Subgroups — C₅×#21681 #11944401

Order 1280.

1230 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#436	C₂	1	6	yes	maximal, complemented
2	C₄×C₅×#127	C₂	1	1	yes	maximal, characteristic
3	C₅×#1954	C₂	1	6	yes	maximal, complemented
4	C₅×#1842	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄²×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₅×M₁₆	C₂²	1	4	yes	—
7	C₂×C₅×#127	C₄	1	1	yes	characteristic
8	C₂×C₄×C₅×D₄	C₂²	1	6	yes	—
9	C₅×#436	—	2	24	no	—
10	C₂×C₅×C₄⋊C₈	C₂²	1	6	yes	—
11	C₂×C₅×C₂²⋊C₈	C₂²	1	12	yes	—
12	C₂×C₄×C₅×C₈	C₂²	1	3	yes	—
13	C₂×C₄×C₅×C₈	C₂²	1	3	yes	—
14	C₅×#390	C₄	1	6	yes	—
15	C₅×#384	C₄	1	1	yes	characteristic
16	C₄³×C₅	C₂²	1	1	yes	characteristic
17	#21681	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂²×C₅×D₄	C₂×C₄	1	6	yes	—
19	C₂³×C₄×C₅	C₂³	1	4	yes	—
20	C₂×C₅×M₁₆	—	2	24	no	—
21	C₂²×C₅×C₈	C₂³	1	4	yes	—
22	C₂²×C₅×C₈	C₂³	1	3	yes	—
23	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
24	C₅×#127	—	2	4	no	—
25	C₄×C₅×D₄	—	2	12	no	—
26	C₄×C₅×D₄	—	2	12	no	—
27	C₂×C₅×C₄⋊C₄	C₂×C₄	1	6	yes	—
28	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	12	yes	—
29	C₂×C₄²×C₅	C₂×C₄	1	3	yes	—
30	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₅	C₂³	1	3	yes	—
32	C₅×C₄⋊C₈	—	2	12	no	—
33	C₅×C₂²⋊C₈	—	2	24	no	—
34	C₄×C₅×C₈	—	2	6	no	—
35	C₄×C₅×C₈	D₄	1	12	yes	complemented
36	C₂×#436	C₂×C₅	1	6	yes	complemented
37	C₄×#127	C₂×C₅	1	1	yes	characteristic
38	#1954	C₂×C₅	1	6	yes	complemented
39	#1842	C₂×C₅	1	1	yes	characteristic, complemented
40	C₄²×C₈	C₂×C₅	1	1	yes	characteristic, complemented
41	C₂⁴×C₅	C₂²×C₄	1	4	yes	—
42	C₂×C₅×D₄	—	2	12	no	—
43	C₂×C₅×D₄	—	2	12	no	—
44	C₂×C₅×D₄	—	2	12	no	—
45	C₂²×C₄×C₅	—	2	12	no	—
46	C₂²×C₄×C₅	—	2	12	no	—
47	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
48	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
49	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
50	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
51	C₅×M₁₆	—	4	16	no	—
52	C₂×C₅×C₈	—	2	6	no	—
53	C₂×C₅×C₈	—	2	12	no	—
54	C₂×C₅×C₈	C₂×D₄	1	6	yes	—
55	C₂×C₅×C₈	C₂×D₄	1	6	yes	—
56	C₅×C₄⋊C₄	—	2	12	no	—
57	C₅×C₂²⋊C₄	—	2	24	no	—
58	C₄²×C₅	—	2	6	no	—
59	C₄²×C₅	M₁₆	1	4	yes	complemented
