Subgroups — #12099713

Order 1280.

1126 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#65395	C₂	1	4	yes	maximal
3	#10613793	C₂	1	2	yes	maximal
4	#10600559	C₂	1	4	yes	maximal, complemented
5	#10600558	C₂	1	2	yes	maximal, complemented
6	#10600557	C₂	1	2	yes	maximal, complemented
7	C₂³×C₅×C₈	C₂²	1	2	yes	—
8	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂×#4072	C₂²	1	4	yes	—
10	C₂×C₄×C₅×C₈	C₄	1	8	yes	complemented
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
14	#65501	—	2	4	no	—
15	#65395	—	2	16	no	—
16	C₂×#3950	C₂²	1	8	yes	—
17	C₂×#3949	C₂²	1	8	yes	—
18	#65225	C₂²	1	4	yes	—
19	#65224	C₂²	1	4	yes	—
20	C₄×C₄⋊Dic₅	—	2	4	no	—
21	#20797	—	5	1	no	maximal, Sylow
22	C₂³×C₄×C₅	D₄	1	2	yes	—
23	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₂²×C₅×C₈	C₂×C₄	1	8	yes	—
25	C₂²×C₅×C₈	D₄	1	8	yes	—
26	C₂²×C₅×C₈	Q₈	1	8	yes	—
27	C₂²×C₅×C₈	C₂³	1	2	yes	—
28	C₂²×C₅×C₈	C₂³	1	2	yes	—
29	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₅	D₄	1	2	yes	—
31	C₂×C₄²×C₅	Q₈	1	4	yes	—
32	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
34	C₂×C₂²⋊Dic₅	—	2	4	no	—
35	#4072	—	2	16	no	—
36	C₂×C₄⋊Dic₅	—	2	8	no	—
37	C₂×C₄⋊Dic₅	C₂³	1	8	yes	—
38	C₂×C₄×Dic₅	—	2	4	no	—
39	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
40	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
41	C₄×C₅×C₈	C₂×C₄	1	8	yes	—
42	#3963	—	2	8	no	—
43	#3950	—	2	16	no	—
44	#3949	—	2	16	no	—
45	C₂×#427	—	5	4	no	—
46	C₂²×C₄×C₈	D₅	1	1	yes	characteristic
47	#2285	—	5	2	no	—
48	#1846	—	5	4	no	—
49	#1845	—	5	2	no	—
50	#1844	—	5	2	no	—
51	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₅	—	2	8	no	—
53	C₂²×C₄×C₅	C₄⋊C₄	1	8	yes	—
54	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
57	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
60	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
61	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
62	C₂²×Dic₅	—	2	8	no	—
63	C₂×C₅×C₈	C₄⋊C₄	1	32	yes	complemented
64	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
65	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
66	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
67	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
68	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
69	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
70	C₄²×C₅	C₄⋊C₄	1	8	yes	—
71	C₄²×C₅	C₂²×C₄	1	2	yes	—
72	C₄²×C₅	C₂²×C₄	1	2	yes	—
73	C₄²×C₅	C₂×Q₈	1	2	yes	—
74	C₄²×C₅	C₂×Q₈	1	2	yes	—
75	C₂²⋊Dic₅	—	4	8	no	—
76	C₄⋊Dic₅	—	2	16	no	—
77	C₄⋊Dic₅	—	4	16	no	—
78	C₄×Dic₅	—	4	8	no	—
79	C₂³×C₈	C₂×D₅	1	2	yes	—
80	C₂×#117	—	5	4	no	—
81	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
