Subgroups — C₅×#33385 #11956105

Order 1280.

1114 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3584	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#504	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2916	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2143	C₂	1	6	yes	maximal, complemented
5	C₂×C₅×#359	C₂	1	6	yes	maximal, complemented
6	C₅×#585	—	2	1	no	—
7	C₅×#578	C₄	1	4	yes	—
8	C₅×#577	—	2	4	no	—
9	C₅×#576	C₄	1	4	yes	—
10	C₅×#575	—	2	3	no	—
11	C₂×C₅×#135	C₂²	1	6	yes	—
12	C₅×#568	—	2	1	no	—
13	C₂×C₅×#131	C₂²	1	1	yes	characteristic
14	C₅×#504	—	2	4	no	—
15	C₂×C₅×M₃₂	C₂²	1	3	yes	complemented
16	C₂²×C₅×C₁₆	C₂²	1	3	yes	complemented
17	C₅×#430	—	2	3	no	—
18	C₂×C₅×#108	C₂²	1	6	yes	complemented
19	C₅×#359	C₂²	1	4	yes	complemented
20	C₅×#359	C₂²	1	12	yes	complemented
21	#33385	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₅×2₋¹⁺⁴	—	2	4	no	—
23	C₅×2₋¹⁺⁴	—	4	1	no	—
24	C₅×H(5,𝔽₂)	—	2	4	no	—
25	C₅×H(5,𝔽₂)	—	4	3	no	—
26	C₂×C₅×#41	—	2	6	no	—
27	C₂×C₅×#41	—	4	4	no	—
28	C₂×C₅×#41	D₄	1	1	yes	characteristic
29	C₅×#137	—	2	12	no	—
30	C₅×AGL(1,ℤ/8)	—	2	12	no	—
31	C₅×#135	—	2	6	no	—
32	C₅×#135	—	2	6	no	—
33	C₅×#135	C₂×C₄	1	12	yes	—
34	C₅×#135	C₂³	1	4	yes	—
35	C₂×C₅×Q₁₆	C₂×C₄	1	6	yes	—
36	C₂×C₅×SD₁₆	—	2	6	no	—
37	C₂×C₅×D₈	C₂×C₄	1	6	yes	—
38	C₅×#131	—	4	4	no	—
39	C₅×#131	C₂×C₄	1	4	yes	—
40	C₅×#131	D₄	1	4	yes	—
41	C₂×C₅×M₁₆	—	2	6	no	—
42	C₂×C₅×M₁₆	D₄	1	3	yes	—
43	C₂²×C₅×C₈	C₂³	1	3	yes	—
44	C₅×M₃₂	—	2	6	no	—
45	C₂×C₅×C₁₆	—	2	6	no	—
46	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
47	C₂×C₅×C₁₆	C₂³	1	3	yes	—
48	C₅×#108	—	2	6	no	—
49	C₅×#108	C₂³	1	4	yes	—
50	#3584	C₂×C₅	1	1	yes	characteristic
51	C₂×#504	C₂×C₅	1	1	yes	characteristic, complemented
52	#2916	C₂×C₅	1	1	yes	characteristic, complemented
53	#2143	C₂×C₅	1	6	yes	complemented
54	C₂×#359	C₂×C₅	1	6	yes	complemented
55	C₅×#41	—	2	12	no	—
56	C₅×#41	—	2	4	no	—
57	C₅×#41	—	4	4	no	—
58	C₅×#41	—	4	12	no	—
59	C₅×#41	—	4	6	no	—
60	C₅×#41	—	4	6	no	—
61	C₅×#41	C₂²⋊C₄	1	4	yes	—
62	C₅×#41	D₈	1	4	yes	—
63	C₂×C₅×Q₈	—	2	6	no	—
64	C₂×C₅×Q₈	—	4	4	no	—
65	C₂×C₅×Q₈	SD₁₆	1	1	yes	characteristic
66	C₂×C₅×D₄	—	2	6	no	—
67	C₂×C₅×D₄	—	4	6	no	—
68	C₂×C₅×D₄	—	4	12	no	—
69	C₂×C₅×D₄	SD₁₆	1	3	yes	—
70	C₂²×C₄×C₅	—	4	6	no	—
71	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
72	C₅×Q₁₆	—	2	6	no	—
73	C₅×Q₁₆	C₂²×C₄	1	4	yes	—
74	C₅×SD₁₆	—	2	12	no	—
75	C₅×SD₁₆	—	2	4	no	—
76	C₅×D₈	—	2	6	no	—
77	C₅×D₈	C₂²×C₄	1	4	yes	—
78	C₅×M₁₆	—	2	4	no	—
79	C₅×M₁₆	—	4	6	no	—
80	C₅×M₁₆	C₂²⋊C₄	1	12	yes	—
81	C₂×C₅×C₈	—	4	6	no	—
82	C₂×C₅×C₈	C₂²×C₄	1	6	yes	—
83	C₂×C₅×C₈	C₂×D₄	1	3	yes	—
84	C₂×C₅×C₈	C₂×D₄	1	6	yes	—
