Subgroups — Dic₅×#709 #13106003

Order 1600.

696 conjugacy classes of subgroups.

The subgroup lattice has 248 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	#11717246	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₅×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
4	Dic₅×C₂²⋊D₅	C₂	1	4	yes	maximal, complemented
5	Dic₅×D₂₀	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	Dic₅×Dic₁₀	C₂	1	2	yes	maximal, complemented
9	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
10	C₂×C₅×C₂²⋊D₅	C₂²	1	2	yes	—
11	C₂²×C₅×Dic₅	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	C₂²⋊#1039	C₂²	1	2	yes	—
18	C₄⋊#1039	C₂²	1	2	yes	—
19	C₄×#1039	C₂²	1	2	yes	—
20	C₄×C₅×Dic₅	C₂²	1	2	yes	—
21	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
22	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
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	C₄×#709	—	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	C₂³	1	4	yes	—
34	C₅×C₂²⋊D₅	C₂×C₄	1	8	yes	complemented
35	C₂×C₅×Dic₅	—	2	4	no	—
36	C₂×C₅×Dic₅	C₂³	1	2	yes	—
37	C₂×C₅×Dic₅	C₂³	1	2	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	C₂×C₄²×C₅	—	5	1	no	—
46	Q₈×Dic₅	—	5	2	no	—
47	D₄×Dic₅	—	5	4	no	—
48	D₄×Dic₅	—	5	2	no	—
49	C₄×C₂²⋊D₅	—	5	4	no	—
50	#4072	—	5	1	no	—
51	#4072	—	5	1	no	—
52	#4072	—	5	2	no	—
53	#4072	—	10	2	no	—
54	C₂×C₄×Dic₅	—	5	2	no	—
55	C₂×C₄×Dic₅	D₅	1	1	yes	characteristic, complemented
56	C₄×D₂₀	—	5	2	no	—
57	#4018	—	5	2	no	—
58	C₄²×D₅	—	5	2	no	—
59	C₄×Dic₁₀	—	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	4	no	—
68	C₅×Dic₅	—	2	2	no	—
69	C₅×Dic₅	C₂²×C₄	1	4	yes	—
70	C₅×Dic₅	#41	1	4	yes	complemented
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	—	5	2	no	—
78	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
80	C₂×C₂²⋊D₅	C₂×D₅	1	2	yes	—
81	C₂²×Dic₅	—	5	4	no	—
82	C₂²×Dic₅	C₂×D₅	1	2	yes	—
83	#709	Dic₅	1	8	yes	direct product factor
84	C₂×D₂₀	C₂×D₅	1	1	yes	characteristic
85	C₂×C₄×D₅	—	5	4	no	—
86	C₂×C₄×D₅	C₂×D₅	1	2	yes	—
87	C₂×Dic₁₀	C₂×D₅	1	1	yes	characteristic
88	C₄²×C₅	—	5	2	no	—
89	C₄²×C₅	—	5	1	no	—
90	C₄²×C₅	—	5	1	no	—
91	C₂²⋊Dic₅	—	5	2	no	—
92	C₂²⋊Dic₅	—	5	2	no	—
93	C₂²⋊Dic₅	—	5	4	no	—
94	C₂²⋊Dic₅	—	10	4	no	—
95	#685	—	5	4	no	—
96	C₄⋊Dic₅	—	5	2	no	—
97	C₄⋊Dic₅	—	5	2	no	—
98	C₄⋊Dic₅	—	5	4	no	—
99	C₄⋊Dic₅	—	10	4	no	—
100	#683	—	5	4	no	—
101	C₄×Dic₅	—	5	2	no	—
102	C₄×Dic₅	—	5	2	no	—
103	C₄×Dic₅	—	5	4	no	—
104	C₄×Dic₅	—	5	4	no	—
105	C₄×Dic₅	—	10	4	no	—
