Subgroups — C₂×C₅×#4033 #13106650

Order 1600.

688 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×Dic₁₀	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅²×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#683	C₂	1	2	yes	maximal
4	C₂×C₄×C₅×Dic₅	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₅×Dic₅	C₂	1	2	yes	maximal
6	C₅×#4033	C₂	1	8	yes	maximal, direct product factor
7	C₂²×C₄×C₅²	C₂²	1	2	yes	—
8	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
9	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
10	C₂²×C₅×Dic₅	C₂²	1	2	yes	—
11	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
12	C₂×C₅×Dic₁₀	C₄	1	8	yes	complemented
13	C₂×C₅×Dic₁₀	C₂²	1	4	yes	—
14	C₅²×C₄⋊C₄	C₂²	1	4	yes	—
15	C₅×#683	C₂²	1	8	yes	—
16	C₄×C₅×Dic₅	C₂²	1	4	yes	—
17	C₄×C₅×Dic₅	C₂²	1	8	yes	—
18	C₂×C₄×C₅×Q₈	—	5	1	no	maximal
19	C₂×#4033	C₅	1	1	yes	maximal, characteristic, direct product factor
20	C₂³×C₅²	C₂³	1	1	yes	characteristic
21	C₂×C₄×C₅²	—	2	4	no	—
22	C₂×C₄×C₅²	C₂×C₄	1	4	yes	—
23	C₂×C₄×C₅²	C₂³	1	4	yes	—
24	C₂×C₄×C₅²	C₂³	1	2	yes	—
25	C₂×C₅×Dic₅	—	2	2	no	—
26	C₂×C₅×Dic₅	C₂×C₄	1	8	yes	—
27	C₂×C₅×Dic₅	Q₈	1	4	yes	complemented
28	C₂×C₅×Dic₅	C₂³	1	2	yes	—
29	C₂×C₅×Dic₅	C₂³	1	4	yes	—
30	C₂×C₅×Dic₅	C₂³	1	2	yes	—
31	C₅×Dic₁₀	C₂×C₄	1	16	yes	complemented
32	C₂²×C₅×Q₈	—	5	1	no	—
33	C₂²×Dic₁₀	C₂×C₅	1	1	yes	characteristic
34	C₄×C₅×Q₈	—	5	8	no	—
35	C₂×C₅×C₄⋊C₄	—	2	2	no	—
36	C₂×C₅×C₄⋊C₄	—	5	2	no	—
37	C₂×C₅×C₄⋊C₄	D₅	1	1	yes	characteristic
38	C₂×C₅×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
39	C₂×C₄²×C₅	—	5	2	no	—
40	C₂×C₄²×C₅	—	5	1	no	—
41	C₂×#683	C₂×C₅	1	2	yes	—
42	C₂×C₄×Dic₅	C₂×C₅	1	1	yes	characteristic
43	C₂×C₄×Dic₅	C₂×C₅	1	2	yes	—
44	#4033	C₂×C₅	1	8	yes	direct product factor
45	C₂²×C₅²	C₂²×C₄	1	2	yes	—
46	C₂²×C₅²	C₂×Q₈	1	2	yes	—
47	C₂²×C₅²	#41	1	2	yes	—
48	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
49	C₄×C₅²	—	2	4	no	—
50	C₄×C₅²	C₂²×C₄	1	4	yes	—
51	C₅×Dic₅	—	2	2	no	—
52	C₅×Dic₅	C₂²×C₄	1	8	yes	—
53	C₅×Dic₅	C₂×Q₈	1	4	yes	complemented
54	C₂×C₅×Q₈	—	5	8	no	—
55	C₂×C₅×Q₈	—	5	4	no	—
56	C₂²×C₄×C₅	—	2	2	no	—
57	C₂²×C₄×C₅	—	2	4	no	—
58	C₂²×C₄×C₅	—	5	1	no	—
59	C₂²×C₄×C₅	—	5	2	no	—
60	C₂²×C₄×C₅	—	5	1	no	—
61	C₂²×C₄×C₅	C₂×D₅	1	2	yes	—
62	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂²×C₅	1	2	yes	—
64	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
65	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic
66	C₂²×Dic₅	C₂²×C₅	1	2	yes	—
67	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic
68	C₂×Dic₁₀	C₄×C₅	1	8	yes	complemented
