Subgroups — C₂²×C₅×#508 #11985348

Order 1280.

2118 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂²×C₄×C₅	D₈
commutator, Frattini, cosocle	C₈	C₂⁵×C₅
socle	C₂³×C₅	C₂×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×#135	C₂	semidirect
2. C₂×C₅×#508	C₂	direct (N × H)
3. C₂²×C₅×Q₃₂	C₂	semidirect
4. C₂²×C₅×SD₃₂	C₂	semidirect
5. C₂²×C₅×D₁₆	C₂	semidirect
6. C₂³×C₅×C₁₆	C₂	semidirect
9. C₂×C₅×#135	C₂²	semidirect
14. C₅×#508	C₂²	direct (N × H)
15. C₂×C₅×Q₃₂	C₂²	semidirect
16. C₂×C₅×SD₃₂	C₂²	semidirect
17. C₂×C₅×D₁₆	C₂²	semidirect
20. C₂²×C₅×C₁₆	C₂²	semidirect
21. C₂²×#508	C₅	direct (N × H)
30. C₅×#135	C₂³	semidirect
40. C₅×Q₃₂	C₂³	semidirect
41. C₅×SD₃₂	C₂³	semidirect
42. C₅×D₁₆	C₂³	semidirect
45. C₂×C₅×C₁₆	C₂³	semidirect
46. C₂²×#135	C₂×C₅	semidirect
47. C₂×#508	C₂×C₅	direct (N × H)
48. C₂²×Q₃₂	C₂×C₅	semidirect
49. C₂²×SD₃₂	C₂×C₅	semidirect
50. C₂²×D₁₆	C₂×C₅	semidirect
51. C₂³×C₁₆	C₂×C₅	semidirect
81. C₂×#135	C₂²×C₅	semidirect
86. #508	C₂²×C₅	direct (N × H)
87. C₂×Q₃₂	C₂²×C₅	semidirect
88. C₂×SD₃₂	C₂²×C₅	semidirect
89. C₂×D₁₆	C₂²×C₅	semidirect
92. C₂²×C₁₆	C₂²×C₅	semidirect
121. #135	C₂³×C₅	semidirect
131. Q₃₂	C₂³×C₅	semidirect
132. SD₃₂	C₂³×C₅	semidirect
133. D₁₆	C₂³×C₅	semidirect
136. C₂×C₁₆	C₂³×C₅	semidirect
140. C₂²×C₅	#508	direct (N × H)
176. C₂×C₅	C₂×#508	direct (N × H)
198. C₅	C₂²×#508	direct (N × H)
202. C₂²	C₅×#508	direct (N × H)
211. C₂	C₂×C₅×#508	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×#135	2
C₂×C₅×#508	2
C₂²×C₅×Q₃₂	2
C₂²×C₅×SD₃₂	2
C₂²×C₅×D₁₆	2
C₂³×C₅×C₁₆	2
C₂²×#508	5

Subgroups — C22×C5×#508 #11985348

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#508 #11985348