Subgroups — C₂×D₅×#683 #13105930

Order 1600.

928 conjugacy classes of subgroups.

The subgroup lattice has 288 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	C₂²×D₅×Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₂²×D₅×Dic₅	C₂	1	1	yes	maximal, characteristic
4	C₂×#87806	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×#683	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#87781	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#87780	C₂	1	1	yes	maximal, characteristic, complemented
8	D₅×#683	C₂	1	8	yes	maximal, direct product factor
9	C₂³×C₅×D₅	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
11	C₂²×#1039	C₂²	1	1	yes	characteristic
12	C₂²×#1039	C₂²	1	1	yes	characteristic
13	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
14	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
15	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₅×D₅	—	2	4	no	—
17	C₂×C₄×C₅×D₅	C₂²	1	4	yes	—
18	C₂×D₅×Dic₅	—	2	4	no	—
19	C₂×D₅×Dic₅	C₄	1	8	yes	complemented
20	C₂×D₅×Dic₅	C₂²	1	4	yes	—
21	C₂×D₅×Dic₅	C₂²	1	4	yes	—
22	#87806	C₂²	1	4	yes	complemented
23	C₅×#683	C₂²	1	4	yes	complemented
24	#87781	C₂²	1	4	yes	complemented
25	#87780	C₂²	1	4	yes	complemented
26	C₂²×#683	—	5	1	no	maximal
27	C₂×D₅×C₄⋊C₄	—	5	1	no	maximal
28	C₂³×C₅²	C₂³	1	1	yes	characteristic
29	C₂²×C₅×D₅	C₂×C₄	1	4	yes	—
30	C₂²×C₅×D₅	D₄	1	4	yes	—
31	C₂²×C₅×D₅	Q₈	1	4	yes	—
32	C₂²×C₅×D₅	C₂³	1	2	yes	—
33	C₂×C₄×C₅²	—	2	2	no	—
34	C₂×C₄×C₅²	C₂³	1	2	yes	—
35	C₂×#1039	—	2	2	no	—
36	C₂×#1039	C₂×C₄	1	4	yes	complemented
37	C₂×#1039	C₂³	1	2	yes	—
38	C₂×#1039	C₂³	1	2	yes	—
39	C₂×C₅×Dic₅	—	2	2	no	—
40	C₂×C₅×Dic₅	—	2	2	no	—
41	C₂×C₅×Dic₅	C₂×C₄	1	4	yes	complemented
42	C₂×C₅×Dic₅	C₂³	1	2	yes	—
43	C₂×C₅×Dic₅	C₂³	1	2	yes	—
44	C₂×C₅×Dic₅	C₂³	1	2	yes	—
45	C₄×C₅×D₅	—	2	8	no	—
46	D₅×Dic₅	—	2	8	no	—
47	D₅×Dic₅	C₂×C₄	1	16	yes	complemented
48	C₂³×C₄×C₅	—	5	1	no	—
49	C₂³×Dic₅	—	5	1	no	—
50	C₂³×Dic₅	—	5	1	no	—
51	C₂²×C₄×D₅	—	5	1	no	—
52	C₂²×C₄×D₅	—	5	1	no	—
53	C₂²×C₄×D₅	D₅	1	1	yes	characteristic
54	C₂×C₅×C₄⋊C₄	—	5	1	no	—
55	C₂×C₄⋊Dic₅	—	5	1	no	—
56	C₂×#683	—	5	1	no	—
57	C₂×#683	—	5	1	no	—
58	C₂×#683	—	5	1	no	—
59	C₂×#683	—	5	1	no	—
60	C₂×#683	—	5	1	no	—
61	C₂×#683	—	5	8	no	—
62	C₂×#683	—	10	2	no	—
63	C₂×#683	D₅	1	1	yes	characteristic, direct product factor
64	D₅×C₄⋊C₄	—	5	8	no	—
65	C₂²×C₅²	C₂²×C₄	1	2	yes	—
66	C₂²×C₅²	C₂×D₄	1	2	yes	—
67	C₂²×C₅²	C₂×Q₈	1	2	yes	—
68	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
69	C₂×C₅×D₅	C₄⋊C₄	1	16	yes	complemented