69	C₂×Dic₁₀	C₂²×C₅	1	4	yes	—
70	C₅×C₄⋊C₄	—	2	8	no	—
71	C₅×C₄⋊C₄	—	5	8	no	—
72	C₅×C₄⋊C₄	C₂×D₅	1	4	yes	—
73	C₅×C₄⋊C₄	C₂²×C₅	1	4	yes	—
74	C₄²×C₅	—	5	8	no	—
75	C₄²×C₅	—	5	4	no	—
76	#683	C₂²×C₅	1	8	yes	—
77	C₄×Dic₅	C₂²×C₅	1	4	yes	—
78	C₄×Dic₅	C₂²×C₅	1	8	yes	—
79	C₂×C₄×Q₈	—	5	1	no	Sylow
80	C₂×C₅²	C₄×Q₈	1	4	yes	complemented
81	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
82	C₂×C₅²	C₂²×Q₈	1	1	yes	characteristic
83	C₂×C₅²	C₂×#41	1	1	yes	characteristic
84	C₂³×C₅	—	2	2	no	—
85	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
86	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
87	C₅×Q₈	—	5	16	no	—
88	C₂×C₄×C₅	—	2	4	no	—
89	C₂×C₄×C₅	—	2	4	no	—
90	C₂×C₄×C₅	—	2	8	no	—
91	C₂×C₄×C₅	—	2	4	no	—
92	C₂×C₄×C₅	—	2	8	no	—
93	C₂×C₄×C₅	—	4	8	no	—
94	C₂×C₄×C₅	—	5	8	no	—
95	C₂×C₄×C₅	—	5	4	no	—
96	C₂×C₄×C₅	—	5	2	no	—
97	C₂×C₄×C₅	—	5	4	no	—
98	C₂×C₄×C₅	—	5	2	no	—
99	C₂×C₄×C₅	—	10	2	no	—
100	C₂×C₄×C₅	C₄×D₅	1	4	yes	—
101	C₂×C₄×C₅	C₂×C₄×C₅	1	4	yes	—
102	C₂×C₄×C₅	C₂²×D₅	1	4	yes	—
103	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
104	C₂×C₄×C₅	C₂³×C₅	1	4	yes	—
105	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
106	C₂×Dic₅	—	2	2	no	—
107	C₂×Dic₅	C₂×C₄×C₅	1	8	yes	—
108	C₂×Dic₅	C₅×Q₈	1	4	yes	complemented
109	C₂×Dic₅	C₂³×C₅	1	2	yes	—
110	C₂×Dic₅	C₂³×C₅	1	4	yes	—
111	C₂×Dic₅	C₂³×C₅	1	2	yes	—
112	Dic₁₀	C₂×C₄×C₅	1	16	yes	complemented
113	C₂²×Q₈	—	5	1	no	—
114	C₄×Q₈	—	5	8	no	—
115	C₂×C₄⋊C₄	—	5	2	no	—
116	C₂×C₄⋊C₄	C₅×D₅	1	1	yes	characteristic
117	C₂×C₄²	—	5	2	no	—
118	C₂×C₄²	—	5	1	no	—
119	C₅²	C₂×C₄×Q₈	1	1	yes	characteristic, complemented, Sylow
120	C₂²×C₅	—	2	4	no	—
121	C₂²×C₅	—	2	4	no	—
122	C₂²×C₅	—	2	4	no	—
123	C₂²×C₅	—	2	2	no	—
124	C₂²×C₅	C₂×C₄×D₅	1	2	yes	—
125	C₂²×C₅	#711	1	2	yes	—
126	C₂²×C₅	Q₈×D₅	1	2	yes	—
127	C₂²×C₅	C₂²×C₄×C₅	1	2	yes	—
128	C₂²×C₅	C₂×C₅×Q₈	1	2	yes	—
129	C₂²×C₅	C₅×#41	1	2	yes	—
130	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
131	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
132	C₄×C₅	—	2	4	no	—
133	C₄×C₅	—	2	4	no	—
134	C₄×C₅	—	2	8	no	—
135	C₄×C₅	—	4	8	no	—
136	C₄×C₅	—	5	8	no	—
137	C₄×C₅	—	5	4	no	—
138	C₄×C₅	—	10	2	no	—
139	C₄×C₅	C₂×C₄×D₅	1	4	yes	—
140	C₄×C₅	C₂²×C₄×C₅	1	4	yes	—
141	Dic₅	—	2	2	no	—
142	Dic₅	C₂²×C₄×C₅	1	8	yes	—
143	Dic₅	C₂×C₅×Q₈	1	4	yes	complemented
144	C₂×Q₈	—	5	8	no	—
145	C₂×Q₈	—	5	4	no	—
146	C₂²×C₄	—	5	1	no	—
147	C₂²×C₄	—	5	2	no	—
148	C₂²×C₄	—	5	1	no	—
149	C₂²×C₄	C₂×C₅×D₅	1	2	yes	—
150	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
151	C₄⋊C₄	—	5	8	no	—
152	C₄⋊C₄	C₂×C₅×D₅	1	4	yes	—
153	C₄²	—	5	8	no	—
