Subgroups — C₅×#69294 #14317257

Order 1920.

316 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×Dic₂₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#800	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×Dic₁₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4696	C₂	1	1	yes	maximal, characteristic
5	C₃×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4509	C₂	1	1	yes	maximal, characteristic
7	C₅×#4503	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2211	—	3	1	no	maximal, Hall
9	C₂×C₅×Dic₁₂	C₂²	1	1	yes	characteristic
10	C₂×C₅×Dic₁₂	C₂²	1	1	yes	characteristic
11	C₂×C₅×D₂₄	C₂²	1	1	yes	characteristic
12	C₂×C₅×#204	—	2	1	no	—
13	C₅×#875	—	2	1	no	—
14	C₄×C₅×Dic₆	—	2	1	no	—
15	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
17	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₅×#818	C₂²	1	1	yes	characteristic
19	C₅×#816	—	2	1	no	—
20	C₅×Dic₂₄	—	2	2	no	—
21	C₅×#800	—	2	2	no	—
22	#69294	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₂×C₅×Q₃₂	—	3	1	no	—
24	C₂×C₅×SD₃₂	—	3	1	no	—
25	C₅×#495	—	3	1	no	—
26	C₄×C₅×Q₁₆	—	3	1	no	—
27	C₅×C₄⋊C₁₆	D₃	1	1	yes	characteristic, complemented
28	C₅×#358	—	3	1	no	—
29	C₅×#357	—	3	1	no	—
30	C₂×C₅×D₁₂	—	2	1	no	—
31	C₂×C₅×Dic₆	—	2	1	no	—
32	C₂×C₅×Dic₆	—	2	1	no	—
33	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
34	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
36	C₅×(C₂×D₃)⋊C₄	—	4	1	no	—
37	C₅×(C₂×D₃)⋊C₄	—	4	1	no	—
38	C₅×C₄⋊Dic₃	—	2	1	no	—
39	C₅×Dic₃⋊C₄	—	4	1	no	—
40	C₄×C₅×Dic₃	—	4	1	no	—
41	C₅×Dic₁₂	—	2	1	no	—
42	C₅×Dic₁₂	—	2	1	no	—
43	C₅×Dic₁₂	—	2	1	no	—
44	C₅×Dic₁₂	D₄	1	2	yes	—
45	C₅×D₂₄	—	2	1	no	—
46	C₅×D₂₄	—	2	1	no	—
47	C₅×#204	—	4	2	no	—
48	C₃×C₅×C₁₆	—	2	1	no	—
49	C₃×C₅×C₁₆	—	2	1	no	—
50	C₂×Dic₂₄	C₂×C₅	1	1	yes	characteristic, complemented
51	C₂×#800	C₂×C₅	1	1	yes	characteristic
52	C₄×Dic₁₂	C₂×C₅	1	1	yes	characteristic, complemented
53	#4696	C₂×C₅	1	1	yes	characteristic
54	C₃×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
55	#4509	C₂×C₅	1	1	yes	characteristic
56	#4503	C₂×C₅	1	1	yes	characteristic, complemented
57	C₂×C₅×Q₁₆	—	3	1	no	—
58	C₂×C₅×Q₁₆	—	3	1	no	—
59	C₂×C₅×SD₁₆	—	6	1	no	—
60	C₂×C₅×D₈	—	3	1	no	—
61	C₅×#124	—	6	1	no	—
62	C₄×C₅×Q₈	—	6	1	no	—
63	C₅×Q₃₂	—	6	2	no	—
64	C₅×SD₃₂	—	6	2	no	—
65	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
66	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
67	C₅×#107	—	3	1	no	—
68	C₅×#103	—	6	1	no	—
69	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
