Subgroups — #13043665

Order 1280.

526 conjugacy classes of subgroups.

The subgroup lattice has 342 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	1	yes	maximal, characteristic, complemented
3	Q₈×#674	C₂	1	1	yes	maximal, characteristic, complemented
4	#10604023	C₂	1	1	yes	maximal, characteristic, complemented
5	#10604022	C₂	1	2	yes	maximal, complemented
6	#10604021	C₂	1	1	yes	maximal, characteristic, complemented
7	#10604021	C₂	1	2	yes	maximal, complemented
8	#10603871	C₂	1	2	yes	maximal, complemented
9	#10603868	C₂	1	2	yes	maximal, complemented
10	#10603831	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₅⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#65914	C₄	1	1	yes	characteristic
13	C₂×#3990	C₂²	1	2	yes	—
14	C₂×D₅⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂×D₅⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₂×D₅⋊C₁₆	C₂²	1	2	yes	—
17	C₅×#446	C₄	1	1	yes	characteristic
18	Q₈×#168	C₂²	1	1	yes	characteristic
19	#65129	C₄	1	2	yes	—
20	#65124	C₄	1	1	yes	characteristic
21	#65123	C₂²	1	2	yes	—
22	#65123	C₂²	1	1	yes	characteristic
23	#64997	C₂²	1	2	yes	—
24	#64995	C₄	1	2	yes	—
25	#64977	C₄	1	1	yes	characteristic
26	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
27	#64887	C₂²	1	2	yes	—
28	#64887	C₂²	1	2	yes	—
29	#64887	C₂²	1	2	yes	—
30	#64886	C₂²	1	2	yes	—
31	#64885	C₂²	1	1	yes	characteristic
32	#64885	C₂²	1	1	yes	characteristic
33	#64859	C₂²	1	2	yes	—
34	C₄⋊#674	C₂²	1	1	yes	characteristic
35	C₄⋊#674	C₂²	1	1	yes	characteristic
36	C₄⋊#674	C₂²	1	2	yes	—
37	C₄⋊#674	C₂²	1	2	yes	—
38	C₄×#674	C₂²	1	1	yes	characteristic
39	C₄×#674	C₂²	1	1	yes	characteristic
40	C₄×#674	C₂²	1	2	yes	—
41	#33480	—	5	1	no	maximal, Sylow
42	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
43	#4093	C₈	1	1	yes	characteristic
44	#4092	C₈	1	1	yes	characteristic
45	C₂×#676	C₂×C₄	1	2	yes	—
46	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
47	C₂×C₈×D₅	C₂³	1	1	yes	characteristic
48	C₂×C₈×D₅	C₂³	1	2	yes	—
49	#4044	C₈	1	2	yes	—
50	#4041	C₈	1	1	yes	characteristic
51	#4040	C₈	1	2	yes	—
52	#4036	C₈	1	1	yes	characteristic
53	C₄×D₂₀	C₂×C₄	1	1	yes	characteristic
54	C₄×D₂₀	C₂×C₄	1	2	yes	—
55	#4018	C₂×C₄	1	2	yes	—
56	C₄²×D₅	C₂×C₄	1	1	yes	characteristic
57	C₂×#674	C₂³	1	1	yes	characteristic
58	C₂×#674	C₂³	1	1	yes	characteristic
59	C₂×#674	C₂³	1	2	yes	—
60	C₂×#674	C₂³	1	2	yes	—
61	C₂×#674	C₂³	1	1	yes	characteristic
62	C₂×#674	C₂³	1	1	yes	characteristic
63	#3990	—	2	4	no	—
64	D₅⋊C₁₆	—	2	2	no	—
65	D₅⋊C₁₆	—	2	4	no	—
66	D₅⋊C₁₆	—	2	2	no	—
67	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
68	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
69	C₅×#97	C₂×C₄	1	2	yes	—
