Subgroups — C₂×C₅×#4097 #13106980

Order 1600.

680 conjugacy classes of subgroups.

The subgroup lattice has 263 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×#41	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4097	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅×#689	C₂	1	2	yes	maximal, complemented
5	C₂×C₅×#687	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#681	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅²×#41	—	2	2	no	—
8	C₅²×#41	C₂²	1	4	yes	—
9	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₅²×D₄	—	2	1	no	—
11	C₂×C₅²×D₄	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₅²	—	2	1	no	—
13	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
14	C₂²×C₅×Dic₅	—	2	1	no	—
15	C₂×C₅×Dic₁₀	—	2	2	no	—
16	C₂×C₅×Dic₁₀	—	2	1	no	—
17	C₂×C₅×Dic₁₀	C₂²	1	2	yes	—
18	C₂×C₅×Dic₁₀	C₂²	1	4	yes	complemented
19	C₅×#689	C₂²	1	8	yes	complemented
20	C₅×#687	C₂²	1	8	yes	complemented
21	C₅×#681	C₂²	1	4	yes	complemented
22	C₂×C₅×#168	C₂²	1	2	yes	—
23	C₂×C₅×#137	—	5	1	no	maximal
24	C₂×#4097	C₅	1	1	yes	maximal, characteristic, direct product factor
25	C₂³×C₅²	—	2	1	no	—
26	C₂³×C₅²	D₄	1	1	yes	characteristic
27	C₅²×Q₈	—	2	1	no	—
28	C₅²×Q₈	C₂³	1	2	yes	—
29	C₅²×D₄	—	2	4	no	—
30	C₅²×D₄	—	2	1	no	—
31	C₅²×D₄	C₂³	1	2	yes	—
32	C₂×C₄×C₅²	—	2	1	no	—
33	C₂×C₄×C₅²	—	2	4	no	—
34	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
35	C₂×C₄×C₅²	D₄	1	2	yes	—
36	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₅²	C₂³	1	2	yes	—
38	C₂×C₅×Dic₅	—	2	2	no	—
39	C₂×C₅×Dic₅	—	2	4	no	—
40	C₅×Dic₁₀	—	2	2	no	—
41	C₅×Dic₁₀	—	2	4	no	—
42	C₅×Dic₁₀	C₂³	1	4	yes	—
43	C₅×#168	C₂³	1	4	yes	—
44	C₂×C₅×#41	—	2	2	no	—
45	C₂×C₅×#41	D₅	1	1	yes	characteristic
46	C₂×C₅×#41	C₂×C₅	1	1	yes	characteristic
47	C₂²×C₅×Q₈	—	5	1	no	—
48	C₂²×Dic₁₀	C₂×C₅	1	1	yes	characteristic, complemented
49	C₅×#137	—	5	8	no	—
50	C₂×C₅×Q₁₆	—	5	2	no	—
51	C₂×C₅×SD₁₆	—	5	2	no	—
52	C₂×C₅×M₁₆	—	5	1	no	—
53	#4097	C₂×C₅	1	8	yes	direct product factor
54	C₂×#689	C₂×C₅	1	2	yes	complemented
55	C₂×#687	C₂×C₅	1	2	yes	complemented
56	C₂×#681	C₂×C₅	1	1	yes	characteristic, complemented
57	C₂²×C₅²	—	2	2	no	—
58	C₂²×C₅²	—	2	2	no	—
59	C₂²×C₅²	—	4	2	no	—
60	C₂²×C₅²	C₂×D₄	1	2	yes	—
61	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
62	C₄×C₅²	—	2	2	no	—
63	C₄×C₅²	C₂×D₄	1	2	yes	—
64	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
65	C₄×C₅²	C₂⁴	1	1	yes	characteristic
66	C₅×Dic₅	—	2	4	no	—
67	C₅×#41	—	2	8	no	—
68	C₅×#41	—	2	2	no	—
69	C₅×#41	—	2	2	no	—
70	C₅×#41	—	2	8	no	—
71	C₅×#41	C₂×D₅	1	4	yes	—
72	C₅×#41	C₂²×C₅	1	4	yes	—
73	C₂×C₅×Q₈	—	2	2	no	—
74	C₂×C₅×Q₈	—	5	2	no	—
75	C₂×C₅×Q₈	—	5	4	no	—
76	C₂×C₅×Q₈	—	10	2	no	—
77	C₂×C₅×Q₈	—	10	1	no	—
78	C₂×C₅×Q₈	C₂×D₅	1	1	yes	characteristic
