Subgroups — C₂×C₅×#1744 #11943081

Order 1280.

1438 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅×C₈	C₂	1	2	yes	maximal
2	C₂×C₄³×C₅	C₂	1	1	yes	maximal, characteristic
3	C₅×#1744	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅×#336	C₂	1	4	yes	maximal
5	C₂³×C₅×C₈	C₂²	1	4	yes	—
6	C₂²×C₄²×C₅	C₄	1	2	yes	—
7	C₂²×C₄²×C₅	C₄	1	2	yes	—
8	C₂²×C₄²×C₅	C₂²	1	2	yes	—
9	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₅×C₈	C₄	1	16	yes	complemented
11	C₂×C₄×C₅×C₈	C₂²	1	8	yes	—
12	C₄³×C₅	C₄	1	4	yes	—
13	C₄³×C₅	C₂²	1	4	yes	—
14	C₅×#336	C₂²	1	16	yes	—
15	C₂×#1744	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂³×C₄×C₅	C₂×C₄	1	4	yes	—
17	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
20	C₂²×C₅×C₈	—	2	16	no	—
21	C₂²×C₅×C₈	C₂×C₄	1	16	yes	—
22	C₂²×C₅×C₈	C₂³	1	8	yes	—
23	C₂×C₄²×C₅	—	2	8	no	—
24	C₂×C₄²×C₅	—	2	8	no	—
25	C₂×C₄²×C₅	C₈	1	16	yes	complemented
26	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₅	C₂×C₄	1	8	yes	—
28	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₅	C₂×C₄	1	8	yes	—
30	C₂×C₄²×C₅	C₂×C₄	1	8	yes	—
31	C₂×C₄²×C₅	C₂³	1	4	yes	—
32	C₂×C₄²×C₅	C₂³	1	2	yes	—
33	C₄×C₅×C₈	C₂×C₄	1	32	yes	complemented
34	C₂²×C₄×C₈	C₂×C₅	1	2	yes	—
35	C₂×C₄³	C₂×C₅	1	1	yes	characteristic
36	#1744	C₂×C₅	1	8	yes	direct product factor
37	C₂×#336	C₂×C₅	1	4	yes	—
38	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅	—	2	8	no	—
40	C₂²×C₄×C₅	—	2	8	no	—
41	C₂²×C₄×C₅	C₄²	1	4	yes	—
42	C₂²×C₄×C₅	C₄²	1	4	yes	—
43	C₂²×C₄×C₅	C₂×C₈	1	16	yes	—
44	C₂²×C₄×C₅	C₂×C₈	1	8	yes	—
45	C₂²×C₄×C₅	M₁₆	1	8	yes	—
46	C₂²×C₄×C₅	C₂²×C₄	1	8	yes	—
47	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
50	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
51	C₂²×C₄×C₅	#41	1	8	yes	—
52	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
53	C₂×C₅×C₈	—	2	32	no	—
54	C₂×C₅×C₈	—	2	16	no	—
55	C₂×C₅×C₈	C₂²×C₄	1	16	yes	—
56	C₄²×C₅	—	2	8	no	—
57	C₄²×C₅	—	2	16	no	—
58	C₄²×C₅	—	2	8	no	—
59	C₄²×C₅	C₄²	1	8	yes	—
60	C₄²×C₅	C₂×C₈	1	32	yes	complemented
61	C₄²×C₅	C₂²×C₄	1	8	yes	—
62	C₂³×C₈	C₂²×C₅	1	4	yes	—
63	C₂²×C₄²	C₄×C₅	1	2	yes	—
64	C₂²×C₄²	C₄×C₅	1	2	yes	—
65	C₂²×C₄²	C₂²×C₅	1	2	yes	—
66	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
67	C₂×C₄×C₈	C₄×C₅	1	16	yes	complemented
68	C₂×C₄×C₈	C₂²×C₅	1	8	yes	—
69	C₄³	C₄×C₅	1	4	yes	—
70	C₄³	C₂²×C₅	1	4	yes	—
71	#336	C₂²×C₅	1	16	yes	—
72	C₂³×C₅	C₂×C₄²	1	2	yes	—
73	C₂³×C₅	#117	1	4	yes	—
74	C₂³×C₅	C₂²×C₈	1	4	yes	—
75	C₂³×C₅	C₂×M₁₆	1	4	yes	—
