Subgroups — C₃×C₅×#2616 #14275650

Order 1920.

1632 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×#41	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×C₂²⋊Q₈	C₂	1	3	yes	maximal, complemented
3	C₃×C₅×#395	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#394	C₂	1	6	yes	maximal, complemented
5	C₃×C₅×#380	C₂	1	3	yes	maximal, complemented
6	C₅×#2616	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×#41	—	2	12	no	—
8	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₅×D₄	C₂²	1	3	yes	—
11	C₂³×C₃×C₄×C₅	C₂²	1	3	yes	—
12	C₃×C₅×C₂²⋊Q₈	—	2	12	no	—
13	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	3	yes	—
14	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	6	yes	—
15	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	6	yes	—
16	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	6	yes	—
17	C₃×C₅×#95	C₂²	1	6	yes	—
18	C₃×#2616	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂²×C₅×#41	C₂×C₃	1	1	yes	characteristic
20	C₂×C₅×C₂²⋊Q₈	C₂×C₃	1	3	yes	complemented
21	C₅×#395	C₂×C₃	1	2	yes	complemented
22	C₅×#394	C₂×C₃	1	6	yes	complemented
23	C₅×#380	C₂×C₃	1	3	yes	complemented
24	C₂⁴×C₃×C₅	C₂³	1	3	yes	—
25	C₃×C₅×#41	—	2	32	no	—
26	C₂×C₃×C₅×Q₈	—	2	6	no	—
27	C₂×C₃×C₅×Q₈	—	2	6	no	—
28	C₂×C₃×C₅×D₄	—	2	12	no	—
29	C₂×C₃×C₅×D₄	—	2	6	no	—
30	C₂²×C₃×C₄×C₅	—	2	12	no	—
31	C₂²×C₃×C₄×C₅	D₄	1	12	yes	—
32	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
33	C₂²×C₃×C₄×C₅	C₂³	1	6	yes	—
34	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
35	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
36	C₃×C₅×C₄⋊C₄	—	2	6	no	—
37	C₃×C₅×C₄⋊C₄	—	2	12	no	—
38	C₃×C₅×C₂²⋊C₄	—	2	12	no	—
39	C₃×C₅×C₂²⋊C₄	—	4	6	no	—
40	C₂²×C₃×#41	C₂×C₅	1	1	yes	characteristic
41	C₂×C₃×C₂²⋊Q₈	C₂×C₅	1	3	yes	complemented
42	C₃×#395	C₂×C₅	1	2	yes	complemented
43	C₃×#394	C₂×C₅	1	6	yes	complemented
44	C₃×#380	C₂×C₅	1	3	yes	complemented
45	C₂×C₅×#41	—	2	12	no	—
46	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
47	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
48	C₂²×C₅×D₄	C₂²×C₃	1	3	yes	—
49	C₂³×C₄×C₅	C₂²×C₃	1	3	yes	—
50	C₅×C₂²⋊Q₈	—	2	12	no	—
51	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	3	yes	—
52	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	6	yes	—
53	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	6	yes	—
54	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	6	yes	—
55	C₅×#95	C₂²×C₃	1	6	yes	—
56	#2616	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
57	C₂³×C₃×C₅	—	2	6	no	—
58	C₂³×C₃×C₅	—	2	3	no	—
59	C₂³×C₃×C₅	—	2	6	no	—
