Subgroups — #12016185

Order 1280.

488 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×#2929	C₂	1	1	yes	maximal, characteristic
2	#10615511	C₂	1	1	yes	maximal, characteristic, complemented
3	#10602800	C₂	1	1	yes	maximal, characteristic, complemented
4	#10602756	C₂	1	2	yes	maximal, complemented
5	#10598163	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#131	C₂²	1	1	yes	characteristic
7	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₅×#525	C₄	1	1	yes	characteristic
9	C₅×#435	—	2	2	no	—
10	C₅×C₈×D₄	—	2	2	no	—
11	C₅×#424	C₄	1	1	yes	characteristic
12	C₅×#408	C₄	1	1	yes	characteristic
13	C₅×#407	C₄	1	1	yes	characteristic
14	C₅×#405	C₂²	1	1	yes	characteristic
15	#65429	—	2	4	no	—
16	C₂×#3944	—	2	2	no	—
17	C₂×#3944	C₂²	1	2	yes	—
18	C₂×#3944	C₂²	1	2	yes	—
19	#64773	—	2	2	no	—
20	C₂²⋊#672	—	2	4	no	—
21	#12981	—	5	1	no	maximal, Sylow
22	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
24	C₅×#131	—	2	2	no	—
25	C₅×#131	—	2	2	no	—
26	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂×C₅×M₁₆	D₄	1	2	yes	—
29	C₂²×C₅×C₈	—	2	2	no	—
30	C₂²×C₅×C₈	—	2	1	no	—
31	C₂²×C₅×C₈	—	2	1	no	—
32	C₂²×C₅×C₈	—	2	2	no	—
33	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₅×C₈	D₄	1	2	yes	—
35	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₅×#123	C₈	1	2	yes	—
37	C₅×C₂²⋊Q₈	—	2	1	no	—
38	C₅×#121	—	2	1	no	—
39	C₅×#120	C₈	1	2	yes	—
40	C₄×C₅×D₄	—	2	2	no	—
41	C₅×#117	C₂×C₄	1	1	yes	characteristic
42	C₅×C₄⋊C₈	—	2	2	no	—
43	C₅×C₂²⋊C₈	—	2	2	no	—
44	C₅×C₂²⋊C₈	—	2	2	no	—
45	C₅×#97	—	2	1	no	—
46	C₄×C₅×C₈	—	2	1	no	—
47	#3944	—	2	4	no	—
48	#3944	—	2	2	no	—
49	#3944	—	2	2	no	—
50	#3944	—	2	2	no	—
51	#3944	—	2	2	no	—
52	C₂×#672	—	2	4	no	—
53	C₂×#672	—	2	2	no	—
54	C₂×#672	D₄	1	4	yes	complemented
55	#3407	—	5	1	no	—
56	#2929	D₅	1	1	yes	characteristic
57	#2115	—	5	1	no	—
58	#2112	—	5	2	no	—
59	#1378	—	5	2	no	—
60	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
61	C₅×#41	—	2	2	no	—
62	C₅×#41	—	2	2	no	—
63	C₂×C₅×Q₈	M₁₆	1	1	yes	characteristic
64	C₂×C₅×D₄	—	2	2	no	—
65	C₂×C₅×D₄	C₂×C₈	1	2	yes	—
66	C₂×C₅×D₄	M₁₆	1	1	yes	characteristic
67	C₂²×C₄×C₅	—	2	1	no	—
68	C₂²×C₄×C₅	—	2	2	no	—
69	C₂²×C₄×C₅	—	2	1	no	—
70	C₂²×C₄×C₅	—	2	2	no	—
71	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
72	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₅×M₁₆	—	2	2	no	—
75	C₅×M₁₆	—	2	4	no	—
76	C₅×M₁₆	—	2	2	no	—
77	C₂×C₅×C₈	—	2	2	no	—
78	C₂×C₅×C₈	—	2	2	no	—
79	C₂×C₅×C₈	—	2	1	no	—
80	C₂×C₅×C₈	—	2	1	no	—
81	C₂×C₅×C₈	—	2	2	no	—
82	C₂×C₅×C₈	—	4	1	no	—
83	C₂×C₅×C₈	—	4	2	no	—
84	C₂×C₅×C₈	—	4	1	no	—
85	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
86	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
87	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
88	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
89	C₂×C₅×C₈	#41	1	2	yes	—
