Subgroups — #11993773

Order 1280.

426 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10610253	C₂	1	1	yes	maximal, characteristic
2	C₅×#1950	C₂	1	1	yes	maximal, characteristic, complemented
3	#10601291	C₂	1	1	yes	maximal, characteristic
4	#10601287	C₂	1	1	yes	maximal, characteristic, complemented
5	#10601224	C₂	1	1	yes	maximal, characteristic
6	#10598194	C₂	1	1	yes	maximal, characteristic, complemented
7	#10598158	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#4053	—	2	1	no	—
9	#65811	C₂²	1	1	yes	characteristic
10	C₂×C₅×#108	—	2	1	no	—
11	C₅×#423	C₂²	1	1	yes	characteristic
12	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
14	#65418	—	2	1	no	—
15	#65214	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
17	C₂×C₄×#168	C₂²	1	1	yes	characteristic
18	#64995	—	2	2	no	—
19	#64977	—	2	1	no	—
20	C₈×D₂₀	—	2	1	no	—
21	C₅×#341	—	2	1	no	—
22	#64705	C₄	1	1	yes	characteristic
23	#64705	C₄	1	1	yes	characteristic
24	#64703	—	2	2	no	—
25	#64702	C₄	1	1	yes	characteristic
26	#64702	C₄	1	1	yes	characteristic
27	#64699	—	2	1	no	—
28	#64699	—	2	1	no	—
29	#8930	—	5	1	no	maximal, Sylow
30	C₂×#709	—	2	1	no	—
31	C₂×C₅×M₁₆	—	2	1	no	—
32	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
33	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
34	C₂²×C₅×C₈	—	2	1	no	—
35	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
36	#4076	—	2	1	no	—
37	#4076	Q₈	1	2	yes	—
38	C₂²×#168	—	2	1	no	—
39	C₂²×#168	—	2	1	no	—
40	#4053	—	4	2	no	—
41	#4053	—	4	2	no	—
42	C₂×#676	—	2	2	no	—
43	C₂×C₈×D₅	—	2	2	no	—
44	#4021	—	2	1	no	—
45	#4020	C₂×C₄	1	1	yes	characteristic
46	C₄×D₂₀	—	2	1	no	—
47	C₄×D₂₀	D₄	1	2	yes	—
48	#4015	C₂×C₄	1	1	yes	characteristic
49	C₅×#108	—	4	2	no	—
50	C₅×C₄⋊C₈	—	2	1	no	—
51	C₅×C₄⋊C₈	—	2	1	no	—
52	C₅×C₄⋊C₈	—	2	1	no	—
53	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
54	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
55	C₅×#97	—	2	1	no	—
56	C₄×C₅×C₈	—	2	1	no	—
57	C₂²⋊#168	—	2	1	no	—
58	#3952	—	2	2	no	—
59	#3952	—	2	2	no	—
60	C₄⋊#168	—	2	1	no	—
61	C₄⋊#168	C₂×C₄	1	1	yes	characteristic
62	C₄⋊#168	C₂×C₄	1	1	yes	characteristic
63	C₄⋊#168	D₄	1	1	yes	characteristic
64	C₄⋊#168	D₄	1	1	yes	characteristic
65	C₄×#168	—	2	1	no	—
66	C₄×#168	—	2	1	no	—
67	C₄×#168	—	2	1	no	—
68	#2935	—	5	1	no	—
69	#1950	D₅	1	1	yes	characteristic, complemented
70	#1571	—	5	1	no	—
71	#1540	—	5	1	no	—
72	#1492	—	5	1	no	—
73	#1278	—	5	1	no	—
74	#1277	—	5	1	no	—
75	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
