₄	—	2	2	no	—
41	C₅×#117	—	2	1	no	—
42	C₅×#117	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₅	—	2	1	no	—
44	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
45	C₅×#97	—	2	1	no	—
46	C₅×#97	—	2	1	no	—
47	C₅×#97	—	2	1	no	—
48	C₅×#97	D₄	1	1	yes	characteristic
49	C₅×#97	D₄	1	1	yes	characteristic
50	C₅×#97	D₄	1	1	yes	characteristic
51	C₅×#97	D₄	1	1	yes	characteristic
52	C₄×C₅×C₈	—	2	1	no	—
53	C₄×C₅×C₈	—	2	1	no	—
54	C₄×C₅×C₈	—	2	1	no	—
55	#3944	—	2	2	no	—
56	#3944	—	2	2	no	—
57	#3944	—	2	2	no	—
58	#3944	C₂×C₄	1	4	yes	complemented
59	C₂×#672	—	2	2	no	—
60	C₂×#672	C₂×C₄	1	4	yes	complemented
61	#2875	D₅	1	1	yes	characteristic
62	#2117	—	5	2	no	—
63	C₂×#347	—	5	1	no	—
64	C₂×#347	—	5	1	no	—
65	#1378	—	5	2	no	—
66	C₅×#41	—	2	1	no	—
67	C₅×#41	—	2	1	no	—
68	C₅×#41	C₄²	1	2	yes	—
69	C₅×#41	M₁₆	1	2	yes	—
70	C₂×C₅×Q₈	M₁₆	1	1	yes	characteristic
71	C₂×C₅×D₄	—	2	1	no	—
72	C₂×C₅×D₄	M₁₆	1	1	yes	characteristic
73	C₂²×C₄×C₅	—	2	1	no	—
74	C₂²×C₄×C₅	—	2	2	no	—
75	C₂²×C₄×C₅	—	2	1	no	—
76	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₅×M₁₆	—	2	4	no	—
78	C₅×M₁₆	—	2	1	no	—
79	C₅×M₁₆	—	2	1	no	—
80	C₅×M₁₆	—	2	4	no	—
81	C₅×M₁₆	C₄²	1	2	yes	—
82	C₅×M₁₆	M₁₆	1	2	yes	—
83	C₂×C₅×C₈	—	2	4	no	—
84	C₂×C₅×C₈	—	2	1	no	—
85	C₂×C₅×C₈	—	2	1	no	—
86	C₂×C₅×C₈	—	2	1	no	—
87	C₂×C₅×C₈	—	2	1	no	—
88	C₂×C₅×C₈	—	2	4	no	—
89	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
92	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
93	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
94	C₂×C₅×C₈	#41	1	2	yes	—
95	C₅×C₄⋊C₄	—	2	2	no	—
96	C₅×C₄⋊C₄	—	2	1	no	—
97	C₅×C₂²⋊C₄	—	2	2	no	—
98	C₅×C₂²⋊C₄	M₁₆	1	2	yes	—
99	C₄²×C₅	—	2	2	no	—
100	C₄²×C₅	—	2	1	no	—
101	C₄²×C₅	—	2	1	no	—
102	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
103	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
104	#672	—	2	4	no	—
105	#672	—	2	4	no	—
106	C₂×#131	Dic₅	1	1	yes	characteristic
107	C₂×#131	C₂×D₅	1	1	yes	characteristic
108	C₄×#41	Dic₅	1	1	yes	characteristic
109	C₂×M₃₂	—	5	2	no	—
110	C₂×M₃₂	—	5	2	no	—
111	#405	—	2	1	no	—
112	#405	—	2	1	no	—
113	#405	Dic₅	1	1	yes	characteristic
114	#405	C₂×D₅	1	1	yes	characteristic
115	C₄×M₁₆	—	2	1	no	—
116	C₄×M₁₆	Dic₅	1	1	yes	characteristic
117	C₂×#97	—	2	1	no	—
118	C₂×#97	C₂×D₅	1	1	yes	characteristic
119	#350	—	5	8	no	—
120	#347	—	10	1	no	—
121	#347	—	10	1	no	—
122	#347	—	10	1	no	—
123	#347	—	10	1	no	—
124	C₂³×C₅	—	2	1	no	—
125	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
126	C₅×Q₈	—	2	1	no	—
127	C₅×Q₈	#97	1	2	yes	—
128	C₅×D₄	—	2	2	no	—
