Subgroups — C₅×#68056 #14287415

Order 1920.

304 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×Dic₁₂	C₂	1	1	yes	maximal, characteristic
2	C₅×#4679	C₂	1	2	yes	maximal
3	C₃×C₅×C₈²	C₂	1	1	yes	maximal, characteristic
4	C₅×#4458	C₂	1	1	yes	maximal, characteristic
5	C₅×#4456	C₂	1	2	yes	maximal
6	C₅×#1591	—	3	1	no	maximal, Hall
7	C₂×C₅×Dic₁₂	C₄	1	1	yes	characteristic
8	C₄×C₅×Dic₆	C₂²	1	2	yes	—
9	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₅×#818	C₄	1	1	yes	characteristic
13	C₅×#816	C₄	1	2	yes	—
14	C₅×#815	—	2	2	no	—
15	C₅×Dic₃⋊C₈	—	2	2	no	—
16	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
17	#68056	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₅×#446	—	3	2	no	—
19	C₄×C₅×Q₁₆	—	3	1	no	—
20	C₅×#335	—	3	1	no	—
21	C₅×Q₈⋊C₈	—	3	2	no	—
22	C₅×C₈²	D₃	1	1	yes	characteristic
23	C₂×C₅×Dic₆	C₂×C₄	1	2	yes	—
24	C₂×C₃×C₅×C₈	—	2	1	no	—
25	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
28	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
29	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
30	C₅×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
31	C₅×Dic₃⋊C₄	—	2	2	no	—
32	C₄×C₅×Dic₃	—	2	2	no	—
33	C₂×C₅×C₃⋊C₈	—	2	2	no	—
34	C₅×Dic₁₂	—	2	2	no	—
35	C₄×Dic₁₂	C₂×C₅	1	1	yes	characteristic
36	#4679	C₂×C₅	1	2	yes	—
37	C₃×C₈²	C₂×C₅	1	1	yes	characteristic
38	#4458	C₂×C₅	1	1	yes	characteristic
39	#4456	C₂×C₅	1	2	yes	—
40	C₂×C₅×Q₁₆	—	3	1	no	—
41	C₄×C₅×Q₈	—	3	2	no	—
42	C₅×#107	—	3	1	no	—
43	C₅×C₄⋊C₈	—	3	2	no	—
44	C₅×C₄⋊C₈	—	6	2	no	—
45	C₅×#103	—	3	2	no	—
46	C₅×#97	—	6	2	no	—
47	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
48	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
49	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
50	#1591	—	3	1	no	Sylow
51	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
54	C₂×C₅×Dic₃	—	2	2	no	—
55	C₂×C₅×Dic₃	—	2	2	no	—
56	C₅×Dic₆	—	2	2	no	—
57	C₅×Dic₆	—	2	2	no	—
58	C₃×C₅×C₈	—	2	2	no	—
59	C₃×C₅×C₈	—	2	1	no	—
60	C₃×C₅×C₈	—	2	1	no	—
61	C₃×C₅×C₈	M₁₆	1	2	yes	—
62	C₃×C₅×C₈	Q₁₆	1	2	yes	complemented
63	C₅×C₃⋊C₈	—	4	2	no	—
64	C₂×Dic₁₂	C₄×C₅	1	1	yes	characteristic
65	C₄×Dic₆	C₂²×C₅	1	2	yes	—
66	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
67	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
68	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
69	#818	C₄×C₅	1	1	yes	characteristic
70	#816	C₄×C₅	1	2	yes	—
71	#815	—	2	2	no	—
72	Dic₃⋊C₈	—	2	2	no	—
73	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	2	yes	—
74	C₂×C₅×Q₈	—	3	2	no	—
75	C₅×Q₁₆	—	6	2	no	—
76	C₂×C₅×C₈	—	2	1	no	—
77	C₂×C₅×C₈	—	6	2	no	—
78	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
79	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
80	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
81	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
82	C₅×C₄⋊C₄	—	3	2	no	—
83	C₅×C₄⋊C₄	—	6	2	no	—
84	C₄²×C₅	—	6	2	no	—
85	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
86	#446	—	3	2	no	—
87	C₄×Q₁₆	—	3	1	no	—
88	#335	—	3	1	no	—
89	Q₈⋊C₈	—	3	2	no	—
90	C₈²	C₅×D₃	1	1	yes	characteristic
91	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
92	C₃×C₄×C₅	—	2	1	no	—
93	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
94	C₃×C₄×C₅	#131	1	1	yes	characteristic
95	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
96	C₃×C₄×C₅	#135	1	1	yes	characteristic
97	C₅×Dic₃	—	4	2	no	—
98	C₅×Dic₃	—	4	2	no	—
99	C₂×Dic₆	C₂×C₄×C₅	1	2	yes	—
100	C₂×C₃×C₈	—	2	1	no	—
101	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
102	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
103	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
104	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
105	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
106	C₄⋊Dic₃	C₂×C₄×C₅	1	2	yes	—
107	Dic₃⋊C₄	—	2	2	no	—
108	C₄×Dic₃	—	2	2	no	—
109	C₂×C₃⋊C₈	—	2	2	no	—
110	Dic₁₂	—	2	2	no	—
111	C₅×Q₈	—	6	2	no	—
112	C₅×Q₈	—	6	2	no	—
113	C₂×C₄×C₅	—	6	2	no	—
114	C₂×C₄×C₅	—	6	2	no	—
115	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
116	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
