Subgroups — C₂×D₁₅×#127 #14479960

Order 1920.

6000 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₄×D₁₅	C₂	1	6	yes	maximal, complemented
2	C₂×C₃×C₅×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#91829	C₂	1	6	yes	maximal, complemented
4	D₁₅×#127	C₂	1	16	yes	maximal, direct product factor
5	C₂×#91763	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄²×D₁₅	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×D₅×#127	—	3	1	no	maximal, Hall
8	C₂⁴×D₁₅	C₂²	1	4	yes	—
9	C₂²×C₃×C₅×D₄	C₂²	1	6	yes	complemented
10	C₂²×#1220	C₂²	1	12	yes	complemented
11	C₂×D₄×D₁₅	—	2	48	no	—
12	C₂×D₄×D₁₅	C₂²	1	48	yes	complemented
13	C₂²×D₆₀	C₂²	1	6	yes	complemented
14	C₂²×C₄×D₁₅	C₂²	1	3	yes	—
15	C₃×C₅×#127	C₂²	1	8	yes	complemented
16	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic, complemented
17	#91829	C₂²	1	48	yes	complemented
18	C₂×C₄×Dic₁₅	C₂²	1	3	yes	complemented
19	#91763	C₂²	1	8	yes	complemented
20	C₄²×D₁₅	C₂²	1	8	yes	complemented
21	C₂×D₃×#127	—	5	1	no	maximal, Hall
22	C₂²×D₄×D₅	—	3	6	no	—
23	C₂×C₅×#127	D₃	1	1	yes	characteristic, complemented
24	C₂×#4086	—	3	6	no	—
25	D₅×#127	—	3	16	no	—
26	C₂×#4020	—	3	1	no	—
27	C₂×C₄²×D₅	—	3	1	no	—
28	C₂⁴×C₃×C₅	C₂³	1	4	yes	—
29	C₂³×D₁₅	—	2	48	no	—
30	C₂³×D₁₅	C₂³	1	4	yes	—
31	C₂³×D₁₅	C₂³	1	16	yes	—
32	C₂³×D₁₅	C₂³	1	1	yes	characteristic
33	C₂×C₃×C₅×D₄	—	2	24	no	—
34	C₂×C₃×C₅×D₄	C₂³	1	24	yes	complemented
35	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
36	C₂×#1220	—	2	48	no	—
37	C₂×#1220	C₂³	1	48	yes	complemented
38	C₂²×Dic₁₅	C₂³	1	3	yes	—
39	D₄×D₁₅	—	2	192	no	—
40	C₂×D₆₀	—	2	24	no	—
41	C₂×D₆₀	C₂³	1	24	yes	complemented
42	C₂×C₄×D₁₅	D₄	1	24	yes	complemented
43	C₂×C₄×D₁₅	C₂³	1	12	yes	—
44	C₃×C₄²×C₅	C₂³	1	4	yes	complemented
45	C₄×Dic₁₅	C₂³	1	12	yes	complemented
46	C₂²×D₃×D₄	—	5	6	no	—
47	C₂×C₃×#127	D₅	1	1	yes	characteristic, complemented
48	C₂×#940	—	5	6	no	—
49	D₃×#127	—	5	16	no	—
50	C₂×#874	—	5	1	no	—
51	C₂×C₄²×D₃	—	5	1	no	—
52	C₂⁴×D₅	—	3	4	no	—
53	C₂²×C₅×D₄	C₂×D₃	1	6	yes	complemented
54	C₂²×C₂²⋊D₅	—	3	12	no	—
55	C₂×D₄×D₅	—	3	48	no	—
56	C₂×D₄×D₅	—	6	48	no	—
57	C₂²×D₂₀	—	3	6	no	—
58	C₂²×C₄×D₅	—	3	3	no	—
59	C₅×#127	C₂×D₃	1	8	yes	complemented
