Subgroups — #11987627

Order 1280.

456 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#4015	C₂	1	1	yes	maximal, characteristic
2	C₅×#1831	C₂	1	1	yes	maximal, characteristic
3	#10600185	C₂	1	1	yes	maximal, characteristic
4	#10597971	C₂	1	4	yes	maximal
5	C₂×#4015	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₁₀	C₄	1	2	yes	—
7	C₂×C₅×C₄⋊C₈	—	2	1	no	—
8	C₄³×C₅	C₂²	1	1	yes	characteristic
9	#65224	C₄	1	1	yes	characteristic
10	C₄×C₄⋊Dic₅	—	2	1	no	—
11	C₄×C₄⋊Dic₅	—	2	1	no	—
12	#65218	C₂²	1	2	yes	—
13	#65217	—	2	1	no	—
14	#65217	—	2	1	no	—
15	C₂×C₄⋊#168	—	2	1	no	—
16	C₅×#336	—	2	1	no	—
17	C₅×#336	C₂²	1	2	yes	—
18	#64744	—	2	1	no	—
19	#64744	C₂²	1	2	yes	—
20	#7653	—	5	1	no	maximal, Sylow
21	C₂²×Dic₁₀	C₂×C₄	1	2	yes	—
22	C₂²×C₅×C₈	—	2	2	no	—
23	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊Dic₅	—	2	1	no	—
31	C₂×C₄⋊Dic₅	—	2	1	no	—
32	C₂×C₄⋊Dic₅	—	2	1	no	—
33	C₂×C₄⋊Dic₅	—	2	1	no	—
34	C₂×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
35	C₂×#683	—	2	2	no	—
36	C₂×C₄×Dic₅	—	2	2	no	—
37	C₂²×#168	—	2	2	no	—
38	#4015	—	2	4	no	—
39	C₄×Dic₁₀	—	4	4	no	—
40	C₅×C₄⋊C₈	—	4	2	no	—
41	#3963	—	2	2	no	—
42	#3963	—	2	2	no	—
43	C₄⋊#168	—	4	2	no	—
44	C₄×C₄⋊Q₈	—	5	1	no	—
45	#1831	—	5	1	no	—
46	#1831	D₅	1	1	yes	characteristic
47	#1272	—	5	4	no	—
48	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂²×Dic₅	—	2	2	no	—
56	C₂²×Dic₅	—	2	2	no	—
57	C₂×Dic₁₀	—	2	2	no	—
58	C₂×Dic₁₀	—	4	2	no	—
59	C₂×Dic₁₀	—	4	2	no	—
60	C₂×Dic₁₀	M₁₆	1	4	yes	—
61	C₂×C₅×C₈	—	4	2	no	—
62	C₂×C₅×C₈	—	4	2	no	—
63	C₂×C₅×C₈	—	4	2	no	—
64	C₄²×C₅	—	2	2	no	—
65	C₄²×C₅	—	2	2	no	—
66	C₄²×C₅	—	4	1	no	—
67	C₄²×C₅	—	4	1	no	—
68	C₄²×C₅	—	4	1	no	—
69	C₄²×C₅	SD₁₆	1	2	yes	—
70	C₄²×C₅	SD₁₆	1	2	yes	—
71	C₄²×C₅	Q₁₆	1	2	yes	—
72	C₄²×C₅	Q₁₆	1	2	yes	—
73	C₄⋊Dic₅	—	2	4	no	—
74	C₄⋊Dic₅	—	2	4	no	—
75	C₄⋊Dic₅	—	4	2	no	—
76	C₄⋊Dic₅	—	4	2	no	—
77	C₄⋊Dic₅	—	4	2	no	—
78	#683	—	4	4	no	—
79	C₄×Dic₅	—	4	4	no	—
80	C₂×#168	—	4	2	no	—
81	C₂×#168	—	4	2	no	—
82	C₂×#168	—	4	2	no	—
83	C₂×C₄⋊Q₈	—	5	1	no	—
84	C₂×C₄×Q₈	—	5	2	no	—
85	C₂×C₄⋊C₈	—	2	1	no	—
86	C₂×C₄⋊C₈	—	10	1	no	—
87	#391	—	5	2	no	—
88	#389	—	10	1	no	—
89	#389	—	10	1	no	—
90	#384	—	5	1	no	—
91	C₄×C₄⋊C₄	—	10	1	no	—
92	C₄×C₄⋊C₄	—	10	1	no	—
93	C₄³	C₂×D₅	1	1	yes	characteristic
94	#336	—	2	1	no	—
95	#336	—	5	2	no	—
96	#336	—	10	1	no	—
97	#336	C₂×D₅	1	2	yes	—
98	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₅	—	2	1	no	—
100	C₂×C₄×C₅	—	2	1	no	—
101	C₂×C₄×C₅	—	2	1	no	—
