Subgroups — C₂×C₅×#4694 #14437641

Order 1920.

1056 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	C₂×C₅×#874	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#869	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#821	C₂	1	4	yes	maximal, complemented
5	C₅×#4694	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₅×#486	—	3	1	no	maximal, Hall
7	C₂²×C₅×D₁₂	—	2	1	no	—
8	C₂²×C₅×D₁₂	—	2	1	no	—
9	C₂²×C₅×D₁₂	C₂²	1	2	yes	—
10	C₂²×C₅×Dic₆	—	2	1	no	—
11	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₂×C₅×C₄⋊Dic₃	—	2	1	no	—
14	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
15	C₅×#874	—	2	2	no	—
16	C₅×#874	C₂²	1	4	yes	—
17	C₅×#869	—	2	2	no	—
18	C₅×#869	C₂²	1	4	yes	complemented
19	C₃×C₄×C₅×C₈	C₂²	1	4	yes	complemented
20	C₅×#821	C₂²	1	16	yes	complemented
21	C₂×#4694	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂×C₅×C₄⋊Q₈	—	3	1	no	—
23	C₂×C₅×#127	—	3	1	no	—
24	C₅×#486	—	3	8	no	—
25	C₂×C₅×#102	—	3	4	no	—
26	C₂×C₄×C₅×C₈	D₃	1	1	yes	characteristic, complemented
27	C₂³×C₅×D₃	—	2	2	no	—
28	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
30	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂²×C₅×Dic₃	—	2	2	no	—
32	C₂×C₅×D₁₂	—	2	8	no	—
33	C₂×C₅×D₁₂	—	2	2	no	—
34	C₂×C₅×D₁₂	—	2	4	no	—
35	C₂×C₅×D₁₂	—	2	4	no	—
36	C₂×C₅×D₁₂	—	4	4	no	—
37	C₂×C₅×D₁₂	—	4	4	no	—
38	C₂×C₅×D₁₂	C₂³	1	4	yes	—
39	C₂×C₅×Dic₆	—	2	4	no	—
40	C₂×C₅×Dic₆	—	4	4	no	—
41	C₂×C₃×C₅×C₈	—	2	4	no	—
42	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
43	C₃×C₄²×C₅	D₄	1	2	yes	—
44	C₃×C₄²×C₅	C₂³	1	2	yes	—
45	C₅×C₄⋊Dic₃	—	2	2	no	—
46	C₅×C₄⋊Dic₃	—	2	4	no	—
47	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
48	C₂×#874	C₂×C₅	1	1	yes	characteristic
49	C₂×#869	C₂×C₅	1	1	yes	characteristic, complemented
50	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic, complemented
51	C₂×#821	C₂×C₅	1	4	yes	complemented
52	#4694	C₂×C₅	1	8	yes	direct product factor
53	C₂²×C₅×Q₈	—	6	1	no	—
54	C₂²×C₅×D₄	—	3	2	no	—
55	C₂²×C₅×D₄	—	6	1	no	—
56	C₂²×C₅×D₄	—	6	1	no	—
57	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
58	C₅×C₄⋊Q₈	—	3	4	no	—
59	C₅×C₄⋊Q₈	—	6	2	no	—
60	C₅×#127	—	3	4	no	—
61	C₅×#127	—	6	2	no	—
62	C₂×C₅×C₄⋊C₄	—	3	2	no	—
63	C₂×C₅×C₄⋊C₄	—	6	1	no	—