82	#427	—	10	16	no	—
83	C₂×#107	—	5	8	no	—
84	C₂×#106	—	5	8	no	—
85	C₂×C₄×C₈	Dic₅	1	8	yes	complemented
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	2	yes	—
89	#384	—	5	4	no	—
90	#382	—	5	4	no	—
91	#380	—	10	4	no	—
92	C₄×C₄⋊C₄	—	10	4	no	—
93	C₂³×C₅	—	2	4	no	—
94	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
95	C₂³×C₅	#127	1	2	yes	—
96	C₂³×C₅	C₄⋊Q₈	1	2	yes	—
97	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₅	—	2	4	no	—
99	C₂×C₄×C₅	—	2	16	no	—
100	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
101	C₂×C₄×C₅	C₂×C₄⋊C₄	1	8	yes	—
102	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
104	C₂×C₄×C₅	#127	1	2	yes	—
105	C₂×C₄×C₅	C₄⋊Q₈	1	8	yes	—
106	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
107	C₂×C₄×C₅	#135	1	8	yes	—
108	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
110	C₂×C₄×C₅	C₂²×Q₈	1	2	yes	—
111	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
112	C₂×Dic₅	—	4	8	no	—
113	C₂×Dic₅	—	4	16	no	—
114	C₅×C₈	C₂×C₄⋊C₄	1	8	yes	—
115	C₅×C₈	C₂×C₄⋊C₄	1	8	yes	—
116	C₂³×C₄	D₂₀	1	2	yes	—
117	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
118	C₂²×C₈	Dic₁₀	1	8	yes	—
119	C₂²×C₈	D₂₀	1	8	yes	—
120	C₂²×C₈	C₂×Dic₅	1	8	yes	—
121	C₂²×C₈	C₂²×D₅	1	2	yes	—
122	C₂²×C₈	C₂²×D₅	1	2	yes	—
123	#117	—	10	16	no	—
124	C₂×C₄⋊C₄	—	5	8	no	—
125	C₂×C₄⋊C₄	—	10	8	no	—
126	C₂×C₂²⋊C₄	—	10	4	no	—
127	C₂×C₄²	—	10	4	no	—
128	C₂×C₄²	Dic₁₀	1	4	yes	—
129	C₂×C₄²	D₂₀	1	2	yes	—
130	C₂×C₄²	C₂×Dic₅	1	4	yes	—
131	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
132	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
133	#107	—	10	16	no	—
134	#106	—	10	16	no	—
135	C₄×C₈	C₂×Dic₅	1	4	yes	—
136	C₄×C₈	C₂×Dic₅	1	4	yes	—
137	C₄×C₈	C₂×Dic₅	1	8	yes	—
138	#95	—	10	8	no	—
139	C₂²×C₅	—	2	4	no	—
140	C₂²×C₅	—	2	4	no	—
141	C₂²×C₅	—	2	4	no	—
142	C₂²×C₅	#384	1	4	yes	—
143	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
144	C₂²×C₅	C₂×#127	1	1	yes	characteristic
145	C₂²×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
146	C₂²×C₅	C₂×#135	1	4	yes	—
147	C₄×C₅	—	2	4	no	—
148	C₄×C₅	#384	1	4	yes	—
149	C₄×C₅	#427	1	8	yes	complemented
150	C₄×C₅	C₂²×C₄⋊C₄	1	2	yes	—
151	C₄×C₅	C₂×C₄⋊Q₈	1	2	yes	—
152	Dic₅	—	8	8	no	—
153	C₂⁴	C₂×D₂₀	1	1	yes	characteristic
154	C₂²×C₄	—	2	8	no	—
155	C₂²×C₄	—	10	8	no	—
156	C₂²×C₄	C₄⋊Dic₅	1	8	yes	—
157	C₂²×C₄	C₂×Dic₁₀	1	2	yes	—
158	C₂²×C₄	C₂×Dic₁₀	1	4	yes	—
159	C₂²×C₄	C₂×D₂₀	1	2	yes	—
160	C₂²×C₄	C₂×D₂₀	1	4	yes	—
161	C₂²×C₄	C₂×D₂₀	1	2	yes	—
162	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
163	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
