Subgroups — C₅×#70516 #14298890

Order 1920.

376 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#423	C₂	1	1	yes	maximal, characteristic
2	C₅×#4924	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₅×#4558	C₂	1	4	yes	maximal
5	C₅×#1837	—	3	1	no	maximal, Hall
6	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₅×#208	C₂²	1	2	yes	—
9	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
10	C₃×C₅×C₄⋊C₈	—	2	2	no	—
11	C₅×C₄⋊(C₃⋊C₈)	—	2	2	no	—
12	C₄×C₅×C₃⋊C₈	C₄	1	2	yes	—
13	C₄×C₅×C₃⋊C₈	C₄	1	2	yes	—
14	#70516	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₅×#423	—	3	1	no	—
16	C₅×#423	D₃	1	1	yes	characteristic
17	C₂×C₄×C₅×C₈	—	3	1	no	—
18	C₅×#341	—	3	4	no	—
19	C₂²×C₃×C₄×C₅	Q₈	1	2	yes	—
20	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
21	C₃×C₅×M₁₆	—	2	4	no	—
22	C₂×C₃×C₅×C₈	—	2	2	no	—
23	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
24	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
25	C₃×C₄²×C₅	D₄	1	2	yes	—
26	C₅×#208	—	2	4	no	—
27	C₂×C₅×C₃⋊C₈	—	2	2	no	—
28	C₂×C₅×C₃⋊C₈	—	2	2	no	—
29	C₂×C₅×C₃⋊C₈	—	2	2	no	—
30	C₂×C₅×C₃⋊C₈	C₂×C₄	1	4	yes	—
31	C₃×#423	C₂×C₅	1	1	yes	characteristic
32	#4924	C₂×C₅	1	1	yes	characteristic
33	C₂×C₄×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic
34	#4558	C₂×C₅	1	4	yes	—
35	C₂×C₅×M₁₆	—	3	2	no	—
36	C₂×C₅×M₁₆	C₂×D₃	1	2	yes	—
37	C₂²×C₅×C₈	—	3	2	no	—
38	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
39	C₅×C₄⋊C₈	—	2	2	no	—
40	C₅×C₄⋊C₈	—	6	2	no	—
41	C₄×C₅×C₈	—	3	2	no	—
42	C₄×C₅×C₈	—	3	2	no	—
43	#1837	—	3	1	no	Sylow
44	C₂³×C₃×C₅	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₃×C₄×C₅	—	2	4	no	—
46	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
47	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
48	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	#41	1	2	yes	—
51	C₂×C₃×C₄×C₅	#41	1	2	yes	—
52	C₃×C₅×C₈	—	4	2	no	—
53	C₅×C₃⋊C₈	—	2	4	no	—
54	C₅×C₃⋊C₈	—	4	2	no	—
55	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
56	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
57	C₂×#208	C₂²×C₅	1	2	yes	—
58	C₂²×C₃⋊C₈	C₂²×C₅	1	2	yes	—
59	C₃×C₄⋊C₈	—	2	2	no	—
60	C₄⋊(C₃⋊C₈)	—	2	2	no	—
61	C₄×C₃⋊C₈	C₄×C₅	1	2	yes	—
62	C₄×C₃⋊C₈	C₄×C₅	1	2	yes	—
63	C₂²×C₄×C₅	Dic₆	1	2	yes	—
64	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
65	C₅×M₁₆	—	2	4	no	—
66	C₅×M₁₆	—	6	4	no	—
67	C₂×C₅×C₈	—	2	2	no	—
68	C₂×C₅×C₈	—	3	4	no	—
69	C₂×C₅×C₈	—	6	2	no	—
70	C₂×C₅×C₈	—	6	2	no	—
71	C₂×C₅×C₈	—	6	2	no	—
72	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
73	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
74	C₄²×C₅	#67	1	2	yes	—
75	#423	—	3	1	no	—
76	#423	C₅×D₃	1	1	yes	characteristic
77	C₂×C₄×C₈	—	3	1	no	—
78	#341	—	3	4	no	—
79	C₂²×C₃×C₅	—	2	2	no	—
80	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₃×C₅	#125	1	2	yes	—
82	C₃×C₄×C₅	—	2	2	no	—
83	C₃×C₄×C₅	#108	1	4	yes	—
84	C₃×C₄×C₅	#117	1	2	yes	—
85	C₃×C₄×C₅	#124	1	1	yes	characteristic
86	C₃×C₄×C₅	#124	1	1	yes	characteristic
87	C₂²×C₃×C₄	C₅×Q₈	1	2	yes	—
88	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
89	C₃×M₁₆	—	2	4	no	—
90	C₂×C₃×C₈	—	2	2	no	—
91	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
92	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
93	C₃×C₄²	C₅×D₄	1	2	yes	—
94	#208	—	2	4	no	—
95	C₂×C₃⋊C₈	—	2	2	no	—
96	C₂×C₃⋊C₈	—	2	2	no	—
97	C₂×C₃⋊C₈	—	2	2	no	—
98	C₂×C₃⋊C₈	C₂×C₄×C₅	1	4	yes	—
99	C₂³×C₅	C₂×Dic₆	1	1	yes	characteristic
100	C₂×C₄×C₅	—	2	4	no	—
101	C₂×C₄×C₅	Dic₃⋊C₄	1	2	yes	—
102	C₂×C₄×C₅	Dic₃⋊C₄	1	2	yes	—
103	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
104	C₂×C₄×C₅	#237	1	2	yes	—
