Subgroups — C₅×#70514 #14307403

Order 1920.

568 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	C₅×#5470	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#423	C₂	1	1	yes	maximal, characteristic
3	C₄²×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₅×#4555	C₂	1	4	yes	maximal
5	C₅×#1994	—	3	1	no	maximal, Hall
6	C₂×C₃×C₅×M₁₆	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×C₄×C₅×Dic₃	—	2	2	no	—
11	C₂×C₄×C₅×Dic₃	C₂²	1	2	yes	—
12	C₅×#870	C₄	1	2	yes	complemented
13	C₅×#869	C₄	1	2	yes	complemented
14	C₄×C₅×Dic₆	—	2	2	no	—
15	C₃×C₅×C₄⋊C₈	—	2	2	no	—
16	#70514	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₅×#533	—	3	1	no	—
18	C₅×#423	D₃	1	1	yes	characteristic
19	C₄³×C₅	—	3	1	no	—
20	C₅×#339	—	3	4	no	—
21	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
22	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₂²×C₅×Dic₃	—	2	2	no	—
24	C₂×C₅×Dic₆	—	2	2	no	—
25	C₃×C₅×M₁₆	—	2	4	no	—
26	C₂×C₃×C₅×C₈	—	2	2	no	—
27	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
28	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
29	C₃×C₄²×C₅	D₄	1	2	yes	—
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	8	no	—
35	C₄×C₅×Dic₃	—	2	2	no	—
36	C₄×C₅×Dic₃	—	2	2	no	—
37	C₄×C₅×Dic₃	Q₈	1	4	yes	complemented
38	#5470	C₂×C₅	1	1	yes	characteristic
39	C₃×#423	C₂×C₅	1	1	yes	characteristic
40	C₄²×Dic₃	C₂×C₅	1	1	yes	characteristic
41	#4555	C₂×C₅	1	4	yes	—
42	C₂×C₅×M₁₆	C₂×D₃	1	2	yes	—
43	C₅×C₄⋊Q₈	—	3	2	no	—
44	C₅×#125	—	3	2	no	—
45	C₅×C₂²⋊Q₈	—	6	2	no	—
46	C₄×C₅×Q₈	—	6	2	no	—
47	C₅×#117	—	3	2	no	—
48	C₂×C₄²×C₅	—	3	2	no	—
49	C₂×C₄²×C₅	—	6	2	no	—
50	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
51	C₅×C₄⋊C₈	—	2	2	no	—
52	#1994	—	3	1	no	Sylow
53	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	—	2	4	no	—
55	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
56	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
57	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
60	C₂×C₃×C₄×C₅	#41	1	2	yes	—
61	C₂×C₅×Dic₃	—	2	8	no	—
62	C₂×C₅×Dic₃	—	2	4	no	—
63	C₂×C₅×Dic₃	—	2	4	no	—
64	C₅×Dic₆	—	4	4	no	—
65	C₃×C₅×C₈	—	4	2	no	—
66	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
67	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
68	#926	C₂²×C₅	1	2	yes	—
69	#924	—	2	2	no	—
70	C₂×C₄×Dic₃	—	2	2	no	—
71	C₂×C₄×Dic₃	C₂²×C₅	1	2	yes	—
72	#870	C₄×C₅	1	2	yes	complemented
73	#869	C₄×C₅	1	2	yes	complemented
74	C₄×Dic₆	—	2	2	no	—
75	C₃×C₄⋊C₈	—	2	2	no	—
76	C₂×C₅×Q₈	—	6	2	no	—
77	C₂²×C₄×C₅	—	6	2	no	—
78	C₂²×C₄×C₅	D₁₂	1	2	yes	—
79	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
80	C₅×M₁₆	—	2	4	no	—
81	C₂×C₅×C₈	—	2	2	no	—
82	C₅×C₄⋊C₄	—	3	4	no	—
83	C₅×C₄⋊C₄	—	6	4	no	—
84	C₅×C₄⋊C₄	—	6	2	no	—
85	C₅×C₂²⋊C₄	—	6	2	no	—
86	C₄²×C₅	—	3	4	no	—
87	C₄²×C₅	—	6	8	no	—
88	C₄²×C₅	—	6	2	no	—
