Subgroups — #12014197

Order 1280.

570 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×#3951	C₂	1	1	yes	maximal, characteristic
3	C₂×#65395	C₂	1	1	yes	maximal, characteristic
4	#10598298	C₂	1	4	yes	maximal, complemented
5	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₂×#4072	—	2	1	no	—
7	C₂²×#681	—	2	1	no	—
8	C₂×C₅×M₃₂	—	2	1	no	—
9	C₂×C₅×M₃₂	—	2	1	no	—
10	C₂×C₅×M₃₂	—	2	2	no	—
11	C₂×C₅×M₃₂	C₄	1	4	yes	complemented
12	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
13	C₂×#3951	—	2	1	no	—
14	C₂×#3951	—	2	2	no	—
15	C₂×#3951	C₄	1	4	yes	complemented
16	C₂×#3951	C₂²	1	2	yes	—
17	#65395	—	2	1	no	—
18	#65395	—	2	1	no	—
19	#65395	C₄	1	2	yes	—
20	#65395	C₂²	1	2	yes	—
21	C₂×#3950	C₄	1	2	yes	—
22	C₂×#3949	—	2	1	no	—
23	#12561	—	5	1	no	maximal, Sylow
24	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
25	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
28	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
29	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
30	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
31	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	D₄	1	2	yes	—
33	C₂²×C₅×C₈	Q₈	1	2	yes	—
34	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₂×C₂²⋊Dic₅	—	4	1	no	—
36	#4072	—	2	2	no	—
37	#4072	—	2	2	no	—
38	#4072	—	4	1	no	—
39	#4072	—	4	1	no	—
40	C₂×C₄⋊Dic₅	—	2	2	no	—
41	C₂×C₄×Dic₅	—	4	1	no	—
42	C₂×#681	—	2	4	no	—
43	C₂×#681	—	2	2	no	—
44	C₂×#681	—	4	1	no	—
45	C₂×#681	—	4	2	no	—
46	C₂²×#168	—	4	1	no	—
47	C₅×M₃₂	—	2	4	no	—
48	C₅×M₃₂	—	2	4	no	—
49	C₂×C₅×C₁₆	—	2	2	no	—
50	C₂×C₅×C₁₆	—	2	2	no	—
51	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	complemented
52	#3951	—	2	4	no	—
53	#3951	—	2	2	no	—
54	#3951	C₂×C₄	1	4	yes	complemented
55	#3950	—	2	2	no	—
56	#3950	C₂×C₄	1	4	yes	—
57	#3949	—	2	2	no	—
58	#3949	—	2	2	no	—
59	C₂²×M₃₂	D₅	1	1	yes	characteristic
60	C₂²×#108	—	5	1	no	—
61	C₂×#427	—	5	1	no	—
62	#1382	—	5	4	no	—
63	C₂⁴×C₅	SD₁₆	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
69	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
70	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
71	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
72	C₂²×C₄×C₅	Q₁₆	1	2	yes	—
73	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
74	C₂²×Dic₅	—	4	2	no	—
75	C₂×C₅×C₈	C₄²	1	2	yes	—
76	C₂×C₅×C₈	C₄²	1	2	yes	—
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	2	yes	—
81	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
82	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
83	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
84	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
85	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
86	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
87	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
91	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
92	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
93	C₂²⋊Dic₅	—	4	2	no	—
94	C₂²⋊Dic₅	—	4	2	no	—
95	C₄⋊Dic₅	—	2	4	no	—
96	C₄⋊Dic₅	—	4	2	no	—
97	C₄×Dic₅	—	4	2	no	—
98	C₄×Dic₅	—	4	2	no	—
99	#681	—	2	4	no	—
100	#681	—	4	4	no	—
101	#681	—	4	2	no	—
102	C₂×#168	—	4	4	no	—
103	C₂×#168	—	4	2	no	—
104	C₅×C₁₆	—	2	4	no	—
105	C₂²×M₁₆	—	10	1	no	—
106	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
107	C₂×#117	—	10	1	no	—
108	C₂×M₃₂	—	2	1	no	—
109	C₂×M₃₂	—	2	1	no	—
110	C₂×M₃₂	—	2	2	no	—
111	C₂×M₃₂	Dic₅	1	4	yes	complemented
