Subgroups — C₂²×C₅×#3935 #72242657

Order 3200.

1858 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄²×C₅²	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₅×#168	C₂	1	2	yes	maximal
3	C₂×C₅×#3935	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₄×C₅²	C₄	1	2	yes	—
5	C₂³×C₄×C₅²	C₂²	1	1	yes	characteristic
6	C₂×C₄²×C₅²	C₄	1	6	yes	—
7	C₂×C₄²×C₅²	C₂²	1	6	yes	—
8	C₂²×C₅×#168	C₄	1	16	yes	complemented
9	C₂²×C₅×#168	C₂²	1	12	yes	—
10	C₅×#3935	C₂²	1	16	yes	direct product factor
11	C₂²×C₅×#97	—	5	1	no	maximal
12	C₂²×#3935	C₅	1	1	yes	maximal, characteristic, direct product factor
13	C₂⁴×C₅²	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₅²	—	2	8	no	—
15	C₂²×C₄×C₅²	C₂×C₄	1	12	yes	—
16	C₂²×C₄×C₅²	C₂×C₄	1	3	yes	—
17	C₂²×C₄×C₅²	C₂×C₄	1	8	yes	—
18	C₂²×C₄×C₅²	C₂³	1	3	yes	—
19	C₄²×C₅²	C₂×C₄	1	12	yes	—
20	C₄²×C₅²	C₂³	1	4	yes	—
21	C₂×C₅×#168	C₂×C₄	1	48	yes	complemented
22	C₂×C₅×#168	C₂³	1	8	yes	—
23	C₂³×C₅×C₈	—	5	2	no	—
24	C₂²×C₄²×C₅	—	2	2	no	—
25	C₂²×C₄²×C₅	D₅	1	1	yes	characteristic
26	C₂²×C₄²×C₅	C₂×C₅	1	1	yes	characteristic
27	C₂³×#168	C₂×C₅	1	2	yes	—
28	C₂×C₅×#97	—	5	12	no	—
29	C₂×#3935	C₂×C₅	1	12	yes	direct product factor
30	C₂³×C₅²	C₄²	1	4	yes	—
31	C₂³×C₅²	M₁₆	1	8	yes	—
32	C₂³×C₅²	C₂²×C₄	1	3	yes	—
33	C₂×C₄×C₅²	—	2	24	no	—
34	C₂×C₄×C₅²	C₄²	1	12	yes	—
35	C₂×C₄×C₅²	C₂²×C₄	1	8	yes	—
36	C₂×C₄×C₅²	C₂²×C₄	1	3	yes	—
37	C₂×C₄×C₅²	C₂²×C₄	1	12	yes	—
38	C₂×C₄×C₅²	C₂⁴	1	1	yes	characteristic
39	C₅×#168	C₂²×C₄	1	16	yes	complemented
40	C₂³×C₄×C₅	—	2	4	no	—
41	C₂³×C₄×C₅	—	2	2	no	—
42	C₂³×C₄×C₅	Dic₅	1	2	yes	—
43	C₂³×C₄×C₅	C₄×C₅	1	2	yes	—
44	C₂³×C₄×C₅	C₂×D₅	1	1	yes	characteristic
45	C₂³×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
46	C₂²×C₅×C₈	—	5	16	no	—
47	C₂²×C₅×C₈	—	5	12	no	—
48	C₂×C₄²×C₅	—	2	12	no	—
49	C₂×C₄²×C₅	—	2	12	no	—
50	C₂×C₄²×C₅	Dic₅	1	6	yes	—
51	C₂×C₄²×C₅	C₄×C₅	1	6	yes	—
52	C₂×C₄²×C₅	C₂×D₅	1	6	yes	—
53	C₂×C₄²×C₅	C₂²×C₅	1	6	yes	—
54	C₂²×#168	C₄×C₅	1	16	yes	complemented
55	C₂²×#168	C₂²×C₅	1	12	yes	—
56	C₅×#97	—	5	16	no	—
57	#3935	C₂²×C₅	1	16	yes	direct product factor
58	C₂²×#97	—	5	1	no	Sylow
59	C₂²×C₅²	#97	1	16	yes	complemented
60	C₂²×C₅²	C₂×C₄²	1	6	yes	—
61	C₂²×C₅²	C₂×M₁₆	1	12	yes	—
62	C₂²×C₅²	C₂³×C₄	1	1	yes	characteristic
63	C₄×C₅²	—	2	8	no	—
64	C₄×C₅²	C₂×C₄²	1	6	yes	—
65	C₄×C₅²	C₂³×C₄	1	2	yes	—
