Subgroups — C₅×#66103 #13106923

Order 1600.

598 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#4149	C₂	1	2	yes	maximal, complemented
2	C₅²×#135	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4095	C₂	1	2	yes	maximal, complemented
4	C₅×#4065	C₂	1	2	yes	maximal, complemented
5	C₅×#4060	C₂	1	4	yes	maximal, complemented
6	C₅×D₅×D₈	C₂	1	2	yes	maximal, complemented
7	C₂×C₅×D₄₀	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#4047	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅²×#41	C₂²	1	2	yes	complemented
10	C₅×#713	C₂²	1	4	yes	complemented
11	C₅×D₄×D₅	—	2	4	no	—
12	C₅×D₄×D₅	C₂²	1	4	yes	complemented
13	C₅×#709	—	2	2	no	—
14	C₅×#709	C₂²	1	1	yes	characteristic
15	C₂×C₅×D₂₀	—	2	2	no	—
16	C₂×C₅×D₂₀	C₂²	1	2	yes	complemented
17	C₅²×Q₁₆	C₂²	1	1	yes	characteristic, complemented
18	C₅²×SD₁₆	C₂²	1	2	yes	complemented
19	C₅²×D₈	C₂²	1	1	yes	characteristic, complemented
20	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic, complemented
21	C₅×#688	C₂²	1	4	yes	complemented
22	C₅×#686	C₂²	1	4	yes	complemented
23	C₅×#681	C₂²	1	1	yes	characteristic, complemented
24	C₅×D₄₀	C₂²	1	2	yes	complemented
25	C₅×D₄₀	C₂²	1	2	yes	complemented
26	C₅×#676	C₂²	1	2	yes	complemented
27	C₅×C₈×D₅	C₂²	1	2	yes	complemented
28	C₅×#576	—	5	1	no	maximal
29	#66103	C₅	1	1	yes	maximal, characteristic, direct product factor
30	C₂²×C₅×D₅	—	2	2	no	—
31	C₂²×C₅×D₅	—	2	4	no	—
32	C₅²×Q₈	C₂³	1	2	yes	—
33	C₅²×D₄	—	2	2	no	—
34	C₅²×D₄	C₂³	1	2	yes	—
35	C₂×C₄×C₅²	—	2	2	no	—
36	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
37	C₅×C₂²⋊D₅	—	2	4	no	—
38	C₅×C₂²⋊D₅	D₄	1	2	yes	—
39	C₅×D₂₀	—	2	4	no	—
40	C₅×D₂₀	—	2	2	no	—
41	C₅×D₂₀	D₄	1	1	yes	characteristic
42	C₅×D₂₀	C₂³	1	4	yes	—
43	C₄×C₅×D₅	—	2	4	no	—
44	C₄×C₅×D₅	C₂³	1	2	yes	—
45	C₅×Dic₁₀	D₄	1	1	yes	characteristic
46	C₅²×C₈	C₂³	1	1	yes	characteristic
47	C₅²×C₈	C₂³	1	1	yes	characteristic
48	C₅×#168	C₂³	1	2	yes	—
49	C₅×H(5,𝔽₂)	—	5	2	no	—
50	#4149	C₂×C₅	1	2	yes	complemented
51	C₅×AGL(1,ℤ/8)	—	5	2	no	—
52	C₅×AGL(1,ℤ/8)	—	5	4	no	—
53	C₅×#135	—	2	2	no	—
54	C₅×#135	—	5	2	no	—
55	C₅×#135	D₅	1	1	yes	characteristic, complemented
56	C₅×#135	C₂×C₅	1	1	yes	characteristic, complemented
