Subgroups — C₅×#69441 #14307443

Order 1920.

488 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5470	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#376	C₂	1	1	yes	maximal, characteristic
3	C₅×#4924	C₂	1	1	yes	maximal, characteristic
4	C₅×#4523	C₂	1	4	yes	maximal
5	C₅×#1995	—	3	1	no	maximal, Hall
6	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₅×#926	C₂²	1	2	yes	—
9	C₅×#924	—	2	2	no	—
10	C₂×C₅×#208	C₂²	1	2	yes	—
11	C₅×#870	C₄	1	2	yes	complemented
12	C₅×#869	C₄	1	2	yes	complemented
13	C₄×C₅×Dic₆	—	2	2	no	—
14	C₃×C₅×#95	—	2	2	no	—
15	C₅×C₄⋊(C₃⋊C₈)	—	2	2	no	—
16	#69441	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₅×#533	—	3	1	no	—
18	C₅×#423	—	3	1	no	—
19	C₅×#376	D₃	1	1	yes	characteristic
20	C₅×#339	—	3	4	no	—
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	C₂×C₅×Dic₆	—	2	2	no	—
25	C₃×C₄²×C₅	—	2	2	no	—
26	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
27	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
28	C₃×C₄²×C₅	D₄	1	2	yes	—
29	C₃×C₄²×C₅	Q₈	1	4	yes	complemented
30	C₅×#217	—	2	2	no	—
31	C₅×C₄⋊Dic₃	—	2	2	no	—
32	C₅×C₄⋊Dic₃	C₂×C₄	1	4	yes	—
33	C₅×Dic₃⋊C₄	—	2	4	no	—
34	C₄×C₅×Dic₃	—	2	2	no	—
35	C₅×#208	—	2	4	no	—
36	C₂×C₅×C₃⋊C₈	—	2	2	no	—
37	#5470	C₂×C₅	1	1	yes	characteristic
38	C₃×#376	C₂×C₅	1	1	yes	characteristic
39	#4924	C₂×C₅	1	1	yes	characteristic
40	#4523	C₂×C₅	1	4	yes	—
41	C₂×C₅×M₁₆	—	3	2	no	—
42	C₅×C₄⋊Q₈	—	3	2	no	—
43	C₅×#125	—	3	2	no	—
44	C₅×C₂²⋊Q₈	—	6	2	no	—
45	C₄×C₅×Q₈	—	6	2	no	—
46	C₅×#117	—	3	2	no	—
47	C₂×C₄²×C₅	C₂×D₃	1	2	yes	—
48	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
49	C₅×C₄⋊C₈	—	6	2	no	—
50	C₅×#95	—	2	2	no	—
51	#1995	—	3	1	no	Sylow
52	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	—	2	4	no	—
54	C₂×C₃×C₄×C₅	—	4	2	no	—
55	C₂×C₃×C₄×C₅	—	4	4	no	—
56	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
57	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
58	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
61	C₂×C₃×C₄×C₅	#41	1	2	yes	—
62	C₂×C₅×Dic₃	—	2	4	no	—
63	C₅×Dic₆	—	4	4	no	—
64	C₅×C₃⋊C₈	—	4	2	no	—
65	C₂×C₃×C₄²	C₂²×C₅	1	2	yes	—
66	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
67	#926	C₂²×C₅	1	2	yes	—
68	#924	—	2	2	no	—
69	C₂×#208	C₂²×C₅	1	2	yes	—
70	#870	C₄×C₅	1	2	yes	complemented
71	#869	C₄×C₅	1	2	yes	complemented
72	C₄×Dic₆	—	2	2	no	—
73	C₃×#95	—	2	2	no	—
74	C₄⋊(C₃⋊C₈)	—	2	2	no	—
75	C₂×C₅×Q₈	—	6	2	no	—
76	C₂²×C₄×C₅	—	2	2	no	—
77	C₂²×C₄×C₅	#67	1	2	yes	—
78	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
79	C₅×M₁₆	—	6	4	no	—
80	C₂×C₅×C₈	—	6	2	no	—
81	C₅×C₄⋊C₄	—	3	4	no	—
82	C₅×C₄⋊C₄	—	6	4	no	—
83	C₅×C₄⋊C₄	—	6	2	no	—
84	C₅×C₂²⋊C₄	—	6	2	no	—
85	C₄²×C₅	—	2	2	no	—
86	C₄²×C₅	—	6	2	no	—
87	C₄²×C₅	Dic₆	1	4	yes	complemented
