Subgroups — C₂×C₃×#64889 #14314405

Order 1920.

336 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#3998	C₂	1	2	yes	maximal
2	C₂×C₃×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₃×#64889	C₂	1	4	yes	maximal, direct product factor
4	C₂×#64889	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂²×C₃×C₅×C₈	C₄	1	1	yes	characteristic
6	C₂×C₃×C₄×Dic₅	C₄	1	1	yes	characteristic
7	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
8	C₃×#3998	—	2	2	no	—
9	C₃×#3998	C₂²	1	4	yes	—
10	C₂×C₃×#674	—	2	2	no	—
11	C₃×C₈×Dic₅	C₄	1	2	yes	—
12	C₃×C₈×Dic₅	C₂²	1	2	yes	—
13	C₂×C₃×#364	—	5	1	no	maximal, Hall
14	C₂×#3998	C₂×C₃	1	2	yes	—
15	C₂×C₈×Dic₅	C₂×C₃	1	1	yes	characteristic
16	#64889	C₂×C₃	1	4	yes	direct product factor
17	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₃×Dic₅	—	2	1	no	—
19	C₂×C₃×C₅×C₈	C₈	1	2	yes	—
20	C₂×C₃×C₅×C₈	C₈	1	2	yes	—
21	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
22	C₃×C₄×Dic₅	—	2	1	no	—
23	C₃×C₄×Dic₅	C₂×C₄	1	2	yes	—
24	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×#168	D₄	1	1	yes	characteristic
26	C₂×C₃×#168	D₄	1	1	yes	characteristic
27	C₂×C₃×#168	Q₈	1	1	yes	characteristic
28	C₂×C₃×#168	Q₈	1	1	yes	characteristic
29	C₂×C₃×#168	C₂³	1	1	yes	characteristic
30	C₃×#674	—	2	4	no	—
31	C₂×C₃×M₃₂	—	5	2	no	—
32	C₂×C₃×C₄×C₈	—	5	1	no	—
33	C₃×#364	—	5	4	no	—
34	C₂²×C₅×C₈	C₃×C₄	1	1	yes	characteristic
35	C₂×C₄×Dic₅	C₃×C₄	1	1	yes	characteristic
36	C₂²×#168	C₂²×C₃	1	1	yes	characteristic
37	#3998	—	2	2	no	—
38	#3998	C₂²×C₃	1	4	yes	—
39	C₂×#674	—	2	2	no	—
40	C₈×Dic₅	C₃×C₄	1	2	yes	—
41	C₈×Dic₅	C₂²×C₃	1	2	yes	—
42	C₂×#364	—	5	1	no	Sylow
43	C₂³×C₃×C₅	M₁₆	1	1	yes	characteristic
44	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
47	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
48	C₂×C₃×C₄×C₅	M₁₆	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₃×Dic₅	—	2	4	no	—
51	C₂×C₃×Dic₅	—	2	2	no	—
52	C₃×C₅×C₈	C₂×C₈	1	4	yes	—
53	C₃×#168	C₄⋊C₄	1	1	yes	characteristic
54	C₃×#168	C₄⋊C₄	1	1	yes	characteristic
55	C₃×#168	C₂×D₄	1	1	yes	characteristic
56	C₃×#168	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₃×C₈	—	5	1	no	—
58	C₂²×C₃×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
59	C₂×C₃×C₄²	—	5	1	no	—
60	C₃×M₃₂	—	5	4	no	—
61	C₃×M₃₂	—	10	2	no	—
62	C₂×C₃×C₁₆	—	10	2	no	—
63	C₃×C₄×C₈	—	5	2	no	—
64	C₃×C₄×C₈	—	5	2	no	—
65	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
66	C₂²×Dic₅	—	2	1	no	—
67	C₂×C₅×C₈	C₃×C₈	1	2	yes	—
68	C₂×C₅×C₈	C₃×C₈	1	2	yes	—
69	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
70	C₄×Dic₅	—	2	1	no	—
71	C₄×Dic₅	C₂×C₃×C₄	1	2	yes	—
72	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
73	C₂×#168	C₃×D₄	1	1	yes	characteristic
74	C₂×#168	C₃×D₄	1	1	yes	characteristic
75	C₂×#168	C₃×Q₈	1	1	yes	characteristic
76	C₂×#168	C₃×Q₈	1	1	yes	characteristic
77	C₂×#168	C₂³×C₃	1	1	yes	characteristic
78	#674	—	2	4	no	—
79	C₂×M₃₂	—	5	2	no	—