70	#2211	—	3	1	no	Sylow
71	C₂²×C₅×D₃	—	4	1	no	—
72	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
75	C₂×C₅×Dic₃	—	4	1	no	—
76	C₂×C₅×Dic₃	—	4	1	no	—
77	C₂×C₅×Dic₃	—	4	1	no	—
78	C₅×D₁₂	—	4	1	no	—
79	C₅×D₁₂	—	4	1	no	—
80	C₅×Dic₆	—	2	2	no	—
81	C₅×Dic₆	—	4	1	no	—
82	C₅×Dic₆	—	4	1	no	—
83	C₅×Dic₆	—	4	1	no	—
84	C₃×C₅×C₈	—	2	1	no	—
85	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
86	C₃×C₅×C₈	#41	1	1	yes	characteristic
87	C₂×Dic₁₂	C₂²×C₅	1	1	yes	characteristic
88	C₂×Dic₁₂	C₂²×C₅	1	1	yes	characteristic
89	C₂×D₂₄	C₂²×C₅	1	1	yes	characteristic
90	C₂×#204	—	2	1	no	—
91	#875	—	2	1	no	—
92	C₄×Dic₆	—	2	1	no	—
93	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
94	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
95	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
96	#818	C₂²×C₅	1	1	yes	characteristic
97	#816	—	2	1	no	—
98	Dic₂₄	—	2	2	no	—
99	#800	—	2	2	no	—
100	C₂×C₅×Q₈	—	6	1	no	—
101	C₂×C₅×Q₈	—	6	1	no	—
102	C₂×C₅×D₄	—	6	1	no	—
103	C₅×Q₁₆	—	3	2	no	—
104	C₅×Q₁₆	—	6	1	no	—
105	C₅×Q₁₆	—	6	1	no	—
106	C₅×Q₁₆	—	6	1	no	—
107	C₅×SD₁₆	—	12	2	no	—
108	C₅×D₈	—	6	1	no	—
109	C₅×D₈	—	6	1	no	—
110	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
111	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
112	C₅×C₄⋊C₄	—	6	1	no	—
113	C₅×C₄⋊C₄	—	12	1	no	—
114	C₅×C₂²⋊C₄	—	12	1	no	—
115	C₅×C₂²⋊C₄	—	12	1	no	—
116	C₄²×C₅	—	12	1	no	—
117	C₄²×C₅	D₁₂	1	1	yes	characteristic
118	C₅×C₁₆	—	2	1	no	—
119	C₅×C₁₆	—	2	1	no	—
120	C₂×Q₃₂	—	3	1	no	—
121	C₂×SD₃₂	—	3	1	no	—
122	#495	—	3	1	no	—
123	C₄×Q₁₆	—	3	1	no	—
124	C₄⋊C₁₆	C₅×D₃	1	1	yes	characteristic, complemented
125	#358	—	3	1	no	—
126	#357	—	3	1	no	—
127	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
128	C₂×C₅×D₃	—	8	1	no	—
129	C₂×C₅×D₃	—	8	1	no	—
130	C₂×C₅×D₃	—	8	1	no	—
131	C₃×C₄×C₅	—	2	1	no	—
132	C₃×C₄×C₅	—	2	1	no	—
133	C₃×C₄×C₅	#121	1	1	yes	characteristic
134	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₅×Dic₃	—	4	2	no	—
136	C₅×Dic₃	—	8	1	no	—
137	C₅×Dic₃	—	8	1	no	—
138	C₂×D₁₂	—	2	1	no	—
139	C₂×Dic₆	—	2	1	no	—
140	C₂×Dic₆	—	2	1	no	—
141	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
142	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
143	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
144	(C₂×D₃)⋊C₄	—	4	1	no	—
145	(C₂×D₃)⋊C₄	—	4	1	no	—
146	C₄⋊Dic₃	—	2	1	no	—
147	Dic₃⋊C₄	—	4	1	no	—
148	C₄×Dic₃	—	4	1	no	—
149	Dic₁₂	—	2	1	no	—
150	Dic₁₂	—	2	1	no	—
151	Dic₁₂	—	2	1	no	—
152	Dic₁₂	C₅×D₄	1	2	yes	—
153	D₂₄	—	2	1	no	—
154	D₂₄	—	2	1	no	—
