Subgroups — C₂×C₅×#2818 #11959405

Order 1280.

3666 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#127	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₅×#126	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2818	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅×#394	C₂	1	6	yes	maximal, complemented
5	C₂×C₅×#392	C₂	1	6	yes	maximal, complemented
6	C₂×C₅×#383	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₅×D₄	C₂²	1	6	yes	—
8	C₂³×C₅×D₄	C₂²	1	4	yes	—
9	C₂×C₅×#127	—	2	8	no	—
10	C₂×C₅×#127	C₂²	1	8	yes	complemented
11	C₂×C₅×#126	—	2	8	no	—
12	C₂×C₅×#126	C₂²	1	8	yes	complemented
13	C₂²×C₅×C₄⋊C₄	C₂²	1	3	yes	—
14	C₂²×C₅×C₂²⋊C₄	C₂²	1	3	yes	—
15	C₂²×C₅×C₂²⋊C₄	C₂²	1	12	yes	—
16	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
17	C₅×#394	C₂²	1	48	yes	complemented
18	C₅×#392	C₂²	1	48	yes	complemented
19	C₅×#383	C₂²	1	8	yes	complemented
20	C₂×C₅×#95	C₂²	1	6	yes	—
21	C₂×#2818	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂⁵×C₅	C₂³	1	1	yes	characteristic
23	C₂⁵×C₅	C₂³	1	4	yes	—
24	C₂²×C₅×D₄	—	2	24	no	—
25	C₂²×C₅×D₄	—	4	24	no	—
26	C₂²×C₅×D₄	—	4	24	no	—
27	C₂²×C₅×D₄	C₂³	1	24	yes	—
28	C₂²×C₅×D₄	C₂³	1	16	yes	—
29	C₂³×C₄×C₅	C₂³	1	4	yes	—
30	C₂³×C₄×C₅	C₂³	1	3	yes	—
31	C₂³×C₄×C₅	C₂³	1	3	yes	—
32	C₅×#127	—	2	32	no	—
33	C₅×#126	—	2	32	no	—
34	C₂×C₅×C₄⋊C₄	—	2	12	no	—
35	C₂×C₅×C₄⋊C₄	C₂³	1	12	yes	—
36	C₂×C₅×C₂²⋊C₄	—	2	12	no	—
37	C₂×C₅×C₂²⋊C₄	—	4	24	no	—
38	C₂×C₅×C₂²⋊C₄	C₂³	1	12	yes	—
39	C₂×C₅×C₂²⋊C₄	C₂³	1	48	yes	—
40	C₂×C₄²×C₅	D₄	1	8	yes	complemented
41	C₂×C₄²×C₅	C₂³	1	4	yes	—
42	C₅×#95	C₂³	1	24	yes	—
43	C₂²×#127	C₂×C₅	1	1	yes	characteristic, complemented
44	C₂²×#126	C₂×C₅	1	1	yes	characteristic, complemented
45	#2818	C₂×C₅	1	16	yes	direct product factor
46	C₂×#394	C₂×C₅	1	6	yes	complemented
47	C₂×#392	C₂×C₅	1	6	yes	complemented
48	C₂×#383	C₂×C₅	1	1	yes	characteristic, complemented
49	C₂⁴×C₅	—	2	2	no	—
50	C₂⁴×C₅	—	2	8	no	—
51	C₂⁴×C₅	—	2	24	no	—
52	C₂⁴×C₅	—	2	24	no	—
53	C₂⁴×C₅	—	2	12	no	—
54	C₂⁴×C₅	C₂⁴	1	2	yes	—
55	C₂⁴×C₅	C₂⁴	1	8	yes	—
56	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
57	C₂×C₅×D₄	—	2	48	no	—
58	C₂×C₅×D₄	—	4	48	no	—
59	C₂×C₅×D₄	—	4	48	no	—
60	C₂×C₅×D₄	—	4	96	no	—
61	C₂×C₅×D₄	—	4	64	no	—
62	C₂²×C₄×C₅	—	2	12	no	—
63	C₂²×C₄×C₅	—	2	6	no	—
64	C₂²×C₄×C₅	—	2	12	no	—
65	C₂²×C₄×C₅	—	2	24	no	—
66	C₂²×C₄×C₅	C₂×D₄	1	12	yes	—
67	C₂²×C₄×C₅	C₂⁴	1	6	yes	—
68	C₂²×C₄×C₅	C₂⁴	1	8	yes	—
69	C₂²×C₄×C₅	C₂⁴	1	6	yes	—
70	C₅×C₄⋊C₄	—	2	24	no	—
71	C₅×C₂²⋊C₄	—	2	24	no	—
72	C₅×C₂²⋊C₄	—	4	48	no	—
