Subgroups — #11999108

Order 1280.

494 conjugacy classes of subgroups.

The subgroup lattice has 216 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	#10602580	C₂	1	2	yes	maximal
3	#10602578	C₂	1	2	yes	maximal
4	#10600555	C₂	1	1	yes	maximal, characteristic
5	C₂×#64674	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	2	no	—
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×#3967	—	2	4	no	—
12	#65224	C₂²	1	2	yes	—
13	C₂×C₄×#168	C₂²	1	1	yes	characteristic
14	C₂×C₄×#168	C₂²	1	1	yes	characteristic
15	#64771	—	2	2	no	—
16	#64749	—	2	4	no	—
17	#64674	—	2	2	no	—
18	#9831	—	5	1	no	maximal, Sylow
19	C₂²×C₅×Q₈	—	2	2	no	—
20	C₂²×C₅×C₈	D₄	1	2	yes	—
21	C₅×C₄⋊Q₈	—	2	2	no	—
22	C₅×C₄⋊Q₈	—	2	2	no	—
23	C₅×C₄⋊Q₈	—	2	2	no	—
24	C₅×C₄⋊Q₈	C₂×C₄	1	4	yes	—
25	C₂×C₅×C₄⋊C₄	—	2	2	no	—
26	C₂×C₅×C₄⋊C₄	—	2	2	no	—
27	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊Dic₅	—	2	2	no	—
29	C₂×C₄⋊Dic₅	—	2	2	no	—
30	C₂×C₄⋊Dic₅	—	2	2	no	—
31	C₂²×#168	D₄	1	2	yes	—
32	C₂²×#168	D₄	1	2	yes	—
33	C₅×#107	—	2	8	no	—
34	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
35	C₄×C₅×C₈	D₄	1	2	yes	—
36	#3967	—	4	8	no	—
37	C₄×#168	—	2	1	no	—
38	C₄×#168	—	2	1	no	—
39	C₄×#168	—	2	1	no	—
40	C₄×#168	—	2	1	no	—
41	C₂×#500	D₅	1	1	yes	characteristic
42	#1990	—	5	2	no	—
43	#1956	—	5	2	no	—
44	#1834	—	5	1	no	—
45	C₂×#322	—	5	1	no	—
46	C₂×C₅×Q₈	—	2	4	no	—
47	C₂×C₅×Q₈	—	4	2	no	—
48	C₂×C₅×Q₈	—	4	4	no	—
49	C₂×C₅×Q₈	—	4	4	no	—
50	C₂²×C₄×C₅	—	4	2	no	—
51	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
54	C₂²×Dic₅	—	4	2	no	—
55	C₂×C₅×C₈	—	2	4	no	—
56	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
57	C₅×C₄⋊C₄	—	2	4	no	—
58	C₅×C₄⋊C₄	—	2	4	no	—
59	C₅×C₄⋊C₄	—	4	2	no	—
60	C₅×C₄⋊C₄	—	4	2	no	—
61	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₅	#41	1	1	yes	characteristic
64	C₄²×C₅	#41	1	1	yes	characteristic
65	C₄⋊Dic₅	—	4	4	no	—
66	C₄⋊Dic₅	—	4	4	no	—
67	C₄⋊Dic₅	—	4	4	no	—
68	C₂×#168	—	2	2	no	—
69	C₂×#168	—	2	2	no	—
70	C₂×#168	—	2	2	no	—
71	C₂×#168	—	2	4	no	—
72	C₂×#168	D₈	1	4	yes	—
73	C₂×C₄⋊Q₈	C₂×D₅	1	2	yes	—
74	#500	—	2	2	no	—
75	#500	Dic₅	1	4	yes	complemented
76	C₂×#107	—	2	2	no	—
77	C₂×#103	—	10	4	no	—
78	C₂×C₄×C₈	—	5	1	no	—
79	C₂×C₄×C₈	—	5	1	no	—
80	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
81	#384	—	5	2	no	—
82	#340	—	10	4	no	—
83	#340	—	10	2	no	—
84	#322	—	10	2	no	—
85	C₂³×C₅	#127	1	1	yes	characteristic
86	C₅×Q₈	—	4	8	no	—
87	C₅×Q₈	—	8	4	no	—
88	C₂×C₄×C₅	—	4	4	no	—
89	C₂×C₄×C₅	—	8	2	no	—
90	C₂×C₄×C₅	—	8	2	no	—
91	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₅	#121	1	1	yes	characteristic
95	C₂×C₄×C₅	#121	1	1	yes	characteristic
96	C₂×C₄×C₅	#124	1	1	yes	characteristic
97	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
98	C₂×C₄×C₅	#135	1	2	yes	—
99	C₂×C₄×C₅	#137	1	4	yes	—
100	C₂×C₄×C₅	#137	1	4	yes	—
101	C₂×Dic₅	—	8	2	no	—
102	C₂×Dic₅	—	8	2	no	—
103	C₂×Dic₅	—	8	2	no	—
104	C₅×C₈	—	2	4	no	—
105	#168	—	2	4	no	—
106	#168	—	4	2	no	—
107	C₂²×Q₈	—	2	2	no	—
108	C₂²×C₈	—	5	2	no	—
109	C₂²×C₈	—	5	2	no	—
110	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
111	C₄⋊Q₈	—	2	2	no	—
112	C₄⋊Q₈	—	2	2	no	—
113	C₄⋊Q₈	—	2	2	no	—
114	C₄⋊Q₈	C₂×Dic₅	1	4	yes	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	—	10	2	no	—
118	C₂×C₄⋊C₄	—	10	2	no	—
