Subgroups — C₂²×C₅×#681 #13106935

Order 1600.

832 conjugacy classes of subgroups.

The subgroup lattice has 149 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×C₅²	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#681	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₅×#168	C₂	1	2	yes	maximal, complemented
4	C₂⁴×C₅²	C₄	1	1	yes	characteristic
5	C₂²×C₄×C₅²	C₄	1	6	yes	—
6	C₂²×C₄×C₅²	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₅²	C₂²	1	6	yes	—
8	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
9	C₅×#681	C₂²	1	16	yes	direct product factor
10	C₂×C₅×#168	C₂²	1	12	yes	complemented
11	C₂²×C₅×M₁₆	—	5	1	no	maximal
12	C₂²×#681	C₅	1	1	yes	maximal, characteristic, direct product factor
13	C₂³×C₅²	—	2	4	no	—
14	C₂³×C₅²	C₂×C₄	1	6	yes	—
15	C₂³×C₅²	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₅²	C₂×C₄	1	12	yes	—
17	C₂×C₄×C₅²	C₂×C₄	1	6	yes	—
18	C₂×C₄×C₅²	C₂×C₄	1	3	yes	—
19	C₂×C₄×C₅²	C₂³	1	3	yes	—
20	C₂×C₄×C₅²	C₂³	1	4	yes	—
21	C₅×#168	C₂³	1	8	yes	complemented
22	C₂³×C₄×C₅	—	2	2	no	—
23	C₂³×C₄×C₅	D₅	1	1	yes	characteristic
24	C₂³×C₄×C₅	C₂×C₅	1	1	yes	characteristic
25	C₂×C₅×M₁₆	—	5	12	no	—
26	C₂²×C₅×C₈	—	5	2	no	—
27	C₂×#681	C₂×C₅	1	12	yes	direct product factor
28	C₂²×#168	C₂×C₅	1	2	yes	complemented
29	C₂²×C₅²	—	2	12	no	—
30	C₂²×C₅²	M₁₆	1	4	yes	complemented
31	C₂²×C₅²	C₂²×C₄	1	4	yes	—
32	C₂²×C₅²	C₂²×C₄	1	3	yes	—
33	C₄×C₅²	C₂²×C₄	1	3	yes	—
34	C₄×C₅²	C₂²×C₄	1	4	yes	—
35	C₄×C₅²	C₂⁴	1	1	yes	characteristic
36	C₂⁴×C₅	—	2	2	no	—
37	C₂⁴×C₅	Dic₅	1	1	yes	characteristic
38	C₂⁴×C₅	C₄×C₅	1	1	yes	characteristic
39	C₂²×C₄×C₅	—	2	12	no	—
40	C₂²×C₄×C₅	—	2	2	no	—
41	C₂²×C₄×C₅	—	2	2	no	—
42	C₂²×C₄×C₅	—	2	12	no	—
43	C₂²×C₄×C₅	Dic₅	1	1	yes	characteristic
44	C₂²×C₄×C₅	Dic₅	1	6	yes	—
45	C₂²×C₄×C₅	C₄×C₅	1	6	yes	—
46	C₂²×C₄×C₅	C₄×C₅	1	1	yes	characteristic
47	C₂²×C₄×C₅	C₂×D₅	1	6	yes	—
48	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
49	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
50	C₂²×C₄×C₅	C₂²×C₅	1	6	yes	—
51	C₅×M₁₆	—	5	16	no	—
52	C₂×C₅×C₈	—	5	12	no	—
53	#681	C₂²×C₅	1	16	yes	direct product factor
54	C₂×#168	C₂²×C₅	1	12	yes	complemented
55	C₂²×M₁₆	—	5	1	no	Sylow
56	C₂×C₅²	—	2	4	no	—
57	C₂×C₅²	C₂×M₁₆	1	6	yes	complemented
58	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
59	C₂³×C₅	—	2	16	no	—
60	C₂³×C₅	—	2	4	no	—
61	C₂³×C₅	—	2	4	no	—
62	C₂³×C₅	—	2	12	no	—
63	C₂³×C₅	—	2	2	no	—
64	C₂³×C₅	C₂×Dic₅	1	6	yes	—
65	C₂³×C₅	C₂×Dic₅	1	1	yes	characteristic
66	C₂³×C₅	C₂×C₄×C₅	1	6	yes	—
67	C₂³×C₅	C₂×C₄×C₅	1	1	yes	characteristic
68	C₂×C₄×C₅	—	2	24	no	—
69	C₂×C₄×C₅	—	2	6	no	—
70	C₂×C₄×C₅	—	2	12	no	—
71	C₂×C₄×C₅	—	2	6	no	—
72	C₂×C₄×C₅	—	2	8	no	—
