Subgroups — C₅×#69285 #14314353

Order 1920.

320 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#798	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×D₃×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₅×#4689	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×Dic₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×Dic₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×C₄⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2151	—	3	1	no	maximal, Hall
9	C₂×C₅×ΓL(1,𝔽₂₅)	C₄	1	1	yes	characteristic
10	C₂×C₅×D₃×C₈	C₂²	1	1	yes	characteristic
11	C₅×#872	C₄	1	1	yes	characteristic
12	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
14	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₅×#815	C₄	1	1	yes	characteristic
16	C₅×C₈×Dic₃	C₂²	1	1	yes	characteristic
17	C₂×C₅×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₅×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
19	C₅×#803	C₄	1	1	yes	characteristic
20	C₅×#798	—	2	2	no	—
21	C₅×D₃×C₁₆	—	2	2	no	—
22	#69285	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₂×C₅×M₃₂	—	3	1	no	—
24	C₂²×C₅×C₁₆	—	3	1	no	—
25	C₅×#405	—	3	1	no	—
26	C₅×C₄⋊C₁₆	—	3	1	no	—
27	C₅×C₄⋊C₁₆	—	3	1	no	—
28	C₅×C₄⋊C₁₆	—	3	1	no	—
29	C₅×C₄⋊C₁₆	D₃	1	1	yes	characteristic, complemented
30	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
34	C₅×(C₂×D₃)⋊C₄	C₈	1	1	yes	characteristic
35	C₅×(C₂×D₃)⋊C₄	C₈	1	1	yes	characteristic
36	C₅×Dic₃⋊C₄	C₈	1	1	yes	characteristic
37	C₅×Dic₃⋊C₄	C₈	1	1	yes	characteristic
38	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
41	C₅×ΓL(1,𝔽₂₅)	—	2	2	no	—
42	C₅×D₃×C₈	D₄	1	2	yes	—
43	C₅×D₃×C₈	Q₈	1	2	yes	—
44	C₃×C₅×C₁₆	—	2	1	no	—
45	C₃×C₅×C₁₆	—	2	1	no	—
46	C₅×C₃⋊C₁₆	—	2	1	no	—
47	C₅×C₃⋊C₁₆	—	2	1	no	—
48	C₂×#798	C₂×C₅	1	1	yes	characteristic
49	C₂×D₃×C₁₆	C₂×C₅	1	1	yes	characteristic
50	#4689	C₂×C₅	1	1	yes	characteristic
51	C₃×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
52	Dic₃⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
53	Dic₃⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
54	C₄⋊(C₃⋊C₁₆)	C₂×C₅	1	1	yes	characteristic, complemented
55	C₂×C₅×M₁₆	—	3	1	no	—
56	C₂²×C₅×C₈	—	3	1	no	—
57	C₅×#117	—	3	1	no	—
58	C₅×M₃₂	—	6	2	no	—
59	C₂×C₅×C₁₆	—	3	1	no	—
60	C₂×C₅×C₁₆	—	3	1	no	—
61	C₂×C₅×C₁₆	—	6	2	no	—
62	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
63	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
64	C₅×#97	—	3	1	no	—
65	C₅×#97	—	3	1	no	—
66	C₄×C₅×C₈	—	3	1	no	—
67	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
68	#2151	—	3	1	no	Sylow
69	C₂²×C₅×D₃	C₂×C₈	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
