Subgroups — C₅×#71420 #14314325

Order 1920.

344 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#812	C₂	1	1	yes	maximal, characteristic
4	C₅×Dic₃⋊C₁₆	C₂	1	4	yes	maximal, complemented
5	C₅×#2150	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₅×Dic₃	C₄	1	1	yes	characteristic
8	C₂²×C₅×C₃⋊C₈	C₄	1	1	yes	characteristic
9	C₃×C₅×M₃₂	—	2	2	no	—
10	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
11	C₅×C₈×Dic₃	C₄	1	2	yes	—
12	C₅×C₈×Dic₃	C₂²	1	2	yes	—
13	C₅×#812	—	2	2	no	—
14	C₂×C₅×C₃⋊C₁₆	C₂²	1	2	yes	—
15	#71420	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂×C₅×M₃₂	—	3	1	no	—
17	C₂×C₅×M₃₂	D₃	1	1	yes	characteristic
18	C₂×C₄×C₅×C₈	—	3	1	no	—
19	C₅×C₄⋊C₁₆	—	3	4	no	—
20	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₅×Dic₃	C₈	1	2	yes	—
22	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
24	C₂×C₃×C₅×C₈	Q₈	1	2	yes	—
25	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
26	C₄×C₅×Dic₃	C₈	1	1	yes	characteristic
27	C₄×C₅×Dic₃	C₈	1	1	yes	characteristic
28	C₄×C₅×Dic₃	C₂×C₄	1	2	yes	—
29	C₂×C₅×C₃⋊C₈	—	2	2	no	—
30	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
31	C₃×C₅×C₁₆	—	2	2	no	—
32	C₅×C₃⋊C₁₆	—	2	2	no	—
33	C₂×C₃×M₃₂	C₂×C₅	1	1	yes	characteristic
34	C₂×C₈×Dic₃	C₂×C₅	1	1	yes	characteristic
35	C₂×#812	C₂×C₅	1	1	yes	characteristic
36	Dic₃⋊C₁₆	C₂×C₅	1	4	yes	complemented
37	C₂²×C₅×C₈	—	3	1	no	—
38	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
39	C₂×C₄²×C₅	—	3	1	no	—
40	C₅×M₃₂	—	2	2	no	—
41	C₅×M₃₂	—	6	2	no	—
42	C₂×C₅×C₁₆	—	3	2	no	—
43	C₂×C₅×C₁₆	C₂×D₃	1	2	yes	—
44	C₄×C₅×C₈	—	3	2	no	—
45	C₄×C₅×C₈	—	3	2	no	—
46	#2150	—	3	1	no	Sylow
47	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
48	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
49	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
51	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₅×Dic₃	—	2	4	no	—
53	C₂×C₅×Dic₃	C₂×C₈	1	2	yes	—
54	C₂×C₅×Dic₃	C₂×C₈	1	2	yes	—
55	C₃×C₅×C₈	C₄⋊C₄	1	2	yes	—
56	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
57	C₃×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
58	C₅×C₃⋊C₈	—	2	2	no	—
59	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
60	C₂×C₄×Dic₃	C₄×C₅	1	1	yes	characteristic
61	C₂²×C₃⋊C₈	C₄×C₅	1	1	yes	characteristic
62	C₃×M₃₂	—	2	2	no	—
63	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	—
64	C₈×Dic₃	C₄×C₅	1	2	yes	—
65	C₈×Dic₃	C₂²×C₅	1	2	yes	—
66	#812	—	2	2	no	—
67	C₂×C₃⋊C₁₆	C₂²×C₅	1	2	yes	—
68	C₂²×C₄×C₅	—	3	2	no	—
69	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
70	C₂×C₅×C₈	—	3	2	no	—
71	C₂×C₅×C₈	—	6	2	no	—
72	C₂×C₅×C₈	Dic₆	1	2	yes	—
73	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
74	C₂×C₅×C₈	#67	1	2	yes	—
