Subgroups — C₅×#68603 #14296131

Order 1920.

584 conjugacy classes of subgroups.

The subgroup lattice has 264 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	C₂×C₅×Dic₃⋊C₈	C₂	1	2	yes	maximal
3	C₄²×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#336	C₂	1	1	yes	maximal, characteristic
5	C₅×#4524	C₂	1	1	yes	maximal, characteristic
6	C₅×#1767	—	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	—	2	4	no	—
15	C₅×Dic₃⋊C₈	C₄	1	8	yes	complemented
16	#68603	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×C₅×C₄⋊C₈	—	3	2	no	—
18	C₂×C₅×#97	—	3	2	no	—
19	C₄³×C₅	—	3	1	no	—
20	C₅×#336	—	3	1	no	—
21	C₅×#336	D₃	1	1	yes	characteristic
22	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
23	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₂²×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₅×Dic₃	C₂×C₄	1	2	yes	—
26	C₂²×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₅×C₈	—	2	2	no	—
28	C₂×C₃×C₅×C₈	—	2	2	no	—
29	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	—
30	C₃×C₄²×C₅	—	2	2	no	—
31	C₄×C₅×Dic₃	—	2	4	no	—
32	C₄×C₅×Dic₃	—	2	4	no	—
33	C₄×C₅×Dic₃	C₂×C₄	1	4	yes	—
34	C₄×C₅×Dic₃	D₄	1	2	yes	—
35	C₄×C₅×Dic₃	Q₈	1	2	yes	—
36	C₂×C₅×C₃⋊C₈	—	2	2	no	—
37	C₂×C₅×C₃⋊C₈	—	2	2	no	—
38	C₂×C₅×C₃⋊C₈	C₂×C₄	1	4	yes	—
39	C₂×#815	C₂×C₅	1	2	yes	—
40	C₂×Dic₃⋊C₈	C₂×C₅	1	2	yes	—
41	C₄²×Dic₃	C₂×C₅	1	1	yes	characteristic
42	C₃×#336	C₂×C₅	1	1	yes	characteristic
43	#4524	C₂×C₅	1	1	yes	characteristic
44	C₂²×C₅×C₈	—	3	2	no	—
45	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
46	C₂×C₄²×C₅	—	3	1	no	—
47	C₂×C₄²×C₅	—	3	1	no	—
48	C₂×C₄²×C₅	—	3	2	no	—
49	C₂×C₄²×C₅	—	3	2	no	—
50	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
51	C₅×C₄⋊C₈	—	3	8	no	—
52	C₅×#97	—	6	4	no	—
53	#1767	—	3	1	no	Sylow
54	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	—	2	2	no	—
56	C₂×C₃×C₄×C₅	—	2	2	no	—
57	C₂×C₃×C₄×C₅	M₁₆	1	4	yes	—
58	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
59	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	#41	1	2	yes	—
62	C₂×C₅×Dic₃	—	2	2	no	—
63	C₂×C₅×Dic₃	—	2	2	no	—
64	C₂×C₅×Dic₃	—	2	2	no	—
65	C₂×C₅×Dic₃	C₄²	1	2	yes	—
66	C₂×C₅×Dic₃	C₄²	1	2	yes	—
67	C₂×C₅×Dic₃	C₄⋊C₄	1	4	yes	—
68	C₂×C₅×Dic₃	M₁₆	1	4	yes	—
69	C₃×C₅×C₈	—	2	4	no	—
70	C₅×C₃⋊C₈	—	2	4	no	—
71	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
72	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
73	C₂×C₄×Dic₃	C₄×C₅	1	2	yes	—
74	C₂×C₄×Dic₃	C₄×C₅	1	2	yes	—
75	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
76	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
77	C₂²×C₃⋊C₈	C₂²×C₅	1	2	yes	—
78	#815	—	2	4	no	—
79	Dic₃⋊C₈	C₄×C₅	1	8	yes	complemented
80	C₂²×C₄×C₅	—	3	1	no	—
81	C₂²×C₄×C₅	—	3	1	no	—
82	C₂²×C₄×C₅	—	3	2	no	—
83	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
84	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
85	C₂×C₅×C₈	—	2	2	no	—
86	C₂×C₅×C₈	—	2	2	no	—
87	C₂×C₅×C₈	—	3	4	no	—
88	C₂×C₅×C₈	—	6	2	no	—
89	C₂×C₅×C₈	—	6	2	no	—
90	C₂×C₅×C₈	C₄×D₃	1	4	yes	—
91	C₄²×C₅	—	2	2	no	—
92	C₄²×C₅	—	3	4	no	—
93	C₄²×C₅	—	3	2	no	—
94	C₄²×C₅	—	3	2	no	—
