Subgroups — C₅×#69552 #14305923

Order 1920.

616 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#869	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#382	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#810	C₂	1	4	yes	maximal
4	C₂×C₅×#803	C₂	1	1	yes	maximal, characteristic
5	C₅×#1960	—	3	1	no	maximal, Hall
6	C₂²×C₅×Dic₆	C₂²	1	2	yes	—
7	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₅×C₄⋊Dic₃	—	2	2	no	—
10	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
11	C₅×#869	—	2	2	no	—
12	C₅×#869	C₄	1	4	yes	complemented
13	C₃×C₅×#95	—	2	2	no	—
14	C₅×#810	—	2	8	no	—
15	C₅×#803	—	2	2	no	—
16	#69552	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×C₅×C₄⋊Q₈	—	3	1	no	—
18	C₂×C₅×#103	—	3	4	no	—
19	C₂×C₅×#97	—	3	1	no	—
20	C₅×#382	D₃	1	1	yes	characteristic
21	C₂²×C₃×C₄×C₅	—	2	2	no	—
22	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
23	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₂²×C₅×Dic₃	—	2	2	no	—
25	C₂×C₅×Dic₆	—	2	4	no	—
26	C₂×C₅×Dic₆	—	2	4	no	—
27	C₂×C₅×Dic₆	—	2	2	no	—
28	C₂×C₅×Dic₆	C₂×C₄	1	4	yes	—
29	C₃×C₅×C₄⋊C₄	—	2	4	no	—
30	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
31	C₃×C₄²×C₅	D₄	1	2	yes	—
32	C₅×C₄⋊Dic₃	—	2	8	no	—
33	C₂×C₅×C₃⋊C₈	—	2	2	no	—
34	C₂×C₅×C₃⋊C₈	—	2	2	no	—
35	C₂×C₅×C₃⋊C₈	D₄	1	4	yes	—
36	C₂×#869	C₂×C₅	1	1	yes	characteristic
37	C₃×#382	C₂×C₅	1	1	yes	characteristic
38	C₂×#810	C₂×C₅	1	4	yes	—
39	C₂×#803	C₂×C₅	1	1	yes	characteristic
40	C₂²×C₅×Q₈	—	3	2	no	—
41	C₂²×C₅×C₈	—	3	2	no	—
42	C₅×C₄⋊Q₈	—	3	4	no	—
43	C₅×C₄⋊Q₈	—	6	2	no	—
44	C₂×C₅×C₄⋊C₄	—	6	2	no	—
45	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
46	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
47	C₅×#103	—	6	8	no	—
48	C₅×#97	—	6	2	no	—
49	C₅×#95	—	2	2	no	—
50	#1960	—	3	1	no	Sylow
51	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	—	2	4	no	—
53	C₂×C₃×C₄×C₅	—	4	2	no	—
54	C₂×C₃×C₄×C₅	—	4	4	no	—
55	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
56	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
58	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	#41	1	2	yes	—
60	C₂×C₅×Dic₃	—	2	4	no	—
61	C₂×C₅×Dic₃	—	4	2	no	—
62	C₂×C₅×Dic₃	—	4	2	no	—
63	C₅×Dic₆	—	2	8	no	—
64	C₅×Dic₆	—	4	4	no	—
65	C₅×C₃⋊C₈	—	2	4	no	—
66	C₂²×Dic₆	C₂²×C₅	1	2	yes	—
67	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
68	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
69	C₂×C₄⋊Dic₃	—	2	2	no	—
70	C₂²×C₃⋊C₈	C₂²×C₅	1	2	yes	—
71	#869	—	2	2	no	—
72	#869	C₄×C₅	1	4	yes	complemented
73	C₃×#95	—	2	2	no	—
74	#810	—	2	8	no	—
75	#803	—	2	2	no	—
76	C₂×C₅×Q₈	—	3	4	no	—
77	C₂×C₅×Q₈	—	6	2	no	—
78	C₂×C₅×Q₈	—	6	4	no	—
79	C₂×C₅×Q₈	—	6	4	no	—
80	C₂²×C₄×C₅	—	2	2	no	—
81	C₂²×C₄×C₅	—	6	2	no	—
82	C₂²×C₄×C₅	#67	1	2	yes	—
