Subgroups — C₅×#33489 #11956209

Order 1280.

614 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₈×D₄	C₂	1	1	yes	maximal, characteristic
2	C₅×#2169	C₂	1	8	yes	maximal, complemented
3	C₅×#2162	C₂	1	2	yes	maximal, complemented
4	C₅×#2112	C₂	1	4	yes	maximal, complemented
5	C₂³×C₅×C₈	C₂²	1	2	yes	—
6	C₂×C₄×C₅×D₄	C₄	1	1	yes	characteristic
7	C₂×C₅×M₃₂	C₂²	1	8	yes	complemented
8	C₅×C₈×D₄	C₄	1	4	yes	—
9	C₅×C₈×D₄	C₂²	1	4	yes	—
10	C₂×C₅×C₄⋊C₈	C₄	1	1	yes	characteristic
11	C₂×C₅×C₂²⋊C₈	C₄	1	2	yes	—
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₅×C₄⋊C₁₆	C₂²	1	4	yes	complemented
14	C₅×C₂²⋊C₁₆	C₂²	1	4	yes	—
15	C₅×C₂²⋊C₁₆	C₂²	1	8	yes	complemented
16	C₄×C₅×C₁₆	C₂²	1	4	yes	complemented
17	#33489	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂²×C₅×D₄	C₈	1	1	yes	characteristic
19	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
20	C₂²×C₅×C₈	—	4	4	no	—
21	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
23	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
24	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
25	C₂²×C₅×C₈	C₂³	1	4	yes	—
26	C₄×C₅×D₄	C₈	1	4	yes	—
27	C₄×C₅×D₄	C₂×C₄	1	4	yes	—
28	C₂×C₅×C₄⋊C₄	C₈	1	1	yes	characteristic
29	C₂×C₅×C₂²⋊C₄	C₈	1	2	yes	—
30	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
31	C₅×M₃₂	—	4	8	no	—
32	C₂×C₅×C₁₆	C₂³	1	8	yes	—
33	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	—
34	C₅×C₂²⋊C₈	C₂×C₄	1	8	yes	—
35	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
36	C₄×C₅×C₈	C₂³	1	2	yes	—
37	C₂×C₈×D₄	C₂×C₅	1	1	yes	characteristic
38	#2169	C₂×C₅	1	8	yes	complemented
39	#2162	C₂×C₅	1	2	yes	complemented
40	#2112	C₂×C₅	1	4	yes	complemented
41	C₂⁴×C₅	C₂×C₈	1	2	yes	—
42	C₂×C₅×D₄	—	2	4	no	—
43	C₂×C₅×D₄	C₂×C₈	1	4	yes	—
44	C₂²×C₄×C₅	—	4	2	no	—
45	C₂²×C₄×C₅	—	4	2	no	—
46	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
49	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
52	C₂×C₅×C₈	—	2	2	no	—
53	C₂×C₅×C₈	—	2	8	no	—
54	C₂×C₅×C₈	—	4	4	no	—
55	C₂×C₅×C₈	—	4	2	no	—
56	C₂×C₅×C₈	—	4	1	no	—
57	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
59	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
60	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
61	C₅×C₄⋊C₄	C₂×C₈	1	4	yes	—
62	C₅×C₂²⋊C₄	C₂×C₈	1	8	yes	—
63	C₄²×C₅	C₂×C₈	1	2	yes	—
64	C₄²×C₅	C₂²×C₄	1	2	yes	—
65	C₅×C₁₆	—	2	8	no	—
66	C₂³×C₈	C₂²×C₅	1	2	yes	—
67	C₂×C₄×D₄	C₄×C₅	1	1	yes	characteristic
68	C₂×M₃₂	C₂²×C₅	1	8	yes	complemented
69	C₈×D₄	C₄×C₅	1	4	yes	—
70	C₈×D₄	C₂²×C₅	1	4	yes	—
71	C₂×C₄⋊C₈	C₄×C₅	1	1	yes	characteristic
72	C₂×C₂²⋊C₈	C₄×C₅	1	2	yes	—
73	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
74	C₄⋊C₁₆	C₂²×C₅	1	4	yes	complemented
