Subgroups — #12016190

Order 1280.

474 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2931	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	#10602799	C₂	1	2	yes	maximal, complemented
5	#10598149	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#131	C₂²	1	1	yes	characteristic
7	C₂×C₅×#131	C₂²	1	1	yes	characteristic
8	C₅×#532	C₄	1	1	yes	characteristic
9	C₅×#436	—	2	2	no	—
10	C₅×C₈×D₄	—	2	2	no	—
11	C₅×#423	C₄	1	1	yes	characteristic
12	C₅×#408	C₄	1	1	yes	characteristic
13	C₅×#408	C₄	1	1	yes	characteristic
14	C₂×C₄×C₅×C₈	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	#64781	—	2	4	no	—
20	#64773	—	2	2	no	—
21	#12982	—	5	1	no	maximal, Sylow
22	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
23	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
24	C₅×#131	—	2	2	no	—
25	C₅×#131	—	2	2	no	—
26	C₅×#131	—	2	2	no	—
27	C₅×#131	—	2	2	no	—
28	C₂×C₅×M₁₆	—	2	1	no	—
29	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
31	C₂×C₅×M₁₆	D₄	1	2	yes	—
32	C₂²×C₅×C₈	—	2	1	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×C₄⋊Q₈	C₈	1	1	yes	characteristic
37	C₅×#127	C₈	1	1	yes	characteristic
38	C₅×#124	C₈	1	1	yes	characteristic
39	C₅×#124	C₈	1	1	yes	characteristic
40	C₅×#121	—	2	2	no	—
41	C₄×C₅×D₄	—	2	2	no	—
42	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
43	C₅×C₄⋊C₈	—	2	2	no	—
44	C₅×C₂²⋊C₈	—	2	2	no	—
45	C₅×C₂²⋊C₈	—	2	2	no	—
46	C₄×C₅×C₈	—	2	1	no	—
47	C₄×C₅×C₈	—	2	1	no	—
48	#3944	—	2	4	no	—
49	#3944	—	2	2	no	—
50	#3944	—	2	2	no	—
51	#3944	—	2	2	no	—
52	#3944	D₄	1	4	yes	complemented
53	C₂×#672	—	2	4	no	—
54	C₂×#672	—	2	2	no	—
55	C₂×#672	—	2	2	no	—
56	#3407	—	5	1	no	—
57	#2931	D₅	1	1	yes	characteristic
58	#2117	—	5	2	no	—
59	#2115	—	5	1	no	—
60	#1376	—	5	2	no	—
61	C₅×#41	—	2	2	no	—
62	C₅×#41	—	2	2	no	—
63	C₅×#41	—	2	2	no	—
64	C₅×#41	—	2	2	no	—
65	C₂×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
66	C₂×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
67	C₂×C₅×D₄	—	2	2	no	—
68	C₂×C₅×D₄	—	2	1	no	—
69	C₂×C₅×D₄	C₂×C₈	1	1	yes	characteristic
70	C₂×C₅×D₄	C₂×C₈	1	1	yes	characteristic
71	C₂×C₅×D₄	M₁₆	1	2	yes	—
72	C₂²×C₄×C₅	—	2	1	no	—
73	C₂²×C₄×C₅	—	2	1	no	—
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	4	no	—
78	C₅×M₁₆	—	2	2	no	—
79	C₅×M₁₆	—	4	2	no	—
80	C₂×C₅×C₈	—	2	2	no	—
81	C₂×C₅×C₈	—	2	2	no	—
82	C₂×C₅×C₈	—	2	1	no	—
83	C₂×C₅×C₈	—	2	1	no	—
84	C₂×C₅×C₈	—	2	2	no	—
85	C₂×C₅×C₈	—	4	1	no	—
86	C₂×C₅×C₈	—	4	1	no	—
87	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
88	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
89	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
91	C₂×C₅×C₈	#41	1	2	yes	—
92	C₅×C₄⋊C₄	—	2	2	no	—
93	C₅×C₂²⋊C₄	—	2	2	no	—
94	C₅×C₂²⋊C₄	—	2	2	no	—
95	C₄²×C₅	—	2	1	no	—
96	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
97	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
