Subgroups — C₃×D₅×#368 #14314662

Order 1920.

288 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×D₅×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₃×D₅×#108	C₂	1	2	yes	maximal
3	C₃×C₅×#368	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#64726	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64713	C₂	1	2	yes	maximal, complemented
6	D₅×#368	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×D₅×M₁₆	—	2	2	no	—
8	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
10	C₃×C₅×#108	C₂²	1	2	yes	—
11	C₃×#3954	—	2	2	no	—
12	C₃×#3951	C₂²	1	2	yes	—
13	C₂×C₃×#672	C₂²	1	1	yes	characteristic
14	C₃×D₅×C₁₆	C₄	1	2	yes	—
15	C₃×D₅×C₁₆	C₄	1	2	yes	—
16	C₂×C₃×#368	—	5	1	no	maximal, Hall
17	C₂×D₅×C₁₆	C₂×C₃	1	1	yes	characteristic
18	D₅×#108	C₂×C₃	1	2	yes	—
19	C₅×#368	C₂×C₃	1	1	yes	characteristic, complemented
20	#64726	C₂×C₃	1	1	yes	characteristic, complemented
21	#64713	C₂×C₃	1	2	yes	complemented
22	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
23	C₃×C₅×M₁₆	—	2	2	no	—
24	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
25	C₃×#681	—	2	2	no	—
26	C₂×C₃×#168	D₄	1	1	yes	characteristic
27	C₃×#676	—	4	2	no	—
28	C₃×C₈×D₅	—	4	2	no	—
29	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
30	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
31	C₃×C₈×D₅	Q₈	1	1	yes	characteristic
32	C₃×C₈×D₅	Q₈	1	1	yes	characteristic
33	C₃×C₅×C₁₆	C₂×C₄	1	2	yes	—
34	C₃×#672	C₂×C₄	1	2	yes	—
35	C₂²×C₃×C₁₆	—	5	1	no	—
36	C₂×C₃×#108	—	5	2	no	—
37	C₃×#368	—	5	1	no	—
38	C₃×#368	—	5	2	no	—
39	C₃×#368	D₅	1	1	yes	characteristic, direct product factor
40	D₅×M₁₆	—	2	2	no	—
41	C₂×C₈×D₅	C₂²×C₃	1	1	yes	characteristic
42	C₂×C₅×C₁₆	C₂²×C₃	1	1	yes	characteristic
43	C₅×#108	C₂²×C₃	1	2	yes	—
44	#3954	—	2	2	no	—
45	#3951	C₂²×C₃	1	2	yes	—
46	C₂×#672	C₂²×C₃	1	1	yes	characteristic
47	D₅×C₁₆	C₃×C₄	1	2	yes	—
48	D₅×C₁₆	C₃×C₄	1	2	yes	—
49	C₂×#368	—	5	1	no	Sylow
50	C₂²×C₃×D₅	Q₁₆	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
52	C₂×C₃×Dic₅	Q₁₆	1	1	yes	characteristic
53	C₃×C₄×D₅	C₄⋊C₄	1	1	yes	characteristic
54	C₃×C₄×D₅	C₄⋊C₄	1	1	yes	characteristic
55	C₃×C₄×D₅	D₈	1	1	yes	characteristic
56	C₃×C₄×D₅	D₈	1	1	yes	characteristic
57	C₃×C₅×C₈	—	4	2	no	—
58	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
59	C₃×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
60	C₃×#168	—	4	2	no	—
61	C₃×#168	C₄⋊C₄	1	1	yes	characteristic
62	C₃×#168	C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₃×M₁₆	—	10	2	no	—
64	C₂²×C₃×C₈	—	5	1	no	—
65	C₂×C₃×C₁₆	—	5	1	no	—
66	C₂×C₃×C₁₆	—	5	2	no	—
67	C₂×C₃×C₁₆	—	5	2	no	—
68	C₂×C₃×C₁₆	C₂×D₅	1	1	yes	characteristic
69	C₃×#108	—	5	2	no	—
70	C₃×#108	—	10	2	no	—
71	C₃×#108	C₂×D₅	1	2	yes	—
72	C₂×C₄×D₅	C₃×D₄	1	1	yes	characteristic
73	C₅×M₁₆	—	2	2	no	—
