Subgroups — #13026495

Order 1280.

542 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂×#64887	C₂	1	2	yes	maximal
3	#10598777	C₂	1	2	yes	maximal, complemented
4	#10598698	C₂	1	2	yes	maximal, complemented
5	C₂²×C₈×D₅	C₄	1	1	yes	characteristic
6	C₂²×C₈×D₅	C₂²	1	1	yes	characteristic
7	C₂×C₄²×D₅	C₄	1	1	yes	characteristic
8	C₂²×#674	C₂²	1	4	yes	—
9	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₅	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
12	C₂×C₄×#168	C₂²	1	1	yes	characteristic
13	C₄×C₈×D₅	—	2	4	no	—
14	#64887	—	2	4	no	—
15	#12844	—	5	1	no	maximal, Sylow
16	C₂²×C₄×D₅	C₈	1	1	yes	characteristic
17	C₂²×C₄×D₅	C₈	1	1	yes	characteristic
18	C₂²×C₄×D₅	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₅	C₈	1	1	yes	characteristic
23	C₂×C₄×Dic₅	C₈	1	1	yes	characteristic
24	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
25	C₂²×#168	C₂×C₄	1	1	yes	characteristic
26	C₂²×#168	C₂³	1	1	yes	characteristic
27	C₂×C₈×D₅	—	2	2	no	—
28	C₂×C₈×D₅	—	2	2	no	—
29	C₂×C₈×D₅	—	2	2	no	—
30	C₂×C₈×D₅	—	2	2	no	—
31	C₂×C₈×D₅	—	2	2	no	—
32	C₂×C₈×D₅	D₄	1	4	yes	—
33	C₄²×D₅	—	2	2	no	—
34	C₄²×D₅	—	2	2	no	—
35	C₂×#674	—	2	4	no	—
36	C₂×#674	—	2	4	no	—
37	C₂×#674	—	2	4	no	—
38	C₄×C₅×C₈	—	2	2	no	—
39	C₈×Dic₅	—	2	2	no	—
40	C₈×Dic₅	—	2	2	no	—
41	C₄×#168	—	2	2	no	—
42	C₂²×C₄×C₈	—	5	1	no	—
43	C₂×C₂²⋊C₁₆	—	5	2	no	—
44	#1374	—	5	2	no	—
45	#1374	—	5	2	no	—
46	C₂³×D₅	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
50	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
51	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
52	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
53	C₂×C₄×D₅	—	2	2	no	—
54	C₂×C₄×D₅	—	2	2	no	—
55	C₂×C₄×D₅	—	2	2	no	—
56	C₂×C₄×D₅	—	2	2	no	—
57	C₂×C₄×D₅	—	2	1	no	—
58	C₂×C₄×D₅	—	2	1	no	—
59	C₂×C₄×D₅	—	2	1	no	—
60	C₂×C₄×D₅	—	2	1	no	—
61	C₂×C₄×D₅	—	2	2	no	—
62	C₂×C₄×D₅	C₁₆	1	4	yes	complemented
63	C₂×C₄×D₅	C₂²⋊C₄	1	2	yes	—
64	C₂×C₄×D₅	M₁₆	1	2	yes	—
65	C₂×C₅×C₈	—	2	1	no	—
66	C₂×C₅×C₈	—	2	1	no	—
67	C₂×C₅×C₈	—	2	1	no	—
68	C₂×C₅×C₈	—	2	1	no	—
69	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
70	C₂×C₅×C₈	M₁₆	1	2	yes	—
71	C₄²×C₅	—	2	1	no	—
72	C₄²×C₅	—	2	1	no	—
73	C₄×Dic₅	—	2	1	no	—
74	C₄×Dic₅	—	2	1	no	—
75	C₄×Dic₅	—	2	2	no	—
