Subgroups — C₃×C₅×#1988 #14275022

Order 1920.

584 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#340	C₂	1	4	yes	maximal
5	C₅×#1988	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×Q₈	—	2	1	no	—
7	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
8	C₃×C₅×C₄⋊Q₈	—	2	2	no	—
9	C₃×C₅×C₄⋊Q₈	C₄	1	4	yes	complemented
10	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
11	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₃×C₄²×C₅	—	2	1	no	—
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₃×C₅×C₄⋊C₈	—	2	2	no	—
16	C₃×C₅×#95	—	2	1	no	—
17	C₃×#1988	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
19	C₂×C₅×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
20	C₄×C₅×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
21	C₅×#340	C₂×C₃	1	4	yes	—
22	C₂×C₃×C₅×Q₈	—	2	4	no	—
23	C₂×C₃×C₅×Q₈	—	4	2	no	—
24	C₂×C₃×C₅×Q₈	—	4	2	no	—
25	C₂²×C₃×C₄×C₅	—	2	2	no	—
26	C₂²×C₃×C₄×C₅	—	2	2	no	—
27	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
30	C₂×C₃×C₅×C₈	—	2	2	no	—
31	C₂×C₃×C₅×C₈	—	2	4	no	—
32	C₃×C₅×C₄⋊C₄	—	2	4	no	—
33	C₃×C₅×C₄⋊C₄	—	2	2	no	—
34	C₃×C₅×C₄⋊C₄	—	2	2	no	—
35	C₃×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
36	C₃×C₅×C₄⋊C₄	Q₈	1	4	yes	complemented
37	C₃×C₄²×C₅	—	4	2	no	—
38	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
39	C₃×C₄²×C₅	D₄	1	2	yes	—
40	C₂×C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
41	C₂×C₃×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic
42	C₃×C₄×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
43	C₃×#340	C₂×C₅	1	4	yes	—
44	C₂²×C₅×Q₈	—	2	1	no	—
45	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
46	C₅×C₄⋊Q₈	—	2	2	no	—
47	C₅×C₄⋊Q₈	C₃×C₄	1	4	yes	complemented
48	C₂×C₅×C₄⋊C₄	—	2	1	no	—
49	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
50	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
51	C₂×C₄²×C₅	—	2	1	no	—
52	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
53	C₅×C₄⋊C₈	—	2	2	no	—
54	C₅×#95	—	2	1	no	—
55	#1988	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
56	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
57	C₃×C₅×Q₈	—	4	8	no	—
58	C₂×C₃×C₄×C₅	—	2	2	no	—
59	C₂×C₃×C₄×C₅	—	2	4	no	—
60	C₂×C₃×C₄×C₅	—	2	4	no	—
61	C₂×C₃×C₄×C₅	—	4	2	no	—
62	C₂×C₃×C₄×C₅	—	4	2	no	—
63	C₂×C₃×C₄×C₅	—	4	2	no	—
64	C₂×C₃×C₄×C₅	—	4	2	no	—
65	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
66	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
67	C₂×C₃×C₄×C₅	D₈	1	2	yes	—
68	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
69	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
72	C₂×C₃×C₄×C₅	#41	1	2	yes	—
73	C₃×C₅×C₈	—	4	2	no	—
74	C₂²×C₃×Q₈	—	2	1	no	—
75	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
76	C₃×C₄⋊Q₈	—	2	2	no	—
77	C₃×C₄⋊Q₈	C₄×C₅	1	4	yes	complemented
78	C₂×C₃×C₄⋊C₄	—	2	1	no	—
79	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
80	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
81	C₂×C₃×C₄²	—	2	1	no	—
82	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
83	C₃×C₄⋊C₈	—	2	2	no	—
84	C₃×#95	—	2	1	no	—
85	C₂×C₅×Q₈	—	2	4	no	—
86	C₂×C₅×Q₈	—	4	2	no	—
87	C₂×C₅×Q₈	—	4	2	no	—
88	C₂²×C₄×C₅	—	2	2	no	—
89	C₂²×C₄×C₅	—	2	2	no	—
90	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
93	C₂×C₅×C₈	—	2	2	no	—
94	C₂×C₅×C₈	—	2	4	no	—
95	C₅×C₄⋊C₄	—	2	4	no	—
96	C₅×C₄⋊C₄	—	2	2	no	—
97	C₅×C₄⋊C₄	—	2	2	no	—
98	C₅×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
99	C₅×C₄⋊C₄	C₃×Q₈	1	4	yes	complemented
100	C₄²×C₅	—	4	2	no	—
