Subgroups — #11990049

Order 1280.

500 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10611008	C₂	1	1	yes	maximal, characteristic
2	C₄×#3957	C₂	1	1	yes	maximal, characteristic
3	M₁₆×#168	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#3947	C₂	1	1	yes	maximal, characteristic, complemented
5	#10600527	C₂	1	1	yes	maximal, characteristic
6	C₅×#1278	C₂	1	1	yes	maximal, characteristic, complemented
7	#10598227	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#709	C₂²	1	1	yes	characteristic
9	C₅×#431	C₂²	1	1	yes	characteristic
10	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₂×#3957	C₄	1	2	yes	—
13	C₂×#3947	—	2	2	no	—
14	C₄²×Dic₅	C₂²	1	1	yes	characteristic
15	#65214	C₂²	1	1	yes	characteristic
16	C₂×C₄×#168	C₂²	1	1	yes	characteristic
17	#65127	C₄	1	2	yes	complemented
18	#65120	C₄	1	2	yes	complemented
19	#65019	—	2	2	no	—
20	C₄×#676	—	2	1	no	—
21	#64974	—	2	1	no	—
22	#64775	C₄	1	2	yes	—
23	#64679	C₄	1	2	yes	complemented
24	#64676	C₄	1	2	yes	complemented
25	#64675	—	2	1	no	—
26	C₈×#168	—	2	1	no	—
27	#8196	—	5	1	no	maximal, Sylow
28	C₂×#709	C₂×C₄	1	1	yes	characteristic
29	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
30	C₂²×C₅×C₈	—	2	1	no	—
31	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
32	C₄×C₂²⋊D₅	—	2	1	no	—
33	C₄×C₂²⋊D₅	—	2	1	no	—
34	#4072	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×Dic₅	—	2	1	no	—
36	C₂×C₄×Dic₅	—	2	1	no	—
37	C₂×C₄×Dic₅	C₂×C₄	1	2	yes	—
38	C₂²×#168	D₄	1	2	yes	—
39	C₂×#676	—	2	2	no	—
40	C₄×D₂₀	C₂×C₄	1	1	yes	characteristic
41	C₄×D₂₀	C₂×C₄	1	1	yes	characteristic
42	#4018	—	2	1	no	—
43	C₄²×D₅	—	2	1	no	—
44	C₄×Dic₁₀	C₂×C₄	1	1	yes	characteristic
45	C₄×Dic₁₀	C₂×C₄	1	1	yes	characteristic
46	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
47	C₅×C₂²⋊C₈	—	2	1	no	—
48	C₅×#97	—	2	1	no	—
49	C₄×C₅×C₈	—	2	1	no	—
50	C₄×C₅×C₈	—	2	1	no	—
51	C₄×C₅×C₈	—	2	1	no	—
52	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
53	C₂²⋊#168	—	2	1	no	—
54	#3957	—	2	4	no	—
55	#3952	—	2	2	no	—
56	#3947	—	2	8	no	—
57	#3947	—	2	2	no	—
58	#3946	—	2	2	no	—
59	C₄⋊#168	C₂×C₄	1	2	yes	—
60	C₄×#168	—	2	1	no	—
61	C₄×#168	C₂×C₄	1	2	yes	—
62	C₄×#168	D₄	1	1	yes	characteristic
63	C₄×#168	D₄	1	1	yes	characteristic
64	#2965	—	5	1	no	—
65	C₄×#104	—	5	1	no	—
66	C₄×#97	—	5	1	no	—
67	#1476	—	5	1	no	—
68	C₈×M₁₆	—	5	1	no	—
69	#1278	D₅	1	1	yes	characteristic, complemented
70	#1276	—	5	1	no	—
