Subgroups — #12254623

Order 1280.

590 conjugacy classes of subgroups.

The subgroup lattice has 268 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	#10610173	C₂	1	1	yes	maximal, characteristic, complemented
3	#10602870	C₂	1	4	yes	maximal, complemented
4	#10602112	C₂	1	4	yes	maximal, complemented
5	#10602104	C₂	1	2	yes	maximal, complemented
6	#10602082	C₂	1	2	yes	maximal, complemented
7	C₂×#64681	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#135	—	2	1	no	—
9	C₂×C₅×#135	C₂²	1	2	yes	—
10	C₅×#532	—	2	1	no	—
11	#65804	—	2	1	no	—
12	C₅×#495	—	2	1	no	—
13	C₅×#495	C₂²	1	2	yes	—
14	C₅×#494	C₂²	1	1	yes	characteristic
15	C₅×#493	C₂²	1	1	yes	characteristic
16	C₅×#492	—	2	1	no	—
17	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	#65395	—	2	1	no	—
19	#65395	C₂²	1	2	yes	—
20	C₂²×#672	C₂²	1	2	yes	—
21	#64972	—	2	1	no	—
22	#64972	C₂²	1	2	yes	—
23	#64971	C₂²	1	2	yes	—
24	#64969	—	2	1	no	—
25	#64784	C₂²	1	8	yes	—
26	#64782	C₂²	1	8	yes	—
27	#64681	C₂²	1	2	yes	—
28	#64681	C₂²	1	2	yes	—
29	#33701	—	5	1	no	maximal, Sylow
30	C₂×C₅×#41	—	2	2	no	—
31	C₅×#135	—	2	8	no	—
32	C₅×#135	—	4	4	no	—
33	C₂×C₅×Q₁₆	—	2	1	no	—
34	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
35	C₂×C₅×SD₁₆	—	2	2	no	—
36	C₂×C₅×SD₁₆	—	2	2	no	—
37	C₂×C₅×D₈	—	2	1	no	—
38	C₂×C₅×D₈	C₂³	1	2	yes	—
39	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
40	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
41	C₅×C₄⋊Q₈	—	2	1	no	—
42	C₅×#127	—	2	1	no	—
43	C₅×#124	—	2	2	no	—
44	C₅×#121	—	4	2	no	—
45	C₄×C₅×D₄	—	4	2	no	—
46	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
47	#4072	—	2	2	no	—
48	#4070	—	4	2	no	—
49	#4016	—	2	2	no	—
50	#4015	—	2	2	no	—
51	C₄×Dic₁₀	—	4	2	no	—
52	C₄×C₅×C₈	D₄	1	1	yes	characteristic
53	C₄×C₅×C₈	D₄	1	1	yes	characteristic
54	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
55	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
56	#3950	—	2	2	no	—
57	#3950	C₂³	1	4	yes	—
58	#3949	—	2	2	no	—
59	#3949	—	2	2	no	—
60	C₂×#672	—	2	4	no	—
61	C₂×#672	C₂³	1	4	yes	—
62	#3160	—	5	1	no	—
63	#3147	D₅	1	1	yes	characteristic
64	#2245	—	5	2	no	—
65	#2244	—	5	2	no	—
66	#2243	—	5	4	no	—
67	#2138	—	5	4	no	—
68	C₂×#346	—	5	1	no	—
69	C₅×#41	—	4	4	no	—
70	C₅×#41	—	4	4	no	—
71	C₂×C₅×Q₈	—	2	2	no	—
72	C₂×C₅×D₄	—	2	2	no	—
73	C₂×C₅×D₄	—	4	2	no	—
74	C₂×C₅×D₄	—	4	2	no	—
75	C₂²×C₄×C₅	—	4	2	no	—
76	C₂²×C₄×C₅	D₈	1	2	yes	—
77	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂×Dic₁₀	—	4	2	no	—
79	C₅×Q₁₆	—	2	4	no	—
80	C₅×Q₁₆	—	4	2	no	—
81	C₅×SD₁₆	—	4	4	no	—
82	C₅×SD₁₆	—	4	4	no	—
83	C₅×D₈	—	2	4	no	—
84	C₅×D₈	—	4	2	no	—
85	C₂×C₅×C₈	SD₁₆	1	4	yes	—
86	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
87	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
88	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
89	C₂×C₅×C₈	#41	1	4	yes	—
