Subgroups — #12117773

Order 1280.

862 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10610847	C₂	1	2	yes	maximal, complemented
2	#10610837	C₂	1	2	yes	maximal
3	C₂×#64989	C₂	1	1	yes	maximal, characteristic
4	C₅×#1844	C₂	1	1	yes	maximal, characteristic, complemented
5	#10601385	C₂	1	2	yes	maximal, complemented
6	#10601365	C₂	1	2	yes	maximal, complemented
7	#10600572	C₂	1	1	yes	maximal, characteristic, complemented
8	#10600385	C₂	1	4	yes	maximal, complemented
9	C₂×#4050	C₂²	1	1	yes	characteristic
10	C₂×#4047	C₂²	1	2	yes	—
11	#65838	C₂²	1	4	yes	—
12	C₂×C₅×#106	C₂²	1	2	yes	—
13	C₂×C₅×#106	C₂²	1	2	yes	—
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₅×#384	C₂²	1	2	yes	—
16	C₂×#3951	C₂²	1	1	yes	characteristic
17	C₂×#3951	C₂²	1	1	yes	characteristic
18	#65395	C₂²	1	1	yes	characteristic
19	#65390	C₂²	1	2	yes	—
20	#65255	C₂²	1	4	yes	—
21	#65253	C₂²	1	4	yes	—
22	C₂×#3937	C₂²	1	2	yes	—
23	#65158	—	2	2	no	—
24	#65150	—	2	4	no	—
25	#65149	—	2	2	no	—
26	D₅×#106	—	2	2	no	—
27	#65142	—	2	2	no	—
28	#65141	—	2	4	no	—
29	#64989	C₄	1	8	yes	complemented
30	#64750	C₂²	1	4	yes	—
31	#64743	C₂²	1	2	yes	—
32	#22689	—	5	1	no	maximal, Sylow
33	C₂×#709	C₂³	1	2	yes	—
34	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₂×C₅×C₄⋊C₄	—	4	2	no	—
37	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
38	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
39	C₄×C₂²⋊D₅	—	4	4	no	—
40	#4072	C₂³	1	2	yes	—
41	C₂×#681	C₂³	1	2	yes	—
42	C₂×#681	C₂³	1	2	yes	—
43	C₂×Dic₂₀	D₄	1	1	yes	characteristic
44	#4050	C₂×C₄	1	4	yes	—
45	#4050	Q₈	1	4	yes	—
46	C₂×D₄₀	D₄	1	1	yes	characteristic
47	C₂×#677	D₄	1	2	yes	—
48	#4047	—	2	4	no	—
49	#4047	C₂×C₄	1	8	yes	complemented
50	C₂×#676	—	2	2	no	—
51	C₂×C₈×D₅	—	2	2	no	—
52	#4039	—	2	4	no	—
53	#4038	—	2	4	no	—
54	D₅×C₄⋊C₄	—	2	4	no	—
55	#4033	—	2	4	no	—
56	C₅×#106	—	2	2	no	—
57	C₅×#106	—	2	2	no	—
58	C₅×#106	—	2	2	no	—
59	C₅×#106	—	2	2	no	—
60	C₄×C₅×C₈	C₂×C₄	1	4	yes	complemented
61	#3951	—	2	2	no	—
62	#3951	C₂×C₄	1	4	yes	complemented
63	#3950	D₄	1	2	yes	—
64	#3949	D₄	1	2	yes	—
65	#3947	—	2	2	no	—
66	C₈×Dic₅	—	2	2	no	—
67	#3942	—	2	4	no	—
68	#3941	—	2	4	no	—
69	#3940	—	2	4	no	—
70	#3939	—	2	4	no	—
71	#3937	C₂×C₄	1	8	yes	complemented
72	C₂×#443	—	5	1	no	—
73	#2952	—	5	2	no	—
74	#2912	—	5	2	no	—
75	#1951	—	5	2	no	—
76	#1859	—	5	4	no	—
77	#1844	D₅	1	1	yes	characteristic, complemented
78	#1815	—	5	2	no	—
79	#1772	—	5	1	no	—
80	C₂²×C₄×C₅	—	2	1	no	—
81	C₂²×C₄×C₅	—	2	4	no	—
82	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
83	C₂×C₂²⋊D₅	—	2	2	no	—
84	#709	—	2	2	no	—
85	#709	—	2	2	no	—
86	#709	C₂²×C₄	1	4	yes	—
87	#709	C₂×Q₈	1	4	yes	—
88	C₂×D₂₀	C₂×D₄	1	2	yes	—
89	C₂×C₄×D₅	—	2	2	no	—
90	C₂×C₄×D₅	—	4	4	no	—
91	C₂×Dic₁₀	C₂×D₄	1	2	yes	—
92	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
93	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
94	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
95	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
96	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