73	C₄²×C₅	C₂×D₄	1	16	yes	complemented
74	C₂³×D₄	C₂²×C₅	1	6	yes	—
75	C₂³×D₄	C₂²×C₅	1	4	yes	—
76	C₂×#127	—	2	8	no	—
77	C₂×#127	C₂²×C₅	1	8	yes	complemented
78	C₂×#126	—	2	8	no	—
79	C₂×#126	C₂²×C₅	1	8	yes	complemented
80	C₂²×C₄⋊C₄	C₂²×C₅	1	3	yes	—
81	C₂²×C₂²⋊C₄	C₂²×C₅	1	3	yes	—
82	C₂²×C₂²⋊C₄	C₂²×C₅	1	12	yes	—
83	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
84	#394	C₂²×C₅	1	48	yes	complemented
85	#392	C₂²×C₅	1	48	yes	complemented
86	#383	C₂²×C₅	1	8	yes	complemented
87	C₂×#95	C₂²×C₅	1	6	yes	—
88	C₂³×C₅	—	2	2	no	—
89	C₂³×C₅	—	2	8	no	—
90	C₂³×C₅	—	2	24	no	—
91	C₂³×C₅	—	2	24	no	—
92	C₂³×C₅	—	2	12	no	—
93	C₂³×C₅	—	4	48	no	—
94	C₂³×C₅	—	4	12	no	—
95	C₂³×C₅	—	4	16	no	—
96	C₂³×C₅	—	4	16	no	—
97	C₂³×C₅	—	4	48	no	—
98	C₂³×C₅	C₂²×D₄	1	2	yes	—
99	C₂³×C₅	H(5,𝔽₂)	1	12	yes	—
100	C₂³×C₅	C₂⁵	1	1	yes	characteristic
101	C₅×D₄	—	4	96	no	—
102	C₅×D₄	—	4	64	no	—
103	C₂×C₄×C₅	—	2	24	no	—
104	C₂×C₄×C₅	—	2	12	no	—
105	C₂×C₄×C₅	—	2	6	no	—
106	C₂×C₄×C₅	—	2	12	no	—
107	C₂×C₄×C₅	—	2	24	no	—
108	C₂×C₄×C₅	—	4	12	no	—
109	C₂×C₄×C₅	—	4	16	no	—
110	C₂×C₄×C₅	C₂²×D₄	1	12	yes	—
111	C₂⁵	C₂³×C₅	1	1	yes	characteristic
112	C₂⁵	C₂³×C₅	1	4	yes	—
113	C₂²×D₄	—	2	24	no	—
114	C₂²×D₄	—	4	24	no	—
115	C₂²×D₄	—	4	24	no	—
116	C₂²×D₄	C₂³×C₅	1	24	yes	—
117	C₂²×D₄	C₂³×C₅	1	16	yes	—
118	C₂³×C₄	C₂³×C₅	1	3	yes	—
119	C₂³×C₄	C₂³×C₅	1	4	yes	—
120	C₂³×C₄	C₂³×C₅	1	3	yes	—
121	#127	—	2	32	no	—
122	#126	—	2	32	no	—
123	C₂×C₄⋊C₄	—	2	12	no	—
124	C₂×C₄⋊C₄	C₂³×C₅	1	12	yes	—
125	C₂×C₂²⋊C₄	—	2	12	no	—
126	C₂×C₂²⋊C₄	—	4	24	no	—
127	C₂×C₂²⋊C₄	C₂³×C₅	1	12	yes	—
128	C₂×C₂²⋊C₄	C₂³×C₅	1	48	yes	—
129	C₂×C₄²	C₅×D₄	1	8	yes	complemented
130	C₂×C₄²	C₂³×C₅	1	4	yes	—
131	#95	C₂³×C₅	1	24	yes	—
132	C₂²×C₅	—	4	24	no	—
133	C₂²×C₅	—	4	6	no	—
134	C₂²×C₅	—	4	8	no	—
135	C₂²×C₅	—	4	8	no	—
136	C₂²×C₅	—	4	24	no	—
137	C₂²×C₅	—	8	32	no	—
138	C₂²×C₅	—	8	8	no	—
139	C₂²×C₅	#535	1	12	yes	—
140	C₂²×C₅	#560	1	16	yes	—
141	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
142	C₂²×C₅	C₂×H(5,𝔽₂)	1	6	yes	—
143	C₄×C₅	—	2	12	no	—
144	C₄×C₅	—	4	6	no	—
145	C₄×C₅	—	4	8	no	—
146	C₂⁴	—	2	2	no	—
147	C₂⁴	—	2	8	no	—
148	C₂⁴	—	2	24	no	—
149	C₂⁴	—	2	24	no	—
150	C₂⁴	—	2	12	no	—
151	C₂⁴	C₂⁴×C₅	1	2	yes	—
152	C₂⁴	C₂⁴×C₅	1	8	yes	—
153	C₂⁴	C₂⁴×C₅	1	1	yes	characteristic
154	C₂×D₄	—	2	48	no	—
155	C₂×D₄	—	4	48	no	—
156	C₂×D₄	—	4	48	no	—
157	C₂×D₄	—	4	96	no	—
158	C₂×D₄	—	4	64	no	—
159	C₂²×C₄	—	2	12	no	—
160	C₂²×C₄	—	2	6	no	—
161	C₂²×C₄	—	2	12	no	—
162	C₂²×C₄	—	2	24	no	—
163	C₂²×C₄	C₂×C₅×D₄	1	12	yes	—
164	C₂²×C₄	C₂⁴×C₅	1	8	yes	—
165	C₂²×C₄	C₂⁴×C₅	1	6	yes	—
