Subgroups — C₂×#10606050 #12257811

Order 1280.

5694 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#4039	C₂	1	6	yes	maximal, complemented
2	C₂²×#4020	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#384	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#65277	C₂	1	6	yes	maximal, complemented
5	C₂×C₄²×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
6	#10606050	C₂	1	16	yes	maximal, direct product factor
7	C₂³×D₂₀	C₂²	1	6	yes	—
8	C₂³×C₄×D₅	C₂²	1	4	yes	—
9	C₂²×C₅×C₄⋊C₄	C₂²	1	6	yes	—
10	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂²×#685	C₂²	1	12	yes	—
12	C₂²×C₄×Dic₅	C₂²	1	3	yes	—
13	C₂²×C₄×Dic₅	C₂²	1	3	yes	—
14	C₂×#4039	—	2	48	no	—
15	C₂×#4039	C₂²	1	48	yes	complemented
16	C₂×#4020	C₄	1	16	yes	complemented
17	C₂×#4020	C₂²	1	8	yes	—
18	C₅×#384	C₂²	1	8	yes	complemented
19	#65277	C₂²	1	48	yes	complemented
20	C₄²×Dic₅	C₂²	1	8	yes	complemented
21	C₂×C₄×#127	—	5	1	no	maximal, Sylow
22	C₂⁴×D₅	C₂³	1	4	yes	—
23	C₂³×C₄×C₅	C₂³	1	4	yes	—
24	C₂³×C₄×C₅	C₂³	1	3	yes	—
25	C₂³×Dic₅	C₂³	1	3	yes	—
26	C₂³×Dic₅	C₂³	1	1	yes	characteristic
27	C₂²×D₂₀	—	2	24	no	—
28	C₂²×D₂₀	—	2	24	no	—
29	C₂²×D₂₀	C₂×C₄	1	48	yes	complemented
30	C₂²×D₂₀	C₂³	1	24	yes	—
31	C₂²×C₄×D₅	—	2	48	no	—
32	C₂²×C₄×D₅	C₂³	1	16	yes	—
33	C₂×C₅×C₄⋊C₄	—	2	24	no	—
34	C₂×C₅×C₄⋊C₄	C₂³	1	24	yes	—
35	C₂×C₄²×C₅	C₂×C₄	1	8	yes	complemented
36	C₂×C₄²×C₅	C₂³	1	4	yes	—
37	C₂×#685	—	2	48	no	—
38	C₂×#685	C₂³	1	48	yes	—
39	C₂×C₄×Dic₅	—	2	12	no	—
40	C₂×C₄×Dic₅	D₄	1	24	yes	complemented
41	C₂×C₄×Dic₅	C₂³	1	12	yes	—
42	C₂×C₄×Dic₅	C₂³	1	12	yes	—
43	#4039	—	2	192	no	—
44	#4020	C₂×C₄	1	64	yes	complemented
45	C₂²×#127	—	5	1	no	—
46	C₂²×C₄×D₄	—	5	6	no	—
47	C₄×#127	—	5	16	no	—
48	C₂×#390	—	5	6	no	—
49	C₂×#384	D₅	1	1	yes	characteristic, complemented
50	C₂×C₄³	—	5	1	no	—
51	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
52	C₂³×D₅	—	2	24	no	—
53	C₂³×D₅	—	2	24	no	—
54	C₂³×D₅	C₂²×C₄	1	16	yes	—
55	C₂³×D₅	C₂⁴	1	8	yes	—
56	C₂²×C₄×C₅	—	2	24	no	—
57	C₂²×C₄×C₅	C₂²×C₄	1	12	yes	—
58	C₂²×C₄×C₅	C₂×D₄	1	12	yes	—
59	C₂²×C₄×C₅	#41	1	12	yes	—
60	C₂²×C₄×C₅	C₂⁴	1	8	yes	—
61	C₂²×C₄×C₅	C₂⁴	1	6	yes	—
62	C₂²×Dic₅	—	2	12	no	—
63	C₂²×Dic₅	C₂×D₄	1	12	yes	—
64	C₂²×Dic₅	C₂×D₄	1	12	yes	—
65	C₂²×Dic₅	C₂⁴	1	6	yes	—
66	C₂²×Dic₅	C₂⁴	1	2	yes	—
67	C₂×D₂₀	—	2	192	no	—
68	C₂×D₂₀	—	2	48	no	—
