Subgroups — #12252433

Order 1280.

520 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2875	C₂	1	1	yes	maximal, characteristic
2	#10602800	C₂	1	2	yes	maximal
3	#10600679	C₂	1	4	yes	maximal, complemented
4	#10600678	C₂	1	2	yes	maximal, complemented
5	#10600657	C₂	1	4	yes	maximal, complemented
6	C₂×#64686	C₂	1	2	yes	maximal, complemented
7	C₂×C₅×#131	C₄	1	1	yes	characteristic
8	C₂×C₅×#131	C₂²	1	1	yes	characteristic
9	C₄×C₅×#41	C₄	1	1	yes	characteristic
10	C₅×#405	C₄	1	2	yes	—
11	C₅×#405	C₄	1	1	yes	characteristic
12	C₅×#405	C₂²	1	2	yes	—
13	C₅×#405	C₂²	1	1	yes	characteristic
14	C₄×C₅×M₁₆	C₄	1	1	yes	characteristic
15	C₄×C₅×M₁₆	C₂²	1	2	yes	—
16	C₂×C₅×#97	C₄	1	2	yes	—
17	C₂×C₅×#97	C₂²	1	1	yes	characteristic
18	C₂×#3944	C₂²	1	4	yes	—
19	#64773	C₂²	1	8	yes	—
20	#64686	C₂²	1	4	yes	—
21	#64686	C₂²	1	4	yes	—
22	#64683	C₂²	1	8	yes	—
23	#33426	—	5	1	no	maximal, Sylow
24	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
25	C₅×#131	—	2	4	no	—
26	C₅×#131	—	2	4	no	—
27	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
29	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂×C₅×M₁₆	C₂³	1	2	yes	—
31	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
32	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
33	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₄×C₅×Q₈	—	2	1	no	—
36	C₄×C₅×D₄	—	2	2	no	—
37	C₄×C₅×D₄	—	2	1	no	—
38	C₅×#117	C₂×C₄	1	2	yes	—
39	C₅×#117	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
42	C₅×#97	C₂×C₄	1	1	yes	characteristic
43	C₅×#97	C₂×C₄	1	2	yes	—
44	C₅×#97	C₂×C₄	1	2	yes	—
45	C₅×#97	C₂×C₄	1	2	yes	—
46	C₅×#97	C₂×C₄	1	1	yes	characteristic
47	C₅×#97	C₂³	1	1	yes	characteristic
48	C₅×#97	C₂³	1	1	yes	characteristic
49	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
50	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
51	C₄×C₅×C₈	C₂³	1	2	yes	—
52	#3944	—	2	4	no	—
53	#3944	C₂³	1	8	yes	—
54	C₂×#672	—	2	4	no	—
55	#2875	D₅	1	1	yes	characteristic
56	#2166	—	5	4	no	—
57	#2165	—	5	4	no	—
58	#2153	—	5	2	no	—
59	#2115	—	5	2	no	—
60	C₂×#347	—	5	2	no	—
61	C₅×#41	—	2	2	no	—
62	C₅×#41	—	2	2	no	—
63	C₂×C₅×Q₈	M₁₆	1	1	yes	characteristic
64	C₂×C₅×D₄	M₁₆	1	2	yes	—
65	C₂×C₅×D₄	M₁₆	1	1	yes	characteristic
66	C₂²×C₄×C₅	—	2	2	no	—
67	C₂²×C₄×C₅	—	2	1	no	—
68	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₅×M₁₆	—	2	2	no	—
71	C₅×M₁₆	—	2	4	no	—
72	C₅×M₁₆	—	2	4	no	—
73	C₅×M₁₆	M₁₆	1	4	yes	—
74	C₂×C₅×C₈	—	2	2	no	—
75	C₂×C₅×C₈	—	2	4	no	—
76	C₂×C₅×C₈	—	2	4	no	—
77	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
80	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
82	C₂×C₅×C₈	#41	1	4	yes	—
83	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
84	C₅×C₄⋊C₄	—	2	2	no	—
85	C₅×C₄⋊C₄	—	2	1	no	—
