Subgroups — C₅×#69291 #14318040

Order 1920.

272 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×D₂₄	C₂	1	1	yes	maximal, characteristic
2	C₅×#4683	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4509	C₂	1	1	yes	maximal, characteristic
5	C₅×#4509	C₂	1	1	yes	maximal, characteristic
6	C₅×#4505	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4504	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2228	—	3	1	no	maximal, Hall
9	C₂×C₅×D₂₄	C₂²	1	1	yes	characteristic
10	C₄×C₅×D₁₂	—	2	1	no	—
11	C₅×#870	—	2	1	no	—
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×#821	—	2	1	no	—
16	C₅×#818	C₂²	1	1	yes	characteristic
17	C₅×#818	C₂²	1	1	yes	characteristic
18	C₅×#818	C₂²	1	1	yes	characteristic
19	C₅×#817	—	2	1	no	—
20	#69291	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₅×#499	—	3	1	no	—
22	C₄×C₅×D₈	—	3	1	no	—
23	C₅×#367	—	3	1	no	—
24	C₅×#366	—	3	1	no	—
25	C₅×C₄⋊C₁₆	D₃	1	1	yes	characteristic, complemented
26	C₅×#357	—	3	1	no	—
27	C₅×#357	—	3	1	no	—
28	C₂×C₅×D₁₂	—	2	1	no	—
29	C₂×C₄×C₅×D₃	—	4	1	no	—
30	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
31	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
32	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
33	C₅×(C₂×D₃)⋊C₄	—	4	1	no	—
34	C₅×C₄⋊Dic₃	—	2	1	no	—
35	C₅×C₄⋊Dic₃	—	2	1	no	—
36	C₅×C₄⋊Dic₃	—	2	1	no	—
37	C₅×Dic₃⋊C₄	—	4	1	no	—
38	C₅×Dic₃⋊C₄	—	4	1	no	—
39	C₅×D₂₄	—	2	1	no	—
40	C₅×D₂₄	Q₈	1	2	yes	—
41	C₃×C₅×C₁₆	—	2	1	no	—
42	C₃×C₅×C₁₆	—	2	1	no	—
43	C₄×D₂₄	C₂×C₅	1	1	yes	characteristic
44	#4683	C₂×C₅	1	1	yes	characteristic, complemented
45	C₃×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
46	#4509	C₂×C₅	1	1	yes	characteristic
47	#4509	C₂×C₅	1	1	yes	characteristic
48	#4505	C₂×C₅	1	1	yes	characteristic, complemented
49	#4504	C₂×C₅	1	1	yes	characteristic, complemented
50	C₂×C₅×D₈	—	3	1	no	—
51	C₅×#125	—	6	1	no	—
52	C₄×C₅×D₄	—	6	1	no	—
53	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
54	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
55	C₅×#107	—	3	1	no	—
56	C₅×#107	—	3	1	no	—
57	C₅×#107	—	3	1	no	—
58	C₅×#106	—	6	1	no	—
59	C₅×#102	—	6	1	no	—
60	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
61	#2228	—	3	1	no	Sylow
62	C₂²×C₅×D₃	—	4	1	no	—
63	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
64	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
66	C₂×C₅×Dic₃	—	4	1	no	—
67	C₂×C₅×Dic₃	—	4	1	no	—
68	C₂×C₅×Dic₃	—	4	1	no	—
69	C₅×D₁₂	—	2	2	no	—
70	C₅×D₁₂	—	4	1	no	—
71	C₄×C₅×D₃	—	8	2	no	—
72	C₃×C₅×C₈	—	2	1	no	—
73	C₃×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
74	C₃×C₅×C₈	#41	1	1	yes	characteristic
75	C₂×D₂₄	C₂²×C₅	1	1	yes	characteristic
76	C₄×D₁₂	—	2	1	no	—
77	#870	—	2	1	no	—
78	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
79	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
80	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
81	#821	—	2	1	no	—
82	#818	C₂²×C₅	1	1	yes	characteristic
