Subgroups — #11992018

Order 1280.

472 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2295	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈⋊#168	C₂	1	2	yes	maximal
3	#10600185	C₂	1	1	yes	maximal, characteristic
4	C₄×C₄⋊#168	C₂	1	1	yes	maximal, characteristic
5	#10598110	C₂	1	2	yes	maximal
6	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
7	C₅×#391	C₄	1	1	yes	characteristic
8	C₅×#391	C₄	1	1	yes	characteristic
9	C₅×#389	C₄	1	1	yes	characteristic
10	C₅×#389	C₄	1	1	yes	characteristic
11	C₅×#387	—	2	1	no	—
12	C₅×#387	—	2	1	no	—
13	C₄×C₅×C₄⋊C₄	—	2	1	no	—
14	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₅×#376	—	2	1	no	—
16	C₄³×C₅	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊#168	—	2	1	no	—
18	C₂×C₄⋊#168	C₂²	1	2	yes	—
19	C₂×C₄×#168	C₂²	1	2	yes	—
20	Q₈⋊#168	—	2	4	no	—
21	#64744	—	2	1	no	—
22	#64744	—	2	1	no	—
23	#64744	—	2	1	no	—
24	#8570	—	5	1	no	maximal, Sylow
25	C₂²×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₅×Q₈	—	2	1	no	—
27	C₄×C₅×Q₈	—	2	1	no	—
28	C₄×C₅×Q₈	D₄	1	2	yes	—
29	C₄×C₅×Q₈	Q₈	1	2	yes	—
30	C₂×C₅×C₄⋊C₄	—	2	1	no	—
31	C₂×C₅×C₄⋊C₄	—	2	1	no	—
32	C₂×C₅×C₄⋊C₄	—	2	1	no	—
33	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
34	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
35	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅	—	2	1	no	—
37	C₂×C₄²×C₅	—	2	1	no	—
38	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
45	C₂²×#168	—	2	2	no	—
46	C₂²×#168	—	2	2	no	—
47	C₅×#95	—	2	1	no	—
48	C₅×#95	—	2	1	no	—
49	C₅×#95	—	2	1	no	—
50	C₅×#95	—	2	1	no	—
51	C₅×#95	—	2	1	no	—
52	C₅×#95	—	2	1	no	—
53	C₅×#95	—	2	1	no	—
54	C₄⋊#168	—	2	4	no	—
55	C₄⋊#168	—	2	2	no	—
56	C₄⋊#168	—	2	2	no	—
57	C₄×#168	—	2	2	no	—
58	C₄×#168	—	2	2	no	—
59	#2295	D₅	1	1	yes	characteristic
60	#1831	—	5	1	no	—
61	C₄×C₄⋊C₈	—	5	1	no	—
62	C₂×Q₈⋊C₈	—	5	2	no	—
63	#1279	—	5	2	no	—
64	C₂×C₅×Q₈	—	2	1	no	—
65	C₂×C₅×Q₈	—	2	2	no	—
66	C₂×C₅×Q₈	—	2	2	no	—
67	C₂×C₅×Q₈	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₅	—	2	1	no	—
69	C₂²×C₄×C₅	—	2	1	no	—
70	C₂²×C₄×C₅	—	2	1	no	—
71	C₂²×C₄×C₅	—	2	1	no	—
72	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₅×C₄⋊C₄	—	2	1	no	—
80	C₅×C₄⋊C₄	—	2	2	no	—
81	C₅×C₄⋊C₄	—	2	2	no	—
82	C₅×C₄⋊C₄	—	2	2	no	—
83	C₅×C₄⋊C₄	—	2	2	no	—
84	C₅×C₄⋊C₄	—	4	2	no	—
85	C₅×C₄⋊C₄	—	4	2	no	—
86	C₅×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
87	C₄²×C₅	—	2	2	no	—
88	C₄²×C₅	—	2	2	no	—
89	C₄²×C₅	—	2	2	no	—
90	C₄²×C₅	—	2	2	no	—
91	C₄²×C₅	—	2	2	no	—
92	C₄²×C₅	—	2	2	no	—
93	C₄²×C₅	—	2	2	no	—
94	C₄²×C₅	—	4	1	no	—
95	C₄²×C₅	—	4	2	no	—
96	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
98	C₄²×C₅	#41	1	1	yes	characteristic
99	C₄²×C₅	#41	1	1	yes	characteristic
100	C₂×#168	—	2	4	no	—
101	C₂×#168	—	4	2	no	—
102	C₂×#168	—	4	2	no	—
103	C₂×#168	—	4	2	no	—
104	C₂×#168	—	4	2	no	—
105	C₂×#168	—	4	2	no	—
106	C₂×C₄×Q₈	C₂×D₅	1	1	yes	characteristic
107	C₂×C₄⋊C₈	—	5	2	no	—
108	C₂×C₄⋊C₈	—	10	1	no	—
109	C₂×C₄×C₈	—	5	2	no	—
110	#391	Dic₅	1	1	yes	characteristic
111	#391	Dic₅	1	1	yes	characteristic
112	#389	Dic₅	1	1	yes	characteristic
113	#389	Dic₅	1	1	yes	characteristic
