Subgroups — #12252443

Order 1280.

528 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10610280	C₂	1	1	yes	maximal, characteristic
2	C₅×#2110	C₂	1	1	yes	maximal, characteristic, complemented
3	#10602228	C₂	1	2	yes	maximal
4	C₂×#64731	C₂	1	2	yes	maximal, complemented
5	#10600677	C₂	1	1	yes	maximal, characteristic, complemented
6	#10599882	C₂	1	4	yes	maximal, complemented
7	#10599881	C₂	1	4	yes	maximal, complemented
8	C₂×#4053	C₄	1	1	yes	characteristic
9	C₂×#4047	C₂²	1	1	yes	characteristic
10	#65856	C₄	1	1	yes	characteristic
11	C₂×C₅×M₃₂	C₂²	1	2	yes	—
12	C₅×#405	C₂²	1	1	yes	characteristic
13	M₁₆×Dic₅	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
15	C₂×#3944	C₂²	1	2	yes	—
16	#65286	C₄	1	1	yes	characteristic
17	#64997	C₂²	1	2	yes	—
18	#64996	C₂²	1	2	yes	—
19	#64978	C₄	1	2	yes	—
20	C₄×C₈×D₅	C₄	1	2	yes	—
21	C₅×#347	C₂²	1	4	yes	—
22	#64734	C₂²	1	8	yes	—
23	#64731	C₂²	1	4	yes	—
24	#64731	C₂²	1	4	yes	—
25	#64686	C₂²	1	4	yes	—
26	#33428	—	5	1	no	maximal, Sylow
27	C₂×#709	C₂×C₄	1	1	yes	characteristic
28	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
30	C₅×#117	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
32	C₂×#681	C₂×C₄	1	1	yes	characteristic
33	C₂²×#168	C₂×C₄	1	1	yes	characteristic
34	#4053	—	2	4	no	—
35	#4047	—	2	2	no	—
36	#4047	—	2	2	no	—
37	C₂×#676	C₂×C₄	1	2	yes	—
38	C₂×#676	C₂³	1	2	yes	—
39	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
41	#4039	—	2	1	no	—
42	#4033	—	2	1	no	—
43	#4027	—	2	2	no	—
44	#4018	C₂×C₄	1	2	yes	—
45	C₄²×D₅	C₂×C₄	1	2	yes	—
46	C₅×M₃₂	—	2	2	no	—
47	C₅×M₃₂	C₂³	1	4	yes	—
48	C₂×C₅×C₁₆	—	2	2	no	—
49	C₅×#97	C₂³	1	2	yes	—
50	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
51	#3947	C₂×C₄	1	2	yes	—
52	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
53	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
54	C₈×Dic₅	C₂×C₄	1	2	yes	—
55	C₈×Dic₅	C₂³	1	1	yes	characteristic
56	C₈×Dic₅	C₂³	1	1	yes	characteristic
57	#3944	—	2	2	no	—
58	#3944	C₂³	1	4	yes	—
59	C₂×#672	—	2	2	no	—
60	#3935	C₂×C₄	1	2	yes	—
61	C₄×#168	C₂×C₄	1	2	yes	—
62	C₄×#131	—	5	1	no	—
63	#2166	—	5	4	no	—
64	#2165	—	5	4	no	—
65	C₂×#364	—	5	2	no	—
66	#2115	—	5	2	no	—
67	#2110	—	5	1	no	—
68	#2110	D₅	1	1	yes	characteristic, complemented
69	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₂²⋊D₅	M₁₆	1	2	yes	—
71	C₂²×Dic₅	—	2	1	no	—
72	#709	—	2	2	no	—
73	#709	—	2	2	no	—
74	C₂×D₂₀	M₁₆	1	1	yes	characteristic
75	C₂×C₄×D₅	—	2	2	no	—
76	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
77	C₂×Dic₁₀	M₁₆	1	1	yes	characteristic
78	C₅×M₁₆	—	2	2	no	—
79	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
80	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₅×C₈	#41	1	2	yes	—
82	C₂×C₅×C₈	#41	1	2	yes	—
83	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
84	C₅×C₄⋊C₄	—	2	1	no	—
85	C₅×C₂²⋊C₄	—	2	1	no	—
86	C₄²×C₅	C₂²×C₄	1	2	yes	—
87	#685	—	2	2	no	—
