Subgroups — #11997825

Order 1280.

318 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#1542	C₂	1	1	yes	maximal, characteristic
2	#10602634	C₂	1	1	yes	maximal, characteristic
3	#10602619	C₂	1	1	yes	maximal, characteristic
4	#10601534	C₂	1	1	yes	maximal, characteristic
5	#10601267	C₂	1	1	yes	maximal, characteristic
6	#10598237	C₂	1	1	yes	maximal, characteristic
7	#10598195	C₂	1	1	yes	maximal, characteristic
8	C₅×#533	C₂²	1	1	yes	characteristic
9	C₅×#431	C₂²	1	1	yes	characteristic
10	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×#3951	—	2	1	no	—
12	C₄×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
13	#65213	C₂²	1	1	yes	characteristic
14	C₄×#681	C₂²	1	1	yes	characteristic
15	C₂×C₄×#168	C₂²	1	1	yes	characteristic
16	C₅×#333	C₄	1	1	yes	characteristic, complemented
17	C₅×#333	C₄	1	1	yes	characteristic, complemented
18	C₅×#332	C₄	1	1	yes	characteristic, complemented
19	C₅×#332	C₄	1	1	yes	characteristic, complemented
20	#64772	—	2	1	no	—
21	Q₈⋊#168	—	2	1	no	—
22	Q₈⋊#168	—	2	1	no	—
23	#64752	—	2	1	no	—
24	#64752	—	2	1	no	—
25	#64675	—	2	1	no	—
26	#64674	—	2	1	no	—
27	#9591	—	5	1	no	maximal, Sylow
28	C₂²×C₅×C₈	—	2	1	no	—
29	C₅×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
30	C₅×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
31	C₅×#125	—	2	1	no	—
32	C₅×C₂²⋊Q₈	—	2	1	no	—
33	C₅×C₂²⋊Q₈	—	2	1	no	—
34	C₄×C₅×Q₈	—	2	1	no	—
35	C₄×C₅×Q₈	—	2	1	no	—
36	C₅×#117	—	2	1	no	—
37	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
38	C₂×C₄⋊Dic₅	—	2	1	no	—
39	C₂×C₄×Dic₅	—	2	1	no	—
40	C₂×#681	—	2	1	no	—
41	C₂×#681	D₄	1	1	yes	characteristic
42	C₂×#681	D₄	1	1	yes	characteristic
43	C₂²×#168	—	2	1	no	—
44	C₅×C₄⋊C₈	—	2	1	no	—
45	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
46	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
47	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
48	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
49	C₅×C₂²⋊C₈	—	2	1	no	—
50	C₄×C₅×C₈	—	2	1	no	—
51	#3963	—	2	1	no	—
52	#3951	—	4	2	no	—
53	C₄⋊#168	—	2	1	no	—
54	C₄⋊#168	—	2	1	no	—
55	#3935	—	2	1	no	—
56	C₄×#168	—	2	1	no	—
57	C₄×#168	—	2	1	no	—
58	C₄×#168	D₄	1	1	yes	characteristic
59	C₄×#168	D₄	1	1	yes	characteristic
60	#1986	—	5	1	no	—
61	#1950	—	5	1	no	—
62	#1542	D₅	1	1	yes	characteristic
63	#1493	—	5	1	no	—
64	#1451	—	5	1	no	—
65	#1303	—	5	1	no	—
66	#1279	—	5	1	no	—
67	C₂×C₅×Q₈	—	2	1	no	—