64	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
65	C₅×#102	—	3	16	no	—
66	C₄×C₅×C₈	C₂×D₃	1	4	yes	complemented
67	C₂×#486	—	3	1	no	Sylow
68	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₅×D₃	—	2	4	no	—
70	C₂²×C₅×D₃	—	4	4	no	—
71	C₂²×C₅×D₃	—	4	4	no	—
72	C₂²×C₅×D₃	—	4	4	no	—
73	C₂×C₃×C₄×C₅	D₈	1	4	yes	—
74	C₂×C₃×C₄×C₅	SD₁₆	1	4	yes	—
75	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
77	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
78	C₂×C₃×C₄×C₅	#41	1	4	yes	—
79	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
80	C₂×C₅×Dic₃	—	2	4	no	—
81	C₂×C₅×Dic₃	—	4	4	no	—
82	C₅×D₁₂	—	2	8	no	—
83	C₅×D₁₂	—	4	4	no	—
84	C₅×D₁₂	—	4	8	no	—
85	C₅×D₁₂	—	4	8	no	—
86	C₅×Dic₆	—	4	8	no	—
87	C₃×C₅×C₈	—	2	4	no	—
88	C₂²×D₁₂	—	2	1	no	—
89	C₂²×D₁₂	—	2	1	no	—
90	C₂²×D₁₂	C₂²×C₅	1	2	yes	—
91	C₂²×Dic₆	—	2	1	no	—
92	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
93	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
94	C₂×C₄⋊Dic₃	—	2	1	no	—
95	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
96	#874	—	2	2	no	—
97	#874	C₂²×C₅	1	4	yes	—
98	#869	—	2	2	no	—
99	#869	C₂²×C₅	1	4	yes	complemented
100	C₃×C₄×C₈	C₂²×C₅	1	4	yes	complemented
101	#821	C₂²×C₅	1	16	yes	complemented
102	C₂⁴×C₅	—	6	2	no	—
103	C₂×C₅×Q₈	—	6	4	no	—
104	C₂×C₅×Q₈	—	12	4	no	—
105	C₂×C₅×D₄	—	3	4	no	—
106	C₂×C₅×D₄	—	6	4	no	—
107	C₂×C₅×D₄	—	6	4	no	—
108	C₂×C₅×D₄	—	6	2	no	—
109	C₂×C₅×D₄	—	6	8	no	—
110	C₂×C₅×D₄	—	12	4	no	—
111	C₂×C₅×D₄	—	12	4	no	—
112	C₂²×C₄×C₅	—	6	2	no	—
113	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
114	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
115	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
116	C₂×C₅×C₈	—	2	4	no	—
117	C₂×C₅×C₈	C₂²×D₃	1	4	yes	—
118	C₅×C₄⋊C₄	—	3	4	no	—
119	C₅×C₄⋊C₄	—	6	2	no	—
120	C₅×C₄⋊C₄	—	6	4	no	—
121	C₄²×C₅	D₁₂	1	2	yes	—
122	C₄²×C₅	C₂²×D₃	1	2	yes	—
123	C₂×C₄⋊Q₈	—	3	1	no	—
124	C₂×#127	—	3	1	no	—
125	#486	—	3	8	no	—
126	C₂×#102	—	3	4	no	—
127	C₂×C₄×C₈	C₅×D₃	1	1	yes	characteristic, complemented
128	C₂²×C₃×C₅	#124	1	2	yes	—
129	C₂²×C₃×C₅	C₂×D₈	1	2	yes	—
130	C₂²×C₃×C₅	C₂×SD₁₆	1	2	yes	—
131	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₅×D₃	—	4	4	no	—
133	C₂×C₅×D₃	—	4	4	no	—
134	C₂×C₅×D₃	—	4	4	no	—
135	C₂×C₅×D₃	—	8	8	no	—
