Subgroups — C₅×#69815 #14298861

Order 1920.

448 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#97	C₂	1	1	yes	maximal, characteristic
2	C₅×#4936	C₂	1	1	yes	maximal, characteristic
3	C₅×#4935	C₂	1	1	yes	maximal, characteristic
4	C₅×#4551	C₂	1	4	yes	maximal
5	C₅×#1836	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₅×C₄⋊Dic₃	—	2	2	no	—
9	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
10	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
11	C₃×C₅×#97	C₄	1	4	yes	complemented
12	C₅×C₃⋊#95	—	2	2	no	—
13	#69815	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂×C₅×#97	D₃	1	1	yes	characteristic
15	C₅×#384	—	3	1	no	—
16	C₅×#382	—	3	1	no	—
17	C₅×#340	—	3	4	no	—
18	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
19	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
20	C₂²×C₅×Dic₃	—	2	2	no	—
21	C₂²×C₅×Dic₃	—	2	2	no	—
22	C₂×C₃×C₅×C₈	—	2	2	no	—
23	C₂×C₃×C₅×C₈	—	2	2	no	—
24	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	—
25	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
26	C₃×C₄²×C₅	Q₈	1	2	yes	—
27	C₅×C₄⋊Dic₃	—	2	4	no	—
28	C₅×C₄⋊Dic₃	—	2	4	no	—
29	C₅×C₄⋊Dic₃	—	4	4	no	—
30	C₂×C₃×#97	C₂×C₅	1	1	yes	characteristic
31	#4936	C₂×C₅	1	1	yes	characteristic
32	#4935	C₂×C₅	1	1	yes	characteristic
33	#4551	C₂×C₅	1	4	yes	—
34	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
35	C₂×C₅×C₄⋊C₄	—	3	2	no	—
36	C₂×C₅×C₄⋊C₄	—	3	2	no	—
37	C₂×C₅×C₄⋊C₄	—	6	2	no	—
38	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
39	C₅×#97	Dic₃	1	4	yes	complemented
40	C₅×#95	—	6	2	no	—
41	#1836	—	3	1	no	Sylow
42	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
43	C₂×C₃×C₄×C₅	—	2	2	no	—
44	C₂×C₃×C₄×C₅	—	2	2	no	—
45	C₂×C₃×C₄×C₅	C₄⋊C₄	1	4	yes	—
46	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₃×C₄×C₅	#41	1	2	yes	—
49	C₂×C₃×C₄×C₅	#41	1	2	yes	—
50	C₂×C₅×Dic₃	—	4	2	no	—
51	C₂×C₅×Dic₃	—	4	2	no	—
52	C₂×C₅×Dic₃	—	4	4	no	—
53	C₂×C₅×Dic₃	—	4	2	no	—
54	C₂×C₅×Dic₃	—	4	2	no	—
55	C₃×C₅×C₈	—	2	4	no	—
56	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
57	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
58	C₂×C₄⋊Dic₃	—	2	2	no	—
59	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
60	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
61	C₃×#97	C₄×C₅	1	4	yes	complemented
62	C₃⋊#95	—	2	2	no	—
63	C₂²×C₄×C₅	—	6	2	no	—
64	C₂²×C₄×C₅	—	6	2	no	—
65	C₂²×C₄×C₅	D₁₂	1	2	yes	—
66	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
67	C₂×C₅×C₈	—	2	2	no	—
68	C₂×C₅×C₈	—	2	2	no	—
69	C₂×C₅×C₈	C₂×Dic₃	1	4	yes	—
70	C₅×C₄⋊C₄	—	6	4	no	—
71	C₅×C₄⋊C₄	—	6	4	no	—
72	C₅×C₄⋊C₄	—	12	4	no	—
73	C₄²×C₅	Dic₆	1	2	yes	—
74	C₄²×C₅	C₂×Dic₃	1	2	yes	—