68	C₂×C₅×Q₈	—	2	1	no	—
69	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	#41	1	1	yes	characteristic
71	C₂²×C₄×C₅	#41	1	1	yes	characteristic
72	C₂²×Dic₅	—	4	1	no	—
73	C₂×C₅×C₈	—	2	1	no	—
74	C₂×C₅×C₈	—	2	1	no	—
75	C₂×C₅×C₈	—	2	1	no	—
76	C₂×C₅×C₈	—	2	1	no	—
77	C₂×C₅×C₈	—	4	1	no	—
78	C₂×C₅×C₈	—	4	1	no	—
79	C₅×C₄⋊C₄	—	2	1	no	—
80	C₅×C₄⋊C₄	—	2	1	no	—
81	C₅×C₄⋊C₄	—	2	1	no	—
82	C₅×C₄⋊C₄	—	2	1	no	—
83	C₅×C₄⋊C₄	—	4	1	no	—
84	C₅×C₄⋊C₄	—	4	1	no	—
85	C₅×C₂²⋊C₄	—	4	1	no	—
86	C₄²×C₅	—	4	1	no	—
87	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₅	#41	1	1	yes	characteristic
90	C₄²×C₅	#41	1	1	yes	characteristic
91	C₄⋊Dic₅	—	2	2	no	—
92	C₄⋊Dic₅	—	4	1	no	—
93	C₄×Dic₅	—	4	1	no	—
94	C₄×Dic₅	—	4	1	no	—
95	#681	—	2	1	no	—
96	#681	—	2	1	no	—
97	#681	—	2	1	no	—
98	#681	—	2	1	no	—
99	#681	—	4	1	no	—
100	#681	—	4	1	no	—
101	C₂×#168	—	2	1	no	—
102	C₂×#168	—	2	1	no	—
103	C₂×#168	—	2	1	no	—
104	C₂×#168	—	2	1	no	—
105	C₂×#168	—	4	2	no	—
106	C₂×#168	—	4	1	no	—
107	#533	C₂×D₅	1	1	yes	characteristic
108	#431	C₂×D₅	1	1	yes	characteristic
109	C₂×#108	—	10	1	no	—
110	#423	—	5	1	no	—
111	C₂×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
112	C₄×M₁₆	—	5	1	no	—
113	C₂×C₄×C₈	—	5	1	no	—
114	C₄×C₄⋊C₄	—	5	1	no	—
115	#341	—	10	1	no	—
116	#339	—	10	1	no	—
117	#339	—	10	1	no	—
118	#333	Dic₅	1	1	yes	characteristic, complemented
119	#333	Dic₅	1	1	yes	characteristic, complemented
120	#332	Dic₅	1	1	yes	characteristic, complemented
121	#332	Dic₅	1	1	yes	characteristic, complemented
122	Q₈⋊C₈	—	10	1	no	—
123	Q₈⋊C₈	—	10	1	no	—
124	#322	—	10	1	no	—
125	#322	—	10	1	no	—
126	C₂³×C₅	#124	1	1	yes	characteristic
127	C₅×Q₈	—	4	1	no	—
128	C₅×Q₈	—	4	1	no	—
129	C₅×Q₈	—	8	1	no	—
130	C₂×C₄×C₅	—	2	1	no	—
131	C₂×C₄×C₅	—	2	1	no	—
132	C₂×C₄×C₅	—	4	1	no	—
133	C₂×C₄×C₅	—	4	1	no	—
134	C₂×C₄×C₅	—	4	1	no	—
135	C₂×C₄×C₅	#117	1	1	yes	characteristic
136	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₅	#121	1	1	yes	characteristic
139	C₂×C₄×C₅	#123	1	1	yes	characteristic
140	C₂×C₄×C₅	#126	1	1	yes	characteristic
141	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₂×C₄×C₅	#137	1	1	yes	characteristic
143	C₂×C₄×C₅	#137	1	1	yes	characteristic
144	C₂×C₄×C₅	#137	1	1	yes	characteristic
145	C₂×Dic₅	—	4	2	no	—
146	C₂×Dic₅	—	8	1	no	—
147	C₂×Dic₅	—	8	1	no	—
148	C₅×C₈	—	4	1	no	—
149	C₅×C₈	—	4	1	no	—
150	C₅×C₈	—	4	1	no	—
151	C₅×C₈	—	4	1	no	—
152	#168	—	2	2	no	—
153	#168	—	4	1	no	—
