Subgroups — C₅×Dic₃×M₃₂ #14312578

Order 1920.

392 conjugacy classes of subgroups.

The subgroup lattice has 224 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×C₅×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#812	C₂	1	1	yes	maximal, characteristic
4	C₅×#4502	C₂	1	2	yes	maximal, complemented
5	C₅×Dic₃×C₁₆	C₂	1	2	yes	maximal, complemented
6	C₄×C₅×M₃₂	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₅×Dic₃	C₄	1	1	yes	characteristic
9	C₂²×C₅×C₃⋊C₈	C₄	1	1	yes	characteristic
10	C₃×C₅×M₃₂	C₄	1	4	yes	complemented
11	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
12	C₅×C₈×Dic₃	C₄	1	1	yes	characteristic
13	C₅×C₈×Dic₃	C₄	1	1	yes	characteristic
14	C₅×C₈×Dic₃	C₂²	1	1	yes	characteristic
15	C₅×C₈×Dic₃	C₂²	1	1	yes	characteristic
16	C₅×#812	C₄	1	4	yes	complemented
17	C₂×C₅×C₃⋊C₁₆	C₂²	1	2	yes	—
18	Dic₃×M₃₂	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂×C₅×M₃₂	—	3	1	no	—
20	C₂×C₅×M₃₂	D₃	1	1	yes	characteristic
21	C₂×C₄×C₅×C₈	—	3	1	no	—
22	C₅×#346	—	3	2	no	—
23	C₄×C₅×C₁₆	—	3	2	no	—
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₅×Dic₃	C₈	1	2	yes	—
26	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
30	C₄×C₅×Dic₃	C₈	1	2	yes	—
31	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₅×C₃⋊C₈	C₈	1	4	yes	—
34	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
36	C₃×C₅×C₁₆	C₂×C₄	1	4	yes	complemented
37	C₅×C₃⋊C₁₆	C₂×C₄	1	4	yes	complemented
38	C₂×C₃×M₃₂	C₂×C₅	1	1	yes	characteristic
39	C₂×C₈×Dic₃	C₂×C₅	1	1	yes	characteristic
40	C₂×#812	C₂×C₅	1	1	yes	characteristic
41	#4502	C₂×C₅	1	2	yes	complemented
42	Dic₃×C₁₆	C₂×C₅	1	2	yes	complemented
43	C₂²×C₅×C₈	—	3	1	no	—
44	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
45	C₂×C₄²×C₅	—	3	1	no	—
46	C₅×M₃₂	—	3	4	no	—
47	C₅×M₃₂	Dic₃	1	4	yes	direct product factor
48	C₂×C₅×C₁₆	—	3	2	no	—
49	C₂×C₅×C₁₆	C₂×D₃	1	2	yes	—
50	C₄×C₅×C₈	—	3	1	no	—
51	C₄×C₅×C₈	—	3	1	no	—
52	C₄×C₅×C₈	—	3	1	no	—
53	C₄×C₅×C₈	—	3	1	no	—
54	C₄×M₃₂	—	3	1	no	Sylow
55	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
57	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
59	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₅×Dic₃	—	2	4	no	—
61	C₂×C₅×Dic₃	C₂×C₈	1	2	yes	—
62	C₂×C₅×Dic₃	C₂×C₈	1	2	yes	—
63	C₃×C₅×C₈	C₄²	1	2	yes	—
64	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
65	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
66	C₅×C₃⋊C₈	C₂×C₈	1	2	yes	—
67	C₅×C₃⋊C₈	C₂×C₈	1	2	yes	—
68	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
69	C₂×C₄×Dic₃	C₄×C₅	1	1	yes	characteristic
70	C₂²×C₃⋊C₈	C₄×C₅	1	1	yes	characteristic
71	C₃×M₃₂	C₄×C₅	1	4	yes	complemented
72	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	—
73	C₈×Dic₃	C₄×C₅	1	1	yes	characteristic
74	C₈×Dic₃	C₄×C₅	1	1	yes	characteristic
75	C₈×Dic₃	C₂²×C₅	1	1	yes	characteristic
76	C₈×Dic₃	C₂²×C₅	1	1	yes	characteristic
77	#812	C₄×C₅	1	4	yes	complemented
78	C₂×C₃⋊C₁₆	C₂²×C₅	1	2	yes	—
79	C₂²×C₄×C₅	—	3	2	no	—
80	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
81	C₂×C₅×C₈	—	3	4	no	—
82	C₂×C₅×C₈	—	3	1	no	—
83	C₂×C₅×C₈	—	3	1	no	—
84	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
85	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
86	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
87	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
88	C₄²×C₅	—	3	2	no	—
89	C₄²×C₅	—	3	1	no	—
90	C₄²×C₅	—	3	1	no	—
91	C₅×C₁₆	—	3	4	no	—
92	C₅×C₁₆	C₂×Dic₃	1	4	yes	complemented
93	C₂×M₃₂	—	3	1	no	—
94	C₂×M₃₂	C₅×D₃	1	1	yes	characteristic
95	C₂×C₄×C₈	—	3	1	no	—
96	#346	—	3	2	no	—
97	C₄×C₁₆	—	3	2	no	—
98	C₂²×C₃×C₅	—	2	2	no	—
99	C₂²×C₃×C₅	C₄×C₈	1	2	yes	—
100	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
101	C₃×C₄×C₅	C₄×C₈	1	2	yes	—
102	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
103	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
104	C₅×Dic₃	—	2	2	no	—
105	C₅×Dic₃	M₃₂	1	4	yes	direct product factor
106	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
107	C₂²×Dic₃	C₅×C₈	1	2	yes	—
108	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
109	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
110	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
111	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
112	C₄×Dic₃	C₅×C₈	1	2	yes	—
113	C₄×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