105	C₂×C₄×C₅	#239	1	2	yes	—
106	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
107	C₅×C₈	—	4	2	no	—
108	C₅×C₈	—	6	4	no	—
109	C₅×C₈	—	12	2	no	—
110	C₂×M₁₆	—	3	2	no	—
111	C₂×M₁₆	C₂×C₅×D₃	1	2	yes	—
112	C₂²×C₈	—	3	2	no	—
113	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
114	C₄⋊C₈	—	2	2	no	—
115	C₄⋊C₈	—	6	2	no	—
116	C₄×C₈	—	3	2	no	—
117	C₄×C₈	—	3	2	no	—
118	C₂×C₃×C₅	—	2	2	no	—
119	C₂×C₃×C₅	#382	1	1	yes	characteristic
120	C₂×C₃×C₅	C₂×#108	1	2	yes	—
121	C₂³×C₃	C₂×C₅×Q₈	1	1	yes	characteristic
122	C₂×C₃×C₄	—	2	4	no	—
123	C₂×C₃×C₄	C₅×C₄⋊C₄	1	2	yes	—
124	C₂×C₃×C₄	C₅×C₄⋊C₄	1	2	yes	—
125	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
126	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
127	C₂×C₃×C₄	C₅×#41	1	2	yes	—
128	C₂×C₃×C₄	C₅×#41	1	2	yes	—
129	C₃×C₈	—	4	2	no	—
130	C₃⋊C₈	—	2	4	no	—
131	C₃⋊C₈	—	4	2	no	—
132	C₂²×C₅	—	2	2	no	—
133	C₂²×C₅	#890	1	2	yes	—
134	C₂²×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
135	C₄×C₅	—	2	2	no	—
136	C₄×C₅	#822	1	4	yes	—
137	C₄×C₅	#892	1	2	yes	—
138	C₄×C₅	#936	1	1	yes	characteristic
139	C₄×C₅	#947	1	1	yes	characteristic
140	C₂²×C₄	C₅×Dic₆	1	2	yes	—
141	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
142	M₁₆	—	2	4	no	—
143	M₁₆	—	6	4	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	3	4	no	—
146	C₂×C₈	—	6	2	no	—
147	C₂×C₈	—	6	2	no	—
148	C₂×C₈	—	6	2	no	—
149	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
150	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
151	C₄²	C₅×#67	1	2	yes	—
152	C₃×C₅	#1837	1	1	yes	characteristic, complemented, Hall
153	C₂²×C₃	—	2	2	no	—
154	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
155	C₂²×C₃	C₅×#125	1	2	yes	—
156	C₃×C₄	—	2	2	no	—
157	C₃×C₄	C₅×#108	1	4	yes	—
158	C₃×C₄	C₅×#117	1	2	yes	—
159	C₃×C₄	C₅×#124	1	1	yes	characteristic
160	C₃×C₄	C₅×#124	1	1	yes	characteristic
161	C₂×C₅	—	2	2	no	—
162	C₂×C₅	#4977	1	1	yes	characteristic
163	C₂×C₅	C₂×#822	1	2	yes	—
164	C₂³	C₂×C₅×Dic₆	1	1	yes	characteristic
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	C₅×Dic₃⋊C₄	1	2	yes	—
167	C₂×C₄	C₅×Dic₃⋊C₄	1	2	yes	—
168	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
169	C₂×C₄	C₅×#237	1	2	yes	—
170	C₂×C₄	C₅×#239	1	2	yes	—
171	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
172	C₈	—	4	2	no	—
173	C₈	—	6	4	no	—
174	C₈	—	12	2	no	—
175	C₂×C₃	—	2	2	no	—
176	C₂×C₃	C₅×#382	1	1	yes	characteristic
177	C₂×C₃	C₂×C₅×#108	1	2	yes	—
178	C₅	#70516	1	1	yes	characteristic, direct product factor, Sylow
179	C₂²	—	2	2	no	—
180	C₂²	C₅×#890	1	2	yes	—
181	C₂²	C₂×C₅×Dic₃⋊C₄	1	1	yes	characteristic
182	C₄	—	2	2	no	—
183	C₄	C₅×#822	1	4	yes	—
184	C₄	C₅×#892	1	2	yes	—
185	C₄	C₅×#936	1	1	yes	characteristic
186	C₄	C₅×#947	1	1	yes	characteristic
187	C₃	C₅×#1837	1	1	yes	characteristic, complemented, Sylow
188	C₂	—	2	2	no	—
189	C₂	C₅×#4977	1	1	yes	characteristic
190	C₂	C₂×C₅×#822	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#423	C₂
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂³×C₃×C₅	C₂×Q₈
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#423	D₃
hypocenter	C₃	C₅×#1837
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₅×#423

Complements

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

N	H ≅ G/N	type
14. #70516	C₅	direct (N × H)
152. C₃×C₅	#1837	semidirect
178. C₅	#70516	direct (N × H)
187. C₃	C₅×#1837	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#423	2
C₅×#4924	2
C₂×C₄×C₅×C₃⋊C₈	2
C₅×#4558	2
C₅×#1837	3
#70516	5

Subgroups — C5×#70516 #14298890

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70516 #14298890