154	#168	—	4	1	no	—
155	#168	—	4	1	no	—
156	#168	—	8	1	no	—
157	C₂×M₁₆	—	5	1	no	—
158	C₂×M₁₆	—	5	1	no	—
159	C₂×M₁₆	—	10	1	no	—
160	C₂²×C₈	—	2	1	no	—
161	C₂²×C₈	—	10	1	no	—
162	C₄⋊Q₈	C₂×Dic₅	1	1	yes	characteristic
163	C₄⋊Q₈	C₂×Dic₅	1	1	yes	characteristic
164	#125	—	2	1	no	—
165	C₂²⋊Q₈	—	2	1	no	—
166	C₂²⋊Q₈	—	2	1	no	—
167	C₄×Q₈	—	2	1	no	—
168	C₄×Q₈	—	2	1	no	—
169	#117	—	2	1	no	—
170	C₂×C₄⋊C₄	—	10	1	no	—
171	C₂×C₄²	—	10	1	no	—
172	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
173	#108	—	20	2	no	—
174	C₄⋊C₈	—	2	1	no	—
175	C₄⋊C₈	—	10	1	no	—
176	C₄⋊C₈	—	10	1	no	—
177	C₄⋊C₈	C₂×Dic₅	1	1	yes	characteristic
178	C₄⋊C₈	C₂×Dic₅	1	1	yes	characteristic
179	C₄⋊C₈	C₂×Dic₅	1	1	yes	characteristic
180	C₄⋊C₈	C₂×Dic₅	1	1	yes	characteristic
181	C₂²⋊C₈	—	2	1	no	—
182	#97	—	10	1	no	—
183	C₄×C₈	—	2	1	no	—
184	C₄×C₈	—	5	1	no	—
185	C₄×C₈	—	5	1	no	—
186	C₄×C₈	—	10	1	no	—
187	C₄×C₈	—	10	1	no	—
188	#95	—	10	1	no	—
189	C₂²×C₅	—	2	1	no	—
190	C₂²×C₅	—	2	1	no	—
191	C₂²×C₅	#388	1	1	yes	characteristic
192	C₂²×C₅	#489	1	1	yes	characteristic
193	C₂²×C₅	#491	1	1	yes	characteristic
194	C₄×C₅	—	2	1	no	—
195	C₄×C₅	—	2	1	no	—
196	C₄×C₅	—	2	1	no	—
197	C₄×C₅	—	8	1	no	—
198	C₄×C₅	—	8	1	no	—
199	C₄×C₅	#440	1	1	yes	characteristic
200	C₄×C₅	#441	1	1	yes	characteristic
201	C₄×C₅	#443	1	1	yes	characteristic
202	C₄×C₅	#444	1	1	yes	characteristic
203	C₄×C₅	#468	1	1	yes	characteristic
204	C₄×C₅	#470	1	1	yes	characteristic
205	Dic₅	—	8	2	no	—
206	C₂×Q₈	—	2	1	no	—
207	C₂×Q₈	—	2	1	no	—
208	C₂²×C₄	—	20	1	no	—
209	C₂²×C₄	#709	1	1	yes	characteristic
210	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
211	C₂²×C₄	#711	1	1	yes	characteristic
212	M₁₆	—	10	1	no	—
213	M₁₆	—	10	1	no	—
214	M₁₆	—	10	1	no	—
215	M₁₆	—	10	1	no	—
216	M₁₆	—	20	1	no	—
217	M₁₆	—	20	1	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	10	1	no	—
225	C₂×C₈	—	10	1	no	—
226	C₂×C₈	—	10	1	no	—
227	C₂×C₈	—	10	1	no	—
228	C₂×C₈	—	20	2	no	—
229	C₂×C₈	—	20	1	no	—
230	C₄⋊C₄	—	2	1	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	4	1	no	—
235	C₄⋊C₄	—	4	1	no	—
236	C₄⋊C₄	—	10	2	no	—
237	C₄⋊C₄	—	20	1	no	—
238	C₂²⋊C₄	—	4	1	no	—
239	C₄²	—	4	1	no	—
240	C₄²	—	20	1	no	—
241	C₄²	—	20	1	no	—
242	C₄²	#709	1	1	yes	characteristic
243	C₄²	D₄×D₅	1	1	yes	characteristic
244	C₄²	#711	1	1	yes	characteristic
245	C₄²	C₂²×Dic₅	1	1	yes	characteristic
246	C₂×C₅	—	2	1	no	—
247	C₂×C₅	—	2	1	no	—
