Subgroups — C₂×#11717425 #13107784

Order 1600.

687 conjugacy classes of subgroups.

The subgroup lattice has 258 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×C₂²⋊#1042	C₂	1	2	yes	maximal
3	#11717425	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊#1042	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×#1042	C₂	1	2	yes	maximal, complemented
6	C₂³×C₅×D₅	C₂²	1	1	yes	characteristic
7	C₂²×#1042	C₂²	1	4	yes	—
8	C₂²×C₄×C₅²	C₄	1	1	yes	characteristic, complemented
9	C₂²×C₅×Dic₅	C₄	1	1	yes	characteristic, complemented
10	C₂×C₄×C₅×D₅	C₄	1	4	yes	complemented
11	C₂×C₄×C₅×D₅	C₄	1	1	yes	characteristic, complemented
12	C₂×C₄×C₅×D₅	C₄	1	1	yes	characteristic, complemented
13	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×D₅	C₂²	1	4	yes	—
16	C₂²⋊#1042	C₂²	1	8	yes	—
17	C₄⋊#1042	C₂²	1	8	yes	complemented
18	C₄×#1042	C₂²	1	8	yes	complemented
19	C₂×#4130	—	5	1	no	maximal
20	C₂×#4072	—	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	2	yes	—
24	C₂²×C₅×D₅	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₅×D₅	C₂³	1	2	yes	—
26	C₂²×C₅×D₅	C₂³	1	1	yes	characteristic
27	C₂×#1042	—	2	8	no	—
28	C₂×#1042	C₂³	1	8	yes	—
29	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×C₅²	C₂×C₄	1	4	yes	complemented
32	C₂×C₅×Dic₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₅×Dic₅	C₂×C₄	1	1	yes	characteristic
34	C₂×C₅×Dic₅	C₂×C₄	1	4	yes	complemented
35	C₄×C₅×D₅	C₂×C₄	1	2	yes	—
36	C₄×C₅×D₅	C₂×C₄	1	2	yes	—
37	C₄×C₅×D₅	C₂×C₄	1	4	yes	complemented
38	C₄×C₅×D₅	C₂×C₄	1	4	yes	complemented
39	C₄×C₅×D₅	C₂³	1	2	yes	—
40	C₄×C₅×D₅	C₂³	1	2	yes	—
41	C₂³×C₄×C₅	—	5	1	no	—
42	C₂²×C₄×D₅	D₅	1	1	yes	characteristic
43	C₂×C₂²⋊AGL(1,𝔽₅)	—	5	2	no	—
44	#4130	—	5	8	no	—
45	C₂×C₄⋊AGL(1,𝔽₅)	—	5	2	no	—
46	C₂×C₄×AGL(1,𝔽₅)	—	5	2	no	—
47	C₂×C₂²⋊Dic₅	—	5	2	no	—
48	#4072	—	5	8	no	—
49	C₂×C₄⋊Dic₅	—	5	2	no	—
50	C₂×C₄×Dic₅	—	5	2	no	—
51	C₂²×C₅²	—	2	2	no	—
52	C₂²×C₅²	C₂²×C₄	1	2	yes	—
53	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₅×D₅	—	2	4	no	—
55	C₂×C₅×D₅	—	2	4	no	—
56	C₂×C₅×D₅	—	2	2	no	—
57	C₂×C₅×D₅	C₂²×C₄	1	2	yes	—
58	C₂×C₅×D₅	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₅×D₅	#41	1	4	yes	—
60	C₂×C₅×D₅	C₂⁴	1	1	yes	characteristic
61	#1042	—	2	8	no	—
62	C₄×C₅²	C₂²×C₄	1	2	yes	—
63	C₄×C₅²	C₂²×C₄	1	2	yes	—
64	C₅×Dic₅	C₂²×C₄	1	2	yes	—
65	C₅×Dic₅	C₂²×C₄	1	2	yes	—
66	C₂⁴×C₅	—	5	1	no	—
