Subgroups — C₅×#69056 #14315192

Order 1920.

292 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#4901	C₂	1	2	yes	maximal, complemented
2	C₅×#4700	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4508	C₂	1	2	yes	maximal, complemented
5	C₅×#4463	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2170	—	3	1	no	maximal, Hall
7	C₅×#904	C₄	1	1	yes	characteristic
8	C₅×#901	C₂²	1	2	yes	—
9	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
11	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₅×#819	C₄	1	1	yes	characteristic
13	C₅×#812	C₂²	1	2	yes	—
14	C₅×#805	C₄	1	2	yes	—
15	C₅×#798	—	2	2	no	—
16	C₅×D₃×C₁₆	—	2	2	no	—
17	#69056	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₅×#504	—	3	2	no	—
19	C₅×#443	—	3	1	no	—
20	C₅×#364	—	3	1	no	—
21	C₅×#350	—	3	2	no	—
22	C₄×C₅×C₁₆	D₃	1	1	yes	characteristic, complemented
23	C₅×#235	C₂×C₄	1	2	yes	—
24	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
26	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
27	C₅×#208	C₂×C₄	1	2	yes	—
28	C₅×Dic₁₂	C₈	1	1	yes	characteristic
29	C₅×D₂₄	C₈	1	1	yes	characteristic
30	C₅×#204	C₈	1	2	yes	—
31	C₅×ΓL(1,𝔽₂₅)	—	2	2	no	—
32	C₅×D₃×C₈	—	2	2	no	—
33	C₃×C₅×C₁₆	—	2	1	no	—
34	C₃×C₅×C₁₆	D₄	1	1	yes	characteristic
35	C₃×C₅×C₁₆	D₄	1	1	yes	characteristic
36	C₅×C₃⋊C₁₆	—	2	2	no	—
37	#4901	C₂×C₅	1	2	yes	complemented
38	#4700	C₂×C₅	1	1	yes	characteristic
39	C₃×C₄×C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
40	#4508	C₂×C₅	1	2	yes	complemented
41	#4463	C₂×C₅	1	1	yes	characteristic, complemented
42	C₅×#135	—	3	1	no	—
43	C₅×#131	—	3	2	no	—
44	C₅×M₃₂	—	3	2	no	—
45	C₅×M₃₂	—	6	2	no	—
46	C₂×C₅×C₁₆	—	6	2	no	—
47	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
48	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
49	C₅×#108	—	3	1	no	—
50	C₅×#104	—	3	2	no	—
51	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
52	#2170	—	3	1	no	Sylow
53	C₂×C₃×C₄×C₅	—	2	1	no	—
54	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
55	C₅×#67	—	2	2	no	—
56	C₅×D₁₂	C₂×C₈	1	2	yes	—
57	C₄×C₅×D₃	—	2	2	no	—
58	C₅×Dic₆	C₂×C₈	1	2	yes	—
59	C₃×C₅×C₈	C₂×C₈	1	1	yes	characteristic
60	C₃×C₅×C₈	C₂×C₈	1	1	yes	characteristic
61	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
62	C₃×C₅×C₈	#41	1	1	yes	characteristic
63	C₅×C₃⋊C₈	—	2	2	no	—
64	#904	C₄×C₅	1	1	yes	characteristic
65	#901	C₂²×C₅	1	2	yes	—
66	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
67	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
68	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
69	#819	C₄×C₅	1	1	yes	characteristic
70	#812	C₂²×C₅	1	2	yes	—
71	#805	C₄×C₅	1	2	yes	—
72	#798	—	2	2	no	—
73	D₃×C₁₆	—	2	2	no	—
