Subgroups — C₅×M₁₆×Dic₅ #13102378

Order 1600.

352 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅×Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#681	C₂	1	1	yes	maximal, characteristic
4	C₅×#3947	C₂	1	2	yes	maximal, complemented
5	C₅×C₈×Dic₅	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
7	C₂²×C₅×Dic₅	C₄	1	2	yes	—
8	C₅²×M₁₆	C₄	1	4	yes	complemented
9	C₂×C₅²×C₈	C₂²	1	2	yes	—
10	C₄×C₅×Dic₅	C₄	1	2	yes	—
11	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
12	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
13	C₅×#681	C₄	1	4	yes	complemented
14	C₂×C₅×#168	C₂²	1	2	yes	—
15	C₄×C₅×M₁₆	—	5	1	no	maximal
16	M₁₆×Dic₅	C₅	1	1	yes	maximal, characteristic, direct product factor
17	C₂³×C₅²	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
22	C₂×C₅×Dic₅	—	2	4	no	—
23	C₂×C₅×Dic₅	C₂×C₄	1	2	yes	—
24	C₂×C₅×Dic₅	C₂×C₄	1	2	yes	—
25	C₅²×C₈	C₂×C₄	1	4	yes	complemented
26	C₅×#168	C₂×C₄	1	4	yes	complemented
27	C₂×C₅×M₁₆	—	2	2	no	—
28	C₂×C₅×M₁₆	—	5	1	no	—
29	C₂×C₅×M₁₆	D₅	1	1	yes	characteristic
30	C₂×C₅×M₁₆	C₂×C₅	1	1	yes	characteristic
31	C₂×C₄²×C₅	—	5	1	no	—
32	C₂×C₄×Dic₅	C₂×C₅	1	1	yes	characteristic
33	C₂×#681	C₂×C₅	1	1	yes	characteristic
34	C₅×#97	—	5	2	no	—
35	C₄×C₅×C₈	—	5	2	no	—
36	#3947	C₂×C₅	1	2	yes	complemented
37	C₈×Dic₅	C₂×C₅	1	2	yes	complemented
38	C₂²×C₅²	—	2	2	no	—
39	C₂²×C₅²	C₄²	1	2	yes	—
40	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
41	C₄×C₅²	C₄²	1	2	yes	—
42	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
43	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
44	C₅×Dic₅	—	2	2	no	—
45	C₅×Dic₅	M₁₆	1	4	yes	direct product factor
46	C₂²×C₄×C₅	—	2	2	no	—
47	C₂²×C₄×C₅	—	5	2	no	—
48	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
49	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
50	C₂²×Dic₅	C₄×C₅	1	2	yes	—
51	C₅×M₁₆	—	2	8	no	—
52	C₅×M₁₆	—	5	4	no	—
53	C₅×M₁₆	Dic₅	1	4	yes	direct product factor
54	C₅×M₁₆	C₄×C₅	1	4	yes	complemented
55	C₂×C₅×C₈	—	2	4	no	—
56	C₂×C₅×C₈	—	5	2	no	—
57	C₂×C₅×C₈	C₂×D₅	1	2	yes	—
58	C₂×C₅×C₈	C₂²×C₅	1	2	yes	—
59	C₄²×C₅	—	5	2	no	—
60	C₄²×C₅	—	5	1	no	—
61	C₄²×C₅	—	5	1	no	—
62	C₄×Dic₅	C₄×C₅	1	2	yes	—
63	C₄×Dic₅	C₂²×C₅	1	1	yes	characteristic
64	C₄×Dic₅	C₂²×C₅	1	1	yes	characteristic
65	#681	C₄×C₅	1	4	yes	complemented
66	C₂×#168	C₂²×C₅	1	2	yes	—
67	C₄×M₁₆	—	5	1	no	Sylow
68	C₂×C₅²	—	2	2	no	—
69	C₂×C₅²	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₅²	C₂×M₁₆	1	2	yes	—
71	C₂³×C₅	—	2	2	no	—
72	C₂³×C₅	C₄×D₅	1	1	yes	characteristic
73	C₂³×C₅	C₂×C₄×C₅	1	1	yes	characteristic
74	C₂×C₄×C₅	—	2	4	no	—
75	C₂×C₄×C₅	—	2	2	no	—
76	C₂×C₄×C₅	—	2	4	no	—
77	C₂×C₄×C₅	—	2	2	no	—
78	C₂×C₄×C₅	—	5	2	no	—
79	C₂×C₄×C₅	—	5	2	no	—
80	C₂×C₄×C₅	—	10	4	no	—
81	C₂×C₄×C₅	C₄×D₅	1	1	yes	characteristic
82	C₂×C₄×C₅	C₄×D₅	1	2	yes	—
83	C₂×C₄×C₅	C₂×Dic₅	1	2	yes	—
84	C₂×C₄×C₅	C₂×C₄×C₅	1	1	yes	characteristic
85	C₂×C₄×C₅	C₂×C₄×C₅	1	2	yes	—