75	C₂×#97	C₅×D₃	1	1	yes	characteristic
76	#384	—	3	1	no	—
77	#382	—	3	1	no	—
78	#340	—	3	4	no	—
79	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₃×C₅	#124	1	1	yes	characteristic
81	C₂²×C₃×C₅	#124	1	1	yes	characteristic
82	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	2	yes	—
83	C₂²×C₃×C₅	#137	1	2	yes	—
84	C₃×C₄×C₅	—	2	4	no	—
85	C₃×C₄×C₅	#117	1	2	yes	—
86	C₃×C₄×C₅	#125	1	2	yes	—
87	C₅×Dic₃	—	8	2	no	—
88	C₅×Dic₃	—	8	2	no	—
89	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
90	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
91	C₂²×Dic₃	—	2	2	no	—
92	C₂²×Dic₃	—	2	2	no	—
93	C₂×C₃×C₈	—	2	2	no	—
94	C₂×C₃×C₈	—	2	2	no	—
95	C₂×C₃×C₈	C₂×C₄×C₅	1	4	yes	—
96	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
97	C₃×C₄²	C₅×Q₈	1	2	yes	—
98	C₄⋊Dic₃	—	2	4	no	—
99	C₄⋊Dic₃	—	2	4	no	—
100	C₄⋊Dic₃	—	4	4	no	—
101	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
102	C₂×C₄×C₅	—	2	2	no	—
103	C₂×C₄×C₅	—	2	2	no	—
104	C₂×C₄×C₅	—	12	2	no	—
105	C₂×C₄×C₅	—	12	2	no	—
106	C₂×C₄×C₅	—	12	4	no	—
107	C₂×C₄×C₅	—	12	2	no	—
108	C₂×C₄×C₅	—	12	2	no	—
109	C₂×C₄×C₅	C₄⋊Dic₃	1	4	yes	—
110	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
111	C₂×C₄×C₅	#235	1	2	yes	—
112	C₂×C₄×C₅	#235	1	2	yes	—
113	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
114	C₅×C₈	—	2	4	no	—
115	C₂²×C₈	C₂×C₅×D₃	1	2	yes	—
116	C₂×C₄⋊C₄	—	3	2	no	—
117	C₂×C₄⋊C₄	—	3	2	no	—
118	C₂×C₄⋊C₄	—	6	2	no	—
119	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
120	#97	C₅×Dic₃	1	4	yes	complemented
121	#95	—	6	2	no	—
122	C₂×C₃×C₅	#382	1	1	yes	characteristic
123	C₂×C₃×C₅	#428	1	2	yes	—
124	C₂×C₃×C₅	#489	1	2	yes	—
125	C₂×C₃×C₅	#490	1	1	yes	characteristic
126	C₂×C₃×C₅	#491	1	1	yes	characteristic
127	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
128	C₂×C₃×C₄	—	2	2	no	—
129	C₂×C₃×C₄	—	2	2	no	—
130	C₂×C₃×C₄	C₅×C₄⋊C₄	1	4	yes	—
131	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
132	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
133	C₂×C₃×C₄	C₅×#41	1	2	yes	—
134	C₂×C₃×C₄	C₅×#41	1	2	yes	—
135	C₂×Dic₃	—	4	2	no	—
136	C₂×Dic₃	—	4	2	no	—
137	C₂×Dic₃	—	4	4	no	—
138	C₂×Dic₃	—	4	2	no	—
139	C₂×Dic₃	—	4	2	no	—
140	C₃×C₈	—	2	4	no	—
141	C₂²×C₅	#875	1	1	yes	characteristic
142	C₂²×C₅	#875	1	1	yes	characteristic
143	C₂²×C₅	#908	1	2	yes	—
144	C₂²×C₅	#909	1	2	yes	—
145	C₂²×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
146	C₄×C₅	—	2	4	no	—
147	C₄×C₅	—	24	2	no	—
148	C₄×C₅	—	24	2	no	—
149	C₄×C₅	#870	1	2	yes	—
150	C₄×C₅	#926	1	2	yes	—
151	C₂²×C₄	—	6	2	no	—
152	C₂²×C₄	—	6	2	no	—
153	C₂²×C₄	C₅×D₁₂	1	2	yes	—
154	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	C₂×C₅×Dic₃	1	4	yes	—