136	C₃×C₄×C₅	#124	1	2	yes	—
137	C₃×C₄×C₅	C₂×D₈	1	4	yes	—
138	C₃×C₄×C₅	C₂×SD₁₆	1	4	yes	—
139	C₃×C₄×C₅	C₂×#41	1	2	yes	—
140	C₅×Dic₃	—	4	4	no	—
141	C₂³×D₃	—	2	2	no	—
142	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
143	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
144	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
145	C₂²×Dic₃	—	2	2	no	—
146	C₂×D₁₂	—	2	8	no	—
147	C₂×D₁₂	—	2	2	no	—
148	C₂×D₁₂	—	2	4	no	—
149	C₂×D₁₂	—	2	4	no	—
150	C₂×D₁₂	—	4	4	no	—
151	C₂×D₁₂	—	4	4	no	—
152	C₂×D₁₂	C₂³×C₅	1	4	yes	—
153	C₂×Dic₆	—	2	4	no	—
154	C₂×Dic₆	—	4	4	no	—
155	C₂×C₃×C₈	—	2	4	no	—
156	C₂×C₃×C₈	C₂³×C₅	1	4	yes	—
157	C₃×C₄²	C₅×D₄	1	2	yes	—
158	C₃×C₄²	C₂³×C₅	1	2	yes	—
159	C₄⋊Dic₃	—	2	2	no	—
160	C₄⋊Dic₃	—	2	4	no	—
161	C₄⋊Dic₃	C₂³×C₅	1	4	yes	—
162	C₂³×C₅	—	6	4	no	—
163	C₂³×C₅	—	12	4	no	—
164	C₂³×C₅	—	12	4	no	—
165	C₂³×C₅	—	12	4	no	—
166	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
167	C₅×Q₈	—	12	8	no	—
168	C₅×D₄	—	6	8	no	—
169	C₅×D₄	—	12	4	no	—
170	C₅×D₄	—	12	8	no	—
171	C₅×D₄	—	12	8	no	—
172	C₂×C₄×C₅	—	6	4	no	—
173	C₂×C₄×C₅	—	12	4	no	—
174	C₂×C₄×C₅	#204	1	4	yes	—
175	C₂×C₄×C₅	D₂₄	1	4	yes	—
176	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
178	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
179	C₂×C₄×C₅	#235	1	4	yes	—
180	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
181	C₅×C₈	—	2	4	no	—
182	C₂²×Q₈	—	6	1	no	—
183	C₂²×D₄	—	3	2	no	—
184	C₂²×D₄	—	6	1	no	—
185	C₂²×D₄	—	6	1	no	—
186	C₂²×C₈	C₂×C₅×D₃	1	2	yes	—
187	C₄⋊Q₈	—	3	4	no	—
188	C₄⋊Q₈	—	6	2	no	—
189	#127	—	3	4	no	—
190	#127	—	6	2	no	—
191	C₂×C₄⋊C₄	—	3	2	no	—
192	C₂×C₄⋊C₄	—	6	1	no	—
193	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
194	#102	—	3	16	no	—
195	C₄×C₈	C₂×C₅×D₃	1	4	yes	complemented
196	C₂×C₃×C₅	#486	1	4	yes	complemented
197	C₂×C₃×C₅	C₂×#124	1	1	yes	characteristic
198	C₂×C₃×C₅	C₂²×D₈	1	1	yes	characteristic
199	C₂×C₃×C₅	C₂²×SD₁₆	1	1	yes	characteristic
200	C₅×D₃	—	8	4	no	—
201	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
202	C₂²×D₃	—	2	4	no	—
203	C₂²×D₃	—	4	4	no	—
204	C₂²×D₃	—	4	4	no	—
205	C₂²×D₃	—	4	4	no	—
206	C₂×C₃×C₄	C₅×D₈	1	4	yes	—
207	C₂×C₃×C₄	C₅×SD₁₆	1	4	yes	—
208	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
210	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
