Subgroups — #13043660

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	#10614857	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	#10604414	C₂	1	1	yes	maximal, characteristic, complemented
5	#10604413	C₂	1	2	yes	maximal, complemented
6	D₄×#674	C₂	1	2	yes	maximal, complemented
7	D₄×#674	C₂	1	1	yes	maximal, characteristic, complemented
8	#10604368	C₂	1	2	yes	maximal, complemented
9	#10604366	C₂	1	2	yes	maximal, complemented
10	#10604152	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×#674	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#529	C₄	1	1	yes	characteristic
13	C₂×#3998	C₂²	1	2	yes	—
14	C₂²×#674	C₂²	1	2	yes	—
15	C₂²×#674	C₂²	1	1	yes	characteristic
16	C₂²×#674	C₂²	1	1	yes	characteristic
17	#65313	C₄	1	1	yes	characteristic
18	Q₈×#168	C₂²	1	1	yes	characteristic
19	#65301	C₄	1	1	yes	characteristic
20	#65300	C₄	1	2	yes	—
21	D₄×#168	C₂²	1	2	yes	—
22	D₄×#168	C₂²	1	1	yes	characteristic
23	#65289	C₄	1	2	yes	—
24	#65286	C₂²	1	2	yes	—
25	#65215	C₄	1	1	yes	characteristic
26	C₂×C₄×#168	C₂²	1	1	yes	characteristic
27	C₂²⋊#674	C₂²	1	2	yes	—
28	C₂²⋊#674	C₂²	1	2	yes	—
29	C₂²⋊#674	C₂²	1	2	yes	—
30	#64859	C₂²	1	2	yes	—
31	#64859	C₂²	1	2	yes	—
32	C₄⋊#674	C₂²	1	2	yes	—
33	C₄⋊#674	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	1	yes	characteristic
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	1	yes	characteristic
41	#33480	—	5	1	no	maximal, Sylow
42	C₅×#126	C₈	1	2	yes	—
43	C₅×#125	C₈	1	1	yes	characteristic
44	C₅×#124	C₈	1	1	yes	characteristic
45	C₅×#123	C₈	1	2	yes	—
46	C₅×C₂²⋊Q₈	C₈	1	1	yes	characteristic
47	C₅×#121	C₈	1	1	yes	characteristic
48	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
49	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
50	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
51	C₅×#117	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
53	C₂×#681	C₂×C₄	1	2	yes	—
54	C₂²×#168	C₂×C₄	1	1	yes	characteristic
55	C₂²×#168	C₂³	1	2	yes	—
56	C₂²×#168	C₂³	1	1	yes	characteristic
57	#3998	—	2	4	no	—
58	C₂×#674	—	2	4	no	—
59	C₂×#674	—	2	2	no	—
60	C₂×#674	—	2	2	no	—
61	C₂×#674	C₂³	1	1	yes	characteristic
62	C₂×#674	C₂³	1	1	yes	characteristic
63	C₂×#674	C₂³	1	2	yes	—
64	C₂×#674	C₂³	1	2	yes	—
65	C₂×#674	C₂³	1	1	yes	characteristic
66	C₂×#674	C₂³	1	1	yes	characteristic
67	C₂²⋊#168	C₂×C₄	1	2	yes	—
68	C₂²⋊#168	C₂×C₄	1	2	yes	—
69	C₂²⋊#168	C₂×C₄	1	2	yes	—
70	C₄⋊#168	C₂×C₄	1	2	yes	—
71	C₄⋊#168	C₂×C₄	1	2	yes	—
72	C₄⋊#168	C₂×C₄	1	1	yes	characteristic
73	C₄⋊#168	C₂×C₄	1	1	yes	characteristic
74	#3935	C₂×C₄	1	2	yes	—
75	#3935	C₂×C₄	1	2	yes	—
76	C₄×#168	C₂×C₄	1	1	yes	characteristic
77	C₄×#168	C₂×C₄	1	1	yes	characteristic
78	C₄×#168	C₂³	1	1	yes	characteristic
79	C₄×#168	C₂³	1	2	yes	—
80	C₄×#168	C₂³	1	1	yes	characteristic
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×C₅×D₄	C₂×C₈	1	2	yes	—
94	C₂×C₅×D₄	C₂×C₈	1	1	yes	characteristic
95	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
96	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
97	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
99	C₅×C₄⋊C₄	C₂×C₈	1	2	yes	—
100	C₅×C₄⋊C₄	C₂×C₈	1	2	yes	—
