Subgroups — C₅×#69076 #14315251

Order 1920.

288 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	C₅×#4901	C₂	1	2	yes	maximal, complemented
2	C₅×#4700	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#346	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×#2171	—	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	#69076	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	C₅×#346	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	2	yes	—
35	C₅×C₃⋊C₁₆	—	2	2	no	—
36	#4901	C₂×C₅	1	2	yes	complemented
37	#4700	C₂×C₅	1	1	yes	characteristic
38	C₃×#346	C₂×C₅	1	1	yes	characteristic, complemented
39	#4508	C₂×C₅	1	2	yes	complemented
40	#4463	C₂×C₅	1	1	yes	characteristic, complemented
41	C₅×#135	—	3	1	no	—
42	C₅×#131	—	3	2	no	—
43	C₅×M₃₂	—	3	2	no	—
44	C₅×M₃₂	—	6	2	no	—
45	C₂×C₅×C₁₆	—	6	2	no	—
46	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
47	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
48	C₅×#108	—	3	1	no	—
49	C₅×#104	—	3	2	no	—
50	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
51	#2171	—	3	1	no	Sylow
52	C₂×C₃×C₄×C₅	—	2	1	no	—
53	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
54	C₅×#67	—	2	2	no	—
55	C₅×D₁₂	C₂×C₈	1	2	yes	—
56	C₄×C₅×D₃	—	2	2	no	—
57	C₅×Dic₆	C₂×C₈	1	2	yes	—
58	C₃×C₅×C₈	C₂×C₈	1	1	yes	characteristic
59	C₃×C₅×C₈	C₂×C₈	1	1	yes	characteristic
60	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
61	C₃×C₅×C₈	#41	1	1	yes	characteristic
62	C₅×C₃⋊C₈	—	2	2	no	—
63	#904	C₄×C₅	1	1	yes	characteristic
64	#901	C₂²×C₅	1	2	yes	—
65	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
66	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
67	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
68	#819	C₄×C₅	1	1	yes	characteristic
69	#812	C₂²×C₅	1	2	yes	—
70	#805	C₄×C₅	1	2	yes	—
71	#798	—	2	2	no	—
72	D₃×C₁₆	—	2	2	no	—
73	C₅×#41	—	3	2	no	—
74	C₅×Q₁₆	—	3	1	no	—
75	C₅×SD₁₆	—	3	2	no	—
76	C₅×D₈	—	3	1	no	—
77	C₅×M₁₆	—	3	2	no	—
78	C₅×M₁₆	—	6	2	no	—
79	C₂×C₅×C₈	—	6	2	no	—
80	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
81	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
82	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
83	C₅×C₁₆	—	2	1	no	—
84	C₅×C₁₆	—	6	2	no	—
85	C₅×C₁₆	D₁₂	1	2	yes	—
86	#504	—	3	2	no	—
87	#443	—	3	1	no	—
88	#364	—	3	1	no	—
89	#350	—	3	2	no	—
90	#346	C₅×D₃	1	1	yes	characteristic, complemented
91	C₂²×C₃×C₅	#131	1	1	yes	characteristic
92	C₂×C₅×D₃	—	2	2	no	—
93	C₃×C₄×C₅	—	4	1	no	—
94	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
95	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
96	C₅×Dic₃	—	2	2	no	—
97	#235	C₂×C₄×C₅	1	2	yes	—
98	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
99	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
100	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
101	#208	C₂×C₄×C₅	1	2	yes	—
102	Dic₁₂	C₅×C₈	1	1	yes	characteristic
103	D₂₄	C₅×C₈	1	1	yes	characteristic
104	#204	C₅×C₈	1	2	yes	—
105	ΓL(1,𝔽₂₅)	—	2	2	no	—
106	D₃×C₈	—	2	2	no	—
107	C₃×C₁₆	—	2	1	no	—
108	C₃×C₁₆	C₅×D₄	1	2	yes	—
109	C₃⋊C₁₆	—	2	2	no	—
110	C₅×Q₈	—	3	2	no	—
111	C₅×D₄	—	3	2	no	—
112	C₅×D₄	—	6	2	no	—
113	C₂×C₄×C₅	—	2	1	no	—
114	C₂×C₄×C₅	—	6	2	no	—
115	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
116	C₅×C₈	—	6	2	no	—
117	C₅×C₈	D₃×C₈	1	1	yes	characteristic
118	C₅×C₈	D₃×C₈	1	1	yes	characteristic
119	C₅×C₈	C₂×D₁₂	1	1	yes	characteristic
120	C₅×C₈	#235	1	1	yes	characteristic
121	#135	—	3	1	no	—
122	#131	—	3	2	no	—
123	M₃₂	—	3	2	no	—
124	M₃₂	—	6	2	no	—