95	C₄²×C₅	—	6	4	no	—
96	C₄²×C₅	—	6	4	no	—
97	C₂×C₄⋊C₈	—	3	2	no	—
98	C₂×#97	—	3	2	no	—
99	C₄³	—	3	1	no	—
100	#336	—	3	1	no	—
101	#336	C₅×D₃	1	1	yes	characteristic
102	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
103	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂²×C₃×C₅	#117	1	1	yes	characteristic
105	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
106	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
107	C₃×C₄×C₅	—	2	4	no	—
108	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
109	C₃×C₄×C₅	C₄×Q₈	1	2	yes	—
110	C₅×Dic₃	—	2	2	no	—
111	C₅×Dic₃	—	2	4	no	—
112	C₅×Dic₃	#97	1	4	yes	complemented
113	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
114	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
115	C₂²×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
116	C₂²×Dic₃	C₂×C₄×C₅	1	2	yes	—
117	C₂²×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
118	C₂×C₃×C₈	—	2	2	no	—
119	C₂×C₃×C₈	—	2	2	no	—
120	C₂×C₃×C₈	C₂×C₄×C₅	1	4	yes	—
121	C₃×C₄²	—	2	2	no	—
122	C₄×Dic₃	—	2	4	no	—
123	C₄×Dic₃	—	2	4	no	—
124	C₄×Dic₃	C₂×C₄×C₅	1	4	yes	—
125	C₄×Dic₃	C₅×D₄	1	2	yes	—
126	C₄×Dic₃	C₅×Q₈	1	2	yes	—
127	C₂×C₃⋊C₈	—	2	2	no	—
128	C₂×C₃⋊C₈	—	2	2	no	—
129	C₂×C₃⋊C₈	C₂×C₄×C₅	1	4	yes	—
130	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
131	C₂×C₄×C₅	—	2	2	no	—
132	C₂×C₄×C₅	—	2	2	no	—
133	C₂×C₄×C₅	—	3	2	no	—
134	C₂×C₄×C₅	—	3	4	no	—
135	C₂×C₄×C₅	—	3	2	no	—
136	C₂×C₄×C₅	—	3	4	no	—
137	C₂×C₄×C₅	—	6	2	no	—
138	C₂×C₄×C₅	—	6	2	no	—
139	C₂×C₄×C₅	—	6	2	no	—
140	C₂×C₄×C₅	ΓL(1,𝔽₂₅)	1	4	yes	—
141	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
142	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
143	C₂×C₄×C₅	#237	1	2	yes	—
144	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
145	C₅×C₈	—	2	4	no	—
146	C₅×C₈	—	6	4	no	—
147	C₂²×C₈	—	3	2	no	—
148	C₂²×C₈	C₂×C₅×D₃	1	2	yes	—
149	C₂×C₄²	—	3	1	no	—
150	C₂×C₄²	—	3	1	no	—
151	C₂×C₄²	—	3	2	no	—
152	C₂×C₄²	—	3	2	no	—
153	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
154	C₄⋊C₈	—	3	8	no	—
155	#97	—	6	4	no	—
156	C₂×C₃×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₃×C₅	C₂×#97	1	1	yes	characteristic
158	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
159	C₂×C₃×C₅	#423	1	2	yes	—
160	C₂×C₃×C₅	#432	1	2	yes	—
161	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
162	C₂×C₃×C₄	—	2	2	no	—
163	C₂×C₃×C₄	—	2	2	no	—
164	C₂×C₃×C₄	C₅×M₁₆	1	4	yes	—
165	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
166	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
167	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
168	C₂×C₃×C₄	C₅×#41	1	2	yes	—
169	C₂×Dic₃	—	2	2	no	—
170	C₂×Dic₃	—	2	2	no	—
171	C₂×Dic₃	—	2	2	no	—
172	C₂×Dic₃	C₄²×C₅	1	2	yes	—
173	C₂×Dic₃	C₄²×C₅	1	2	yes	—
174	C₂×Dic₃	C₅×C₄⋊C₄	1	4	yes	—
175	C₂×Dic₃	C₅×M₁₆	1	4	yes	—
176	C₃×C₈	—	2	4	no	—
177	C₃⋊C₈	—	2	4	no	—
178	C₂²×C₅	C₄²×D₃	1	1	yes	characteristic
179	C₂²×C₅	#877	1	1	yes	characteristic
180	C₂²×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
181	C₂²×C₅	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
182	C₂²×C₅	D₃×M₁₆	1	2	yes	—