75	C₂²⋊C₁₆	C₂²×C₅	1	4	yes	—
76	C₂²⋊C₁₆	C₂²×C₅	1	8	yes	complemented
77	C₄×C₁₆	C₂²×C₅	1	4	yes	complemented
78	C₂³×C₅	—	2	4	no	—
79	C₂³×C₅	—	4	2	no	—
80	C₂³×C₅	—	4	2	no	—
81	C₂³×C₅	C₂²×C₈	1	4	yes	—
82	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
83	C₅×D₄	—	2	8	no	—
84	C₂×C₄×C₅	—	2	1	no	—
85	C₂×C₄×C₅	—	2	4	no	—
86	C₂×C₄×C₅	—	2	1	no	—
87	C₂×C₄×C₅	—	2	4	no	—
88	C₂×C₄×C₅	—	4	4	no	—
89	C₂×C₄×C₅	—	4	2	no	—
90	C₂×C₄×C₅	—	4	1	no	—
91	C₂×C₄×C₅	M₃₂	1	4	yes	complemented
92	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
93	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
94	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
95	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
96	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
97	C₅×C₈	—	2	1	no	—
98	C₅×C₈	—	2	4	no	—
99	C₅×C₈	—	2	2	no	—
100	C₅×C₈	H(5,𝔽₂)	1	2	yes	—
101	C₂²×D₄	C₅×C₈	1	1	yes	characteristic
102	C₂³×C₄	C₂×C₄×C₅	1	2	yes	—
103	C₂²×C₈	—	4	4	no	—
104	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
105	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
106	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
107	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
108	C₂²×C₈	C₂³×C₅	1	4	yes	—
109	C₄×D₄	C₅×C₈	1	4	yes	—
110	C₄×D₄	C₂×C₄×C₅	1	4	yes	—
111	C₂×C₄⋊C₄	C₅×C₈	1	1	yes	characteristic
112	C₂×C₂²⋊C₄	C₅×C₈	1	2	yes	—
113	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
114	M₃₂	—	4	8	no	—
115	C₂×C₁₆	C₂³×C₅	1	8	yes	—
116	C₄⋊C₈	C₂×C₄×C₅	1	4	yes	—
117	C₂²⋊C₈	C₂×C₄×C₅	1	8	yes	—
118	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
119	C₄×C₈	C₂³×C₅	1	2	yes	—
120	C₂²×C₅	—	2	1	no	—
121	C₂²×C₅	—	2	1	no	—
122	C₂²×C₅	—	2	4	no	—
123	C₂²×C₅	—	2	4	no	—
124	C₂²×C₅	—	2	4	no	—
125	C₂²×C₅	—	4	8	no	—
126	C₂²×C₅	C₂×M₃₂	1	2	yes	—
127	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
128	C₄×C₅	—	2	2	no	—
129	C₄×C₅	—	2	1	no	—
130	C₄×C₅	—	2	4	no	—
131	C₄×C₅	C₂×M₃₂	1	4	yes	—
132	C₄×C₅	#518	1	1	yes	characteristic
133	C₄×C₅	#568	1	1	yes	characteristic
134	C₂⁴	C₂×C₅×C₈	1	2	yes	—
135	C₂×D₄	—	2	4	no	—
136	C₂×D₄	C₂×C₅×C₈	1	4	yes	—
137	C₂²×C₄	—	4	2	no	—
138	C₂²×C₄	—	4	2	no	—
139	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
140	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
141	C₂²×C₄	C₂×C₅×C₈	1	4	yes	—
142	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
143	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
144	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
145	C₂×C₈	—	2	8	no	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	4	4	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	4	1	no	—
150	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
