Subgroups — #12252445

Order 1280.

528 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#131	C₂	1	1	yes	maximal, characteristic
2	#10602800	C₂	1	2	yes	maximal
3	#10600679	C₂	1	4	yes	maximal, complemented
4	#10600677	C₂	1	2	yes	maximal, complemented
5	#10600657	C₂	1	4	yes	maximal, complemented
6	C₂×#64683	C₂	1	2	yes	maximal, complemented
7	C₂×C₅×#131	C₄	1	1	yes	characteristic
8	C₂×C₅×#131	C₂²	1	1	yes	characteristic
9	C₄×C₅×#41	C₄	1	1	yes	characteristic
10	C₅×#405	C₄	1	2	yes	—
11	C₅×#405	C₄	1	1	yes	characteristic
12	C₅×#405	C₂²	1	2	yes	—
13	C₅×#405	C₂²	1	1	yes	characteristic
14	C₄×C₅×M₁₆	C₄	1	1	yes	characteristic
15	C₄×C₅×M₁₆	C₂²	1	2	yes	—
16	C₂×C₄×C₅×C₈	C₄	1	2	yes	—
17	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₂×#3944	C₂²	1	4	yes	—
19	#64773	C₂²	1	8	yes	—
20	#64686	C₂²	1	8	yes	—
21	#64683	C₂²	1	4	yes	—
22	#64683	C₂²	1	4	yes	—
23	#33428	—	5	1	no	maximal, Sylow
24	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
25	C₅×#131	—	2	4	no	—
26	C₅×#131	—	2	2	no	—
27	C₅×#131	—	2	2	no	—
28	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
31	C₂×C₅×M₁₆	C₂³	1	2	yes	—
32	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
33	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
34	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₄×C₅×Q₈	—	2	1	no	—
37	C₄×C₅×D₄	—	2	2	no	—
38	C₄×C₅×D₄	—	2	1	no	—
39	C₅×#117	C₂×C₄	1	2	yes	—
40	C₅×#117	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
43	C₅×#97	C₂×C₄	1	2	yes	—
44	C₅×#97	C₂×C₄	1	2	yes	—
45	C₅×#97	C₂³	1	2	yes	—
46	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
47	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
48	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
49	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
50	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
51	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
52	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
53	#3944	—	2	4	no	—
54	#3944	C₂³	1	8	yes	—
55	C₂×#672	—	2	4	no	—
56	C₄×#131	D₅	1	1	yes	characteristic
57	#2166	—	5	4	no	—
58	#2165	—	5	4	no	—
59	C₂×#364	—	5	2	no	—
60	#2115	—	5	2	no	—
61	#2110	—	5	2	no	—
62	C₅×#41	—	2	2	no	—
63	C₅×#41	—	2	2	no	—
64	C₂×C₅×Q₈	M₁₆	1	1	yes	characteristic
65	C₂×C₅×D₄	M₁₆	1	2	yes	—
66	C₂×C₅×D₄	M₁₆	1	1	yes	characteristic
67	C₂²×C₄×C₅	—	2	2	no	—
68	C₂²×C₄×C₅	—	2	1	no	—
69	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₅×M₁₆	—	2	2	no	—
72	C₅×M₁₆	—	2	2	no	—
73	C₅×M₁₆	—	2	4	no	—
74	C₅×M₁₆	—	2	4	no	—
75	C₂×C₅×C₈	—	2	2	no	—
76	C₂×C₅×C₈	—	2	2	no	—
77	C₂×C₅×C₈	—	2	4	no	—
78	C₂×C₅×C₈	M₁₆	1	4	yes	—
79	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
80	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
81	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₅×C₈	#41	1	2	yes	—
85	C₂×C₅×C₈	#41	1	2	yes	—
86	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