73	C₂×C₄×C₅	C₂×Dic₅	1	12	yes	—
74	C₂×C₄×C₅	C₂×Dic₅	1	6	yes	—
75	C₂×C₄×C₅	C₂×Dic₅	1	3	yes	—
76	C₂×C₄×C₅	C₂×C₄×C₅	1	12	yes	—
77	C₂×C₄×C₅	C₂×C₄×C₅	1	3	yes	—
78	C₂×C₄×C₅	C₂×C₄×C₅	1	6	yes	—
79	C₂×C₄×C₅	C₂²×D₅	1	3	yes	—
80	C₂×C₄×C₅	C₂²×D₅	1	4	yes	—
81	C₂×C₄×C₅	C₂³×C₅	1	3	yes	—
82	C₂×C₄×C₅	C₂³×C₅	1	4	yes	—
83	C₅×C₈	—	5	8	no	—
84	#168	C₂³×C₅	1	8	yes	complemented
85	C₂³×C₄	C₅×D₅	1	1	yes	characteristic
86	C₂×M₁₆	—	5	12	no	—
87	C₂²×C₈	—	5	2	no	—
88	C₅²	C₂²×M₁₆	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₅	—	2	48	no	—
90	C₂²×C₅	—	2	12	no	—
91	C₂²×C₅	—	2	12	no	—
92	C₂²×C₅	—	2	8	no	—
93	C₂²×C₅	—	2	6	no	—
94	C₂²×C₅	—	2	8	no	—
95	C₂²×C₅	#681	1	4	yes	direct product factor
96	C₂²×C₅	C₅×M₁₆	1	4	yes	complemented
97	C₂²×C₅	C₂²×Dic₅	1	4	yes	—
98	C₂²×C₅	C₂²×Dic₅	1	3	yes	—
99	C₂²×C₅	C₂²×C₄×C₅	1	4	yes	—
100	C₂²×C₅	C₂²×C₄×C₅	1	3	yes	—
101	C₄×C₅	—	2	6	no	—
102	C₄×C₅	—	2	8	no	—
103	C₄×C₅	—	2	2	no	—
104	C₄×C₅	C₂²×Dic₅	1	4	yes	—
105	C₄×C₅	C₂²×Dic₅	1	3	yes	—
106	C₄×C₅	C₂²×C₄×C₅	1	3	yes	—
107	C₄×C₅	C₂²×C₄×C₅	1	4	yes	—
108	C₄×C₅	C₂³×D₅	1	1	yes	characteristic
109	C₄×C₅	C₂⁴×C₅	1	1	yes	characteristic
110	C₂⁴	C₅×Dic₅	1	1	yes	characteristic
111	C₂²×C₄	C₅×Dic₅	1	6	yes	—
112	C₂²×C₄	C₅×Dic₅	1	1	yes	characteristic
113	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
114	C₂²×C₄	C₂×C₅×D₅	1	6	yes	—
115	M₁₆	—	5	16	no	—
116	C₂×C₈	—	5	12	no	—
117	C₂×C₅	—	2	16	no	—
118	C₂×C₅	—	2	4	no	—
119	C₂×C₅	—	2	4	no	—
120	C₂×C₅	—	2	12	no	—
121	C₂×C₅	—	2	2	no	—
122	C₂×C₅	C₂×#681	1	6	yes	direct product factor
123	C₂×C₅	C₂×C₅×M₁₆	1	6	yes	complemented
124	C₂×C₅	C₂³×Dic₅	1	1	yes	characteristic
125	C₂×C₅	C₂³×C₄×C₅	1	1	yes	characteristic
126	C₂³	—	2	4	no	—
127	C₂³	C₂×C₅×Dic₅	1	6	yes	—
128	C₂³	C₂×C₅×Dic₅	1	1	yes	characteristic
129	C₂×C₄	C₂×C₅×Dic₅	1	12	yes	—
130	C₂×C₄	C₂×C₅×Dic₅	1	3	yes	—
131	C₂×C₄	C₂×C₅×Dic₅	1	6	yes	—
132	C₂×C₄	C₂²×C₅×D₅	1	3	yes	—
133	C₂×C₄	C₂²×C₅×D₅	1	4	yes	—
134	C₈	—	5	8	no	—
135	C₅	—	2	2	no	—
136	C₅	C₂²×#681	1	1	yes	characteristic, direct product factor
137	C₅	C₂²×C₅×M₁₆	1	1	yes	characteristic, complemented
138	C₂²	—	2	12	no	—
139	C₂²	C₅×#681	1	4	yes	direct product factor
140	C₂²	C₂²×C₅×Dic₅	1	4	yes	—
141	C₂²	C₂²×C₅×Dic₅	1	3	yes	—
142	C₄	C₂²×C₅×Dic₅	1	3	yes	—
143	C₄	C₂²×C₅×Dic₅	1	4	yes	—
144	C₄	C₂³×C₅×D₅	1	1	yes	characteristic
145	C₂	—	2	4	no	—
146	C₂	C₂×C₅×#681	1	6	yes	direct product factor
147	C₂	C₂³×C₅×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₅
commutator	C₂×C₅	C₂³×C₄×C₅
Frattini	C₄	C₂³×C₅×D₅
Fitting	C₂³×C₄×C₅²	C₂
socle	C₂³×C₅²	C₂×C₄
prime radical	C₂⁴×C₅²	C₄
cosocle	C₄×C₅	C₂⁴×C₅
hypercenter	C₂³×C₄×C₅	D₅
hypocenter	C₅	C₂²×C₅×M₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂³×C₄×C₅