90	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
91	C₅×C₄⋊C₄	—	4	2	no	—
92	C₅×C₂²⋊C₄	—	4	4	no	—
93	C₄²×C₅	D₈	1	2	yes	—
94	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
96	C₂²⋊Dic₅	—	4	2	no	—
97	C₄⋊Dic₅	—	2	4	no	—
98	C₄⋊Dic₅	—	4	2	no	—
99	#683	—	4	4	no	—
100	C₄×Dic₅	—	4	2	no	—
101	#672	—	2	4	no	—
102	C₂×#135	—	2	1	no	—
103	C₂×#135	C₂×D₅	1	2	yes	—
104	#533	—	10	1	no	—
105	#532	—	2	1	no	—
106	C₂²×C₁₆	—	5	2	no	—
107	#500	—	5	2	no	—
108	#499	—	5	2	no	—
109	#499	—	10	1	no	—
110	#498	—	10	1	no	—
111	#495	—	2	1	no	—
112	#495	C₂×D₅	1	2	yes	—
113	#494	C₂×D₅	1	1	yes	characteristic
114	#493	C₂×D₅	1	1	yes	characteristic
115	#492	—	2	1	no	—
116	#427	—	5	2	no	—
117	#427	—	10	1	no	—
118	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
119	#358	—	5	8	no	—
120	#357	—	5	8	no	—
121	#346	—	5	2	no	—
122	#346	—	5	2	no	—
123	C₂³×C₅	—	4	2	no	—
124	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
125	C₅×Q₈	—	4	2	no	—
126	C₅×Q₈	—	4	2	no	—
127	C₅×D₄	—	4	2	no	—
128	C₅×D₄	—	4	2	no	—
129	C₅×D₄	—	8	4	no	—
130	C₅×D₄	—	8	4	no	—
131	C₂×C₄×C₅	—	4	2	no	—
132	C₂×C₄×C₅	—	4	2	no	—
133	C₂×C₄×C₅	—	8	4	no	—
134	C₂×C₄×C₅	#124	1	2	yes	—
135	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
137	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
139	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
140	C₂×Dic₅	—	4	4	no	—
141	Dic₁₀	—	8	4	no	—
142	C₅×C₈	#124	1	2	yes	—
143	C₅×C₈	C₂×SD₁₆	1	2	yes	—
144	C₅×C₈	C₂×SD₁₆	1	2	yes	—
145	C₅×C₈	C₂×#41	1	2	yes	—
146	C₂×#41	—	2	2	no	—
147	#135	—	2	8	no	—
148	#135	—	4	4	no	—
149	C₂×Q₁₆	—	2	1	no	—
150	C₂×Q₁₆	C₂²×D₅	1	2	yes	—
151	C₂×SD₁₆	—	2	2	no	—
152	C₂×SD₁₆	—	2	2	no	—
153	C₂×D₈	—	2	1	no	—
154	C₂×D₈	C₂²×D₅	1	2	yes	—
155	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
156	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
157	C₄⋊Q₈	—	2	1	no	—
158	C₄⋊Q₈	—	10	2	no	—
159	#127	—	2	1	no	—
160	#125	—	10	2	no	—
161	#124	—	2	2	no	—
162	C₂²⋊Q₈	—	20	2	no	—
163	#121	—	4	2	no	—
164	C₄×Q₈	—	20	2	no	—
165	C₄×D₄	—	4	2	no	—
166	#117	—	10	2	no	—
167	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
168	C₂×C₁₆	—	5	4	no	—
169	C₂×C₁₆	—	10	4	no	—
170	#107	—	5	4	no	—
171	#107	—	10	2	no	—
172	#106	—	10	2	no	—
173	#106	—	10	2	no	—
174	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
175	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
176	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
177	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
178	C₂²×C₅	—	2	2	no	—
179	C₂²×C₅	—	8	2	no	—
180	C₂²×C₅	—	8	4	no	—
181	C₂²×C₅	#486	1	2	yes	—
182	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
183	C₄×C₅	—	2	2	no	—
184	C₄×C₅	—	2	2	no	—
185	C₄×C₅	—	8	2	no	—
186	C₄×C₅	#486	1	2	yes	—
187	C₄×C₅	C₂×#124	1	1	yes	characteristic
188	C₄×C₅	C₂²×SD₁₆	1	1	yes	characteristic
189	Dic₅	—	8	4	no	—
190	#41	—	4	4	no	—
191	#41	—	4	4	no	—
