Subgroups — C₅×#69990 #14296916

Order 1920.

1128 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×Dic₆	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#407	C₂	1	1	yes	maximal, characteristic
3	C₅×#5207	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#824	C₂	1	4	yes	maximal, complemented
5	C₅×#1784	—	3	1	no	maximal, Hall
6	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₂³×C₅×Dic₃	C₄	1	2	yes	—
8	C₂²×C₅×Dic₆	—	2	2	no	—
9	C₂²×C₅×Dic₆	—	2	2	no	—
10	C₂²×C₅×Dic₆	—	2	8	no	—
11	C₂²×C₅×Dic₆	C₄	1	2	yes	—
12	C₂²×C₅×Dic₆	C₂²	1	2	yes	—
13	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
14	C₂×C₅×#208	C₂²	1	2	yes	—
15	C₃×C₅×C₂²⋊C₈	—	2	2	no	—
16	C₅×#830	—	2	2	no	—
17	C₅×#824	—	2	8	no	—
18	#69990	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂³×C₅×Q₈	—	3	1	no	—
20	C₂×C₅×#101	—	3	4	no	—
21	C₅×#407	—	3	1	no	—
22	C₅×#407	D₃	1	1	yes	characteristic
23	C₂⁴×C₃×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃×C₄×C₅	—	2	4	no	—
25	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
27	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
28	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂²×C₅×Dic₃	—	2	4	no	—
30	C₂²×C₅×Dic₃	—	2	8	no	—
31	C₂²×C₅×Dic₃	C₂×C₄	1	4	yes	—
32	C₂×C₅×Dic₆	—	2	16	no	—
33	C₂×C₅×Dic₆	—	2	16	no	—
34	C₂×C₅×Dic₆	—	2	4	no	—
35	C₂×C₅×Dic₆	—	2	2	no	—
36	C₂×C₅×Dic₆	—	2	4	no	—
37	C₂×C₅×Dic₆	—	2	2	no	—
38	C₂×C₅×Dic₆	—	2	8	no	—
39	C₂×C₅×Dic₆	D₄	1	8	yes	—
40	C₃×C₅×M₁₆	—	2	4	no	—
41	C₂×C₃×C₅×C₈	—	2	2	no	—
42	C₅×#208	—	2	4	no	—
43	C₂×C₅×C₃⋊C₈	—	2	2	no	—
44	C₂³×Dic₆	C₂×C₅	1	1	yes	characteristic
45	C₃×#407	C₂×C₅	1	1	yes	characteristic
46	#5207	C₂×C₅	1	1	yes	characteristic
47	C₂×#824	C₂×C₅	1	4	yes	complemented
48	C₂²×C₅×Q₈	—	3	2	no	—
49	C₂²×C₅×Q₈	—	3	2	no	—
50	C₂²×C₅×Q₈	—	6	8	no	—
51	C₂²×C₅×Q₈	—	6	2	no	—
52	C₂²×C₅×Q₈	—	6	2	no	—
53	C₂³×C₄×C₅	—	3	2	no	—
54	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
55	C₂×C₅×M₁₆	—	3	2	no	—
56	C₂×C₅×M₁₆	C₂×D₃	1	2	yes	—
57	C₅×#101	—	6	8	no	—
58	C₅×C₂²⋊C₈	—	2	2	no	—
59	C₅×C₂²⋊C₈	—	6	2	no	—
60	#1784	—	3	1	no	Sylow
61	C₂³×C₃×C₅	—	2	4	no	—
62	C₂³×C₃×C₅	—	2	2	no	—
63	C₂³×C₃×C₅	C₂²⋊C₄	1	2	yes	—
64	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	—	2	2	no	—
66	C₂×C₃×C₄×C₅	—	2	4	no	—
67	C₂×C₃×C₄×C₅	—	2	4	no	—
68	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
69	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
71	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂×C₅×Dic₃	—	2	16	no	—
73	C₂×C₅×Dic₃	—	2	4	no	—
74	C₂×C₅×Dic₃	—	2	4	no	—
