Subgroups — #11990366

Order 1280.

482 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10610772	C₂	1	1	yes	maximal, characteristic
2	#10601565	C₂	1	1	yes	maximal, characteristic
3	#10601533	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄⋊C₄×#168	C₂	1	1	yes	maximal, characteristic, complemented
5	#10600262	C₂	1	1	yes	maximal, characteristic
6	C₅×#1303	C₂	1	1	yes	maximal, characteristic, complemented
7	#10598205	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#4053	—	2	1	no	—
9	#65811	C₂²	1	1	yes	characteristic
10	C₅×#431	C₂²	1	1	yes	characteristic
11	C₅×#423	C₂²	1	1	yes	characteristic
12	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	#65418	—	2	1	no	—
14	#65214	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
16	C₂×C₄×#168	—	2	1	no	—
17	C₂×C₄×#168	C₂²	1	1	yes	characteristic
18	#65127	—	2	1	no	—
19	#65127	C₄	1	2	yes	complemented
20	#65123	—	2	1	no	—
21	#65123	C₄	1	2	yes	complemented
22	#64744	—	2	1	no	—
23	#64706	C₄	1	2	yes	complemented
24	#64705	C₄	1	2	yes	complemented
25	#64674	—	2	1	no	—
26	C₈×#168	—	2	1	no	—
27	#64671	C₄	1	4	yes	—
28	#8251	—	5	1	no	maximal, Sylow
29	C₂×#709	—	2	1	no	—
30	C₂×C₅×M₁₆	—	2	1	no	—
31	C₂×C₅×C₄⋊C₄	—	2	1	no	—
32	C₂×C₄²×C₅	—	2	1	no	—
33	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
34	#4076	—	2	1	no	—
35	#4076	C₂×C₄	1	2	yes	—
36	C₂²×#168	—	2	1	no	—
37	C₂²×#168	D₄	1	1	yes	characteristic
38	C₂²×#168	D₄	1	1	yes	characteristic
39	#4053	—	4	2	no	—
40	#4053	—	4	2	no	—
41	C₂×#676	—	2	2	no	—
42	C₂×C₈×D₅	—	2	2	no	—
43	#4021	—	2	1	no	—
44	#4020	C₂×C₄	1	1	yes	characteristic
45	C₄×D₂₀	—	2	1	no	—
46	C₄×D₂₀	C₂×C₄	1	2	yes	—
47	#4015	C₂×C₄	1	1	yes	characteristic
48	C₅×C₄⋊C₈	—	2	1	no	—
49	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
50	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
51	C₅×C₂²⋊C₈	—	2	1	no	—
52	C₄×C₅×C₈	—	2	1	no	—
53	C₅×#95	C₂×C₄	1	2	yes	—
54	C₂²⋊#168	C₂×C₄	1	2	yes	—
55	#3952	—	2	2	no	—
56	#3952	—	2	2	no	—
57	C₄⋊#168	—	2	1	no	—
58	C₄⋊#168	—	2	1	no	—
59	C₄⋊#168	C₂×C₄	1	2	yes	—
60	C₄×#168	—	2	1	no	—
61	C₄×#168	—	4	2	no	—
62	C₄×#168	C₂×C₄	1	1	yes	characteristic
63	C₄×#168	C₂×C₄	1	1	yes	characteristic
64	C₄×#168	D₄	1	1	yes	characteristic
65	C₄×#168	D₄	1	1	yes	characteristic
66	#2931	—	5	1	no	—
67	C₈×C₄⋊C₄	—	5	1	no	—
68	#1540	—	5	1	no	—
69	#1502	—	5	1	no	—
70	#1450	—	5	1	no	—
71	#1303	D₅	1	1	yes	characteristic, complemented
72	#1301	—	5	1	no	—
73	C₂²×C₄×C₅	—	2	2	no	—
74	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	#41	1	1	yes	characteristic
