Subgroups — #11999112

Order 1280.

694 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#64984	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#322	C₂	1	1	yes	maximal, characteristic, complemented
3	#10600565	C₂	1	2	yes	maximal, complemented
4	#10600565	C₂	1	2	yes	maximal, complemented
5	#10600555	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄₀	—	2	2	no	—
7	C₂×#4020	C₂²	1	2	yes	—
8	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×#3953	—	2	4	no	—
12	C₂×#3953	—	2	4	no	—
13	#65224	C₂²	1	2	yes	—
14	#64984	—	2	2	no	—
15	#64984	C₄	1	4	yes	complemented
16	C₅×#322	—	2	2	no	—
17	#64771	—	2	2	no	—
18	#9832	—	5	1	no	maximal, Sylow
19	C₂²×D₂₀	—	2	2	no	—
20	C₂²×D₂₀	—	2	2	no	—
21	C₂²×D₂₀	—	2	2	no	—
22	C₂²×C₅×C₈	D₄	1	2	yes	—
23	C₂²×C₅×C₈	D₄	1	2	yes	—
24	C₂²×C₅×C₈	D₄	1	2	yes	—
25	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
26	C₂×C₄⋊Dic₅	—	2	2	no	—
27	C₂×C₄⋊Dic₅	—	2	2	no	—
28	C₂×C₄⋊Dic₅	—	2	2	no	—
29	C₂×D₄₀	—	2	8	no	—
30	C₂×D₄₀	—	4	8	no	—
31	#4020	—	2	2	no	—
32	#4020	—	2	2	no	—
33	#4020	—	2	2	no	—
34	#4020	C₂×C₄	1	4	yes	—
35	C₄×C₅×C₈	—	2	1	no	—
36	C₄×C₅×C₈	—	2	1	no	—
37	C₄×C₅×C₈	—	2	1	no	—
38	C₄×C₅×C₈	—	2	1	no	—
39	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
40	C₄×C₅×C₈	D₄	1	2	yes	—
41	#3953	—	4	8	no	—
42	#3953	—	4	8	no	—
43	C₂×#493	—	5	1	no	—
44	#1957	—	5	2	no	—
45	#1957	—	5	2	no	—
46	#1834	—	5	1	no	—
47	C₂×#322	D₅	1	1	yes	characteristic, complemented
48	C₂³×D₅	—	4	2	no	—
49	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
52	C₂²×Dic₅	—	4	2	no	—
53	C₂×D₂₀	—	2	4	no	—
54	C₂×D₂₀	—	2	4	no	—
55	C₂×D₂₀	—	2	4	no	—
56	C₂×D₂₀	—	4	4	no	—
57	C₂×D₂₀	—	4	4	no	—
58	C₂×D₂₀	—	4	4	no	—
59	C₂×D₂₀	—	4	4	no	—
60	C₂×D₂₀	—	4	4	no	—
61	C₂×D₂₀	—	4	4	no	—
62	C₂×D₂₀	—	4	2	no	—
63	C₂×D₂₀	—	4	2	no	—
64	C₂×D₂₀	—	4	2	no	—
65	C₂×C₅×C₈	—	2	2	no	—
66	C₂×C₅×C₈	—	2	2	no	—
67	C₂×C₅×C₈	—	2	4	no	—
68	C₂×C₅×C₈	—	2	2	no	—
69	C₂×C₅×C₈	—	2	4	no	—
70	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
71	C₂×C₅×C₈	D₈	1	4	yes	complemented
72	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₅	#41	1	1	yes	characteristic
75	C₄²×C₅	#41	1	1	yes	characteristic
76	C₄⋊Dic₅	—	4	4	no	—
77	C₄⋊Dic₅	—	4	4	no	—
78	C₄⋊Dic₅	—	4	4	no	—
79	D₄₀	—	4	16	no	—
80	C₂²×D₈	—	10	2	no	—
81	C₂×#127	—	5	2	no	—
82	#493	—	5	4	no	—
83	#493	—	10	2	no	—
84	C₂×#102	—	10	4	no	—
85	C₂×#102	—	10	4	no	—
86	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
87	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
88	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
89	#384	—	5	2	no	—
90	#340	—	10	2	no	—
