Subgroups — C₃×Dic₅×SD₃₂ #14315356

Order 1920.

324 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×SD₃₂	C₂	1	1	yes	maximal, characteristic
2	C₃×Q₁₆×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×D₈×Dic₅	C₂	1	1	yes	maximal, characteristic
4	C₃×#64784	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64782	C₂	1	1	yes	maximal, characteristic
6	C₃×#64725	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₁₆×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
8	Dic₅×SD₃₂	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₅×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₅×D₈	C₂²	1	1	yes	characteristic
11	C₃×Q₈×Dic₅	—	2	1	no	—
12	C₃×D₄×Dic₅	—	2	1	no	—
13	C₃×C₅×SD₃₂	C₄	1	4	yes	complemented
14	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
15	C₃×#3967	—	2	1	no	—
16	C₃×#3964	—	2	1	no	—
17	C₃×#3950	C₂²	1	1	yes	characteristic
18	C₃×#3950	C₂²	1	1	yes	characteristic
19	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
20	C₂×C₃×#672	C₂²	1	1	yes	characteristic
21	C₃×C₄×SD₃₂	—	5	1	no	maximal, Hall
22	C₂×C₅×SD₃₂	C₂×C₃	1	1	yes	characteristic
23	Q₁₆×Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
24	D₈×Dic₅	C₂×C₃	1	1	yes	characteristic
25	#64784	C₂×C₃	1	1	yes	characteristic, complemented
26	#64782	C₂×C₃	1	1	yes	characteristic
27	#64725	C₂×C₃	1	1	yes	characteristic, complemented
28	C₁₆×Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂×C₃×C₅×Q₈	—	2	1	no	—
30	C₂×C₃×C₅×D₄	—	2	1	no	—
31	C₂²×C₃×Dic₅	—	4	1	no	—
32	C₃×C₅×Q₁₆	—	2	1	no	—
33	C₃×C₅×Q₁₆	C₂×C₄	1	2	yes	complemented
34	C₃×C₅×D₈	—	2	1	no	—
35	C₃×C₅×D₈	C₂×C₄	1	2	yes	—
36	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
37	C₃×C₂²⋊Dic₅	—	4	1	no	—
38	C₃×C₄⋊Dic₅	—	2	1	no	—
39	C₃×C₄⋊Dic₅	—	2	1	no	—
40	C₃×C₄⋊Dic₅	—	4	1	no	—
41	C₃×C₄×Dic₅	—	4	1	no	—
42	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
43	C₂×C₃×#168	D₄	1	1	yes	characteristic
44	C₃×C₅×C₁₆	C₂×C₄	1	2	yes	complemented
45	C₃×#672	—	2	1	no	—
46	C₂×C₃×SD₃₂	D₅	1	1	yes	characteristic
47	C₃×C₄×Q₁₆	—	5	1	no	—
48	C₃×C₄×D₈	—	5	1	no	—
49	C₃×#367	—	5	1	no	—
50	C₃×#358	—	5	1	no	—
51	C₃×#357	—	5	1	no	—
52	C₃×C₄×C₁₆	—	5	1	no	—
53	C₂×C₅×Q₁₆	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₅×D₈	C₂²×C₃	1	1	yes	characteristic
55	Q₈×Dic₅	—	2	1	no	—
56	D₄×Dic₅	—	2	1	no	—
57	C₅×SD₃₂	C₃×C₄	1	4	yes	complemented
58	C₂×C₅×C₁₆	C₂²×C₃	1	1	yes	characteristic
59	#3967	—	2	1	no	—
60	#3964	—	2	1	no	—
61	#3950	C₂²×C₃	1	1	yes	characteristic
62	#3950	C₂²×C₃	1	1	yes	characteristic
63	C₈×Dic₅	C₂²×C₃	1	1	yes	characteristic
64	C₂×#672	C₂²×C₃	1	1	yes	characteristic
