Subgroups — #12009708

Order 1280.

394 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3169	C₂	1	1	yes	maximal, characteristic
2	C₄×#3950	C₂	1	1	yes	maximal, characteristic, complemented
3	#10600537	C₂	1	1	yes	maximal, characteristic, complemented
4	#10598265	C₂	1	4	yes	maximal, complemented
5	C₅×#532	—	2	1	no	—
6	C₅×#500	C₄	1	1	yes	characteristic, complemented
7	C₅×#493	C₄	1	1	yes	characteristic, complemented
8	C₅×#486	—	2	1	no	—
9	C₅×#466	—	2	1	no	—
10	C₅×#466	C₂²	1	2	yes	—
11	C₄×C₅×D₈	—	2	1	no	—
12	C₄×C₅×D₈	C₄	1	2	yes	complemented
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₂×#3950	—	2	1	no	—
15	C₄×C₄⋊Dic₅	—	2	1	no	—
16	#64771	C₂²	1	2	yes	—
17	C₂²⋊#672	C₂²	1	2	yes	—
18	C₄⋊#672	—	2	1	no	—
19	C₄×#672	—	2	1	no	—
20	#11742	—	5	1	no	maximal, Sylow
21	C₂×C₅×#41	—	4	1	no	—
22	C₂×C₅×D₈	—	2	1	no	—
23	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
24	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
25	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
26	C₅×C₄⋊Q₈	—	2	1	no	—
27	C₅×#127	—	2	1	no	—
28	C₅×#124	—	4	1	no	—
29	C₅×#121	—	2	2	no	—
30	C₅×#121	—	4	1	no	—
31	C₄×C₅×D₄	—	2	2	no	—
32	C₄×C₅×D₄	—	4	1	no	—
33	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
34	C₂×C₄⋊Dic₅	—	2	2	no	—
35	C₂×C₄×Dic₅	—	4	1	no	—
36	C₅×#107	—	2	1	no	—
37	C₅×#107	C₂×C₄	1	2	yes	—
38	C₅×#102	—	2	2	no	—
39	C₅×#102	—	2	2	no	—
40	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
41	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
42	C₄×C₅×C₈	D₄	1	1	yes	characteristic
43	C₄×C₅×C₈	D₄	1	1	yes	characteristic
44	#3963	—	4	1	no	—
45	#3950	—	2	4	no	—
46	C₂×#672	—	2	2	no	—
47	#3169	D₅	1	1	yes	characteristic
48	#2162	—	5	1	no	—
49	C₄×#107	—	5	1	no	—
50	#1347	—	5	4	no	—
51	C₅×#41	—	8	2	no	—
52	C₅×#41	—	8	2	no	—
53	C₂×C₅×Q₈	—	4	1	no	—
54	C₂×C₅×D₄	—	2	2	no	—
55	C₂×C₅×D₄	—	4	2	no	—
56	C₂×C₅×D₄	—	4	1	no	—
57	C₂×C₅×D₄	—	4	1	no	—
58	C₂²×C₄×C₅	—	4	2	no	—
59	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂²×Dic₅	—	4	2	no	—
63	C₅×D₈	—	4	2	no	—
64	C₅×D₈	—	4	2	no	—
65	C₂×C₅×C₈	—	2	2	no	—
66	C₂×C₅×C₈	—	4	1	no	—
67	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
71	C₅×C₄⋊C₄	—	2	2	no	—
72	C₅×C₄⋊C₄	—	4	1	no	—
73	C₅×C₂²⋊C₄	—	4	2	no	—
74	C₅×C₂²⋊C₄	—	4	2	no	—
75	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
77	C₄²×C₅	SD₁₆	1	1	yes	characteristic
78	C₄²×C₅	SD₁₆	1	1	yes	characteristic
79	C₄⋊Dic₅	—	2	4	no	—
80	C₄⋊Dic₅	—	4	2	no	—
81	C₄×Dic₅	—	8	2	no	—
82	#672	—	4	2	no	—
83	#532	—	2	1	no	—
84	#500	Dic₅	1	1	yes	characteristic, complemented
85	#493	Dic₅	1	1	yes	characteristic, complemented
86	#486	—	2	1	no	—
87	#466	—	2	1	no	—
88	#466	C₂×D₅	1	2	yes	—
89	C₄×D₈	—	2	1	no	—
90	C₄×D₈	Dic₅	1	2	yes	complemented
91	C₂×#107	—	10	1	no	—
92	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
93	C₄×C₄⋊C₄	—	10	1	no	—