101	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
102	C₄²×C₅	C₃×D₄	1	2	yes	—
103	C₂×C₄⋊Q₈	C₂×C₃×C₅	1	1	yes	characteristic
104	C₂×C₄⋊C₈	C₂×C₃×C₅	1	1	yes	characteristic
105	C₄×C₄⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic
106	#340	C₂×C₃×C₅	1	4	yes	—
107	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂²×C₃×C₅	C₂²⋊Q₈	1	2	yes	—
109	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
110	C₂²×C₃×C₅	C₂×SD₁₆	1	1	yes	characteristic
111	C₂²×C₃×C₅	#137	1	2	yes	—
112	C₃×C₄×C₅	—	2	2	no	—
113	C₃×C₄×C₅	—	4	4	no	—
114	C₃×C₄×C₅	—	4	4	no	—
115	C₃×C₄×C₅	#102	1	4	yes	—
116	C₃×C₄×C₅	C₄×Q₈	1	2	yes	—
117	C₃×C₄×C₅	#124	1	1	yes	characteristic
118	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
119	C₂×C₃×Q₈	—	2	4	no	—
120	C₂×C₃×Q₈	—	4	2	no	—
121	C₂×C₃×Q₈	—	4	2	no	—
122	C₂²×C₃×C₄	—	2	2	no	—
123	C₂²×C₃×C₄	—	2	2	no	—
124	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
125	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
126	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
127	C₂×C₃×C₈	—	2	2	no	—
128	C₂×C₃×C₈	—	2	4	no	—
129	C₃×C₄⋊C₄	—	2	4	no	—
130	C₃×C₄⋊C₄	—	2	2	no	—
131	C₃×C₄⋊C₄	—	2	2	no	—
132	C₃×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
133	C₃×C₄⋊C₄	C₅×Q₈	1	4	yes	complemented
134	C₃×C₄²	—	4	2	no	—
135	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
136	C₃×C₄²	C₅×D₄	1	2	yes	—
137	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
138	C₅×Q₈	—	4	8	no	—
139	C₂×C₄×C₅	—	2	2	no	—
140	C₂×C₄×C₅	—	2	4	no	—
141	C₂×C₄×C₅	—	2	4	no	—
142	C₂×C₄×C₅	—	4	2	no	—
143	C₂×C₄×C₅	—	4	2	no	—
144	C₂×C₄×C₅	—	4	2	no	—
145	C₂×C₄×C₅	—	4	2	no	—
146	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
147	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
148	C₂×C₄×C₅	C₃×D₈	1	2	yes	—
149	C₂×C₄×C₅	C₃×SD₁₆	1	2	yes	—
150	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
153	C₂×C₄×C₅	C₃×#41	1	2	yes	—
154	C₅×C₈	—	4	2	no	—
155	C₂²×Q₈	—	2	1	no	—
156	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
157	C₄⋊Q₈	—	2	2	no	—
158	C₄⋊Q₈	C₃×C₄×C₅	1	4	yes	complemented
159	C₂×C₄⋊C₄	—	2	1	no	—
160	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
161	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
162	C₂×C₄²	—	2	1	no	—
163	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
164	C₄⋊C₈	—	2	2	no	—
165	#95	—	2	1	no	—
166	C₂×C₃×C₅	#391	1	1	yes	characteristic
167	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
168	C₂×C₃×C₅	#419	1	1	yes	characteristic
169	C₂×C₃×C₅	#474	1	2	yes	—
170	C₂×C₃×C₅	#475	1	2	yes	—
171	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
172	C₃×Q₈	—	4	8	no	—
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	—	2	4	no	—
175	C₂×C₃×C₄	—	2	4	no	—
176	C₂×C₃×C₄	—	4	2	no	—
177	C₂×C₃×C₄	—	4	2	no	—
178	C₂×C₃×C₄	—	4	2	no	—
179	C₂×C₃×C₄	—	4	2	no	—
180	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
181	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
182	C₂×C₃×C₄	C₅×D₈	1	2	yes	—
183	C₂×C₃×C₄	C₅×SD₁₆	1	2	yes	—
184	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
187	C₂×C₃×C₄	C₅×#41	1	2	yes	—
188	C₃×C₈	—	4	2	no	—
189	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
190	C₂²×C₅	C₃×C₂²⋊Q₈	1	2	yes	—
191	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
192	C₂²×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
193	C₂²×C₅	C₃×#137	1	2	yes	—
194	C₄×C₅	—	2	2	no	—
195	C₄×C₅	—	4	4	no	—
196	C₄×C₅	—	4	4	no	—
197	C₄×C₅	C₃×#102	1	4	yes	—
198	C₄×C₅	C₃×C₄×Q₈	1	2	yes	—
199	C₄×C₅	C₃×#124	1	1	yes	characteristic
