Subgroups — C₅×#68597 #14294892

Order 1920.

544 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#815	C₂	1	2	yes	maximal
2	C₄²×C₅×Dic₃	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×#1745	—	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	C₄	1	2	yes	—
10	C₂×C₄×C₅×Dic₃	C₄	1	2	yes	—
11	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
13	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
14	C₅×#815	C₄	1	8	yes	complemented
15	#68597	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂×C₅×#97	—	3	2	no	—
17	C₄³×C₅	—	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	C₂×C₄	1	4	yes	—
29	C₃×C₄²×C₅	—	2	2	no	—
30	C₄×C₅×Dic₃	—	2	4	no	—
31	C₄×C₅×Dic₃	—	2	1	no	—
32	C₄×C₅×Dic₃	—	2	1	no	—
33	C₄×C₅×Dic₃	—	2	4	no	—
34	C₄×C₅×Dic₃	C₂×C₄	1	4	yes	—
35	C₂×C₅×C₃⋊C₈	—	2	2	no	—
36	C₂×C₅×C₃⋊C₈	—	2	2	no	—
37	C₂×C₅×C₃⋊C₈	C₂×C₄	1	4	yes	—
38	C₂×#815	C₂×C₅	1	2	yes	—
39	C₄²×Dic₃	C₂×C₅	1	1	yes	characteristic
40	C₃×#336	C₂×C₅	1	1	yes	characteristic
41	#4550	C₂×C₅	1	2	yes	—
42	#4524	C₂×C₅	1	1	yes	characteristic
43	C₂²×C₅×C₈	—	3	2	no	—
44	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
45	C₂×C₄²×C₅	—	3	1	no	—
46	C₂×C₄²×C₅	—	3	2	no	—
47	C₂×C₄²×C₅	—	3	2	no	—
48	C₂×C₄²×C₅	—	3	1	no	—
49	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
50	C₅×#97	—	3	8	no	—
51	#1745	—	3	1	no	Sylow
52	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	—	2	2	no	—
54	C₂×C₃×C₄×C₅	—	2	2	no	—
55	C₂×C₃×C₄×C₅	M₁₆	1	4	yes	—
56	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
57	C₂×C₃×C₄×C₅	#41	1	2	yes	—
58	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
60	C₂×C₅×Dic₃	—	2	2	no	—
61	C₂×C₅×Dic₃	—	2	2	no	—
62	C₂×C₅×Dic₃	—	2	2	no	—
63	C₂×C₅×Dic₃	—	2	2	no	—
64	C₂×C₅×Dic₃	C₄²	1	2	yes	—
65	C₂×C₅×Dic₃	C₄²	1	2	yes	—
66	C₂×C₅×Dic₃	M₁₆	1	4	yes	—
67	C₃×C₅×C₈	—	2	4	no	—
68	C₅×C₃⋊C₈	—	2	4	no	—
69	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
70	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
71	C₂×C₄×Dic₃	C₄×C₅	1	2	yes	—
72	C₂×C₄×Dic₃	C₄×C₅	1	2	yes	—
73	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
74	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
75	C₂²×C₃⋊C₈	C₂²×C₅	1	2	yes	—
76	#815	C₄×C₅	1	8	yes	complemented
77	C₂²×C₄×C₅	—	3	1	no	—
78	C₂²×C₄×C₅	—	3	1	no	—
79	C₂²×C₄×C₅	—	3	2	no	—
80	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
81	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
82	C₂×C₅×C₈	—	2	2	no	—
83	C₂×C₅×C₈	—	2	2	no	—
84	C₂×C₅×C₈	—	3	4	no	—
85	C₂×C₅×C₈	—	6	2	no	—
86	C₂×C₅×C₈	—	6	2	no	—
87	C₂×C₅×C₈	C₄×D₃	1	4	yes	—
88	C₄²×C₅	—	2	2	no	—
89	C₄²×C₅	—	3	4	no	—
90	C₄²×C₅	—	6	4	no	—
91	C₄²×C₅	—	6	4	no	—
92	C₄²×C₅	—	6	1	no	—
93	C₄²×C₅	—	6	1	no	—
94	C₂×#97	—	3	2	no	—
95	C₄³	—	3	1	no	—
96	#336	—	3	2	no	—
97	#336	—	3	1	no	—
98	#336	C₅×D₃	1	1	yes	characteristic
99	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