94	C₄⋊C₁₆	—	10	1	no	—
95	C₂²⋊C₁₆	—	5	2	no	—
96	C₄×C₁₆	—	10	1	no	—
97	#340	—	5	2	no	—
98	C₂³×C₅	—	4	2	no	—
99	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
100	C₅×Q₈	—	8	2	no	—
101	C₅×D₄	—	4	2	no	—
102	C₅×D₄	—	4	2	no	—
103	C₅×D₄	—	8	2	no	—
104	C₅×D₄	—	8	2	no	—
105	C₅×D₄	—	8	2	no	—
106	C₅×D₄	—	8	2	no	—
107	C₂×C₄×C₅	—	2	1	no	—
108	C₂×C₄×C₅	—	2	1	no	—
109	C₂×C₄×C₅	—	4	1	no	—
110	C₂×C₄×C₅	—	4	2	no	—
111	C₂×C₄×C₅	—	8	4	no	—
112	C₂×C₄×C₅	#102	1	1	yes	characteristic
113	C₂×C₄×C₅	#102	1	1	yes	characteristic
114	C₂×C₄×C₅	#102	1	1	yes	characteristic
115	C₂×C₄×C₅	#102	1	1	yes	characteristic
116	C₂×C₄×C₅	D₁₆	1	2	yes	—
117	C₂×C₄×C₅	SD₃₂	1	2	yes	—
118	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×Dic₅	—	4	4	no	—
121	C₂×Dic₅	—	8	2	no	—
122	C₂×Dic₅	—	8	2	no	—
123	C₅×C₈	—	2	1	no	—
124	C₅×C₈	—	2	1	no	—
125	C₅×C₈	—	2	1	no	—
126	C₅×C₈	#104	1	2	yes	—
127	C₂×#41	—	4	1	no	—
128	C₂×D₈	—	2	1	no	—
129	C₂×D₈	C₂×Dic₅	1	2	yes	—
130	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
131	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
132	C₄⋊Q₈	—	2	1	no	—
133	#127	—	2	1	no	—
134	#124	—	4	1	no	—
135	#121	—	2	2	no	—
136	#121	—	4	1	no	—
137	C₄×D₄	—	2	2	no	—
138	C₄×D₄	—	4	1	no	—
139	C₂×C₄⋊C₄	—	10	2	no	—
140	C₂×C₄²	—	20	1	no	—
141	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
142	C₂×C₁₆	—	10	2	no	—
143	#107	—	2	1	no	—
144	#107	—	10	4	no	—
145	#107	C₂×Dic₅	1	2	yes	—
146	#102	—	2	2	no	—
147	#102	—	2	2	no	—
148	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
149	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
150	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
151	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
152	#95	—	20	1	no	—
153	C₂²×C₅	—	2	1	no	—
154	C₂²×C₅	—	2	1	no	—
155	C₂²×C₅	—	8	2	no	—
156	C₂²×C₅	—	8	2	no	—
157	C₂²×C₅	—	8	2	no	—
158	C₂²×C₅	C₂×#102	1	1	yes	characteristic
159	C₂²×C₅	C₂×D₁₆	1	1	yes	characteristic
160	C₂²×C₅	C₂×SD₃₂	1	1	yes	characteristic
161	C₄×C₅	—	2	1	no	—
162	C₄×C₅	—	2	1	no	—
163	C₄×C₅	—	2	1	no	—
164	C₄×C₅	—	8	2	no	—
165	C₄×C₅	#357	1	2	yes	—
166	C₄×C₅	#357	1	2	yes	—
167	C₄×C₅	#410	1	1	yes	characteristic
168	C₄×C₅	C₂×#104	1	1	yes	characteristic
169	Dic₅	—	8	4	no	—
170	#41	—	8	2	no	—
171	#41	—	8	2	no	—
172	C₂×Q₈	—	4	1	no	—
173	C₂×D₄	—	2	2	no	—
174	C₂×D₄	—	4	2	no	—
175	C₂×D₄	—	4	1	no	—
176	C₂×D₄	—	4	1	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	—	20	2	no	—
179	C₂²×C₄	#686	1	1	yes	characteristic
180	C₂²×C₄	#686	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
182	D₈	—	4	2	no	—
183	D₈	—	4	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	4	1	no	—
186	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
187	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
188	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
189	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	4	1	no	—