65	C₄×SD₃₂	—	5	1	no	Sylow
66	C₂³×C₃×C₅	—	4	1	no	—
67	C₃×C₅×Q₈	—	2	2	no	—
68	C₃×C₅×Q₈	—	4	1	no	—
69	C₃×C₅×D₄	—	2	2	no	—
70	C₃×C₅×D₄	—	4	1	no	—
71	C₂×C₃×C₄×C₅	—	4	1	no	—
72	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
73	C₂×C₃×Dic₅	—	4	1	no	—
74	C₂×C₃×Dic₅	—	4	1	no	—
75	C₂×C₃×Dic₅	—	8	2	no	—
76	C₂×C₃×Dic₅	D₈	1	1	yes	characteristic
77	C₂×C₃×Dic₅	D₈	1	1	yes	characteristic
78	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
79	C₃×C₅×C₈	#41	1	1	yes	characteristic
80	C₃×#168	—	2	1	no	—
81	C₂×C₃×Q₁₆	C₂×D₅	1	1	yes	characteristic
82	C₂×C₃×D₈	C₂×D₅	1	1	yes	characteristic
83	C₃×C₄×Q₈	—	10	1	no	—
84	C₃×C₄×D₄	—	10	1	no	—
85	C₃×SD₃₂	Dic₅	1	4	yes	direct product factor
86	C₂×C₃×C₁₆	—	5	1	no	—
87	C₂×C₃×C₁₆	C₂×D₅	1	1	yes	characteristic
88	C₃×#107	—	5	1	no	—
89	C₃×#107	—	5	1	no	—
90	C₃×#103	—	10	1	no	—
91	C₃×#102	—	10	1	no	—
92	C₃×C₄×C₈	—	5	1	no	—
93	C₂×C₅×Q₈	—	2	1	no	—
94	C₂×C₅×D₄	—	2	1	no	—
95	C₂²×Dic₅	—	4	1	no	—
96	C₅×Q₁₆	—	2	1	no	—
97	C₅×Q₁₆	C₂×C₃×C₄	1	2	yes	complemented
98	C₅×D₈	—	2	1	no	—
99	C₅×D₈	C₂×C₃×C₄	1	2	yes	—
100	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
101	C₂²⋊Dic₅	—	4	1	no	—
102	C₄⋊Dic₅	—	2	1	no	—
103	C₄⋊Dic₅	—	2	1	no	—
104	C₄⋊Dic₅	—	4	1	no	—
105	C₄×Dic₅	—	4	1	no	—
106	C₄×Dic₅	C₃×D₄	1	1	yes	characteristic
107	C₂×#168	C₃×D₄	1	1	yes	characteristic
108	C₅×C₁₆	C₂×C₃×C₄	1	2	yes	complemented
109	#672	—	2	1	no	—
110	C₂×SD₃₂	C₃×D₅	1	1	yes	characteristic
111	C₄×Q₁₆	—	5	1	no	—
112	C₄×D₈	—	5	1	no	—
113	#367	—	5	1	no	—
114	#358	—	5	1	no	—
115	#357	—	5	1	no	—
116	C₄×C₁₆	—	5	1	no	—
117	C₂²×C₃×C₅	—	4	2	no	—
118	C₂²×C₃×C₅	—	8	1	no	—
119	C₂²×C₃×C₅	—	8	1	no	—
120	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
121	C₃×C₄×C₅	—	4	2	no	—
122	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
123	C₃×C₄×C₅	#135	1	1	yes	characteristic
124	C₃×Dic₅	—	2	1	no	—
125	C₃×Dic₅	—	8	1	no	—
126	C₃×Dic₅	—	8	1	no	—
127	C₃×Dic₅	SD₃₂	1	2	yes	direct product factor
128	C₂×C₃×Q₈	—	2	1	no	—
129	C₂×C₃×D₄	—	2	1	no	—
130	C₂²×C₃×C₄	—	20	1	no	—
131	C₃×Q₁₆	—	2	1	no	—
132	C₃×Q₁₆	C₂×Dic₅	1	2	yes	complemented
133	C₃×D₈	—	2	1	no	—
134	C₃×D₈	C₂×Dic₅	1	2	yes	—
135	C₂×C₃×C₈	—	5	1	no	—
136	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
137	C₃×C₄⋊C₄	—	10	1	no	—
138	C₃×C₄⋊C₄	—	10	1	no	—
139	C₃×C₄⋊C₄	—	20	1	no	—
140	C₃×C₂²⋊C₄	—	20	1	no	—
141	C₃×C₄²	—	5	1	no	—
142	C₃×C₄²	—	20	1	no	—
143	C₃×C₁₆	—	10	1	no	—
144	C₃×C₁₆	C₂×Dic₅	1	2	yes	complemented
145	C₂³×C₅	—	4	1	no	—
146	C₅×Q₈	—	2	2	no	—
147	C₅×Q₈	—	4	1	no	—
148	C₅×D₄	—	2	2	no	—
149	C₅×D₄	—	4	1	no	—
150	C₂×C₄×C₅	—	4	1	no	—
