Subgroups — #11998679

Order 1280.

502 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3147	C₂	1	1	yes	maximal, characteristic
2	#10602607	C₂	1	4	yes	maximal, complemented
3	#10600559	C₂	1	1	yes	maximal, characteristic, complemented
4	#10600517	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×#135	—	2	1	no	—
6	C₂×C₅×#135	—	2	1	no	—
7	C₅×#532	C₂²	1	2	yes	—
8	C₅×#495	—	2	1	no	—
9	C₅×#495	—	2	1	no	—
10	C₅×#494	C₄	1	1	yes	characteristic, complemented
11	C₅×#493	C₄	1	1	yes	characteristic, complemented
12	C₅×#492	C₄	1	2	yes	complemented
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₂×#3950	—	2	1	no	—
15	C₂×#3949	—	2	1	no	—
16	#65225	C₂²	1	2	yes	—
17	#65214	C₂²	1	2	yes	—
18	#64757	—	2	8	no	—
19	#64675	—	2	1	no	—
20	C₈×#168	—	2	1	no	—
21	#9756	—	5	1	no	maximal, Sylow
22	C₂×C₅×#41	—	2	2	no	—
23	C₅×#135	—	4	2	no	—
24	C₅×#135	—	4	2	no	—
25	C₅×#135	—	4	2	no	—
26	C₅×#135	—	4	2	no	—
27	C₂×C₅×Q₁₆	—	2	1	no	—
28	C₂×C₅×Q₁₆	—	2	1	no	—
29	C₂×C₅×SD₁₆	—	2	2	no	—
30	C₂×C₅×SD₁₆	—	2	2	no	—
31	C₂×C₅×D₈	—	2	1	no	—
32	C₂×C₅×D₈	—	2	1	no	—
33	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
34	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
35	C₅×C₄⋊Q₈	C₂×C₄	1	2	yes	—
36	C₅×#127	C₂×C₄	1	2	yes	—
37	C₅×#124	—	2	2	no	—
38	C₅×#121	—	2	4	no	—
39	C₄×C₅×D₄	—	2	4	no	—
40	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
41	C₂×C₄⋊Dic₅	—	2	2	no	—
42	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
43	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
44	C₄×C₅×C₈	D₄	1	1	yes	characteristic
45	C₄×C₅×C₈	D₄	1	1	yes	characteristic
46	#3963	—	2	4	no	—
47	C₂²⋊#168	D₄	1	4	yes	—
48	#3950	—	4	2	no	—
49	#3949	—	4	2	no	—
50	C₄⋊#168	—	2	2	no	—
51	C₄×#168	—	2	2	no	—
52	#3147	D₅	1	1	yes	characteristic
53	#1846	—	5	1	no	—
54	#1517	—	5	4	no	—
55	#1450	—	5	1	no	—
56	C₅×#41	—	4	2	no	—
57	C₅×#41	—	4	2	no	—
58	C₅×#41	—	4	2	no	—
59	C₅×#41	—	4	2	no	—
60	C₂×C₅×Q₈	—	2	2	no	—
61	C₂×C₅×D₄	—	2	2	no	—
62	C₂×C₅×D₄	—	2	4	no	—
63	C₂×C₅×D₄	—	4	2	no	—
64	C₂²×C₄×C₅	—	4	2	no	—
65	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
67	C₂²×Dic₅	—	4	2	no	—
68	C₅×Q₁₆	—	4	2	no	—
69	C₅×Q₁₆	—	4	2	no	—
70	C₅×SD₁₆	—	4	4	no	—
71	C₅×SD₁₆	—	4	4	no	—
72	C₅×D₈	—	4	2	no	—
73	C₅×D₈	—	4	2	no	—
74	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₅×C₈	D₈	1	2	yes	—
79	C₂×C₅×C₈	D₈	1	2	yes	—
80	C₂×C₅×C₈	SD₁₆	1	2	yes	—
81	C₂×C₅×C₈	SD₁₆	1	2	yes	—
82	C₅×C₄⋊C₄	—	2	4	no	—
83	C₅×C₂²⋊C₄	—	4	4	no	—
84	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₅	#41	1	2	yes	—
87	C₄⋊Dic₅	—	4	2	no	—
88	C₄⋊Dic₅	—	4	2	no	—
89	C₂×#168	—	2	4	no	—
90	C₂×#135	—	2	1	no	—
91	C₂×#135	—	2	1	no	—
92	#532	C₂×D₅	1	2	yes	—
93	#495	—	2	1	no	—
94	#495	—	2	1	no	—
95	#494	Dic₅	1	1	yes	characteristic, complemented
