Subgroups — #12082374

Order 1280.

714 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#494	C₂	1	1	yes	maximal, characteristic, complemented
2	#10614946	C₂	1	2	yes	maximal
3	#10610189	C₂	1	1	yes	maximal, characteristic
4	#10602566	C₂	1	2	yes	maximal, complemented
5	#10602099	C₂	1	4	yes	maximal, complemented
6	#10602098	C₂	1	4	yes	maximal, complemented
7	#10600522	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₅×Q₁₆	—	2	1	no	—
9	C₂²×C₅×Q₁₆	—	2	1	no	—
10	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂×#4047	—	2	1	no	—
12	#65811	C₂²	1	1	yes	characteristic
13	C₅×#494	—	2	1	no	—
14	C₅×#494	—	2	1	no	—
15	C₅×#494	C₂²	1	2	yes	—
16	C₅×#494	C₂²	1	2	yes	—
17	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	#65399	—	2	1	no	—
19	#65377	—	2	2	no	—
20	#65376	C₂²	1	8	yes	—
21	#65367	—	2	2	no	—
22	#65256	—	2	4	no	—
23	#65214	C₂²	1	2	yes	—
24	#65213	C₂²	1	1	yes	characteristic
25	#64977	—	2	1	no	—
26	#64977	C₂²	1	2	yes	—
27	C₈×D₂₀	—	2	1	no	—
28	C₈×D₂₀	C₂²	1	2	yes	—
29	#64767	C₂²	1	8	yes	—
30	#64678	C₂²	1	2	yes	—
31	#64677	C₂²	1	2	yes	—
32	#19188	—	5	1	no	maximal, Sylow
33	C₂²×C₅×Q₈	—	2	2	no	—
34	C₂×#709	—	2	1	no	—
35	C₂×C₅×Q₁₆	—	2	2	no	—
36	C₂×C₅×Q₁₆	—	2	2	no	—
37	C₂×C₅×Q₁₆	—	2	2	no	—
38	C₂×C₅×Q₁₆	—	2	2	no	—
39	C₂×C₅×Q₁₆	—	4	4	no	—
40	C₂×C₅×Q₁₆	—	4	4	no	—
41	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
42	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
43	C₅×C₄⋊Q₈	—	2	2	no	—
44	C₅×C₄⋊Q₈	—	2	2	no	—
45	C₅×C₄⋊Q₈	—	2	2	no	—
46	C₅×C₄⋊Q₈	C₂³	1	4	yes	—
47	C₂×C₅×C₄⋊C₄	—	2	2	no	—
48	C₂×C₅×C₄⋊C₄	—	2	2	no	—
49	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
50	#4076	—	2	1	no	—
51	#4076	D₄	1	2	yes	—
52	C₂×#681	—	2	1	no	—
53	#4047	—	4	4	no	—
54	C₂×#676	—	2	2	no	—
55	C₂×C₈×D₅	—	2	2	no	—
56	#4021	—	2	1	no	—
57	#4020	D₄	1	1	yes	characteristic
58	C₄×D₂₀	—	2	1	no	—
59	C₄×D₂₀	C₂³	1	2	yes	—
60	#4015	D₄	1	1	yes	characteristic
61	C₄×C₅×C₈	D₄	1	1	yes	characteristic
62	C₄×C₅×C₈	D₄	1	1	yes	characteristic
63	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
64	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
65	C₂²⋊#168	—	2	2	no	—
66	#3952	—	2	2	no	—
67	#3952	—	2	2	no	—
68	#3941	—	2	8	no	—
69	#3939	—	2	8	no	—
70	C₄⋊#168	—	2	1	no	—
71	C₄⋊#168	C₂³	1	4	yes	—
72	C₄⋊#168	C₂³	1	2	yes	—
73	C₄×#168	—	2	2	no	—
74	#3226	—	5	2	no	—
75	C₂×#494	D₅	1	1	yes	characteristic, complemented
76	#2931	—	5	1	no	—
77	#1689	—	5	4	no	—
78	#1669	—	5	4	no	—
79	#1564	—	5	1	no	—
80	#1546	—	5	2	no	—