119	C₂×C₄⋊C₄	—	10	2	no	—
120	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
121	#107	—	2	8	no	—
122	#103	—	20	8	no	—
123	C₄×C₈	—	10	1	no	—
124	C₄×C₈	—	10	1	no	—
125	C₄×C₈	—	10	1	no	—
126	C₄×C₈	—	10	1	no	—
127	C₄×C₈	C₂×Dic₅	1	2	yes	—
128	C₄×C₈	C₂²⋊D₅	1	2	yes	—
129	C₂²×C₅	#390	1	1	yes	characteristic
130	C₂²×C₅	#493	1	1	yes	characteristic
131	C₂²×C₅	#495	1	1	yes	characteristic
132	C₂²×C₅	#497	1	2	yes	—
133	C₂²×C₅	#497	1	2	yes	—
134	C₄×C₅	—	8	4	no	—
135	C₄×C₅	#419	1	2	yes	—
136	C₄×C₅	C₄×D₈	1	2	yes	—
137	C₄×C₅	#441	1	2	yes	—
138	C₄×C₅	#466	1	2	yes	—
139	C₄×C₅	#470	1	2	yes	—
140	C₄×C₅	#491	1	2	yes	—
141	Dic₅	—	16	2	no	—
142	C₂×Q₈	—	2	4	no	—
143	C₂×Q₈	—	4	2	no	—
144	C₂×Q₈	—	4	4	no	—
145	C₂×Q₈	—	4	4	no	—
146	C₂²×C₄	—	4	2	no	—
147	C₂²×C₄	—	20	2	no	—
148	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
149	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
150	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	5	4	no	—
153	C₂×C₈	—	10	2	no	—
154	C₂×C₈	—	10	2	no	—
155	C₂×C₈	—	10	4	no	—
156	C₂×C₈	—	10	2	no	—
157	C₂×C₈	C₂²⋊Dic₅	1	4	yes	—
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄⋊C₄	—	20	4	no	—
163	C₄⋊C₄	—	20	4	no	—
164	C₄⋊C₄	—	20	4	no	—
165	C₄²	#711	1	1	yes	characteristic
166	C₄²	#711	1	1	yes	characteristic
167	C₄²	C₂²×Dic₅	1	1	yes	characteristic
168	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
169	C₂×C₅	#1718	1	2	yes	—
170	C₂×C₅	#1722	1	2	yes	—
171	C₂×C₅	#1960	1	1	yes	characteristic
172	C₂×C₅	#1970	1	1	yes	characteristic
173	C₂×C₅	#1971	1	1	yes	characteristic
174	C₂³	#4086	1	1	yes	characteristic
175	Q₈	—	4	8	no	—
176	Q₈	—	8	4	no	—
177	C₂×C₄	—	4	4	no	—
178	C₂×C₄	—	8	2	no	—
179	C₂×C₄	—	8	2	no	—
180	C₂×C₄	—	40	2	no	—
181	C₂×C₄	—	40	2	no	—
182	C₂×C₄	—	40	2	no	—
183	C₂×C₄	D₅×D₈	1	2	yes	—
184	C₂×C₄	#4058	1	2	yes	—
185	C₂×C₄	#4064	1	4	yes	—
186	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
187	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
188	C₂×C₄	#4082	1	1	yes	characteristic
189	C₂×C₄	#4084	1	1	yes	characteristic
190	C₂×C₄	#4084	1	1	yes	characteristic
191	C₂×C₄	#4088	1	4	yes	—
192	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
193	C₈	—	2	4	no	—
194	C₈	—	10	4	no	—
195	C₈	—	20	2	no	—
196	C₅	#9831	1	1	yes	characteristic, complemented, Sylow
197	C₂²	#65375	1	2	yes	—
198	C₂²	#65440	1	1	yes	characteristic
199	C₂²	#65444	1	1	yes	characteristic
200	C₂²	#65479	1	2	yes	—
201	C₂²	#65510	1	1	yes	characteristic
202	C₄	—	8	4	no	—
203	C₄	—	80	2	no	—
204	C₄	#65371	1	2	yes	—
205	C₄	D₈×Dic₅	1	2	yes	—
206	C₄	#65446	1	2	yes	—
207	C₄	#65473	1	2	yes	—
208	C₄	#65478	1	2	yes	—
209	C₄	#65514	1	2	yes	—
210	C₂	#10602157	1	2	yes	—
211	C₂	#10602159	1	2	yes	—
212	C₂	#10602579	1	1	yes	characteristic
213	C₂	#10602831	1	1	yes	characteristic
214	C₂	#10602860	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#4086
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₂×C₅×#500	C₂
socle, prime radical	C₂³×C₅	#127
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₂×#500	D₅
hypocenter	C₅	#9831
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
75. #500	Dic₅	semidirect
196. C₅	#9831	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
#10602580	2
#10602578	2
#10600555	2
C₂×#64674	2
#9831	5