Subgroups — C₂×C₅×#4454 #14280327

Order 1920.

496 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	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₅×#4454	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₅×#322	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₅×C₈	C₄	1	2	yes	—
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂²×C₅×C₃⋊C₈	C₄	1	2	yes	—
9	C₂²×C₅×C₃⋊C₈	C₄	1	2	yes	—
10	C₃×C₄×C₅×C₈	C₄	1	2	yes	—
11	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
12	C₄×C₅×C₃⋊C₈	C₄	1	2	yes	—
13	C₄×C₅×C₃⋊C₈	C₄	1	2	yes	—
14	C₄×C₅×C₃⋊C₈	C₂²	1	2	yes	—
15	C₄×C₅×C₃⋊C₈	C₂²	1	2	yes	—
16	C₂×#4454	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×C₄×C₅×C₈	—	3	1	no	—
18	C₂×C₄×C₅×C₈	—	3	1	no	—
19	C₂×C₄×C₅×C₈	D₃	1	1	yes	characteristic
20	C₅×#322	—	3	4	no	—
21	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₅×C₈	—	2	4	no	—
25	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	—
26	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
27	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
28	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
29	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
30	C₂×C₅×C₃⋊C₈	—	2	4	no	—
31	C₂×C₅×C₃⋊C₈	C₈	1	8	yes	complemented
32	C₂×C₅×C₃⋊C₈	C₂×C₄	1	4	yes	—
33	C₂×C₅×C₃⋊C₈	C₂×C₄	1	4	yes	—
34	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic
35	C₂×C₄×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic
36	C₂×C₄×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic
37	#4454	C₂×C₅	1	4	yes	direct product factor
38	C₂²×C₅×C₈	—	3	2	no	—
39	C₂²×C₅×C₈	—	3	2	no	—
40	C₂²×C₅×C₈	Dic₃	1	2	yes	—
41	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
42	C₄×C₅×C₈	—	3	2	no	—
43	C₄×C₅×C₈	—	3	2	no	—
44	C₄×C₅×C₈	—	3	2	no	—
45	C₄×C₅×C₈	—	3	2	no	—
46	C₄×C₅×C₈	Dic₃	1	2	yes	—
47	C₄×C₅×C₈	C₂×D₃	1	2	yes	—
48	C₂×#322	—	3	1	no	Sylow
49	C₂³×C₃×C₅	C₄²	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂×C₈	1	4	yes	—
54	C₂×C₃×C₄×C₅	M₁₆	1	4	yes	—
55	C₂×C₃×C₄×C₅	M₁₆	1	4	yes	—
56	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₃×C₅×C₈	—	2	4	no	—
60	C₅×C₃⋊C₈	—	2	4	no	—
61	C₅×C₃⋊C₈	C₂×C₈	1	8	yes	complemented
62	C₂²×C₃×C₈	C₄×C₅	1	2	yes	—
63	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
64	C₂²×C₃⋊C₈	C₄×C₅	1	2	yes	—
65	C₂²×C₃⋊C₈	C₄×C₅	1	2	yes	—
66	C₃×C₄×C₈	C₄×C₅	1	2	yes	—
67	C₃×C₄×C₈	C₂²×C₅	1	2	yes	—
68	C₄×C₃⋊C₈	C₄×C₅	1	2	yes	—
69	C₄×C₃⋊C₈	C₄×C₅	1	2	yes	—
70	C₄×C₃⋊C₈	C₂²×C₅	1	2	yes	—
71	C₄×C₃⋊C₈	C₂²×C₅	1	2	yes	—
72	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
75	C₂×C₅×C₈	—	2	4	no	—
76	C₂×C₅×C₈	—	3	8	no	—
77	C₂×C₅×C₈	—	3	4	no	—
78	C₂×C₅×C₈	—	3	4	no	—
79	C₂×C₅×C₈	—	6	4	no	—
80	C₂×C₅×C₈	C₂×Dic₃	1	4	yes	—
81	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
82	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
83	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
84	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
