Subgroups — C₅×#71511 #14315629

Order 1920.

292 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#508	C₂	1	1	yes	maximal, characteristic
2	C₅×#5196	C₂	1	2	yes	maximal, complemented
3	C₅×#4567	C₂	1	2	yes	maximal, complemented
4	C₅×#4506	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×Dic₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2178	—	3	1	no	maximal, Hall
7	C₃×C₅×#135	C₂²	1	2	yes	—
8	C₅×#948	—	2	2	no	—
9	C₃×C₅×Q₃₂	C₄	1	1	yes	characteristic, complemented
10	C₃×C₅×SD₃₂	C₄	1	2	yes	complemented
11	C₃×C₅×D₁₆	C₄	1	1	yes	characteristic, complemented
12	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
13	C₅×#837	—	2	2	no	—
14	C₅×#819	C₂²	1	2	yes	—
15	C₅×C₈×Dic₃	C₂²	1	1	yes	characteristic
16	C₂×C₅×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
17	#71511	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₅×#508	D₃	1	1	yes	characteristic
19	C₅×#443	—	3	2	no	—
20	C₅×#368	—	3	1	no	—
21	C₅×#359	—	3	2	no	—
22	C₄×C₅×C₁₆	—	3	1	no	—
23	C₃×C₅×#41	—	2	2	no	—
24	C₃×C₅×Q₁₆	C₂×C₄	1	2	yes	—
25	C₃×C₅×SD₁₆	—	2	2	no	—
26	C₃×C₅×D₈	C₂×C₄	1	2	yes	—
27	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
28	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
29	C₅×#208	—	2	2	no	—
30	C₅×#208	—	4	2	no	—
31	C₂×C₅×C₃⋊C₈	—	4	2	no	—
32	C₂×C₅×C₃⋊C₈	D₄	1	1	yes	characteristic
33	C₃×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
34	C₃×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
35	C₅×C₃⋊C₁₆	—	2	1	no	—
36	C₃×#508	C₂×C₅	1	1	yes	characteristic
37	#5196	C₂×C₅	1	2	yes	complemented
38	#4567	C₂×C₅	1	2	yes	complemented
39	#4506	C₂×C₅	1	1	yes	characteristic, complemented
40	Dic₃×C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
41	C₅×#135	C₂×D₃	1	2	yes	—
42	C₅×#131	—	6	2	no	—
43	C₅×Q₃₂	Dic₃	1	1	yes	characteristic, complemented
44	C₅×SD₃₂	Dic₃	1	2	yes	complemented
45	C₅×D₁₆	Dic₃	1	1	yes	characteristic, complemented
46	C₂×C₅×C₁₆	—	3	1	no	—
47	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
48	C₅×#108	—	3	2	no	—
49	C₅×#104	—	6	2	no	—
50	C₄×C₅×C₈	—	3	1	no	—
51	#2178	—	3	1	no	Sylow
52	C₃×C₅×Q₈	—	2	2	no	—
53	C₃×C₅×D₄	—	2	2	no	—
54	C₃×C₅×D₄	—	4	2	no	—
55	C₂×C₃×C₄×C₅	—	4	2	no	—
56	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
57	C₂×C₅×Dic₃	—	2	1	no	—
58	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
59	C₃×C₅×C₈	#41	1	1	yes	characteristic
60	C₅×C₃⋊C₈	—	4	2	no	—
61	C₅×C₃⋊C₈	D₈	1	1	yes	characteristic
62	C₅×C₃⋊C₈	D₈	1	1	yes	characteristic
63	C₃×#135	C₂²×C₅	1	2	yes	—
64	#948	—	2	2	no	—
65	C₃×Q₃₂	C₄×C₅	1	1	yes	characteristic, complemented
66	C₃×SD₃₂	C₄×C₅	1	2	yes	complemented
67	C₃×D₁₆	C₄×C₅	1	1	yes	characteristic, complemented
68	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
69	#837	—	2	2	no	—
70	#819	C₂²×C₅	1	2	yes	—
71	C₈×Dic₃	C₂²×C₅	1	1	yes	characteristic
72	C₂×C₃⋊C₁₆	C₂²×C₅	1	1	yes	characteristic
73	C₅×#41	—	2	2	no	—
74	C₅×Q₁₆	C₂×Dic₃	1	2	yes	—
75	C₅×SD₁₆	—	2	2	no	—
76	C₅×D₈	C₂×Dic₃	1	2	yes	—
77	C₅×M₁₆	—	6	2	no	—
78	C₅×M₁₆	—	12	2	no	—
79	C₂×C₅×C₈	—	3	1	no	—
80	C₂×C₅×C₈	—	12	2	no	—
81	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
82	C₄²×C₅	—	3	1	no	—
83	C₅×C₁₆	—	6	1	no	—
84	C₅×C₁₆	C₂×Dic₃	1	1	yes	characteristic
85	C₅×C₁₆	C₂×Dic₃	1	1	yes	characteristic
86	#508	C₅×D₃	1	1	yes	characteristic
87	#443	—	3	2	no	—
88	#368	—	3	1	no	—
89	#359	—	3	2	no	—
90	C₄×C₁₆	—	3	1	no	—
91	C₂²×C₃×C₅	—	4	2	no	—
92	C₂²×C₃×C₅	#135	1	1	yes	characteristic
93	C₃×C₄×C₅	—	4	2	no	—
94	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
95	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
