Subgroups — #12251951

Order 1280.

818 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#65196	C₂	1	2	yes	maximal
2	C₄×#4047	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#10602212	C₂	1	2	yes	maximal, complemented
5	C₂×#64681	C₂	1	1	yes	maximal, characteristic, complemented
6	#10599864	C₂	1	2	yes	maximal, complemented
7	D₅×#346	C₂	1	2	yes	maximal, complemented
8	#10599844	C₂	1	4	yes	maximal, complemented
9	C₂×#4047	C₄	1	1	yes	characteristic
10	C₂×#4047	C₂²	1	1	yes	characteristic
11	C₄×#709	C₄	1	1	yes	characteristic
12	C₂×C₅×M₃₂	C₂²	1	2	yes	—
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	#65390	C₄	1	1	yes	characteristic
15	#65390	C₂²	1	1	yes	characteristic
16	C₂²×#672	C₂²	1	2	yes	—
17	C₄×#681	C₄	1	1	yes	characteristic
18	#65196	C₄	1	16	yes	complemented
19	C₂×#3930	C₂²	1	4	yes	—
20	C₂×D₅×C₁₆	C₂²	1	4	yes	—
21	#64978	C₄	1	2	yes	—
22	#64978	C₂²	1	2	yes	—
23	C₄×#676	C₄	1	2	yes	—
24	C₄×C₈×D₅	C₂²	1	2	yes	—
25	C₅×#346	C₂²	1	2	yes	—
26	C₄×C₅×C₁₆	C₂²	1	2	yes	—
27	#64722	C₂²	1	4	yes	—
28	C₁₆×Dic₅	C₂²	1	4	yes	—
29	#64681	C₂²	1	2	yes	—
30	#64681	C₂²	1	1	yes	characteristic
31	#64681	C₂²	1	1	yes	characteristic
32	#33326	—	5	1	no	maximal, Sylow
33	C₂×#709	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
37	C₄×C₂²⋊D₅	C₈	1	4	yes	—
38	#4072	C₂×C₄	1	1	yes	characteristic
39	C₂×#681	C₂×C₄	1	1	yes	characteristic
40	C₂×#681	C₂×C₄	1	1	yes	characteristic
41	#4047	C₈	1	8	yes	—
42	#4047	C₂×C₄	1	8	yes	—
43	C₂×#676	C₂×C₄	1	2	yes	—
44	C₂×#676	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
46	C₂×C₈×D₅	C₂³	1	2	yes	—
47	C₄×D₂₀	C₈	1	2	yes	—
48	#4018	C₂×C₄	1	2	yes	—
49	C₄²×D₅	C₂×C₄	1	2	yes	—
50	C₄×Dic₁₀	C₈	1	2	yes	—
51	C₅×M₃₂	C₂×C₄	1	8	yes	complemented
52	C₂×C₅×C₁₆	C₂³	1	4	yes	—
53	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
54	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
55	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
56	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
57	#3947	C₂×C₄	1	2	yes	—
58	#3947	C₂×C₄	1	2	yes	—
59	C₈×Dic₅	C₂×C₄	1	2	yes	—
60	C₈×Dic₅	C₂³	1	2	yes	—
61	C₂×#672	C₂×C₄	1	8	yes	complemented
62	C₂×#672	C₂³	1	2	yes	—
63	C₂×#672	C₂³	1	2	yes	—
64	#3935	C₂×C₄	1	2	yes	—
65	C₄×#168	C₂×C₄	1	2	yes	—
66	#3930	C₂×C₄	1	16	yes	complemented
67	D₅×C₁₆	C₂×C₄	1	16	yes	complemented
68	C₂×#504	—	5	2	no	—
69	C₄×#131	—	5	1	no	—
70	#2108	—	5	2	no	—
71	#2108	—	5	4	no	—
72	C₄×M₃₂	—	5	2	no	—
73	C₄×M₃₂	D₅	1	1	yes	characteristic, complemented
74	C₂×#346	—	5	1	no	—
75	C₂×#346	—	5	2	no	—
76	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
77	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₂²⋊D₅	C₂×C₈	1	2	yes	—
79	#709	C₄²	1	4	yes	—
80	#709	C₂×C₈	1	4	yes	—
81	C₂×D₂₀	C₂×C₈	1	1	yes	characteristic
82	C₂×C₄×D₅	C₂×C₈	1	4	yes	—
83	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
84	C₂×Dic₁₀	C₂×C₈	1	1	yes	characteristic
85	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
86	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
88	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
91	C₄²×C₅	C₂×C₈	1	2	yes	—
92	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
