Subgroups — #11989971

Order 1280.

530 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#4053	C₂	1	1	yes	maximal, characteristic
2	#10601547	C₂	1	1	yes	maximal, characteristic
3	#10601533	C₂	1	1	yes	maximal, characteristic, complemented
4	#10600527	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈×#168	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1278	C₂	1	1	yes	maximal, characteristic, complemented
7	#10598158	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#4053	C₄	1	2	yes	—
9	C₄×#709	C₂²	1	1	yes	characteristic
10	C₅×#431	C₂²	1	1	yes	characteristic
11	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	#65411	C₄	1	2	yes	—
14	#65214	C₂²	1	1	yes	characteristic
15	C₂×C₄×#168	C₂²	1	1	yes	characteristic
16	C₂×C₄×#168	C₂²	1	1	yes	characteristic
17	#64997	—	2	2	no	—
18	C₄×#676	—	2	1	no	—
19	C₄×C₈×D₅	—	2	1	no	—
20	#64704	C₄	1	2	yes	—
21	#64703	C₄	1	2	yes	—
22	#64679	C₄	1	2	yes	—
23	#64676	C₄	1	2	yes	—
24	#64674	—	2	1	no	—
25	C₈×#168	—	2	1	no	—
26	C₈×#168	—	2	2	no	—
27	#8184	—	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	C₂²×#168	—	2	1	no	—
36	C₂²×#168	C₂×C₄	1	2	yes	—
37	#4053	—	2	4	no	—
38	#4053	C₈	1	8	yes	complemented
39	C₂×#676	—	2	2	no	—
40	C₂×C₈×D₅	—	2	2	no	—
41	C₄×D₂₀	C₂×C₄	1	1	yes	characteristic
42	C₄×D₂₀	C₂×C₄	1	1	yes	characteristic
43	#4018	—	2	1	no	—
44	C₄²×D₅	—	2	1	no	—
45	C₄×Dic₁₀	C₂×C₄	1	1	yes	characteristic
46	C₄×Dic₁₀	C₂×C₄	1	1	yes	characteristic
47	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
48	C₅×C₂²⋊C₈	—	2	1	no	—
49	C₅×#97	—	2	1	no	—
50	C₄×C₅×C₈	—	2	1	no	—
51	C₄×C₅×C₈	—	2	1	no	—
52	C₄×C₅×C₈	—	2	1	no	—
53	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
54	C₂²⋊#168	—	2	1	no	—
55	#3947	—	2	2	no	—
56	C₈×Dic₅	—	2	2	no	—
57	C₄⋊#168	C₂×C₄	1	2	yes	—
58	C₄×#168	—	2	1	no	—
59	C₄×#168	—	2	1	no	—
60	C₄×#168	C₂×C₄	1	2	yes	—
61	C₄×#168	D₄	1	1	yes	characteristic
62	C₄×#168	D₄	1	2	yes	—
63	C₄×#168	D₄	1	1	yes	characteristic
64	C₄×#131	—	5	1	no	—
65	#1476	—	5	1	no	—
66	#1476	—	5	1	no	—
67	#1450	—	5	1	no	—
68	C₂×C₈²	—	5	1	no	—
69	#1278	—	5	1	no	—
70	#1278	D₅	1	1	yes	characteristic, complemented
71	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	#41	1	1	yes	characteristic
73	C₂²×C₄×C₅	#41	1	1	yes	characteristic
74	C₂×C₂²⋊D₅	—	2	1	no	—
75	#709	—	2	2	no	—
76	#709	C₂×C₈	1	4	yes	—
77	C₂×D₂₀	C₄²	1	1	yes	characteristic
78	C₂×C₄×D₅	—	2	1	no	—
79	C₂×C₄×D₅	—	2	1	no	—
80	C₂×C₄×D₅	—	2	1	no	—
81	C₂×Dic₁₀	C₄²	1	1	yes	characteristic
82	C₅×M₁₆	—	2	2	no	—
83	C₅×M₁₆	C₂×C₈	1	4	yes	—
84	C₂×C₅×C₈	—	2	4	no	—
85	C₂×C₅×C₈	—	2	2	no	—
86	C₂×C₅×C₈	—	2	2	no	—
87	C₂×C₅×C₈	—	2	2	no	—
88	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
92	C₂²⋊Dic₅	—	2	1	no	—
93	#685	—	2	1	no	—
94	#685	—	2	1	no	—
95	C₄⋊Dic₅	C₄²	1	1	yes	characteristic
96	C₄⋊Dic₅	C₄²	1	1	yes	characteristic
97	#683	—	2	1	no	—
98	#683	—	2	1	no	—
99	C₄×Dic₅	—	2	1	no	—
100	C₄×Dic₅	—	2	1	no	—
101	C₄×Dic₅	—	2	1	no	—
102	C₂×#168	—	2	4	no	—
