Subgroups — C₅×#65804 #13106624

Order 1600.

552 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄²×C₅²	C₂	1	1	yes	maximal, characteristic
2	C₅×#4072	C₂	1	2	yes	maximal
3	C₅×#4070	C₂	1	4	yes	maximal
4	C₅×#4016	C₂	1	2	yes	maximal, complemented
5	C₅×#4015	C₂	1	2	yes	maximal, complemented
6	C₄×C₅×Dic₁₀	C₂	1	4	yes	maximal, complemented
7	C₂²×C₄×C₅²	C₂²	1	2	yes	—
8	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
9	C₂×C₅×Dic₁₀	C₂²	1	4	yes	—
10	C₄²×C₅²	C₂²	1	2	yes	—
11	C₄²×C₅²	C₂²	1	1	yes	characteristic
12	C₄²×C₅²	C₂²	1	1	yes	characteristic
13	C₅×C₂²⋊Dic₅	C₂²	1	4	yes	—
14	C₅×C₄⋊Dic₅	C₂²	1	4	yes	—
15	C₅×C₄⋊Dic₅	C₂²	1	4	yes	—
16	C₅×#683	C₂²	1	8	yes	—
17	C₄×C₅×Dic₅	C₂²	1	4	yes	—
18	C₅×#533	—	5	1	no	maximal
19	#65804	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	Q₈	1	4	yes	complemented
23	C₂×C₄×C₅²	C₂³	1	2	yes	—
24	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
25	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
26	C₂×C₄×C₅²	C₂³	1	2	yes	—
27	C₂×C₅×Dic₅	C₂³	1	8	yes	—
28	C₅×Dic₁₀	—	2	8	no	—
29	C₅×C₄⋊Q₈	—	5	2	no	—
30	C₅×#125	—	5	2	no	—
31	C₅×C₂²⋊Q₈	—	5	4	no	—
32	C₄×C₅×Q₈	—	5	4	no	—
33	C₅×#117	—	5	2	no	—
34	C₂×C₄²×C₅	—	2	2	no	—
35	C₂×C₄²×C₅	D₅	1	1	yes	characteristic
36	C₂×C₄²×C₅	C₂×C₅	1	1	yes	characteristic
37	#4072	C₂×C₅	1	2	yes	—
38	#4070	C₂×C₅	1	4	yes	—
39	#4016	C₂×C₅	1	2	yes	complemented
40	#4015	C₂×C₅	1	2	yes	complemented
41	C₄×Dic₁₀	C₂×C₅	1	4	yes	complemented
42	C₂²×C₅²	—	2	2	no	—
43	C₂²×C₅²	C₂×Q₈	1	2	yes	—
44	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
45	C₄×C₅²	—	2	2	no	—
46	C₄×C₅²	C₂×Q₈	1	2	yes	—
47	C₄×C₅²	C₂×Q₈	1	2	yes	—
48	C₄×C₅²	#41	1	4	yes	—
49	C₅×Dic₅	—	2	8	no	—
50	C₂×C₅×Q₈	—	5	4	no	—
51	C₂²×C₄×C₅	—	2	4	no	—
52	C₂²×C₄×C₅	—	2	2	no	—
53	C₂²×C₄×C₅	C₂×D₅	1	2	yes	—
54	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₂²×C₅	1	2	yes	—
56	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
57	C₂×Dic₁₀	C₂²×C₅	1	4	yes	—
58	C₅×C₄⋊C₄	—	5	4	no	—
59	C₅×C₄⋊C₄	—	5	8	no	—
60	C₅×C₄⋊C₄	—	5	4	no	—
61	C₅×C₂²⋊C₄	—	5	4	no	—
62	C₄²×C₅	—	2	4	no	—
63	C₄²×C₅	—	2	2	no	—
64	C₄²×C₅	—	2	2	no	—
65	C₄²×C₅	—	5	4	no	—
66	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
67	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
68	C₄²×C₅	C₂×D₅	1	2	yes	—
69	C₄²×C₅	C₂²×C₅	1	2	yes	—
70	C₄²×C₅	C₂²×C₅	1	1	yes	characteristic
71	C₄²×C₅	C₂²×C₅	1	1	yes	characteristic
72	C₂²⋊Dic₅	C₂²×C₅	1	4	yes	—
73	C₄⋊Dic₅	C₂²×C₅	1	4	yes	—
74	C₄⋊Dic₅	C₂²×C₅	1	4	yes	—
75	#683	C₂²×C₅	1	8	yes	—
76	C₄×Dic₅	C₂²×C₅	1	4	yes	—
77	#533	—	5	1	no	Sylow
78	C₂×C₅²	—	2	2	no	—
79	C₂×C₅²	C₂²×Q₈	1	1	yes	characteristic
80	C₂×C₅²	C₂×#41	1	2	yes	—
81	C₂³×C₅	—	2	2	no	—
82	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
83	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
84	C₅×Q₈	—	10	8	no	—
85	C₂×C₄×C₅	—	2	16	no	—
86	C₂×C₄×C₅	—	2	4	no	—
87	C₂×C₄×C₅	—	2	4	no	—
88	C₂×C₄×C₅	—	2	8	no	—
89	C₂×C₄×C₅	—	2	4	no	—
90	C₂×C₄×C₅	—	2	2	no	—
91	C₂×C₄×C₅	—	2	2	no	—
92	C₂×C₄×C₅	—	2	4	no	—
93	C₂×C₄×C₅	—	5	8	no	—
94	C₂×C₄×C₅	Dic₁₀	1	4	yes	complemented
95	C₂×C₄×C₅	C₅×Q₈	1	4	yes	complemented
96	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
97	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
98	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
99	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
100	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
101	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