88	#683	—	2	2	no	—
89	C₄×Dic₅	—	2	1	no	—
90	C₄×Dic₅	—	2	2	no	—
91	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
92	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
93	#681	—	2	2	no	—
94	C₂×#168	M₁₆	1	4	yes	—
95	C₂×#168	C₂²×C₄	1	2	yes	—
96	C₂×#168	C₂²×C₄	1	1	yes	characteristic
97	C₂×#168	C₂²×C₄	1	1	yes	characteristic
98	#676	—	2	4	no	—
99	#676	—	2	4	no	—
100	C₈×D₅	—	2	2	no	—
101	C₈×D₅	—	2	2	no	—
102	C₈×D₅	—	2	4	no	—
103	C₅×C₁₆	—	2	4	no	—
104	#672	—	2	4	no	—
105	C₂×#131	—	5	1	no	—
106	C₂×#131	—	5	1	no	—
107	C₄×#41	—	5	1	no	—
108	C₂×M₃₂	—	5	2	no	—
109	C₂×M₃₂	C₂×D₅	1	2	yes	—
110	#405	—	5	1	no	—
111	#405	—	5	2	no	—
112	#405	—	5	2	no	—
113	#405	C₂×D₅	1	1	yes	characteristic
114	C₄×M₁₆	—	5	1	no	—
115	C₄×M₁₆	—	5	2	no	—
116	C₂×C₄×C₈	—	5	1	no	—
117	C₂×C₄×C₈	—	5	2	no	—
118	#364	—	5	4	no	—
119	#364	—	5	4	no	—
120	#349	—	5	8	no	—
121	#347	—	5	4	no	—
122	#347	C₂×D₅	1	4	yes	—
123	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
124	C₂²×D₅	C₂×M₁₆	1	2	yes	—
125	C₂×C₄×C₅	—	2	2	no	—
126	C₂×C₄×C₅	#117	1	2	yes	—
127	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
128	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
129	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
130	C₂²⋊D₅	—	2	4	no	—
131	C₂×Dic₅	—	2	1	no	—
132	C₂×Dic₅	—	2	1	no	—
133	C₂×Dic₅	—	4	2	no	—
134	C₂×Dic₅	C₂×M₁₆	1	2	yes	—
135	D₂₀	—	2	2	no	—
136	C₄×D₅	—	2	2	no	—
137	C₄×D₅	—	2	2	no	—
138	C₄×D₅	—	4	4	no	—
139	Dic₁₀	—	2	2	no	—
140	C₅×C₈	—	2	2	no	—
141	C₅×C₈	#117	1	1	yes	characteristic
142	C₅×C₈	#117	1	1	yes	characteristic
143	C₅×C₈	C₂×#41	1	2	yes	—
144	#168	—	2	2	no	—
145	#168	C₂×M₁₆	1	2	yes	—
146	#168	C₂×M₁₆	1	2	yes	—
147	C₂×#41	—	5	1	no	—
148	#131	—	10	4	no	—
149	#131	—	10	2	no	—
150	#131	—	10	2	no	—
151	C₂×M₁₆	—	5	1	no	—
152	C₂×M₁₆	—	5	2	no	—
153	C₂×M₁₆	—	5	2	no	—
154	C₂×M₁₆	C₄×D₅	1	1	yes	characteristic
155	C₂²×C₈	—	5	1	no	—
156	C₂²×C₈	—	5	2	no	—
157	C₂²×C₈	—	5	2	no	—
158	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
159	C₄×Q₈	—	10	1	no	—
160	C₄×D₄	—	10	2	no	—
161	C₄×D₄	—	10	1	no	—
162	#117	—	5	2	no	—
163	#117	C₄×D₅	1	1	yes	characteristic
164	C₂×C₄²	—	5	1	no	—
165	C₂×C₄²	—	5	2	no	—
166	M₃₂	—	2	2	no	—
167	M₃₂	—	5	4	no	—
168	M₃₂	—	10	2	no	—
169	M₃₂	C₂²×D₅	1	4	yes	—
170	C₂×C₁₆	—	2	2	no	—
171	C₂×C₁₆	—	10	2	no	—
172	#97	—	5	2	no	—
173	#97	—	5	2	no	—
174	#97	C₂²×D₅	1	2	yes	—
175	C₄×C₈	—	5	1	no	—
176	C₄×C₈	—	5	2	no	—
177	C₄×C₈	—	5	1	no	—
178	C₄×C₈	—	5	1	no	—
179	C₄×C₈	—	5	1	no	—
180	C₄×C₈	—	5	2	no	—
181	C₄×C₈	C₄×D₅	1	2	yes	—
182	C₂²×C₅	—	4	1	no	—
183	C₂²×C₅	#432	1	2	yes	—
184	C₂²×C₅	C₂²×M₁₆	1	1	yes	characteristic
185	C₂×D₅	—	2	2	no	—
186	C₂×D₅	—	4	2	no	—
187	C₄×C₅	—	4	2	no	—
188	C₄×C₅	#432	1	2	yes	—
189	C₄×C₅	C₂×#117	1	1	yes	characteristic
190	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
191	Dic₅	—	2	2	no	—
192	Dic₅	—	2	2	no	—
193	Dic₅	—	4	1	no	—
194	#41	—	10	2	no	—