70	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
71	#3952	C₂×C₄	1	2	yes	—
72	#3952	C₂×C₄	1	2	yes	—
73	#3952	C₂×C₄	1	2	yes	—
74	#3947	C₂×C₄	1	2	yes	—
75	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
76	C₈×Dic₅	C₂³	1	1	yes	characteristic
77	C₄⋊#168	C₂×C₄	1	1	yes	characteristic
78	C₄⋊#168	C₂×C₄	1	2	yes	—
79	C₄×#168	C₂³	1	1	yes	characteristic
80	C₄×#168	C₂³	1	2	yes	—
81	#2966	—	5	1	no	—
82	#2183	—	5	1	no	—
83	Q₈×C₁₆	—	5	1	no	—
84	#2169	—	5	1	no	—
85	#2168	—	5	2	no	—
86	D₄×C₁₆	—	5	1	no	—
87	D₄×C₁₆	—	5	2	no	—
88	#2164	—	5	2	no	—
89	#2163	—	5	1	no	—
90	#2108	—	5	2	no	—
91	C₂×C₄×C₁₆	—	5	1	no	—
92	C₂×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
93	C₂×D₂₀	C₂×C₈	1	2	yes	—
94	C₂×D₂₀	C₂×C₈	1	1	yes	characteristic
95	C₂×C₄×D₅	C₂×C₈	1	1	yes	characteristic
96	C₂×C₄×D₅	C₂×C₈	1	1	yes	characteristic
97	C₂×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
99	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
101	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
102	C₅×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
103	C₅×C₄⋊C₄	C₂×C₈	1	2	yes	—
104	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₅	C₂²×C₄	1	2	yes	—
106	#685	C₂×C₈	1	2	yes	—
107	#685	C₂×C₈	1	2	yes	—
108	#685	C₂×C₈	1	2	yes	—
109	#685	C₂×C₈	1	2	yes	—
110	#685	C₂×C₈	1	2	yes	—
111	C₄⋊Dic₅	C₂×C₈	1	2	yes	—
112	C₄⋊Dic₅	C₂×C₈	1	1	yes	characteristic
113	#683	C₂×C₈	1	2	yes	—
114	#683	C₂×C₈	1	2	yes	—
115	C₄×Dic₅	C₂×C₈	1	1	yes	characteristic
116	C₄×Dic₅	C₂×C₈	1	1	yes	characteristic
117	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
118	C₂×#168	C₂²×C₄	1	1	yes	characteristic
119	C₂×#168	C₂²×C₄	1	2	yes	—
120	C₂×#168	C₂⁴	1	1	yes	characteristic
121	#676	—	2	4	no	—
122	C₈×D₅	—	2	2	no	—
123	C₈×D₅	—	2	1	no	—
124	C₈×D₅	—	2	1	no	—
125	C₈×D₅	—	2	2	no	—
126	C₈×D₅	—	2	2	no	—
127	#674	—	2	1	no	—
128	#674	—	2	1	no	—
129	#674	—	2	2	no	—
130	#674	—	2	2	no	—
131	#674	#41	1	2	yes	—
132	#674	#41	1	2	yes	—
133	#529	—	5	1	no	—
134	C₂×M₃₂	—	5	2	no	—
135	C₂²×C₁₆	—	5	1	no	—
136	C₂²×C₁₆	—	5	1	no	—
137	C₂²×C₁₆	—	5	2	no	—
138	#446	AGL(1,𝔽₅)	1	1	yes	characteristic
139	C₈×Q₈	—	5	1	no	—
140	#436	—	5	1	no	—
141	#435	—	5	2	no	—
142	C₈×D₄	—	5	2	no	—
143	C₈×D₄	—	5	1	no	—
144	#433	—	5	2	no	—
145	#432	—	5	1	no	—
146	#405	—	5	2	no	—
147	C₂×C₄×C₈	—	5	1	no	—
148	C₄⋊C₁₆	—	5	1	no	—
149	C₄⋊C₁₆	—	5	1	no	—
150	C₄⋊C₁₆	—	5	2	no	—
151	C₄⋊C₁₆	—	5	2	no	—
152	C₂²⋊C₁₆	—	5	2	no	—
153	C₂²⋊C₁₆	—	5	2	no	—
154	C₂²⋊C₁₆	—	5	2	no	—
155	#346	—	5	2	no	—
156	#346	—	5	2	no	—
157	C₄×C₁₆	—	5	1	no	—
158	C₄×C₁₆	—	5	1	no	—
159	C₄×C₁₆	—	5	1	no	—
160	C₄×C₁₆	—	5	1	no	—
161	C₄×C₁₆	—	5	2	no	—
162	C₂²×D₅	C₂²×C₈	1	2	yes	—
163	C₂²×D₅	C₂²×C₈	1	1	yes	characteristic