106	C₄×Dic₅	C₂×D₅	1	1	yes	characteristic
107	C₄×Dic₅	C₂×D₅	1	1	yes	characteristic
108	C₄×Dic₅	C₂×D₅	1	2	yes	—
109	C₄×#41	—	25	1	no	Sylow
110	C₂×C₅²	—	2	2	no	—
111	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₅²	C₂×#41	1	2	yes	—
113	C₅×D₅	—	2	4	no	—
114	C₂³×C₅	—	2	2	no	—
115	C₂³×C₅	—	5	2	no	—
116	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
117	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
118	C₂²×D₅	C₂²×D₅	1	2	yes	—
119	C₅×Q₈	—	5	4	no	—
120	C₅×D₄	—	5	8	no	—
121	C₅×D₄	—	5	4	no	—
122	C₂×C₄×C₅	—	2	8	no	—
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	—	2	2	no	—
125	C₂×C₄×C₅	—	5	2	no	—
126	C₂×C₄×C₅	—	5	2	no	—
127	C₂×C₄×C₅	—	5	8	no	—
128	C₂×C₄×C₅	—	5	2	no	—
129	C₂×C₄×C₅	—	10	4	no	—
130	C₂×C₄×C₅	C₄×D₅	1	4	yes	complemented
131	C₂×C₄×C₅	C₂×Dic₅	1	4	yes	complemented
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	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
136	C₂²⋊D₅	C₂×Dic₅	1	8	yes	complemented
137	C₂×Dic₅	—	2	4	no	—
138	C₂×Dic₅	—	5	4	no	—
139	C₂×Dic₅	—	5	4	no	—
140	C₂×Dic₅	—	10	8	no	—
141	C₂×Dic₅	—	10	8	no	—
142	C₂×Dic₅	C₂²×D₅	1	2	yes	—
143	C₂×Dic₅	C₂²×D₅	1	2	yes	—
144	C₂×Dic₅	C₂²×D₅	1	2	yes	—
145	D₂₀	C₂×Dic₅	1	4	yes	complemented
146	C₄×D₅	—	10	8	no	—
147	C₄×D₅	C₂×Dic₅	1	8	yes	complemented
148	Dic₁₀	C₂×Dic₅	1	4	yes	complemented
149	C₂×#41	—	5	1	no	—
150	C₄×Q₈	—	25	2	no	—
151	C₄×D₄	—	25	2	no	—
152	C₄×D₄	—	25	4	no	—
153	#117	—	25	2	no	—
154	#117	—	25	1	no	—
155	C₂×C₄²	—	5	1	no	—
156	C₂×C₄²	—	25	2	no	—
157	C₅²	C₄×#41	1	1	yes	characteristic, complemented, Sylow
158	C₂²×C₅	—	2	2	no	—
159	C₂²×C₅	—	2	2	no	—
160	C₂²×C₅	—	2	4	no	—
161	C₂²×C₅	—	2	2	no	—
162	C₂²×C₅	—	4	4	no	—
163	C₂²×C₅	—	5	4	no	—
164	C₂²×C₅	—	10	4	no	—
165	C₂²×C₅	C₂×C₄×D₅	1	2	yes	—
166	C₂²×C₅	C₂²×Dic₅	1	2	yes	—
167	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
168	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
169	C₂×D₅	—	2	4	no	—
170	C₂×D₅	C₂²×Dic₅	1	4	yes	—
171	C₄×C₅	—	2	4	no	—
172	C₄×C₅	—	2	4	no	—
173	C₄×C₅	—	5	4	no	—
174	C₄×C₅	—	5	4	no	—
175	C₄×C₅	—	10	2	no	—
176	C₄×C₅	C₂×C₄×D₅	1	2	yes	—
177	C₄×C₅	C₂×C₄×D₅	1	2	yes	—
178	C₄×C₅	C₂²×Dic₅	1	2	yes	—
179	C₄×C₅	C₂²×Dic₅	1	2	yes	—
180	Dic₅	—	2	2	no	—
181	Dic₅	—	10	4	no	—
182	Dic₅	—	10	4	no	—
183	Dic₅	—	20	8	no	—
184	Dic₅	#709	1	4	yes	direct product factor
185	Dic₅	C₂²×Dic₅	1	4	yes	—
186	#41	—	5	8	no	—
187	C₂×Q₈	—	5	1	no	—
188	C₂×D₄	—	5	2	no	—
189	C₂×D₄	—	5	1	no	—
190	C₂²×C₄	—	5	2	no	—
191	C₂²×C₄	—	5	2	no	—