67	C₂³×D₅	C₂×D₅	1	1	yes	characteristic
68	C₂²×AGL(1,𝔽₅)	—	5	4	no	—
69	C₂²×C₄×C₅	—	4	1	no	—
70	C₂²×C₄×C₅	—	5	1	no	—
71	C₂²×C₄×C₅	—	5	1	no	—
72	C₂²×C₄×C₅	—	5	1	no	—
73	C₂²×C₄×C₅	—	5	1	no	—
74	C₂²×C₄×C₅	—	5	1	no	—
75	C₂²×C₄×C₅	—	5	4	no	—
76	C₂²×C₄×C₅	—	5	4	no	—
77	C₂²×C₄×C₅	Dic₅	1	1	yes	characteristic, complemented
78	C₂²×C₄×C₅	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
79	C₂²×Dic₅	—	5	4	no	—
80	C₂²×Dic₅	Dic₅	1	1	yes	characteristic, complemented
81	C₂×C₄×D₅	Dic₅	1	4	yes	complemented
82	C₂×C₄×D₅	Dic₅	1	1	yes	characteristic, complemented
83	C₂×C₄×D₅	Dic₅	1	1	yes	characteristic, complemented
84	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
85	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
86	C₂×C₄×D₅	C₂×D₅	1	4	yes	—
87	C₂²⋊AGL(1,𝔽₅)	—	5	8	no	—
88	C₄⋊AGL(1,𝔽₅)	—	5	8	no	—
89	C₄×AGL(1,𝔽₅)	—	5	8	no	—
90	C₂²⋊Dic₅	—	5	8	no	—
91	C₄⋊Dic₅	—	5	8	no	—
92	C₄×Dic₅	—	5	8	no	—
93	C₂×#117	—	25	1	no	Sylow
94	C₂×C₅²	—	2	2	no	—
95	C₂×C₅²	#117	1	2	yes	complemented
96	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
97	C₅×D₅	—	2	2	no	—
98	C₅×D₅	#117	1	2	yes	complemented
99	C₅×D₅	C₂×#41	1	2	yes	—
100	C₂³×C₅	—	4	1	no	—
101	C₂³×C₅	—	5	1	no	—
102	C₂³×C₅	—	5	2	no	—
103	C₂³×C₅	—	5	1	no	—
104	C₂³×C₅	—	5	2	no	—
105	C₂³×C₅	—	10	4	no	—
106	C₂³×C₅	C₂×Dic₅	1	1	yes	characteristic
107	C₂³×C₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
108	C₂²×D₅	—	2	4	no	—
109	C₂²×D₅	C₂×Dic₅	1	1	yes	characteristic
110	C₂²×D₅	C₂×Dic₅	1	2	yes	—
111	C₂²×D₅	C₂²×D₅	1	2	yes	—
112	C₂²×D₅	C₂²×D₅	1	1	yes	characteristic
113	C₂×AGL(1,𝔽₅)	—	5	8	no	—
114	C₂×AGL(1,𝔽₅)	—	10	8	no	—
115	C₂×C₄×C₅	—	4	1	no	—
116	C₂×C₄×C₅	—	4	1	no	—
117	C₂×C₄×C₅	—	4	4	no	—
118	C₂×C₄×C₅	—	5	1	no	—
119	C₂×C₄×C₅	—	5	1	no	—
120	C₂×C₄×C₅	—	5	4	no	—
121	C₂×C₄×C₅	—	5	2	no	—
122	C₂×C₄×C₅	—	5	2	no	—
123	C₂×C₄×C₅	—	5	4	no	—
124	C₂×C₄×C₅	—	5	4	no	—
125	C₂×C₄×C₅	—	5	2	no	—
126	C₂×C₄×C₅	—	5	2	no	—
127	C₂×C₄×C₅	C₂×Dic₅	1	1	yes	characteristic
128	C₂×C₄×C₅	C₂×Dic₅	1	1	yes	characteristic
129	C₂×C₄×C₅	C₂×Dic₅	1	4	yes	complemented
130	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	4	yes	complemented
133	C₂×Dic₅	—	5	8	no	—
134	C₂×Dic₅	—	10	8	no	—
135	C₂×Dic₅	C₂×Dic₅	1	1	yes	characteristic
136	C₂×Dic₅	C₂×Dic₅	1	1	yes	characteristic
137	C₂×Dic₅	C₂×Dic₅	1	4	yes	complemented
138	C₄×D₅	C₂×Dic₅	1	2	yes	—
139	C₄×D₅	C₂×Dic₅	1	2	yes	—