71	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₅×Dic₃	C₂×C₈	1	1	yes	characteristic
74	C₂×C₅×Dic₃	C₂×C₈	1	1	yes	characteristic
75	C₂×C₅×Dic₃	C₂×C₈	1	1	yes	characteristic
76	C₄×C₅×D₃	C₄⋊C₄	1	2	yes	—
77	C₄×C₅×D₃	M₁₆	1	2	yes	—
78	C₃×C₅×C₈	—	2	1	no	—
79	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
80	C₃×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
81	C₅×C₃⋊C₈	—	2	1	no	—
82	C₅×C₃⋊C₈	C₄⋊C₄	1	2	yes	—
83	C₂×ΓL(1,𝔽₂₅)	C₄×C₅	1	1	yes	characteristic
84	C₂×D₃×C₈	C₂²×C₅	1	1	yes	characteristic
85	#872	C₄×C₅	1	1	yes	characteristic
86	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
87	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
88	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
89	#815	C₄×C₅	1	1	yes	characteristic
90	C₈×Dic₃	C₂²×C₅	1	1	yes	characteristic
91	C₂×C₃⋊C₁₆	C₂²×C₅	1	1	yes	characteristic
92	C₂×C₃⋊C₁₆	C₂²×C₅	1	1	yes	characteristic
93	#803	C₄×C₅	1	1	yes	characteristic
94	#798	—	2	2	no	—
95	D₃×C₁₆	—	2	2	no	—
96	C₂²×C₄×C₅	—	3	1	no	—
97	C₅×M₁₆	—	6	2	no	—
98	C₂×C₅×C₈	—	3	1	no	—
99	C₂×C₅×C₈	—	3	2	no	—
100	C₂×C₅×C₈	—	3	2	no	—
101	C₂×C₅×C₈	—	3	1	no	—
102	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
103	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
104	C₅×C₄⋊C₄	—	3	1	no	—
105	C₅×C₄⋊C₄	—	3	1	no	—
106	C₅×C₂²⋊C₄	—	3	1	no	—
107	C₅×C₂²⋊C₄	—	3	1	no	—
108	C₄²×C₅	—	3	1	no	—
109	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
110	C₅×C₁₆	—	2	1	no	—
111	C₅×C₁₆	—	2	1	no	—
112	C₅×C₁₆	—	6	1	no	—
113	C₅×C₁₆	—	6	1	no	—
114	C₂×M₃₂	—	3	1	no	—
115	C₂²×C₁₆	—	3	1	no	—
116	#405	—	3	1	no	—
117	C₄⋊C₁₆	—	3	1	no	—
118	C₄⋊C₁₆	—	3	1	no	—
119	C₄⋊C₁₆	—	3	1	no	—
120	C₄⋊C₁₆	C₅×D₃	1	1	yes	characteristic, complemented
121	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₅×D₃	—	2	1	no	—
123	C₂×C₅×D₃	—	2	1	no	—
124	C₂×C₅×D₃	C₄⋊C₈	1	2	yes	—
125	C₃×C₄×C₅	—	2	1	no	—
126	C₃×C₄×C₅	—	2	1	no	—
127	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
129	C₅×Dic₃	—	2	1	no	—
130	C₅×Dic₃	—	2	1	no	—
131	C₅×Dic₃	C₄⋊C₈	1	2	yes	—
132	C₂×C₄×D₃	C₂×C₄×C₅	1	1	yes	characteristic
133	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
134	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
135	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
136	(C₂×D₃)⋊C₄	C₅×C₈	1	1	yes	characteristic
137	(C₂×D₃)⋊C₄	C₅×C₈	1	1	yes	characteristic
138	Dic₃⋊C₄	C₅×C₈	1	1	yes	characteristic
139	Dic₃⋊C₄	C₅×C₈	1	1	yes	characteristic
140	C₄×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
141	C₂×C₃⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
142	C₂×C₃⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
143	ΓL(1,𝔽₂₅)	—	2	2	no	—