71	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₅	#41	1	1	yes	characteristic
74	C₂×C₂²⋊D₅	—	2	1	no	—
75	C₂²×Dic₅	—	2	2	no	—
76	#709	—	2	2	no	—
77	#709	—	2	2	no	—
78	C₂×D₂₀	C₄²	1	1	yes	characteristic
79	C₂×C₄×D₅	—	2	1	no	—
80	C₂×C₄×D₅	—	2	1	no	—
81	C₂×C₄×D₅	—	2	1	no	—
82	C₂×Dic₁₀	C₄²	1	1	yes	characteristic
83	C₅×M₁₆	—	2	2	no	—
84	C₅×M₁₆	M₁₆	1	4	yes	complemented
85	C₂×C₅×C₈	—	2	4	no	—
86	C₂×C₅×C₈	—	2	2	no	—
87	C₂×C₅×C₈	—	2	2	no	—
88	C₂×C₅×C₈	—	2	2	no	—
89	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₅	#41	1	1	yes	characteristic
93	C₂²⋊Dic₅	—	2	1	no	—
94	#685	—	2	1	no	—
95	#685	—	2	1	no	—
96	C₄⋊Dic₅	C₄²	1	1	yes	characteristic
97	C₄⋊Dic₅	C₄²	1	1	yes	characteristic
98	#683	—	2	1	no	—
99	#683	—	2	1	no	—
100	C₄×Dic₅	—	2	4	no	—
101	C₄×Dic₅	—	2	1	no	—
102	C₄×Dic₅	—	2	1	no	—
103	C₄×Dic₅	—	2	1	no	—
104	C₄×Dic₅	—	2	2	no	—
105	C₄×Dic₅	—	2	2	no	—
106	C₄×Dic₅	—	4	2	no	—
107	C₂×#168	—	2	4	no	—
108	C₂×#168	—	2	2	no	—
109	C₂×#168	C₂²⋊C₄	1	2	yes	—
110	C₂×#168	C₂²⋊C₄	1	2	yes	—
111	#676	—	4	4	no	—
112	C₄×#41	—	5	1	no	—
113	#446	—	5	2	no	—
114	#436	—	5	2	no	—
115	#435	—	10	2	no	—
116	#431	—	5	1	no	—
117	#431	—	10	1	no	—
118	#431	C₂×D₅	1	1	yes	characteristic
119	C₂×#104	—	5	2	no	—
120	C₄×M₁₆	—	10	1	no	—
121	C₄×M₁₆	C₂×D₅	1	1	yes	characteristic
122	C₂×#97	—	10	2	no	—
123	C₂×C₄×C₈	—	5	1	no	—
124	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
125	C₄³	—	5	1	no	—
126	#339	—	5	2	no	—
127	Q₈⋊C₈	—	5	2	no	—
128	D₄⋊C₈	—	5	2	no	—
129	#322	—	10	1	no	—
130	C₈²	—	10	1	no	—
131	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
132	C₂²×D₅	—	2	1	no	—
133	C₂×C₄×C₅	—	2	2	no	—
134	C₂×C₄×C₅	—	2	2	no	—
135	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₅	#117	1	1	yes	characteristic
138	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
142	C₂×C₄×C₅	#131	1	2	yes	—
143	C₂²⋊D₅	—	4	2	no	—
144	C₂×Dic₅	—	2	4	no	—
145	C₂×Dic₅	—	2	1	no	—
146	C₂×Dic₅	—	2	1	no	—
147	C₂×Dic₅	—	2	1	no	—
148	C₂×Dic₅	—	4	2	no	—
149	C₂×Dic₅	—	4	2	no	—
150	D₂₀	—	2	1	no	—
151	D₂₀	—	2	1	no	—
152	C₄×D₅	—	4	2	no	—
153	C₄×D₅	—	4	2	no	—
154	C₄×D₅	—	4	2	no	—
155	Dic₁₀	—	2	1	no	—
156	Dic₁₀	—	2	1	no	—
157	C₅×C₈	—	2	4	no	—
158	C₅×C₈	—	2	4	no	—
159	C₅×C₈	—	4	2	no	—
160	#168	—	2	2	no	—
161	#168	—	4	2	no	—
162	#168	#104	1	4	yes	complemented
163	C₂×#41	—	5	1	no	—
164	C₂×M₁₆	—	10	2	no	—
165	C₂×M₁₆	C₄×D₅	1	2	yes	—
166	C₂²×C₈	—	2	1	no	—
167	C₂²×C₈	—	5	2	no	—