164	C₅×Q₈	—	2	2	no	—
165	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
166	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
167	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
168	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
169	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
170	C₂×Dic₅	C₂²×C₈	1	1	yes	characteristic
171	C₂×Dic₅	C₂²×C₈	1	1	yes	characteristic
172	C₂×Dic₅	C₂²×C₈	1	1	yes	characteristic
173	C₂×Dic₅	C₂²×C₈	1	2	yes	—
174	D₂₀	—	2	4	no	—
175	D₂₀	—	2	2	no	—
176	C₄×D₅	—	2	2	no	—
177	C₄×D₅	—	2	1	no	—
178	C₄×D₅	—	2	2	no	—
179	C₄×D₅	—	2	1	no	—
180	C₄×D₅	—	2	2	no	—
181	C₄×D₅	—	2	2	no	—
182	C₅×C₈	—	2	1	no	—
183	C₅×C₈	—	2	2	no	—
184	C₅×C₈	#131	1	2	yes	—
185	#168	—	2	1	no	—
186	#168	—	2	2	no	—
187	#168	C₂×#41	1	1	yes	characteristic
188	#168	C₂×#41	1	1	yes	characteristic
189	C₂×M₁₆	—	5	2	no	—
190	C₂²×C₈	—	5	1	no	—
191	C₂²×C₈	—	5	2	no	—
192	C₂²×C₈	—	5	1	no	—
193	#126	—	5	2	no	—
194	#125	—	5	1	no	—
195	#124	—	5	1	no	—
196	#123	—	5	2	no	—
197	C₂²⋊Q₈	—	5	1	no	—
198	#121	—	5	1	no	—
199	C₄×Q₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
200	C₄×D₄	—	5	2	no	—
201	C₄×D₄	—	5	1	no	—
202	#117	—	5	2	no	—
203	C₂×C₄²	—	5	1	no	—
204	M₃₂	—	10	4	no	—
205	C₂×C₁₆	—	5	1	no	—
206	C₂×C₁₆	—	5	1	no	—
207	C₂×C₁₆	—	5	1	no	—
208	C₂×C₁₆	—	5	1	no	—
209	C₂×C₁₆	—	5	2	no	—
210	C₂×C₁₆	—	5	2	no	—
211	C₂×C₁₆	—	10	2	no	—
212	C₂×C₁₆	—	10	4	no	—
213	C₂×C₁₆	—	10	2	no	—
214	C₄⋊C₈	—	5	1	no	—
215	C₄⋊C₈	—	5	2	no	—
216	C₄⋊C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
217	C₄⋊C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
218	C₂²⋊C₈	—	5	2	no	—
219	C₂²⋊C₈	—	5	2	no	—
220	C₂²⋊C₈	—	5	2	no	—
221	#97	—	5	2	no	—
222	#97	C₂×AGL(1,𝔽₅)	1	2	yes	—
223	C₄×C₈	—	5	1	no	—
224	C₄×C₈	—	5	1	no	—
225	C₄×C₈	—	5	1	no	—
226	C₄×C₈	—	5	2	no	—
227	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
228	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
229	C₂×D₅	—	2	1	no	—
230	C₂×D₅	—	2	1	no	—
231	C₂×D₅	—	2	2	no	—
232	C₂×D₅	—	2	2	no	—
233	C₂×D₅	—	2	2	no	—
234	C₂×D₅	#504	1	2	yes	—
235	C₄×C₅	—	2	1	no	—
236	C₄×C₅	—	2	2	no	—
237	C₄×C₅	—	2	2	no	—
238	C₄×C₅	#504	1	2	yes	—
239	C₄×C₅	C₄×#41	1	1	yes	characteristic
240	C₄×C₅	C₂×#131	1	1	yes	characteristic
241	Dic₅	—	2	1	no	—
242	Dic₅	—	2	1	no	—
243	Dic₅	—	2	1	no	—
244	Dic₅	—	2	2	no	—
245	C₂×Q₈	D₅⋊C₈	1	1	yes	characteristic
246	C₂×D₄	—	5	2	no	—
247	C₂×D₄	—	5	1	no	—
248	C₂²×C₄	—	5	1	no	—
249	C₂²×C₄	—	5	1	no	—
250	C₂²×C₄	—	5	1	no	—
251	C₂²×C₄	—	5	2	no	—
252	M₁₆	—	10	4	no	—
253	C₂×C₈	—	5	1	no	—
254	C₂×C₈	—	5	1	no	—
255	C₂×C₈	—	5	2	no	—
256	C₂×C₈	—	10	1	no	—
257	C₂×C₈	—	10	1	no	—
258	C₂×C₈	—	10	2	no	—
259	C₂×C₈	—	10	2	no	—
260	C₂×C₈	—	10	2	no	—