144	D₃×C₈	C₅×D₄	1	2	yes	—
145	D₃×C₈	C₅×Q₈	1	2	yes	—
146	C₃×C₁₆	—	2	1	no	—
147	C₃×C₁₆	—	2	1	no	—
148	C₃⋊C₁₆	—	2	1	no	—
149	C₃⋊C₁₆	—	2	1	no	—
150	C₂³×C₅	—	3	1	no	—
151	C₂×C₄×C₅	—	3	1	no	—
152	C₂×C₄×C₅	—	3	2	no	—
153	C₂×C₄×C₅	—	3	2	no	—
154	C₂×C₄×C₅	—	3	1	no	—
155	C₂×C₄×C₅	—	3	1	no	—
156	C₂×C₄×C₅	D₃×C₈	1	1	yes	characteristic
157	C₂×C₄×C₅	D₃×C₈	1	1	yes	characteristic
158	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
159	C₅×C₈	—	2	1	no	—
160	C₅×C₈	—	3	2	no	—
161	C₅×C₈	—	6	1	no	—
162	C₅×C₈	D₃×D₄	1	1	yes	characteristic
163	C₅×C₈	D₃×Q₈	1	1	yes	characteristic
164	C₂×M₁₆	—	3	1	no	—
165	C₂²×C₈	—	3	1	no	—
166	#117	—	3	1	no	—
167	M₃₂	—	6	2	no	—
168	C₂×C₁₆	—	3	1	no	—
169	C₂×C₁₆	—	3	1	no	—
170	C₂×C₁₆	—	6	2	no	—
171	C₂×C₁₆	C₂×C₅×D₃	1	1	yes	characteristic
172	C₂×C₁₆	C₂×C₅×D₃	1	1	yes	characteristic
173	#97	—	3	1	no	—
174	#97	—	3	1	no	—
175	C₄×C₈	—	3	1	no	—
176	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
177	C₂×C₃×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
178	C₂×C₃×C₅	#504	1	1	yes	characteristic
179	C₂×C₃×C₅	#504	1	1	yes	characteristic
180	C₅×D₃	—	2	2	no	—
181	C₂²×D₃	C₂×C₅×C₈	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂×C₅×C₈	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂×C₅×C₈	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
185	C₂×Dic₃	C₂×C₅×C₈	1	1	yes	characteristic
186	C₂×Dic₃	C₂×C₅×C₈	1	1	yes	characteristic
187	C₂×Dic₃	C₂×C₅×C₈	1	1	yes	characteristic
188	C₄×D₃	C₅×C₄⋊C₄	1	2	yes	—
189	C₄×D₃	C₅×M₁₆	1	2	yes	—
190	C₃×C₈	—	2	1	no	—
191	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
192	C₃×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
193	C₃⋊C₈	—	2	1	no	—
194	C₃⋊C₈	C₅×C₄⋊C₄	1	2	yes	—
195	C₂²×C₅	—	3	2	no	—
196	C₂²×C₅	—	6	1	no	—
197	C₂²×C₅	—	6	1	no	—
198	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
199	C₄×C₅	—	2	1	no	—
200	C₄×C₅	—	2	1	no	—
201	C₄×C₅	—	3	2	no	—
202	C₄×C₅	—	6	1	no	—
203	C₄×C₅	—	6	1	no	—
204	C₄×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
205	C₄×C₅	D₃×M₁₆	1	1	yes	characteristic
206	C₂²×C₄	—	3	1	no	—
207	M₁₆	—	6	2	no	—
208	C₂×C₈	—	3	1	no	—
209	C₂×C₈	—	3	2	no	—
210	C₂×C₈	—	3	2	no	—
211	C₂×C₈	—	3	1	no	—
212	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
213	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
214	C₄⋊C₄	—	3	1	no	—
215	C₄⋊C₄	—	3	1	no	—
216	C₂²⋊C₄	—	3	1	no	—
217	C₂²⋊C₄	—	3	1	no	—
218	C₄²	—	3	1	no	—
219	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
220	C₁₆	—	2	1	no	—
221	C₁₆	—	2	1	no	—
222	C₁₆	—	6	1	no	—
223	C₁₆	—	6	1	no	—
224	C₃×C₅	#2151	1	1	yes	characteristic, complemented, Hall
225	C₂²×C₃	C₂²×C₅×C₈	1	1	yes	characteristic