75	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
76	C₄²×C₅	—	3	1	no	—
77	C₄²×C₅	—	3	1	no	—
78	C₄²×C₅	—	3	2	no	—
79	C₅×C₁₆	—	2	2	no	—
80	C₅×C₁₆	—	6	2	no	—
81	C₂×M₃₂	—	3	1	no	—
82	C₂×M₃₂	C₅×D₃	1	1	yes	characteristic
83	C₂×C₄×C₈	—	3	1	no	—
84	C₄⋊C₁₆	—	3	4	no	—
85	C₂²×C₃×C₅	—	2	2	no	—
86	C₂²×C₃×C₅	C₄⋊C₈	1	2	yes	—
87	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
88	C₃×C₄×C₅	C₄⋊C₈	1	2	yes	—
89	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
91	C₅×Dic₃	—	2	2	no	—
92	C₅×Dic₃	M₃₂	1	4	yes	complemented
93	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
94	C₂²×Dic₃	C₅×C₈	1	2	yes	—
95	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
96	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
97	C₂×C₃×C₈	C₅×Q₈	1	2	yes	—
98	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
99	C₄×Dic₃	C₅×C₈	1	1	yes	characteristic
100	C₄×Dic₃	C₅×C₈	1	1	yes	characteristic
101	C₄×Dic₃	C₂×C₄×C₅	1	2	yes	—
102	C₂×C₃⋊C₈	—	2	2	no	—
103	C₂×C₃⋊C₈	C₂×C₄×C₅	1	2	yes	—
104	C₃×C₁₆	—	2	2	no	—
105	C₃⋊C₁₆	—	2	2	no	—
106	C₂³×C₅	D₃×C₈	1	1	yes	characteristic
107	C₂×C₄×C₅	—	3	2	no	—
108	C₂×C₄×C₅	—	3	2	no	—
109	C₂×C₄×C₅	—	6	4	no	—
110	C₂×C₄×C₅	D₃×C₈	1	1	yes	characteristic
111	C₂×C₄×C₅	ΓL(1,𝔽₂₅)	1	2	yes	—
112	C₂×C₄×C₅	Dic₃⋊C₄	1	2	yes	—
113	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
114	C₅×C₈	—	6	2	no	—
115	C₅×C₈	Dic₃⋊C₄	1	2	yes	—
116	C₅×C₈	C₂×Dic₆	1	1	yes	characteristic
117	C₅×C₈	C₂×#67	1	1	yes	characteristic
118	C₂²×C₈	—	3	1	no	—
119	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
120	C₂×C₄²	—	3	1	no	—
121	M₃₂	—	2	2	no	—
122	M₃₂	—	6	2	no	—
123	C₂×C₁₆	—	3	2	no	—
124	C₂×C₁₆	C₂×C₅×D₃	1	2	yes	—
125	C₄×C₈	—	3	2	no	—
126	C₄×C₈	—	3	2	no	—
127	C₂×C₃×C₅	—	2	2	no	—
128	C₂×C₃×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
129	C₂×C₃×C₅	C₂×M₃₂	1	2	yes	—
130	C₂³×C₃	C₂×C₅×C₈	1	1	yes	characteristic
131	C₂×C₃×C₄	C₅×C₄⋊C₄	1	2	yes	—
132	C₂×C₃×C₄	C₂×C₅×C₈	1	1	yes	characteristic
133	C₂×C₃×C₄	C₅×M₁₆	1	2	yes	—
134	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
135	C₂×Dic₃	—	2	4	no	—
136	C₂×Dic₃	C₂×C₅×C₈	1	2	yes	—
137	C₂×Dic₃	C₂×C₅×C₈	1	2	yes	—
138	C₃×C₈	C₅×C₄⋊C₄	1	2	yes	—
139	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
140	C₃×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
141	C₃⋊C₈	—	2	2	no	—
142	C₂²×C₅	—	2	2	no	—
143	C₂²×C₅	Dic₃⋊C₈	1	2	yes	—
144	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
145	C₄×C₅	—	3	4	no	—
146	C₄×C₅	—	6	2	no	—
147	C₄×C₅	Dic₃⋊C₈	1	2	yes	—
148	C₄×C₅	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
149	C₄×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
150	C₂²×C₄	—	3	2	no	—
151	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