76	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
77	C₂×C₄×C₅	—	2	8	no	—
78	C₂×C₄×C₅	—	2	8	no	—
79	C₂×C₄×C₅	—	2	32	no	—
80	C₂×C₄×C₅	C₄×C₈	1	16	yes	—
81	C₂×C₄×C₅	#97	1	16	yes	—
82	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
83	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
84	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
85	C₂×C₄×C₅	#117	1	16	yes	—
86	C₂×C₄×C₅	#117	1	4	yes	—
87	C₂×C₄×C₅	C₂²×C₈	1	16	yes	—
88	C₂×C₄×C₅	C₂²×C₈	1	8	yes	—
89	C₂×C₄×C₅	C₂×M₁₆	1	8	yes	—
90	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
91	C₂×C₄×C₅	C₂×#41	1	4	yes	—
92	C₅×C₈	—	2	16	no	—
93	C₂³×C₄	C₂×C₄×C₅	1	4	yes	—
94	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
95	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
96	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
97	C₂²×C₈	—	2	16	no	—
98	C₂²×C₈	C₂×C₄×C₅	1	16	yes	—
99	C₂²×C₈	C₂³×C₅	1	8	yes	—
100	C₂×C₄²	—	2	8	no	—
101	C₂×C₄²	—	2	8	no	—
102	C₂×C₄²	C₅×C₈	1	16	yes	complemented
103	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
104	C₂×C₄²	C₂×C₄×C₅	1	8	yes	—
105	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
106	C₂×C₄²	C₂×C₄×C₅	1	8	yes	—
107	C₂×C₄²	C₂×C₄×C₅	1	8	yes	—
108	C₂×C₄²	C₂³×C₅	1	4	yes	—
109	C₂×C₄²	C₂³×C₅	1	2	yes	—
110	C₄×C₈	C₂×C₄×C₅	1	32	yes	complemented
111	C₂²×C₅	#376	1	4	yes	—
112	C₂²×C₅	C₂×C₄×C₈	1	4	yes	—
113	C₂²×C₅	C₂×#97	1	4	yes	—
114	C₂²×C₅	#431	1	16	yes	—
115	C₂²×C₅	C₂²×C₄²	1	1	yes	characteristic
116	C₂²×C₅	C₂×#117	1	2	yes	—
117	C₂²×C₅	C₂³×C₈	1	2	yes	—
118	C₂²×C₅	C₂²×M₁₆	1	2	yes	—
119	C₄×C₅	—	2	16	no	—
120	C₄×C₅	#376	1	4	yes	—
121	C₄×C₅	C₂×C₄×C₈	1	8	yes	—
122	C₄×C₅	C₂×#97	1	8	yes	—
123	C₄×C₅	C₂×#117	1	4	yes	—
124	C₂⁴	C₂²×C₄×C₅	1	1	yes	characteristic
125	C₂²×C₄	—	2	8	no	—
126	C₂²×C₄	—	2	8	no	—
127	C₂²×C₄	C₄²×C₅	1	4	yes	—
128	C₂²×C₄	C₄²×C₅	1	4	yes	—
129	C₂²×C₄	C₂×C₅×C₈	1	16	yes	—
130	C₂²×C₄	C₂×C₅×C₈	1	8	yes	—
131	C₂²×C₄	C₅×M₁₆	1	8	yes	—
132	C₂²×C₄	C₂²×C₄×C₅	1	8	yes	—
133	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
134	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
135	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
136	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
137	C₂²×C₄	C₅×#41	1	8	yes	—
138	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
139	C₂×C₈	—	2	32	no	—
140	C₂×C₈	—	2	16	no	—
141	C₂×C₈	C₂²×C₄×C₅	1	16	yes	—
142	C₄²	—	2	8	no	—
143	C₄²	—	2	16	no	—
144	C₄²	—	2	8	no	—
145	C₄²	C₄²×C₅	1	8	yes	—