98	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
99	C₂×C₅×C₈	#41	1	2	yes	—
100	C₅×C₄⋊C₄	—	2	4	no	—
101	C₅×C₄⋊C₄	—	2	4	no	—
102	C₅×C₄⋊C₄	—	8	4	no	—
103	C₄²×C₅	—	2	1	no	—
104	C₄²×C₅	C₂²×C₄	1	2	yes	—
105	C₂²⋊Dic₅	—	2	2	no	—
106	#685	—	4	4	no	—
107	C₄⋊Dic₅	C₂×D₄	1	4	yes	—
108	#683	—	4	4	no	—
109	C₄×Dic₅	—	2	2	no	—
110	C₄×Dic₅	—	4	4	no	—
111	#681	—	2	2	no	—
112	#681	—	2	2	no	—
113	#681	—	2	2	no	—
114	#681	C₂²×C₄	1	4	yes	—
115	C₂×#168	—	2	2	no	—
116	C₂×#168	—	2	2	no	—
117	Dic₂₀	C₄⋊C₄	1	4	yes	complemented
118	D₄₀	C₄⋊C₄	1	4	yes	complemented
119	#677	C₄⋊C₄	1	8	yes	complemented
120	#676	—	2	8	no	—
121	C₈×D₅	—	2	8	no	—
122	C₂×#135	—	5	1	no	—
123	C₂×#131	—	5	2	no	—
124	#520	—	5	4	no	—
125	#443	—	5	8	no	—
126	#442	—	10	2	no	—
127	#441	—	10	2	no	—
128	C₄×SD₁₆	—	10	4	no	—
129	C₂×#108	—	5	1	no	—
130	C₂×#108	—	5	1	no	—
131	#428	—	5	4	no	—
132	#428	—	10	4	no	—
133	#427	—	5	1	no	—
134	#427	—	10	2	no	—
135	C₂×#106	—	10	2	no	—
136	C₂×#106	C₂×D₅	1	2	yes	—
137	C₂×#106	C₂×D₅	1	2	yes	—
138	C₂×#104	—	5	2	no	—
139	#417	—	5	4	no	—
140	#405	—	5	2	no	—
141	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
142	#384	C₂×D₅	1	2	yes	—
143	#343	—	5	4	no	—
144	#339	—	5	2	no	—
145	C₂³×C₅	H(5,𝔽₂)	1	1	yes	characteristic
146	C₂²×D₅	—	2	2	no	—
147	C₂×C₄×C₅	—	2	4	no	—
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	—	4	4	no	—
150	C₂×C₄×C₅	—	8	4	no	—
151	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
155	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
156	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
157	C₂²⋊D₅	—	2	4	no	—
158	C₂²⋊D₅	—	2	4	no	—
159	C₂×Dic₅	—	2	2	no	—
160	C₂×Dic₅	—	2	4	no	—
161	D₂₀	—	2	2	no	—
162	D₂₀	C₂×C₄⋊C₄	1	4	yes	—
163	C₄×D₅	—	2	4	no	—
164	C₄×D₅	—	2	4	no	—
165	C₄×D₅	—	16	4	no	—
166	Dic₁₀	—	2	2	no	—
167	Dic₁₀	C₂×C₄⋊C₄	1	4	yes	—
168	C₅×C₈	C₂×C₄⋊C₄	1	2	yes	—
169	C₅×C₈	C₂×C₄⋊C₄	1	2	yes	—
170	C₅×C₈	C₄×D₄	1	2	yes	—
171	C₅×C₈	C₄×D₄	1	2	yes	—
172	#168	—	2	4	no	—
173	#168	—	4	2	no	—
174	C₂×#41	—	5	2	no	—
175	#135	—	5	4	no	—
176	#135	—	5	4	no	—
177	C₂×Q₁₆	—	5	1	no	—
178	C₂×SD₁₆	—	5	2	no	—
179	C₂×D₈	—	5	1	no	—
180	#131	—	5	8	no	—
181	#131	—	10	4	no	—
182	C₂×M₁₆	—	5	2	no	—
183	C₂×M₁₆	—	5	2	no	—
184	C₂×M₁₆	—	10	2	no	—
185	C₂²×C₈	—	10	2	no	—
186	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
187	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
188	C₄×Q₈	—	10	4	no	—
189	C₄×D₄	—	10	4	no	—
190	C₄×D₄	—	20	4	no	—
191	#117	—	5	2	no	—
192	#117	—	10	4	no	—
193	C₂×C₄⋊C₄	—	4	2	no	—
194	C₂×C₄⋊C₄	—	10	4	no	—
195	C₂×C₄⋊C₄	C₂²×D₅	1	4	yes	—
196	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
197	#108	—	5	4	no	—
198	#108	—	10	2	no	—
199	#107	—	5	2	no	—
200	#107	—	10	4	no	—
201	#106	—	2	2	no	—
202	#106	—	2	2	no	—
203	#106	—	2	2	no	—
204	#106	—	2	2	no	—
205	#106	—	5	2	no	—
206	#106	—	10	4	no	—
207	#104	—	5	8	no	—
208	#103	—	10	4	no	—
209	#102	—	10	4	no	—
210	#97	—	10	2	no	—
211	C₄×C₈	—	10	2	no	—
212	C₄×C₈	C₄×D₅	1	4	yes	complemented
213	C₂²×C₅	—	2	2	no	—
214	C₂²×C₅	#520	1	1	yes	characteristic