192	C₂²×C₄	—	25	4	no	—
193	C₂²×C₄	D₅²	1	1	yes	characteristic
194	C₄⋊C₄	—	25	4	no	—
195	C₄⋊C₄	—	25	2	no	—
196	C₂²⋊C₄	—	25	4	no	—
197	C₂²⋊C₄	—	25	2	no	—
198	C₄²	—	5	2	no	—
199	C₄²	—	5	1	no	—
200	C₄²	—	5	1	no	—
201	C₄²	—	25	4	no	—
202	C₄²	—	25	2	no	—
203	C₂×C₅	—	2	2	no	—
204	C₂×C₅	—	2	2	no	—
205	C₂×C₅	—	2	4	no	—
206	C₂×C₅	—	2	2	no	—
207	C₂×C₅	—	4	4	no	—
208	C₂×C₅	—	10	4	no	—
209	C₂×C₅	C₂²×C₄×D₅	1	1	yes	characteristic
210	C₂×C₅	C₂×#709	1	2	yes	—
211	C₂×C₅	D₅×#41	1	2	yes	—
212	C₂×C₅	C₂³×Dic₅	1	1	yes	characteristic
213	D₅	—	2	4	no	—
214	C₂³	—	5	2	no	—
215	C₂³	C₂×D₅²	1	1	yes	characteristic
216	Q₈	—	5	4	no	—
217	D₄	—	5	8	no	—
218	D₄	—	5	4	no	—
219	C₂×C₄	—	5	2	no	—
220	C₂×C₄	—	5	2	no	—
221	C₂×C₄	—	5	8	no	—
222	C₂×C₄	—	5	2	no	—
223	C₂×C₄	—	10	4	no	—
224	C₂×C₄	—	25	4	no	—
225	C₂×C₄	—	50	8	no	—
226	C₂×C₄	D₅×Dic₅	1	4	yes	complemented
227	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
228	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
229	C₅	—	2	2	no	—
230	C₅	C₄×#709	1	1	yes	characteristic, complemented
231	C₅	#41×Dic₅	1	1	yes	characteristic, complemented
232	C₂²	—	2	2	no	—
233	C₂²	—	5	4	no	—
234	C₂²	—	10	4	no	—
235	C₂²	C₂×D₅×Dic₅	1	2	yes	—
236	C₂²	C₂²×D₅²	1	1	yes	characteristic
237	C₄	—	5	4	no	—
238	C₄	—	5	4	no	—
239	C₄	—	10	2	no	—
240	C₄	—	50	4	no	—
241	C₄	C₂×D₅×Dic₅	1	2	yes	—
242	C₄	C₂×D₅×Dic₅	1	2	yes	—
243	C₂	—	2	2	no	—
244	C₂	—	10	4	no	—
245	C₂	D₅×#709	1	2	yes	—
246	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. #11717246	C₂	semidirect
3. C₂×C₄×C₅×Dic₅	C₂	semidirect
4. Dic₅×C₂²⋊D₅	C₂	semidirect
5. Dic₅×D₂₀	C₂	semidirect
6. C₄×D₅×Dic₅	C₂	semidirect
7. #11716973	C₂	semidirect
8. Dic₅×Dic₁₀	C₂	semidirect
12. C₅×#709	C₄	semidirect
30. C₂×C₄×C₅²	C₂×C₄	semidirect
34. 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
55. C₂×C₄×Dic₅	D₅	semidirect
70. C₅×Dic₅	#41	semidirect
83. #709	Dic₅	direct (N × H)
130. C₂×C₄×C₅	C₄×D₅	semidirect
131. C₂×C₄×C₅	C₂×Dic₅	semidirect
136. C₂²⋊D₅	C₂×Dic₅	semidirect
145. D₂₀	C₂×Dic₅	semidirect
147. C₄×D₅	C₂×Dic₅	semidirect
148. Dic₁₀	C₂×Dic₅	semidirect
157. C₅²	C₄×#41	semidirect
184. Dic₅	#709	direct (N × H)
226. C₂×C₄	D₅×Dic₅	semidirect
230. C₅	C₄×#709	semidirect
231. 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
#11717246	2
C₂×C₄×C₅×Dic₅	2
Dic₅×C₂²⋊D₅	2
Dic₅×D₂₀	2
C₄×D₅×Dic₅	2
#11716973	2
Dic₅×Dic₁₀	2
#41×Dic₅	5
C₄×#709	5

Subgroups — Dic5×#709 #13106003

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₅×#709 #13106003