Subgroups — #12252400

Order 1280.

582 conjugacy classes of subgroups.

The subgroup lattice has 266 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	#10602797	C₂	1	2	yes	maximal
3	#10600676	C₂	1	4	yes	maximal, complemented
4	#10600675	C₂	1	2	yes	maximal, complemented
5	#10600538	C₂	1	4	yes	maximal, complemented
6	C₄×#3944	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#64681	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#131	C₄	1	1	yes	characteristic
9	C₂×C₅×#131	C₂²	1	1	yes	characteristic
10	C₄×C₅×#41	C₄	1	1	yes	characteristic
11	C₅×#405	C₄	1	2	yes	—
12	C₅×#405	C₄	1	1	yes	characteristic
13	C₅×#405	C₂²	1	1	yes	characteristic
14	C₅×#405	C₂²	1	2	yes	—
15	C₄×C₅×M₁₆	C₄	1	2	yes	—
16	C₄×C₅×M₁₆	C₄	1	1	yes	characteristic
17	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
19	C₂×#3944	C₂²	1	2	yes	—
20	C₂²×#672	C₂²	1	2	yes	—
21	C₂²⋊#672	C₂²	1	8	yes	—
22	C₄⋊#672	C₂²	1	8	yes	—
23	#64681	C₂²	1	2	yes	—
24	#64681	C₂²	1	2	yes	—
25	#64681	C₂²	1	2	yes	—
26	C₄×#672	C₂²	1	2	yes	—
27	#33421	—	5	1	no	maximal, Sylow
28	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
29	C₅×#131	—	2	4	no	—
30	C₅×#131	—	2	4	no	—
31	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
32	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
33	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
34	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
35	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
36	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₅×C₈	C₂³	1	2	yes	—
38	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
39	C₄×C₅×Q₈	C₈	1	2	yes	—
40	C₄×C₅×D₄	C₈	1	4	yes	—
41	C₄×C₅×D₄	C₈	1	2	yes	—
42	C₅×#117	C₂×C₄	1	2	yes	—
43	C₅×#117	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
46	C₅×#97	C₂×C₄	1	2	yes	—
47	C₅×#97	C₂×C₄	1	2	yes	—
48	C₅×#97	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	C₂×C₄	1	2	yes	—
52	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
53	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
54	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
55	C₄×C₅×C₈	C₂³	1	2	yes	—
56	#3944	—	2	4	no	—
57	C₂×#672	—	2	4	no	—
58	C₂×#672	C₂³	1	4	yes	—
59	C₂×#672	C₂³	1	4	yes	—
60	C₄×#131	D₅	1	1	yes	characteristic
61	#2164	—	5	4	no	—
62	#2163	—	5	4	no	—
63	#2151	—	5	2	no	—
64	#2113	—	5	2	no	—
65	C₄×M₃₂	—	5	1	no	—
66	C₂×#346	—	5	1	no	—
67	C₅×#41	—	2	2	no	—
68	C₅×#41	—	2	2	no	—
69	C₂×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
70	C₂×C₅×D₄	C₂×C₈	1	2	yes	—
71	C₂×C₅×D₄	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
73	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
74	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₅×M₁₆	—	2	2	no	—
77	C₅×M₁₆	—	2	2	no	—
78	C₅×M₁₆	—	2	4	no	—
79	C₅×M₁₆	—	2	4	no	—
80	C₂×C₅×C₈	—	2	4	no	—
81	C₂×C₅×C₈	—	2	4	no	—
82	C₂×C₅×C₈	M₁₆	1	4	yes	—
83	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
84	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
85	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₅×C₈	#41	1	4	yes	—
