Subgroups — C₂×C₃×C₅×#365 #14275188

Order 1920.

304 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×#108	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₅×#106	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#365	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₅×#365	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×C₅×M₁₆	—	2	1	no	—
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
9	C₃×C₅×M₃₂	C₄	1	2	yes	complemented
10	C₃×C₅×M₃₂	C₂²	1	2	yes	—
11	C₂×C₃×C₅×C₁₆	C₄	1	2	yes	complemented
12	C₃×C₅×#108	—	2	1	no	—
13	C₃×C₅×#108	C₂²	1	2	yes	—
14	C₃×C₅×#106	—	2	1	no	—
15	C₃×C₅×#106	C₂²	1	2	yes	—
16	C₂×C₃×#365	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×C₅×M₃₂	C₂×C₃	1	1	yes	characteristic
18	C₂×C₅×#108	C₂×C₃	1	1	yes	characteristic
19	C₂×C₅×#106	C₂×C₃	1	1	yes	characteristic
20	C₅×#365	C₂×C₃	1	4	yes	direct product factor
21	C₂²×C₃×C₄×C₅	—	4	1	no	—
22	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
23	C₃×C₅×M₁₆	—	2	2	no	—
24	C₃×C₅×M₁₆	—	4	1	no	—
25	C₂×C₃×C₅×C₈	—	4	1	no	—
26	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
29	C₂×C₃×C₅×C₈	Q₈	1	1	yes	characteristic
30	C₂×C₃×C₅×C₈	Q₈	1	1	yes	characteristic
31	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
32	C₃×C₅×C₄⋊C₄	—	2	2	no	—
33	C₃×C₅×C₄⋊C₄	—	4	1	no	—
34	C₃×C₅×C₁₆	C₂×C₄	1	4	yes	complemented
35	C₂×C₃×M₃₂	C₂×C₅	1	1	yes	characteristic
36	C₂×C₃×#108	C₂×C₅	1	1	yes	characteristic
37	C₂×C₃×#106	C₂×C₅	1	1	yes	characteristic
38	C₃×#365	C₂×C₅	1	4	yes	direct product factor
39	C₂×C₅×M₁₆	—	2	1	no	—
40	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
41	C₂×C₅×C₄⋊C₄	—	2	1	no	—
42	C₅×M₃₂	C₃×C₄	1	2	yes	complemented
43	C₅×M₃₂	C₂²×C₃	1	2	yes	—
44	C₂×C₅×C₁₆	C₃×C₄	1	2	yes	complemented
45	C₅×#108	—	2	1	no	—
46	C₅×#108	C₂²×C₃	1	2	yes	—
47	C₅×#106	—	2	1	no	—
48	C₅×#106	C₂²×C₃	1	2	yes	—
49	C₂×#365	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
50	C₂³×C₃×C₅	SD₁₆	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	—	4	2	no	—
52	C₂×C₃×C₄×C₅	—	8	2	no	—
53	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	SD₁₆	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	SD₁₆	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	SD₁₆	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
59	C₃×C₅×C₈	—	4	2	no	—
60	C₃×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
61	C₃×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
62	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
63	C₃×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₃×M₁₆	—	2	1	no	—
65	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
66	C₂×C₃×C₄⋊C₄	—	2	1	no	—
67	C₃×M₃₂	C₄×C₅	1	2	yes	complemented
68	C₃×M₃₂	C₂²×C₅	1	2	yes	—
69	C₂×C₃×C₁₆	C₄×C₅	1	2	yes	complemented
70	C₃×#108	—	2	1	no	—
71	C₃×#108	C₂²×C₅	1	2	yes	—
72	C₃×#106	—	2	1	no	—
73	C₃×#106	C₂²×C₅	1	2	yes	—
74	C₂²×C₄×C₅	—	4	1	no	—
75	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
76	C₅×M₁₆	—	2	2	no	—
77	C₅×M₁₆	—	4	1	no	—
78	C₂×C₅×C₈	—	4	1	no	—
79	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
80	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
81	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
82	C₂×C₅×C₈	C₃×Q₈	1	1	yes	characteristic
83	C₂×C₅×C₈	C₃×Q₈	1	1	yes	characteristic
84	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
85	C₅×C₄⋊C₄	—	2	2	no	—
86	C₅×C₄⋊C₄	—	4	1	no	—
87	C₅×C₁₆	C₂×C₃×C₄	1	4	yes	complemented
88	C₂×M₃₂	C₂×C₃×C₅	1	1	yes	characteristic
89	C₂×#108	C₂×C₃×C₅	1	1	yes	characteristic
