Subgroups — C₂²×C₄×#6036 #72242660

Order 3200.

7362 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic
2	C₂³×#6036	C₂	1	2	yes	maximal
3	C₂²×C₄×C₅×Dic₅	C₂	1	2	yes	maximal, complemented
4	C₂²×Dic₅²	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×#6036	C₂	1	24	yes	maximal, direct product factor
6	C₂⁴×C₅⋊D₅	C₂²	1	1	yes	characteristic
7	C₂³×C₄×C₅²	C₂²	1	1	yes	characteristic
8	C₂³×#1039	C₂²	1	1	yes	characteristic
9	C₂²×C₄×C₅⋊D₅	C₄	1	16	yes	complemented
10	C₂²×C₄×C₅⋊D₅	C₂²	1	12	yes	—
11	C₂³×C₅×Dic₅	C₂²	1	4	yes	—
12	C₂²×#6036	C₄	1	32	yes	direct product factor
13	C₂²×#6036	C₂²	1	24	yes	—
14	C₂×C₄×C₅×Dic₅	C₂²	1	24	yes	complemented
15	C₂×Dic₅²	C₂²	1	24	yes	complemented
16	C₄×#6036	C₂²	1	64	yes	direct product factor
17	C₂²×C₄²×D₅	—	5	2	no	maximal
18	C₂⁴×C₅²	C₂³	1	1	yes	characteristic
19	C₂³×C₅⋊D₅	C₂×C₄	1	16	yes	—
20	C₂³×C₅⋊D₅	C₂×C₄	1	8	yes	—
21	C₂³×C₅⋊D₅	C₂³	1	6	yes	—
22	C₂²×C₄×C₅²	C₂×C₄	1	8	yes	complemented
23	C₂²×C₄×C₅²	C₂³	1	6	yes	—
24	C₂²×#1039	C₂×C₄	1	8	yes	complemented
25	C₂²×#1039	C₂³	1	6	yes	—
26	C₂×C₄×C₅⋊D₅	C₂×C₄	1	96	yes	complemented
27	C₂×C₄×C₅⋊D₅	C₂³	1	16	yes	—
28	C₂²×C₅×Dic₅	C₂×C₄	1	32	yes	complemented
29	C₂²×C₅×Dic₅	C₂³	1	24	yes	—
30	C₂×#6036	C₂×C₄	1	192	yes	direct product factor
31	C₂×#6036	C₂³	1	32	yes	—
32	C₄×C₅×Dic₅	C₂³	1	32	yes	complemented
33	Dic₅²	C₂³	1	32	yes	complemented
34	C₂³×C₄×D₅	—	5	2	no	—
35	C₂³×C₄×D₅	—	5	4	no	—
36	C₂³×C₄×D₅	—	10	2	no	—
37	C₂²×C₄²×C₅	—	5	2	no	—
38	C₂²×C₄×Dic₅	—	5	4	no	—
39	C₂²×C₄×Dic₅	D₅	1	2	yes	complemented
40	C₂×C₄²×D₅	—	5	48	no	—
41	C₂³×C₅²	C₂²×C₄	1	8	yes	—
42	C₂³×C₅²	C₂²×C₄	1	4	yes	—
43	C₂³×C₅²	C₂⁴	1	3	yes	—
44	C₂²×C₅⋊D₅	C₄²	1	64	yes	complemented
45	C₂²×C₅⋊D₅	C₂²×C₄	1	48	yes	—
46	C₂²×C₅⋊D₅	C₂²×C₄	1	24	yes	—
47	C₂²×C₅⋊D₅	C₂⁴	1	4	yes	—
48	C₂×C₄×C₅²	C₂²×C₄	1	24	yes	complemented
49	C₂×C₄×C₅²	C₂⁴	1	4	yes	—
50	C₂×#1039	C₂²×C₄	1	24	yes	complemented
51	C₂×#1039	C₂⁴	1	4	yes	—
52	C₄×C₅⋊D₅	C₂²×C₄	1	64	yes	complemented
53	C₂×C₅×Dic₅	C₂²×C₄	1	96	yes	complemented
54	C₂×C₅×Dic₅	C₂⁴	1	16	yes	—
55	#6036	C₂²×C₄	1	128	yes	direct product factor
56	C₂⁴×D₅	—	5	2	no	—
57	C₂⁴×D₅	—	10	2	no	—
58	C₂³×C₄×C₅	—	2	2	no	—
59	C₂³×C₄×C₅	—	5	4	no	—
60	C₂³×C₄×C₅	C₂×D₅	1	2	yes	—
61	C₂³×Dic₅	—	5	2	no	—
62	C₂³×Dic₅	—	10	2	no	—
63	C₂³×Dic₅	C₂×D₅	1	4	yes	—
64	C₂²×C₄×D₅	—	5	32	no	—