66	C₂⁴×C₅	—	2	2	no	—
67	C₂⁴×C₅	C₂×Dic₅	1	1	yes	characteristic
68	C₂⁴×C₅	C₂×C₄×C₅	1	1	yes	characteristic
69	C₂²×C₄×C₅	—	2	16	no	—
70	C₂²×C₄×C₅	—	2	16	no	—
71	C₂²×C₄×C₅	—	2	8	no	—
72	C₂²×C₄×C₅	—	2	8	no	—
73	C₂²×C₄×C₅	—	2	16	no	—
74	C₂²×C₄×C₅	—	2	6	no	—
75	C₂²×C₄×C₅	—	2	24	no	—
76	C₂²×C₄×C₅	—	2	6	no	—
77	C₂²×C₄×C₅	C₄×D₅	1	8	yes	—
78	C₂²×C₄×C₅	C₂×Dic₅	1	12	yes	—
79	C₂²×C₄×C₅	C₂×Dic₅	1	3	yes	—
80	C₂²×C₄×C₅	C₂×C₄×C₅	1	8	yes	—
81	C₂²×C₄×C₅	C₂×C₄×C₅	1	12	yes	—
82	C₂²×C₄×C₅	C₂×C₄×C₅	1	3	yes	—
83	C₂²×C₄×C₅	C₂²×D₅	1	3	yes	—
84	C₂²×C₄×C₅	C₂³×C₅	1	3	yes	—
85	C₂×C₅×C₈	—	5	48	no	—
86	C₂×C₅×C₈	—	5	8	no	—
87	C₄²×C₅	—	2	24	no	—
88	C₄²×C₅	—	2	8	no	—
89	C₄²×C₅	C₂×Dic₅	1	12	yes	—
90	C₄²×C₅	C₂×C₄×C₅	1	12	yes	—
91	C₄²×C₅	C₂²×D₅	1	4	yes	—
92	C₄²×C₅	C₂³×C₅	1	4	yes	—
93	C₂×#168	C₂×C₄×C₅	1	48	yes	complemented
94	C₂×#168	C₂³×C₅	1	8	yes	—
95	C₂³×C₈	—	5	2	no	—
96	C₂²×C₄²	C₅×D₅	1	1	yes	characteristic
97	C₂×#97	—	5	12	no	—
98	C₂×C₅²	C₂×#97	1	12	yes	complemented
99	C₂×C₅²	C₂²×C₄²	1	1	yes	characteristic
100	C₂×C₅²	C₂²×M₁₆	1	2	yes	—
101	C₂³×C₅	—	2	6	no	—
102	C₂³×C₅	—	2	16	no	—
103	C₂³×C₅	—	2	8	no	—
104	C₂³×C₅	#681	1	8	yes	—
105	C₂³×C₅	C₄×Dic₅	1	4	yes	—
106	C₂³×C₅	C₄²×C₅	1	4	yes	—
107	C₂³×C₅	C₅×M₁₆	1	8	yes	—
108	C₂³×C₅	C₂²×Dic₅	1	3	yes	—
109	C₂³×C₅	C₂²×C₄×C₅	1	3	yes	—
110	C₂×C₄×C₅	—	2	48	no	—
111	C₂×C₄×C₅	—	2	48	no	—
112	C₂×C₄×C₅	—	2	24	no	—
113	C₂×C₄×C₅	—	2	24	no	—
114	C₂×C₄×C₅	—	2	24	no	—
115	C₂×C₄×C₅	—	2	6	no	—
116	C₂×C₄×C₅	—	2	16	no	—
117	C₂×C₄×C₅	—	2	2	no	—
118	C₂×C₄×C₅	—	2	24	no	—
119	C₂×C₄×C₅	C₄×Dic₅	1	12	yes	—
120	C₂×C₄×C₅	C₄²×C₅	1	12	yes	—
121	C₂×C₄×C₅	C₂×C₄×D₅	1	12	yes	—
122	C₂×C₄×C₅	C₂²×Dic₅	1	8	yes	—
123	C₂×C₄×C₅	C₂²×Dic₅	1	3	yes	—
124	C₂×C₄×C₅	C₂²×C₄×C₅	1	12	yes	—
125	C₂×C₄×C₅	C₂²×C₄×C₅	1	8	yes	—
126	C₂×C₄×C₅	C₂²×C₄×C₅	1	3	yes	—
127	C₂×C₄×C₅	C₂³×D₅	1	1	yes	characteristic
128	C₂×C₄×C₅	C₂⁴×C₅	1	1	yes	characteristic
129	C₅×C₈	—	5	16	no	—
130	#168	C₂²×C₄×C₅	1	16	yes	complemented
131	C₂³×C₄	C₅×Dic₅	1	2	yes	—
132	C₂³×C₄	C₂×C₅×D₅	1	1	yes	characteristic
133	C₂²×C₈	—	5	16	no	—
134	C₂²×C₈	—	5	12	no	—
135	C₂×C₄²	C₅×Dic₅	1	6	yes	—
136	C₂×C₄²	C₂×C₅×D₅	1	6	yes	—
137	#97	—	5	16	no	—
138	C₅²	C₂²×#97	1	1	yes	characteristic, complemented, Sylow
139	C₂²×C₅	—	2	32	no	—
140	C₂²×C₅	—	2	24	no	—
141	C₂²×C₅	—	2	12	no	—
142	C₂²×C₅	—	2	2	no	—
