Subgroups — C₅×#87139 #14495603

Order 1920.

1860 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	C₃×C₅×#564	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#5700	C₂	1	5	yes	maximal, complemented
3	C₅×#5699	C₂	1	10	yes	maximal, complemented
4	C₅×#5691	C₂	1	10	yes	maximal, complemented
5	C₅×#5688	C₂	1	5	yes	maximal, complemented
6	C₅×#3531	—	3	1	no	maximal, Hall
7	C₃×C₅×#125	C₂²	1	5	yes	—
8	C₃×C₅×#125	C₂²	1	10	yes	—
9	C₅×#898	C₂²	1	20	yes	—
10	C₅×#897	C₂²	1	10	yes	—
11	C₅×#896	C₂²	1	20	yes	—
12	C₅×#895	C₂²	1	10	yes	—
13	C₅×#894	C₂²	1	20	yes	—
14	C₅×#893	C₂²	1	10	yes	—
15	C₅×#892	C₂²	1	10	yes	—
16	C₅×#888	C₂²	1	10	yes	—
17	C₅×#887	C₂²	1	10	yes	—
18	C₅×#875	C₂²	1	10	yes	—
19	C₄²×C₅×D₃	C₂²	1	5	yes	—
20	C₅×#870	C₂²	1	5	yes	—
21	#87139	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₅×#564	D₃	1	1	yes	characteristic, complemented
23	C₅×#563	—	3	10	no	—
24	C₅×#562	—	3	5	no	—
25	C₅×#533	—	3	5	no	—
26	C₅×#529	—	3	10	no	—
27	C₂×C₅×D₁₂	C₂³	1	10	yes	—
28	C₂×C₄×C₅×D₃	C₂³	1	5	yes	—
29	C₂×C₄×C₅×D₃	C₂³	1	10	yes	—
30	C₂×C₅×Dic₆	C₂³	1	10	yes	—
31	C₃×C₅×C₄⋊C₄	C₂³	1	10	yes	—
32	C₃×C₅×C₄⋊C₄	C₂³	1	10	yes	—
33	C₃×C₅×C₄⋊C₄	C₂³	1	10	yes	—
34	C₃×C₄²×C₅	C₂³	1	5	yes	—
35	C₅×(C₂×D₃)⋊C₄	C₂³	1	20	yes	—
36	C₅×(C₂×D₃)⋊C₄	C₂³	1	20	yes	—
37	C₅×C₄⋊Dic₃	C₂³	1	10	yes	—
38	C₅×Dic₃⋊C₄	C₂³	1	20	yes	—
39	C₄×C₅×Dic₃	C₂³	1	10	yes	—
40	C₄×C₅×Dic₃	C₂³	1	5	yes	—
41	C₃×#564	C₂×C₅	1	1	yes	characteristic, complemented
42	#5700	C₂×C₅	1	5	yes	complemented
43	#5699	C₂×C₅	1	10	yes	complemented
44	#5691	C₂×C₅	1	10	yes	complemented
45	#5688	C₂×C₅	1	5	yes	complemented
46	C₅×C₄⋊Q₈	—	3	10	no	—
47	C₅×#126	—	3	20	no	—
48	C₅×#125	—	3	5	no	—
49	C₅×#125	C₂×D₃	1	10	yes	—
50	C₅×#125	C₂×D₃	1	5	yes	—
51	C₅×#124	—	3	10	no	—
52	C₅×#123	—	3	20	no	—
53	C₅×C₂²⋊Q₈	—	3	20	no	—
54	C₅×C₂²⋊Q₈	—	3	10	no	—
55	C₅×#121	—	3	10	no	—
56	C₄×C₅×Q₈	—	3	10	no	—
57	C₄×C₅×D₄	—	3	10	no	—
58	C₅×#117	—	3	10	no	—
59	C₂×C₄²×C₅	—	3	5	no	—
60	#3531	—	3	1	no	Sylow
61	C₂²×C₅×D₃	C₂⁴	1	5	yes	—
62	C₂×C₃×C₄×C₅	C₂⁴	1	10	yes	—
63	C₂×C₃×C₄×C₅	C₂⁴	1	5	yes	—
64	C₂×C₅×Dic₃	C₂⁴	1	10	yes	—
65	C₂×C₅×Dic₃	C₂⁴	1	1	yes	characteristic
66	C₅×D₁₂	—	4	10	no	—
67	C₄×C₅×D₃	—	2	10	no	—
68	C₄×C₅×D₃	—	4	10	no	—
69	C₅×Dic₆	—	4	10	no	—
70	C₃×#125	C₂²×C₅	1	5	yes	—
71	C₃×#125	C₂²×C₅	1	10	yes	—
72	#898	C₂²×C₅	1	20	yes	—
73	#897	C₂²×C₅	1	10	yes	—
74	#896	C₂²×C₅	1	20	yes	—
75	#895	C₂²×C₅	1	10	yes	—