89	C₄²×C₅	—	6	2	no	—
90	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
91	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
92	C₄²×C₅	#67	1	2	yes	—
93	#533	—	3	1	no	—
94	#423	C₅×D₃	1	1	yes	characteristic
95	C₄³	—	3	1	no	—
96	#339	—	3	4	no	—
97	C₂²×C₃×C₅	—	2	2	no	—
98	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂²×C₃×C₅	C₂²⋊Q₈	1	2	yes	—
100	C₃×C₄×C₅	—	2	2	no	—
101	C₃×C₄×C₅	#104	1	4	yes	—
102	C₃×C₄×C₅	C₄×Q₈	1	2	yes	—
103	C₃×C₄×C₅	#124	1	1	yes	characteristic
104	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
105	C₅×Dic₃	—	2	8	no	—
106	C₅×Dic₃	—	4	4	no	—
107	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
108	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
109	C₂²×Dic₃	—	2	2	no	—
110	C₂×Dic₆	—	2	2	no	—
111	C₃×M₁₆	—	2	4	no	—
112	C₂×C₃×C₈	—	2	2	no	—
113	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
114	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
115	C₃×C₄²	C₅×D₄	1	2	yes	—
116	#217	—	2	2	no	—
117	C₄⋊Dic₃	—	2	2	no	—
118	C₄⋊Dic₃	C₂×C₄×C₅	1	4	yes	—
119	Dic₃⋊C₄	—	2	4	no	—
120	C₄×Dic₃	—	2	8	no	—
121	C₄×Dic₃	—	2	2	no	—
122	C₄×Dic₃	—	2	2	no	—
123	C₄×Dic₃	C₅×Q₈	1	4	yes	complemented
124	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
125	C₅×Q₈	—	12	4	no	—
126	C₂×C₄×C₅	—	2	4	no	—
127	C₂×C₄×C₅	—	6	8	no	—
128	C₂×C₄×C₅	—	6	4	no	—
129	C₂×C₄×C₅	—	6	4	no	—
130	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
131	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
132	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
133	C₂×C₄×C₅	#237	1	2	yes	—
134	C₂×C₄×C₅	D₃×Q₈	1	2	yes	—
135	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
136	C₅×C₈	—	4	2	no	—
137	C₂×M₁₆	C₂×C₅×D₃	1	2	yes	—
138	C₄⋊Q₈	—	3	2	no	—
139	#125	—	3	2	no	—
140	C₂²⋊Q₈	—	6	2	no	—
141	C₄×Q₈	—	6	2	no	—
142	#117	—	3	2	no	—
143	C₂×C₄²	—	3	2	no	—
144	C₂×C₄²	—	6	2	no	—
145	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
146	C₄⋊C₈	—	2	2	no	—
147	C₂×C₃×C₅	—	2	2	no	—
148	C₂×C₃×C₅	#391	1	1	yes	characteristic
149	C₂×C₃×C₅	C₂×#104	1	2	yes	—
150	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
151	C₂×C₃×C₄	—	2	4	no	—
152	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
153	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
154	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
157	C₂×C₃×C₄	C₅×#41	1	2	yes	—
158	C₂×Dic₃	—	2	8	no	—
159	C₂×Dic₃	—	2	4	no	—
160	C₂×Dic₃	—	2	4	no	—
161	Dic₆	—	4	4	no	—
162	C₃×C₈	—	4	2	no	—
163	C₂²×C₅	—	2	2	no	—
164	C₂²×C₅	#897	1	2	yes	—
165	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
166	C₄×C₅	—	2	2	no	—
167	C₄×C₅	—	6	8	no	—
168	C₄×C₅	—	12	4	no	—
169	C₄×C₅	#825	1	4	yes	—
170	C₄×C₅	#887	1	2	yes	—
171	C₄×C₅	#936	1	1	yes	characteristic
172	C₄×C₅	#944	1	1	yes	characteristic
173	C₂×Q₈	—	6	2	no	—
174	C₂²×C₄	—	6	2	no	—