77	C₂×C₂²⋊D₅	—	2	2	no	—
78	#709	—	4	1	no	—
79	#709	—	4	1	no	—
80	#709	—	4	1	no	—
81	#709	—	4	1	no	—
82	C₂×D₂₀	—	2	1	no	—
83	C₂×D₂₀	—	2	1	no	—
84	C₂×D₂₀	C₄²	1	2	yes	—
85	C₂×C₄×D₅	—	2	2	no	—
86	C₂×Dic₁₀	—	2	1	no	—
87	C₅×M₁₆	—	4	1	no	—
88	C₅×M₁₆	—	4	1	no	—
89	C₂×C₅×C₈	—	2	2	no	—
90	C₂×C₅×C₈	—	2	2	no	—
91	C₅×C₄⋊C₄	—	2	4	no	—
92	C₄²×C₅	—	4	1	no	—
93	C₄²×C₅	—	4	1	no	—
94	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₅	#41	1	1	yes	characteristic
98	#685	—	2	2	no	—
99	#685	—	2	2	no	—
100	C₄⋊Dic₅	—	2	1	no	—
101	C₄⋊Dic₅	C₄²	1	2	yes	—
102	C₂×#168	—	2	1	no	—
103	C₂×#168	—	2	1	no	—
104	C₂×#168	—	2	2	no	—
105	C₂×#168	—	2	2	no	—
106	C₂×#168	—	4	1	no	—
107	C₂×#168	—	4	1	no	—
108	C₂×#168	C₂²⋊C₄	1	1	yes	characteristic
109	C₂×#168	C₂²⋊C₄	1	1	yes	characteristic
110	C₂×#168	C₂²⋊C₄	1	1	yes	characteristic
111	C₂×#168	C₂²⋊C₄	1	1	yes	characteristic
112	C₂×#168	D₈	1	2	yes	—
113	C₂×#168	SD₁₆	1	2	yes	—
114	#676	—	4	4	no	—
115	C₈×D₅	—	4	4	no	—
116	C₂×#131	—	10	1	no	—
117	#532	—	5	1	no	—
118	#436	—	5	2	no	—
119	#436	—	10	1	no	—
120	C₈×D₄	—	5	2	no	—
121	C₈×D₄	—	10	1	no	—
122	#431	—	5	1	no	—
123	#431	C₂×D₅	1	1	yes	characteristic
124	#423	C₂×D₅	1	1	yes	characteristic
125	C₂×C₄⋊C₈	—	5	1	no	—
126	#408	—	10	1	no	—
127	C₂×C₄×C₈	—	5	1	no	—
128	C₂×C₄×C₈	—	10	1	no	—
129	C₄×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
130	#336	—	10	1	no	—
131	#332	—	5	2	no	—
132	#331	—	5	2	no	—
133	#327	—	5	4	no	—
134	#322	—	10	1	no	—
135	C₈²	—	10	1	no	—
136	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
137	C₂²×D₅	—	2	2	no	—
138	C₂×C₄×C₅	—	2	2	no	—
139	C₂×C₄×C₅	—	2	4	no	—
140	C₂×C₄×C₅	—	4	1	no	—
141	C₂×C₄×C₅	—	4	2	no	—
142	C₂×C₄×C₅	—	4	2	no	—
143	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₅	#117	1	1	yes	characteristic
146	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
151	C₂×C₄×C₅	#135	1	1	yes	characteristic
152	C₂×C₄×C₅	#135	1	1	yes	characteristic
153	C₂²⋊D₅	—	4	2	no	—
154	C₂²⋊D₅	—	4	2	no	—
155	C₂×Dic₅	—	2	2	no	—
156	D₂₀	—	4	2	no	—
157	D₂₀	—	4	1	no	—
158	D₂₀	—	4	1	no	—
159	D₂₀	—	4	2	no	—
160	C₄×D₅	—	4	2	no	—
161	C₄×D₅	—	4	2	no	—
162	Dic₁₀	—	4	1	no	—
163	Dic₁₀	—	4	1	no	—
164	C₅×C₈	—	4	2	no	—
165	C₅×C₈	—	4	2	no	—
166	#168	—	2	1	no	—
167	#168	—	2	1	no	—
168	#168	—	4	2	no	—
169	#168	—	4	2	no	—
170	#168	#102	1	2	yes	—
171	#168	#102	1	2	yes	—
172	C₂×#41	—	10	1	no	—
173	#131	—	20	2	no	—
174	#131	—	20	2	no	—
175	C₂×M₁₆	—	2	1	no	—
176	C₂×M₁₆	—	10	2	no	—
177	C₂²×C₈	—	5	1	no	—
178	C₂²×C₈	—	5	1	no	—
179	C₂²×C₈	—	10	2	no	—
