Subgroups — C₂×C₅×#3964 #13102476

Order 1600.

640 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅²×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₅×#168	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#3964	C₂	1	4	yes	maximal, direct product factor
5	C₂⁴×C₅²	—	2	1	no	—
6	C₂×C₅²×D₄	—	2	1	no	—
7	C₂×C₅²×D₄	—	2	2	no	—
8	C₂×C₅²×D₄	C₄	1	4	yes	complemented
9	C₂×C₅²×D₄	C₂²	1	2	yes	—
10	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
11	C₂²×C₅×Dic₅	—	2	1	no	—
12	C₅×C₄⋊Dic₅	—	2	1	no	—
13	C₅×C₄⋊Dic₅	C₂²	1	2	yes	complemented
14	C₂×C₅×#168	—	2	2	no	—
15	C₂×C₅×#168	C₂²	1	2	yes	complemented
16	C₂×C₅×#102	—	5	1	no	maximal
17	C₂×#3964	C₅	1	1	yes	maximal, characteristic, direct product factor
18	C₂³×C₅²	—	2	4	no	—
19	C₂³×C₅²	—	2	2	no	—
20	C₂³×C₅²	—	4	2	no	—
21	C₂³×C₅²	—	4	2	no	—
22	C₂³×C₅²	D₄	1	1	yes	characteristic
23	C₅²×D₄	—	2	4	no	—
24	C₅²×D₄	—	2	2	no	—
25	C₅²×D₄	C₂×C₄	1	4	yes	complemented
26	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
27	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
28	C₂×C₄×C₅²	D₄	1	2	yes	—
29	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
30	C₂×C₅×Dic₅	—	2	2	no	—
31	C₂×C₅×Dic₅	—	4	2	no	—
32	C₅×#168	—	2	2	no	—
33	C₂²×C₅×D₄	—	2	2	no	—
34	C₂²×C₅×D₄	D₅	1	1	yes	characteristic
35	C₂²×C₅×D₄	C₂×C₅	1	1	yes	characteristic
36	C₂²×C₅×C₈	—	5	1	no	—
37	C₂×C₅×C₄⋊C₄	—	5	1	no	—
38	C₂×C₄⋊Dic₅	C₂×C₅	1	1	yes	characteristic, complemented
39	C₂²×#168	C₂×C₅	1	1	yes	characteristic, complemented
40	C₅×#102	—	5	4	no	—
41	#3964	C₂×C₅	1	4	yes	direct product factor
42	C₂²×C₅²	—	2	4	no	—
43	C₂²×C₅²	—	4	8	no	—
44	C₂²×C₅²	—	4	2	no	—
45	C₂²×C₅²	—	4	2	no	—
46	C₂²×C₅²	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₅²	D₈	1	2	yes	—
48	C₂²×C₅²	SD₁₆	1	2	yes	—
49	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
50	C₄×C₅²	C₂²⋊C₄	1	2	yes	—
51	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
52	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
53	C₅×Dic₅	—	4	2	no	—
54	C₂⁴×C₅	—	2	4	no	—
55	C₂⁴×C₅	—	2	1	no	—
56	C₂⁴×C₅	—	2	1	no	—
57	C₂×C₅×D₄	—	2	8	no	—
58	C₂×C₅×D₄	—	2	4	no	—
59	C₂×C₅×D₄	—	2	1	no	—
60	C₂×C₅×D₄	—	2	2	no	—