98	#672	—	2	4	no	—
99	#672	—	4	2	no	—
100	C₂×#131	C₂×D₅	1	1	yes	characteristic
101	C₂×#131	C₂×D₅	1	1	yes	characteristic
102	#532	Dic₅	1	1	yes	characteristic
103	#504	—	10	4	no	—
104	C₂×M₃₂	—	5	2	no	—
105	C₂×M₃₂	—	5	2	no	—
106	C₂×M₃₂	—	10	2	no	—
107	#436	—	2	2	no	—
108	C₈×D₄	—	2	2	no	—
109	#423	Dic₅	1	1	yes	characteristic
110	#408	Dic₅	1	1	yes	characteristic
111	#408	Dic₅	1	1	yes	characteristic
112	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
113	#350	—	10	4	no	—
114	#349	—	10	2	no	—
115	C₂³×C₅	—	2	1	no	—
116	C₂³×C₅	C₂²⋊C₈	1	2	yes	—
117	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
118	C₅×Q₈	—	2	2	no	—
119	C₅×Q₈	—	2	2	no	—
120	C₅×D₄	—	2	4	no	—
121	C₅×D₄	—	2	2	no	—
122	C₅×D₄	—	2	2	no	—
123	C₅×D₄	—	4	4	no	—
124	C₅×D₄	—	4	2	no	—
125	C₂×C₄×C₅	—	2	1	no	—
126	C₂×C₄×C₅	—	2	2	no	—
127	C₂×C₄×C₅	—	2	2	no	—
128	C₂×C₄×C₅	—	2	1	no	—
129	C₂×C₄×C₅	—	2	1	no	—
130	C₂×C₄×C₅	—	4	2	no	—
131	C₂×C₄×C₅	—	4	1	no	—
132	C₂×C₄×C₅	—	4	1	no	—
133	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
134	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
136	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
137	C₂×C₄×C₅	#131	1	2	yes	—
138	C₅×C₈	—	2	2	no	—
139	C₅×C₈	—	4	1	no	—
140	C₅×C₈	—	4	1	no	—
141	C₅×C₈	#120	1	1	yes	characteristic
142	C₅×C₈	#121	1	2	yes	—
143	C₅×C₈	#123	1	1	yes	characteristic
144	C₂×#41	C₂×Dic₅	1	1	yes	characteristic
145	C₂×#41	C₂×Dic₅	1	1	yes	characteristic
146	#131	—	2	2	no	—
147	#131	—	2	2	no	—
148	#131	—	2	2	no	—
149	#131	—	2	2	no	—
150	C₂×M₁₆	—	2	1	no	—
151	C₂×M₁₆	C₂×Dic₅	1	1	yes	characteristic
152	C₂×M₁₆	C₂×Dic₅	1	1	yes	characteristic
153	C₂×M₁₆	C₂²⋊D₅	1	2	yes	—
154	C₂²×C₈	—	2	1	no	—
155	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
156	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
157	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
158	C₄⋊Q₈	#168	1	1	yes	characteristic
159	#127	#168	1	1	yes	characteristic
160	#124	#168	1	1	yes	characteristic
161	#124	#168	1	1	yes	characteristic
162	#121	—	2	2	no	—
163	C₄×D₄	—	2	2	no	—
164	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
165	M₃₂	—	5	4	no	—
166	M₃₂	—	10	4	no	—
167	M₃₂	—	10	2	no	—
168	M₃₂	—	10	2	no	—
169	M₃₂	—	10	2	no	—
170	C₂×C₁₆	—	10	4	no	—
171	C₂×C₁₆	—	10	2	no	—
172	C₂×C₁₆	—	10	2	no	—
173	C₄⋊C₈	—	2	2	no	—
174	C₂²⋊C₈	—	2	2	no	—
175	C₂²⋊C₈	—	2	2	no	—
176	C₄×C₈	—	2	1	no	—
177	C₄×C₈	—	2	1	no	—
178	C₂²×C₅	—	2	2	no	—
179	C₂²×C₅	—	4	1	no	—
180	C₂²×C₅	—	4	2	no	—
181	C₂²×C₅	—	4	2	no	—
182	C₂²×C₅	—	4	1	no	—
183	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
184	C₂²×C₅	#435	1	2	yes	—
185	C₄×C₅	—	2	2	no	—
186	C₄×C₅	—	2	2	no	—
187	C₄×C₅	—	4	1	no	—
188	C₄×C₅	—	4	1	no	—
189	C₄×C₅	#386	1	1	yes	characteristic
190	C₄×C₅	#408	1	1	yes	characteristic
191	C₄×C₅	C₈×D₄	1	2	yes	—
192	#41	—	2	2	no	—