80	C₂×C₄×C₈	—	5	1	no	—
81	#364	—	5	4	no	—
82	C₂²×C₃×C₅	—	2	2	no	—
83	C₂²×C₃×C₅	C₄⋊C₈	1	1	yes	characteristic
84	C₂²×C₃×C₅	C₄⋊C₈	1	1	yes	characteristic
85	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
86	C₃×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
87	C₃×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
88	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
90	C₃×Dic₅	—	2	4	no	—
91	C₂²×C₃×C₄	—	10	1	no	—
92	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
93	C₂×C₃×C₈	—	5	1	no	—
94	C₂×C₃×C₈	—	5	1	no	—
95	C₂×C₃×C₈	—	5	1	no	—
96	C₂×C₃×C₈	—	5	1	no	—
97	C₂×C₃×C₈	—	5	1	no	—
98	C₂×C₃×C₈	—	5	1	no	—
99	C₂×C₃×C₈	#170	1	2	yes	—
100	C₂×C₃×C₈	#170	1	2	yes	—
101	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
102	C₃×C₄²	—	5	2	no	—
103	C₃×C₄²	—	10	1	no	—
104	C₃×C₁₆	—	10	4	no	—
105	C₂³×C₅	C₃×M₁₆	1	1	yes	characteristic
106	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
109	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
110	C₂×C₄×C₅	C₃×M₁₆	1	1	yes	characteristic
111	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
112	C₂×Dic₅	—	2	4	no	—
113	C₂×Dic₅	—	2	2	no	—
114	C₅×C₈	C₂×C₃×C₈	1	4	yes	—
115	#168	C₃×C₄⋊C₄	1	1	yes	characteristic
116	#168	C₃×C₄⋊C₄	1	1	yes	characteristic
117	#168	C₂×C₃×D₄	1	1	yes	characteristic
118	#168	C₂×C₃×Q₈	1	1	yes	characteristic
119	C₂²×C₈	—	5	1	no	—
120	C₂²×C₈	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
121	C₂×C₄²	—	5	1	no	—
122	M₃₂	—	5	4	no	—
123	M₃₂	—	10	2	no	—
124	C₂×C₁₆	—	10	2	no	—
125	C₄×C₈	—	5	2	no	—
126	C₄×C₈	—	5	2	no	—
127	C₂×C₃×C₅	—	2	1	no	—
128	C₂×C₃×C₅	—	2	1	no	—
129	C₂×C₃×C₅	#364	1	2	yes	complemented
130	C₂×C₃×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
131	C₂³×C₃	#700	1	1	yes	characteristic
132	C₂×C₃×C₄	—	10	4	no	—
133	C₂×C₃×C₄	—	10	2	no	—
134	C₂×C₃×C₄	#700	1	1	yes	characteristic
135	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
136	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
137	C₂×C₃×C₄	C₂×#170	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂×#170	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
140	C₃×C₈	—	5	1	no	—
141	C₃×C₈	—	5	1	no	—
142	C₃×C₈	—	5	1	no	—
143	C₃×C₈	—	5	1	no	—
144	C₃×C₈	C₂×#170	1	4	yes	—
145	C₂²×C₅	—	2	2	no	—
146	C₂²×C₅	C₃×C₄⋊C₈	1	1	yes	characteristic
147	C₂²×C₅	C₃×C₄⋊C₈	1	1	yes	characteristic
148	C₂²×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
149	C₄×C₅	C₃×C₄⋊C₈	1	1	yes	characteristic
150	C₄×C₅	C₃×C₄⋊C₈	1	1	yes	characteristic
151	C₄×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
152	C₄×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
153	Dic₅	—	2	4	no	—
154	C₂²×C₄	—	10	1	no	—
155	C₂²×C₄	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
156	C₂×C₈	—	5	1	no	—
157	C₂×C₈	—	5	1	no	—
158	C₂×C₈	—	5	1	no	—
159	C₂×C₈	—	5	1	no	—
160	C₂×C₈	—	5	1	no	—
161	C₂×C₈	—	5	1	no	—
162	C₂×C₈	C₃×#170	1	2	yes	—
163	C₂×C₈	C₃×#170	1	2	yes	—