61	C₂×C₅×D₄	—	2	1	no	—
62	C₂×C₅×D₄	—	2	2	no	—
63	C₂×C₅×D₄	—	2	8	no	—
64	C₂×C₅×D₄	—	2	4	no	—
65	C₂×C₅×D₄	Dic₅	1	4	yes	complemented
66	C₂×C₅×D₄	C₄×C₅	1	4	yes	complemented
67	C₂×C₅×D₄	C₂×D₅	1	2	yes	—
68	C₂×C₅×D₄	C₂²×C₅	1	2	yes	—
69	C₂²×C₄×C₅	—	2	2	no	—
70	C₂²×C₄×C₅	—	10	1	no	—
71	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
73	C₂²×Dic₅	—	2	1	no	—
74	C₂×C₅×C₈	—	5	2	no	—
75	C₂×C₅×C₈	—	10	2	no	—
76	C₅×C₄⋊C₄	—	5	2	no	—
77	C₅×C₄⋊C₄	—	10	1	no	—
78	C₄⋊Dic₅	—	2	1	no	—
79	C₄⋊Dic₅	C₂²×C₅	1	2	yes	complemented
80	C₂×#168	—	2	2	no	—
81	C₂×#168	C₂²×C₅	1	2	yes	complemented
82	C₂×#102	—	5	1	no	Sylow
83	C₂×C₅²	—	4	4	no	—
84	C₂×C₅²	#102	1	4	yes	complemented
85	C₂×C₅²	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₅²	C₂×D₈	1	1	yes	characteristic
87	C₂×C₅²	C₂×SD₁₆	1	1	yes	characteristic
88	C₂³×C₅	—	2	8	no	—
89	C₂³×C₅	—	2	16	no	—
90	C₂³×C₅	—	2	4	no	—
91	C₂³×C₅	—	2	4	no	—
92	C₂³×C₅	—	2	2	no	—
93	C₂³×C₅	—	2	2	no	—
94	C₂³×C₅	—	2	2	no	—
95	C₂³×C₅	—	4	8	no	—
96	C₂³×C₅	—	4	8	no	—
97	C₂³×C₅	—	4	2	no	—
98	C₂³×C₅	—	4	2	no	—
99	C₂³×C₅	—	4	2	no	—
100	C₂³×C₅	—	4	2	no	—
101	C₂³×C₅	C₂²⋊D₅	1	1	yes	characteristic
102	C₂³×C₅	C₅×D₄	1	1	yes	characteristic
103	C₅×D₄	—	2	8	no	—
104	C₅×D₄	—	2	16	no	—
105	C₅×D₄	—	2	4	no	—
106	C₅×D₄	—	2	4	no	—
107	C₅×D₄	—	2	2	no	—
108	C₅×D₄	—	2	2	no	—
109	C₅×D₄	—	2	8	no	—
110	C₅×D₄	C₂×Dic₅	1	4	yes	complemented
111	C₅×D₄	C₂×C₄×C₅	1	4	yes	complemented
112	C₂×C₄×C₅	—	2	4	no	—
113	C₂×C₄×C₅	—	2	4	no	—
114	C₂×C₄×C₅	—	2	2	no	—
115	C₂×C₄×C₅	—	2	2	no	—
116	C₂×C₄×C₅	—	10	2	no	—
117	C₂×C₄×C₅	—	20	2	no	—
118	C₂×C₄×C₅	C₂×Dic₅	1	2	yes	—
119	C₂×C₄×C₅	C₂²⋊D₅	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂²⋊D₅	1	2	yes	—
121	C₂×C₄×C₅	C₂×C₄×C₅	1	2	yes	—
122	C₂×C₄×C₅	C₅×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅	C₅×D₄	1	2	yes	—
124	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
126	C₂×Dic₅	—	2	2	no	—
127	C₂×Dic₅	—	4	2	no	—
128	C₅×C₈	—	10	2	no	—
129	#168	—	2	2	no	—
130	C₂²×D₄	C₅×D₅	1	1	yes	characteristic
131	C₂²×C₈	—	5	1	no	—
132	C₂×C₄⋊C₄	—	5	1	no	—
133	#102	—	5	4	no	—
134	C₅²	C₂×#102	1	1	yes	characteristic, complemented, Sylow
135	C₂²×C₅	—	2	16	no	—
136	C₂²×C₅	—	2	4	no	—
