Subgroups — C₅×C₈×#208 #14280366

Order 1920.

448 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₅×#4925	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×#208	C₂	1	1	yes	maximal, characteristic
4	C₅×#4454	C₂	1	2	yes	maximal, complemented
5	C₅×C₈×C₃⋊C₈	C₂	1	2	yes	maximal, complemented
6	C₅×C₈×M₁₆	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅×C₈	C₄	1	2	yes	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₅×#208	C₄	1	2	yes	—
10	C₃×C₄×C₅×C₈	C₄	1	2	yes	—
11	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₅×#830	C₄	1	2	yes	—
14	C₅×C₄⋊(C₃⋊C₈)	C₄	1	2	yes	—
15	C₅×#803	C₄	1	2	yes	—
16	C₄×C₅×C₃⋊C₈	C₂²	1	2	yes	—
17	C₄×C₅×C₃⋊C₈	C₂²	1	2	yes	—
18	C₈×#208	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₅×#431	—	3	1	no	—
20	C₄×C₅×M₁₆	—	3	1	no	—
21	C₂×C₄×C₅×C₈	D₃	1	1	yes	characteristic
22	C₅×#322	—	3	2	no	—
23	C₅×C₈²	—	3	2	no	—
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₅×C₈	—	2	4	no	—
28	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
30	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
31	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
32	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
33	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
34	C₅×#208	C₈	1	8	yes	direct product factor
35	C₂×C₅×C₃⋊C₈	C₂×C₄	1	4	yes	—
36	C₂×C₅×C₃⋊C₈	C₂×C₄	1	4	yes	—
37	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic
38	#4925	C₂×C₅	1	1	yes	characteristic
39	C₄×#208	C₂×C₅	1	1	yes	characteristic
40	#4454	C₂×C₅	1	2	yes	complemented
41	C₈×C₃⋊C₈	C₂×C₅	1	2	yes	complemented
42	C₂×C₅×M₁₆	—	3	2	no	—
43	C₂²×C₅×C₈	Dic₃	1	2	yes	—
44	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
45	C₅×C₄⋊C₈	—	3	2	no	—
46	C₅×C₂²⋊C₈	—	3	2	no	—
47	C₅×#97	—	3	2	no	—
48	C₄×C₅×C₈	—	3	2	no	—
49	C₄×C₅×C₈	—	3	2	no	—
50	C₄×C₅×C₈	Dic₃	1	2	yes	—
51	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
52	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
53	C₈×M₁₆	—	3	1	no	Sylow
54	C₂³×C₃×C₅	C₄²	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	—	2	2	no	—
56	C₂×C₃×C₄×C₅	—	2	2	no	—
57	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂×C₈	1	4	yes	—
61	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
64	C₃×C₅×C₈	—	2	2	no	—
65	C₃×C₅×C₈	M₁₆	1	4	yes	complemented
66	C₅×C₃⋊C₈	—	2	4	no	—
67	C₅×C₃⋊C₈	C₂×C₈	1	8	yes	complemented
68	C₂²×C₃×C₈	C₄×C₅	1	2	yes	—
69	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
70	C₂×#208	C₄×C₅	1	2	yes	—
71	C₃×C₄×C₈	C₄×C₅	1	2	yes	—
72	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
73	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
74	#830	C₄×C₅	1	2	yes	—
75	C₄⋊(C₃⋊C₈)	C₄×C₅	1	2	yes	—
76	#803	C₄×C₅	1	2	yes	—
77	C₄×C₃⋊C₈	C₂²×C₅	1	2	yes	—
78	C₄×C₃⋊C₈	C₂²×C₅	1	2	yes	—
79	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
82	C₅×M₁₆	—	3	8	no	—
83	C₂×C₅×C₈	—	2	4	no	—
84	C₂×C₅×C₈	—	3	4	no	—
85	C₂×C₅×C₈	—	3	4	no	—
86	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
87	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
88	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
89	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
90	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
91	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