226	C₂×D₃	—	2	1	no	—
227	C₂×D₃	—	2	1	no	—
228	C₂×D₃	C₅×C₄⋊C₈	1	2	yes	—
229	C₃×C₄	—	2	1	no	—
230	C₃×C₄	—	2	1	no	—
231	C₃×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
232	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
233	Dic₃	—	2	1	no	—
234	Dic₃	—	2	1	no	—
235	Dic₃	C₅×C₄⋊C₈	1	2	yes	—
236	C₂×C₅	—	6	2	no	—
237	C₂×C₅	D₃×C₄⋊C₈	1	1	yes	characteristic
238	C₂×C₅	#4901	1	1	yes	characteristic
239	C₂×C₅	#4907	1	1	yes	characteristic
240	C₂³	—	3	1	no	—
241	C₂×C₄	—	3	1	no	—
242	C₂×C₄	—	3	2	no	—
243	C₂×C₄	—	3	2	no	—
244	C₂×C₄	—	3	1	no	—
245	C₂×C₄	—	3	1	no	—
246	C₂×C₄	C₅×D₃×C₈	1	1	yes	characteristic
247	C₂×C₄	C₅×D₃×C₈	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
249	C₈	—	2	1	no	—
250	C₈	—	3	2	no	—
251	C₈	—	6	1	no	—
252	C₈	C₅×D₃×D₄	1	1	yes	characteristic
253	C₈	C₅×D₃×Q₈	1	1	yes	characteristic
254	C₂×C₃	C₂×C₅×C₄⋊C₈	1	1	yes	characteristic
255	C₂×C₃	C₅×#504	1	1	yes	characteristic
256	C₂×C₃	C₅×#504	1	1	yes	characteristic
257	D₃	—	2	2	no	—
258	C₅	#69285	1	1	yes	characteristic, direct product factor, Sylow
259	C₂²	—	3	2	no	—
260	C₂²	—	6	1	no	—
261	C₂²	—	6	1	no	—
262	C₂²	C₂×C₅×D₃×C₈	1	1	yes	characteristic
263	C₄	—	2	1	no	—
264	C₄	—	2	1	no	—
265	C₄	—	3	2	no	—
266	C₄	—	6	1	no	—
267	C₄	—	6	1	no	—
268	C₄	C₅×D₃×C₄⋊C₄	1	1	yes	characteristic
269	C₄	C₅×D₃×M₁₆	1	1	yes	characteristic
270	C₃	C₅×#2151	1	1	yes	characteristic, complemented, Sylow
271	C₂	—	6	2	no	—
272	C₂	C₅×D₃×C₄⋊C₈	1	1	yes	characteristic
273	C₂	C₅×#4901	1	1	yes	characteristic
274	C₂	C₅×#4907	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅×C₈	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₅×C₈
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₃×C₅×C₄⋊C₁₆	C₂
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂²×C₅×D₃	C₂×C₈
cosocle	C₂×C₃×C₈	C₂³×C₅
hypercenter	C₅×C₄⋊C₁₆	D₃
hypocenter	C₃	C₅×#2151
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅×C₈ ◁ C₅×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×C₄⋊C₁₆	C₂	semidirect
5. C₅×Dic₃⋊C₁₆	C₂	semidirect
6. C₅×Dic₃⋊C₁₆	C₂	semidirect
7. C₅×C₄⋊(C₃⋊C₁₆)	C₂	semidirect
22. #69285	C₅	direct (N × H)
29. C₅×C₄⋊C₁₆	D₃	semidirect
51. C₃×C₄⋊C₁₆	C₂×C₅	semidirect
52. Dic₃⋊C₁₆	C₂×C₅	semidirect
53. Dic₃⋊C₁₆	C₂×C₅	semidirect
54. C₄⋊(C₃⋊C₁₆)	C₂×C₅	semidirect
120. C₄⋊C₁₆	C₅×D₃	semidirect
224. C₃×C₅	#2151	semidirect
258. C₅	#69285	direct (N × H)
270. C₃	C₅×#2151	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#798	2
C₂×C₅×D₃×C₁₆	2
C₅×#4689	2
C₃×C₅×C₄⋊C₁₆	2
C₅×Dic₃⋊C₁₆	2
C₅×C₄⋊(C₃⋊C₁₆)	2
C₅×#2151	3
#69285	5

Subgroups — C5×#69285 #14314353

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69285 #14314353