Subgroups — C₅×#67731 #14278918

Order 1920.

248 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	C₂×C₃×C₅×#108	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#819	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#812	C₂	1	1	yes	maximal, characteristic
4	C₅×#1385	—	3	1	no	maximal, Hall
5	C₂×C₃×C₅×M₁₆	—	2	1	no	—
6	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₅×#208	—	2	1	no	—
8	C₃×C₅×#108	—	2	1	no	—
9	C₃×C₅×#108	C₄	1	2	yes	—
10	C₅×#819	—	2	1	no	—
11	C₅×#819	C₄	1	2	yes	—
12	C₅×#812	—	2	1	no	—
13	C₅×#812	—	2	1	no	—
14	C₂×C₅×C₃⋊C₁₆	C₄	1	2	yes	—
15	#67731	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂×C₅×M₃₂	—	3	1	no	—
17	C₂×C₅×#108	—	3	1	no	—
18	C₂×C₅×#108	D₃	1	1	yes	characteristic
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×#208	—	2	2	no	—
30	C₅×#208	—	4	1	no	—
31	C₂×C₅×C₃⋊C₈	—	4	1	no	—
32	C₅×C₃⋊C₁₆	—	2	2	no	—
33	C₂×C₃×#108	C₂×C₅	1	1	yes	characteristic
34	C₂×#819	C₂×C₅	1	1	yes	characteristic
35	C₂×#812	C₂×C₅	1	1	yes	characteristic
36	C₂×C₅×M₁₆	—	2	1	no	—
37	C₂×C₅×M₁₆	—	6	1	no	—
38	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
39	C₅×M₃₂	—	6	1	no	—
40	C₅×M₃₂	—	6	1	no	—
41	C₂×C₅×C₁₆	—	3	2	no	—
42	C₅×#108	—	2	1	no	—
43	C₅×#108	—	3	2	no	—
44	C₅×#108	—	6	1	no	—
45	C₅×#108	Dic₃	1	2	yes	—
46	#1385	—	3	1	no	Sylow
47	C₂³×C₃×C₅	Q₁₆	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	SD₁₆	1	1	yes	characteristic
54	C₃×C₅×C₈	—	4	2	no	—
55	C₃×C₅×C₈	C₄²	1	1	yes	characteristic
56	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
57	C₃×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
58	C₃×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
59	C₅×C₃⋊C₈	—	4	2	no	—
60	C₂×C₃×M₁₆	—	2	1	no	—
61	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
62	C₂×#208	—	2	1	no	—
63	C₃×#108	—	2	1	no	—
64	C₃×#108	C₄×C₅	1	2	yes	—
65	#819	—	2	1	no	—
66	#819	C₄×C₅	1	2	yes	—
67	#812	—	2	1	no	—
68	#812	—	2	1	no	—
69	C₂×C₃⋊C₁₆	C₄×C₅	1	2	yes	—
70	C₂²×C₄×C₅	#67	1	1	yes	characteristic
71	C₅×M₁₆	—	2	2	no	—
72	C₅×M₁₆	—	4	1	no	—
73	C₅×M₁₆	—	6	2	no	—
74	C₅×M₁₆	—	12	1	no	—
75	C₂×C₅×C₈	—	4	1	no	—
76	C₂×C₅×C₈	—	12	1	no	—
77	C₂×C₅×C₈	Dic₆	1	1	yes	characteristic
78	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
79	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
80	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
81	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
82	C₂×C₅×C₈	#67	1	1	yes	characteristic
83	C₅×C₁₆	—	6	2	no	—
84	C₂×M₃₂	—	3	1	no	—
85	C₂×#108	—	3	1	no	—
86	C₂×#108	C₅×D₃	1	1	yes	characteristic
87	C₂²×C₃×C₅	—	4	1	no	—
88	C₂²×C₃×C₅	#103	1	1	yes	characteristic
89	C₂²×C₃×C₅	#103	1	1	yes	characteristic
90	C₂²×C₃×C₅	#107	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	#102	1	1	yes	characteristic
94	C₃×C₄×C₅	#106	1	1	yes	characteristic
95	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
96	C₃×M₁₆	—	2	2	no	—
97	C₃×M₁₆	—	4	1	no	—
98	C₂×C₃×C₈	—	4	1	no	—
99	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
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	C₅×D₄	1	1	yes	characteristic
103	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
104	C₂×C₃×C₈	C₅×Q₈	1	1	yes	characteristic
105	#208	—	2	2	no	—
106	#208	—	4	1	no	—
107	C₂×C₃⋊C₈	—	4	1	no	—
108	C₃⋊C₁₆	—	2	2	no	—
109	C₂³×C₅	#216	1	1	yes	characteristic
110	C₂×C₄×C₅	Dic₃⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₅	#213	1	1	yes	characteristic
113	C₂×C₄×C₅	#214	1	1	yes	characteristic
114	C₂×C₄×C₅	#215	1	1	yes	characteristic
115	C₂×C₄×C₅	#217	1	1	yes	characteristic
116	C₅×C₈	—	4	2	no	—
117	C₅×C₈	—	12	2	no	—
118	C₅×C₈	C₄×Dic₃	1	1	yes	characteristic
119	C₅×C₈	Dic₃⋊C₄	1	1	yes	characteristic
120	C₅×C₈	C₄⋊Dic₃	1	1	yes	characteristic
121	C₅×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
122	C₂×M₁₆	—	2	1	no	—
123	C₂×M₁₆	—	6	1	no	—
124	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
125	M₃₂	—	6	1	no	—
