Subgroups — #11999134

Order 1280.

510 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#64971	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#322	C₂	1	1	yes	maximal, characteristic
3	#10600570	C₂	1	2	yes	maximal
4	#10600564	C₂	1	2	yes	maximal
5	#10600557	C₂	1	1	yes	maximal, characteristic
6	C₂×#4015	C₂²	1	2	yes	—
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂×#3950	—	2	2	no	—
11	C₂×#3949	—	2	2	no	—
12	C₂×#3948	—	2	4	no	—
13	#65224	C₂²	1	2	yes	—
14	#64971	—	2	2	no	—
15	#64971	C₄	1	4	yes	complemented
16	C₅×#322	—	2	2	no	—
17	#64771	—	2	4	no	—
18	#9836	—	5	1	no	maximal, Sylow
19	C₂²×Dic₁₀	—	2	2	no	—
20	C₂²×C₅×C₈	D₄	1	2	yes	—
21	C₂²×C₅×C₈	D₄	1	2	yes	—
22	C₂²×C₅×C₈	D₄	1	2	yes	—
23	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊Dic₅	—	2	2	no	—
25	C₂×C₄⋊Dic₅	—	2	2	no	—
26	C₂×C₄⋊Dic₅	—	2	2	no	—
27	C₂×C₄⋊Dic₅	—	2	2	no	—
28	C₂×C₄⋊Dic₅	—	2	2	no	—
29	#4015	—	2	2	no	—
30	#4015	—	2	2	no	—
31	#4015	—	2	2	no	—
32	#4015	C₂×C₄	1	4	yes	—
33	C₄×C₅×C₈	—	2	1	no	—
34	C₄×C₅×C₈	—	2	1	no	—
35	C₄×C₅×C₈	—	2	1	no	—
36	C₄×C₅×C₈	—	2	1	no	—
37	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
38	C₄×C₅×C₈	D₄	1	2	yes	—
39	#3950	—	2	8	no	—
40	#3949	—	4	4	no	—
41	#3948	—	4	8	no	—
42	C₂×#500	—	5	1	no	—
43	#1990	—	5	2	no	—
44	#1956	—	5	2	no	—
45	#1844	—	5	1	no	—
46	C₂×#322	D₅	1	1	yes	characteristic
47	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
50	C₂²×Dic₅	—	4	2	no	—
51	C₂²×Dic₅	—	4	2	no	—
52	C₂×Dic₁₀	—	2	4	no	—
53	C₂×Dic₁₀	—	4	4	no	—
54	C₂×Dic₁₀	—	4	4	no	—
55	C₂×Dic₁₀	—	4	2	no	—
56	C₂×C₅×C₈	—	2	2	no	—
57	C₂×C₅×C₈	—	2	2	no	—
58	C₂×C₅×C₈	—	2	4	no	—
59	C₂×C₅×C₈	—	2	4	no	—
60	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
61	C₂×C₅×C₈	SD₁₆	1	4	yes	—
62	C₂×C₅×C₈	Q₁₆	1	4	yes	—
63	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₅	#41	1	1	yes	characteristic
66	C₄²×C₅	#41	1	1	yes	characteristic
67	C₄⋊Dic₅	—	2	4	no	—
68	C₄⋊Dic₅	—	2	4	no	—
69	C₄⋊Dic₅	—	4	2	no	—
70	C₄⋊Dic₅	—	4	2	no	—
71	C₄⋊Dic₅	—	4	4	no	—
72	C₄⋊Dic₅	—	4	4	no	—
73	C₄⋊Dic₅	—	4	4	no	—
74	C₂×C₄⋊Q₈	—	5	2	no	—
75	#500	—	5	4	no	—
76	#500	—	10	2	no	—
77	C₂×#107	—	10	2	no	—
78	C₂×#106	—	10	2	no	—
79	C₂×#103	—	10	4	no	—
80	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
81	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
82	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
83	#384	—	5	2	no	—
84	#340	—	10	4	no	—
85	#322	—	2	2	no	—
86	C₂³×C₅	#127	1	1	yes	characteristic
87	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₅	#121	1	1	yes	characteristic
91	C₂×C₄×C₅	#121	1	1	yes	characteristic
92	C₂×C₄×C₅	#124	1	1	yes	characteristic
93	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
94	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
95	C₂×C₄×C₅	AGL(1,ℤ/8)	1	4	yes	—
96	C₂×C₄×C₅	#137	1	4	yes	—
97	C₂×Dic₅	—	4	4	no	—
98	C₂×Dic₅	—	8	2	no	—
99	C₂×Dic₅	—	8	2	no	—
100	C₂×Dic₅	—	8	2	no	—
101	C₂×Dic₅	—	8	2	no	—
102	C₂×Dic₅	—	8	2	no	—
103	Dic₁₀	—	4	8	no	—
104	Dic₁₀	—	8	4	no	—
105	C₅×C₈	—	4	2	no	—
106	C₅×C₈	—	4	2	no	—
107	C₅×C₈	#103	1	8	yes	complemented
108	C₂²×Q₈	—	10	2	no	—
109	C₂²×C₈	D₂₀	1	2	yes	—
110	C₂²×C₈	D₂₀	1	2	yes	—
111	C₂²×C₈	D₂₀	1	2	yes	—
112	C₄⋊Q₈	—	5	4	no	—
113	C₄⋊Q₈	—	10	2	no	—
114	C₄⋊Q₈	—	10	2	no	—
115	C₄⋊Q₈	—	10	2	no	—
116	C₂×C₄⋊C₄	—	10	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×C₄⋊C₄	—	10	2	no	—