87	C₅×C₄⋊C₄	—	2	2	no	—
88	C₅×C₄⋊C₄	—	2	1	no	—
89	C₅×C₂²⋊C₄	—	2	2	no	—
90	C₅×C₂²⋊C₄	—	2	1	no	—
91	C₄²×C₅	—	2	2	no	—
92	C₄²×C₅	—	2	1	no	—
93	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₅	C₂²×C₄	1	2	yes	—
95	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
96	#672	—	2	8	no	—
97	C₂×#131	Dic₅	1	1	yes	characteristic
98	C₂×#131	C₂×D₅	1	1	yes	characteristic
99	C₄×#41	Dic₅	1	1	yes	characteristic
100	C₂×M₃₂	—	5	4	no	—
101	#405	Dic₅	1	1	yes	characteristic
102	#405	Dic₅	1	2	yes	—
103	#405	C₂×D₅	1	2	yes	—
104	#405	C₂×D₅	1	1	yes	characteristic
105	C₄×M₁₆	Dic₅	1	1	yes	characteristic
106	C₄×M₁₆	C₂×D₅	1	2	yes	—
107	C₂×C₄×C₈	Dic₅	1	2	yes	—
108	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
109	#364	—	5	4	no	—
110	#364	—	5	4	no	—
111	#349	—	5	8	no	—
112	#347	—	5	8	no	—
113	C₂³×C₅	C₂×M₁₆	1	2	yes	—
114	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
115	C₅×Q₈	—	2	2	no	—
116	C₅×D₄	—	2	4	no	—
117	C₅×D₄	—	2	2	no	—
118	C₂×C₄×C₅	—	2	1	no	—
119	C₂×C₄×C₅	—	2	2	no	—
120	C₂×C₄×C₅	—	2	2	no	—
121	C₂×C₄×C₅	—	2	2	no	—
122	C₂×C₄×C₅	—	2	1	no	—
123	C₂×C₄×C₅	—	4	2	no	—
124	C₂×C₄×C₅	—	4	4	no	—
125	C₂×C₄×C₅	#117	1	2	yes	—
126	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
127	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
128	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
129	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
130	C₅×C₈	—	2	2	no	—
131	C₅×C₈	—	2	2	no	—
132	C₅×C₈	#117	1	1	yes	characteristic
133	C₅×C₈	#117	1	1	yes	characteristic
134	C₅×C₈	C₂×M₁₆	1	2	yes	—
135	C₅×C₈	C₂×M₁₆	1	2	yes	—
136	C₅×C₈	C₂×#41	1	2	yes	—
137	C₂×#41	C₂×Dic₅	1	1	yes	characteristic
138	#131	—	2	4	no	—
139	#131	—	2	2	no	—
140	#131	—	2	2	no	—
141	C₂×M₁₆	C₂×Dic₅	1	1	yes	characteristic
142	C₂×M₁₆	C₂×Dic₅	1	2	yes	—
143	C₂×M₁₆	C₂×Dic₅	1	1	yes	characteristic
144	C₂×M₁₆	C₂²×D₅	1	2	yes	—
145	C₂²×C₈	C₂×Dic₅	1	2	yes	—
146	C₂²×C₈	C₂×Dic₅	1	2	yes	—
147	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
148	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
149	C₄×Q₈	—	2	1	no	—
150	C₄×D₄	—	2	2	no	—
151	C₄×D₄	—	2	1	no	—
152	#117	C₂×Dic₅	1	2	yes	—
153	#117	C₂×Dic₅	1	1	yes	characteristic
154	C₂×C₄²	C₂×Dic₅	1	2	yes	—
155	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
156	M₃₂	—	5	8	no	—
157	M₃₂	—	10	4	no	—
158	C₂×C₁₆	—	10	4	no	—
159	#97	C₂×Dic₅	1	2	yes	—
160	#97	C₂×Dic₅	1	2	yes	—
161	#97	C₂²×D₅	1	2	yes	—
162	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
163	C₄×C₈	C₂×Dic₅	1	2	yes	—
164	C₄×C₈	C₂×Dic₅	1	2	yes	—
165	C₄×C₈	C₂×Dic₅	1	2	yes	—
166	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
167	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
168	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
169	C₂²×C₅	—	2	2	no	—
170	C₂²×C₅	—	4	1	no	—
171	C₂²×C₅	—	4	2	no	—
172	C₂²×C₅	#432	1	2	yes	—
173	C₂²×C₅	C₂²×M₁₆	1	1	yes	characteristic
174	C₄×C₅	—	2	2	no	—