140	C₄×D₅	C₂×Dic₅	1	4	yes	complemented
141	C₄×D₅	C₂×Dic₅	1	4	yes	complemented
142	C₄×D₅	C₂²×D₅	1	2	yes	—
143	C₄×D₅	C₂²×D₅	1	2	yes	—
144	C₂³×C₄	—	5	1	no	—
145	#117	—	25	8	no	—
146	C₂×C₄⋊C₄	—	25	2	no	—
147	C₂×C₂²⋊C₄	—	25	2	no	—
148	C₂×C₄²	—	25	2	no	—
149	C₅²	C₂×#117	1	1	yes	characteristic, complemented, Sylow
150	C₂²×C₅	—	2	2	no	—
151	C₂²×C₅	—	2	2	no	—
152	C₂²×C₅	—	4	4	no	—
153	C₂²×C₅	—	4	2	no	—
154	C₂²×C₅	—	4	1	no	—
155	C₂²×C₅	—	5	4	no	—
156	C₂²×C₅	—	5	1	no	—
157	C₂²×C₅	—	5	1	no	—
158	C₂²×C₅	—	5	2	no	—
159	C₂²×C₅	—	10	4	no	—
160	C₂²×C₅	—	10	4	no	—
161	C₂²×C₅	—	10	2	no	—
162	C₂²×C₅	C₂²×Dic₅	1	2	yes	—
163	C₂²×C₅	C₂²×Dic₅	1	1	yes	characteristic
164	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	2	yes	—
165	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
166	C₂×D₅	—	2	2	no	—
167	C₂×D₅	—	2	4	no	—
168	C₂×D₅	—	2	4	no	—
169	C₂×D₅	#709	1	4	yes	—
170	C₂×D₅	C₂²×Dic₅	1	2	yes	—
171	C₂×D₅	C₂²×Dic₅	1	1	yes	characteristic
172	C₂×D₅	C₂³×D₅	1	1	yes	characteristic
173	AGL(1,𝔽₅)	—	10	8	no	—
174	C₄×C₅	—	4	2	no	—
175	C₄×C₅	—	4	2	no	—
176	C₄×C₅	—	5	2	no	—
177	C₄×C₅	—	5	2	no	—
178	C₄×C₅	C₂²×Dic₅	1	2	yes	—
179	C₄×C₅	C₂²×Dic₅	1	2	yes	—
180	C₄×C₅	C₂²×AGL(1,𝔽₅)	1	2	yes	—
181	C₄×C₅	C₂²×AGL(1,𝔽₅)	1	2	yes	—
182	Dic₅	—	10	8	no	—
183	Dic₅	C₂²×Dic₅	1	2	yes	—
184	Dic₅	C₂²×Dic₅	1	2	yes	—
185	C₂⁴	—	5	1	no	—
186	C₂²×C₄	—	5	1	no	—
187	C₂²×C₄	—	5	1	no	—
188	C₂²×C₄	—	5	1	no	—
189	C₂²×C₄	—	5	1	no	—
190	C₂²×C₄	—	5	1	no	—
191	C₂²×C₄	—	5	4	no	—
192	C₂²×C₄	—	5	4	no	—
193	C₂²×C₄	—	25	4	no	—
194	C₂²×C₄	#1042	1	1	yes	characteristic, complemented
195	C₄⋊C₄	—	25	8	no	—
196	C₂²⋊C₄	—	25	8	no	—
197	C₄²	—	25	8	no	—
198	C₂×C₅	—	2	2	no	—
199	C₂×C₅	—	2	2	no	—
200	C₂×C₅	—	4	4	no	—
201	C₂×C₅	—	4	2	no	—
202	C₂×C₅	—	4	1	no	—
203	C₂×C₅	—	5	2	no	—
204	C₂×C₅	—	5	2	no	—
205	C₂×C₅	—	10	2	no	—
206	C₂×C₅	#4072	1	2	yes	complemented
207	C₂×C₅	#4130	1	2	yes	complemented
208	C₂×C₅	C₂³×Dic₅	1	1	yes	characteristic
209	C₂×C₅	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
210	D₅	—	2	2	no	—
211	D₅	#4072	1	2	yes	complemented
212	D₅	C₂×#709	1	2	yes	—
213	C₂³	—	5	1	no	—
214	C₂³	—	5	2	no	—
215	C₂³	—	5	2	no	—
216	C₂³	—	5	1	no	—
217	C₂³	—	10	4	no	—
218	C₂³	C₂×#1042	1	1	yes	characteristic
219	C₂×C₄	—	5	1	no	—
220	C₂×C₄	—	5	1	no	—
221	C₂×C₄	—	5	4	no	—
222	C₂×C₄	—	5	2	no	—
223	C₂×C₄	—	5	2	no	—
224	C₂×C₄	—	5	2	no	—
225	C₂×C₄	—	5	2	no	—
