Subgroups — C₂×C₅×#4094 #13106972

Order 1600.

696 conjugacy classes of subgroups.

The subgroup lattice has 155 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×#41	C₂	1	1	yes	maximal, characteristic
2	C₅×#4094	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₅×#681	C₂	1	3	yes	maximal, complemented
4	C₂²×C₅×#168	C₂	1	3	yes	maximal, complemented
5	C₅²×#41	C₄	1	4	yes	—
6	C₅²×#41	C₂²	1	4	yes	—
7	C₂×C₅²×Q₈	C₄	1	1	yes	characteristic
8	C₂×C₅²×D₄	C₄	1	3	yes	—
9	C₂²×C₄×C₅²	C₂²	1	3	yes	—
10	C₅×#681	C₂²	1	12	yes	complemented
11	C₂×C₅×#168	C₂²	1	1	yes	characteristic
12	C₂×C₅×#168	C₂²	1	3	yes	—
13	C₂×C₅×#168	C₂²	1	12	yes	complemented
14	C₂×C₅×#131	—	5	1	no	maximal
15	C₂×#4094	C₅	1	1	yes	maximal, characteristic, direct product factor
16	C₂³×C₅²	C₂×C₄	1	3	yes	—
17	C₅²×Q₈	C₂×C₄	1	4	yes	—
18	C₅²×D₄	C₂×C₄	1	12	yes	—
19	C₂×C₄×C₅²	C₂×C₄	1	3	yes	—
20	C₂×C₄×C₅²	C₂×C₄	1	6	yes	—
21	C₂×C₄×C₅²	C₂³	1	6	yes	—
22	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
23	C₅×#168	C₂³	1	2	yes	—
24	C₅×#168	C₂³	1	6	yes	—
25	C₂×C₅×#41	—	2	2	no	—
26	C₂×C₅×#41	D₅	1	1	yes	characteristic
27	C₂×C₅×#41	C₂×C₅	1	1	yes	characteristic
28	C₅×#131	—	5	8	no	—
29	C₂×C₅×M₁₆	—	5	3	no	—
30	C₂²×C₅×C₈	—	5	3	no	—
31	#4094	C₂×C₅	1	8	yes	direct product factor
32	C₂×#681	C₂×C₅	1	3	yes	complemented
33	C₂²×#168	C₂×C₅	1	3	yes	complemented
34	C₂²×C₅²	—	2	6	no	—
35	C₂²×C₅²	C₂²×C₄	1	6	yes	—
36	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
37	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
38	C₄×C₅²	C₂²×C₄	1	6	yes	—
39	C₄×C₅²	C₂⁴	1	1	yes	characteristic
40	C₅×#41	—	2	8	no	—
41	C₅×#41	—	2	8	no	—
42	C₅×#41	Dic₅	1	4	yes	—
43	C₅×#41	C₄×C₅	1	4	yes	—
44	C₅×#41	C₂×D₅	1	4	yes	—
45	C₅×#41	C₂²×C₅	1	4	yes	—
46	C₂×C₅×Q₈	—	2	2	no	—
47	C₂×C₅×Q₈	Dic₅	1	1	yes	characteristic
48	C₂×C₅×Q₈	C₄×C₅	1	1	yes	characteristic
49	C₂×C₅×D₄	—	2	6	no	—
50	C₂×C₅×D₄	Dic₅	1	3	yes	—
51	C₂×C₅×D₄	C₄×C₅	1	3	yes	—
52	C₂²×C₄×C₅	—	2	6	no	—
53	C₂²×C₄×C₅	C₂×D₅	1	3	yes	—
54	C₂²×C₄×C₅	C₂²×C₅	1	3	yes	—
55	C₅×M₁₆	—	5	12	no	—
56	C₂×C₅×C₈	—	5	12	no	—
57	C₂×C₅×C₈	—	5	1	no	—
58	C₂×C₅×C₈	—	5	3	no	—
59	#681	C₂²×C₅	1	12	yes	complemented
60	C₂×#168	C₂²×C₅	1	1	yes	characteristic
61	C₂×#168	C₂²×C₅	1	3	yes	—
62	C₂×#168	C₂²×C₅	1	12	yes	complemented
63	C₂×#131	—	5	1	no	Sylow
64	C₂×C₅²	—	2	6	no	—
65	C₂×C₅²	#131	1	2	yes	complemented
66	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
67	C₂³×C₅	—	2	6	no	—
68	C₂³×C₅	C₂×Dic₅	1	3	yes	—
69	C₂³×C₅	C₂×C₄×C₅	1	3	yes	—
70	C₅×Q₈	—	2	8	no	—
71	C₅×Q₈	C₂×Dic₅	1	4	yes	—
72	C₅×Q₈	C₂×C₄×C₅	1	4	yes	—
73	C₅×D₄	—	2	24	no	—
74	C₅×D₄	C₂×Dic₅	1	12	yes	—
75	C₅×D₄	C₂×C₄×C₅	1	12	yes	—
76	C₂×C₄×C₅	—	2	6	no	—
77	C₂×C₄×C₅	—	2	12	no	—
78	C₂×C₄×C₅	—	2	2	no	—
79	C₂×C₄×C₅	—	2	12	no	—
80	C₂×C₄×C₅	C₂×Dic₅	1	3	yes	—
81	C₂×C₄×C₅	C₂×Dic₅	1	6	yes	—
82	C₂×C₄×C₅	C₂×C₄×C₅	1	3	yes	—
83	C₂×C₄×C₅	C₂×C₄×C₅	1	6	yes	—
84	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
85	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
86	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
87	C₂×C₄×C₅	C₂³×C₅	1	6	yes	—
88	C₅×C₈	—	5	2	no	—
89	C₅×C₈	—	5	6	no	—
90	#168	C₂³×C₅	1	2	yes	—
91	#168	C₂³×C₅	1	6	yes	—
