Subgroups — #12252439

Order 1280.

524 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2873	C₂	1	1	yes	maximal, characteristic
2	C₂×#64773	C₂	1	2	yes	maximal
3	#10600679	C₂	1	4	yes	maximal, complemented
4	#10600678	C₂	1	2	yes	maximal, complemented
5	#10600677	C₂	1	2	yes	maximal, complemented
6	#10600657	C₂	1	4	yes	maximal, complemented
7	C₂²×C₅×M₁₆	C₄	1	1	yes	characteristic
8	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₅×#117	C₄	1	1	yes	characteristic
10	C₅×#405	C₄	1	1	yes	characteristic
11	C₅×#405	C₄	1	1	yes	characteristic
12	C₅×#405	C₄	1	2	yes	—
13	C₅×#405	C₂²	1	2	yes	—
14	C₅×#405	C₂²	1	1	yes	characteristic
15	C₅×#405	C₂²	1	1	yes	characteristic
16	C₄×C₅×M₁₆	C₄	1	2	yes	—
17	C₄×C₅×M₁₆	C₂²	1	2	yes	—
18	C₂×#3944	C₂²	1	4	yes	—
19	#64773	C₂²	1	4	yes	—
20	#64773	C₂²	1	4	yes	—
21	#64686	C₂²	1	8	yes	complemented
22	#64683	C₂²	1	8	yes	complemented
23	#33427	—	5	1	no	maximal, Sylow
24	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₅×M₁₆	—	2	4	no	—
26	C₂×C₅×M₁₆	—	2	4	no	—
27	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
29	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
31	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
32	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
33	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
34	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
36	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
37	C₅×#117	—	2	1	no	—
38	C₅×#117	—	2	1	no	—
39	C₅×#117	C₂×C₄	1	2	yes	—
40	C₅×#117	C₂×C₄	1	1	yes	characteristic
41	C₅×#117	C₂×C₄	1	1	yes	characteristic
42	C₂×C₅×C₄⋊C₄	—	2	1	no	—
43	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
44	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
45	C₅×#97	C₂×C₄	1	2	yes	—
46	C₅×#97	C₂×C₄	1	2	yes	—
47	C₅×#97	C₂×C₄	1	2	yes	—
48	C₅×#97	C₂³	1	2	yes	—
49	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
50	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
51	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
52	C₄×C₅×C₈	C₂³	1	2	yes	—
53	#3944	—	2	4	no	—
54	#3944	C₂³	1	8	yes	—
55	C₂×#672	—	2	4	no	—
56	#2873	D₅	1	1	yes	characteristic
57	#2166	—	5	4	no	—
58	#2165	—	5	4	no	—
59	#2153	—	5	2	no	—
60	C₂×#349	—	5	2	no	—
61	#2110	—	5	2	no	—
62	C₂⁴×C₅	M₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₅	—	2	2	no	—
64	C₂²×C₄×C₅	—	2	2	no	—
65	C₂²×C₄×C₅	—	2	2	no	—
66	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
67	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
68	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₅×M₁₆	—	2	2	no	—
73	C₅×M₁₆	—	2	2	no	—
74	C₅×M₁₆	—	2	2	no	—
75	C₅×M₁₆	—	2	4	no	—
76	C₅×M₁₆	—	2	2	no	—
77	C₅×M₁₆	—	2	2	no	—
78	C₅×M₁₆	M₁₆	1	4	yes	—
79	C₂×C₅×C₈	—	2	2	no	—
80	C₂×C₅×C₈	—	2	4	no	—
81	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
82	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
86	C₂×C₅×C₈	#41	1	4	yes	—
87	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
