Subgroups — C₂×C₅×#5155 #14438832

Order 1920.

1184 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×D₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#936	C₂	1	2	yes	maximal, complemented
3	C₂²×C₅×#214	C₂	1	2	yes	maximal, complemented
4	C₂²×C₅×Dic₁₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#5155	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₅×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅×#495	—	3	1	no	maximal, Hall
8	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
9	C₂²×C₅×Dic₆	C₂²	1	2	yes	—
10	C₂×C₃×C₅×D₈	—	2	4	no	—
11	C₂×C₃×C₅×D₈	C₂²	1	4	yes	—
12	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₅×#217	—	2	4	no	—
14	C₅×#936	—	2	4	no	—
15	C₅×#936	C₂²	1	8	yes	complemented
16	C₂×C₅×#214	—	2	8	no	—
17	C₂×C₅×#214	C₂²	1	8	yes	complemented
18	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
19	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₅×Dic₁₂	—	2	4	no	—
21	C₂×C₅×Dic₁₂	C₂²	1	4	yes	complemented
22	C₅×C₈×Dic₃	C₂²	1	4	yes	complemented
23	C₂×#5155	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
24	C₂²×C₅×Q₁₆	—	3	1	no	—
25	C₂²×C₅×SD₁₆	—	3	2	no	—
26	C₂²×C₅×D₈	D₃	1	1	yes	characteristic
27	C₂×C₅×#124	—	3	2	no	—
28	C₅×#495	—	3	8	no	—
29	C₂×C₄×C₅×C₈	—	3	1	no	—
30	C₂⁴×C₃×C₅	—	2	2	no	—
31	C₂×C₃×C₅×D₄	—	2	4	no	—
32	C₂×C₃×C₅×D₄	—	2	4	no	—
33	C₂×C₃×C₅×D₄	—	2	2	no	—
34	C₂×C₃×C₅×D₄	C₂³	1	4	yes	—
35	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
36	C₂²×C₅×Dic₃	—	2	2	no	—
37	C₂²×C₅×Dic₃	D₄	1	2	yes	—
38	C₂×C₅×Dic₆	—	2	4	no	—
39	C₂×C₅×Dic₆	—	2	4	no	—
40	C₂×C₅×Dic₆	—	2	2	no	—
41	C₂×C₅×Dic₆	C₂³	1	4	yes	—
42	C₃×C₅×D₈	—	2	8	no	—
43	C₂×C₃×C₅×C₈	D₄	1	4	yes	—
44	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
45	C₅×#217	—	2	16	no	—
46	C₅×#214	—	2	16	no	—
47	C₄×C₅×Dic₃	D₄	1	2	yes	—
48	C₄×C₅×Dic₃	C₂³	1	2	yes	—
49	C₂×C₅×C₃⋊C₈	D₄	1	4	yes	—
50	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
51	C₅×Dic₁₂	—	2	8	no	—
52	C₂²×C₃×D₈	C₂×C₅	1	1	yes	characteristic
53	C₂×#936	C₂×C₅	1	2	yes	complemented
54	C₂²×#214	C₂×C₅	1	2	yes	complemented
55	C₂²×Dic₁₂	C₂×C₅	1	1	yes	characteristic, complemented
56	#5155	C₂×C₅	1	8	yes	direct product factor
57	C₂×C₈×Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
58	C₂²×C₅×Q₈	—	3	2	no	—
59	C₂²×C₅×D₄	C₂×D₃	1	2	yes	—
60	C₂×C₅×Q₁₆	—	3	4	no	—
61	C₂×C₅×Q₁₆	—	6	4	no	—
62	C₂×C₅×SD₁₆	—	3	8	no	—
63	C₂×C₅×SD₁₆	—	6	8	no	—
64	C₂×C₅×D₈	—	2	4	no	—
65	C₂×C₅×D₈	C₂×D₃	1	4	yes	—
66	C₂²×C₅×C₈	—	3	1	no	—
67	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
68	C₅×#124	—	3	8	no	—
69	C₅×#124	—	6	4	no	—
70	C₂×C₅×C₂²⋊C₄	—	6	4	no	—
71	C₂×C₄²×C₅	—	3	1	no	—
72	C₄×C₅×C₈	—	3	4	no	—
73	C₂×#495	—	3	1	no	Sylow
74	C₂³×C₃×C₅	—	2	4	no	—
75	C₂³×C₃×C₅	—	4	4	no	—
76	C₂³×C₃×C₅	—	4	8	no	—
77	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
78	C₃×C₅×D₄	—	2	4	no	—
79	C₃×C₅×D₄	—	2	4	no	—
80	C₃×C₅×D₄	—	4	4	no	—