96	#493	Dic₅	1	1	yes	characteristic, complemented
97	#492	Dic₅	1	2	yes	complemented
98	#431	—	5	2	no	—
99	C₂×#107	—	10	1	no	—
100	C₂×#106	—	10	1	no	—
101	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
102	#382	—	5	2	no	—
103	#327	—	10	8	no	—
104	#322	—	10	1	no	—
105	C₈²	—	10	1	no	—
106	C₂³×C₅	—	4	2	no	—
107	C₂³×C₅	#127	1	1	yes	characteristic
108	C₅×Q₈	—	4	2	no	—
109	C₅×Q₈	—	4	2	no	—
110	C₅×D₄	—	4	2	no	—
111	C₅×D₄	—	4	2	no	—
112	C₅×D₄	—	8	4	no	—
113	C₅×D₄	—	8	2	no	—
114	C₅×D₄	—	8	2	no	—
115	C₂×C₄×C₅	—	4	2	no	—
116	C₂×C₄×C₅	—	4	2	no	—
117	C₂×C₄×C₅	—	8	2	no	—
118	C₂×C₄×C₅	—	8	2	no	—
119	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₅	#121	1	2	yes	—
122	C₂×C₄×C₅	#124	1	1	yes	characteristic
123	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×Dic₅	—	8	4	no	—
128	C₂×Dic₅	—	8	2	no	—
129	C₅×C₈	#102	1	2	yes	—
130	C₅×C₈	#102	1	2	yes	—
131	C₅×C₈	#102	1	2	yes	—
132	C₅×C₈	#102	1	2	yes	—
133	#168	—	4	4	no	—
134	C₂×#41	—	2	2	no	—
135	#135	—	4	2	no	—
136	#135	—	4	2	no	—
137	#135	—	4	2	no	—
138	#135	—	4	2	no	—
139	C₂×Q₁₆	—	2	1	no	—
140	C₂×Q₁₆	—	2	1	no	—
141	C₂×SD₁₆	—	2	2	no	—
142	C₂×SD₁₆	—	2	2	no	—
143	C₂×D₈	—	2	1	no	—
144	C₂×D₈	—	2	1	no	—
145	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
146	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
147	C₄⋊Q₈	C₂×Dic₅	1	2	yes	—
148	#127	C₂×Dic₅	1	2	yes	—
149	#124	—	2	2	no	—
150	#121	—	2	4	no	—
151	C₄×D₄	—	2	4	no	—
152	C₂×C₄⋊C₄	—	10	2	no	—
153	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
154	#107	—	20	2	no	—
155	#106	—	20	2	no	—
156	C₄⋊C₈	—	10	2	no	—
157	C₂²⋊C₈	—	5	4	no	—
158	C₄×C₈	—	10	2	no	—
159	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
160	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
161	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
162	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
163	#95	—	10	4	no	—
164	C₂²×C₅	—	2	2	no	—
165	C₂²×C₅	—	8	2	no	—
166	C₂²×C₅	—	8	4	no	—
167	C₂²×C₅	#390	1	1	yes	characteristic
168	C₂²×C₅	#492	1	1	yes	characteristic
169	C₂²×C₅	#493	1	1	yes	characteristic
170	C₄×C₅	—	2	2	no	—
171	C₄×C₅	—	2	2	no	—
172	C₄×C₅	—	8	2	no	—
173	C₄×C₅	C₂×#102	1	1	yes	characteristic
174	C₄×C₅	C₂×#102	1	1	yes	characteristic
175	C₄×C₅	#486	1	1	yes	characteristic
176	C₄×C₅	#486	1	1	yes	characteristic
177	Dic₅	—	16	2	no	—
178	#41	—	4	2	no	—
179	#41	—	4	2	no	—
180	#41	—	4	2	no	—
181	#41	—	4	2	no	—
182	C₂×Q₈	—	2	2	no	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	—	2	4	no	—
185	C₂×D₄	—	4	2	no	—
186	C₂²×C₄	—	4	2	no	—
187	C₂²×C₄	—	20	2	no	—
188	C₂²×C₄	D₄×D₅	1	2	yes	—
189	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
190	Q₁₆	—	4	2	no	—
191	Q₁₆	—	4	2	no	—
192	SD₁₆	—	4	4	no	—
193	SD₁₆	—	4	4	no	—
194	D₈	—	4	2	no	—
195	D₈	—	4	2	no	—
196	C₂×C₈	—	10	4	no	—