91	#322	—	2	2	no	—
92	C₂³×C₅	#127	1	1	yes	characteristic
93	C₂²×D₅	—	4	4	no	—
94	C₂²×D₅	—	8	2	no	—
95	C₂²×D₅	—	8	2	no	—
96	C₂²×D₅	—	8	2	no	—
97	C₂²×D₅	—	8	2	no	—
98	C₂²×D₅	—	8	2	no	—
99	C₂²×D₅	—	8	2	no	—
100	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₅	#121	1	1	yes	characteristic
104	C₂×C₄×C₅	#121	1	1	yes	characteristic
105	C₂×C₄×C₅	#124	1	1	yes	characteristic
106	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
107	C₂×C₄×C₅	#135	1	2	yes	—
108	C₂×C₄×C₅	AGL(1,ℤ/8)	1	4	yes	—
109	C₂×C₄×C₅	AGL(1,ℤ/8)	1	4	yes	—
110	C₂×Dic₅	—	8	2	no	—
111	C₂×Dic₅	—	8	2	no	—
112	C₂×Dic₅	—	8	2	no	—
113	D₂₀	—	4	8	no	—
114	D₂₀	—	4	8	no	—
115	D₂₀	—	4	8	no	—
116	D₂₀	—	8	4	no	—
117	D₂₀	—	8	4	no	—
118	D₂₀	—	8	4	no	—
119	C₅×C₈	—	2	4	no	—
120	C₅×C₈	—	2	4	no	—
121	C₅×C₈	—	4	2	no	—
122	C₂²×D₄	—	10	2	no	—
123	C₂²×D₄	—	10	2	no	—
124	C₂²×D₄	—	10	2	no	—
125	C₂×D₈	—	10	8	no	—
126	C₂×D₈	—	20	8	no	—
127	C₂²×C₈	D₂₀	1	2	yes	—
128	C₂²×C₈	D₂₀	1	2	yes	—
129	C₂²×C₈	D₂₀	1	2	yes	—
130	#127	—	5	4	no	—
131	#127	—	10	2	no	—
132	#127	—	10	2	no	—
133	#127	—	10	2	no	—
134	C₂×C₄⋊C₄	—	10	2	no	—
135	C₂×C₄⋊C₄	—	10	2	no	—
136	C₂×C₄⋊C₄	—	10	2	no	—
137	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
138	#102	—	20	8	no	—
139	#102	—	20	8	no	—
140	C₄×C₈	—	2	1	no	—
141	C₄×C₈	—	2	1	no	—
142	C₄×C₈	—	2	1	no	—
143	C₄×C₈	—	2	1	no	—
144	C₄×C₈	C₄×D₅	1	2	yes	—
145	C₄×C₈	C₂²⋊D₅	1	2	yes	—
146	C₂²×C₅	#390	1	1	yes	characteristic
147	C₂²×C₅	#493	1	1	yes	characteristic
148	C₂²×C₅	#495	1	1	yes	characteristic
149	C₂²×C₅	#496	1	2	yes	—
150	C₂²×C₅	#496	1	2	yes	—
151	C₂×D₅	—	8	4	no	—
152	C₂×D₅	—	8	4	no	—
153	C₂×D₅	—	8	4	no	—
154	C₂×D₅	—	16	2	no	—
155	C₂×D₅	—	16	2	no	—
156	C₂×D₅	—	16	2	no	—
157	C₂×D₅	—	16	2	no	—
158	C₄×C₅	#418	1	2	yes	—
159	C₄×C₅	C₄×D₈	1	2	yes	—
160	C₄×C₅	#442	1	2	yes	—
161	C₄×C₅	#466	1	2	yes	—
162	C₄×C₅	#469	1	2	yes	—
163	C₄×C₅	#490	1	2	yes	—
164	Dic₅	—	16	2	no	—
165	C₂⁴	—	20	2	no	—
166	C₂×D₄	—	10	4	no	—
167	C₂×D₄	—	10	4	no	—
168	C₂×D₄	—	10	4	no	—
169	C₂×D₄	—	20	2	no	—
170	C₂×D₄	—	20	4	no	—
171	C₂×D₄	—	20	4	no	—
172	C₂×D₄	—	20	4	no	—
173	C₂×D₄	—	20	4	no	—
174	C₂×D₄	—	20	4	no	—
175	C₂×D₄	—	20	4	no	—
176	C₂×D₄	—	20	2	no	—
177	C₂×D₄	—	20	2	no	—
178	C₂²×C₄	—	20	2	no	—
179	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
181	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
182	D₈	—	20	16	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	D₄₀	1	4	yes	complemented
189	C₂×C₈	#685	1	4	yes	—
190	C₄⋊C₄	—	20	4	no	—
191	C₄⋊C₄	—	20	4	no	—
192	C₄⋊C₄	—	20	4	no	—
193	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