88	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
89	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
90	C₄²×C₅	D₁₂	1	2	yes	—
91	#533	—	3	1	no	—
92	#423	—	3	1	no	—
93	#376	C₅×D₃	1	1	yes	characteristic
94	#339	—	3	4	no	—
95	C₂²×C₃×C₅	—	2	2	no	—
96	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂²×C₃×C₅	C₂²⋊Q₈	1	2	yes	—
98	C₃×C₄×C₅	—	2	2	no	—
99	C₃×C₄×C₅	—	2	2	no	—
100	C₃×C₄×C₅	—	4	4	no	—
101	C₃×C₄×C₅	C₄×Q₈	1	2	yes	—
102	C₃×C₄×C₅	#124	1	1	yes	characteristic
103	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
104	C₅×Dic₃	—	4	4	no	—
105	C₂²×C₃×C₄	—	2	2	no	—
106	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
107	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
108	C₂×Dic₆	—	2	2	no	—
109	C₃×C₄²	—	2	2	no	—
110	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
111	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
112	C₃×C₄²	C₅×D₄	1	2	yes	—
113	C₃×C₄²	C₅×Q₈	1	4	yes	complemented
114	#217	—	2	2	no	—
115	C₄⋊Dic₃	—	2	2	no	—
116	C₄⋊Dic₃	C₂×C₄×C₅	1	4	yes	—
117	Dic₃⋊C₄	—	2	4	no	—
118	C₄×Dic₃	—	2	2	no	—
119	#208	—	2	4	no	—
120	C₂×C₃⋊C₈	—	2	2	no	—
121	C₂³×C₅	C₂×#67	1	1	yes	characteristic
122	C₅×Q₈	—	12	4	no	—
123	C₂×C₄×C₅	—	2	4	no	—
124	C₂×C₄×C₅	—	4	2	no	—
125	C₂×C₄×C₅	—	4	4	no	—
126	C₂×C₄×C₅	—	6	4	no	—
127	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
128	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
129	C₂×C₄×C₅	C₂×Dic₆	1	2	yes	—
130	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
132	C₂×C₄×C₅	#235	1	2	yes	—
133	C₅×C₈	—	12	2	no	—
134	C₂×M₁₆	—	3	2	no	—
135	C₄⋊Q₈	—	3	2	no	—
136	#125	—	3	2	no	—
137	C₂²⋊Q₈	—	6	2	no	—
138	C₄×Q₈	—	6	2	no	—
139	#117	—	3	2	no	—
140	C₂×C₄²	C₂×C₅×D₃	1	2	yes	—
141	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
142	C₄⋊C₈	—	6	2	no	—
143	#95	—	2	2	no	—
144	C₂×C₃×C₅	—	2	2	no	—
145	C₂×C₃×C₅	#391	1	1	yes	characteristic
146	C₂×C₃×C₅	#421	1	2	yes	—
147	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
148	C₂×C₃×C₄	—	2	4	no	—
149	C₂×C₃×C₄	—	4	2	no	—
150	C₂×C₃×C₄	—	4	4	no	—
151	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
152	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	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×Q₈	1	2	yes	—
156	C₂×C₃×C₄	C₅×#41	1	2	yes	—
157	C₂×Dic₃	—	2	4	no	—
158	Dic₆	—	4	4	no	—
159	C₃⋊C₈	—	4	2	no	—
160	C₂²×C₅	—	2	2	no	—
161	C₂²×C₅	#924	1	2	yes	—
162	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
163	C₄×C₅	—	2	2	no	—
164	C₄×C₅	—	2	2	no	—
165	C₄×C₅	—	4	4	no	—
166	C₄×C₅	—	12	4	no	—
167	C₄×C₅	C₄×Dic₆	1	2	yes	—
168	C₄×C₅	#869	1	1	yes	characteristic
169	C₄×C₅	#875	1	1	yes	characteristic
170	C₂×Q₈	—	6	2	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	C₅×#67	1	2	yes	—
173	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