155	#204	—	4	2	no	—
156	C₃×C₁₆	—	2	1	no	—
157	C₃×C₁₆	—	2	1	no	—
158	C₂³×C₅	—	12	1	no	—
159	C₅×Q₈	—	6	2	no	—
160	C₅×Q₈	—	12	1	no	—
161	C₅×Q₈	—	12	1	no	—
162	C₅×Q₈	—	12	1	no	—
163	C₅×D₄	—	12	1	no	—
164	C₅×D₄	—	12	1	no	—
165	C₂×C₄×C₅	—	12	1	no	—
166	C₂×C₄×C₅	—	12	1	no	—
167	C₂×C₄×C₅	—	12	1	no	—
168	C₂×C₄×C₅	D₂₄	1	1	yes	characteristic
169	C₂×C₄×C₅	D₂₄	1	1	yes	characteristic
170	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
171	C₅×C₈	—	2	1	no	—
172	C₅×C₈	D₃×D₄	1	1	yes	characteristic
173	C₅×C₈	#239	1	1	yes	characteristic
174	C₂×Q₁₆	—	3	1	no	—
175	C₂×Q₁₆	—	3	1	no	—
176	C₂×SD₁₆	—	6	1	no	—
177	C₂×D₈	—	3	1	no	—
178	#124	—	6	1	no	—
179	C₄×Q₈	—	6	1	no	—
180	Q₃₂	—	6	2	no	—
181	SD₃₂	—	6	2	no	—
182	C₂×C₁₆	C₂×C₅×D₃	1	1	yes	characteristic
183	C₂×C₁₆	C₂×C₅×D₃	1	1	yes	characteristic
184	#107	—	3	1	no	—
185	#103	—	6	1	no	—
186	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
187	C₂×C₃×C₅	#459	1	1	yes	characteristic
188	C₂×C₃×C₅	#508	1	1	yes	characteristic
189	C₂×C₃×C₅	#510	1	1	yes	characteristic
190	C₅×D₃	—	16	1	no	—
191	C₂²×D₃	—	4	1	no	—
192	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
193	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
195	C₂×Dic₃	—	4	1	no	—
196	C₂×Dic₃	—	4	1	no	—
197	C₂×Dic₃	—	4	1	no	—
198	D₁₂	—	4	1	no	—
199	D₁₂	—	4	1	no	—
200	Dic₆	—	2	2	no	—
201	Dic₆	—	4	1	no	—
202	Dic₆	—	4	1	no	—
203	Dic₆	—	4	1	no	—
204	C₃×C₈	—	2	1	no	—
205	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
206	C₃×C₈	C₅×#41	1	1	yes	characteristic
207	C₂²×C₅	—	24	1	no	—
208	C₂²×C₅	—	24	1	no	—
209	C₂²×C₅	—	24	1	no	—
210	C₂²×C₅	C₂×D₂₄	1	1	yes	characteristic
211	C₄×C₅	—	2	1	no	—
212	C₄×C₅	—	2	1	no	—
213	C₄×C₅	—	12	2	no	—
214	C₄×C₅	—	24	1	no	—
215	C₄×C₅	—	24	1	no	—
216	C₄×C₅	#895	1	1	yes	characteristic
217	C₄×C₅	#908	1	1	yes	characteristic
218	C₂×Q₈	—	6	1	no	—
219	C₂×Q₈	—	6	1	no	—
220	C₂×D₄	—	6	1	no	—
221	Q₁₆	—	3	2	no	—
222	Q₁₆	—	6	1	no	—
223	Q₁₆	—	6	1	no	—
224	Q₁₆	—	6	1	no	—
225	SD₁₆	—	12	2	no	—
226	D₈	—	6	1	no	—
227	D₈	—	6	1	no	—
228	C₂×C₈	C₅×D₁₂	1	1	yes	characteristic
229	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
230	C₄⋊C₄	—	6	1	no	—
231	C₄⋊C₄	—	12	1	no	—
232	C₂²⋊C₄	—	12	1	no	—
233	C₂²⋊C₄	—	12	1	no	—
234	C₄²	—	12	1	no	—
235	C₄²	C₅×D₁₂	1	1	yes	characteristic
236	C₁₆	—	2	1	no	—
237	C₁₆	—	2	1	no	—
238	C₃×C₅	#2211	1	1	yes	characteristic, complemented, Hall
239	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
240	C₂×D₃	—	8	1	no	—
241	C₂×D₃	—	8	1	no	—
242	C₂×D₃	—	8	1	no	—
243	C₃×C₄	—	2	1	no	—