77	C₂²×C₄×C₅	#41	1	1	yes	characteristic
78	C₂×C₂²⋊D₅	—	2	2	no	—
79	#709	—	4	1	no	—
80	#709	—	4	1	no	—
81	#709	—	4	1	no	—
82	#709	—	4	1	no	—
83	C₂×D₂₀	—	2	1	no	—
84	C₂×D₂₀	—	2	1	no	—
85	C₂×D₂₀	C₄⋊C₄	1	2	yes	—
86	C₂×C₄×D₅	—	2	2	no	—
87	C₂×Dic₁₀	—	2	1	no	—
88	C₅×M₁₆	—	2	1	no	—
89	C₅×M₁₆	—	2	1	no	—
90	C₅×M₁₆	—	2	1	no	—
91	C₅×M₁₆	—	4	1	no	—
92	C₅×M₁₆	—	4	1	no	—
93	C₅×M₁₆	D₈	1	2	yes	—
94	C₂×C₅×C₈	—	2	1	no	—
95	C₂×C₅×C₈	—	2	1	no	—
96	C₂×C₅×C₈	—	2	1	no	—
97	C₂×C₅×C₈	—	2	1	no	—
98	C₂×C₅×C₈	—	4	1	no	—
99	C₂×C₅×C₈	—	4	1	no	—
100	C₂×C₅×C₈	—	4	1	no	—
101	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₅	#41	1	1	yes	characteristic
105	#685	—	2	2	no	—
106	#685	—	2	2	no	—
107	C₄⋊Dic₅	—	2	1	no	—
108	C₄⋊Dic₅	C₄⋊C₄	1	2	yes	—
109	C₂×#168	—	2	2	no	—
110	C₂×#168	—	2	1	no	—
111	C₂×#168	—	2	1	no	—
112	C₂×#168	—	2	1	no	—
113	C₂×#168	—	2	1	no	—
114	C₂×#168	—	4	1	no	—
115	C₂×#168	—	4	1	no	—
116	C₂×#168	—	4	1	no	—
117	C₂×#168	—	4	1	no	—
118	#676	—	4	4	no	—
119	C₈×D₅	—	4	4	no	—
120	C₂×#131	—	10	1	no	—
121	#532	—	5	1	no	—
122	#436	—	10	1	no	—
123	#435	—	10	2	no	—
124	C₈×D₄	—	10	1	no	—
125	#431	—	5	1	no	—
126	C₂×#108	—	2	1	no	—
127	#423	C₂×D₅	1	1	yes	characteristic
128	C₂×C₄⋊C₈	—	5	1	no	—
129	C₂×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
130	#408	—	10	1	no	—
131	C₄×M₁₆	C₂×D₅	1	1	yes	characteristic
132	C₂×C₄×C₈	—	5	1	no	—
133	#341	—	2	1	no	—
134	#335	—	10	1	no	—
135	#335	—	10	1	no	—
136	#332	—	5	1	no	—
137	#332	—	5	1	no	—
138	#331	—	5	1	no	—
139	#331	—	5	1	no	—
140	D₄⋊C₈	—	10	2	no	—
141	C₂³×C₅	C₂²⋊Q₈	1	1	yes	characteristic
142	C₂²×D₅	—	2	2	no	—
143	C₂×C₄×C₅	—	2	1	no	—
144	C₂×C₄×C₅	—	2	1	no	—
145	C₂×C₄×C₅	—	4	1	no	—
146	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
149	C₂×C₄×C₅	#120	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
151	C₂×C₄×C₅	#123	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
153	C₂×C₄×C₅	#135	1	1	yes	characteristic
154	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
155	C₂×C₄×C₅	#137	1	1	yes	characteristic
156	C₂²⋊D₅	—	4	2	no	—
157	C₂²⋊D₅	—	4	2	no	—
158	C₂×Dic₅	—	2	2	no	—
159	D₂₀	—	4	2	no	—
160	D₂₀	—	4	1	no	—
161	D₂₀	—	4	1	no	—
162	D₂₀	—	4	1	no	—
163	D₂₀	—	4	1	no	—
164	C₄×D₅	—	4	2	no	—
165	C₄×D₅	—	4	2	no	—
166	Dic₁₀	—	4	1	no	—
167	Dic₁₀	—	4	1	no	—
168	C₅×C₈	—	2	2	no	—
169	C₅×C₈	—	4	1	no	—
170	C₅×C₈	—	4	1	no	—
171	C₅×C₈	—	4	1	no	—
172	C₅×C₈	—	8	1	no	—
173	#168	—	4	2	no	—
174	#168	—	4	1	no	—
175	#168	—	4	1	no	—
176	#168	—	4	1	no	—
177	#168	—	4	1	no	—
178	C₂×#41	—	10	1	no	—
179	#131	—	20	2	no	—