79	C₂×C₅×Q₈	C₂²×C₅	1	1	yes	characteristic
80	C₂×C₅×D₄	—	2	4	no	—
81	C₂×C₅×D₄	—	2	1	no	—
82	C₂×C₅×D₄	—	2	1	no	—
83	C₂×C₅×D₄	—	2	2	no	—
84	C₂×C₅×D₄	C₂×D₅	1	1	yes	characteristic
85	C₂×C₅×D₄	C₂²×C₅	1	1	yes	characteristic
86	C₂²×C₄×C₅	—	2	4	no	—
87	C₂²×C₄×C₅	—	2	1	no	—
88	C₂²×C₄×C₅	—	2	1	no	—
89	C₂²×C₄×C₅	—	2	2	no	—
90	C₂²×C₄×C₅	—	10	1	no	—
91	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
92	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
93	C₂²×Dic₅	—	2	1	no	—
94	C₂×Dic₁₀	—	2	2	no	—
95	C₂×Dic₁₀	—	2	1	no	—
96	C₂×Dic₁₀	C₂²×C₅	1	2	yes	—
97	C₂×Dic₁₀	C₂²×C₅	1	4	yes	complemented
98	C₅×Q₁₆	—	5	8	no	—
99	C₅×SD₁₆	—	5	8	no	—
100	C₅×M₁₆	—	5	4	no	—
101	C₂×C₅×C₈	—	5	2	no	—
102	#689	C₂²×C₅	1	8	yes	complemented
103	#687	C₂²×C₅	1	8	yes	complemented
104	#681	C₂²×C₅	1	4	yes	complemented
105	C₂×#168	C₂²×C₅	1	2	yes	—
106	C₂×#137	—	5	1	no	Sylow
107	C₂×C₅²	—	2	2	no	—
108	C₂×C₅²	—	4	2	no	—
109	C₂×C₅²	#137	1	2	yes	complemented
110	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
111	C₂³×C₅	—	2	4	no	—
112	C₂³×C₅	—	2	1	no	—
113	C₂³×C₅	—	2	1	no	—
114	C₂³×C₅	—	2	2	no	—
115	C₂³×C₅	C₂²⋊D₅	1	1	yes	characteristic
116	C₂³×C₅	C₅×D₄	1	1	yes	characteristic
117	C₅×Q₈	—	2	4	no	—
118	C₅×Q₈	—	2	1	no	—
119	C₅×Q₈	—	2	1	no	—
120	C₅×Q₈	—	2	4	no	—
121	C₅×Q₈	—	5	4	no	—
122	C₅×Q₈	—	10	4	no	—
123	C₅×Q₈	—	10	2	no	—
124	C₅×Q₈	C₂²×D₅	1	2	yes	—
125	C₅×Q₈	C₂³×C₅	1	2	yes	—
126	C₅×D₄	—	2	16	no	—
127	C₅×D₄	—	2	4	no	—
128	C₅×D₄	—	2	4	no	—
129	C₅×D₄	—	2	4	no	—
130	C₅×D₄	—	2	1	no	—
131	C₅×D₄	—	2	1	no	—
132	C₅×D₄	—	2	4	no	—
133	C₅×D₄	C₂²×D₅	1	2	yes	—
134	C₅×D₄	C₂³×C₅	1	2	yes	—
135	C₂×C₄×C₅	—	2	4	no	—
136	C₂×C₄×C₅	—	2	16	no	—
137	C₂×C₄×C₅	—	2	1	no	—
138	C₂×C₄×C₅	—	2	1	no	—
139	C₂×C₄×C₅	—	2	4	no	—
140	C₂×C₄×C₅	—	2	4	no	—
141	C₂×C₄×C₅	—	2	2	no	—
142	C₂×C₄×C₅	—	2	4	no	—
143	C₂×C₄×C₅	—	2	2	no	—
144	C₂×C₄×C₅	—	2	4	no	—
145	C₂×C₄×C₅	—	10	4	no	—
146	C₂×C₄×C₅	—	10	2	no	—
147	C₂×C₄×C₅	C₂²⋊D₅	1	2	yes	—
148	C₂×C₄×C₅	C₂²⋊D₅	1	1	yes	characteristic
149	C₂×C₄×C₅	C₅×D₄	1	1	yes	characteristic
150	C₂×C₄×C₅	C₅×D₄	1	2	yes	—
151	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
152	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
155	C₂×Dic₅	—	2	2	no	—
156	C₂×Dic₅	—	2	4	no	—
157	Dic₁₀	—	2	2	no	—
158	Dic₁₀	—	2	4	no	—
159	Dic₁₀	C₂³×C₅	1	4	yes	—
160	C₅×C₈	—	5	4	no	—
161	#168	C₂³×C₅	1	4	yes	—
162	C₂×#41	C₅×D₅	1	1	yes	characteristic
163	C₂²×Q₈	—	5	1	no	—
164	#137	—	5	8	no	—
165	C₂×Q₁₆	—	5	2	no	—
166	C₂×SD₁₆	—	5	2	no	—
167	C₂×M₁₆	—	5	1	no	—
168	C₅²	C₂×#137	1	1	yes	characteristic, complemented, Sylow
169	C₂²×C₅	—	2	2	no	—
170	C₂²×C₅	—	2	2	no	—
171	C₂²×C₅	—	2	8	no	—
172	C₂²×C₅	—	2	2	no	—
173	C₂²×C₅	—	2	2	no	—
174	C₂²×C₅	—	2	2	no	—
175	C₂²×C₅	—	2	4	no	—
176	C₂²×C₅	—	4	4	no	—
177	C₂²×C₅	—	4	8	no	—
178	C₂²×C₅	—	4	2	no	—