57	C₂×C₅×D₈	—	5	2	no	—
58	C₂×C₅×D₈	—	5	1	no	—
59	C₅×#131	—	5	1	no	—
60	#4095	C₂×C₅	1	2	yes	complemented
61	#4065	C₂×C₅	1	2	yes	complemented
62	#4060	C₂×C₅	1	4	yes	complemented
63	D₅×D₈	C₂×C₅	1	2	yes	complemented
64	C₂×D₄₀	C₂×C₅	1	1	yes	characteristic, complemented
65	#4047	C₂×C₅	1	1	yes	characteristic, complemented
66	C₂²×C₅²	—	2	2	no	—
67	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
68	C₂×C₅×D₅	—	2	4	no	—
69	C₂×C₅×D₅	—	8	2	no	—
70	C₂×C₅×D₅	—	8	4	no	—
71	C₂×C₅×D₅	C₂×D₄	1	2	yes	—
72	C₄×C₅²	—	2	2	no	—
73	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
74	C₄×C₅²	C₂⁴	1	1	yes	characteristic
75	C₅×Dic₅	C₂×D₄	1	2	yes	—
76	C₅×#41	—	2	4	no	—
77	C₅×#41	—	5	1	no	—
78	C₅×#41	—	5	4	no	—
79	C₅×#41	—	10	2	no	—
80	C₅×#41	C₂×D₅	1	2	yes	complemented
81	C₅×#41	C₂²×C₅	1	2	yes	complemented
82	C₂×C₅×D₄	—	5	4	no	—
83	C₂×C₅×D₄	—	5	2	no	—
84	C₂×C₅×D₄	—	10	2	no	—
85	C₂×C₅×D₄	—	10	4	no	—
86	#713	C₂²×C₅	1	4	yes	complemented
87	D₄×D₅	—	2	4	no	—
88	D₄×D₅	C₂²×C₅	1	4	yes	complemented
89	#709	—	2	2	no	—
90	#709	C₂²×C₅	1	1	yes	characteristic
91	C₂×D₂₀	—	2	2	no	—
92	C₂×D₂₀	C₂²×C₅	1	2	yes	complemented
93	C₅×Q₁₆	—	2	2	no	—
94	C₅×Q₁₆	C₂×D₅	1	1	yes	characteristic, complemented
95	C₅×Q₁₆	C₂²×C₅	1	1	yes	characteristic, complemented
96	C₅×SD₁₆	—	2	4	no	—
97	C₅×SD₁₆	—	5	4	no	—
98	C₅×SD₁₆	C₂×D₅	1	2	yes	complemented
99	C₅×SD₁₆	C₂²×C₅	1	2	yes	complemented
100	C₅×D₈	—	2	2	no	—
101	C₅×D₈	—	5	4	no	—
102	C₅×D₈	—	5	2	no	—
103	C₅×D₈	—	5	2	no	—
104	C₅×D₈	C₂×D₅	1	1	yes	characteristic, complemented
105	C₅×D₈	C₂²×C₅	1	1	yes	characteristic, complemented
106	C₅×M₁₆	—	5	1	no	—
107	C₅×M₁₆	—	5	2	no	—
108	C₂×C₅×C₈	—	2	2	no	—
109	C₂×C₅×C₈	—	5	2	no	—
110	C₂×C₅×C₈	C₂×D₅	1	1	yes	characteristic, complemented
111	C₂×C₅×C₈	C₂²×C₅	1	1	yes	characteristic, complemented
112	#688	C₂²×C₅	1	4	yes	complemented
113	#686	C₂²×C₅	1	4	yes	complemented
114	#681	C₂²×C₅	1	1	yes	characteristic, complemented
115	D₄₀	C₂²×C₅	1	2	yes	complemented
116	D₄₀	C₂²×C₅	1	2	yes	complemented
117	#676	C₂²×C₅	1	2	yes	complemented
118	C₈×D₅	C₂²×C₅	1	2	yes	complemented
119	#576	—	5	1	no	Sylow
120	C₂×C₅²	—	2	1	no	—
121	C₂×C₅²	—	4	2	no	—
