Subgroups — C₃×C₅×#2214 #14275248

Order 1920.

300 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	C₃×C₅×#510	C₂	1	2	yes	maximal
2	C₃×C₅×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#443	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#364	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#362	C₂	1	2	yes	maximal, complemented
6	C₅×#2214	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×C₅×#135	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₅×Q₁₆	—	2	1	no	—
9	C₂×C₃×C₅×Q₁₆	C₂²	1	2	yes	—
10	C₃×C₅×#131	—	2	1	no	—
11	C₃×C₅×C₄⋊Q₈	—	2	1	no	—
12	C₃×C₅×Q₃₂	—	2	2	no	—
13	C₃×C₅×SD₃₂	—	2	2	no	—
14	C₃×C₅×M₃₂	C₂²	1	2	yes	—
15	C₃×C₅×#108	C₂²	1	1	yes	characteristic
16	C₃×C₅×#104	—	2	1	no	—
17	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₃×#2214	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₅×#510	C₂×C₃	1	2	yes	—
20	C₅×#494	C₂×C₃	1	1	yes	characteristic, complemented
21	C₅×#443	C₂×C₃	1	1	yes	characteristic
22	C₅×#364	C₂×C₃	1	1	yes	characteristic, complemented
23	C₅×#362	C₂×C₃	1	2	yes	complemented
24	C₃×C₅×#41	—	2	1	no	—
25	C₂×C₃×C₅×Q₈	—	2	2	no	—
26	C₃×C₅×Q₁₆	—	2	2	no	—
27	C₃×C₅×Q₁₆	—	2	2	no	—
28	C₃×C₅×Q₁₆	—	4	1	no	—
29	C₃×C₅×Q₁₆	—	4	1	no	—
30	C₃×C₅×Q₁₆	D₄	1	1	yes	characteristic
31	C₃×C₅×SD₁₆	—	2	1	no	—
32	C₃×C₅×D₈	D₄	1	1	yes	characteristic
33	C₃×C₅×M₁₆	—	2	1	no	—
34	C₃×C₅×M₁₆	—	4	1	no	—
35	C₂×C₃×C₅×C₈	—	4	1	no	—
36	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
37	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
38	C₃×C₅×C₄⋊C₄	—	4	2	no	—
39	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
40	C₃×C₅×C₁₆	—	2	2	no	—
41	C₃×#510	C₂×C₅	1	2	yes	—
42	C₃×#494	C₂×C₅	1	1	yes	characteristic, complemented
43	C₃×#443	C₂×C₅	1	1	yes	characteristic
44	C₃×#364	C₂×C₅	1	1	yes	characteristic, complemented
45	C₃×#362	C₂×C₅	1	2	yes	complemented
46	C₅×#135	C₂²×C₃	1	1	yes	characteristic
47	C₂×C₅×Q₁₆	—	2	1	no	—
48	C₂×C₅×Q₁₆	C₂²×C₃	1	2	yes	—
49	C₅×#131	—	2	1	no	—
50	C₅×C₄⋊Q₈	—	2	1	no	—
51	C₅×Q₃₂	—	2	2	no	—
52	C₅×SD₃₂	—	2	2	no	—
53	C₅×M₃₂	C₂²×C₃	1	2	yes	—
54	C₅×#108	C₂²×C₃	1	1	yes	characteristic
55	C₅×#104	—	2	1	no	—
56	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
57	#2214	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
58	C₃×C₅×Q₈	—	2	1	no	—
59	C₃×C₅×Q₈	—	4	2	no	—
60	C₃×C₅×Q₈	—	4	2	no	—
61	C₃×C₅×D₄	—	2	1	no	—
62	C₃×C₅×D₄	—	4	1	no	—
63	C₂×C₃×C₄×C₅	—	2	1	no	—
64	C₂×C₃×C₄×C₅	—	4	1	no	—
65	C₂×C₃×C₄×C₅	—	4	2	no	—
66	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
67	C₃×C₅×C₈	—	4	1	no	—
68	C₃×C₅×C₈	D₈	1	1	yes	characteristic
69	C₃×C₅×C₈	D₈	1	1	yes	characteristic
70	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
71	C₃×C₅×C₈	#41	1	1	yes	characteristic
72	C₃×#135	C₂²×C₅	1	1	yes	characteristic
73	C₂×C₃×Q₁₆	—	2	1	no	—
74	C₂×C₃×Q₁₆	C₂²×C₅	1	2	yes	—
75	C₃×#131	—	2	1	no	—
76	C₃×C₄⋊Q₈	—	2	1	no	—
77	C₃×Q₃₂	—	2	2	no	—
78	C₃×SD₃₂	—	2	2	no	—
79	C₃×M₃₂	C₂²×C₅	1	2	yes	—
80	C₃×#108	C₂²×C₅	1	1	yes	characteristic
81	C₃×#104	—	2	1	no	—
