Subgroups — #13043671

Order 1280.

508 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10611670	C₂	1	1	yes	maximal, characteristic
2	#10604448	C₂	1	2	yes	maximal, complemented
3	#10604023	C₂	1	2	yes	maximal, complemented
4	#10604022	C₂	1	2	yes	maximal, complemented
5	#10604022	C₂	1	2	yes	maximal, complemented
6	#10603871	C₂	1	2	yes	maximal, complemented
7	#10603868	C₂	1	2	yes	maximal, complemented
8	#10603831	C₂	1	1	yes	maximal, characteristic, complemented
9	#10603825	C₂	1	1	yes	maximal, characteristic, complemented
10	#65914	C₄	1	1	yes	characteristic
11	C₂×#3990	C₂²	1	2	yes	—
12	C₂×#3990	C₂²	1	2	yes	—
13	C₂×D₅⋊C₁₆	C₂²	1	2	yes	—
14	C₅×#446	C₄	1	1	yes	characteristic
15	Q₈×#168	C₂²	1	1	yes	characteristic
16	#65129	C₄	1	2	yes	—
17	#65124	C₄	1	1	yes	characteristic
18	#65123	C₂²	1	1	yes	characteristic
19	#65123	C₂²	1	2	yes	—
20	#64997	C₂²	1	2	yes	—
21	#64995	C₄	1	2	yes	—
22	#64977	C₄	1	1	yes	characteristic
23	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
24	#64887	C₂²	1	1	yes	characteristic
25	#64887	C₂²	1	1	yes	characteristic
26	#64887	C₂²	1	2	yes	—
27	#64887	C₂²	1	2	yes	—
28	#64886	C₂²	1	2	yes	—
29	#64885	C₂²	1	1	yes	characteristic
30	#64885	C₂²	1	1	yes	characteristic
31	#64859	C₂²	1	2	yes	—
32	#64859	C₂²	1	2	yes	—
33	C₄⋊#674	C₂²	1	2	yes	—
34	C₄⋊#674	C₂²	1	2	yes	—
35	C₄⋊#674	C₂²	1	2	yes	—
36	C₄×#674	C₂²	1	2	yes	—
37	#33481	—	5	1	no	maximal, Sylow
38	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
39	#4093	C₈	1	1	yes	characteristic
40	#4092	C₈	1	1	yes	characteristic
41	C₂×#676	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₅	C₂³	1	2	yes	—
44	C₂×C₈×D₅	C₂³	1	1	yes	characteristic
45	#4044	C₈	1	2	yes	—
46	#4041	C₈	1	1	yes	characteristic
47	#4040	C₈	1	2	yes	—
48	#4036	C₈	1	1	yes	characteristic
49	C₄×D₂₀	C₂×C₄	1	1	yes	characteristic
50	C₄×D₂₀	C₂×C₄	1	2	yes	—
51	#4018	C₂×C₄	1	2	yes	—
52	C₄²×D₅	C₂×C₄	1	1	yes	characteristic
53	C₂×#674	C₂³	1	2	yes	—
54	C₂×#674	C₂³	1	2	yes	—
55	C₂×#674	C₂³	1	2	yes	—
56	C₂×#674	C₂³	1	2	yes	—
57	#3990	—	2	4	no	—
58	#3990	—	2	4	no	—
59	D₅⋊C₁₆	—	2	4	no	—
60	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
61	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
62	C₅×#97	C₂×C₄	1	2	yes	—
63	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
64	#3952	C₂×C₄	1	2	yes	—
65	#3952	C₂×C₄	1	2	yes	—
66	#3952	C₂×C₄	1	2	yes	—
67	#3947	C₂×C₄	1	2	yes	—
68	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
69	C₈×Dic₅	C₂³	1	1	yes	characteristic