129	C₅×D₄	—	2	2	no	—
130	C₅×D₄	—	2	1	no	—
131	C₅×D₄	#97	1	2	yes	—
132	C₂×C₄×C₅	—	2	1	no	—
133	C₂×C₄×C₅	—	2	2	no	—
134	C₂×C₄×C₅	—	2	2	no	—
135	C₂×C₄×C₅	—	2	1	no	—
136	C₂×C₄×C₅	—	2	2	no	—
137	C₂×C₄×C₅	—	2	1	no	—
138	C₂×C₄×C₅	—	8	1	no	—
139	C₂×C₄×C₅	—	8	2	no	—
140	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₅	#117	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
146	C₅×C₈	—	2	2	no	—
147	C₅×C₈	—	2	1	no	—
148	C₅×C₈	—	2	2	no	—
149	C₅×C₈	—	2	1	no	—
150	C₅×C₈	C₄×D₄	1	2	yes	—
151	C₅×C₈	C₄×D₄	1	2	yes	—
152	C₂×#41	C₂×Dic₅	1	1	yes	characteristic
153	#131	—	2	2	no	—
154	#131	—	2	2	no	—
155	#131	—	2	2	no	—
156	#131	C₄×D₅	1	4	yes	—
157	C₂×M₁₆	—	2	1	no	—
158	C₂×M₁₆	—	2	1	no	—
159	C₂×M₁₆	C₂×Dic₅	1	1	yes	characteristic
160	C₂×M₁₆	C₂×Dic₅	1	1	yes	characteristic
161	C₂²×C₈	—	2	1	no	—
162	C₂²×C₈	—	2	1	no	—
163	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
164	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
165	C₄×Q₈	—	2	1	no	—
166	C₄×D₄	—	2	1	no	—
167	C₄×D₄	—	2	2	no	—
168	#117	—	2	1	no	—
169	#117	C₂×Dic₅	1	1	yes	characteristic
170	C₂×C₄²	—	2	1	no	—
171	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
172	M₃₂	—	5	4	no	—
173	M₃₂	—	10	2	no	—
174	M₃₂	—	10	2	no	—
175	M₃₂	—	10	2	no	—
176	C₂×C₁₆	—	5	4	no	—
177	C₂×C₁₆	—	10	2	no	—
178	#97	—	2	1	no	—
179	#97	—	2	1	no	—
180	#97	—	2	1	no	—
181	#97	C₂²⋊D₅	1	1	yes	characteristic
182	#97	C₂²⋊D₅	1	1	yes	characteristic
183	#97	C₂²⋊D₅	1	1	yes	characteristic
184	#97	C₂²⋊D₅	1	1	yes	characteristic
185	C₄×C₈	—	2	1	no	—
186	C₄×C₈	—	2	1	no	—
187	C₄×C₈	—	2	1	no	—
188	C₂²×C₅	—	2	2	no	—
189	C₂²×C₅	—	4	1	no	—
190	C₂²×C₅	—	4	2	no	—
191	C₂²×C₅	C₄×M₁₆	1	1	yes	characteristic
192	C₂²×C₅	#407	1	1	yes	characteristic
193	C₂²×C₅	#432	1	1	yes	characteristic
194	C₄×C₅	—	2	2	no	—
195	C₄×C₅	—	4	2	no	—
196	C₄×C₅	—	4	1	no	—
197	C₄×C₅	—	4	1	no	—
198	C₄×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
199	C₄×C₅	C₂×#97	1	1	yes	characteristic
200	C₄×C₅	#407	1	1	yes	characteristic
201	C₄×C₅	#432	1	1	yes	characteristic
202	#41	—	2	1	no	—
203	#41	—	2	1	no	—
204	#41	#681	1	2	yes	—
205	#41	C₄×Dic₅	1	2	yes	—
206	C₂×Q₈	#681	1	1	yes	characteristic
207	C₂×D₄	—	2	1	no	—
208	C₂×D₄	#681	1	1	yes	characteristic
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
213	M₁₆	—	2	4	no	—
214	M₁₆	—	2	1	no	—
215	M₁₆	—	2	1	no	—
216	M₁₆	—	2	4	no	—
217	M₁₆	#681	1	2	yes	—
218	M₁₆	C₄×Dic₅	1	2	yes	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
226	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