114	#387	—	2	1	no	—
115	#387	—	2	1	no	—
116	C₄×C₄⋊C₄	—	2	1	no	—
117	C₄×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
118	#376	—	2	1	no	—
119	C₄³	C₂×D₅	1	1	yes	characteristic
120	#336	—	10	1	no	—
121	#336	—	10	1	no	—
122	#336	—	10	1	no	—
123	Q₈⋊C₈	—	10	4	no	—
124	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₅×Q₈	—	2	4	no	—
126	C₅×Q₈	—	4	2	no	—
127	C₂×C₄×C₅	—	2	1	no	—
128	C₂×C₄×C₅	—	2	1	no	—
129	C₂×C₄×C₅	—	2	1	no	—
130	C₂×C₄×C₅	—	2	1	no	—
131	C₂×C₄×C₅	—	2	2	no	—
132	C₂×C₄×C₅	—	2	2	no	—
133	C₂×C₄×C₅	—	2	1	no	—
134	C₂×C₄×C₅	—	2	1	no	—
135	C₂×C₄×C₅	—	2	1	no	—
136	C₂×C₄×C₅	—	2	1	no	—
137	C₂×C₄×C₅	—	2	1	no	—
138	C₂×C₄×C₅	—	2	1	no	—
139	C₂×C₄×C₅	—	2	1	no	—
140	C₂×C₄×C₅	—	2	1	no	—
141	C₂×C₄×C₅	—	2	1	no	—
142	C₂×C₄×C₅	—	4	1	no	—
143	C₂×C₄×C₅	—	4	1	no	—
144	C₂×C₄×C₅	—	4	1	no	—
145	C₂×C₄×C₅	—	4	1	no	—
146	C₂×C₄×C₅	—	4	1	no	—
147	C₂×C₄×C₅	—	4	1	no	—
148	C₂×C₄×C₅	—	4	1	no	—
149	C₂×C₄×C₅	—	4	1	no	—
150	C₂×C₄×C₅	—	4	1	no	—
151	C₂×C₄×C₅	—	4	1	no	—
152	C₂×C₄×C₅	#104	1	2	yes	—
153	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₂×C₄×C₅	#117	1	1	yes	characteristic
155	C₂×C₄×C₅	#121	1	1	yes	characteristic
156	C₂×C₄×C₅	#121	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
158	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
159	C₂×C₄×C₅	#131	1	2	yes	—
160	C₂×C₄×C₅	#131	1	2	yes	—
161	C₂×C₄×C₅	#135	1	2	yes	—
162	C₂×C₄×C₅	#137	1	2	yes	—
163	#168	—	4	4	no	—
164	#168	—	8	2	no	—
165	C₂²×Q₈	C₂×Dic₅	1	1	yes	characteristic
166	C₂²×C₈	—	10	2	no	—
167	C₂²×C₈	—	10	2	no	—
168	C₄×Q₈	—	2	1	no	—
169	C₄×Q₈	—	2	1	no	—
170	C₄×Q₈	Dic₁₀	1	2	yes	—
171	C₄×Q₈	D₂₀	1	2	yes	—
172	C₂×C₄⋊C₄	—	2	1	no	—
173	C₂×C₄⋊C₄	—	2	1	no	—
174	C₂×C₄⋊C₄	—	2	1	no	—
175	C₂×C₄⋊C₄	C₂×Dic₅	1	1	yes	characteristic
176	C₂×C₄⋊C₄	C₂×Dic₅	1	1	yes	characteristic
177	C₂×C₄⋊C₄	C₂×Dic₅	1	1	yes	characteristic
178	C₂×C₄²	—	2	1	no	—
179	C₂×C₄²	—	2	1	no	—
180	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
181	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
182	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
183	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
184	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
185	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
186	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
187	C₄⋊C₈	—	10	4	no	—
188	C₄⋊C₈	—	10	2	no	—
189	C₄⋊C₈	—	10	2	no	—
190	C₄×C₈	—	10	2	no	—
191	C₄×C₈	—	10	2	no	—
192	#95	—	2	1	no	—
193	#95	—	2	1	no	—
194	#95	—	2	1	no	—
195	#95	—	2	1	no	—
196	#95	—	2	1	no	—
197	#95	—	2	1	no	—
198	#95	—	2	1	no	—
199	C₂²×C₅	#380	1	1	yes	characteristic
200	C₂²×C₅	#408	1	1	yes	characteristic
201	C₂²×C₅	#408	1	1	yes	characteristic
202	C₂²×C₅	#416	1	1	yes	characteristic
203	C₂²×C₅	#419	1	1	yes	characteristic
204	C₂²×C₅	C₂×#104	1	1	yes	characteristic
205	C₂²×C₅	#421	1	1	yes	characteristic
206	C₄×C₅	—	2	1	no	—
207	C₄×C₅	—	2	1	no	—
208	C₄×C₅	—	2	2	no	—
209	C₄×C₅	—	4	2	no	—
210	C₄×C₅	—	4	2	no	—
211	C₄×C₅	—	4	1	no	—
212	C₄×C₅	—	4	1	no	—
213	C₄×C₅	—	4	1	no	—
214	C₄×C₅	—	8	1	no	—
215	C₄×C₅	—	8	1	no	—
216	C₄×C₅	#424	1	2	yes	—
217	C₄×C₅	#433	1	2	yes	—
