Subgroups — C₃×C₅×#2115 #14275149

Order 1920.

440 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#131	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×M₃₂	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#349	C₂	1	4	yes	maximal, complemented
4	C₅×#2115	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₃×C₅×#41	C₄	1	1	yes	characteristic
6	C₃×C₅×#131	—	2	2	no	—
7	C₃×C₅×#131	—	2	2	no	—
8	C₂×C₃×C₅×M₁₆	C₄	1	1	yes	characteristic
9	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
10	C₂²×C₃×C₅×C₈	C₄	1	2	yes	—
11	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×M₃₂	—	2	2	no	—
13	C₃×C₅×M₃₂	C₂²	1	4	yes	—
14	C₂×C₃×C₅×C₁₆	—	2	2	no	—
15	C₃×#2115	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂×C₅×#131	C₂×C₃	1	1	yes	characteristic
17	C₂×C₅×M₃₂	C₂×C₃	1	2	yes	complemented
18	C₅×#349	C₂×C₃	1	4	yes	complemented
19	C₃×C₅×#41	—	2	2	no	—
20	C₃×C₅×#41	—	2	2	no	—
21	C₂×C₃×C₅×Q₈	C₈	1	1	yes	characteristic
22	C₂×C₃×C₅×D₄	C₈	1	2	yes	—
23	C₂×C₃×C₅×D₄	C₈	1	1	yes	characteristic
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
25	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
26	C₃×C₅×M₁₆	—	2	2	no	—
27	C₃×C₅×M₁₆	—	2	4	no	—
28	C₂×C₃×C₅×C₈	—	2	2	no	—
29	C₂×C₃×C₅×C₈	—	2	2	no	—
30	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
33	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
34	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₃×C₅×C₁₆	—	2	4	no	—
36	C₂×C₃×#131	C₂×C₅	1	1	yes	characteristic
37	C₂×C₃×M₃₂	C₂×C₅	1	2	yes	complemented
38	C₃×#349	C₂×C₅	1	4	yes	complemented
39	C₂×C₅×#41	C₃×C₄	1	1	yes	characteristic
40	C₅×#131	—	2	2	no	—
41	C₅×#131	—	2	2	no	—
42	C₂×C₅×M₁₆	C₃×C₄	1	1	yes	characteristic
43	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
44	C₂²×C₅×C₈	C₃×C₄	1	2	yes	—
45	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
46	C₅×M₃₂	—	2	2	no	—
47	C₅×M₃₂	C₂²×C₃	1	4	yes	—
48	C₂×C₅×C₁₆	—	2	2	no	—
49	#2115	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
50	C₂³×C₃×C₅	C₂×C₈	1	2	yes	—
51	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
52	C₃×C₅×Q₈	—	2	2	no	—
53	C₃×C₅×D₄	—	2	4	no	—
54	C₃×C₅×D₄	—	2	2	no	—
55	C₂×C₃×C₄×C₅	—	2	2	no	—
56	C₂×C₃×C₄×C₅	—	2	2	no	—
57	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
58	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
59	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
61	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₃×C₅×C₈	—	2	2	no	—
63	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
64	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
65	C₃×C₅×C₈	C₂×D₄	1	2	yes	—
66	C₂×C₃×#41	C₄×C₅	1	1	yes	characteristic
67	C₃×#131	—	2	2	no	—
68	C₃×#131	—	2	2	no	—
69	C₂×C₃×M₁₆	C₄×C₅	1	1	yes	characteristic
70	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
71	C₂²×C₃×C₈	C₄×C₅	1	2	yes	—
72	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
73	C₃×M₃₂	—	2	2	no	—
74	C₃×M₃₂	C₂²×C₅	1	4	yes	—
75	C₂×C₃×C₁₆	—	2	2	no	—
76	C₅×#41	—	2	2	no	—
77	C₅×#41	—	2	2	no	—
78	C₂×C₅×Q₈	C₃×C₈	1	1	yes	characteristic
79	C₂×C₅×D₄	C₃×C₈	1	2	yes	—
80	C₂×C₅×D₄	C₃×C₈	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
82	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
83	C₅×M₁₆	—	2	2	no	—
84	C₅×M₁₆	—	2	4	no	—
85	C₂×C₅×C₈	—	2	2	no	—
86	C₂×C₅×C₈	—	2	2	no	—
87	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