112	C₂²×C₁₆	C₂×D₅	1	2	yes	—
113	C₂×#108	—	5	4	no	—
114	C₂×#108	—	5	2	no	—
115	C₂×#108	—	10	1	no	—
116	C₂×#108	—	10	2	no	—
117	#427	—	5	2	no	—
118	#427	—	5	2	no	—
119	#427	—	10	1	no	—
120	#427	—	10	1	no	—
121	C₂×#107	—	5	2	no	—
122	C₂×#106	—	10	1	no	—
123	C₂³×C₅	—	4	2	no	—
124	C₂³×C₅	#102	1	1	yes	characteristic
125	C₂³×C₅	#102	1	1	yes	characteristic
126	C₂³×C₅	#103	1	2	yes	—
127	C₂³×C₅	#106	1	2	yes	—
128	C₂³×C₅	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₄×C₅	#95	1	2	yes	—
130	C₂×C₄×C₅	#95	1	2	yes	—
131	C₂×C₄×C₅	#102	1	1	yes	characteristic
132	C₂×C₄×C₅	#102	1	1	yes	characteristic
133	C₂×C₄×C₅	#102	1	2	yes	—
134	C₂×C₄×C₅	#102	1	2	yes	—
135	C₂×C₄×C₅	#103	1	2	yes	—
136	C₂×C₄×C₅	#103	1	2	yes	—
137	C₂×C₄×C₅	#103	1	2	yes	—
138	C₂×C₄×C₅	#106	1	2	yes	—
139	C₂×C₄×C₅	#107	1	2	yes	—
140	C₂×C₄×C₅	#107	1	2	yes	—
141	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
147	C₂×Dic₅	—	4	4	no	—
148	C₂×Dic₅	—	8	4	no	—
149	C₅×C₈	#95	1	2	yes	—
150	C₅×C₈	#95	1	2	yes	—
151	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
152	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
153	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
155	#168	—	4	4	no	—
156	C₂³×C₄	D₂₀	1	1	yes	characteristic
157	C₂×M₁₆	—	10	4	no	—
158	C₂×M₁₆	—	10	2	no	—
159	C₂×M₁₆	—	20	2	no	—
160	C₂×M₁₆	—	20	1	no	—
161	C₂²×C₈	—	20	1	no	—
162	C₂²×C₈	Dic₁₀	1	2	yes	—
163	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
164	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
165	C₂²×C₈	C₄×D₅	1	2	yes	—
166	C₂²×C₈	D₂₀	1	1	yes	characteristic
167	C₂²×C₈	C₂×Dic₅	1	2	yes	—
168	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
169	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
170	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
171	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
172	#117	—	10	2	no	—
173	#117	—	10	2	no	—
174	#117	—	20	1	no	—
175	#117	—	20	1	no	—
176	C₂×C₄⋊C₄	—	10	2	no	—
177	C₂×C₂²⋊C₄	—	20	1	no	—
178	C₂×C₄²	—	20	1	no	—
179	M₃₂	—	2	4	no	—
180	M₃₂	—	2	4	no	—
181	C₂×C₁₆	—	2	2	no	—
182	C₂×C₁₆	—	2	2	no	—
183	C₂×C₁₆	C₂×Dic₅	1	4	yes	complemented
184	#108	—	5	4	no	—
185	#108	—	10	4	no	—
186	#108	—	10	2	no	—
187	#107	—	5	4	no	—
188	#107	—	10	2	no	—
189	#106	—	10	2	no	—
190	#106	—	10	2	no	—
191	C₂²×C₅	—	2	2	no	—
192	C₂²×C₅	—	2	2	no	—
193	C₂²×C₅	—	2	2	no	—
194	C₂²×C₅	—	4	2	no	—
195	C₂²×C₅	—	4	2	no	—
196	C₂²×C₅	#340	1	2	yes	—
197	C₂²×C₅	#340	1	2	yes	—
198	C₂²×C₅	C₂×#102	1	1	yes	characteristic
199	C₂²×C₅	C₂×#103	1	1	yes	characteristic
200	C₂²×C₅	C₂×#106	1	1	yes	characteristic
201	C₄×C₅	#340	1	2	yes	—
202	C₄×C₅	#340	1	2	yes	—
203	C₄×C₅	C₂×#95	1	1	yes	characteristic
204	C₄×C₅	C₂×#102	1	1	yes	characteristic
205	C₄×C₅	C₂×#103	1	1	yes	characteristic
206	C₄×C₅	C₂×#107	1	1	yes	characteristic
207	Dic₅	—	8	4	no	—
208	C₂⁴	#677	1	1	yes	characteristic
209	C₂²×C₄	—	20	2	no	—
210	C₂²×C₄	#677	1	1	yes	characteristic
211	C₂²×C₄	#677	1	2	yes	—
212	C₂²×C₄	D₄₀	1	1	yes	characteristic
213	C₂²×C₄	D₄₀	1	1	yes	characteristic
214	C₂²×C₄	Dic₂₀	1	2	yes	—
215	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
216	C₂²×C₄	#685	1	1	yes	characteristic
217	C₂²×C₄	#685	1	1	yes	characteristic
218	C₂²×C₄	#685	1	2	yes	—
219	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
220	M₁₆	—	10	4	no	—
221	M₁₆	—	20	4	no	—
222	M₁₆	—	20	2	no	—
223	C₂×C₈	—	20	4	no	—
224	C₂×C₈	—	20	2	no	—
225	C₂×C₈	C₄×Dic₅	1	2	yes	—
226	C₂×C₈	C₄×Dic₅	1	2	yes	—