158	C₄⋊C₄	—	6	4	no	—
159	C₄⋊C₄	—	6	4	no	—
160	C₄⋊C₄	—	12	4	no	—
161	C₄²	C₅×Dic₆	1	2	yes	—
162	C₄²	C₂×C₅×Dic₃	1	2	yes	—
163	C₃×C₅	#1836	1	1	yes	characteristic, complemented, Hall
164	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
165	C₂²×C₃	C₅×#124	1	1	yes	characteristic
166	C₂²×C₃	C₅×#124	1	1	yes	characteristic
167	C₂²×C₃	C₅×AGL(1,ℤ/8)	1	2	yes	—
168	C₂²×C₃	C₅×#137	1	2	yes	—
169	C₃×C₄	—	2	4	no	—
170	C₃×C₄	C₅×#117	1	2	yes	—
171	C₃×C₄	C₅×#125	1	2	yes	—
172	Dic₃	—	8	2	no	—
173	Dic₃	—	8	2	no	—
174	C₂×C₅	#4703	1	1	yes	characteristic
175	C₂×C₅	#4713	1	1	yes	characteristic
176	C₂×C₅	#4714	1	2	yes	—
177	C₂×C₅	#4936	1	1	yes	characteristic
178	C₂×C₅	#5121	1	2	yes	—
179	C₂³	C₂×C₅×D₁₂	1	1	yes	characteristic
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	12	2	no	—
183	C₂×C₄	—	12	2	no	—
184	C₂×C₄	—	12	4	no	—
185	C₂×C₄	—	12	2	no	—
186	C₂×C₄	—	12	2	no	—
187	C₂×C₄	C₅×C₄⋊Dic₃	1	4	yes	—
188	C₂×C₄	C₂×C₅×Dic₆	1	1	yes	characteristic
189	C₂×C₄	C₅×#235	1	2	yes	—
190	C₂×C₄	C₅×#235	1	2	yes	—
191	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
192	C₈	—	2	4	no	—
193	C₂×C₃	C₅×#382	1	1	yes	characteristic
194	C₂×C₃	C₅×#428	1	2	yes	—
195	C₂×C₃	C₅×#489	1	2	yes	—
196	C₂×C₃	C₅×#490	1	1	yes	characteristic
197	C₂×C₃	C₅×#491	1	1	yes	characteristic
198	C₅	#69815	1	1	yes	characteristic, direct product factor, Sylow
199	C₂²	C₅×#875	1	1	yes	characteristic
200	C₂²	C₅×#875	1	1	yes	characteristic
201	C₂²	C₅×#908	1	2	yes	—
202	C₂²	C₅×#909	1	2	yes	—
203	C₂²	C₂×C₅×C₄⋊Dic₃	1	1	yes	characteristic
204	C₄	—	2	4	no	—
205	C₄	—	24	2	no	—
206	C₄	—	24	2	no	—
207	C₄	C₅×#870	1	2	yes	—
208	C₄	C₅×#926	1	2	yes	—
209	C₃	C₅×#1836	1	1	yes	characteristic, complemented, Sylow
210	C₂	C₅×#4703	1	1	yes	characteristic
211	C₂	C₅×#4713	1	1	yes	characteristic
212	C₂	C₅×#4714	1	2	yes	—
213	C₂	C₅×#4936	1	1	yes	characteristic
214	C₂	C₅×#5121	1	2	yes	—

	subgroup	G/H
center	C₂³×C₅	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₂×C₃×C₅×#97	C₂
socle, prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₂×C₅×#97	D₃
hypocenter	C₃	C₅×#1836
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₅×#97

Complements

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

N	H ≅ G/N	type
11. C₃×C₅×#97	C₄	semidirect
13. #69815	C₅	direct (N × H)
39. C₅×#97	Dic₃	semidirect
61. C₃×#97	C₄×C₅	semidirect
120. #97	C₅×Dic₃	semidirect
163. C₃×C₅	#1836	semidirect
198. C₅	#69815	direct (N × H)
209. C₃	C₅×#1836	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×#97	2
C₅×#4936	2
C₅×#4935	2
C₅×#4551	2
C₅×#1836	3
#69815	5

Subgroups — C5×#69815 #14298861

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69815 #14298861