Subgroups — C₂×C₅×#3937 #13102198

Order 1600.

448 conjugacy classes of subgroups.

The subgroup lattice has 260 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×C₄²×C₅²	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#681	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#3937	C₂	1	4	yes	maximal, direct product factor
5	C₂²×C₄×C₅²	—	2	1	no	—
6	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
7	C₂×C₅×C₂²⋊D₅	—	2	1	no	—
8	C₅×#709	—	2	1	no	—
9	C₅×#709	—	2	1	no	—
10	C₅×#709	C₄	1	2	yes	complemented
11	C₅×#709	C₂²	1	2	yes	—
12	C₂×C₅×D₂₀	C₄	1	1	yes	characteristic, complemented
13	C₂×C₄×C₅×D₅	—	2	1	no	—
14	C₂×C₅×Dic₁₀	C₄	1	1	yes	characteristic, complemented
15	C₄²×C₅²	—	2	1	no	—
16	C₄²×C₅²	C₂²	1	2	yes	complemented
17	C₅×#681	—	2	1	no	—
18	C₅×#681	C₂²	1	2	yes	complemented
19	C₂×C₅×#168	—	2	1	no	—
20	C₂×C₅×#104	—	5	1	no	maximal
21	C₂×#3937	C₅	1	1	yes	maximal, characteristic, direct product factor
22	C₂³×C₅²	D₄	1	1	yes	characteristic
23	C₂²×C₅×D₅	—	2	1	no	—
24	C₂×C₄×C₅²	—	2	4	no	—
25	C₂×C₄×C₅²	—	2	2	no	—
26	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
29	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
30	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
31	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
32	C₅×C₂²⋊D₅	—	2	2	no	—
33	C₅×C₂²⋊D₅	—	2	2	no	—
34	C₂×C₅×Dic₅	—	2	1	no	—
35	C₅×D₂₀	—	2	1	no	—
36	C₅×D₂₀	C₂×C₄	1	2	yes	complemented
37	C₄×C₅×D₅	—	2	2	no	—
38	C₄×C₅×D₅	—	2	2	no	—
39	C₅×Dic₁₀	—	2	1	no	—
40	C₅×Dic₁₀	C₂×C₄	1	2	yes	complemented
41	C₅×#168	—	2	2	no	—
42	C₂×C₅×#41	—	5	1	no	—
43	C₂×#709	C₂×C₅	1	1	yes	characteristic
44	C₂×C₅×M₁₆	—	5	1	no	—
45	C₂×C₄²×C₅	—	2	2	no	—
46	C₂×C₄²×C₅	D₅	1	1	yes	characteristic, complemented
47	C₂×C₄²×C₅	C₂×C₅	1	1	yes	characteristic, complemented
48	C₂×#681	C₂×C₅	1	1	yes	characteristic, complemented
49	C₅×#104	—	5	4	no	—
50	#3937	C₂×C₅	1	4	yes	direct product factor
51	C₂²×C₅²	—	2	2	no	—
52	C₂²×C₅²	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₅²	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
55	C₂×C₅×D₅	—	2	2	no	—
56	C₂×C₅×D₅	—	4	2	no	—
57	C₄×C₅²	—	2	4	no	—
58	C₄×C₅²	C₂²⋊C₄	1	1	yes	characteristic
59	C₄×C₅²	C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
61	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
62	C₅×Dic₅	—	2	2	no	—
63	C₅×#41	—	5	2	no	—
64	C₅×#41	—	5	2	no	—
65	C₅×#41	—	10	1	no	—
66	C₅×#41	—	10	1	no	—
67	C₂×C₅×Q₈	—	5	1	no	—
68	C₂×C₅×D₄	—	5	1	no	—
69	C₂×C₅×D₄	—	10	1	no	—
70	C₂²×C₄×C₅	—	2	4	no	—
71	C₂²×C₄×C₅	—	2	1	no	—
72	C₂²×C₄×C₅	—	2	1	no	—
73	C₂²×C₄×C₅	—	2	2	no	—
74	C₂²×C₄×C₅	—	10	1	no	—
75	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
77	C₂×C₂²⋊D₅	—	2	1	no	—
78	#709	—	2	1	no	—
79	#709	—	2	1	no	—
80	#709	C₄×C₅	1	2	yes	complemented
81	#709	C₂²×C₅	1	2	yes	—
82	C₂×D₂₀	C₄×C₅	1	1	yes	characteristic, complemented