75	C₂×C₅×Dic₃	C₂²⋊C₄	1	8	yes	—
76	C₅×Dic₆	—	2	16	no	—
77	C₅×Dic₆	—	2	8	no	—
78	C₅×Dic₆	—	2	8	no	—
79	C₃×C₅×C₈	—	4	2	no	—
80	C₅×C₃⋊C₈	—	4	2	no	—
81	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
82	C₂³×Dic₃	C₄×C₅	1	2	yes	—
83	C₂²×Dic₆	—	2	2	no	—
84	C₂²×Dic₆	—	2	2	no	—
85	C₂²×Dic₆	—	2	8	no	—
86	C₂²×Dic₆	C₄×C₅	1	2	yes	—
87	C₂²×Dic₆	C₂²×C₅	1	2	yes	—
88	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
89	C₂×#208	C₂²×C₅	1	2	yes	—
90	C₃×C₂²⋊C₈	—	2	2	no	—
91	#830	—	2	2	no	—
92	#824	—	2	8	no	—
93	C₂⁴×C₅	C₄×D₃	1	1	yes	characteristic
94	C₂×C₅×Q₈	—	3	8	no	—
95	C₂×C₅×Q₈	—	6	8	no	—
96	C₂×C₅×Q₈	—	6	16	no	—
97	C₂×C₅×Q₈	—	6	16	no	—
98	C₂×C₅×Q₈	—	6	4	no	—
99	C₂×C₅×Q₈	—	6	2	no	—
100	C₂×C₅×Q₈	—	6	4	no	—
101	C₂×C₅×Q₈	—	6	2	no	—
102	C₂²×C₄×C₅	—	2	4	no	—
103	C₂²×C₄×C₅	—	3	4	no	—
104	C₂²×C₄×C₅	—	6	8	no	—
105	C₂²×C₄×C₅	—	6	4	no	—
106	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
107	C₂²×C₄×C₅	D₁₂	1	2	yes	—
108	C₂²×C₄×C₅	#67	1	2	yes	—
109	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
110	C₅×M₁₆	—	2	4	no	—
111	C₅×M₁₆	—	6	4	no	—
112	C₂×C₅×C₈	—	2	2	no	—
113	C₂×C₅×C₈	—	6	2	no	—
114	C₂³×Q₈	—	3	1	no	—
115	C₂×#101	—	3	4	no	—
116	#407	—	3	1	no	—
117	#407	C₅×D₃	1	1	yes	characteristic
118	C₂²×C₃×C₅	—	2	8	no	—
119	C₂²×C₃×C₅	—	2	2	no	—
120	C₂²×C₃×C₅	—	2	2	no	—
121	C₂²×C₃×C₅	—	2	2	no	—
122	C₂²×C₃×C₅	#101	1	4	yes	complemented
123	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	2	yes	—
124	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₃×C₄×C₅	—	2	2	no	—
126	C₃×C₄×C₅	#120	1	2	yes	—
127	C₃×C₄×C₅	#120	1	2	yes	—
128	C₅×Dic₃	—	2	8	no	—
129	C₂⁴×C₃	C₂×C₄×C₅	1	1	yes	characteristic
130	C₂²×C₃×C₄	—	2	4	no	—
131	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
132	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
133	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
134	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
135	C₂²×Dic₃	—	2	4	no	—
136	C₂²×Dic₃	—	2	8	no	—
137	C₂²×Dic₃	C₂×C₄×C₅	1	4	yes	—
138	C₂×Dic₆	—	2	16	no	—
139	C₂×Dic₆	—	2	16	no	—
140	C₂×Dic₆	—	2	4	no	—
141	C₂×Dic₆	—	2	2	no	—
142	C₂×Dic₆	—	2	4	no	—
143	C₂×Dic₆	—	2	2	no	—
144	C₂×Dic₆	—	2	8	no	—
145	C₂×Dic₆	C₅×D₄	1	8	yes	—
146	C₃×M₁₆	—	2	4	no	—
147	C₂×C₃×C₈	—	2	2	no	—
148	#208	—	2	4	no	—
149	C₂×C₃⋊C₈	—	2	2	no	—
150	C₂³×C₅	—	2	4	no	—
151	C₂³×C₅	—	2	2	no	—
152	C₂³×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
153	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
154	C₅×Q₈	—	6	16	no	—
155	C₅×Q₈	—	6	8	no	—
156	C₅×Q₈	—	6	8	no	—
157	C₂×C₄×C₅	—	2	2	no	—