151	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
152	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
153	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
154	C₄⋊C₄	C₂×C₅×C₈	1	4	yes	—
155	C₂²⋊C₄	C₂×C₅×C₈	1	8	yes	—
156	C₄²	C₂×C₅×C₈	1	2	yes	—
157	C₄²	C₂²×C₄×C₅	1	2	yes	—
158	C₁₆	—	2	8	no	—
159	C₂×C₅	—	2	2	no	—
160	C₂×C₅	—	4	4	no	—
161	C₂×C₅	#2972	1	1	yes	characteristic
162	C₂×C₅	C₂²×M₃₂	1	1	yes	characteristic
163	C₂×C₅	#3407	1	1	yes	characteristic
164	C₂³	—	2	4	no	—
165	C₂³	—	4	2	no	—
166	C₂³	—	4	2	no	—
167	C₂³	C₂²×C₅×C₈	1	4	yes	—
168	C₂³	C₂²×C₅×C₈	1	1	yes	characteristic
169	D₄	—	2	8	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	4	4	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	4	1	no	—
177	C₂×C₄	C₅×M₃₂	1	4	yes	complemented
178	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
179	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
180	C₂×C₄	C₂²×C₅×C₈	1	4	yes	—
181	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
182	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
183	C₈	—	2	1	no	—
184	C₈	—	2	4	no	—
185	C₈	—	2	2	no	—
186	C₈	C₅×H(5,𝔽₂)	1	2	yes	—
187	C₅	#33489	1	1	yes	characteristic, direct product factor, Sylow
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	—	2	4	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	2	4	no	—
193	C₂²	—	4	8	no	—
194	C₂²	C₂×C₅×M₃₂	1	2	yes	—
195	C₂²	C₂³×C₅×C₈	1	1	yes	characteristic
196	C₄	—	2	2	no	—
197	C₄	—	2	1	no	—
198	C₄	—	2	4	no	—
199	C₄	C₂×C₅×M₃₂	1	4	yes	—
200	C₄	C₅×#518	1	1	yes	characteristic
201	C₄	C₅×#568	1	1	yes	characteristic
202	C₂	—	2	2	no	—
203	C₂	—	4	4	no	—
204	C₂	C₅×#2972	1	1	yes	characteristic
205	C₂	C₂²×C₅×M₃₂	1	1	yes	characteristic
206	C₂	C₅×#3407	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅×C₈	C₂⁴
commutator	C₂²	C₂³×C₅×C₈
Frattini, cosocle	C₂×C₈	C₂⁴×C₅
socle	C₂²×C₅	C₂³×C₈
prime radical	C₂²×C₅×D₄	C₈
LCS	G ▷ C₂² ▷ 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×#2169	C₂	semidirect
3. C₅×#2162	C₂	semidirect
4. C₅×#2112	C₂	semidirect
7. C₂×C₅×M₃₂	C₂²	semidirect
13. C₅×C₄⋊C₁₆	C₂²	semidirect
15. C₅×C₂²⋊C₁₆	C₂²	semidirect
16. C₄×C₅×C₁₆	C₂²	semidirect
17. #33489	C₅	direct (N × H)
38. #2169	C₂×C₅	semidirect
39. #2162	C₂×C₅	semidirect
40. #2112	C₂×C₅	semidirect
68. C₂×M₃₂	C₂²×C₅	semidirect
74. C₄⋊C₁₆	C₂²×C₅	semidirect
76. C₂²⋊C₁₆	C₂²×C₅	semidirect
77. C₄×C₁₆	C₂²×C₅	semidirect
91. C₂×C₄×C₅	M₃₂	semidirect
177. C₂×C₄	C₅×M₃₂	semidirect
187. C₅	#33489	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×C₈×D₄	2
C₅×#2169	2
C₅×#2162	2
C₅×#2112	2
#33489	5

Subgroups — C5×#33489 #11956209

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33489 #11956209