76	#894	C₂²×C₅	1	20	yes	—
77	#893	C₂²×C₅	1	10	yes	—
78	#892	C₂²×C₅	1	10	yes	—
79	#888	C₂²×C₅	1	10	yes	—
80	#887	C₂²×C₅	1	10	yes	—
81	#875	C₂²×C₅	1	10	yes	—
82	C₄²×D₃	C₂²×C₅	1	5	yes	—
83	#870	C₂²×C₅	1	5	yes	—
84	C₂×C₅×Q₈	—	3	10	no	—
85	C₂×C₅×D₄	—	3	10	no	—
86	C₂²×C₄×C₅	—	3	5	no	—
87	C₂²×C₄×C₅	—	3	10	no	—
88	C₅×C₄⋊C₄	—	3	10	no	—
89	C₅×C₄⋊C₄	—	3	20	no	—
90	C₅×C₄⋊C₄	C₂²×D₃	1	10	yes	—
91	C₅×C₄⋊C₄	C₂²×D₃	1	10	yes	—
92	C₅×C₄⋊C₄	C₂²×D₃	1	10	yes	—
93	C₅×C₂²⋊C₄	—	3	20	no	—
94	C₅×C₂²⋊C₄	—	3	20	no	—
95	C₄²×C₅	—	3	10	no	—
96	C₄²×C₅	—	3	5	no	—
97	C₄²×C₅	C₂²×D₃	1	5	yes	—
98	#564	C₅×D₃	1	1	yes	characteristic, complemented
99	#563	—	3	10	no	—
100	#562	—	3	5	no	—
101	#533	—	3	5	no	—
102	#529	—	3	10	no	—
103	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
104	C₂×C₅×D₃	—	2	5	no	—
105	C₂×C₅×D₃	—	2	10	no	—
106	C₃×C₄×C₅	—	2	5	no	—
107	C₃×C₄×C₅	—	2	10	no	—
108	C₅×Dic₃	—	2	10	no	—
109	C₅×Dic₃	2₋¹⁺⁴	1	2	yes	—
110	C₂×D₁₂	C₂³×C₅	1	10	yes	—
111	C₂×C₄×D₃	C₂³×C₅	1	5	yes	—
112	C₂×C₄×D₃	C₂³×C₅	1	10	yes	—
113	C₂×Dic₆	C₂³×C₅	1	10	yes	—
114	C₃×C₄⋊C₄	C₂³×C₅	1	10	yes	—
115	C₃×C₄⋊C₄	C₂³×C₅	1	10	yes	—
116	C₃×C₄⋊C₄	C₂³×C₅	1	10	yes	—
117	C₃×C₄²	C₂³×C₅	1	5	yes	—
118	(C₂×D₃)⋊C₄	C₂³×C₅	1	20	yes	—
119	(C₂×D₃)⋊C₄	C₂³×C₅	1	20	yes	—
120	C₄⋊Dic₃	C₂³×C₅	1	10	yes	—
121	Dic₃⋊C₄	C₂³×C₅	1	20	yes	—
122	C₄×Dic₃	C₂³×C₅	1	10	yes	—
123	C₄×Dic₃	C₂³×C₅	1	5	yes	—
124	C₂³×C₅	—	3	5	no	—
125	C₅×Q₈	—	12	10	no	—
126	C₅×D₄	—	12	10	no	—
127	C₂×C₄×C₅	—	3	1	no	—
128	C₂×C₄×C₅	—	3	10	no	—
129	C₂×C₄×C₅	—	6	10	no	—
130	C₂×C₄×C₅	—	12	10	no	—
131	C₂×C₄×C₅	C₂³×D₃	1	5	yes	—
132	C₂×C₄×C₅	C₂³×D₃	1	10	yes	—
133	C₄⋊Q₈	—	3	10	no	—
134	#126	—	3	20	no	—
135	#125	—	3	5	no	—
136	#125	C₂×C₅×D₃	1	10	yes	—
137	#125	C₂×C₅×D₃	1	5	yes	—
138	#124	—	3	10	no	—
139	#123	—	3	20	no	—
140	C₂²⋊Q₈	—	3	10	no	—
141	C₂²⋊Q₈	—	3	20	no	—
142	#121	—	3	10	no	—
143	C₄×Q₈	—	3	10	no	—
144	C₄×D₄	—	3	10	no	—
145	#117	—	3	10	no	—
146	C₂×C₄²	—	3	5	no	—
147	C₂×C₃×C₅	C₂×2₋¹⁺⁴	1	1	yes	characteristic
148	C₂×C₃×C₅	#585	1	2	yes	—
149	C₅×D₃	—	4	5	no	—
150	C₂²×D₃	C₂⁴×C₅	1	5	yes	—
151	C₂×C₃×C₄	C₂⁴×C₅	1	10	yes	—
152	C₂×C₃×C₄	C₂⁴×C₅	1	5	yes	—
153	C₂×Dic₃	C₂⁴×C₅	1	10	yes	—
154	C₂×Dic₃	C₂⁴×C₅	1	1	yes	characteristic
155	D₁₂	—	4	10	no	—
156	C₄×D₃	—	2	10	no	—
157	C₄×D₃	—	4	10	no	—
158	Dic₆	—	4	10	no	—
159	C₂²×C₅	—	6	5	no	—
160	C₂²×C₅	—	6	10	no	—
161	C₂²×C₅	C₂⁴×D₃	1	1	yes	characteristic
162	C₄×C₅	—	2	5	no	—
163	C₄×C₅	—	2	10	no	—