143	C₂²×C₅	#3935	1	16	yes	direct product factor
144	C₂²×C₅	C₅×#97	1	16	yes	complemented
145	C₂²×C₅	C₂×#681	1	12	yes	—
146	C₂²×C₅	C₂×C₄×Dic₅	1	6	yes	—
147	C₂²×C₅	C₂×C₄²×C₅	1	6	yes	—
148	C₂²×C₅	C₂×C₅×M₁₆	1	12	yes	—
149	C₂²×C₅	C₂³×Dic₅	1	1	yes	characteristic
150	C₂²×C₅	C₂³×C₄×C₅	1	1	yes	characteristic
151	C₄×C₅	—	2	16	no	—
152	C₄×C₅	—	2	16	no	—
153	C₄×C₅	—	2	8	no	—
154	C₄×C₅	—	2	8	no	—
155	C₄×C₅	—	2	12	no	—
156	C₄×C₅	—	2	4	no	—
157	C₄×C₅	C₂×C₄×Dic₅	1	6	yes	—
158	C₄×C₅	C₂×C₄²×C₅	1	6	yes	—
159	C₄×C₅	C₂²×C₄×D₅	1	2	yes	—
160	C₄×C₅	C₂³×C₄×C₅	1	2	yes	—
161	C₂⁴	C₂×C₅×Dic₅	1	1	yes	characteristic
162	C₂²×C₄	—	2	8	no	—
163	C₂²×C₄	C₄×C₅×D₅	1	8	yes	—
164	C₂²×C₄	C₂×C₅×Dic₅	1	12	yes	—
165	C₂²×C₄	C₂×C₅×Dic₅	1	3	yes	—
166	C₂²×C₄	C₂²×C₅×D₅	1	3	yes	—
167	C₂×C₈	—	5	48	no	—
168	C₂×C₈	—	5	8	no	—
169	C₄²	C₂×C₅×Dic₅	1	12	yes	—
170	C₄²	C₂²×C₅×D₅	1	4	yes	—
171	C₂×C₅	—	2	24	no	—
172	C₂×C₅	—	2	4	no	—
173	C₂×C₅	—	2	2	no	—
174	C₂×C₅	C₂×#3935	1	12	yes	direct product factor
175	C₂×C₅	C₂×C₅×#97	1	12	yes	complemented
176	C₂×C₅	C₂²×#681	1	2	yes	—
177	C₂×C₅	C₂²×C₄×Dic₅	1	1	yes	characteristic
178	C₂×C₅	C₂²×C₄²×C₅	1	1	yes	characteristic
179	C₂×C₅	C₂²×C₅×M₁₆	1	2	yes	—
180	C₂³	C₅×#681	1	8	yes	—
181	C₂³	C₄×C₅×Dic₅	1	4	yes	—
182	C₂³	C₂²×C₅×Dic₅	1	3	yes	—
183	C₂×C₄	—	2	24	no	—
184	C₂×C₄	C₄×C₅×Dic₅	1	12	yes	—
185	C₂×C₄	C₂×C₄×C₅×D₅	1	12	yes	—
186	C₂×C₄	C₂²×C₅×Dic₅	1	8	yes	—
187	C₂×C₄	C₂²×C₅×Dic₅	1	3	yes	—
188	C₂×C₄	C₂³×C₅×D₅	1	1	yes	characteristic
189	C₈	—	5	16	no	—
190	C₅	—	2	2	no	—
191	C₅	C₂²×#3935	1	1	yes	characteristic, direct product factor
192	C₅	C₂²×C₅×#97	1	1	yes	characteristic, complemented
193	C₂²	C₅×#3935	1	16	yes	direct product factor
194	C₂²	C₂×C₅×#681	1	12	yes	—
195	C₂²	C₂×C₄×C₅×Dic₅	1	6	yes	—
196	C₂²	C₂³×C₅×Dic₅	1	1	yes	characteristic
197	C₄	—	2	8	no	—
198	C₄	C₂×C₄×C₅×Dic₅	1	6	yes	—
199	C₄	C₂²×C₄×C₅×D₅	1	2	yes	—
200	C₂	C₂×C₅×#3935	1	12	yes	direct product factor
201	C₂	C₂²×C₅×#681	1	2	yes	—
202	C₂	C₂²×C₄×C₅×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄×C₅	C₂×D₅
commutator	C₂×C₅	C₂²×C₄²×C₅
Frattini	C₂×C₄	C₂³×C₅×D₅
Fitting	C₂²×C₄²×C₅²	C₂
socle, prime radical	C₂⁴×C₅²	C₂×C₄
cosocle	C₂×C₄×C₅	C₂⁴×C₅
hypercenter	C₂²×C₄²×C₅	D₅
hypocenter	C₅	C₂²×C₅×#97
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₄×C₅ ◁ C₂²×C₄²×C₅