218	C₄×C₅	#464	1	2	yes	—
219	C₄×C₅	#479	1	2	yes	—
220	C₂×Q₈	—	2	1	no	—
221	C₂×Q₈	—	2	2	no	—
222	C₂×Q₈	—	2	2	no	—
223	C₂×Q₈	C₄⋊Dic₅	1	2	yes	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
229	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
230	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
231	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
232	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
235	C₂×C₈	—	10	4	no	—
236	C₂×C₈	—	20	2	no	—
237	C₂×C₈	—	20	2	no	—
238	C₂×C₈	—	20	2	no	—
239	C₂×C₈	—	20	2	no	—
240	C₂×C₈	—	20	2	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	1	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₄⋊C₄	C₄⋊Dic₅	1	2	yes	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	4	1	no	—
257	C₄²	—	4	2	no	—
258	C₄²	C₂×Dic₁₀	1	1	yes	characteristic
259	C₄²	C₂×D₂₀	1	1	yes	characteristic
260	C₄²	#709	1	1	yes	characteristic
261	C₄²	#709	1	1	yes	characteristic
262	C₂×C₅	#1482	1	1	yes	characteristic
263	C₂×C₅	#1485	1	1	yes	characteristic
264	C₂×C₅	#1494	1	1	yes	characteristic
265	C₂×C₅	#1497	1	1	yes	characteristic
266	C₂×C₅	#1798	1	1	yes	characteristic
267	C₂×C₅	#1805	1	1	yes	characteristic
268	C₂×C₅	#1806	1	1	yes	characteristic
269	C₂³	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
270	Q₈	—	2	4	no	—
271	Q₈	—	4	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	#3969	1	2	yes	—
298	C₂×C₄	#4070	1	1	yes	characteristic
299	C₂×C₄	#4070	1	1	yes	characteristic
300	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
301	C₂×C₄	#4072	1	1	yes	characteristic
302	C₂×C₄	#4076	1	1	yes	characteristic
303	C₂×C₄	#4076	1	1	yes	characteristic
304	C₂×C₄	#4088	1	2	yes	—
305	C₂×C₄	#4094	1	2	yes	—
306	C₂×C₄	#4094	1	2	yes	—
307	C₂×C₄	#4096	1	2	yes	—
308	C₈	—	20	4	no	—
309	C₈	—	40	2	no	—
310	C₅	#8570	1	1	yes	characteristic, complemented, Sylow
311	C₂²	#65501	1	1	yes	characteristic
312	C₂²	#65514	1	1	yes	characteristic
313	C₂²	#65522	1	1	yes	characteristic
314	C₂²	#65523	1	1	yes	characteristic
315	C₂²	#65523	1	1	yes	characteristic
316	C₂²	C₂×#3969	1	1	yes	characteristic
317	C₂²	#65525	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	4	2	no	—
325	C₄	—	4	2	no	—
326	C₄	—	8	1	no	—
327	C₄	—	8	1	no	—
328	C₄	#65288	1	2	yes	—
329	C₄	#65289	1	2	yes	—
330	C₄	#65311	1	2	yes	—
331	C₄	#65317	1	2	yes	—
332	C₂	#10602376	1	1	yes	characteristic
333	C₂	#10602420	1	1	yes	characteristic
334	C₂	#10602439	1	1	yes	characteristic
335	C₂	#10602543	1	1	yes	characteristic
336	C₂	#10602545	1	1	yes	characteristic
337	C₂	#10602618	1	1	yes	characteristic
338	C₂	#10602620	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅
commutator	C₂²×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#2295	C₂
socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#2295	D₅
hypocenter	C₅	#8570
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #2295

Complements

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

N	H ≅ G/N	type
310. C₅	#8570	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2295	2
C₂×Q₈⋊#168	2
#10600185	2
C₄×C₄⋊#168	2
#10598110	2
#8570	5