Subgroups — C₅×#71797 #14283979

Order 1920.

540 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	C₃×C₅×#532	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅×#208	C₂	1	2	yes	maximal, complemented
3	C₅×#4540	C₂	1	4	yes	maximal, complemented
4	C₅×#1525	—	3	1	no	maximal, Hall
5	C₂×C₃×C₅×#41	C₄	1	2	yes	—
6	C₃×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic
7	C₃×C₅×#127	C₄	1	1	yes	characteristic
8	C₃×C₅×#124	C₂²	1	2	yes	—
9	C₃×C₅×#121	—	2	2	no	—
10	C₃×C₄×C₅×D₄	—	2	2	no	—
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₅×#208	—	2	2	no	—
13	C₅×#803	C₂²	1	4	yes	—
14	C₄×C₅×C₃⋊C₈	—	2	2	no	—
15	#71797	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₅×#532	D₃	1	1	yes	characteristic
17	C₄×C₅×M₁₆	—	3	2	no	—
18	C₅×#329	—	3	4	no	—
19	C₃×C₅×#41	—	4	4	no	—
20	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
21	C₂×C₃×C₅×D₄	—	2	2	no	—
22	C₂×C₃×C₅×D₄	—	2	2	no	—
23	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
24	C₂²×C₃×C₄×C₅	—	2	2	no	—
25	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
27	C₃×C₅×C₄⋊C₄	—	2	2	no	—
28	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
29	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
30	C₃×C₄²×C₅	D₄	1	2	yes	—
31	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
32	C₅×#208	—	2	8	no	—
33	C₂×C₅×C₃⋊C₈	—	2	4	no	—
34	C₃×#532	C₂×C₅	1	1	yes	characteristic
35	C₄×#208	C₂×C₅	1	2	yes	complemented
36	#4540	C₂×C₅	1	4	yes	complemented
37	C₂×C₅×#41	Dic₃	1	2	yes	—
38	C₂×C₅×M₁₆	—	6	2	no	—
39	C₅×C₄⋊Q₈	Dic₃	1	1	yes	characteristic
40	C₅×#127	Dic₃	1	1	yes	characteristic
41	C₅×#124	C₂×D₃	1	2	yes	—
42	C₅×#121	—	2	2	no	—
43	C₄×C₅×D₄	—	2	2	no	—
44	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
45	C₅×#97	—	3	4	no	—
46	C₄×C₅×C₈	—	6	2	no	—
47	#1525	—	3	1	no	Sylow
48	C₂³×C₃×C₅	—	2	2	no	—
49	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
50	C₃×C₅×Q₈	—	4	2	no	—
51	C₃×C₅×D₄	—	4	2	no	—
52	C₃×C₅×D₄	—	4	4	no	—
53	C₃×C₅×D₄	—	4	4	no	—
54	C₂×C₃×C₄×C₅	—	2	2	no	—
55	C₂×C₃×C₄×C₅	—	2	4	no	—
56	C₂×C₃×C₄×C₅	—	2	2	no	—
57	C₂×C₃×C₄×C₅	—	4	4	no	—
58	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
60	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
62	C₅×C₃⋊C₈	—	2	8	no	—
63	C₂×C₃×#41	C₄×C₅	1	2	yes	—
64	C₃×C₄⋊Q₈	C₄×C₅	1	1	yes	characteristic
65	C₃×#127	C₄×C₅	1	1	yes	characteristic
66	C₃×#124	C₂²×C₅	1	2	yes	—
67	C₃×#121	—	2	2	no	—
68	C₃×C₄×D₄	—	2	2	no	—
69	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
70	C₂×#208	—	2	2	no	—
71	#803	C₂²×C₅	1	4	yes	—
72	C₄×C₃⋊C₈	—	2	2	no	—
73	C₅×#41	—	4	4	no	—
74	C₂×C₅×Q₈	C₂×Dic₃	1	2	yes	—
75	C₂×C₅×D₄	—	2	2	no	—
76	C₂×C₅×D₄	—	2	2	no	—
77	C₂×C₅×D₄	C₂×Dic₃	1	2	yes	—
78	C₂²×C₄×C₅	—	2	2	no	—
79	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