200	C₄×C₅	C₃×C₄⋊Q₈	1	1	yes	characteristic
201	C₂×Q₈	—	2	4	no	—
202	C₂×Q₈	—	4	2	no	—
203	C₂×Q₈	—	4	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
207	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
208	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	4	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	C₂×C₃×C₄×C₅	1	4	yes	—
215	C₄⋊C₄	C₃×C₅×Q₈	1	4	yes	complemented
216	C₄²	—	4	2	no	—
217	C₄²	C₂×C₃×C₄×C₅	1	2	yes	—
218	C₄²	C₃×C₅×D₄	1	2	yes	—
219	C₃×C₅	#1988	1	1	yes	characteristic, direct product factor, Hall
220	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
221	C₂²×C₃	C₅×C₂²⋊Q₈	1	2	yes	—
222	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
223	C₂²×C₃	C₂×C₅×SD₁₆	1	1	yes	characteristic
224	C₂²×C₃	C₅×#137	1	2	yes	—
225	C₃×C₄	—	2	2	no	—
226	C₃×C₄	—	4	4	no	—
227	C₃×C₄	—	4	4	no	—
228	C₃×C₄	C₅×#102	1	4	yes	—
229	C₃×C₄	C₄×C₅×Q₈	1	2	yes	—
230	C₃×C₄	C₅×#124	1	1	yes	characteristic
231	C₃×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
232	C₂×C₅	C₃×#391	1	1	yes	characteristic
233	C₂×C₅	C₂×C₃×#102	1	1	yes	characteristic
234	C₂×C₅	C₃×#419	1	1	yes	characteristic
235	C₂×C₅	C₃×#474	1	2	yes	—
236	C₂×C₅	C₃×#475	1	2	yes	—
237	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
238	Q₈	—	4	8	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
247	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
248	C₂×C₄	C₃×C₅×D₈	1	2	yes	—
249	C₂×C₄	C₃×C₅×SD₁₆	1	2	yes	—
250	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
251	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
252	C₂×C₄	C₂×C₃×C₅×Q₈	1	2	yes	—
253	C₂×C₄	C₃×C₅×#41	1	2	yes	—
254	C₈	—	4	2	no	—
255	C₂×C₃	C₅×#391	1	1	yes	characteristic
256	C₂×C₃	C₂×C₅×#102	1	1	yes	characteristic
257	C₂×C₃	C₅×#419	1	1	yes	characteristic
258	C₂×C₃	C₅×#474	1	2	yes	—
259	C₂×C₃	C₅×#475	1	2	yes	—
260	C₅	C₃×#1988	1	1	yes	characteristic, direct product factor, Sylow
261	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
262	C₂²	C₃×C₅×C₂²⋊Q₈	1	2	yes	—
263	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
264	C₂²	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
265	C₂²	C₃×C₅×#137	1	2	yes	—
266	C₄	—	2	2	no	—
267	C₄	—	4	4	no	—
268	C₄	—	4	4	no	—
269	C₄	C₃×C₅×#102	1	4	yes	—
270	C₄	C₃×C₄×C₅×Q₈	1	2	yes	—
271	C₄	C₃×C₅×#124	1	1	yes	characteristic
272	C₄	C₃×C₅×C₄⋊Q₈	1	1	yes	characteristic
273	C₃	C₅×#1988	1	1	yes	characteristic, direct product factor, Sylow
274	C₂	C₃×C₅×#391	1	1	yes	characteristic
275	C₂	C₂×C₃×C₅×#102	1	1	yes	characteristic
276	C₂	C₃×C₅×#419	1	1	yes	characteristic
277	C₂	C₃×C₅×#474	1	2	yes	—
278	C₂	C₃×C₅×#475	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃×C₅	C₂×D₄
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂³×C₃×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₅ ◁ C₂×C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₅×#1988	C₃	direct (N × H)
9. C₃×C₅×C₄⋊Q₈	C₄	semidirect
17. C₃×#1988	C₅	direct (N × H)
36. C₃×C₅×C₄⋊C₄	Q₈	semidirect
47. C₅×C₄⋊Q₈	C₃×C₄	semidirect
55. #1988	C₃×C₅	direct (N × H)
77. C₃×C₄⋊Q₈	C₄×C₅	semidirect
99. C₅×C₄⋊C₄	C₃×Q₈	semidirect
133. C₃×C₄⋊C₄	C₅×Q₈	semidirect
158. C₄⋊Q₈	C₃×C₄×C₅	semidirect
215. C₄⋊C₄	C₃×C₅×Q₈	semidirect
219. C₃×C₅	#1988	direct (N × H)
260. C₅	C₃×#1988	direct (N × H)
273. C₃	C₅×#1988	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×C₄⋊Q₈	2
C₂×C₃×C₅×C₄⋊C₈	2
C₃×C₄×C₅×C₄⋊C₄	2
C₃×C₅×#340	2
C₅×#1988	3
C₃×#1988	5

Subgroups — C3×C5×#1988 #14275022

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1988 #14275022