Subgroups — C₅×#68606 #14298736

Order 1920.

432 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	C₂×C₅×Dic₃⋊C₈	C₂	1	2	yes	maximal
2	C₅×#4932	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#336	C₂	1	1	yes	maximal, characteristic
4	C₅×#4550	C₂	1	2	yes	maximal
5	C₅×#4524	C₂	1	1	yes	maximal, characteristic
6	C₅×#1832	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
11	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
12	C₅×C₃⋊#95	C₄	1	2	yes	—
13	C₅×C₃⋊#95	C₄	1	2	yes	—
14	C₅×Dic₃⋊C₈	—	2	4	no	—
15	#68606	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂×C₅×C₄⋊C₈	—	3	2	no	—
17	C₅×#376	—	3	1	no	—
18	C₅×#336	—	3	2	no	—
19	C₅×#336	—	3	1	no	—
20	C₅×#336	D₃	1	1	yes	characteristic
21	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
22	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₂²×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₅×Dic₃	C₂×C₄	1	2	yes	—
26	C₂×C₃×C₅×C₈	—	2	2	no	—
27	C₂×C₃×C₅×C₈	—	2	2	no	—
28	C₂×C₃×C₅×C₈	—	2	2	no	—
29	C₃×C₄²×C₅	—	2	2	no	—
30	C₄×C₅×Dic₃	—	2	2	no	—
31	C₄×C₅×Dic₃	D₄	1	2	yes	—
32	C₄×C₅×Dic₃	Q₈	1	2	yes	—
33	C₂×C₅×C₃⋊C₈	—	2	2	no	—
34	C₂×C₅×C₃⋊C₈	—	2	2	no	—
35	C₂×C₅×C₃⋊C₈	—	2	2	no	—
36	C₂×Dic₃⋊C₈	C₂×C₅	1	2	yes	—
37	#4932	C₂×C₅	1	1	yes	characteristic
38	C₃×#336	C₂×C₅	1	1	yes	characteristic
39	#4550	C₂×C₅	1	2	yes	—
40	#4524	C₂×C₅	1	1	yes	characteristic
41	C₂²×C₅×C₈	—	3	2	no	—
42	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
43	C₂×C₄²×C₅	—	3	1	no	—
44	C₂×C₄²×C₅	—	3	1	no	—
45	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
46	C₅×C₄⋊C₈	—	6	4	no	—
47	C₅×#95	—	3	2	no	—
48	C₅×#95	—	3	2	no	—
49	#1832	—	3	1	no	Sylow
50	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	—	2	2	no	—
52	C₂×C₃×C₄×C₅	—	2	2	no	—
53	C₂×C₃×C₄×C₅	—	2	2	no	—
54	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	#41	1	2	yes	—
57	C₂×C₃×C₄×C₅	#41	1	2	yes	—
58	C₂×C₅×Dic₃	—	2	2	no	—
59	C₂×C₅×Dic₃	—	2	2	no	—
60	C₂×C₅×Dic₃	—	2	1	no	—
61	C₂×C₅×Dic₃	—	2	1	no	—
62	C₂×C₅×Dic₃	—	2	2	no	—
63	C₂×C₅×Dic₃	—	2	2	no	—
64	C₂×C₅×Dic₃	C₄⋊C₄	1	4	yes	—
65	C₃×C₅×C₈	—	4	2	no	—
66	C₅×C₃⋊C₈	—	4	2	no	—
67	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
68	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
69	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
70	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
71	C₂²×C₃⋊C₈	C₂²×C₅	1	2	yes	—
72	C₃⋊#95	C₄×C₅	1	2	yes	—
73	C₃⋊#95	C₄×C₅	1	2	yes	—
74	Dic₃⋊C₈	—	2	4	no	—
75	C₂²×C₄×C₅	—	3	2	no	—
76	C₂²×C₄×C₅	—	3	1	no	—
77	C₂²×C₄×C₅	—	3	1	no	—
78	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
79	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
80	C₂×C₅×C₈	—	2	2	no	—
81	C₂×C₅×C₈	—	2	2	no	—
82	C₂×C₅×C₈	—	2	2	no	—
83	C₂×C₅×C₈	—	6	2	no	—
84	C₂×C₅×C₈	—	6	2	no	—
85	C₂×C₅×C₈	—	6	2	no	—
