Subgroups — C₃×C₄×C₅×SD₃₂ #14275207

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×C₄×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄×C₅×D₈	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#367	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#358	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₅×#357	C₂	1	1	yes	maximal, characteristic
7	C₃×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₅×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×C₄×C₅×Q₈	—	2	1	no	—
12	C₃×C₄×C₅×D₄	—	2	1	no	—
13	C₃×C₅×SD₃₂	C₄	1	4	yes	direct product factor
14	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
16	C₃×C₅×#107	C₂²	1	1	yes	characteristic
17	C₃×C₅×#107	C₂²	1	1	yes	characteristic
18	C₃×C₅×#103	—	2	1	no	—
19	C₃×C₅×#102	—	2	1	no	—
20	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	C₃×C₄×SD₃₂	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂×C₅×SD₃₂	C₂×C₃	1	1	yes	characteristic
23	C₄×C₅×Q₁₆	C₂×C₃	1	1	yes	characteristic, complemented
24	C₄×C₅×D₈	C₂×C₃	1	1	yes	characteristic
25	C₅×#367	C₂×C₃	1	1	yes	characteristic, complemented
26	C₅×#358	C₂×C₃	1	1	yes	characteristic, complemented
27	C₅×#357	C₂×C₃	1	1	yes	characteristic
28	C₄×C₅×C₁₆	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×C₄×C₅	—	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	D₄	1	1	yes	characteristic
37	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
38	C₃×C₅×C₄⋊C₄	—	2	1	no	—
39	C₃×C₅×C₄⋊C₄	—	2	1	no	—
40	C₃×C₅×C₄⋊C₄	—	4	1	no	—
41	C₃×C₅×C₂²⋊C₄	—	4	1	no	—
42	C₃×C₄²×C₅	—	4	1	no	—
43	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
44	C₃×C₅×C₁₆	—	2	1	no	—
45	C₃×C₅×C₁₆	C₂×C₄	1	2	yes	complemented
46	C₂×C₃×SD₃₂	C₂×C₅	1	1	yes	characteristic
47	C₃×C₄×Q₁₆	C₂×C₅	1	1	yes	characteristic, complemented
48	C₃×C₄×D₈	C₂×C₅	1	1	yes	characteristic
49	C₃×#367	C₂×C₅	1	1	yes	characteristic, complemented
50	C₃×#358	C₂×C₅	1	1	yes	characteristic, complemented
51	C₃×#357	C₂×C₅	1	1	yes	characteristic
52	C₃×C₄×C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
53	C₂×C₅×Q₁₆	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₅×D₈	C₂²×C₃	1	1	yes	characteristic
55	C₄×C₅×Q₈	—	2	1	no	—
56	C₄×C₅×D₄	—	2	1	no	—
57	C₅×SD₃₂	C₃×C₄	1	4	yes	direct product factor
58	C₂×C₅×C₁₆	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₅×C₁₆	C₂²×C₃	1	1	yes	characteristic
60	C₅×#107	C₂²×C₃	1	1	yes	characteristic
61	C₅×#107	C₂²×C₃	1	1	yes	characteristic
62	C₅×#103	—	2	1	no	—
63	C₅×#102	—	2	1	no	—
64	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
65	C₄×SD₃₂	C₃×C₅	1	1	yes	characteristic, direct product factor, 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	—	4	1	no	—
73	C₂×C₃×C₄×C₅	—	4	1	no	—
74	C₂×C₃×C₄×C₅	—	8	2	no	—
75	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₃×C₅×C₈	—	2	1	no	—
79	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
80	C₃×C₅×C₈	#41	1	1	yes	characteristic
81	C₂×C₃×Q₁₆	C₂²×C₅	1	1	yes	characteristic
82	C₂×C₃×D₈	C₂²×C₅	1	1	yes	characteristic
83	C₃×C₄×Q₈	—	2	1	no	—
84	C₃×C₄×D₄	—	2	1	no	—
85	C₃×SD₃₂	C₄×C₅	1	4	yes	direct product factor
86	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
87	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
88	C₃×#107	C₂²×C₅	1	1	yes	characteristic
89	C₃×#107	C₂²×C₅	1	1	yes	characteristic
90	C₃×#103	—	2	1	no	—
91	C₃×#102	—	2	1	no	—
92	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
93	C₂×C₅×Q₈	—	2	1	no	—
94	C₂×C₅×D₄	—	2	1	no	—
95	C₂²×C₄×C₅	—	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	C₃×D₄	1	1	yes	characteristic