197	C₂×C₈	#686	1	2	yes	—
198	C₂×C₈	#686	1	2	yes	—
199	C₂×C₈	#687	1	2	yes	—
200	C₂×C₈	#687	1	2	yes	—
201	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
202	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
203	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
204	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	20	2	no	—
207	C₄⋊C₄	—	20	2	no	—
208	C₂²⋊C₄	—	4	4	no	—
209	C₄²	#711	1	2	yes	—
210	C₄²	C₂²×Dic₅	1	1	yes	characteristic
211	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
212	C₂×C₅	—	2	1	no	—
213	C₂×C₅	—	2	1	no	—
214	C₂×C₅	—	16	2	no	—
215	C₂×C₅	#1957	1	1	yes	characteristic
216	C₂×C₅	#1976	1	2	yes	—
217	C₂³	—	4	2	no	—
218	C₂³	#4086	1	1	yes	characteristic
219	Q₈	—	4	2	no	—
220	Q₈	—	4	2	no	—
221	D₄	—	4	2	no	—
222	D₄	—	4	2	no	—
223	D₄	—	8	4	no	—
224	D₄	—	8	2	no	—
225	D₄	—	8	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	8	2	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	—	40	4	no	—
231	C₂×C₄	—	40	2	no	—
232	C₂×C₄	C₂×#686	1	1	yes	characteristic
233	C₂×C₄	C₂×#686	1	1	yes	characteristic
234	C₂×C₄	C₂×#687	1	1	yes	characteristic
235	C₂×C₄	C₂×#687	1	1	yes	characteristic
236	C₂×C₄	D₄×Dic₅	1	2	yes	—
237	C₂×C₄	#4082	1	1	yes	characteristic
238	C₂×C₄	#4084	1	2	yes	—
239	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
240	C₈	—	20	4	no	—
241	C₈	#3964	1	2	yes	—
242	C₈	#3964	1	2	yes	—
243	C₈	#3964	1	2	yes	—
244	C₈	#3964	1	2	yes	—
245	C₅	#9756	1	1	yes	characteristic, complemented, Sylow
246	C₂²	—	2	2	no	—
247	C₂²	—	8	2	no	—
248	C₂²	—	8	4	no	—
249	C₂²	#65362	1	1	yes	characteristic
250	C₂²	#65365	1	1	yes	characteristic
251	C₂²	#65510	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	8	2	no	—
255	C₄	—	80	2	no	—
256	C₄	#65359	1	1	yes	characteristic
257	C₄	#65359	1	1	yes	characteristic
258	C₄	C₂×#3964	1	1	yes	characteristic
259	C₄	C₂×#3964	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	16	2	no	—
263	C₂	#10602557	1	1	yes	characteristic
264	C₂	#10602879	1	2	yes	—

	subgroup	G/H
center	C₂²	#65510
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting, prime radical	C₅×#3147	C₂
socle	C₂²×C₅	#390
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#3147	D₅
hypocenter	C₅	#9756
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3147

Complements

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

N	H ≅ G/N	type
2. #10602607	C₂	semidirect
3. #10600559	C₂	semidirect
4. #10600517	C₂	semidirect
10. C₅×#494	C₄	semidirect
11. C₅×#493	C₄	semidirect
12. C₅×#492	C₄	semidirect
95. #494	Dic₅	semidirect
96. #493	Dic₅	semidirect
97. #492	Dic₅	semidirect
245. C₅	#9756	semidirect

Maximal subgroups

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

subgroup	index
C₅×#3147	2
#10602607	2
#10600559	2
#10600517	2
#9756	5