Subgroups — C₅×#68077 #14286442

Order 1920.

384 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×D₂₄	C₂	1	1	yes	maximal, characteristic
2	C₅×C₈×D₁₂	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#322	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4459	C₂	1	2	yes	maximal, complemented
5	C₅×#4457	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1580	—	3	1	no	maximal, Hall
7	C₂×C₅×D₂₄	C₄	1	1	yes	characteristic
8	C₂×C₅×D₃×C₈	—	2	2	no	—
9	C₄×C₅×D₁₂	C₂²	1	2	yes	—
10	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₅×#821	C₄	1	2	yes	—
14	C₅×(C₂×D₃)⋊C₈	—	2	2	no	—
15	C₅×#818	C₄	1	1	yes	characteristic
16	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
17	#68077	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₄×C₅×D₈	—	3	1	no	—
19	C₅×C₈×D₄	—	3	2	no	—
20	C₅×#334	—	3	1	no	—
21	C₅×D₄⋊C₈	—	3	2	no	—
22	C₅×#322	D₃	1	1	yes	characteristic, complemented
23	C₂×C₅×D₁₂	C₂×C₄	1	2	yes	—
24	C₂×C₄×C₅×D₃	—	2	2	no	—
25	C₂×C₃×C₅×C₈	—	2	1	no	—
26	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
29	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
30	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
31	C₅×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
32	C₂×C₅×C₃⋊C₈	—	2	2	no	—
33	C₅×D₂₄	C₈	1	4	yes	complemented
34	C₅×D₃×C₈	—	4	4	no	—
35	C₄×D₂₄	C₂×C₅	1	1	yes	characteristic
36	C₈×D₁₂	C₂×C₅	1	2	yes	complemented
37	C₃×#322	C₂×C₅	1	1	yes	characteristic, complemented
38	#4459	C₂×C₅	1	2	yes	complemented
39	#4457	C₂×C₅	1	1	yes	characteristic, complemented
40	C₂×C₅×D₈	—	3	1	no	—
41	C₂²×C₅×C₈	—	6	2	no	—
42	C₄×C₅×D₄	—	3	2	no	—
43	C₅×#107	—	3	1	no	—
44	C₅×C₄⋊C₈	—	3	2	no	—
45	C₅×#102	—	3	2	no	—
46	C₅×C₂²⋊C₈	—	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	#1580	—	3	1	no	Sylow
51	C₂²×C₅×D₃	—	2	2	no	—
52	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
55	C₂×C₅×Dic₃	—	2	2	no	—
56	C₅×D₁₂	—	2	2	no	—
57	C₅×D₁₂	C₂×C₈	1	4	yes	complemented
58	C₄×C₅×D₃	—	4	2	no	—
59	C₄×C₅×D₃	—	4	2	no	—
60	C₃×C₅×C₈	—	2	1	no	—
61	C₃×C₅×C₈	—	2	2	no	—
62	C₃×C₅×C₈	—	4	1	no	—
63	C₃×C₅×C₈	C₂×C₈	1	2	yes	complemented
64	C₅×C₃⋊C₈	—	4	2	no	—
65	C₂×D₂₄	C₄×C₅	1	1	yes	characteristic
66	C₂×D₃×C₈	—	2	2	no	—
67	C₄×D₁₂	C₂²×C₅	1	2	yes	—
68	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
69	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
70	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
71	#821	C₄×C₅	1	2	yes	—
72	(C₂×D₃)⋊C₈	—	2	2	no	—
73	#818	C₄×C₅	1	1	yes	characteristic
74	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	2	yes	—
75	C₂×C₅×D₄	—	3	2	no	—
76	C₂²×C₄×C₅	—	6	2	no	—
77	C₅×D₈	—	3	4	no	—
78	C₂×C₅×C₈	—	2	1	no	—
79	C₂×C₅×C₈	—	6	2	no	—
80	C₂×C₅×C₈	—	12	4	no	—
81	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
82	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
83	C₂×C₅×C₈	D₁₂	1	2	yes	—
84	C₅×C₄⋊C₄	—	3	2	no	—
85	C₅×C₂²⋊C₄	—	6	2	no	—
86	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
87	C₄×D₈	—	3	1	no	—
88	C₈×D₄	—	3	2	no	—
89	#334	—	3	1	no	—
90	D₄⋊C₈	—	3	2	no	—
91	#322	C₅×D₃	1	1	yes	characteristic, complemented
92	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
93	C₂×C₅×D₃	—	2	4	no	—
94	C₂×C₅×D₃	—	4	2	no	—
95	C₂×C₅×D₃	—	4	2	no	—
96	C₃×C₄×C₅	—	2	1	no	—
97	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
98	C₃×C₄×C₅	#131	1	1	yes	characteristic
99	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
100	C₅×Dic₃	—	4	2	no	—
101	C₂×D₁₂	C₂×C₄×C₅	1	2	yes	—
102	C₂×C₄×D₃	—	2	2	no	—
103	C₂×C₃×C₈	—	2	1	no	—
104	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
105	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
106	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
107	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
108	(C₂×D₃)⋊C₄	—	2	2	no	—
109	C₄⋊Dic₃	C₂×C₄×C₅	1	2	yes	—
110	C₂×C₃⋊C₈	—	2	2	no	—
111	D₂₄	C₅×C₈	1	4	yes	complemented
112	D₃×C₈	—	4	4	no	—
