Subgroups — #11998678

Order 1280.

502 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10610198	C₂	1	1	yes	maximal, characteristic
2	C₅×#1450	C₂	1	1	yes	maximal, characteristic, complemented
3	#10601583	C₂	1	4	yes	maximal, complemented
4	#10600559	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#4050	—	2	1	no	—
6	C₂×#4050	—	2	1	no	—
7	#65811	C₂²	1	2	yes	—
8	C₅×#431	C₂²	1	2	yes	—
9	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂×#3950	—	2	1	no	—
11	C₂×#3949	—	2	1	no	—
12	#65225	C₂²	1	2	yes	—
13	#64988	C₄	1	1	yes	characteristic, complemented
14	#64985	—	2	1	no	—
15	#64985	—	2	1	no	—
16	#64984	C₄	1	1	yes	characteristic, complemented
17	#64982	C₄	1	2	yes	complemented
18	C₅×#322	—	2	1	no	—
19	C₅×C₈²	—	2	1	no	—
20	#64690	—	2	8	no	—
21	#9756	—	5	1	no	maximal, Sylow
22	C₂×#709	—	2	2	no	—
23	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
24	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
26	#4076	—	2	4	no	—
27	C₂×C₄⋊Dic₅	—	2	2	no	—
28	C₂×Dic₂₀	—	2	1	no	—
29	C₂×Dic₂₀	—	2	1	no	—
30	#4050	—	4	2	no	—
31	#4050	—	4	2	no	—
32	#4050	—	4	2	no	—
33	#4050	—	4	2	no	—
34	C₂×D₄₀	—	2	1	no	—
35	C₂×D₄₀	—	2	1	no	—
36	C₂×#677	—	2	2	no	—
37	C₂×#677	—	2	2	no	—
38	#4021	—	2	2	no	—
39	#4020	C₂×C₄	1	2	yes	—
40	C₄×D₂₀	—	2	4	no	—
41	#4015	C₂×C₄	1	2	yes	—
42	C₅×C₄⋊C₈	—	2	2	no	—
43	C₅×C₂²⋊C₈	D₄	1	4	yes	—
44	C₄×C₅×C₈	—	2	2	no	—
45	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
46	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
47	C₄×C₅×C₈	D₄	1	1	yes	characteristic
48	C₄×C₅×C₈	D₄	1	1	yes	characteristic
49	#3963	—	2	4	no	—
50	#3950	—	4	2	no	—
51	#3949	—	4	2	no	—
52	#3147	—	5	1	no	—
53	#1846	—	5	1	no	—
54	#1517	—	5	4	no	—
55	#1450	D₅	1	1	yes	characteristic, complemented
56	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
58	C₂×C₂²⋊D₅	—	4	2	no	—
59	C₂²×Dic₅	—	4	2	no	—
60	#709	—	4	2	no	—
61	#709	—	4	2	no	—
62	#709	—	4	2	no	—
63	#709	—	4	2	no	—
64	C₂×D₂₀	—	2	2	no	—
65	C₂×D₂₀	—	2	4	no	—
66	C₂×C₄×D₅	—	4	2	no	—
67	C₂×Dic₁₀	—	2	2	no	—
68	C₂×C₅×C₈	—	2	4	no	—
69	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₅×C₈	D₈	1	2	yes	—
74	C₂×C₅×C₈	D₈	1	2	yes	—
75	C₂×C₅×C₈	SD₁₆	1	2	yes	—
76	C₂×C₅×C₈	SD₁₆	1	2	yes	—
77	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₅	#41	1	2	yes	—
80	#685	—	4	4	no	—
81	C₄⋊Dic₅	—	2	4	no	—
82	C₄⋊Dic₅	—	4	2	no	—
83	C₄⋊Dic₅	—	4	2	no	—
84	Dic₂₀	—	4	2	no	—
85	Dic₂₀	—	4	2	no	—
86	D₄₀	—	4	2	no	—
87	D₄₀	—	4	2	no	—
88	#677	—	4	4	no	—
89	#677	—	4	4	no	—
90	C₂×#135	—	10	1	no	—
91	C₂×#135	—	10	1	no	—
92	#532	—	5	2	no	—
93	#495	—	10	1	no	—
94	#495	—	10	1	no	—
95	#494	—	5	1	no	—
96	#493	—	5	1	no	—
97	#492	—	5	2	no	—
98	#431	C₂×D₅	1	2	yes	—
