Subgroups — C₃×C₅×#2139 #14275173

Order 1920.

464 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#135	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₅×#107	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#358	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#357	C₂	1	2	yes	maximal, complemented
6	C₅×#2139	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×#41	—	2	1	no	—
8	C₃×C₅×#135	—	2	2	no	—
9	C₃×C₅×#135	C₄	1	4	yes	complemented
10	C₂×C₃×C₅×Q₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₅×SD₁₆	—	2	1	no	—
12	C₂×C₃×C₅×D₈	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
15	C₃×C₅×M₃₂	—	2	2	no	—
16	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
17	C₃×C₅×#107	—	2	1	no	—
18	C₃×C₅×#107	C₂²	1	2	yes	—
19	C₃×#2139	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₂×C₅×#135	C₂×C₃	1	1	yes	characteristic
21	C₂×C₅×M₃₂	C₂×C₃	1	1	yes	characteristic, complemented
22	C₂×C₅×#107	C₂×C₃	1	1	yes	characteristic, complemented
23	C₅×#358	C₂×C₃	1	2	yes	complemented
24	C₅×#357	C₂×C₃	1	2	yes	complemented
25	C₃×C₅×#41	—	2	4	no	—
26	C₃×C₅×#41	—	4	2	no	—
27	C₂×C₃×C₅×Q₈	—	2	1	no	—
28	C₂×C₃×C₅×D₄	—	2	1	no	—
29	C₂×C₃×C₅×D₄	—	4	1	no	—
30	C₂²×C₃×C₄×C₅	—	4	1	no	—
31	C₂²×C₃×C₄×C₅	—	4	1	no	—
32	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
33	C₃×C₅×Q₁₆	—	2	1	no	—
34	C₃×C₅×Q₁₆	C₂×C₄	1	2	yes	complemented
35	C₃×C₅×SD₁₆	—	2	4	no	—
36	C₃×C₅×D₈	—	2	1	no	—
37	C₃×C₅×D₈	C₂×C₄	1	2	yes	complemented
38	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
39	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
40	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
41	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
42	C₃×C₅×C₄⋊C₄	—	2	2	no	—
43	C₃×C₅×C₄⋊C₄	—	4	1	no	—
44	C₃×C₅×C₁₆	—	2	2	no	—
45	C₂×C₃×#135	C₂×C₅	1	1	yes	characteristic
46	C₂×C₃×M₃₂	C₂×C₅	1	1	yes	characteristic, complemented
47	C₂×C₃×#107	C₂×C₅	1	1	yes	characteristic, complemented
48	C₃×#358	C₂×C₅	1	2	yes	complemented
49	C₃×#357	C₂×C₅	1	2	yes	complemented
50	C₂×C₅×#41	—	2	1	no	—
51	C₅×#135	—	2	2	no	—
52	C₅×#135	C₃×C₄	1	4	yes	complemented
53	C₂×C₅×Q₁₆	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₅×SD₁₆	—	2	1	no	—
55	C₂×C₅×D₈	C₂²×C₃	1	1	yes	characteristic
56	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₅×C₄⋊C₄	—	2	1	no	—
58	C₅×M₃₂	—	2	2	no	—
59	C₂×C₅×C₁₆	C₂²×C₃	1	2	yes	—
60	C₅×#107	—	2	1	no	—
61	C₅×#107	C₂²×C₃	1	2	yes	—
62	#2139	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
63	C₂³×C₃×C₅	—	4	1	no	—
64	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
65	C₃×C₅×Q₈	—	2	2	no	—
66	C₃×C₅×Q₈	—	4	1	no	—
67	C₃×C₅×D₄	—	2	2	no	—
68	C₃×C₅×D₄	—	4	4	no	—
69	C₃×C₅×D₄	—	4	1	no	—
70	C₂×C₃×C₄×C₅	—	4	1	no	—
71	C₂×C₃×C₄×C₅	—	4	4	no	—
72	C₂×C₃×C₄×C₅	—	4	2	no	—
73	C₂×C₃×C₄×C₅	—	8	2	no	—
74	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
75	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
77	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
