Subgroups — C₂×C₅×#3969 #13102493

Order 1600.

504 conjugacy classes of subgroups.

The subgroup lattice has 276 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	C₂×C₄×C₅×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#681	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#3969	C₂	1	4	yes	maximal, direct product factor
5	C₅²×#41	—	2	1	no	—
6	C₅²×#41	—	2	1	no	—
7	C₅²×#41	C₄	1	2	yes	complemented
8	C₅²×#41	C₂²	1	2	yes	—
9	C₂×C₅²×Q₈	C₄	1	1	yes	characteristic, complemented
10	C₂×C₅²×D₄	—	2	1	no	—
11	C₂×C₅²×D₄	C₄	1	1	yes	characteristic, complemented
12	C₂²×C₄×C₅²	—	2	1	no	—
13	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
14	C₂²×C₅×Dic₅	—	2	1	no	—
15	C₄×C₅×Dic₅	—	2	1	no	—
16	C₄×C₅×Dic₅	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×#3969	C₅	1	1	yes	maximal, characteristic, direct product factor
22	C₂³×C₅²	—	2	1	no	—
23	C₂³×C₅²	D₄	1	1	yes	characteristic
24	C₅²×Q₈	—	2	1	no	—
25	C₅²×Q₈	C₂×C₄	1	2	yes	complemented
26	C₅²×D₄	—	2	2	no	—
27	C₅²×D₄	—	2	2	no	—
28	C₅²×D₄	—	2	1	no	—
29	C₅²×D₄	C₂×C₄	1	2	yes	complemented
30	C₂×C₄×C₅²	—	2	1	no	—
31	C₂×C₄×C₅²	—	2	2	no	—
32	C₂×C₄×C₅²	—	2	2	no	—
33	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
36	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
37	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
38	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
39	C₂×C₅×Dic₅	—	2	4	no	—
40	C₂×C₅×Dic₅	—	2	2	no	—
41	C₅×#168	—	2	2	no	—
42	C₂×C₅×#41	—	2	2	no	—
43	C₂×C₅×#41	D₅	1	1	yes	characteristic
44	C₂×C₅×#41	C₂×C₅	1	1	yes	characteristic
45	C₂×C₅×M₁₆	—	5	1	no	—
46	C₂×C₄²×C₅	—	5	1	no	—
47	C₂×C₄×Dic₅	C₂×C₅	1	1	yes	characteristic, complemented
48	C₂×#681	C₂×C₅	1	1	yes	characteristic, complemented
49	C₅×#104	—	5	4	no	—
50	#3969	C₂×C₅	1	4	yes	direct product factor
51	C₂²×C₅²	—	2	2	no	—
52	C₂²×C₅²	—	2	2	no	—
53	C₂²×C₅²	—	4	2	no	—
54	C₂²×C₅²	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₅²	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
57	C₄×C₅²	—	2	2	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	4	no	—
63	C₅×#41	—	2	4	no	—
64	C₅×#41	—	2	4	no	—
65	C₅×#41	—	2	1	no	—
66	C₅×#41	—	2	1	no	—
67	C₅×#41	—	2	1	no	—
68	C₅×#41	—	2	1	no	—
69	C₅×#41	—	2	4	no	—
70	C₅×#41	—	2	4	no	—
71	C₅×#41	Dic₅	1	2	yes	complemented
72	C₅×#41	C₄×C₅	1	2	yes	complemented
73	C₅×#41	C₂×D₅	1	2	yes	—
74	C₅×#41	C₂²×C₅	1	2	yes	—
75	C₂×C₅×Q₈	—	2	2	no	—
76	C₂×C₅×Q₈	Dic₅	1	1	yes	characteristic, complemented
77	C₂×C₅×Q₈	C₄×C₅	1	1	yes	characteristic, complemented
78	C₂×C₅×D₄	—	2	4	no	—
79	C₂×C₅×D₄	—	2	1	no	—
80	C₂×C₅×D₄	—	2	1	no	—
81	C₂×C₅×D₄	—	2	2	no	—