100	C₂²×C₃×C₅	#117	1	1	yes	characteristic
101	C₂²×C₃×C₅	#117	1	1	yes	characteristic
102	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
103	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
104	C₃×C₄×C₅	—	2	4	no	—
105	C₃×C₄×C₅	#117	1	2	yes	—
106	C₃×C₄×C₅	#117	1	2	yes	—
107	C₅×Dic₃	—	2	2	no	—
108	C₅×Dic₃	—	2	2	no	—
109	C₅×Dic₃	—	2	4	no	—
110	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
111	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
112	C₂²×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
113	C₂²×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
114	C₂²×Dic₃	C₂×C₄×C₅	1	2	yes	—
115	C₂×C₃×C₈	—	2	2	no	—
116	C₂×C₃×C₈	—	2	2	no	—
117	C₂×C₃×C₈	C₂×C₄×C₅	1	4	yes	—
118	C₃×C₄²	—	2	2	no	—
119	C₄×Dic₃	—	2	4	no	—
120	C₄×Dic₃	—	2	1	no	—
121	C₄×Dic₃	—	2	1	no	—
122	C₄×Dic₃	—	2	4	no	—
123	C₄×Dic₃	C₂×C₄×C₅	1	4	yes	—
124	C₂×C₃⋊C₈	—	2	2	no	—
125	C₂×C₃⋊C₈	—	2	2	no	—
126	C₂×C₃⋊C₈	C₂×C₄×C₅	1	4	yes	—
127	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
128	C₂×C₄×C₅	—	2	2	no	—
129	C₂×C₄×C₅	—	2	2	no	—
130	C₂×C₄×C₅	—	3	2	no	—
131	C₂×C₄×C₅	—	3	2	no	—
132	C₂×C₄×C₅	—	3	4	no	—
133	C₂×C₄×C₅	—	6	2	no	—
134	C₂×C₄×C₅	—	6	2	no	—
135	C₂×C₄×C₅	—	6	2	no	—
136	C₂×C₄×C₅	—	6	2	no	—
137	C₂×C₄×C₅	ΓL(1,𝔽₂₅)	1	4	yes	—
138	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
139	C₂×C₄×C₅	#237	1	1	yes	characteristic
140	C₂×C₄×C₅	#237	1	2	yes	—
141	C₂×C₄×C₅	#239	1	1	yes	characteristic
142	C₅×C₈	—	2	4	no	—
143	C₅×C₈	—	6	4	no	—
144	C₂²×C₈	—	3	2	no	—
145	C₂²×C₈	C₂×C₅×D₃	1	2	yes	—
146	C₂×C₄²	—	3	1	no	—
147	C₂×C₄²	—	3	2	no	—
148	C₂×C₄²	—	3	2	no	—
149	C₂×C₄²	—	3	1	no	—
150	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
151	#97	—	3	8	no	—
152	C₂×C₃×C₅	#376	1	1	yes	characteristic
153	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
154	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
155	C₂×C₃×C₅	#432	1	2	yes	—
156	C₂×C₃×C₅	#432	1	2	yes	—
157	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
158	C₂×C₃×C₄	—	2	2	no	—
159	C₂×C₃×C₄	—	2	2	no	—
160	C₂×C₃×C₄	C₅×M₁₆	1	4	yes	—
161	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
162	C₂×C₃×C₄	C₅×#41	1	2	yes	—
163	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
164	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
165	C₂×Dic₃	—	2	2	no	—
166	C₂×Dic₃	—	2	2	no	—
167	C₂×Dic₃	—	2	2	no	—
168	C₂×Dic₃	—	2	2	no	—
169	C₂×Dic₃	C₄²×C₅	1	2	yes	—
170	C₂×Dic₃	C₄²×C₅	1	2	yes	—
171	C₂×Dic₃	C₅×M₁₆	1	4	yes	—
172	C₃×C₈	—	2	4	no	—
173	C₃⋊C₈	—	2	4	no	—
174	C₂²×C₅	C₄²×D₃	1	1	yes	characteristic
175	C₂²×C₅	#877	1	1	yes	characteristic
176	C₂²×C₅	#892	1	1	yes	characteristic
177	C₂²×C₅	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
178	C₂²×C₅	D₃×M₁₆	1	2	yes	—
179	C₄×C₅	—	2	4	no	—
180	C₄×C₅	—	6	2	no	—
181	C₄×C₅	—	6	2	no	—
182	C₄×C₅	—	6	4	no	—
183	C₄×C₅	#877	1	2	yes	—
184	C₄×C₅	#892	1	2	yes	—
185	C₂²×C₄	—	3	1	no	—