60	C₄²×C₅	C₂×D₄	1	6	yes	—
61	C₄²×C₅	#41	1	6	yes	—
62	C₂²×M₁₆	C₂²×C₅	1	4	yes	—
63	C₂×#127	C₄×C₅	1	1	yes	characteristic
64	C₂×C₄×D₄	C₂²×C₅	1	6	yes	—
65	#436	—	2	24	no	—
66	C₂×C₄⋊C₈	C₂²×C₅	1	6	yes	—
67	C₂×C₂²⋊C₈	C₂²×C₅	1	12	yes	—
68	C₂×C₄×C₈	C₂²×C₅	1	3	yes	—
69	C₂×C₄×C₈	C₂²×C₅	1	3	yes	—
70	#390	C₄×C₅	1	6	yes	—
71	#384	C₄×C₅	1	1	yes	characteristic
72	C₄³	C₂²×C₅	1	1	yes	characteristic
73	C₂³×C₅	—	2	12	no	—
74	C₂³×C₅	—	2	4	no	—
75	C₂³×C₅	—	2	12	no	—
76	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
77	C₅×D₄	—	4	24	no	—
78	C₂×C₄×C₅	—	2	6	no	—
79	C₂×C₄×C₅	—	2	12	no	—
80	C₂×C₄×C₅	—	4	4	no	—
81	C₂×C₄×C₅	—	4	12	no	—
82	C₂×C₄×C₅	C₄×D₄	1	6	yes	—
83	C₂×C₄×C₅	C₄×D₄	1	6	yes	—
84	C₂×C₄×C₅	C₂×M₁₆	1	6	yes	—
85	C₂×C₄×C₅	#131	1	6	yes	—
86	C₂×C₄×C₅	C₂²×D₄	1	3	yes	—
87	C₂×C₄×C₅	C₂×#41	1	3	yes	—
88	C₅×C₈	—	2	6	no	—
89	C₅×C₈	—	4	4	no	—
90	C₅×C₈	#127	1	4	yes	complemented
91	C₂²×D₄	C₂×C₄×C₅	1	6	yes	—
92	C₂³×C₄	C₂³×C₅	1	4	yes	—
93	C₂×M₁₆	—	2	24	no	—
94	C₂²×C₈	C₂³×C₅	1	4	yes	—
95	C₂²×C₈	C₂³×C₅	1	3	yes	—
96	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
97	#127	—	2	4	no	—
98	C₄×D₄	—	2	12	no	—
99	C₄×D₄	—	2	12	no	—
100	C₂×C₄⋊C₄	C₂×C₄×C₅	1	6	yes	—
101	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	12	yes	—
102	C₂×C₄²	C₂×C₄×C₅	1	3	yes	—
103	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
104	C₂×C₄²	C₂³×C₅	1	3	yes	—
105	C₄⋊C₈	—	2	12	no	—
106	C₂²⋊C₈	—	2	24	no	—
107	C₄×C₈	—	2	6	no	—
108	C₄×C₈	C₅×D₄	1	12	yes	complemented
109	C₂²×C₅	—	4	4	no	—
110	C₂²×C₅	—	4	12	no	—
111	C₂²×C₅	—	4	12	no	—
112	C₂²×C₅	C₂×C₄×D₄	1	3	yes	—
113	C₂²×C₅	C₂²×M₁₆	1	1	yes	characteristic
114	C₂²×C₅	C₂×#131	1	3	yes	—
115	C₄×C₅	—	2	6	no	—
116	C₄×C₅	—	4	4	no	—
117	C₄×C₅	#436	1	12	yes	complemented
118	C₄×C₅	C₂×#127	1	1	yes	characteristic
119	C₄×C₅	#532	1	3	yes	—
120	C₂⁴	C₂²×C₄×C₅	1	4	yes	—
121	C₂×D₄	—	2	12	no	—
122	C₂×D₄	—	2	12	no	—
123	C₂×D₄	—	2	12	no	—
124	C₂²×C₄	—	2	12	no	—
125	C₂²×C₄	—	2	12	no	—
126	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
127	C₂²×C₄	C₂²×C₄×C₅	1	3	yes	—
128	C₂²×C₄	C₂²×C₄×C₅	1	3	yes	—
129	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
130	M₁₆	—	4	16	no	—
131	C₂×C₈	—	2	6	no	—
132	C₂×C₈	—	2	12	no	—
133	C₂×C₈	C₂×C₅×D₄	1	6	yes	—