193	#41	—	2	2	no	—
194	#41	—	2	2	no	—
195	#41	—	2	2	no	—
196	C₂×Q₈	C₂×#168	1	1	yes	characteristic
197	C₂×Q₈	C₂×#168	1	1	yes	characteristic
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	C₂×#168	1	1	yes	characteristic
201	C₂×D₄	C₂×#168	1	1	yes	characteristic
202	C₂×D₄	#681	1	2	yes	—
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
206	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
207	M₁₆	—	2	2	no	—
208	M₁₆	—	2	4	no	—
209	M₁₆	—	2	2	no	—
210	M₁₆	—	4	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	2	1	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	4	1	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
219	C₂×C₈	D₄×D₅	1	2	yes	—
220	C₂×C₈	#711	1	2	yes	—
221	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
222	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
223	C₄⋊C₄	—	2	2	no	—
224	C₂²⋊C₄	—	2	2	no	—
225	C₂²⋊C₄	—	2	2	no	—
226	C₄²	—	2	1	no	—
227	C₄²	C₂×#168	1	1	yes	characteristic
228	C₄²	C₂×#168	1	1	yes	characteristic
229	C₁₆	—	10	4	no	—
230	C₁₆	—	20	2	no	—
231	C₂×C₅	—	2	1	no	—
232	C₂×C₅	—	2	2	no	—
233	C₂×C₅	—	4	2	no	—
234	C₂×C₅	—	8	1	no	—
235	C₂×C₅	#1869	1	1	yes	characteristic
236	C₂³	—	2	1	no	—
237	C₂³	C₂²⋊#168	1	2	yes	—
238	C₂³	C₂×#681	1	1	yes	characteristic
239	Q₈	—	2	2	no	—
240	Q₈	—	2	2	no	—
241	D₄	—	2	4	no	—
242	D₄	—	2	2	no	—
243	D₄	—	2	2	no	—
244	D₄	—	4	4	no	—
245	D₄	—	4	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	C₂²⋊#168	1	2	yes	—
255	C₂×C₄	C₂²×#168	1	1	yes	characteristic
256	C₂×C₄	D₄×Dic₅	1	2	yes	—
257	C₂×C₄	#4094	1	2	yes	—
258	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
259	C₈	—	2	2	no	—
260	C₈	—	4	1	no	—
261	C₈	—	4	1	no	—
262	C₈	#4081	1	1	yes	characteristic
263	C₈	#4083	1	1	yes	characteristic
264	C₈	#4085	1	2	yes	—
265	C₅	#12982	1	1	yes	characteristic, complemented, Sylow
266	C₂²	—	2	2	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	4	1	no	—
271	C₂²	#65300	1	2	yes	—
272	C₂²	C₂×C₂²⋊#168	1	1	yes	characteristic
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	—	4	1	no	—
276	C₄	—	4	1	no	—
277	C₄	D₄×#168	1	2	yes	—
278	C₄	#65509	1	1	yes	characteristic
279	C₄	#65523	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	2	no	—
282	C₂	—	4	2	no	—
283	C₂	—	8	1	no	—
284	C₂	#10602403	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#4083
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#2931	C₂
socle	C₂×C₅	#1869
prime radical	C₅×#532	C₄
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#2931	D₅
hypocenter	C₅	#12982
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2931

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. #10602799	C₂	semidirect
5. #10598149	C₂	semidirect
52. #3944	D₄	semidirect
265. C₅	#12982	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2931	2
#10615511	2
#10602800	2
#10602799	2
#10598149	2
#12982	5