60	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic, complemented
61	#4086	—	3	48	no	—
62	C₂×C₄×Dic₅	—	3	3	no	—
63	#4020	—	3	8	no	—
64	C₄²×D₅	—	3	8	no	—
65	C₂²×#127	—	15	1	no	Sylow
66	C₂³×C₃×C₅	—	2	24	no	—
67	C₂³×C₃×C₅	C₂⁴	1	8	yes	—
68	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
69	C₂²×D₁₅	—	2	24	no	—
70	C₂²×D₁₅	—	2	96	no	—
71	C₂²×D₁₅	—	4	64	no	—
72	C₂²×D₁₅	C₂×D₄	1	12	yes	—
73	C₂²×D₁₅	C₂⁴	1	8	yes	—
74	C₂²×D₁₅	C₂⁴	1	2	yes	—
75	C₃×C₅×D₄	—	2	48	no	—
76	C₂×C₃×C₄×C₅	C₂×D₄	1	12	yes	complemented
77	C₂×C₃×C₄×C₅	C₂⁴	1	6	yes	—
78	#1220	—	2	96	no	—
79	C₂×Dic₁₅	C₂×D₄	1	12	yes	complemented
80	C₂×Dic₁₅	C₂⁴	1	6	yes	—
81	D₆₀	—	2	48	no	—
82	C₄×D₁₅	C₂×D₄	1	48	yes	complemented
83	C₂⁴×D₃	—	5	4	no	—
84	C₂²×C₃×D₄	C₂×D₅	1	6	yes	complemented
85	C₂²×#67	—	5	12	no	—
86	C₂×D₃×D₄	—	5	48	no	—
87	C₂×D₃×D₄	—	10	48	no	—
88	C₂²×D₁₂	—	5	6	no	—
89	C₂²×C₄×D₃	—	5	3	no	—
90	C₃×#127	C₂×D₅	1	8	yes	complemented
91	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic, complemented
92	#940	—	5	48	no	—
93	C₂×C₄×Dic₃	—	5	3	no	—
94	#874	—	5	8	no	—
95	C₄²×D₃	—	5	8	no	—
96	C₂⁴×C₅	C₂²×D₃	1	4	yes	—
97	C₂³×D₅	—	3	1	no	—
98	C₂³×D₅	—	3	4	no	—
99	C₂³×D₅	—	3	16	no	—
100	C₂³×D₅	—	6	48	no	—
101	C₂×C₅×D₄	—	2	24	no	—
102	C₂×C₅×D₄	C₂²×D₃	1	24	yes	complemented
103	C₂²×C₄×C₅	C₂²×D₃	1	3	yes	—
104	C₂×C₂²⋊D₅	—	3	48	no	—
105	C₂×C₂²⋊D₅	—	6	48	no	—
106	C₂²×Dic₅	—	3	3	no	—
107	D₄×D₅	—	6	192	no	—
108	C₂×D₂₀	—	3	24	no	—
109	C₂×D₂₀	—	6	24	no	—
110	C₂×C₄×D₅	—	3	24	no	—
111	C₂×C₄×D₅	—	3	12	no	—
112	C₄²×C₅	C₂²×D₃	1	4	yes	complemented
113	C₄×Dic₅	—	3	12	no	—
114	C₂³×D₄	—	15	6	no	—
115	C₂×#127	—	15	6	no	—
116	C₂×#127	—	15	16	no	—
117	C₂×#127	—	15	1	no	—
118	C₂×#127	D₁₅	1	1	yes	characteristic, direct product factor
119	C₂²×C₄²	—	15	1	no	—
120	C₂²×C₃×C₅	—	2	24	no	—
121	C₂²×C₃×C₅	—	4	16	no	—
122	C₂²×C₃×C₅	C₂²×D₄	1	6	yes	—
123	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
124	C₂×D₁₅	—	2	24	no	—
125	C₂×D₁₅	—	4	16	no	—
126	C₂×D₁₅	—	4	64	no	—
127	C₂×D₁₅	#127	1	16	yes	direct product factor
128	C₂×D₁₅	C₂²×D₄	1	12	yes	—
129	C₃×C₄×C₅	C₂²×D₄	1	12	yes	complemented
130	Dic₁₅	C₂²×D₄	1	12	yes	complemented
131	C₂⁴×C₃	C₂²×D₅	1	4	yes	—
132	C₂³×D₃	—	5	1	no	—
133	C₂³×D₃	—	5	4	no	—
134	C₂³×D₃	—	5	16	no	—
135	C₂³×D₃	—	10	48	no	—
