Subgroups — #11998580

Order 1280.

442 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#500	C₂	1	1	yes	maximal, characteristic
2	#10602572	C₂	1	2	yes	maximal
3	C₂×#64766	C₂	1	2	yes	maximal
4	#10600550	C₂	1	1	yes	maximal, characteristic
5	C₂×#64677	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₅×#500	—	2	2	no	—
8	C₅×#500	C₄	1	4	yes	complemented
9	C₂×C₅×#107	—	2	1	no	—
10	C₂×C₅×#107	—	2	1	no	—
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₂×#3947	—	2	1	no	—
13	C₄×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊#168	C₂²	1	2	yes	—
15	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
16	#64770	—	2	1	no	—
17	#64766	—	2	4	no	—
18	#64749	—	2	4	no	—
19	#64677	—	2	2	no	—
20	#9739	—	5	1	no	maximal, Sylow
21	C₂²×C₅×Q₈	—	2	2	no	—
22	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
24	C₅×C₄⋊Q₈	—	2	2	no	—
25	C₅×C₄⋊Q₈	—	2	2	no	—
26	C₅×C₄⋊Q₈	—	2	2	no	—
27	C₅×C₄⋊Q₈	C₂×C₄	1	4	yes	—
28	C₂×C₅×C₄⋊C₄	—	2	2	no	—
29	C₂×C₅×C₄⋊C₄	—	2	2	no	—
30	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
31	C₂×C₄⋊Dic₅	D₄	1	2	yes	—
32	C₂×C₄×Dic₅	—	2	1	no	—
33	C₂²×#168	—	2	1	no	—
34	C₂²×#168	—	2	2	no	—
35	C₅×#107	—	2	4	no	—
36	C₅×#107	—	2	4	no	—
37	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
38	C₄×C₅×C₈	D₄	1	2	yes	—
39	#3963	—	2	1	no	—
40	#3947	—	4	2	no	—
41	C₄⋊#168	—	2	1	no	—
42	C₄⋊#168	—	2	2	no	—
43	C₄⋊#168	—	2	2	no	—
44	C₄⋊#168	D₄	1	2	yes	—
45	C₂×#500	D₅	1	1	yes	characteristic
46	#1987	—	5	2	no	—
47	#1770	—	5	1	no	—
48	C₂×#334	—	5	1	no	—
49	C₂×#333	—	5	2	no	—
50	C₂×C₅×Q₈	—	2	4	no	—
51	C₂×C₅×Q₈	—	4	2	no	—
52	C₂×C₅×Q₈	—	8	4	no	—
53	C₂²×C₄×C₅	—	4	2	no	—
54	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅	#41	1	1	yes	characteristic
57	C₂²×Dic₅	—	2	2	no	—
58	C₂×C₅×C₈	—	2	1	no	—
59	C₂×C₅×C₈	—	2	1	no	—
60	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
61	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
62	C₂×C₅×C₈	SD₁₆	1	2	yes	—
63	C₂×C₅×C₈	Q₁₆	1	2	yes	—
64	C₅×C₄⋊C₄	—	2	4	no	—
65	C₅×C₄⋊C₄	—	2	4	no	—
66	C₅×C₄⋊C₄	—	4	2	no	—
67	C₅×C₄⋊C₄	—	4	2	no	—
68	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₅	#41	1	1	yes	characteristic
72	C₄⋊Dic₅	—	2	4	no	—
73	C₄×Dic₅	—	4	1	no	—
74	C₄×Dic₅	—	4	1	no	—
75	C₂×#168	—	2	2	no	—
76	C₂×#168	—	4	1	no	—
77	C₂×#168	—	4	1	no	—
78	C₂×#168	—	4	2	no	—
79	C₂×#168	—	4	4	no	—
80	C₂×C₄⋊Q₈	C₂×D₅	1	2	yes	—
81	#500	—	2	2	no	—
82	#500	Dic₅	1	4	yes	complemented
83	C₂×#107	—	2	1	no	—
84	C₂×#107	—	2	1	no	—
85	C₂×C₄⋊C₈	—	5	1	no	—
86	C₂×C₄⋊C₈	—	5	2	no	—
87	C₂×#97	—	10	1	no	—
88	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
89	C₄×C₄⋊C₄	—	5	1	no	—
90	#340	—	10	4	no	—
91	#336	—	10	1	no	—
92	#334	—	10	2	no	—
93	#333	—	10	4	no	—
94	C₂³×C₅	#121	1	1	yes	characteristic
95	C₅×Q₈	—	8	4	no	—
96	C₅×Q₈	—	8	4	no	—
97	C₂×C₄×C₅	—	2	2	no	—
98	C₂×C₄×C₅	—	4	4	no	—
99	C₂×C₄×C₅	—	8	4	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	#124	1	1	yes	characteristic
105	C₂×C₄×C₅	#127	1	1	yes	characteristic
106	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
108	C₂×C₄×C₅	#137	1	4	yes	—
109	C₂×C₄×C₅	#137	1	1	yes	characteristic
110	C₂×C₄×C₅	#137	1	1	yes	characteristic
111	C₂×Dic₅	—	4	2	no	—
112	C₂×Dic₅	—	4	2	no	—
113	C₂×Dic₅	—	4	2	no	—
114	C₅×C₈	—	2	2	no	—
115	C₅×C₈	#103	1	4	yes	—
116	#168	—	4	2	no	—
117	#168	—	8	2	no	—
118	C₂²×Q₈	—	2	2	no	—
119	C₂²×C₈	—	10	1	no	—
120	C₂²×C₈	—	10	2	no	—
121	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
122	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
123	C₄⋊Q₈	—	2	2	no	—
124	C₄⋊Q₈	—	2	2	no	—
125	C₄⋊Q₈	—	2	2	no	—
126	C₄⋊Q₈	C₂×Dic₅	1	4	yes	—
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	5	2	no	—
