Subgroups — C₂×C₅×#3407 #11985343

Order 1280.

2074 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#131	C₂	1	1	yes	maximal, characteristic
2	C₅×#3407	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₅×#504	C₂	1	8	yes	maximal, complemented
4	C₂²×C₅×M₃₂	C₂	1	6	yes	maximal, complemented
5	C₂²×C₅×#41	C₄	1	1	yes	characteristic
6	C₂×C₅×#131	C₄	1	4	yes	—
7	C₂×C₅×#131	C₄	1	8	yes	—
8	C₂×C₅×#131	C₂²	1	4	yes	—
9	C₂×C₅×#131	C₂²	1	8	yes	—
10	C₂²×C₅×M₁₆	C₄	1	3	yes	—
11	C₂³×C₅×C₈	C₂²	1	3	yes	—
12	C₅×#504	C₂²	1	64	yes	complemented
13	C₂×C₅×M₃₂	C₂²	1	4	yes	—
14	C₂×C₅×M₃₂	C₂²	1	12	yes	complemented
15	C₂×C₅×M₃₂	C₂²	1	48	yes	complemented
16	C₂²×C₅×C₁₆	C₂²	1	12	yes	complemented
17	C₂×#3407	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂×C₅×#41	C₈	1	4	yes	—
19	C₂×C₅×#41	C₈	1	8	yes	—
20	C₂×C₅×#41	C₂×C₄	1	4	yes	—
21	C₂×C₅×#41	C₂×C₄	1	8	yes	—
22	C₂²×C₅×Q₈	C₈	1	1	yes	characteristic
23	C₂²×C₅×D₄	C₈	1	3	yes	—
24	C₂³×C₄×C₅	C₂×C₄	1	3	yes	—
25	C₅×#131	C₂×C₄	1	16	yes	—
26	C₅×#131	C₂×C₄	1	32	yes	—
27	C₅×#131	C₂³	1	16	yes	—
28	C₂×C₅×M₁₆	C₂×C₄	1	12	yes	—
29	C₂×C₅×M₁₆	C₂×C₄	1	24	yes	—
30	C₂²×C₅×C₈	C₂×C₄	1	3	yes	—
31	C₂²×C₅×C₈	C₂×C₄	1	6	yes	—
32	C₂²×C₅×C₈	C₂×C₄	1	12	yes	—
33	C₂²×C₅×C₈	C₂³	1	12	yes	—
34	C₂²×C₅×C₈	C₂³	1	6	yes	—
35	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₅×M₃₂	C₂³	1	16	yes	—
37	C₅×M₃₂	C₂³	1	48	yes	complemented
38	C₂×C₅×C₁₆	C₂³	1	8	yes	—
39	C₂×C₅×C₁₆	C₂³	1	48	yes	complemented
40	C₂²×#131	C₂×C₅	1	1	yes	characteristic
41	#3407	C₂×C₅	1	16	yes	direct product factor
42	C₂×#504	C₂×C₅	1	8	yes	complemented
43	C₂²×M₃₂	C₂×C₅	1	6	yes	complemented
44	C₂⁴×C₅	C₂×C₈	1	3	yes	—
45	C₅×#41	C₂×C₈	1	16	yes	—
46	C₅×#41	C₂×C₈	1	32	yes	—
47	C₅×#41	C₂²×C₄	1	16	yes	—
48	C₂×C₅×Q₈	C₂×C₈	1	8	yes	—
49	C₂×C₅×Q₈	C₂×C₈	1	4	yes	—
50	C₂×C₅×D₄	C₂×C₈	1	12	yes	—
51	C₂×C₅×D₄	C₂×C₈	1	24	yes	—
52	C₂²×C₄×C₅	C₂×C₈	1	3	yes	—
53	C₂²×C₄×C₅	C₂×C₈	1	6	yes	—
54	C₂²×C₄×C₅	C₂×C₈	1	12	yes	—
55	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅	C₂²×C₄	1	12	yes	—
57	C₂²×C₄×C₅	C₂²×C₄	1	6	yes	—
58	C₅×M₁₆	C₂²×C₄	1	48	yes	—
59	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
61	C₂×C₅×C₈	C₂²×C₄	1	12	yes	—
62	C₂×C₅×C₈	C₂²×C₄	1	6	yes	—
63	C₂×C₅×C₈	C₂²×C₄	1	24	yes	—
64	C₂×C₅×C₈	C₂²×C₄	1	12	yes	—
65	C₂×C₅×C₈	C₂⁴	1	2	yes	—
66	C₂×C₅×C₈	C₂⁴	1	12	yes	—
67	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
68	C₅×C₁₆	C₂⁴	1	16	yes	—
69	C₂²×#41	C₄×C₅	1	1	yes	characteristic
70	C₂×#131	C₄×C₅	1	4	yes	—
71	C₂×#131	C₄×C₅	1	8	yes	—
72	C₂×#131	C₂²×C₅	1	4	yes	—
73	C₂×#131	C₂²×C₅	1	8	yes	—
74	C₂²×M₁₆	C₄×C₅	1	3	yes	—
75	C₂³×C₈	C₂²×C₅	1	3	yes	—