158	C₂×C₄×C₅	—	2	4	no	—
159	C₂×C₄×C₅	—	2	4	no	—
160	C₂×C₄×C₅	—	3	8	no	—
161	C₂×C₄×C₅	—	6	16	no	—
162	C₂×C₄×C₅	—	6	4	no	—
163	C₂×C₄×C₅	—	6	4	no	—
164	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
165	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
166	C₂×C₄×C₅	D₃×D₄	1	4	yes	—
167	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
168	C₅×C₈	—	4	2	no	—
169	C₅×C₈	—	12	2	no	—
170	C₂²×Q₈	—	3	2	no	—
171	C₂²×Q₈	—	3	2	no	—
172	C₂²×Q₈	—	6	8	no	—
173	C₂²×Q₈	—	6	2	no	—
174	C₂²×Q₈	—	6	2	no	—
175	C₂³×C₄	—	3	2	no	—
176	C₂³×C₄	C₂×C₅×D₃	1	1	yes	characteristic
177	C₂×M₁₆	—	3	2	no	—
178	C₂×M₁₆	C₂×C₅×D₃	1	2	yes	—
179	#101	—	6	8	no	—
180	C₂²⋊C₈	—	2	2	no	—
181	C₂²⋊C₈	—	6	2	no	—
182	C₂×C₃×C₅	—	2	4	no	—
183	C₂×C₃×C₅	—	2	2	no	—
184	C₂×C₃×C₅	#379	1	1	yes	characteristic
185	C₂×C₃×C₅	C₂×#101	1	2	yes	—
186	C₂³×C₃	—	2	2	no	—
187	C₂³×C₃	—	2	4	no	—
188	C₂³×C₃	C₅×C₂²⋊C₄	1	2	yes	—
189	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
190	C₂×C₃×C₄	—	2	2	no	—
191	C₂×C₃×C₄	—	2	4	no	—
192	C₂×C₃×C₄	—	2	4	no	—
193	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
194	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
195	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
196	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
197	C₂×Dic₃	—	2	16	no	—
198	C₂×Dic₃	—	2	4	no	—
199	C₂×Dic₃	—	2	4	no	—
200	C₂×Dic₃	C₅×C₂²⋊C₄	1	8	yes	—
201	Dic₆	—	2	16	no	—
202	Dic₆	—	2	8	no	—
203	Dic₆	—	2	8	no	—
204	C₃×C₈	—	4	2	no	—
205	C₃⋊C₈	—	4	2	no	—
206	C₂²×C₅	—	2	8	no	—
207	C₂²×C₅	—	2	2	no	—
208	C₂²×C₅	—	2	2	no	—
209	C₂²×C₅	—	2	2	no	—
210	C₂²×C₅	#824	1	4	yes	complemented
211	C₂²×C₅	D₃×C₂²⋊C₄	1	2	yes	—
212	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
213	C₄×C₅	—	2	2	no	—
214	C₄×C₅	—	6	8	no	—
215	C₄×C₅	#882	1	2	yes	—
216	C₄×C₅	#937	1	2	yes	—
217	C₂⁴	C₄×C₅×D₃	1	1	yes	characteristic
218	C₂×Q₈	—	3	8	no	—
219	C₂×Q₈	—	6	8	no	—
220	C₂×Q₈	—	6	16	no	—
221	C₂×Q₈	—	6	16	no	—
222	C₂×Q₈	—	6	4	no	—
223	C₂×Q₈	—	6	2	no	—
224	C₂×Q₈	—	6	4	no	—
225	C₂×Q₈	—	6	2	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	3	4	no	—
228	C₂²×C₄	—	6	8	no	—
229	C₂²×C₄	—	6	4	no	—
230	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
231	C₂²×C₄	C₅×D₁₂	1	2	yes	—
232	C₂²×C₄	C₅×#67	1	2	yes	—
233	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
234	M₁₆	—	2	4	no	—
235	M₁₆	—	6	4	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	6	2	no	—
238	C₃×C₅	#1784	1	1	yes	characteristic, complemented, Hall
239	C₂²×C₃	—	2	8	no	—
240	C₂²×C₃	—	2	2	no	—
241	C₂²×C₃	—	2	2	no	—
242	C₂²×C₃	—	2	2	no	—