113	C₂³×C₅	—	6	2	no	—
114	C₅×D₄	—	3	4	no	—
115	C₅×D₄	—	6	2	no	—
116	C₂×C₄×C₅	—	6	2	no	—
117	C₂×C₄×C₅	—	12	2	no	—
118	C₂×C₄×C₅	—	12	2	no	—
119	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
121	C₂×C₄×C₅	#235	1	1	yes	characteristic
122	C₅×C₈	—	2	1	no	—
123	C₅×C₈	—	2	2	no	—
124	C₅×C₈	—	4	1	no	—
125	C₅×C₈	—	12	2	no	—
126	C₅×C₈	D₃×C₈	1	2	yes	complemented
127	C₂×D₈	—	3	1	no	—
128	C₂²×C₈	—	6	2	no	—
129	C₄×D₄	—	3	2	no	—
130	#107	—	3	1	no	—
131	C₄⋊C₈	—	3	2	no	—
132	#102	—	3	2	no	—
133	C₂²⋊C₈	—	6	2	no	—
134	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
135	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
136	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
137	C₂×C₃×C₅	C₈×D₄	1	1	yes	characteristic
138	C₂×C₃×C₅	#442	1	1	yes	characteristic
139	C₂×C₃×C₅	#444	1	1	yes	characteristic
140	C₅×D₃	—	4	4	no	—
141	C₂²×D₃	—	2	2	no	—
142	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
144	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
145	C₂×Dic₃	—	2	2	no	—
146	D₁₂	—	2	2	no	—
147	D₁₂	C₂×C₅×C₈	1	4	yes	complemented
148	C₄×D₃	—	4	2	no	—
149	C₄×D₃	—	4	2	no	—
150	C₃×C₈	—	2	1	no	—
151	C₃×C₈	—	2	2	no	—
152	C₃×C₈	—	4	1	no	—
153	C₃×C₈	C₂×C₅×C₈	1	2	yes	complemented
154	C₃⋊C₈	—	4	2	no	—
155	C₂²×C₅	—	6	4	no	—
156	C₂²×C₅	—	12	2	no	—
157	C₂²×C₅	—	12	2	no	—
158	C₂²×C₅	C₄×D₁₂	1	1	yes	characteristic
159	C₄×C₅	—	2	1	no	—
160	C₄×C₅	—	12	2	no	—
161	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
162	C₄×C₅	#901	1	1	yes	characteristic
163	C₄×C₅	#908	1	2	yes	—
164	C₂×D₄	—	3	2	no	—
165	C₂²×C₄	—	6	2	no	—
166	D₈	—	3	4	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	6	2	no	—
169	C₂×C₈	—	12	4	no	—
170	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
171	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
172	C₂×C₈	C₅×D₁₂	1	2	yes	—
173	C₄⋊C₄	—	3	2	no	—
174	C₂²⋊C₄	—	6	2	no	—
175	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
176	C₃×C₅	#1580	1	1	yes	characteristic, complemented, Hall
177	C₂²×C₃	C₄×C₅×D₄	1	1	yes	characteristic
178	C₂×D₃	—	2	4	no	—
179	C₂×D₃	—	4	2	no	—
180	C₂×D₃	—	4	2	no	—
181	C₃×C₄	—	2	1	no	—
182	C₃×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
183	C₃×C₄	C₅×#131	1	1	yes	characteristic
184	C₃×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
185	Dic₃	—	4	2	no	—
186	C₂×C₅	—	12	4	no	—
187	C₂×C₅	C₈×D₁₂	1	1	yes	characteristic
188	C₂×C₅	#4711	1	1	yes	characteristic
189	C₂×C₅	#4717	1	1	yes	characteristic
190	C₂³	—	6	2	no	—
191	D₄	—	3	4	no	—
192	D₄	—	6	2	no	—
193	C₂×C₄	—	6	2	no	—
194	C₂×C₄	—	12	2	no	—
195	C₂×C₄	—	12	2	no	—
196	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
197	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
198	C₂×C₄	C₅×#235	1	1	yes	characteristic
199	C₈	—	2	1	no	—
200	C₈	—	2	2	no	—
201	C₈	—	4	1	no	—
202	C₈	—	12	2	no	—
203	C₈	C₅×D₃×C₈	1	2	yes	complemented
204	C₂×C₃	C₅×C₈×D₄	1	1	yes	characteristic
205	C₂×C₃	C₅×#442	1	1	yes	characteristic
206	C₂×C₃	C₅×#444	1	1	yes	characteristic
207	D₃	—	4	4	no	—
208	C₅	#68077	1	1	yes	characteristic, direct product factor, Sylow
209	C₂²	—	6	4	no	—
210	C₂²	—	12	2	no	—
211	C₂²	—	12	2	no	—
212	C₂²	C₄×C₅×D₁₂	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	12	2	no	—
215	C₄	C₂×C₅×D₃×C₈	1	1	yes	characteristic
216	C₄	C₅×#901	1	1	yes	characteristic
217	C₄	C₅×#908	1	2	yes	—
218	C₃	C₅×#1580	1	1	yes	characteristic, complemented, Sylow
219	C₂	—	12	4	no	—
220	C₂	C₅×C₈×D₁₂	1	1	yes	characteristic
221	C₂	C₅×#4711	1	1	yes	characteristic
222	C₂	C₅×#4717	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₁₂
commutator	C₃×C₄	C₂²×C₅×C₈
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#322	C₂
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂×C₅×D₂₄	C₄
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×#322	D₃
hypocenter	C₃	C₅×#1580
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₄×C₅×C₈ ◁ C₅×#322