122	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
123	C₅×D₅	—	2	2	no	—
124	C₅×D₅	—	4	4	no	—
125	C₂³×C₅	—	10	4	no	—
126	C₂³×C₅	—	10	2	no	—
127	C₂²×D₅	—	2	2	no	—
128	C₂²×D₅	—	2	4	no	—
129	C₅×Q₈	—	2	4	no	—
130	C₅×Q₈	—	5	1	no	—
131	C₅×Q₈	C₂²×D₅	1	2	yes	—
132	C₅×Q₈	C₂³×C₅	1	2	yes	—
133	C₅×D₄	—	2	2	no	—
134	C₅×D₄	—	2	2	no	—
135	C₅×D₄	—	2	4	no	—
136	C₅×D₄	—	4	4	no	—
137	C₅×D₄	—	5	2	no	—
138	C₅×D₄	—	5	1	no	—
139	C₅×D₄	—	5	4	no	—
140	C₅×D₄	—	10	4	no	—
141	C₅×D₄	—	10	4	no	—
142	C₅×D₄	—	10	2	no	—
143	C₅×D₄	C₂²×D₅	1	2	yes	—
144	C₅×D₄	C₂³×C₅	1	2	yes	—
145	C₂×C₄×C₅	—	2	2	no	—
146	C₂×C₄×C₅	—	2	2	no	—
147	C₂×C₄×C₅	—	2	2	no	—
148	C₂×C₄×C₅	—	4	4	no	—
149	C₂×C₄×C₅	—	5	2	no	—
150	C₂×C₄×C₅	—	10	4	no	—
151	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
153	C₂²⋊D₅	—	2	4	no	—
154	C₂²⋊D₅	C₅×D₄	1	2	yes	—
155	D₂₀	—	2	4	no	—
156	D₂₀	—	2	2	no	—
157	D₂₀	C₅×D₄	1	1	yes	characteristic
158	D₂₀	C₂³×C₅	1	4	yes	—
159	C₄×D₅	—	2	4	no	—
160	C₄×D₅	C₂³×C₅	1	2	yes	—
161	Dic₁₀	C₅×D₄	1	1	yes	characteristic
162	C₅×C₈	—	2	2	no	—
163	C₅×C₈	—	2	2	no	—
164	C₅×C₈	—	5	2	no	—
165	C₅×C₈	C₂²×D₅	1	1	yes	characteristic
166	C₅×C₈	C₂²×D₅	1	1	yes	characteristic
167	C₅×C₈	C₂³×C₅	1	1	yes	characteristic
168	C₅×C₈	C₂³×C₅	1	1	yes	characteristic
169	#168	C₂³×C₅	1	2	yes	—
170	H(5,𝔽₂)	—	5	2	no	—
171	AGL(1,ℤ/8)	—	5	2	no	—
172	AGL(1,ℤ/8)	—	5	4	no	—
173	#135	—	5	2	no	—
174	#135	C₅×D₅	1	1	yes	characteristic, complemented
175	C₂×D₈	—	5	2	no	—
176	C₂×D₈	—	5	1	no	—
177	#131	—	5	1	no	—
178	C₅²	#576	1	1	yes	characteristic, complemented, Sylow
179	C₂²×C₅	—	2	2	no	—
180	C₂²×C₅	—	2	2	no	—
181	C₂²×C₅	—	2	2	no	—
182	C₂²×C₅	—	4	4	no	—
183	C₂²×C₅	—	5	2	no	—
184	C₂²×C₅	—	10	4	no	—
185	C₂²×C₅	—	40	2	no	—
186	C₂²×C₅	—	40	4	no	—
187	C₂²×C₅	D₄×D₅	1	1	yes	characteristic
188	C₂²×C₅	C₂×C₅×D₄	1	1	yes	characteristic
189	C₂×D₅	—	2	4	no	—
190	C₂×D₅	—	8	4	no	—
191	C₂×D₅	—	8	2	no	—
192	C₂×D₅	C₂×C₅×D₄	1	2	yes	—
193	C₄×C₅	—	2	2	no	—
194	C₄×C₅	—	2	2	no	—
195	C₄×C₅	—	2	2	no	—
196	C₄×C₅	—	2	2	no	—
197	C₄×C₅	—	4	4	no	—
198	C₄×C₅	—	5	2	no	—
199	C₄×C₅	D₄×D₅	1	1	yes	characteristic
200	C₄×C₅	C₂×C₅×D₄	1	1	yes	characteristic
201	C₄×C₅	C₂³×D₅	1	1	yes	characteristic
