Subgroups — C₂×C₅×#1746 #11943083

Order 1280.

1134 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#97	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#376	C₂	1	1	yes	maximal, characteristic
4	C₅×#1746	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅×#336	C₂	1	4	yes	maximal
6	C₂³×C₅×C₈	C₂²	1	2	yes	—
7	C₂³×C₅×C₈	C₂²	1	2	yes	—
8	C₂²×C₄²×C₅	C₂²	1	2	yes	—
9	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₅×#97	C₄	1	8	yes	complemented
11	C₂×C₅×#97	C₂²	1	4	yes	—
12	C₂×C₄×C₅×C₈	C₄	1	8	yes	complemented
13	C₂×C₄×C₅×C₈	C₂²	1	4	yes	—
14	C₅×#376	C₄	1	4	yes	—
15	C₅×#376	C₂²	1	4	yes	—
16	C₂×C₅×#95	C₄	1	4	yes	—
17	C₅×#336	C₂²	1	16	yes	—
18	C₂×#1746	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂³×C₄×C₅	C₂×C₄	1	4	yes	—
20	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
22	C₂²×C₅×C₈	—	2	8	no	—
23	C₂²×C₅×C₈	—	2	8	no	—
24	C₂²×C₅×C₈	C₂×C₄	1	8	yes	—
25	C₂²×C₅×C₈	C₂×C₄	1	8	yes	—
26	C₂²×C₅×C₈	C₂³	1	4	yes	—
27	C₂²×C₅×C₈	C₂³	1	4	yes	—
28	C₂×C₄²×C₅	—	2	8	no	—
29	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₅	C₂³	1	4	yes	—
34	C₂×C₄²×C₅	C₂³	1	2	yes	—
35	C₅×#97	C₂×C₄	1	16	yes	complemented
36	C₄×C₅×C₈	C₂×C₄	1	16	yes	complemented
37	C₅×#95	C₂×C₄	1	16	yes	—
38	C₂²×#97	C₂×C₅	1	1	yes	characteristic
39	C₂²×C₄×C₈	C₂×C₅	1	1	yes	characteristic
40	C₂×#376	C₂×C₅	1	1	yes	characteristic
41	#1746	C₂×C₅	1	8	yes	direct product factor
42	C₂×#336	C₂×C₅	1	4	yes	—
43	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅	—	2	8	no	—
45	C₂²×C₄×C₅	—	2	8	no	—
46	C₂²×C₄×C₅	—	2	16	no	—
47	C₂²×C₄×C₅	C₄²	1	8	yes	—
48	C₂²×C₄×C₅	C₂²×C₄	1	8	yes	—
49	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
50	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₅	#41	1	8	yes	—
54	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
55	C₂×C₅×C₈	—	2	16	no	—
56	C₂×C₅×C₈	—	2	16	no	—
57	C₂×C₅×C₈	—	2	8	no	—
58	C₂×C₅×C₈	—	2	8	no	—
59	C₂×C₅×C₈	C₂²×C₄	1	8	yes	—
60	C₂×C₅×C₈	C₂²×C₄	1	8	yes	—
61	C₄²×C₅	—	2	8	no	—
62	C₄²×C₅	—	2	8	no	—
63	C₄²×C₅	C₄²	1	8	yes	—
64	C₄²×C₅	C₂²×C₄	1	4	yes	—
65	C₄²×C₅	C₂²×C₄	1	4	yes	—
66	C₂³×C₈	C₂²×C₅	1	2	yes	—
67	C₂³×C₈	C₂²×C₅	1	2	yes	—
68	C₂²×C₄²	C₂²×C₅	1	2	yes	—
69	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
70	C₂×#97	C₄×C₅	1	8	yes	complemented
71	C₂×#97	C₂²×C₅	1	4	yes	—
72	C₂×C₄×C₈	C₄×C₅	1	8	yes	complemented
73	C₂×C₄×C₈	C₂²×C₅	1	4	yes	—
74	#376	C₄×C₅	1	4	yes	—
75	#376	C₂²×C₅	1	4	yes	—
76	C₂×#95	C₄×C₅	1	4	yes	—
77	#336	C₂²×C₅	1	16	yes	—
78	C₂³×C₅	C₂×C₄²	1	2	yes	—
79	C₂³×C₅	#117	1	4	yes	—
80	C₂³×C₅	#131	1	8	yes	—
81	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
82	C₂×C₄×C₅	—	2	8	no	—
83	C₂×C₄×C₅	—	2	16	no	—
84	C₂×C₄×C₅	—	2	8	no	—
85	C₂×C₄×C₅	—	2	16	no	—
86	C₂×C₄×C₅	—	4	16	no	—
87	C₂×C₄×C₅	C₂×C₄²	1	8	yes	—
88	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
89	C₂×C₄×C₅	#117	1	8	yes	—
90	C₂×C₄×C₅	#117	1	8	yes	—
91	C₂×C₄×C₅	#117	1	4	yes	—
92	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂×#41	1	4	yes	—
95	C₅×C₈	—	2	8	no	—
96	C₅×C₈	—	2	8	no	—
97	C₂³×C₄	C₂×C₄×C₅	1	4	yes	—
98	C₂³×C₄	C₂×C₄×C₅	1	2	yes	—
99	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
100	C₂²×C₈	—	2	8	no	—
101	C₂²×C₈	—	2	8	no	—
102	C₂²×C₈	C₂×C₄×C₅	1	8	yes	—
103	C₂²×C₈	C₂×C₄×C₅	1	8	yes	—
104	C₂²×C₈	C₂³×C₅	1	4	yes	—
105	C₂²×C₈	C₂³×C₅	1	4	yes	—
106	C₂×C₄²	—	2	8	no	—
107	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
108	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