154	C₄²	—	5	4	no	—
155	C₂×C₅	—	2	8	no	—
156	C₂×C₅	—	2	2	no	—
157	C₂×C₅	—	2	2	no	—
158	C₂×C₅	—	2	2	no	—
159	C₂×C₅	#4033	1	4	yes	direct product factor
160	C₂×C₅	C₄×C₅×Q₈	1	4	yes	complemented
161	C₂×C₅	C₂²×C₄×D₅	1	1	yes	characteristic
162	C₂×C₅	C₂×#711	1	1	yes	characteristic
163	C₂×C₅	C₂×Q₈×D₅	1	1	yes	characteristic
164	C₂×C₅	C₂³×C₄×C₅	1	1	yes	characteristic
165	C₂×C₅	C₂²×C₅×Q₈	1	1	yes	characteristic
166	C₂×C₅	C₂×C₅×#41	1	1	yes	characteristic
167	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
168	Q₈	—	5	16	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	5	8	no	—
171	C₂×C₄	—	5	4	no	—
172	C₂×C₄	—	5	2	no	—
173	C₂×C₄	—	5	4	no	—
174	C₂×C₄	—	5	2	no	—
175	C₂×C₄	—	10	2	no	—
176	C₂×C₄	C₄×C₅×D₅	1	4	yes	—
177	C₂×C₄	C₂²×C₅×D₅	1	4	yes	—
178	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
179	C₅	—	2	2	no	—
180	C₅	C₂×#4033	1	1	yes	characteristic, direct product factor
181	C₅	C₂×C₄×C₅×Q₈	1	1	yes	characteristic, complemented
182	C₂²	C₂×C₄×C₅×D₅	1	2	yes	—
183	C₂²	C₅×#711	1	2	yes	—
184	C₂²	C₅×Q₈×D₅	1	2	yes	—
185	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
186	C₄	—	2	4	no	—
187	C₄	—	5	8	no	—
188	C₄	—	5	4	no	—
189	C₄	—	10	2	no	—
190	C₄	C₂×C₄×C₅×D₅	1	4	yes	—
191	C₂	C₅×#4033	1	4	yes	direct product factor
192	C₂	C₂²×C₄×C₅×D₅	1	1	yes	characteristic
193	C₂	C₂×C₅×#711	1	1	yes	characteristic
194	C₂	C₂×C₅×Q₈×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	C₂²×D₅
commutator	C₂×C₅	C₂³×C₄×C₅
Frattini	C₂²	C₂³×C₅×D₅
Fitting	C₂×C₅²×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₅²	C₂³
cosocle	C₂²×C₅	C₂⁴×C₅
hypercenter	C₂×C₅×C₄⋊C₄	D₅
hypocenter	C₅	C₂×C₄×C₅×Q₈
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₅×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
6. C₅×#4033	C₂	direct (N × H)
12. C₂×C₅×Dic₁₀	C₄	semidirect
19. C₂×#4033	C₅	direct (N × H)
27. C₂×C₅×Dic₅	Q₈	semidirect
31. C₅×Dic₁₀	C₂×C₄	semidirect
44. #4033	C₂×C₅	direct (N × H)
53. C₅×Dic₅	C₂×Q₈	semidirect
68. C₂×Dic₁₀	C₄×C₅	semidirect
80. C₂×C₅²	C₄×Q₈	semidirect
108. C₂×Dic₅	C₅×Q₈	semidirect
112. Dic₁₀	C₂×C₄×C₅	semidirect
119. C₅²	C₂×C₄×Q₈	semidirect
143. Dic₅	C₂×C₅×Q₈	semidirect
159. C₂×C₅	#4033	direct (N × H)
160. C₂×C₅	C₄×C₅×Q₈	semidirect
180. C₅	C₂×#4033	direct (N × H)
181. C₅	C₂×C₄×C₅×Q₈	semidirect
191. C₂	C₅×#4033	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×C₅×Dic₁₀	2
C₂×C₅²×C₄⋊C₄	2
C₂×C₅×#683	2
C₂×C₄×C₅×Dic₅	2
C₅×#4033	2
C₂×C₄×C₅×Q₈	5
C₂×#4033	5

Subgroups — C2×C5×#4033 #13106650

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4033 #13106650