85	C₂×C₄×C₈	—	3	1	no	—
86	C₂×C₄×C₈	—	3	1	no	—
87	C₂×C₄×C₈	C₅×D₃	1	1	yes	characteristic
88	#322	—	3	4	no	—
89	C₂²×C₃×C₅	C₄×C₈	1	2	yes	—
90	C₂²×C₃×C₅	#97	1	2	yes	—
91	C₂²×C₃×C₅	#97	1	2	yes	—
92	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
93	C₃×C₄×C₅	C₄×C₈	1	2	yes	—
94	C₃×C₄×C₅	#97	1	2	yes	—
95	C₃×C₄×C₅	#97	1	2	yes	—
96	C₃×C₄×C₅	C₂²×C₈	1	2	yes	—
97	C₃×C₄×C₅	C₂×M₁₆	1	2	yes	—
98	C₃×C₄×C₅	C₂×M₁₆	1	2	yes	—
99	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
100	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
101	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
102	C₂×C₃×C₈	—	2	4	no	—
103	C₂×C₃×C₈	C₂×C₄×C₅	1	4	yes	—
104	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
105	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
106	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
107	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
108	C₂×C₃⋊C₈	—	2	4	no	—
109	C₂×C₃⋊C₈	C₅×C₈	1	8	yes	complemented
110	C₂×C₃⋊C₈	C₂×C₄×C₅	1	4	yes	—
111	C₂×C₃⋊C₈	C₂×C₄×C₅	1	4	yes	—
112	C₂³×C₅	C₄×Dic₃	1	1	yes	characteristic
113	C₂×C₄×C₅	D₃×C₈	1	4	yes	—
114	C₂×C₄×C₅	ΓL(1,𝔽₂₅)	1	4	yes	—
115	C₂×C₄×C₅	#208	1	4	yes	—
116	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
117	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
118	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
122	C₅×C₈	—	2	4	no	—
123	C₅×C₈	—	3	8	no	—
124	C₅×C₈	—	6	4	no	—
125	C₂²×C₈	—	3	2	no	—
126	C₂²×C₈	—	3	2	no	—
127	C₂²×C₈	C₅×Dic₃	1	2	yes	—
128	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
129	C₄×C₈	—	3	2	no	—
130	C₄×C₈	—	3	2	no	—
131	C₄×C₈	—	3	2	no	—
132	C₄×C₈	—	3	2	no	—
133	C₄×C₈	C₅×Dic₃	1	2	yes	—
134	C₄×C₈	C₂×C₅×D₃	1	2	yes	—
135	C₂×C₃×C₅	#322	1	4	yes	complemented
136	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
137	C₂×C₃×C₅	C₂×#97	1	1	yes	characteristic
138	C₂×C₃×C₅	C₂×#97	1	1	yes	characteristic
139	C₂³×C₃	C₄²×C₅	1	1	yes	characteristic
140	C₂×C₃×C₄	C₄²×C₅	1	1	yes	characteristic
141	C₂×C₃×C₄	C₄²×C₅	1	1	yes	characteristic
142	C₂×C₃×C₄	C₄²×C₅	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂×C₅×C₈	1	4	yes	—
144	C₂×C₃×C₄	C₅×M₁₆	1	4	yes	—
145	C₂×C₃×C₄	C₅×M₁₆	1	4	yes	—
146	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
147	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
148	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
149	C₃×C₈	—	2	4	no	—
150	C₃⋊C₈	—	2	4	no	—
151	C₃⋊C₈	C₂×C₅×C₈	1	8	yes	complemented
152	C₂²×C₅	#803	1	2	yes	—
153	C₂²×C₅	C₈×Dic₃	1	2	yes	—
154	C₂²×C₅	#815	1	2	yes	—
155	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
156	C₄×C₅	#803	1	2	yes	—
157	C₄×C₅	C₈×Dic₃	1	2	yes	—
158	C₄×C₅	#815	1	2	yes	—
159	C₄×C₅	C₂×D₃×C₈	1	2	yes	—
160	C₄×C₅	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
161	C₄×C₅	C₂×#208	1	2	yes	—
162	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
163	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