109	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
110	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
111	C₂×C₄²	C₂³×C₅	1	4	yes	—
112	C₂×C₄²	C₂³×C₅	1	2	yes	—
113	#97	C₂×C₄×C₅	1	16	yes	complemented
114	C₄×C₈	C₂×C₄×C₅	1	16	yes	complemented
115	#95	C₂×C₄×C₅	1	16	yes	—
116	C₂²×C₅	#376	1	4	yes	—
117	C₂²×C₅	#405	1	8	yes	—
118	C₂²×C₅	#433	1	16	yes	—
119	C₂²×C₅	C₂²×C₄²	1	1	yes	characteristic
120	C₂²×C₅	C₂×#117	1	2	yes	—
121	C₂²×C₅	C₂×#131	1	4	yes	—
122	C₄×C₅	—	2	8	no	—
123	C₄×C₅	—	4	8	no	—
124	C₄×C₅	#376	1	4	yes	—
125	C₄×C₅	C₂×#117	1	2	yes	—
126	C₄×C₅	C₂×#117	1	2	yes	—
127	C₂⁴	C₂²×C₄×C₅	1	1	yes	characteristic
128	C₂²×C₄	—	2	8	no	—
129	C₂²×C₄	—	2	8	no	—
130	C₂²×C₄	—	2	16	no	—
131	C₂²×C₄	C₄²×C₅	1	8	yes	—
132	C₂²×C₄	C₂²×C₄×C₅	1	8	yes	—
133	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
134	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
135	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
136	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
137	C₂²×C₄	C₅×#41	1	8	yes	—
138	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
139	C₂×C₈	—	2	16	no	—
140	C₂×C₈	—	2	16	no	—
141	C₂×C₈	—	2	8	no	—
142	C₂×C₈	—	2	8	no	—
143	C₂×C₈	C₂²×C₄×C₅	1	8	yes	—
144	C₂×C₈	C₂²×C₄×C₅	1	8	yes	—
145	C₄²	—	2	8	no	—
146	C₄²	—	2	8	no	—
147	C₄²	C₄²×C₅	1	8	yes	—
148	C₄²	C₂²×C₄×C₅	1	4	yes	—
149	C₄²	C₂²×C₄×C₅	1	4	yes	—
150	C₂×C₅	#1746	1	8	yes	direct product factor
151	C₂×C₅	C₂×#376	1	1	yes	characteristic
152	C₂×C₅	C₂×#405	1	2	yes	—
153	C₂×C₅	C₂×#433	1	4	yes	—
154	C₂³	C₂×C₄²×C₅	1	2	yes	—
155	C₂³	C₅×#117	1	4	yes	—
156	C₂³	C₅×#131	1	8	yes	—
157	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
158	C₂×C₄	—	2	16	no	—
159	C₂×C₄	—	2	8	no	—
160	C₂×C₄	—	2	8	no	—
161	C₂×C₄	—	2	16	no	—
162	C₂×C₄	—	4	16	no	—
163	C₂×C₄	C₂×C₄²×C₅	1	8	yes	—
164	C₂×C₄	C₂×C₄²×C₅	1	2	yes	—
165	C₂×C₄	C₅×#117	1	4	yes	—
166	C₂×C₄	C₅×#117	1	8	yes	—
167	C₂×C₄	C₅×#117	1	8	yes	—
168	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
169	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
170	C₂×C₄	C₂×C₅×#41	1	4	yes	—
171	C₈	—	2	8	no	—
172	C₈	—	2	8	no	—
173	C₅	C₂×#1746	1	1	yes	characteristic, direct product factor, Sylow
174	C₂²	C₅×#376	1	4	yes	—
175	C₂²	C₅×#405	1	8	yes	—
176	C₂²	C₅×#433	1	16	yes	—
177	C₂²	C₂²×C₄²×C₅	1	1	yes	characteristic
178	C₂²	C₂×C₅×#117	1	2	yes	—
179	C₂²	C₂×C₅×#131	1	4	yes	—
180	C₄	—	2	8	no	—
181	C₄	—	4	8	no	—
182	C₄	C₅×#376	1	4	yes	—
183	C₄	C₂×C₅×#117	1	2	yes	—
184	C₄	C₂×C₅×#117	1	2	yes	—
185	C₂	C₅×#1746	1	8	yes	direct product factor
186	C₂	C₂×C₅×#376	1	1	yes	characteristic
187	C₂	C₂×C₅×#405	1	2	yes	—
188	C₂	C₂×C₅×#433	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄×C₅	C₂³
commutator	C₂²	C₂²×C₄²×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle, prime radical	C₂⁴×C₅	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#1746	C₂	direct (N × H)
10. C₂×C₅×#97	C₄	semidirect
12. C₂×C₄×C₅×C₈	C₄	semidirect
18. C₂×#1746	C₅	direct (N × H)
35. C₅×#97	C₂×C₄	semidirect
36. C₄×C₅×C₈	C₂×C₄	semidirect
41. #1746	C₂×C₅	direct (N × H)
70. C₂×#97	C₄×C₅	semidirect
72. C₂×C₄×C₈	C₄×C₅	semidirect
113. #97	C₂×C₄×C₅	semidirect
114. C₄×C₈	C₂×C₄×C₅	semidirect
150. C₂×C₅	#1746	direct (N × H)
173. C₅	C₂×#1746	direct (N × H)
185. C₂	C₅×#1746	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×#97	2
C₂²×C₄×C₅×C₈	2
C₂×C₅×#376	2
C₅×#1746	2
C₂×C₅×#336	2
C₂×#1746	5

Subgroups — C2×C5×#1746 #11943083

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1746 #11943083