86	C₂×C₄×C₅	C₂×C₄×C₅	1	2	yes	—
87	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
88	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
89	C₂×Dic₅	—	2	4	no	—
90	C₂×Dic₅	C₂×C₄×C₅	1	2	yes	—
91	C₂×Dic₅	C₂×C₄×C₅	1	2	yes	—
92	C₅×C₈	—	2	8	no	—
93	C₅×C₈	—	5	4	no	—
94	C₅×C₈	C₂×Dic₅	1	4	yes	complemented
95	C₅×C₈	C₂×C₄×C₅	1	4	yes	complemented
96	#168	C₂×C₄×C₅	1	4	yes	complemented
97	C₂×M₁₆	—	5	1	no	—
98	C₂×M₁₆	C₅×D₅	1	1	yes	characteristic
99	C₂×C₄²	—	5	1	no	—
100	#97	—	5	2	no	—
101	C₄×C₈	—	5	2	no	—
102	C₅²	C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
103	C₂²×C₅	—	2	2	no	—
104	C₂²×C₅	—	2	2	no	—
105	C₂²×C₅	—	2	2	no	—
106	C₂²×C₅	—	2	4	no	—
107	C₂²×C₅	—	4	4	no	—
108	C₂²×C₅	C₄×Dic₅	1	2	yes	—
109	C₂²×C₅	C₄²×C₅	1	2	yes	—
110	C₂²×C₅	C₂×C₄×D₅	1	1	yes	characteristic
111	C₂²×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
112	C₄×C₅	—	2	2	no	—
113	C₄×C₅	—	2	2	no	—
114	C₄×C₅	—	2	4	no	—
115	C₄×C₅	—	5	4	no	—
116	C₄×C₅	—	10	2	no	—
117	C₄×C₅	C₄×Dic₅	1	2	yes	—
118	C₄×C₅	C₄²×C₅	1	2	yes	—
119	C₄×C₅	C₂×C₄×D₅	1	1	yes	characteristic
120	C₄×C₅	C₂²×Dic₅	1	1	yes	characteristic
121	C₄×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
122	C₄×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
123	Dic₅	—	2	2	no	—
124	Dic₅	C₅×M₁₆	1	4	yes	direct product factor
125	C₂²×C₄	—	5	2	no	—
126	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
127	M₁₆	—	5	4	no	—
128	M₁₆	C₅×Dic₅	1	4	yes	direct product factor
129	C₂×C₈	—	5	2	no	—
130	C₂×C₈	C₂×C₅×D₅	1	2	yes	—
131	C₄²	—	5	2	no	—
132	C₄²	—	5	1	no	—
133	C₄²	—	5	1	no	—
134	C₂×C₅	—	2	2	no	—
135	C₂×C₅	—	2	2	no	—
136	C₂×C₅	—	2	4	no	—
137	C₂×C₅	—	2	2	no	—
138	C₂×C₅	—	4	4	no	—
139	C₂×C₅	D₅×M₁₆	1	2	yes	—
140	C₂×C₅	C₂×C₄×Dic₅	1	1	yes	characteristic
141	C₂×C₅	C₂×C₄²×C₅	1	1	yes	characteristic
142	C₂×C₅	C₂×C₅×M₁₆	1	2	yes	—
143	C₂³	C₄×C₅×D₅	1	1	yes	characteristic
144	C₂×C₄	—	5	2	no	—
145	C₂×C₄	—	5	2	no	—
146	C₂×C₄	—	10	4	no	—
147	C₂×C₄	C₄×C₅×D₅	1	2	yes	—
148	C₂×C₄	C₄×C₅×D₅	1	1	yes	characteristic
149	C₂×C₄	C₂×C₅×Dic₅	1	2	yes	—
150	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
151	C₈	—	5	4	no	—
152	C₈	C₂×C₅×Dic₅	1	4	yes	complemented
153	C₅	—	2	2	no	—
154	C₅	M₁₆×Dic₅	1	1	yes	characteristic, direct product factor
155	C₅	C₄×C₅×M₁₆	1	1	yes	characteristic, complemented
156	C₂²	—	2	2	no	—
157	C₂²	C₄×C₅×Dic₅	1	2	yes	—
158	C₂²	C₂×C₄×C₅×D₅	1	1	yes	characteristic
159	C₄	—	5	4	no	—
160	C₄	—	10	2	no	—
161	C₄	C₄×C₅×Dic₅	1	2	yes	—
162	C₄	C₂×C₄×C₅×D₅	1	1	yes	characteristic
163	C₄	C₂²×C₅×Dic₅	1	1	yes	characteristic
164	C₂	—	2	2	no	—
165	C₂	C₅×D₅×M₁₆	1	2	yes	—
166	C₂	C₂×C₄×C₅×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₅
commutator	C₂×C₅	C₂×C₄²×C₅
Frattini	C₂×C₄	C₂²×C₅×D₅
Fitting	C₂×C₅²×M₁₆	C₂
socle	C₂²×C₅²	C₂²×C₄
prime radical	C₂³×C₅²	C₂×C₄
cosocle	C₂×C₄×C₅	C₂³×C₅
hypercenter	C₂×C₅×M₁₆	D₅
hypocenter	C₅	C₄×C₅×M₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₅×M₁₆

