Subgroups — #13106022

Order 1600.

728 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#709	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#1039	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4072	C₂	1	1	yes	maximal, characteristic, complemented
4	#11717091	C₂	1	4	yes	maximal, complemented
5	#11717017	C₂	1	2	yes	maximal, complemented
6	C₄×D₅×Dic₅	C₂	1	2	yes	maximal, complemented
7	#11716973	C₂	1	2	yes	maximal, complemented
8	#11716961	C₂	1	2	yes	maximal, complemented
9	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
10	C₂²×#1039	C₂²	1	2	yes	—
11	C₂×C₅×C₂²⋊D₅	C₂²	1	2	yes	—
12	C₅×#709	C₄	1	8	yes	complemented
13	C₂×C₅×D₂₀	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₅×D₅	C₂²	1	2	yes	—
15	C₂×C₅×Dic₁₀	C₂²	1	1	yes	characteristic
16	C₂×D₅×Dic₅	C₂²	1	4	yes	—
17	C₄×#1039	C₂²	1	2	yes	—
18	C₄×#1039	C₂²	1	1	yes	characteristic
19	C₄×#1039	C₂²	1	1	yes	characteristic
20	C₅×C₂²⋊Dic₅	C₂²	1	2	yes	—
21	C₅×C₄⋊Dic₅	C₂²	1	2	yes	—
22	C₄×C₅×Dic₅	C₂²	1	2	yes	—
23	#87780	C₂²	1	4	yes	—
24	#87778	C₂²	1	4	yes	—
25	Dic₅²	C₂²	1	4	yes	—
26	#41×Dic₅	—	5	1	no	maximal
27	#65856	—	5	1	no	maximal
28	C₂³×C₅²	C₂³	1	1	yes	characteristic
29	C₂²×C₅×D₅	C₂³	1	2	yes	—
30	C₂×C₄×C₅²	C₂×C₄	1	4	yes	complemented
31	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
32	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
33	C₂×#1039	—	2	4	no	—
34	C₂×#1039	C₂³	1	2	yes	—
35	C₂×#1039	C₂³	1	2	yes	—
36	C₅×C₂²⋊D₅	C₂×C₄	1	8	yes	complemented
37	C₂×C₅×Dic₅	C₂³	1	4	yes	—
38	C₂×C₅×Dic₅	C₂³	1	2	yes	—
39	C₅×D₂₀	C₂×C₄	1	4	yes	complemented
40	C₄×C₅×D₅	C₂×C₄	1	8	yes	complemented
41	C₅×Dic₁₀	C₂×C₄	1	4	yes	complemented
42	D₅×Dic₅	—	2	8	no	—
43	C₂×C₅×#41	—	5	1	no	—
44	C₂×#709	D₅	1	1	yes	characteristic
45	C₅×#117	—	5	1	no	—
46	Q₈×Dic₅	—	5	2	no	—
47	D₄×Dic₅	—	5	4	no	—
48	D₄×Dic₅	—	5	2	no	—
49	#4072	—	5	2	no	—
50	#4072	D₅	1	1	yes	characteristic, complemented
51	C₂×C₄×Dic₅	—	5	1	no	—
52	C₂×C₄×Dic₅	—	5	1	no	—
53	C₂×C₄×Dic₅	—	5	2	no	—
54	C₂×C₄×Dic₅	—	10	2	no	—
55	#4039	—	5	2	no	—
56	#4033	—	5	2	no	—
57	#4027	—	5	4	no	—
58	#4018	—	5	2	no	—
