Subgroups — C₅×#12832 #11935552

Order 1280.

358 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#504	C₂	1	1	yes	maximal, characteristic
2	C₅×#2155	C₂	1	2	yes	maximal, complemented
3	C₅×#1813	C₂	1	1	yes	maximal, characteristic
4	C₅×#1812	C₂	1	1	yes	maximal, characteristic
5	C₅×#1382	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#131	C₂²	1	1	yes	characteristic
7	C₅×#504	—	2	4	no	—
8	C₂×C₅×M₃₂	C₄	1	2	yes	complemented
9	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
10	C₂²×C₅×C₁₆	C₄	1	2	yes	complemented
11	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₅×#108	C₂²	1	2	yes	—
13	C₅×#427	C₂²	1	2	yes	—
14	C₅×#413	—	2	1	no	—
15	C₅×#410	—	2	1	no	—
16	C₅×#365	—	2	4	no	—
17	C₅×#338	—	2	1	no	—
18	C₅×H(3,ℤ/4)	—	2	1	no	—
19	#12832	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂×C₅×#41	D₄	1	1	yes	characteristic
21	C₅×#131	—	2	2	no	—
22	C₅×#131	—	2	2	no	—
23	C₂×C₅×M₁₆	—	2	2	no	—
24	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
25	C₂×C₅×M₁₆	Q₈	1	2	yes	—
26	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
27	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
28	C₅×#117	—	2	2	no	—
29	C₅×M₃₂	—	2	4	no	—
30	C₅×M₃₂	D₄	1	4	yes	—
31	C₂×C₅×C₁₆	—	2	2	no	—
32	C₂×C₅×C₁₆	—	2	4	no	—
33	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
34	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
35	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
36	C₅×#108	—	2	2	no	—
37	C₅×#108	—	2	2	no	—
38	C₅×#107	—	2	2	no	—
39	C₅×#106	—	2	2	no	—
40	C₅×#101	—	4	1	no	—
41	C₅×#100	—	4	1	no	—
42	C₅×#99	—	4	2	no	—
43	C₂×#504	C₂×C₅	1	1	yes	characteristic
44	#2155	C₂×C₅	1	2	yes	complemented
45	#1813	C₂×C₅	1	1	yes	characteristic
46	#1812	C₂×C₅	1	1	yes	characteristic
47	#1382	C₂×C₅	1	2	yes	complemented
48	C₅×#41	—	2	2	no	—
49	C₅×#41	—	2	2	no	—
50	C₂×C₅×Q₈	SD₁₆	1	1	yes	characteristic
51	C₂×C₅×D₄	SD₁₆	1	2	yes	—
52	C₂×C₅×D₄	SD₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
55	C₅×M₁₆	—	2	2	no	—
56	C₅×M₁₆	—	2	4	no	—
57	C₅×M₁₆	—	4	2	no	—
58	C₅×M₁₆	—	4	2	no	—
59	C₂×C₅×C₈	—	2	2	no	—
60	C₂×C₅×C₈	—	2	2	no	—
61	C₂×C₅×C₈	—	4	2	no	—
62	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
63	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
65	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₅×C₈	#41	1	2	yes	—
67	C₅×C₄⋊C₄	—	4	2	no	—
68	C₅×C₂²⋊C₄	—	4	2	no	—
69	C₄²×C₅	—	4	2	no	—
70	C₅×C₁₆	—	2	1	no	—
71	C₅×C₁₆	—	2	1	no	—
72	C₅×C₁₆	C₂²⋊C₄	1	4	yes	—
73	C₂×#131	C₂²×C₅	1	1	yes	characteristic
74	#504	—	2	4	no	—
75	C₂×M₃₂	C₄×C₅	1	2	yes	complemented
76	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
77	C₂²×C₁₆	C₄×C₅	1	2	yes	complemented
78	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
79	C₂×#108	C₂²×C₅	1	2	yes	—
80	#427	C₂²×C₅	1	2	yes	—
81	#413	—	2	1	no	—
82	#410	—	2	1	no	—
83	#365	—	2	4	no	—
84	#338	—	2	1	no	—
85	H(3,ℤ/4)	—	2	1	no	—
86	C₂³×C₅	#106	1	2	yes	—
87	C₂³×C₅	C₂×SD₁₆	1	1	yes	characteristic
88	C₅×Q₈	—	2	2	no	—
89	C₅×D₄	—	2	4	no	—
90	C₅×D₄	—	2	2	no	—
91	C₂×C₄×C₅	—	2	2	no	—
92	C₂×C₄×C₅	—	2	2	no	—
93	C₂×C₄×C₅	—	8	2	no	—
94	C₂×C₄×C₅	#106	1	2	yes	—
95	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅	#121	1	2	yes	—
97	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
98	C₂×C₄×C₅	#135	1	2	yes	—
99	C₅×C₈	—	2	2	no	—
100	C₅×C₈	—	8	2	no	—
101	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₅×C₈	#117	1	1	yes	characteristic
103	C₅×C₈	C₂²⋊Q₈	1	2	yes	—
104	C₂×#41	C₅×D₄	1	1	yes	characteristic
105	#131	—	2	2	no	—