92	#431	—	3	1	no	—
93	C₄×M₁₆	—	3	1	no	—
94	C₂×C₄×C₈	C₅×D₃	1	1	yes	characteristic
95	#322	—	3	2	no	—
96	C₈²	—	3	2	no	—
97	C₂²×C₃×C₅	—	2	1	no	—
98	C₂²×C₃×C₅	—	2	1	no	—
99	C₂²×C₃×C₅	C₄×C₈	1	2	yes	—
100	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
101	C₃×C₄×C₅	—	2	1	no	—
102	C₃×C₄×C₅	—	2	1	no	—
103	C₃×C₄×C₅	C₄×C₈	1	2	yes	—
104	C₃×C₄×C₅	C₂²×C₈	1	2	yes	—
105	C₃×C₄×C₅	C₂×M₁₆	1	2	yes	—
106	C₃×C₄×C₅	#131	1	2	yes	—
107	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
108	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
109	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
110	C₂×C₃×C₈	—	2	4	no	—
111	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
112	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
113	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
114	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
115	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
116	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
117	#208	C₅×C₈	1	8	yes	direct product factor
118	C₂×C₃⋊C₈	C₂×C₄×C₅	1	4	yes	—
119	C₂×C₃⋊C₈	C₂×C₄×C₅	1	4	yes	—
120	C₂³×C₅	C₄×Dic₃	1	1	yes	characteristic
121	C₂×C₄×C₅	—	2	2	no	—
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	D₃×C₈	1	4	yes	—
124	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
125	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
126	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
128	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
129	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
130	C₅×C₈	—	2	2	no	—
131	C₅×C₈	—	3	8	no	—
132	C₅×C₈	—	6	4	no	—
133	C₅×C₈	#208	1	4	yes	direct product factor
134	C₂×M₁₆	—	3	2	no	—
135	C₂²×C₈	C₅×Dic₃	1	2	yes	—
136	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
137	C₄⋊C₈	—	3	2	no	—
138	C₂²⋊C₈	—	3	2	no	—
139	#97	—	3	2	no	—
140	C₄×C₈	—	3	2	no	—
141	C₄×C₈	—	3	2	no	—
142	C₄×C₈	C₅×Dic₃	1	2	yes	—
143	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
144	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
145	C₂×C₃×C₅	—	2	2	no	—
146	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
147	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
148	C₂×C₃×C₅	#405	1	1	yes	characteristic
149	C₂³×C₃	C₄²×C₅	1	1	yes	characteristic
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	C₄²×C₅	1	1	yes	characteristic
153	C₂×C₃×C₄	C₄²×C₅	1	1	yes	characteristic
154	C₂×C₃×C₄	C₄²×C₅	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂×C₅×C₈	1	4	yes	—
156	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
157	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
159	C₃×C₈	—	2	2	no	—
160	C₃×C₈	C₅×M₁₆	1	4	yes	complemented
161	C₃⋊C₈	—	2	4	no	—
162	C₃⋊C₈	C₂×C₅×C₈	1	8	yes	complemented
163	C₂²×C₅	—	2	1	no	—
164	C₂²×C₅	—	2	1	no	—
165	C₂²×C₅	C₈×Dic₃	1	2	yes	—
166	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
167	C₄×C₅	—	2	1	no	—
168	C₄×C₅	—	2	1	no	—
169	C₄×C₅	C₈×Dic₃	1	2	yes	—
170	C₄×C₅	C₂×D₃×C₈	1	2	yes	—
171	C₄×C₅	#901	1	2	yes	—
172	C₄×C₅	C₂×#208	1	2	yes	—
173	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
174	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
175	C₂²×C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
176	M₁₆	—	3	8	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	3	4	no	—