88	C₅×C₄⋊C₄	—	2	2	no	—
89	C₅×C₄⋊C₄	—	2	1	no	—
90	C₅×C₄⋊C₄	—	2	1	no	—
91	C₅×C₂²⋊C₄	—	2	2	no	—
92	C₅×C₂²⋊C₄	—	2	1	no	—
93	C₅×C₂²⋊C₄	—	2	1	no	—
94	C₄²×C₅	—	2	2	no	—
95	C₄²×C₅	C₂²×C₄	1	2	yes	—
96	C₄²×C₅	C₂²×C₄	1	2	yes	—
97	#672	—	2	8	no	—
98	C₂²×M₁₆	Dic₅	1	1	yes	characteristic
99	C₂²×M₁₆	C₂×D₅	1	1	yes	characteristic
100	C₂×#117	Dic₅	1	1	yes	characteristic
101	C₂×M₃₂	—	5	4	no	—
102	#405	Dic₅	1	1	yes	characteristic
103	#405	Dic₅	1	2	yes	—
104	#405	Dic₅	1	1	yes	characteristic
105	#405	C₂×D₅	1	2	yes	—
106	#405	C₂×D₅	1	1	yes	characteristic
107	#405	C₂×D₅	1	1	yes	characteristic
108	C₄×M₁₆	Dic₅	1	2	yes	—
109	C₄×M₁₆	C₂×D₅	1	2	yes	—
110	#364	—	5	8	no	—
111	#349	—	5	4	no	—
112	#349	—	5	4	no	—
113	#347	—	5	8	no	—
114	C₂³×C₅	—	2	2	no	—
115	C₂³×C₅	—	8	1	no	—
116	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
117	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
118	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
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	2	no	—
125	C₂×C₄×C₅	—	2	2	no	—
126	C₂×C₄×C₅	—	2	2	no	—
127	C₂×C₄×C₅	—	2	2	no	—
128	C₂×C₄×C₅	—	8	2	no	—
129	C₂×C₄×C₅	#117	1	2	yes	—
130	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
133	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
135	C₅×C₈	—	2	1	no	—
136	C₅×C₈	—	2	1	no	—
137	C₅×C₈	—	2	2	no	—
138	C₅×C₈	#117	1	2	yes	—
139	C₅×C₈	C₂×M₁₆	1	4	yes	—
140	C₅×C₈	C₂×#41	1	2	yes	—
141	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
142	C₂×M₁₆	—	2	4	no	—
143	C₂×M₁₆	—	2	4	no	—
144	C₂×M₁₆	C₂×Dic₅	1	2	yes	—
145	C₂×M₁₆	C₂×Dic₅	1	1	yes	characteristic
146	C₂×M₁₆	C₂×Dic₅	1	1	yes	characteristic
147	C₂×M₁₆	C₂×Dic₅	1	1	yes	characteristic
148	C₂×M₁₆	C₂×Dic₅	1	1	yes	characteristic
149	C₂×M₁₆	C₂²×D₅	1	1	yes	characteristic
150	C₂×M₁₆	C₂²×D₅	1	1	yes	characteristic
151	C₂²×C₈	C₂×Dic₅	1	2	yes	—
152	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
153	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
154	#117	—	2	1	no	—
155	#117	—	2	1	no	—
156	#117	C₂×Dic₅	1	2	yes	—
157	#117	C₂×Dic₅	1	1	yes	characteristic
158	#117	C₂×Dic₅	1	1	yes	characteristic
159	C₂×C₄⋊C₄	—	2	1	no	—
160	C₂×C₂²⋊C₄	—	2	1	no	—
161	C₂×C₄²	C₂×Dic₅	1	2	yes	—
162	M₃₂	—	5	8	no	—
163	M₃₂	—	10	4	no	—
164	C₂×C₁₆	—	10	4	no	—
165	#97	C₂×Dic₅	1	2	yes	—
166	#97	C₂×Dic₅	1	2	yes	—
167	#97	C₂×Dic₅	1	2	yes	—
168	#97	C₂²×D₅	1	2	yes	—
169	C₄×C₈	C₂×Dic₅	1	2	yes	—
170	C₄×C₈	C₂×Dic₅	1	2	yes	—
171	C₄×C₈	C₂×Dic₅	1	2	yes	—
172	C₄×C₈	C₂²×D₅	1	2	yes	—
173	C₂²×C₅	—	2	1	no	—
174	C₂²×C₅	—	2	1	no	—
175	C₂²×C₅	—	4	1	no	—
176	C₂²×C₅	—	4	1	no	—
177	C₂²×C₅	—	4	1	no	—
178	C₂²×C₅	—	8	2	no	—
179	C₂²×C₅	#432	1	2	yes	—
180	C₂²×C₅	C₂²×M₁₆	1	1	yes	characteristic
181	C₄×C₅	—	2	1	no	—
182	C₄×C₅	—	2	1	no	—
183	C₄×C₅	—	4	2	no	—
184	C₄×C₅	—	4	2	no	—
185	C₄×C₅	#432	1	2	yes	—
186	C₄×C₅	C₂×#117	1	1	yes	characteristic
187	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
188	C₂⁴	#681	1	1	yes	characteristic
