Subgroups — C₂×C₅×#1818 #11943401

Order 1280.

1526 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#117	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×C₂²⋊C₈	C₂	1	2	yes	maximal
3	C₅×#1818	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅×#336	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₄×C₅	C₂²	1	1	yes	characteristic
6	C₂³×C₅×C₈	C₂²	1	4	yes	—
7	C₂×C₅×#117	—	2	8	no	—
8	C₂×C₅×#117	C₄	1	4	yes	—
9	C₂×C₅×#117	C₂²	1	4	yes	—
10	C₂²×C₅×C₄⋊C₄	C₄	1	2	yes	—
11	C₂²×C₅×C₂²⋊C₄	C₄	1	2	yes	—
12	C₂²×C₄²×C₅	C₂²	1	2	yes	—
13	C₂×C₅×C₂²⋊C₈	—	2	8	no	—
14	C₂×C₅×C₂²⋊C₈	C₂²	1	8	yes	—
15	C₅×#336	C₂²	1	16	yes	complemented
16	C₂×#1818	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂⁵×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₅	—	2	8	no	—
19	C₂³×C₄×C₅	C₂×C₄	1	4	yes	—
20	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
22	C₂³×C₄×C₅	D₄	1	8	yes	—
23	C₂³×C₄×C₅	C₂³	1	2	yes	—
24	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂²×C₅×C₈	—	2	8	no	—
26	C₂²×C₅×C₈	—	2	16	no	—
27	C₂²×C₅×C₈	C₂³	1	8	yes	—
28	C₅×#117	—	2	16	no	—
29	C₅×#117	—	2	16	no	—
30	C₂×C₅×C₄⋊C₄	—	2	8	no	—
31	C₂×C₅×C₄⋊C₄	C₂×C₄	1	8	yes	—
32	C₂×C₅×C₂²⋊C₄	—	2	8	no	—
33	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	8	yes	—
34	C₂×C₄²×C₅	—	2	8	no	—
35	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₅	C₂³	1	4	yes	—
37	C₅×C₂²⋊C₈	—	2	16	no	—
38	C₂²×#117	C₂×C₅	1	1	yes	characteristic
39	C₂²×C₂²⋊C₈	C₂×C₅	1	2	yes	—
40	#1818	C₂×C₅	1	8	yes	direct product factor
41	C₂×#336	C₂×C₅	1	4	yes	complemented
42	C₂⁴×C₅	—	2	4	no	—
43	C₂⁴×C₅	—	2	8	no	—
44	C₂⁴×C₅	C₂²⋊C₄	1	4	yes	—
45	C₂⁴×C₅	C₂²×C₄	1	2	yes	—
46	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅	—	2	4	no	—
48	C₂²×C₄×C₅	—	2	8	no	—
49	C₂²×C₄×C₅	—	2	32	no	—
50	C₂²×C₄×C₅	—	2	8	no	—
51	C₂²×C₄×C₅	—	2	8	no	—
52	C₂²×C₄×C₅	—	2	16	no	—
53	C₂²×C₄×C₅	C₂²⋊C₄	1	4	yes	—
54	C₂²×C₄×C₅	C₂²⋊C₄	1	8	yes	—
55	C₂²×C₄×C₅	C₂²×C₄	1	8	yes	—
56	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅	C₂×D₄	1	8	yes	—
59	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
60	C₂²×C₄×C₅	#41	1	8	yes	—
61	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
62	C₂×C₅×C₈	—	2	8	no	—
63	C₂×C₅×C₈	—	2	16	no	—
64	C₂×C₅×C₈	—	4	16	no	—
65	C₅×C₄⋊C₄	—	2	16	no	—
66	C₅×C₂²⋊C₄	—	2	16	no	—
67	C₄²×C₅	—	2	8	no	—
68	C₄²×C₅	—	2	8	no	—
69	C₂⁴×C₄	C₂²×C₅	1	1	yes	characteristic
70	C₂³×C₈	C₂²×C₅	1	4	yes	—
71	C₂×#117	—	2	8	no	—
72	C₂×#117	C₄×C₅	1	4	yes	—
73	C₂×#117	C₂²×C₅	1	4	yes	—
74	C₂²×C₄⋊C₄	C₄×C₅	1	2	yes	—
75	C₂²×C₂²⋊C₄	C₄×C₅	1	2	yes	—