136	C₂×C₃×D₄	—	2	24	no	—
137	C₂×C₃×D₄	C₂²×D₅	1	24	yes	complemented
138	C₂²×C₃×C₄	C₂²×D₅	1	3	yes	—
139	C₂×#67	—	5	48	no	—
140	C₂×#67	—	10	48	no	—
141	C₂²×Dic₃	—	5	3	no	—
142	D₃×D₄	—	10	192	no	—
143	C₂×D₁₂	—	5	24	no	—
144	C₂×D₁₂	—	10	24	no	—
145	C₂×C₄×D₃	—	5	24	no	—
146	C₂×C₄×D₃	—	5	12	no	—
147	C₃×C₄²	C₂²×D₅	1	4	yes	complemented
148	C₄×Dic₃	—	5	12	no	—
149	C₂³×C₅	—	2	24	no	—
150	C₂³×C₅	C₂³×D₃	1	8	yes	—
151	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
152	C₂²×D₅	—	3	12	no	—
153	C₂²×D₅	—	3	2	no	—
154	C₂²×D₅	—	3	8	no	—
155	C₂²×D₅	—	6	96	no	—
156	C₂²×D₅	—	6	24	no	—
157	C₂²×D₅	—	12	64	no	—
158	C₅×D₄	—	2	48	no	—
159	C₂×C₄×C₅	D₃×D₄	1	12	yes	complemented
160	C₂×C₄×C₅	C₂³×D₃	1	6	yes	—
161	C₂²⋊D₅	—	6	96	no	—
162	C₂×Dic₅	—	3	12	no	—
163	C₂×Dic₅	—	3	6	no	—
164	D₂₀	—	6	48	no	—
165	C₄×D₅	—	3	48	no	—
166	C₂⁵	—	15	4	no	—
167	C₂²×D₄	—	15	12	no	—
168	C₂²×D₄	—	15	48	no	—
169	C₂²×D₄	—	15	6	no	—
170	C₂²×D₄	—	30	48	no	—
171	C₂²×D₄	C₂×D₁₅	1	6	yes	complemented
172	C₂³×C₄	—	15	3	no	—
173	#127	—	15	48	no	—
174	#127	—	15	8	no	—
175	#127	C₂×D₁₅	1	8	yes	direct product factor
176	C₂×C₄²	—	15	3	no	—
177	C₂×C₄²	—	15	8	no	—
178	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic, complemented
179	C₂×C₃×C₅	—	4	8	no	—
180	C₂×C₃×C₅	C₂×#127	1	4	yes	complemented
181	C₂×C₃×C₅	C₂³×D₄	1	3	yes	—
182	D₁₅	—	4	8	no	—
183	D₁₅	C₂×#127	1	8	yes	direct product factor
184	C₂³×C₃	—	2	24	no	—
185	C₂³×C₃	C₂³×D₅	1	8	yes	—
186	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
187	C₂²×D₃	—	5	12	no	—
188	C₂²×D₃	—	5	2	no	—
189	C₂²×D₃	—	5	8	no	—
190	C₂²×D₃	—	10	96	no	—
191	C₂²×D₃	—	10	24	no	—
192	C₂²×D₃	—	20	64	no	—
193	C₃×D₄	—	2	48	no	—
194	C₂×C₃×C₄	D₄×D₅	1	12	yes	complemented
195	C₂×C₃×C₄	C₂³×D₅	1	6	yes	—
196	#67	—	10	96	no	—
197	C₂×Dic₃	—	5	12	no	—
198	C₂×Dic₃	—	5	6	no	—
199	D₁₂	—	10	48	no	—
200	C₄×D₃	—	5	48	no	—
201	C₂²×C₅	—	2	24	no	—
202	C₂²×C₅	—	4	16	no	—
203	C₂²×C₅	C₂×D₃×D₄	1	6	yes	—
204	C₂²×C₅	C₂⁴×D₃	1	1	yes	characteristic
205	C₂×D₅	—	3	12	no	—
206	C₂×D₅	—	3	16	no	—
207	C₂×D₅	—	6	24	no	—
208	C₂×D₅	—	12	64	no	—
209	C₂×D₅	—	12	16	no	—
210	C₄×C₅	C₂×D₃×D₄	1	12	yes	complemented
211	Dic₅	—	3	12	no	—
212	C₂⁴	—	15	1	no	—
213	C₂⁴	—	15	16	no	—
