Subgroups — C₂×C₅×#3415 #11985352

Order 1280.

2126 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#576	C₂	1	2	yes	maximal, complemented
2	C₅×#3415	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₅×#509	C₂	1	6	yes	maximal, complemented
4	C₂×C₅×#508	C₂	1	3	yes	maximal, complemented
5	C₂²×C₅×D₁₆	C₂	1	3	yes	maximal, complemented
6	C₂×C₅×#504	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅×H(5,𝔽₂)	—	2	2	no	—
8	C₅×#576	—	2	8	no	—
9	C₅×#576	C₂²	1	16	yes	complemented
10	C₂×C₅×AGL(1,ℤ/8)	—	2	6	no	—
11	C₂×C₅×#135	C₂²	1	6	yes	complemented
12	C₂²×C₅×D₈	C₂²	1	6	yes	complemented
13	C₂×C₅×#131	C₂²	1	1	yes	characteristic
14	C₅×#509	C₂²	1	48	yes	complemented
15	C₅×#508	C₂²	1	24	yes	complemented
16	C₂×C₅×Q₃₂	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₅×SD₃₂	C₂²	1	6	yes	complemented
18	C₂×C₅×D₁₆	C₂²	1	3	yes	complemented
19	C₂×C₅×D₁₆	C₂²	1	6	yes	complemented
20	C₂×C₅×D₁₆	C₂²	1	24	yes	complemented
21	C₅×#504	C₂²	1	8	yes	complemented
22	C₂×C₅×M₃₂	C₂²	1	3	yes	complemented
23	C₂²×C₅×C₁₆	C₂²	1	3	yes	complemented
24	C₂×#3415	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₅×H(5,𝔽₂)	—	2	16	no	—
26	C₅×H(5,𝔽₂)	—	4	8	no	—
27	C₂×C₅×#41	—	2	6	no	—
28	C₂×C₅×#41	—	4	2	no	—
29	C₂×C₅×#41	D₄	1	1	yes	characteristic
30	C₂²×C₅×D₄	—	2	6	no	—
31	C₂²×C₅×D₄	—	4	6	no	—
32	C₅×AGL(1,ℤ/8)	—	2	48	no	—
33	C₅×#135	—	2	12	no	—
34	C₅×#135	C₂³	1	24	yes	complemented
35	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
36	C₂×C₅×SD₁₆	—	2	6	no	—
37	C₂×C₅×D₈	—	2	6	no	—
38	C₂×C₅×D₈	—	2	12	no	—
39	C₂×C₅×D₈	C₂³	1	6	yes	—
40	C₂×C₅×D₈	C₂³	1	24	yes	complemented
41	C₅×#131	D₄	1	4	yes	—
42	C₅×#131	C₂³	1	4	yes	—
43	C₂×C₅×M₁₆	D₄	1	3	yes	—
44	C₂²×C₅×C₈	C₂³	1	3	yes	—
45	C₅×Q₃₂	C₂³	1	4	yes	complemented
46	C₅×SD₃₂	C₂³	1	24	yes	complemented
47	C₅×D₁₆	C₂³	1	12	yes	complemented
48	C₅×D₁₆	C₂³	1	24	yes	complemented
49	C₅×M₃₂	C₂³	1	12	yes	complemented
50	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
51	C₂×C₅×C₁₆	C₂³	1	3	yes	—
52	C₂×C₅×C₁₆	C₂³	1	12	yes	complemented
53	C₂×#576	C₂×C₅	1	2	yes	complemented
54	#3415	C₂×C₅	1	16	yes	direct product factor
55	C₂×#509	C₂×C₅	1	6	yes	complemented
56	C₂×#508	C₂×C₅	1	3	yes	complemented
57	C₂²×D₁₆	C₂×C₅	1	3	yes	complemented
58	C₂×#504	C₂×C₅	1	1	yes	characteristic, complemented
59	C₂⁴×C₅	—	4	6	no	—
60	C₅×#41	—	2	24	no	—
61	C₅×#41	—	4	16	no	—
62	C₅×#41	—	4	12	no	—
63	C₅×#41	D₈	1	4	yes	—
64	C₅×#41	C₂×D₄	1	4	yes	—