85	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
86	C₅×C₁₆	—	2	1	no	—
87	C₅×C₁₆	—	2	3	no	—
88	#585	—	2	1	no	—
89	#578	C₄×C₅	1	4	yes	—
90	#577	—	2	4	no	—
91	#576	C₄×C₅	1	4	yes	—
92	#575	—	2	3	no	—
93	C₂×#135	C₂²×C₅	1	6	yes	—
94	#568	—	2	1	no	—
95	C₂×#131	C₂²×C₅	1	1	yes	characteristic
96	#504	—	2	4	no	—
97	C₂×M₃₂	C₂²×C₅	1	3	yes	complemented
98	C₂²×C₁₆	C₂²×C₅	1	3	yes	complemented
99	#430	—	2	3	no	—
100	C₂×#108	C₂²×C₅	1	6	yes	complemented
101	#359	C₂²×C₅	1	4	yes	complemented
102	#359	C₂²×C₅	1	12	yes	complemented
103	C₂³×C₅	—	4	6	no	—
104	C₂³×C₅	C₂×SD₁₆	1	3	yes	—
105	C₅×Q₈	—	2	4	no	—
106	C₅×Q₈	—	4	6	no	—
107	C₅×Q₈	#102	1	4	yes	—
108	C₅×D₄	—	2	4	no	—
109	C₅×D₄	—	4	4	no	—
110	C₅×D₄	—	4	12	no	—
111	C₅×D₄	—	4	6	no	—
112	C₅×D₄	#102	1	12	yes	—
113	C₂×C₄×C₅	—	4	6	no	—
114	C₂×C₄×C₅	—	4	4	no	—
115	C₂×C₄×C₅	—	4	12	no	—
116	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	6	yes	—
117	C₂×C₄×C₅	C₂×D₈	1	6	yes	—
118	C₂×C₄×C₅	C₂×SD₁₆	1	3	yes	—
119	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
120	C₅×C₈	—	4	4	no	—
121	C₅×C₈	C₂×C₂²⋊C₄	1	6	yes	—
122	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
123	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
124	2₋¹⁺⁴	—	2	4	no	—
125	2₋¹⁺⁴	—	4	1	no	—
126	H(5,𝔽₂)	—	2	4	no	—
127	H(5,𝔽₂)	—	4	3	no	—
128	C₂×#41	—	2	6	no	—
129	C₂×#41	—	4	4	no	—
130	C₂×#41	C₅×D₄	1	1	yes	characteristic
131	#137	—	2	12	no	—
132	AGL(1,ℤ/8)	—	2	12	no	—
133	#135	—	2	6	no	—
134	#135	—	2	6	no	—
135	#135	C₂×C₄×C₅	1	12	yes	—
136	#135	C₂³×C₅	1	4	yes	—
137	C₂×Q₁₆	C₂×C₄×C₅	1	6	yes	—
138	C₂×SD₁₆	—	2	6	no	—
139	C₂×D₈	C₂×C₄×C₅	1	6	yes	—
140	#131	—	4	4	no	—
141	#131	C₂×C₄×C₅	1	4	yes	—
142	#131	C₅×D₄	1	4	yes	—
143	C₂×M₁₆	—	2	6	no	—
144	C₂×M₁₆	C₅×D₄	1	3	yes	—
145	C₂²×C₈	C₂³×C₅	1	3	yes	—
146	M₃₂	—	2	6	no	—
147	C₂×C₁₆	—	2	6	no	—
148	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
149	C₂×C₁₆	C₂³×C₅	1	3	yes	—
150	#108	—	2	6	no	—
151	#108	C₂³×C₅	1	4	yes	—
152	C₂²×C₅	—	4	3	no	—
153	C₂²×C₅	—	4	4	no	—
154	C₂²×C₅	—	16	6	no	—
155	C₂²×C₅	C₂×#102	1	6	yes	—
156	C₂²×C₅	C₂²×SD₁₆	1	1	yes	characteristic
157	C₄×C₅	—	4	4	no	—
158	C₄×C₅	C₂×#102	1	6	yes	—
159	C₄×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
160	C₄×C₅	C₂²×D₈	1	1	yes	characteristic
161	#41	—	2	12	no	—
162	#41	—	2	4	no	—
163	#41	—	4	4	no	—
164	#41	—	4	12	no	—
165	#41	—	4	6	no	—
166	#41	—	4	6	no	—
167	#41	C₅×C₂²⋊C₄	1	4	yes	—
168	#41	C₅×D₈	1	4	yes	—
169	C₂×Q₈	—	2	6	no	—
170	C₂×Q₈	—	4	4	no	—
171	C₂×Q₈	C₅×SD₁₆	1	1	yes	characteristic
172	C₂×D₄	—	2	6	no	—
173	C₂×D₄	—	4	6	no	—
174	C₂×D₄	—	4	12	no	—
175	C₂×D₄	C₅×SD₁₆	1	3	yes	—
176	C₂²×C₄	—	4	6	no	—