86	C₄²×C₅	—	2	2	no	—
87	C₄²×C₅	—	3	2	no	—
88	C₄²×C₅	—	3	2	no	—
89	C₄²×C₅	—	6	2	no	—
90	C₂×C₄⋊C₈	—	3	2	no	—
91	#376	—	3	1	no	—
92	#336	—	3	2	no	—
93	#336	—	3	1	no	—
94	#336	C₅×D₃	1	1	yes	characteristic
95	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂²×C₃×C₅	#117	1	1	yes	characteristic
97	C₂²×C₃×C₅	#117	1	1	yes	characteristic
98	C₂²×C₃×C₅	#131	1	2	yes	—
99	C₂²×C₃×C₅	#131	1	2	yes	—
100	C₃×C₄×C₅	—	4	2	no	—
101	C₃×C₄×C₅	#124	1	2	yes	—
102	C₃×C₄×C₅	#125	1	2	yes	—
103	C₅×Dic₃	—	2	4	no	—
104	C₅×Dic₃	—	4	1	no	—
105	C₅×Dic₃	—	4	1	no	—
106	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
107	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
108	C₂²×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
109	C₂²×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
110	C₂²×Dic₃	C₂×C₄×C₅	1	2	yes	—
111	C₂×C₃×C₈	—	2	2	no	—
112	C₂×C₃×C₈	—	2	2	no	—
113	C₂×C₃×C₈	—	2	2	no	—
114	C₃×C₄²	—	2	2	no	—
115	C₄×Dic₃	—	2	2	no	—
116	C₄×Dic₃	C₅×D₄	1	2	yes	—
117	C₄×Dic₃	C₅×Q₈	1	2	yes	—
118	C₂×C₃⋊C₈	—	2	2	no	—
119	C₂×C₃⋊C₈	—	2	2	no	—
120	C₂×C₃⋊C₈	—	2	2	no	—
121	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	—	2	2	no	—
125	C₂×C₄×C₅	—	3	4	no	—
126	C₂×C₄×C₅	—	6	2	no	—
127	C₂×C₄×C₅	—	6	2	no	—
128	C₂×C₄×C₅	—	6	1	no	—
129	C₂×C₄×C₅	—	6	1	no	—
130	C₂×C₄×C₅	—	6	2	no	—
131	C₂×C₄×C₅	—	6	2	no	—
132	C₂×C₄×C₅	#235	1	2	yes	—
133	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
134	C₂×C₄×C₅	#237	1	2	yes	—
135	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
136	C₅×C₈	—	4	2	no	—
137	C₅×C₈	—	12	2	no	—
138	C₂²×C₈	—	3	2	no	—
139	C₂²×C₈	C₂×C₅×D₃	1	2	yes	—
140	C₂×C₄²	—	3	1	no	—
141	C₂×C₄²	—	3	1	no	—
142	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
143	C₄⋊C₈	—	6	4	no	—
144	#95	—	3	2	no	—
145	#95	—	3	2	no	—
146	C₂×C₃×C₅	#382	1	1	yes	characteristic
147	C₂×C₃×C₅	#424	1	2	yes	—
148	C₂×C₃×C₅	#433	1	1	yes	characteristic
149	C₂×C₃×C₅	#433	1	1	yes	characteristic
150	C₂×C₃×C₅	#433	1	2	yes	—
151	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
152	C₂×C₃×C₄	—	2	2	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
157	C₂×C₃×C₄	C₅×#41	1	2	yes	—
158	C₂×C₃×C₄	C₅×#41	1	2	yes	—
159	C₂×Dic₃	—	2	2	no	—
160	C₂×Dic₃	—	2	2	no	—
161	C₂×Dic₃	—	2	1	no	—
162	C₂×Dic₃	—	2	1	no	—
163	C₂×Dic₃	—	2	2	no	—
164	C₂×Dic₃	—	2	2	no	—
165	C₂×Dic₃	C₅×C₄⋊C₄	1	4	yes	—
166	C₃×C₈	—	4	2	no	—
167	C₃⋊C₈	—	4	2	no	—
168	C₂²×C₅	#872	1	1	yes	characteristic
169	C₂²×C₅	#877	1	1	yes	characteristic
170	C₂²×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
171	C₂²×C₅	#901	1	2	yes	—
172	C₂²×C₅	#907	1	2	yes	—
173	C₄×C₅	—	4	2	no	—
174	C₄×C₅	—	6	4	no	—
175	C₄×C₅	—	12	1	no	—
176	C₄×C₅	—	12	1	no	—
177	C₄×C₅	#885	1	2	yes	—
178	C₄×C₅	#889	1	2	yes	—
179	C₂²×C₄	—	3	2	no	—
180	C₂²×C₄	—	3	1	no	—
181	C₂²×C₄	—	3	1	no	—