180	C₂²×C₈	—	10	1	no	—
181	C₄⋊Q₈	—	5	1	no	—
182	#127	—	5	1	no	—
183	#124	—	10	1	no	—
184	#121	—	5	2	no	—
185	#121	—	10	1	no	—
186	C₄×D₄	—	5	2	no	—
187	C₄×D₄	—	10	1	no	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄²	—	2	1	no	—
190	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
191	C₄⋊C₈	—	2	1	no	—
192	C₄⋊C₈	—	5	2	no	—
193	C₄⋊C₈	—	10	1	no	—
194	C₄⋊C₈	—	10	1	no	—
195	C₄⋊C₈	C₄×D₅	1	2	yes	—
196	C₄⋊C₈	C₄×D₅	1	2	yes	—
197	C₂²⋊C₈	—	2	1	no	—
198	C₂²⋊C₈	—	5	2	no	—
199	C₂²⋊C₈	—	10	2	no	—
200	C₂²⋊C₈	—	10	2	no	—
201	C₄×C₈	—	2	1	no	—
202	C₄×C₈	—	5	1	no	—
203	C₄×C₈	—	5	1	no	—
204	C₄×C₈	—	5	1	no	—
205	C₄×C₈	—	5	1	no	—
206	C₄×C₈	—	10	1	no	—
207	C₄×C₈	—	20	2	no	—
208	#95	C₄×D₅	1	2	yes	—
209	C₂²×C₅	—	2	1	no	—
210	C₂²×C₅	—	2	1	no	—
211	C₂²×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
212	C₂²×C₅	C₄×D₈	1	1	yes	characteristic
213	C₂²×C₅	C₄×SD₁₆	1	1	yes	characteristic
214	C₂×D₅	—	4	2	no	—
215	C₂×D₅	—	4	2	no	—
216	C₂×D₅	—	4	2	no	—
217	C₄×C₅	—	2	1	no	—
218	C₄×C₅	—	2	1	no	—
219	C₄×C₅	—	2	1	no	—
220	C₄×C₅	—	4	4	no	—
221	C₄×C₅	C₂×#102	1	1	yes	characteristic
222	C₄×C₅	#416	1	1	yes	characteristic
223	C₄×C₅	C₄×D₈	1	1	yes	characteristic
224	C₄×C₅	C₄×SD₁₆	1	1	yes	characteristic
225	C₄×C₅	#443	1	2	yes	—
226	Dic₅	—	4	2	no	—
227	#41	—	20	1	no	—
228	#41	—	20	1	no	—
229	#41	—	20	1	no	—
230	#41	—	20	1	no	—
231	C₂×Q₈	—	10	1	no	—
232	C₂×D₄	—	5	2	no	—
233	C₂×D₄	—	10	2	no	—
234	C₂×D₄	—	10	1	no	—
235	C₂×D₄	—	10	1	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	10	2	no	—
238	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
239	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
240	C₂²×C₄	#711	1	1	yes	characteristic
241	M₁₆	—	4	1	no	—
242	M₁₆	—	4	1	no	—
243	M₁₆	—	20	4	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	5	1	no	—
247	C₂×C₈	—	5	1	no	—
248	C₂×C₈	—	5	2	no	—
249	C₂×C₈	—	5	2	no	—
250	C₂×C₈	—	5	1	no	—
251	C₂×C₈	—	5	1	no	—
252	C₂×C₈	—	10	1	no	—
253	C₂×C₈	—	10	1	no	—
254	C₂×C₈	—	10	2	no	—
255	C₂×C₈	—	10	2	no	—
256	C₂×C₈	—	20	4	no	—
257	C₂×C₈	—	20	1	no	—
258	C₂×C₈	—	20	1	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	5	2	no	—
261	C₄⋊C₄	—	10	1	no	—
262	C₂²⋊C₄	—	10	2	no	—
263	C₂²⋊C₄	—	10	2	no	—
264	C₄²	—	4	1	no	—
265	C₄²	—	4	1	no	—
266	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
267	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
268	C₄²	D₄×D₅	1	1	yes	characteristic
269	C₄²	#713	1	1	yes	characteristic
270	C₂×C₅	—	2	1	no	—
271	C₂×C₅	—	2	1	no	—
272	C₂×C₅	#1757	1	1	yes	characteristic