76	#504	C₂²×C₅	1	64	yes	complemented
77	C₂×M₃₂	C₂²×C₅	1	4	yes	—
78	C₂×M₃₂	C₂²×C₅	1	12	yes	complemented
79	C₂×M₃₂	C₂²×C₅	1	48	yes	complemented
80	C₂²×C₁₆	C₂²×C₅	1	12	yes	complemented
81	C₂³×C₅	—	4	6	no	—
82	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
83	C₂³×C₅	C₂²×C₈	1	6	yes	—
84	C₂³×C₅	C₂²×C₈	1	12	yes	—
85	C₅×Q₈	C₂²×C₈	1	16	yes	—
86	C₅×D₄	C₂²×C₈	1	48	yes	—
87	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
88	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₅	C₂²×C₈	1	12	yes	—
90	C₂×C₄×C₅	C₂²×C₈	1	6	yes	—
91	C₂×C₄×C₅	C₂²×C₈	1	24	yes	—
92	C₂×C₄×C₅	C₂²×C₈	1	12	yes	—
93	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
94	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₅	C₂³×C₄	1	12	yes	—
96	C₅×C₈	C₂³×C₄	1	2	yes	—
97	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
98	C₅×C₈	C₂³×C₄	1	12	yes	—
99	C₅×C₈	C₂⁵	1	1	yes	characteristic
100	C₂×#41	C₅×C₈	1	4	yes	—
101	C₂×#41	C₅×C₈	1	8	yes	—
102	C₂×#41	C₂×C₄×C₅	1	4	yes	—
103	C₂×#41	C₂×C₄×C₅	1	8	yes	—
104	C₂²×Q₈	C₅×C₈	1	1	yes	characteristic
105	C₂²×D₄	C₅×C₈	1	3	yes	—
106	C₂³×C₄	C₂×C₄×C₅	1	3	yes	—
107	#131	C₂×C₄×C₅	1	16	yes	—
108	#131	C₂×C₄×C₅	1	32	yes	—
109	#131	C₂³×C₅	1	16	yes	—
110	C₂×M₁₆	C₂×C₄×C₅	1	12	yes	—
111	C₂×M₁₆	C₂×C₄×C₅	1	24	yes	—
112	C₂²×C₈	C₂×C₄×C₅	1	3	yes	—
113	C₂²×C₈	C₂×C₄×C₅	1	6	yes	—
114	C₂²×C₈	C₂×C₄×C₅	1	12	yes	—
115	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
116	C₂²×C₈	C₂³×C₅	1	6	yes	—
117	C₂²×C₈	C₂³×C₅	1	12	yes	—
118	M₃₂	C₂³×C₅	1	16	yes	—
119	M₃₂	C₂³×C₅	1	48	yes	complemented
120	C₂×C₁₆	C₂³×C₅	1	8	yes	—
121	C₂×C₁₆	C₂³×C₅	1	48	yes	complemented
122	C₂²×C₅	—	2	2	no	—
123	C₂²×C₅	—	2	12	no	—
124	C₂²×C₅	—	4	12	no	—
125	C₂²×C₅	C₂³×C₈	1	2	yes	—
126	C₂²×C₅	C₂³×C₈	1	12	yes	—
127	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
128	C₄×C₅	C₂³×C₈	1	2	yes	—
129	C₄×C₅	C₂³×C₈	1	1	yes	characteristic
130	C₄×C₅	C₂³×C₈	1	12	yes	—
131	C₄×C₅	C₂⁴×C₄	1	1	yes	characteristic
132	C₂⁴	C₂×C₅×C₈	1	3	yes	—
133	#41	C₂×C₅×C₈	1	16	yes	—
134	#41	C₂×C₅×C₈	1	32	yes	—
135	#41	C₂²×C₄×C₅	1	16	yes	—
136	C₂×Q₈	C₂×C₅×C₈	1	8	yes	—
137	C₂×Q₈	C₂×C₅×C₈	1	4	yes	—
138	C₂×D₄	C₂×C₅×C₈	1	12	yes	—
139	C₂×D₄	C₂×C₅×C₈	1	24	yes	—
140	C₂²×C₄	C₂×C₅×C₈	1	3	yes	—
141	C₂²×C₄	C₂×C₅×C₈	1	6	yes	—
142	C₂²×C₄	C₂×C₅×C₈	1	12	yes	—
143	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
144	C₂²×C₄	C₂²×C₄×C₅	1	6	yes	—
145	C₂²×C₄	C₂²×C₄×C₅	1	12	yes	—
146	M₁₆	C₂²×C₄×C₅	1	48	yes	—
147	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
148	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
149	C₂×C₈	C₂²×C₄×C₅	1	6	yes	—
150	C₂×C₈	C₂²×C₄×C₅	1	12	yes	—