88	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
89	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
90	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
91	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
92	C₅×C₁₆	—	2	4	no	—
93	C₂×#131	C₂×C₃×C₅	1	1	yes	characteristic
94	C₂×M₃₂	C₂×C₃×C₅	1	2	yes	complemented
95	#349	C₂×C₃×C₅	1	4	yes	complemented
96	C₂²×C₃×C₅	—	2	2	no	—
97	C₂²×C₃×C₅	—	4	1	no	—
98	C₂²×C₃×C₅	—	4	2	no	—
99	C₂²×C₃×C₅	C₂²⋊C₈	1	2	yes	—
100	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
101	C₃×C₄×C₅	—	2	2	no	—
102	C₃×C₄×C₅	C₂²⋊C₈	1	2	yes	—
103	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
105	C₃×#41	—	2	2	no	—
106	C₃×#41	—	2	2	no	—
107	C₂×C₃×Q₈	C₅×C₈	1	1	yes	characteristic
108	C₂×C₃×D₄	C₅×C₈	1	2	yes	—
109	C₂×C₃×D₄	C₅×C₈	1	1	yes	characteristic
110	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
111	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
112	C₃×M₁₆	—	2	2	no	—
113	C₃×M₁₆	—	2	4	no	—
114	C₂×C₃×C₈	—	2	2	no	—
115	C₂×C₃×C₈	—	2	2	no	—
116	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
117	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
118	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
119	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
120	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
121	C₃×C₁₆	—	2	4	no	—
122	C₂³×C₅	C₂×C₃×C₈	1	2	yes	—
123	C₂³×C₅	C₂×C₃×C₈	1	1	yes	characteristic
124	C₅×Q₈	—	2	2	no	—
125	C₅×D₄	—	2	4	no	—
126	C₅×D₄	—	2	2	no	—
127	C₂×C₄×C₅	—	2	2	no	—
128	C₂×C₄×C₅	—	2	2	no	—
129	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
130	C₂×C₄×C₅	C₂×C₃×C₈	1	2	yes	—
131	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
132	C₂×C₄×C₅	C₃×M₁₆	1	2	yes	—
133	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
134	C₅×C₈	—	2	2	no	—
135	C₅×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
136	C₅×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
137	C₅×C₈	C₂×C₃×D₄	1	2	yes	—
138	C₂×#41	C₃×C₄×C₅	1	1	yes	characteristic
139	#131	—	2	2	no	—
140	#131	—	2	2	no	—
141	C₂×M₁₆	C₃×C₄×C₅	1	1	yes	characteristic
142	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
143	C₂²×C₈	C₃×C₄×C₅	1	2	yes	—
144	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
145	M₃₂	—	2	2	no	—
146	M₃₂	C₂²×C₃×C₅	1	4	yes	—
147	C₂×C₁₆	—	2	2	no	—
148	C₂×C₃×C₅	—	2	1	no	—
149	C₂×C₃×C₅	—	2	2	no	—
150	C₂×C₃×C₅	—	4	2	no	—
151	C₂×C₃×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
152	C₂³×C₃	C₂×C₅×C₈	1	2	yes	—
153	C₂³×C₃	C₂×C₅×C₈	1	1	yes	characteristic
154	C₃×Q₈	—	2	2	no	—
155	C₃×D₄	—	2	4	no	—
156	C₃×D₄	—	2	2	no	—
157	C₂×C₃×C₄	—	2	2	no	—
158	C₂×C₃×C₄	—	2	2	no	—
159	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
160	C₂×C₃×C₄	C₂×C₅×C₈	1	2	yes	—
161	C₂×C₃×C₄	C₂×C₅×C₈	1	1	yes	characteristic
162	C₂×C₃×C₄	C₅×M₁₆	1	2	yes	—
163	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
164	C₃×C₈	—	2	2	no	—
165	C₃×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
166	C₃×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
167	C₃×C₈	C₂×C₅×D₄	1	2	yes	—
168	C₂²×C₅	—	2	2	no	—
169	C₂²×C₅	—	4	1	no	—
170	C₂²×C₅	—	4	2	no	—
171	C₂²×C₅	C₃×C₂²⋊C₈	1	2	yes	—
172	C₂²×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
173	C₄×C₅	—	2	2	no	—
174	C₄×C₅	C₃×C₂²⋊C₈	1	2	yes	—
175	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