80	C₂²×C₄×C₅	#67	1	2	yes	—
81	C₅×M₁₆	—	6	8	no	—
82	C₂×C₅×C₈	—	6	4	no	—
83	C₅×C₄⋊C₄	—	2	2	no	—
84	C₅×C₂²⋊C₄	—	2	4	no	—
85	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
86	C₄²×C₅	#67	1	2	yes	—
87	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
88	#532	C₅×D₃	1	1	yes	characteristic
89	C₄×M₁₆	—	3	2	no	—
90	#329	—	3	4	no	—
91	C₂²×C₃×C₅	—	2	1	no	—
92	C₂²×C₃×C₅	—	2	2	no	—
93	C₂²×C₃×C₅	—	4	2	no	—
94	C₂²×C₃×C₅	—	4	4	no	—
95	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₃×C₄×C₅	—	2	1	no	—
97	C₃×C₄×C₅	—	2	2	no	—
98	C₃×C₄×C₅	—	4	2	no	—
99	C₃×C₄×C₅	#100	1	2	yes	—
100	C₃×C₄×C₅	#104	1	4	yes	—
101	C₃×#41	—	4	4	no	—
102	C₂×C₃×Q₈	C₂×C₄×C₅	1	2	yes	—
103	C₂×C₃×D₄	—	2	2	no	—
104	C₂×C₃×D₄	—	2	2	no	—
105	C₂×C₃×D₄	C₂×C₄×C₅	1	2	yes	—
106	C₂²×C₃×C₄	—	2	2	no	—
107	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
108	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
109	C₃×C₄⋊C₄	—	2	2	no	—
110	C₃×C₂²⋊C₄	—	2	4	no	—
111	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
112	C₃×C₄²	C₅×D₄	1	2	yes	—
113	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
114	#208	—	2	8	no	—
115	C₂×C₃⋊C₈	—	2	4	no	—
116	C₂³×C₅	—	2	2	no	—
117	C₂³×C₅	#217	1	1	yes	characteristic
118	C₅×Q₈	—	4	2	no	—
119	C₅×D₄	—	4	2	no	—
120	C₅×D₄	—	4	4	no	—
121	C₅×D₄	—	4	4	no	—
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	—	2	4	no	—
124	C₂×C₄×C₅	—	2	2	no	—
125	C₂×C₄×C₅	—	4	4	no	—
126	C₂×C₄×C₅	#217	1	2	yes	—
127	C₂×C₄×C₅	#217	1	1	yes	characteristic
128	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
129	C₂×C₄×C₅	C₂×#67	1	2	yes	—
130	C₅×C₈	—	6	8	no	—
131	C₂×#41	C₅×Dic₃	1	2	yes	—
132	C₂×M₁₆	—	6	2	no	—
133	C₄⋊Q₈	C₅×Dic₃	1	1	yes	characteristic
134	#127	C₅×Dic₃	1	1	yes	characteristic
135	#124	C₂×C₅×D₃	1	2	yes	—
136	#121	—	2	2	no	—
137	C₄×D₄	—	2	2	no	—
138	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
139	#97	—	3	4	no	—
140	C₄×C₈	—	6	2	no	—
141	C₂×C₃×C₅	—	4	1	no	—
142	C₂×C₃×C₅	—	8	2	no	—
143	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
144	C₂×C₃×C₅	C₂×#104	1	2	yes	—
145	C₂³×C₃	—	2	2	no	—
146	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
147	C₃×Q₈	—	4	2	no	—
148	C₃×D₄	—	4	2	no	—
149	C₃×D₄	—	4	4	no	—
150	C₃×D₄	—	4	4	no	—
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	—	2	4	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	4	4	no	—
155	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
156	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
157	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
159	C₃⋊C₈	—	2	8	no	—
160	C₂²×C₅	—	2	1	no	—
161	C₂²×C₅	—	2	2	no	—
162	C₂²×C₅	—	4	2	no	—
163	C₂²×C₅	—	4	4	no	—
164	C₂²×C₅	C₂×#217	1	1	yes	characteristic
165	C₄×C₅	—	2	1	no	—
166	C₄×C₅	—	2	2	no	—