60	C₂³×C₃×C₅	—	2	3	no	—
61	C₂³×C₃×C₅	C₂×D₄	1	6	yes	—
62	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
63	C₃×C₅×Q₈	—	2	8	no	—
64	C₃×C₅×Q₈	—	4	4	no	—
65	C₃×C₅×D₄	—	2	24	no	—
66	C₂×C₃×C₄×C₅	—	2	6	no	—
67	C₂×C₃×C₄×C₅	—	2	24	no	—
68	C₂×C₃×C₄×C₅	—	2	6	no	—
69	C₂×C₃×C₄×C₅	—	2	6	no	—
70	C₂×C₃×C₄×C₅	—	2	6	no	—
71	C₂×C₃×C₄×C₅	—	2	6	no	—
72	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
73	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
74	C₂×C₃×#41	—	2	12	no	—
75	C₂²×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
76	C₂²×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
77	C₂²×C₃×D₄	C₂²×C₅	1	3	yes	—
78	C₂³×C₃×C₄	C₂²×C₅	1	3	yes	—
79	C₃×C₂²⋊Q₈	—	2	12	no	—
80	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	3	yes	—
81	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	6	yes	—
82	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	6	yes	—
83	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	6	yes	—
84	C₃×#95	C₂²×C₅	1	6	yes	—
85	C₂⁴×C₅	C₂³×C₃	1	3	yes	—
86	C₅×#41	—	2	32	no	—
87	C₂×C₅×Q₈	—	2	6	no	—
88	C₂×C₅×Q₈	—	2	6	no	—
89	C₂×C₅×D₄	—	2	12	no	—
90	C₂×C₅×D₄	—	2	6	no	—
91	C₂²×C₄×C₅	—	2	12	no	—
92	C₂²×C₄×C₅	C₃×D₄	1	12	yes	—
93	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
94	C₂²×C₄×C₅	C₂³×C₃	1	6	yes	—
95	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
96	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
97	C₅×C₄⋊C₄	—	2	6	no	—
98	C₅×C₄⋊C₄	—	2	12	no	—
99	C₅×C₂²⋊C₄	—	2	12	no	—
100	C₅×C₂²⋊C₄	—	4	6	no	—
101	C₂²×#41	C₂×C₃×C₅	1	1	yes	characteristic
102	C₂×C₂²⋊Q₈	C₂×C₃×C₅	1	3	yes	complemented
103	#395	C₂×C₃×C₅	1	2	yes	complemented
104	#394	C₂×C₃×C₅	1	6	yes	complemented
105	#380	C₂×C₃×C₅	1	3	yes	complemented
106	C₂²×C₃×C₅	—	2	6	no	—
107	C₂²×C₃×C₅	—	2	6	no	—
108	C₂²×C₃×C₅	—	2	6	no	—
109	C₂²×C₃×C₅	—	4	6	no	—
110	C₂²×C₃×C₅	—	4	6	no	—
111	C₂²×C₃×C₅	C₂²×D₄	1	3	yes	—
112	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
113	C₂²×C₃×C₅	2₋¹⁺⁴	1	3	yes	—
114	C₃×C₄×C₅	—	2	6	no	—
115	C₃×C₄×C₅	—	4	2	no	—
116	C₃×C₄×C₅	—	4	6	no	—
117	C₃×C₄×C₅	#120	1	4	yes	—
118	C₂⁴×C₃	C₂³×C₅	1	3	yes	—
119	C₃×#41	—	2	32	no	—
120	C₂×C₃×Q₈	—	2	6	no	—
121	C₂×C₃×Q₈	—	2	6	no	—
122	C₂×C₃×D₄	—	2	12	no	—
123	C₂×C₃×D₄	—	2	6	no	—
124	C₂²×C₃×C₄	—	2	12	no	—
125	C₂²×C₃×C₄	C₅×D₄	1	12	yes	—
126	C₂²×C₃×C₄	C₂³×C₅	1	6	yes	—
127	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
128	C₂²×C₃×C₄	C₂³×C₅	1	3	yes	—
129	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
130	C₃×C₄⋊C₄	—	2	6	no	—
131	C₃×C₄⋊C₄	—	2	12	no	—
132	C₃×C₂²⋊C₄	—	2	12	no	—
133	C₃×C₂²⋊C₄	—	4	6	no	—