168	C₄×Q₈	—	5	1	no	—
169	C₄×Q₈	—	5	1	no	—
170	C₄×D₄	—	5	1	no	—
171	C₄×D₄	—	5	1	no	—
172	C₄×D₄	—	10	1	no	—
173	C₄×D₄	—	10	1	no	—
174	#117	—	5	1	no	—
175	#117	—	10	1	no	—
176	C₂×C₄²	—	5	2	no	—
177	C₂×C₄²	—	10	1	no	—
178	C₂×C₄²	—	10	1	no	—
179	C₂×C₄²	—	10	1	no	—
180	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
181	C₄⋊C₈	—	5	2	no	—
182	C₄⋊C₈	—	10	2	no	—
183	C₄⋊C₈	C₄×D₅	1	2	yes	—
184	#104	—	10	4	no	—
185	C₂²⋊C₈	—	2	1	no	—
186	C₂²⋊C₈	—	10	2	no	—
187	C₂²⋊C₈	—	10	1	no	—
188	#97	—	2	1	no	—
189	#97	—	10	2	no	—
190	#97	—	10	8	no	—
191	C₄×C₈	—	2	1	no	—
192	C₄×C₈	—	2	1	no	—
193	C₄×C₈	—	2	1	no	—
194	C₄×C₈	—	5	2	no	—
195	C₄×C₈	—	5	1	no	—
196	C₄×C₈	—	5	1	no	—
197	C₄×C₈	—	10	1	no	—
198	C₄×C₈	C₄×D₅	1	2	yes	—
199	C₂²×C₅	—	2	1	no	—
200	C₂²×C₅	—	2	1	no	—
201	C₂²×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
202	C₂²×C₅	#436	1	2	yes	—
203	C₂×D₅	—	4	1	no	—
204	C₂×D₅	—	4	1	no	—
205	C₂×D₅	—	4	1	no	—
206	C₄×C₅	—	2	1	no	—
207	C₄×C₅	—	2	1	no	—
208	C₄×C₅	C₄×M₁₆	1	1	yes	characteristic
209	C₄×C₅	#405	1	1	yes	characteristic
210	C₄×C₅	C₂×#104	1	2	yes	—
211	C₄×C₅	#435	1	2	yes	—
212	C₄×C₅	#444	1	2	yes	—
213	Dic₅	—	4	4	no	—
214	Dic₅	—	4	1	no	—
215	Dic₅	—	4	1	no	—
216	Dic₅	—	4	1	no	—
217	#41	—	10	2	no	—
218	#41	—	10	2	no	—
219	C₂×Q₈	—	5	1	no	—
220	C₂×D₄	—	5	1	no	—
221	C₂×D₄	—	10	1	no	—
222	C₂²×C₄	—	10	2	no	—
223	C₂²×C₄	—	10	1	no	—
224	C₂²×C₄	—	10	1	no	—
225	C₂²×C₄	—	10	1	no	—
226	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
227	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
228	C₂²×C₄	#713	1	1	yes	characteristic
229	M₁₆	—	2	2	no	—
230	M₁₆	—	20	4	no	—
231	M₁₆	#676	1	4	yes	complemented
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	5	2	no	—
237	C₂×C₈	—	5	2	no	—
238	C₂×C₈	—	10	4	no	—
239	C₂×C₈	—	10	2	no	—
240	C₄⋊C₄	—	5	1	no	—
241	C₄⋊C₄	—	5	1	no	—
242	C₄⋊C₄	—	10	1	no	—
243	C₄⋊C₄	—	10	1	no	—
244	C₂²⋊C₄	—	10	1	no	—
245	C₂²⋊C₄	—	10	1	no	—
246	C₂²⋊C₄	—	10	1	no	—
247	C₄²	—	10	4	no	—
248	C₄²	—	10	1	no	—
249	C₄²	—	10	1	no	—
250	C₄²	—	10	1	no	—
251	C₄²	—	10	2	no	—
252	C₄²	—	10	2	no	—
253	C₄²	—	20	2	no	—
254	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
255	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
256	C₄²	D₄×D₅	1	1	yes	characteristic
257	C₄²	#711	1	1	yes	characteristic
258	C₂×C₅	—	2	2	no	—
259	C₂×C₅	#1753	1	1	yes	characteristic
260	C₂×C₅	C₄×#104	1	1	yes	characteristic
261	C₂×C₅	#1765	1	1	yes	characteristic