70	C₄⋊#168	C₂×C₄	1	1	yes	characteristic
71	C₄⋊#168	C₂×C₄	1	2	yes	—
72	C₄×#168	C₂³	1	2	yes	—
73	C₄×#168	C₂³	1	1	yes	characteristic
74	#2966	—	5	1	no	—
75	#2183	—	5	2	no	—
76	#2169	—	5	2	no	—
77	#2168	—	5	2	no	—
78	#2168	—	5	2	no	—
79	#2164	—	5	2	no	—
80	#2163	—	5	1	no	—
81	#2108	—	5	2	no	—
82	C₄×M₃₂	—	5	1	no	—
83	C₂×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
84	C₂×D₂₀	C₂×C₈	1	2	yes	—
85	C₂×D₂₀	C₂×C₈	1	1	yes	characteristic
86	C₂×C₄×D₅	C₂×C₈	1	1	yes	characteristic
87	C₂×C₄×D₅	C₂×C₈	1	1	yes	characteristic
88	C₂×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
90	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
92	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
93	C₅×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
94	C₅×C₄⋊C₄	C₂×C₈	1	2	yes	—
95	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₅	C₂²×C₄	1	2	yes	—
97	#685	C₂×C₈	1	2	yes	—
98	#685	C₂×C₈	1	2	yes	—
99	#685	C₂×C₈	1	2	yes	—
100	#685	C₂×C₈	1	2	yes	—
101	#685	C₂×C₈	1	2	yes	—
102	C₄⋊Dic₅	C₂×C₈	1	2	yes	—
103	C₄⋊Dic₅	C₂×C₈	1	1	yes	characteristic
104	#683	C₂×C₈	1	2	yes	—
105	#683	C₂×C₈	1	2	yes	—
106	C₄×Dic₅	C₂×C₈	1	1	yes	characteristic
107	C₄×Dic₅	C₂×C₈	1	1	yes	characteristic
108	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
109	C₂×#168	C₂²×C₄	1	1	yes	characteristic
110	C₂×#168	C₂²×C₄	1	2	yes	—
111	C₂×#168	C₂⁴	1	1	yes	characteristic
112	#676	—	2	4	no	—
113	C₈×D₅	—	2	2	no	—
114	C₈×D₅	—	2	2	no	—
115	C₈×D₅	—	2	2	no	—
116	C₈×D₅	—	2	2	no	—
117	#674	—	2	2	no	—
118	#674	—	2	2	no	—
119	#674	—	2	2	no	—
120	#674	#41	1	4	yes	—
121	#529	—	5	1	no	—
122	C₂×M₃₂	—	5	2	no	—
123	C₂×M₃₂	—	5	2	no	—
124	C₂²×C₁₆	—	5	2	no	—
125	#446	AGL(1,𝔽₅)	1	1	yes	characteristic
126	C₈×Q₈	—	5	1	no	—
127	#436	—	5	1	no	—
128	#435	—	5	2	no	—
129	C₈×D₄	—	5	1	no	—
130	C₈×D₄	—	5	2	no	—
131	#433	—	5	2	no	—
132	#432	—	5	1	no	—
133	#405	—	5	2	no	—
134	C₂×C₄×C₈	—	5	1	no	—
135	C₄⋊C₁₆	—	5	2	no	—
136	C₄⋊C₁₆	—	5	2	no	—
137	C₄⋊C₁₆	—	5	2	no	—
138	C₂²⋊C₁₆	—	5	1	no	—
139	C₂²⋊C₁₆	—	5	1	no	—
140	C₂²⋊C₁₆	—	5	2	no	—
141	C₂²⋊C₁₆	—	5	2	no	—
142	#346	—	5	2	no	—
143	#346	—	5	2	no	—
144	#346	—	5	2	no	—
145	C₄×C₁₆	—	5	1	no	—
146	C₄×C₁₆	—	5	1	no	—
147	C₄×C₁₆	—	5	2	no	—
148	C₂²×D₅	C₂²×C₈	1	2	yes	—
149	C₂²×D₅	C₂²×C₈	1	1	yes	characteristic
150	C₅×Q₈	—	4	1	no	—
151	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
154	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
155	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
