Subgroups — C₅×#67729 #14278854

Order 1920.

264 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#108	C₂	1	1	yes	maximal, characteristic
2	C₅×#5106	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₅×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₅×#1382	—	3	1	no	maximal, Hall
5	C₂×C₃×C₅×M₁₆	—	2	1	no	—
6	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₅×#926	—	2	1	no	—
8	C₃×C₅×#108	—	2	1	no	—
9	C₃×C₅×#108	C₄	1	2	yes	complemented
10	C₅×#818	C₄	1	2	yes	—
11	C₅×#817	—	2	1	no	—
12	C₂×C₅×C₃⋊C₁₆	—	2	1	no	—
13	C₂×C₅×C₃⋊C₁₆	—	2	1	no	—
14	C₂×C₅×C₃⋊C₁₆	C₄	1	2	yes	complemented
15	#67729	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂²×C₅×C₁₆	—	3	1	no	—
17	C₂×C₅×#108	D₃	1	1	yes	characteristic
18	C₅×#427	—	3	1	no	—
19	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
20	C₃×C₅×M₁₆	—	2	2	no	—
21	C₃×C₅×M₁₆	—	4	1	no	—
22	C₂×C₃×C₅×C₈	—	4	1	no	—
23	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
27	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
28	C₂×C₃×C₅×C₈	Q₈	1	1	yes	characteristic
29	C₅×#217	—	4	1	no	—
30	C₅×C₄⋊Dic₃	—	2	2	no	—
31	C₄×C₅×Dic₃	—	4	1	no	—
32	C₅×C₃⋊C₁₆	—	2	2	no	—
33	C₂×C₃×#108	C₂×C₅	1	1	yes	characteristic
34	#5106	C₂×C₅	1	1	yes	characteristic
35	C₂²×C₃⋊C₁₆	C₂×C₅	1	1	yes	characteristic
36	C₂×C₅×M₁₆	—	2	1	no	—
37	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
38	C₅×#117	—	6	1	no	—
39	C₂×C₅×C₁₆	—	3	2	no	—
40	C₂×C₅×C₁₆	—	6	1	no	—
41	C₂×C₅×C₁₆	—	6	1	no	—
42	C₅×#108	—	2	1	no	—
43	C₅×#108	Dic₃	1	2	yes	complemented
44	C₅×#107	—	3	2	no	—
45	C₅×#106	—	6	1	no	—
46	#1382	—	3	1	no	Sylow
47	C₂³×C₃×C₅	SD₁₆	1	1	yes	characteristic
48	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	SD₁₆	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	Q₁₆	1	1	yes	characteristic
54	C₂×C₅×Dic₃	—	4	2	no	—
55	C₃×C₅×C₈	—	4	2	no	—
56	C₃×C₅×C₈	C₄²	1	1	yes	characteristic
57	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
58	C₃×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
59	C₃×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₃×M₁₆	—	2	1	no	—
61	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
62	#926	—	2	1	no	—
63	C₃×#108	—	2	1	no	—
64	C₃×#108	C₄×C₅	1	2	yes	complemented
65	#818	C₄×C₅	1	2	yes	—
66	#817	—	2	1	no	—
67	C₂×C₃⋊C₁₆	—	2	1	no	—
68	C₂×C₃⋊C₁₆	—	2	1	no	—
69	C₂×C₃⋊C₁₆	C₄×C₅	1	2	yes	complemented
70	C₂²×C₄×C₅	#67	1	1	yes	characteristic
71	C₅×M₁₆	—	2	2	no	—
72	C₅×M₁₆	—	4	1	no	—
73	C₂×C₅×C₈	—	4	1	no	—
74	C₂×C₅×C₈	Dic₆	1	1	yes	characteristic
75	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
76	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
77	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
78	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
79	C₂×C₅×C₈	#67	1	1	yes	characteristic
80	C₅×C₄⋊C₄	—	6	2	no	—
81	C₅×C₂²⋊C₄	—	12	1	no	—
82	C₄²×C₅	—	12	1	no	—
83	C₅×C₁₆	—	6	2	no	—
84	C₂²×C₁₆	—	3	1	no	—