179	C₂×C₈	—	3	4	no	—
180	C₂×C₈	C₂×C₅×Dic₃	1	2	yes	—
181	C₂×C₈	C₂×C₅×Dic₃	1	2	yes	—
182	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
183	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
184	C₄²	C₂×C₅×Dic₃	1	1	yes	characteristic
185	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
186	C₃×C₅	C₈×M₁₆	1	1	yes	characteristic, complemented, Hall
187	C₂²×C₃	—	2	1	no	—
188	C₂²×C₃	—	2	1	no	—
189	C₂²×C₃	C₄×C₅×C₈	1	2	yes	—
190	C₂²×C₃	C₂×C₄²×C₅	1	1	yes	characteristic
191	C₃×C₄	—	2	1	no	—
192	C₃×C₄	—	2	1	no	—
193	C₃×C₄	C₄×C₅×C₈	1	2	yes	—
194	C₃×C₄	C₂²×C₅×C₈	1	2	yes	—
195	C₃×C₄	C₂×C₅×M₁₆	1	2	yes	—
196	C₃×C₄	C₅×#131	1	2	yes	—
197	C₂×C₅	—	2	2	no	—
198	C₂×C₅	C₄×#208	1	1	yes	characteristic
199	C₂×C₅	C₂×C₈×Dic₃	1	1	yes	characteristic
200	C₂×C₅	#5101	1	1	yes	characteristic
201	C₂³	C₄×C₅×Dic₃	1	1	yes	characteristic
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	C₅×D₃×C₈	1	4	yes	—
205	C₂×C₄	C₄×C₅×Dic₃	1	1	yes	characteristic
206	C₂×C₄	C₄×C₅×Dic₃	1	1	yes	characteristic
207	C₂×C₄	C₄×C₅×Dic₃	1	1	yes	characteristic
208	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
209	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
210	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
211	C₈	—	2	2	no	—
212	C₈	—	3	8	no	—
213	C₈	—	6	4	no	—
214	C₈	C₅×#208	1	4	yes	direct product factor
215	C₂×C₃	—	2	2	no	—
216	C₂×C₃	C₂×C₄×C₅×C₈	1	1	yes	characteristic
217	C₂×C₃	C₄×C₅×M₁₆	1	1	yes	characteristic
218	C₂×C₃	C₅×#405	1	1	yes	characteristic
219	C₅	C₈×#208	1	1	yes	characteristic, direct product factor, Sylow
220	C₂²	—	2	1	no	—
221	C₂²	—	2	1	no	—
222	C₂²	C₅×C₈×Dic₃	1	2	yes	—
223	C₂²	C₂×C₄×C₅×Dic₃	1	1	yes	characteristic
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	C₅×C₈×Dic₃	1	2	yes	—
227	C₄	C₂×C₅×D₃×C₈	1	2	yes	—
228	C₄	C₅×#901	1	2	yes	—
229	C₄	C₂×C₅×#208	1	2	yes	—
230	C₃	C₅×C₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
231	C₂	—	2	2	no	—
232	C₂	C₄×C₅×#208	1	1	yes	characteristic
233	C₂	C₂×C₅×C₈×Dic₃	1	1	yes	characteristic
234	C₂	C₅×#5101	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅×C₈	C₂×D₃
commutator	C₂×C₃	C₂×C₄×C₅×C₈
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₂×C₃×C₄×C₅×C₈	C₂
socle	C₂²×C₃×C₅	C₂×C₄²
prime radical	C₂³×C₃×C₅	C₄²
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₂×C₄×C₅×C₈	D₃
hypocenter	C₃	C₅×C₈×M₁₆
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅×C₈ ◁ C₂×C₄×C₅×C₈

Complements

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

N	H ≅ G/N	type
4. C₅×#4454	C₂	semidirect
5. C₅×C₈×C₃⋊C₈	C₂	semidirect
18. C₈×#208	C₅	direct (N × H)
34. C₅×#208	C₈	direct (N × H)
40. #4454	C₂×C₅	semidirect
41. C₈×C₃⋊C₈	C₂×C₅	semidirect
65. C₃×C₅×C₈	M₁₆	semidirect
67. C₅×C₃⋊C₈	C₂×C₈	semidirect
117. #208	C₅×C₈	direct (N × H)
133. C₅×C₈	#208	direct (N × H)
160. C₃×C₈	C₅×M₁₆	semidirect
162. C₃⋊C₈	C₂×C₅×C₈	semidirect
186. C₃×C₅	C₈×M₁₆	semidirect
214. C₈	C₅×#208	direct (N × H)
219. C₅	C₈×#208	direct (N × H)
230. C₃	C₅×C₈×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₅×C₈	2
C₅×#4925	2
C₄×C₅×#208	2
C₅×#4454	2
C₅×C₈×C₃⋊C₈	2
C₅×C₈×M₁₆	3
C₈×#208	5

Subgroups — C5×C8×#208 #14280366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₈×#208 #14280366