175	C₄×C₅	—	2	2	no	—
176	C₄×C₅	—	4	2	no	—
177	C₄×C₅	—	4	1	no	—
178	C₄×C₅	#432	1	2	yes	—
179	C₄×C₅	C₂×#117	1	1	yes	characteristic
180	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
181	#41	—	2	2	no	—
182	#41	—	2	2	no	—
183	C₂×Q₈	#681	1	1	yes	characteristic
184	C₂×D₄	#681	1	2	yes	—
185	C₂×D₄	#681	1	1	yes	characteristic
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
189	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
190	M₁₆	—	2	2	no	—
191	M₁₆	—	2	2	no	—
192	M₁₆	—	2	4	no	—
193	M₁₆	—	2	4	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	#681	1	4	yes	—
198	C₂×C₈	#709	1	2	yes	—
199	C₂×C₈	#709	1	2	yes	—
200	C₂×C₈	C₂²×Dic₅	1	2	yes	—
201	C₂×C₈	C₂²×Dic₅	1	2	yes	—
202	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
203	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
204	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
205	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	1	no	—
208	C₂²⋊C₄	—	2	2	no	—
209	C₂²⋊C₄	—	2	1	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	1	no	—
212	C₄²	C₂²×Dic₅	1	1	yes	characteristic
213	C₄²	C₂²×Dic₅	1	2	yes	—
214	C₄²	C₂²×Dic₅	1	1	yes	characteristic
215	C₁₆	—	10	8	no	—
216	C₂×C₅	—	2	1	no	—
217	C₂×C₅	—	2	2	no	—
218	C₂×C₅	—	4	2	no	—
219	C₂×C₅	C₂×#432	1	1	yes	characteristic
220	C₂³	C₂×#681	1	2	yes	—
221	C₂³	C₂×#681	1	1	yes	characteristic
222	Q₈	—	2	2	no	—
223	D₄	—	2	4	no	—
224	D₄	—	2	2	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	C₂×#681	1	2	yes	—
233	C₂×C₄	C₂×#681	1	2	yes	—
234	C₂×C₄	C₂×#681	1	1	yes	characteristic
235	C₂×C₄	#4072	1	2	yes	—
236	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	—	2	2	no	—
239	C₈	C₂×#681	1	2	yes	—
240	C₈	C₂×#681	1	2	yes	—
241	C₈	#4072	1	1	yes	characteristic
242	C₈	#4072	1	1	yes	characteristic
243	C₈	C₂×#709	1	2	yes	—
244	C₅	#33428	1	1	yes	characteristic, complemented, Sylow
245	C₂²	—	2	2	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	2	no	—
248	C₂²	#65215	1	2	yes	—
249	C₂²	C₂²×#681	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	4	2	no	—
253	C₄	—	4	1	no	—
254	C₄	#65215	1	2	yes	—
255	C₄	C₂²×#681	1	1	yes	characteristic
256	C₄	C₂×#4072	1	1	yes	characteristic
257	C₂	—	2	1	no	—
258	C₂	—	2	2	no	—
259	C₂	—	4	2	no	—
260	C₂	C₂×#65215	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#4072
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₄×C₅×#131	C₂
socle	C₂×C₅	C₂×#432
prime radical	C₂×C₅×#41	C₂×C₄
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	C₄×#131	D₅
hypocenter	C₅	#33428
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₈ ◁ #405 ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
3. #10600679	C₂	semidirect
4. #10600677	C₂	semidirect
5. #10600657	C₂	semidirect
6. C₂×#64683	C₂	semidirect
244. C₅	#33428	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×#131	2
#10602800	2
#10600679	2
#10600677	2
#10600657	2
C₂×#64683	2
#33428	5