152	C₂×C₈	—	3	2	no	—
153	C₂×C₈	—	6	2	no	—
154	C₂×C₈	C₅×Dic₆	1	2	yes	—
155	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
156	C₂×C₈	C₅×#67	1	2	yes	—
157	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
158	C₄²	—	3	1	no	—
159	C₄²	—	3	1	no	—
160	C₄²	—	3	2	no	—
161	C₁₆	—	2	2	no	—
162	C₁₆	—	6	2	no	—
163	C₃×C₅	#2150	1	1	yes	characteristic, complemented, Hall
164	C₂²×C₃	—	2	2	no	—
165	C₂²×C₃	C₅×C₄⋊C₈	1	2	yes	—
166	C₂²×C₃	C₂²×C₅×C₈	1	1	yes	characteristic
167	C₃×C₄	C₅×C₄⋊C₈	1	2	yes	—
168	C₃×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
169	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
170	Dic₃	—	2	2	no	—
171	Dic₃	C₅×M₃₂	1	4	yes	complemented
172	C₂×C₅	—	2	2	no	—
173	C₂×C₅	D₃×M₃₂	1	2	yes	—
174	C₂×C₅	C₂×Dic₃⋊C₈	1	1	yes	characteristic
175	C₂³	C₅×D₃×C₈	1	1	yes	characteristic
176	C₂×C₄	—	3	2	no	—
177	C₂×C₄	—	3	2	no	—
178	C₂×C₄	—	6	4	no	—
179	C₂×C₄	C₅×D₃×C₈	1	1	yes	characteristic
180	C₂×C₄	C₅×ΓL(1,𝔽₂₅)	1	2	yes	—
181	C₂×C₄	C₅×Dic₃⋊C₄	1	2	yes	—
182	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
183	C₈	—	6	2	no	—
184	C₈	C₅×Dic₃⋊C₄	1	2	yes	—
185	C₈	C₂×C₅×Dic₆	1	1	yes	characteristic
186	C₈	C₂×C₅×#67	1	1	yes	characteristic
187	C₂×C₃	—	2	2	no	—
188	C₂×C₃	C₂×C₅×C₄⋊C₈	1	1	yes	characteristic
189	C₂×C₃	C₂×C₅×M₃₂	1	2	yes	—
190	C₅	#71420	1	1	yes	characteristic, direct product factor, Sylow
191	C₂²	—	2	2	no	—
192	C₂²	C₅×Dic₃⋊C₈	1	2	yes	—
193	C₂²	C₂×C₅×D₃×C₈	1	1	yes	characteristic
194	C₄	—	3	4	no	—
195	C₄	—	6	2	no	—
196	C₄	C₅×Dic₃⋊C₈	1	2	yes	—
197	C₄	C₂×C₅×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
198	C₄	C₂×C₅×Dic₃⋊C₄	1	1	yes	characteristic
199	C₃	C₅×#2150	1	1	yes	characteristic, complemented, Sylow
200	C₂	—	2	2	no	—
201	C₂	C₅×D₃×M₃₂	1	2	yes	—
202	C₂	C₂×C₅×Dic₃⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅×C₈	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₅×C₈
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₂×C₃×C₅×M₃₂	C₂
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂³×C₃×C₅	C₂×C₈
cosocle	C₂×C₃×C₈	C₂³×C₅
hypercenter	C₂×C₅×M₃₂	D₃
hypocenter	C₃	C₅×#2150
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅×C₈ ◁ C₂×C₅×M₃₂

Complements

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

N	H ≅ G/N	type
4. C₅×Dic₃⋊C₁₆	C₂	semidirect
15. #71420	C₅	direct (N × H)
36. Dic₃⋊C₁₆	C₂×C₅	semidirect
92. C₅×Dic₃	M₃₂	semidirect
163. C₃×C₅	#2150	semidirect
171. Dic₃	C₅×M₃₂	semidirect
190. C₅	#71420	direct (N × H)
199. C₃	C₅×#2150	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×M₃₂	2
C₂×C₅×C₈×Dic₃	2
C₂×C₅×#812	2
C₅×Dic₃⋊C₁₆	2
C₅×#2150	3
#71420	5

Subgroups — C5×#71420 #14314325

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71420 #14314325