102	C₂×C₄×C₅	—	2	1	no	—
103	C₂×C₄×C₅	—	2	1	no	—
104	C₂×C₄×C₅	—	2	1	no	—
105	C₂×C₄×C₅	—	2	1	no	—
106	C₂×C₄×C₅	—	2	1	no	—
107	C₂×C₄×C₅	—	2	1	no	—
108	C₂×C₄×C₅	—	2	1	no	—
109	C₂×C₄×C₅	—	2	1	no	—
110	C₂×C₄×C₅	—	2	1	no	—
111	C₂×C₄×C₅	—	2	1	no	—
112	C₂×C₄×C₅	#100	1	2	yes	—
113	C₂×C₄×C₅	#103	1	2	yes	—
114	C₂×C₄×C₅	#103	1	2	yes	—
115	C₂×C₄×C₅	#103	1	2	yes	—
116	C₂×C₄×C₅	#103	1	2	yes	—
117	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
118	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
122	C₂×Dic₅	—	2	4	no	—
123	C₂×Dic₅	—	4	2	no	—
124	C₂×Dic₅	—	4	2	no	—
125	C₂×Dic₅	—	4	2	no	—
126	C₂×Dic₅	—	4	2	no	—
127	C₂×Dic₅	—	4	2	no	—
128	Dic₁₀	—	4	4	no	—
129	Dic₁₀	—	4	4	no	—
130	C₅×C₈	—	8	2	no	—
131	#168	—	8	2	no	—
132	C₂²×Q₈	—	5	2	no	—
133	C₂²×C₈	—	2	2	no	—
134	C₂²×C₈	—	10	2	no	—
135	C₄⋊Q₈	—	10	4	no	—
136	C₄×Q₈	—	20	4	no	—
137	C₂×C₄⋊C₄	—	5	2	no	—
138	C₂×C₄⋊C₄	—	10	1	no	—
139	C₂×C₄⋊C₄	—	10	1	no	—
140	C₂×C₄⋊C₄	—	10	2	no	—
141	C₂×C₄⋊C₄	—	10	1	no	—
142	C₂×C₄⋊C₄	—	10	1	no	—
143	C₂×C₄²	—	10	2	no	—
144	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
145	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
146	C₂×C₄²	D₂₀	1	1	yes	characteristic
147	C₂×C₄²	D₂₀	1	1	yes	characteristic
148	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
149	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
150	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
151	C₄⋊C₈	—	4	2	no	—
152	C₄⋊C₈	—	20	2	no	—
153	#95	—	10	2	no	—
154	#95	—	10	2	no	—
155	C₂²×C₅	#407	1	1	yes	characteristic
156	C₂²×C₅	#410	1	1	yes	characteristic
157	C₂²×C₅	C₂×#100	1	1	yes	characteristic
158	C₂²×C₅	C₂×#103	1	1	yes	characteristic
159	C₂²×C₅	C₂×#103	1	1	yes	characteristic
160	C₂²×C₅	#421	1	1	yes	characteristic
161	C₂²×C₅	#421	1	1	yes	characteristic
162	C₄×C₅	—	2	2	no	—
163	C₄×C₅	—	2	2	no	—
164	C₄×C₅	—	2	2	no	—
165	C₄×C₅	—	2	1	no	—
166	C₄×C₅	—	2	1	no	—
167	C₄×C₅	—	2	1	no	—
168	C₄×C₅	—	2	1	no	—
169	C₄×C₅	—	4	1	no	—
170	C₄×C₅	#333	1	4	yes	—
171	Dic₅	—	4	4	no	—
172	Dic₅	—	8	2	no	—
173	C₂×Q₈	—	5	4	no	—
174	C₂×Q₈	—	10	2	no	—
175	C₂×Q₈	—	20	2	no	—
176	C₂×Q₈	—	20	2	no	—
177	C₂²×C₄	—	10	2	no	—
178	C₂²×C₄	—	10	2	no	—
179	C₂²×C₄	#685	1	1	yes	characteristic
180	C₂²×C₄	#685	1	1	yes	characteristic
181	C₂²×C₄	#685	1	1	yes	characteristic
182	C₂²×C₄	#685	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
184	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	20	2	no	—
190	C₂×C₈	—	20	2	no	—
191	C₂×C₈	—	20	2	no	—
