Subgroups — #12015790

Order 1280.

376 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10613738	C₂	1	1	yes	maximal, characteristic
2	C₅×#2153	C₂	1	1	yes	maximal, characteristic, complemented
3	#10602231	C₂	1	1	yes	maximal, characteristic
4	#10602213	C₂	1	1	yes	maximal, characteristic, complemented
5	#10600680	C₂	1	1	yes	maximal, characteristic
6	#10598301	C₂	1	1	yes	maximal, characteristic, complemented
7	#10598163	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#4047	C₂²	1	1	yes	characteristic
9	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
11	C₅×#405	C₂²	1	1	yes	characteristic
12	C₂×#3957	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
14	C₂×#3944	C₂²	1	1	yes	characteristic
15	C₂×#3944	C₂²	1	1	yes	characteristic
16	#65291	C₄	1	1	yes	characteristic
17	#65196	—	2	2	no	—
18	D₅×M₃₂	—	2	2	no	—
19	#65190	—	2	2	no	—
20	C₅×#364	—	2	2	no	—
21	#64778	C₄	1	1	yes	characteristic
22	#64752	C₄	1	1	yes	characteristic
23	#64735	—	2	2	no	—
24	#64721	—	2	2	no	—
25	#12918	—	5	1	no	maximal, Sylow
26	C₂×#709	C₂×C₄	1	1	yes	characteristic
27	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
29	C₅×#117	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
31	C₂×#681	C₂×C₄	1	1	yes	characteristic
32	C₂²×#168	C₂×C₄	1	1	yes	characteristic
33	#4047	—	2	1	no	—
34	#4047	—	2	1	no	—
35	#4047	D₄	1	2	yes	—
36	#4047	Q₈	1	2	yes	—
37	C₂×#676	—	2	1	no	—
38	C₂×C₈×D₅	—	2	1	no	—
39	C₅×M₃₂	—	2	1	no	—
40	C₅×M₃₂	—	2	1	no	—
41	C₅×M₃₂	—	2	1	no	—
42	C₅×M₃₂	D₄	1	2	yes	—
43	C₂×C₅×C₁₆	—	2	1	no	—
44	C₂×C₅×C₁₆	—	2	1	no	—
45	C₅×#97	—	2	1	no	—
46	C₄×C₅×C₈	—	2	1	no	—
47	#3957	C₈	1	2	yes	—
48	#3957	C₈	1	2	yes	—
49	C₈×Dic₅	—	2	1	no	—
50	C₈×Dic₅	—	2	1	no	—
51	#3944	—	2	1	no	—
52	#3944	—	2	1	no	—
53	#3944	—	2	1	no	—
54	#3944	—	2	1	no	—
55	C₂×#672	—	2	1	no	—
56	C₂×#672	D₄	1	2	yes	—
57	#3942	—	2	1	no	—
58	#3941	—	2	1	no	—
59	#3930	—	2	4	no	—
60	D₅×C₁₆	—	2	4	no	—
61	#3407	—	5	1	no	—
62	#2153	D₅	1	1	yes	characteristic, complemented
63	#2150	—	5	1	no	—
64	#2117	—	5	1	no	—
65	#1764	—	5	1	no	—
66	#1378	—	5	1	no	—
67	#1376	—	5	1	no	—
68	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₂²⋊D₅	—	2	1	no	—
70	C₂²×Dic₅	—	2	1	no	—
71	#709	—	2	1	no	—
72	#709	—	2	1	no	—
73	#709	C₄⋊C₄	1	2	yes	—
74	#709	C₂×C₈	1	2	yes	—
75	C₂×D₂₀	M₁₆	1	1	yes	characteristic
76	C₂×C₄×D₅	—	2	1	no	—
77	C₂×Dic₁₀	M₁₆	1	1	yes	characteristic
78	C₅×M₁₆	—	2	1	no	—
79	C₅×M₁₆	C₂×C₈	1	2	yes	—
80	C₂×C₅×C₈	—	2	1	no	—
81	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
82	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
83	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
84	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₅×C₈	#41	1	1	yes	characteristic
86	C₂×C₅×C₈	#41	1	1	yes	characteristic
87	C₅×C₄⋊C₄	—	2	1	no	—
88	C₅×C₂²⋊C₄	—	2	1	no	—
89	C₄²×C₅	—	2	1	no	—
90	C₄×Dic₅	—	2	1	no	—
91	C₄×Dic₅	C₂×C₈	1	1	yes	characteristic
92	C₄×Dic₅	C₂×C₈	1	1	yes	characteristic
93	#681	—	2	1	no	—
94	#681	C₄⋊C₄	1	2	yes	—
95	C₂×#168	—	2	1	no	—