65	C₂×C₅×Q₈	—	2	2	no	—
66	C₂×C₅×Q₈	D₈	1	1	yes	characteristic
67	C₂×C₅×D₄	—	2	6	no	—
68	C₂×C₅×D₄	—	2	24	no	—
69	C₂×C₅×D₄	—	4	6	no	—
70	C₂×C₅×D₄	—	4	48	no	—
71	C₂×C₅×D₄	—	4	6	no	—
72	C₂×C₅×D₄	—	4	12	no	—
73	C₂×C₅×D₄	D₈	1	3	yes	—
74	C₂²×C₄×C₅	—	4	6	no	—
75	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
76	C₅×Q₁₆	—	2	2	no	—
77	C₅×Q₁₆	C₂⁴	1	4	yes	—
78	C₅×SD₁₆	—	2	24	no	—
79	C₅×D₈	—	2	6	no	—
80	C₅×D₈	—	2	24	no	—
81	C₅×D₈	C₂⁴	1	12	yes	—
82	C₅×M₁₆	C₂×D₄	1	12	yes	—
83	C₂×C₅×C₈	C₂×D₄	1	3	yes	—
84	C₂×C₅×C₈	C₂×D₄	1	6	yes	—
85	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
86	C₂×C₅×C₈	C₂⁴	1	6	yes	—
87	C₅×C₁₆	C₂⁴	1	2	yes	—
88	C₅×C₁₆	C₂⁴	1	6	yes	—
89	C₂×H(5,𝔽₂)	—	2	2	no	—
90	#576	—	2	8	no	—
91	#576	C₂²×C₅	1	16	yes	complemented
92	C₂×AGL(1,ℤ/8)	—	2	6	no	—
93	C₂×#135	C₂²×C₅	1	6	yes	complemented
94	C₂²×D₈	C₂²×C₅	1	6	yes	complemented
95	C₂×#131	C₂²×C₅	1	1	yes	characteristic
96	#509	C₂²×C₅	1	48	yes	complemented
97	#508	C₂²×C₅	1	24	yes	complemented
98	C₂×Q₃₂	C₂²×C₅	1	1	yes	characteristic, complemented
99	C₂×SD₃₂	C₂²×C₅	1	6	yes	complemented
100	C₂×D₁₆	C₂²×C₅	1	3	yes	complemented
101	C₂×D₁₆	C₂²×C₅	1	6	yes	complemented
102	C₂×D₁₆	C₂²×C₅	1	24	yes	complemented
103	#504	C₂²×C₅	1	8	yes	complemented
104	C₂×M₃₂	C₂²×C₅	1	3	yes	complemented
105	C₂²×C₁₆	C₂²×C₅	1	3	yes	complemented
106	C₂³×C₅	—	4	6	no	—
107	C₂³×C₅	—	4	24	no	—
108	C₂³×C₅	—	16	12	no	—
109	C₂³×C₅	C₂×D₈	1	3	yes	—
110	C₅×Q₈	—	2	4	no	—
111	C₅×Q₈	—	4	2	no	—
112	C₅×Q₈	C₂×D₈	1	4	yes	—
113	C₅×D₄	—	2	12	no	—
114	C₅×D₄	—	4	24	no	—
115	C₅×D₄	—	4	24	no	—
116	C₅×D₄	—	4	6	no	—
117	C₅×D₄	C₂×D₈	1	12	yes	—
118	C₂×C₄×C₅	—	4	2	no	—
119	C₂×C₄×C₅	—	4	24	no	—
120	C₂×C₄×C₅	C₂×D₈	1	6	yes	—
121	C₂×C₄×C₅	C₂×D₈	1	3	yes	—
122	C₂×C₄×C₅	C₂²×D₄	1	6	yes	—
123	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
124	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
125	C₅×C₈	C₂²×D₄	1	6	yes	—
126	C₅×C₈	C₂⁵	1	1	yes	characteristic
127	H(5,𝔽₂)	—	2	16	no	—
128	H(5,𝔽₂)	—	4	8	no	—
129	C₂×#41	—	2	6	no	—
130	C₂×#41	—	4	2	no	—
131	C₂×#41	C₅×D₄	1	1	yes	characteristic
132	C₂²×D₄	—	2	6	no	—
133	C₂²×D₄	—	4	6	no	—
134	AGL(1,ℤ/8)	—	2	48	no	—
135	#135	—	2	12	no	—
136	#135	C₂³×C₅	1	24	yes	complemented
137	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
138	C₂×SD₁₆	—	2	6	no	—
139	C₂×D₈	—	2	6	no	—
140	C₂×D₈	—	2	12	no	—
141	C₂×D₈	C₂³×C₅	1	6	yes	—
142	C₂×D₈	C₂³×C₅	1	24	yes	complemented
143	#131	C₅×D₄	1	4	yes	—