69	C₂×D₂₀	—	2	48	no	—
70	C₂×D₂₀	C₂²×C₄	1	96	yes	complemented
71	C₂×C₄×D₅	—	2	96	no	—
72	C₂×C₄×D₅	—	4	64	no	—
73	C₅×C₄⋊C₄	—	2	48	no	—
74	C₄²×C₅	C₂²×C₄	1	16	yes	complemented
75	#685	—	2	96	no	—
76	C₄×Dic₅	—	2	24	no	—
77	C₄×Dic₅	C₂×D₄	1	48	yes	complemented
78	C₂³×D₄	—	5	6	no	—
79	C₂⁴×C₄	—	5	4	no	—
80	C₂×#127	—	5	16	no	—
81	C₂×#127	—	5	8	no	—
82	C₂×C₄×D₄	—	5	48	no	—
83	C₂×C₄×D₄	—	10	48	no	—
84	C₂²×C₄⋊C₄	C₂×D₅	1	6	yes	—
85	C₂²×C₂²⋊C₄	—	5	12	no	—
86	C₂²×C₄²	—	5	3	no	—
87	C₂²×C₄²	—	5	3	no	—
88	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
89	#390	—	5	48	no	—
90	#384	C₂×D₅	1	8	yes	complemented
91	C₄³	—	5	8	no	—
92	C₂³×C₅	C₂³×C₄	1	2	yes	—
93	C₂³×C₅	C₂²×D₄	1	6	yes	—
94	C₂³×C₅	C₂×#41	1	6	yes	—
95	C₂³×C₅	C₂⁵	1	1	yes	characteristic
96	C₂²×D₅	—	2	96	no	—
97	C₂²×D₅	—	2	24	no	—
98	C₂²×D₅	—	2	24	no	—
99	C₂²×D₅	—	4	16	no	—
100	C₂²×D₅	—	4	48	no	—
101	C₂²×D₅	C₂³×C₄	1	16	yes	—
102	C₂×C₄×C₅	—	2	24	no	—
103	C₂×C₄×C₅	—	4	16	no	—
104	C₂×C₄×C₅	C₄×D₄	1	48	yes	complemented
105	C₂×C₄×C₅	C₂³×C₄	1	12	yes	—
106	C₂×C₄×C₅	C₂²×D₄	1	12	yes	—
107	C₂×C₄×C₅	C₂×#41	1	12	yes	—
108	C₂×Dic₅	—	2	24	no	—
109	C₂×Dic₅	—	2	12	no	—
110	C₂×Dic₅	#127	1	16	yes	complemented
111	C₂×Dic₅	C₂²×D₄	1	12	yes	—
112	C₂×Dic₅	C₂²×D₄	1	12	yes	—
113	D₂₀	—	2	192	no	—
114	C₄×D₅	—	4	64	no	—
115	C₂⁵	—	5	4	no	—
116	C₂²×D₄	—	5	48	no	—
117	C₂²×D₄	—	5	24	no	—
118	C₂²×D₄	—	10	24	no	—
119	C₂²×D₄	—	10	24	no	—
120	C₂³×C₄	—	5	1	no	—
121	C₂³×C₄	—	5	3	no	—
122	C₂³×C₄	—	5	16	no	—
123	C₂³×C₄	—	10	48	no	—
124	C₂³×C₄	C₂²×D₅	1	4	yes	—
125	C₂³×C₄	C₂²×D₅	1	3	yes	—
126	#127	—	5	64	no	—
127	C₄×D₄	—	10	192	no	—
128	C₂×C₄⋊C₄	—	2	24	no	—
129	C₂×C₄⋊C₄	C₂²×D₅	1	24	yes	—
130	C₂×C₂²⋊C₄	—	5	48	no	—
131	C₂×C₂²⋊C₄	—	10	48	no	—
132	C₂×C₄²	—	5	24	no	—
133	C₂×C₄²	—	5	12	no	—
134	C₂×C₄²	—	5	12	no	—
135	C₂×C₄²	—	10	12	no	—
136	C₂×C₄²	C₄×D₅	1	8	yes	complemented
137	C₂×C₄²	C₂²×D₅	1	4	yes	—
138	C₂²×C₅	C₂×C₄×D₄	1	12	yes	—
139	C₂²×C₅	C₂×#127	1	4	yes	—
140	C₂²×C₅	#532	1	12	yes	—
141	C₂²×C₅	C₂⁴×C₄	1	1	yes	characteristic
142	C₂²×C₅	C₂³×D₄	1	3	yes	—
143	C₂²×C₅	C₂²×#41	1	3	yes	—
144	C₂×D₅	—	2	48	no	—
145	C₂×D₅	—	4	64	no	—
146	C₂×D₅	—	4	8	no	—
147	C₂×D₅	—	4	24	no	—
148	C₄×C₅	—	4	8	no	—
149	C₄×C₅	C₂×C₄×D₄	1	24	yes	complemented
150	Dic₅	—	2	12	no	—
151	Dic₅	C₂×#127	1	8	yes	complemented
152	C₂⁴	—	5	16	no	—
153	C₂⁴	—	5	8	no	—