179	C₂²×C₅	—	4	2	no	—
180	C₂²×C₅	C₂×C₂²⋊D₅	1	2	yes	—
181	C₂²×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
182	C₂²×C₅	C₂×C₅×D₄	1	2	yes	—
183	C₂²×C₅	C₂×C₅×D₄	1	1	yes	characteristic
184	C₄×C₅	—	2	8	no	—
185	C₄×C₅	—	2	2	no	—
186	C₄×C₅	—	2	2	no	—
187	C₄×C₅	—	2	4	no	—
188	C₄×C₅	—	2	2	no	—
189	C₄×C₅	—	2	2	no	—
190	C₄×C₅	—	10	4	no	—
191	C₄×C₅	C₂×C₂²⋊D₅	1	2	yes	—
192	C₄×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
193	C₄×C₅	C₂×C₅×D₄	1	2	yes	—
194	C₄×C₅	C₂×C₅×D₄	1	1	yes	characteristic
195	C₄×C₅	C₂³×D₅	1	1	yes	characteristic
196	C₄×C₅	C₂⁴×C₅	1	1	yes	characteristic
197	Dic₅	—	2	4	no	—
198	#41	—	2	2	no	—
199	#41	C₂×C₅×D₅	1	4	yes	—
200	C₂×Q₈	—	5	2	no	—
201	C₂×Q₈	—	5	4	no	—
202	C₂×Q₈	—	10	1	no	—
203	C₂×Q₈	—	10	2	no	—
204	C₂×Q₈	C₂×C₅×D₅	1	1	yes	characteristic
205	C₂×D₄	—	2	1	no	—
206	C₂×D₄	C₂×C₅×D₅	1	1	yes	characteristic
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	10	1	no	—
209	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
210	Q₁₆	—	5	8	no	—
211	SD₁₆	—	5	8	no	—
212	M₁₆	—	5	4	no	—
213	C₂×C₈	—	5	2	no	—
214	C₂×C₅	—	2	2	no	—
215	C₂×C₅	—	2	2	no	—
216	C₂×C₅	—	2	4	no	—
217	C₂×C₅	—	2	2	no	—
218	C₂×C₅	—	4	4	no	—
219	C₂×C₅	—	4	8	no	—
220	C₂×C₅	—	4	2	no	—
221	C₂×C₅	—	4	2	no	—
222	C₂×C₅	#4097	1	2	yes	direct product factor
223	C₂×C₅	C₅×#137	1	2	yes	complemented
224	C₂×C₅	C₂²×C₂²⋊D₅	1	1	yes	characteristic
225	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
226	C₂³	—	2	1	no	—
227	C₂³	C₅×C₂²⋊D₅	1	1	yes	characteristic
228	Q₈	—	2	1	no	—
229	Q₈	—	5	4	no	—
230	Q₈	—	10	4	no	—
231	Q₈	—	10	2	no	—
232	Q₈	C₂²×C₅×D₅	1	2	yes	—
233	D₄	—	2	4	no	—
234	D₄	—	2	1	no	—
235	D₄	C₂²×C₅×D₅	1	2	yes	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	10	4	no	—
239	C₂×C₄	—	10	2	no	—
240	C₂×C₄	C₅×C₂²⋊D₅	1	1	yes	characteristic
241	C₂×C₄	C₅×C₂²⋊D₅	1	2	yes	—
242	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
243	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
244	C₈	—	5	4	no	—
245	C₅	—	2	2	no	—
246	C₅	C₂×#4097	1	1	yes	characteristic, direct product factor
247	C₅	C₂×C₅×#137	1	1	yes	characteristic, complemented
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	C₂×C₅×C₂²⋊D₅	1	2	yes	—
252	C₂²	C₂×C₅×C₂²⋊D₅	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	10	4	no	—
255	C₄	C₂×C₅×C₂²⋊D₅	1	2	yes	—
256	C₄	C₂×C₅×C₂²⋊D₅	1	1	yes	characteristic
257	C₄	C₂³×C₅×D₅	1	1	yes	characteristic
258	C₂	—	2	2	no	—
259	C₂	—	4	2	no	—
260	C₂	C₅×#4097	1	2	yes	direct product factor
261	C₂	C₂²×C₅×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂×C₂²⋊D₅
commutator, cosocle	C₄×C₅	C₂⁴×C₅
Frattini	C₄	C₂³×C₅×D₅
Fitting, prime radical	C₂×C₅²×#41	C₂
socle	C₂²×C₅²	C₂×D₄
hypercenter	C₂×C₅×#41	D₅
hypocenter	C₅	C₂×C₅×#137
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#41