114	C₄×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
115	C₂×C₃⋊C₈	C₅×C₈	1	4	yes	—
116	C₂×C₃⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
117	C₂×C₃⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
118	C₃×C₁₆	C₂×C₄×C₅	1	4	yes	complemented
119	C₃⋊C₁₆	C₂×C₄×C₅	1	4	yes	complemented
120	C₂³×C₅	D₃×C₈	1	1	yes	characteristic
121	C₂×C₄×C₅	—	3	2	no	—
122	C₂×C₄×C₅	—	3	2	no	—
123	C₂×C₄×C₅	—	6	4	no	—
124	C₂×C₄×C₅	D₃×C₈	1	1	yes	characteristic
125	C₂×C₄×C₅	D₃×C₈	1	2	yes	—
126	C₂×C₄×C₅	C₄×Dic₃	1	2	yes	—
127	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
128	C₅×C₈	—	3	2	no	—
129	C₅×C₈	—	3	2	no	—
130	C₅×C₈	C₄×Dic₃	1	2	yes	—
131	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
132	C₅×C₈	C₂²×Dic₃	1	1	yes	characteristic
133	C₂²×C₈	—	3	1	no	—
134	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
135	C₂×C₄²	—	3	1	no	—
136	M₃₂	—	3	4	no	—
137	M₃₂	C₅×Dic₃	1	4	yes	direct product factor
138	C₂×C₁₆	—	3	2	no	—
139	C₂×C₁₆	C₂×C₅×D₃	1	2	yes	—
140	C₄×C₈	—	3	1	no	—
141	C₄×C₈	—	3	1	no	—
142	C₄×C₈	—	3	1	no	—
143	C₄×C₈	—	3	1	no	—
144	C₂×C₃×C₅	—	2	2	no	—
145	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
146	C₂×C₃×C₅	C₂×M₃₂	1	2	yes	—
147	C₂³×C₃	C₂×C₅×C₈	1	1	yes	characteristic
148	C₂×C₃×C₄	C₄²×C₅	1	2	yes	—
149	C₂×C₃×C₄	C₂×C₅×C₈	1	2	yes	—
150	C₂×C₃×C₄	C₂×C₅×C₈	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
152	C₂×Dic₃	—	2	4	no	—
153	C₂×Dic₃	C₂×C₅×C₈	1	2	yes	—
154	C₂×Dic₃	C₂×C₅×C₈	1	2	yes	—
155	C₃×C₈	C₄²×C₅	1	2	yes	—
156	C₃×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
157	C₃×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
158	C₃⋊C₈	C₂×C₅×C₈	1	2	yes	—
159	C₃⋊C₈	C₂×C₅×C₈	1	2	yes	—
160	C₂²×C₅	—	2	2	no	—
161	C₂²×C₅	C₈×Dic₃	1	2	yes	—
162	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
163	C₄×C₅	—	3	4	no	—
164	C₄×C₅	—	6	2	no	—
165	C₄×C₅	C₈×Dic₃	1	2	yes	—
166	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
167	C₄×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
168	C₂²×C₄	—	3	2	no	—
169	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
170	C₂×C₈	—	3	4	no	—
171	C₂×C₈	—	3	1	no	—
172	C₂×C₈	—	3	1	no	—
173	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
174	C₂×C₈	C₄×C₅×D₃	1	2	yes	—
175	C₂×C₈	C₂×C₅×Dic₃	1	2	yes	—
176	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
177	C₄²	—	3	2	no	—
178	C₄²	—	3	1	no	—
179	C₄²	—	3	1	no	—
180	C₁₆	—	3	4	no	—
181	C₁₆	C₂×C₅×Dic₃	1	4	yes	complemented
182	C₃×C₅	C₄×M₃₂	1	1	yes	characteristic, complemented, Hall
183	C₂²×C₃	—	2	2	no	—
184	C₂²×C₃	C₄×C₅×C₈	1	2	yes	—
185	C₂²×C₃	C₂²×C₅×C₈	1	1	yes	characteristic
186	C₃×C₄	C₄×C₅×C₈	1	2	yes	—
187	C₃×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
188	C₃×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
189	Dic₃	—	2	2	no	—
190	Dic₃	C₅×M₃₂	1	4	yes	direct product factor
191	C₂×C₅	—	2	2	no	—
192	C₂×C₅	D₃×M₃₂	1	2	yes	—
193	C₂×C₅	C₂×C₈×Dic₃	1	1	yes	characteristic
194	C₂³	C₅×D₃×C₈	1	1	yes	characteristic
195	C₂×C₄	—	3	2	no	—
196	C₂×C₄	—	3	2	no	—
197	C₂×C₄	—	6	4	no	—
198	C₂×C₄	C₅×D₃×C₈	1	2	yes	—
199	C₂×C₄	C₅×D₃×C₈	1	1	yes	characteristic
200	C₂×C₄	C₄×C₅×Dic₃	1	2	yes	—
201	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
202	C₈	—	3	2	no	—
203	C₈	—	3	2	no	—
204	C₈	C₄×C₅×Dic₃	1	2	yes	—
205	C₈	C₂×C₄×C₅×D₃	1	1	yes	characteristic
206	C₈	C₂²×C₅×Dic₃	1	1	yes	characteristic
207	C₂×C₃	—	2	2	no	—
208	C₂×C₃	C₂×C₄×C₅×C₈	1	1	yes	characteristic
209	C₂×C₃	C₂×C₅×M₃₂	1	2	yes	—
210	C₅	Dic₃×M₃₂	1	1	yes	characteristic, direct product factor, Sylow
211	C₂²	—	2	2	no	—
212	C₂²	C₅×C₈×Dic₃	1	2	yes	—
213	C₂²	C₂×C₅×D₃×C₈	1	1	yes	characteristic
214	C₄	—	3	4	no	—
215	C₄	—	6	2	no	—
216	C₄	C₅×C₈×Dic₃	1	2	yes	—
217	C₄	C₂×C₅×D₃×C₈	1	1	yes	characteristic
218	C₄	C₂×C₄×C₅×Dic₃	1	1	yes	characteristic
219	C₃	C₄×C₅×M₃₂	1	1	yes	characteristic, complemented, Sylow
220	C₂	—	2	2	no	—
221	C₂	C₅×D₃×M₃₂	1	2	yes	—
222	C₂	C₂×C₅×C₈×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅×C₈	C₂²×D₃
commutator	C₂×C₃	C₂×C₄×C₅×C₈
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₂×C₃×C₅×M₃₂	C₂
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂³×C₃×C₅	C₂×C₈
cosocle	C₂×C₃×C₈	C₂³×C₅
hypercenter	C₂×C₅×M₃₂	D₃
hypocenter	C₃	C₄×C₅×M₃₂
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅×C₈ ◁ C₂×C₅×M₃₂