81	C₂×C₅×Q₈	—	2	4	no	—
82	C₂×C₅×Q₈	—	4	2	no	—
83	C₂×C₅×Q₈	—	4	4	no	—
84	C₂×C₅×Q₈	—	4	4	no	—
85	C₂²×C₄×C₅	—	4	2	no	—
86	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₅	#41	1	1	yes	characteristic
89	C₂×C₂²⋊D₅	—	2	2	no	—
90	#709	—	4	2	no	—
91	#709	—	4	2	no	—
92	C₂×D₂₀	—	2	1	no	—
93	C₂×D₂₀	—	2	1	no	—
94	C₂×D₂₀	C₂×D₄	1	2	yes	—
95	C₂×C₄×D₅	—	2	2	no	—
96	C₂×Dic₁₀	—	2	1	no	—
97	C₅×Q₁₆	—	4	4	no	—
98	C₅×Q₁₆	—	4	4	no	—
99	C₅×Q₁₆	—	4	4	no	—
100	C₅×Q₁₆	—	4	4	no	—
101	C₂×C₅×C₈	—	2	2	no	—
102	C₂×C₅×C₈	D₈	1	4	yes	—
103	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
104	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
105	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
106	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
107	C₅×C₄⋊C₄	—	2	4	no	—
108	C₅×C₄⋊C₄	—	2	4	no	—
109	C₅×C₄⋊C₄	—	4	2	no	—
110	C₅×C₄⋊C₄	—	4	2	no	—
111	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₅	#41	1	1	yes	characteristic
114	C₄²×C₅	C₂⁴	1	1	yes	characteristic
115	#685	—	2	2	no	—
116	#685	—	2	2	no	—
117	C₄⋊Dic₅	—	2	1	no	—
118	C₄⋊Dic₅	C₂×D₄	1	2	yes	—
119	#681	—	4	2	no	—
120	C₂×#168	—	2	4	no	—
121	C₂×#168	—	2	2	no	—
122	#676	—	4	4	no	—
123	C₈×D₅	—	4	4	no	—
124	C₂²×Q₁₆	—	2	1	no	—
125	C₂²×Q₁₆	—	2	1	no	—
126	C₂×#131	—	10	1	no	—
127	#532	—	5	1	no	—
128	C₂×C₄⋊Q₈	C₂×D₅	1	2	yes	—
129	#500	—	10	2	no	—
130	#494	—	2	1	no	—
131	#494	—	2	1	no	—
132	#494	C₂×D₅	1	2	yes	—
133	#494	C₂×D₅	1	2	yes	—
134	#486	—	10	2	no	—
135	#480	—	10	4	no	—
136	#474	—	5	8	no	—
137	#436	—	5	2	no	—
138	#436	—	10	1	no	—
139	C₈×D₄	—	5	2	no	—
140	C₈×D₄	—	10	1	no	—
141	#431	—	5	2	no	—
142	#423	—	5	1	no	—
143	#408	—	10	1	no	—
144	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
145	#335	—	5	2	no	—
146	#334	—	5	2	no	—
147	#332	—	5	8	no	—
148	C₂³×C₅	#121	1	1	yes	characteristic
149	C₂²×D₅	—	2	2	no	—
150	C₅×Q₈	—	4	4	no	—
151	C₅×Q₈	—	4	4	no	—
152	C₅×Q₈	—	8	4	no	—
153	C₂×C₄×C₅	—	2	2	no	—
154	C₂×C₄×C₅	—	2	2	no	—
155	C₂×C₄×C₅	—	4	4	no	—
156	C₂×C₄×C₅	—	8	4	no	—
157	C₂×C₄×C₅	#121	1	1	yes	characteristic
158	C₂×C₄×C₅	#121	1	1	yes	characteristic
159	C₂×C₄×C₅	#121	1	1	yes	characteristic
160	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
161	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
162	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
165	C₂²⋊D₅	—	4	4	no	—
166	C₂×Dic₅	—	2	2	no	—
167	D₂₀	—	2	4	no	—
168	D₂₀	—	4	2	no	—
169	D₂₀	—	4	2	no	—
170	C₄×D₅	—	4	2	no	—
171	C₄×D₅	—	4	2	no	—
172	Dic₁₀	—	4	2	no	—
173	C₅×C₈	—	2	1	no	—
174	C₅×C₈	—	2	1	no	—