103	C₂×#168	—	2	2	no	—
104	C₂×#168	—	2	2	no	—
105	C₂×#168	C₂²⋊C₄	1	2	yes	—
106	C₂×#168	C₂²⋊C₄	1	2	yes	—
107	C₂×#168	C₂×C₈	1	4	yes	—
108	C₂×#168	M₁₆	1	4	yes	—
109	#676	—	2	8	no	—
110	C₈×D₅	—	2	8	no	—
111	C₂×#131	—	5	2	no	—
112	C₄×#41	—	5	1	no	—
113	#431	—	5	1	no	—
114	#431	C₂×D₅	1	1	yes	characteristic
115	#405	—	5	2	no	—
116	#405	—	10	2	no	—
117	C₄×M₁₆	—	10	1	no	—
118	C₄×M₁₆	C₂×D₅	1	1	yes	characteristic
119	C₂×C₄×C₈	—	5	1	no	—
120	C₂×C₄×C₈	—	5	1	no	—
121	C₂×C₄×C₈	—	10	1	no	—
122	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
123	Q₈⋊C₈	—	5	2	no	—
124	Q₈⋊C₈	—	5	2	no	—
125	D₄⋊C₈	—	5	2	no	—
126	D₄⋊C₈	—	5	2	no	—
127	#322	—	10	1	no	—
128	C₈²	—	10	2	no	—
129	C₈²	—	10	1	no	—
130	C₂³×C₅	#117	1	1	yes	characteristic
131	C₂²×D₅	—	2	1	no	—
132	C₂×C₄×C₅	—	2	2	no	—
133	C₂×C₄×C₅	—	2	2	no	—
134	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₅	C₄×D₄	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	C₂²×C₈	1	2	yes	—
141	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
142	C₂²⋊D₅	—	2	4	no	—
143	C₂×Dic₅	—	2	1	no	—
144	C₂×Dic₅	—	2	1	no	—
145	C₂×Dic₅	—	2	1	no	—
146	D₂₀	—	2	1	no	—
147	D₂₀	C₄×C₈	1	2	yes	—
148	C₄×D₅	—	2	4	no	—
149	C₄×D₅	—	4	2	no	—
150	C₄×D₅	—	4	2	no	—
151	Dic₁₀	—	2	1	no	—
152	Dic₁₀	C₄×C₈	1	2	yes	—
153	C₅×C₈	—	2	4	no	—
154	C₅×C₈	—	2	4	no	—
155	C₅×C₈	—	4	2	no	—
156	#168	—	2	4	no	—
157	#168	—	4	2	no	—
158	#168	C₂²⋊C₈	1	4	yes	—
159	#168	C₂²⋊C₈	1	4	yes	—
160	C₂×#41	—	5	1	no	—
161	#131	—	5	8	no	—
162	#131	—	10	4	no	—
163	C₂×M₁₆	—	10	2	no	—
164	C₂×M₁₆	C₄×D₅	1	2	yes	—
165	C₂²×C₈	—	2	1	no	—
166	C₂²×C₈	—	5	2	no	—
167	C₂²×C₈	—	10	2	no	—
168	C₂²×C₈	—	10	1	no	—
169	C₄×Q₈	—	5	1	no	—
170	C₄×Q₈	—	5	1	no	—
171	C₄×D₄	—	5	1	no	—
172	C₄×D₄	—	5	1	no	—
173	C₄×D₄	—	10	1	no	—
174	C₄×D₄	—	10	1	no	—
175	#117	—	5	1	no	—
176	#117	—	10	1	no	—
177	C₂×C₄²	—	10	1	no	—
178	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
179	C₄⋊C₈	—	5	2	no	—
180	C₄⋊C₈	C₄×D₅	1	2	yes	—
181	C₂²⋊C₈	—	2	1	no	—
182	C₂²⋊C₈	—	10	1	no	—
183	#97	—	2	1	no	—
184	#97	—	10	2	no	—
185	C₄×C₈	—	2	1	no	—
186	C₄×C₈	—	2	1	no	—
187	C₄×C₈	—	2	1	no	—
188	C₄×C₈	—	5	2	no	—
189	C₄×C₈	—	5	2	no	—
190	C₄×C₈	—	5	1	no	—
191	C₄×C₈	—	5	1	no	—
192	C₄×C₈	—	10	2	no	—
193	C₄×C₈	—	10	1	no	—
194	C₄×C₈	—	10	1	no	—
195	C₄×C₈	C₄×D₅	1	2	yes	—
196	C₂²×C₅	—	2	1	no	—
197	C₂²×C₅	—	2	1	no	—
198	C₂²×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
199	C₂²×C₅	#431	1	2	yes	—
200	C₂×D₅	—	2	2	no	—
201	C₂×D₅	—	4	1	no	—
202	C₂×D₅	—	4	1	no	—
203	C₄×C₅	—	2	1	no	—
204	C₄×C₅	—	2	1	no	—
205	C₄×C₅	C₂×C₄×C₈	1	1	yes	characteristic
206	C₄×C₅	C₄×M₁₆	1	1	yes	characteristic
207	C₄×C₅	C₂×C₂²⋊C₈	1	2	yes	—
208	C₄×C₅	#443	1	2	yes	—
209	C₄×C₅	#443	1	2	yes	—
210	Dic₅	—	2	2	no	—
211	Dic₅	—	4	1	no	—
212	Dic₅	—	4	1	no	—
213	#41	—	5	4	no	—
214	#41	—	10	2	no	—
215	C₂×Q₈	—	5	1	no	—
216	C₂×D₄	—	5	1	no	—
217	C₂×D₄	—	10	1	no	—
218	C₂²×C₄	—	10	1	no	—
219	C₂²×C₄	—	10	1	no	—
220	C₂²×C₄	—	10	1	no	—
221	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