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
83	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
84	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
85	C₂×C₅×Dic₃	—	2	4	no	—
86	C₂×C₅×Dic₃	—	2	4	no	—
87	C₂×C₅×Dic₃	—	4	4	no	—
88	C₂×C₅×Dic₃	C₂×D₄	1	4	yes	—
89	C₅×Dic₆	—	2	4	no	—
90	C₅×Dic₆	—	2	4	no	—
91	C₅×Dic₆	—	4	4	no	—
92	C₃×C₅×C₈	C₂×D₄	1	4	yes	—
93	C₅×C₃⋊C₈	C₂×D₄	1	4	yes	—
94	C₂²×C₃×D₄	C₂²×C₅	1	2	yes	—
95	C₂²×Dic₆	C₂²×C₅	1	2	yes	—
96	C₂×C₃×D₈	—	2	4	no	—
97	C₂×C₃×D₈	C₂²×C₅	1	4	yes	—
98	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
99	C₂×#217	—	2	4	no	—
100	#936	—	2	4	no	—
101	#936	C₂²×C₅	1	8	yes	complemented
102	C₂×#214	—	2	8	no	—
103	C₂×#214	C₂²×C₅	1	8	yes	complemented
104	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
105	C₂²×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
106	C₂×Dic₁₂	—	2	4	no	—
107	C₂×Dic₁₂	C₂²×C₅	1	4	yes	complemented
108	C₈×Dic₃	C₂²×C₅	1	4	yes	complemented
109	C₂⁴×C₅	—	2	2	no	—
110	C₂×C₅×Q₈	—	3	4	no	—
111	C₂×C₅×Q₈	—	6	2	no	—
112	C₂×C₅×Q₈	—	6	4	no	—
113	C₂×C₅×Q₈	—	6	4	no	—
114	C₂×C₅×D₄	—	2	4	no	—
115	C₂×C₅×D₄	—	2	2	no	—
116	C₂×C₅×D₄	—	2	4	no	—
117	C₂×C₅×D₄	C₂²×D₃	1	4	yes	—
118	C₂²×C₄×C₅	—	3	2	no	—
119	C₂²×C₄×C₅	—	6	2	no	—
120	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
121	C₅×Q₁₆	—	6	8	no	—
122	C₅×SD₁₆	—	6	16	no	—
123	C₅×D₈	—	2	8	no	—
124	C₂×C₅×C₈	—	3	4	no	—
125	C₂×C₅×C₈	—	3	2	no	—
126	C₂×C₅×C₈	#67	1	4	yes	—
127	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
128	C₅×C₂²⋊C₄	—	6	16	no	—
129	C₄²×C₅	—	3	2	no	—
130	C₄²×C₅	—	3	2	no	—
131	C₂²×Q₁₆	—	3	1	no	—
132	C₂²×SD₁₆	—	3	2	no	—
133	C₂²×D₈	C₅×D₃	1	1	yes	characteristic
134	C₂×#124	—	3	2	no	—
135	#495	—	3	8	no	—
136	C₂×C₄×C₈	—	3	1	no	—
137	C₂²×C₃×C₅	—	4	4	no	—
138	C₂²×C₃×C₅	—	4	8	no	—
139	C₂²×C₃×C₅	—	8	8	no	—
140	C₂²×C₃×C₅	#127	1	2	yes	—
141	C₂²×C₃×C₅	#135	1	4	yes	—
142	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
143	C₃×C₄×C₅	#127	1	2	yes	—
144	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
145	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
146	C₅×Dic₃	—	2	4	no	—
147	C₅×Dic₃	—	4	4	no	—
148	C₂⁴×C₃	—	2	2	no	—
149	C₂×C₃×D₄	—	2	4	no	—
150	C₂×C₃×D₄	—	2	4	no	—
151	C₂×C₃×D₄	—	2	2	no	—
152	C₂×C₃×D₄	C₂³×C₅	1	4	yes	—
153	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
154	C₂²×Dic₃	—	2	2	no	—
155	C₂²×Dic₃	C₅×D₄	1	2	yes	—
156	C₂×Dic₆	—	2	4	no	—
157	C₂×Dic₆	—	2	4	no	—
158	C₂×Dic₆	—	2	2	no	—
159	C₂×Dic₆	C₂³×C₅	1	4	yes	—
160	C₃×D₈	—	2	8	no	—
161	C₂×C₃×C₈	C₅×D₄	1	4	yes	—
162	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
163	#217	—	2	16	no	—
164	#214	—	2	16	no	—
165	C₄×Dic₃	C₅×D₄	1	2	yes	—
166	C₄×Dic₃	C₂³×C₅	1	2	yes	—
167	C₂×C₃⋊C₈	C₅×D₄	1	4	yes	—
168	C₂×C₃⋊C₈	C₂³×C₅	1	2	yes	—
169	Dic₁₂	—	2	8	no	—
170	C₂³×C₅	—	2	4	no	—
171	C₂³×C₅	—	4	8	no	—
172	C₂³×C₅	—	4	4	no	—
173	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