94	C₂²⋊Dic₅	C₂×C₈	1	2	yes	—
95	#685	C₂×C₈	1	4	yes	—
96	C₄⋊Dic₅	C₂×C₈	1	2	yes	—
97	#683	C₂×C₈	1	4	yes	—
98	C₄×Dic₅	C₂×C₈	1	4	yes	—
99	C₄×Dic₅	C₂²×C₄	1	2	yes	—
100	#681	C₄²	1	4	yes	—
101	#681	C₂×C₈	1	4	yes	—
102	C₂×#168	C₂²×C₄	1	2	yes	—
103	C₂×#168	C₂²×C₄	1	2	yes	—
104	#676	C₄²	1	8	yes	—
105	#676	C₂×C₈	1	8	yes	—
106	C₈×D₅	C₂×C₈	1	8	yes	—
107	C₈×D₅	C₂²×C₄	1	8	yes	—
108	C₅×C₁₆	C₂²×C₄	1	8	yes	—
109	#672	C₂²×C₄	1	4	yes	—
110	#672	C₂²×C₄	1	4	yes	—
111	C₂×#131	—	5	1	no	—
112	C₂×#131	—	5	1	no	—
113	C₄×#41	—	5	1	no	—
114	#504	—	5	16	no	—
115	C₂×M₃₂	—	5	4	no	—
116	C₂×M₃₂	C₂×D₅	1	2	yes	—
117	C₂²×C₁₆	—	5	2	no	—
118	C₂²×C₁₆	—	5	4	no	—
119	#405	—	5	1	no	—
120	#405	—	5	1	no	—
121	#405	—	5	2	no	—
122	#405	—	5	2	no	—
123	C₄×M₁₆	—	5	1	no	—
124	C₄×M₁₆	—	5	2	no	—
125	C₂×C₄×C₈	—	5	2	no	—
126	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
127	#346	—	5	4	no	—
128	#346	—	5	1	no	—
129	#346	—	5	1	no	—
130	#346	—	5	2	no	—
131	#346	C₂×D₅	1	2	yes	—
132	C₄×C₁₆	—	5	4	no	—
133	C₄×C₁₆	C₂×D₅	1	2	yes	—
134	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
135	C₂²×D₅	C₂²×C₈	1	2	yes	—
136	C₂×C₄×C₅	—	4	2	no	—
137	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
138	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
139	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
140	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
141	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
143	C₂²⋊D₅	C₄×C₈	1	8	yes	—
144	C₂×Dic₅	C₂²×C₈	1	4	yes	—
145	C₂×Dic₅	C₂²×C₈	1	2	yes	—
146	D₂₀	C₄×C₈	1	4	yes	—
147	C₄×D₅	—	4	4	no	—
148	C₄×D₅	C₂×C₄²	1	4	yes	—
149	C₄×D₅	C₂²×C₈	1	4	yes	—
150	Dic₁₀	C₄×C₈	1	4	yes	—
151	C₅×C₈	C₂×C₄²	1	2	yes	—
152	C₅×C₈	C₂²×C₈	1	2	yes	—
153	C₅×C₈	C₂²×C₈	1	2	yes	—
154	C₅×C₈	C₂³×C₄	1	2	yes	—
155	#168	C₂×C₄²	1	4	yes	—
156	#168	C₂²×C₈	1	4	yes	—
157	C₂×#41	—	5	1	no	—
158	#131	—	5	8	no	—
159	#131	—	5	8	no	—
160	C₂×M₁₆	—	5	1	no	—
161	C₂×M₁₆	—	5	1	no	—
162	C₂×M₁₆	—	5	2	no	—
163	C₂×M₁₆	—	5	2	no	—
164	C₂²×C₈	—	5	2	no	—
165	C₂²×C₈	—	5	2	no	—
166	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
167	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
168	C₄×Q₈	—	5	2	no	—
169	C₄×D₄	—	5	4	no	—
170	C₄×D₄	—	5	2	no	—
171	#117	—	5	1	no	—
172	#117	—	5	2	no	—
173	C₂×C₄²	—	5	2	no	—
174	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
175	M₃₂	—	5	16	no	—
176	M₃₂	C₄×D₅	1	8	yes	complemented
177	C₂×C₁₆	—	5	8	no	—
178	C₂×C₁₆	—	5	2	no	—
179	C₂×C₁₆	—	5	2	no	—
180	C₂×C₁₆	—	5	16	no	—
181	C₂×C₁₆	C₂²×D₅	1	4	yes	—
182	#97	—	5	2	no	—
183	#97	—	5	2	no	—
184	#97	—	5	2	no	—
185	C₄×C₈	—	5	2	no	—
186	C₄×C₈	—	5	2	no	—
187	C₄×C₈	—	5	2	no	—
188	C₄×C₈	C₄×D₅	1	1	yes	characteristic
189	C₄×C₈	C₄×D₅	1	1	yes	characteristic
190	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
191	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
192	C₂²×C₅	—	2	2	no	—
193	C₂²×C₅	C₂×C₄×C₈	1	2	yes	—
194	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
195	C₂×D₅	—	2	4	no	—
196	C₂×D₅	C₂×C₄×C₈	1	4	yes	—
197	C₄×C₅	—	2	2	no	—
198	C₄×C₅	—	2	2	no	—
199	C₄×C₅	C₂×C₄×C₈	1	2	yes	—
200	C₄×C₅	C₂²×C₄²	1	1	yes	characteristic
201	C₄×C₅	C₂³×C₈	1	1	yes	characteristic