126	M₃₂	—	6	1	no	—
127	C₂×C₁₆	—	3	2	no	—
128	#108	—	2	1	no	—
129	#108	—	3	2	no	—
130	#108	—	6	1	no	—
131	#108	C₅×Dic₃	1	2	yes	—
132	C₂×C₃×C₅	—	2	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	C₂³×C₃	C₅×Q₁₆	1	1	yes	characteristic
137	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
138	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
139	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
141	C₂×C₃×C₄	C₅×SD₁₆	1	1	yes	characteristic
142	C₂×C₃×C₄	C₅×SD₁₆	1	1	yes	characteristic
143	C₃×C₈	—	4	2	no	—
144	C₃×C₈	C₄²×C₅	1	1	yes	characteristic
145	C₃×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
146	C₃×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
147	C₃×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
148	C₃⋊C₈	—	4	2	no	—
149	C₂²×C₅	—	4	1	no	—
150	C₂²×C₅	#807	1	1	yes	characteristic
151	C₂²×C₅	#810	1	1	yes	characteristic
152	C₂²×C₅	#835	1	1	yes	characteristic
153	C₄×C₅	#808	1	1	yes	characteristic
154	C₄×C₅	#809	1	1	yes	characteristic
155	C₄×C₅	C₃⋊#95	1	1	yes	characteristic
156	C₄×C₅	#832	1	1	yes	characteristic
157	C₂²×C₄	C₅×#67	1	1	yes	characteristic
158	M₁₆	—	2	2	no	—
159	M₁₆	—	4	1	no	—
160	M₁₆	—	6	2	no	—
161	M₁₆	—	12	1	no	—
162	C₂×C₈	—	4	1	no	—
163	C₂×C₈	—	12	1	no	—
164	C₂×C₈	C₅×Dic₆	1	1	yes	characteristic
165	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
166	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
167	C₂×C₈	C₅×D₁₂	1	1	yes	characteristic
168	C₂×C₈	C₂×C₅×Dic₃	1	1	yes	characteristic
169	C₂×C₈	C₅×#67	1	1	yes	characteristic
170	C₁₆	—	6	2	no	—
171	C₃×C₅	#1385	1	1	yes	characteristic, complemented, Hall
172	C₂²×C₃	—	4	1	no	—
173	C₂²×C₃	C₅×#103	1	1	yes	characteristic
174	C₂²×C₃	C₅×#103	1	1	yes	characteristic
175	C₂²×C₃	C₅×#107	1	1	yes	characteristic
176	C₃×C₄	C₅×#95	1	1	yes	characteristic
177	C₃×C₄	C₅×#102	1	1	yes	characteristic
178	C₃×C₄	C₅×#102	1	1	yes	characteristic
179	C₃×C₄	C₅×#106	1	1	yes	characteristic
180	C₂×C₅	—	2	1	no	—
181	C₂×C₅	—	2	1	no	—
182	C₂×C₅	—	2	1	no	—
183	C₂×C₅	#4529	1	1	yes	characteristic
184	C₂³	C₅×#216	1	1	yes	characteristic
185	C₂×C₄	C₅×Dic₃⋊C₄	1	1	yes	characteristic
186	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
187	C₂×C₄	C₅×#213	1	1	yes	characteristic
188	C₂×C₄	C₅×#214	1	1	yes	characteristic
189	C₂×C₄	C₅×#215	1	1	yes	characteristic
190	C₂×C₄	C₅×#217	1	1	yes	characteristic
191	C₈	—	4	2	no	—
192	C₈	—	12	2	no	—
193	C₈	C₄×C₅×Dic₃	1	1	yes	characteristic
194	C₈	C₅×Dic₃⋊C₄	1	1	yes	characteristic
195	C₈	C₅×C₄⋊Dic₃	1	1	yes	characteristic
196	C₈	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
197	C₂×C₃	—	2	1	no	—
198	C₂×C₃	—	2	1	no	—
199	C₂×C₃	—	2	1	no	—
200	C₂×C₃	C₅×#340	1	1	yes	characteristic
201	C₅	#67731	1	1	yes	characteristic, direct product factor, Sylow
202	C₂²	—	4	1	no	—
203	C₂²	C₅×#807	1	1	yes	characteristic
204	C₂²	C₅×#810	1	1	yes	characteristic
205	C₂²	C₅×#835	1	1	yes	characteristic
206	C₄	C₅×#808	1	1	yes	characteristic
207	C₄	C₅×#809	1	1	yes	characteristic
208	C₄	C₅×C₃⋊#95	1	1	yes	characteristic
209	C₄	C₅×#832	1	1	yes	characteristic
210	C₃	C₅×#1385	1	1	yes	characteristic, complemented, Sylow
211	C₂	—	2	1	no	—
212	C₂	—	2	1	no	—
213	C₂	—	2	1	no	—
214	C₂	C₅×#4529	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	#808
commutator	C₃×C₈	C₄²×C₅
Frattini	C₂²×C₈	C₂×C₅×D₃
Fitting	C₂×C₃×C₅×#108	C₂
socle	C₂×C₃×C₅	#340
prime radical	C₂³×C₃×C₅	Q₁₆
cosocle	C₂²×C₃×C₈	C₂²×C₅
hypercenter	C₂×C₅×#108	D₃
hypocenter	C₃	C₅×#1385
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ 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
15. #67731	C₅	direct (N × H)
171. C₃×C₅	#1385	semidirect
201. C₅	#67731	direct (N × H)
210. C₃	C₅×#1385	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
C₂×C₅×#819	2
C₂×C₅×#812	2
C₅×#1385	3
#67731	5

Subgroups — C5×#67731 #14278918

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#67731 #14278918