Subgroups — #12005140

Order 1280.

802 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2285	C₂	1	1	yes	maximal, characteristic
2	C₂×#65213	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#681	C₂	1	1	yes	maximal, characteristic
4	C₂×#64753	C₂	1	4	yes	maximal
5	C₂×C₅×#117	C₂²	1	2	yes	—
6	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₂²×#681	C₂²	1	2	yes	—
8	C₂²×#681	C₂²	1	2	yes	—
9	C₅×#384	—	2	1	no	—
10	C₅×#382	—	2	1	no	—
11	C₅×#380	C₄	1	4	yes	—
12	C₄×C₅×C₄⋊C₄	—	2	2	no	—
13	#65213	—	2	1	no	—
14	#65213	—	2	1	no	—
15	#65213	C₄	1	4	yes	complemented
16	C₂×C₄⋊#168	—	2	2	no	—
17	C₄×#681	—	2	2	no	—
18	C₄×#681	—	2	2	no	—
19	C₂×#3935	C₄	1	2	yes	complemented
20	C₂×C₄×#168	C₄	1	2	yes	complemented
21	#64753	—	2	8	no	—
22	#10848	—	5	1	no	maximal, Sylow
23	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₅×#117	—	2	4	no	—
26	C₅×#117	—	2	4	no	—
27	C₂×C₅×C₄⋊C₄	—	2	4	no	—
28	C₂×C₅×C₄⋊C₄	—	2	2	no	—
29	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₅	—	2	2	no	—
31	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₅	D₄	1	2	yes	—
35	C₂×C₄²×C₅	D₄	1	2	yes	—
36	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
39	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
40	C₂×#681	—	2	2	no	—
41	C₂×#681	—	2	2	no	—
42	C₂×#681	—	2	2	no	—
43	C₂×#681	—	2	2	no	—
44	C₂×#681	—	2	2	no	—
45	C₂×#681	—	2	4	no	—
46	C₂×#681	—	2	4	no	—
47	C₂×#681	C₂×C₄	1	4	yes	—
48	C₂×#681	D₄	1	4	yes	—
49	C₂×#681	Q₈	1	4	yes	—
50	C₂²×#168	—	2	2	no	—
51	C₂²×#168	C₂×C₄	1	4	yes	—
52	C₅×#95	—	2	4	no	—
53	C₄⋊#168	—	2	4	no	—
54	C₄⋊#168	—	2	4	no	—
55	#3935	—	2	4	no	—
56	C₄×#168	—	2	4	no	—
57	C₂×#423	—	5	1	no	—
58	C₂×C₄×M₁₆	—	5	1	no	—
59	#2285	D₅	1	1	yes	characteristic
60	C₂×#344	—	5	4	no	—
61	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	—	2	4	no	—
63	C₂²×C₄×C₅	—	4	4	no	—
64	C₂²×C₄×C₅	C₄²	1	2	yes	—
65	C₂²×C₄×C₅	C₄²	1	2	yes	—
66	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₅	#41	1	2	yes	—
79	C₂²×C₄×C₅	#41	1	2	yes	—
80	C₅×C₄⋊C₄	—	4	8	no	—
81	C₅×C₄⋊C₄	—	4	2	no	—
82	C₅×C₄⋊C₄	—	4	2	no	—
83	C₅×C₂²⋊C₄	—	2	8	no	—
84	C₄²×C₅	—	4	4	no	—
85	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
86	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
87	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
88	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
89	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
90	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
91	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
92	C₄²×C₅	C₄⋊C₄	1	2	yes	—
93	C₄²×C₅	C₄⋊C₄	1	2	yes	—
94	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
95	#681	—	2	4	no	—
96	#681	—	2	4	no	—
97	#681	—	2	4	no	—
98	#681	—	2	4	no	—
99	#681	—	2	8	no	—
100	#681	—	4	2	no	—
101	#681	—	4	2	no	—
102	C₂×#168	—	2	4	no	—
103	C₂×#168	—	2	4	no	—
104	C₂×#168	—	2	4	no	—
105	C₂×#168	—	4	4	no	—
106	C₂×#168	C₄⋊C₄	1	8	yes	complemented
107	C₂²×M₁₆	—	5	2	no	—
108	C₂²×M₁₆	—	5	2	no	—
109	C₂×#117	C₂×D₅	1	2	yes	—
110	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
111	#423	—	5	4	no	—
112	#423	—	10	1	no	—
113	#423	—	10	1	no	—
114	C₂×C₄⋊C₈	—	10	2	no	—
115	C₄×M₁₆	—	10	2	no	—
116	C₄×M₁₆	—	10	2	no	—
117	C₂×#97	—	5	2	no	—
118	C₂×C₄×C₈	—	5	2	no	—
119	#384	—	2	1	no	—
120	#382	—	2	1	no	—
121	#380	Dic₅	1	4	yes	—
122	C₄×C₄⋊C₄	—	2	2	no	—
123	#344	—	10	8	no	—
124	C₂³×C₅	—	4	2	no	—
125	C₂³×C₅	C₂×C₄²	1	1	yes	characteristic
