Subgroups — C₅×#69067 #14315632

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	C₅×#4904	C₂	1	1	yes	maximal, characteristic
2	C₅×#4700	C₂	1	2	yes	maximal, complemented
3	C₃×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4510	C₂	1	2	yes	maximal, complemented
5	C₅×#4506	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2178	—	3	1	no	maximal, Hall
7	C₅×#904	C₂²	1	2	yes	—
8	C₅×#901	—	2	2	no	—
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	C₂²	1	2	yes	—
13	C₅×#805	—	2	2	no	—
14	C₅×Dic₂₄	C₄	1	1	yes	characteristic, complemented
15	C₅×#800	C₄	1	2	yes	complemented
16	C₅×D₄₈	C₄	1	1	yes	characteristic, complemented
17	#69067	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₅×#508	—	3	1	no	—
19	C₅×#443	—	3	2	no	—
20	C₅×#368	—	3	1	no	—
21	C₅×#359	—	3	2	no	—
22	C₄×C₅×C₁₆	D₃	1	1	yes	characteristic, complemented
23	C₅×#235	—	2	2	no	—
24	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
25	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
26	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
27	C₅×#208	—	2	2	no	—
28	C₅×Dic₁₂	C₂×C₄	1	2	yes	—
29	C₅×D₂₄	C₂×C₄	1	2	yes	—
30	C₅×#204	—	2	2	no	—
31	C₅×ΓL(1,𝔽₂₅)	—	4	2	no	—
32	C₅×D₃×C₈	—	4	2	no	—
33	C₃×C₅×C₁₆	—	2	1	no	—
34	C₃×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
35	C₃×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
36	#4904	C₂×C₅	1	1	yes	characteristic
37	#4700	C₂×C₅	1	2	yes	complemented
38	C₃×C₄×C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
39	#4510	C₂×C₅	1	2	yes	complemented
40	#4506	C₂×C₅	1	1	yes	characteristic, complemented
41	C₅×#135	—	3	2	no	—
42	C₅×#131	—	6	2	no	—
43	C₅×Q₃₂	—	3	1	no	—
44	C₅×SD₃₂	—	3	2	no	—
45	C₅×D₁₆	—	3	1	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	2	no	—
49	C₅×#104	—	6	2	no	—
50	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
51	#2178	—	3	1	no	Sylow
52	C₂×C₃×C₄×C₅	—	2	1	no	—
53	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
54	C₅×#67	—	4	2	no	—
55	C₅×D₁₂	—	2	2	no	—
56	C₄×C₅×D₃	—	4	2	no	—
57	C₅×Dic₆	—	2	2	no	—
58	C₃×C₅×C₈	D₈	1	1	yes	characteristic
59	C₃×C₅×C₈	D₈	1	1	yes	characteristic
60	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
61	C₃×C₅×C₈	#41	1	1	yes	characteristic
62	C₅×C₃⋊C₈	—	4	2	no	—
63	#904	C₂²×C₅	1	2	yes	—
64	#901	—	2	2	no	—
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	C₂²×C₅	1	2	yes	—
69	#805	—	2	2	no	—
70	Dic₂₄	C₄×C₅	1	1	yes	characteristic, complemented
71	#800	C₄×C₅	1	2	yes	complemented
72	D₄₈	C₄×C₅	1	1	yes	characteristic, complemented
73	C₅×#41	—	6	2	no	—
74	C₅×Q₁₆	—	3	2	no	—