96	C₂×#168	—	2	1	no	—
97	C₂×#168	—	2	1	no	—
98	C₂×#168	—	2	1	no	—
99	#676	—	2	2	no	—
100	#676	—	2	2	no	—
101	C₈×D₅	—	2	2	no	—
102	C₈×D₅	—	2	2	no	—
103	C₅×C₁₆	—	2	2	no	—
104	C₅×C₁₆	—	4	1	no	—
105	#672	—	2	2	no	—
106	#672	—	4	1	no	—
107	C₂×#131	—	5	1	no	—
108	#504	—	10	2	no	—
109	#504	—	10	2	no	—
110	C₂×M₃₂	—	5	1	no	—
111	C₂×M₃₂	—	5	1	no	—
112	C₂×M₃₂	—	10	2	no	—
113	C₂×M₃₂	C₂×D₅	1	1	yes	characteristic
114	C₂×M₃₂	C₂×D₅	1	1	yes	characteristic
115	C₂×#104	—	5	1	no	—
116	#416	—	5	1	no	—
117	#405	C₂×D₅	1	1	yes	characteristic
118	C₂×C₄×C₈	—	5	1	no	—
119	#364	—	2	2	no	—
120	C₄⋊C₁₆	—	10	2	no	—
121	#350	—	10	2	no	—
122	#341	—	5	1	no	—
123	#339	—	5	1	no	—
124	C₂³×C₅	#131	1	1	yes	characteristic
125	C₂²×D₅	—	2	1	no	—
126	C₂×C₄×C₅	—	4	1	no	—
127	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
132	C₂×C₄×C₅	#131	1	1	yes	characteristic
133	C₂²⋊D₅	—	2	2	no	—
134	C₂²⋊D₅	—	2	2	no	—
135	C₂×Dic₅	—	2	1	no	—
136	C₂×Dic₅	—	2	1	no	—
137	C₂×Dic₅	—	2	1	no	—
138	C₂×Dic₅	—	4	2	no	—
139	D₂₀	—	2	1	no	—
140	D₂₀	C₄⋊C₈	1	2	yes	—
141	C₄×D₅	—	2	2	no	—
142	C₄×D₅	—	2	2	no	—
143	Dic₁₀	—	2	1	no	—
144	Dic₁₀	C₄⋊C₈	1	2	yes	—
145	C₅×C₈	—	2	2	no	—
146	C₅×C₈	#120	1	1	yes	characteristic
147	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
148	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
149	C₅×C₈	#123	1	1	yes	characteristic
150	#168	—	2	2	no	—
151	#168	—	4	1	no	—
152	C₂×#41	—	5	1	no	—
153	#131	—	5	2	no	—
154	#131	—	5	2	no	—
155	#131	—	10	1	no	—
156	#131	—	10	1	no	—
157	C₂×M₁₆	—	5	1	no	—
158	C₂×M₁₆	—	10	1	no	—
159	C₂×M₁₆	C₄×D₅	1	1	yes	characteristic
160	C₂²×C₈	—	5	1	no	—
161	C₂²×C₈	—	10	1	no	—
162	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
163	#117	C₄×D₅	1	1	yes	characteristic
164	C₂×C₄²	—	5	1	no	—
165	M₃₂	—	2	1	no	—
166	M₃₂	—	2	1	no	—
167	M₃₂	—	2	1	no	—
168	M₃₂	—	10	4	no	—
169	M₃₂	—	10	1	no	—
170	M₃₂	—	10	1	no	—
171	M₃₂	—	10	1	no	—
172	M₃₂	—	10	1	no	—
173	M₃₂	C₂²⋊D₅	1	2	yes	—
174	C₂×C₁₆	—	2	1	no	—
175	C₂×C₁₆	—	2	1	no	—
176	C₂×C₁₆	—	5	2	no	—
177	C₂×C₁₆	—	10	4	no	—
178	C₂×C₁₆	—	10	1	no	—
179	#104	—	5	2	no	—
180	#104	—	5	2	no	—
181	#103	—	10	1	no	—
182	#102	—	10	1	no	—
183	#97	—	2	1	no	—
184	C₄×C₈	—	2	1	no	—
185	C₄×C₈	—	10	1	no	—
186	C₄×C₈	—	10	1	no	—
187	C₂²×C₅	—	4	1	no	—
188	C₂²×C₅	#424	1	1	yes	characteristic
189	C₂²×C₅	C₈×D₄	1	1	yes	characteristic
190	C₂²×C₅	#435	1	1	yes	characteristic
191	C₂×D₅	—	2	2	no	—
192	C₂×D₅	—	8	1	no	—
193	C₄×C₅	—	8	1	no	—
194	C₄×C₅	#385	1	1	yes	characteristic
195	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
196	C₄×C₅	C₈×D₄	1	1	yes	characteristic
197	C₄×C₅	#435	1	1	yes	characteristic
198	Dic₅	—	2	2	no	—
199	Dic₅	—	2	2	no	—
200	Dic₅	—	4	1	no	—
201	#41	—	5	2	no	—
202	#41	—	5	2	no	—
203	#41	—	10	1	no	—
204	#41	—	10	1	no	—
205	C₂×Q₈	—	5	1	no	—
206	C₂×D₄	—	5	1	no	—
207	C₂×D₄	—	10	1	no	—
208	C₂²×C₄	—	10	1	no	—
209	C₂²×C₄	—	10	1	no	—
210	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
211	M₁₆	—	2	1	no	—
212	M₁₆	—	5	2	no	—
213	M₁₆	—	10	1	no	—