180	#131	—	20	2	no	—
181	C₂×M₁₆	—	2	1	no	—
182	C₂×M₁₆	—	10	2	no	—
183	C₂×M₁₆	C₂²⋊D₅	1	1	yes	characteristic
184	C₂×M₁₆	C₂²⋊D₅	1	1	yes	characteristic
185	C₂²×C₈	—	2	1	no	—
186	C₂²×C₈	—	10	2	no	—
187	C₂²×C₈	—	10	1	no	—
188	C₂²×C₈	—	10	1	no	—
189	C₄⋊Q₈	—	5	1	no	—
190	#127	—	5	1	no	—
191	#124	—	10	1	no	—
192	#121	—	5	2	no	—
193	#121	—	10	1	no	—
194	C₄×D₄	—	5	2	no	—
195	C₄×D₄	—	10	1	no	—
196	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
197	#108	—	4	2	no	—
198	C₄⋊C₈	—	2	1	no	—
199	C₄⋊C₈	—	2	1	no	—
200	C₄⋊C₈	—	2	1	no	—
201	C₄⋊C₈	—	5	1	no	—
202	C₄⋊C₈	—	5	1	no	—
203	C₄⋊C₈	—	5	1	no	—
204	C₄⋊C₈	—	5	1	no	—
205	C₄⋊C₈	—	10	1	no	—
206	C₄⋊C₈	C₄×D₅	1	1	yes	characteristic
207	C₄⋊C₈	C₄×D₅	1	1	yes	characteristic
208	C₂²⋊C₈	—	10	2	no	—
209	C₂²⋊C₈	—	10	2	no	—
210	C₂²⋊C₈	—	10	1	no	—
211	#97	—	2	1	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	—	10	1	no	—
214	C₄×C₈	—	10	1	no	—
215	C₄×C₈	—	10	1	no	—
216	C₂²×C₅	—	2	1	no	—
217	C₂²×C₅	—	2	1	no	—
218	C₂²×C₅	#385	1	1	yes	characteristic
219	C₂²×C₅	#474	1	1	yes	characteristic
220	C₂²×C₅	#478	1	1	yes	characteristic
221	C₂×D₅	—	4	2	no	—
222	C₂×D₅	—	4	2	no	—
223	C₂×D₅	—	4	2	no	—
224	C₄×C₅	—	2	1	no	—
225	C₄×C₅	—	2	1	no	—
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	C₄×D₈	1	1	yes	characteristic
228	C₄×C₅	#440	1	1	yes	characteristic
229	C₄×C₅	#443	1	1	yes	characteristic
230	C₄×C₅	#444	1	1	yes	characteristic
231	C₄×C₅	#447	1	1	yes	characteristic
232	C₄×C₅	#452	1	1	yes	characteristic
233	Dic₅	—	4	2	no	—
234	#41	—	20	1	no	—
235	#41	—	20	1	no	—
236	#41	—	20	1	no	—
237	#41	—	20	1	no	—
238	C₂×Q₈	—	10	1	no	—
239	C₂×D₄	—	5	2	no	—
240	C₂×D₄	—	10	2	no	—
241	C₂×D₄	—	10	1	no	—
242	C₂×D₄	—	10	1	no	—
243	C₂²×C₄	—	10	2	no	—
244	C₂²×C₄	Q₈×D₅	1	1	yes	characteristic
245	C₂²×C₄	#713	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
247	M₁₆	—	2	1	no	—
248	M₁₆	—	2	1	no	—
249	M₁₆	—	2	1	no	—
250	M₁₆	—	4	1	no	—
251	M₁₆	—	4	1	no	—
252	M₁₆	—	20	4	no	—
253	M₁₆	#686	1	2	yes	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	—	10	2	no	—
262	C₂×C₈	—	10	1	no	—
263	C₂×C₈	—	10	1	no	—
264	C₂×C₈	—	10	1	no	—
265	C₂×C₈	—	10	1	no	—
266	C₂×C₈	—	20	4	no	—
267	C₂×C₈	—	20	1	no	—
268	C₂×C₈	—	20	1	no	—
269	C₂×C₈	—	20	1	no	—
270	C₂×C₈	—	20	1	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	C₂×C₄×D₅	1	1	yes	characteristic
276	C₄²	#709	1	1	yes	characteristic
277	C₄²	D₄×D₅	1	1	yes	characteristic
278	C₄²	D₄×D₅	1	1	yes	characteristic
279	C₂×C₅	—	2	1	no	—
280	C₂×C₅	—	2	1	no	—
281	C₂×C₅	#1858	1	1	yes	characteristic