75	C₅×SD₁₆	—	6	2	no	—
76	C₅×D₈	—	3	2	no	—
77	C₅×M₁₆	—	6	2	no	—
78	C₅×M₁₆	—	12	2	no	—
79	C₂×C₅×C₈	—	12	2	no	—
80	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
81	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
82	C₄²×C₅	D₁₂	1	1	yes	characteristic
83	C₅×C₁₆	—	2	1	no	—
84	C₅×C₁₆	C₄×D₃	1	1	yes	characteristic
85	C₅×C₁₆	C₄×D₃	1	1	yes	characteristic
86	#508	—	3	1	no	—
87	#443	—	3	2	no	—
88	#368	—	3	1	no	—
89	#359	—	3	2	no	—
90	C₄×C₁₆	C₅×D₃	1	1	yes	characteristic, complemented
91	C₂²×C₃×C₅	#135	1	1	yes	characteristic
92	C₂×C₅×D₃	—	4	2	no	—
93	C₃×C₄×C₅	—	2	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	—	4	2	no	—
97	#235	—	2	2	no	—
98	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
99	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
100	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
101	#208	—	2	2	no	—
102	Dic₁₂	C₂×C₄×C₅	1	2	yes	—
103	D₂₄	C₂×C₄×C₅	1	2	yes	—
104	#204	—	2	2	no	—
105	ΓL(1,𝔽₂₅)	—	4	2	no	—
106	D₃×C₈	—	4	2	no	—
107	C₃×C₁₆	—	2	1	no	—
108	C₃×C₁₆	C₂×C₄×C₅	1	1	yes	characteristic
109	C₃×C₁₆	C₂×C₄×C₅	1	1	yes	characteristic
110	C₅×Q₈	—	6	2	no	—
111	C₅×D₄	—	6	2	no	—
112	C₅×D₄	—	12	2	no	—
113	C₂×C₄×C₅	—	2	1	no	—
114	C₂×C₄×C₅	—	12	2	no	—
115	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
116	C₅×C₈	—	12	2	no	—
117	C₅×C₈	D₂₄	1	1	yes	characteristic
118	C₅×C₈	D₂₄	1	1	yes	characteristic
119	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
120	C₅×C₈	#235	1	1	yes	characteristic
121	#135	—	3	2	no	—
122	#131	—	6	2	no	—
123	Q₃₂	—	3	1	no	—
124	SD₃₂	—	3	2	no	—
125	D₁₆	—	3	1	no	—
126	C₂×C₁₆	C₂×C₅×D₃	1	1	yes	characteristic
127	C₂×C₁₆	C₂×C₅×D₃	1	1	yes	characteristic
128	#108	—	3	2	no	—
129	#104	—	6	2	no	—
130	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
131	C₂×C₃×C₅	—	2	1	no	—
132	C₂×C₃×C₅	C₄×D₈	1	1	yes	characteristic
133	C₅×D₃	—	8	2	no	—
134	C₂×C₃×C₄	—	2	1	no	—
135	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
136	#67	—	4	2	no	—
137	D₁₂	—	2	2	no	—
138	C₄×D₃	—	4	2	no	—
139	Dic₆	—	2	2	no	—
140	C₃×C₈	C₅×D₈	1	1	yes	characteristic
141	C₃×C₈	C₅×D₈	1	1	yes	characteristic
142	C₃×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
143	C₃×C₈	C₅×#41	1	1	yes	characteristic
144	C₃⋊C₈	—	4	2	no	—
145	C₂²×C₅	—	12	2	no	—
146	C₂²×C₅	#904	1	1	yes	characteristic
147	C₄×C₅	—	2	2	no	—
148	C₄×C₅	—	12	2	no	—
149	C₄×C₅	C₄×D₁₂	1	1	yes	characteristic
150	C₄×C₅	C₂×D₂₄	1	1	yes	characteristic
151	#41	—	6	2	no	—
152	Q₁₆	—	3	2	no	—
153	SD₁₆	—	6	2	no	—
154	D₈	—	3	2	no	—
155	M₁₆	—	6	2	no	—
156	M₁₆	—	12	2	no	—
157	C₂×C₈	—	12	2	no	—