262	D₅	—	8	1	no	—
263	C₂³	—	10	1	no	—
264	C₂³	#4039	1	1	yes	characteristic
265	Q₈	—	10	1	no	—
266	Q₈	—	10	1	no	—
267	D₄	—	10	1	no	—
268	D₄	—	10	1	no	—
269	D₄	—	20	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	10	4	no	—
273	C₂×C₄	—	10	1	no	—
274	C₂×C₄	—	10	1	no	—
275	C₂×C₄	—	10	1	no	—
276	C₂×C₄	—	20	2	no	—
277	C₂×C₄	—	20	2	no	—
278	C₂×C₄	—	20	2	no	—
279	C₂×C₄	—	20	2	no	—
280	C₂×C₄	—	20	2	no	—
281	C₂×C₄	C₄²×D₅	1	1	yes	characteristic
282	C₂×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
283	C₂×C₄	#4027	1	1	yes	characteristic
284	C₂×C₄	#4027	1	1	yes	characteristic
285	C₂×C₄	#4038	1	1	yes	characteristic
286	C₂×C₄	#4039	1	1	yes	characteristic
287	C₂×C₄	C₂×#676	1	2	yes	—
288	C₂×C₄	#4053	1	2	yes	—
289	C₈	—	2	4	no	—
290	C₈	—	2	4	no	—
291	C₈	—	4	2	no	—
292	C₈	—	5	4	no	—
293	C₈	—	10	2	no	—
294	C₈	—	20	2	no	—
295	C₅	#8196	1	1	yes	characteristic, complemented, Sylow
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	20	1	no	—
299	C₂²	—	20	1	no	—
300	C₂²	—	20	1	no	—
301	C₂²	#64955	1	1	yes	characteristic
302	C₂²	#65127	1	2	yes	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	20	4	no	—
306	C₄	—	20	1	no	—
307	C₄	—	20	1	no	—
308	C₄	—	20	1	no	—
309	C₄	C₄×#676	1	1	yes	characteristic
310	C₄	#64997	1	1	yes	characteristic
311	C₄	#65019	1	2	yes	—
312	C₄	D₅×#104	1	2	yes	—
313	C₄	#65183	1	2	yes	—
314	C₂	—	2	2	no	—
315	C₂	—	40	1	no	—
316	C₂	#10599442	1	1	yes	characteristic
317	C₂	#10599521	1	1	yes	characteristic
318	C₂	#10599720	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₅
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1278	C₂
socle	C₂²×C₅	C₄×C₂²⋊C₄
prime radical	C₂×#709	C₂×C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1278	D₅
hypocenter	C₅	#8196
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1278

Complements

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

N	H ≅ G/N	type
3. M₁₆×#168	C₂	semidirect
4. C₄×#3947	C₂	semidirect
6. C₅×#1278	C₂	semidirect
7. #10598227	C₂	semidirect
17. #65127	C₄	semidirect
18. #65120	C₄	semidirect
23. #64679	C₄	semidirect
24. #64676	C₄	semidirect
69. #1278	D₅	semidirect
84. C₅×M₁₆	M₁₆	semidirect
162. #168	#104	semidirect
231. M₁₆	#676	semidirect
295. C₅	#8196	semidirect

Maximal subgroups

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

subgroup	index
#10611008	2
C₄×#3957	2
M₁₆×#168	2
C₄×#3947	2
#10600527	2
C₅×#1278	2
#10598227	2
#8196	5