Subgroups — #11998674

Order 1280.

394 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10610173	C₂	1	1	yes	maximal, characteristic
2	C₅×#1450	C₂	1	1	yes	maximal, characteristic
3	#10601586	C₂	1	4	yes	maximal
4	#10600559	C₂	1	1	yes	maximal, characteristic
5	#65804	C₂²	1	2	yes	—
6	C₅×#431	C₂²	1	2	yes	—
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	#65395	—	2	1	no	—
9	#65395	—	2	1	no	—
10	C₂×#3950	—	2	1	no	—
11	C₂×#3949	—	2	1	no	—
12	#65225	C₂²	1	2	yes	—
13	#64972	—	2	1	no	—
14	#64972	—	2	1	no	—
15	#64971	C₄	1	2	yes	complemented
16	#64969	C₄	1	2	yes	complemented
17	C₅×#322	—	2	1	no	—
18	C₅×C₈²	—	2	1	no	—
19	#64687	—	2	8	no	—
20	#9755	—	5	1	no	maximal, Sylow
21	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
22	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
24	#4072	—	2	2	no	—
25	C₂×C₄⋊Dic₅	—	2	2	no	—
26	#4070	—	2	4	no	—
27	#4016	—	2	2	no	—
28	#4015	C₂×C₄	1	4	yes	—
29	C₄×Dic₁₀	—	2	4	no	—
30	C₅×C₄⋊C₈	—	2	2	no	—
31	C₅×C₂²⋊C₈	D₄	1	4	yes	—
32	C₄×C₅×C₈	—	2	2	no	—
33	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
34	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
35	C₄×C₅×C₈	D₄	1	1	yes	characteristic
36	C₄×C₅×C₈	D₄	1	1	yes	characteristic
37	#3963	—	2	4	no	—
38	#3950	—	2	2	no	—
39	#3950	—	2	2	no	—
40	#3950	—	4	2	no	—
41	#3949	—	2	2	no	—
42	#3949	—	2	2	no	—
43	#3949	—	4	2	no	—
44	#3160	—	5	1	no	—
45	#1846	—	5	1	no	—
46	#1518	—	5	4	no	—
47	#1450	D₅	1	1	yes	characteristic
48	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
50	C₂²×Dic₅	—	4	2	no	—
51	C₂×Dic₁₀	—	2	4	no	—
52	C₂×C₅×C₈	—	2	4	no	—
53	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₅×C₈	SD₁₆	1	2	yes	—
58	C₂×C₅×C₈	SD₁₆	1	2	yes	—
59	C₂×C₅×C₈	Q₁₆	1	2	yes	—
60	C₂×C₅×C₈	Q₁₆	1	2	yes	—
61	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₅	#41	1	2	yes	—
64	C₂²⋊Dic₅	—	4	2	no	—
65	C₄⋊Dic₅	—	2	4	no	—
66	C₄⋊Dic₅	—	2	4	no	—
67	C₄⋊Dic₅	—	4	2	no	—
68	C₄⋊Dic₅	—	4	2	no	—
69	#683	—	4	4	no	—
70	C₄×Dic₅	—	4	2	no	—
71	#533	—	5	2	no	—
72	#500	—	5	2	no	—
73	#499	—	10	1	no	—
74	#499	—	10	1	no	—
75	#498	—	5	2	no	—
76	#431	C₂×D₅	1	2	yes	—
77	#427	—	10	1	no	—
78	#427	—	10	1	no	—
79	C₂×#107	—	10	1	no	—
80	C₂×#106	—	10	1	no	—
81	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
82	#382	—	5	2	no	—
83	#328	—	10	8	no	—
84	#322	—	2	1	no	—
85	C₈²	—	2	1	no	—
86	C₂³×C₅	#127	1	1	yes	characteristic
87	C₂×C₄×C₅	—	4	2	no	—
88	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
90	C₂×C₄×C₅	#121	1	2	yes	—
91	C₂×C₄×C₅	#124	1	1	yes	characteristic
92	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
93	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
95	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
96	C₂×Dic₅	—	4	4	no	—
97	C₂×Dic₅	—	8	4	no	—
98	C₂×Dic₅	—	8	2	no	—
99	Dic₁₀	—	8	4	no	—
100	C₅×C₈	—	4	4	no	—
101	C₅×C₈	#103	1	2	yes	—
102	C₅×C₈	#103	1	2	yes	—
103	C₅×C₈	#103	1	2	yes	—
104	C₅×C₈	#103	1	2	yes	—
105	C₂²×C₈	D₂₀	1	1	yes	characteristic
106	C₂²×C₈	D₂₀	1	1	yes	characteristic
107	C₄⋊Q₈	—	5	4	no	—
108	#125	—	10	2	no	—
109	C₂²⋊Q₈	—	10	4	no	—
110	C₄×Q₈	—	10	4	no	—
111	#117	—	10	2	no	—
112	C₂×C₄⋊C₄	—	10	2	no	—