82	C₂×C₅×D₄	Dic₅	1	1	yes	characteristic, complemented
83	C₂×C₅×D₄	C₄×C₅	1	1	yes	characteristic, complemented
84	C₂²×C₄×C₅	—	2	4	no	—
85	C₂²×C₄×C₅	—	2	1	no	—
86	C₂²×C₄×C₅	—	2	1	no	—
87	C₂²×C₄×C₅	—	2	2	no	—
88	C₂²×C₄×C₅	—	10	1	no	—
89	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
90	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
91	C₂²×Dic₅	—	2	1	no	—
92	C₅×M₁₆	—	5	2	no	—
93	C₅×M₁₆	—	10	1	no	—
94	C₂×C₅×C₈	—	10	1	no	—
95	C₄²×C₅	—	5	2	no	—
96	C₄²×C₅	—	10	1	no	—
97	C₄×Dic₅	—	2	1	no	—
98	C₄×Dic₅	C₂²×C₅	1	2	yes	complemented
99	#681	—	2	1	no	—
100	#681	C₂²×C₅	1	2	yes	complemented
101	C₂×#168	—	2	1	no	—
102	C₂×#104	—	5	1	no	Sylow
103	C₂×C₅²	—	2	1	no	—
104	C₂×C₅²	—	2	1	no	—
105	C₂×C₅²	—	4	2	no	—
106	C₂×C₅²	#104	1	2	yes	complemented
107	C₂×C₅²	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂³×C₅	—	2	4	no	—
109	C₂³×C₅	—	2	1	no	—
110	C₂³×C₅	—	2	1	no	—
111	C₂³×C₅	—	2	2	no	—
112	C₂³×C₅	C₂²⋊D₅	1	1	yes	characteristic
113	C₂³×C₅	C₅×D₄	1	1	yes	characteristic
114	C₅×Q₈	—	2	4	no	—
115	C₅×Q₈	—	2	1	no	—
116	C₅×Q₈	—	2	1	no	—
117	C₅×Q₈	—	2	4	no	—
118	C₅×Q₈	C₂×Dic₅	1	2	yes	complemented
119	C₅×Q₈	C₂×C₄×C₅	1	2	yes	complemented
120	C₅×D₄	—	2	8	no	—
121	C₅×D₄	—	2	8	no	—
122	C₅×D₄	—	2	4	no	—
123	C₅×D₄	—	2	2	no	—
124	C₅×D₄	—	2	2	no	—
125	C₅×D₄	—	2	2	no	—
126	C₅×D₄	—	2	2	no	—
127	C₅×D₄	—	2	1	no	—
128	C₅×D₄	—	2	1	no	—
129	C₅×D₄	—	2	4	no	—
130	C₅×D₄	C₂×Dic₅	1	2	yes	complemented
131	C₅×D₄	C₂×C₄×C₅	1	2	yes	complemented
132	C₂×C₄×C₅	—	2	4	no	—
133	C₂×C₄×C₅	—	2	8	no	—
134	C₂×C₄×C₅	—	2	8	no	—
135	C₂×C₄×C₅	—	2	1	no	—
136	C₂×C₄×C₅	—	2	1	no	—
137	C₂×C₄×C₅	—	2	2	no	—
138	C₂×C₄×C₅	—	2	2	no	—
139	C₂×C₄×C₅	—	2	2	no	—
140	C₂×C₄×C₅	—	2	2	no	—
141	C₂×C₄×C₅	—	2	2	no	—
142	C₂×C₄×C₅	—	2	2	no	—
143	C₂×C₄×C₅	—	2	2	no	—
144	C₂×C₄×C₅	—	2	2	no	—
145	C₂×C₄×C₅	—	2	2	no	—
146	C₂×C₄×C₅	—	2	2	no	—
147	C₂×C₄×C₅	—	10	4	no	—
148	C₂×C₄×C₅	—	10	2	no	—
149	C₂×C₄×C₅	C₂×Dic₅	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂×Dic₅	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂²⋊D₅	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂²⋊D₅	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂²⋊D₅	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂×C₄×C₅	1	1	yes	characteristic
155	C₂×C₄×C₅	C₂×C₄×C₅	1	1	yes	characteristic
156	C₂×C₄×C₅	C₅×D₄	1	1	yes	characteristic
157	C₂×C₄×C₅	C₅×D₄	1	1	yes	characteristic
158	C₂×C₄×C₅	C₅×D₄	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
160	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
161	C₂×Dic₅	—	2	4	no	—
162	C₂×Dic₅	—	2	2	no	—
163	C₅×C₈	—	10	2	no	—
164	#168	—	2	2	no	—
