Subgroups — C₅×#71401 #14314356

Order 1920.

320 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₅×#5101	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×#2151	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₅×#926	C₄	1	1	yes	characteristic
8	C₂×C₅×#208	C₄	1	1	yes	characteristic
9	C₂×C₃×C₅×C₁₆	—	2	2	no	—
10	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
11	C₅×#815	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	#71401	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂×C₅×M₃₂	—	3	1	no	—
17	C₂²×C₅×C₁₆	D₃	1	1	yes	characteristic
18	C₅×#405	—	3	1	no	—
19	C₅×C₄⋊C₁₆	—	3	4	no	—
20	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
23	C₂×C₃×C₅×C₈	Q₈	1	2	yes	—
24	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
25	C₅×#217	C₈	1	2	yes	—
26	C₅×C₄⋊Dic₃	C₈	1	2	yes	—
27	C₄×C₅×Dic₃	C₂×C₄	1	2	yes	—
28	C₅×#208	—	2	2	no	—
29	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
30	C₃×C₅×C₁₆	—	2	2	no	—
31	C₅×C₃⋊C₁₆	—	2	2	no	—
32	C₂²×C₃×C₁₆	C₂×C₅	1	1	yes	characteristic
33	#5101	C₂×C₅	1	1	yes	characteristic
34	C₂×#812	C₂×C₅	1	1	yes	characteristic
35	Dic₃⋊C₁₆	C₂×C₅	1	4	yes	complemented
36	C₂×C₅×M₁₆	—	3	1	no	—
37	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
38	C₅×#117	—	3	1	no	—
39	C₅×M₃₂	—	6	2	no	—
40	C₂×C₅×C₁₆	—	2	2	no	—
41	C₂×C₅×C₁₆	—	3	2	no	—
42	C₂×C₅×C₁₆	C₂×D₃	1	2	yes	—
43	C₅×#97	—	3	2	no	—
44	C₄×C₅×C₈	—	3	2	no	—
45	#2151	—	3	1	no	Sylow
46	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
47	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
48	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
50	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₅×Dic₃	C₂×C₈	1	4	yes	—
52	C₃×C₅×C₈	C₄⋊C₄	1	2	yes	—
53	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
54	C₃×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
55	C₅×C₃⋊C₈	—	2	2	no	—
56	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
57	#926	C₄×C₅	1	1	yes	characteristic
58	C₂×#208	C₄×C₅	1	1	yes	characteristic
59	C₂×C₃×C₁₆	—	2	2	no	—
60	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	—
61	#815	C₄×C₅	1	2	yes	—
62	C₈×Dic₃	C₂²×C₅	1	2	yes	—
63	#812	—	2	2	no	—
64	C₂×C₃⋊C₁₆	C₂²×C₅	1	2	yes	—
65	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
66	C₅×M₁₆	—	6	2	no	—
67	C₂×C₅×C₈	—	3	2	no	—
68	C₂×C₅×C₈	Dic₆	1	2	yes	—
69	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
70	C₂×C₅×C₈	#67	1	2	yes	—
71	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
72	C₅×C₄⋊C₄	—	3	2	no	—
73	C₅×C₂²⋊C₄	—	3	2	no	—
74	C₄²×C₅	—	3	2	no	—
75	C₅×C₁₆	—	2	2	no	—
76	C₅×C₁₆	—	6	2	no	—
77	C₂×M₃₂	—	3	1	no	—
78	C₂²×C₁₆	C₅×D₃	1	1	yes	characteristic
79	#405	—	3	1	no	—
80	C₄⋊C₁₆	—	3	4	no	—
81	C₂²×C₃×C₅	—	2	2	no	—
82	C₂²×C₃×C₅	C₄⋊C₈	1	2	yes	—
83	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
84	C₃×C₄×C₅	C₄⋊C₈	1	2	yes	—
85	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
87	C₅×Dic₃	—	2	4	no	—
88	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
89	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
90	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
91	C₂×C₃×C₈	C₅×Q₈	1	2	yes	—
92	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
93	#217	C₅×C₈	1	2	yes	—
94	C₄⋊Dic₃	C₅×C₈	1	2	yes	—
95	C₄×Dic₃	C₂×C₄×C₅	1	2	yes	—
96	#208	—	2	2	no	—
97	C₂×C₃⋊C₈	C₂×C₄×C₅	1	2	yes	—
98	C₃×C₁₆	—	2	2	no	—
99	C₃⋊C₁₆	—	2	2	no	—
100	C₂³×C₅	D₃×C₈	1	1	yes	characteristic
101	C₂×C₄×C₅	—	3	4	no	—
102	C₂×C₄×C₅	D₃×C₈	1	1	yes	characteristic
103	C₂×C₄×C₅	ΓL(1,𝔽₂₅)	1	2	yes	—
104	C₂×C₄×C₅	Dic₃⋊C₄	1	2	yes	—
105	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
106	C₅×C₈	—	6	2	no	—
107	C₅×C₈	Dic₃⋊C₄	1	2	yes	—