90	C₅×C₄⋊C₄	—	2	2	no	—
91	C₅×C₄⋊C₄	—	2	1	no	—
92	C₅×C₂²⋊C₄	—	2	2	no	—
93	C₅×C₂²⋊C₄	—	2	2	no	—
94	C₅×C₂²⋊C₄	C₂×C₈	1	2	yes	—
95	C₄²×C₅	—	2	1	no	—
96	#672	—	2	4	no	—
97	#672	—	4	2	no	—
98	C₂×#131	C₂×D₅	1	1	yes	characteristic
99	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
100	#525	Dic₅	1	1	yes	characteristic
101	#504	—	10	4	no	—
102	C₂×M₃₂	—	5	2	no	—
103	C₂×M₃₂	—	5	2	no	—
104	C₂×M₃₂	—	10	2	no	—
105	#435	—	2	2	no	—
106	C₈×D₄	—	2	2	no	—
107	#424	Dic₅	1	1	yes	characteristic
108	#408	Dic₅	1	1	yes	characteristic
109	#407	Dic₅	1	1	yes	characteristic
110	#405	C₂×D₅	1	1	yes	characteristic
111	#349	—	10	2	no	—
112	C₂²⋊C₁₆	—	10	4	no	—
113	C₂³×C₅	—	2	1	no	—
114	C₂³×C₅	—	2	2	no	—
115	C₂³×C₅	—	4	2	no	—
116	C₂³×C₅	C₂²⋊C₈	1	2	yes	—
117	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
118	C₅×Q₈	—	2	2	no	—
119	C₅×D₄	—	2	4	no	—
120	C₅×D₄	—	2	2	no	—
121	C₅×D₄	—	4	4	no	—
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	—	2	2	no	—
125	C₂×C₄×C₅	—	2	1	no	—
126	C₂×C₄×C₅	—	2	2	no	—
127	C₂×C₄×C₅	—	4	1	no	—
128	C₂×C₄×C₅	—	4	1	no	—
129	C₂×C₄×C₅	—	4	2	no	—
130	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
131	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
133	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
134	C₂×C₄×C₅	#131	1	2	yes	—
135	C₅×C₈	—	2	2	no	—
136	C₅×C₈	—	4	1	no	—
137	C₅×C₈	—	4	1	no	—
138	C₅×C₈	#120	1	1	yes	characteristic
139	C₅×C₈	#121	1	2	yes	—
140	C₅×C₈	#123	1	1	yes	characteristic
141	C₂×#41	C₂×Dic₅	1	1	yes	characteristic
142	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
143	#131	—	2	2	no	—
144	#131	—	2	2	no	—
145	C₂×M₁₆	C₂×Dic₅	1	1	yes	characteristic
146	C₂×M₁₆	C₂×Dic₅	1	1	yes	characteristic
147	C₂×M₁₆	C₂²⋊D₅	1	2	yes	—
148	C₂²×C₈	—	2	2	no	—
149	C₂²×C₈	—	2	2	no	—
150	C₂²×C₈	—	2	1	no	—
151	C₂²×C₈	—	2	1	no	—
152	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
153	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
154	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
155	#123	#168	1	2	yes	—
156	C₂²⋊Q₈	—	2	1	no	—
157	#121	—	2	1	no	—
158	#120	#168	1	2	yes	—
159	C₄×D₄	—	2	2	no	—
160	#117	C₂×Dic₅	1	1	yes	characteristic
161	M₃₂	—	10	4	no	—
162	M₃₂	—	10	2	no	—
163	M₃₂	—	10	2	no	—
164	M₃₂	—	10	2	no	—
165	M₃₂	—	10	2	no	—
166	C₂×C₁₆	—	5	4	no	—
167	C₂×C₁₆	—	10	4	no	—
168	C₂×C₁₆	—	10	2	no	—
169	C₄⋊C₈	—	2	2	no	—
170	C₂²⋊C₈	—	2	2	no	—
171	C₂²⋊C₈	—	2	2	no	—
172	#97	—	2	1	no	—
173	C₄×C₈	—	2	1	no	—
174	C₂²×C₅	—	2	2	no	—
175	C₂²×C₅	—	2	4	no	—
176	C₂²×C₅	—	2	2	no	—
177	C₂²×C₅	—	4	1	no	—
178	C₂²×C₅	—	4	2	no	—
179	C₂²×C₅	—	4	1	no	—
180	C₂²×C₅	—	4	2	no	—
181	C₂²×C₅	—	4	1	no	—
182	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
183	C₂²×C₅	C₈×D₄	1	2	yes	—
184	C₄×C₅	—	2	2	no	—
185	C₄×C₅	—	4	2	no	—
186	C₄×C₅	—	4	1	no	—
187	C₄×C₅	#386	1	1	yes	characteristic
188	C₄×C₅	#408	1	1	yes	characteristic