214	M₁₆	—	10	2	no	—
215	M₁₆	—	10	2	no	—
216	M₁₆	C₈×D₅	1	2	yes	—
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	10	1	no	—
219	C₂×C₈	—	10	2	no	—
220	C₂×C₈	—	10	2	no	—
221	C₂×C₈	—	10	1	no	—
222	C₂×C₈	—	10	1	no	—
223	C₂×C₈	—	10	1	no	—
224	C₂×C₈	#709	1	1	yes	characteristic
225	C₂×C₈	D₄×D₅	1	1	yes	characteristic
226	C₂×C₈	D₄×D₅	1	1	yes	characteristic
227	C₂×C₈	Q₈×D₅	1	1	yes	characteristic
228	C₂×C₈	#713	1	1	yes	characteristic
229	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
230	C₄⋊C₄	—	2	1	no	—
231	C₂²⋊C₄	—	2	1	no	—
232	C₄²	—	2	1	no	—
233	C₄²	—	5	1	no	—
234	C₄²	—	5	1	no	—
235	C₄²	—	10	1	no	—
236	C₁₆	—	2	2	no	—
237	C₁₆	—	4	1	no	—
238	C₁₆	—	10	2	no	—
239	C₁₆	—	20	1	no	—
240	C₂×C₅	—	2	1	no	—
241	C₂×C₅	—	2	1	no	—
242	C₂×C₅	—	2	1	no	—
243	C₂×C₅	#1850	1	1	yes	characteristic
244	D₅	—	4	2	no	—
245	C₂³	—	10	1	no	—
246	C₂³	#4053	1	1	yes	characteristic
247	Q₈	—	5	2	no	—
248	Q₈	—	10	1	no	—
249	D₄	—	5	2	no	—
250	D₄	—	10	2	no	—
251	D₄	—	10	2	no	—
252	D₄	—	10	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	10	1	no	—
255	C₂×C₄	—	10	1	no	—
256	C₂×C₄	—	10	2	no	—
257	C₂×C₄	—	10	2	no	—
258	C₂×C₄	—	10	1	no	—
259	C₂×C₄	—	20	2	no	—
260	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
261	C₂×C₄	#4039	1	1	yes	characteristic
262	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
263	C₂×C₄	#4047	1	1	yes	characteristic
264	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
265	C₂×C₄	C₄×C₂²⋊D₅	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₈	—	10	2	no	—
268	C₈	—	20	1	no	—
269	C₈	#4040	1	1	yes	characteristic
270	C₈	#4042	1	1	yes	characteristic
271	C₈	#4083	1	1	yes	characteristic
272	C₈	#4092	1	1	yes	characteristic
273	C₅	#12918	1	1	yes	characteristic, complemented, Sylow
274	C₂²	—	4	1	no	—
275	C₂²	—	10	2	no	—
276	C₂²	—	40	1	no	—
277	C₂²	#65123	1	1	yes	characteristic
278	C₂²	#65128	1	1	yes	characteristic
279	C₂²	#65416	1	1	yes	characteristic
280	C₄	—	8	1	no	—
281	C₄	—	10	2	no	—
282	C₄	—	10	2	no	—
283	C₄	—	20	1	no	—
284	C₄	D₅×C₄⋊C₈	1	1	yes	characteristic
285	C₄	#65125	1	1	yes	characteristic
286	C₄	#65278	1	1	yes	characteristic
287	C₄	C₈×C₂²⋊D₅	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	—	2	1	no	—
291	C₂	—	20	2	no	—
292	C₂	#10601270	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#4083
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#2153	C₂
socle	C₂×C₅	#1850
prime radical	C₂×#709	C₂×C₄
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#2153	D₅
hypocenter	C₅	#12918
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2153

Complements

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

N	H ≅ G/N	type
2. C₅×#2153	C₂	semidirect
4. #10602213	C₂	semidirect
6. #10598301	C₂	semidirect
7. #10598163	C₂	semidirect
62. #2153	D₅	semidirect
273. C₅	#12918	semidirect

Maximal subgroups

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

subgroup	index
#10613738	2
C₅×#2153	2
#10602231	2
#10602213	2
#10600680	2
#10598301	2
#10598163	2
#12918	5