146	C₄²	C₂×C₅×C₈	1	32	yes	complemented
147	C₄²	C₂²×C₄×C₅	1	8	yes	—
148	C₂×C₅	#1744	1	8	yes	direct product factor
149	C₂×C₅	C₂×#376	1	1	yes	characteristic
150	C₂×C₅	C₂²×C₄×C₈	1	1	yes	characteristic
151	C₂×C₅	C₂²×#97	1	1	yes	characteristic
152	C₂×C₅	C₂×#431	1	4	yes	—
153	C₂³	C₂×C₄²×C₅	1	2	yes	—
154	C₂³	C₅×#117	1	4	yes	—
155	C₂³	C₂²×C₅×C₈	1	4	yes	—
156	C₂³	C₂×C₅×M₁₆	1	4	yes	—
157	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
158	C₂×C₄	—	2	32	no	—
159	C₂×C₄	—	2	8	no	—
160	C₂×C₄	—	2	8	no	—
161	C₂×C₄	C₄×C₅×C₈	1	16	yes	—
162	C₂×C₄	C₅×#97	1	16	yes	—
163	C₂×C₄	C₂×C₄²×C₅	1	2	yes	—
164	C₂×C₄	C₂×C₄²×C₅	1	4	yes	—
165	C₂×C₄	C₂×C₄²×C₅	1	4	yes	—
166	C₂×C₄	C₅×#117	1	4	yes	—
167	C₂×C₄	C₅×#117	1	16	yes	—
168	C₂×C₄	C₂²×C₅×C₈	1	16	yes	—
169	C₂×C₄	C₂²×C₅×C₈	1	8	yes	—
170	C₂×C₄	C₂×C₅×M₁₆	1	8	yes	—
171	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
172	C₂×C₄	C₂×C₅×#41	1	4	yes	—
173	C₈	—	2	16	no	—
174	C₅	C₂×#1744	1	1	yes	characteristic, direct product factor, Sylow
175	C₂²	C₅×#376	1	4	yes	—
176	C₂²	C₂×C₄×C₅×C₈	1	4	yes	—
177	C₂²	C₂×C₅×#97	1	4	yes	—
178	C₂²	C₅×#431	1	16	yes	—
179	C₂²	C₂²×C₄²×C₅	1	1	yes	characteristic
180	C₂²	C₂×C₅×#117	1	2	yes	—
181	C₂²	C₂³×C₅×C₈	1	2	yes	—
182	C₂²	C₂²×C₅×M₁₆	1	2	yes	—
183	C₄	—	2	16	no	—
184	C₄	C₅×#376	1	4	yes	—
185	C₄	C₂×C₄×C₅×C₈	1	8	yes	—
186	C₄	C₂×C₅×#97	1	8	yes	—
187	C₄	C₂×C₅×#117	1	4	yes	—
188	C₂	C₅×#1744	1	8	yes	direct product factor
189	C₂	C₂×C₅×#376	1	1	yes	characteristic
190	C₂	C₂²×C₄×C₅×C₈	1	1	yes	characteristic
191	C₂	C₂²×C₅×#97	1	1	yes	characteristic
192	C₂	C₂×C₅×#431	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄²×C₅	C₂²
commutator	C₂	C₂²×C₄×C₅×C₈
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle, prime radical	C₂⁴×C₅	C₂²×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1744	C₂	direct (N × H)
10. C₂×C₄×C₅×C₈	C₄	semidirect
15. C₂×#1744	C₅	direct (N × H)
25. C₂×C₄²×C₅	C₈	semidirect
33. C₄×C₅×C₈	C₂×C₄	semidirect
36. #1744	C₂×C₅	direct (N × H)
60. C₄²×C₅	C₂×C₈	semidirect
67. C₂×C₄×C₈	C₄×C₅	semidirect
102. C₂×C₄²	C₅×C₈	semidirect
110. C₄×C₈	C₂×C₄×C₅	semidirect
146. C₄²	C₂×C₅×C₈	semidirect
148. C₂×C₅	#1744	direct (N × H)
174. C₅	C₂×#1744	direct (N × H)
188. C₂	C₅×#1744	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×C₄×C₅×C₈	2
C₂×C₄³×C₅	2
C₅×#1744	2
C₂×C₅×#336	2
C₂×#1744	5

Subgroups — C2×C5×#1744 #11943081

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1744 #11943081