89	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
90	C₅×C₄⋊C₄	C₂×C₈	1	4	yes	—
91	C₅×C₄⋊C₄	C₂×C₈	1	2	yes	—
92	C₅×C₂²⋊C₄	C₂×C₈	1	4	yes	—
93	C₅×C₂²⋊C₄	C₂×C₈	1	2	yes	—
94	C₄²×C₅	C₂×C₈	1	4	yes	—
95	C₄²×C₅	C₂×C₈	1	2	yes	—
96	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₅	C₂²×C₄	1	2	yes	—
99	#672	—	2	4	no	—
100	#672	—	2	4	no	—
101	C₂×#131	Dic₅	1	1	yes	characteristic
102	C₂×#131	C₂×D₅	1	1	yes	characteristic
103	C₄×#41	Dic₅	1	1	yes	characteristic
104	C₂×M₃₂	—	5	2	no	—
105	C₂²×C₁₆	—	5	2	no	—
106	#405	Dic₅	1	2	yes	—
107	#405	Dic₅	1	1	yes	characteristic
108	#405	C₂×D₅	1	1	yes	characteristic
109	#405	C₂×D₅	1	2	yes	—
110	C₄×M₁₆	Dic₅	1	1	yes	characteristic
111	C₄×M₁₆	Dic₅	1	2	yes	—
112	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
113	C₂×C₄×C₈	C₂×D₅	1	2	yes	—
114	C₄⋊C₁₆	—	5	8	no	—
115	C₂²⋊C₁₆	—	5	8	no	—
116	#346	—	5	2	no	—
117	#346	—	5	2	no	—
118	#346	—	5	2	no	—
119	C₄×C₁₆	—	5	2	no	—
120	C₂³×C₅	C₂²×C₈	1	2	yes	—
121	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
122	C₅×Q₈	—	2	2	no	—
123	C₅×D₄	—	2	4	no	—
124	C₅×D₄	—	2	2	no	—
125	C₂×C₄×C₅	—	2	2	no	—
126	C₂×C₄×C₅	—	2	2	no	—
127	C₂×C₄×C₅	—	4	2	no	—
128	C₂×C₄×C₅	—	4	4	no	—
129	C₂×C₄×C₅	#117	1	2	yes	—
130	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
131	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
132	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
133	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
134	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
136	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
137	C₅×C₈	—	2	2	no	—
138	C₅×C₈	—	2	2	no	—
139	C₅×C₈	#117	1	2	yes	—
140	C₅×C₈	C₂×M₁₆	1	2	yes	—
141	C₅×C₈	C₂×M₁₆	1	2	yes	—
142	C₅×C₈	C₂×#41	1	2	yes	—
143	C₂×#41	C₂×Dic₅	1	1	yes	characteristic
144	#131	—	2	4	no	—
145	#131	—	2	4	no	—
146	C₂×M₁₆	C₂×Dic₅	1	1	yes	characteristic
147	C₂×M₁₆	C₂×Dic₅	1	2	yes	—
148	C₂×M₁₆	C₂×Dic₅	1	1	yes	characteristic
149	C₂×M₁₆	C₂×Dic₅	1	2	yes	—
150	C₂²×C₈	C₂×Dic₅	1	2	yes	—
151	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
152	C₂²×C₈	C₂²×D₅	1	2	yes	—
153	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
154	C₄×Q₈	#168	1	2	yes	—
155	C₄×D₄	#168	1	4	yes	—
156	C₄×D₄	#168	1	2	yes	—
157	#117	C₂×Dic₅	1	2	yes	—
158	#117	C₂×Dic₅	1	1	yes	characteristic
159	C₂×C₄²	C₂×Dic₅	1	2	yes	—
160	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
161	M₃₂	—	10	4	no	—
162	C₂×C₁₆	—	5	4	no	—
163	C₂×C₁₆	—	5	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	C₄×C₈	C₂×Dic₅	1	2	yes	—
169	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
170	C₄×C₈	C₂×Dic₅	1	2	yes	—
171	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
172	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
173	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
174	C₄×C₈	C₂²×D₅	1	2	yes	—
175	C₂²×C₅	—	2	2	no	—
176	C₂²×C₅	—	2	4	no	—
177	C₂²×C₅	—	2	2	no	—
178	C₂²×C₅	#431	1	2	yes	—
179	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
180	C₄×C₅	—	2	2	no	—