117	C₂×C₄×C₅	#235	1	1	yes	characteristic
118	C₅×C₈	—	2	2	no	—
119	C₅×C₈	—	2	1	no	—
120	C₅×C₈	—	2	1	no	—
121	C₅×C₈	—	12	2	no	—
122	C₅×C₈	ΓL(1,𝔽₂₅)	1	2	yes	—
123	C₅×C₈	Dic₁₂	1	2	yes	complemented
124	C₂×Q₁₆	—	3	1	no	—
125	C₄×Q₈	—	3	2	no	—
126	#107	—	3	1	no	—
127	C₄⋊C₈	—	3	2	no	—
128	C₄⋊C₈	—	6	2	no	—
129	#103	—	3	2	no	—
130	#97	—	6	2	no	—
131	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
132	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
133	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
134	C₂×C₃×C₅	#436	1	1	yes	characteristic
135	C₂×C₃×C₅	C₄×Q₁₆	1	1	yes	characteristic
136	C₂×C₃×C₅	#443	1	1	yes	characteristic
137	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
139	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
140	C₂×Dic₃	—	2	2	no	—
141	C₂×Dic₃	—	2	2	no	—
142	Dic₆	—	2	2	no	—
143	Dic₆	—	2	2	no	—
144	C₃×C₈	—	2	2	no	—
145	C₃×C₈	—	2	1	no	—
146	C₃×C₈	—	2	1	no	—
147	C₃×C₈	C₅×M₁₆	1	2	yes	—
148	C₃×C₈	C₅×Q₁₆	1	2	yes	complemented
149	C₃⋊C₈	—	4	2	no	—
150	C₂²×C₅	C₄×D₁₂	1	1	yes	characteristic
151	C₄×C₅	—	2	1	no	—
152	C₄×C₅	—	12	2	no	—
153	C₄×C₅	—	12	2	no	—
154	C₄×C₅	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
155	C₄×C₅	#901	1	1	yes	characteristic
156	C₄×C₅	#904	1	1	yes	characteristic
157	C₄×C₅	C₂×Dic₁₂	1	1	yes	characteristic
158	C₂×Q₈	—	3	2	no	—
159	Q₁₆	—	6	2	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	6	2	no	—
162	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
163	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
164	C₂×C₈	C₅×D₁₂	1	1	yes	characteristic
165	C₂×C₈	C₅×D₁₂	1	1	yes	characteristic
166	C₄⋊C₄	—	3	2	no	—
167	C₄⋊C₄	—	6	2	no	—
168	C₄²	—	6	2	no	—
169	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
170	C₃×C₅	#1591	1	1	yes	characteristic, complemented, Hall
171	C₂²×C₃	C₄×C₅×D₄	1	1	yes	characteristic
172	C₃×C₄	—	2	1	no	—
173	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
174	C₃×C₄	C₅×#131	1	1	yes	characteristic
175	C₃×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
176	C₃×C₄	C₅×#135	1	1	yes	characteristic
177	Dic₃	—	4	2	no	—
178	Dic₃	—	4	2	no	—
179	C₂×C₅	#4688	1	1	yes	characteristic
180	C₂×C₅	C₄×Dic₁₂	1	1	yes	characteristic
181	C₂×C₅	#4700	1	1	yes	characteristic
182	Q₈	—	6	2	no	—
183	Q₈	—	6	2	no	—
184	C₂×C₄	—	6	2	no	—
185	C₂×C₄	—	6	2	no	—
186	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
187	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
188	C₂×C₄	C₅×#235	1	1	yes	characteristic
189	C₈	—	2	2	no	—
190	C₈	—	2	1	no	—
191	C₈	—	2	1	no	—
192	C₈	—	12	2	no	—
193	C₈	C₅×ΓL(1,𝔽₂₅)	1	2	yes	—
194	C₈	C₅×Dic₁₂	1	2	yes	complemented
195	C₂×C₃	C₅×#436	1	1	yes	characteristic
196	C₂×C₃	C₄×C₅×Q₁₆	1	1	yes	characteristic
197	C₂×C₃	C₅×#443	1	1	yes	characteristic
198	C₅	#68056	1	1	yes	characteristic, direct product factor, Sylow
199	C₂²	C₄×C₅×D₁₂	1	1	yes	characteristic
200	C₄	—	2	1	no	—
201	C₄	—	12	2	no	—
202	C₄	—	12	2	no	—
203	C₄	C₂×C₅×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
204	C₄	C₅×#901	1	1	yes	characteristic
205	C₄	C₅×#904	1	1	yes	characteristic
206	C₄	C₂×C₅×Dic₁₂	1	1	yes	characteristic
207	C₃	C₅×#1591	1	1	yes	characteristic, complemented, Sylow
208	C₂	C₅×#4688	1	1	yes	characteristic
209	C₂	C₄×C₅×Dic₁₂	1	1	yes	characteristic
210	C₂	C₅×#4700	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×C₈²	C₂
socle, prime radical	C₂²×C₃×C₅	C₄×D₄
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×C₈²	D₃
hypocenter	C₃	C₅×#1591
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₄×C₅×C₈ ◁ C₅×C₈²

Complements

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

N	H ≅ G/N	type
17. #68056	C₅	direct (N × H)
62. C₃×C₅×C₈	Q₁₆	semidirect
123. C₅×C₈	Dic₁₂	semidirect
148. C₃×C₈	C₅×Q₁₆	semidirect
170. C₃×C₅	#1591	semidirect
194. C₈	C₅×Dic₁₂	semidirect
198. C₅	#68056	direct (N × H)
207. C₃	C₅×#1591	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×Dic₁₂	2
C₅×#4679	2
C₃×C₅×C₈²	2
C₅×#4458	2
C₅×#4456	2
C₅×#1591	3
#68056	5

Subgroups — C5×#68056 #14287415

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68056 #14287415