96	C₅×Dic₃	—	2	2	no	—
97	C₃×#41	—	2	2	no	—
98	C₃×Q₁₆	C₂×C₄×C₅	1	2	yes	—
99	C₃×SD₁₆	—	2	2	no	—
100	C₃×D₈	C₂×C₄×C₅	1	2	yes	—
101	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
102	C₄×Dic₃	C₅×D₄	1	1	yes	characteristic
103	#208	—	2	2	no	—
104	#208	—	4	2	no	—
105	C₂×C₃⋊C₈	—	4	2	no	—
106	C₂×C₃⋊C₈	C₅×D₄	1	1	yes	characteristic
107	C₃×C₁₆	C₂×C₄×C₅	1	1	yes	characteristic
108	C₃×C₁₆	C₂×C₄×C₅	1	1	yes	characteristic
109	C₃⋊C₁₆	—	2	1	no	—
110	C₅×Q₈	—	2	2	no	—
111	C₅×D₄	—	2	2	no	—
112	C₅×D₄	—	4	2	no	—
113	C₂×C₄×C₅	—	4	2	no	—
114	C₂×C₄×C₅	—	6	1	no	—
115	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
116	C₅×C₈	—	3	1	no	—
117	C₅×C₈	—	3	1	no	—
118	C₅×C₈	—	12	2	no	—
119	C₅×C₈	#237	1	1	yes	characteristic
120	C₅×C₈	C₂²×Dic₃	1	1	yes	characteristic
121	#135	C₂×C₅×D₃	1	2	yes	—
122	#131	—	6	2	no	—
123	Q₃₂	C₅×Dic₃	1	1	yes	characteristic, complemented
124	SD₃₂	C₅×Dic₃	1	2	yes	complemented
125	D₁₆	C₅×Dic₃	1	1	yes	characteristic, complemented
126	C₂×C₁₆	—	3	1	no	—
127	C₂×C₁₆	C₂×C₅×D₃	1	1	yes	characteristic
128	#108	—	3	2	no	—
129	#104	—	6	2	no	—
130	C₄×C₈	—	3	1	no	—
131	C₂×C₃×C₅	—	2	1	no	—
132	C₂×C₃×C₅	—	8	2	no	—
133	C₂×C₃×C₅	C₄×D₈	1	1	yes	characteristic
134	C₃×Q₈	—	2	2	no	—
135	C₃×D₄	—	2	2	no	—
136	C₃×D₄	—	4	2	no	—
137	C₂×C₃×C₄	—	4	2	no	—
138	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
139	C₂×Dic₃	—	2	1	no	—
140	C₃×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
141	C₃×C₈	C₅×#41	1	1	yes	characteristic
142	C₃⋊C₈	—	4	2	no	—
143	C₃⋊C₈	C₅×D₈	1	1	yes	characteristic
144	C₃⋊C₈	C₅×D₈	1	1	yes	characteristic
145	C₂²×C₅	—	4	2	no	—
146	C₂²×C₅	#912	1	1	yes	characteristic
147	C₄×C₅	—	4	2	no	—
148	C₄×C₅	—	6	2	no	—
149	C₄×C₅	D₃×D₈	1	1	yes	characteristic
150	C₄×C₅	D₄×Dic₃	1	1	yes	characteristic
151	#41	—	2	2	no	—
152	Q₁₆	C₂×C₅×Dic₃	1	2	yes	—
153	SD₁₆	—	2	2	no	—
154	D₈	C₂×C₅×Dic₃	1	2	yes	—
155	M₁₆	—	6	2	no	—
156	M₁₆	—	12	2	no	—
157	C₂×C₈	—	3	1	no	—
158	C₂×C₈	—	12	2	no	—
159	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
160	C₄²	—	3	1	no	—
161	C₁₆	—	6	1	no	—
162	C₁₆	C₂×C₅×Dic₃	1	1	yes	characteristic
163	C₁₆	C₂×C₅×Dic₃	1	1	yes	characteristic
164	C₃×C₅	#2178	1	1	yes	characteristic, complemented, Hall
165	C₂²×C₃	—	4	2	no	—
166	C₂²×C₃	C₅×#135	1	1	yes	characteristic
167	C₃×C₄	—	4	2	no	—
168	C₃×C₄	C₄×C₅×D₄	1	1	yes	characteristic
169	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
170	Dic₃	—	2	2	no	—
171	C₂×C₅	—	2	1	no	—
172	C₂×C₅	—	8	2	no	—
173	C₂×C₅	Dic₃×D₈	1	1	yes	characteristic
174	Q₈	—	2	2	no	—
175	D₄	—	2	2	no	—
176	D₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	6	1	no	—
179	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
180	C₈	—	3	1	no	—
181	C₈	—	3	1	no	—
182	C₈	—	12	2	no	—
183	C₈	C₅×#237	1	1	yes	characteristic
184	C₈	C₂²×C₅×Dic₃	1	1	yes	characteristic
185	C₂×C₃	—	2	1	no	—
186	C₂×C₃	—	8	2	no	—
187	C₂×C₃	C₄×C₅×D₈	1	1	yes	characteristic
188	C₅	#71511	1	1	yes	characteristic, direct product factor, Sylow
189	C₂²	—	4	2	no	—
190	C₂²	C₅×#912	1	1	yes	characteristic
191	C₄	—	4	2	no	—
192	C₄	—	6	2	no	—
193	C₄	C₅×D₃×D₈	1	1	yes	characteristic
194	C₄	C₅×D₄×Dic₃	1	1	yes	characteristic
195	C₃	C₅×#2178	1	1	yes	characteristic, complemented, Sylow
196	C₂	—	2	1	no	—
197	C₂	—	8	2	no	—
198	C₂	C₅×Dic₃×D₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	D₃×D₈
commutator	C₃×C₈	C₂²×C₄×C₅
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting, prime radical	C₃×C₅×#508	C₂
socle	C₂×C₃×C₅	C₄×D₈
cosocle	C₂×C₃×C₈	C₂³×C₅
hypercenter	C₅×#508	D₃
hypocenter	C₃	C₅×#2178
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₅×#508