282	C₂×C₅	#1862	1	1	yes	characteristic
283	C₂×C₅	#1866	1	1	yes	characteristic
284	D₅	—	8	2	no	—
285	C₂³	—	10	2	no	—
286	C₂³	#4092	1	1	yes	characteristic
287	Q₈	—	20	1	no	—
288	Q₈	—	20	1	no	—
289	D₄	—	20	2	no	—
290	D₄	—	20	2	no	—
291	D₄	—	20	2	no	—
292	D₄	—	20	1	no	—
293	D₄	—	20	1	no	—
294	D₄	—	20	1	no	—
295	D₄	—	20	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	10	2	no	—
300	C₂×C₄	—	20	2	no	—
301	C₂×C₄	—	20	2	no	—
302	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
303	C₂×C₄	#4039	1	1	yes	characteristic
304	C₂×C₄	#4040	1	1	yes	characteristic
305	C₂×C₄	#4042	1	1	yes	characteristic
306	C₂×C₄	C₄×C₂²⋊D₅	1	1	yes	characteristic
307	C₂×C₄	C₂×#686	1	1	yes	characteristic
308	C₂×C₄	#4083	1	1	yes	characteristic
309	C₂×C₄	#4088	1	1	yes	characteristic
310	C₂×C₄	#4095	1	1	yes	characteristic
311	C₂×C₄	#4096	1	1	yes	characteristic
312	C₈	—	2	2	no	—
313	C₈	—	4	1	no	—
314	C₈	—	4	1	no	—
315	C₈	—	4	1	no	—
316	C₈	—	8	1	no	—
317	C₈	—	20	2	no	—
318	C₈	—	20	1	no	—
319	C₈	—	20	1	no	—
320	C₈	—	20	1	no	—
321	C₈	—	20	1	no	—
322	C₅	#8930	1	1	yes	characteristic, complemented, Sylow
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	20	2	no	—
326	C₂²	—	20	2	no	—
327	C₂²	—	20	2	no	—
328	C₂²	#65278	1	1	yes	characteristic
329	C₂²	#65355	1	1	yes	characteristic
330	C₂²	#65376	1	1	yes	characteristic
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	20	2	no	—
335	C₄	#65183	1	1	yes	characteristic
336	C₄	#65184	1	1	yes	characteristic
337	C₄	C₄×#686	1	1	yes	characteristic
338	C₄	#65315	1	1	yes	characteristic
339	C₄	#65325	1	1	yes	characteristic
340	C₄	#65336	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	40	2	no	—
344	C₂	#10600418	1	1	yes	characteristic
345	C₂	#10601515	1	1	yes	characteristic
346	C₂	#10601527	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#4083
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1950	C₂
socle	C₂²×C₅	#385
prime radical	#65811	C₂²
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1950	D₅
hypocenter	C₅	#8930
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1950

Complements

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

N	H ≅ G/N	type
2. C₅×#1950	C₂	semidirect
4. #10601287	C₂	semidirect
6. #10598194	C₂	semidirect
7. #10598158	C₂	semidirect
69. #1950	D₅	semidirect
322. C₅	#8930	semidirect

Maximal subgroups

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

subgroup	index
#10610253	2
C₅×#1950	2
#10601291	2
#10601287	2
#10601224	2
#10598194	2
#10598158	2
#8930	5