183	C₄×C₅	—	2	4	no	—
184	C₄×C₅	—	3	4	no	—
185	C₄×C₅	—	6	2	no	—
186	C₄×C₅	—	6	4	no	—
187	C₄×C₅	#67⋊C₄	1	2	yes	—
188	C₄×C₅	#887	1	2	yes	—
189	C₂²×C₄	—	3	1	no	—
190	C₂²×C₄	—	3	1	no	—
191	C₂²×C₄	—	3	2	no	—
192	C₂²×C₄	C₄×C₅×D₃	1	2	yes	—
193	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	3	4	no	—
197	C₂×C₈	—	6	2	no	—
198	C₂×C₈	—	6	2	no	—
199	C₂×C₈	C₄×C₅×D₃	1	4	yes	—
200	C₄²	—	2	2	no	—
201	C₄²	—	3	4	no	—
202	C₄²	—	3	2	no	—
203	C₄²	—	3	2	no	—
204	C₄²	—	6	4	no	—
205	C₄²	—	6	4	no	—
206	C₃×C₅	#1767	1	1	yes	characteristic, complemented, Hall
207	C₂²×C₃	C₂×C₄²×C₅	1	1	yes	characteristic
208	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
209	C₂²×C₃	C₅×#117	1	1	yes	characteristic
210	C₂²×C₃	C₂×C₅×M₁₆	1	2	yes	—
211	C₂²×C₃	C₂×C₅×M₁₆	1	2	yes	—
212	C₃×C₄	—	2	4	no	—
213	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
214	C₃×C₄	C₄×C₅×Q₈	1	2	yes	—
215	Dic₃	—	2	2	no	—
216	Dic₃	—	2	4	no	—
217	Dic₃	C₅×#97	1	4	yes	complemented
218	C₂×C₅	#4649	1	1	yes	characteristic
219	C₂×C₅	C₄×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
220	C₂×C₅	D₃×#97	1	1	yes	characteristic
221	C₂×C₅	#4719	1	2	yes	—
222	C₂×C₅	#4837	1	2	yes	—
223	C₂³	C₂×C₄×C₅×D₃	1	1	yes	characteristic
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	3	2	no	—
227	C₂×C₄	—	3	4	no	—
228	C₂×C₄	—	3	2	no	—
229	C₂×C₄	—	3	4	no	—
230	C₂×C₄	—	6	2	no	—
231	C₂×C₄	—	6	2	no	—
232	C₂×C₄	—	6	2	no	—
233	C₂×C₄	C₅×ΓL(1,𝔽₂₅)	1	4	yes	—
234	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
235	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
236	C₂×C₄	C₅×#237	1	2	yes	—
237	C₂×C₄	C₅×D₃×Q₈	1	1	yes	characteristic
238	C₈	—	2	4	no	—
239	C₈	—	6	4	no	—
240	C₂×C₃	C₄×C₅×C₄⋊C₄	1	1	yes	characteristic
241	C₂×C₃	C₂×C₅×#97	1	1	yes	characteristic
242	C₂×C₃	C₄×C₅×M₁₆	1	1	yes	characteristic
243	C₂×C₃	C₅×#423	1	2	yes	—
244	C₂×C₃	C₅×#432	1	2	yes	—
245	C₅	#68603	1	1	yes	characteristic, direct product factor, Sylow
246	C₂²	C₄²×C₅×D₃	1	1	yes	characteristic
247	C₂²	C₅×#877	1	1	yes	characteristic
248	C₂²	C₅×D₃×C₄⋊C₄	1	1	yes	characteristic
249	C₂²	C₂×C₅×ΓL(1,𝔽₂₅)	1	2	yes	—
250	C₂²	C₅×D₃×M₁₆	1	2	yes	—
251	C₄	—	2	4	no	—
252	C₄	—	3	4	no	—
253	C₄	—	6	2	no	—
254	C₄	—	6	4	no	—
255	C₄	C₅×#67⋊C₄	1	2	yes	—
256	C₄	C₅×#887	1	2	yes	—
257	C₃	C₅×#1767	1	1	yes	characteristic, complemented, Sylow
258	C₂	C₅×#4649	1	1	yes	characteristic
259	C₂	C₄×C₅×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
260	C₂	C₅×D₃×#97	1	1	yes	characteristic
261	C₂	C₅×#4719	1	2	yes	—
262	C₂	C₅×#4837	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×#1767
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
15. C₅×Dic₃⋊C₈	C₄	semidirect
16. #68603	C₅	direct (N × H)
79. Dic₃⋊C₈	C₄×C₅	semidirect
112. C₅×Dic₃	#97	semidirect
206. C₃×C₅	#1767	semidirect
217. Dic₃	C₅×#97	semidirect
245. C₅	#68603	direct (N × H)
257. C₃	C₅×#1767	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
C₂×C₅×Dic₃⋊C₈	2
C₄²×C₅×Dic₃	2
C₃×C₅×#336	2
C₅×#4524	2
C₅×#1767	3
#68603	5

Subgroups — C5×#68603 #14296131

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68603 #14296131