Complements

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

N	H ≅ G/N	type
2. C₅×#5196	C₂	semidirect
3. C₅×#4567	C₂	semidirect
4. C₅×#4506	C₂	semidirect
5. C₅×Dic₃×C₁₆	C₂	semidirect
9. C₃×C₅×Q₃₂	C₄	semidirect
10. C₃×C₅×SD₃₂	C₄	semidirect
11. C₃×C₅×D₁₆	C₄	semidirect
17. #71511	C₅	direct (N × H)
37. #5196	C₂×C₅	semidirect
38. #4567	C₂×C₅	semidirect
39. #4506	C₂×C₅	semidirect
40. Dic₃×C₁₆	C₂×C₅	semidirect
43. C₅×Q₃₂	Dic₃	semidirect
44. C₅×SD₃₂	Dic₃	semidirect
45. C₅×D₁₆	Dic₃	semidirect
65. C₃×Q₃₂	C₄×C₅	semidirect
66. C₃×SD₃₂	C₄×C₅	semidirect
67. C₃×D₁₆	C₄×C₅	semidirect
123. Q₃₂	C₅×Dic₃	semidirect
124. SD₃₂	C₅×Dic₃	semidirect
125. D₁₆	C₅×Dic₃	semidirect
164. C₃×C₅	#2178	semidirect
188. C₅	#71511	direct (N × H)
195. C₃	C₅×#2178	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#508	2
C₅×#5196	2
C₅×#4567	2
C₅×#4506	2
C₅×Dic₃×C₁₆	2
C₅×#2178	3
#71511	5

Subgroups — C5×#71511 #14315629

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71511 #14315629