182	C₂²×C₄	C₄×C₅×D₃	1	2	yes	—
183	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	6	2	no	—
188	C₂×C₈	—	6	2	no	—
189	C₂×C₈	—	6	2	no	—
190	C₄²	—	2	2	no	—
191	C₄²	—	3	2	no	—
192	C₄²	—	3	2	no	—
193	C₄²	—	6	2	no	—
194	C₃×C₅	#1832	1	1	yes	characteristic, complemented, Hall
195	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
196	C₂²×C₃	C₅×#117	1	1	yes	characteristic
197	C₂²×C₃	C₅×#117	1	1	yes	characteristic
198	C₂²×C₃	C₅×#131	1	2	yes	—
199	C₂²×C₃	C₅×#131	1	2	yes	—
200	C₃×C₄	—	4	2	no	—
201	C₃×C₄	C₅×#124	1	2	yes	—
202	C₃×C₄	C₅×#125	1	2	yes	—
203	Dic₃	—	2	4	no	—
204	Dic₃	—	4	1	no	—
205	Dic₃	—	4	1	no	—
206	C₂×C₅	#4650	1	1	yes	characteristic
207	C₂×C₅	#4690	1	1	yes	characteristic
208	C₂×C₅	#4709	1	1	yes	characteristic
209	C₂×C₅	#4720	1	2	yes	—
210	C₂×C₅	#4839	1	2	yes	—
211	C₂³	C₂×C₄×C₅×D₃	1	1	yes	characteristic
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	3	4	no	—
216	C₂×C₄	—	6	2	no	—
217	C₂×C₄	—	6	2	no	—
218	C₂×C₄	—	6	1	no	—
219	C₂×C₄	—	6	1	no	—
220	C₂×C₄	—	6	2	no	—
221	C₂×C₄	—	6	2	no	—
222	C₂×C₄	C₅×#235	1	2	yes	—
223	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
224	C₂×C₄	C₅×#237	1	2	yes	—
225	C₂×C₄	C₅×D₃×Q₈	1	1	yes	characteristic
226	C₈	—	4	2	no	—
227	C₈	—	12	2	no	—
228	C₂×C₃	C₅×#382	1	1	yes	characteristic
229	C₂×C₃	C₅×#424	1	2	yes	—
230	C₂×C₃	C₅×#433	1	1	yes	characteristic
231	C₂×C₃	C₅×#433	1	1	yes	characteristic
232	C₂×C₃	C₅×#433	1	2	yes	—
233	C₅	#68606	1	1	yes	characteristic, direct product factor, Sylow
234	C₂²	C₅×#872	1	1	yes	characteristic
235	C₂²	C₅×#877	1	1	yes	characteristic
236	C₂²	C₅×D₃×C₄⋊C₄	1	1	yes	characteristic
237	C₂²	C₅×#901	1	2	yes	—
238	C₂²	C₅×#907	1	2	yes	—
239	C₄	—	4	2	no	—
240	C₄	—	6	4	no	—
241	C₄	—	12	1	no	—
242	C₄	—	12	1	no	—
243	C₄	C₅×#885	1	2	yes	—
244	C₄	C₅×#889	1	2	yes	—
245	C₃	C₅×#1832	1	1	yes	characteristic, complemented, Sylow
246	C₂	C₅×#4650	1	1	yes	characteristic
247	C₂	C₅×#4690	1	1	yes	characteristic
248	C₂	C₅×#4709	1	1	yes	characteristic
249	C₂	C₅×#4720	1	2	yes	—
250	C₂	C₅×#4839	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂²×D₃
commutator	C₂³×C₃	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#336	C₂
socle, prime radical	C₂³×C₃×C₅	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#336	D₃
hypocenter	C₃	C₅×#1832
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₅×#336

Complements

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

N	H ≅ G/N	type
15. #68606	C₅	direct (N × H)
194. C₃×C₅	#1832	semidirect
233. C₅	#68606	direct (N × H)
245. C₃	C₅×#1832	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×Dic₃⋊C₈	2
C₅×#4932	2
C₃×C₅×#336	2
C₅×#4550	2
C₅×#4524	2
C₅×#1832	3
#68606	5

Subgroups — C5×#68606 #14298736

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68606 #14298736