156	C₂×Dic₅	C₂²×C₈	1	1	yes	characteristic
157	C₂×Dic₅	C₂²×C₈	1	1	yes	characteristic
158	C₂×Dic₅	C₂²×C₈	1	1	yes	characteristic
159	C₂×Dic₅	C₂²×C₈	1	2	yes	—
160	D₂₀	—	4	1	no	—
161	D₂₀	—	4	2	no	—
162	C₄×D₅	—	2	2	no	—
163	C₄×D₅	—	2	1	no	—
164	C₄×D₅	—	2	1	no	—
165	C₄×D₅	—	2	2	no	—
166	C₄×D₅	—	4	1	no	—
167	C₄×D₅	—	4	1	no	—
168	C₅×C₈	—	2	2	no	—
169	C₅×C₈	—	2	1	no	—
170	C₅×C₈	#131	1	2	yes	—
171	#168	—	2	1	no	—
172	#168	—	2	2	no	—
173	#168	C₂×#41	1	2	yes	—
174	C₂×M₁₆	—	5	2	no	—
175	C₂²×C₈	—	5	1	no	—
176	C₂²×C₈	—	5	2	no	—
177	C₂²×C₈	—	5	1	no	—
178	#126	—	5	2	no	—
179	#125	—	5	1	no	—
180	#124	—	5	1	no	—
181	#123	—	5	2	no	—
182	C₂²⋊Q₈	—	5	1	no	—
183	#121	—	5	1	no	—
184	C₄×Q₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
185	C₄×D₄	—	5	2	no	—
186	C₄×D₄	—	5	1	no	—
187	#117	—	5	2	no	—
188	C₂×C₄²	—	5	1	no	—
189	M₃₂	—	10	4	no	—
190	M₃₂	—	10	4	no	—
191	C₂×C₁₆	—	5	2	no	—
192	C₂×C₁₆	—	5	2	no	—
193	C₂×C₁₆	—	5	2	no	—
194	C₂×C₁₆	—	5	2	no	—
195	C₂×C₁₆	—	10	4	no	—
196	C₄⋊C₈	—	5	1	no	—
197	C₄⋊C₈	—	5	2	no	—
198	C₄⋊C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
199	C₄⋊C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
200	C₂²⋊C₈	—	5	2	no	—
201	C₂²⋊C₈	—	5	2	no	—
202	C₂²⋊C₈	—	5	2	no	—
203	#97	—	5	2	no	—
204	#97	C₂×AGL(1,𝔽₅)	1	2	yes	—
205	C₄×C₈	—	5	1	no	—
206	C₄×C₈	—	5	1	no	—
207	C₄×C₈	—	5	2	no	—
208	C₄×C₈	—	5	1	no	—
209	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
210	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
211	C₂×D₅	—	2	1	no	—
212	C₂×D₅	—	2	1	no	—
213	C₂×D₅	—	2	2	no	—
214	C₂×D₅	—	2	2	no	—
215	C₂×D₅	—	2	1	no	—
216	C₂×D₅	—	2	2	no	—
217	C₄×C₅	—	2	1	no	—
218	C₄×C₅	—	2	1	no	—
219	C₄×C₅	—	2	2	no	—
220	C₄×C₅	—	2	2	no	—
221	C₄×C₅	C₄×#41	1	1	yes	characteristic
222	C₄×C₅	C₂×#131	1	1	yes	characteristic
223	Dic₅	—	2	1	no	—
224	Dic₅	—	2	1	no	—
225	Dic₅	—	2	1	no	—
226	Dic₅	—	2	2	no	—
227	C₂×Q₈	D₅⋊C₈	1	1	yes	characteristic
228	C₂×D₄	—	5	2	no	—
229	C₂×D₄	—	5	1	no	—
230	C₂²×C₄	—	5	1	no	—
231	C₂²×C₄	—	5	2	no	—
232	C₂²×C₄	—	5	1	no	—
233	C₂²×C₄	—	5	1	no	—
234	M₁₆	—	10	4	no	—
235	C₂×C₈	—	5	1	no	—
236	C₂×C₈	—	5	1	no	—
237	C₂×C₈	—	5	2	no	—
238	C₂×C₈	—	10	2	no	—
239	C₂×C₈	—	10	2	no	—
240	C₂×C₈	—	10	2	no	—
241	C₂×C₈	—	10	2	no	—
242	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
243	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	2	yes	—
244	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
245	C₄⋊C₄	—	5	2	no	—
246	C₄⋊C₄	—	5	2	no	—
247	C₄⋊C₄	—	5	1	no	—
248	C₄⋊C₄	—	5	2	no	—