177	C₂²×C₄	C₂×C₅×D₄	1	3	yes	—
178	Q₁₆	—	2	6	no	—
179	Q₁₆	C₂²×C₄×C₅	1	4	yes	—
180	SD₁₆	—	2	12	no	—
181	SD₁₆	—	2	4	no	—
182	D₈	—	2	6	no	—
183	D₈	C₂²×C₄×C₅	1	4	yes	—
184	M₁₆	—	2	4	no	—
185	M₁₆	—	4	6	no	—
186	M₁₆	C₅×C₂²⋊C₄	1	12	yes	—
187	C₂×C₈	—	4	6	no	—
188	C₂×C₈	C₂²×C₄×C₅	1	6	yes	—
189	C₂×C₈	C₂×C₅×D₄	1	3	yes	—
190	C₂×C₈	C₂×C₅×D₄	1	6	yes	—
191	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
192	C₁₆	—	2	1	no	—
193	C₁₆	—	2	3	no	—
194	C₂×C₅	—	2	1	no	—
195	C₂×C₅	—	2	6	no	—
196	C₂×C₅	—	8	4	no	—
197	C₂×C₅	C₂²×#102	1	1	yes	characteristic
198	C₂³	—	4	6	no	—
199	C₂³	C₂×C₅×SD₁₆	1	3	yes	—
200	Q₈	—	2	4	no	—
201	Q₈	—	4	6	no	—
202	Q₈	C₅×#102	1	4	yes	—
203	D₄	—	2	4	no	—
204	D₄	—	4	12	no	—
205	D₄	—	4	4	no	—
206	D₄	—	4	6	no	—
207	D₄	C₅×#102	1	12	yes	—
208	C₂×C₄	—	4	6	no	—
209	C₂×C₄	—	4	12	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	6	yes	—
212	C₂×C₄	C₂×C₅×D₈	1	6	yes	—
213	C₂×C₄	C₂×C₅×SD₁₆	1	3	yes	—
214	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
215	C₈	—	4	4	no	—
216	C₈	C₂×C₅×C₂²⋊C₄	1	6	yes	—
217	C₈	C₂³×C₄×C₅	1	1	yes	characteristic
218	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
219	C₅	#33385	1	1	yes	characteristic, direct product factor, Sylow
220	C₂²	—	4	3	no	—
221	C₂²	—	4	4	no	—
222	C₂²	—	16	6	no	—
223	C₂²	C₂×C₅×#102	1	6	yes	—
224	C₂²	C₂²×C₅×SD₁₆	1	1	yes	characteristic
225	C₄	—	4	4	no	—
226	C₄	C₂×C₅×#102	1	6	yes	—
227	C₄	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
228	C₄	C₂²×C₅×D₈	1	1	yes	characteristic
229	C₂	—	2	1	no	—
230	C₂	—	2	6	no	—
231	C₂	—	8	4	no	—
232	C₂	C₂²×C₅×#102	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	C₂²×D₈
commutator	C₈	C₂³×C₄×C₅
Frattini, cosocle	C₂×C₈	C₂⁴×C₅
socle	C₂×C₅	C₂²×#102
prime radical	C₅×#3584	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₅×#41 ◁ C₂×C₅×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#504	C₂	semidirect
3. C₅×#2916	C₂	semidirect
4. C₅×#2143	C₂	semidirect
5. C₂×C₅×#359	C₂	semidirect
15. C₂×C₅×M₃₂	C₂²	semidirect
16. C₂²×C₅×C₁₆	C₂²	semidirect
18. C₂×C₅×#108	C₂²	semidirect
19. C₅×#359	C₂²	semidirect
20. C₅×#359	C₂²	semidirect
21. #33385	C₅	direct (N × H)
51. C₂×#504	C₂×C₅	semidirect
52. #2916	C₂×C₅	semidirect
53. #2143	C₂×C₅	semidirect
54. C₂×#359	C₂×C₅	semidirect
97. C₂×M₃₂	C₂²×C₅	semidirect
98. C₂²×C₁₆	C₂²×C₅	semidirect
100. C₂×#108	C₂²×C₅	semidirect
101. #359	C₂²×C₅	semidirect
102. #359	C₂²×C₅	semidirect
219. C₅	#33385	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×#3584	2
C₂×C₅×#504	2
C₅×#2916	2
C₅×#2143	2
C₂×C₅×#359	2
#33385	5

Subgroups — C5×#33385 #11956105

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33385 #11956105