192	C₂×Q₈	—	2	2	no	—
193	C₂×Q₈	—	20	2	no	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	—	4	2	no	—
196	C₂×D₄	—	4	2	no	—
197	C₂²×C₄	—	4	2	no	—
198	C₂²×C₄	#686	1	2	yes	—
199	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
200	Q₁₆	—	2	4	no	—
201	Q₁₆	—	4	2	no	—
202	SD₁₆	—	4	4	no	—
203	SD₁₆	—	4	4	no	—
204	D₈	—	2	4	no	—
205	D₈	—	4	2	no	—
206	C₂×C₈	#687	1	4	yes	—
207	C₂×C₈	#711	1	4	yes	—
208	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
209	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
210	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
211	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
212	C₄⋊C₄	—	4	2	no	—
213	C₄⋊C₄	—	10	4	no	—
214	C₄⋊C₄	—	20	4	no	—
215	C₄⋊C₄	—	20	2	no	—
216	C₂²⋊C₄	—	4	4	no	—
217	C₂²⋊C₄	—	20	2	no	—
218	C₄²	—	20	2	no	—
219	C₄²	#686	1	2	yes	—
220	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
221	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
222	C₁₆	—	10	4	no	—
223	C₂×C₅	—	2	2	no	—
224	C₂×C₅	—	16	2	no	—
225	C₂×C₅	C₂×#486	1	1	yes	characteristic
226	C₂×C₅	#3414	1	2	yes	—
227	C₂³	—	4	2	no	—
228	C₂³	C₂×#686	1	1	yes	characteristic
229	Q₈	—	4	2	no	—
230	Q₈	—	4	2	no	—
231	Q₈	—	40	4	no	—
232	D₄	—	4	2	no	—
233	D₄	—	4	2	no	—
234	D₄	—	8	4	no	—
235	D₄	—	8	4	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	8	4	no	—
239	C₂×C₄	—	20	4	no	—
240	C₂×C₄	C₂×#686	1	2	yes	—
241	C₂×C₄	C₂×#686	1	2	yes	—
242	C₂×C₄	C₂×#686	1	1	yes	characteristic
243	C₂×C₄	C₂×#687	1	2	yes	—
244	C₂×C₄	#4082	1	2	yes	—
245	C₂×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
246	C₈	C₂×#687	1	2	yes	—
247	C₈	C₂×#687	1	2	yes	—
248	C₈	#4082	1	2	yes	—
249	C₈	C₂×#711	1	2	yes	—
250	C₅	#33701	1	1	yes	characteristic, complemented, Sylow
251	C₂²	—	2	2	no	—
252	C₂²	—	8	4	no	—
253	C₂²	—	8	2	no	—
254	C₂²	#65359	1	2	yes	—
255	C₂²	C₂²×#686	1	1	yes	characteristic
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	8	2	no	—
259	C₄	—	40	4	no	—
260	C₄	#65359	1	2	yes	—
261	C₄	C₂²×#687	1	1	yes	characteristic
262	C₄	C₂×#4082	1	1	yes	characteristic
263	C₂	—	2	2	no	—
264	C₂	—	16	2	no	—
265	C₂	C₂×#65359	1	1	yes	characteristic
266	C₂	#10615625	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#686
commutator, cosocle	C₂×C₅×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₅
Fitting, prime radical	C₅×#3147	C₂
socle	C₂²×C₅	C₂²×D₈
hypercenter	#3147	D₅
hypocenter	C₅	#33701
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3147

Complements

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

N	H ≅ G/N	type
2. #10610173	C₂	semidirect
3. #10602870	C₂	semidirect
4. #10602112	C₂	semidirect
5. #10602104	C₂	semidirect
6. #10602082	C₂	semidirect
7. C₂×#64681	C₂	semidirect
250. C₅	#33701	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
#10610173	2
#10602870	2
#10602112	2
#10602104	2
#10602082	2
C₂×#64681	2
#33701	5