106	#131	—	2	2	no	—
107	C₂×M₁₆	—	2	2	no	—
108	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
109	C₂×M₁₆	C₅×Q₈	1	2	yes	—
110	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
111	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
112	#117	—	2	2	no	—
113	M₃₂	—	2	4	no	—
114	M₃₂	C₅×D₄	1	4	yes	—
115	C₂×C₁₆	—	2	2	no	—
116	C₂×C₁₆	—	2	4	no	—
117	C₂×C₁₆	C₂×C₄×C₅	1	1	yes	characteristic
118	C₂×C₁₆	C₂×C₄×C₅	1	1	yes	characteristic
119	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
120	#108	—	2	2	no	—
121	#108	—	2	2	no	—
122	#107	—	2	2	no	—
123	#106	—	2	2	no	—
124	#101	—	4	1	no	—
125	#100	—	4	1	no	—
126	#99	—	4	2	no	—
127	C₂²×C₅	—	2	2	no	—
128	C₂²×C₅	—	4	1	no	—
129	C₂²×C₅	—	4	2	no	—
130	C₂²×C₅	C₂×#106	1	1	yes	characteristic
131	C₂²×C₅	#465	1	2	yes	—
132	C₄×C₅	—	2	2	no	—
133	C₄×C₅	—	16	2	no	—
134	C₄×C₅	#380	1	1	yes	characteristic
135	C₄×C₅	#427	1	1	yes	characteristic
136	C₄×C₅	#465	1	2	yes	—
137	#41	—	2	2	no	—
138	#41	—	2	2	no	—
139	C₂×Q₈	C₅×SD₁₆	1	1	yes	characteristic
140	C₂×D₄	C₅×SD₁₆	1	1	yes	characteristic
141	C₂×D₄	C₅×SD₁₆	1	2	yes	—
142	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
143	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
144	M₁₆	—	2	2	no	—
145	M₁₆	—	2	4	no	—
146	M₁₆	—	4	2	no	—
147	M₁₆	—	4	2	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
152	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
153	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
154	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
155	C₂×C₈	C₅×#41	1	2	yes	—
156	C₄⋊C₄	—	4	2	no	—
157	C₂²⋊C₄	—	4	2	no	—
158	C₄²	—	4	2	no	—
159	C₁₆	—	2	1	no	—
160	C₁₆	—	2	1	no	—
161	C₁₆	C₅×C₂²⋊C₄	1	4	yes	—
162	C₂×C₅	—	2	2	no	—
163	C₂×C₅	—	2	1	no	—
164	C₂×C₅	—	4	2	no	—
165	C₂×C₅	#1807	1	1	yes	characteristic
166	C₂³	C₅×#106	1	2	yes	—
167	C₂³	C₂×C₅×SD₁₆	1	1	yes	characteristic
168	Q₈	—	2	2	no	—
169	D₄	—	2	4	no	—
170	D₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	C₅×#106	1	2	yes	—
175	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
176	C₂×C₄	C₅×#121	1	2	yes	—
177	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
178	C₂×C₄	C₅×#135	1	2	yes	—
179	C₈	—	2	2	no	—
180	C₈	—	8	2	no	—
181	C₈	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
182	C₈	C₅×#117	1	1	yes	characteristic
183	C₈	C₅×C₂²⋊Q₈	1	2	yes	—
184	C₅	#12832	1	1	yes	characteristic, direct product factor, Sylow
185	C₂²	—	2	2	no	—
186	C₂²	—	4	1	no	—
187	C₂²	—	4	2	no	—
188	C₂²	C₂×C₅×#106	1	1	yes	characteristic
189	C₂²	C₅×#465	1	2	yes	—
190	C₄	—	2	2	no	—
191	C₄	—	16	2	no	—
192	C₄	C₅×#380	1	1	yes	characteristic
193	C₄	C₅×#427	1	1	yes	characteristic
194	C₄	C₅×#465	1	2	yes	—
195	C₂	—	2	2	no	—
196	C₂	—	2	1	no	—
197	C₂	—	4	2	no	—
198	C₂	C₅×#1807	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	#427
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#1807
prime radical	C₂×C₅×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₅×M₁₆ ◁ C₂×C₅×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2155	C₂	semidirect
5. C₅×#1382	C₂	semidirect
8. C₂×C₅×M₃₂	C₄	semidirect
10. C₂²×C₅×C₁₆	C₄	semidirect
19. #12832	C₅	direct (N × H)
44. #2155	C₂×C₅	semidirect
47. #1382	C₂×C₅	semidirect
75. C₂×M₃₂	C₄×C₅	semidirect
77. C₂²×C₁₆	C₄×C₅	semidirect
184. C₅	#12832	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×#504	2
C₅×#2155	2
C₅×#1813	2
C₅×#1812	2
C₅×#1382	2
#12832	5

Subgroups — C5×#12832 #11935552

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12832 #11935552