Subgroups — C₅×#33340 #11956060

Order 1280.

1114 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3573	C₂	1	1	yes	maximal, characteristic
2	C₅×#3407	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#504	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2117	C₂	1	6	yes	maximal, complemented
5	C₂×C₅×#350	C₂	1	6	yes	maximal, complemented
6	C₅×#585	C₄	1	1	yes	characteristic
7	C₅×#568	—	2	4	no	—
8	C₅×#568	C₄	1	6	yes	—
9	C₅×#568	C₄	1	1	yes	characteristic
10	C₂×C₅×#131	—	2	4	no	—
11	C₂×C₅×#131	C₂²	1	6	yes	—
12	C₂×C₅×#131	C₂²	1	1	yes	characteristic
13	C₅×#504	—	2	4	no	—
14	C₅×#504	—	2	4	no	—
15	C₂×C₅×M₃₂	C₂²	1	6	yes	complemented
16	C₂×C₅×M₃₂	C₂²	1	3	yes	complemented
17	C₂²×C₅×C₁₆	C₂²	1	3	yes	complemented
18	C₅×#350	C₂²	1	4	yes	complemented
19	C₅×#350	C₂²	1	12	yes	complemented
20	#33340	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₅×2₋¹⁺⁴	—	2	1	no	—
22	C₅×2₋¹⁺⁴	C₈	1	4	yes	—
23	C₅×H(5,𝔽₂)	—	2	3	no	—
24	C₅×H(5,𝔽₂)	C₈	1	4	yes	—
25	C₂×C₅×#41	—	2	4	no	—
26	C₂×C₅×#41	C₂×C₄	1	6	yes	—
27	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
28	C₅×#131	—	2	4	no	—
29	C₅×#131	—	2	12	no	—
30	C₅×#131	—	2	6	no	—
31	C₅×#131	—	2	6	no	—
32	C₅×#131	C₂×C₄	1	12	yes	—
33	C₅×#131	D₄	1	4	yes	—
34	C₅×#131	D₄	1	4	yes	—
35	C₅×#131	C₂³	1	4	yes	—
36	C₂×C₅×M₁₆	—	2	12	no	—
37	C₂×C₅×M₁₆	—	2	6	no	—
38	C₂×C₅×M₁₆	C₂×C₄	1	6	yes	—
39	C₂×C₅×M₁₆	C₂×C₄	1	3	yes	—
40	C₂²×C₅×C₈	—	2	6	no	—
41	C₂²×C₅×C₈	C₂³	1	3	yes	—
42	C₅×M₃₂	—	2	6	no	—
43	C₅×M₃₂	—	2	6	no	—
44	C₅×M₃₂	C₂³	1	4	yes	—
45	C₂×C₅×C₁₆	—	2	6	no	—
46	C₂×C₅×C₁₆	—	2	6	no	—
47	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
48	C₂×C₅×C₁₆	C₂³	1	3	yes	—
49	#3573	C₂×C₅	1	1	yes	characteristic
50	#3407	C₂×C₅	1	1	yes	characteristic, complemented
51	C₂×#504	C₂×C₅	1	1	yes	characteristic, complemented
52	#2117	C₂×C₅	1	6	yes	complemented
53	C₂×#350	C₂×C₅	1	6	yes	complemented
54	C₅×#41	—	2	4	no	—
55	C₅×#41	—	2	12	no	—
56	C₅×#41	—	2	6	no	—
57	C₅×#41	—	2	6	no	—
58	C₅×#41	C₂²⋊C₄	1	4	yes	—
59	C₅×#41	C₂×C₈	1	12	yes	—
60	C₅×#41	C₂×C₈	1	4	yes	—
61	C₅×#41	C₂²×C₄	1	4	yes	—
62	C₂×C₅×Q₈	—	2	4	no	—
63	C₂×C₅×Q₈	C₂×C₈	1	6	yes	—
64	C₂×C₅×Q₈	M₁₆	1	1	yes	characteristic
65	C₂×C₅×D₄	—	2	6	no	—
66	C₂×C₅×D₄	—	2	12	no	—
67	C₂×C₅×D₄	C₂×C₈	1	6	yes	—
68	C₂×C₅×D₄	M₁₆	1	3	yes	—
69	C₂²×C₄×C₅	—	2	6	no	—
70	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
71	C₅×M₁₆	—	2	6	no	—
72	C₅×M₁₆	—	2	12	no	—
73	C₅×M₁₆	—	2	4	no	—
74	C₅×M₁₆	C₂²⋊C₄	1	12	yes	—
75	C₅×M₁₆	C₂²×C₄	1	4	yes	—
76	C₂×C₅×C₈	—	2	6	no	—
77	C₂×C₅×C₈	—	2	12	no	—
78	C₂×C₅×C₈	—	2	4	no	—
79	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
80	C₂×C₅×C₈	C₂×D₄	1	6	yes	—
81	C₂×C₅×C₈	C₂×D₄	1	6	yes	—