174	C₅×Q₈	—	6	4	no	—
175	C₅×Q₈	—	6	4	no	—
176	C₅×Q₈	—	12	4	no	—
177	C₅×D₄	—	2	4	no	—
178	C₅×D₄	—	2	4	no	—
179	C₅×D₄	—	4	4	no	—
180	C₂×C₄×C₅	—	3	4	no	—
181	C₂×C₄×C₅	—	6	4	no	—
182	C₂×C₄×C₅	—	6	4	no	—
183	C₂×C₄×C₅	—	12	4	no	—
184	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
185	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
186	C₂×C₄×C₅	C₂×#67	1	2	yes	—
187	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
188	C₅×C₈	—	3	4	no	—
189	C₅×C₈	C₂×#67	1	4	yes	—
190	C₂²×Q₈	—	3	2	no	—
191	C₂²×D₄	C₂×C₅×D₃	1	2	yes	—
192	C₂×Q₁₆	—	3	4	no	—
193	C₂×Q₁₆	—	6	4	no	—
194	C₂×SD₁₆	—	3	8	no	—
195	C₂×SD₁₆	—	6	8	no	—
196	C₂×D₈	—	2	4	no	—
197	C₂×D₈	C₂×C₅×D₃	1	4	yes	—
198	C₂²×C₈	—	3	1	no	—
199	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
200	#124	—	3	8	no	—
201	#124	—	6	4	no	—
202	C₂×C₂²⋊C₄	—	6	4	no	—
203	C₂×C₄²	—	3	1	no	—
204	C₄×C₈	—	3	4	no	—
205	C₂×C₃×C₅	—	8	4	no	—
206	C₂×C₃×C₅	#495	1	4	yes	complemented
207	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
208	C₂×C₃×C₅	C₂×#135	1	2	yes	—
209	C₂³×C₃	—	2	4	no	—
210	C₂³×C₃	—	4	4	no	—
211	C₂³×C₃	—	4	8	no	—
212	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
213	C₃×D₄	—	2	4	no	—
214	C₃×D₄	—	2	4	no	—
215	C₃×D₄	—	4	4	no	—
216	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
217	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
218	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
219	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
220	C₂×Dic₃	—	2	4	no	—
221	C₂×Dic₃	—	2	4	no	—
222	C₂×Dic₃	—	4	4	no	—
223	C₂×Dic₃	C₂×C₅×D₄	1	4	yes	—
224	Dic₆	—	2	4	no	—
225	Dic₆	—	2	4	no	—
226	Dic₆	—	4	4	no	—
227	C₃×C₈	C₂×C₅×D₄	1	4	yes	—
228	C₃⋊C₈	C₂×C₅×D₄	1	4	yes	—
229	C₂²×C₅	—	4	8	no	—
230	C₂²×C₅	—	4	4	no	—
231	C₂²×C₅	—	8	8	no	—
232	C₂²×C₅	#912	1	4	yes	—
233	C₂²×C₅	#940	1	2	yes	—
234	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
235	C₄×C₅	—	6	4	no	—
236	C₄×C₅	—	12	4	no	—
237	C₄×C₅	#940	1	2	yes	—
238	C₄×C₅	C₂×D₃×D₄	1	1	yes	characteristic
239	C₄×C₅	C₂²×#67	1	1	yes	characteristic
240	C₂⁴	—	2	2	no	—
241	C₂×Q₈	—	3	4	no	—
242	C₂×Q₈	—	6	2	no	—
243	C₂×Q₈	—	6	4	no	—
244	C₂×Q₈	—	6	4	no	—
245	C₂×D₄	—	2	4	no	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	—	2	4	no	—
248	C₂×D₄	C₂²×C₅×D₃	1	4	yes	—
249	C₂²×C₄	—	3	2	no	—
250	C₂²×C₄	—	6	2	no	—
251	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
252	Q₁₆	—	6	8	no	—
253	SD₁₆	—	6	16	no	—
254	D₈	—	2	8	no	—
255	C₂×C₈	—	3	4	no	—
256	C₂×C₈	—	3	2	no	—
257	C₂×C₈	C₅×#67	1	4	yes	—
258	C₂×C₈	C₂²×C₅×D₃	1	2	yes	—
259	C₂²⋊C₄	—	6	16	no	—
260	C₄²	—	3	2	no	—
261	C₄²	—	3	2	no	—
262	C₃×C₅	C₂×#495	1	1	yes	characteristic, complemented, Hall
263	C₂²×C₃	—	4	4	no	—
264	C₂²×C₃	—	4	8	no	—
265	C₂²×C₃	—	8	8	no	—
266	C₂²×C₃	C₅×#127	1	2	yes	—
267	C₂²×C₃	C₅×#135	1	4	yes	—
268	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
269	C₃×C₄	C₅×#127	1	2	yes	—