166	C₂²×C₄	C₂⁴×C₅	1	6	yes	—
167	C₄⋊C₄	—	2	24	no	—
168	C₂²⋊C₄	—	2	24	no	—
169	C₂²⋊C₄	—	4	48	no	—
170	C₄²	C₂×C₅×D₄	1	16	yes	complemented
171	C₂×C₅	—	8	8	no	—
172	C₂×C₅	—	8	2	no	—
173	C₂×C₅	#2818	1	8	yes	direct product factor
174	C₂×C₅	C₂×#535	1	3	yes	—
175	C₂×C₅	C₂×#560	1	4	yes	—
176	C₂³	—	2	2	no	—
177	C₂³	—	2	8	no	—
178	C₂³	—	2	24	no	—
179	C₂³	—	2	24	no	—
180	C₂³	—	2	12	no	—
181	C₂³	—	4	48	no	—
182	C₂³	—	4	12	no	—
183	C₂³	—	4	16	no	—
184	C₂³	—	4	16	no	—
185	C₂³	—	4	48	no	—
186	C₂³	C₂²×C₅×D₄	1	2	yes	—
187	C₂³	C₅×H(5,𝔽₂)	1	12	yes	—
188	C₂³	C₂⁵×C₅	1	1	yes	characteristic
189	D₄	—	4	96	no	—
190	D₄	—	4	64	no	—
191	C₂×C₄	—	2	24	no	—
192	C₂×C₄	—	2	12	no	—
193	C₂×C₄	—	2	6	no	—
194	C₂×C₄	—	2	12	no	—
195	C₂×C₄	—	2	24	no	—
196	C₂×C₄	—	4	12	no	—
197	C₂×C₄	—	4	16	no	—
198	C₂×C₄	C₂²×C₅×D₄	1	12	yes	—
199	C₅	C₂×#2818	1	1	yes	characteristic, direct product factor, Sylow
200	C₂²	—	4	24	no	—
201	C₂²	—	4	6	no	—
202	C₂²	—	4	8	no	—
203	C₂²	—	4	8	no	—
204	C₂²	—	4	24	no	—
205	C₂²	—	8	32	no	—
206	C₂²	—	8	8	no	—
207	C₂²	C₅×#535	1	12	yes	—
208	C₂²	C₅×#560	1	16	yes	—
209	C₂²	C₂³×C₅×D₄	1	1	yes	characteristic
210	C₂²	C₂×C₅×H(5,𝔽₂)	1	6	yes	—
211	C₄	—	2	12	no	—
212	C₄	—	4	6	no	—
213	C₄	—	4	8	no	—
214	C₂	—	8	8	no	—
215	C₂	—	8	2	no	—
216	C₂	C₅×#2818	1	8	yes	direct product factor
217	C₂	C₂×C₅×#535	1	3	yes	—
218	C₂	C₂×C₅×#560	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×#127	C₂	semidirect
2. C₂²×C₅×#126	C₂	semidirect
3. C₅×#2818	C₂	direct (N × H)
4. C₂×C₅×#394	C₂	semidirect
5. C₂×C₅×#392	C₂	semidirect
6. C₂×C₅×#383	C₂	semidirect
10. C₂×C₅×#127	C₂²	semidirect
12. C₂×C₅×#126	C₂²	semidirect
17. C₅×#394	C₂²	semidirect
18. C₅×#392	C₂²	semidirect
19. C₅×#383	C₂²	semidirect
21. C₂×#2818	C₅	direct (N × H)
40. C₂×C₄²×C₅	D₄	semidirect
43. C₂²×#127	C₂×C₅	semidirect
44. C₂²×#126	C₂×C₅	semidirect
45. #2818	C₂×C₅	direct (N × H)
46. C₂×#394	C₂×C₅	semidirect
47. C₂×#392	C₂×C₅	semidirect
48. C₂×#383	C₂×C₅	semidirect
73. C₄²×C₅	C₂×D₄	semidirect
77. C₂×#127	C₂²×C₅	semidirect
79. C₂×#126	C₂²×C₅	semidirect
84. #394	C₂²×C₅	semidirect
85. #392	C₂²×C₅	semidirect
86. #383	C₂²×C₅	semidirect
129. C₂×C₄²	C₅×D₄	semidirect
170. C₄²	C₂×C₅×D₄	semidirect
173. C₂×C₅	#2818	direct (N × H)
199. C₅	C₂×#2818	direct (N × H)
216. C₂	C₅×#2818	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×C₅×#127	2
C₂²×C₅×#126	2
C₅×#2818	2
C₂×C₅×#394	2
C₂×C₅×#392	2
C₂×C₅×#383	2
C₂×#2818	5

Subgroups — C2×C5×#2818 #11959405

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2818 #11959405