76	C₂²×C₄²	C₂²×C₅	1	2	yes	—
77	C₂×C₂²⋊C₈	—	2	8	no	—
78	C₂×C₂²⋊C₈	C₂²×C₅	1	8	yes	—
79	#336	C₂²×C₅	1	16	yes	complemented
80	C₂³×C₅	—	2	4	no	—
81	C₂³×C₅	—	2	8	no	—
82	C₂³×C₅	—	2	8	no	—
83	C₂³×C₅	—	2	16	no	—
84	C₂³×C₅	—	4	16	no	—
85	C₂³×C₅	C₂×C₂²⋊C₄	1	4	yes	—
86	C₂³×C₅	C₂×C₂²⋊C₄	1	2	yes	—
87	C₂³×C₅	#117	1	4	yes	—
88	C₂³×C₅	#131	1	8	yes	—
89	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
90	C₂×C₄×C₅	—	2	4	no	—
91	C₂×C₄×C₅	—	2	8	no	—
92	C₂×C₄×C₅	—	2	16	no	—
93	C₂×C₄×C₅	—	2	16	no	—
94	C₂×C₄×C₅	—	2	8	no	—
95	C₂×C₄×C₅	—	2	16	no	—
96	C₂×C₄×C₅	—	4	16	no	—
97	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	4	yes	—
98	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
99	C₂×C₄×C₅	#117	1	4	yes	—
100	C₂×C₄×C₅	#123	1	16	yes	—
101	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
102	C₂×C₄×C₅	C₂×#41	1	4	yes	—
103	C₅×C₈	—	4	8	no	—
104	C₂⁵	C₂×C₄×C₅	1	1	yes	characteristic
105	C₂³×C₄	—	2	8	no	—
106	C₂³×C₄	C₂×C₄×C₅	1	4	yes	—
107	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
108	C₂³×C₄	C₂×C₄×C₅	1	2	yes	—
109	C₂³×C₄	C₅×D₄	1	8	yes	—
110	C₂³×C₄	C₂³×C₅	1	2	yes	—
111	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
112	C₂²×C₈	—	2	8	no	—
113	C₂²×C₈	—	2	16	no	—
114	C₂²×C₈	C₂³×C₅	1	8	yes	—
115	#117	—	2	16	no	—
116	#117	—	2	16	no	—
117	C₂×C₄⋊C₄	—	2	8	no	—
118	C₂×C₄⋊C₄	C₂×C₄×C₅	1	8	yes	—
119	C₂×C₂²⋊C₄	—	2	8	no	—
120	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	8	yes	—
121	C₂×C₄²	—	2	8	no	—
122	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
123	C₂×C₄²	C₂³×C₅	1	4	yes	—
124	C₂²⋊C₈	—	2	16	no	—
125	C₂²×C₅	—	2	8	no	—
126	C₂²×C₅	—	2	4	no	—
127	C₂²×C₅	—	4	16	no	—
128	C₂²×C₅	—	4	8	no	—
129	C₂²×C₅	#381	1	4	yes	—
130	C₂²×C₅	#408	1	8	yes	—
131	C₂²×C₅	#433	1	16	yes	—
132	C₂²×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
133	C₂²×C₅	C₂×#117	1	2	yes	—
134	C₂²×C₅	C₂×#131	1	4	yes	—
135	C₄×C₅	—	2	4	no	—
136	C₄×C₅	—	4	8	no	—
137	C₄×C₅	#381	1	4	yes	—
138	C₄×C₅	C₂×#123	1	4	yes	—
139	C₂⁴	—	2	4	no	—
140	C₂⁴	—	2	8	no	—
141	C₂⁴	C₅×C₂²⋊C₄	1	4	yes	—
142	C₂⁴	C₂²×C₄×C₅	1	2	yes	—
143	C₂⁴	C₂²×C₄×C₅	1	1	yes	characteristic
144	C₂²×C₄	—	2	4	no	—
145	C₂²×C₄	—	2	8	no	—
146	C₂²×C₄	—	2	32	no	—
147	C₂²×C₄	—	2	8	no	—
148	C₂²×C₄	—	2	8	no	—
149	C₂²×C₄	—	2	16	no	—
150	C₂²×C₄	C₅×C₂²⋊C₄	1	4	yes	—
151	C₂²×C₄	C₅×C₂²⋊C₄	1	8	yes	—
152	C₂²×C₄	C₂²×C₄×C₅	1	8	yes	—
153	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
154	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
155	C₂²×C₄	C₂×C₅×D₄	1	8	yes	—