202	C₄×C₅	C₂⁴×C₅	1	1	yes	characteristic
203	Dic₅	C₂×C₅×D₄	1	2	yes	—
204	#41	—	5	1	no	—
205	#41	—	5	4	no	—
206	#41	—	10	2	no	—
207	#41	C₂×C₅×D₅	1	2	yes	complemented
208	C₂×D₄	—	5	4	no	—
209	C₂×D₄	—	5	2	no	—
210	C₂×D₄	—	10	2	no	—
211	C₂×D₄	—	10	4	no	—
212	Q₁₆	C₂×C₅×D₅	1	1	yes	characteristic, complemented
213	SD₁₆	—	5	4	no	—
214	SD₁₆	C₂×C₅×D₅	1	2	yes	complemented
215	D₈	—	5	4	no	—
216	D₈	—	5	2	no	—
217	D₈	—	5	2	no	—
218	D₈	C₂×C₅×D₅	1	1	yes	characteristic, complemented
219	M₁₆	—	5	1	no	—
220	M₁₆	—	5	2	no	—
221	C₂×C₈	—	5	2	no	—
222	C₂×C₈	C₂×C₅×D₅	1	1	yes	characteristic, complemented
223	C₂×C₅	—	2	1	no	—
224	C₂×C₅	—	2	1	no	—
225	C₂×C₅	—	2	2	no	—
226	C₂×C₅	—	4	2	no	—
227	C₂×C₅	—	4	2	no	—
228	C₂×C₅	—	4	2	no	—
229	C₂×C₅	—	8	4	no	—
230	C₂×C₅	—	10	2	no	—
231	C₂×C₅	—	20	4	no	—
232	C₂×C₅	C₂×D₄×D₅	1	1	yes	characteristic
233	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
234	D₅	—	2	2	no	—
235	D₅	—	4	4	no	—
236	C₂³	—	10	4	no	—
237	C₂³	—	10	2	no	—
238	Q₈	—	5	1	no	—
239	Q₈	C₂²×C₅×D₅	1	2	yes	—
240	D₄	—	2	2	no	—
241	D₄	—	5	2	no	—
242	D₄	—	5	1	no	—
243	D₄	—	5	4	no	—
244	D₄	—	10	4	no	—
245	D₄	—	10	4	no	—
246	D₄	—	10	2	no	—
247	D₄	C₂²×C₅×D₅	1	2	yes	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	5	2	no	—
250	C₂×C₄	—	10	4	no	—
251	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
252	C₈	—	5	2	no	—
253	C₈	C₂²×C₅×D₅	1	1	yes	characteristic
254	C₈	C₂²×C₅×D₅	1	1	yes	characteristic
255	C₅	—	2	2	no	—
256	C₅	#66103	1	1	yes	characteristic, direct product factor
257	C₅	C₅×#576	1	1	yes	characteristic, complemented
258	C₂²	—	2	2	no	—
259	C₂²	—	5	2	no	—
260	C₂²	—	10	4	no	—
261	C₂²	—	40	2	no	—
262	C₂²	—	40	4	no	—
263	C₂²	C₅×D₄×D₅	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	5	2	no	—
266	C₄	C₅×D₄×D₅	1	1	yes	characteristic
267	C₄	C₂³×C₅×D₅	1	1	yes	characteristic
268	C₂	—	2	1	no	—
269	C₂	—	4	2	no	—
270	C₂	—	10	2	no	—
271	C₂	—	20	4	no	—
272	C₂	C₂×C₅×D₄×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅	C₂×D₄×D₅
commutator, cosocle	C₄×C₅	C₂⁴×C₅
Frattini	C₄	C₂³×C₅×D₅
Fitting	C₅²×#135	C₂
socle	C₂×C₅²	C₂²×D₄
hypercenter	C₅×#135	D₅
hypocenter	C₅	C₅×#576
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#135