85	C₂×#108	C₅×D₃	1	1	yes	characteristic
86	#427	—	3	1	no	—
87	C₂²×C₃×C₅	—	2	2	no	—
88	C₂²×C₃×C₅	#102	1	1	yes	characteristic
89	C₂²×C₃×C₅	#103	1	1	yes	characteristic
90	C₂²×C₃×C₅	#106	1	1	yes	characteristic
91	C₃×C₄×C₅	#95	1	1	yes	characteristic
92	C₃×C₄×C₅	#102	1	1	yes	characteristic
93	C₃×C₄×C₅	#103	1	1	yes	characteristic
94	C₃×C₄×C₅	#107	1	1	yes	characteristic
95	C₅×Dic₃	—	8	2	no	—
96	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
97	C₃×M₁₆	—	2	2	no	—
98	C₃×M₁₆	—	4	1	no	—
99	C₂×C₃×C₈	—	4	1	no	—
100	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
101	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
102	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
103	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
104	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
105	C₂×C₃×C₈	C₅×Q₈	1	1	yes	characteristic
106	#217	—	4	1	no	—
107	C₄⋊Dic₃	—	2	2	no	—
108	C₄×Dic₃	—	4	1	no	—
109	C₃⋊C₁₆	—	2	2	no	—
110	C₂³×C₅	#215	1	1	yes	characteristic
111	C₂×C₄×C₅	—	12	2	no	—
112	C₂×C₄×C₅	Dic₃⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅	#213	1	1	yes	characteristic
115	C₂×C₄×C₅	#214	1	1	yes	characteristic
116	C₂×C₄×C₅	#216	1	1	yes	characteristic
117	C₂×C₄×C₅	#217	1	1	yes	characteristic
118	C₅×C₈	—	4	2	no	—
119	C₅×C₈	C₄×Dic₃	1	1	yes	characteristic
120	C₅×C₈	Dic₃⋊C₄	1	1	yes	characteristic
121	C₅×C₈	C₄⋊Dic₃	1	1	yes	characteristic
122	C₅×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
123	C₂×M₁₆	—	2	1	no	—
124	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
125	#117	—	6	1	no	—
126	C₂×C₁₆	—	3	2	no	—
127	C₂×C₁₆	—	6	1	no	—
128	C₂×C₁₆	—	6	1	no	—
129	#108	—	2	1	no	—
130	#108	C₅×Dic₃	1	2	yes	complemented
131	#107	—	3	2	no	—
132	#106	—	6	1	no	—
133	C₂×C₃×C₅	—	2	1	no	—
134	C₂×C₃×C₅	—	2	1	no	—
135	C₂×C₃×C₅	#340	1	1	yes	characteristic
136	C₂×C₃×C₅	#359	1	2	yes	—
137	C₂³×C₃	C₅×SD₁₆	1	1	yes	characteristic
138	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
139	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
142	C₂×C₃×C₄	C₅×SD₁₆	1	1	yes	characteristic
143	C₂×C₃×C₄	C₅×Q₁₆	1	1	yes	characteristic
144	C₂×Dic₃	—	4	2	no	—
145	C₃×C₈	—	4	2	no	—
146	C₃×C₈	C₄²×C₅	1	1	yes	characteristic
147	C₃×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
148	C₃×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
149	C₃×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
150	C₂²×C₅	—	2	2	no	—
151	C₂²×C₅	#808	1	1	yes	characteristic
152	C₂²×C₅	#809	1	1	yes	characteristic
153	C₂²×C₅	#835	1	1	yes	characteristic
154	C₄×C₅	—	24	2	no	—
155	C₄×C₅	#807	1	1	yes	characteristic
156	C₄×C₅	#810	1	1	yes	characteristic
157	C₄×C₅	C₃⋊#95	1	1	yes	characteristic
158	C₄×C₅	#832	1	1	yes	characteristic
159	C₂²×C₄	C₅×#67	1	1	yes	characteristic
160	M₁₆	—	2	2	no	—
161	M₁₆	—	4	1	no	—
162	C₂×C₈	—	4	1	no	—
163	C₂×C₈	C₅×Dic₆	1	1	yes	characteristic
164	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