215	C₂²×C₅	D₄²	1	1	yes	characteristic
216	C₂²×C₅	#554	1	1	yes	characteristic
217	C₂×D₅	—	2	4	no	—
218	C₂×D₅	—	4	4	no	—
219	C₄×C₅	—	2	4	no	—
220	C₄×C₅	—	8	4	no	—
221	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
222	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
223	C₄×C₅	#547	1	1	yes	characteristic
224	C₄×C₅	D₄×Q₈	1	1	yes	characteristic
225	Dic₅	—	2	4	no	—
226	Dic₅	—	4	4	no	—
227	#41	—	5	4	no	—
228	#41	—	5	4	no	—
229	#41	—	10	2	no	—
230	#41	—	10	2	no	—
231	C₂×Q₈	—	5	2	no	—
232	C₂×D₄	—	5	2	no	—
233	C₂×D₄	—	10	2	no	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	10	2	no	—
237	C₂²×C₄	—	20	4	no	—
238	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
239	Q₁₆	—	5	4	no	—
240	SD₁₆	—	5	8	no	—
241	D₈	—	5	4	no	—
242	M₁₆	—	5	4	no	—
243	M₁₆	—	10	2	no	—
244	M₁₆	—	10	8	no	—
245	M₁₆	—	10	2	no	—
246	M₁₆	—	10	2	no	—
247	C₂×C₈	—	10	2	no	—
248	C₂×C₈	—	10	8	no	—
249	C₂×C₈	—	10	2	no	—
250	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
251	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
252	C₂×C₈	D₄×D₅	1	1	yes	characteristic
253	C₂×C₈	D₄×D₅	1	1	yes	characteristic
254	C₂×C₈	D₄×D₅	1	1	yes	characteristic
255	C₂×C₈	D₄×D₅	1	1	yes	characteristic
256	C₂×C₈	Q₈×D₅	1	2	yes	—
257	C₂×C₈	#713	1	2	yes	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	5	4	no	—
261	C₄⋊C₄	—	8	4	no	—
262	C₄⋊C₄	—	20	4	no	—
263	C₂²⋊C₄	—	10	2	no	—
264	C₂²⋊C₄	—	20	4	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	10	2	no	—
267	C₄²	—	20	4	no	—
268	C₄²	C₂×C₄×D₅	1	2	yes	—
269	C₂×C₅	—	2	1	no	—
270	C₂×C₅	—	2	1	no	—
271	C₂×C₅	D₄×C₄⋊C₄	1	1	yes	characteristic
272	C₂×C₅	#3291	1	2	yes	—
273	D₅	—	4	4	no	—
274	C₂³	—	10	2	no	—
275	C₂³	#4144	1	1	yes	characteristic
276	Q₈	—	5	4	no	—
277	Q₈	—	10	2	no	—
278	D₄	—	5	4	no	—
279	D₄	—	10	4	no	—
280	D₄	—	10	4	no	—
281	D₄	—	10	2	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	8	4	no	—
286	C₂×C₄	—	10	2	no	—
287	C₂×C₄	—	10	4	no	—
288	C₂×C₄	—	10	4	no	—
289	C₂×C₄	—	10	4	no	—
290	C₂×C₄	—	80	4	no	—
291	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
292	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
293	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
294	C₂×C₄	C₂×Q₈×D₅	1	1	yes	characteristic
295	C₂×C₄	C₂×#713	1	1	yes	characteristic
296	C₂×C₄	#4147	1	1	yes	characteristic
297	C₈	—	10	4	no	—
298	C₈	—	20	2	no	—
299	C₈	D₅×C₄⋊C₄	1	2	yes	—
300	C₈	D₅×C₄⋊C₄	1	2	yes	—
301	C₈	#4039	1	2	yes	—
302	C₈	#4039	1	2	yes	—
303	C₅	#22689	1	1	yes	characteristic, complemented, Sylow
304	C₂²	—	2	2	no	—
305	C₂²	—	10	4	no	—
306	C₂²	—	20	4	no	—
307	C₂²	#65838	1	1	yes	characteristic
308	C₂²	#66051	1	1	yes	characteristic
309	C₂²	#66064	1	1	yes	characteristic
310	C₄	—	2	4	no	—
311	C₄	—	8	4	no	—
312	C₄	—	10	4	no	—
313	C₄	—	20	4	no	—
314	C₄	C₂×D₅×C₄⋊C₄	1	1	yes	characteristic
315	C₄	C₂×#4039	1	1	yes	characteristic
316	C₄	#66060	1	1	yes	characteristic
317	C₄	#66061	1	1	yes	characteristic
318	C₂	—	2	1	no	—
319	C₂	—	2	1	no	—
320	C₂	—	20	4	no	—
321	C₂	#10606091	1	1	yes	characteristic
322	C₂	#10613505	1	2	yes	—