144	#131	C₂³×C₅	1	4	yes	—
145	C₂×M₁₆	C₅×D₄	1	3	yes	—
146	C₂²×C₈	C₂³×C₅	1	3	yes	—
147	Q₃₂	C₂³×C₅	1	4	yes	complemented
148	SD₃₂	C₂³×C₅	1	24	yes	complemented
149	D₁₆	C₂³×C₅	1	12	yes	complemented
150	D₁₆	C₂³×C₅	1	24	yes	complemented
151	M₃₂	C₂³×C₅	1	12	yes	complemented
152	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
153	C₂×C₁₆	C₂³×C₅	1	3	yes	—
154	C₂×C₁₆	C₂³×C₅	1	12	yes	complemented
155	C₂²×C₅	—	2	6	no	—
156	C₂²×C₅	—	4	12	no	—
157	C₂²×C₅	—	8	12	no	—
158	C₂²×C₅	—	16	24	no	—
159	C₂²×C₅	C₂²×D₈	1	6	yes	—
160	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
161	C₄×C₅	—	4	4	no	—
162	C₄×C₅	C₂²×D₈	1	6	yes	—
163	C₄×C₅	C₂²×D₈	1	1	yes	characteristic
164	C₄×C₅	C₂³×D₄	1	1	yes	characteristic
165	C₂⁴	—	4	6	no	—
166	#41	—	2	24	no	—
167	#41	—	4	16	no	—
168	#41	—	4	12	no	—
169	#41	C₅×D₈	1	4	yes	—
170	#41	C₂×C₅×D₄	1	4	yes	—
171	C₂×Q₈	—	2	2	no	—
172	C₂×Q₈	C₅×D₈	1	1	yes	characteristic
173	C₂×D₄	—	2	6	no	—
174	C₂×D₄	—	2	24	no	—
175	C₂×D₄	—	4	6	no	—
176	C₂×D₄	—	4	48	no	—
177	C₂×D₄	—	4	6	no	—
178	C₂×D₄	—	4	12	no	—
179	C₂×D₄	C₅×D₈	1	3	yes	—
180	C₂²×C₄	—	4	6	no	—
181	C₂²×C₄	C₂×C₅×D₄	1	3	yes	—
182	Q₁₆	—	2	2	no	—
183	Q₁₆	C₂⁴×C₅	1	4	yes	—
184	SD₁₆	—	2	24	no	—
185	D₈	—	2	6	no	—
186	D₈	—	2	24	no	—
187	D₈	C₂⁴×C₅	1	12	yes	—
188	M₁₆	C₂×C₅×D₄	1	12	yes	—
189	C₂×C₈	C₂×C₅×D₄	1	3	yes	—
190	C₂×C₈	C₂×C₅×D₄	1	6	yes	—
191	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
192	C₂×C₈	C₂⁴×C₅	1	6	yes	—
193	C₁₆	C₂⁴×C₅	1	2	yes	—
194	C₁₆	C₂⁴×C₅	1	6	yes	—
195	C₂×C₅	—	2	6	no	—
196	C₂×C₅	—	8	12	no	—
197	C₂×C₅	#3415	1	2	yes	direct product factor
198	C₂×C₅	C₂³×D₈	1	1	yes	characteristic
199	C₂³	—	4	6	no	—
200	C₂³	—	4	24	no	—
201	C₂³	—	16	12	no	—
202	C₂³	C₂×C₅×D₈	1	3	yes	—
203	Q₈	—	2	4	no	—
204	Q₈	—	4	2	no	—
205	Q₈	C₂×C₅×D₈	1	4	yes	—
206	D₄	—	2	12	no	—
207	D₄	—	4	24	no	—
208	D₄	—	4	24	no	—
209	D₄	—	4	6	no	—
210	D₄	C₂×C₅×D₈	1	12	yes	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	24	no	—
213	C₂×C₄	C₂×C₅×D₈	1	3	yes	—
214	C₂×C₄	C₂×C₅×D₈	1	6	yes	—
215	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
216	C₂×C₄	C₂²×C₅×D₄	1	6	yes	—
217	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
218	C₈	C₂²×C₅×D₄	1	6	yes	—
219	C₈	C₂⁵×C₅	1	1	yes	characteristic
220	C₅	C₂×#3415	1	1	yes	characteristic, direct product factor, Sylow
221	C₂²	—	2	6	no	—
222	C₂²	—	4	12	no	—
223	C₂²	—	8	12	no	—
224	C₂²	—	16	24	no	—
225	C₂²	C₂²×C₅×D₈	1	6	yes	—