79	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
80	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×#41	—	2	1	no	—
82	C₃×#135	—	2	2	no	—
83	C₃×#135	C₄×C₅	1	4	yes	complemented
84	C₂×C₃×Q₁₆	C₂²×C₅	1	1	yes	characteristic
85	C₂×C₃×SD₁₆	—	2	1	no	—
86	C₂×C₃×D₈	C₂²×C₅	1	1	yes	characteristic
87	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
88	C₂×C₃×C₄⋊C₄	—	2	1	no	—
89	C₃×M₃₂	—	2	2	no	—
90	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	—
91	C₃×#107	—	2	1	no	—
92	C₃×#107	C₂²×C₅	1	2	yes	—
93	C₅×#41	—	2	4	no	—
94	C₅×#41	—	4	2	no	—
95	C₂×C₅×Q₈	—	2	1	no	—
96	C₂×C₅×D₄	—	2	1	no	—
97	C₂×C₅×D₄	—	4	1	no	—
98	C₂²×C₄×C₅	—	4	1	no	—
99	C₂²×C₄×C₅	—	4	1	no	—
100	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
101	C₅×Q₁₆	—	2	1	no	—
102	C₅×Q₁₆	C₂×C₃×C₄	1	2	yes	complemented
103	C₅×SD₁₆	—	2	4	no	—
104	C₅×D₈	—	2	1	no	—
105	C₅×D₈	C₂×C₃×C₄	1	2	yes	complemented
106	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
107	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
108	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
109	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
110	C₅×C₄⋊C₄	—	2	2	no	—
111	C₅×C₄⋊C₄	—	4	1	no	—
112	C₅×C₁₆	—	2	2	no	—
113	C₂×#135	C₂×C₃×C₅	1	1	yes	characteristic
114	C₂×M₃₂	C₂×C₃×C₅	1	1	yes	characteristic, complemented
115	C₂×#107	C₂×C₃×C₅	1	1	yes	characteristic, complemented
116	#358	C₂×C₃×C₅	1	2	yes	complemented
117	#357	C₂×C₃×C₅	1	2	yes	complemented
118	C₂²×C₃×C₅	—	2	1	no	—
119	C₂²×C₃×C₅	—	2	1	no	—
120	C₂²×C₃×C₅	—	4	2	no	—
121	C₂²×C₃×C₅	—	8	1	no	—
122	C₂²×C₃×C₅	—	8	1	no	—
123	C₂²×C₃×C₅	#102	1	2	yes	—
124	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
125	C₃×C₄×C₅	—	4	2	no	—
126	C₃×C₄×C₅	—	8	2	no	—
127	C₃×C₄×C₅	#102	1	2	yes	—
128	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
130	C₃×#41	—	2	4	no	—
131	C₃×#41	—	4	2	no	—
132	C₂×C₃×Q₈	—	2	1	no	—
133	C₂×C₃×D₄	—	2	1	no	—
134	C₂×C₃×D₄	—	4	1	no	—
135	C₂²×C₃×C₄	—	4	1	no	—
136	C₂²×C₃×C₄	—	4	1	no	—
137	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
138	C₃×Q₁₆	—	2	1	no	—
139	C₃×Q₁₆	C₂×C₄×C₅	1	2	yes	complemented
140	C₃×SD₁₆	—	2	4	no	—
141	C₃×D₈	—	2	1	no	—
142	C₃×D₈	C₂×C₄×C₅	1	2	yes	complemented
143	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
144	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
145	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
146	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
147	C₃×C₄⋊C₄	—	2	2	no	—
148	C₃×C₄⋊C₄	—	4	1	no	—
149	C₃×C₁₆	—	2	2	no	—
150	C₂³×C₅	—	4	1	no	—
151	C₂³×C₅	C₃×D₈	1	1	yes	characteristic
152	C₅×Q₈	—	2	2	no	—
153	C₅×Q₈	—	4	1	no	—
154	C₅×D₄	—	2	2	no	—
155	C₅×D₄	—	4	4	no	—
156	C₅×D₄	—	4	1	no	—
157	C₂×C₄×C₅	—	4	1	no	—
158	C₂×C₄×C₅	—	4	4	no	—
159	C₂×C₄×C₅	—	4	2	no	—
160	C₂×C₄×C₅	—	8	2	no	—
161	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
162	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
163	C₂×C₄×C₅	C₃×SD₁₆	1	2	yes	—
164	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