154	C₂⁴	—	10	24	no	—
155	C₂⁴	—	10	24	no	—
156	C₂⁴	C₂³×D₅	1	1	yes	characteristic
157	C₂×D₄	—	5	96	no	—
158	C₂×D₄	—	10	48	no	—
159	C₂×D₄	—	10	48	no	—
160	C₂×D₄	—	10	192	no	—
161	C₂²×C₄	—	2	24	no	—
162	C₂²×C₄	—	5	12	no	—
163	C₂²×C₄	—	5	2	no	—
164	C₂²×C₄	—	5	12	no	—
165	C₂²×C₄	—	5	6	no	—
166	C₂²×C₄	—	10	96	no	—
167	C₂²×C₄	—	10	12	no	—
168	C₂²×C₄	—	20	64	no	—
169	C₂²×C₄	C₂×C₄×D₅	1	12	yes	—
170	C₂²×C₄	D₄×D₅	1	12	yes	—
171	C₂²×C₄	#713	1	12	yes	—
172	C₂²×C₄	C₂³×D₅	1	8	yes	—
173	C₂²×C₄	C₂³×D₅	1	6	yes	—
174	C₄⋊C₄	—	2	48	no	—
175	C₂²⋊C₄	—	10	96	no	—
176	C₄²	—	5	48	no	—
177	C₄²	—	10	24	no	—
178	C₄²	C₂×C₄×D₅	1	16	yes	complemented
179	C₂×C₅	C₄×#127	1	8	yes	complemented
180	C₂×C₅	C₂²×C₄×D₄	1	3	yes	—
181	C₂×C₅	C₂²×#127	1	1	yes	characteristic
182	C₂×C₅	C₂×#532	1	3	yes	—
183	D₅	—	4	16	no	—
184	C₂³	—	5	16	no	—
185	C₂³	—	10	96	no	—
186	C₂³	—	10	24	no	—
187	C₂³	—	10	24	no	—
188	C₂³	—	20	16	no	—
189	C₂³	—	20	48	no	—
190	C₂³	C₂²×C₄×D₅	1	2	yes	—
191	C₂³	C₂×D₄×D₅	1	6	yes	—
192	C₂³	C₂×#713	1	6	yes	—
193	C₂³	C₂⁴×D₅	1	1	yes	characteristic
194	D₄	—	10	192	no	—
195	C₂×C₄	—	2	24	no	—
196	C₂×C₄	—	4	16	no	—
197	C₂×C₄	—	5	16	no	—
198	C₂×C₄	—	5	12	no	—
199	C₂×C₄	—	5	12	no	—
200	C₂×C₄	—	10	24	no	—
201	C₂×C₄	—	10	12	no	—
202	C₂×C₄	—	20	64	no	—
203	C₂×C₄	#4039	1	48	yes	complemented
204	C₂×C₄	C₂²×C₄×D₅	1	12	yes	—
205	C₂×C₄	C₂×D₄×D₅	1	12	yes	—
206	C₂×C₄	C₂×#713	1	12	yes	—
207	C₅	C₂×C₄×#127	1	1	yes	characteristic, complemented, Sylow
208	C₂²	—	10	48	no	—
209	C₂²	—	20	64	no	—
210	C₂²	—	20	8	no	—
211	C₂²	—	20	24	no	—
212	C₂²	C₂×#4039	1	12	yes	—
213	C₂²	D₅×#127	1	4	yes	—
214	C₂²	#66059	1	12	yes	—
215	C₂²	C₂³×C₄×D₅	1	1	yes	characteristic
216	C₂²	C₂²×D₄×D₅	1	3	yes	—
217	C₂²	C₂²×#713	1	3	yes	—
218	C₄	—	4	8	no	—
219	C₄	—	5	8	no	—
220	C₄	—	10	12	no	—
221	C₄	C₂×#4039	1	24	yes	complemented
222	C₂	—	20	16	no	—
223	C₂	#10606050	1	8	yes	direct product factor
224	C₂	C₂²×#4039	1	3	yes	—
225	C₂	C₂×D₅×#127	1	1	yes	characteristic
226	C₂	C₂×#66059	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₅
commutator	C₂²×C₅	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₅
Fitting	C₂×C₅×#384	C₂
socle	C₂⁴×C₅	C₂⁴
prime radical	C₂²×#4020	C₂
cosocle	C₂³×C₅	C₂⁵
hypercenter	C₂×#384	D₅
hypocenter	C₅	C₂×C₄×#127
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂⁴ ◁ C₂×#384