83	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
84	C₂×C₅×C₈	—	3	4	no	—
85	C₂×C₅×C₈	—	6	2	no	—
86	C₂×C₅×C₈	—	6	2	no	—
87	C₅×C₄⋊C₄	—	2	4	no	—
88	C₅×C₄⋊C₄	—	6	8	no	—
89	C₄²×C₅	C₄×D₃	1	2	yes	—
90	C₄²×C₅	D₁₂	1	2	yes	—
91	C₂×C₄⋊Q₈	—	3	1	no	—
92	C₂×#103	—	3	4	no	—
93	C₂×#97	—	3	1	no	—
94	#382	C₅×D₃	1	1	yes	characteristic
95	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂²×C₃×C₅	#124	1	1	yes	characteristic
97	C₂²×C₃×C₅	#127	1	1	yes	characteristic
98	C₂²×C₃×C₅	#137	1	4	yes	—
99	C₃×C₄×C₅	—	2	4	no	—
100	C₃×C₄×C₅	—	8	2	no	—
101	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
102	C₃×C₄×C₅	#121	1	2	yes	—
103	C₅×Dic₃	—	4	4	no	—
104	C₂²×C₃×C₄	—	2	2	no	—
105	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
106	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
107	C₂²×Dic₃	—	2	2	no	—
108	C₂×Dic₆	—	2	4	no	—
109	C₂×Dic₆	—	2	4	no	—
110	C₂×Dic₆	—	2	2	no	—
111	C₂×Dic₆	C₂×C₄×C₅	1	4	yes	—
112	C₃×C₄⋊C₄	—	2	4	no	—
113	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
114	C₃×C₄²	C₅×D₄	1	2	yes	—
115	C₄⋊Dic₃	—	2	8	no	—
116	C₂×C₃⋊C₈	—	2	2	no	—
117	C₂×C₃⋊C₈	—	2	2	no	—
118	C₂×C₃⋊C₈	C₅×D₄	1	4	yes	—
119	C₂³×C₅	C₂×#67	1	1	yes	characteristic
120	C₅×Q₈	—	6	8	no	—
121	C₅×Q₈	—	12	4	no	—
122	C₂×C₄×C₅	—	2	4	no	—
123	C₂×C₄×C₅	—	4	2	no	—
124	C₂×C₄×C₅	—	4	4	no	—
125	C₂×C₄×C₅	—	6	4	no	—
126	C₂×C₄×C₅	—	12	2	no	—
127	C₂×C₄×C₅	—	12	2	no	—
128	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	4	yes	—
129	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
131	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
132	C₂×C₄×C₅	#239	1	2	yes	—
133	C₅×C₈	—	6	4	no	—
134	C₂²×Q₈	—	3	2	no	—
135	C₂²×C₈	—	3	2	no	—
136	C₄⋊Q₈	—	3	4	no	—
137	C₄⋊Q₈	—	6	2	no	—
138	C₂×C₄⋊C₄	—	6	2	no	—
139	C₂×C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
140	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
141	#103	—	6	8	no	—
142	#97	—	6	2	no	—
143	#95	—	2	2	no	—
144	C₂×C₃×C₅	#390	1	1	yes	characteristic
145	C₂×C₃×C₅	#419	1	2	yes	—
146	C₂×C₃×C₅	#491	1	2	yes	—
147	C₂×C₃×C₅	#497	1	2	yes	—
148	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
149	C₂×C₃×C₄	—	2	4	no	—
150	C₂×C₃×C₄	—	4	2	no	—
151	C₂×C₃×C₄	—	4	4	no	—
152	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	4	yes	—
153	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
156	C₂×C₃×C₄	C₅×#41	1	2	yes	—
157	C₂×Dic₃	—	2	4	no	—
158	C₂×Dic₃	—	4	2	no	—
159	C₂×Dic₃	—	4	2	no	—
160	Dic₆	—	2	8	no	—
161	Dic₆	—	4	4	no	—
162	C₃⋊C₈	—	2	4	no	—
163	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
164	C₂²×C₅	#940	1	1	yes	characteristic
165	C₂²×C₅	#947	1	1	yes	characteristic
166	C₂²×C₅	#951	1	4	yes	—
167	C₄×C₅	—	2	4	no	—
168	C₄×C₅	—	8	2	no	—
169	C₄×C₅	—	12	4	no	—
170	C₄×C₅	#893	1	2	yes	—
171	C₄×C₅	#895	1	2	yes	—
172	C₂×Q₈	—	3	4	no	—