174	M₁₆	—	6	4	no	—
175	C₂×C₈	—	6	2	no	—
176	C₄⋊C₄	—	3	4	no	—
177	C₄⋊C₄	—	6	4	no	—
178	C₄⋊C₄	—	6	2	no	—
179	C₂²⋊C₄	—	6	2	no	—
180	C₄²	—	2	2	no	—
181	C₄²	—	6	2	no	—
182	C₄²	C₅×Dic₆	1	4	yes	complemented
183	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
184	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
185	C₄²	C₅×D₁₂	1	2	yes	—
186	C₃×C₅	#1995	1	1	yes	characteristic, complemented, Hall
187	C₂²×C₃	—	2	2	no	—
188	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
189	C₂²×C₃	C₅×C₂²⋊Q₈	1	2	yes	—
190	C₃×C₄	—	2	2	no	—
191	C₃×C₄	—	2	2	no	—
192	C₃×C₄	—	4	4	no	—
193	C₃×C₄	C₄×C₅×Q₈	1	2	yes	—
194	C₃×C₄	C₅×#124	1	1	yes	characteristic
195	C₃×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
196	Dic₃	—	4	4	no	—
197	C₂×C₅	—	2	2	no	—
198	C₂×C₅	#4929	1	1	yes	characteristic
199	C₂×C₅	#5005	1	2	yes	—
200	C₂³	C₂×C₅×#67	1	1	yes	characteristic
201	Q₈	—	12	4	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	—	6	4	no	—
206	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	2	yes	—
207	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	2	yes	—
208	C₂×C₄	C₂×C₅×Dic₆	1	2	yes	—
209	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
210	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
211	C₂×C₄	C₅×#235	1	2	yes	—
212	C₈	—	12	2	no	—
213	C₂×C₃	—	2	2	no	—
214	C₂×C₃	C₅×#391	1	1	yes	characteristic
215	C₂×C₃	C₅×#421	1	2	yes	—
216	C₅	#69441	1	1	yes	characteristic, direct product factor, Sylow
217	C₂²	—	2	2	no	—
218	C₂²	C₅×#924	1	2	yes	—
219	C₂²	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
220	C₄	—	2	2	no	—
221	C₄	—	2	2	no	—
222	C₄	—	4	4	no	—
223	C₄	—	12	4	no	—
224	C₄	C₄×C₅×Dic₆	1	2	yes	—
225	C₄	C₅×#869	1	1	yes	characteristic
226	C₄	C₅×#875	1	1	yes	characteristic
227	C₃	C₅×#1995	1	1	yes	characteristic, complemented, Sylow
228	C₂	—	2	2	no	—
229	C₂	C₅×#4929	1	1	yes	characteristic
230	C₂	C₅×#5005	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#376	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#376	D₃
hypocenter	C₃	C₅×#1995
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₅×#376

Complements

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

N	H ≅ G/N	type
11. C₅×#870	C₄	semidirect
12. C₅×#869	C₄	semidirect
16. #69441	C₅	direct (N × H)
29. C₃×C₄²×C₅	Q₈	semidirect
70. #870	C₄×C₅	semidirect
71. #869	C₄×C₅	semidirect
87. C₄²×C₅	Dic₆	semidirect
113. C₃×C₄²	C₅×Q₈	semidirect
182. C₄²	C₅×Dic₆	semidirect
186. C₃×C₅	#1995	semidirect
216. C₅	#69441	direct (N × H)
227. C₃	C₅×#1995	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5470	2
C₃×C₅×#376	2
C₅×#4924	2
C₅×#4523	2
C₅×#1995	3
#69441	5

Subgroups — C5×#69441 #14307443

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69441 #14307443