Subgroups — C₂×C₅×#5720 #14495178

Order 1920.

2688 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#127	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#939	C₂	1	6	yes	maximal, complemented
3	C₂×C₅×#937	C₂	1	3	yes	maximal, complemented
4	C₂×C₅×#935	C₂	1	3	yes	maximal, complemented
5	C₅×#5720	C₂	1	16	yes	maximal, direct product factor
6	C₂×C₅×#876	C₂	1	2	yes	maximal, complemented
7	C₂×C₅×#560	—	3	1	no	maximal, Hall
8	C₂⁴×C₅×D₃	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₅×D₄	C₂²	1	3	yes	—
10	C₂²×C₃×C₅×D₄	C₂²	1	3	yes	complemented
11	C₂²×C₅×#67	C₂²	1	6	yes	—
12	C₂³×C₅×Dic₃	C₂²	1	3	yes	complemented
13	C₃×C₅×#127	C₂²	1	8	yes	complemented
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic, complemented
15	C₂×C₅×#217	C₂²	1	3	yes	—
16	C₂×C₅×#217	C₂²	1	3	yes	complemented
17	C₅×#939	C₂²	1	48	yes	complemented
18	C₅×#937	C₂²	1	24	yes	complemented
19	C₅×#935	C₂²	1	24	yes	complemented
20	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	6	yes	—
21	C₂×C₅×Dic₃⋊C₄	C₂²	1	6	yes	complemented
22	C₅×#876	C₂²	1	16	yes	complemented
23	C₂×#5720	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
24	C₅×#560	—	3	16	no	—
25	C₂×C₅×#127	D₃	1	1	yes	characteristic, complemented
26	C₂×C₅×#126	—	3	2	no	—
27	C₂×C₅×#123	—	3	3	no	—
28	C₂×C₅×#121	—	3	6	no	—
29	C₂×C₅×#120	—	3	3	no	—
30	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
31	C₂⁴×C₃×C₅	C₂³	1	3	yes	—
32	C₂³×C₅×D₃	—	4	6	no	—
33	C₂³×C₅×D₃	C₂³	1	2	yes	—
34	C₂³×C₅×D₃	C₂³	1	4	yes	—
35	C₂×C₃×C₅×D₄	—	4	6	no	—
36	C₂×C₃×C₅×D₄	—	4	6	no	—
37	C₂×C₃×C₅×D₄	C₂³	1	12	yes	—
38	C₂×C₃×C₅×D₄	C₂³	1	12	yes	complemented
39	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
40	C₂×C₅×#67	—	4	12	no	—
41	C₂×C₅×#67	C₂³	1	24	yes	—
42	C₂²×C₅×Dic₃	—	4	6	no	—
43	C₂²×C₅×Dic₃	C₂³	1	6	yes	—
44	C₂²×C₅×Dic₃	C₂³	1	12	yes	complemented
45	C₃×C₄²×C₅	C₂³	1	4	yes	complemented
46	C₅×#217	C₂³	1	12	yes	—
47	C₅×#217	C₂³	1	12	yes	complemented
48	C₅×(C₂×D₃)⋊C₄	C₂³	1	24	yes	—
49	C₅×Dic₃⋊C₄	C₂³	1	24	yes	complemented