86	C₅×C₂²⋊C₄	—	2	2	no	—
87	C₅×C₂²⋊C₄	—	2	1	no	—
88	C₄²×C₅	—	2	2	no	—
89	C₄²×C₅	—	2	1	no	—
90	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₅	C₂²×C₄	1	2	yes	—
93	#672	—	2	8	no	—
94	C₂×#131	Dic₅	1	1	yes	characteristic
95	C₂×#131	C₂×D₅	1	1	yes	characteristic
96	C₄×#41	Dic₅	1	1	yes	characteristic
97	C₂×M₃₂	—	5	4	no	—
98	#405	Dic₅	1	2	yes	—
99	#405	Dic₅	1	1	yes	characteristic
100	#405	C₂×D₅	1	2	yes	—
101	#405	C₂×D₅	1	1	yes	characteristic
102	C₄×M₁₆	Dic₅	1	1	yes	characteristic
103	C₄×M₁₆	C₂×D₅	1	2	yes	—
104	C₂×#97	Dic₅	1	2	yes	—
105	C₂×#97	C₂×D₅	1	1	yes	characteristic
106	#364	—	5	8	no	—
107	#349	—	5	8	no	—
108	#347	—	5	4	no	—
109	#347	—	5	4	no	—
110	C₂³×C₅	C₂×M₁₆	1	2	yes	—
111	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
112	C₅×Q₈	—	2	2	no	—
113	C₅×D₄	—	2	4	no	—
114	C₅×D₄	—	2	2	no	—
115	C₂×C₄×C₅	—	2	2	no	—
116	C₂×C₄×C₅	—	2	2	no	—
117	C₂×C₄×C₅	—	2	1	no	—
118	C₂×C₄×C₅	—	2	1	no	—
119	C₂×C₄×C₅	—	2	2	no	—
120	C₂×C₄×C₅	—	8	1	no	—
121	C₂×C₄×C₅	—	8	2	no	—
122	C₂×C₄×C₅	#117	1	2	yes	—
123	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
124	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
125	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
126	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
127	C₅×C₈	—	2	2	no	—
128	C₅×C₈	—	2	1	no	—
129	C₅×C₈	—	2	1	no	—
130	C₅×C₈	#117	1	2	yes	—
131	C₅×C₈	C₂×M₁₆	1	4	yes	—
132	C₅×C₈	C₂×#41	1	2	yes	—
133	C₂×#41	C₂×Dic₅	1	1	yes	characteristic
134	#131	—	2	4	no	—
135	#131	—	2	4	no	—
136	C₂×M₁₆	C₂×Dic₅	1	2	yes	—
137	C₂×M₁₆	C₂×Dic₅	1	1	yes	characteristic
138	C₂×M₁₆	C₂×Dic₅	1	1	yes	characteristic
139	C₂×M₁₆	C₂²×D₅	1	2	yes	—
140	C₂²×C₈	C₂×Dic₅	1	2	yes	—
141	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
142	C₂²×C₈	C₂×Dic₅	1	2	yes	—
143	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
144	C₄×Q₈	—	2	1	no	—
145	C₄×D₄	—	2	2	no	—
146	C₄×D₄	—	2	1	no	—
147	#117	C₂×Dic₅	1	2	yes	—
148	#117	C₂×Dic₅	1	1	yes	characteristic
149	C₂×C₄²	C₂×Dic₅	1	2	yes	—
150	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
151	M₃₂	—	5	8	no	—
152	M₃₂	—	10	4	no	—
153	C₂×C₁₆	—	10	4	no	—
154	#97	C₂×Dic₅	1	1	yes	characteristic
155	#97	C₂×Dic₅	1	2	yes	—
156	#97	C₂×Dic₅	1	2	yes	—
157	#97	C₂×Dic₅	1	2	yes	—
158	#97	C₂×Dic₅	1	1	yes	characteristic
159	#97	C₂²×D₅	1	1	yes	characteristic
160	#97	C₂²×D₅	1	1	yes	characteristic
161	C₄×C₈	C₂×Dic₅	1	2	yes	—
162	C₄×C₈	C₂×Dic₅	1	2	yes	—
163	C₄×C₈	C₂²×D₅	1	2	yes	—
164	C₂²×C₅	—	2	2	no	—
165	C₂²×C₅	—	4	1	no	—
166	C₂²×C₅	—	4	2	no	—
167	C₂²×C₅	#432	1	2	yes	—
168	C₂²×C₅	C₂²×M₁₆	1	1	yes	characteristic
169	C₄×C₅	—	2	2	no	—
170	C₄×C₅	—	4	1	no	—
171	C₄×C₅	—	4	2	no	—
172	C₄×C₅	—	4	1	no	—
173	C₄×C₅	#432	1	2	yes	—
174	C₄×C₅	C₂×#117	1	1	yes	characteristic