121	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
122	#107	—	10	8	no	—
123	#106	—	20	4	no	—
124	#103	—	20	8	no	—
125	C₄×C₈	—	2	1	no	—
126	C₄×C₈	—	2	1	no	—
127	C₄×C₈	—	2	1	no	—
128	C₄×C₈	—	2	1	no	—
129	C₄×C₈	C₄×D₅	1	2	yes	—
130	C₄×C₈	C₂²⋊D₅	1	2	yes	—
131	C₂²×C₅	#390	1	1	yes	characteristic
132	C₂²×C₅	#492	1	1	yes	characteristic
133	C₂²×C₅	#494	1	1	yes	characteristic
134	C₂²×C₅	#496	1	2	yes	—
135	C₂²×C₅	#497	1	2	yes	—
136	C₄×C₅	C₂×#103	1	2	yes	—
137	C₄×C₅	#441	1	2	yes	—
138	C₄×C₅	#442	1	2	yes	—
139	C₄×C₅	#469	1	2	yes	—
140	C₄×C₅	#470	1	2	yes	—
141	C₄×C₅	#487	1	2	yes	—
142	Dic₅	—	8	4	no	—
143	Dic₅	—	16	2	no	—
144	C₂×Q₈	—	10	4	no	—
145	C₂×Q₈	—	20	2	no	—
146	C₂×Q₈	—	20	4	no	—
147	C₂×Q₈	—	20	4	no	—
148	C₂²×C₄	—	20	2	no	—
149	C₂²×C₄	—	20	2	no	—
150	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
151	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
152	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	—	2	4	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	#677	1	4	yes	—
158	C₂×C₈	Dic₂₀	1	4	yes	—
159	C₂×C₈	#685	1	4	yes	—
160	C₄⋊C₄	—	10	4	no	—
161	C₄⋊C₄	—	10	4	no	—
162	C₄⋊C₄	—	20	2	no	—
163	C₄⋊C₄	—	20	2	no	—
164	C₄⋊C₄	—	20	4	no	—
165	C₄⋊C₄	—	20	4	no	—
166	C₄⋊C₄	—	20	4	no	—
167	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
168	C₄²	#709	1	1	yes	characteristic
169	C₄²	#709	1	1	yes	characteristic
170	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
171	C₂×C₅	#1714	1	2	yes	—
172	C₂×C₅	#1723	1	2	yes	—
173	C₂×C₅	#1956	1	1	yes	characteristic
174	C₂×C₅	#1971	1	1	yes	characteristic
175	C₂×C₅	#1972	1	1	yes	characteristic
176	C₂³	#4020	1	1	yes	characteristic
177	Q₈	—	20	8	no	—
178	Q₈	—	40	4	no	—
179	C₂×C₄	—	20	4	no	—
180	C₂×C₄	—	40	2	no	—
181	C₂×C₄	—	40	2	no	—
182	C₂×C₄	—	40	2	no	—
183	C₂×C₄	—	40	2	no	—
184	C₂×C₄	—	40	2	no	—
185	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
186	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
187	C₂×C₄	#4021	1	1	yes	characteristic
188	C₂×C₄	C₂×#677	1	2	yes	—
189	C₂×C₄	C₂×Dic₂₀	1	2	yes	—
190	C₂×C₄	#4054	1	4	yes	—
191	C₂×C₄	#4055	1	4	yes	—
192	C₂×C₄	C₂×#685	1	1	yes	characteristic
193	C₂×C₄	#4076	1	1	yes	characteristic
194	C₂×C₄	#4076	1	1	yes	characteristic
195	C₈	—	4	2	no	—
196	C₈	—	4	2	no	—
197	C₈	#3948	1	8	yes	complemented
198	C₅	#9836	1	1	yes	characteristic, complemented, Sylow
199	C₂²	#64982	1	1	yes	characteristic
200	C₂²	#64988	1	1	yes	characteristic
201	C₂²	#65001	1	2	yes	—
202	C₂²	#65004	1	2	yes	—
203	C₂²	#65228	1	1	yes	characteristic
204	C₄	—	40	4	no	—
205	C₄	—	80	2	no	—
206	C₄	#64970	1	2	yes	—
207	C₄	#65000	1	2	yes	—
208	C₄	#65005	1	2	yes	—
209	C₄	C₂×#3948	1	2	yes	—
210	C₄	#65424	1	2	yes	—
211	C₄	#65426	1	2	yes	—
212	C₂	#10598883	1	2	yes	—
213	C₂	#10598892	1	2	yes	—
214	C₂	#10600564	1	1	yes	characteristic
215	C₂	#10600617	1	1	yes	characteristic
216	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, prime radical	C₂³×C₅	#127
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₂×#322	D₅
hypocenter	C₅	#9836
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
15. #64971	C₄	semidirect
107. C₅×C₈	#103	semidirect
197. C₈	#3948	semidirect
198. C₅	#9836	semidirect

Maximal subgroups

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

subgroup	index
C₂×#64971	2
C₂×C₅×#322	2
#10600570	2
#10600564	2
#10600557	2
#9836	5