	subgroup	G/H
center	C₂²	#66051
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅
Fitting	C₅×#1844	C₂
socle	C₂²×C₅	D₄²
prime radical	C₂×#4050	C₂²
cosocle	C₂²×C₄×C₅	C₂⁴
hypercenter	#1844	D₅
hypocenter	C₅	#22689
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1844

Complements

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

N	H ≅ G/N	type
1. #10610847	C₂	semidirect
4. C₅×#1844	C₂	semidirect
5. #10601385	C₂	semidirect
6. #10601365	C₂	semidirect
7. #10600572	C₂	semidirect
8. #10600385	C₂	semidirect
29. #64989	C₄	semidirect
49. #4047	C₂×C₄	semidirect
60. C₄×C₅×C₈	C₂×C₄	semidirect
62. #3951	C₂×C₄	semidirect
71. #3937	C₂×C₄	semidirect
77. #1844	D₅	semidirect
117. Dic₂₀	C₄⋊C₄	semidirect
118. D₄₀	C₄⋊C₄	semidirect
119. #677	C₄⋊C₄	semidirect
212. C₄×C₈	C₄×D₅	semidirect
303. C₅	#22689	semidirect

Maximal subgroups

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

subgroup	index
#10610847	2
#10610837	2
C₂×#64989	2
C₅×#1844	2
#10601385	2
#10601365	2
#10600572	2
#10600385	2
#22689	5