167	C₄×C₅	—	4	2	no	—
168	C₄×C₅	#833	1	2	yes	—
169	C₄×C₅	#837	1	4	yes	—
170	#41	—	4	4	no	—
171	C₂×Q₈	C₂×C₅×Dic₃	1	2	yes	—
172	C₂×D₄	—	2	2	no	—
173	C₂×D₄	—	2	2	no	—
174	C₂×D₄	C₂×C₅×Dic₃	1	2	yes	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
177	C₂²×C₄	C₅×#67	1	2	yes	—
178	M₁₆	—	6	8	no	—
179	C₂×C₈	—	6	4	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₂²⋊C₄	—	2	4	no	—
182	C₄²	C₂×C₅×Dic₃	1	1	yes	characteristic
183	C₄²	C₅×#67	1	2	yes	—
184	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
185	C₃×C₅	#1525	1	1	yes	characteristic, complemented, Hall
186	C₂²×C₃	—	2	1	no	—
187	C₂²×C₃	—	2	2	no	—
188	C₂²×C₃	—	4	2	no	—
189	C₂²×C₃	—	4	4	no	—
190	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
191	C₃×C₄	—	2	1	no	—
192	C₃×C₄	—	2	2	no	—
193	C₃×C₄	—	4	2	no	—
194	C₃×C₄	C₅×#100	1	2	yes	—
195	C₃×C₄	C₅×#104	1	4	yes	—
196	C₂×C₅	—	4	1	no	—
197	C₂×C₅	—	8	2	no	—
198	C₂×C₅	C₂×#833	1	1	yes	characteristic
199	C₂×C₅	C₂×#837	1	2	yes	—
200	C₂³	—	2	2	no	—
201	C₂³	C₅×#217	1	1	yes	characteristic
202	Q₈	—	4	2	no	—
203	D₄	—	4	2	no	—
204	D₄	—	4	4	no	—
205	D₄	—	4	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	C₅×#217	1	1	yes	characteristic
211	C₂×C₄	C₅×#217	1	2	yes	—
212	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
213	C₂×C₄	C₂×C₅×#67	1	2	yes	—
214	C₈	—	6	8	no	—
215	C₂×C₃	—	4	1	no	—
216	C₂×C₃	—	8	2	no	—
217	C₂×C₃	C₂×C₅×#100	1	1	yes	characteristic
218	C₂×C₃	C₂×C₅×#104	1	2	yes	—
219	C₅	#71797	1	1	yes	characteristic, direct product factor, Sylow
220	C₂²	—	2	1	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	4	2	no	—
223	C₂²	—	4	4	no	—
224	C₂²	C₂×C₅×#217	1	1	yes	characteristic
225	C₄	—	2	1	no	—
226	C₄	—	2	2	no	—
227	C₄	—	4	2	no	—
228	C₄	C₅×#833	1	2	yes	—
229	C₄	C₅×#837	1	4	yes	—
230	C₃	C₅×#1525	1	1	yes	characteristic, complemented, Sylow
231	C₂	—	4	1	no	—
232	C₂	—	8	2	no	—
233	C₂	C₂×C₅×#833	1	1	yes	characteristic
234	C₂	C₂×C₅×#837	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	#217
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting, prime radical	C₃×C₅×#532	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×#532	D₃
hypocenter	C₃	C₅×#1525
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₅×#532

Complements

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

N	H ≅ G/N	type
2. C₄×C₅×#208	C₂	semidirect
3. C₅×#4540	C₂	semidirect
15. #71797	C₅	direct (N × H)
35. C₄×#208	C₂×C₅	semidirect
36. #4540	C₂×C₅	semidirect
185. C₃×C₅	#1525	semidirect
219. C₅	#71797	direct (N × H)
230. C₃	C₅×#1525	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#532	2
C₄×C₅×#208	2
C₅×#4540	2
C₅×#1525	3
#71797	5

Subgroups — C5×#71797 #14283979

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71797 #14283979