90	C₂×#106	C₂×C₃×C₅	1	1	yes	characteristic
91	#365	C₂×C₃×C₅	1	4	yes	direct product factor
92	C₂²×C₃×C₅	—	2	2	no	—
93	C₂²×C₃×C₅	#106	1	1	yes	characteristic
94	C₂²×C₃×C₅	#106	1	1	yes	characteristic
95	C₂²×C₃×C₅	C₂×SD₁₆	1	1	yes	characteristic
96	C₃×C₄×C₅	—	8	2	no	—
97	C₃×C₄×C₅	#106	1	1	yes	characteristic
98	C₃×C₄×C₅	#106	1	1	yes	characteristic
99	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
101	C₂²×C₃×C₄	—	4	1	no	—
102	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
103	C₃×M₁₆	—	2	2	no	—
104	C₃×M₁₆	—	4	1	no	—
105	C₂×C₃×C₈	—	4	1	no	—
106	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
107	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
108	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
109	C₂×C₃×C₈	C₅×Q₈	1	1	yes	characteristic
110	C₂×C₃×C₈	C₅×Q₈	1	1	yes	characteristic
111	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
112	C₃×C₄⋊C₄	—	2	2	no	—
113	C₃×C₄⋊C₄	—	4	1	no	—
114	C₃×C₁₆	C₂×C₄×C₅	1	4	yes	complemented
115	C₂³×C₅	C₃×SD₁₆	1	1	yes	characteristic
116	C₂×C₄×C₅	—	4	2	no	—
117	C₂×C₄×C₅	—	8	2	no	—
118	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₅	C₃×SD₁₆	1	1	yes	characteristic
121	C₂×C₄×C₅	C₃×SD₁₆	1	1	yes	characteristic
122	C₂×C₄×C₅	C₃×SD₁₆	1	1	yes	characteristic
123	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
124	C₅×C₈	—	4	2	no	—
125	C₅×C₈	C₃×C₄⋊C₄	1	1	yes	characteristic
126	C₅×C₈	C₃×C₄⋊C₄	1	1	yes	characteristic
127	C₅×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
128	C₅×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
129	C₂×M₁₆	—	2	1	no	—
130	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
131	C₂×C₄⋊C₄	—	2	1	no	—
132	M₃₂	C₃×C₄×C₅	1	2	yes	complemented
133	M₃₂	C₂²×C₃×C₅	1	2	yes	—
134	C₂×C₁₆	C₃×C₄×C₅	1	2	yes	complemented
135	#108	—	2	1	no	—
136	#108	C₂²×C₃×C₅	1	2	yes	—
137	#106	—	2	1	no	—
138	#106	C₂²×C₃×C₅	1	2	yes	—
139	C₂×C₃×C₅	—	2	1	no	—
140	C₂×C₃×C₅	—	2	1	no	—
141	C₂×C₃×C₅	#365	1	2	yes	direct product factor
142	C₂×C₃×C₅	C₂×#106	1	1	yes	characteristic
143	C₂³×C₃	C₅×SD₁₆	1	1	yes	characteristic
144	C₂×C₃×C₄	—	4	2	no	—
145	C₂×C₃×C₄	—	8	2	no	—
146	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₃×C₄	C₅×SD₁₆	1	1	yes	characteristic
149	C₂×C₃×C₄	C₅×SD₁₆	1	1	yes	characteristic
150	C₂×C₃×C₄	C₅×SD₁₆	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
152	C₃×C₈	—	4	2	no	—
153	C₃×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
154	C₃×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
155	C₃×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
156	C₃×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
157	C₂²×C₅	—	2	2	no	—
158	C₂²×C₅	C₃×#106	1	1	yes	characteristic
159	C₂²×C₅	C₃×#106	1	1	yes	characteristic
160	C₂²×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
161	C₄×C₅	—	8	2	no	—
162	C₄×C₅	C₃×#106	1	1	yes	characteristic
163	C₄×C₅	C₃×#106	1	1	yes	characteristic
164	C₄×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
165	C₄×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
166	C₂²×C₄	—	4	1	no	—
167	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
168	M₁₆	—	2	2	no	—
169	M₁₆	—	4	1	no	—
170	C₂×C₈	—	4	1	no	—
171	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
172	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
173	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
174	C₂×C₈	C₃×C₅×Q₈	1	1	yes	characteristic
175	C₂×C₈	C₃×C₅×Q₈	1	1	yes	characteristic
176	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	4	1	no	—