164	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
165	C₂×C₈	C₄⋊Dic₅	1	32	yes	complemented
166	C₂×C₈	C₂×Dic₁₀	1	4	yes	—
167	C₂×C₈	C₂×Dic₁₀	1	4	yes	—
168	C₂×C₈	C₂×D₂₀	1	4	yes	—
169	C₂×C₈	C₂×D₂₀	1	4	yes	—
170	C₂×C₈	C₂²×Dic₅	1	4	yes	—
171	C₂×C₈	C₂²×Dic₅	1	4	yes	—
172	C₄⋊C₄	—	10	16	no	—
173	C₄⋊C₄	—	20	16	no	—
174	C₂²⋊C₄	—	20	8	no	—
175	C₄²	—	20	8	no	—
176	C₄²	C₄⋊Dic₅	1	8	yes	—
177	C₄²	C₂×Dic₁₀	1	2	yes	—
178	C₄²	C₂×Dic₁₀	1	2	yes	—
179	C₄²	C₂²×Dic₅	1	2	yes	—
180	C₄²	C₂²×Dic₅	1	2	yes	—
181	C₂×C₅	—	2	4	no	—
182	C₂×C₅	C₂×#384	1	1	yes	characteristic
183	C₂×C₅	C₂×#427	1	2	yes	—
184	C₂×C₅	#3147	1	2	yes	—
185	C₂×C₅	#3160	1	2	yes	—
186	C₂³	—	2	4	no	—
187	C₂³	#4015	1	2	yes	—
188	C₂³	#4020	1	2	yes	—
189	C₂³	C₂×C₄⋊Dic₅	1	2	yes	—
190	C₂³	C₂²×D₂₀	1	1	yes	characteristic
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	16	no	—
193	C₂×C₄	—	20	8	no	—
194	C₂×C₄	—	20	16	no	—
195	C₂×C₄	#4015	1	8	yes	—
196	C₂×C₄	#4015	1	2	yes	—
197	C₂×C₄	#4020	1	2	yes	—
198	C₂×C₄	#4050	1	8	yes	—
199	C₂×C₄	C₂×C₄⋊Dic₅	1	4	yes	—
200	C₂×C₄	C₂×C₄⋊Dic₅	1	8	yes	—
201	C₂×C₄	C₂×C₄⋊Dic₅	1	4	yes	—
202	C₂×C₄	C₂×C₄⋊Dic₅	1	2	yes	—
203	C₂×C₄	C₂²×Dic₁₀	1	1	yes	characteristic
204	C₂×C₄	C₂²×Dic₁₀	1	2	yes	—
205	C₂×C₄	C₂²×D₂₀	1	2	yes	—
206	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
207	C₈	C₂×C₄⋊Dic₅	1	8	yes	—
208	C₈	C₂×C₄⋊Dic₅	1	8	yes	—
209	C₅	#20797	1	1	yes	characteristic, complemented, Sylow
210	C₂²	—	2	4	no	—
211	C₂²	—	2	4	no	—
212	C₂²	—	2	4	no	—
213	C₂²	#65224	1	4	yes	—
214	C₂²	C₂×#4015	1	1	yes	characteristic
215	C₂²	C₂×#4020	1	1	yes	characteristic
216	C₂²	C₂×#4050	1	4	yes	—
217	C₂²	C₂²×C₄⋊Dic₅	1	1	yes	characteristic
218	C₄	—	2	4	no	—
219	C₄	—	40	8	no	—
220	C₄	#65224	1	4	yes	—
221	C₄	#65395	1	8	yes	complemented
222	C₄	C₂×#4015	1	2	yes	—
223	C₄	C₂²×C₄⋊Dic₅	1	2	yes	—
224	C₂	—	2	4	no	—
225	C₂	#10610173	1	2	yes	—
226	C₂	#10610198	1	2	yes	—
227	C₂	C₂×#65224	1	1	yes	characteristic
228	C₂	C₂×#65395	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₀
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅
Fitting	C₂²×C₄×C₅×C₈	C₂
socle	C₂³×C₅	C₂²×D₄
prime radical	C₂⁴×C₅	C₂×D₄
cosocle	C₂²×C₄×C₅	C₂⁴
hypercenter	C₂²×C₄×C₈	D₅
hypocenter	C₅	#20797
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. #10600559	C₂	semidirect
5. #10600558	C₂	semidirect
6. #10600557	C₂	semidirect
10. C₂×C₄×C₅×C₈	C₄	semidirect
63. C₂×C₅×C₈	C₄⋊C₄	semidirect
85. C₂×C₄×C₈	Dic₅	semidirect
149. C₄×C₅	#427	semidirect
165. C₂×C₈	C₄⋊Dic₅	semidirect
209. C₅	#20797	semidirect
221. C₄	#65395	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₅×C₈	2
C₂×#65395	2
#10613793	2
#10600559	2
#10600558	2
#10600557	2
#20797	5