273	C₂×C₅	C₄×#102	1	1	yes	characteristic
274	C₂×C₅	#1764	1	1	yes	characteristic
275	D₅	—	8	2	no	—
276	C₂³	—	10	2	no	—
277	C₂³	#4027	1	1	yes	characteristic
278	Q₈	—	20	1	no	—
279	Q₈	—	20	1	no	—
280	D₄	—	20	2	no	—
281	D₄	—	20	2	no	—
282	D₄	—	20	2	no	—
283	D₄	—	20	2	no	—
284	D₄	—	20	1	no	—
285	D₄	—	20	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	10	2	no	—
292	C₂×C₄	—	20	2	no	—
293	C₂×C₄	—	20	2	no	—
294	C₂×C₄	C₄²×D₅	1	1	yes	characteristic
295	C₂×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
296	C₂×C₄	#4027	1	1	yes	characteristic
297	C₂×C₄	#4038	1	1	yes	characteristic
298	C₂×C₄	#4039	1	1	yes	characteristic
299	C₂×C₄	#4039	1	1	yes	characteristic
300	C₂×C₄	D₅×D₈	1	1	yes	characteristic
301	C₂×C₄	D₅×SD₁₆	1	1	yes	characteristic
302	C₂×C₄	#4062	1	1	yes	characteristic
303	C₂×C₄	#4065	1	1	yes	characteristic
304	C₈	—	4	2	no	—
305	C₈	—	4	2	no	—
306	C₈	—	5	2	no	—
307	C₈	—	5	2	no	—
308	C₈	—	10	1	no	—
309	C₈	—	10	1	no	—
310	C₈	—	20	2	no	—
311	C₈	—	20	2	no	—
312	C₅	#8251	1	1	yes	characteristic, complemented, Sylow
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	20	2	no	—
316	C₂²	—	20	2	no	—
317	C₂²	—	20	2	no	—
318	C₂²	#64955	1	1	yes	characteristic
319	C₂²	#65045	1	1	yes	characteristic
320	C₂²	#65077	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	4	no	—
325	C₄	—	20	2	no	—
326	C₄	D₅×#102	1	1	yes	characteristic
327	C₄	#65093	1	1	yes	characteristic
328	C₄	#65113	1	2	yes	—
329	C₄	#65141	1	1	yes	characteristic
330	C₄	#65161	1	1	yes	characteristic
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	40	2	no	—
334	C₂	#10599532	1	1	yes	characteristic
335	C₂	#10599573	1	1	yes	characteristic
336	C₂	#10599807	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₅×C₂²⋊C₄
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1303	C₂
socle	C₂²×C₅	C₄×C₂²⋊C₄
prime radical	#65811	C₂²
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1303	D₅
hypocenter	C₅	#8251
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1303

Complements

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

N	H ≅ G/N	type
3. #10601533	C₂	semidirect
4. C₄⋊C₄×#168	C₂	semidirect
6. C₅×#1303	C₂	semidirect
7. #10598205	C₂	semidirect
19. #65127	C₄	semidirect
21. #65123	C₄	semidirect
23. #64706	C₄	semidirect
24. #64705	C₄	semidirect
71. #1303	D₅	semidirect
312. C₅	#8251	semidirect

Maximal subgroups

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

subgroup	index
#10610772	2
#10601565	2
#10601533	2
C₄⋊C₄×#168	2
#10600262	2
C₅×#1303	2
#10598205	2
#8251	5