134	C₂³×C₅	—	2	6	no	—
135	C₂³×C₅	—	2	3	no	—
136	C₂³×C₅	—	2	6	no	—
137	C₂³×C₅	—	2	3	no	—
138	C₂³×C₅	C₂×C₃×D₄	1	6	yes	—
139	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
140	C₅×Q₈	—	2	8	no	—
141	C₅×Q₈	—	4	4	no	—
142	C₅×D₄	—	2	24	no	—
143	C₂×C₄×C₅	—	2	6	no	—
144	C₂×C₄×C₅	—	2	24	no	—
145	C₂×C₄×C₅	—	2	6	no	—
146	C₂×C₄×C₅	—	2	6	no	—
147	C₂×C₄×C₅	—	2	6	no	—
148	C₂×C₄×C₅	—	2	6	no	—
149	C₂×C₄×C₅	C₂×C₃×D₄	1	6	yes	—
150	C₂×C₄×C₅	C₂×C₃×D₄	1	6	yes	—
151	C₂×#41	—	2	12	no	—
152	C₂²×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
153	C₂²×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
154	C₂²×D₄	C₂²×C₃×C₅	1	3	yes	—
155	C₂³×C₄	C₂²×C₃×C₅	1	3	yes	—
156	C₂²⋊Q₈	—	2	12	no	—
157	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	3	yes	—
158	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	6	yes	—
159	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	6	yes	—
160	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	6	yes	—
161	#95	C₂²×C₃×C₅	1	6	yes	—
162	C₂×C₃×C₅	—	4	6	no	—
163	C₂×C₃×C₅	C₂×#120	1	1	yes	characteristic
164	C₂×C₃×C₅	#536	1	3	yes	—
165	C₂×C₃×C₅	#537	1	3	yes	—
166	C₂³×C₃	—	2	6	no	—
167	C₂³×C₃	—	2	3	no	—
168	C₂³×C₃	—	2	6	no	—
169	C₂³×C₃	—	2	3	no	—
170	C₂³×C₃	C₂×C₅×D₄	1	6	yes	—
171	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
172	C₃×Q₈	—	2	8	no	—
173	C₃×Q₈	—	4	4	no	—
174	C₃×D₄	—	2	24	no	—
175	C₂×C₃×C₄	—	2	6	no	—
176	C₂×C₃×C₄	—	2	24	no	—
177	C₂×C₃×C₄	—	2	6	no	—
178	C₂×C₃×C₄	—	2	6	no	—
179	C₂×C₃×C₄	—	2	6	no	—
180	C₂×C₃×C₄	—	2	6	no	—
181	C₂×C₃×C₄	C₂×C₅×D₄	1	6	yes	—
182	C₂×C₃×C₄	C₂×C₅×D₄	1	6	yes	—
183	C₂²×C₅	—	2	6	no	—
184	C₂²×C₅	—	2	6	no	—
185	C₂²×C₅	—	2	6	no	—
186	C₂²×C₅	—	4	6	no	—
187	C₂²×C₅	—	4	6	no	—
188	C₂²×C₅	C₂²×C₃×D₄	1	3	yes	—
189	C₂²×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
190	C₂²×C₅	C₃×2₋¹⁺⁴	1	3	yes	—
191	C₄×C₅	—	2	6	no	—
192	C₄×C₅	—	4	2	no	—
193	C₄×C₅	—	4	6	no	—
194	C₄×C₅	C₃×#120	1	4	yes	—
195	C₂⁴	C₂³×C₃×C₅	1	3	yes	—
196	#41	—	2	32	no	—
197	C₂×Q₈	—	2	6	no	—
198	C₂×Q₈	—	2	6	no	—
199	C₂×D₄	—	2	12	no	—
200	C₂×D₄	—	2	6	no	—
201	C₂²×C₄	—	2	12	no	—
202	C₂²×C₄	C₃×C₅×D₄	1	12	yes	—
203	C₂²×C₄	C₂³×C₃×C₅	1	6	yes	—
204	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
205	C₂²×C₄	C₂³×C₃×C₅	1	3	yes	—
206	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
207	C₄⋊C₄	—	2	6	no	—
208	C₄⋊C₄	—	2	12	no	—
209	C₂²⋊C₄	—	2	12	no	—
210	C₂²⋊C₄	—	4	6	no	—
211	C₃×C₅	#2616	1	1	yes	characteristic, direct product factor, Hall
212	C₂²×C₃	—	2	6	no	—
213	C₂²×C₃	—	2	6	no	—
214	C₂²×C₃	—	2	6	no	—
215	C₂²×C₃	—	4	6	no	—
216	C₂²×C₃	—	4	6	no	—