50	C₂×C₃×#127	C₂×C₅	1	1	yes	characteristic, complemented
51	C₂×#939	C₂×C₅	1	6	yes	complemented
52	C₂×#937	C₂×C₅	1	3	yes	complemented
53	C₂×#935	C₂×C₅	1	3	yes	complemented
54	#5720	C₂×C₅	1	16	yes	direct product factor
55	C₂×#876	C₂×C₅	1	2	yes	complemented
56	C₂⁵×C₅	—	3	1	no	—
57	C₂²×C₅×D₄	—	3	6	no	—
58	C₂²×C₅×D₄	C₂×D₃	1	3	yes	—
59	C₂²×C₅×D₄	C₂×D₃	1	3	yes	complemented
60	C₂³×C₄×C₅	—	3	3	no	—
61	C₅×#127	C₂×D₃	1	8	yes	complemented
62	C₅×#126	—	3	16	no	—
63	C₅×#123	—	3	24	no	—
64	C₅×#121	—	3	48	no	—
65	C₅×#120	—	3	24	no	—
66	C₂×C₅×C₄⋊C₄	—	3	6	no	—
67	C₂×C₅×C₂²⋊C₄	—	3	3	no	—
68	C₂×C₅×C₂²⋊C₄	—	3	3	no	—
69	C₂×C₅×C₂²⋊C₄	—	3	6	no	—
70	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic, complemented
71	C₂×#560	—	3	1	no	Sylow
72	C₂³×C₃×C₅	—	2	6	no	—
73	C₂³×C₃×C₅	—	2	12	no	—
74	C₂³×C₃×C₅	—	2	6	no	—
75	C₂³×C₃×C₅	C₂⁴	1	2	yes	—
76	C₂³×C₃×C₅	C₂⁴	1	6	yes	—
77	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
78	C₂²×C₅×D₃	—	2	12	no	—
79	C₂²×C₅×D₃	—	4	12	no	—
80	C₂²×C₅×D₃	—	4	16	no	—
81	C₂²×C₅×D₃	C₂⁴	1	4	yes	—
82	C₃×C₅×D₄	—	4	12	no	—
83	C₃×C₅×D₄	—	4	12	no	—
84	C₂×C₃×C₄×C₅	—	2	6	no	—
85	C₂×C₃×C₄×C₅	C₂⁴	1	6	yes	—
86	C₅×#67	—	4	24	no	—
87	C₂×C₅×Dic₃	—	2	12	no	—
88	C₂×C₅×Dic₃	—	4	12	no	—
89	C₂×C₅×Dic₃	C₂⁴	1	12	yes	—
90	C₂⁴×D₃	C₂²×C₅	1	1	yes	characteristic
91	C₂²×C₃×D₄	C₂²×C₅	1	3	yes	—
92	C₂²×C₃×D₄	C₂²×C₅	1	3	yes	complemented
93	C₂²×#67	C₂²×C₅	1	6	yes	—
94	C₂³×Dic₃	C₂²×C₅	1	3	yes	complemented
95	C₃×#127	C₂²×C₅	1	8	yes	complemented
96	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic, complemented
97	C₂×#217	C₂²×C₅	1	3	yes	—
98	C₂×#217	C₂²×C₅	1	3	yes	complemented
99	#939	C₂²×C₅	1	48	yes	complemented
100	#937	C₂²×C₅	1	24	yes	complemented
101	#935	C₂²×C₅	1	24	yes	complemented
102	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	6	yes	—
103	C₂×Dic₃⋊C₄	C₂²×C₅	1	6	yes	complemented
104	#876	C₂²×C₅	1	16	yes	complemented
105	C₂⁴×C₅	—	3	2	no	—
106	C₂⁴×C₅	—	3	4	no	—
107	C₂⁴×C₅	—	12	6	no	—
108	C₂⁴×C₅	C₂²×D₃	1	1	yes	characteristic
109	C₂⁴×C₅	C₂²×D₃	1	3	yes	—