74	C₅×#41	—	3	2	no	—
75	C₅×Q₁₆	—	3	1	no	—
76	C₅×SD₁₆	—	3	2	no	—
77	C₅×D₈	—	3	1	no	—
78	C₅×M₁₆	—	3	2	no	—
79	C₅×M₁₆	—	6	2	no	—
80	C₂×C₅×C₈	—	6	2	no	—
81	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
82	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
83	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
84	C₅×C₁₆	—	2	1	no	—
85	C₅×C₁₆	—	6	2	no	—
86	C₅×C₁₆	D₁₂	1	1	yes	characteristic
87	C₅×C₁₆	D₁₂	1	1	yes	characteristic
88	#504	—	3	2	no	—
89	#443	—	3	1	no	—
90	#364	—	3	1	no	—
91	#350	—	3	2	no	—
92	C₄×C₁₆	C₅×D₃	1	1	yes	characteristic, complemented
93	C₂²×C₃×C₅	#131	1	1	yes	characteristic
94	C₂×C₅×D₃	—	2	2	no	—
95	C₃×C₄×C₅	—	2	2	no	—
96	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
97	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
98	C₅×Dic₃	—	2	2	no	—
99	#235	C₂×C₄×C₅	1	2	yes	—
100	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
101	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
102	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
103	#208	C₂×C₄×C₅	1	2	yes	—
104	Dic₁₂	C₅×C₈	1	1	yes	characteristic
105	D₂₄	C₅×C₈	1	1	yes	characteristic
106	#204	C₅×C₈	1	2	yes	—
107	ΓL(1,𝔽₂₅)	—	2	2	no	—
108	D₃×C₈	—	2	2	no	—
109	C₃×C₁₆	—	2	1	no	—
110	C₃×C₁₆	C₅×D₄	1	1	yes	characteristic
111	C₃×C₁₆	C₅×D₄	1	1	yes	characteristic
112	C₃⋊C₁₆	—	2	2	no	—
113	C₅×Q₈	—	3	2	no	—
114	C₅×D₄	—	3	2	no	—
115	C₅×D₄	—	6	2	no	—
116	C₂×C₄×C₅	—	2	1	no	—
117	C₂×C₄×C₅	—	6	2	no	—
118	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
119	C₅×C₈	—	6	2	no	—
120	C₅×C₈	D₃×C₈	1	1	yes	characteristic
121	C₅×C₈	D₃×C₈	1	1	yes	characteristic
122	C₅×C₈	C₂×D₁₂	1	1	yes	characteristic
123	C₅×C₈	#235	1	1	yes	characteristic
124	#135	—	3	1	no	—
125	#131	—	3	2	no	—
126	M₃₂	—	3	2	no	—
127	M₃₂	—	6	2	no	—
128	C₂×C₁₆	—	6	2	no	—
129	C₂×C₁₆	C₂×C₅×D₃	1	1	yes	characteristic
130	C₂×C₁₆	C₂×C₅×D₃	1	1	yes	characteristic
131	#108	—	3	1	no	—
132	#104	—	3	2	no	—
133	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
134	C₂×C₃×C₅	—	2	1	no	—
135	C₂×C₃×C₅	C₈×D₄	1	1	yes	characteristic
136	C₅×D₃	—	4	2	no	—
137	C₂×C₃×C₄	—	2	1	no	—
138	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
139	#67	—	2	2	no	—
140	D₁₂	C₂×C₅×C₈	1	2	yes	—
141	C₄×D₃	—	2	2	no	—
142	Dic₆	C₂×C₅×C₈	1	2	yes	—
143	C₃×C₈	C₂×C₅×C₈	1	1	yes	characteristic
144	C₃×C₈	C₂×C₅×C₈	1	1	yes	characteristic
145	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
146	C₃×C₈	C₅×#41	1	1	yes	characteristic
147	C₃⋊C₈	—	2	2	no	—
148	C₂²×C₅	—	6	2	no	—
149	C₂²×C₅	#901	1	1	yes	characteristic
150	C₄×C₅	—	2	2	no	—
151	C₄×C₅	—	6	2	no	—
152	C₄×C₅	C₄×D₁₂	1	1	yes	characteristic
153	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
154	#41	—	3	2	no	—