101	C₅×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
102	C₅×C₄⋊C₄	C₂×C₈	1	2	yes	—
103	C₅×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
104	C₅×C₄⋊C₄	C₂×C₈	1	2	yes	—
105	C₅×C₂²⋊C₄	C₂×C₈	1	2	yes	—
106	C₅×C₂²⋊C₄	C₂×C₈	1	2	yes	—
107	C₅×C₂²⋊C₄	C₂×C₈	1	2	yes	—
108	C₅×C₂²⋊C₄	C₂×C₈	1	2	yes	—
109	C₅×C₂²⋊C₄	C₂×C₈	1	2	yes	—
110	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
111	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
112	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₅	C₂²×C₄	1	2	yes	—
115	#681	—	2	4	no	—
116	C₂×#168	—	2	2	no	—
117	C₂×#168	—	2	2	no	—
118	C₂×#168	—	2	1	no	—
119	C₂×#168	—	2	1	no	—
120	C₂×#168	—	2	2	no	—
121	C₂×#168	C₂²×C₄	1	1	yes	characteristic
122	C₂×#168	C₂²×C₄	1	1	yes	characteristic
123	C₂×#168	C₂²×C₄	1	2	yes	—
124	C₂×#168	C₂²×C₄	1	2	yes	—
125	C₂×#168	C₂²×C₄	1	1	yes	characteristic
126	C₂×#168	C₂⁴	1	1	yes	characteristic
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	AGL(1,𝔽₅)	1	1	yes	characteristic
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	—	5	1	no	—
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	C₂³×C₅	C₂²×C₈	1	2	yes	—
163	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
164	C₅×Q₈	—	2	2	no	—
165	C₅×D₄	—	2	4	no	—
166	C₅×D₄	—	2	2	no	—
167	C₂×C₄×C₅	—	2	2	no	—
168	C₂×C₄×C₅	—	2	1	no	—
169	C₂×C₄×C₅	—	2	2	no	—
170	C₂×C₄×C₅	—	2	1	no	—
171	C₂×C₄×C₅	—	2	2	no	—
172	C₂×C₄×C₅	—	2	2	no	—
173	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
175	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
176	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
178	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
179	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
180	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
182	#168	—	2	1	no	—
183	#168	—	2	2	no	—
184	#168	—	2	1	no	—
185	#168	—	2	2	no	—
186	#168	#131	1	2	yes	—
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	#170	1	2	yes	—
194	#125	#170	1	1	yes	characteristic
195	#124	#170	1	1	yes	characteristic
196	#123	#170	1	2	yes	—
197	C₂²⋊Q₈	#170	1	1	yes	characteristic
198	#121	#170	1	1	yes	characteristic
199	C₄×Q₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
200	C₄×D₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
201	C₄×D₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
202	#117	C₂×AGL(1,𝔽₅)	1	2	yes	—
203	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
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	1	no	—
216	C₄⋊C₈	—	5	2	no	—
217	C₄⋊C₈	—	5	2	no	—
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	—	5	2	no	—
223	C₄×C₈	—	5	1	no	—
224	C₄×C₈	—	5	1	no	—
225	C₄×C₈	—	5	2	no	—
226	C₄×C₈	—	5	1	no	—
227	C₄×C₈	—	5	1	no	—
228	C₂²×C₅	—	2	1	no	—
229	C₂²×C₅	—	2	1	no	—
230	C₂²×C₅	—	2	2	no	—
231	C₂²×C₅	—	2	2	no	—
232	C₂²×C₅	—	2	2	no	—
233	C₂²×C₅	#504	1	2	yes	—
234	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
235	C₄×C₅	—	2	1	no	—
236	C₄×C₅	—	2	1	no	—
237	C₄×C₅	—	2	1	no	—
238	C₄×C₅	—	2	2	no	—
239	C₄×C₅	—	2	2	no	—
240	C₄×C₅	—	2	1	no	—
241	C₄×C₅	—	2	2	no	—
242	C₄×C₅	#504	1	2	yes	—
243	C₄×C₅	C₄×#41	1	1	yes	characteristic