110	C₂×C₅×D₄	—	3	24	no	—
111	C₂×C₅×D₄	—	4	6	no	—
112	C₂×C₅×D₄	—	4	6	no	—
113	C₂×C₅×D₄	—	12	12	no	—
114	C₂×C₅×D₄	C₂²×D₃	1	12	yes	—
115	C₂×C₅×D₄	C₂²×D₃	1	12	yes	complemented
116	C₂²×C₄×C₅	—	3	6	no	—
117	C₂²×C₄×C₅	—	3	12	no	—
118	C₂²×C₄×C₅	—	12	6	no	—
119	C₂²×C₄×C₅	C₂²×D₃	1	3	yes	—
120	C₅×C₄⋊C₄	—	3	24	no	—
121	C₅×C₂²⋊C₄	—	3	12	no	—
122	C₅×C₂²⋊C₄	—	3	12	no	—
123	C₅×C₂²⋊C₄	—	3	24	no	—
124	C₄²×C₅	C₂²×D₃	1	4	yes	complemented
125	#560	—	3	16	no	—
126	C₂×#127	C₅×D₃	1	1	yes	characteristic, complemented
127	C₂×#126	—	3	2	no	—
128	C₂×#123	—	3	3	no	—
129	C₂×#121	—	3	6	no	—
130	C₂×#120	—	3	3	no	—
131	C₂²×C₃×C₅	—	2	6	no	—
132	C₂²×C₃×C₅	—	2	12	no	—
133	C₂²×C₃×C₅	—	2	6	no	—
134	C₂²×C₃×C₅	—	4	4	no	—
135	C₂²×C₃×C₅	—	4	12	no	—
136	C₂²×C₃×C₅	H(5,𝔽₂)	1	6	yes	—
137	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
138	C₂×C₅×D₃	—	2	12	no	—
139	C₂×C₅×D₃	—	4	16	no	—
140	C₂×C₅×D₃	—	4	8	no	—
141	C₃×C₄×C₅	—	2	6	no	—
142	C₅×Dic₃	—	2	12	no	—
143	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
144	C₂⁴×C₃	C₂³×C₅	1	3	yes	—
145	C₂³×D₃	—	4	6	no	—
146	C₂³×D₃	C₂³×C₅	1	2	yes	—
147	C₂³×D₃	C₂³×C₅	1	4	yes	—
148	C₂×C₃×D₄	—	4	6	no	—
149	C₂×C₃×D₄	—	4	6	no	—
150	C₂×C₃×D₄	C₂³×C₅	1	12	yes	—
151	C₂×C₃×D₄	C₂³×C₅	1	12	yes	complemented
152	C₂²×C₃×C₄	C₂³×C₅	1	3	yes	—
153	C₂×#67	—	4	12	no	—
154	C₂×#67	C₂³×C₅	1	24	yes	—
155	C₂²×Dic₃	—	4	6	no	—
156	C₂²×Dic₃	C₂³×C₅	1	6	yes	—
157	C₂²×Dic₃	C₂³×C₅	1	12	yes	complemented
158	C₃×C₄²	C₂³×C₅	1	4	yes	complemented
159	#217	C₂³×C₅	1	12	yes	—
160	#217	C₂³×C₅	1	12	yes	complemented
161	(C₂×D₃)⋊C₄	C₂³×C₅	1	24	yes	—
162	Dic₃⋊C₄	C₂³×C₅	1	24	yes	complemented
163	C₂³×C₅	—	2	6	no	—
164	C₂³×C₅	—	2	12	no	—
165	C₂³×C₅	—	2	6	no	—
166	C₂³×C₅	—	3	4	no	—
167	C₂³×C₅	—	6	12	no	—
168	C₂³×C₅	—	12	12	no	—
169	C₂³×C₅	—	12	16	no	—
170	C₂³×C₅	C₂³×D₃	1	2	yes	—
171	C₂³×C₅	C₂³×D₃	1	6	yes	—
172	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
173	C₅×D₄	—	4	12	no	—
174	C₅×D₄	—	4	12	no	—
175	C₅×D₄	—	12	24	no	—
176	C₂×C₄×C₅	—	2	6	no	—
177	C₂×C₄×C₅	—	3	12	no	—