Complements

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

N	H ≅ G/N	type
1. C₅×#4149	C₂	semidirect
2. C₅²×#135	C₂	semidirect
3. C₅×#4095	C₂	semidirect
4. C₅×#4065	C₂	semidirect
5. C₅×#4060	C₂	semidirect
6. C₅×D₅×D₈	C₂	semidirect
7. C₂×C₅×D₄₀	C₂	semidirect
8. C₅×#4047	C₂	semidirect
9. C₅²×#41	C₂²	semidirect
10. C₅×#713	C₂²	semidirect
12. C₅×D₄×D₅	C₂²	semidirect
16. C₂×C₅×D₂₀	C₂²	semidirect
17. C₅²×Q₁₆	C₂²	semidirect
18. C₅²×SD₁₆	C₂²	semidirect
19. C₅²×D₈	C₂²	semidirect
20. C₂×C₅²×C₈	C₂²	semidirect
21. C₅×#688	C₂²	semidirect
22. C₅×#686	C₂²	semidirect
23. C₅×#681	C₂²	semidirect
24. C₅×D₄₀	C₂²	semidirect
25. C₅×D₄₀	C₂²	semidirect
26. C₅×#676	C₂²	semidirect
27. C₅×C₈×D₅	C₂²	semidirect
29. #66103	C₅	direct (N × H)
50. #4149	C₂×C₅	semidirect
55. C₅×#135	D₅	semidirect
56. C₅×#135	C₂×C₅	semidirect
60. #4095	C₂×C₅	semidirect
61. #4065	C₂×C₅	semidirect
62. #4060	C₂×C₅	semidirect
63. D₅×D₈	C₂×C₅	semidirect
64. C₂×D₄₀	C₂×C₅	semidirect
65. #4047	C₂×C₅	semidirect
80. C₅×#41	C₂×D₅	semidirect
81. C₅×#41	C₂²×C₅	semidirect
86. #713	C₂²×C₅	semidirect
88. D₄×D₅	C₂²×C₅	semidirect
92. C₂×D₂₀	C₂²×C₅	semidirect
94. C₅×Q₁₆	C₂×D₅	semidirect
95. C₅×Q₁₆	C₂²×C₅	semidirect
98. C₅×SD₁₆	C₂×D₅	semidirect
99. C₅×SD₁₆	C₂²×C₅	semidirect
104. C₅×D₈	C₂×D₅	semidirect
105. C₅×D₈	C₂²×C₅	semidirect
110. C₂×C₅×C₈	C₂×D₅	semidirect
111. C₂×C₅×C₈	C₂²×C₅	semidirect
112. #688	C₂²×C₅	semidirect
113. #686	C₂²×C₅	semidirect
114. #681	C₂²×C₅	semidirect
115. D₄₀	C₂²×C₅	semidirect
116. D₄₀	C₂²×C₅	semidirect
117. #676	C₂²×C₅	semidirect
118. C₈×D₅	C₂²×C₅	semidirect
174. #135	C₅×D₅	semidirect
178. C₅²	#576	semidirect
207. #41	C₂×C₅×D₅	semidirect
212. Q₁₆	C₂×C₅×D₅	semidirect
214. SD₁₆	C₂×C₅×D₅	semidirect
218. D₈	C₂×C₅×D₅	semidirect
222. C₂×C₈	C₂×C₅×D₅	semidirect
256. C₅	#66103	direct (N × H)
257. C₅	C₅×#576	semidirect

Maximal subgroups

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

subgroup	index
C₅×#4149	2
C₅²×#135	2
C₅×#4095	2
C₅×#4065	2
C₅×#4060	2
C₅×D₅×D₈	2
C₂×C₅×D₄₀	2
C₅×#4047	2
C₅×#576	5
#66103	5

Subgroups — C5×#66103 #13106923

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#66103 #13106923