195	#41	—	10	2	no	—
196	C₂×Q₈	—	5	1	no	—
197	C₂×D₄	—	5	2	no	—
198	C₂×D₄	—	5	1	no	—
199	C₂²×C₄	—	5	2	no	—
200	C₂²×C₄	—	10	2	no	—
201	C₂²×C₄	—	10	1	no	—
202	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
203	M₁₆	—	2	2	no	—
204	M₁₆	—	10	4	no	—
205	M₁₆	—	10	2	no	—
206	M₁₆	—	10	4	no	—
207	C₂×C₈	—	5	1	no	—
208	C₂×C₈	—	5	1	no	—
209	C₂×C₈	—	5	4	no	—
210	C₂×C₈	—	5	2	no	—
211	C₂×C₈	—	10	4	no	—
212	C₂×C₈	—	10	2	no	—
213	C₂×C₈	—	10	2	no	—
214	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
215	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
216	C₂×C₈	#709	1	2	yes	—
217	C₂×C₈	#709	1	2	yes	—
218	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
219	C₄⋊C₄	—	2	1	no	—
220	C₄⋊C₄	—	10	2	no	—
221	C₂²⋊C₄	—	2	1	no	—
222	C₂²⋊C₄	—	10	2	no	—
223	C₄²	—	5	1	no	—
224	C₄²	—	5	1	no	—
225	C₄²	—	10	2	no	—
226	C₄²	—	10	1	no	—
227	C₄²	C₂×C₄×D₅	1	2	yes	—
228	C₁₆	—	2	4	no	—
229	C₁₆	—	10	4	no	—
230	C₂×C₅	—	2	1	no	—
231	C₂×C₅	—	2	2	no	—
232	C₂×C₅	C₂×#432	1	1	yes	characteristic
233	D₅	—	4	2	no	—
234	C₂³	—	5	2	no	—
235	C₂³	D₅×M₁₆	1	1	yes	characteristic
236	Q₈	—	10	2	no	—
237	D₄	—	10	4	no	—
238	D₄	—	10	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	5	2	no	—
241	C₂×C₄	—	10	1	no	—
242	C₂×C₄	—	10	2	no	—
243	C₂×C₄	—	10	2	no	—
244	C₂×C₄	—	10	1	no	—
245	C₂×C₄	—	20	4	no	—
246	C₂×C₄	—	20	2	no	—
247	C₂×C₄	#4018	1	2	yes	—
248	C₂×C₄	D₅×M₁₆	1	2	yes	—
249	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
250	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	5	2	no	—
253	C₈	—	5	2	no	—
254	C₈	—	10	2	no	—
255	C₈	#4018	1	1	yes	characteristic
256	C₈	#4018	1	1	yes	characteristic
257	C₈	C₂×#709	1	2	yes	—
258	C₅	#33428	1	1	yes	characteristic, complemented, Sylow
259	C₂²	—	4	1	no	—
260	C₂²	—	10	2	no	—
261	C₂²	—	20	2	no	—
262	C₂²	#64994	1	2	yes	—
263	C₂²	C₂×D₅×M₁₆	1	1	yes	characteristic
264	C₄	—	4	2	no	—
265	C₄	—	10	2	no	—
266	C₄	—	10	2	no	—
267	C₄	—	20	1	no	—
268	C₄	#64994	1	2	yes	—
269	C₄	C₂×#4018	1	1	yes	characteristic
270	C₄	C₂×D₅×M₁₆	1	1	yes	characteristic
271	C₂	—	2	1	no	—
272	C₂	—	2	2	no	—
273	C₂	—	20	2	no	—
274	C₂	C₂×#64994	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#4018
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₅×#2110	C₂
socle	C₂×C₅	C₂×#432
prime radical	C₂×#709	C₂×C₄
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	#2110	D₅
hypocenter	C₅	#33428
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₈ ◁ #405 ◁ #2110

Complements

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

N	H ≅ G/N	type
2. C₅×#2110	C₂	semidirect
4. C₂×#64731	C₂	semidirect
5. #10600677	C₂	semidirect
6. #10599882	C₂	semidirect
7. #10599881	C₂	semidirect
68. #2110	D₅	semidirect
258. C₅	#33428	semidirect

Maximal subgroups

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

subgroup	index
#10610280	2
C₅×#2110	2
#10602228	2
C₂×#64731	2
#10600677	2
#10599882	2
#10599881	2
#33428	5