165	C₂×#41	C₅×D₅	1	1	yes	characteristic
166	C₂×M₁₆	—	5	1	no	—
167	C₂×C₄²	—	5	1	no	—
168	#104	—	5	4	no	—
169	C₅²	C₂×#104	1	1	yes	characteristic, complemented, Sylow
170	C₂²×C₅	—	2	2	no	—
171	C₂²×C₅	—	2	2	no	—
172	C₂²×C₅	—	2	8	no	—
173	C₂²×C₅	—	2	2	no	—
174	C₂²×C₅	—	2	2	no	—
175	C₂²×C₅	—	2	2	no	—
176	C₂²×C₅	—	2	2	no	—
177	C₂²×C₅	—	2	2	no	—
178	C₂²×C₅	—	4	4	no	—
179	C₂²×C₅	—	4	8	no	—
180	C₂²×C₅	—	4	2	no	—
181	C₂²×C₅	—	4	2	no	—
182	C₂²×C₅	C₂²⋊Dic₅	1	1	yes	characteristic
183	C₂²×C₅	C₂²⋊Dic₅	1	1	yes	characteristic
184	C₂²×C₅	C₅×C₂²⋊C₄	1	1	yes	characteristic
185	C₂²×C₅	C₅×C₂²⋊C₄	1	1	yes	characteristic
186	C₂²×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
187	C₂²×C₅	C₂×C₅×D₄	1	1	yes	characteristic
188	C₄×C₅	—	2	8	no	—
189	C₄×C₅	—	2	2	no	—
190	C₄×C₅	—	2	2	no	—
191	C₄×C₅	—	2	2	no	—
192	C₄×C₅	—	2	2	no	—
193	C₄×C₅	—	2	2	no	—
194	C₄×C₅	—	2	2	no	—
195	C₄×C₅	—	10	4	no	—
196	C₄×C₅	C₂²⋊Dic₅	1	1	yes	characteristic
197	C₄×C₅	C₂²⋊Dic₅	1	1	yes	characteristic
198	C₄×C₅	C₅×C₂²⋊C₄	1	1	yes	characteristic
199	C₄×C₅	C₅×C₂²⋊C₄	1	1	yes	characteristic
200	C₄×C₅	C₂²×Dic₅	1	1	yes	characteristic
201	C₄×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
202	C₄×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
203	C₄×C₅	C₂×C₅×D₄	1	1	yes	characteristic
204	Dic₅	—	2	4	no	—
205	#41	—	2	1	no	—
206	#41	—	2	1	no	—
207	#41	C₅×Dic₅	1	2	yes	complemented
208	#41	C₂×C₅×D₅	1	2	yes	—
209	C₂×Q₈	C₅×Dic₅	1	1	yes	characteristic, complemented
210	C₂×D₄	—	2	1	no	—
211	C₂×D₄	C₅×Dic₅	1	1	yes	characteristic, complemented
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	10	1	no	—
214	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
215	M₁₆	—	5	2	no	—
216	M₁₆	—	10	1	no	—
217	C₂×C₈	—	10	1	no	—
218	C₄²	—	5	2	no	—
219	C₄²	—	10	1	no	—
220	C₂×C₅	—	2	1	no	—
221	C₂×C₅	—	2	1	no	—
222	C₂×C₅	—	2	1	no	—
223	C₂×C₅	—	2	1	no	—
224	C₂×C₅	—	2	4	no	—
225	C₂×C₅	—	2	2	no	—
226	C₂×C₅	—	4	2	no	—
227	C₂×C₅	—	4	2	no	—
228	C₂×C₅	—	4	8	no	—
229	C₂×C₅	—	4	2	no	—
230	C₂×C₅	—	4	2	no	—
231	C₂×C₅	#3969	1	2	yes	direct product factor
232	C₂×C₅	C₅×#104	1	2	yes	complemented
233	C₂×C₅	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
234	C₂×C₅	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
235	C₂³	—	2	1	no	—
236	C₂³	C₅×C₂²⋊D₅	1	1	yes	characteristic
237	Q₈	—	2	1	no	—
238	Q₈	C₂×C₅×Dic₅	1	2	yes	complemented
239	D₄	—	2	2	no	—
240	D₄	—	2	2	no	—
241	D₄	—	2	1	no	—
242	D₄	C₂×C₅×Dic₅	1	2	yes	complemented
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	10	4	no	—
247	C₂×C₄	—	10	2	no	—
248	C₂×C₄	C₂×C₅×Dic₅	1	1	yes	characteristic