82	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
83	C₅×#41	—	2	1	no	—
84	C₂×C₅×Q₈	—	2	2	no	—
85	C₅×Q₁₆	—	2	2	no	—
86	C₅×Q₁₆	—	2	2	no	—
87	C₅×Q₁₆	—	4	1	no	—
88	C₅×Q₁₆	—	4	1	no	—
89	C₅×Q₁₆	C₃×D₄	1	1	yes	characteristic
90	C₅×SD₁₆	—	2	1	no	—
91	C₅×D₈	C₃×D₄	1	1	yes	characteristic
92	C₅×M₁₆	—	2	1	no	—
93	C₅×M₁₆	—	4	1	no	—
94	C₂×C₅×C₈	—	4	1	no	—
95	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
96	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
97	C₅×C₄⋊C₄	—	4	2	no	—
98	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
99	C₅×C₁₆	—	2	2	no	—
100	#510	C₂×C₃×C₅	1	2	yes	—
101	#494	C₂×C₃×C₅	1	1	yes	characteristic, complemented
102	#443	C₂×C₃×C₅	1	1	yes	characteristic
103	#364	C₂×C₃×C₅	1	1	yes	characteristic, complemented
104	#362	C₂×C₃×C₅	1	2	yes	complemented
105	C₂²×C₃×C₅	—	4	1	no	—
106	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
107	C₃×C₄×C₅	—	2	2	no	—
108	C₃×C₄×C₅	—	4	1	no	—
109	C₃×C₄×C₅	—	8	2	no	—
110	C₃×C₄×C₅	#121	1	1	yes	characteristic
111	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
112	C₃×#41	—	2	1	no	—
113	C₂×C₃×Q₈	—	2	2	no	—
114	C₃×Q₁₆	—	2	2	no	—
115	C₃×Q₁₆	—	2	2	no	—
116	C₃×Q₁₆	—	4	1	no	—
117	C₃×Q₁₆	—	4	1	no	—
118	C₃×Q₁₆	C₅×D₄	1	1	yes	characteristic
119	C₃×SD₁₆	—	2	1	no	—
120	C₃×D₈	C₅×D₄	1	1	yes	characteristic
121	C₃×M₁₆	—	2	1	no	—
122	C₃×M₁₆	—	4	1	no	—
123	C₂×C₃×C₈	—	4	1	no	—
124	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
125	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
126	C₃×C₄⋊C₄	—	4	2	no	—
127	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
128	C₃×C₁₆	—	2	2	no	—
129	C₅×Q₈	—	2	1	no	—
130	C₅×Q₈	—	4	2	no	—
131	C₅×Q₈	—	4	2	no	—
132	C₅×D₄	—	2	1	no	—
133	C₅×D₄	—	4	1	no	—
134	C₂×C₄×C₅	—	2	1	no	—
135	C₂×C₄×C₅	—	4	1	no	—
136	C₂×C₄×C₅	—	4	2	no	—
137	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
138	C₅×C₈	—	4	1	no	—
139	C₅×C₈	C₃×D₈	1	1	yes	characteristic
140	C₅×C₈	C₃×D₈	1	1	yes	characteristic
141	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
142	C₅×C₈	C₃×#41	1	1	yes	characteristic
143	#135	C₂²×C₃×C₅	1	1	yes	characteristic
144	C₂×Q₁₆	—	2	1	no	—
145	C₂×Q₁₆	C₂²×C₃×C₅	1	2	yes	—
146	#131	—	2	1	no	—
147	C₄⋊Q₈	—	2	1	no	—
148	Q₃₂	—	2	2	no	—
149	SD₃₂	—	2	2	no	—
150	M₃₂	C₂²×C₃×C₅	1	2	yes	—
151	#108	C₂²×C₃×C₅	1	1	yes	characteristic
152	#104	—	2	1	no	—
153	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
154	C₂×C₃×C₅	—	2	1	no	—
155	C₂×C₃×C₅	—	8	1	no	—
156	C₂×C₃×C₅	#459	1	1	yes	characteristic
157	C₃×Q₈	—	2	1	no	—
158	C₃×Q₈	—	4	2	no	—
159	C₃×Q₈	—	4	2	no	—
160	C₃×D₄	—	2	1	no	—
161	C₃×D₄	—	4	1	no	—
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	4	1	no	—
164	C₂×C₃×C₄	—	4	2	no	—
165	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
166	C₃×C₈	—	4	1	no	—
167	C₃×C₈	C₅×D₈	1	1	yes	characteristic
168	C₃×C₈	C₅×D₈	1	1	yes	characteristic
169	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
170	C₃×C₈	C₅×#41	1	1	yes	characteristic
171	C₂²×C₅	—	4	1	no	—