214	C₂⁴	—	15	4	no	—
215	C₂⁴	—	30	48	no	—
216	C₂⁴	C₂²×D₁₅	1	4	yes	—
217	C₂×D₄	—	2	24	no	—
218	C₂×D₄	—	15	48	no	—
219	C₂×D₄	—	15	24	no	—
220	C₂×D₄	—	30	192	no	—
221	C₂×D₄	—	30	48	no	—
222	C₂×D₄	—	30	24	no	—
223	C₂×D₄	C₂²×D₁₅	1	24	yes	complemented
224	C₂²×C₄	—	15	3	no	—
225	C₂²×C₄	—	15	24	no	—
226	C₂²×C₄	—	15	12	no	—
227	C₂²×C₄	C₂²×D₁₅	1	3	yes	—
228	C₄²	—	15	12	no	—
229	C₄²	C₂²×D₁₅	1	4	yes	complemented
230	C₃×C₅	C₂²×#127	1	1	yes	characteristic, complemented, Hall
231	C₂²×C₃	—	2	24	no	—
232	C₂²×C₃	—	4	16	no	—
233	C₂²×C₃	C₂×D₄×D₅	1	6	yes	—
234	C₂²×C₃	C₂⁴×D₅	1	1	yes	characteristic
235	C₂×D₃	—	5	16	no	—
236	C₂×D₃	—	5	12	no	—
237	C₂×D₃	—	10	24	no	—
238	C₂×D₃	—	20	64	no	—
239	C₂×D₃	—	20	16	no	—
240	C₃×C₄	C₂×D₄×D₅	1	12	yes	complemented
241	Dic₃	—	5	12	no	—
242	C₂×C₅	—	4	8	no	—
243	C₂×C₅	D₃×#127	1	4	yes	complemented
244	C₂×C₅	C₂²×D₃×D₄	1	3	yes	—
245	D₅	—	3	8	no	—
246	D₅	—	12	8	no	—
247	C₂³	—	2	24	no	—
248	C₂³	—	15	12	no	—
249	C₂³	—	15	2	no	—
250	C₂³	—	15	8	no	—
251	C₂³	—	30	96	no	—
252	C₂³	—	30	24	no	—
253	C₂³	—	60	64	no	—
254	C₂³	C₂³×D₁₅	1	8	yes	—
255	C₂³	C₂³×D₁₅	1	1	yes	characteristic
256	D₄	—	2	48	no	—
257	D₄	—	30	96	no	—
258	D₄	—	30	48	no	—
259	C₂×C₄	—	15	12	no	—
260	C₂×C₄	—	15	6	no	—
261	C₂×C₄	—	15	48	no	—
262	C₂×C₄	D₄×D₁₅	1	12	yes	complemented
263	C₂×C₄	C₂³×D₁₅	1	6	yes	—
264	C₂×C₃	—	4	8	no	—
265	C₂×C₃	D₅×#127	1	4	yes	complemented
266	C₂×C₃	C₂²×D₄×D₅	1	3	yes	—
267	D₃	—	5	8	no	—
268	D₃	—	20	8	no	—
269	C₅	C₂×D₃×#127	1	1	yes	characteristic, complemented, Sylow
270	C₂²	—	2	24	no	—
271	C₂²	—	4	16	no	—
272	C₂²	—	15	12	no	—
273	C₂²	—	15	16	no	—
274	C₂²	—	30	24	no	—
275	C₂²	—	60	64	no	—
276	C₂²	—	60	16	no	—
277	C₂²	C₂×D₄×D₁₅	1	6	yes	—
278	C₂²	C₂⁴×D₁₅	1	1	yes	characteristic
279	C₄	—	15	12	no	—
280	C₄	C₂×D₄×D₁₅	1	12	yes	complemented
281	C₃	C₂×D₅×#127	1	1	yes	characteristic, complemented, Sylow
282	C₂	—	4	8	no	—
283	C₂	—	15	8	no	—
284	C₂	—	60	8	no	—
285	C₂	D₁₅×#127	1	4	yes	direct product factor
286	C₂	C₂²×D₄×D₁₅	1	3	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₁₅
commutator, cosocle	C₂²×C₃×C₅	C₂⁵
Frattini	C₂²	C₂⁴×D₁₅
Fitting	C₂×C₃×C₅×#127	C₂
socle	C₂³×C₃×C₅	C₂⁴
hypercenter	C₂×#127	D₁₅
hypocenter	C₃×C₅	C₂²×#127