217	C₂²×C₃	C₂²×C₅×D₄	1	3	yes	—
218	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
219	C₂²×C₃	C₅×2₋¹⁺⁴	1	3	yes	—
220	C₃×C₄	—	2	6	no	—
221	C₃×C₄	—	4	2	no	—
222	C₃×C₄	—	4	6	no	—
223	C₃×C₄	C₅×#120	1	4	yes	—
224	C₂×C₅	—	4	6	no	—
225	C₂×C₅	C₂×C₃×#120	1	1	yes	characteristic
226	C₂×C₅	C₃×#536	1	3	yes	—
227	C₂×C₅	C₃×#537	1	3	yes	—
228	C₂³	—	2	6	no	—
229	C₂³	—	2	3	no	—
230	C₂³	—	2	6	no	—
231	C₂³	—	2	3	no	—
232	C₂³	C₂×C₃×C₅×D₄	1	6	yes	—
233	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
234	Q₈	—	2	8	no	—
235	Q₈	—	4	4	no	—
236	D₄	—	2	24	no	—
237	C₂×C₄	—	2	6	no	—
238	C₂×C₄	—	2	24	no	—
239	C₂×C₄	—	2	6	no	—
240	C₂×C₄	—	2	6	no	—
241	C₂×C₄	—	2	6	no	—
242	C₂×C₄	—	2	6	no	—
243	C₂×C₄	C₂×C₃×C₅×D₄	1	6	yes	—
244	C₂×C₄	C₂×C₃×C₅×D₄	1	6	yes	—
245	C₂×C₃	—	4	6	no	—
246	C₂×C₃	C₂×C₅×#120	1	1	yes	characteristic
247	C₂×C₃	C₅×#536	1	3	yes	—
248	C₂×C₃	C₅×#537	1	3	yes	—
249	C₅	C₃×#2616	1	1	yes	characteristic, direct product factor, Sylow
250	C₂²	—	2	6	no	—
251	C₂²	—	2	6	no	—
252	C₂²	—	2	6	no	—
253	C₂²	—	4	6	no	—
254	C₂²	—	4	6	no	—
255	C₂²	C₂²×C₃×C₅×D₄	1	3	yes	—
256	C₂²	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic
257	C₂²	C₃×C₅×2₋¹⁺⁴	1	3	yes	—
258	C₄	—	2	6	no	—
259	C₄	—	4	2	no	—
260	C₄	—	4	6	no	—
261	C₄	C₃×C₅×#120	1	4	yes	—
262	C₃	C₅×#2616	1	1	yes	characteristic, direct product factor, Sylow
263	C₂	—	4	6	no	—
264	C₂	C₂×C₃×C₅×#120	1	1	yes	characteristic
265	C₂	C₃×C₅×#536	1	3	yes	—
266	C₂	C₃×C₅×#537	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₃×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃×C₅
prime radical	C₂²×C₃×C₅×#41	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×C₂²⋊Q₈	C₂	semidirect
3. C₃×C₅×#395	C₂	semidirect
4. C₃×C₅×#394	C₂	semidirect
5. C₃×C₅×#380	C₂	semidirect
6. C₅×#2616	C₃	direct (N × H)
18. C₃×#2616	C₅	direct (N × H)
20. C₂×C₅×C₂²⋊Q₈	C₂×C₃	semidirect
21. C₅×#395	C₂×C₃	semidirect
22. C₅×#394	C₂×C₃	semidirect
23. C₅×#380	C₂×C₃	semidirect
41. C₂×C₃×C₂²⋊Q₈	C₂×C₅	semidirect
42. C₃×#395	C₂×C₅	semidirect
43. C₃×#394	C₂×C₅	semidirect
44. C₃×#380	C₂×C₅	semidirect
56. #2616	C₃×C₅	direct (N × H)
102. C₂×C₂²⋊Q₈	C₂×C₃×C₅	semidirect
103. #395	C₂×C₃×C₅	semidirect
104. #394	C₂×C₃×C₅	semidirect
105. #380	C₂×C₃×C₅	semidirect
211. C₃×C₅	#2616	direct (N × H)
249. C₅	C₃×#2616	direct (N × H)
262. C₃	C₅×#2616	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃×C₅×#41	2
C₂×C₃×C₅×C₂²⋊Q₈	2
C₃×C₅×#395	2
C₃×C₅×#394	2
C₃×C₅×#380	2
C₅×#2616	3
C₃×#2616	5

Subgroups — C3×C5×#2616 #14275650

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2616 #14275650