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#681	C₂	direct (N × H)
3. C₂²×C₅×#168	C₂	semidirect
9. C₅×#681	C₂²	direct (N × H)
10. C₂×C₅×#168	C₂²	semidirect
12. C₂²×#681	C₅	direct (N × H)
21. C₅×#168	C₂³	semidirect
27. C₂×#681	C₂×C₅	direct (N × H)
28. C₂²×#168	C₂×C₅	semidirect
30. C₂²×C₅²	M₁₆	semidirect
53. #681	C₂²×C₅	direct (N × H)
54. C₂×#168	C₂²×C₅	semidirect
57. C₂×C₅²	C₂×M₁₆	semidirect
84. #168	C₂³×C₅	semidirect
88. C₅²	C₂²×M₁₆	semidirect
95. C₂²×C₅	#681	direct (N × H)
96. C₂²×C₅	C₅×M₁₆	semidirect
122. C₂×C₅	C₂×#681	direct (N × H)
123. C₂×C₅	C₂×C₅×M₁₆	semidirect
136. C₅	C₂²×#681	direct (N × H)
137. C₅	C₂²×C₅×M₁₆	semidirect
139. C₂²	C₅×#681	direct (N × H)
146. C₂	C₂×C₅×#681	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×C₅²	2
C₂×C₅×#681	2
C₂²×C₅×#168	2
C₂²×C₅×M₁₆	5
C₂²×#681	5

Subgroups — C22×C5×#681 #13106935

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#681 #13106935