82	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
83	C₅×C₁₆	—	2	1	no	—
84	C₅×C₁₆	—	2	3	no	—
85	C₅×C₁₆	—	2	4	no	—
86	#585	C₄×C₅	1	1	yes	characteristic
87	#568	—	2	4	no	—
88	#568	C₄×C₅	1	6	yes	—
89	#568	C₄×C₅	1	1	yes	characteristic
90	C₂×#131	—	2	4	no	—
91	C₂×#131	C₂²×C₅	1	6	yes	—
92	C₂×#131	C₂²×C₅	1	1	yes	characteristic
93	#504	—	2	4	no	—
94	#504	—	2	4	no	—
95	C₂×M₃₂	C₂²×C₅	1	6	yes	complemented
96	C₂×M₃₂	C₂²×C₅	1	3	yes	complemented
97	C₂²×C₁₆	C₂²×C₅	1	3	yes	complemented
98	#350	C₂²×C₅	1	4	yes	complemented
99	#350	C₂²×C₅	1	12	yes	complemented
100	C₂³×C₅	—	2	6	no	—
101	C₂³×C₅	C₂×M₁₆	1	3	yes	—
102	C₅×Q₈	—	2	6	no	—
103	C₅×Q₈	C₂²⋊C₈	1	4	yes	—
104	C₅×Q₈	C₂²×C₈	1	4	yes	—
105	C₅×D₄	—	2	4	no	—
106	C₅×D₄	—	2	12	no	—
107	C₅×D₄	—	2	6	no	—
108	C₅×D₄	C₂²⋊C₈	1	12	yes	—
109	C₅×D₄	C₂²×C₈	1	4	yes	—
110	C₂×C₄×C₅	—	2	6	no	—
111	C₂×C₄×C₅	—	2	4	no	—
112	C₂×C₄×C₅	—	2	12	no	—
113	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	6	yes	—
114	C₂×C₄×C₅	C₂²×C₈	1	6	yes	—
115	C₂×C₄×C₅	C₂×M₁₆	1	3	yes	—
116	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
117	C₅×C₈	—	2	4	no	—
118	C₅×C₈	C₂×C₂²⋊C₄	1	6	yes	—
119	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
120	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
121	2₋¹⁺⁴	—	2	1	no	—
122	2₋¹⁺⁴	C₅×C₈	1	4	yes	—
123	H(5,𝔽₂)	—	2	3	no	—
124	H(5,𝔽₂)	C₅×C₈	1	4	yes	—
125	C₂×#41	—	2	4	no	—
126	C₂×#41	C₂×C₄×C₅	1	6	yes	—
127	C₂×#41	C₂×C₄×C₅	1	1	yes	characteristic
128	#131	—	2	4	no	—
129	#131	—	2	12	no	—
130	#131	—	2	6	no	—
131	#131	—	2	6	no	—
132	#131	C₂×C₄×C₅	1	12	yes	—
133	#131	C₅×D₄	1	4	yes	—
134	#131	C₅×D₄	1	4	yes	—
135	#131	C₂³×C₅	1	4	yes	—
136	C₂×M₁₆	—	2	12	no	—
137	C₂×M₁₆	—	2	6	no	—
138	C₂×M₁₆	C₂×C₄×C₅	1	6	yes	—
139	C₂×M₁₆	C₂×C₄×C₅	1	3	yes	—
140	C₂²×C₈	—	2	6	no	—
141	C₂²×C₈	C₂³×C₅	1	3	yes	—
142	M₃₂	—	2	6	no	—
143	M₃₂	—	2	6	no	—
144	M₃₂	C₂³×C₅	1	4	yes	—
145	C₂×C₁₆	—	2	6	no	—
146	C₂×C₁₆	—	2	6	no	—
147	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
148	C₂×C₁₆	C₂³×C₅	1	3	yes	—
149	C₂²×C₅	—	2	4	no	—
150	C₂²×C₅	—	4	3	no	—
151	C₂²×C₅	—	8	6	no	—
152	C₂²×C₅	C₂×C₂²⋊C₈	1	6	yes	—
153	C₂²×C₅	C₂²×M₁₆	1	1	yes	characteristic
154	C₄×C₅	—	2	4	no	—
155	C₄×C₅	C₂×C₂²⋊C₈	1	6	yes	—
156	C₄×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
157	C₄×C₅	C₂³×C₈	1	1	yes	characteristic
158	#41	—	2	4	no	—
159	#41	—	2	12	no	—
160	#41	—	2	6	no	—
161	#41	—	2	6	no	—
162	#41	C₅×C₂²⋊C₄	1	4	yes	—
163	#41	C₂×C₅×C₈	1	12	yes	—
164	#41	C₂×C₅×C₈	1	4	yes	—
165	#41	C₂²×C₄×C₅	1	4	yes	—
166	C₂×Q₈	—	2	4	no	—
167	C₂×Q₈	C₂×C₅×C₈	1	6	yes	—
168	C₂×Q₈	C₅×M₁₆	1	1	yes	characteristic
169	C₂×D₄	—	2	6	no	—
170	C₂×D₄	—	2	12	no	—
171	C₂×D₄	C₂×C₅×C₈	1	6	yes	—
172	C₂×D₄	C₅×M₁₆	1	3	yes	—
173	C₂²×C₄	—	2	6	no	—