Complements

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

N	H ≅ G/N	type
2. C₅×C₈×D₁₂	C₂	semidirect
3. C₃×C₅×#322	C₂	semidirect
4. C₅×#4459	C₂	semidirect
5. C₅×#4457	C₂	semidirect
17. #68077	C₅	direct (N × H)
22. C₅×#322	D₃	semidirect
33. C₅×D₂₄	C₈	semidirect
36. C₈×D₁₂	C₂×C₅	semidirect
37. C₃×#322	C₂×C₅	semidirect
38. #4459	C₂×C₅	semidirect
39. #4457	C₂×C₅	semidirect
57. C₅×D₁₂	C₂×C₈	semidirect
63. C₃×C₅×C₈	C₂×C₈	semidirect
91. #322	C₅×D₃	semidirect
111. D₂₄	C₅×C₈	semidirect
126. C₅×C₈	D₃×C₈	semidirect
147. D₁₂	C₂×C₅×C₈	semidirect
153. C₃×C₈	C₂×C₅×C₈	semidirect
176. C₃×C₅	#1580	semidirect
203. C₈	C₅×D₃×C₈	semidirect
208. C₅	#68077	direct (N × H)
218. C₃	C₅×#1580	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×D₂₄	2
C₅×C₈×D₁₂	2
C₃×C₅×#322	2
C₅×#4459	2
C₅×#4457	2
C₅×#1580	3
#68077	5

Subgroups — C5×#68077 #14286442

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68077 #14286442