211	C₂×C₃×C₄	C₅×#41	1	4	yes	—
212	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
213	C₂×Dic₃	—	2	4	no	—
214	C₂×Dic₃	—	4	4	no	—
215	D₁₂	—	2	8	no	—
216	D₁₂	—	4	4	no	—
217	D₁₂	—	4	8	no	—
218	D₁₂	—	4	8	no	—
219	Dic₆	—	4	8	no	—
220	C₃×C₈	—	2	4	no	—
221	C₂²×C₅	—	12	4	no	—
222	C₂²×C₅	—	12	4	no	—
223	C₂²×C₅	—	12	4	no	—
224	C₂²×C₅	—	24	8	no	—
225	C₂²×C₅	#875	1	2	yes	—
226	C₂²×C₅	C₂×#204	1	2	yes	—
227	C₂²×C₅	C₂×D₂₄	1	2	yes	—
228	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
229	C₄×C₅	—	12	4	no	—
230	C₄×C₅	#875	1	2	yes	—
231	C₄×C₅	C₂×#204	1	4	yes	—
232	C₄×C₅	C₂×D₂₄	1	4	yes	—
233	C₄×C₅	C₂×#235	1	2	yes	—
234	C₂⁴	—	6	2	no	—
235	C₂×Q₈	—	6	4	no	—
236	C₂×Q₈	—	12	4	no	—
237	C₂×D₄	—	3	4	no	—
238	C₂×D₄	—	6	4	no	—
239	C₂×D₄	—	6	4	no	—
240	C₂×D₄	—	6	2	no	—
241	C₂×D₄	—	6	8	no	—
242	C₂×D₄	—	12	4	no	—
243	C₂×D₄	—	12	4	no	—
244	C₂²×C₄	—	6	2	no	—
245	C₂²×C₄	C₅×D₁₂	1	1	yes	characteristic
246	C₂²×C₄	C₅×D₁₂	1	1	yes	characteristic
247	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	C₂²×C₅×D₃	1	4	yes	—
250	C₄⋊C₄	—	3	4	no	—
251	C₄⋊C₄	—	6	2	no	—
252	C₄⋊C₄	—	6	4	no	—
253	C₄²	C₅×D₁₂	1	2	yes	—
254	C₄²	C₂²×C₅×D₃	1	2	yes	—
255	C₃×C₅	C₂×#486	1	1	yes	characteristic, complemented, Hall
256	C₂²×C₃	C₅×#124	1	2	yes	—
257	C₂²×C₃	C₂×C₅×D₈	1	2	yes	—
258	C₂²×C₃	C₂×C₅×SD₁₆	1	2	yes	—
259	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
260	C₂×D₃	—	4	4	no	—
261	C₂×D₃	—	4	4	no	—
262	C₂×D₃	—	4	4	no	—
263	C₂×D₃	—	8	8	no	—
264	C₃×C₄	C₅×#124	1	2	yes	—
265	C₃×C₄	C₂×C₅×D₈	1	4	yes	—
266	C₃×C₄	C₂×C₅×SD₁₆	1	4	yes	—
267	C₃×C₄	C₂×C₅×#41	1	2	yes	—
268	Dic₃	—	4	4	no	—
269	C₂×C₅	—	24	4	no	—
270	C₂×C₅	#4694	1	4	yes	direct product factor
271	C₂×C₅	C₂×#875	1	1	yes	characteristic
272	C₂×C₅	C₂²×#204	1	1	yes	characteristic
273	C₂×C₅	C₂²×D₂₄	1	1	yes	characteristic
274	C₂³	—	6	4	no	—
275	C₂³	—	12	4	no	—
276	C₂³	—	12	4	no	—
277	C₂³	—	12	4	no	—
278	C₂³	C₂×C₅×D₁₂	1	1	yes	characteristic
279	Q₈	—	12	8	no	—
280	D₄	—	6	8	no	—
281	D₄	—	12	4	no	—
282	D₄	—	12	8	no	—
283	D₄	—	12	8	no	—
284	C₂×C₄	—	6	4	no	—
285	C₂×C₄	—	12	4	no	—
286	C₂×C₄	C₅×#204	1	4	yes	—
287	C₂×C₄	C₅×D₂₄	1	4	yes	—
288	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
289	C₂×C₄	C₂×C₅×D₁₂	1	2	yes	—