101	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
102	C₅×C₄⋊C₄	—	2	1	no	—
103	C₅×C₄⋊C₄	—	2	1	no	—
104	C₅×C₄⋊C₄	—	4	1	no	—
105	C₅×C₂²⋊C₄	—	4	1	no	—
106	C₄²×C₅	—	4	1	no	—
107	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
108	C₅×C₁₆	—	2	1	no	—
109	C₅×C₁₆	C₂×C₃×C₄	1	2	yes	complemented
110	C₂×SD₃₂	C₂×C₃×C₅	1	1	yes	characteristic
111	C₄×Q₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
112	C₄×D₈	C₂×C₃×C₅	1	1	yes	characteristic
113	#367	C₂×C₃×C₅	1	1	yes	characteristic, complemented
114	#358	C₂×C₃×C₅	1	1	yes	characteristic, complemented
115	#357	C₂×C₃×C₅	1	1	yes	characteristic
116	C₄×C₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
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	—	2	1	no	—
122	C₃×C₄×C₅	—	4	2	no	—
123	C₃×C₄×C₅	—	8	1	no	—
124	C₃×C₄×C₅	—	8	1	no	—
125	C₃×C₄×C₅	SD₃₂	1	2	yes	direct product factor
126	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
127	C₃×C₄×C₅	#135	1	1	yes	characteristic
128	C₂×C₃×Q₈	—	2	1	no	—
129	C₂×C₃×D₄	—	2	1	no	—
130	C₂²×C₃×C₄	—	4	1	no	—
131	C₃×Q₁₆	—	2	1	no	—
132	C₃×Q₁₆	C₂×C₄×C₅	1	2	yes	complemented
133	C₃×D₈	—	2	1	no	—
134	C₃×D₈	C₂×C₄×C₅	1	2	yes	—
135	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
136	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
137	C₃×C₄⋊C₄	—	2	1	no	—
138	C₃×C₄⋊C₄	—	2	1	no	—
139	C₃×C₄⋊C₄	—	4	1	no	—
140	C₃×C₂²⋊C₄	—	4	1	no	—
141	C₃×C₄²	—	4	1	no	—
142	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
143	C₃×C₁₆	—	2	1	no	—
144	C₃×C₁₆	C₂×C₄×C₅	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	—	4	1	no	—
152	C₂×C₄×C₅	—	4	1	no	—
153	C₂×C₄×C₅	—	8	2	no	—
154	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
155	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
156	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
157	C₅×C₈	—	2	1	no	—
158	C₅×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
159	C₅×C₈	C₃×#41	1	1	yes	characteristic
160	C₂×Q₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
161	C₂×D₈	C₂²×C₃×C₅	1	1	yes	characteristic
162	C₄×Q₈	—	2	1	no	—
163	C₄×D₄	—	2	1	no	—
164	SD₃₂	C₃×C₄×C₅	1	4	yes	direct product factor
165	C₂×C₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
166	C₂×C₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
167	#107	C₂²×C₃×C₅	1	1	yes	characteristic
168	#107	C₂²×C₃×C₅	1	1	yes	characteristic
169	#103	—	2	1	no	—
170	#102	—	2	1	no	—
171	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
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	—	4	1	no	—
183	C₂×C₃×C₄	—	4	1	no	—
184	C₂×C₃×C₄	—	8	2	no	—
185	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
186	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
188	C₃×C₈	—	2	1	no	—
189	C₃×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
190	C₃×C₈	C₅×#41	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	—	2	1	no	—
196	C₄×C₅	—	4	2	no	—
197	C₄×C₅	—	8	1	no	—
198	C₄×C₅	—	8	1	no	—
199	C₄×C₅	C₃×SD₃₂	1	2	yes	direct product factor
200	C₄×C₅	C₃×C₄×D₄	1	1	yes	characteristic
201	C₄×C₅	C₃×#135	1	1	yes	characteristic
202	C₂×Q₈	—	2	1	no	—
203	C₂×D₄	—	2	1	no	—
204	C₂²×C₄	—	4	1	no	—
205	Q₁₆	—	2	1	no	—
206	Q₁₆	C₂×C₃×C₄×C₅	1	2	yes	complemented
207	D₈	—	2	1	no	—
208	D₈	C₂×C₃×C₄×C₅	1	2	yes	—
209	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
210	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
211	C₄⋊C₄	—	2	1	no	—
212	C₄⋊C₄	—	2	1	no	—
213	C₄⋊C₄	—	4	1	no	—
214	C₂²⋊C₄	—	4	1	no	—
215	C₄²	—	4	1	no	—