174	C₂²×C₄	C₂²×C₄×C₅	1	3	yes	—
175	M₁₆	—	2	6	no	—
176	M₁₆	—	2	12	no	—
177	M₁₆	—	2	4	no	—
178	M₁₆	C₅×C₂²⋊C₄	1	12	yes	—
179	M₁₆	C₂²×C₄×C₅	1	4	yes	—
180	C₂×C₈	—	2	6	no	—
181	C₂×C₈	—	2	12	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	C₂²×C₄×C₅	1	3	yes	—
184	C₂×C₈	C₂×C₅×D₄	1	6	yes	—
185	C₂×C₈	C₂×C₅×D₄	1	6	yes	—
186	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
187	C₁₆	—	2	1	no	—
188	C₁₆	—	2	4	no	—
189	C₁₆	—	2	3	no	—
190	C₂×C₅	—	2	1	no	—
191	C₂×C₅	—	2	6	no	—
192	C₂×C₅	—	4	4	no	—
193	C₂×C₅	C₂²×C₂²⋊C₈	1	1	yes	characteristic
194	C₂³	—	2	6	no	—
195	C₂³	C₂×C₅×M₁₆	1	3	yes	—
196	Q₈	—	2	6	no	—
197	Q₈	C₅×C₂²⋊C₈	1	4	yes	—
198	Q₈	C₂²×C₅×C₈	1	4	yes	—
199	D₄	—	2	12	no	—
200	D₄	—	2	4	no	—
201	D₄	—	2	6	no	—
202	D₄	C₅×C₂²⋊C₈	1	12	yes	—
203	D₄	C₂²×C₅×C₈	1	4	yes	—
204	C₂×C₄	—	2	6	no	—
205	C₂×C₄	—	2	12	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	6	yes	—
208	C₂×C₄	C₂²×C₅×C₈	1	6	yes	—
209	C₂×C₄	C₂×C₅×M₁₆	1	3	yes	—
210	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
211	C₈	—	2	4	no	—
212	C₈	C₂×C₅×C₂²⋊C₄	1	6	yes	—
213	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
214	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
215	C₅	#33340	1	1	yes	characteristic, direct product factor, Sylow
216	C₂²	—	2	4	no	—
217	C₂²	—	4	3	no	—
218	C₂²	—	8	6	no	—
219	C₂²	C₂×C₅×C₂²⋊C₈	1	6	yes	—
220	C₂²	C₂²×C₅×M₁₆	1	1	yes	characteristic
221	C₄	—	2	4	no	—
222	C₄	C₂×C₅×C₂²⋊C₈	1	6	yes	—
223	C₄	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
224	C₄	C₂³×C₅×C₈	1	1	yes	characteristic
225	C₂	—	2	1	no	—
226	C₂	—	2	6	no	—
227	C₂	—	4	4	no	—
228	C₂	C₂²×C₅×C₂²⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₅×C₈	C₂²×D₄
commutator	C₄	C₂³×C₅×C₈
Frattini, cosocle	C₂×C₈	C₂⁴×C₅
socle	C₂×C₅	C₂²×C₂²⋊C₈
prime radical	C₅×#585	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₅×C₈ ◁ C₂×C₅×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#3407	C₂	semidirect
3. C₂×C₅×#504	C₂	semidirect
4. C₅×#2117	C₂	semidirect
5. C₂×C₅×#350	C₂	semidirect
15. C₂×C₅×M₃₂	C₂²	semidirect
16. C₂×C₅×M₃₂	C₂²	semidirect
17. C₂²×C₅×C₁₆	C₂²	semidirect
18. C₅×#350	C₂²	semidirect
19. C₅×#350	C₂²	semidirect
20. #33340	C₅	direct (N × H)
50. #3407	C₂×C₅	semidirect
51. C₂×#504	C₂×C₅	semidirect
52. #2117	C₂×C₅	semidirect
53. C₂×#350	C₂×C₅	semidirect
95. C₂×M₃₂	C₂²×C₅	semidirect
96. C₂×M₃₂	C₂²×C₅	semidirect
97. C₂²×C₁₆	C₂²×C₅	semidirect
98. #350	C₂²×C₅	semidirect
99. #350	C₂²×C₅	semidirect
215. C₅	#33340	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#3573	2
C₅×#3407	2
C₂×C₅×#504	2
C₅×#2117	2
C₂×C₅×#350	2
#33340	5

Subgroups — C5×#33340 #11956060

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33340 #11956060