192	C₄⋊C₄	—	10	4	no	—
193	C₄⋊C₄	—	20	2	no	—
194	C₂²⋊C₄	—	4	2	no	—
195	C₂²⋊C₄	—	4	2	no	—
196	C₄²	—	40	2	no	—
197	C₄²	#687	1	1	yes	characteristic
198	C₄²	#687	1	1	yes	characteristic
199	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
200	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
201	C₁₆	—	20	2	no	—
202	C₂×C₅	—	2	1	no	—
203	C₂×C₅	—	2	1	no	—
204	C₂×C₅	—	16	2	no	—
205	C₂×C₅	#1502	1	1	yes	characteristic
206	C₂×C₅	C₂×#357	1	1	yes	characteristic
207	C₂×C₅	#2145	1	1	yes	characteristic
208	C₂³	—	4	2	no	—
209	C₂³	C₂×#686	1	1	yes	characteristic
210	Q₈	—	8	2	no	—
211	D₄	—	4	2	no	—
212	D₄	—	4	2	no	—
213	D₄	—	8	2	no	—
214	D₄	—	8	2	no	—
215	D₄	—	8	2	no	—
216	D₄	—	8	2	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	4	1	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	8	4	no	—
222	C₂×C₄	—	20	4	no	—
223	C₂×C₄	—	40	2	no	—
224	C₂×C₄	—	40	2	no	—
225	C₂×C₄	#3958	1	2	yes	—
226	C₂×C₄	#3959	1	2	yes	—
227	C₂×C₄	#3964	1	1	yes	characteristic
228	C₂×C₄	#3964	1	1	yes	characteristic
229	C₂×C₄	#3964	1	1	yes	characteristic
230	C₂×C₄	#3964	1	1	yes	characteristic
231	C₂×C₄	C₂×#687	1	1	yes	characteristic
232	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
233	C₈	—	2	1	no	—
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	#3969	1	2	yes	—
237	C₅	#11742	1	1	yes	characteristic, complemented, Sylow
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	8	2	no	—
241	C₂²	—	8	2	no	—
242	C₂²	—	8	2	no	—
243	C₂²	C₂×#3958	1	1	yes	characteristic
244	C₂²	C₂×#3959	1	1	yes	characteristic
245	C₂²	C₂×#3964	1	1	yes	characteristic
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	8	2	no	—
250	C₄	—	40	4	no	—
251	C₄	#64782	1	2	yes	—
252	C₄	#64782	1	2	yes	—
253	C₄	C₂×#3969	1	1	yes	characteristic
254	C₄	#65529	1	1	yes	characteristic
255	C₂	—	2	1	no	—
256	C₂	—	2	1	no	—
257	C₂	—	16	2	no	—
258	C₂	#10602604	1	1	yes	characteristic
259	C₂	C₂×#64782	1	1	yes	characteristic
260	C₂	#10602869	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#3964
commutator	C₂×C₅×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅
Fitting, prime radical	C₅×#3169	C₂
socle	C₂²×C₅	C₂×#102
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#3169	D₅
hypocenter	C₅	#11742
LCS	G ▷ C₂×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3169

Complements

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

N	H ≅ G/N	type
2. C₄×#3950	C₂	semidirect
3. #10600537	C₂	semidirect
4. #10598265	C₂	semidirect
6. C₅×#500	C₄	semidirect
7. C₅×#493	C₄	semidirect
12. C₄×C₅×D₈	C₄	semidirect
84. #500	Dic₅	semidirect
85. #493	Dic₅	semidirect
90. C₄×D₈	Dic₅	semidirect
237. C₅	#11742	semidirect

Maximal subgroups

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

subgroup	index
C₅×#3169	2
C₄×#3950	2
#10600537	2
#10598265	2
#11742	5