216	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
217	C₁₆	—	2	1	no	—
218	C₁₆	C₂×C₃×C₄×C₅	1	2	yes	complemented
219	C₃×C₅	C₄×SD₃₂	1	1	yes	characteristic, direct product factor, Hall
220	C₂²×C₃	—	4	2	no	—
221	C₂²×C₃	—	8	1	no	—
222	C₂²×C₃	—	8	1	no	—
223	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
224	C₃×C₄	—	2	1	no	—
225	C₃×C₄	—	4	2	no	—
226	C₃×C₄	—	8	1	no	—
227	C₃×C₄	—	8	1	no	—
228	C₃×C₄	C₅×SD₃₂	1	2	yes	direct product factor
229	C₃×C₄	C₄×C₅×D₄	1	1	yes	characteristic
230	C₃×C₄	C₅×#135	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	—	4	1	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	8	2	no	—
244	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
245	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
246	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
247	C₈	—	2	1	no	—
248	C₈	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
249	C₈	C₃×C₅×#41	1	1	yes	characteristic
250	C₂×C₃	—	8	2	no	—
251	C₂×C₃	C₄×C₅×D₈	1	1	yes	characteristic
252	C₂×C₃	C₂×C₅×SD₃₂	1	1	yes	characteristic
253	C₂×C₃	C₅×#508	1	1	yes	characteristic
254	C₅	C₃×C₄×SD₃₂	1	1	yes	characteristic, direct product factor, Sylow
255	C₂²	—	4	2	no	—
256	C₂²	—	8	1	no	—
257	C₂²	—	8	1	no	—
258	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
259	C₄	—	2	1	no	—
260	C₄	—	4	2	no	—
261	C₄	—	8	1	no	—
262	C₄	—	8	1	no	—
263	C₄	C₃×C₅×SD₃₂	1	2	yes	direct product factor
264	C₄	C₃×C₄×C₅×D₄	1	1	yes	characteristic
265	C₄	C₃×C₅×#135	1	1	yes	characteristic
266	C₃	C₄×C₅×SD₃₂	1	1	yes	characteristic, direct product factor, Sylow
267	C₂	—	8	2	no	—
268	C₂	C₃×C₄×C₅×D₈	1	1	yes	characteristic
269	C₂	C₂×C₃×C₅×SD₃₂	1	1	yes	characteristic
270	C₂	C₃×C₅×#508	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄×C₅	D₈
commutator	C₈	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
socle	C₂²×C₃×C₅	C₂×D₈
prime radical	C₂×C₃×C₅×D₈	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄×C₅ ◁ C₃×C₄²×C₅ ◁ C₃×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×C₅×Q₁₆	C₂	semidirect
4. C₃×C₅×#367	C₂	semidirect
5. C₃×C₅×#358	C₂	semidirect
7. C₃×C₄×C₅×C₁₆	C₂	semidirect
8. C₄×C₅×SD₃₂	C₃	direct (N × H)
13. C₃×C₅×SD₃₂	C₄	direct (N × H)
21. C₃×C₄×SD₃₂	C₅	direct (N × H)
23. C₄×C₅×Q₁₆	C₂×C₃	semidirect
25. C₅×#367	C₂×C₃	semidirect
26. C₅×#358	C₂×C₃	semidirect
28. C₄×C₅×C₁₆	C₂×C₃	semidirect
33. C₃×C₅×Q₁₆	C₂×C₄	semidirect
45. C₃×C₅×C₁₆	C₂×C₄	semidirect
47. C₃×C₄×Q₁₆	C₂×C₅	semidirect
49. C₃×#367	C₂×C₅	semidirect
50. C₃×#358	C₂×C₅	semidirect
52. C₃×C₄×C₁₆	C₂×C₅	semidirect
57. C₅×SD₃₂	C₃×C₄	direct (N × H)
65. C₄×SD₃₂	C₃×C₅	direct (N × H)
85. C₃×SD₃₂	C₄×C₅	direct (N × H)
97. C₅×Q₁₆	C₂×C₃×C₄	semidirect
109. C₅×C₁₆	C₂×C₃×C₄	semidirect
111. C₄×Q₁₆	C₂×C₃×C₅	semidirect
113. #367	C₂×C₃×C₅	semidirect
114. #358	C₂×C₃×C₅	semidirect
116. C₄×C₁₆	C₂×C₃×C₅	semidirect
125. C₃×C₄×C₅	SD₃₂	direct (N × H)
132. C₃×Q₁₆	C₂×C₄×C₅	semidirect
144. C₃×C₁₆	C₂×C₄×C₅	semidirect
164. SD₃₂	C₃×C₄×C₅	direct (N × H)
199. C₄×C₅	C₃×SD₃₂	direct (N × H)
206. Q₁₆	C₂×C₃×C₄×C₅	semidirect
218. C₁₆	C₂×C₃×C₄×C₅	semidirect
219. C₃×C₅	C₄×SD₃₂	direct (N × H)
228. C₃×C₄	C₅×SD₃₂	direct (N × H)
254. C₅	C₃×C₄×SD₃₂	direct (N × H)
263. C₄	C₃×C₅×SD₃₂	direct (N × H)
266. C₃	C₄×C₅×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×C₄×C₅×Q₁₆	2
C₃×C₄×C₅×D₈	2
C₃×C₅×#367	2
C₃×C₅×#358	2
C₃×C₅×#357	2
C₃×C₄×C₅×C₁₆	2
C₄×C₅×SD₃₂	3
C₃×C₄×SD₃₂	5

Subgroups — C3×C4×C5×SD32 #14275207

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×C₅×SD₃₂ #14275207