249	C₄⋊C₄	D₅⋊C₈	1	1	yes	characteristic
250	C₄⋊C₄	D₅⋊C₈	1	2	yes	—
251	C₂²⋊C₄	—	5	2	no	—
252	C₂²⋊C₄	—	5	2	no	—
253	C₂²⋊C₄	—	5	2	no	—
254	C₂²⋊C₄	—	5	2	no	—
255	C₂²⋊C₄	—	5	2	no	—
256	C₄²	—	5	1	no	—
257	C₄²	—	5	1	no	—
258	C₄²	—	5	1	no	—
259	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
260	C₄²	C₂²×AGL(1,𝔽₅)	1	2	yes	—
261	C₁₆	—	5	4	no	—
262	C₁₆	—	10	2	no	—
263	C₁₆	—	10	2	no	—
264	C₁₆	—	10	2	no	—
265	C₂×C₅	C₈×#41	1	1	yes	characteristic
266	C₂×C₅	#3407	1	1	yes	characteristic
267	C₂×C₅	#3407	1	1	yes	characteristic
268	D₅	—	4	1	no	—
269	D₅	—	4	2	no	—
270	C₂³	—	5	2	no	—
271	C₂³	—	5	1	no	—
272	Q₈	—	4	1	no	—
273	D₄	—	20	1	no	—
274	D₄	—	20	2	no	—
275	C₂×C₄	—	5	1	no	—
276	C₂×C₄	—	5	1	no	—
277	C₂×C₄	—	5	1	no	—
278	C₂×C₄	—	5	2	no	—
279	C₂×C₄	—	10	1	no	—
280	C₂×C₄	—	10	2	no	—
281	C₂×C₄	—	10	2	no	—
282	C₂×C₄	—	10	1	no	—
283	C₂×C₄	—	20	1	no	—
284	C₂×C₄	—	20	1	no	—
285	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
286	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
287	C₂×C₄	C₂×D₅⋊C₈	1	2	yes	—
288	C₂×C₄	C₂×D₅⋊C₈	1	2	yes	—
289	C₂×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
290	C₈	—	2	2	no	—
291	C₈	—	2	1	no	—
292	C₈	—	5	2	no	—
293	C₈	—	10	1	no	—
294	C₈	—	10	2	no	—
295	C₈	#4133	1	2	yes	—
296	C₅	#33481	1	1	yes	characteristic, complemented, Sylow
297	C₂²	—	10	1	no	—
298	C₂²	—	10	2	no	—
299	C₂²	—	10	1	no	—
300	C₂²	—	10	2	no	—
301	C₂²	—	10	1	no	—
302	C₂²	—	10	2	no	—
303	C₂²	C₂²×D₅⋊C₈	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	2	no	—
308	C₄	—	10	1	no	—
309	C₄	—	10	1	no	—
310	C₄	—	10	1	no	—
311	C₄	—	10	2	no	—
312	C₄	C₂×#4133	1	1	yes	characteristic
313	C₄	#41×AGL(1,𝔽₅)	1	1	yes	characteristic
314	C₂	—	20	1	no	—
315	C₂	—	20	2	no	—
316	C₂	#10617195	1	1	yes	characteristic
317	C₂	#10617404	1	1	yes	characteristic
318	C₂	#10617413	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂³×C₈
Fitting	C₅×#446	C₄
prime radical	#4041	C₈
cosocle	C₂×#168	C₂⁴
hypercenter	#446	AGL(1,𝔽₅)
hypocenter	C₅	#33481
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #10604448	C₂	semidirect
3. #10604023	C₂	semidirect
4. #10604022	C₂	semidirect
5. #10604022	C₂	semidirect
6. #10603871	C₂	semidirect
7. #10603868	C₂	semidirect
8. #10603831	C₂	semidirect
9. #10603825	C₂	semidirect
296. C₅	#33481	semidirect

Maximal subgroups

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

subgroup	index
#10611670	2
#10604448	2
#10604023	2
#10604022	2
#10603871	2
#10603868	2
#10603831	2
#10603825	2
#33481	5