Complements

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

N	H ≅ G/N	type
3. C₂×C₅×#3935	C₂	direct (N × H)
8. C₂²×C₅×#168	C₄	semidirect
10. C₅×#3935	C₂²	direct (N × H)
12. C₂²×#3935	C₅	direct (N × H)
21. C₂×C₅×#168	C₂×C₄	semidirect
29. C₂×#3935	C₂×C₅	direct (N × H)
39. C₅×#168	C₂²×C₄	semidirect
54. C₂²×#168	C₄×C₅	semidirect
57. #3935	C₂²×C₅	direct (N × H)
59. C₂²×C₅²	#97	semidirect
93. C₂×#168	C₂×C₄×C₅	semidirect
98. C₂×C₅²	C₂×#97	semidirect
130. #168	C₂²×C₄×C₅	semidirect
138. C₅²	C₂²×#97	semidirect
143. C₂²×C₅	#3935	direct (N × H)
144. C₂²×C₅	C₅×#97	semidirect
174. C₂×C₅	C₂×#3935	direct (N × H)
175. C₂×C₅	C₂×C₅×#97	semidirect
191. C₅	C₂²×#3935	direct (N × H)
192. C₅	C₂²×C₅×#97	semidirect
193. C₂²	C₅×#3935	direct (N × H)
200. C₂	C₂×C₅×#3935	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×C₄²×C₅²	2
C₂³×C₅×#168	2
C₂×C₅×#3935	2
C₂²×C₅×#97	5
C₂²×#3935	5

Subgroups — C22×C5×#3935 #72242657

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#3935 #72242657