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	C₅×#121	1	1	yes	characteristic
246	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
247	Dic₃	—	4	2	no	—
248	Dic₃	—	8	1	no	—
249	Dic₃	—	8	1	no	—
250	C₂×C₅	—	48	1	no	—
251	C₂×C₅	#4843	1	1	yes	characteristic
252	C₂×C₅	#4904	1	1	yes	characteristic
253	C₂×C₅	#4909	1	1	yes	characteristic
254	C₂³	—	12	1	no	—
255	Q₈	—	6	2	no	—
256	Q₈	—	12	1	no	—
257	Q₈	—	12	1	no	—
258	Q₈	—	12	1	no	—
259	D₄	—	12	1	no	—
260	D₄	—	12	1	no	—
261	C₂×C₄	—	12	1	no	—
262	C₂×C₄	—	12	1	no	—
263	C₂×C₄	—	12	1	no	—
264	C₂×C₄	C₅×D₂₄	1	1	yes	characteristic
265	C₂×C₄	C₅×D₂₄	1	1	yes	characteristic
266	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
267	C₈	—	2	1	no	—
268	C₈	C₅×D₃×D₄	1	1	yes	characteristic
269	C₈	C₅×#239	1	1	yes	characteristic
270	C₂×C₃	C₅×#459	1	1	yes	characteristic
271	C₂×C₃	C₅×#508	1	1	yes	characteristic
272	C₂×C₃	C₅×#510	1	1	yes	characteristic
273	D₃	—	16	1	no	—
274	C₅	#69294	1	1	yes	characteristic, direct product factor, Sylow
275	C₂²	—	24	1	no	—
276	C₂²	—	24	1	no	—
277	C₂²	—	24	1	no	—
278	C₂²	C₂×C₅×D₂₄	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	12	2	no	—
282	C₄	—	24	1	no	—
283	C₄	—	24	1	no	—
284	C₄	C₅×#895	1	1	yes	characteristic
285	C₄	C₅×#908	1	1	yes	characteristic
286	C₃	C₅×#2211	1	1	yes	characteristic, complemented, Sylow
287	C₂	—	48	1	no	—
288	C₂	C₅×#4843	1	1	yes	characteristic
289	C₂	C₅×#4904	1	1	yes	characteristic
290	C₂	C₅×#4909	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂×D₂₄
commutator, cosocle	C₂×C₃×C₈	C₂³×C₅
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₃×C₅×C₄⋊C₁₆	C₂
socle	C₂²×C₃×C₅	C₂×D₈
prime radical	C₂×C₅×D₂₄	C₂²
hypercenter	C₅×C₄⋊C₁₆	D₃
hypocenter	C₃	C₅×#2211
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₄×C₅×C₈ ◁ C₅×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×Dic₂₄	C₂	semidirect
3. C₄×C₅×Dic₁₂	C₂	semidirect
5. C₃×C₅×C₄⋊C₁₆	C₂	semidirect
7. C₅×#4503	C₂	semidirect
22. #69294	C₅	direct (N × H)
27. C₅×C₄⋊C₁₆	D₃	semidirect
50. C₂×Dic₂₄	C₂×C₅	semidirect
52. C₄×Dic₁₂	C₂×C₅	semidirect
54. C₃×C₄⋊C₁₆	C₂×C₅	semidirect
56. #4503	C₂×C₅	semidirect
124. C₄⋊C₁₆	C₅×D₃	semidirect
238. C₃×C₅	#2211	semidirect
274. C₅	#69294	direct (N × H)
286. C₃	C₅×#2211	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×Dic₂₄	2
C₂×C₅×#800	2
C₄×C₅×Dic₁₂	2
C₅×#4696	2
C₃×C₅×C₄⋊C₁₆	2
C₅×#4509	2
C₅×#4503	2
C₅×#2211	3
#69294	5

Subgroups — C5×#69294 #14317257

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69294 #14317257