113	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
114	#107	—	10	2	no	—
115	#107	—	10	2	no	—
116	#107	—	20	2	no	—
117	#106	—	10	2	no	—
118	#106	—	10	2	no	—
119	#106	—	20	2	no	—
120	C₄⋊C₈	—	2	2	no	—
121	C₂²⋊C₈	D₂₀	1	4	yes	—
122	C₄×C₈	—	2	2	no	—
123	C₄×C₈	C₄×D₅	1	1	yes	characteristic
124	C₄×C₈	C₄×D₅	1	1	yes	characteristic
125	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
126	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
127	#95	—	10	4	no	—
128	C₂²×C₅	—	2	2	no	—
129	C₂²×C₅	#390	1	1	yes	characteristic
130	C₂²×C₅	#492	1	1	yes	characteristic
131	C₂²×C₅	#494	1	1	yes	characteristic
132	C₄×C₅	—	2	2	no	—
133	C₄×C₅	—	2	2	no	—
134	C₄×C₅	C₂×#103	1	1	yes	characteristic
135	C₄×C₅	C₂×#103	1	1	yes	characteristic
136	C₄×C₅	#487	1	1	yes	characteristic
137	C₄×C₅	#487	1	1	yes	characteristic
138	Dic₅	—	8	4	no	—
139	Dic₅	—	16	2	no	—
140	C₂×Q₈	—	10	4	no	—
141	C₂²×C₄	—	20	2	no	—
142	C₂²×C₄	C₂×D₂₀	1	2	yes	—
143	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	#677	1	2	yes	—
146	C₂×C₈	#677	1	2	yes	—
147	C₂×C₈	Dic₂₀	1	2	yes	—
148	C₂×C₈	Dic₂₀	1	2	yes	—
149	C₂×C₈	#685	1	1	yes	characteristic
150	C₂×C₈	#685	1	1	yes	characteristic
151	C₂×C₈	#685	1	1	yes	characteristic
152	C₂×C₈	#685	1	1	yes	characteristic
153	C₄⋊C₄	—	10	4	no	—
154	C₄⋊C₄	—	10	4	no	—
155	C₄⋊C₄	—	20	4	no	—
156	C₄⋊C₄	—	20	2	no	—
157	C₄⋊C₄	—	20	2	no	—
158	C₂²⋊C₄	—	20	2	no	—
159	C₄²	—	20	2	no	—
160	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
161	C₄²	#709	1	2	yes	—
162	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
163	C₂×C₅	—	2	1	no	—
164	C₂×C₅	—	2	1	no	—
165	C₂×C₅	#1956	1	1	yes	characteristic
166	C₂×C₅	#1976	1	2	yes	—
167	C₂³	#4020	1	1	yes	characteristic
168	Q₈	—	40	4	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	20	4	no	—
171	C₂×C₄	—	40	4	no	—
172	C₂×C₄	—	40	2	no	—
173	C₂×C₄	C₄×D₂₀	1	2	yes	—
174	C₂×C₄	#4021	1	1	yes	characteristic
175	C₂×C₄	C₂×#677	1	1	yes	characteristic
176	C₂×C₄	C₂×#677	1	1	yes	characteristic
177	C₂×C₄	C₂×Dic₂₀	1	1	yes	characteristic
178	C₂×C₄	C₂×Dic₂₀	1	1	yes	characteristic
179	C₂×C₄	C₂×#685	1	1	yes	characteristic
180	C₂×C₄	#4076	1	2	yes	—
181	C₈	—	4	4	no	—
182	C₈	#3948	1	2	yes	—
183	C₈	#3948	1	2	yes	—
184	C₈	#3948	1	2	yes	—
185	C₈	#3948	1	2	yes	—
186	C₅	#9755	1	1	yes	characteristic, complemented, Sylow
187	C₂²	—	2	2	no	—
188	C₂²	#64982	1	1	yes	characteristic
189	C₂²	#64988	1	1	yes	characteristic
190	C₂²	#65228	1	1	yes	characteristic
191	C₄	—	2	2	no	—
192	C₄	—	2	2	no	—
193	C₄	—	40	4	no	—
194	C₄	—	80	2	no	—
195	C₄	#64970	1	1	yes	characteristic
196	C₄	#64970	1	1	yes	characteristic
197	C₄	C₂×#3948	1	1	yes	characteristic
198	C₄	C₂×#3948	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	#10600564	1	1	yes	characteristic
202	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	C₂³×C₅	#127
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1450	D₅
hypocenter	C₅	#9755
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
15. #64971	C₄	semidirect
16. #64969	C₄	semidirect
186. C₅	#9755	semidirect

Maximal subgroups

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

subgroup	index
#10610173	2
C₅×#1450	2
#10601586	2
#10600559	2
#9755	5