137	C₂²×C₅	—	2	4	no	—
138	C₂²×C₅	—	2	2	no	—
139	C₂²×C₅	—	2	4	no	—
140	C₂²×C₅	—	2	4	no	—
141	C₂²×C₅	—	2	4	no	—
142	C₂²×C₅	—	4	8	no	—
143	C₂²×C₅	—	4	8	no	—
144	C₂²×C₅	—	4	32	no	—
145	C₂²×C₅	—	4	2	no	—
146	C₂²×C₅	—	4	2	no	—
147	C₂²×C₅	—	4	8	no	—
148	C₂²×C₅	—	4	8	no	—
149	C₂²×C₅	—	4	2	no	—
150	C₂²×C₅	—	4	2	no	—
151	C₂²×C₅	#686	1	2	yes	—
152	C₂²×C₅	#687	1	2	yes	—
153	C₂²×C₅	C₂²⋊Dic₅	1	2	yes	—
154	C₂²×C₅	C₅×C₂²⋊C₄	1	2	yes	—
155	C₂²×C₅	C₅×D₈	1	2	yes	—
156	C₂²×C₅	C₅×SD₁₆	1	2	yes	—
157	C₂²×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
158	C₂²×C₅	C₂×C₅×D₄	1	1	yes	characteristic
159	C₄×C₅	—	2	2	no	—
160	C₄×C₅	—	2	2	no	—
161	C₄×C₅	—	2	4	no	—
162	C₄×C₅	—	20	2	no	—
163	C₄×C₅	C₂²⋊Dic₅	1	2	yes	—
164	C₄×C₅	C₅×C₂²⋊C₄	1	2	yes	—
165	C₄×C₅	C₂²×Dic₅	1	1	yes	characteristic
166	C₄×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
167	C₄×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
168	C₄×C₅	C₂×C₅×D₄	1	1	yes	characteristic
169	Dic₅	—	4	2	no	—
170	C₂⁴	—	2	1	no	—
171	C₂×D₄	—	2	2	no	—
172	C₂×D₄	—	2	1	no	—
173	C₂×D₄	C₅×Dic₅	1	4	yes	complemented
174	C₂×D₄	C₂×C₅×D₅	1	2	yes	—
175	C₂²×C₄	—	10	1	no	—
176	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
177	C₂×C₈	—	5	2	no	—
178	C₂×C₈	—	10	2	no	—
179	C₄⋊C₄	—	5	2	no	—
180	C₄⋊C₄	—	10	1	no	—
181	C₂×C₅	—	2	2	no	—
182	C₂×C₅	—	2	2	no	—
183	C₂×C₅	—	2	8	no	—
184	C₂×C₅	—	2	2	no	—
185	C₂×C₅	—	4	16	no	—
186	C₂×C₅	—	4	4	no	—
187	C₂×C₅	—	4	4	no	—
188	C₂×C₅	#3964	1	4	yes	direct product factor
189	C₂×C₅	C₅×#102	1	4	yes	complemented
190	C₂×C₅	C₂×#686	1	1	yes	characteristic
191	C₂×C₅	C₂×#687	1	1	yes	characteristic
192	C₂×C₅	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
193	C₂×C₅	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
194	C₂×C₅	C₂×C₅×D₈	1	1	yes	characteristic
195	C₂×C₅	C₂×C₅×SD₁₆	1	1	yes	characteristic
196	C₂³	—	2	2	no	—
197	C₂³	—	2	4	no	—
198	C₂³	—	4	2	no	—
199	C₂³	—	4	2	no	—
200	C₂³	C₅×C₂²⋊D₅	1	1	yes	characteristic
201	D₄	—	2	2	no	—
202	D₄	—	2	4	no	—
203	D₄	C₂×C₅×Dic₅	1	4	yes	complemented
204	C₂×C₄	—	10	2	no	—
205	C₂×C₄	—	20	2	no	—
206	C₂×C₄	C₂×C₅×Dic₅	1	2	yes	—
207	C₂×C₄	C₅×C₂²⋊D₅	1	2	yes	—