Complements

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

N	H ≅ G/N	type
2. C₂²×C₅×Dic₁₀	C₂	semidirect
3. C₅×#4097	C₂	direct (N × H)
4. C₂×C₅×#689	C₂	semidirect
5. C₂×C₅×#687	C₂	semidirect
6. C₂×C₅×#681	C₂	semidirect
18. C₂×C₅×Dic₁₀	C₂²	semidirect
19. C₅×#689	C₂²	semidirect
20. C₅×#687	C₂²	semidirect
21. C₅×#681	C₂²	semidirect
24. C₂×#4097	C₅	direct (N × H)
48. C₂²×Dic₁₀	C₂×C₅	semidirect
53. #4097	C₂×C₅	direct (N × H)
54. C₂×#689	C₂×C₅	semidirect
55. C₂×#687	C₂×C₅	semidirect
56. C₂×#681	C₂×C₅	semidirect
97. C₂×Dic₁₀	C₂²×C₅	semidirect
102. #689	C₂²×C₅	semidirect
103. #687	C₂²×C₅	semidirect
104. #681	C₂²×C₅	semidirect
109. C₂×C₅²	#137	semidirect
168. C₅²	C₂×#137	semidirect
222. C₂×C₅	#4097	direct (N × H)
223. C₂×C₅	C₅×#137	semidirect
246. C₅	C₂×#4097	direct (N × H)
247. C₅	C₂×C₅×#137	semidirect
260. C₂	C₅×#4097	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅²×#41	2
C₂²×C₅×Dic₁₀	2
C₅×#4097	2
C₂×C₅×#689	2
C₂×C₅×#687	2
C₂×C₅×#681	2
C₂×C₅×#137	5
C₂×#4097	5

Subgroups — C2×C5×#4097 #13106980

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4097 #13106980