165	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
166	C₂×C₈	C₅×D₁₂	1	1	yes	characteristic
167	C₂×C₈	C₂×C₅×Dic₃	1	1	yes	characteristic
168	C₂×C₈	C₅×#67	1	1	yes	characteristic
169	C₄⋊C₄	—	6	2	no	—
170	C₂²⋊C₄	—	12	1	no	—
171	C₄²	—	12	1	no	—
172	C₁₆	—	6	2	no	—
173	C₃×C₅	#1382	1	1	yes	characteristic, complemented, Hall
174	C₂²×C₃	—	2	2	no	—
175	C₂²×C₃	C₅×#102	1	1	yes	characteristic
176	C₂²×C₃	C₅×#103	1	1	yes	characteristic
177	C₂²×C₃	C₅×#106	1	1	yes	characteristic
178	C₃×C₄	C₅×#95	1	1	yes	characteristic
179	C₃×C₄	C₅×#102	1	1	yes	characteristic
180	C₃×C₄	C₅×#103	1	1	yes	characteristic
181	C₃×C₄	C₅×#107	1	1	yes	characteristic
182	Dic₃	—	8	2	no	—
183	C₂×C₅	—	2	1	no	—
184	C₂×C₅	—	2	1	no	—
185	C₂×C₅	#4497	1	2	yes	—
186	C₂×C₅	#4529	1	1	yes	characteristic
187	C₂³	C₅×#215	1	1	yes	characteristic
188	C₂×C₄	—	12	2	no	—
189	C₂×C₄	C₅×Dic₃⋊C₄	1	1	yes	characteristic
190	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
191	C₂×C₄	C₅×#213	1	1	yes	characteristic
192	C₂×C₄	C₅×#214	1	1	yes	characteristic
193	C₂×C₄	C₅×#216	1	1	yes	characteristic
194	C₂×C₄	C₅×#217	1	1	yes	characteristic
195	C₈	—	4	2	no	—
196	C₈	C₄×C₅×Dic₃	1	1	yes	characteristic
197	C₈	C₅×Dic₃⋊C₄	1	1	yes	characteristic
198	C₈	C₅×C₄⋊Dic₃	1	1	yes	characteristic
199	C₈	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
200	C₂×C₃	—	2	1	no	—
201	C₂×C₃	—	2	1	no	—
202	C₂×C₃	C₅×#340	1	1	yes	characteristic
203	C₂×C₃	C₅×#359	1	2	yes	—
204	C₅	#67729	1	1	yes	characteristic, direct product factor, Sylow
205	C₂²	—	2	2	no	—
206	C₂²	C₅×#808	1	1	yes	characteristic
207	C₂²	C₅×#809	1	1	yes	characteristic
208	C₂²	C₅×#835	1	1	yes	characteristic
209	C₄	—	24	2	no	—
210	C₄	C₅×#807	1	1	yes	characteristic
211	C₄	C₅×#810	1	1	yes	characteristic
212	C₄	C₅×C₃⋊#95	1	1	yes	characteristic
213	C₄	C₅×#832	1	1	yes	characteristic
214	C₃	C₅×#1382	1	1	yes	characteristic, complemented, Sylow
215	C₂	—	2	1	no	—
216	C₂	—	2	1	no	—
217	C₂	C₅×#4497	1	2	yes	—
218	C₂	C₅×#4529	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	#213
commutator	C₃×C₈	C₄²×C₅
Frattini	C₂²×C₈	C₂×C₅×D₃
Fitting	C₂×C₃×C₅×#108	C₂
socle	C₂²×C₃×C₅	#102
prime radical	C₂³×C₃×C₅	SD₁₆
cosocle	C₂²×C₃×C₈	C₂²×C₅
hypercenter	C₂×C₅×#108	D₃
hypocenter	C₃	C₅×#1382
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂²×C₅×C₈ ◁ C₂×C₅×#108

Complements

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

N	H ≅ G/N	type
9. C₃×C₅×#108	C₄	semidirect
14. C₂×C₅×C₃⋊C₁₆	C₄	semidirect
15. #67729	C₅	direct (N × H)
43. C₅×#108	Dic₃	semidirect
64. C₃×#108	C₄×C₅	semidirect
69. C₂×C₃⋊C₁₆	C₄×C₅	semidirect
130. #108	C₅×Dic₃	semidirect
173. C₃×C₅	#1382	semidirect
204. C₅	#67729	direct (N × H)
214. C₃	C₅×#1382	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×#108	2
C₅×#5106	2
C₂²×C₅×C₃⋊C₁₆	2
C₅×#1382	3
#67729	5

Subgroups — C5×#67729 #14278854

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#67729 #14278854