74	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
75	#681	—	2	2	no	—
76	C₂×#168	C₃×D₄	1	1	yes	characteristic
77	#676	—	4	2	no	—
78	C₈×D₅	—	4	2	no	—
79	C₈×D₅	C₂×C₃×C₄	1	1	yes	characteristic
80	C₈×D₅	C₂×C₃×C₄	1	1	yes	characteristic
81	C₈×D₅	C₃×Q₈	1	1	yes	characteristic
82	C₈×D₅	C₃×Q₈	1	1	yes	characteristic
83	C₅×C₁₆	C₂×C₃×C₄	1	2	yes	—
84	#672	C₂×C₃×C₄	1	2	yes	—
85	C₂²×C₁₆	—	5	1	no	—
86	C₂×#108	—	5	2	no	—
87	#368	—	5	1	no	—
88	#368	—	5	2	no	—
89	#368	C₃×D₅	1	1	yes	characteristic, direct product factor
90	C₂²×C₃×C₅	C₂×Q₁₆	1	1	yes	characteristic
91	C₂×C₃×D₅	—	2	2	no	—
92	C₂×C₃×D₅	#107	1	1	yes	characteristic
93	C₂×C₃×D₅	#107	1	1	yes	characteristic
94	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
96	C₃×Dic₅	#107	1	1	yes	characteristic
97	C₃×Dic₅	#107	1	1	yes	characteristic
98	C₂²×C₃×C₄	—	5	1	no	—
99	C₃×M₁₆	—	2	2	no	—
100	C₃×M₁₆	—	10	2	no	—
101	C₃×M₁₆	—	20	2	no	—
102	C₂×C₃×C₈	—	5	1	no	—
103	C₂×C₃×C₈	—	5	1	no	—
104	C₂×C₃×C₈	—	5	1	no	—
105	C₂×C₃×C₈	—	5	1	no	—
106	C₂×C₃×C₈	—	5	1	no	—
107	C₂×C₃×C₈	—	20	2	no	—
108	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
109	C₃×C₁₆	—	5	2	no	—
110	C₃×C₁₆	C₄×D₅	1	2	yes	—
111	C₂²×D₅	C₃×Q₁₆	1	1	yes	characteristic
112	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
113	C₂×Dic₅	C₃×Q₁₆	1	1	yes	characteristic
114	C₄×D₅	C₃×C₄⋊C₄	1	1	yes	characteristic
115	C₄×D₅	C₃×C₄⋊C₄	1	1	yes	characteristic
116	C₄×D₅	C₃×D₈	1	1	yes	characteristic
117	C₄×D₅	C₃×D₈	1	1	yes	characteristic
118	C₅×C₈	—	4	2	no	—
119	C₅×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
120	C₅×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
121	#168	—	4	2	no	—
122	#168	C₃×C₄⋊C₄	1	1	yes	characteristic
123	#168	C₃×C₄⋊C₄	1	1	yes	characteristic
124	C₂×M₁₆	—	10	2	no	—
125	C₂²×C₈	—	5	1	no	—
126	C₂×C₁₆	—	5	1	no	—
127	C₂×C₁₆	—	5	2	no	—
128	C₂×C₁₆	—	5	2	no	—
129	C₂×C₁₆	C₂×C₃×D₅	1	1	yes	characteristic
130	#108	—	5	2	no	—
131	#108	—	10	2	no	—
132	#108	C₂×C₃×D₅	1	2	yes	—
133	C₂×C₃×C₅	—	2	1	no	—
134	C₂×C₃×C₅	C₂×#107	1	1	yes	characteristic
135	C₃×D₅	—	2	1	no	—
136	C₃×D₅	#368	1	2	yes	direct product factor
137	C₂³×C₃	—	5	1	no	—
138	C₂×C₃×C₄	—	5	1	no	—
139	C₂×C₃×C₄	—	5	1	no	—
140	C₂×C₃×C₄	—	5	1	no	—
141	C₂×C₃×C₄	—	5	1	no	—
142	C₂×C₃×C₄	—	5	1	no	—
143	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
144	C₃×C₈	—	4	2	no	—
145	C₃×C₈	—	5	1	no	—
146	C₃×C₈	—	5	1	no	—
147	C₃×C₈	—	20	2	no	—
148	C₃×C₈	C₂×C₄×D₅	1	1	yes	characteristic
149	C₃×C₈	Q₈×D₅	1	1	yes	characteristic
150	C₂²×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
151	C₂×D₅	—	2	2	no	—
152	C₂×D₅	C₃×#107	1	1	yes	characteristic
153	C₂×D₅	C₃×#107	1	1	yes	characteristic
154	C₄×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
155	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