155	Q₁₆	—	3	1	no	—
156	SD₁₆	—	3	2	no	—
157	D₈	—	3	1	no	—
158	M₁₆	—	3	2	no	—
159	M₁₆	—	6	2	no	—
160	C₂×C₈	—	6	2	no	—
161	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
162	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
163	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
164	C₁₆	—	2	1	no	—
165	C₁₆	—	6	2	no	—
166	C₁₆	C₅×D₁₂	1	1	yes	characteristic
167	C₁₆	C₅×D₁₂	1	1	yes	characteristic
168	C₃×C₅	#2170	1	1	yes	characteristic, complemented, Hall
169	C₂²×C₃	C₅×#131	1	1	yes	characteristic
170	C₂×D₃	—	2	2	no	—
171	C₃×C₄	—	2	2	no	—
172	C₃×C₄	C₄×C₅×D₄	1	1	yes	characteristic
173	C₃×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
174	Dic₃	—	2	2	no	—
175	C₂×C₅	—	2	1	no	—
176	C₂×C₅	—	12	2	no	—
177	C₂×C₅	C₈×D₁₂	1	1	yes	characteristic
178	Q₈	—	3	2	no	—
179	D₄	—	3	2	no	—
180	D₄	—	6	2	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	6	2	no	—
183	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
184	C₈	—	6	2	no	—
185	C₈	C₅×D₃×C₈	1	1	yes	characteristic
186	C₈	C₅×D₃×C₈	1	1	yes	characteristic
187	C₈	C₂×C₅×D₁₂	1	1	yes	characteristic
188	C₈	C₅×#235	1	1	yes	characteristic
189	C₂×C₃	—	2	1	no	—
190	C₂×C₃	C₅×C₈×D₄	1	1	yes	characteristic
191	D₃	—	4	2	no	—
192	C₅	#69056	1	1	yes	characteristic, direct product factor, Sylow
193	C₂²	—	6	2	no	—
194	C₂²	C₅×#901	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	6	2	no	—
197	C₄	C₄×C₅×D₁₂	1	1	yes	characteristic
198	C₄	C₂×C₅×D₃×C₈	1	1	yes	characteristic
199	C₃	C₅×#2170	1	1	yes	characteristic, complemented, Sylow
200	C₂	—	2	1	no	—
201	C₂	—	12	2	no	—
202	C₂	C₅×C₈×D₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₅×C₁₆	D₁₂
commutator	C₃×C₄	C₂²×C₅×C₈
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₃×C₄×C₅×C₁₆	C₂
socle	C₂×C₃×C₅	C₈×D₄
prime radical	C₅×#904	C₄
cosocle	C₂×C₃×C₈	C₂³×C₅
hypercenter	C₄×C₅×C₁₆	D₃
hypocenter	C₃	C₅×#2170
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₅×C₁₆ ◁ C₂×C₅×C₁₆ ◁ C₄×C₅×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₅×#4901	C₂	semidirect
3. C₃×C₄×C₅×C₁₆	C₂	semidirect
4. C₅×#4508	C₂	semidirect
5. C₅×#4463	C₂	semidirect
17. #69056	C₅	direct (N × H)
22. C₄×C₅×C₁₆	D₃	semidirect
37. #4901	C₂×C₅	semidirect
39. C₃×C₄×C₁₆	C₂×C₅	semidirect
40. #4508	C₂×C₅	semidirect
41. #4463	C₂×C₅	semidirect
92. C₄×C₁₆	C₅×D₃	semidirect
168. C₃×C₅	#2170	semidirect
192. C₅	#69056	direct (N × H)
199. C₃	C₅×#2170	semidirect

Maximal subgroups

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

subgroup	index
C₅×#4901	2
C₅×#4700	2
C₃×C₄×C₅×C₁₆	2
C₅×#4508	2
C₅×#4463	2
C₅×#2170	3
#69056	5

Subgroups — C5×#69056 #14315192

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69056 #14315192