208	C₂×C₄	C₅×C₂²⋊D₅	1	1	yes	characteristic
209	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
210	C₈	—	10	2	no	—
211	C₅	—	2	2	no	—
212	C₅	C₂×#3964	1	1	yes	characteristic, direct product factor
213	C₅	C₂×C₅×#102	1	1	yes	characteristic, complemented
214	C₂²	—	2	4	no	—
215	C₂²	—	4	2	no	—
216	C₂²	—	4	2	no	—
217	C₂²	—	4	8	no	—
218	C₂²	C₅×#686	1	2	yes	—
219	C₂²	C₅×#687	1	2	yes	—
220	C₂²	C₅×C₂²⋊Dic₅	1	2	yes	—
221	C₂²	C₂×C₅×C₂²⋊D₅	1	1	yes	characteristic
222	C₄	—	20	2	no	—
223	C₄	C₅×C₂²⋊Dic₅	1	2	yes	—
224	C₄	C₂²×C₅×Dic₅	1	1	yes	characteristic
225	C₄	C₂×C₅×C₂²⋊D₅	1	1	yes	characteristic
226	C₂	—	4	4	no	—
227	C₂	C₅×#3964	1	4	yes	direct product factor
228	C₂	C₂×C₅×#686	1	1	yes	characteristic
229	C₂	C₂×C₅×#687	1	1	yes	characteristic
230	C₂	C₂×C₅×C₂²⋊Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	C₂²⋊D₅
commutator	C₄×C₅	C₂²×C₄×C₅
Frattini	C₂×C₄	C₂²×C₅×D₅
Fitting, prime radical	C₂²×C₅²×D₄	C₂
socle	C₂³×C₅²	D₄
cosocle	C₂×C₄×C₅	C₂³×C₅
hypercenter	C₂²×C₅×D₄	D₅
hypocenter	C₅	C₂×C₅×#102
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₅ ◁ C₂²×C₄×C₅ ◁ C₂²×C₅×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×C₄⋊Dic₅	C₂	semidirect
3. C₂²×C₅×#168	C₂	semidirect
4. C₅×#3964	C₂	direct (N × H)
8. C₂×C₅²×D₄	C₄	semidirect
13. C₅×C₄⋊Dic₅	C₂²	semidirect
15. C₂×C₅×#168	C₂²	semidirect
17. C₂×#3964	C₅	direct (N × H)
25. C₅²×D₄	C₂×C₄	semidirect
38. C₂×C₄⋊Dic₅	C₂×C₅	semidirect
39. C₂²×#168	C₂×C₅	semidirect
41. #3964	C₂×C₅	direct (N × H)
65. C₂×C₅×D₄	Dic₅	semidirect
66. C₂×C₅×D₄	C₄×C₅	semidirect
79. C₄⋊Dic₅	C₂²×C₅	semidirect
81. C₂×#168	C₂²×C₅	semidirect
84. C₂×C₅²	#102	semidirect
110. C₅×D₄	C₂×Dic₅	semidirect
111. C₅×D₄	C₂×C₄×C₅	semidirect
134. C₅²	C₂×#102	semidirect
173. C₂×D₄	C₅×Dic₅	semidirect
188. C₂×C₅	#3964	direct (N × H)
189. C₂×C₅	C₅×#102	semidirect
203. D₄	C₂×C₅×Dic₅	semidirect
212. C₅	C₂×#3964	direct (N × H)
213. C₅	C₂×C₅×#102	semidirect
227. C₂	C₅×#3964	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×C₅²×D₄	2
C₂×C₅×C₄⋊Dic₅	2
C₂²×C₅×#168	2
C₅×#3964	2
C₂×C₅×#102	5
C₂×#3964	5

Subgroups — C2×C5×#3964 #13102476

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#3964 #13102476