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	#681	1	1	yes	characteristic
193	C₂²×C₄	#681	1	1	yes	characteristic
194	C₂²×C₄	#681	1	1	yes	characteristic
195	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
196	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
197	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
198	M₁₆	—	2	2	no	—
199	M₁₆	—	2	2	no	—
200	M₁₆	—	2	2	no	—
201	M₁₆	—	2	4	no	—
202	M₁₆	—	2	2	no	—
203	M₁₆	—	2	2	no	—
204	M₁₆	#681	1	4	yes	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	#709	1	4	yes	—
208	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
209	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
210	C₂×C₈	C₂²×Dic₅	1	2	yes	—
211	C₂×C₈	C₂²×Dic₅	1	2	yes	—
212	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
213	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	—	2	1	no	—
216	C₄⋊C₄	—	2	1	no	—
217	C₂²⋊C₄	—	2	2	no	—
218	C₂²⋊C₄	—	2	1	no	—
219	C₂²⋊C₄	—	2	1	no	—
220	C₄²	—	2	2	no	—
221	C₄²	C₂²×Dic₅	1	2	yes	—
222	C₄²	C₂²×Dic₅	1	2	yes	—
223	C₁₆	—	10	8	no	—
224	C₂×C₅	—	2	1	no	—
225	C₂×C₅	—	2	2	no	—
226	C₂×C₅	—	4	1	no	—
227	C₂×C₅	—	4	1	no	—
228	C₂×C₅	C₂×#432	1	1	yes	characteristic
229	C₂³	—	2	2	no	—
230	C₂³	—	8	1	no	—
231	C₂³	C₂×#681	1	1	yes	characteristic
232	C₂³	C₂×#681	1	1	yes	characteristic
233	C₂³	C₂×#681	1	1	yes	characteristic
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	8	2	no	—
244	C₂×C₄	C₂×#681	1	1	yes	characteristic
245	C₂×C₄	C₂×#681	1	1	yes	characteristic
246	C₂×C₄	C₂×#681	1	2	yes	—
247	C₂×C₄	C₂×#681	1	1	yes	characteristic
248	C₂×C₄	#4072	1	2	yes	—
249	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
250	C₈	—	2	2	no	—
251	C₈	—	2	1	no	—
252	C₈	—	2	1	no	—
253	C₈	C₂×#681	1	4	yes	—
254	C₈	#4072	1	2	yes	—
255	C₈	C₂×#709	1	2	yes	—
256	C₅	#33427	1	1	yes	characteristic, complemented, Sylow
257	C₂²	—	2	1	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	8	2	no	—
263	C₂²	#65215	1	2	yes	—
264	C₂²	C₂²×#681	1	1	yes	characteristic
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	4	2	no	—
268	C₄	—	4	2	no	—
269	C₄	#65215	1	2	yes	—
270	C₄	C₂²×#681	1	1	yes	characteristic
271	C₄	C₂×#4072	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	2	2	no	—
274	C₂	—	4	1	no	—
275	C₂	—	4	1	no	—
276	C₂	C₂×#65215	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#4072
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₅×#2873	C₂
socle	C₂×C₅	C₂×#432
prime radical	C₂⁴×C₅	M₁₆
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	#2873	D₅
hypocenter	C₅	#33427
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ #405 ◁ #2873

Complements

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

N	H ≅ G/N	type
3. #10600679	C₂	semidirect
4. #10600678	C₂	semidirect
5. #10600677	C₂	semidirect
6. #10600657	C₂	semidirect
21. #64686	C₂²	semidirect
22. #64683	C₂²	semidirect
256. C₅	#33427	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2873	2
C₂×#64773	2
#10600679	2
#10600678	2
#10600677	2
#10600657	2
#33427	5