Subgroups — C₂×D₅⋊#681 #13107781

Order 1600.

623 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂×#87853	C₂	1	2	yes	maximal, complemented
3	D₅⋊#681	C₂	1	8	yes	maximal, direct product factor
4	C₂×#87849	C₂	1	2	yes	maximal, complemented
5	C₂×D₅⋊#168	C₂	1	2	yes	maximal, complemented
6	C₂³×C₅×D₅	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₅²	C₄	1	1	yes	characteristic
8	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₅×D₅	C₄	1	4	yes	—
10	C₂×C₄×C₅×D₅	C₄	1	1	yes	characteristic
11	C₂×C₄×C₅×D₅	C₄	1	1	yes	characteristic
12	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₅×D₅	C₂²	1	4	yes	—
15	#87853	C₂²	1	8	yes	complemented
16	C₂×#6032	C₂²	1	4	yes	complemented
17	#87849	C₂²	1	8	yes	complemented
18	D₅⋊#168	C₂²	1	8	yes	complemented
19	C₂×D₅⋊M₁₆	—	5	1	no	maximal
20	C₂²×#681	—	5	1	no	maximal
21	C₂³×C₅²	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₅×D₅	—	2	4	no	—
23	C₂²×C₅×D₅	C₂×C₄	1	4	yes	—
24	C₂²×C₅×D₅	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₅×D₅	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₅²	C₂×C₄	1	4	yes	—
27	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
29	C₂×C₅×Dic₅	C₂×C₄	1	2	yes	—
30	C₂×C₅×Dic₅	C₂×C₄	1	1	yes	characteristic
31	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
32	C₂×C₅×Dic₅	C₂³	1	2	yes	—
33	C₄×C₅×D₅	C₂×C₄	1	2	yes	—
34	C₄×C₅×D₅	C₂×C₄	1	4	yes	—
35	C₄×C₅×D₅	C₂×C₄	1	2	yes	—
36	C₄×C₅×D₅	C₂×C₄	1	4	yes	—
37	C₄×C₅×D₅	C₂³	1	2	yes	—
38	C₄×C₅×D₅	C₂³	1	2	yes	—
39	#6032	C₂³	1	8	yes	complemented
40	C₂³×C₄×C₅	—	5	1	no	—
41	C₂²×C₄×D₅	D₅	1	1	yes	characteristic
42	C₂×#704	—	5	2	no	—
43	D₅⋊M₁₆	—	5	8	no	—
44	C₂×#700	—	5	2	no	—
45	C₂×D₅⋊C₈	—	5	2	no	—
46	C₂×#681	—	5	2	no	—
47	C₂×#681	—	5	8	no	—
48	C₂×#681	—	5	2	no	—
49	C₂²×#168	—	5	2	no	—
50	C₂²×C₅²	—	2	2	no	—
51	C₂²×C₅²	C₂²×C₄	1	2	yes	—
52	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₅×D₅	—	2	8	no	—
54	C₂×C₅×D₅	—	2	2	no	—
55	C₂×C₅×D₅	M₁₆	1	4	yes	complemented
56	C₂×C₅×D₅	C₂²×C₄	1	2	yes	—
57	C₂×C₅×D₅	C₂²×C₄	1	2	yes	—
58	C₄×C₅²	C₂²×C₄	1	2	yes	—
59	C₄×C₅²	C₂²×C₄	1	2	yes	—
60	C₅×Dic₅	C₂²×C₄	1	1	yes	characteristic