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂×#127

Complements

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

N	H ≅ G/N	type
1. C₂²×D₄×D₁₅	C₂	semidirect
2. C₂×C₃×C₅×#127	C₂	semidirect
3. C₂×#91829	C₂	semidirect
4. D₁₅×#127	C₂	direct (N × H)
5. C₂×#91763	C₂	semidirect
6. C₂×C₄²×D₁₅	C₂	semidirect
9. C₂²×C₃×C₅×D₄	C₂²	semidirect
10. C₂²×#1220	C₂²	semidirect
12. C₂×D₄×D₁₅	C₂²	semidirect
13. C₂²×D₆₀	C₂²	semidirect
15. C₃×C₅×#127	C₂²	semidirect
16. C₂×C₃×C₄²×C₅	C₂²	semidirect
17. #91829	C₂²	semidirect
18. C₂×C₄×Dic₁₅	C₂²	semidirect
19. #91763	C₂²	semidirect
20. C₄²×D₁₅	C₂²	semidirect
23. C₂×C₅×#127	D₃	semidirect
34. C₂×C₃×C₅×D₄	C₂³	semidirect
37. C₂×#1220	C₂³	semidirect
41. C₂×D₆₀	C₂³	semidirect
42. C₂×C₄×D₁₅	D₄	semidirect
44. C₃×C₄²×C₅	C₂³	semidirect
45. C₄×Dic₁₅	C₂³	semidirect
47. C₂×C₃×#127	D₅	semidirect
53. C₂²×C₅×D₄	C₂×D₃	semidirect
59. C₅×#127	C₂×D₃	semidirect
60. C₂×C₄²×C₅	C₂×D₃	semidirect
76. C₂×C₃×C₄×C₅	C₂×D₄	semidirect
79. C₂×Dic₁₅	C₂×D₄	semidirect
82. C₄×D₁₅	C₂×D₄	semidirect
84. C₂²×C₃×D₄	C₂×D₅	semidirect
90. C₃×#127	C₂×D₅	semidirect
91. C₂×C₃×C₄²	C₂×D₅	semidirect
102. C₂×C₅×D₄	C₂²×D₃	semidirect
112. C₄²×C₅	C₂²×D₃	semidirect
118. C₂×#127	D₁₅	direct (N × H)
127. C₂×D₁₅	#127	direct (N × H)
129. C₃×C₄×C₅	C₂²×D₄	semidirect
130. Dic₁₅	C₂²×D₄	semidirect
137. C₂×C₃×D₄	C₂²×D₅	semidirect
147. C₃×C₄²	C₂²×D₅	semidirect
159. C₂×C₄×C₅	D₃×D₄	semidirect
171. C₂²×D₄	C₂×D₁₅	semidirect
175. #127	C₂×D₁₅	direct (N × H)
178. C₂×C₄²	C₂×D₁₅	semidirect
180. C₂×C₃×C₅	C₂×#127	semidirect
183. D₁₅	C₂×#127	direct (N × H)
194. C₂×C₃×C₄	D₄×D₅	semidirect
210. C₄×C₅	C₂×D₃×D₄	semidirect
223. C₂×D₄	C₂²×D₁₅	semidirect
229. C₄²	C₂²×D₁₅	semidirect
230. C₃×C₅	C₂²×#127	semidirect
240. C₃×C₄	C₂×D₄×D₅	semidirect
243. C₂×C₅	D₃×#127	semidirect
262. C₂×C₄	D₄×D₁₅	semidirect
265. C₂×C₃	D₅×#127	semidirect
269. C₅	C₂×D₃×#127	semidirect
280. C₄	C₂×D₄×D₁₅	semidirect
281. C₃	C₂×D₅×#127	semidirect
285. C₂	D₁₅×#127	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×D₄×D₁₅	2
C₂×C₃×C₅×#127	2
C₂×#91829	2
D₁₅×#127	2
C₂×#91763	2
C₂×C₄²×D₁₅	2
C₂×D₅×#127	3
C₂×D₃×#127	5

Subgroups — C2×D15×#127 #14479960

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₁₅×#127 #14479960