65	C₂²×C₄×D₅	—	5	24	no	—
66	C₂²×C₄×D₅	—	5	64	no	—
67	C₂²×C₄×D₅	—	5	48	no	—
68	C₂²×C₄×D₅	—	10	32	no	—
69	C₂²×C₄×D₅	—	10	24	no	—
70	C₂×C₄²×C₅	—	5	24	no	—
71	C₂×C₄×Dic₅	—	5	48	no	—
72	C₂×C₄×Dic₅	C₂×D₅	1	24	yes	complemented
73	C₄²×D₅	—	5	128	no	—
74	C₂³×C₄²	—	25	1	no	Sylow
75	C₂²×C₅²	C₂×C₄²	1	16	yes	complemented
76	C₂²×C₅²	C₂³×C₄	1	12	yes	—
77	C₂²×C₅²	C₂³×C₄	1	6	yes	—
78	C₂²×C₅²	C₂⁵	1	1	yes	characteristic
79	C₂×C₅⋊D₅	C₂×C₄²	1	96	yes	complemented
80	C₂×C₅⋊D₅	C₂³×C₄	1	16	yes	—
81	C₂×C₅⋊D₅	C₂³×C₄	1	8	yes	—
82	C₄×C₅²	C₂³×C₄	1	8	yes	complemented
83	#1039	C₂³×C₄	1	8	yes	complemented
84	C₅×Dic₅	C₂³×C₄	1	32	yes	complemented
85	C₂⁴×C₅	—	2	2	no	—
86	C₂⁴×C₅	C₂²×D₅	1	2	yes	—
87	C₂³×D₅	—	5	32	no	—
88	C₂³×D₅	—	5	16	no	—
89	C₂³×D₅	—	5	12	no	—
90	C₂³×D₅	—	10	16	no	—
91	C₂³×D₅	—	10	12	no	—
92	C₂³×D₅	—	10	32	no	—
93	C₂²×C₄×C₅	—	2	16	no	—
94	C₂²×C₄×C₅	—	2	12	no	—
95	C₂²×C₄×C₅	—	5	32	no	—
96	C₂²×C₄×C₅	—	5	24	no	—
97	C₂²×C₄×C₅	C₄×D₅	1	16	yes	complemented
98	C₂²×C₄×C₅	C₂²×D₅	1	12	yes	—
99	C₂²×Dic₅	—	5	16	no	—
100	C₂²×Dic₅	—	5	12	no	—
101	C₂²×Dic₅	—	10	16	no	—
102	C₂²×Dic₅	—	10	12	no	—
103	C₂²×Dic₅	C₄×D₅	1	32	yes	complemented
104	C₂²×Dic₅	C₂²×D₅	1	24	yes	—
105	C₂×C₄×D₅	—	5	192	no	—
106	C₂×C₄×D₅	—	5	32	no	—
107	C₂×C₄×D₅	—	5	384	no	—
108	C₂×C₄×D₅	—	5	64	no	—
109	C₂×C₄×D₅	—	10	192	no	—
110	C₂×C₄×D₅	—	10	32	no	—
111	C₄²×C₅	—	5	32	no	—
112	C₄×Dic₅	—	5	64	no	—
113	C₄×Dic₅	C₂²×D₅	1	32	yes	complemented
114	C₂⁴×C₄	—	25	2	no	—
115	C₂⁴×C₄	—	25	1	no	—
116	C₂²×C₄²	—	5	2	no	—
117	C₂²×C₄²	—	25	24	no	—
118	C₂²×C₄²	—	25	2	no	—
119	C₂×C₅²	C₂²×C₄²	1	12	yes	complemented
120	C₂×C₅²	C₂⁴×C₄	1	2	yes	—
121	C₂×C₅²	C₂⁴×C₄	1	1	yes	characteristic
122	C₅⋊D₅	C₂²×C₄²	1	16	yes	complemented
123	C₂³×C₅	—	2	16	no	—
124	C₂³×C₅	—	2	8	no	—
125	C₂³×C₅	—	2	6	no	—
126	C₂³×C₅	C₂×C₄×D₅	1	16	yes	—
127	C₂³×C₅	C₂×C₄×D₅	1	8	yes	—
128	C₂³×C₅	C₂³×D₅	1	6	yes	—
129	C₂²×D₅	—	5	96	no	—
130	C₂²×D₅	—	5	48	no	—
131	C₂²×D₅	—	5	8	no	—
132	C₂²×D₅	—	5	128	no	—
133	C₂²×D₅	—	10	128	no	—
134	C₂²×D₅	—	10	48	no	—
135	C₂²×D₅	—	10	96	no	—
136	C₂²×D₅	—	10	8	no	—
137	C₂×C₄×C₅	—	2	48	no	—
138	C₂×C₄×C₅	—	2	8	no	—
139	C₂×C₄×C₅	—	5	96	no	—
140	C₂×C₄×C₅	—	5	16	no	—
141	C₂×C₄×C₅	C₂×C₄×D₅	1	48	yes	complemented
142	C₂×C₄×C₅	C₂³×D₅	1	8	yes	—
143	C₂×Dic₅	—	5	48	no	—
144	C₂×Dic₅	—	5	8	no	—