175	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
176	#41	—	2	2	no	—
177	#41	—	2	2	no	—
178	C₂×Q₈	#681	1	1	yes	characteristic
179	C₂×D₄	#681	1	2	yes	—
180	C₂×D₄	#681	1	1	yes	characteristic
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
184	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
185	M₁₆	—	2	2	no	—
186	M₁₆	—	2	4	no	—
187	M₁₆	—	2	4	no	—
188	M₁₆	#681	1	4	yes	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	#709	1	4	yes	—
193	C₂×C₈	C₂²×Dic₅	1	2	yes	—
194	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
195	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
196	C₂×C₈	C₂²×Dic₅	1	2	yes	—
197	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
198	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	2	1	no	—
201	C₂²⋊C₄	—	2	2	no	—
202	C₂²⋊C₄	—	2	1	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	2	1	no	—
205	C₄²	C₂²×Dic₅	1	1	yes	characteristic
206	C₄²	C₂²×Dic₅	1	2	yes	—
207	C₄²	C₂²×Dic₅	1	1	yes	characteristic
208	C₁₆	—	10	8	no	—
209	C₂×C₅	—	2	1	no	—
210	C₂×C₅	—	2	2	no	—
211	C₂×C₅	—	4	2	no	—
212	C₂×C₅	C₂×#432	1	1	yes	characteristic
213	C₂³	C₂×#681	1	2	yes	—
214	C₂³	C₂×#681	1	1	yes	characteristic
215	Q₈	—	2	2	no	—
216	D₄	—	2	4	no	—
217	D₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	8	1	no	—
224	C₂×C₄	—	8	2	no	—
225	C₂×C₄	C₂×#681	1	2	yes	—
226	C₂×C₄	C₂×#681	1	2	yes	—
227	C₂×C₄	C₂×#681	1	1	yes	characteristic
228	C₂×C₄	#4072	1	2	yes	—
229	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
230	C₈	—	2	1	no	—
231	C₈	—	2	1	no	—
232	C₈	—	2	2	no	—
233	C₈	C₂×#681	1	4	yes	—
234	C₈	#4072	1	2	yes	—
235	C₈	C₂×#709	1	2	yes	—
236	C₅	#33426	1	1	yes	characteristic, complemented, Sylow
237	C₂²	—	2	2	no	—
238	C₂²	—	4	1	no	—
239	C₂²	—	4	2	no	—
240	C₂²	#65215	1	2	yes	—
241	C₂²	C₂²×#681	1	1	yes	characteristic
242	C₄	—	2	2	no	—
243	C₄	—	4	1	no	—
244	C₄	—	4	1	no	—
245	C₄	—	4	2	no	—
246	C₄	#65215	1	2	yes	—
247	C₄	C₂²×#681	1	1	yes	characteristic
248	C₄	C₂×#4072	1	1	yes	characteristic
249	C₂	—	2	1	no	—
250	C₂	—	2	2	no	—
251	C₂	—	4	2	no	—
252	C₂	C₂×#65215	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#4072
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₅×#2875	C₂
socle	C₂×C₅	C₂×#432
prime radical	C₂×C₅×#41	C₂×C₄
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	#2875	D₅
hypocenter	C₅	#33426
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×#97 ◁ #2875

Complements

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

N	H ≅ G/N	type
3. #10600679	C₂	semidirect
4. #10600678	C₂	semidirect
5. #10600657	C₂	semidirect
6. C₂×#64686	C₂	semidirect
236. C₅	#33426	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2875	2
#10602800	2
#10600679	2
#10600678	2
#10600657	2
C₂×#64686	2
#33426	5