99	C₂×#107	—	10	1	no	—
100	C₂×#106	—	10	1	no	—
101	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
102	#382	—	5	2	no	—
103	#327	—	10	8	no	—
104	#322	—	2	1	no	—
105	C₈²	—	2	1	no	—
106	C₂³×C₅	#127	1	1	yes	characteristic
107	C₂²×D₅	—	4	2	no	—
108	C₂×C₄×C₅	—	4	2	no	—
109	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₅	#121	1	2	yes	—
112	C₂×C₄×C₅	#124	1	1	yes	characteristic
113	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
116	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
117	C₂²⋊D₅	—	8	2	no	—
118	C₂²⋊D₅	—	8	2	no	—
119	C₂×Dic₅	—	4	2	no	—
120	C₂×Dic₅	—	8	4	no	—
121	C₂×Dic₅	—	8	2	no	—
122	D₂₀	—	4	2	no	—
123	D₂₀	—	4	2	no	—
124	D₂₀	—	8	4	no	—
125	C₄×D₅	—	8	2	no	—
126	C₄×D₅	—	8	2	no	—
127	Dic₁₀	—	4	2	no	—
128	Dic₁₀	—	4	2	no	—
129	C₅×C₈	—	4	4	no	—
130	C₅×C₈	#102	1	2	yes	—
131	C₅×C₈	#102	1	2	yes	—
132	C₅×C₈	#102	1	2	yes	—
133	C₅×C₈	#102	1	2	yes	—
134	C₂×#41	—	10	2	no	—
135	#135	—	20	2	no	—
136	#135	—	20	2	no	—
137	#135	—	20	2	no	—
138	#135	—	20	2	no	—
139	C₂×Q₁₆	—	10	1	no	—
140	C₂×Q₁₆	—	10	1	no	—
141	C₂×SD₁₆	—	10	2	no	—
142	C₂×SD₁₆	—	10	2	no	—
143	C₂×D₈	—	10	1	no	—
144	C₂×D₈	—	10	1	no	—
145	C₂²×C₈	D₂₀	1	1	yes	characteristic
146	C₂²×C₈	D₂₀	1	1	yes	characteristic
147	C₄⋊Q₈	—	5	2	no	—
148	#127	—	5	2	no	—
149	#124	—	10	2	no	—
150	#121	—	10	4	no	—
151	C₄×D₄	—	10	4	no	—
152	C₂×C₄⋊C₄	—	10	2	no	—
153	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
154	#107	—	20	2	no	—
155	#106	—	20	2	no	—
156	C₄⋊C₈	—	2	2	no	—
157	C₂²⋊C₈	D₂₀	1	4	yes	—
158	C₄×C₈	—	2	2	no	—
159	C₄×C₈	C₄×D₅	1	1	yes	characteristic
160	C₄×C₈	C₄×D₅	1	1	yes	characteristic
161	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
162	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
163	#95	—	10	4	no	—
164	C₂²×C₅	—	2	2	no	—
165	C₂²×C₅	#390	1	1	yes	characteristic
166	C₂²×C₅	#492	1	1	yes	characteristic
167	C₂²×C₅	#493	1	1	yes	characteristic
168	C₂×D₅	—	8	2	no	—
169	C₂×D₅	—	8	4	no	—
170	C₄×C₅	—	2	2	no	—
171	C₄×C₅	—	2	2	no	—
172	C₄×C₅	C₂×#102	1	1	yes	characteristic
173	C₄×C₅	C₂×#102	1	1	yes	characteristic
174	C₄×C₅	#486	1	1	yes	characteristic
175	C₄×C₅	#486	1	1	yes	characteristic
176	Dic₅	—	8	2	no	—
177	Dic₅	—	16	2	no	—
178	#41	—	20	2	no	—
179	#41	—	20	2	no	—
180	#41	—	20	2	no	—
181	#41	—	20	2	no	—
182	C₂×Q₈	—	10	2	no	—
183	C₂×D₄	—	10	4	no	—
184	C₂×D₄	—	10	2	no	—
185	C₂×D₄	—	20	2	no	—
186	C₂²×C₄	—	20	2	no	—
187	C₂²×C₄	—	20	2	no	—
188	C₂²×C₄	C₂×D₂₀	1	2	yes	—
189	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
190	Q₁₆	—	20	2	no	—
191	Q₁₆	—	20	2	no	—
192	SD₁₆	—	20	4	no	—
193	SD₁₆	—	20	4	no	—
194	D₈	—	20	2	no	—
195	D₈	—	20	2	no	—
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	#677	1	2	yes	—