145	C₂×Dic₅	—	10	48	no	—
146	C₂×Dic₅	—	10	8	no	—
147	C₂×Dic₅	C₂×C₄×D₅	1	96	yes	complemented
148	C₂×Dic₅	C₂³×D₅	1	16	yes	—
149	C₄×D₅	—	5	128	no	—
150	C₄×D₅	—	5	256	no	—
151	C₄×D₅	—	10	128	no	—
152	C₂⁵	—	25	1	no	—
153	C₂³×C₄	—	5	4	no	—
154	C₂³×C₄	—	25	32	no	—
155	C₂³×C₄	—	25	24	no	—
156	C₂³×C₄	—	25	1	no	—
157	C₂³×C₄	—	25	16	no	—
158	C₂³×C₄	—	25	12	no	—
159	C₂³×C₄	D₅²	1	1	yes	characteristic
160	C₂×C₄²	—	5	24	no	—
161	C₂×C₄²	—	25	64	no	—
162	C₂×C₄²	—	25	24	no	—
163	C₅²	C₂³×C₄²	1	1	yes	characteristic, complemented, Sylow
164	C₂²×C₅	—	2	24	no	—
165	C₂²×C₅	—	2	12	no	—
166	C₂²×C₅	—	2	32	no	—
167	C₂²×C₅	—	2	2	no	—
168	C₂²×C₅	C₄²×D₅	1	32	yes	complemented
169	C₂²×C₅	C₂²×C₄×D₅	1	24	yes	—
170	C₂²×C₅	C₂²×C₄×D₅	1	12	yes	—
171	C₂²×C₅	C₂⁴×D₅	1	2	yes	—
172	C₂×D₅	—	5	192	no	—
173	C₂×D₅	—	5	32	no	—
174	C₂×D₅	—	5	16	no	—
175	C₂×D₅	—	10	192	no	—
176	C₂×D₅	—	10	16	no	—
177	C₂×D₅	—	10	32	no	—
178	C₄×C₅	—	2	16	no	—
179	C₄×C₅	—	5	32	no	—
180	C₄×C₅	C₂²×C₄×D₅	1	16	yes	complemented
181	Dic₅	—	5	16	no	—
182	Dic₅	—	10	16	no	—
183	Dic₅	C₂²×C₄×D₅	1	32	yes	complemented
184	C₂⁴	—	25	16	no	—
185	C₂⁴	—	25	8	no	—
186	C₂⁴	—	25	6	no	—
187	C₂⁴	C₂×D₅²	1	1	yes	characteristic
188	C₂²×C₄	—	5	32	no	—
189	C₂²×C₄	—	5	24	no	—
190	C₂²×C₄	—	25	32	no	—
191	C₂²×C₄	—	25	192	no	—
192	C₂²×C₄	—	25	8	no	—
193	C₂²×C₄	—	25	6	no	—
194	C₂²×C₄	—	25	96	no	—
195	C₂²×C₄	—	25	16	no	—
196	C₂²×C₄	#6036	1	8	yes	direct product factor
197	C₂²×C₄	C₂×D₅²	1	6	yes	—
198	C₄²	—	5	32	no	—
199	C₄²	—	25	32	no	—
200	C₂×C₅	—	2	24	no	—
201	C₂×C₅	—	2	4	no	—
202	C₂×C₅	—	2	2	no	—
203	C₂×C₅	C₂×C₄²×D₅	1	24	yes	complemented
204	C₂×C₅	C₂³×C₄×D₅	1	4	yes	—
205	C₂×C₅	C₂³×C₄×D₅	1	2	yes	—
206	D₅	—	5	32	no	—
207	D₅	—	10	32	no	—
208	C₂³	—	25	48	no	—
209	C₂³	—	25	24	no	—
210	C₂³	—	25	64	no	—
211	C₂³	—	25	4	no	—
212	C₂³	C₄×D₅²	1	8	yes	—
213	C₂³	C₂×#6036	1	4	yes	—
214	C₂³	C₂²×D₅²	1	3	yes	—
215	C₂×C₄	—	5	96	no	—
216	C₂×C₄	—	5	16	no	—
217	C₂×C₄	—	25	128	no	—
218	C₂×C₄	—	25	24	no	—
219	C₂×C₄	—	25	4	no	—
220	C₂×C₄	—	25	64	no	—
221	C₂×C₄	C₂×#6036	1	24	yes	direct product factor
222	C₂×C₄	C₂²×D₅²	1	4	yes	—
223	C₅	—	2	2	no	—
224	C₅	C₂²×C₄²×D₅	1	2	yes	complemented
225	C₂²	—	25	96	no	—
226	C₂²	—	25	16	no	—
227	C₂²	—	25	8	no	—
228	C₂²	C₄×#6036	1	16	yes	direct product factor