192	C₄⋊C₄	—	10	4	no	—
193	C₄⋊C₄	—	10	4	no	—
194	C₄⋊C₄	—	20	2	no	—
195	C₄⋊C₄	—	20	4	no	—
196	C₄⋊C₄	—	20	2	no	—
197	C₄⋊C₄	—	20	2	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	4	1	no	—
201	C₄²	—	4	1	no	—
202	C₄²	—	4	1	no	—
203	C₄²	—	20	4	no	—
204	C₄²	#677	1	2	yes	—
205	C₄²	Dic₂₀	1	2	yes	—
206	C₄²	#687	1	2	yes	—
207	C₄²	#689	1	2	yes	—
208	C₂×C₅	#1466	1	1	yes	characteristic
209	C₂×C₅	#1487	1	1	yes	characteristic
210	C₂×C₅	#1487	1	1	yes	characteristic
211	C₂×C₅	#1518	1	1	yes	characteristic
212	C₂×C₅	#1518	1	1	yes	characteristic
213	C₂×C₅	#1536	1	1	yes	characteristic
214	C₂×C₅	C₂×#333	1	1	yes	characteristic
215	C₂³	C₂×#685	1	1	yes	characteristic
216	Q₈	—	20	4	no	—
217	Q₈	—	20	4	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	10	4	no	—
232	C₂×C₄	—	20	2	no	—
233	C₂×C₄	—	20	2	no	—
234	C₂×C₄	—	20	2	no	—
235	C₂×C₄	—	20	2	no	—
236	C₂×C₄	—	20	2	no	—
237	C₂×C₄	#3942	1	2	yes	—
238	C₂×C₄	#3942	1	2	yes	—
239	C₂×C₄	#3948	1	2	yes	—
240	C₂×C₄	#3948	1	2	yes	—
241	C₂×C₄	#3955	1	2	yes	—
242	C₂×C₄	C₂×#677	1	1	yes	characteristic
243	C₂×C₄	C₂×Dic₂₀	1	1	yes	characteristic
244	C₂×C₄	D₅×M₁₆	1	2	yes	—
245	C₂×C₄	C₂×#687	1	1	yes	characteristic
246	C₂×C₄	C₂×#689	1	1	yes	characteristic
247	C₈	—	8	2	no	—
248	C₈	—	40	2	no	—
249	C₅	#7653	1	1	yes	characteristic, complemented, Sylow
250	C₂²	C₂×#3942	1	1	yes	characteristic
251	C₂²	#65294	1	1	yes	characteristic
252	C₂²	#65295	1	1	yes	characteristic
253	C₂²	C₂×#3948	1	1	yes	characteristic
254	C₂²	#65415	1	1	yes	characteristic
255	C₂²	C₂×#3955	1	1	yes	characteristic
256	C₂²	#65428	1	1	yes	characteristic
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	2	no	—
264	C₄	—	4	1	no	—
265	C₄	—	20	4	no	—
266	C₄	—	40	2	no	—
267	C₄	#64701	1	4	yes	—
268	C₂	#10600339	1	1	yes	characteristic
269	C₂	C₂×#64701	1	1	yes	characteristic
270	C₂	#10601293	1	1	yes	characteristic
271	C₂	#10601563	1	1	yes	characteristic
272	C₂	#10601586	1	1	yes	characteristic
273	C₂	#10601604	1	1	yes	characteristic
274	C₂	#10601620	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#685
commutator	C₂²×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1831	C₂
socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1831	D₅
hypocenter	C₅	#7653
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
249. C₅	#7653	semidirect

Maximal subgroups

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

subgroup	index
C₄×#4015	2
C₅×#1831	2
#10600185	2
#10597971	2
#7653	5