198	C₂×C₈	#677	1	2	yes	—
199	C₂×C₈	D₄₀	1	2	yes	—
200	C₂×C₈	D₄₀	1	2	yes	—
201	C₂×C₈	#685	1	1	yes	characteristic
202	C₂×C₈	#685	1	1	yes	characteristic
203	C₂×C₈	#685	1	1	yes	characteristic
204	C₂×C₈	#685	1	1	yes	characteristic
205	C₄⋊C₄	—	10	4	no	—
206	C₄⋊C₄	—	20	2	no	—
207	C₄⋊C₄	—	20	2	no	—
208	C₂²⋊C₄	—	20	4	no	—
209	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
210	C₄²	#709	1	2	yes	—
211	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
212	C₂×C₅	—	2	1	no	—
213	C₂×C₅	—	2	1	no	—
214	C₂×C₅	#1957	1	1	yes	characteristic
215	C₂×C₅	#1976	1	2	yes	—
216	D₅	—	16	2	no	—
217	C₂³	—	20	2	no	—
218	C₂³	#4020	1	1	yes	characteristic
219	Q₈	—	20	2	no	—
220	Q₈	—	20	2	no	—
221	D₄	—	20	2	no	—
222	D₄	—	20	2	no	—
223	D₄	—	40	4	no	—
224	D₄	—	40	2	no	—
225	D₄	—	40	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	20	2	no	—
228	C₂×C₄	—	40	2	no	—
229	C₂×C₄	—	40	2	no	—
230	C₂×C₄	—	40	4	no	—
231	C₂×C₄	—	40	2	no	—
232	C₂×C₄	C₄×D₂₀	1	2	yes	—
233	C₂×C₄	#4021	1	1	yes	characteristic
234	C₂×C₄	C₂×#677	1	1	yes	characteristic
235	C₂×C₄	C₂×#677	1	1	yes	characteristic
236	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
237	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
238	C₂×C₄	C₂×#685	1	1	yes	characteristic
239	C₂×C₄	#4076	1	2	yes	—
240	C₈	—	4	4	no	—
241	C₈	#3953	1	2	yes	—
242	C₈	#3953	1	2	yes	—
243	C₈	#3953	1	2	yes	—
244	C₈	#3953	1	2	yes	—
245	C₅	#9756	1	1	yes	characteristic, complemented, Sylow
246	C₂²	—	2	2	no	—
247	C₂²	—	40	2	no	—
248	C₂²	—	40	4	no	—
249	C₂²	#64982	1	1	yes	characteristic
250	C₂²	#64984	1	1	yes	characteristic
251	C₂²	#65228	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	40	2	no	—
255	C₄	—	80	2	no	—
256	C₄	#64983	1	1	yes	characteristic
257	C₄	#64983	1	1	yes	characteristic
258	C₄	C₂×#3953	1	1	yes	characteristic
259	C₄	C₂×#3953	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	80	2	no	—
263	C₂	#10600565	1	1	yes	characteristic
264	C₂	#10600690	1	2	yes	—

	subgroup	G/H
center	C₂²	#65228
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1450	C₂
socle	C₂²×C₅	#390
prime radical	#10610198	C₂
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1450	D₅
hypocenter	C₅	#9756
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

Complements

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

N	H ≅ G/N	type
2. C₅×#1450	C₂	semidirect
3. #10601583	C₂	semidirect
4. #10600559	C₂	semidirect
13. #64988	C₄	semidirect
16. #64984	C₄	semidirect
17. #64982	C₄	semidirect
55. #1450	D₅	semidirect
245. C₅	#9756	semidirect

Maximal subgroups

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

subgroup	index
#10610198	2
C₅×#1450	2
#10601583	2
#10600559	2
#9756	5