83	C₂×C₄×D₅	—	2	1	no	—
84	C₂×Dic₁₀	C₄×C₅	1	1	yes	characteristic, complemented
85	C₅×M₁₆	—	5	2	no	—
86	C₅×M₁₆	—	10	1	no	—
87	C₂×C₅×C₈	—	10	1	no	—
88	C₄²×C₅	—	2	4	no	—
89	C₄²×C₅	—	2	1	no	—
90	C₄²×C₅	—	2	1	no	—
91	C₄²×C₅	—	2	4	no	—
92	C₄²×C₅	C₂×D₅	1	2	yes	complemented
93	C₄²×C₅	C₂²×C₅	1	2	yes	complemented
94	#681	—	2	1	no	—
95	#681	C₂²×C₅	1	2	yes	complemented
96	C₂×#168	—	2	1	no	—
97	C₂×#104	—	5	1	no	Sylow
98	C₂×C₅²	—	2	1	no	—
99	C₂×C₅²	—	2	1	no	—
100	C₂×C₅²	#104	1	2	yes	complemented
101	C₂×C₅²	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₅×D₅	—	4	2	no	—
103	C₂³×C₅	—	2	2	no	—
104	C₂³×C₅	—	10	1	no	—
105	C₂³×C₅	C₂²⋊D₅	1	1	yes	characteristic
106	C₂³×C₅	C₅×D₄	1	1	yes	characteristic
107	C₂²×D₅	—	2	1	no	—
108	C₅×Q₈	—	5	2	no	—
109	C₅×Q₈	—	10	1	no	—
110	C₅×D₄	—	5	2	no	—
111	C₅×D₄	—	10	2	no	—
112	C₅×D₄	—	10	2	no	—
113	C₅×D₄	—	10	1	no	—
114	C₂×C₄×C₅	—	2	8	no	—
115	C₂×C₄×C₅	—	2	16	no	—
116	C₂×C₄×C₅	—	2	4	no	—
117	C₂×C₄×C₅	—	2	4	no	—
118	C₂×C₄×C₅	—	2	2	no	—
119	C₂×C₄×C₅	—	2	2	no	—
120	C₂×C₄×C₅	—	2	2	no	—
121	C₂×C₄×C₅	—	2	2	no	—
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	—	2	2	no	—
125	C₂×C₄×C₅	—	2	2	no	—
126	C₂×C₄×C₅	—	10	1	no	—
127	C₂×C₄×C₅	—	10	2	no	—
128	C₂×C₄×C₅	—	10	2	no	—
129	C₂×C₄×C₅	C₄×D₅	1	1	yes	characteristic
130	C₂×C₄×C₅	C₄×D₅	1	1	yes	characteristic
131	C₂×C₄×C₅	D₂₀	1	1	yes	characteristic
132	C₂×C₄×C₅	D₂₀	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂²⋊D₅	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂×C₄×C₅	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂×C₄×C₅	1	1	yes	characteristic
136	C₂×C₄×C₅	C₅×D₄	1	1	yes	characteristic
137	C₂×C₄×C₅	C₅×D₄	1	1	yes	characteristic
138	C₂×C₄×C₅	C₅×D₄	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
141	C₂²⋊D₅	—	2	2	no	—
142	C₂²⋊D₅	—	2	2	no	—
143	C₂×Dic₅	—	2	1	no	—
144	D₂₀	—	2	1	no	—
145	D₂₀	C₂×C₄×C₅	1	2	yes	complemented
146	C₄×D₅	—	2	2	no	—
147	C₄×D₅	—	2	2	no	—
148	Dic₁₀	—	2	1	no	—
149	Dic₁₀	C₂×C₄×C₅	1	2	yes	complemented
150	C₅×C₈	—	10	2	no	—
151	#168	—	2	2	no	—
152	C₂×#41	—	5	1	no	—
153	C₂×M₁₆	—	5	1	no	—
154	C₂×C₄²	C₅×D₅	1	1	yes	characteristic, complemented
155	#104	—	5	4	no	—
156	C₅²	C₂×#104	1	1	yes	characteristic, complemented, Sylow
157	C₂²×C₅	—	2	8	no	—
158	C₂²×C₅	—	2	2	no	—
159	C₂²×C₅	—	2	2	no	—
160	C₂²×C₅	—	2	2	no	—
161	C₂²×C₅	—	2	2	no	—
162	C₂²×C₅	—	2	2	no	—
163	C₂²×C₅	—	10	2	no	—
164	C₂²×C₅	—	20	2	no	—
165	C₂²×C₅	#685	1	1	yes	characteristic
166	C₂²×C₅	#685	1	1	yes	characteristic
167	C₂²×C₅	C₅×C₂²⋊C₄	1	1	yes	characteristic
168	C₂²×C₅	C₅×C₂²⋊C₄	1	1	yes	characteristic
169	C₂²×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
170	C₂²×C₅	C₂×C₅×D₄	1	1	yes	characteristic
171	C₂×D₅	—	2	2	no	—
172	C₂×D₅	—	4	2	no	—
173	C₄×C₅	—	2	16	no	—
174	C₄×C₅	—	2	4	no	—
175	C₄×C₅	—	2	4	no	—
176	C₄×C₅	—	2	2	no	—