175	C₅×C₈	C₂×D₈	1	2	yes	—
176	C₅×C₈	C₂×D₈	1	2	yes	—
177	#168	—	4	2	no	—
178	#168	—	4	4	no	—
179	C₂×#41	—	10	1	no	—
180	C₂²×Q₈	—	2	2	no	—
181	C₂×Q₁₆	—	2	2	no	—
182	C₂×Q₁₆	—	2	2	no	—
183	C₂×Q₁₆	—	2	2	no	—
184	C₂×Q₁₆	—	2	2	no	—
185	C₂×Q₁₆	—	4	4	no	—
186	C₂×Q₁₆	—	4	4	no	—
187	#131	—	20	4	no	—
188	C₂×M₁₆	—	10	1	no	—
189	C₂×M₁₆	—	10	2	no	—
190	C₂²×C₈	—	10	2	no	—
191	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
192	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
193	C₄⋊Q₈	—	2	2	no	—
194	C₄⋊Q₈	—	2	2	no	—
195	C₄⋊Q₈	—	2	2	no	—
196	C₄⋊Q₈	—	5	1	no	—
197	C₄⋊Q₈	C₂²×D₅	1	4	yes	—
198	#127	—	5	1	no	—
199	#124	—	10	1	no	—
200	#121	—	5	2	no	—
201	#121	—	10	1	no	—
202	C₄×D₄	—	5	2	no	—
203	C₄×D₄	—	10	1	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
207	#107	—	10	8	no	—
208	C₄⋊C₈	—	5	2	no	—
209	C₄⋊C₈	—	5	4	no	—
210	C₄⋊C₈	—	10	1	no	—
211	#102	—	10	8	no	—
212	C₂²⋊C₈	—	10	2	no	—
213	C₂²⋊C₈	—	10	2	no	—
214	C₂²⋊C₈	—	10	2	no	—
215	C₄×C₈	—	10	2	no	—
216	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
217	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
218	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
219	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
220	C₂²×C₅	—	2	1	no	—
221	C₂²×C₅	—	2	1	no	—
222	C₂²×C₅	#459	1	2	yes	—
223	C₂²×C₅	C₂×#121	1	1	yes	characteristic
224	C₂×D₅	—	4	2	no	—
225	C₂×D₅	—	4	2	no	—
226	C₂×D₅	—	4	2	no	—
227	C₄×C₅	—	2	1	no	—
228	C₄×C₅	—	2	1	no	—
229	C₄×C₅	—	8	4	no	—
230	C₄×C₅	#459	1	2	yes	—
231	C₄×C₅	#473	1	2	yes	—
232	C₄×C₅	C₂²×D₈	1	1	yes	characteristic
233	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
234	C₄×C₅	#578	1	2	yes	—
235	Dic₅	—	4	2	no	—
236	#41	—	20	2	no	—
237	#41	—	20	2	no	—
238	C₂×Q₈	—	2	4	no	—
239	C₂×Q₈	—	4	2	no	—
240	C₂×Q₈	—	4	4	no	—
241	C₂×Q₈	—	4	4	no	—
242	C₂×Q₈	—	10	1	no	—
243	C₂×D₄	—	5	2	no	—
244	C₂×D₄	—	10	2	no	—
245	C₂×D₄	—	10	1	no	—
246	C₂×D₄	—	10	1	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	—	10	2	no	—
249	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
250	C₂²×C₄	#711	1	1	yes	characteristic
251	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
252	Q₁₆	—	4	4	no	—
253	Q₁₆	—	4	4	no	—
254	Q₁₆	—	4	4	no	—
255	Q₁₆	—	4	4	no	—
256	M₁₆	—	20	2	no	—
257	M₁₆	—	20	4	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	10	4	no	—
260	C₂×C₈	—	10	2	no	—
261	C₂×C₈	—	20	4	no	—
262	C₂×C₈	#686	1	4	yes	—
263	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
264	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