83	#818	C₂²×C₅	1	1	yes	characteristic
84	#818	C₂²×C₅	1	1	yes	characteristic
85	#817	—	2	1	no	—
86	C₂×C₅×D₄	—	6	1	no	—
87	C₂²×C₄×C₅	—	12	1	no	—
88	C₅×D₈	—	3	2	no	—
89	C₅×D₈	—	6	1	no	—
90	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
91	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
92	C₅×C₄⋊C₄	—	6	1	no	—
93	C₅×C₄⋊C₄	—	6	1	no	—
94	C₅×C₄⋊C₄	—	6	1	no	—
95	C₅×C₄⋊C₄	—	12	1	no	—
96	C₅×C₄⋊C₄	—	12	1	no	—
97	C₅×C₂²⋊C₄	—	12	1	no	—
98	C₄²×C₅	D₁₂	1	1	yes	characteristic
99	C₅×C₁₆	—	2	1	no	—
100	C₅×C₁₆	—	2	1	no	—
101	#499	—	3	1	no	—
102	C₄×D₈	—	3	1	no	—
103	#367	—	3	1	no	—
104	#366	—	3	1	no	—
105	C₄⋊C₁₆	C₅×D₃	1	1	yes	characteristic, complemented
106	#357	—	3	1	no	—
107	#357	—	3	1	no	—
108	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
109	C₂×C₅×D₃	—	4	2	no	—
110	C₂×C₅×D₃	—	8	1	no	—
111	C₂×C₅×D₃	—	8	1	no	—
112	C₃×C₄×C₅	—	2	1	no	—
113	C₃×C₄×C₅	—	2	1	no	—
114	C₃×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
115	C₃×C₄×C₅	#137	1	1	yes	characteristic
116	C₅×Dic₃	—	8	1	no	—
117	C₅×Dic₃	—	8	1	no	—
118	C₅×Dic₃	—	8	1	no	—
119	C₂×D₁₂	—	2	1	no	—
120	C₂×C₄×D₃	—	4	1	no	—
121	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
122	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
123	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
124	(C₂×D₃)⋊C₄	—	4	1	no	—
125	C₄⋊Dic₃	—	2	1	no	—
126	C₄⋊Dic₃	—	2	1	no	—
127	C₄⋊Dic₃	—	2	1	no	—
128	Dic₃⋊C₄	—	4	1	no	—
129	Dic₃⋊C₄	—	4	1	no	—
130	D₂₄	—	2	1	no	—
131	D₂₄	C₅×Q₈	1	2	yes	—
132	C₃×C₁₆	—	2	1	no	—
133	C₃×C₁₆	—	2	1	no	—
134	C₂³×C₅	—	12	1	no	—
135	C₅×D₄	—	6	2	no	—
136	C₅×D₄	—	12	1	no	—
137	C₂×C₄×C₅	—	12	1	no	—
138	C₂×C₄×C₅	—	12	1	no	—
139	C₂×C₄×C₅	—	12	1	no	—
140	C₂×C₄×C₅	—	24	2	no	—
141	C₂×C₄×C₅	D₂₄	1	1	yes	characteristic
142	C₂×C₄×C₅	D₂₄	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
144	C₅×C₈	—	2	1	no	—
145	C₅×C₈	#237	1	1	yes	characteristic
146	C₅×C₈	D₃×Q₈	1	1	yes	characteristic
147	C₂×D₈	—	3	1	no	—
148	#125	—	6	1	no	—
149	C₄×D₄	—	6	1	no	—
150	C₂×C₁₆	C₂×C₅×D₃	1	1	yes	characteristic
151	C₂×C₁₆	C₂×C₅×D₃	1	1	yes	characteristic
152	#107	—	3	1	no	—
153	#107	—	3	1	no	—
154	#107	—	3	1	no	—
155	#106	—	6	1	no	—
156	#102	—	6	1	no	—
157	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
158	C₂×C₃×C₅	#474	1	1	yes	characteristic
159	C₂×C₃×C₅	#508	1	1	yes	characteristic
160	C₂×C₃×C₅	#509	1	1	yes	characteristic
161	C₅×D₃	—	8	2	no	—
162	C₂²×D₃	—	4	1	no	—
163	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
164	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
166	C₂×Dic₃	—	4	1	no	—
167	C₂×Dic₃	—	4	1	no	—
168	C₂×Dic₃	—	4	1	no	—
169	D₁₂	—	2	2	no	—
170	D₁₂	—	4	1	no	—
171	C₄×D₃	—	8	2	no	—
172	C₃×C₈	—	2	1	no	—
173	C₃×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
174	C₃×C₈	C₅×#41	1	1	yes	characteristic
175	C₂²×C₅	—	12	2	no	—
176	C₂²×C₅	—	24	1	no	—
177	C₂²×C₅	—	24	1	no	—
178	C₂²×C₅	C₂×D₂₄	1	1	yes	characteristic
179	C₄×C₅	—	2	1	no	—