177	C₄×C₅	—	2	2	no	—
178	C₄×C₅	—	2	2	no	—
179	C₄×C₅	—	2	2	no	—
180	C₄×C₅	—	10	2	no	—
181	C₄×C₅	#685	1	1	yes	characteristic
182	C₄×C₅	#685	1	1	yes	characteristic
183	C₄×C₅	C₅×C₂²⋊C₄	1	1	yes	characteristic
184	C₄×C₅	C₅×C₂²⋊C₄	1	1	yes	characteristic
185	C₄×C₅	C₂×C₄×D₅	1	1	yes	characteristic
186	C₄×C₅	C₂×D₂₀	1	1	yes	characteristic
187	C₄×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
188	C₄×C₅	C₂×C₅×D₄	1	1	yes	characteristic
189	Dic₅	—	2	2	no	—
190	#41	—	5	2	no	—
191	#41	—	5	2	no	—
192	#41	—	10	1	no	—
193	#41	—	10	1	no	—
194	C₂×Q₈	—	5	1	no	—
195	C₂×D₄	—	5	1	no	—
196	C₂×D₄	—	10	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	10	1	no	—
199	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
200	M₁₆	—	5	2	no	—
201	M₁₆	—	10	1	no	—
202	C₂×C₈	—	10	1	no	—
203	C₄²	—	2	1	no	—
204	C₄²	C₂×C₅×D₅	1	2	yes	complemented
205	C₂×C₅	—	2	4	no	—
206	C₂×C₅	—	2	4	no	—
207	C₂×C₅	—	2	1	no	—
208	C₂×C₅	—	2	1	no	—
209	C₂×C₅	—	2	1	no	—
210	C₂×C₅	—	2	1	no	—
211	C₂×C₅	—	2	4	no	—
212	C₂×C₅	—	2	2	no	—
213	C₂×C₅	—	20	2	no	—
214	C₂×C₅	#3937	1	2	yes	direct product factor
215	C₂×C₅	C₅×#104	1	2	yes	complemented
216	C₂×C₅	C₂×#685	1	1	yes	characteristic
217	C₂×C₅	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
218	D₅	—	4	2	no	—
219	C₂³	—	10	1	no	—
220	C₂³	C₅×C₂²⋊D₅	1	1	yes	characteristic
221	Q₈	—	5	2	no	—
222	Q₈	—	10	1	no	—
223	D₄	—	5	2	no	—
224	D₄	—	10	2	no	—
225	D₄	—	10	2	no	—
226	D₄	—	10	1	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	10	1	no	—
230	C₂×C₄	—	10	2	no	—
231	C₂×C₄	—	10	2	no	—
232	C₂×C₄	C₄×C₅×D₅	1	1	yes	characteristic
233	C₂×C₄	C₄×C₅×D₅	1	1	yes	characteristic
234	C₂×C₄	C₅×D₂₀	1	1	yes	characteristic
235	C₂×C₄	C₅×D₂₀	1	1	yes	characteristic
236	C₂×C₄	C₅×C₂²⋊D₅	1	1	yes	characteristic
237	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
238	C₈	—	10	2	no	—
239	C₅	—	2	2	no	—
240	C₅	C₂×#3937	1	1	yes	characteristic, direct product factor
241	C₅	C₂×C₅×#104	1	1	yes	characteristic, complemented
242	C₂²	—	2	2	no	—
243	C₂²	—	10	2	no	—
244	C₂²	—	20	2	no	—
245	C₂²	C₅×#685	1	1	yes	characteristic
246	C₂²	C₅×#685	1	1	yes	characteristic
247	C₂²	C₂×C₅×C₂²⋊D₅	1	1	yes	characteristic
248	C₄	—	2	4	no	—
249	C₄	—	10	2	no	—
250	C₄	C₅×#685	1	1	yes	characteristic
251	C₄	C₅×#685	1	1	yes	characteristic
252	C₄	C₂×C₄×C₅×D₅	1	1	yes	characteristic
253	C₄	C₂×C₅×D₂₀	1	1	yes	characteristic
254	C₂	—	2	1	no	—
255	C₂	—	2	1	no	—
256	C₂	—	20	2	no	—
257	C₂	C₅×#3937	1	2	yes	direct product factor
258	C₂	C₂×C₅×#685	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	D₂₀
commutator	C₄×C₅	C₂²×C₄×C₅
Frattini	C₂×C₄	C₂²×C₅×D₅
Fitting	C₂×C₄²×C₅²	C₂
socle	C₂²×C₅²	C₂²⋊C₄
prime radical	C₂×C₅×#709	C₂
cosocle	C₂×C₄×C₅	C₂³×C₅
hypercenter	C₂×C₄²×C₅	D₅
hypocenter	C₅	C₂×C₅×#104
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₄²×C₅