178	C₂×C₄×C₅	—	6	12	no	—
179	C₂×C₄×C₅	—	12	12	no	—
180	C₂×C₄×C₅	C₂³×D₃	1	6	yes	—
181	C₂⁵	—	3	1	no	—
182	C₂²×D₄	—	3	6	no	—
183	C₂²×D₄	C₂×C₅×D₃	1	3	yes	—
184	C₂²×D₄	C₂×C₅×D₃	1	3	yes	complemented
185	C₂³×C₄	—	3	3	no	—
186	#127	C₂×C₅×D₃	1	8	yes	complemented
187	#126	—	3	16	no	—
188	#123	—	3	24	no	—
189	#121	—	3	48	no	—
190	#120	—	3	24	no	—
191	C₂×C₄⋊C₄	—	3	6	no	—
192	C₂×C₂²⋊C₄	—	3	3	no	—
193	C₂×C₂²⋊C₄	—	3	3	no	—
194	C₂×C₂²⋊C₄	—	3	6	no	—
195	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic, complemented
196	C₂×C₃×C₅	—	4	2	no	—
197	C₂×C₃×C₅	—	4	6	no	—
198	C₂×C₃×C₅	#560	1	4	yes	complemented
199	C₂×C₃×C₅	C₂×H(5,𝔽₂)	1	3	yes	—
200	C₅×D₃	—	4	4	no	—
201	C₂³×C₃	—	2	6	no	—
202	C₂³×C₃	—	2	12	no	—
203	C₂³×C₃	—	2	6	no	—
204	C₂³×C₃	C₂⁴×C₅	1	2	yes	—
205	C₂³×C₃	C₂⁴×C₅	1	6	yes	—
206	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
207	C₂²×D₃	—	2	12	no	—
208	C₂²×D₃	—	4	12	no	—
209	C₂²×D₃	—	4	16	no	—
210	C₂²×D₃	C₂⁴×C₅	1	4	yes	—
211	C₃×D₄	—	4	12	no	—
212	C₃×D₄	—	4	12	no	—
213	C₂×C₃×C₄	—	2	6	no	—
214	C₂×C₃×C₄	C₂⁴×C₅	1	6	yes	—
215	#67	—	4	24	no	—
216	C₂×Dic₃	—	2	12	no	—
217	C₂×Dic₃	—	4	12	no	—
218	C₂×Dic₃	C₂⁴×C₅	1	12	yes	—
219	C₂²×C₅	—	2	6	no	—
220	C₂²×C₅	—	2	12	no	—
221	C₂²×C₅	—	2	6	no	—
222	C₂²×C₅	—	4	12	no	—
223	C₂²×C₅	—	4	4	no	—
224	C₂²×C₅	—	6	12	no	—
225	C₂²×C₅	—	12	16	no	—
226	C₂²×C₅	—	12	8	no	—
227	C₂²×C₅	#1004	1	6	yes	—
228	C₂²×C₅	C₂⁴×D₃	1	1	yes	characteristic
229	C₄×C₅	—	2	6	no	—
230	C₄×C₅	—	6	12	no	—
231	C₂⁴	—	3	2	no	—
232	C₂⁴	—	3	4	no	—
233	C₂⁴	—	12	6	no	—
234	C₂⁴	C₂²×C₅×D₃	1	1	yes	characteristic
235	C₂⁴	C₂²×C₅×D₃	1	3	yes	—
236	C₂×D₄	—	3	24	no	—
237	C₂×D₄	—	4	6	no	—
238	C₂×D₄	—	4	6	no	—
239	C₂×D₄	—	12	12	no	—
240	C₂×D₄	C₂²×C₅×D₃	1	12	yes	—
241	C₂×D₄	C₂²×C₅×D₃	1	12	yes	complemented
242	C₂²×C₄	—	3	6	no	—
243	C₂²×C₄	—	3	12	no	—
244	C₂²×C₄	—	12	6	no	—
245	C₂²×C₄	C₂²×C₅×D₃	1	3	yes	—
246	C₄⋊C₄	—	3	24	no	—
247	C₂²⋊C₄	—	3	12	no	—
248	C₂²⋊C₄	—	3	12	no	—
249	C₂²⋊C₄	—	3	24	no	—
250	C₄²	C₂²×C₅×D₃	1	4	yes	complemented
251	C₃×C₅	C₂×#560	1	1	yes	characteristic, complemented, Hall