125	C₂×C₁₆	—	6	2	no	—
126	C₂×C₁₆	C₂×C₅×D₃	1	1	yes	characteristic
127	C₂×C₁₆	C₂×C₅×D₃	1	1	yes	characteristic
128	#108	—	3	1	no	—
129	#104	—	3	2	no	—
130	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
131	C₂×C₃×C₅	—	2	1	no	—
132	C₂×C₃×C₅	C₈×D₄	1	1	yes	characteristic
133	C₅×D₃	—	4	2	no	—
134	C₂×C₃×C₄	—	2	1	no	—
135	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
136	#67	—	2	2	no	—
137	D₁₂	C₂×C₅×C₈	1	2	yes	—
138	C₄×D₃	—	2	2	no	—
139	Dic₆	C₂×C₅×C₈	1	2	yes	—
140	C₃×C₈	C₂×C₅×C₈	1	1	yes	characteristic
141	C₃×C₈	C₂×C₅×C₈	1	1	yes	characteristic
142	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
143	C₃×C₈	C₅×#41	1	1	yes	characteristic
144	C₃⋊C₈	—	2	2	no	—
145	C₂²×C₅	—	6	2	no	—
146	C₂²×C₅	#901	1	1	yes	characteristic
147	C₄×C₅	—	4	1	no	—
148	C₄×C₅	—	6	2	no	—
149	C₄×C₅	C₄×D₁₂	1	1	yes	characteristic
150	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
151	#41	—	3	2	no	—
152	Q₁₆	—	3	1	no	—
153	SD₁₆	—	3	2	no	—
154	D₈	—	3	1	no	—
155	M₁₆	—	3	2	no	—
156	M₁₆	—	6	2	no	—
157	C₂×C₈	—	6	2	no	—
158	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
159	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
160	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
161	C₁₆	—	2	1	no	—
162	C₁₆	—	6	2	no	—
163	C₁₆	C₅×D₁₂	1	2	yes	—
164	C₃×C₅	#2171	1	1	yes	characteristic, complemented, Hall
165	C₂²×C₃	C₅×#131	1	1	yes	characteristic
166	C₂×D₃	—	2	2	no	—
167	C₃×C₄	—	4	1	no	—
168	C₃×C₄	C₄×C₅×D₄	1	1	yes	characteristic
169	C₃×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
170	Dic₃	—	2	2	no	—
171	C₂×C₅	—	2	1	no	—
172	C₂×C₅	—	12	2	no	—
173	C₂×C₅	C₈×D₁₂	1	1	yes	characteristic
174	Q₈	—	3	2	no	—
175	D₄	—	3	2	no	—
176	D₄	—	6	2	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	6	2	no	—
179	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
180	C₈	—	6	2	no	—
181	C₈	C₅×D₃×C₈	1	1	yes	characteristic
182	C₈	C₅×D₃×C₈	1	1	yes	characteristic
183	C₈	C₂×C₅×D₁₂	1	1	yes	characteristic
184	C₈	C₅×#235	1	1	yes	characteristic
185	C₂×C₃	—	2	1	no	—
186	C₂×C₃	C₅×C₈×D₄	1	1	yes	characteristic
187	D₃	—	4	2	no	—
188	C₅	#69076	1	1	yes	characteristic, direct product factor, Sylow
189	C₂²	—	6	2	no	—
190	C₂²	C₅×#901	1	1	yes	characteristic
191	C₄	—	4	1	no	—
192	C₄	—	6	2	no	—
193	C₄	C₄×C₅×D₁₂	1	1	yes	characteristic
194	C₄	C₂×C₅×D₃×C₈	1	1	yes	characteristic
195	C₃	C₅×#2171	1	1	yes	characteristic, complemented, Sylow
196	C₂	—	2	1	no	—
197	C₂	—	12	2	no	—
198	C₂	C₅×C₈×D₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₅×C₈	C₂×D₁₂
commutator	C₃×C₄	C₂²×C₅×C₈
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₃×C₅×#346	C₂
socle	C₂×C₃×C₅	C₈×D₄
prime radical	C₅×#904	C₄
cosocle	C₂×C₃×C₈	C₂³×C₅
hypercenter	C₅×#346	D₃
hypocenter	C₃	C₅×#2171
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₅×C₈ ◁ C₂×C₅×C₁₆ ◁ C₅×#346

Complements

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

N	H ≅ G/N	type
1. C₅×#4901	C₂	semidirect
3. C₃×C₅×#346	C₂	semidirect
4. C₅×#4508	C₂	semidirect
5. C₅×#4463	C₂	semidirect
17. #69076	C₅	direct (N × H)
22. C₅×#346	D₃	semidirect
36. #4901	C₂×C₅	semidirect
38. C₃×#346	C₂×C₅	semidirect
39. #4508	C₂×C₅	semidirect
40. #4463	C₂×C₅	semidirect
90. #346	C₅×D₃	semidirect
164. C₃×C₅	#2171	semidirect
188. C₅	#69076	direct (N × H)
195. C₃	C₅×#2171	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×C₅×#346	2
C₅×#4508	2
C₅×#4463	2
C₅×#2171	3
#69076	5

Subgroups — C5×#69076 #14315251

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69076 #14315251