102	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
103	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
104	C₂×Dic₅	C₂³×C₅	1	8	yes	—
105	Dic₁₀	—	2	8	no	—
106	C₄⋊Q₈	—	5	2	no	—
107	#125	—	5	2	no	—
108	C₂²⋊Q₈	—	5	4	no	—
109	C₄×Q₈	—	5	4	no	—
110	#117	—	5	2	no	—
111	C₂×C₄²	C₅×D₅	1	1	yes	characteristic
112	C₅²	#533	1	1	yes	characteristic, complemented, Sylow
113	C₂²×C₅	—	2	8	no	—
114	C₂²×C₅	—	2	2	no	—
115	C₂²×C₅	—	2	2	no	—
116	C₂²×C₅	—	2	4	no	—
117	C₂²×C₅	—	2	2	no	—
118	C₂²×C₅	C₂×Dic₁₀	1	2	yes	—
119	C₂²×C₅	C₂×C₅×Q₈	1	2	yes	—
120	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
121	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
122	C₄×C₅	—	2	8	no	—
123	C₄×C₅	—	2	2	no	—
124	C₄×C₅	—	2	2	no	—
125	C₄×C₅	—	2	4	no	—
126	C₄×C₅	—	2	4	no	—
127	C₄×C₅	—	2	8	no	—
128	C₄×C₅	—	10	8	no	—
129	C₄×C₅	C₂×Dic₁₀	1	2	yes	—
130	C₄×C₅	C₂×Dic₁₀	1	2	yes	—
131	C₄×C₅	#709	1	4	yes	—
132	C₄×C₅	C₂×C₅×Q₈	1	2	yes	—
133	C₄×C₅	C₂×C₅×Q₈	1	2	yes	—
134	C₄×C₅	C₅×#41	1	4	yes	—
135	Dic₅	—	2	8	no	—
136	C₂×Q₈	—	5	4	no	—
137	C₂²×C₄	C₂×C₅×D₅	1	2	yes	—
138	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
139	C₄⋊C₄	—	5	4	no	—
140	C₄⋊C₄	—	5	8	no	—
141	C₄⋊C₄	—	5	4	no	—
142	C₂²⋊C₄	—	5	4	no	—
143	C₄²	—	5	4	no	—
144	C₄²	C₂×C₅×D₅	1	2	yes	—
145	C₄²	C₂×C₅×D₅	1	1	yes	characteristic
146	C₄²	C₂×C₅×D₅	1	1	yes	characteristic
147	C₂×C₅	—	2	8	no	—
148	C₂×C₅	—	2	2	no	—
149	C₂×C₅	—	2	2	no	—
150	C₂×C₅	—	2	2	no	—
151	C₂×C₅	—	2	4	no	—
152	C₂×C₅	C₂²×Dic₁₀	1	1	yes	characteristic
153	C₂×C₅	C₂×#709	1	2	yes	—
154	C₂×C₅	C₂²×C₅×Q₈	1	1	yes	characteristic
155	C₂×C₅	C₂×C₅×#41	1	2	yes	—
156	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
157	Q₈	—	10	8	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	5	8	no	—
160	C₂×C₄	C₅×Dic₁₀	1	4	yes	complemented
161	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
162	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
163	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
164	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
165	C₅	—	2	2	no	—
166	C₅	#65804	1	1	yes	characteristic, direct product factor
167	C₅	C₅×#533	1	1	yes	characteristic, complemented
168	C₂²	—	2	2	no	—
169	C₂²	C₂×C₅×Dic₁₀	1	2	yes	—
170	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
171	C₄	—	2	2	no	—
172	C₄	—	10	8	no	—
173	C₄	C₂×C₅×Dic₁₀	1	2	yes	—
174	C₄	C₂×C₅×Dic₁₀	1	2	yes	—
175	C₄	C₅×#709	1	4	yes	—
176	C₂	—	2	2	no	—
177	C₂	C₂²×C₅×Dic₁₀	1	1	yes	characteristic
178	C₂	C₂×C₅×#709	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₅
Fitting	C₂×C₄²×C₅²	C₂
socle	C₂²×C₅²	C₂⁴
prime radical	C₂³×C₅²	C₂³
hypercenter	C₂×C₄²×C₅	D₅
hypocenter	C₅	C₅×#533
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅

Complements

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

N	H ≅ G/N	type
4. C₅×#4016	C₂	semidirect
5. C₅×#4015	C₂	semidirect
6. C₄×C₅×Dic₁₀	C₂	semidirect
19. #65804	C₅	direct (N × H)
22. C₂×C₄×C₅²	Q₈	semidirect
39. #4016	C₂×C₅	semidirect
40. #4015	C₂×C₅	semidirect
41. C₄×Dic₁₀	C₂×C₅	semidirect
94. C₂×C₄×C₅	Dic₁₀	semidirect
95. C₂×C₄×C₅	C₅×Q₈	semidirect
112. C₅²	#533	semidirect
160. C₂×C₄	C₅×Dic₁₀	semidirect
166. C₅	#65804	direct (N × H)
167. C₅	C₅×#533	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄²×C₅²	2
C₅×#4072	2
C₅×#4070	2
C₅×#4016	2
C₅×#4015	2
C₄×C₅×Dic₁₀	2
C₅×#533	5
#65804	5

Subgroups — C5×#65804 #13106624

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#65804 #13106624