186	C₂²×C₄	—	3	1	no	—
187	C₂²×C₄	—	3	2	no	—
188	C₂²×C₄	C₄×C₅×D₃	1	2	yes	—
189	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	3	4	no	—
193	C₂×C₈	—	6	2	no	—
194	C₂×C₈	—	6	2	no	—
195	C₂×C₈	C₄×C₅×D₃	1	4	yes	—
196	C₄²	—	2	2	no	—
197	C₄²	—	3	4	no	—
198	C₄²	—	6	4	no	—
199	C₄²	—	6	4	no	—
200	C₄²	—	6	1	no	—
201	C₄²	—	6	1	no	—
202	C₃×C₅	#1745	1	1	yes	characteristic, complemented, Hall
203	C₂²×C₃	C₂×C₄²×C₅	1	1	yes	characteristic
204	C₂²×C₃	C₅×#117	1	1	yes	characteristic
205	C₂²×C₃	C₅×#117	1	1	yes	characteristic
206	C₂²×C₃	C₂×C₅×M₁₆	1	2	yes	—
207	C₂²×C₃	C₂×C₅×M₁₆	1	2	yes	—
208	C₃×C₄	—	2	4	no	—
209	C₃×C₄	C₅×#117	1	2	yes	—
210	C₃×C₄	C₅×#117	1	2	yes	—
211	Dic₃	—	2	2	no	—
212	Dic₃	—	2	2	no	—
213	Dic₃	—	2	4	no	—
214	C₂×C₅	#4648	1	1	yes	characteristic
215	C₂×C₅	C₄×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
216	C₂×C₅	#4707	1	1	yes	characteristic
217	C₂×C₅	#4719	1	2	yes	—
218	C₂×C₅	#4835	1	2	yes	—
219	C₂³	C₂×C₄×C₅×D₃	1	1	yes	characteristic
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	3	2	no	—
223	C₂×C₄	—	3	2	no	—
224	C₂×C₄	—	3	4	no	—
225	C₂×C₄	—	6	2	no	—
226	C₂×C₄	—	6	2	no	—
227	C₂×C₄	—	6	2	no	—
228	C₂×C₄	—	6	2	no	—
229	C₂×C₄	C₅×ΓL(1,𝔽₂₅)	1	4	yes	—
230	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
231	C₂×C₄	C₅×#237	1	1	yes	characteristic
232	C₂×C₄	C₅×#237	1	2	yes	—
233	C₂×C₄	C₅×#239	1	1	yes	characteristic
234	C₈	—	2	4	no	—
235	C₈	—	6	4	no	—
236	C₂×C₃	C₅×#376	1	1	yes	characteristic
237	C₂×C₃	C₄×C₅×M₁₆	1	1	yes	characteristic
238	C₂×C₃	C₄×C₅×M₁₆	1	1	yes	characteristic
239	C₂×C₃	C₅×#432	1	2	yes	—
240	C₂×C₃	C₅×#432	1	2	yes	—
241	C₅	#68597	1	1	yes	characteristic, direct product factor, Sylow
242	C₂²	C₄²×C₅×D₃	1	1	yes	characteristic
243	C₂²	C₅×#877	1	1	yes	characteristic
244	C₂²	C₅×#892	1	1	yes	characteristic
245	C₂²	C₂×C₅×ΓL(1,𝔽₂₅)	1	2	yes	—
246	C₂²	C₅×D₃×M₁₆	1	2	yes	—
247	C₄	—	2	4	no	—
248	C₄	—	6	2	no	—
249	C₄	—	6	2	no	—
250	C₄	—	6	4	no	—
251	C₄	C₅×#877	1	2	yes	—
252	C₄	C₅×#892	1	2	yes	—
253	C₃	C₅×#1745	1	1	yes	characteristic, complemented, Sylow
254	C₂	C₅×#4648	1	1	yes	characteristic
255	C₂	C₄×C₅×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
256	C₂	C₅×#4707	1	1	yes	characteristic
257	C₂	C₅×#4719	1	2	yes	—
258	C₂	C₅×#4835	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂²×D₃
commutator	C₂²×C₃	C₂×C₄²×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×#1745
LCS	G ▷ C₂²×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
14. C₅×#815	C₄	semidirect
15. #68597	C₅	direct (N × H)
76. #815	C₄×C₅	semidirect
202. C₃×C₅	#1745	semidirect
241. C₅	#68597	direct (N × H)
253. C₃	C₅×#1745	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#815	2
C₄²×C₅×Dic₃	2
C₃×C₅×#336	2
C₅×#4550	2
C₅×#4524	2
C₅×#1745	3
#68597	5

Subgroups — C5×#68597 #14294892

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68597 #14294892