151	C₂×C₈	C₂²×C₄×C₅	1	12	yes	—
152	C₂×C₈	C₂²×C₄×C₅	1	24	yes	—
153	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
154	C₂×C₈	C₂⁴×C₅	1	12	yes	—
155	C₂×C₈	C₂⁴×C₅	1	2	yes	—
156	C₁₆	C₂⁴×C₅	1	16	yes	—
157	C₂×C₅	—	2	2	no	—
158	C₂×C₅	—	2	12	no	—
159	C₂×C₅	#3407	1	2	yes	direct product factor
160	C₂×C₅	C₂⁴×C₈	1	1	yes	characteristic
161	C₂³	—	4	6	no	—
162	C₂³	C₂²×C₅×C₈	1	1	yes	characteristic
163	C₂³	C₂²×C₅×C₈	1	6	yes	—
164	C₂³	C₂²×C₅×C₈	1	12	yes	—
165	Q₈	C₂²×C₅×C₈	1	16	yes	—
166	D₄	C₂²×C₅×C₈	1	48	yes	—
167	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
168	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
169	C₂×C₄	C₂²×C₅×C₈	1	12	yes	—
170	C₂×C₄	C₂²×C₅×C₈	1	6	yes	—
171	C₂×C₄	C₂²×C₅×C₈	1	24	yes	—
172	C₂×C₄	C₂²×C₅×C₈	1	12	yes	—
173	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
174	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
175	C₂×C₄	C₂³×C₄×C₅	1	12	yes	—
176	C₈	C₂³×C₄×C₅	1	1	yes	characteristic
177	C₈	C₂³×C₄×C₅	1	2	yes	—
178	C₈	C₂³×C₄×C₅	1	12	yes	—
179	C₈	C₂⁵×C₅	1	1	yes	characteristic
180	C₅	C₂×#3407	1	1	yes	characteristic, direct product factor, Sylow
181	C₂²	—	2	2	no	—
182	C₂²	—	2	12	no	—
183	C₂²	—	4	12	no	—
184	C₂²	C₂³×C₅×C₈	1	2	yes	—
185	C₂²	C₂³×C₅×C₈	1	12	yes	—
186	C₂²	C₂³×C₅×C₈	1	1	yes	characteristic
187	C₄	C₂³×C₅×C₈	1	1	yes	characteristic
188	C₄	C₂³×C₅×C₈	1	2	yes	—
189	C₄	C₂³×C₅×C₈	1	12	yes	—
190	C₄	C₂⁴×C₄×C₅	1	1	yes	characteristic
191	C₂	—	2	2	no	—
192	C₂	—	2	12	no	—
193	C₂	C₅×#3407	1	2	yes	direct product factor
194	C₂	C₂⁴×C₅×C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅×C₈	C₂⁴
commutator	C₂	C₂⁴×C₅×C₈
Frattini, cosocle	C₈	C₂⁵×C₅
socle	C₂²×C₅	C₂³×C₈
prime radical	C₂²×C₅×#41	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#3407	C₂	direct (N × H)
3. C₂×C₅×#504	C₂	semidirect
4. C₂²×C₅×M₃₂	C₂	semidirect
12. C₅×#504	C₂²	semidirect
14. C₂×C₅×M₃₂	C₂²	semidirect
15. C₂×C₅×M₃₂	C₂²	semidirect
16. C₂²×C₅×C₁₆	C₂²	semidirect
17. C₂×#3407	C₅	direct (N × H)
37. C₅×M₃₂	C₂³	semidirect
39. C₂×C₅×C₁₆	C₂³	semidirect
41. #3407	C₂×C₅	direct (N × H)
42. C₂×#504	C₂×C₅	semidirect
43. C₂²×M₃₂	C₂×C₅	semidirect
76. #504	C₂²×C₅	semidirect
78. C₂×M₃₂	C₂²×C₅	semidirect
79. C₂×M₃₂	C₂²×C₅	semidirect
80. C₂²×C₁₆	C₂²×C₅	semidirect
119. M₃₂	C₂³×C₅	semidirect
121. C₂×C₁₆	C₂³×C₅	semidirect
159. C₂×C₅	#3407	direct (N × H)
180. C₅	C₂×#3407	direct (N × H)
193. C₂	C₅×#3407	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×#131	2
C₅×#3407	2
C₂×C₅×#504	2
C₂²×C₅×M₃₂	2
C₂×#3407	5

Subgroups — C2×C5×#3407 #11985343

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#3407 #11985343