175	C₂²×C₄	C₅×D₁₂	1	2	yes	—
176	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
177	M₁₆	—	2	4	no	—
178	C₂×C₈	—	2	2	no	—
179	C₄⋊C₄	—	3	4	no	—
180	C₄⋊C₄	—	6	4	no	—
181	C₄⋊C₄	—	6	2	no	—
182	C₂²⋊C₄	—	6	2	no	—
183	C₄²	—	3	4	no	—
184	C₄²	—	6	8	no	—
185	C₄²	—	6	2	no	—
186	C₄²	—	6	2	no	—
187	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
188	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
189	C₄²	C₅×#67	1	2	yes	—
190	C₃×C₅	#1994	1	1	yes	characteristic, complemented, Hall
191	C₂²×C₃	—	2	2	no	—
192	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
193	C₂²×C₃	C₅×C₂²⋊Q₈	1	2	yes	—
194	C₃×C₄	—	2	2	no	—
195	C₃×C₄	C₅×#104	1	4	yes	—
196	C₃×C₄	C₄×C₅×Q₈	1	2	yes	—
197	C₃×C₄	C₅×#124	1	1	yes	characteristic
198	C₃×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
199	Dic₃	—	2	8	no	—
200	Dic₃	—	4	4	no	—
201	C₂×C₅	—	2	2	no	—
202	C₂×C₅	#4973	1	1	yes	characteristic
203	C₂×C₅	C₂×#825	1	2	yes	—
204	C₂³	C₂×C₅×D₁₂	1	1	yes	characteristic
205	Q₈	—	12	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	6	8	no	—
208	C₂×C₄	—	6	4	no	—
209	C₂×C₄	—	6	4	no	—
210	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	2	yes	—
211	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	2	yes	—
212	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
213	C₂×C₄	C₅×#237	1	2	yes	—
214	C₂×C₄	C₅×D₃×Q₈	1	2	yes	—
215	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
216	C₈	—	4	2	no	—
217	C₂×C₃	—	2	2	no	—
218	C₂×C₃	C₅×#391	1	1	yes	characteristic
219	C₂×C₃	C₂×C₅×#104	1	2	yes	—
220	C₅	#70514	1	1	yes	characteristic, direct product factor, Sylow
221	C₂²	—	2	2	no	—
222	C₂²	C₅×#897	1	2	yes	—
223	C₂²	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
224	C₄	—	2	2	no	—
225	C₄	—	6	8	no	—
226	C₄	—	12	4	no	—
227	C₄	C₅×#825	1	4	yes	—
228	C₄	C₅×#887	1	2	yes	—
229	C₄	C₅×#936	1	1	yes	characteristic
230	C₄	C₅×#944	1	1	yes	characteristic
231	C₃	C₅×#1994	1	1	yes	characteristic, complemented, Sylow
232	C₂	—	2	2	no	—
233	C₂	C₅×#4973	1	1	yes	characteristic
234	C₂	C₂×C₅×#825	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#423	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#423	D₃
hypocenter	C₃	C₅×#1994
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₅×#423

Complements

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

N	H ≅ G/N	type
12. C₅×#870	C₄	semidirect
13. C₅×#869	C₄	semidirect
16. #70514	C₅	direct (N × H)
37. C₄×C₅×Dic₃	Q₈	semidirect
72. #870	C₄×C₅	semidirect
73. #869	C₄×C₅	semidirect
123. C₄×Dic₃	C₅×Q₈	semidirect
190. C₃×C₅	#1994	semidirect
220. C₅	#70514	direct (N × H)
231. C₃	C₅×#1994	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×#423	2
C₄²×C₅×Dic₃	2
C₅×#4555	2
C₅×#1994	3
#70514	5

Subgroups — C5×#70514 #14307403

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70514 #14307403