173	C₂×Q₈	—	6	2	no	—
174	C₂×Q₈	—	6	4	no	—
175	C₂×Q₈	—	6	4	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	6	2	no	—
178	C₂²×C₄	C₅×#67	1	2	yes	—
179	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
180	C₂×C₈	—	3	4	no	—
181	C₂×C₈	—	6	2	no	—
182	C₂×C₈	—	6	2	no	—
183	C₄⋊C₄	—	2	4	no	—
184	C₄⋊C₄	—	6	8	no	—
185	C₄²	C₄×C₅×D₃	1	2	yes	—
186	C₄²	C₅×D₁₂	1	2	yes	—
187	C₃×C₅	#1960	1	1	yes	characteristic, complemented, Hall
188	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
189	C₂²×C₃	C₅×#124	1	1	yes	characteristic
190	C₂²×C₃	C₅×#127	1	1	yes	characteristic
191	C₂²×C₃	C₅×#137	1	4	yes	—
192	C₃×C₄	—	2	4	no	—
193	C₃×C₄	—	8	2	no	—
194	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
195	C₃×C₄	C₅×#121	1	2	yes	—
196	Dic₃	—	4	4	no	—
197	C₂×C₅	#4988	1	1	yes	characteristic
198	C₂×C₅	#5004	1	2	yes	—
199	C₂×C₅	#5060	1	2	yes	—
200	C₂×C₅	#5069	1	2	yes	—
201	C₂³	C₂×C₅×#67	1	1	yes	characteristic
202	Q₈	—	6	8	no	—
203	Q₈	—	12	4	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	6	4	no	—
208	C₂×C₄	—	12	2	no	—
209	C₂×C₄	—	12	2	no	—
210	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	4	yes	—
211	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
212	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
213	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
214	C₂×C₄	C₅×#239	1	2	yes	—
215	C₈	—	6	4	no	—
216	C₂×C₃	C₅×#390	1	1	yes	characteristic
217	C₂×C₃	C₅×#419	1	2	yes	—
218	C₂×C₃	C₅×#491	1	2	yes	—
219	C₂×C₃	C₅×#497	1	2	yes	—
220	C₅	#69552	1	1	yes	characteristic, direct product factor, Sylow
221	C₂²	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
222	C₂²	C₅×#940	1	1	yes	characteristic
223	C₂²	C₅×#947	1	1	yes	characteristic
224	C₂²	C₅×#951	1	4	yes	—
225	C₄	—	2	4	no	—
226	C₄	—	8	2	no	—
227	C₄	—	12	4	no	—
228	C₄	C₅×#893	1	2	yes	—
229	C₄	C₅×#895	1	2	yes	—
230	C₃	C₅×#1960	1	1	yes	characteristic, complemented, Sylow
231	C₂	C₅×#4988	1	1	yes	characteristic
232	C₂	C₅×#5004	1	2	yes	—
233	C₂	C₅×#5060	1	2	yes	—
234	C₂	C₅×#5069	1	2	yes	—

	subgroup	G/H
center	C₂³×C₅	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#382	C₂
socle, prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#382	D₃
hypocenter	C₃	C₅×#1960
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ C₅×#382

Complements

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

N	H ≅ G/N	type
12. C₅×#869	C₄	semidirect
16. #69552	C₅	direct (N × H)
72. #869	C₄×C₅	semidirect
187. C₃×C₅	#1960	semidirect
220. C₅	#69552	direct (N × H)
230. C₃	C₅×#1960	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#869	2
C₃×C₅×#382	2
C₂×C₅×#810	2
C₂×C₅×#803	2
C₅×#1960	3
#69552	5

Subgroups — C5×#69552 #14305923

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69552 #14305923