92	C₂×#41	C₅×D₅	1	1	yes	characteristic
93	#131	—	5	8	no	—
94	C₂×M₁₆	—	5	3	no	—
95	C₂²×C₈	—	5	3	no	—
96	C₅²	C₂×#131	1	1	yes	characteristic, complemented, Sylow
97	C₂²×C₅	—	2	6	no	—
98	C₂²×C₅	—	2	6	no	—
99	C₂²×C₅	—	2	2	no	—
100	C₂²×C₅	—	2	12	no	—
101	C₂²×C₅	—	4	12	no	—
102	C₂²×C₅	C₂²×Dic₅	1	6	yes	—
103	C₂²×C₅	C₂²×Dic₅	1	1	yes	characteristic
104	C₂²×C₅	C₂²×C₄×C₅	1	6	yes	—
105	C₂²×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
106	C₄×C₅	—	2	2	no	—
107	C₄×C₅	—	2	12	no	—
108	C₄×C₅	—	2	2	no	—
109	C₄×C₅	C₂²×Dic₅	1	1	yes	characteristic
110	C₄×C₅	C₂²×Dic₅	1	6	yes	—
111	C₄×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
112	C₄×C₅	C₂²×C₄×C₅	1	6	yes	—
113	C₄×C₅	C₂³×D₅	1	1	yes	characteristic
114	C₄×C₅	C₂⁴×C₅	1	1	yes	characteristic
115	#41	C₅×Dic₅	1	4	yes	—
116	#41	C₂×C₅×D₅	1	4	yes	—
117	C₂×Q₈	C₅×Dic₅	1	1	yes	characteristic
118	C₂×D₄	C₅×Dic₅	1	3	yes	—
119	C₂²×C₄	C₂×C₅×D₅	1	3	yes	—
120	M₁₆	—	5	12	no	—
121	C₂×C₈	—	5	1	no	—
122	C₂×C₈	—	5	3	no	—
123	C₂×C₈	—	5	12	no	—
124	C₂×C₅	—	2	6	no	—
125	C₂×C₅	—	2	6	no	—
126	C₂×C₅	—	2	4	no	—
127	C₂×C₅	—	2	2	no	—
128	C₂×C₅	—	4	12	no	—
129	C₂×C₅	#4094	1	2	yes	direct product factor
130	C₂×C₅	C₅×#131	1	2	yes	complemented
131	C₂×C₅	C₂³×Dic₅	1	1	yes	characteristic
132	C₂×C₅	C₂³×C₄×C₅	1	1	yes	characteristic
133	C₂³	C₂×C₅×Dic₅	1	3	yes	—
134	Q₈	C₂×C₅×Dic₅	1	4	yes	—
135	D₄	C₂×C₅×Dic₅	1	12	yes	—
136	C₂×C₄	C₂×C₅×Dic₅	1	3	yes	—
137	C₂×C₄	C₂×C₅×Dic₅	1	6	yes	—
138	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
139	C₂×C₄	C₂²×C₅×D₅	1	6	yes	—
140	C₈	—	5	2	no	—
141	C₈	—	5	6	no	—
142	C₅	—	2	2	no	—
143	C₅	C₂×#4094	1	1	yes	characteristic, direct product factor
144	C₅	C₂×C₅×#131	1	1	yes	characteristic, complemented
145	C₂²	—	2	6	no	—
146	C₂²	C₂²×C₅×Dic₅	1	6	yes	—
147	C₂²	C₂²×C₅×Dic₅	1	1	yes	characteristic
148	C₄	C₂²×C₅×Dic₅	1	1	yes	characteristic
149	C₄	C₂²×C₅×Dic₅	1	6	yes	—
150	C₄	C₂³×C₅×D₅	1	1	yes	characteristic
151	C₂	—	2	6	no	—
152	C₂	C₅×#4094	1	2	yes	direct product factor
153	C₂	C₂³×C₅×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₅
commutator	C₂×C₅	C₂³×C₄×C₅
Frattini	C₄	C₂³×C₅×D₅
Fitting, prime radical	C₂×C₅²×#41	C₂
socle	C₂²×C₅²	C₂²×C₄
cosocle	C₄×C₅	C₂⁴×C₅
hypercenter	C₂×C₅×#41	D₅
hypocenter	C₅	C₂×C₅×#131
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#41

Complements

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

N	H ≅ G/N	type
2. C₅×#4094	C₂	direct (N × H)
3. C₂×C₅×#681	C₂	semidirect
4. C₂²×C₅×#168	C₂	semidirect
10. C₅×#681	C₂²	semidirect
13. C₂×C₅×#168	C₂²	semidirect
15. C₂×#4094	C₅	direct (N × H)
31. #4094	C₂×C₅	direct (N × H)
32. C₂×#681	C₂×C₅	semidirect
33. C₂²×#168	C₂×C₅	semidirect
59. #681	C₂²×C₅	semidirect
62. C₂×#168	C₂²×C₅	semidirect
65. C₂×C₅²	#131	semidirect
96. C₅²	C₂×#131	semidirect
129. C₂×C₅	#4094	direct (N × H)
130. C₂×C₅	C₅×#131	semidirect
143. C₅	C₂×#4094	direct (N × H)
144. C₅	C₂×C₅×#131	semidirect
152. C₂	C₅×#4094	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅²×#41	2
C₅×#4094	2
C₂×C₅×#681	2
C₂²×C₅×#168	2
C₂×C₅×#131	5
C₂×#4094	5

Subgroups — C2×C5×#4094 #13106972

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4094 #13106972