158	C₂×C₈	C₅×D₁₂	1	1	yes	characteristic
159	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
160	C₄²	C₅×D₁₂	1	1	yes	characteristic
161	C₁₆	—	2	1	no	—
162	C₁₆	C₄×C₅×D₃	1	1	yes	characteristic
163	C₁₆	C₄×C₅×D₃	1	1	yes	characteristic
164	C₃×C₅	#2178	1	1	yes	characteristic, complemented, Hall
165	C₂²×C₃	C₅×#135	1	1	yes	characteristic
166	C₂×D₃	—	4	2	no	—
167	C₃×C₄	—	2	2	no	—
168	C₃×C₄	C₄×C₅×D₄	1	1	yes	characteristic
169	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
170	Dic₃	—	4	2	no	—
171	C₂×C₅	—	2	1	no	—
172	C₂×C₅	—	24	2	no	—
173	C₂×C₅	C₄×D₂₄	1	1	yes	characteristic
174	Q₈	—	6	2	no	—
175	D₄	—	6	2	no	—
176	D₄	—	12	2	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	12	2	no	—
179	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
180	C₈	—	12	2	no	—
181	C₈	C₅×D₂₄	1	1	yes	characteristic
182	C₈	C₅×D₂₄	1	1	yes	characteristic
183	C₈	C₂×C₄×C₅×D₃	1	1	yes	characteristic
184	C₈	C₅×#235	1	1	yes	characteristic
185	C₂×C₃	—	2	1	no	—
186	C₂×C₃	C₄×C₅×D₈	1	1	yes	characteristic
187	D₃	—	8	2	no	—
188	C₅	#69067	1	1	yes	characteristic, direct product factor, Sylow
189	C₂²	—	12	2	no	—
190	C₂²	C₅×#904	1	1	yes	characteristic
191	C₄	—	2	2	no	—
192	C₄	—	12	2	no	—
193	C₄	C₄×C₅×D₁₂	1	1	yes	characteristic
194	C₄	C₂×C₅×D₂₄	1	1	yes	characteristic
195	C₃	C₅×#2178	1	1	yes	characteristic, complemented, Sylow
196	C₂	—	2	1	no	—
197	C₂	—	24	2	no	—
198	C₂	C₄×C₅×D₂₄	1	1	yes	characteristic

	subgroup	G/H
center	C₅×C₈	D₂₄
commutator	C₃×C₈	C₂²×C₄×C₅
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₃×C₄×C₅×C₁₆	C₂
socle	C₂×C₃×C₅	C₄×D₈
prime radical	C₅×#4904	C₂
cosocle	C₂×C₃×C₈	C₂³×C₅
hypercenter	C₄×C₅×C₁₆	D₃
hypocenter	C₃	C₅×#2178
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₅×C₈ ◁ C₂×C₅×C₈ ◁ C₄×C₅×C₈ ◁ C₄×C₅×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₅×#4700	C₂	semidirect
3. C₃×C₄×C₅×C₁₆	C₂	semidirect
4. C₅×#4510	C₂	semidirect
5. C₅×#4506	C₂	semidirect
14. C₅×Dic₂₄	C₄	semidirect
15. C₅×#800	C₄	semidirect
16. C₅×D₄₈	C₄	semidirect
17. #69067	C₅	direct (N × H)
22. C₄×C₅×C₁₆	D₃	semidirect
37. #4700	C₂×C₅	semidirect
38. C₃×C₄×C₁₆	C₂×C₅	semidirect
39. #4510	C₂×C₅	semidirect
40. #4506	C₂×C₅	semidirect
70. Dic₂₄	C₄×C₅	semidirect
71. #800	C₄×C₅	semidirect
72. D₄₈	C₄×C₅	semidirect
90. C₄×C₁₆	C₅×D₃	semidirect
164. C₃×C₅	#2178	semidirect
188. C₅	#69067	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
C₅×#4904	2
C₅×#4700	2
C₃×C₄×C₅×C₁₆	2
C₅×#4510	2
C₅×#4506	2
C₅×#2178	3
#69067	5

Subgroups — C5×#69067 #14315632

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69067 #14315632