134	C₂×C₈	C₂×C₅×D₄	1	6	yes	—
135	C₄⋊C₄	—	2	12	no	—
136	C₂²⋊C₄	—	2	24	no	—
137	C₄²	—	2	6	no	—
138	C₄²	C₅×M₁₆	1	4	yes	complemented
139	C₄²	C₂×C₅×D₄	1	6	yes	—
140	C₄²	C₅×#41	1	6	yes	—
141	C₂×C₅	—	8	4	no	—
142	C₂×C₅	C₄×#127	1	1	yes	characteristic
143	C₂×C₅	C₂×#436	1	3	yes	—
144	C₂×C₅	#2931	1	3	yes	—
145	C₂³	—	2	12	no	—
146	C₂³	—	2	4	no	—
147	C₂³	—	2	12	no	—
148	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
149	D₄	—	4	24	no	—
150	C₂×C₄	—	2	6	no	—
151	C₂×C₄	—	2	12	no	—
152	C₂×C₄	—	4	4	no	—
153	C₂×C₄	—	4	12	no	—
154	C₂×C₄	C₄×C₅×D₄	1	6	yes	—
155	C₂×C₄	C₄×C₅×D₄	1	6	yes	—
156	C₂×C₄	C₂×C₅×M₁₆	1	6	yes	—
157	C₂×C₄	C₅×#131	1	6	yes	—
158	C₂×C₄	C₂²×C₅×D₄	1	3	yes	—
159	C₂×C₄	C₂×C₅×#41	1	3	yes	—
160	C₈	—	2	6	no	—
161	C₈	—	4	4	no	—
162	C₈	C₅×#127	1	4	yes	complemented
163	C₅	#21681	1	1	yes	characteristic, direct product factor, Sylow
164	C₂²	—	4	4	no	—
165	C₂²	—	4	12	no	—
166	C₂²	—	4	12	no	—
167	C₂²	C₂×C₄×C₅×D₄	1	3	yes	—
168	C₂²	C₂²×C₅×M₁₆	1	1	yes	characteristic
169	C₂²	C₂×C₅×#131	1	3	yes	—
170	C₄	—	2	6	no	—
171	C₄	—	4	4	no	—
172	C₄	C₅×#436	1	12	yes	complemented
173	C₄	C₂×C₅×#127	1	1	yes	characteristic
174	C₄	C₅×#532	1	3	yes	—
175	C₂	—	8	4	no	—
176	C₂	C₄×C₅×#127	1	1	yes	characteristic
177	C₂	C₂×C₅×#436	1	3	yes	—
178	C₂	C₅×#2931	1	3	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂⁴
commutator	C₂³	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂³×C₅	C₂³×C₄
prime radical	C₂×C₅×#127	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#436	C₂	semidirect
3. C₅×#1954	C₂	semidirect
4. C₅×#1842	C₂	semidirect
5. C₄²×C₅×C₈	C₂	semidirect
17. #21681	C₅	direct (N × H)
35. C₄×C₅×C₈	D₄	semidirect
36. C₂×#436	C₂×C₅	semidirect
38. #1954	C₂×C₅	semidirect
39. #1842	C₂×C₅	semidirect
40. C₄²×C₈	C₂×C₅	semidirect
59. C₄²×C₅	M₁₆	semidirect
90. C₅×C₈	#127	semidirect
108. C₄×C₈	C₅×D₄	semidirect
117. C₄×C₅	#436	semidirect
138. C₄²	C₅×M₁₆	semidirect
162. C₈	C₅×#127	semidirect
163. C₅	#21681	direct (N × H)
172. C₄	C₅×#436	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#436	2
C₄×C₅×#127	2
C₅×#1954	2
C₅×#1842	2
C₄²×C₅×C₈	2
#21681	5

Subgroups — C5×#21681 #11944401

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21681 #11944401