244	C₄×C₅	C₂×#131	1	1	yes	characteristic
245	C₂×Q₈	C₂×#170	1	1	yes	characteristic
246	C₂×D₄	C₂×#170	1	2	yes	—
247	C₂×D₄	C₂×#170	1	1	yes	characteristic
248	C₂²×C₄	C₂×#170	1	1	yes	characteristic
249	C₂²×C₄	C₂×#170	1	1	yes	characteristic
250	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
251	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
252	M₁₆	—	10	4	no	—
253	C₂×C₈	—	5	1	no	—
254	C₂×C₈	—	5	1	no	—
255	C₂×C₈	—	5	1	no	—
256	C₂×C₈	—	5	1	no	—
257	C₂×C₈	—	5	2	no	—
258	C₂×C₈	—	5	2	no	—
259	C₂×C₈	—	10	1	no	—
260	C₂×C₈	—	10	1	no	—
261	C₂×C₈	—	10	2	no	—
262	C₂×C₈	—	10	2	no	—
263	C₂×C₈	—	10	2	no	—
264	C₄⋊C₄	C₂×#170	1	2	yes	—
265	C₄⋊C₄	C₂×#170	1	2	yes	—
266	C₄⋊C₄	C₂×#170	1	1	yes	characteristic
267	C₄⋊C₄	C₂×#170	1	2	yes	—
268	C₄⋊C₄	C₂×#170	1	1	yes	characteristic
269	C₄⋊C₄	C₂×#170	1	2	yes	—
270	C₂²⋊C₄	C₂×#170	1	2	yes	—
271	C₂²⋊C₄	C₂×#170	1	2	yes	—
272	C₂²⋊C₄	C₂×#170	1	2	yes	—
273	C₂²⋊C₄	C₂×#170	1	2	yes	—
274	C₂²⋊C₄	C₂×#170	1	2	yes	—
275	C₄²	C₂×#170	1	1	yes	characteristic
276	C₄²	C₂×#170	1	1	yes	characteristic
277	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
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	—	2	2	no	—
287	C₂×C₅	—	4	2	no	—
288	C₂×C₅	C₈×#41	1	1	yes	characteristic
289	C₂×C₅	C₂×#504	1	1	yes	characteristic
290	C₂×C₅	C₂×#504	1	1	yes	characteristic
291	C₂³	C₂²×#170	1	2	yes	—
292	C₂³	C₂²×#170	1	1	yes	characteristic
293	Q₈	—	2	2	no	—
294	D₄	—	2	4	no	—
295	D₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	C₂²×#170	1	1	yes	characteristic
303	C₂×C₄	C₂²×#170	1	1	yes	characteristic
304	C₂×C₄	C₂²×#170	1	1	yes	characteristic
305	C₂×C₄	C₂²×#170	1	1	yes	characteristic
306	C₂×C₄	C₂²×#170	1	2	yes	—
307	C₂×C₄	C₂²×#170	1	2	yes	—
308	C₂×C₄	C₂²×#170	1	2	yes	—
309	C₂×C₄	C₂²×#170	1	1	yes	characteristic
310	C₂×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
311	C₈	—	5	2	no	—
312	C₈	—	5	1	no	—
313	C₈	—	5	1	no	—
314	C₈	—	10	1	no	—
315	C₈	—	10	2	no	—
316	C₈	—	10	1	no	—
317	C₈	—	10	2	no	—
318	C₅	#33480	1	1	yes	characteristic, complemented, Sylow
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	#65791	1	2	yes	—
325	C₂²	C₂³×#170	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	2	no	—
333	C₄	#65791	1	2	yes	—
334	C₄	D₅⋊#131	1	1	yes	characteristic
335	C₄	#41×AGL(1,𝔽₅)	1	1	yes	characteristic
336	C₂	—	2	2	no	—
337	C₂	—	4	2	no	—
338	C₂	C₂×#65791	1	1	yes	characteristic
339	C₂	C₂×#65791	1	1	yes	characteristic
340	C₂	#41×#170	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂³×C₈
Fitting	C₅×#529	C₄
prime radical	C₅×#121	C₈
cosocle	C₂×#168	C₂⁴
hypercenter	#529	AGL(1,𝔽₅)
hypocenter	C₅	#33480
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #529

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. #10604414	C₂	semidirect
5. #10604413	C₂	semidirect
6. D₄×#674	C₂	semidirect
7. D₄×#674	C₂	semidirect
8. #10604368	C₂	semidirect
9. #10604366	C₂	semidirect
10. #10604152	C₂	semidirect
11. C₂×C₄×#674	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
#10614857	2
#10604448	2
Q₈×#674	2
#10604414	2
#10604413	2
D₄×#674	2
#10604368	2
#10604366	2
#10604152	2
C₂×C₄×#674	2
#33480	5