165	C₅×C₈	C₃×C₂²⋊C₄	1	2	yes	—
166	C₅×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
167	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
168	C₂×#41	—	2	1	no	—
169	#135	—	2	2	no	—
170	#135	C₃×C₄×C₅	1	4	yes	complemented
171	C₂×Q₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
172	C₂×SD₁₆	—	2	1	no	—
173	C₂×D₈	C₂²×C₃×C₅	1	1	yes	characteristic
174	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
175	C₂×C₄⋊C₄	—	2	1	no	—
176	M₃₂	—	2	2	no	—
177	C₂×C₁₆	C₂²×C₃×C₅	1	2	yes	—
178	#107	—	2	1	no	—
179	#107	C₂²×C₃×C₅	1	2	yes	—
180	C₂×C₃×C₅	—	2	2	no	—
181	C₂×C₃×C₅	—	8	2	no	—
182	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
183	C₂×C₃×C₅	#509	1	1	yes	characteristic
184	C₂×C₃×C₅	#510	1	1	yes	characteristic
185	C₂³×C₃	—	4	1	no	—
186	C₂³×C₃	C₅×D₈	1	1	yes	characteristic
187	C₃×Q₈	—	2	2	no	—
188	C₃×Q₈	—	4	1	no	—
189	C₃×D₄	—	2	2	no	—
190	C₃×D₄	—	4	4	no	—
191	C₃×D₄	—	4	1	no	—
192	C₂×C₃×C₄	—	4	1	no	—
193	C₂×C₃×C₄	—	4	4	no	—
194	C₂×C₃×C₄	—	4	2	no	—
195	C₂×C₃×C₄	—	8	2	no	—
196	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
197	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
198	C₂×C₃×C₄	C₅×SD₁₆	1	2	yes	—
199	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
200	C₃×C₈	C₅×C₂²⋊C₄	1	2	yes	—
201	C₃×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
202	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
203	C₂²×C₅	—	2	1	no	—
204	C₂²×C₅	—	2	1	no	—
205	C₂²×C₅	—	4	2	no	—
206	C₂²×C₅	—	8	1	no	—
207	C₂²×C₅	—	8	1	no	—
208	C₂²×C₅	C₃×#102	1	2	yes	—
209	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
210	C₄×C₅	—	4	2	no	—
211	C₄×C₅	—	8	2	no	—
212	C₄×C₅	C₃×#102	1	2	yes	—
213	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
214	C₄×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
215	#41	—	2	4	no	—
216	#41	—	4	2	no	—
217	C₂×Q₈	—	2	1	no	—
218	C₂×D₄	—	2	1	no	—
219	C₂×D₄	—	4	1	no	—
220	C₂²×C₄	—	4	1	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
223	Q₁₆	—	2	1	no	—
224	Q₁₆	C₂×C₃×C₄×C₅	1	2	yes	complemented
225	SD₁₆	—	2	4	no	—
226	D₈	—	2	1	no	—
227	D₈	C₂×C₃×C₄×C₅	1	2	yes	complemented
228	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
229	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
230	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
231	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	4	1	no	—
234	C₁₆	—	2	2	no	—
235	C₃×C₅	#2139	1	1	yes	characteristic, direct product factor, Hall
236	C₂²×C₃	—	2	1	no	—
237	C₂²×C₃	—	2	1	no	—
238	C₂²×C₃	—	4	2	no	—
239	C₂²×C₃	—	8	1	no	—
240	C₂²×C₃	—	8	1	no	—
241	C₂²×C₃	C₅×#102	1	2	yes	—
242	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
243	C₃×C₄	—	4	2	no	—
244	C₃×C₄	—	8	2	no	—
245	C₃×C₄	C₅×#102	1	2	yes	—
246	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
247	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
248	C₂×C₅	—	2	2	no	—
249	C₂×C₅	—	8	2	no	—
250	C₂×C₅	C₂×C₃×#102	1	1	yes	characteristic