164	C₂²×C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
165	C₂×C₈	—	2	4	no	—
166	C₂×C₈	—	3	8	no	—
167	C₂×C₈	—	3	4	no	—
168	C₂×C₈	—	3	4	no	—
169	C₂×C₈	—	6	4	no	—
170	C₂×C₈	C₂×C₅×Dic₃	1	4	yes	—
171	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
172	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
173	C₄²	C₂×C₅×Dic₃	1	1	yes	characteristic
174	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
175	C₃×C₅	C₂×#322	1	1	yes	characteristic, complemented, Hall
176	C₂²×C₃	C₄×C₅×C₈	1	2	yes	—
177	C₂²×C₃	C₅×#97	1	2	yes	—
178	C₂²×C₃	C₅×#97	1	2	yes	—
179	C₂²×C₃	C₂×C₄²×C₅	1	1	yes	characteristic
180	C₃×C₄	C₄×C₅×C₈	1	2	yes	—
181	C₃×C₄	C₅×#97	1	2	yes	—
182	C₃×C₄	C₅×#97	1	2	yes	—
183	C₃×C₄	C₂²×C₅×C₈	1	2	yes	—
184	C₃×C₄	C₂×C₅×M₁₆	1	2	yes	—
185	C₃×C₄	C₂×C₅×M₁₆	1	2	yes	—
186	C₂×C₅	#4454	1	4	yes	direct product factor
187	C₂×C₅	C₂×#803	1	1	yes	characteristic
188	C₂×C₅	C₂×C₈×Dic₃	1	1	yes	characteristic
189	C₂×C₅	C₂×#815	1	1	yes	characteristic
190	C₂³	C₄×C₅×Dic₃	1	1	yes	characteristic
191	C₂×C₄	C₅×D₃×C₈	1	4	yes	—
192	C₂×C₄	C₅×ΓL(1,𝔽₂₅)	1	4	yes	—
193	C₂×C₄	C₅×#208	1	4	yes	—
194	C₂×C₄	C₄×C₅×Dic₃	1	1	yes	characteristic
195	C₂×C₄	C₄×C₅×Dic₃	1	1	yes	characteristic
196	C₂×C₄	C₄×C₅×Dic₃	1	1	yes	characteristic
197	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
199	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
200	C₈	—	2	4	no	—
201	C₈	—	3	8	no	—
202	C₈	—	6	4	no	—
203	C₂×C₃	C₅×#322	1	4	yes	complemented
204	C₂×C₃	C₂×C₄×C₅×C₈	1	1	yes	characteristic
205	C₂×C₃	C₂×C₅×#97	1	1	yes	characteristic
206	C₂×C₃	C₂×C₅×#97	1	1	yes	characteristic
207	C₅	C₂×#4454	1	1	yes	characteristic, direct product factor, Sylow
208	C₂²	C₅×#803	1	2	yes	—
209	C₂²	C₅×C₈×Dic₃	1	2	yes	—
210	C₂²	C₅×#815	1	2	yes	—
211	C₂²	C₂×C₄×C₅×Dic₃	1	1	yes	characteristic
212	C₄	C₅×#803	1	2	yes	—
213	C₄	C₅×C₈×Dic₃	1	2	yes	—
214	C₄	C₅×#815	1	2	yes	—
215	C₄	C₂×C₅×D₃×C₈	1	2	yes	—
216	C₄	C₂×C₅×ΓL(1,𝔽₂₅)	1	2	yes	—
217	C₄	C₂×C₅×#208	1	2	yes	—
218	C₃	C₂×C₅×#322	1	1	yes	characteristic, complemented, Sylow
219	C₂	C₅×#4454	1	4	yes	direct product factor
220	C₂	C₂×C₅×#803	1	1	yes	characteristic
221	C₂	C₂×C₅×C₈×Dic₃	1	1	yes	characteristic
222	C₂	C₂×C₅×#815	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₅×#4454	C₂	direct (N × H)
16. C₂×#4454	C₅	direct (N × H)
31. C₂×C₅×C₃⋊C₈	C₈	semidirect
37. #4454	C₂×C₅	direct (N × H)
61. C₅×C₃⋊C₈	C₂×C₈	semidirect
109. C₂×C₃⋊C₈	C₅×C₈	semidirect
135. C₂×C₃×C₅	#322	semidirect
151. C₃⋊C₈	C₂×C₅×C₈	semidirect
175. C₃×C₅	C₂×#322	semidirect
186. C₂×C₅	#4454	direct (N × H)
203. C₂×C₃	C₅×#322	semidirect
207. C₅	C₂×#4454	direct (N × H)
218. C₃	C₂×C₅×#322	semidirect
219. C₂	C₅×#4454	direct (N × H)

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
C₂×C₄×C₅×C₃⋊C₈	2
C₅×#4454	2
C₂×C₅×#322	3
C₂×#4454	5

Subgroups — C2×C5×#4454 #14280327

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4454 #14280327