76	C₄×Dic₅	—	2	2	no	—
77	C₂×#168	—	2	1	no	—
78	C₂×#168	—	2	1	no	—
79	C₂×#168	M₁₆	1	2	yes	—
80	C₂×#168	C₂×D₄	1	2	yes	—
81	C₂×#168	#41	1	2	yes	—
82	C₂×#168	#41	1	2	yes	—
83	C₈×D₅	—	2	8	no	—
84	C₈×D₅	—	2	8	no	—
85	#674	—	4	4	no	—
86	C₂³×C₈	—	5	1	no	—
87	C₂³×C₈	—	5	1	no	—
88	C₂²×C₄²	—	5	1	no	—
89	C₂²×C₁₆	—	5	4	no	—
90	C₂×C₄×C₈	—	5	1	no	—
91	C₂×C₄×C₈	—	5	1	no	—
92	C₂×C₄×C₈	—	5	1	no	—
93	C₂×C₄×C₈	—	10	4	no	—
94	C₂×C₄×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
95	C₂²⋊C₁₆	—	10	4	no	—
96	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
97	C₂²×D₅	—	2	1	no	—
98	C₂²×D₅	—	2	1	no	—
99	C₂²×D₅	—	2	1	no	—
100	C₂²×D₅	—	2	1	no	—
101	C₂²×D₅	C₂²⋊C₈	1	2	yes	—
102	C₂²×D₅	C₂×C₁₆	1	2	yes	—
103	C₂²×D₅	M₃₂	1	2	yes	—
104	C₂×C₄×C₅	—	2	1	no	—
105	C₂×C₄×C₅	—	2	1	no	—
106	C₂×C₄×C₅	—	2	1	no	—
107	C₂×C₄×C₅	—	2	1	no	—
108	C₂×C₄×C₅	C₂×C₁₆	1	2	yes	—
109	C₂×C₄×C₅	M₃₂	1	2	yes	—
110	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₅	#117	1	1	yes	characteristic
112	C₂×C₄×C₅	#117	1	1	yes	characteristic
113	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
116	C₂×Dic₅	—	2	1	no	—
117	C₂×Dic₅	—	2	1	no	—
118	C₂×Dic₅	—	2	1	no	—
119	C₂×Dic₅	—	2	1	no	—
120	C₂×Dic₅	—	2	1	no	—
121	C₂×Dic₅	—	2	1	no	—
122	C₂×Dic₅	C₂²⋊C₈	1	2	yes	—
123	C₂×Dic₅	C₂×C₁₆	1	2	yes	—
124	C₂×Dic₅	M₃₂	1	2	yes	—
125	C₄×D₅	—	2	8	no	—
126	C₄×D₅	—	2	8	no	—
127	C₄×D₅	—	2	2	no	—
128	C₄×D₅	—	2	2	no	—
129	C₄×D₅	—	2	2	no	—
130	C₄×D₅	—	2	2	no	—
131	C₅×C₈	—	2	2	no	—
132	C₅×C₈	—	2	2	no	—
133	#168	—	2	2	no	—
134	#168	#123	1	4	yes	—
135	C₂³×C₄	—	5	1	no	—
136	C₂³×C₄	—	5	1	no	—
137	C₂³×C₄	—	5	1	no	—
138	C₂²×C₈	—	5	1	no	—
139	C₂²×C₈	—	5	1	no	—
140	C₂²×C₈	—	5	4	no	—
141	C₂²×C₈	—	10	2	no	—
142	C₂²×C₈	—	10	2	no	—
143	C₂²×C₈	—	10	2	no	—
144	C₂²×C₈	—	10	2	no	—
145	C₂²×C₈	—	10	2	no	—
146	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
147	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
148	C₂×C₄²	—	5	1	no	—
149	C₂×C₄²	—	5	1	no	—
150	C₂×C₄²	—	5	1	no	—
151	C₂×C₄²	—	10	2	no	—
152	C₂×C₄²	—	10	2	no	—
153	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
154	C₂×C₁₆	—	10	4	no	—
155	C₂×C₁₆	—	10	4	no	—
156	C₂×C₁₆	—	10	4	no	—
157	C₄×C₈	—	2	2	no	—
158	C₄×C₈	—	10	2	no	—
159	C₄×C₈	—	10	2	no	—
160	C₄×C₈	—	10	2	no	—