Complements

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

N	H ≅ G/N	type
2. C₂×C₄²×C₅²	C₂	semidirect
3. C₂×C₅×#681	C₂	semidirect
4. C₅×#3937	C₂	direct (N × H)
10. C₅×#709	C₄	semidirect
12. C₂×C₅×D₂₀	C₄	semidirect
14. C₂×C₅×Dic₁₀	C₄	semidirect
16. C₄²×C₅²	C₂²	semidirect
18. C₅×#681	C₂²	semidirect
21. C₂×#3937	C₅	direct (N × H)
36. C₅×D₂₀	C₂×C₄	semidirect
40. C₅×Dic₁₀	C₂×C₄	semidirect
46. C₂×C₄²×C₅	D₅	semidirect
47. C₂×C₄²×C₅	C₂×C₅	semidirect
48. C₂×#681	C₂×C₅	semidirect
50. #3937	C₂×C₅	direct (N × H)
80. #709	C₄×C₅	semidirect
82. C₂×D₂₀	C₄×C₅	semidirect
84. C₂×Dic₁₀	C₄×C₅	semidirect
92. C₄²×C₅	C₂×D₅	semidirect
93. C₄²×C₅	C₂²×C₅	semidirect
95. #681	C₂²×C₅	semidirect
100. C₂×C₅²	#104	semidirect
145. D₂₀	C₂×C₄×C₅	semidirect
149. Dic₁₀	C₂×C₄×C₅	semidirect
154. C₂×C₄²	C₅×D₅	semidirect
156. C₅²	C₂×#104	semidirect
204. C₄²	C₂×C₅×D₅	semidirect
214. C₂×C₅	#3937	direct (N × H)
215. C₂×C₅	C₅×#104	semidirect
240. C₅	C₂×#3937	direct (N × H)
241. C₅	C₂×C₅×#104	semidirect
257. C₂	C₅×#3937	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×C₅×#709	2
C₂×C₄²×C₅²	2
C₂×C₅×#681	2
C₅×#3937	2
C₂×C₅×#104	5
C₂×#3937	5

Subgroups — C2×C5×#3937 #13102198

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#3937 #13102198