61	C₅×Dic₅	C₂²×C₄	1	2	yes	—
62	C₅×Dic₅	C₂⁴	1	1	yes	characteristic
63	C₂⁴×C₅	—	5	1	no	—
64	C₂³×D₅	Dic₅	1	1	yes	characteristic
65	C₂²×C₄×C₅	—	4	1	no	—
66	C₂²×C₄×C₅	—	5	1	no	—
67	C₂²×C₄×C₅	—	5	1	no	—
68	C₂²×C₄×C₅	—	5	4	no	—
69	C₂²×C₄×C₅	—	5	1	no	—
70	C₂²×C₄×C₅	—	5	1	no	—
71	C₂²×C₄×C₅	—	5	1	no	—
72	C₂²×C₄×C₅	—	5	4	no	—
73	C₂²×C₄×C₅	Dic₅	1	1	yes	characteristic
74	C₂²×C₄×C₅	AGL(1,𝔽₅)	1	1	yes	characteristic
75	C₂²×Dic₅	C₂×D₅	1	1	yes	characteristic
76	C₂×C₄×D₅	Dic₅	1	4	yes	—
77	C₂×C₄×D₅	Dic₅	1	1	yes	characteristic
78	C₂×C₄×D₅	Dic₅	1	1	yes	characteristic
79	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
80	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
81	C₂×C₄×D₅	C₂×D₅	1	4	yes	—
82	#704	—	5	8	no	—
83	C₂×#170	—	5	4	no	—
84	#700	—	5	8	no	—
85	D₅⋊C₈	—	5	8	no	—
86	#681	—	5	8	no	—
87	#681	—	5	8	no	—
88	C₂×#168	—	5	8	no	—
89	C₂×#168	—	5	4	no	—
90	C₂²×M₁₆	—	25	1	no	Sylow
91	C₂×C₅²	—	2	2	no	—
92	C₂×C₅²	C₂×M₁₆	1	2	yes	complemented
93	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
94	C₅×D₅	—	2	2	no	—
95	C₅×D₅	C₂×M₁₆	1	4	yes	complemented
96	C₂³×C₅	—	4	1	no	—
97	C₂³×C₅	—	5	1	no	—
98	C₂³×C₅	—	5	4	no	—
99	C₂³×C₅	—	5	1	no	—
100	C₂³×C₅	—	10	4	no	—
101	C₂³×C₅	C₂×Dic₅	1	1	yes	characteristic
102	C₂³×C₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
103	C₂²×D₅	—	2	4	no	—
104	C₂²×D₅	C₂×Dic₅	1	1	yes	characteristic
105	C₂²×D₅	C₂×Dic₅	1	4	yes	—
106	C₂²×D₅	C₂×Dic₅	1	1	yes	characteristic
107	C₂×C₄×C₅	—	4	1	no	—
108	C₂×C₄×C₅	—	4	4	no	—
109	C₂×C₄×C₅	—	4	1	no	—
110	C₂×C₄×C₅	—	5	2	no	—
111	C₂×C₄×C₅	—	5	1	no	—
112	C₂×C₄×C₅	—	5	1	no	—
113	C₂×C₄×C₅	—	5	2	no	—
114	C₂×C₄×C₅	—	5	2	no	—
115	C₂×C₄×C₅	—	5	4	no	—
116	C₂×C₄×C₅	—	5	2	no	—
117	C₂×C₄×C₅	—	5	4	no	—
118	C₂×C₄×C₅	—	5	2	no	—
119	C₂×C₄×C₅	—	5	2	no	—
120	C₂×C₄×C₅	C₂×Dic₅	1	4	yes	—
121	C₂×C₄×C₅	C₂×Dic₅	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂×Dic₅	1	1	yes	characteristic
123	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	4	yes	—
124	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
126	C₂×Dic₅	C₂×Dic₅	1	2	yes	—
127	C₂×Dic₅	C₂×Dic₅	1	1	yes	characteristic