126	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂³×C₅	C₄×D₄	1	2	yes	—
129	C₂³×C₅	C₄×Q₈	1	2	yes	—
130	C₂×C₄×C₅	—	2	4	no	—
131	C₂×C₄×C₅	—	2	4	no	—
132	C₂×C₄×C₅	—	2	4	no	—
133	C₂×C₄×C₅	—	4	2	no	—
134	C₂×C₄×C₅	—	4	2	no	—
135	C₂×C₄×C₅	—	4	2	no	—
136	C₂×C₄×C₅	—	4	8	no	—
137	C₂×C₄×C₅	—	4	4	no	—
138	C₂×C₄×C₅	#95	1	2	yes	—
139	C₂×C₄×C₅	#95	1	2	yes	—
140	C₂×C₄×C₅	#95	1	2	yes	—
141	C₂×C₄×C₅	#95	1	2	yes	—
142	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
145	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₅	#117	1	2	yes	—
148	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
149	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
150	C₂×C₄×C₅	#121	1	2	yes	—
151	C₂×C₄×C₅	#121	1	2	yes	—
152	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
153	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
154	C₂×C₄×C₅	#124	1	1	yes	characteristic
155	C₂×C₄×C₅	#124	1	1	yes	characteristic
156	C₂×C₄×C₅	#125	1	2	yes	—
157	C₂×C₄×C₅	#127	1	1	yes	characteristic
158	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
159	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
160	#168	—	2	8	no	—
161	#168	—	4	4	no	—
162	C₂³×C₄	C₂×Dic₅	1	2	yes	—
163	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
164	C₂×M₁₆	—	5	4	no	—
165	C₂×M₁₆	—	5	4	no	—
166	C₂×M₁₆	—	5	4	no	—
167	C₂×M₁₆	—	10	2	no	—
168	C₂×M₁₆	—	10	2	no	—
169	C₂×M₁₆	—	10	2	no	—
170	C₂×M₁₆	—	10	2	no	—
171	C₂×M₁₆	—	10	2	no	—
172	C₂×M₁₆	—	10	4	no	—
173	C₂×M₁₆	—	10	4	no	—
174	C₂²×C₈	—	5	4	no	—
175	C₂²×C₈	—	10	2	no	—
176	#117	—	2	4	no	—
177	#117	—	2	4	no	—
178	C₂×C₄⋊C₄	—	2	4	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₂²⋊C₄	C₂×Dic₅	1	4	yes	—
181	C₂×C₄²	—	2	2	no	—
182	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
183	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
184	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
185	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
186	C₂×C₄²	C₄×D₅	1	2	yes	—
187	C₂×C₄²	D₂₀	1	2	yes	—
188	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
189	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
190	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
191	C₄⋊C₈	—	10	4	no	—
192	C₄⋊C₈	—	10	4	no	—
193	#97	—	10	4	no	—
194	C₄×C₈	—	10	4	no	—
195	#95	—	2	4	no	—
196	C₂²×C₅	—	2	2	no	—
197	C₂²×C₅	—	2	2	no	—
198	C₂²×C₅	—	2	2	no	—
199	C₂²×C₅	—	4	2	no	—
200	C₂²×C₅	—	4	2	no	—
201	C₂²×C₅	C₂×#95	1	1	yes	characteristic
202	C₂²×C₅	C₄×C₄⋊C₄	1	2	yes	—
203	C₂²×C₅	#389	1	2	yes	—
204	C₂²×C₅	#390	1	1	yes	characteristic
205	C₂²×C₅	#391	1	1	yes	characteristic
206	C₄×C₅	—	2	2	no	—
207	C₄×C₅	—	2	2	no	—
208	C₄×C₅	—	2	2	no	—
209	C₄×C₅	—	8	4	no	—
210	C₄×C₅	#344	1	4	yes	complemented
211	C₄×C₅	#380	1	2	yes	—
212	C₄×C₅	#382	1	1	yes	characteristic
213	C₄×C₅	#384	1	1	yes	characteristic
214	C₄×C₅	#389	1	2	yes	—
215	C₄×C₅	#390	1	1	yes	characteristic
216	C₄×C₅	#391	1	1	yes	characteristic
217	C₂⁴	C₂²×Dic₅	1	1	yes	characteristic
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	4	4	no	—
220	C₂²×C₄	C₄×Dic₅	1	2	yes	—
221	C₂²×C₄	C₄×Dic₅	1	2	yes	—
222	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
223	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
224	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
225	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
226	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
230	C₂²×C₄	D₄×D₅	1	2	yes	—
231	C₂²×C₄	#711	1	2	yes	—
232	C₂²×C₄	Q₈×D₅	1	2	yes	—
233	C₂²×C₄	#713	1	2	yes	—