164	C₄×C₅	—	3	2	no	—
165	C₄×C₅	—	6	10	no	—
166	C₂×Q₈	—	3	10	no	—
167	C₂×D₄	—	3	10	no	—
168	C₂²×C₄	—	3	5	no	—
169	C₂²×C₄	—	3	10	no	—
170	C₄⋊C₄	—	3	10	no	—
171	C₄⋊C₄	—	3	20	no	—
172	C₄⋊C₄	C₂²×C₅×D₃	1	10	yes	—
173	C₄⋊C₄	C₂²×C₅×D₃	1	10	yes	—
174	C₄⋊C₄	C₂²×C₅×D₃	1	10	yes	—
175	C₂²⋊C₄	—	3	20	no	—
176	C₂²⋊C₄	—	3	20	no	—
177	C₄²	—	3	10	no	—
178	C₄²	—	3	5	no	—
179	C₄²	C₂²×C₅×D₃	1	5	yes	—
180	C₃×C₅	#3531	1	1	yes	characteristic, complemented, Hall
181	C₂²×C₃	C₂⁵×C₅	1	1	yes	characteristic
182	C₂×D₃	—	2	5	no	—
183	C₂×D₃	—	2	10	no	—
184	C₃×C₄	—	2	5	no	—
185	C₃×C₄	—	2	10	no	—
186	Dic₃	—	2	10	no	—
187	Dic₃	C₅×2₋¹⁺⁴	1	2	yes	—
188	C₂×C₅	—	12	5	no	—
189	C₂×C₅	D₃×2₋¹⁺⁴	1	1	yes	characteristic
190	C₂×C₅	#5968	1	2	yes	—
191	C₂³	—	3	5	no	—
192	Q₈	—	12	10	no	—
193	D₄	—	12	10	no	—
194	C₂×C₄	—	3	1	no	—
195	C₂×C₄	—	3	10	no	—
196	C₂×C₄	—	6	10	no	—
197	C₂×C₄	—	12	10	no	—
198	C₂×C₄	C₂³×C₅×D₃	1	5	yes	—
199	C₂×C₄	C₂³×C₅×D₃	1	10	yes	—
200	C₂×C₃	C₂×C₅×2₋¹⁺⁴	1	1	yes	characteristic
201	C₂×C₃	C₅×#585	1	2	yes	—
202	D₃	—	4	5	no	—
203	C₅	#87139	1	1	yes	characteristic, direct product factor, Sylow
204	C₂²	—	6	5	no	—
205	C₂²	—	6	10	no	—
206	C₂²	C₂⁴×C₅×D₃	1	1	yes	characteristic
207	C₄	—	2	5	no	—
208	C₄	—	2	10	no	—
209	C₄	—	3	2	no	—
210	C₄	—	6	10	no	—
211	C₃	C₅×#3531	1	1	yes	characteristic, complemented, Sylow
212	C₂	—	12	5	no	—
213	C₂	C₅×D₃×2₋¹⁺⁴	1	1	yes	characteristic
214	C₂	C₅×#5968	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅	C₂⁴×D₃
commutator, cosocle	C₂²×C₃	C₂⁵×C₅
Frattini	C₂²	C₂⁴×C₅×D₃
Fitting	C₃×C₅×#564	C₂
socle	C₂²×C₃×C₅	C₂⁵
hypercenter	C₅×#564	D₃
hypocenter	C₃	C₅×#3531
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×#564

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#564	C₂	semidirect
2. C₅×#5700	C₂	semidirect
3. C₅×#5699	C₂	semidirect
4. C₅×#5691	C₂	semidirect
5. C₅×#5688	C₂	semidirect
21. #87139	C₅	direct (N × H)
22. C₅×#564	D₃	semidirect
41. C₃×#564	C₂×C₅	semidirect
42. #5700	C₂×C₅	semidirect
43. #5699	C₂×C₅	semidirect
44. #5691	C₂×C₅	semidirect
45. #5688	C₂×C₅	semidirect
98. #564	C₅×D₃	semidirect
180. C₃×C₅	#3531	semidirect
203. C₅	#87139	direct (N × H)
211. C₃	C₅×#3531	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#564	2
C₅×#5700	2
C₅×#5699	2
C₅×#5691	2
C₅×#5688	2
C₅×#3531	3
#87139	5

Subgroups — C5×#87139 #14495603

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#87139 #14495603