Complements

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

N	H ≅ G/N	type
4. C₅×#4502	C₂	semidirect
5. C₅×Dic₃×C₁₆	C₂	semidirect
10. C₃×C₅×M₃₂	C₄	semidirect
16. C₅×#812	C₄	semidirect
18. Dic₃×M₃₂	C₅	direct (N × H)
36. C₃×C₅×C₁₆	C₂×C₄	semidirect
37. C₅×C₃⋊C₁₆	C₂×C₄	semidirect
41. #4502	C₂×C₅	semidirect
42. Dic₃×C₁₆	C₂×C₅	semidirect
47. C₅×M₃₂	Dic₃	direct (N × H)
71. C₃×M₃₂	C₄×C₅	semidirect
77. #812	C₄×C₅	semidirect
92. C₅×C₁₆	C₂×Dic₃	semidirect
105. C₅×Dic₃	M₃₂	direct (N × H)
118. C₃×C₁₆	C₂×C₄×C₅	semidirect
119. C₃⋊C₁₆	C₂×C₄×C₅	semidirect
137. M₃₂	C₅×Dic₃	direct (N × H)
181. C₁₆	C₂×C₅×Dic₃	semidirect
182. C₃×C₅	C₄×M₃₂	semidirect
190. Dic₃	C₅×M₃₂	direct (N × H)
210. C₅	Dic₃×M₃₂	direct (N × H)
219. C₃	C₄×C₅×M₃₂	semidirect

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×C₅×C₈×Dic₃	2
C₂×C₅×#812	2
C₅×#4502	2
C₅×Dic₃×C₁₆	2
C₄×C₅×M₃₂	3
Dic₃×M₃₂	5

Subgroups — C5×Dic3×M32 #14312578

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×Dic₃×M₃₂ #14312578