172	C₂²×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
173	C₄×C₅	—	2	2	no	—
174	C₄×C₅	—	4	1	no	—
175	C₄×C₅	—	8	2	no	—
176	C₄×C₅	C₃×#121	1	1	yes	characteristic
177	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
178	#41	—	2	1	no	—
179	C₂×Q₈	—	2	2	no	—
180	Q₁₆	—	2	2	no	—
181	Q₁₆	—	2	2	no	—
182	Q₁₆	—	4	1	no	—
183	Q₁₆	—	4	1	no	—
184	Q₁₆	C₃×C₅×D₄	1	1	yes	characteristic
185	SD₁₆	—	2	1	no	—
186	D₈	C₃×C₅×D₄	1	1	yes	characteristic
187	M₁₆	—	2	1	no	—
188	M₁₆	—	4	1	no	—
189	C₂×C₈	—	4	1	no	—
190	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
191	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
192	C₄⋊C₄	—	4	2	no	—
193	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
194	C₁₆	—	2	2	no	—
195	C₃×C₅	#2214	1	1	yes	characteristic, direct product factor, Hall
196	C₂²×C₃	—	4	1	no	—
197	C₂²×C₃	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
198	C₃×C₄	—	2	2	no	—
199	C₃×C₄	—	4	1	no	—
200	C₃×C₄	—	8	2	no	—
201	C₃×C₄	C₅×#121	1	1	yes	characteristic
202	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
203	C₂×C₅	—	2	1	no	—
204	C₂×C₅	—	8	1	no	—
205	C₂×C₅	C₃×#459	1	1	yes	characteristic
206	Q₈	—	2	1	no	—
207	Q₈	—	4	2	no	—
208	Q₈	—	4	2	no	—
209	D₄	—	2	1	no	—
210	D₄	—	4	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
215	C₈	—	4	1	no	—
216	C₈	C₃×C₅×D₈	1	1	yes	characteristic
217	C₈	C₃×C₅×D₈	1	1	yes	characteristic
218	C₈	C₂×C₃×C₅×D₄	1	1	yes	characteristic
219	C₈	C₃×C₅×#41	1	1	yes	characteristic
220	C₂×C₃	—	2	1	no	—
221	C₂×C₃	—	8	1	no	—
222	C₂×C₃	C₅×#459	1	1	yes	characteristic
223	C₅	C₃×#2214	1	1	yes	characteristic, direct product factor, Sylow
224	C₂²	—	4	1	no	—
225	C₂²	C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
226	C₄	—	2	2	no	—
227	C₄	—	4	1	no	—
228	C₄	—	8	2	no	—
229	C₄	C₃×C₅×#121	1	1	yes	characteristic
230	C₄	C₂×C₃×C₅×D₈	1	1	yes	characteristic
231	C₃	C₅×#2214	1	1	yes	characteristic, direct product factor, Sylow
232	C₂	—	2	1	no	—
233	C₂	—	8	1	no	—
234	C₂	C₃×C₅×#459	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃×C₅	#459
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
prime radical	C₃×C₅×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₅ ◁ C₂×C₃×C₄×C₅ ◁ C₃×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#494	C₂	semidirect
4. C₃×C₅×#364	C₂	semidirect
5. C₃×C₅×#362	C₂	semidirect
6. C₅×#2214	C₃	direct (N × H)
18. C₃×#2214	C₅	direct (N × H)
20. C₅×#494	C₂×C₃	semidirect
22. C₅×#364	C₂×C₃	semidirect
23. C₅×#362	C₂×C₃	semidirect
42. C₃×#494	C₂×C₅	semidirect
44. C₃×#364	C₂×C₅	semidirect
45. C₃×#362	C₂×C₅	semidirect
57. #2214	C₃×C₅	direct (N × H)
101. #494	C₂×C₃×C₅	semidirect
103. #364	C₂×C₃×C₅	semidirect
104. #362	C₂×C₃×C₅	semidirect
195. C₃×C₅	#2214	direct (N × H)
223. C₅	C₃×#2214	direct (N × H)
231. C₃	C₅×#2214	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅×#510	2
C₃×C₅×#494	2
C₃×C₅×#443	2
C₃×C₅×#364	2
C₃×C₅×#362	2
C₅×#2214	3
C₃×#2214	5

Subgroups — C3×C5×#2214 #14275248

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2214 #14275248