252	C₂²×C₃	—	2	6	no	—
253	C₂²×C₃	—	2	12	no	—
254	C₂²×C₃	—	2	6	no	—
255	C₂²×C₃	—	4	4	no	—
256	C₂²×C₃	—	4	12	no	—
257	C₂²×C₃	C₅×H(5,𝔽₂)	1	6	yes	—
258	C₂²×C₃	C₂⁵×C₅	1	1	yes	characteristic
259	C₂×D₃	—	2	12	no	—
260	C₂×D₃	—	4	16	no	—
261	C₂×D₃	—	4	8	no	—
262	C₃×C₄	—	2	6	no	—
263	Dic₃	—	2	12	no	—
264	C₂×C₅	—	4	2	no	—
265	C₂×C₅	—	4	6	no	—
266	C₂×C₅	—	12	4	no	—
267	C₂×C₅	#5720	1	4	yes	direct product factor
268	C₂×C₅	C₂×#1004	1	3	yes	—
269	C₂³	—	2	6	no	—
270	C₂³	—	2	12	no	—
271	C₂³	—	2	6	no	—
272	C₂³	—	3	4	no	—
273	C₂³	—	6	12	no	—
274	C₂³	—	12	12	no	—
275	C₂³	—	12	16	no	—
276	C₂³	C₂³×C₅×D₃	1	2	yes	—
277	C₂³	C₂³×C₅×D₃	1	6	yes	—
278	C₂³	C₂³×C₅×D₃	1	1	yes	characteristic
279	D₄	—	4	12	no	—
280	D₄	—	4	12	no	—
281	D₄	—	12	24	no	—
282	C₂×C₄	—	2	6	no	—
283	C₂×C₄	—	3	12	no	—
284	C₂×C₄	—	6	12	no	—
285	C₂×C₄	—	12	12	no	—
286	C₂×C₄	C₂³×C₅×D₃	1	6	yes	—
287	C₂×C₃	—	4	2	no	—
288	C₂×C₃	—	4	6	no	—
289	C₂×C₃	C₅×#560	1	4	yes	complemented
290	C₂×C₃	C₂×C₅×H(5,𝔽₂)	1	3	yes	—
291	D₃	—	4	4	no	—
292	C₅	C₂×#5720	1	1	yes	characteristic, direct product factor, Sylow
293	C₂²	—	2	6	no	—
294	C₂²	—	2	12	no	—
295	C₂²	—	2	6	no	—
296	C₂²	—	4	4	no	—
297	C₂²	—	4	12	no	—
298	C₂²	—	6	12	no	—
299	C₂²	—	12	16	no	—
300	C₂²	—	12	8	no	—
301	C₂²	C₅×#1004	1	6	yes	—
302	C₂²	C₂⁴×C₅×D₃	1	1	yes	characteristic
303	C₄	—	2	6	no	—
304	C₄	—	6	12	no	—
305	C₃	C₂×C₅×#560	1	1	yes	characteristic, complemented, Sylow
306	C₂	—	4	2	no	—
307	C₂	—	4	6	no	—
308	C₂	—	12	4	no	—
309	C₂	C₅×#5720	1	4	yes	direct product factor
310	C₂	C₂×C₅×#1004	1	3	yes	—

	subgroup	G/H
center	C₂³×C₅	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁵×C₅
Frattini	C₂²	C₂⁴×C₅×D₃
Fitting	C₂×C₃×C₅×#127	C₂
socle	C₂³×C₃×C₅	C₂⁴
hypercenter	C₂×C₅×#127	D₃
hypocenter	C₃	C₂×C₅×#560
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₅×#127

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×#127	C₂	semidirect
2. C₂×C₅×#939	C₂	semidirect
3. C₂×C₅×#937	C₂	semidirect
4. C₂×C₅×#935	C₂	semidirect
5. C₅×#5720	C₂	direct (N × H)