161	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
162	C₂²×C₅	#431	1	1	yes	characteristic
163	C₂²×C₅	#431	1	1	yes	characteristic
164	C₂²×C₅	C₂²×C₁₆	1	1	yes	characteristic
165	C₂²×C₅	C₂×M₃₂	1	1	yes	characteristic
166	C₂²×C₅	C₂×M₃₂	1	1	yes	characteristic
167	C₂²×C₅	C₂×M₃₂	1	1	yes	characteristic
168	C₂×D₅	—	2	2	no	—
169	C₂×D₅	—	2	2	no	—
170	C₂×D₅	—	2	2	no	—
171	C₂×D₅	—	2	2	no	—
172	C₂×D₅	—	2	2	no	—
173	C₂×D₅	—	2	2	no	—
174	C₂×D₅	C₂²⋊C₁₆	1	4	yes	—
175	C₄×C₅	—	2	2	no	—
176	C₄×C₅	—	2	2	no	—
177	C₄×C₅	#381	1	1	yes	characteristic
178	C₄×C₅	#407	1	1	yes	characteristic
179	C₄×C₅	#432	1	1	yes	characteristic
180	C₄×C₅	#432	1	1	yes	characteristic
181	Dic₅	—	2	2	no	—
182	Dic₅	—	2	2	no	—
183	Dic₅	—	2	2	no	—
184	C₂⁴	—	5	1	no	—
185	C₂²×C₄	—	5	1	no	—
186	C₂²×C₄	—	5	1	no	—
187	C₂²×C₄	—	5	1	no	—
188	C₂²×C₄	—	5	2	no	—
189	C₂²×C₄	—	5	2	no	—
190	C₂²×C₄	—	5	4	no	—
191	C₂²×C₄	—	10	2	no	—
192	C₂²×C₄	—	10	2	no	—
193	C₂²×C₄	—	10	2	no	—
194	C₂²×C₄	—	10	2	no	—
195	C₂²×C₄	—	10	1	no	—
196	C₂²×C₄	—	10	1	no	—
197	C₂²×C₄	—	10	1	no	—
198	C₂²×C₄	—	10	1	no	—
199	C₂²×C₄	—	10	2	no	—
200	C₂²×C₄	D₅⋊C₈	1	1	yes	characteristic
201	C₂²×C₄	D₅⋊C₈	1	1	yes	characteristic
202	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	—	5	2	no	—
208	C₂×C₈	—	5	2	no	—
209	C₂×C₈	—	5	2	no	—
210	C₂×C₈	—	5	2	no	—
211	C₂×C₈	—	10	8	no	—
212	C₂×C₈	—	10	8	no	—
213	C₂×C₈	—	10	1	no	—
214	C₂×C₈	—	10	1	no	—
215	C₂×C₈	#700	1	2	yes	—
216	C₂×C₈	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
217	C₄²	—	2	1	no	—
218	C₄²	—	2	1	no	—
219	C₄²	—	10	2	no	—
220	C₄²	—	10	2	no	—
221	C₄²	—	10	1	no	—
222	C₄²	—	10	1	no	—
223	C₁₆	—	20	4	no	—
224	C₂×C₅	#1817	1	1	yes	characteristic
225	C₂×C₅	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
226	C₂×C₅	#2112	1	1	yes	characteristic
227	C₂×C₅	#2162	1	1	yes	characteristic
228	C₂×C₅	#2162	1	1	yes	characteristic
229	C₂×C₅	#2163	1	1	yes	characteristic
230	C₂×C₅	#2163	1	1	yes	characteristic
231	D₅	—	2	4	no	—
232	C₂³	—	5	2	no	—
233	C₂³	—	5	2	no	—
234	C₂³	—	5	2	no	—
235	C₂³	—	10	1	no	—
236	C₂³	—	10	1	no	—
237	C₂³	—	10	1	no	—
238	C₂³	—	10	1	no	—
239	C₂³	C₂×D₅⋊C₈	1	1	yes	characteristic
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	5	2	no	—
245	C₂×C₄	—	5	2	no	—