251	C₂×C₅	C₃×#509	1	1	yes	characteristic
252	C₂×C₅	C₃×#510	1	1	yes	characteristic
253	C₂³	—	4	1	no	—
254	C₂³	C₃×C₅×D₈	1	1	yes	characteristic
255	Q₈	—	2	2	no	—
256	Q₈	—	4	1	no	—
257	D₄	—	2	2	no	—
258	D₄	—	4	4	no	—
259	D₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	8	2	no	—
264	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
265	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
266	C₂×C₄	C₃×C₅×SD₁₆	1	2	yes	—
267	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
268	C₈	C₃×C₅×C₂²⋊C₄	1	2	yes	—
269	C₈	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
270	C₈	C₂×C₃×C₅×D₄	1	1	yes	characteristic
271	C₂×C₃	—	2	2	no	—
272	C₂×C₃	—	8	2	no	—
273	C₂×C₃	C₂×C₅×#102	1	1	yes	characteristic
274	C₂×C₃	C₅×#509	1	1	yes	characteristic
275	C₂×C₃	C₅×#510	1	1	yes	characteristic
276	C₅	C₃×#2139	1	1	yes	characteristic, direct product factor, Sylow
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	8	1	no	—
281	C₂²	—	8	1	no	—
282	C₂²	C₃×C₅×#102	1	2	yes	—
283	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
284	C₄	—	4	2	no	—
285	C₄	—	8	2	no	—
286	C₄	C₃×C₅×#102	1	2	yes	—
287	C₄	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
288	C₄	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
289	C₃	C₅×#2139	1	1	yes	characteristic, direct product factor, Sylow
290	C₂	—	2	2	no	—
291	C₂	—	8	2	no	—
292	C₂	C₂×C₃×C₅×#102	1	1	yes	characteristic
293	C₂	C₃×C₅×#509	1	1	yes	characteristic
294	C₂	C₃×C₅×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂×D₈
commutator	C₈	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
prime radical	C₂×C₃×C₅×#135	C₂
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
2. C₂×C₃×C₅×M₃₂	C₂	semidirect
3. C₂×C₃×C₅×#107	C₂	semidirect
4. C₃×C₅×#358	C₂	semidirect
5. C₃×C₅×#357	C₂	semidirect
6. C₅×#2139	C₃	direct (N × H)
9. C₃×C₅×#135	C₄	semidirect
19. C₃×#2139	C₅	direct (N × H)
21. C₂×C₅×M₃₂	C₂×C₃	semidirect
22. C₂×C₅×#107	C₂×C₃	semidirect
23. C₅×#358	C₂×C₃	semidirect
24. C₅×#357	C₂×C₃	semidirect
34. C₃×C₅×Q₁₆	C₂×C₄	semidirect
37. C₃×C₅×D₈	C₂×C₄	semidirect
46. C₂×C₃×M₃₂	C₂×C₅	semidirect
47. C₂×C₃×#107	C₂×C₅	semidirect
48. C₃×#358	C₂×C₅	semidirect
49. C₃×#357	C₂×C₅	semidirect
52. C₅×#135	C₃×C₄	semidirect
62. #2139	C₃×C₅	direct (N × H)
83. C₃×#135	C₄×C₅	semidirect
102. C₅×Q₁₆	C₂×C₃×C₄	semidirect
105. C₅×D₈	C₂×C₃×C₄	semidirect
114. C₂×M₃₂	C₂×C₃×C₅	semidirect
115. C₂×#107	C₂×C₃×C₅	semidirect
116. #358	C₂×C₃×C₅	semidirect
117. #357	C₂×C₃×C₅	semidirect
139. C₃×Q₁₆	C₂×C₄×C₅	semidirect
142. C₃×D₈	C₂×C₄×C₅	semidirect
170. #135	C₃×C₄×C₅	semidirect
224. Q₁₆	C₂×C₃×C₄×C₅	semidirect
227. D₈	C₂×C₃×C₄×C₅	semidirect
235. C₃×C₅	#2139	direct (N × H)
276. C₅	C₃×#2139	direct (N × H)
289. C₃	C₅×#2139	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×#135	2
C₂×C₃×C₅×M₃₂	2
C₂×C₃×C₅×#107	2
C₃×C₅×#358	2
C₃×C₅×#357	2
C₅×#2139	3
C₃×#2139	5

Subgroups — C3×C5×#2139 #14275173

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2139 #14275173