176	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
177	#41	—	2	2	no	—
178	#41	—	2	2	no	—
179	C₂×Q₈	C₃×C₅×C₈	1	1	yes	characteristic
180	C₂×D₄	C₃×C₅×C₈	1	2	yes	—
181	C₂×D₄	C₃×C₅×C₈	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₃×C₄×C₅	1	2	yes	—
183	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
184	M₁₆	—	2	2	no	—
185	M₁₆	—	2	4	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
189	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
190	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
191	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
192	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
193	C₁₆	—	2	4	no	—
194	C₃×C₅	#2115	1	1	yes	characteristic, direct product factor, Hall
195	C₂²×C₃	—	2	2	no	—
196	C₂²×C₃	—	4	1	no	—
197	C₂²×C₃	—	4	2	no	—
198	C₂²×C₃	C₅×C₂²⋊C₈	1	2	yes	—
199	C₂²×C₃	C₂²×C₅×C₈	1	1	yes	characteristic
200	C₃×C₄	—	2	2	no	—
201	C₃×C₄	C₅×C₂²⋊C₈	1	2	yes	—
202	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
203	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
204	C₂×C₅	—	2	1	no	—
205	C₂×C₅	—	2	2	no	—
206	C₂×C₅	—	4	2	no	—
207	C₂×C₅	C₂×C₃×C₂²⋊C₈	1	1	yes	characteristic
208	C₂³	C₂×C₃×C₅×C₈	1	2	yes	—
209	C₂³	C₂×C₃×C₅×C₈	1	1	yes	characteristic
210	Q₈	—	2	2	no	—
211	D₄	—	2	4	no	—
212	D₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
216	C₂×C₄	C₂×C₃×C₅×C₈	1	2	yes	—
217	C₂×C₄	C₂×C₃×C₅×C₈	1	1	yes	characteristic
218	C₂×C₄	C₃×C₅×M₁₆	1	2	yes	—
219	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
220	C₈	—	2	2	no	—
221	C₈	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
222	C₈	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
223	C₈	C₂×C₃×C₅×D₄	1	2	yes	—
224	C₂×C₃	—	2	1	no	—
225	C₂×C₃	—	2	2	no	—
226	C₂×C₃	—	4	2	no	—
227	C₂×C₃	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
228	C₅	C₃×#2115	1	1	yes	characteristic, direct product factor, Sylow
229	C₂²	—	2	2	no	—
230	C₂²	—	4	1	no	—
231	C₂²	—	4	2	no	—
232	C₂²	C₃×C₅×C₂²⋊C₈	1	2	yes	—
233	C₂²	C₂²×C₃×C₅×C₈	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	C₃×C₅×C₂²⋊C₈	1	2	yes	—
236	C₄	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
237	C₄	C₂×C₃×C₅×M₁₆	1	1	yes	characteristic
238	C₃	C₅×#2115	1	1	yes	characteristic, direct product factor, Sylow
239	C₂	—	2	1	no	—
240	C₂	—	2	2	no	—
241	C₂	—	4	2	no	—
242	C₂	C₂×C₃×C₅×C₂²⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₅×C₈	C₂²⋊C₄
commutator	C₂²	C₂²×C₃×C₅×C₈
Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
socle	C₂×C₃×C₅	C₂×C₂²⋊C₈
prime radical	C₂×C₃×C₅×#41	C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₅×C₈ ◁ C₂²×C₃×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×M₃₂	C₂	semidirect
3. C₃×C₅×#349	C₂	semidirect
4. C₅×#2115	C₃	direct (N × H)
15. C₃×#2115	C₅	direct (N × H)
17. C₂×C₅×M₃₂	C₂×C₃	semidirect
18. C₅×#349	C₂×C₃	semidirect
37. C₂×C₃×M₃₂	C₂×C₅	semidirect
38. C₃×#349	C₂×C₅	semidirect
49. #2115	C₃×C₅	direct (N × H)
94. C₂×M₃₂	C₂×C₃×C₅	semidirect
95. #349	C₂×C₃×C₅	semidirect
194. C₃×C₅	#2115	direct (N × H)
228. C₅	C₃×#2115	direct (N × H)
238. C₃	C₅×#2115	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×C₅×#131	2
C₂×C₃×C₅×M₃₂	2
C₃×C₅×#349	2
C₅×#2115	3
C₃×#2115	5

Subgroups — C3×C5×#2115 #14275149

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2115 #14275149