164	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
165	C₄²	—	5	2	no	—
166	C₄²	—	10	1	no	—
167	C₁₆	—	10	4	no	—
168	C₃×C₅	C₂×#364	1	1	yes	characteristic, complemented, Hall
169	C₂²×C₃	—	2	2	no	—
170	C₂²×C₃	C₄⋊#170	1	1	yes	characteristic
171	C₂²×C₃	C₄⋊#170	1	1	yes	characteristic
172	C₂²×C₃	C₂×#700	1	1	yes	characteristic
173	C₃×C₄	—	10	4	no	—
174	C₃×C₄	C₄⋊#170	1	1	yes	characteristic
175	C₃×C₄	C₄⋊#170	1	1	yes	characteristic
176	C₃×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
177	C₃×C₄	C₂²×#170	1	1	yes	characteristic
178	C₂×C₅	—	2	1	no	—
179	C₂×C₅	—	2	1	no	—
180	C₂×C₅	C₃×#364	1	2	yes	complemented
181	C₂×C₅	C₂×C₃×C₄⋊C₈	1	1	yes	characteristic
182	C₂³	C₃×#700	1	1	yes	characteristic
183	C₂×C₄	—	10	4	no	—
184	C₂×C₄	—	10	2	no	—
185	C₂×C₄	C₃×#700	1	1	yes	characteristic
186	C₂×C₄	C₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
187	C₂×C₄	C₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
188	C₂×C₄	C₂×C₃×#170	1	1	yes	characteristic
189	C₂×C₄	C₂×C₃×#170	1	1	yes	characteristic
190	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
191	C₈	—	5	1	no	—
192	C₈	—	5	1	no	—
193	C₈	—	5	1	no	—
194	C₈	—	5	1	no	—
195	C₈	C₂×C₃×#170	1	4	yes	—
196	C₂×C₃	—	2	1	no	—
197	C₂×C₃	—	2	1	no	—
198	C₂×C₃	#64889	1	2	yes	direct product factor
199	C₂×C₃	C₂×C₄⋊#170	1	1	yes	characteristic
200	C₅	C₂×C₃×#364	1	1	yes	characteristic, complemented, Sylow
201	C₂²	—	2	2	no	—
202	C₂²	C₃×C₄⋊#170	1	1	yes	characteristic
203	C₂²	C₃×C₄⋊#170	1	1	yes	characteristic
204	C₂²	C₂×C₃×#700	1	1	yes	characteristic
205	C₄	—	10	4	no	—
206	C₄	C₃×C₄⋊#170	1	1	yes	characteristic
207	C₄	C₃×C₄⋊#170	1	1	yes	characteristic
208	C₄	C₂×C₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
209	C₄	C₂²×C₃×#170	1	1	yes	characteristic
210	C₃	C₂×#64889	1	1	yes	characteristic, direct product factor, Sylow
211	C₂	—	2	1	no	—
212	C₂	—	2	1	no	—
213	C₂	C₃×#64889	1	2	yes	direct product factor
214	C₂	C₂×C₃×C₄⋊#170	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)
commutator	C₄×C₅	C₂²×C₃×C₈
Frattini	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₂²×C₃×C₅×C₈	C₄
socle	C₂²×C₃×C₅	C₄⋊C₈
prime radical	C₂³×C₃×C₅	M₁₆
cosocle	C₂×#168	C₂³×C₃
hypercenter	C₂²×C₃×C₈	AGL(1,𝔽₅)
hypocenter	C₅	C₂×C₃×#364
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×C₈

Complements

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

N	H ≅ G/N	type
3. C₃×#64889	C₂	direct (N × H)
4. C₂×#64889	C₃	direct (N × H)
16. #64889	C₂×C₃	direct (N × H)
129. C₂×C₃×C₅	#364	semidirect
168. C₃×C₅	C₂×#364	semidirect
180. C₂×C₅	C₃×#364	semidirect
198. C₂×C₃	#64889	direct (N × H)
200. C₅	C₂×C₃×#364	semidirect
210. C₃	C₂×#64889	direct (N × H)
213. C₂	C₃×#64889	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×#3998	2
C₂×C₃×C₈×Dic₅	2
C₃×#64889	2
C₂×#64889	3
C₂×C₃×#364	5

Subgroups — C2×C3×#64889 #14314405

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#64889 #14314405