261	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
262	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	2	yes	—
263	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
264	C₄⋊C₄	—	5	2	no	—
265	C₄⋊C₄	—	5	2	no	—
266	C₄⋊C₄	—	5	1	no	—
267	C₄⋊C₄	—	5	2	no	—
268	C₄⋊C₄	D₅⋊C₈	1	1	yes	characteristic
269	C₄⋊C₄	D₅⋊C₈	1	2	yes	—
270	C₂²⋊C₄	—	5	2	no	—
271	C₂²⋊C₄	—	5	2	no	—
272	C₂²⋊C₄	—	5	2	no	—
273	C₂²⋊C₄	—	5	2	no	—
274	C₂²⋊C₄	—	5	2	no	—
275	C₄²	—	5	1	no	—
276	C₄²	—	5	1	no	—
277	C₄²	—	5	1	no	—
278	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
279	C₄²	C₂²×AGL(1,𝔽₅)	1	2	yes	—
280	C₁₆	—	5	2	no	—
281	C₁₆	—	5	2	no	—
282	C₁₆	—	10	1	no	—
283	C₁₆	—	10	1	no	—
284	C₁₆	—	10	2	no	—
285	C₁₆	—	10	2	no	—
286	C₂×C₅	C₈×#41	1	1	yes	characteristic
287	C₂×C₅	C₂×#504	1	1	yes	characteristic
288	C₂×C₅	C₂×#504	1	1	yes	characteristic
289	D₅	—	2	2	no	—
290	D₅	—	4	2	no	—
291	C₂³	—	5	1	no	—
292	C₂³	—	5	2	no	—
293	Q₈	—	2	2	no	—
294	D₄	—	10	4	no	—
295	D₄	—	10	2	no	—
296	C₂×C₄	—	5	1	no	—
297	C₂×C₄	—	5	1	no	—
298	C₂×C₄	—	5	1	no	—
299	C₂×C₄	—	5	2	no	—
300	C₂×C₄	—	10	2	no	—
301	C₂×C₄	—	10	2	no	—
302	C₂×C₄	—	10	1	no	—
303	C₂×C₄	—	10	2	no	—
304	C₂×C₄	—	10	1	no	—
305	C₂×C₄	—	10	2	no	—
306	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
307	C₂×C₄	C₂×D₅⋊C₈	1	2	yes	—
308	C₂×C₄	C₂×D₅⋊C₈	1	2	yes	—
309	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
310	C₂×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	2	no	—
313	C₈	—	5	1	no	—
314	C₈	—	5	1	no	—
315	C₈	—	10	1	no	—
316	C₈	—	10	2	no	—
317	C₈	#4133	1	2	yes	—
318	C₅	#33480	1	1	yes	characteristic, complemented, Sylow
319	C₂²	—	5	2	no	—
320	C₂²	—	10	1	no	—
321	C₂²	—	10	1	no	—
322	C₂²	—	10	2	no	—
323	C₂²	—	10	2	no	—
324	C₂²	—	10	2	no	—
325	C₂²	C₂²×D₅⋊C₈	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	2	no	—
329	C₄	—	10	1	no	—
330	C₄	—	10	1	no	—
331	C₄	—	10	1	no	—
332	C₄	—	10	2	no	—
333	C₄	#65725	1	2	yes	—
334	C₄	C₂×#4133	1	1	yes	characteristic
335	C₄	#41×AGL(1,𝔽₅)	1	1	yes	characteristic
336	C₂	—	10	2	no	—
337	C₂	—	20	2	no	—
338	C₂	#10617195	1	1	yes	characteristic
339	C₂	C₂×#65725	1	1	yes	characteristic
340	C₂	D₅⋊#504	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	#33480
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. Q₈×#674	C₂	semidirect
4. #10604023	C₂	semidirect
5. #10604022	C₂	semidirect
6. #10604021	C₂	semidirect
7. #10604021	C₂	semidirect
8. #10603871	C₂	semidirect
9. #10603868	C₂	semidirect
10. #10603831	C₂	semidirect
11. C₄×D₅⋊C₁₆	C₂	semidirect
318. C₅	#33480	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
Q₈×#674	2
#10604023	2
#10604022	2
#10604021	2
#10603871	2
#10603868	2
#10603831	2
C₄×D₅⋊C₁₆	2
#33480	5