70	C₂×C₅×D₅	C₂²×C₄	1	4	yes	—
71	C₂×C₅×D₅	C₂×D₄	1	4	yes	—
72	C₂×C₅×D₅	C₂×Q₈	1	4	yes	—
73	C₄×C₅²	—	2	2	no	—
74	#1039	—	2	2	no	—
75	#1039	C₂²×C₄	1	4	yes	complemented
76	C₅×Dic₅	—	2	2	no	—
77	C₅×Dic₅	—	2	2	no	—
78	C₅×Dic₅	C₂²×C₄	1	4	yes	complemented
79	C₂⁴×C₅	—	5	1	no	—
80	C₂³×D₅	C₂×D₅	1	1	yes	characteristic
81	C₂²×C₄×C₅	—	2	2	no	—
82	C₂²×C₄×C₅	—	5	1	no	—
83	C₂²×C₄×C₅	—	5	1	no	—
84	C₂²×C₄×C₅	—	5	1	no	—
85	C₂²×C₄×C₅	—	5	4	no	—
86	C₂²×C₄×C₅	—	10	4	no	—
87	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
89	C₂²×Dic₅	—	5	1	no	—
90	C₂²×Dic₅	—	5	1	no	—
91	C₂²×Dic₅	—	5	8	no	—
92	C₂²×Dic₅	—	5	4	no	—
93	C₂²×Dic₅	—	5	1	no	—
94	C₂²×Dic₅	—	5	1	no	—
95	C₂²×Dic₅	—	5	4	no	—
96	C₂²×Dic₅	—	10	2	no	—
97	C₂²×Dic₅	—	10	2	no	—
98	C₂²×Dic₅	—	10	4	no	—
99	C₂²×Dic₅	C₂×D₅	1	1	yes	characteristic
100	C₂²×Dic₅	C₂×D₅	1	1	yes	characteristic
101	C₂²×Dic₅	C₂×D₅	1	1	yes	characteristic
102	C₂×C₄×D₅	—	2	4	no	—
103	C₂×C₄×D₅	—	5	8	no	—
104	C₂×C₄×D₅	—	5	4	no	—
105	C₂×C₄×D₅	—	5	4	no	—
106	C₂×C₄×D₅	—	10	4	no	—
107	C₂×C₄×D₅	C₂×D₅	1	4	yes	—
108	C₅×C₄⋊C₄	—	5	4	no	—
109	C₄⋊Dic₅	—	5	4	no	—
110	#683	—	5	4	no	—
111	#683	—	5	4	no	—
112	#683	—	5	4	no	—
113	#683	—	5	4	no	—
114	#683	—	5	4	no	—
115	#683	—	10	8	no	—
116	#683	C₂×D₅	1	4	yes	direct product factor
117	C₂²×C₄⋊C₄	—	25	1	no	Sylow
118	C₂×C₅²	C₂×C₄⋊C₄	1	4	yes	complemented
119	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₅²	C₂²×Q₈	1	1	yes	characteristic
122	C₅×D₅	C₂×C₄⋊C₄	1	8	yes	complemented
123	C₂³×C₅	—	2	2	no	—
124	C₂³×C₅	—	5	4	no	—
125	C₂³×C₅	—	5	4	no	—
126	C₂³×C₅	—	5	4	no	—
127	C₂³×C₅	—	5	2	no	—
128	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
129	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
130	C₂²×D₅	Dic₁₀	1	4	yes	—
131	C₂²×D₅	C₄×D₅	1	4	yes	—
132	C₂²×D₅	C₂²⋊D₅	1	4	yes	—
133	C₂²×D₅	C₂²×D₅	1	2	yes	—
134	C₂×C₄×C₅	—	2	2	no	—
135	C₂×C₄×C₅	—	2	2	no	—
136	C₂×C₄×C₅	—	2	4	no	—
137	C₂×C₄×C₅	—	4	4	no	—
138	C₂×C₄×C₅	—	5	4	no	—
139	C₂×C₄×C₅	—	5	2	no	—
140	C₂×C₄×C₅	—	5	2	no	—
141	C₂×C₄×C₅	—	5	2	no	—
142	C₂×C₄×C₅	—	10	8	no	—
143	C₂×C₄×C₅	—	10	2	no	—
144	C₂×C₄×C₅	—	10	2	no	—
145	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
146	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
147	C₂×Dic₅	—	2	2	no	—
148	C₂×Dic₅	—	2	2	no	—
149	C₂×Dic₅	—	5	4	no	—
150	C₂×Dic₅	—	5	4	no	—
151	C₂×Dic₅	—	5	2	no	—
152	C₂×Dic₅	—	5	2	no	—
153	C₂×Dic₅	—	5	16	no	—
154	C₂×Dic₅	—	5	2	no	—
155	C₂×Dic₅	—	5	2	no	—
156	C₂×Dic₅	—	10	8	no	—
157	C₂×Dic₅	—	10	4	no	—