130	C₂×C₄²	—	10	1	no	—
131	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
132	#107	—	2	4	no	—
133	#107	—	2	4	no	—
134	C₄⋊C₈	—	5	2	no	—
135	C₄⋊C₈	—	10	2	no	—
136	C₄⋊C₈	—	10	1	no	—
137	C₄⋊C₈	—	10	2	no	—
138	#97	—	20	2	no	—
139	C₄×C₈	C₂×Dic₅	1	2	yes	—
140	C₄×C₈	C₂²⋊D₅	1	2	yes	—
141	#95	—	10	1	no	—
142	C₂²×C₅	#390	1	1	yes	characteristic
143	C₂²×C₅	#461	1	1	yes	characteristic
144	C₂²×C₅	#462	1	1	yes	characteristic
145	C₂²×C₅	#470	1	2	yes	—
146	C₂²×C₅	#472	1	2	yes	—
147	C₄×C₅	—	2	1	no	—
148	C₄×C₅	—	2	1	no	—
149	C₄×C₅	—	8	4	no	—
150	C₄×C₅	C₂×#103	1	1	yes	characteristic
151	C₄×C₅	#419	1	1	yes	characteristic
152	C₄×C₅	#441	1	2	yes	—
153	C₄×C₅	#461	1	1	yes	characteristic
154	C₄×C₅	#462	1	1	yes	characteristic
155	C₄×C₅	#497	1	2	yes	—
156	Dic₅	—	8	2	no	—
157	C₂×Q₈	—	2	4	no	—
158	C₂×Q₈	—	4	2	no	—
159	C₂×Q₈	—	8	4	no	—
160	C₂²×C₄	—	4	2	no	—
161	C₂²×C₄	—	10	2	no	—
162	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
163	C₂²×C₄	#711	1	1	yes	characteristic
164	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	10	2	no	—
168	C₂×C₈	—	20	1	no	—
169	C₂×C₈	—	20	1	no	—
170	C₂×C₈	—	20	4	no	—
171	C₂×C₈	—	20	2	no	—
172	C₂×C₈	#688	1	2	yes	—
173	C₂×C₈	#689	1	2	yes	—
174	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
175	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	10	4	no	—
181	C₄²	—	20	1	no	—
182	C₄²	—	20	1	no	—
183	C₄²	D₄×D₅	1	1	yes	characteristic
184	C₄²	#711	1	1	yes	characteristic
185	C₄²	C₂²×Dic₅	1	1	yes	characteristic
186	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
187	C₂×C₅	#1695	1	2	yes	—
188	C₂×C₅	#1696	1	2	yes	—
189	C₂×C₅	#1971	1	1	yes	characteristic
190	C₂×C₅	#1978	1	1	yes	characteristic
191	C₂×C₅	#1981	1	1	yes	characteristic
192	C₂³	#4084	1	1	yes	characteristic
193	Q₈	—	8	4	no	—
194	Q₈	—	8	4	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	—	8	4	no	—
198	C₂×C₄	—	20	2	no	—
199	C₂×C₄	—	20	2	no	—
200	C₂×C₄	—	20	2	no	—
201	C₂×C₄	#4064	1	4	yes	—
202	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
203	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
204	C₂×C₄	#4082	1	1	yes	characteristic
205	C₂×C₄	#4084	1	1	yes	characteristic
206	C₂×C₄	#4086	1	1	yes	characteristic
207	C₂×C₄	C₂×#688	1	1	yes	characteristic
208	C₂×C₄	#4088	1	1	yes	characteristic
209	C₂×C₄	#4088	1	1	yes	characteristic
210	C₂×C₄	C₂×#689	1	1	yes	characteristic
211	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
212	C₈	—	2	2	no	—
213	C₈	—	20	2	no	—
214	C₈	—	40	2	no	—
215	C₈	#3967	1	4	yes	—
216	C₅	#9739	1	1	yes	characteristic, complemented, Sylow
217	C₂²	#65372	1	1	yes	characteristic
218	C₂²	#65379	1	1	yes	characteristic
219	C₂²	#65449	1	2	yes	—
220	C₂²	#65478	1	2	yes	—
221	C₂²	#65510	1	1	yes	characteristic
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	8	4	no	—
225	C₄	—	40	2	no	—
226	C₄	#65372	1	1	yes	characteristic
227	C₄	#65379	1	1	yes	characteristic
228	C₄	#65473	1	2	yes	—
229	C₄	#65479	1	2	yes	—
230	C₄	C₂×#3967	1	1	yes	characteristic
231	C₄	#65514	1	1	yes	characteristic
232	C₂	#10602167	1	2	yes	—
233	C₂	#10602173	1	2	yes	—
234	C₂	#10602571	1	1	yes	characteristic
235	C₂	#10602834	1	1	yes	characteristic
236	C₂	#10602860	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#4084
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₂×C₅×#500	C₂
socle, prime radical	C₂³×C₅	#121
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₂×#500	D₅
hypocenter	C₅	#9739
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
8. C₅×#500	C₄	semidirect
82. #500	Dic₅	semidirect
216. C₅	#9739	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#500	2
#10602572	2
C₂×#64766	2
#10600550	2
C₂×#64677	2
#9739	5