59	C₄²×D₅	—	5	2	no	—
60	C₂²×C₅²	—	2	2	no	—
61	C₂²×C₅²	C₂²×C₄	1	2	yes	—
62	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
63	C₂×C₅×D₅	—	2	4	no	—
64	C₂×C₅×D₅	C₂²×C₄	1	4	yes	—
65	C₄×C₅²	C₂²×C₄	1	2	yes	—
66	C₄×C₅²	C₂²×C₄	1	2	yes	—
67	#1039	—	2	2	no	—
68	#1039	#41	1	4	yes	complemented
69	C₅×Dic₅	—	2	4	no	—
70	C₅×Dic₅	C₂²×C₄	1	4	yes	—
71	C₅×#41	—	5	8	no	—
72	C₂×C₅×Q₈	—	5	1	no	—
73	C₂×C₅×D₄	—	5	2	no	—
74	C₂×C₅×D₄	—	5	1	no	—
75	C₂²×C₄×C₅	—	2	2	no	—
76	C₂²×C₄×C₅	—	5	2	no	—
77	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
79	C₂×C₂²⋊D₅	C₂×D₅	1	2	yes	—
80	C₂²×Dic₅	—	5	2	no	—
81	C₂²×Dic₅	—	5	2	no	—
82	C₂²×Dic₅	—	5	4	no	—
83	C₂²×Dic₅	—	10	4	no	—
84	#709	Dic₅	1	8	yes	complemented
85	C₂×D₂₀	C₂×D₅	1	1	yes	characteristic
86	C₂×C₄×D₅	—	5	4	no	—
87	C₂×C₄×D₅	C₂×D₅	1	2	yes	—
88	C₂×Dic₁₀	C₂×D₅	1	1	yes	characteristic
89	C₅×C₄⋊C₄	—	5	2	no	—
90	C₅×C₂²⋊C₄	—	5	2	no	—
91	C₄²×C₅	—	5	2	no	—
92	C₂²⋊Dic₅	—	5	4	no	—
93	C₂²⋊Dic₅	C₂×D₅	1	2	yes	—
94	#685	—	5	4	no	—
95	C₄⋊Dic₅	—	5	4	no	—
96	C₄⋊Dic₅	C₂×D₅	1	2	yes	—
97	#683	—	5	4	no	—
98	C₄×Dic₅	—	5	1	no	—
99	C₄×Dic₅	—	5	1	no	—
100	C₄×Dic₅	—	5	2	no	—
101	C₄×Dic₅	—	5	1	no	—
102	C₄×Dic₅	—	5	1	no	—
103	C₄×Dic₅	—	5	2	no	—
104	C₄×Dic₅	—	5	4	no	—
105	C₄×Dic₅	—	5	4	no	—
106	C₄×Dic₅	—	10	4	no	—
107	C₄×Dic₅	—	10	2	no	—
108	C₄×Dic₅	—	10	2	no	—
109	C₄×Dic₅	C₂×D₅	1	2	yes	—
110	C₄×#41	—	25	1	no	Sylow
111	C₂×C₅²	—	2	2	no	—
112	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₅²	C₂×#41	1	2	yes	—
114	C₅×D₅	—	2	4	no	—
115	C₂³×C₅	—	2	2	no	—
116	C₂³×C₅	—	5	2	no	—
117	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
118	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
119	C₂²×D₅	C₂²×D₅	1	2	yes	—
120	C₅×Q₈	—	5	4	no	—
121	C₅×D₄	—	5	8	no	—
122	C₅×D₄	—	5	4	no	—
123	C₂×C₄×C₅	—	2	8	no	—
124	C₂×C₄×C₅	—	2	2	no	—
125	C₂×C₄×C₅	—	2	2	no	—
126	C₂×C₄×C₅	—	5	2	no	—
127	C₂×C₄×C₅	—	5	8	no	—
128	C₂×C₄×C₅	—	5	4	no	—
129	C₂×C₄×C₅	C₄×D₅	1	4	yes	complemented
130	C₂×C₄×C₅	C₂×Dic₅	1	4	yes	complemented
131	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
135	C₂²⋊D₅	C₂×Dic₅	1	8	yes	complemented
136	C₂×Dic₅	—	5	2	no	—