194	C₄²	#709	1	1	yes	characteristic
195	C₄²	#709	1	1	yes	characteristic
196	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
197	C₂×C₅	#1717	1	2	yes	—
198	C₂×C₅	#1721	1	2	yes	—
199	C₂×C₅	#1961	1	1	yes	characteristic
200	C₂×C₅	#1970	1	1	yes	characteristic
201	C₂×C₅	#1972	1	1	yes	characteristic
202	D₅	—	16	4	no	—
203	C₂³	—	20	4	no	—
204	C₂³	—	40	2	no	—
205	C₂³	—	40	2	no	—
206	C₂³	—	40	2	no	—
207	C₂³	—	40	2	no	—
208	C₂³	—	40	2	no	—
209	C₂³	—	40	2	no	—
210	C₂³	#4020	1	1	yes	characteristic
211	D₄	—	20	8	no	—
212	D₄	—	20	8	no	—
213	D₄	—	20	8	no	—
214	D₄	—	40	4	no	—
215	D₄	—	40	4	no	—
216	D₄	—	40	4	no	—
217	C₂×C₄	—	40	2	no	—
218	C₂×C₄	—	40	2	no	—
219	C₂×C₄	—	40	2	no	—
220	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
221	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
222	C₂×C₄	#4021	1	1	yes	characteristic
223	C₂×C₄	C₂×D₄₀	1	2	yes	—
224	C₂×C₄	#4050	1	2	yes	—
225	C₂×C₄	#4054	1	4	yes	—
226	C₂×C₄	#4054	1	4	yes	—
227	C₂×C₄	C₂×#685	1	1	yes	characteristic
228	C₂×C₄	#4076	1	1	yes	characteristic
229	C₂×C₄	#4076	1	1	yes	characteristic
230	C₈	—	2	4	no	—
231	C₈	—	2	4	no	—
232	C₈	—	4	2	no	—
233	C₅	#9832	1	1	yes	characteristic, complemented, Sylow
234	C₂²	—	40	4	no	—
235	C₂²	—	40	4	no	—
236	C₂²	—	40	4	no	—
237	C₂²	—	80	2	no	—
238	C₂²	—	80	2	no	—
239	C₂²	—	80	2	no	—
240	C₂²	—	80	2	no	—
241	C₂²	#64984	1	1	yes	characteristic
242	C₂²	#64985	1	1	yes	characteristic
243	C₂²	#65001	1	2	yes	—
244	C₂²	#65001	1	2	yes	—
245	C₂²	#65228	1	1	yes	characteristic
246	C₄	—	80	2	no	—
247	C₄	C₄×D₄₀	1	2	yes	—
248	C₄	#65000	1	2	yes	—
249	C₄	#65002	1	2	yes	—
250	C₄	#65404	1	2	yes	—
251	C₄	#65420	1	2	yes	—
252	C₄	#65424	1	2	yes	—
253	C₂	—	80	4	no	—
254	C₂	#10598885	1	2	yes	—
255	C₂	#10598887	1	2	yes	—
256	C₂	#10600569	1	1	yes	characteristic
257	C₂	#10600616	1	1	yes	characteristic
258	C₂	#10600618	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#4020
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₂×C₅×#322	C₂
socle	C₂³×C₅	#127
prime radical	C₂×#64984	C₂
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₂×#322	D₅
hypocenter	C₅	#9832
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#322	C₂	semidirect
3. #10600565	C₂	semidirect
4. #10600565	C₂	semidirect
5. #10600555	C₂	semidirect
15. #64984	C₄	semidirect
47. C₂×#322	D₅	semidirect
71. C₂×C₅×C₈	D₈	semidirect
188. C₂×C₈	D₄₀	semidirect
233. C₅	#9832	semidirect

Maximal subgroups

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

subgroup	index
C₂×#64984	2
C₂×C₅×#322	2
#10600565	2
#10600555	2
#9832	5