222	C₂²×C₄	#711	1	1	yes	characteristic
223	C₂²×C₄	#713	1	1	yes	characteristic
224	M₁₆	—	2	2	no	—
225	M₁₆	—	10	8	no	—
226	M₁₆	C₈×D₅	1	4	yes	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	5	2	no	—
232	C₂×C₈	—	5	2	no	—
233	C₂×C₈	—	5	4	no	—
234	C₂×C₈	—	5	4	no	—
235	C₂×C₈	—	10	8	no	—
236	C₂×C₈	—	10	4	no	—
237	C₂×C₈	—	10	2	no	—
238	C₂×C₈	—	10	2	no	—
239	C₄⋊C₄	—	5	1	no	—
240	C₄⋊C₄	—	5	1	no	—
241	C₄⋊C₄	—	10	1	no	—
242	C₄⋊C₄	—	10	1	no	—
243	C₂²⋊C₄	—	10	1	no	—
244	C₂²⋊C₄	—	10	1	no	—
245	C₂²⋊C₄	—	10	1	no	—
246	C₄²	—	10	1	no	—
247	C₄²	—	10	1	no	—
248	C₄²	—	10	1	no	—
249	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
250	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
251	C₄²	D₄×D₅	1	1	yes	characteristic
252	C₄²	D₄×D₅	1	1	yes	characteristic
253	C₂×C₅	—	2	2	no	—
254	C₂×C₅	C₄×C₂²⋊C₈	1	1	yes	characteristic
255	C₂×C₅	#1764	1	1	yes	characteristic
256	C₂×C₅	#1764	1	1	yes	characteristic
257	D₅	—	4	2	no	—
258	C₂³	—	10	1	no	—
259	C₂³	#4038	1	1	yes	characteristic
260	Q₈	—	5	2	no	—
261	Q₈	—	10	1	no	—
262	D₄	—	5	2	no	—
263	D₄	—	10	4	no	—
264	D₄	—	10	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	10	1	no	—
268	C₂×C₄	—	10	4	no	—
269	C₂×C₄	—	10	1	no	—
270	C₂×C₄	—	10	1	no	—
271	C₂×C₄	—	20	2	no	—
272	C₂×C₄	—	20	2	no	—
273	C₂×C₄	C₄²×D₅	1	1	yes	characteristic
274	C₂×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
275	C₂×C₄	#4027	1	1	yes	characteristic
276	C₂×C₄	#4027	1	1	yes	characteristic
277	C₂×C₄	#4039	1	1	yes	characteristic
278	C₂×C₄	#4039	1	1	yes	characteristic
279	C₂×C₄	C₂×C₈×D₅	1	2	yes	—
280	C₂×C₄	D₅×M₁₆	1	2	yes	—
281	C₈	—	2	4	no	—
282	C₈	—	2	4	no	—
283	C₈	—	4	2	no	—
284	C₈	—	5	4	no	—
285	C₈	—	5	4	no	—
286	C₈	—	10	4	no	—
287	C₈	—	20	2	no	—
288	C₅	#8184	1	1	yes	characteristic, complemented, Sylow
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	10	2	no	—
292	C₂²	—	20	1	no	—
293	C₂²	—	20	1	no	—
294	C₂²	#64955	1	1	yes	characteristic
295	C₂²	#65122	1	2	yes	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	10	2	no	—
299	C₄	—	20	1	no	—
300	C₄	—	20	1	no	—
301	C₄	C₄×C₈×D₅	1	1	yes	characteristic
302	C₄	#64996	1	1	yes	characteristic
303	C₄	D₅×C₂²⋊C₈	1	2	yes	—
304	C₄	#65113	1	2	yes	—
305	C₄	#65184	1	2	yes	—
306	C₂	—	2	2	no	—
307	C₂	—	20	2	no	—
308	C₂	#10599392	1	1	yes	characteristic
309	C₂	#10599532	1	1	yes	characteristic
310	C₂	#10599719	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₅
commutator	C₄×C₅	C₂×C₄×C₈
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	#8184
LCS	G ▷ 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. #10601533	C₂	semidirect
5. C₂×C₈×#168	C₂	semidirect
6. C₅×#1278	C₂	semidirect
7. #10598158	C₂	semidirect
38. #4053	C₈	semidirect
70. #1278	D₅	semidirect
288. C₅	#8184	semidirect

Maximal subgroups

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

subgroup	index
C₄×#4053	2
#10601547	2
#10601533	2
#10600527	2
C₂×C₈×#168	2
C₅×#1278	2
#10598158	2
#8184	5