227	C₂×C₈	#683	1	2	yes	—
228	C₂×C₈	#683	1	2	yes	—
229	C₂×C₈	#683	1	2	yes	—
230	C₂×C₈	#683	1	2	yes	—
231	C₂×C₈	C₄⋊Dic₅	1	2	yes	—
232	C₂×C₈	#685	1	1	yes	characteristic
233	C₂×C₈	#685	1	1	yes	characteristic
234	C₂×C₈	#685	1	2	yes	—
235	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
236	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
237	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
238	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
239	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
240	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
241	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
242	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
243	C₄⋊C₄	—	10	4	no	—
244	C₄⋊C₄	—	20	2	no	—
245	C₂²⋊C₄	—	20	2	no	—
246	C₂²⋊C₄	—	20	2	no	—
247	C₄²	—	20	2	no	—
248	C₄²	—	20	2	no	—
249	C₁₆	—	2	4	no	—
250	C₂×C₅	—	2	2	no	—
251	C₂×C₅	—	2	2	no	—
252	C₂×C₅	—	2	1	no	—
253	C₂×C₅	—	2	1	no	—
254	C₂×C₅	C₂×#340	1	1	yes	characteristic
255	C₂×C₅	#2143	1	2	yes	—
256	C₂³	—	4	2	no	—
257	C₂³	#3948	1	2	yes	—
258	C₂³	#3949	1	2	yes	—
259	C₂³	#3953	1	1	yes	characteristic
260	C₂³	#3953	1	1	yes	characteristic
261	C₂³	C₂×#677	1	1	yes	characteristic
262	C₂×C₄	—	20	4	no	—
263	C₂×C₄	—	40	4	no	—
264	C₂×C₄	#3948	1	2	yes	—
265	C₂×C₄	#3948	1	2	yes	—
266	C₂×C₄	#3948	1	2	yes	—
267	C₂×C₄	#3949	1	2	yes	—
268	C₂×C₄	#3950	1	2	yes	—
269	C₂×C₄	#3950	1	2	yes	—
270	C₂×C₄	#3953	1	1	yes	characteristic
271	C₂×C₄	#3953	1	1	yes	characteristic
272	C₂×C₄	#3953	1	2	yes	—
273	C₂×C₄	#3953	1	2	yes	—
274	C₂×C₄	#3963	1	2	yes	—
275	C₂×C₄	#3963	1	2	yes	—
276	C₂×C₄	C₂×#677	1	1	yes	characteristic
277	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
278	C₂×C₄	C₂×Dic₂₀	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
280	C₂×C₄	C₂×#685	1	1	yes	characteristic
281	C₂×C₄	C₂×#685	1	1	yes	characteristic
282	C₈	—	20	4	no	—
283	C₈	#3963	1	2	yes	—
284	C₈	#3963	1	2	yes	—
285	C₈	C₂×C₄×Dic₅	1	1	yes	characteristic
286	C₈	C₂×#683	1	1	yes	characteristic
287	C₈	C₂×#683	1	1	yes	characteristic
288	C₈	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
289	C₅	#12561	1	1	yes	characteristic, complemented, Sylow
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	4	2	no	—
294	C₂²	—	4	2	no	—
295	C₂²	#64771	1	2	yes	—
296	C₂²	#64771	1	2	yes	—
297	C₂²	C₂×#3948	1	1	yes	characteristic
298	C₂²	C₂×#3949	1	1	yes	characteristic
299	C₂²	C₂×#3953	1	1	yes	characteristic
300	C₄	—	40	4	no	—
301	C₄	#64771	1	2	yes	—
302	C₄	#64771	1	2	yes	—
303	C₄	C₂×#3948	1	1	yes	characteristic
304	C₄	C₂×#3950	1	1	yes	characteristic
305	C₄	C₂×#3953	1	1	yes	characteristic
306	C₄	C₂×#3963	1	1	yes	characteristic
307	C₂	—	2	2	no	—
308	C₂	—	2	2	no	—
309	C₂	—	2	1	no	—
310	C₂	—	2	1	no	—
311	C₂	#10602241	1	2	yes	—
312	C₂	C₂×#64771	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄₀
commutator	C₅×C₈	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂²×C₅×M₃₂	C₂
socle	C₂²×C₅	C₂×#102
prime radical	C₂⁴×C₅	SD₁₆
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂²×M₃₂	D₅
hypocenter	C₅	#12561
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
4. #10598298	C₂	semidirect
11. C₂×C₅×M₃₂	C₄	semidirect
15. C₂×#3951	C₄	semidirect
51. C₂×C₅×C₁₆	C₂×C₄	semidirect
54. #3951	C₂×C₄	semidirect
111. C₂×M₃₂	Dic₅	semidirect
183. C₂×C₁₆	C₂×Dic₅	semidirect
289. C₅	#12561	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅×M₃₂	2
C₂²×#3951	2
C₂×#65395	2
#10598298	2
#12561	5