6. C₂×C₅×#876	C₂	semidirect
10. C₂²×C₃×C₅×D₄	C₂²	semidirect
12. C₂³×C₅×Dic₃	C₂²	semidirect
13. C₃×C₅×#127	C₂²	semidirect
14. C₂×C₃×C₄²×C₅	C₂²	semidirect
16. C₂×C₅×#217	C₂²	semidirect
17. C₅×#939	C₂²	semidirect
18. C₅×#937	C₂²	semidirect
19. C₅×#935	C₂²	semidirect
21. C₂×C₅×Dic₃⋊C₄	C₂²	semidirect
22. C₅×#876	C₂²	semidirect
23. C₂×#5720	C₅	direct (N × H)
25. C₂×C₅×#127	D₃	semidirect
38. C₂×C₃×C₅×D₄	C₂³	semidirect
44. C₂²×C₅×Dic₃	C₂³	semidirect
45. C₃×C₄²×C₅	C₂³	semidirect
47. C₅×#217	C₂³	semidirect
49. C₅×Dic₃⋊C₄	C₂³	semidirect
50. C₂×C₃×#127	C₂×C₅	semidirect
51. C₂×#939	C₂×C₅	semidirect
52. C₂×#937	C₂×C₅	semidirect
53. C₂×#935	C₂×C₅	semidirect
54. #5720	C₂×C₅	direct (N × H)
55. C₂×#876	C₂×C₅	semidirect
59. C₂²×C₅×D₄	C₂×D₃	semidirect
61. C₅×#127	C₂×D₃	semidirect
70. C₂×C₄²×C₅	C₂×D₃	semidirect
92. C₂²×C₃×D₄	C₂²×C₅	semidirect
94. C₂³×Dic₃	C₂²×C₅	semidirect
95. C₃×#127	C₂²×C₅	semidirect
96. C₂×C₃×C₄²	C₂²×C₅	semidirect
98. C₂×#217	C₂²×C₅	semidirect
99. #939	C₂²×C₅	semidirect
100. #937	C₂²×C₅	semidirect
101. #935	C₂²×C₅	semidirect
103. C₂×Dic₃⋊C₄	C₂²×C₅	semidirect
104. #876	C₂²×C₅	semidirect
115. C₂×C₅×D₄	C₂²×D₃	semidirect
124. C₄²×C₅	C₂²×D₃	semidirect
126. C₂×#127	C₅×D₃	semidirect
151. C₂×C₃×D₄	C₂³×C₅	semidirect
157. C₂²×Dic₃	C₂³×C₅	semidirect
158. C₃×C₄²	C₂³×C₅	semidirect
160. #217	C₂³×C₅	semidirect
162. Dic₃⋊C₄	C₂³×C₅	semidirect
184. C₂²×D₄	C₂×C₅×D₃	semidirect
186. #127	C₂×C₅×D₃	semidirect
195. C₂×C₄²	C₂×C₅×D₃	semidirect
198. C₂×C₃×C₅	#560	semidirect
241. C₂×D₄	C₂²×C₅×D₃	semidirect
250. C₄²	C₂²×C₅×D₃	semidirect
251. C₃×C₅	C₂×#560	semidirect
267. C₂×C₅	#5720	direct (N × H)
289. C₂×C₃	C₅×#560	semidirect
292. C₅	C₂×#5720	direct (N × H)
305. C₃	C₂×C₅×#560	semidirect
309. C₂	C₅×#5720	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×#127	2
C₂×C₅×#939	2
C₂×C₅×#937	2
C₂×C₅×#935	2
C₅×#5720	2
C₂×C₅×#876	2
C₂×C₅×#560	3
C₂×#5720	5

Subgroups — C2×C5×#5720 #14495178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#5720 #14495178