151	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
152	C₂×Dic₅	—	4	1	no	—
153	C₂×Dic₅	—	4	1	no	—
154	C₂×Dic₅	—	8	2	no	—
155	C₂×Dic₅	C₃×D₈	1	1	yes	characteristic
156	C₂×Dic₅	C₃×D₈	1	1	yes	characteristic
157	C₅×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
158	C₅×C₈	C₃×#41	1	1	yes	characteristic
159	#168	—	2	1	no	—
160	C₂×Q₁₆	C₂×C₃×D₅	1	1	yes	characteristic
161	C₂×D₈	C₂×C₃×D₅	1	1	yes	characteristic
162	C₄×Q₈	—	10	1	no	—
163	C₄×D₄	—	10	1	no	—
164	SD₃₂	C₃×Dic₅	1	4	yes	direct product factor
165	C₂×C₁₆	—	5	1	no	—
166	C₂×C₁₆	C₂×C₃×D₅	1	1	yes	characteristic
167	#107	—	5	1	no	—
168	#107	—	5	1	no	—
169	#103	—	10	1	no	—
170	#102	—	10	1	no	—
171	C₄×C₈	—	5	1	no	—
172	C₂×C₃×C₅	—	8	2	no	—
173	C₂×C₃×C₅	C₄×D₈	1	1	yes	characteristic
174	C₂×C₃×C₅	C₂×SD₃₂	1	1	yes	characteristic
175	C₂×C₃×C₅	#508	1	1	yes	characteristic
176	C₂³×C₃	—	4	1	no	—
177	C₃×Q₈	—	2	2	no	—
178	C₃×Q₈	—	4	1	no	—
179	C₃×D₄	—	2	2	no	—
180	C₃×D₄	—	4	1	no	—
181	C₂×C₃×C₄	—	4	1	no	—
182	C₂×C₃×C₄	—	5	1	no	—
183	C₂×C₃×C₄	—	5	1	no	—
184	C₂×C₃×C₄	—	20	1	no	—
185	C₂×C₃×C₄	—	20	1	no	—
186	C₂×C₃×C₄	—	40	2	no	—
187	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
188	C₃×C₈	—	10	1	no	—
189	C₃×C₈	#711	1	1	yes	characteristic
190	C₃×C₈	C₂²×Dic₅	1	1	yes	characteristic
191	C₂²×C₅	—	4	2	no	—
192	C₂²×C₅	—	8	1	no	—
193	C₂²×C₅	—	8	1	no	—
194	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
195	C₄×C₅	—	4	2	no	—
196	C₄×C₅	C₃×C₄×D₄	1	1	yes	characteristic
197	C₄×C₅	C₃×#135	1	1	yes	characteristic
198	Dic₅	—	2	1	no	—
199	Dic₅	—	8	1	no	—
200	Dic₅	—	8	1	no	—
201	Dic₅	C₃×SD₃₂	1	2	yes	direct product factor
202	C₂×Q₈	—	2	1	no	—
203	C₂×D₄	—	2	1	no	—
204	C₂²×C₄	—	20	1	no	—
205	Q₁₆	—	2	1	no	—
206	Q₁₆	C₂×C₃×Dic₅	1	2	yes	complemented
207	D₈	—	2	1	no	—
208	D₈	C₂×C₃×Dic₅	1	2	yes	—
209	C₂×C₈	—	5	1	no	—
210	C₂×C₈	C₂²×C₃×D₅	1	1	yes	characteristic
211	C₄⋊C₄	—	10	1	no	—
212	C₄⋊C₄	—	10	1	no	—
213	C₄⋊C₄	—	20	1	no	—
214	C₂²⋊C₄	—	20	1	no	—
215	C₄²	—	5	1	no	—
216	C₄²	—	20	1	no	—
217	C₁₆	—	10	1	no	—
218	C₁₆	C₂×C₃×Dic₅	1	2	yes	complemented
219	C₃×C₅	C₄×SD₃₂	1	1	yes	characteristic, complemented, Hall
220	C₂²×C₃	—	4	2	no	—
221	C₂²×C₃	—	8	1	no	—
222	C₂²×C₃	—	8	1	no	—
223	C₂²×C₃	D₅×D₈	1	1	yes	characteristic
224	C₃×C₄	—	4	2	no	—
225	C₃×C₄	—	5	2	no	—
226	C₃×C₄	—	10	1	no	—
227	C₃×C₄	—	40	1	no	—
228	C₃×C₄	—	40	1	no	—
229	C₃×C₄	#4058	1	1	yes	characteristic
230	C₃×C₄	D₄×Dic₅	1	1	yes	characteristic
231	C₂×C₅	—	8	2	no	—
232	C₂×C₅	C₃×C₄×D₈	1	1	yes	characteristic
233	C₂×C₅	C₂×C₃×SD₃₂	1	1	yes	characteristic
234	C₂×C₅	C₃×#508	1	1	yes	characteristic