229	C₂²	C₂×C₄×D₅²	1	12	yes	—
230	C₂²	C₂²×#6036	1	6	yes	—
231	C₂²	C₂³×D₅²	1	1	yes	characteristic
232	C₄	—	5	32	no	—
233	C₄	—	25	8	no	—
234	C₄	C₂²×#6036	1	8	yes	direct product factor
235	C₂	—	25	16	no	—
236	C₂	C₂×C₄×#6036	1	12	yes	direct product factor
237	C₂	C₂²×C₄×D₅²	1	2	yes	—
238	C₂	C₂³×#6036	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₄	D₅²
commutator, hypocenter	C₅²	C₂³×C₄²
Frattini	C₂²	C₂³×D₅²
Fitting	C₂³×C₄×C₅²	C₂²
socle	C₂⁴×C₅²	C₂³
prime radical	C₂⁴×C₅⋊D₅	C₂²
cosocle	C₂²×C₅²	C₂⁵
LCS	G ▷ C₅²
UCS	C₁ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
3. C₂²×C₄×C₅×Dic₅	C₂	semidirect
4. C₂²×Dic₅²	C₂	semidirect
5. C₂×C₄×#6036	C₂	direct (N × H)
9. C₂²×C₄×C₅⋊D₅	C₄	semidirect
12. C₂²×#6036	C₄	direct (N × H)
14. C₂×C₄×C₅×Dic₅	C₂²	semidirect
15. C₂×Dic₅²	C₂²	semidirect
16. C₄×#6036	C₂²	direct (N × H)
22. C₂²×C₄×C₅²	C₂×C₄	semidirect
24. C₂²×#1039	C₂×C₄	semidirect
26. C₂×C₄×C₅⋊D₅	C₂×C₄	semidirect
28. C₂²×C₅×Dic₅	C₂×C₄	semidirect
30. C₂×#6036	C₂×C₄	direct (N × H)
32. C₄×C₅×Dic₅	C₂³	semidirect
33. Dic₅²	C₂³	semidirect
39. C₂²×C₄×Dic₅	D₅	semidirect
44. C₂²×C₅⋊D₅	C₄²	semidirect
48. C₂×C₄×C₅²	C₂²×C₄	semidirect
50. C₂×#1039	C₂²×C₄	semidirect
52. C₄×C₅⋊D₅	C₂²×C₄	semidirect
53. C₂×C₅×Dic₅	C₂²×C₄	semidirect
55. #6036	C₂²×C₄	direct (N × H)
72. C₂×C₄×Dic₅	C₂×D₅	semidirect
75. C₂²×C₅²	C₂×C₄²	semidirect
79. C₂×C₅⋊D₅	C₂×C₄²	semidirect
82. C₄×C₅²	C₂³×C₄	semidirect
83. #1039	C₂³×C₄	semidirect
84. C₅×Dic₅	C₂³×C₄	semidirect
97. C₂²×C₄×C₅	C₄×D₅	semidirect
103. C₂²×Dic₅	C₄×D₅	semidirect
113. C₄×Dic₅	C₂²×D₅	semidirect
119. C₂×C₅²	C₂²×C₄²	semidirect
122. C₅⋊D₅	C₂²×C₄²	semidirect
141. C₂×C₄×C₅	C₂×C₄×D₅	semidirect
147. C₂×Dic₅	C₂×C₄×D₅	semidirect
163. C₅²	C₂³×C₄²	semidirect
168. C₂²×C₅	C₄²×D₅	semidirect
180. C₄×C₅	C₂²×C₄×D₅	semidirect
183. Dic₅	C₂²×C₄×D₅	semidirect
196. C₂²×C₄	#6036	direct (N × H)
203. C₂×C₅	C₂×C₄²×D₅	semidirect
221. C₂×C₄	C₂×#6036	direct (N × H)
224. C₅	C₂²×C₄²×D₅	semidirect
228. C₂²	C₄×#6036	direct (N × H)
234. C₄	C₂²×#6036	direct (N × H)
236. C₂	C₂×C₄×#6036	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄×C₅⋊D₅	2
C₂³×#6036	2
C₂²×C₄×C₅×Dic₅	2
C₂²×Dic₅²	2
C₂×C₄×#6036	2
C₂²×C₄²×D₅	5

Subgroups — C22×C4×#6036 #72242660

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×#6036 #72242660