158	C₂×Dic₅	—	10	4	no	—
159	C₂×Dic₅	—	10	8	no	—
160	C₂×Dic₅	—	10	2	no	—
161	C₂×Dic₅	—	10	2	no	—
162	C₂×Dic₅	—	20	4	no	—
163	C₂×Dic₅	C₄×D₅	1	4	yes	complemented
164	C₂×Dic₅	C₂²×D₅	1	2	yes	—
165	C₂×Dic₅	C₂²×D₅	1	2	yes	—
166	C₂×Dic₅	C₂²×D₅	1	2	yes	—
167	C₄×D₅	—	2	8	no	—
168	C₄×D₅	—	5	16	no	—
169	C₄×D₅	—	10	8	no	—
170	C₂³×C₄	—	5	1	no	—
171	C₂³×C₄	—	25	1	no	—
172	C₂³×C₄	—	25	1	no	—
173	C₂×C₄⋊C₄	—	5	1	no	—
174	C₂×C₄⋊C₄	—	25	8	no	—
175	C₂×C₄⋊C₄	—	25	1	no	—
176	C₂×C₄⋊C₄	—	25	1	no	—
177	C₂×C₄⋊C₄	—	25	1	no	—
178	C₅²	C₂²×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
179	C₂²×C₅	—	2	4	no	—
180	C₂²×C₅	—	2	4	no	—
181	C₂²×C₅	—	2	4	no	—
182	C₂²×C₅	—	2	2	no	—
183	C₂²×C₅	—	5	4	no	—
184	C₂²×C₅	—	5	4	no	—
185	C₂²×C₅	—	5	4	no	—
186	C₂²×C₅	—	5	16	no	—
187	C₂²×C₅	C₂×Dic₁₀	1	2	yes	—
188	C₂²×C₅	C₂×C₄×D₅	1	2	yes	—
189	C₂²×C₅	C₂×C₄×D₅	1	2	yes	—
190	C₂²×C₅	D₄×D₅	1	2	yes	—
191	C₂²×C₅	Q₈×D₅	1	2	yes	—
192	C₂²×C₅	C₂×C₂²⋊D₅	1	2	yes	—
193	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
194	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
195	C₂×D₅	#683	1	16	yes	direct product factor
196	C₂×D₅	C₂×Dic₁₀	1	4	yes	—
197	C₂×D₅	C₂×C₄×D₅	1	4	yes	—
198	C₂×D₅	C₂×C₂²⋊D₅	1	4	yes	—
199	C₄×C₅	—	2	2	no	—
200	C₄×C₅	—	2	2	no	—
201	C₄×C₅	—	4	4	no	—
202	C₄×C₅	—	5	4	no	—
203	C₄×C₅	—	10	2	no	—
204	C₄×C₅	—	10	2	no	—
205	Dic₅	—	2	2	no	—
206	Dic₅	—	2	2	no	—
207	Dic₅	—	5	4	no	—
208	Dic₅	—	5	4	no	—
209	Dic₅	—	10	8	no	—
210	Dic₅	—	10	2	no	—
211	Dic₅	—	10	2	no	—
212	Dic₅	—	20	4	no	—
213	Dic₅	C₂×C₄×D₅	1	4	yes	complemented
214	C₂⁴	—	5	1	no	—
215	C₂²×C₄	—	5	1	no	—
216	C₂²×C₄	—	5	1	no	—
217	C₂²×C₄	—	5	1	no	—
218	C₂²×C₄	—	5	4	no	—
219	C₂²×C₄	—	10	4	no	—
220	C₂²×C₄	—	25	4	no	—
221	C₂²×C₄	—	25	8	no	—
222	C₂²×C₄	—	25	4	no	—
223	C₂²×C₄	—	25	1	no	—
224	C₂²×C₄	—	25	1	no	—
225	C₂²×C₄	—	50	4	no	—
226	C₂²×C₄	D₅²	1	1	yes	characteristic
227	C₄⋊C₄	—	5	4	no	—
228	C₄⋊C₄	—	25	4	no	—
229	C₄⋊C₄	—	25	4	no	—
230	C₄⋊C₄	—	25	4	no	—
231	C₂×C₅	—	2	2	no	—
232	C₂×C₅	—	2	2	no	—
233	C₂×C₅	—	2	2	no	—
234	C₂×C₅	—	2	8	no	—
235	C₂×C₅	—	5	8	no	—
236	C₂×C₅	D₅×C₄⋊C₄	1	4	yes	complemented
237	C₂×C₅	C₂×#683	1	4	yes	complemented
238	C₂×C₅	C₂²×Dic₁₀	1	1	yes	characteristic
239	C₂×C₅	C₂²×C₄×D₅	1	1	yes	characteristic
240	C₂×C₅	C₂²×C₄×D₅	1	1	yes	characteristic
241	C₂×C₅	C₂×D₄×D₅	1	1	yes	characteristic
242	C₂×C₅	C₂×Q₈×D₅	1	1	yes	characteristic
243	C₂×C₅	C₂²×C₂²⋊D₅	1	1	yes	characteristic
244	D₅	C₂×#683	1	8	yes	direct product factor
245	C₂³	—	5	4	no	—
246	C₂³	—	5	4	no	—