248	C₂×C₅	#1929	1	1	yes	characteristic
249	C₂×C₅	#1932	1	1	yes	characteristic
250	C₂×C₅	#1935	1	1	yes	characteristic
251	C₂³	#4031	1	1	yes	characteristic
252	Q₈	—	4	1	no	—
253	Q₈	—	4	1	no	—
254	Q₈	—	8	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	20	2	no	—
261	C₂×C₄	—	40	1	no	—
262	C₂×C₄	—	40	1	no	—
263	C₂×C₄	#4025	1	1	yes	characteristic
264	C₂×C₄	#4029	1	1	yes	characteristic
265	C₂×C₄	#4030	1	1	yes	characteristic
266	C₂×C₄	#4060	1	1	yes	characteristic
267	C₂×C₄	#4061	1	1	yes	characteristic
268	C₂×C₄	#4064	1	1	yes	characteristic
269	C₂×C₄	#4064	1	1	yes	characteristic
270	C₂×C₄	#4072	1	1	yes	characteristic
271	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
272	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
273	C₈	—	4	1	no	—
274	C₈	—	4	1	no	—
275	C₈	—	4	1	no	—
276	C₈	—	4	1	no	—
277	C₈	—	10	2	no	—
278	C₈	—	20	1	no	—
279	C₈	—	20	1	no	—
280	C₈	—	20	1	no	—
281	C₈	—	40	1	no	—
282	C₅	#9591	1	1	yes	characteristic, complemented, Sylow
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	#65084	1	1	yes	characteristic
286	C₂²	#65086	1	1	yes	characteristic
287	C₂²	#65240	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	8	1	no	—
292	C₄	—	8	1	no	—
293	C₄	—	40	2	no	—
294	C₄	#65101	1	1	yes	characteristic
295	C₄	#65105	1	1	yes	characteristic
296	C₄	#65453	1	1	yes	characteristic
297	C₄	#65473	1	1	yes	characteristic
298	C₄	#65485	1	1	yes	characteristic
299	C₄	#65486	1	1	yes	characteristic
300	C₂	—	2	1	no	—
301	C₂	—	2	1	no	—
302	C₂	#10600923	1	1	yes	characteristic
303	C₂	#10601048	1	1	yes	characteristic
304	C₂	#10601084	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#4030
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1542	C₂
socle	C₂²×C₅	#388
prime radical	C₂³×C₅	#124
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1542	D₅
hypocenter	C₅	#9591
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1542

Complements

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

N	H ≅ G/N	type
16. C₅×#333	C₄	semidirect
17. C₅×#333	C₄	semidirect
18. C₅×#332	C₄	semidirect
19. C₅×#332	C₄	semidirect
118. #333	Dic₅	semidirect
119. #333	Dic₅	semidirect
120. #332	Dic₅	semidirect
121. #332	Dic₅	semidirect
282. C₅	#9591	semidirect

Maximal subgroups

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

subgroup	index
C₅×#1542	2
#10602634	2
#10602619	2
#10601534	2
#10601267	2
#10598237	2
#10598195	2
#9591	5