235	C₂³	—	4	1	no	—
236	Q₈	—	2	2	no	—
237	Q₈	—	4	1	no	—
238	D₄	—	2	2	no	—
239	D₄	—	4	1	no	—
240	C₂×C₄	—	4	1	no	—
241	C₂×C₄	—	5	1	no	—
242	C₂×C₄	—	5	1	no	—
243	C₂×C₄	—	20	1	no	—
244	C₂×C₄	—	20	1	no	—
245	C₂×C₄	—	40	2	no	—
246	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
247	C₈	—	10	1	no	—
248	C₈	C₃×#711	1	1	yes	characteristic
249	C₈	C₂²×C₃×Dic₅	1	1	yes	characteristic
250	C₂×C₃	—	8	2	no	—
251	C₂×C₃	D₅×SD₃₂	1	1	yes	characteristic
252	C₂×C₃	#65205	1	1	yes	characteristic
253	C₂×C₃	D₈×Dic₅	1	1	yes	characteristic
254	C₅	C₃×C₄×SD₃₂	1	1	yes	characteristic, complemented, Sylow
255	C₂²	—	4	2	no	—
256	C₂²	—	8	1	no	—
257	C₂²	—	8	1	no	—
258	C₂²	C₃×D₅×D₈	1	1	yes	characteristic
259	C₄	—	4	2	no	—
260	C₄	—	5	2	no	—
261	C₄	—	10	1	no	—
262	C₄	—	40	1	no	—
263	C₄	—	40	1	no	—
264	C₄	C₃×#4058	1	1	yes	characteristic
265	C₄	C₃×D₄×Dic₅	1	1	yes	characteristic
266	C₃	Dic₅×SD₃₂	1	1	yes	characteristic, direct product factor, Sylow
267	C₂	—	8	2	no	—
268	C₂	C₃×D₅×SD₃₂	1	1	yes	characteristic
269	C₂	C₃×#65205	1	1	yes	characteristic
270	C₂	C₃×D₈×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	D₅×D₈
commutator	C₅×C₈	C₂²×C₃×C₄
Frattini	C₂×C₈	C₂²×C₃×D₅
Fitting	C₂×C₃×C₅×SD₃₂	C₂
socle	C₂²×C₃×C₅	C₂×D₈
prime radical	C₂×C₃×C₅×D₈	C₂²
cosocle	C₂×C₅×C₈	C₂³×C₃
hypercenter	C₂×C₃×SD₃₂	D₅
hypocenter	C₅	C₃×C₄×SD₃₂
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈ ◁ C₂×C₃×SD₃₂

Complements

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

N	H ≅ G/N	type
2. C₃×Q₁₆×Dic₅	C₂	semidirect
4. C₃×#64784	C₂	semidirect
6. C₃×#64725	C₂	semidirect
7. C₃×C₁₆×Dic₅	C₂	semidirect
8. Dic₅×SD₃₂	C₃	direct (N × H)
13. C₃×C₅×SD₃₂	C₄	semidirect
23. Q₁₆×Dic₅	C₂×C₃	semidirect
25. #64784	C₂×C₃	semidirect
27. #64725	C₂×C₃	semidirect
28. C₁₆×Dic₅	C₂×C₃	semidirect
33. C₃×C₅×Q₁₆	C₂×C₄	semidirect
44. C₃×C₅×C₁₆	C₂×C₄	semidirect
57. C₅×SD₃₂	C₃×C₄	semidirect
85. C₃×SD₃₂	Dic₅	direct (N × H)
97. C₅×Q₁₆	C₂×C₃×C₄	semidirect
108. C₅×C₁₆	C₂×C₃×C₄	semidirect
127. C₃×Dic₅	SD₃₂	direct (N × H)
132. C₃×Q₁₆	C₂×Dic₅	semidirect
144. C₃×C₁₆	C₂×Dic₅	semidirect
164. SD₃₂	C₃×Dic₅	direct (N × H)
201. Dic₅	C₃×SD₃₂	direct (N × H)
206. Q₁₆	C₂×C₃×Dic₅	semidirect
218. C₁₆	C₂×C₃×Dic₅	semidirect
219. C₃×C₅	C₄×SD₃₂	semidirect
254. C₅	C₃×C₄×SD₃₂	semidirect
266. C₃	Dic₅×SD₃₂	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×SD₃₂	2
C₃×Q₁₆×Dic₅	2
C₃×D₈×Dic₅	2
C₃×#64784	2
C₃×#64782	2
C₃×#64725	2
C₃×C₁₆×Dic₅	2
Dic₅×SD₃₂	3
C₃×C₄×SD₃₂	5

Subgroups — C3×Dic5×SD32 #14315356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×Dic₅×SD₃₂ #14315356