Complements

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

N	H ≅ G/N	type
4. C₅×#3947	C₂	semidirect
5. C₅×C₈×Dic₅	C₂	semidirect
8. C₅²×M₁₆	C₄	semidirect
13. C₅×#681	C₄	semidirect
16. M₁₆×Dic₅	C₅	direct (N × H)
25. C₅²×C₈	C₂×C₄	semidirect
26. C₅×#168	C₂×C₄	semidirect
36. #3947	C₂×C₅	semidirect
37. C₈×Dic₅	C₂×C₅	semidirect
45. C₅×Dic₅	M₁₆	direct (N × H)
53. C₅×M₁₆	Dic₅	direct (N × H)
54. C₅×M₁₆	C₄×C₅	semidirect
65. #681	C₄×C₅	semidirect
94. C₅×C₈	C₂×Dic₅	semidirect
95. C₅×C₈	C₂×C₄×C₅	semidirect
96. #168	C₂×C₄×C₅	semidirect
102. C₅²	C₄×M₁₆	semidirect
124. Dic₅	C₅×M₁₆	direct (N × H)
128. M₁₆	C₅×Dic₅	direct (N × H)
152. C₈	C₂×C₅×Dic₅	semidirect
154. C₅	M₁₆×Dic₅	direct (N × H)
155. C₅	C₄×C₅×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅²×M₁₆	2
C₂×C₄×C₅×Dic₅	2
C₂×C₅×#681	2
C₅×#3947	2
C₅×C₈×Dic₅	2
C₄×C₅×M₁₆	5
M₁₆×Dic₅	5

Subgroups — C5×M16×Dic5 #13102378

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×M₁₆×Dic₅ #13102378