137	C₂×Dic₅	—	5	2	no	—
138	C₂×Dic₅	—	5	2	no	—
139	C₂×Dic₅	—	5	2	no	—
140	C₂×Dic₅	—	10	4	no	—
141	C₂×Dic₅	—	10	4	no	—
142	C₂×Dic₅	—	10	8	no	—
143	C₂×Dic₅	—	10	4	no	—
144	C₂×Dic₅	—	10	4	no	—
145	C₂×Dic₅	—	10	16	no	—
146	C₂×Dic₅	C₂²×D₅	1	4	yes	—
147	C₂×Dic₅	C₂²×D₅	1	2	yes	—
148	D₂₀	C₂×Dic₅	1	4	yes	complemented
149	C₄×D₅	—	10	8	no	—
150	C₄×D₅	C₂×Dic₅	1	8	yes	complemented
151	Dic₁₀	C₂×Dic₅	1	4	yes	complemented
152	C₂×#41	—	5	1	no	—
153	C₄×Q₈	—	25	2	no	—
154	C₄×D₄	—	25	4	no	—
155	C₄×D₄	—	25	2	no	—
156	#117	—	5	1	no	—
157	#117	—	25	2	no	—
158	C₂×C₄²	—	25	2	no	—
159	C₂×C₄²	—	25	1	no	—
160	C₅²	C₄×#41	1	1	yes	characteristic, complemented, Sylow
161	C₂²×C₅	—	2	8	no	—
162	C₂²×C₅	—	2	2	no	—
163	C₂²×C₅	—	2	2	no	—
164	C₂²×C₅	—	2	4	no	—
165	C₂²×C₅	—	2	2	no	—
166	C₂²×C₅	—	5	4	no	—
167	C₂²×C₅	—	10	4	no	—
168	C₂²×C₅	C₂×C₄×D₅	1	2	yes	—
169	C₂²×C₅	C₂²×Dic₅	1	2	yes	—
170	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
171	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
172	C₂×D₅	—	2	4	no	—
173	C₂×D₅	C₂²×Dic₅	1	4	yes	—
174	C₄×C₅	—	2	4	no	—
175	C₄×C₅	—	2	4	no	—
176	C₄×C₅	—	5	4	no	—
177	C₄×C₅	—	10	4	no	—
178	C₄×C₅	C₂×C₄×D₅	1	2	yes	—
179	C₄×C₅	C₂×C₄×D₅	1	2	yes	—
180	C₄×C₅	C₂²×Dic₅	1	2	yes	—
181	C₄×C₅	C₂²×Dic₅	1	2	yes	—
182	Dic₅	—	2	4	no	—
183	Dic₅	—	5	4	no	—
184	Dic₅	—	5	4	no	—
185	Dic₅	—	10	8	no	—
186	Dic₅	—	10	8	no	—
187	Dic₅	—	10	2	no	—
188	Dic₅	—	10	2	no	—
189	Dic₅	C₂²×Dic₅	1	4	yes	—
190	#41	—	5	8	no	—
191	C₂×Q₈	—	5	1	no	—
192	C₂×D₄	—	5	2	no	—
193	C₂×D₄	—	5	1	no	—
194	C₂²×C₄	—	5	2	no	—
195	C₂²×C₄	—	25	4	no	—
196	C₂²×C₄	—	25	2	no	—
197	C₂²×C₄	D₅²	1	1	yes	characteristic
198	C₄⋊C₄	—	5	2	no	—
199	C₄⋊C₄	—	25	4	no	—
200	C₂²⋊C₄	—	5	2	no	—
201	C₂²⋊C₄	—	25	4	no	—
202	C₄²	—	5	2	no	—
203	C₄²	—	25	4	no	—
204	C₄²	—	25	2	no	—
205	C₄²	—	25	1	no	—
206	C₄²	—	25	1	no	—
207	C₂×C₅	—	2	8	no	—
208	C₂×C₅	—	2	2	no	—
209	C₂×C₅	—	2	2	no	—
210	C₂×C₅	—	2	4	no	—
211	C₂×C₅	—	2	2	no	—
212	C₂×C₅	—	10	4	no	—
213	C₂×C₅	C₂²×C₄×D₅	1	1	yes	characteristic
214	C₂×C₅	D₅×#41	1	2	yes	—
215	C₂×C₅	D₅×#41	1	2	yes	—
216	C₂×C₅	C₂³×Dic₅	1	1	yes	characteristic