202	Dic₅	—	2	4	no	—
203	Dic₅	C₂×C₄×C₈	1	4	yes	—
204	#41	—	5	4	no	—
205	#41	—	5	4	no	—
206	C₂×Q₈	—	5	1	no	—
207	C₂×D₄	—	5	2	no	—
208	C₂×D₄	—	5	1	no	—
209	C₂²×C₄	—	5	2	no	—
210	C₂²×C₄	—	5	4	no	—
211	C₂²×C₄	C₈×D₅	1	2	yes	—
212	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
213	M₁₆	—	5	4	no	—
214	M₁₆	—	5	4	no	—
215	M₁₆	—	5	8	no	—
216	M₁₆	—	5	8	no	—
217	C₂×C₈	—	5	2	no	—
218	C₂×C₈	—	5	2	no	—
219	C₂×C₈	—	5	8	no	—
220	C₂×C₈	—	5	8	no	—
221	C₂×C₈	C₈×D₅	1	4	yes	—
222	C₂×C₈	C₂×C₄×D₅	1	4	yes	—
223	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
224	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
225	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
226	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
227	C₄⋊C₄	—	5	4	no	—
228	C₄⋊C₄	—	5	2	no	—
229	C₂²⋊C₄	—	5	2	no	—
230	C₂²⋊C₄	—	5	4	no	—
231	C₄²	—	5	2	no	—
232	C₄²	—	5	4	no	—
233	C₄²	C₈×D₅	1	2	yes	—
234	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
235	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
236	C₁₆	—	5	4	no	—
237	C₁₆	—	5	4	no	—
238	C₁₆	C₂×C₄×D₅	1	8	yes	—
239	C₂×C₅	—	2	2	no	—
240	C₂×C₅	C₂²×C₄×C₈	1	1	yes	characteristic
241	C₂×C₅	#3407	1	2	yes	—
242	D₅	—	2	4	no	—
243	C₂³	—	5	2	no	—
244	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
245	Q₈	—	5	4	no	—
246	D₄	—	5	8	no	—
247	D₄	—	5	4	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	5	2	no	—
250	C₂×C₄	—	5	4	no	—
251	C₂×C₄	—	5	4	no	—
252	C₂×C₄	—	5	4	no	—
253	C₂×C₄	—	20	4	no	—
254	C₂×C₄	C₄²×D₅	1	2	yes	—
255	C₂×C₄	C₂×C₈×D₅	1	2	yes	—
256	C₂×C₄	C₂×C₈×D₅	1	2	yes	—
257	C₂×C₄	C₂×C₈×D₅	1	2	yes	—
258	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
260	C₈	—	5	4	no	—
261	C₈	—	5	4	no	—
262	C₈	C₄²×D₅	1	2	yes	—
263	C₈	C₂×C₈×D₅	1	2	yes	—
264	C₈	C₂×C₈×D₅	1	2	yes	—
265	C₈	C₂²×C₄×D₅	1	2	yes	—
266	C₅	#33326	1	1	yes	characteristic, complemented, Sylow
267	C₂²	—	2	2	no	—
268	C₂²	—	5	4	no	—
269	C₂²	—	10	4	no	—
270	C₂²	C₄×C₈×D₅	1	2	yes	—
271	C₂²	C₂²×C₈×D₅	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	5	4	no	—
275	C₄	—	10	4	no	—
276	C₄	C₄×C₈×D₅	1	2	yes	—
277	C₄	C₂×C₄²×D₅	1	1	yes	characteristic
278	C₄	C₂²×C₈×D₅	1	1	yes	characteristic
279	C₂	—	2	2	no	—
280	C₂	—	10	4	no	—
281	C₂	C₂×C₄×C₈×D₅	1	1	yes	characteristic
282	C₂	#10613738	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅
commutator	C₂×C₅	C₂²×C₄×C₈
Fitting	C₄×C₅×M₃₂	C₂
socle	C₂²×C₅	C₂³×C₈
prime radical	C₂×#709	C₂×C₄
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	C₄×M₃₂	D₅
hypocenter	C₅	#33326
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. C₄×C₅×M₃₂	C₂	semidirect
4. #10602212	C₂	semidirect
5. C₂×#64681	C₂	semidirect
6. #10599864	C₂	semidirect
7. D₅×#346	C₂	semidirect
8. #10599844	C₂	semidirect
18. #65196	C₄	semidirect
51. C₅×M₃₂	C₂×C₄	semidirect
61. C₂×#672	C₂×C₄	semidirect
66. #3930	C₂×C₄	semidirect
67. D₅×C₁₆	C₂×C₄	semidirect
73. C₄×M₃₂	D₅	semidirect
176. M₃₂	C₄×D₅	semidirect
266. C₅	#33326	semidirect

Maximal subgroups

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

subgroup	index
C₂×#65196	2
C₄×#4047	2
C₄×C₅×M₃₂	2
#10602212	2
C₂×#64681	2
#10599864	2
D₅×#346	2
#10599844	2
#33326	5