189	C₄×C₅	#435	1	2	yes	—
190	C₂⁴	C₂×#168	1	1	yes	characteristic
191	#41	—	2	2	no	—
192	#41	—	2	2	no	—
193	C₂×Q₈	#681	1	1	yes	characteristic
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	C₂×#168	1	2	yes	—
196	C₂×D₄	#681	1	1	yes	characteristic
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	C₂×#168	1	1	yes	characteristic
202	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
203	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
204	M₁₆	—	2	2	no	—
205	M₁₆	—	2	4	no	—
206	M₁₆	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	4	1	no	—
213	C₂×C₈	—	4	1	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
216	C₂×C₈	D₄×D₅	1	2	yes	—
217	C₂×C₈	#711	1	2	yes	—
218	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
219	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	1	no	—
222	C₂²⋊C₄	—	2	2	no	—
223	C₂²⋊C₄	—	2	2	no	—
224	C₂²⋊C₄	C₂×#168	1	2	yes	—
225	C₄²	—	2	1	no	—
226	C₁₆	—	10	4	no	—
227	C₁₆	—	20	2	no	—
228	C₂×C₅	—	2	1	no	—
229	C₂×C₅	—	2	2	no	—
230	C₂×C₅	—	4	2	no	—
231	C₂×C₅	—	4	2	no	—
232	C₂×C₅	#1869	1	1	yes	characteristic
233	C₂³	—	2	2	no	—
234	C₂³	—	2	1	no	—
235	C₂³	—	4	2	no	—
236	C₂³	C₂²⋊#168	1	2	yes	—
237	C₂³	C₂²×#168	1	1	yes	characteristic
238	Q₈	—	2	2	no	—
239	D₄	—	2	4	no	—
240	D₄	—	2	2	no	—
241	D₄	—	4	4	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	C₂²⋊#168	1	2	yes	—
251	C₂×C₄	C₂×#681	1	1	yes	characteristic
252	C₂×C₄	D₄×Dic₅	1	2	yes	—
253	C₂×C₄	#4094	1	2	yes	—
254	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
255	C₈	—	2	2	no	—
256	C₈	—	4	1	no	—
257	C₈	—	4	1	no	—
258	C₈	#4081	1	1	yes	characteristic
259	C₈	#4083	1	1	yes	characteristic
260	C₈	#4085	1	2	yes	—
261	C₅	#12981	1	1	yes	characteristic, complemented, Sylow
262	C₂²	—	2	2	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	1	no	—
270	C₂²	D₄×#168	1	2	yes	—
271	C₂²	C₂×C₂²⋊#168	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	—	4	2	no	—
274	C₄	—	4	1	no	—
275	C₄	#65300	1	2	yes	—
276	C₄	#65509	1	1	yes	characteristic
277	C₄	#65523	1	1	yes	characteristic
278	C₂	—	2	1	no	—
279	C₂	—	2	2	no	—
280	C₂	—	4	2	no	—
281	C₂	—	4	2	no	—
282	C₂	#10602403	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#4083
commutator	C₂³×C₅	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#2929	C₂
socle	C₂×C₅	#1869
prime radical	C₅×#525	C₄
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#2929	D₅
hypocenter	C₅	#12981
LCS	G ▷ C₂³×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2929

Complements

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

N	H ≅ G/N	type
2. #10615511	C₂	semidirect
3. #10602800	C₂	semidirect
4. #10602756	C₂	semidirect
5. #10598163	C₂	semidirect
54. C₂×#672	D₄	semidirect
261. C₅	#12981	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2929	2
#10615511	2
#10602800	2
#10602756	2
#10598163	2
#12981	5