181	C₄×C₅	—	2	2	no	—
182	C₄×C₅	—	2	4	no	—
183	C₄×C₅	—	2	2	no	—
184	C₄×C₅	#431	1	2	yes	—
185	C₄×C₅	C₂×#117	1	1	yes	characteristic
186	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
187	#41	—	2	2	no	—
188	#41	—	2	2	no	—
189	C₂×Q₈	C₂×#168	1	1	yes	characteristic
190	C₂×D₄	C₂×#168	1	2	yes	—
191	C₂×D₄	C₂×#168	1	1	yes	characteristic
192	C₂²×C₄	C₂×#168	1	4	yes	—
193	C₂²×C₄	C₂×#168	1	2	yes	—
194	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
195	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
196	M₁₆	—	2	2	no	—
197	M₁₆	—	2	2	no	—
198	M₁₆	—	2	4	no	—
199	M₁₆	—	2	4	no	—
200	C₂×C₈	—	2	4	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	#681	1	4	yes	—
203	C₂×C₈	#709	1	4	yes	—
204	C₂×C₈	C₂²×Dic₅	1	2	yes	—
205	C₂×C₈	C₂²×Dic₅	1	2	yes	—
206	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
207	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
208	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
209	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
210	C₄⋊C₄	C₂×#168	1	4	yes	—
211	C₄⋊C₄	C₂×#168	1	2	yes	—
212	C₂²⋊C₄	C₂×#168	1	4	yes	—
213	C₂²⋊C₄	C₂×#168	1	2	yes	—
214	C₄²	C₂×#168	1	4	yes	—
215	C₄²	C₂×#168	1	2	yes	—
216	C₄²	C₂²×Dic₅	1	1	yes	characteristic
217	C₄²	C₂²×Dic₅	1	1	yes	characteristic
218	C₄²	C₂²×Dic₅	1	2	yes	—
219	C₁₆	—	10	4	no	—
220	C₁₆	—	10	4	no	—
221	C₂×C₅	—	2	2	no	—
222	C₂×C₅	—	4	2	no	—
223	C₂×C₅	C₂×#431	1	1	yes	characteristic
224	C₂×C₅	#3407	1	2	yes	—
225	C₂³	C₂²×#168	1	2	yes	—
226	C₂³	C₂²×#168	1	1	yes	characteristic
227	Q₈	—	2	2	no	—
228	D₄	—	2	4	no	—
229	D₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	C₂²×#168	1	2	yes	—
235	C₂×C₄	C₂²×#168	1	2	yes	—
236	C₂×C₄	C₂²×#168	1	4	yes	—
237	C₂×C₄	C₂²×#168	1	2	yes	—
238	C₂×C₄	C₂²×#168	1	1	yes	characteristic
239	C₂×C₄	C₂×#681	1	2	yes	—
240	C₂×C₄	#4072	1	2	yes	—
241	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
242	C₈	—	2	2	no	—
243	C₈	—	2	2	no	—
244	C₈	C₂×#681	1	2	yes	—
245	C₈	C₂×#681	1	2	yes	—
246	C₈	#4072	1	2	yes	—
247	C₈	C₂×#709	1	2	yes	—
248	C₅	#33421	1	1	yes	characteristic, complemented, Sylow
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	4	no	—
252	C₂²	#65214	1	2	yes	—
253	C₂²	C₂³×#168	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	2	4	no	—
258	C₄	#65214	1	2	yes	—
259	C₄	C₂²×#681	1	1	yes	characteristic
260	C₄	C₂×#4072	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	4	2	no	—
263	C₂	C₂×#65214	1	1	yes	characteristic
264	C₂	#10615511	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₈
Fitting	C₄×C₅×#131	C₂
prime radical	C₂×C₅×#41	C₂×C₄
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	C₄×#131	D₅
hypocenter	C₅	#33421
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
3. #10600676	C₂	semidirect
4. #10600675	C₂	semidirect
5. #10600538	C₂	semidirect
6. C₄×#3944	C₂	semidirect
7. C₂×#64681	C₂	semidirect
248. C₅	#33421	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
#10602797	2
#10600676	2
#10600675	2
#10600538	2
C₄×#3944	2
C₂×#64681	2
#33421	5