180	C₄×C₅	—	2	1	no	—
181	C₄×C₅	—	24	1	no	—
182	C₄×C₅	—	24	1	no	—
183	C₄×C₅	—	24	1	no	—
184	C₄×C₅	#897	1	1	yes	characteristic
185	C₄×C₅	#909	1	1	yes	characteristic
186	C₂×D₄	—	6	1	no	—
187	C₂²×C₄	—	12	1	no	—
188	D₈	—	3	2	no	—
189	D₈	—	6	1	no	—
190	C₂×C₈	C₅×D₁₂	1	1	yes	characteristic
191	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
192	C₄⋊C₄	—	6	1	no	—
193	C₄⋊C₄	—	6	1	no	—
194	C₄⋊C₄	—	6	1	no	—
195	C₄⋊C₄	—	12	1	no	—
196	C₄⋊C₄	—	12	1	no	—
197	C₂²⋊C₄	—	12	1	no	—
198	C₄²	C₅×D₁₂	1	1	yes	characteristic
199	C₁₆	—	2	1	no	—
200	C₁₆	—	2	1	no	—
201	C₃×C₅	#2228	1	1	yes	characteristic, complemented, Hall
202	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
203	C₂×D₃	—	4	2	no	—
204	C₂×D₃	—	8	1	no	—
205	C₂×D₃	—	8	1	no	—
206	C₃×C₄	—	2	1	no	—
207	C₃×C₄	—	2	1	no	—
208	C₃×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
209	C₃×C₄	C₅×#137	1	1	yes	characteristic
210	Dic₃	—	8	1	no	—
211	Dic₃	—	8	1	no	—
212	Dic₃	—	8	1	no	—
213	C₂×C₅	—	24	2	no	—
214	C₂×C₅	#4846	1	1	yes	characteristic
215	C₂×C₅	#4904	1	1	yes	characteristic
216	C₂×C₅	#4908	1	1	yes	characteristic
217	C₂³	—	12	1	no	—
218	D₄	—	6	2	no	—
219	D₄	—	12	1	no	—
220	C₂×C₄	—	12	1	no	—
221	C₂×C₄	—	12	1	no	—
222	C₂×C₄	—	12	1	no	—
223	C₂×C₄	—	24	2	no	—
224	C₂×C₄	C₅×D₂₄	1	1	yes	characteristic
225	C₂×C₄	C₅×D₂₄	1	1	yes	characteristic
226	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
227	C₈	—	2	1	no	—
228	C₈	C₅×#237	1	1	yes	characteristic
229	C₈	C₅×D₃×Q₈	1	1	yes	characteristic
230	C₂×C₃	C₅×#474	1	1	yes	characteristic
231	C₂×C₃	C₅×#508	1	1	yes	characteristic
232	C₂×C₃	C₅×#509	1	1	yes	characteristic
233	D₃	—	8	2	no	—
234	C₅	#69291	1	1	yes	characteristic, direct product factor, Sylow
235	C₂²	—	12	2	no	—
236	C₂²	—	24	1	no	—
237	C₂²	—	24	1	no	—
238	C₂²	C₂×C₅×D₂₄	1	1	yes	characteristic
239	C₄	—	2	1	no	—
240	C₄	—	2	1	no	—
241	C₄	—	24	1	no	—
242	C₄	—	24	1	no	—
243	C₄	—	24	1	no	—
244	C₄	C₅×#897	1	1	yes	characteristic
245	C₄	C₅×#909	1	1	yes	characteristic
246	C₃	C₅×#2228	1	1	yes	characteristic, complemented, Sylow
247	C₂	—	24	2	no	—
248	C₂	C₅×#4846	1	1	yes	characteristic
249	C₂	C₅×#4904	1	1	yes	characteristic
250	C₂	C₅×#4908	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×#2228
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
2. C₅×#4683	C₂	semidirect
3. C₃×C₅×C₄⋊C₁₆	C₂	semidirect
6. C₅×#4505	C₂	semidirect
7. C₅×#4504	C₂	semidirect
20. #69291	C₅	direct (N × H)
25. C₅×C₄⋊C₁₆	D₃	semidirect
44. #4683	C₂×C₅	semidirect
45. C₃×C₄⋊C₁₆	C₂×C₅	semidirect
48. #4505	C₂×C₅	semidirect
49. #4504	C₂×C₅	semidirect
105. C₄⋊C₁₆	C₅×D₃	semidirect
201. C₃×C₅	#2228	semidirect
234. C₅	#69291	direct (N × H)
246. C₃	C₅×#2228	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×D₂₄	2
C₅×#4683	2
C₃×C₅×C₄⋊C₁₆	2
C₅×#4509	2
C₅×#4505	2
C₅×#4504	2
C₅×#2228	3
#69291	5

Subgroups — C5×#69291 #14318040

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69291 #14318040