265	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
266	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	5	2	no	—
272	C₄⋊C₄	—	10	1	no	—
273	C₂²⋊C₄	—	10	2	no	—
274	C₂²⋊C₄	—	10	2	no	—
275	C₄²	D₄×D₅	1	1	yes	characteristic
276	C₄²	#711	1	1	yes	characteristic
277	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
278	C₄²	C₂³×D₅	1	1	yes	characteristic
279	C₂×C₅	—	2	2	no	—
280	C₂×C₅	C₂×#459	1	1	yes	characteristic
281	C₂×C₅	#3059	1	1	yes	characteristic
282	C₂×C₅	C₂×#473	1	1	yes	characteristic
283	D₅	—	8	2	no	—
284	C₂³	—	10	2	no	—
285	C₂³	#4084	1	1	yes	characteristic
286	Q₈	—	4	4	no	—
287	Q₈	—	4	4	no	—
288	Q₈	—	8	4	no	—
289	Q₈	—	20	2	no	—
290	D₄	—	10	4	no	—
291	D₄	—	20	4	no	—
292	D₄	—	20	2	no	—
293	D₄	—	20	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	8	4	no	—
298	C₂×C₄	—	10	2	no	—
299	C₂×C₄	—	20	2	no	—
300	C₂×C₄	—	20	2	no	—
301	C₂×C₄	C₂×#686	1	2	yes	—
302	C₂×C₄	#4078	1	2	yes	—
303	C₂×C₄	#4084	1	1	yes	characteristic
304	C₂×C₄	#4084	1	1	yes	characteristic
305	C₂×C₄	#4084	1	1	yes	characteristic
306	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
307	C₂×C₄	C₂×#711	1	1	yes	characteristic
308	C₂×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	20	4	no	—
312	C₈	—	20	2	no	—
313	C₈	C₂×#686	1	2	yes	—
314	C₈	C₂×#686	1	2	yes	—
315	C₅	#19188	1	1	yes	characteristic, complemented, Sylow
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	20	2	no	—
319	C₂²	—	20	2	no	—
320	C₂²	—	20	2	no	—
321	C₂²	#65361	1	2	yes	—
322	C₂²	C₂×#4084	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	8	4	no	—
326	C₄	—	20	2	no	—
327	C₄	#65361	1	2	yes	—
328	C₄	#65481	1	2	yes	—
329	C₄	#66100	1	2	yes	—
330	C₄	C₂²×#686	1	1	yes	characteristic
331	C₄	C₂×#4078	1	1	yes	characteristic
332	C₂	—	2	2	no	—
333	C₂	—	40	2	no	—
334	C₂	C₂×#65361	1	1	yes	characteristic
335	C₂	#10615618	1	1	yes	characteristic
336	C₂	C₂×#65481	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#4084
commutator, cosocle	C₄²×C₅	C₂⁴
Frattini	C₄²	C₂³×D₅
Fitting	C₂×C₅×#494	C₂
socle	C₂²×C₅	C₂×#121
prime radical	#65811	C₂²
hypercenter	C₂×#494	D₅
hypocenter	C₅	#19188
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#494	C₂	semidirect
4. #10602566	C₂	semidirect
5. #10602099	C₂	semidirect
6. #10602098	C₂	semidirect
7. #10600522	C₂	semidirect
75. C₂×#494	D₅	semidirect
315. C₅	#19188	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#494	2
#10614946	2
#10610189	2
#10602566	2
#10602099	2
#10602098	2
#10600522	2
#19188	5