128	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
129	C₂×Dic₅	C₂²×D₅	1	2	yes	—
130	C₄×D₅	C₂×Dic₅	1	2	yes	—
131	C₄×D₅	C₂×Dic₅	1	4	yes	—
132	C₄×D₅	C₂×Dic₅	1	2	yes	—
133	C₄×D₅	C₂×Dic₅	1	4	yes	—
134	C₄×D₅	C₂²×D₅	1	2	yes	—
135	C₄×D₅	C₂²×D₅	1	2	yes	—
136	#170	—	5	8	no	—
137	#168	—	5	8	no	—
138	C₂³×C₄	—	5	1	no	—
139	C₂×M₁₆	—	25	2	no	—
140	C₂×M₁₆	—	25	8	no	—
141	C₂×M₁₆	—	25	2	no	—
142	C₂²×C₈	—	25	2	no	—
143	C₅²	C₂²×M₁₆	1	1	yes	characteristic, complemented, Sylow
144	C₂²×C₅	—	2	2	no	—
145	C₂²×C₅	—	2	2	no	—
146	C₂²×C₅	—	4	4	no	—
147	C₂²×C₅	—	4	2	no	—
148	C₂²×C₅	—	4	1	no	—
149	C₂²×C₅	—	5	4	no	—
150	C₂²×C₅	—	5	2	no	—
151	C₂²×C₅	—	5	2	no	—
152	C₂²×C₅	—	10	8	no	—
153	C₂²×C₅	—	10	2	no	—
154	C₂²×C₅	C₂²×Dic₅	1	2	yes	—
155	C₂²×C₅	C₂²×Dic₅	1	1	yes	characteristic
156	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	2	yes	—
157	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
158	C₂×D₅	—	2	2	no	—
159	C₂×D₅	—	2	8	no	—
160	C₂×D₅	#681	1	4	yes	complemented
161	C₂×D₅	C₂²×Dic₅	1	2	yes	—
162	C₂×D₅	C₂²×Dic₅	1	2	yes	—
163	C₄×C₅	—	4	2	no	—
164	C₄×C₅	—	4	2	no	—
165	C₄×C₅	—	5	1	no	—
166	C₄×C₅	—	5	2	no	—
167	C₄×C₅	—	5	1	no	—
168	C₄×C₅	C₂²×Dic₅	1	2	yes	—
169	C₄×C₅	C₂²×Dic₅	1	2	yes	—
170	C₄×C₅	C₂²×AGL(1,𝔽₅)	1	2	yes	—
171	C₄×C₅	C₂²×AGL(1,𝔽₅)	1	2	yes	—
172	Dic₅	C₂²×Dic₅	1	1	yes	characteristic
173	Dic₅	C₂²×Dic₅	1	2	yes	—
174	Dic₅	C₂³×D₅	1	1	yes	characteristic
175	C₂⁴	—	5	1	no	—
176	C₂²×C₄	—	5	1	no	—
177	C₂²×C₄	—	5	1	no	—
178	C₂²×C₄	—	5	4	no	—
179	C₂²×C₄	—	5	1	no	—
180	C₂²×C₄	—	5	4	no	—
181	C₂²×C₄	—	5	1	no	—
182	C₂²×C₄	—	5	1	no	—
183	C₂²×C₄	#1042	1	1	yes	characteristic
184	M₁₆	—	25	8	no	—
185	M₁₆	—	25	8	no	—
186	C₂×C₈	—	25	8	no	—
187	C₂×C₈	—	25	4	no	—
188	C₂×C₅	—	2	2	no	—
189	C₂×C₅	—	2	2	no	—
190	C₂×C₅	—	4	4	no	—
191	C₂×C₅	—	4	2	no	—
192	C₂×C₅	—	4	1	no	—
193	C₂×C₅	—	5	4	no	—
194	C₂×C₅	—	10	2	no	—
195	C₂×C₅	C₂×#681	1	2	yes	complemented
196	C₂×C₅	D₅⋊M₁₆	1	2	yes	complemented
197	C₂×C₅	C₂³×Dic₅	1	1	yes	characteristic
198	C₂×C₅	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
199	D₅	—	2	2	no	—
200	D₅	C₂×#681	1	4	yes	complemented
201	C₂³	—	5	1	no	—