156	Dic₅	C₃×#107	1	1	yes	characteristic
157	Dic₅	C₃×#107	1	1	yes	characteristic
158	C₂²×C₄	—	5	1	no	—
159	M₁₆	—	2	2	no	—
160	M₁₆	—	10	2	no	—
161	M₁₆	—	20	2	no	—
162	C₂×C₈	—	5	1	no	—
163	C₂×C₈	—	5	1	no	—
164	C₂×C₈	—	5	1	no	—
165	C₂×C₈	—	5	1	no	—
166	C₂×C₈	—	5	1	no	—
167	C₂×C₈	—	20	2	no	—
168	C₂×C₈	C₂²×C₃×D₅	1	1	yes	characteristic
169	C₁₆	—	5	2	no	—
170	C₁₆	C₃×C₄×D₅	1	2	yes	—
171	C₃×C₅	C₂×#368	1	1	yes	characteristic, complemented, Hall
172	C₂²×C₃	—	5	1	no	—
173	C₂²×C₃	—	5	1	no	—
174	C₂²×C₃	—	10	2	no	—
175	C₂²×C₃	D₅×Q₁₆	1	1	yes	characteristic
176	C₃×C₄	—	5	1	no	—
177	C₃×C₄	—	5	1	no	—
178	C₃×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
179	C₃×C₄	D₅×D₈	1	1	yes	characteristic
180	C₂×C₅	—	2	1	no	—
181	C₂×C₅	C₂×C₃×#107	1	1	yes	characteristic
182	D₅	—	2	1	no	—
183	D₅	C₃×#368	1	2	yes	direct product factor
184	C₂³	—	5	1	no	—
185	C₂×C₄	—	5	1	no	—
186	C₂×C₄	—	5	1	no	—
187	C₂×C₄	—	5	1	no	—
188	C₂×C₄	—	5	1	no	—
189	C₂×C₄	—	5	1	no	—
190	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
191	C₈	—	4	2	no	—
192	C₈	—	5	1	no	—
193	C₈	—	5	1	no	—
194	C₈	—	20	2	no	—
195	C₈	C₂×C₃×C₄×D₅	1	1	yes	characteristic
196	C₈	C₃×Q₈×D₅	1	1	yes	characteristic
197	C₂×C₃	—	2	1	no	—
198	C₂×C₃	—	5	2	no	—
199	C₂×C₃	—	10	1	no	—
200	C₂×C₃	D₅×#107	1	1	yes	characteristic
201	C₅	C₂×C₃×#368	1	1	yes	characteristic, complemented, Sylow
202	C₂²	—	5	1	no	—
203	C₂²	—	5	1	no	—
204	C₂²	—	10	2	no	—
205	C₂²	C₃×D₅×Q₁₆	1	1	yes	characteristic
206	C₄	—	5	1	no	—
207	C₄	—	5	1	no	—
208	C₄	C₃×D₅×C₄⋊C₄	1	1	yes	characteristic
209	C₄	C₃×D₅×D₈	1	1	yes	characteristic
210	C₃	D₅×#368	1	1	yes	characteristic, direct product factor, Sylow
211	C₂	—	2	1	no	—
212	C₂	—	5	2	no	—
213	C₂	—	10	1	no	—
214	C₂	C₃×D₅×#107	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	D₅×D₈
commutator	C₅×C₈	C₂²×C₃×C₄
Frattini	C₂×C₈	C₂²×C₃×D₅
Fitting	C₃×C₅×#368	C₂
socle	C₂×C₃×C₅	C₂×#107
prime radical	C₂²×C₃×D₅	Q₁₆
cosocle	C₂×C₅×C₈	C₂³×C₃
hypercenter	C₃×#368	D₅
hypocenter	C₅	C₂×C₃×#368
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈ ◁ C₃×#368

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#368	C₂	semidirect
4. C₃×#64726	C₂	semidirect
5. C₃×#64713	C₂	semidirect
6. D₅×#368	C₃	direct (N × H)
19. C₅×#368	C₂×C₃	semidirect
20. #64726	C₂×C₃	semidirect
21. #64713	C₂×C₃	semidirect
39. C₃×#368	D₅	direct (N × H)
89. #368	C₃×D₅	direct (N × H)
136. C₃×D₅	#368	direct (N × H)
171. C₃×C₅	C₂×#368	semidirect
183. D₅	C₃×#368	direct (N × H)
201. C₅	C₂×C₃×#368	semidirect
210. C₃	D₅×#368	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×D₅×C₁₆	2
C₃×D₅×#108	2
C₃×C₅×#368	2
C₃×#64726	2
C₃×#64713	2
D₅×#368	3
C₂×C₃×#368	5

Subgroups — C3×D5×#368 #14314662

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₅×#368 #14314662