227	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
228	C₂×C₈	#709	1	2	yes	—
229	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
230	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	1	no	—
233	C₂²⋊C₄	—	2	2	no	—
234	C₂²⋊C₄	#681	1	2	yes	—
235	C₄²	—	2	2	no	—
236	C₄²	—	2	1	no	—
237	C₄²	—	2	1	no	—
238	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
239	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
240	C₁₆	—	10	4	no	—
241	C₁₆	—	10	4	no	—
242	C₂×C₅	—	2	1	no	—
243	C₂×C₅	—	2	1	no	—
244	C₂×C₅	—	2	1	no	—
245	C₂×C₅	—	4	2	no	—
246	C₂×C₅	#1749	1	1	yes	characteristic
247	C₂³	—	2	1	no	—
248	C₂³	C₂×#681	1	1	yes	characteristic
249	Q₈	—	2	1	no	—
250	Q₈	#3935	1	2	yes	—
251	D₄	—	2	2	no	—
252	D₄	—	2	2	no	—
253	D₄	—	2	1	no	—
254	D₄	#3935	1	2	yes	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	8	1	no	—
262	C₂×C₄	—	8	2	no	—
263	C₂×C₄	C₂×#681	1	1	yes	characteristic
264	C₂×C₄	C₂×#681	1	1	yes	characteristic
265	C₂×C₄	C₂×#681	1	1	yes	characteristic
266	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
267	C₂×C₄	#4072	1	1	yes	characteristic
268	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
269	C₈	—	2	1	no	—
270	C₈	—	2	2	no	—
271	C₈	—	2	1	no	—
272	C₈	—	2	2	no	—
273	C₈	C₄×C₂²⋊D₅	1	2	yes	—
274	C₈	C₄×C₂²⋊D₅	1	2	yes	—
275	C₅	#12609	1	1	yes	characteristic, complemented, Sylow
276	C₂²	—	2	2	no	—
277	C₂²	—	4	1	no	—
278	C₂²	—	4	2	no	—
279	C₂²	C₄×#681	1	1	yes	characteristic
280	C₂²	#65215	1	1	yes	characteristic
281	C₂²	#65496	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	4	1	no	—
284	C₄	—	4	2	no	—
285	C₄	—	4	1	no	—
286	C₄	C₂×#3935	1	1	yes	characteristic
287	C₄	#65215	1	1	yes	characteristic
288	C₄	#65496	1	1	yes	characteristic
289	C₄	C₄×C₂²⋊Dic₅	1	1	yes	characteristic
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	2	1	no	—
293	C₂	—	4	2	no	—
294	C₂	#10602322	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×C₂²⋊Dic₅
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#2875	C₂
socle	C₂×C₅	#1749
prime radical	C₂×C₅×#41	C₂×C₄
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#2875	D₅
hypocenter	C₅	#12609
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×#97 ◁ #2875

Complements

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

N	H ≅ G/N	type
3. C₂×#64686	C₂	semidirect
4. C₂×#64686	C₂	semidirect
5. #10598163	C₂	semidirect
19. #64781	C₄	semidirect
58. #3944	C₂×C₄	semidirect
60. C₂×#672	C₂×C₄	semidirect
275. C₅	#12609	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2875	2
#10602799	2
C₂×#64686	2
#10598163	2
#12609	5