234	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
235	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
236	M₁₆	—	10	4	no	—
237	M₁₆	—	10	4	no	—
238	M₁₆	—	10	4	no	—
239	M₁₆	—	10	4	no	—
240	M₁₆	—	10	8	no	—
241	M₁₆	—	20	2	no	—
242	M₁₆	—	20	2	no	—
243	C₂×C₈	—	5	8	no	—
244	C₂×C₈	—	10	4	no	—
245	C₂×C₈	—	10	4	no	—
246	C₂×C₈	—	10	4	no	—
247	C₂×C₈	—	20	4	no	—
248	C₄⋊C₄	—	4	8	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₂²⋊C₄	—	2	8	no	—
252	C₄²	—	4	4	no	—
253	C₄²	#683	1	2	yes	—
254	C₄²	#683	1	1	yes	characteristic
255	C₄²	#683	1	1	yes	characteristic
256	C₄²	#683	1	1	yes	characteristic
257	C₄²	#683	1	2	yes	—
258	C₄²	#683	1	1	yes	characteristic
259	C₄²	#685	1	2	yes	—
260	C₄²	#685	1	2	yes	—
261	C₄²	#685	1	2	yes	—
262	C₄²	#685	1	2	yes	—
263	C₂×C₅	—	2	1	no	—
264	C₂×C₅	—	2	2	no	—
265	C₂×C₅	—	2	2	no	—
266	C₂×C₅	—	2	1	no	—
267	C₂×C₅	C₄⋊#95	1	1	yes	characteristic
268	C₂×C₅	C₂×#344	1	2	yes	—
269	C₂³	—	4	2	no	—
270	C₂³	C₂×C₄×Dic₅	1	1	yes	characteristic
271	C₂³	C₂×C₄⋊Dic₅	1	1	yes	characteristic
272	C₂³	D₄×Dic₅	1	2	yes	—
273	C₂³	Q₈×Dic₅	1	2	yes	—
274	C₂³	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	8	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	#3963	1	2	yes	—
284	C₂×C₄	#3963	1	2	yes	—
285	C₂×C₄	#3963	1	2	yes	—
286	C₂×C₄	#3963	1	2	yes	—
287	C₂×C₄	#4033	1	2	yes	—
288	C₂×C₄	#4034	1	2	yes	—
289	C₂×C₄	#4036	1	2	yes	—
290	C₂×C₄	D₅×C₄⋊C₄	1	2	yes	—
291	C₂×C₄	#4038	1	2	yes	—
292	C₂×C₄	#4039	1	2	yes	—
293	C₂×C₄	#4041	1	2	yes	—
294	C₂×C₄	#4043	1	2	yes	—
295	C₂×C₄	C₂×#683	1	1	yes	characteristic
296	C₂×C₄	C₂×#683	1	1	yes	characteristic
297	C₂×C₄	C₂×#685	1	1	yes	characteristic
298	C₂×C₄	C₂×#685	1	1	yes	characteristic
299	C₂×C₄	#4082	1	1	yes	characteristic
300	C₂×C₄	#4084	1	2	yes	—
301	C₂×C₄	#4086	1	1	yes	characteristic
302	C₂×C₄	#4090	1	1	yes	characteristic
303	C₂×C₄	#4092	1	2	yes	—
304	C₂×C₄	#4093	1	1	yes	characteristic
305	C₈	—	10	8	no	—
306	C₈	—	20	4	no	—
307	C₅	#10848	1	1	yes	characteristic, complemented, Sylow
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	4	2	no	—
312	C₂²	—	4	2	no	—
313	C₂²	C₄⋊C₄×Dic₅	1	2	yes	—
314	C₂²	#65274	1	2	yes	—
315	C₂²	C₂×#3963	1	1	yes	characteristic
316	C₂²	#65510	1	1	yes	characteristic
317	C₂²	#65519	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	2	2	no	—
321	C₄	—	8	4	no	—
322	C₄	#64753	1	4	yes	complemented
323	C₄	#65262	1	2	yes	—
324	C₄	#65263	1	1	yes	characteristic
325	C₄	#65265	1	1	yes	characteristic
326	C₄	#65266	1	1	yes	characteristic
327	C₄	#65276	1	2	yes	—
328	C₄	#65277	1	1	yes	characteristic
329	C₂	—	2	1	no	—
330	C₂	—	2	1	no	—
331	C₂	—	2	2	no	—
332	C₂	—	2	2	no	—
333	C₂	#10602356	1	1	yes	characteristic
334	C₂	C₂×#64753	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#685
commutator	C₂³×C₅	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅
Fitting	C₅×#2285	C₂
socle	C₂²×C₅	C₂×#95
prime radical	C₂⁴×C₅	C₂²×C₄
cosocle	C₂³×C₄×C₅	C₂³
hypercenter	#2285	D₅
hypocenter	C₅	#10848
LCS	G ▷ C₂³×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ #2285

Complements

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

N	H ≅ G/N	type
15. #65213	C₄	semidirect
19. C₂×#3935	C₄	semidirect
20. C₂×C₄×#168	C₄	semidirect
106. C₂×#168	C₄⋊C₄	semidirect
210. C₄×C₅	#344	semidirect
307. C₅	#10848	semidirect
322. C₄	#64753	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2285	2
C₂×#65213	2
C₂×C₄×#681	2
C₂×#64753	2
#10848	5