156	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
157	C₂²×C₄	C₅×#41	1	8	yes	—
158	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
159	C₂×C₈	—	2	8	no	—
160	C₂×C₈	—	2	16	no	—
161	C₂×C₈	—	4	16	no	—
162	C₄⋊C₄	—	2	16	no	—
163	C₂²⋊C₄	—	2	16	no	—
164	C₄²	—	2	8	no	—
165	C₄²	—	2	8	no	—
166	C₂×C₅	—	4	4	no	—
167	C₂×C₅	#1818	1	8	yes	direct product factor
168	C₂×C₅	C₂×#381	1	1	yes	characteristic
169	C₂×C₅	C₂×#408	1	2	yes	—
170	C₂×C₅	C₂×#433	1	4	yes	—
171	C₂³	—	2	8	no	—
172	C₂³	—	2	16	no	—
173	C₂³	—	2	4	no	—
174	C₂³	—	2	8	no	—
175	C₂³	—	4	16	no	—
176	C₂³	C₂×C₅×C₂²⋊C₄	1	4	yes	—
177	C₂³	C₂×C₅×C₂²⋊C₄	1	2	yes	—
178	C₂³	C₅×#117	1	4	yes	—
179	C₂³	C₅×#131	1	8	yes	—
180	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
181	C₂×C₄	—	2	8	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	16	no	—
184	C₂×C₄	—	2	16	no	—
185	C₂×C₄	—	2	8	no	—
186	C₂×C₄	—	2	16	no	—
187	C₂×C₄	—	4	16	no	—
188	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	4	yes	—
189	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
190	C₂×C₄	C₅×#117	1	4	yes	—
191	C₂×C₄	C₅×#123	1	16	yes	—
192	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
193	C₂×C₄	C₂×C₅×#41	1	4	yes	—
194	C₈	—	4	8	no	—
195	C₅	C₂×#1818	1	1	yes	characteristic, direct product factor, Sylow
196	C₂²	—	2	4	no	—
197	C₂²	—	2	8	no	—
198	C₂²	—	4	16	no	—
199	C₂²	—	4	8	no	—
200	C₂²	C₅×#381	1	4	yes	—
201	C₂²	C₅×#408	1	8	yes	—
202	C₂²	C₅×#433	1	16	yes	—
203	C₂²	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
204	C₂²	C₂×C₅×#117	1	2	yes	—
205	C₂²	C₂×C₅×#131	1	4	yes	—
206	C₄	—	2	4	no	—
207	C₄	—	4	8	no	—
208	C₄	C₅×#381	1	4	yes	—
209	C₄	C₂×C₅×#123	1	4	yes	—
210	C₂	—	4	4	no	—
211	C₂	C₅×#1818	1	8	yes	direct product factor
212	C₂	C₂×C₅×#381	1	1	yes	characteristic
213	C₂	C₂×C₅×#408	1	2	yes	—
214	C₂	C₂×C₅×#433	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄×C₅	C₂³
commutator	C₂³	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂⁴×C₅	C₂²×C₄
prime radical	C₂⁵×C₅	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1818	C₂	direct (N × H)
4. C₂×C₅×#336	C₂	semidirect
15. C₅×#336	C₂²	semidirect
16. C₂×#1818	C₅	direct (N × H)
40. #1818	C₂×C₅	direct (N × H)
41. C₂×#336	C₂×C₅	semidirect
79. #336	C₂²×C₅	semidirect
167. C₂×C₅	#1818	direct (N × H)
195. C₅	C₂×#1818	direct (N × H)
211. C₂	C₅×#1818	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×#117	2
C₂²×C₅×C₂²⋊C₈	2
C₅×#1818	2
C₂×C₅×#336	2
C₂×#1818	5

Subgroups — C2×C5×#1818 #11943401

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1818 #11943401