270	C₃×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
271	C₃×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
272	Dic₃	—	2	4	no	—
273	Dic₃	—	4	4	no	—
274	C₂×C₅	—	8	4	no	—
275	C₂×C₅	#5155	1	4	yes	direct product factor
276	C₂×C₅	C₂×#912	1	2	yes	—
277	C₂×C₅	C₂×#940	1	1	yes	characteristic
278	C₂³	—	2	4	no	—
279	C₂³	—	4	8	no	—
280	C₂³	—	4	4	no	—
281	C₂³	C₅×D₃×D₄	1	1	yes	characteristic
282	Q₈	—	6	4	no	—
283	Q₈	—	6	4	no	—
284	Q₈	—	12	4	no	—
285	D₄	—	2	4	no	—
286	D₄	—	2	4	no	—
287	D₄	—	4	4	no	—
288	C₂×C₄	—	3	4	no	—
289	C₂×C₄	—	6	4	no	—
290	C₂×C₄	—	6	4	no	—
291	C₂×C₄	—	12	4	no	—
292	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
293	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
294	C₂×C₄	C₂×C₅×#67	1	2	yes	—
295	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
296	C₈	—	3	4	no	—
297	C₈	C₂×C₅×#67	1	4	yes	—
298	C₂×C₃	—	8	4	no	—
299	C₂×C₃	C₅×#495	1	4	yes	complemented
300	C₂×C₃	C₂×C₅×#127	1	1	yes	characteristic
301	C₂×C₃	C₂×C₅×#135	1	2	yes	—
302	C₅	C₂×#5155	1	1	yes	characteristic, direct product factor, Sylow
303	C₂²	—	4	8	no	—
304	C₂²	—	4	4	no	—
305	C₂²	—	8	8	no	—
306	C₂²	C₅×#912	1	4	yes	—
307	C₂²	C₅×#940	1	2	yes	—
308	C₂²	C₂×C₅×D₃×D₄	1	1	yes	characteristic
309	C₄	—	6	4	no	—
310	C₄	—	12	4	no	—
311	C₄	C₅×#940	1	2	yes	—
312	C₄	C₂×C₅×D₃×D₄	1	1	yes	characteristic
313	C₄	C₂²×C₅×#67	1	1	yes	characteristic
314	C₃	C₂×C₅×#495	1	1	yes	characteristic, complemented, Sylow
315	C₂	—	8	4	no	—
316	C₂	C₅×#5155	1	4	yes	direct product factor
317	C₂	C₂×C₅×#912	1	2	yes	—
318	C₂	C₂×C₅×#940	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting, prime radical	C₂²×C₃×C₅×D₈	C₂
socle	C₂³×C₃×C₅	C₂×D₄
hypercenter	C₂²×C₅×D₈	D₃
hypocenter	C₃	C₂×C₅×#495
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂²×C₄×C₅ ◁ C₂²×C₅×D₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#936	C₂	semidirect
3. C₂²×C₅×#214	C₂	semidirect
4. C₂²×C₅×Dic₁₂	C₂	semidirect
5. C₅×#5155	C₂	direct (N × H)
6. C₂×C₅×C₈×Dic₃	C₂	semidirect
15. C₅×#936	C₂²	semidirect
17. C₂×C₅×#214	C₂²	semidirect
21. C₂×C₅×Dic₁₂	C₂²	semidirect
22. C₅×C₈×Dic₃	C₂²	semidirect
23. C₂×#5155	C₅	direct (N × H)
53. C₂×#936	C₂×C₅	semidirect
54. C₂²×#214	C₂×C₅	semidirect
55. C₂²×Dic₁₂	C₂×C₅	semidirect
56. #5155	C₂×C₅	direct (N × H)
57. C₂×C₈×Dic₃	C₂×C₅	semidirect
101. #936	C₂²×C₅	semidirect
103. C₂×#214	C₂²×C₅	semidirect
107. C₂×Dic₁₂	C₂²×C₅	semidirect
108. C₈×Dic₃	C₂²×C₅	semidirect
206. C₂×C₃×C₅	#495	semidirect
262. C₃×C₅	C₂×#495	semidirect
275. C₂×C₅	#5155	direct (N × H)
299. C₂×C₃	C₅×#495	semidirect
302. C₅	C₂×#5155	direct (N × H)
314. C₃	C₂×C₅×#495	semidirect
316. C₂	C₅×#5155	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×D₈	2
C₂×C₅×#936	2
C₂²×C₅×#214	2
C₂²×C₅×Dic₁₂	2
C₅×#5155	2
C₂×C₅×C₈×Dic₃	2
C₂×C₅×#495	3
C₂×#5155	5

Subgroups — C2×C5×#5155 #14438832

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#5155 #14438832