Complements

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

N	H ≅ G/N	type
1. C₂²×#4039	C₂	semidirect
3. C₂×C₅×#384	C₂	semidirect
4. C₂×#65277	C₂	semidirect
5. C₂×C₄²×Dic₅	C₂	semidirect
6. #10606050	C₂	direct (N × H)
15. C₂×#4039	C₂²	semidirect
16. C₂×#4020	C₄	semidirect
18. C₅×#384	C₂²	semidirect
19. #65277	C₂²	semidirect
20. C₄²×Dic₅	C₂²	semidirect
29. C₂²×D₂₀	C₂×C₄	semidirect
35. C₂×C₄²×C₅	C₂×C₄	semidirect
40. C₂×C₄×Dic₅	D₄	semidirect
44. #4020	C₂×C₄	semidirect
49. C₂×#384	D₅	semidirect
70. C₂×D₂₀	C₂²×C₄	semidirect
74. C₄²×C₅	C₂²×C₄	semidirect
77. C₄×Dic₅	C₂×D₄	semidirect
90. #384	C₂×D₅	semidirect
104. C₂×C₄×C₅	C₄×D₄	semidirect
110. C₂×Dic₅	#127	semidirect
136. C₂×C₄²	C₄×D₅	semidirect
149. C₄×C₅	C₂×C₄×D₄	semidirect
151. Dic₅	C₂×#127	semidirect
178. C₄²	C₂×C₄×D₅	semidirect
179. C₂×C₅	C₄×#127	semidirect
203. C₂×C₄	#4039	semidirect
207. C₅	C₂×C₄×#127	semidirect
221. C₄	C₂×#4039	semidirect
223. C₂	#10606050	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#4039	2
C₂²×#4020	2
C₂×C₅×#384	2
C₂×#65277	2
C₂×C₄²×Dic₅	2
#10606050	2
C₂×C₄×#127	5

Subgroups — C2×#10606050 #12257811

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10606050 #12257811