202	C₂³	—	5	4	no	—
203	C₂³	—	5	1	no	—
204	C₂³	—	10	4	no	—
205	C₂³	C₂×#1042	1	1	yes	characteristic
206	C₂×C₄	—	5	1	no	—
207	C₂×C₄	—	5	2	no	—
208	C₂×C₄	—	5	2	no	—
209	C₂×C₄	—	5	1	no	—
210	C₂×C₄	—	5	2	no	—
211	C₂×C₄	—	5	2	no	—
212	C₂×C₄	—	5	4	no	—
213	C₂×C₄	—	5	2	no	—
214	C₂×C₄	—	5	2	no	—
215	C₂×C₄	—	5	4	no	—
216	C₂×C₄	C₂×#1042	1	1	yes	characteristic
217	C₂×C₄	C₂×#1042	1	4	yes	—
218	C₂×C₄	C₂×#1042	1	1	yes	characteristic
219	C₈	—	25	8	no	—
220	C₅	—	4	1	no	—
221	C₅	C₂²×#681	1	1	yes	characteristic, complemented
222	C₅	C₂×D₅⋊M₁₆	1	1	yes	characteristic, complemented
223	C₂²	—	2	2	no	—
224	C₂²	—	5	4	no	—
225	C₂²	—	5	2	no	—
226	C₂²	—	5	2	no	—
227	C₂²	—	10	8	no	—
228	C₂²	—	10	2	no	—
229	C₂²	C₂²×#1042	1	2	yes	—
230	C₂²	C₂²×#1042	1	1	yes	characteristic
231	C₄	—	5	1	no	—
232	C₄	—	5	1	no	—
233	C₄	—	5	2	no	—
234	C₄	C₂²×#1042	1	2	yes	—
235	C₄	C₂²×#1042	1	2	yes	—
236	C₂	—	2	2	no	—
237	C₂	—	5	4	no	—
238	C₂	—	10	2	no	—
239	C₂	D₅⋊#681	1	2	yes	direct product factor
240	C₂	C₂³×#1042	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#1042
commutator	C₂×C₅²	C₂³×C₄
Frattini	C₂	C₂³×#1042
Fitting	C₂²×C₄×C₅²	C₄
socle	C₂²×C₅²	C₂²×C₄
prime radical	C₂³×C₅×D₅	C₄
cosocle	C₅×Dic₅	C₂⁴
hypercenter	C₂²×C₄	#1042
hypocenter	C₅²	C₂²×M₁₆
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×#87853	C₂	semidirect
3. D₅⋊#681	C₂	direct (N × H)
4. C₂×#87849	C₂	semidirect
5. C₂×D₅⋊#168	C₂	semidirect
15. #87853	C₂²	semidirect
16. C₂×#6032	C₂²	semidirect
17. #87849	C₂²	semidirect
18. D₅⋊#168	C₂²	semidirect
39. #6032	C₂³	semidirect
55. C₂×C₅×D₅	M₁₆	semidirect
92. C₂×C₅²	C₂×M₁₆	semidirect
95. C₅×D₅	C₂×M₁₆	semidirect
143. C₅²	C₂²×M₁₆	semidirect
160. C₂×D₅	#681	semidirect
195. C₂×C₅	C₂×#681	semidirect
196. C₂×C₅	D₅⋊M₁₆	semidirect
200. D₅	C₂×#681	semidirect
221. C₅	C₂²×#681	semidirect
222. C₅	C₂×D₅⋊M₁₆	semidirect
239. C₂	D₅⋊#681	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×C₄×C₅×D₅	2
C₂×#87853	2
D₅⋊#681	2
C₂×#87849	2
C₂×D₅⋊#168	2
C₂×D₅⋊M₁₆	5
C₂²×#681	5

Subgroups — C2×D5⋊#681 #13107781

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₅⋊#681 #13107781