226	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
227	C₄	—	4	4	no	—
228	C₄	C₂²×C₅×D₈	1	6	yes	—
229	C₄	C₂²×C₅×D₈	1	1	yes	characteristic
230	C₄	C₂³×C₅×D₄	1	1	yes	characteristic
231	C₂	—	2	6	no	—
232	C₂	—	8	12	no	—
233	C₂	C₅×#3415	1	2	yes	direct product factor
234	C₂	C₂³×C₅×D₈	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator, Frattini, cosocle	C₈	C₂⁵×C₅
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₅×#41 ◁ C₂×C₅×#131 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#576	C₂	semidirect
2. C₅×#3415	C₂	direct (N × H)
3. C₂×C₅×#509	C₂	semidirect
4. C₂×C₅×#508	C₂	semidirect
5. C₂²×C₅×D₁₆	C₂	semidirect
6. C₂×C₅×#504	C₂	semidirect
9. C₅×#576	C₂²	semidirect
11. C₂×C₅×#135	C₂²	semidirect
12. C₂²×C₅×D₈	C₂²	semidirect
14. C₅×#509	C₂²	semidirect
15. C₅×#508	C₂²	semidirect
16. C₂×C₅×Q₃₂	C₂²	semidirect
17. C₂×C₅×SD₃₂	C₂²	semidirect
18. C₂×C₅×D₁₆	C₂²	semidirect
19. C₂×C₅×D₁₆	C₂²	semidirect
20. C₂×C₅×D₁₆	C₂²	semidirect
21. C₅×#504	C₂²	semidirect
22. C₂×C₅×M₃₂	C₂²	semidirect
23. C₂²×C₅×C₁₆	C₂²	semidirect
24. C₂×#3415	C₅	direct (N × H)
34. C₅×#135	C₂³	semidirect
40. C₂×C₅×D₈	C₂³	semidirect
45. C₅×Q₃₂	C₂³	semidirect
46. C₅×SD₃₂	C₂³	semidirect
47. C₅×D₁₆	C₂³	semidirect
48. C₅×D₁₆	C₂³	semidirect
49. C₅×M₃₂	C₂³	semidirect
52. C₂×C₅×C₁₆	C₂³	semidirect
53. C₂×#576	C₂×C₅	semidirect
54. #3415	C₂×C₅	direct (N × H)
55. C₂×#509	C₂×C₅	semidirect
56. C₂×#508	C₂×C₅	semidirect
57. C₂²×D₁₆	C₂×C₅	semidirect
58. C₂×#504	C₂×C₅	semidirect
91. #576	C₂²×C₅	semidirect
93. C₂×#135	C₂²×C₅	semidirect
94. C₂²×D₈	C₂²×C₅	semidirect
96. #509	C₂²×C₅	semidirect
97. #508	C₂²×C₅	semidirect
98. C₂×Q₃₂	C₂²×C₅	semidirect
99. C₂×SD₃₂	C₂²×C₅	semidirect
100. C₂×D₁₆	C₂²×C₅	semidirect
101. C₂×D₁₆	C₂²×C₅	semidirect
102. C₂×D₁₆	C₂²×C₅	semidirect
103. #504	C₂²×C₅	semidirect
104. C₂×M₃₂	C₂²×C₅	semidirect
105. C₂²×C₁₆	C₂²×C₅	semidirect
136. #135	C₂³×C₅	semidirect
142. C₂×D₈	C₂³×C₅	semidirect
147. Q₃₂	C₂³×C₅	semidirect
148. SD₃₂	C₂³×C₅	semidirect
149. D₁₆	C₂³×C₅	semidirect
150. D₁₆	C₂³×C₅	semidirect
151. M₃₂	C₂³×C₅	semidirect
154. C₂×C₁₆	C₂³×C₅	semidirect
197. C₂×C₅	#3415	direct (N × H)
220. C₅	C₂×#3415	direct (N × H)
233. C₂	C₅×#3415	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#576	2
C₅×#3415	2
C₂×C₅×#509	2
C₂×C₅×#508	2
C₂²×C₅×D₁₆	2
C₂×C₅×#504	2
C₂×#3415	5

Subgroups — C2×C5×#3415 #11985352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#3415 #11985352