290	C₂×C₄	C₂×C₅×D₁₂	1	2	yes	—
291	C₂×C₄	C₅×#235	1	4	yes	—
292	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
293	C₈	—	2	4	no	—
294	C₂×C₃	C₅×#486	1	4	yes	complemented
295	C₂×C₃	C₂×C₅×#124	1	1	yes	characteristic
296	C₂×C₃	C₂²×C₅×D₈	1	1	yes	characteristic
297	C₂×C₃	C₂²×C₅×SD₁₆	1	1	yes	characteristic
298	D₃	—	8	4	no	—
299	C₅	C₂×#4694	1	1	yes	characteristic, direct product factor, Sylow
300	C₂²	—	12	4	no	—
301	C₂²	—	12	4	no	—
302	C₂²	—	12	4	no	—
303	C₂²	—	24	8	no	—
304	C₂²	C₅×#875	1	2	yes	—
305	C₂²	C₂×C₅×#204	1	2	yes	—
306	C₂²	C₂×C₅×D₂₄	1	2	yes	—
307	C₂²	C₂²×C₅×D₁₂	1	1	yes	characteristic
308	C₄	—	12	4	no	—
309	C₄	C₅×#875	1	2	yes	—
310	C₄	C₂×C₅×#204	1	4	yes	—
311	C₄	C₂×C₅×D₂₄	1	4	yes	—
312	C₄	C₂×C₅×#235	1	2	yes	—
313	C₃	C₂×C₅×#486	1	1	yes	characteristic, complemented, Sylow
314	C₂	—	24	4	no	—
315	C₂	C₅×#4694	1	4	yes	direct product factor
316	C₂	C₂×C₅×#875	1	1	yes	characteristic
317	C₂	C₂²×C₅×#204	1	1	yes	characteristic
318	C₂	C₂²×C₅×D₂₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₂×C₃×C₄×C₅×C₈	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂×C₅×#874	C₂
hypercenter	C₂×C₄×C₅×C₈	D₃
hypocenter	C₃	C₂×C₅×#486
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#869	C₂	semidirect
3. C₂×C₃×C₄×C₅×C₈	C₂	semidirect
4. C₂×C₅×#821	C₂	semidirect
5. C₅×#4694	C₂	direct (N × H)
18. C₅×#869	C₂²	semidirect
19. C₃×C₄×C₅×C₈	C₂²	semidirect
20. C₅×#821	C₂²	semidirect
21. C₂×#4694	C₅	direct (N × H)
26. C₂×C₄×C₅×C₈	D₃	semidirect
49. C₂×#869	C₂×C₅	semidirect
50. C₂×C₃×C₄×C₈	C₂×C₅	semidirect
51. C₂×#821	C₂×C₅	semidirect
52. #4694	C₂×C₅	direct (N × H)
66. C₄×C₅×C₈	C₂×D₃	semidirect
99. #869	C₂²×C₅	semidirect
100. C₃×C₄×C₈	C₂²×C₅	semidirect
101. #821	C₂²×C₅	semidirect
127. C₂×C₄×C₈	C₅×D₃	semidirect
195. C₄×C₈	C₂×C₅×D₃	semidirect
196. C₂×C₃×C₅	#486	semidirect
255. C₃×C₅	C₂×#486	semidirect
270. C₂×C₅	#4694	direct (N × H)
294. C₂×C₃	C₅×#486	semidirect
299. C₅	C₂×#4694	direct (N × H)
313. C₃	C₂×C₅×#486	semidirect
315. C₂	C₅×#4694	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×C₅×#874	2
C₂×C₅×#869	2
C₂×C₃×C₄×C₅×C₈	2
C₂×C₅×#821	2
C₅×#4694	2
C₂×C₅×#486	3
C₂×#4694	5

Subgroups — C2×C5×#4694 #14437641

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4694 #14437641