Subgroups — #12083178

Order 1280.

606 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	C₂×C₅×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	#10614179	C₂	1	2	yes	maximal
3	#10610206	C₂	1	1	yes	maximal, characteristic
4	#10601739	C₂	1	4	yes	maximal, complemented
5	#10601737	C₂	1	2	yes	maximal, complemented
6	#10601735	C₂	1	2	yes	maximal, complemented
7	#10601232	C₂	1	2	yes	maximal
8	#10600525	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
10	#65804	C₂²	1	2	yes	—
11	C₅×C₈×Q₈	—	2	1	no	—
12	C₅×C₈×Q₈	—	2	1	no	—
13	C₅×C₈×Q₈	C₂²	1	2	yes	—
14	C₅×C₈×Q₈	C₂²	1	2	yes	—
15	C₂×C₅×C₄⋊C₈	—	2	1	no	—
16	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₅×C₈	—	2	1	no	—
18	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
19	#65517	—	2	2	no	—
20	#65405	—	2	1	no	—
21	#65405	—	2	1	no	—
22	#65320	C₂²	1	4	yes	—
23	#65319	—	2	4	no	—
24	#65309	C₂²	1	4	yes	—
25	#65260	—	2	2	no	—
26	#65215	C₂²	1	2	yes	—
27	#64988	C₂²	1	1	yes	characteristic
28	C₄×Dic₂₀	—	2	1	no	—
29	C₄×Dic₂₀	—	2	1	no	—
30	#64971	C₂²	1	1	yes	characteristic
31	#64970	C₂²	1	2	yes	—
32	#64706	C₂²	1	4	yes	—
33	#64702	C₂²	1	4	yes	—
34	#64678	C₂²	1	2	yes	—
35	#64677	C₂²	1	2	yes	—
36	#19265	—	5	1	no	maximal, Sylow
37	C₂²×C₅×Q₈	D₄	1	1	yes	characteristic
38	C₂²×C₅×C₈	—	2	1	no	—
39	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
40	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
41	C₄×C₅×Q₈	—	2	1	no	—
42	C₄×C₅×Q₈	—	2	1	no	—
43	C₄×C₅×Q₈	D₄	1	2	yes	—
44	C₄×C₅×Q₈	C₂³	1	2	yes	—
45	C₂×C₅×C₄⋊C₄	—	2	1	no	—
46	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₅	—	2	1	no	—
48	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
49	C₂×#689	—	2	4	no	—
50	#4072	—	2	2	no	—
51	#4070	—	2	2	no	—
52	#4070	—	2	2	no	—
53	C₂×Dic₂₀	—	2	1	no	—
54	C₂×Dic₂₀	—	2	1	no	—
55	#4016	—	2	2	no	—
56	#4015	C₂³	1	2	yes	—
57	#4015	C₂³	1	2	yes	—
58	C₄×Dic₁₀	—	2	2	no	—
59	C₄×Dic₁₀	—	2	2	no	—
60	C₅×C₄⋊C₈	—	2	2	no	—
61	C₅×C₄⋊C₈	—	2	2	no	—
62	C₅×C₄⋊C₈	—	2	1	no	—
63	C₅×C₄⋊C₈	C₂³	1	2	yes	—
64	C₄×C₅×C₈	—	2	2	no	—
65	C₄×C₅×C₈	—	2	2	no	—
66	C₄×C₅×C₈	D₄	1	1	yes	characteristic
67	C₄×C₅×C₈	D₄	1	1	yes	characteristic
68	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
69	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
70	#3967	—	2	4	no	—
71	C₂²⋊#168	—	2	2	no	—
72	#3950	—	2	1	no	—
73	#3950	—	2	1	no	—
74	#3948	—	2	2	no	—
75	#3948	—	2	2	no	—
76	#3942	—	2	4	no	—
77	#3940	—	2	4	no	—
78	C₄⋊#168	C₂³	1	4	yes	—
79	#3935	—	2	2	no	—
80	#3170	—	5	1	no	—
81	#3113	—	5	2	no	—
82	C₂×C₈×Q₈	D₅	1	1	yes	characteristic
83	#1680	—	5	4	no	—
84	#1674	—	5	2	no	—
85	#1670	—	5	2	no	—
86	#1568	—	5	1	no	—
87	#1552	—	5	2	no	—
88	C₂×C₅×Q₈	—	2	4	no	—
89	C₂×C₅×Q₈	—	2	1	no	—
90	C₂×C₅×Q₈	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₅	—	2	1	no	—
92	C₂²×C₄×C₅	—	2	1	no	—
93	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₅	#41	1	1	yes	characteristic
96	C₂×Dic₁₀	—	2	2	no	—
97	C₂×Dic₁₀	—	2	2	no	—
98	C₂×C₅×C₈	—	2	2	no	—
99	C₂×C₅×C₈	—	2	2	no	—
100	C₂×C₅×C₈	—	4	1	no	—
101	C₂×C₅×C₈	—	4	2	no	—
102	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
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	C₅×C₄⋊C₄	—	2	2	no	—
107	C₅×C₄⋊C₄	—	2	1	no	—