226	C₂×C₄	—	5	4	no	—
227	C₂×C₄	—	5	4	no	—
228	C₂×C₄	—	25	8	no	—
229	C₂×C₄	—	50	8	no	—
230	C₂×C₄	C₂×#1042	1	1	yes	characteristic
231	C₂×C₄	C₂×#1042	1	1	yes	characteristic
232	C₂×C₄	C₂×#1042	1	4	yes	complemented
233	C₅	—	4	1	no	—
234	C₅	C₂×#4072	1	1	yes	characteristic, complemented
235	C₅	C₂×#4130	1	1	yes	characteristic, complemented
236	C₂²	—	2	2	no	—
237	C₂²	—	5	4	no	—
238	C₂²	—	5	1	no	—
239	C₂²	—	5	1	no	—
240	C₂²	—	5	2	no	—
241	C₂²	—	10	4	no	—
242	C₂²	—	10	4	no	—
243	C₂²	—	10	2	no	—
244	C₂²	C₂²×#1042	1	2	yes	—
245	C₂²	C₂²×#1042	1	1	yes	characteristic
246	C₄	—	5	2	no	—
247	C₄	—	5	2	no	—
248	C₄	—	50	8	no	—
249	C₄	C₂²×#1042	1	2	yes	—
250	C₄	C₂²×#1042	1	2	yes	—
251	C₂	—	2	2	no	—
252	C₂	—	5	2	no	—
253	C₂	—	5	2	no	—
254	C₂	—	10	2	no	—
255	C₂	#11717425	1	2	yes	direct product factor
256	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	C₂²
cosocle	C₂×C₅×D₅	C₂⁴
hypercenter	C₂²×C₄	#1042
hypocenter	C₅²	C₂×#117
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
3. #11717425	C₂	direct (N × H)
4. C₂×C₄⋊#1042	C₂	semidirect
5. C₂×C₄×#1042	C₂	semidirect
8. C₂²×C₄×C₅²	C₄	semidirect
9. C₂²×C₅×Dic₅	C₄	semidirect
10. C₂×C₄×C₅×D₅	C₄	semidirect
11. C₂×C₄×C₅×D₅	C₄	semidirect
12. C₂×C₄×C₅×D₅	C₄	semidirect
17. C₄⋊#1042	C₂²	semidirect
18. C₄×#1042	C₂²	semidirect
31. C₂×C₄×C₅²	C₂×C₄	semidirect
34. C₂×C₅×Dic₅	C₂×C₄	semidirect
37. C₄×C₅×D₅	C₂×C₄	semidirect
38. C₄×C₅×D₅	C₂×C₄	semidirect
77. C₂²×C₄×C₅	Dic₅	semidirect
78. C₂²×C₄×C₅	AGL(1,𝔽₅)	semidirect
80. C₂²×Dic₅	Dic₅	semidirect
81. C₂×C₄×D₅	Dic₅	semidirect
82. C₂×C₄×D₅	Dic₅	semidirect
83. C₂×C₄×D₅	Dic₅	semidirect
95. C₂×C₅²	#117	semidirect
98. C₅×D₅	#117	semidirect
129. C₂×C₄×C₅	C₂×Dic₅	semidirect
132. C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	semidirect
137. C₂×Dic₅	C₂×Dic₅	semidirect
140. C₄×D₅	C₂×Dic₅	semidirect
141. C₄×D₅	C₂×Dic₅	semidirect
149. C₅²	C₂×#117	semidirect
194. C₂²×C₄	#1042	semidirect
206. C₂×C₅	#4072	semidirect
207. C₂×C₅	#4130	semidirect
211. D₅	#4072	semidirect
232. C₂×C₄	C₂×#1042	semidirect
234. C₅	C₂×#4072	semidirect
235. C₅	C₂×#4130	semidirect
255. C₂	#11717425	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×C₂²⋊#1042	2
#11717425	2
C₂×C₄⋊#1042	2
C₂×C₄×#1042	2
C₂×#4130	5
C₂×#4072	5

Subgroups — C2×#11717425 #13107784

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11717425 #13107784