243	C₂²×C₃	C₅×#101	1	4	yes	complemented
244	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	2	yes	—
245	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
246	C₃×C₄	—	2	2	no	—
247	C₃×C₄	C₅×#120	1	2	yes	—
248	C₃×C₄	C₅×#120	1	2	yes	—
249	Dic₃	—	2	8	no	—
250	C₂×C₅	—	2	4	no	—
251	C₂×C₅	—	2	2	no	—
252	C₂×C₅	#4955	1	1	yes	characteristic
253	C₂×C₅	C₂×#824	1	2	yes	—
254	C₂³	—	2	2	no	—
255	C₂³	—	2	4	no	—
256	C₂³	C₅×(C₂×D₃)⋊C₄	1	2	yes	—
257	C₂³	C₂×C₄×C₅×D₃	1	1	yes	characteristic
258	Q₈	—	6	16	no	—
259	Q₈	—	6	8	no	—
260	Q₈	—	6	8	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	3	8	no	—
265	C₂×C₄	—	6	16	no	—
266	C₂×C₄	—	6	4	no	—
267	C₂×C₄	—	6	4	no	—
268	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	2	yes	—
269	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
270	C₂×C₄	C₅×D₃×D₄	1	4	yes	—
271	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
272	C₈	—	4	2	no	—
273	C₈	—	12	2	no	—
274	C₂×C₃	—	2	2	no	—
275	C₂×C₃	—	2	4	no	—
276	C₂×C₃	C₅×#379	1	1	yes	characteristic
277	C₂×C₃	C₂×C₅×#101	1	2	yes	—
278	C₅	#69990	1	1	yes	characteristic, direct product factor, Sylow
279	C₂²	—	2	8	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	C₅×#824	1	4	yes	complemented
284	C₂²	C₅×D₃×C₂²⋊C₄	1	2	yes	—
285	C₂²	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
286	C₄	—	2	2	no	—
287	C₄	—	6	8	no	—
288	C₄	C₅×#882	1	2	yes	—
289	C₄	C₅×#937	1	2	yes	—
290	C₃	C₅×#1784	1	1	yes	characteristic, complemented, Sylow
291	C₂	—	2	2	no	—
292	C₂	—	2	4	no	—
293	C₂	C₅×#4955	1	1	yes	characteristic
294	C₂	C₂×C₅×#824	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅	C₂×(C₂×D₃)⋊C₄
commutator	C₂³×C₃	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#407	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₃×C₅	C₂×C₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#407	D₃
hypocenter	C₃	C₅×#1784
LCS	G ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ C₅×#407

Complements

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

N	H ≅ G/N	type
4. C₂×C₅×#824	C₂	semidirect
18. #69990	C₅	direct (N × H)
47. C₂×#824	C₂×C₅	semidirect
122. C₂²×C₃×C₅	#101	semidirect
210. C₂²×C₅	#824	semidirect
238. C₃×C₅	#1784	semidirect
243. C₂²×C₃	C₅×#101	semidirect
278. C₅	#69990	direct (N × H)
283. C₂²	C₅×#824	semidirect
290. C₃	C₅×#1784	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₅×Dic₆	2
C₃×C₅×#407	2
C₅×#5207	2
C₂×C₅×#824	2
C₅×#1784	3
#69990	5

Subgroups — C5×#69990 #14296916

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69990 #14296916