246	C₂×C₄	—	5	2	no	—
247	C₂×C₄	—	10	8	no	—
248	C₂×C₄	—	10	8	no	—
249	C₂×C₄	—	10	1	no	—
250	C₂×C₄	—	10	1	no	—
251	C₂×C₄	—	10	1	no	—
252	C₂×C₄	—	10	1	no	—
253	C₂×C₄	—	10	1	no	—
254	C₂×C₄	—	10	1	no	—
255	C₂×C₄	—	10	2	no	—
256	C₂×C₄	—	10	2	no	—
257	C₂×C₄	—	10	2	no	—
258	C₂×C₄	—	10	2	no	—
259	C₂×C₄	D₅⋊C₁₆	1	2	yes	—
260	C₂×C₄	#3990	1	2	yes	—
261	C₂×C₄	C₂×#700	1	1	yes	characteristic
262	C₂×C₄	D₅⋊M₁₆	1	1	yes	characteristic
263	C₂×C₄	D₅⋊M₁₆	1	1	yes	characteristic
264	C₂×C₄	#4130	1	1	yes	characteristic
265	C₂×C₄	#4130	1	1	yes	characteristic
266	C₂×C₄	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
267	C₈	—	2	2	no	—
268	C₈	—	2	2	no	—
269	C₈	—	5	4	no	—
270	C₈	—	10	2	no	—
271	C₅	#12844	1	1	yes	characteristic, complemented, Sylow
272	C₂²	—	5	4	no	—
273	C₂²	—	10	2	no	—
274	C₂²	—	10	2	no	—
275	C₂²	—	10	2	no	—
276	C₂²	—	10	2	no	—
277	C₂²	—	10	2	no	—
278	C₂²	—	10	2	no	—
279	C₂²	#65678	1	1	yes	characteristic
280	C₂²	#65680	1	1	yes	characteristic
281	C₂²	C₂×D₅⋊C₁₆	1	1	yes	characteristic
282	C₂²	C₂×#3990	1	1	yes	characteristic
283	C₂²	D₅⋊M₃₂	1	1	yes	characteristic
284	C₂²	D₅⋊M₃₂	1	1	yes	characteristic
285	C₂²	D₅⋊(C₂²⋊C₈)	1	1	yes	characteristic
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₄	—	10	2	no	—
289	C₄	—	10	2	no	—
290	C₄	—	10	2	no	—
291	C₄	#65683	1	1	yes	characteristic
292	C₄	#65684	1	1	yes	characteristic
293	C₄	#65756	1	1	yes	characteristic
294	C₄	#65764	1	1	yes	characteristic
295	C₂	—	10	4	no	—
296	C₂	#10603828	1	1	yes	characteristic
297	C₂	#10603829	1	1	yes	characteristic
298	C₂	#10603830	1	1	yes	characteristic
299	C₂	#10603831	1	1	yes	characteristic
300	C₂	#10604184	1	1	yes	characteristic
301	C₂	D₅⋊(C₂²⋊C₁₆)	1	1	yes	characteristic
302	C₂	#10604706	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×AGL(1,𝔽₅)
commutator	C₂²×C₅	C₂²×C₁₆
Fitting	C₂×C₄×C₅×C₈	C₄
socle	C₂³×C₅	C₂²×C₈
prime radical	C₂³×D₅	C₂×C₈
cosocle	C₂²×#168	C₂³
hypercenter	C₂×C₄×C₈	AGL(1,𝔽₅)
hypocenter	C₅	#12844
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #10598777	C₂	semidirect
4. #10598698	C₂	semidirect
62. C₂×C₄×D₅	C₁₆	semidirect
271. C₅	#12844	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₈×D₅	2
C₂×#64887	2
#10598777	2
#10598698	2
#12844	5