108	C₅×C₄⋊C₄	—	2	2	no	—
109	C₅×C₄⋊C₄	C₂×D₄	1	2	yes	—
110	C₄²×C₅	—	2	2	no	—
111	C₄²×C₅	—	2	2	no	—
112	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₅	#41	1	1	yes	characteristic
115	C₄²×C₅	C₂⁴	1	1	yes	characteristic
116	C₂²⋊Dic₅	—	4	2	no	—
117	#689	—	4	8	no	—
118	C₄⋊Dic₅	—	2	2	no	—
119	C₄⋊Dic₅	—	2	2	no	—
120	C₄⋊Dic₅	—	2	2	no	—
121	C₄⋊Dic₅	—	2	2	no	—
122	#683	—	4	2	no	—
123	#683	—	4	2	no	—
124	C₄×Dic₅	—	4	2	no	—
125	C₂×#168	—	2	4	no	—
126	Dic₂₀	—	4	2	no	—
127	Dic₂₀	—	4	2	no	—
128	#533	—	5	2	no	—
129	C₂×C₄×Q₈	C₂×D₅	1	1	yes	characteristic
130	#500	—	5	1	no	—
131	#494	—	5	1	no	—
132	#487	—	5	2	no	—
133	#483	—	10	2	no	—
134	#475	—	5	4	no	—
135	#467	—	10	1	no	—
136	#467	—	10	1	no	—
137	#462	—	5	4	no	—
138	#451	—	10	2	no	—
139	C₈×Q₈	—	2	1	no	—
140	C₈×Q₈	—	2	1	no	—
141	C₈×Q₈	C₂×D₅	1	2	yes	—
142	C₈×Q₈	C₂×D₅	1	2	yes	—
143	#441	—	10	4	no	—
144	C₄×Q₁₆	—	10	1	no	—
145	C₄×Q₁₆	—	10	1	no	—
146	#432	—	5	2	no	—
147	C₂×C₄⋊C₈	—	2	1	no	—
148	C₂×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
149	C₂×C₄×C₈	—	2	1	no	—
150	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
151	#335	—	5	2	no	—
152	#334	—	5	2	no	—
153	#332	—	5	4	no	—
154	#332	—	5	4	no	—
155	C₂³×C₅	#121	1	1	yes	characteristic
156	C₅×Q₈	—	2	4	no	—
157	C₅×Q₈	—	2	2	no	—
158	C₅×Q₈	#135	1	4	yes	—
159	C₂×C₄×C₅	—	2	4	no	—
160	C₂×C₄×C₅	—	2	2	no	—
161	C₂×C₄×C₅	—	2	2	no	—
162	C₂×C₄×C₅	—	2	2	no	—
163	C₂×C₄×C₅	—	2	1	no	—
164	C₂×C₄×C₅	—	2	1	no	—
165	C₂×C₄×C₅	—	2	2	no	—
166	C₂×C₄×C₅	—	4	2	no	—
167	C₂×C₄×C₅	#121	1	1	yes	characteristic
168	C₂×C₄×C₅	#121	1	1	yes	characteristic
169	C₂×C₄×C₅	#121	1	1	yes	characteristic
170	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
173	C₂×Dic₅	—	4	2	no	—
174	C₂×Dic₅	—	4	2	no	—
175	Dic₁₀	—	4	4	no	—
176	Dic₁₀	—	4	2	no	—
177	Dic₁₀	—	4	2	no	—
178	C₅×C₈	—	2	1	no	—
179	C₅×C₈	—	2	1	no	—
180	C₅×C₈	—	2	1	no	—
181	C₅×C₈	—	4	2	no	—
182	C₅×C₈	#137	1	2	yes	—
183	#168	—	4	4	no	—
184	C₂²×Q₈	D₂₀	1	1	yes	characteristic
185	C₂×Q₁₆	—	10	4	no	—
186	C₂×Q₁₆	—	10	1	no	—
187	C₂×Q₁₆	—	10	1	no	—
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
190	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
191	C₄⋊Q₈	—	5	2	no	—
192	C₄⋊Q₈	—	5	2	no	—
193	#125	—	10	2	no	—
194	C₂²⋊Q₈	—	10	2	no	—
195	C₂²⋊Q₈	—	10	2	no	—
196	C₄×Q₈	—	2	1	no	—
197	C₄×Q₈	—	2	1	no	—
198	C₄×Q₈	—	10	2	no	—
199	C₄×Q₈	—	10	2	no	—
200	C₄×Q₈	D₂₀	1	2	yes	—
201	C₄×Q₈	C₂²×D₅	1	2	yes	—
202	#117	—	10	2	no	—
203	C₂×C₄⋊C₄	—	2	1	no	—
204	C₂×C₄⋊C₄	D₂₀	1	1	yes	characteristic
205	C₂×C₄²	—	2	1	no	—
206	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
207	#107	—	10	1	no	—
208	#107	—	10	1	no	—
209	#106	—	10	4	no	—
210	C₄⋊C₈	—	2	2	no	—
211	C₄⋊C₈	—	2	2	no	—
212	C₄⋊C₈	—	2	1	no	—
213	C₄⋊C₈	—	5	4	no	—
214	C₄⋊C₈	C₂²×D₅	1	2	yes	—
215	#103	—	10	4	no	—
216	#103	—	10	2	no	—
217	#103	—	10	2	no	—
218	#103	—	10	4	no	—
219	C₂²⋊C₈	—	10	2	no	—
220	#97	—	10	2	no	—
221	C₄×C₈	—	2	2	no	—
222	C₄×C₈	—	2	2	no	—
223	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
224	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
225	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
226	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
227	C₂²×C₅	—	2	1	no	—