249	C₂×C₄	C₂×C₅×Dic₅	1	1	yes	characteristic
250	C₂×C₄	C₅×C₂²⋊D₅	1	1	yes	characteristic
251	C₂×C₄	C₅×C₂²⋊D₅	1	1	yes	characteristic
252	C₂×C₄	C₅×C₂²⋊D₅	1	1	yes	characteristic
253	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
254	C₈	—	10	2	no	—
255	C₅	—	2	2	no	—
256	C₅	C₂×#3969	1	1	yes	characteristic, direct product factor
257	C₅	C₂×C₅×#104	1	1	yes	characteristic, complemented
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	C₅×C₂²⋊Dic₅	1	1	yes	characteristic
262	C₂²	C₅×C₂²⋊Dic₅	1	1	yes	characteristic
263	C₂²	C₂×C₅×C₂²⋊D₅	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	10	4	no	—
266	C₄	C₅×C₂²⋊Dic₅	1	1	yes	characteristic
267	C₄	C₅×C₂²⋊Dic₅	1	1	yes	characteristic
268	C₄	C₂²×C₅×Dic₅	1	1	yes	characteristic
269	C₄	C₂×C₅×C₂²⋊D₅	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	—	4	2	no	—
273	C₂	C₅×#3969	1	2	yes	direct product factor
274	C₂	C₂×C₅×C₂²⋊Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂²⋊D₅
commutator	C₄×C₅	C₂²×C₄×C₅
Frattini	C₂×C₄	C₂²×C₅×D₅
Fitting, prime radical	C₂×C₅²×#41	C₂
socle	C₂²×C₅²	C₂²⋊C₄
cosocle	C₂×C₄×C₅	C₂³×C₅
hypercenter	C₂×C₅×#41	D₅
hypocenter	C₅	C₂×C₅×#104
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#41

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×C₅×Dic₅	C₂	semidirect
3. C₂×C₅×#681	C₂	semidirect
4. C₅×#3969	C₂	direct (N × H)
7. C₅²×#41	C₄	semidirect
9. C₂×C₅²×Q₈	C₄	semidirect
11. C₂×C₅²×D₄	C₄	semidirect
16. C₄×C₅×Dic₅	C₂²	semidirect
18. C₅×#681	C₂²	semidirect
21. C₂×#3969	C₅	direct (N × H)
25. C₅²×Q₈	C₂×C₄	semidirect
29. C₅²×D₄	C₂×C₄	semidirect
47. C₂×C₄×Dic₅	C₂×C₅	semidirect
48. C₂×#681	C₂×C₅	semidirect
50. #3969	C₂×C₅	direct (N × H)
71. C₅×#41	Dic₅	semidirect
72. C₅×#41	C₄×C₅	semidirect
76. C₂×C₅×Q₈	Dic₅	semidirect
77. C₂×C₅×Q₈	C₄×C₅	semidirect
82. C₂×C₅×D₄	Dic₅	semidirect
83. C₂×C₅×D₄	C₄×C₅	semidirect
98. C₄×Dic₅	C₂²×C₅	semidirect
100. #681	C₂²×C₅	semidirect
106. C₂×C₅²	#104	semidirect
118. C₅×Q₈	C₂×Dic₅	semidirect
119. C₅×Q₈	C₂×C₄×C₅	semidirect
130. C₅×D₄	C₂×Dic₅	semidirect
131. C₅×D₄	C₂×C₄×C₅	semidirect
169. C₅²	C₂×#104	semidirect
207. #41	C₅×Dic₅	semidirect
209. C₂×Q₈	C₅×Dic₅	semidirect
211. C₂×D₄	C₅×Dic₅	semidirect
231. C₂×C₅	#3969	direct (N × H)
232. C₂×C₅	C₅×#104	semidirect
238. Q₈	C₂×C₅×Dic₅	semidirect
242. D₄	C₂×C₅×Dic₅	semidirect
256. C₅	C₂×#3969	direct (N × H)
257. C₅	C₂×C₅×#104	semidirect
273. C₂	C₅×#3969	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
C₂×C₄×C₅×Dic₅	2
C₂×C₅×#681	2
C₅×#3969	2
C₂×C₅×#104	5
C₂×#3969	5

Subgroups — C2×C5×#3969 #13102493

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#3969 #13102493