247	C₂³	—	5	4	no	—
248	C₂³	—	5	2	no	—
249	C₂³	C₂×D₅²	1	1	yes	characteristic
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	5	4	no	—
252	C₂×C₄	—	5	2	no	—
253	C₂×C₄	—	5	2	no	—
254	C₂×C₄	—	5	2	no	—
255	C₂×C₄	—	10	8	no	—
256	C₂×C₄	—	10	2	no	—
257	C₂×C₄	—	10	2	no	—
258	C₂×C₄	—	25	16	no	—
259	C₂×C₄	—	25	4	no	—
260	C₂×C₄	—	25	2	no	—
261	C₂×C₄	—	25	2	no	—
262	C₂×C₄	—	50	8	no	—
263	C₂×C₄	—	50	2	no	—
264	C₂×C₄	C₂×D₅²	1	2	yes	—
265	C₅	—	2	2	no	—
266	C₅	C₂×D₅×C₄⋊C₄	1	1	yes	characteristic, complemented
267	C₅	C₂²×#683	1	1	yes	characteristic, complemented
268	C₂²	—	5	4	no	—
269	C₂²	—	5	4	no	—
270	C₂²	—	5	4	no	—
271	C₂²	—	5	16	no	—
272	C₂²	D₅×Dic₁₀	1	2	yes	—
273	C₂²	C₄×D₅²	1	2	yes	—
274	C₂²	D₅×C₂²⋊D₅	1	2	yes	—
275	C₂²	C₂²×D₅²	1	1	yes	characteristic
276	C₄	—	2	2	no	—
277	C₄	—	5	4	no	—
278	C₄	—	10	2	no	—
279	C₄	—	10	2	no	—
280	C₄	—	25	4	no	—
281	C₄	—	50	2	no	—
282	C₂	—	5	8	no	—
283	C₂	D₅×#683	1	4	yes	direct product factor
284	C₂	C₂×D₅×Dic₁₀	1	1	yes	characteristic
285	C₂	C₂×C₄×D₅²	1	1	yes	characteristic
286	C₂	C₂×D₅×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₅²
commutator	C₂×C₅²	C₂³×C₄
Frattini	C₂²	C₂²×D₅²
Fitting	C₂²×C₄×C₅²	C₂²
socle	C₂³×C₅²	C₂³
prime radical	C₂³×C₅×D₅	C₂²
cosocle	C₂²×C₅²	C₂⁴
hypercenter	C₂²×C₄	D₅²
hypocenter	C₅²	C₂²×C₄⋊C₄
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
4. C₂×#87806	C₂	semidirect
5. C₂×C₅×#683	C₂	semidirect
6. C₂×#87781	C₂	semidirect
7. C₂×#87780	C₂	semidirect
8. D₅×#683	C₂	direct (N × H)
19. C₂×D₅×Dic₅	C₄	semidirect
22. #87806	C₂²	semidirect
23. C₅×#683	C₂²	semidirect
24. #87781	C₂²	semidirect
25. #87780	C₂²	semidirect
36. C₂×#1039	C₂×C₄	semidirect
41. C₂×C₅×Dic₅	C₂×C₄	semidirect
47. D₅×Dic₅	C₂×C₄	semidirect
63. C₂×#683	D₅	direct (N × H)
69. C₂×C₅×D₅	C₄⋊C₄	semidirect
75. #1039	C₂²×C₄	semidirect
78. C₅×Dic₅	C₂²×C₄	semidirect
116. #683	C₂×D₅	direct (N × H)
118. C₂×C₅²	C₂×C₄⋊C₄	semidirect
122. C₅×D₅	C₂×C₄⋊C₄	semidirect
163. C₂×Dic₅	C₄×D₅	semidirect
178. C₅²	C₂²×C₄⋊C₄	semidirect
195. C₂×D₅	#683	direct (N × H)
213. Dic₅	C₂×C₄×D₅	semidirect
236. C₂×C₅	D₅×C₄⋊C₄	semidirect
237. C₂×C₅	C₂×#683	semidirect
244. D₅	C₂×#683	direct (N × H)
266. C₅	C₂×D₅×C₄⋊C₄	semidirect
267. C₅	C₂²×#683	semidirect
283. C₂	D₅×#683	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
C₂²×D₅×Dic₅	2
C₂×#87806	2
C₂×C₅×#683	2
C₂×#87781	2
C₂×#87780	2
D₅×#683	2
C₂²×#683	5
C₂×D₅×C₄⋊C₄	5

Subgroups — C2×D5×#683 #13105930

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₅×#683 #13105930