228	C₂²×C₅	—	2	1	no	—
229	C₂²×C₅	#473	1	2	yes	—
230	C₂²×C₅	C₂×#121	1	1	yes	characteristic
231	C₄×C₅	—	2	1	no	—
232	C₄×C₅	—	2	4	no	—
233	C₄×C₅	—	2	1	no	—
234	C₄×C₅	—	2	1	no	—
235	C₄×C₅	—	2	1	no	—
236	C₄×C₅	—	4	2	no	—
237	C₄×C₅	C₂×#135	1	2	yes	—
238	C₄×C₅	C₂×#137	1	1	yes	characteristic
239	C₄×C₅	#575	1	1	yes	characteristic
240	Dic₅	—	8	2	no	—
241	Dic₅	—	8	2	no	—
242	C₂×Q₈	—	2	4	no	—
243	C₂×Q₈	—	2	1	no	—
244	C₂×Q₈	—	10	2	no	—
245	C₂×Q₈	—	10	2	no	—
246	C₂×Q₈	C₂×D₂₀	1	2	yes	—
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
250	C₂²×C₄	#709	1	1	yes	characteristic
251	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
252	Q₁₆	—	20	2	no	—
253	Q₁₆	—	20	8	no	—
254	Q₁₆	—	20	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	4	2	no	—
259	C₂×C₈	—	10	4	no	—
260	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
261	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
262	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
263	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	10	2	no	—
268	C₄⋊C₄	—	10	2	no	—
269	C₄⋊C₄	—	10	2	no	—
270	C₄⋊C₄	—	10	2	no	—
271	C₄⋊C₄	—	20	2	no	—
272	C₄⋊C₄	—	20	2	no	—
273	C₄⋊C₄	C₂×D₂₀	1	2	yes	—
274	C₂²⋊C₄	—	20	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	—	20	2	no	—
278	C₄²	C₂×D₂₀	1	1	yes	characteristic
279	C₄²	#709	1	1	yes	characteristic
280	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
281	C₄²	C₂³×D₅	1	1	yes	characteristic
282	C₂×C₅	—	2	2	no	—
283	C₂×C₅	#3037	1	1	yes	characteristic
284	C₂×C₅	#3050	1	1	yes	characteristic
285	C₂×C₅	C₂×#473	1	1	yes	characteristic
286	C₂³	#4076	1	1	yes	characteristic
287	Q₈	—	2	4	no	—
288	Q₈	—	2	2	no	—
289	Q₈	—	20	4	no	—
290	Q₈	—	20	2	no	—
291	Q₈	—	20	2	no	—
292	Q₈	#4050	1	4	yes	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	20	2	no	—
302	C₂×C₄	—	20	2	no	—
303	C₂×C₄	#4076	1	1	yes	characteristic
304	C₂×C₄	#4076	1	1	yes	characteristic
305	C₂×C₄	#4076	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
307	C₂×C₄	C₂×#709	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	—	2	1	no	—
312	C₈	—	4	2	no	—
313	C₈	—	20	4	no	—
314	C₈	#4088	1	2	yes	—
315	C₅	#19265	1	1	yes	characteristic, complemented, Sylow
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	#65432	1	2	yes	—
319	C₂²	C₂×#4076	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	4	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	4	2	no	—
326	C₄	—	40	2	no	—
327	C₄	—	40	2	no	—
328	C₄	C₂×#4050	1	2	yes	—
329	C₄	C₂×#4088	1	1	yes	characteristic
330	C₄	#66156	1	1	yes	characteristic
331	C₂	—	2	2	no	—
332	C₂	#10614172	1	1	yes	characteristic
333	C₂	#10615473	1	1	yes	characteristic
334	C₂	C₂×#65432	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#4076
commutator, cosocle	C₄²×C₅	C₂⁴
Frattini	C₄²	C₂³×D₅
Fitting	C₂×C₅×C₈×Q₈	C₂
socle	C₂²×C₅	C₂×#121
prime radical	C₂³×C₅	#121
hypercenter	C₂×C₈×Q₈	D₅
hypocenter	C₅	#19265
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
4. #10601739	C₂	semidirect
5. #10601737	C₂	semidirect
6. #10601735	C₂	semidirect
315. C₅	#19265	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×C₈×Q₈	2
#10614179	2
#10610206	2
#10601739	2
#10601737	2
#10601735	2
#10601232	2
#10600525	2
#19265	5