217	D₅	—	2	4	no	—
218	C₂³	—	5	2	no	—
219	C₂³	C₂×D₅²	1	1	yes	characteristic
220	Q₈	—	5	4	no	—
221	D₄	—	5	8	no	—
222	D₄	—	5	4	no	—
223	C₂×C₄	—	5	2	no	—
224	C₂×C₄	—	5	8	no	—
225	C₂×C₄	—	5	4	no	—
226	C₂×C₄	—	25	2	no	—
227	C₂×C₄	—	25	2	no	—
228	C₂×C₄	—	50	8	no	—
229	C₂×C₄	—	50	4	no	—
230	C₂×C₄	D₅×Dic₅	1	4	yes	complemented
231	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
232	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
233	C₅	—	2	2	no	—
234	C₅	#65856	1	1	yes	characteristic, complemented
235	C₅	#41×Dic₅	1	1	yes	characteristic, complemented
236	C₂²	—	2	2	no	—
237	C₂²	—	5	4	no	—
238	C₂²	—	10	4	no	—
239	C₂²	C₂×D₅×Dic₅	1	2	yes	—
240	C₂²	C₂²×D₅²	1	1	yes	characteristic
241	C₄	—	5	4	no	—
242	C₄	—	10	4	no	—
243	C₄	—	25	4	no	—
244	C₄	—	50	2	no	—
245	C₄	C₂×D₅×Dic₅	1	2	yes	—
246	C₄	C₂×D₅×Dic₅	1	2	yes	—
247	C₂	—	2	2	no	—
248	C₂	—	10	4	no	—
249	C₂	#11717471	1	2	yes	—
250	C₂	C₂²×D₅×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₅²
commutator	C₂×C₅²	C₂³×C₄
Frattini	C₂²	C₂²×D₅²
Fitting	C₂²×C₄×C₅²	C₂²
socle, cosocle	C₂²×C₅²	C₂⁴
prime radical	C₂×C₅×#709	C₂
hypercenter	C₂²×C₄	D₅²
hypocenter	C₅²	C₄×#41
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×C₄×#1039	C₂	semidirect
3. C₅×#4072	C₂	semidirect
4. #11717091	C₂	semidirect
5. #11717017	C₂	semidirect
6. C₄×D₅×Dic₅	C₂	semidirect
7. #11716973	C₂	semidirect
8. #11716961	C₂	semidirect
12. C₅×#709	C₄	semidirect
30. C₂×C₄×C₅²	C₂×C₄	semidirect
36. C₅×C₂²⋊D₅	C₂×C₄	semidirect
39. C₅×D₂₀	C₂×C₄	semidirect
40. C₄×C₅×D₅	C₂×C₄	semidirect
41. C₅×Dic₁₀	C₂×C₄	semidirect
50. #4072	D₅	semidirect
68. #1039	#41	semidirect
84. #709	Dic₅	semidirect
129. C₂×C₄×C₅	C₄×D₅	semidirect
130. C₂×C₄×C₅	C₂×Dic₅	semidirect
135. C₂²⋊D₅	C₂×Dic₅	semidirect
148. D₂₀	C₂×Dic₅	semidirect
150. C₄×D₅	C₂×Dic₅	semidirect
151. Dic₁₀	C₂×Dic₅	semidirect
160. C₅²	C₄×#41	semidirect
230. C₂×C₄	D₅×Dic₅	semidirect
234. C₅	#65856	semidirect
235. C₅	#41×Dic₅	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#709	2
C₂×C₄×#1039	2
C₅×#4072	2
#11717091	2
#11717017	2
C₄×D₅×Dic₅	2
#11716973	2
#11716961	2
#41×Dic₅	5
#65856	5