108	C₅×C₈	C₂×Dic₆	1	1	yes	characteristic
109	C₅×C₈	C₂×#67	1	1	yes	characteristic
110	C₂×M₁₆	—	3	1	no	—
111	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
112	#117	—	3	1	no	—
113	M₃₂	—	6	2	no	—
114	C₂×C₁₆	—	2	2	no	—
115	C₂×C₁₆	—	3	2	no	—
116	C₂×C₁₆	C₂×C₅×D₃	1	2	yes	—
117	#97	—	3	2	no	—
118	C₄×C₈	—	3	2	no	—
119	C₂×C₃×C₅	—	2	2	no	—
120	C₂×C₃×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
121	C₂×C₃×C₅	#504	1	2	yes	—
122	C₂³×C₃	C₂×C₅×C₈	1	1	yes	characteristic
123	C₂×C₃×C₄	C₅×C₄⋊C₄	1	2	yes	—
124	C₂×C₃×C₄	C₂×C₅×C₈	1	1	yes	characteristic
125	C₂×C₃×C₄	C₅×M₁₆	1	2	yes	—
126	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
127	C₂×Dic₃	C₂×C₅×C₈	1	4	yes	—
128	C₃×C₈	C₅×C₄⋊C₄	1	2	yes	—
129	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
130	C₃×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
131	C₃⋊C₈	—	2	2	no	—
132	C₂²×C₅	—	2	2	no	—
133	C₂²×C₅	Dic₃⋊C₈	1	2	yes	—
134	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
135	C₄×C₅	—	6	4	no	—
136	C₄×C₅	Dic₃⋊C₈	1	2	yes	—
137	C₄×C₅	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
138	C₄×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
139	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
140	M₁₆	—	6	2	no	—
141	C₂×C₈	—	3	2	no	—
142	C₂×C₈	C₅×Dic₆	1	2	yes	—
143	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
144	C₂×C₈	C₅×#67	1	2	yes	—
145	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
146	C₄⋊C₄	—	3	2	no	—
147	C₂²⋊C₄	—	3	2	no	—
148	C₄²	—	3	2	no	—
149	C₁₆	—	2	2	no	—
150	C₁₆	—	6	2	no	—
151	C₃×C₅	#2151	1	1	yes	characteristic, complemented, Hall
152	C₂²×C₃	—	2	2	no	—
153	C₂²×C₃	C₅×C₄⋊C₈	1	2	yes	—
154	C₂²×C₃	C₂²×C₅×C₈	1	1	yes	characteristic
155	C₃×C₄	C₅×C₄⋊C₈	1	2	yes	—
156	C₃×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
157	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
158	Dic₃	—	2	4	no	—
159	C₂×C₅	—	2	2	no	—
160	C₂×C₅	#4901	1	2	yes	—
161	C₂×C₅	C₂×Dic₃⋊C₈	1	1	yes	characteristic
162	C₂³	C₅×D₃×C₈	1	1	yes	characteristic
163	C₂×C₄	—	3	4	no	—
164	C₂×C₄	C₅×D₃×C₈	1	1	yes	characteristic
165	C₂×C₄	C₅×ΓL(1,𝔽₂₅)	1	2	yes	—
166	C₂×C₄	C₅×Dic₃⋊C₄	1	2	yes	—
167	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
168	C₈	—	6	2	no	—
169	C₈	C₅×Dic₃⋊C₄	1	2	yes	—
170	C₈	C₂×C₅×Dic₆	1	1	yes	characteristic
171	C₈	C₂×C₅×#67	1	1	yes	characteristic
172	C₂×C₃	—	2	2	no	—
173	C₂×C₃	C₂×C₅×C₄⋊C₈	1	1	yes	characteristic
174	C₂×C₃	C₅×#504	1	2	yes	—
175	C₅	#71401	1	1	yes	characteristic, direct product factor, Sylow
176	C₂²	—	2	2	no	—
177	C₂²	C₅×Dic₃⋊C₈	1	2	yes	—
178	C₂²	C₂×C₅×D₃×C₈	1	1	yes	characteristic
179	C₄	—	6	4	no	—
180	C₄	C₅×Dic₃⋊C₈	1	2	yes	—
181	C₄	C₂×C₅×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
182	C₄	C₂×C₅×Dic₃⋊C₄	1	1	yes	characteristic
183	C₃	C₅×#2151	1	1	yes	characteristic, complemented, Sylow
184	C₂	—	2	2	no	—
185	C₂	C₅×#4901	1	2	yes	—
186	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	C₂²×C₃×C₅×C₁₆	C₂
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂³×C₃×C₅	C₂×C₈
cosocle	C₂×C₃×C₈	C₂³×C₅
hypercenter	C₂²×C₅×C₁₆	D₃
hypocenter	C₃	C₅×#2151
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅×C₈ ◁ C₂²×C₅×C₁₆

Complements

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

N	H ≅ G/N	type
4. C₅×Dic₃⋊C₁₆	C₂	semidirect
15. #71401	C₅	direct (N × H)
35. Dic₃⋊C₁₆	C₂×C₅	semidirect
151. C₃×C₅	#2151	semidirect
175. C₅	#71401	direct (N × H)
183. C₃	C₅×#2151	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₅×C₁₆	2
C₅×#5101	2
C₂×C₅×#812	2
C₅×Dic₃⋊C₁₆	2
C₅×#2151	3
#71401	5

Subgroups — C5×#71401 #14314356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71401 #14314356