179	C₁₆	C₂×C₃×C₄×C₅	1	4	yes	complemented
180	C₃×C₅	C₂×#365	1	1	yes	characteristic, direct product factor, Hall
181	C₂²×C₃	—	2	2	no	—
182	C₂²×C₃	C₅×#106	1	1	yes	characteristic
183	C₂²×C₃	C₅×#106	1	1	yes	characteristic
184	C₂²×C₃	C₂×C₅×SD₁₆	1	1	yes	characteristic
185	C₃×C₄	—	8	2	no	—
186	C₃×C₄	C₅×#106	1	1	yes	characteristic
187	C₃×C₄	C₅×#106	1	1	yes	characteristic
188	C₃×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
189	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
190	C₂×C₅	—	2	1	no	—
191	C₂×C₅	—	2	1	no	—
192	C₂×C₅	C₃×#365	1	2	yes	direct product factor
193	C₂×C₅	C₂×C₃×#106	1	1	yes	characteristic
194	C₂³	C₃×C₅×SD₁₆	1	1	yes	characteristic
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	8	2	no	—
197	C₂×C₄	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
198	C₂×C₄	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
199	C₂×C₄	C₃×C₅×SD₁₆	1	1	yes	characteristic
200	C₂×C₄	C₃×C₅×SD₁₆	1	1	yes	characteristic
201	C₂×C₄	C₃×C₅×SD₁₆	1	1	yes	characteristic
202	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
203	C₈	—	4	2	no	—
204	C₈	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
205	C₈	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
206	C₈	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
207	C₈	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
208	C₂×C₃	—	2	1	no	—
209	C₂×C₃	—	2	1	no	—
210	C₂×C₃	C₅×#365	1	2	yes	direct product factor
211	C₂×C₃	C₂×C₅×#106	1	1	yes	characteristic
212	C₅	C₂×C₃×#365	1	1	yes	characteristic, direct product factor, Sylow
213	C₂²	—	2	2	no	—
214	C₂²	C₃×C₅×#106	1	1	yes	characteristic
215	C₂²	C₃×C₅×#106	1	1	yes	characteristic
216	C₂²	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
217	C₄	—	8	2	no	—
218	C₄	C₃×C₅×#106	1	1	yes	characteristic
219	C₄	C₃×C₅×#106	1	1	yes	characteristic
220	C₄	C₂×C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
221	C₄	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
222	C₃	C₂×C₅×#365	1	1	yes	characteristic, direct product factor, Sylow
223	C₂	—	2	1	no	—
224	C₂	—	2	1	no	—
225	C₂	C₃×C₅×#365	1	2	yes	direct product factor
226	C₂	C₂×C₃×C₅×#106	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	#106
commutator	C₈	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
prime radical	C₂³×C₃×C₅	SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂²×C₃×C₄×C₅ ◁ C₂²×C₃×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×#365	C₂	direct (N × H)
5. C₂×C₅×#365	C₃	direct (N × H)
9. C₃×C₅×M₃₂	C₄	semidirect
11. C₂×C₃×C₅×C₁₆	C₄	semidirect
16. C₂×C₃×#365	C₅	direct (N × H)
20. C₅×#365	C₂×C₃	direct (N × H)
34. C₃×C₅×C₁₆	C₂×C₄	semidirect
38. C₃×#365	C₂×C₅	direct (N × H)
42. C₅×M₃₂	C₃×C₄	semidirect
44. C₂×C₅×C₁₆	C₃×C₄	semidirect
49. C₂×#365	C₃×C₅	direct (N × H)
67. C₃×M₃₂	C₄×C₅	semidirect
69. C₂×C₃×C₁₆	C₄×C₅	semidirect
87. C₅×C₁₆	C₂×C₃×C₄	semidirect
91. #365	C₂×C₃×C₅	direct (N × H)
114. C₃×C₁₆	C₂×C₄×C₅	semidirect
132. M₃₂	C₃×C₄×C₅	semidirect
134. C₂×C₁₆	C₃×C₄×C₅	semidirect
141. C₂×C₃×C₅	#365	direct (N × H)
179. C₁₆	C₂×C₃×C₄×C₅	semidirect
180. C₃×C₅	C₂×#365	direct (N × H)
192. C₂×C₅	C₃×#365	direct (N × H)
210. C₂×C₃	C₅×#365	direct (N × H)
212. C₅	C₂×C₃×#365	direct (N × H)
222. C₃	C₂×C₅×#365	direct (N × H)
225. C₂	C₃×C₅×#365	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×M₃₂	2
C₂×C₃×C₅×#108	2
C₂×C₃×C₅×#106	2
C₃×C₅×#365	2
C₂×C₅×#365	3
C₂×C₃×#365	5

Subgroups — C2×C3×C5×#365 #14275188

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×#365 #14275188