Subgroups — C₅×#71404 #14314593

Order 1920.

304 conjugacy classes of subgroups.

The subgroup lattice has 168 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×#5106	C₂	1	2	yes	maximal
3	C₅×#4505	C₂	1	2	yes	maximal, complemented
4	C₅×#4504	C₂	1	2	yes	maximal, complemented
5	C₅×#2158	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₅×#926	—	2	2	no	—
8	C₂×C₃×C₅×C₁₆	C₄	1	4	yes	complemented
9	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
11	C₅×#818	C₂²	1	4	yes	—
12	C₅×#817	—	2	2	no	—
13	#71404	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂²×C₅×C₁₆	D₃	1	1	yes	characteristic
15	C₅×#427	—	3	2	no	—
16	C₅×#367	—	3	2	no	—
17	C₅×#366	—	3	2	no	—
18	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
19	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
20	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
21	C₂×C₃×C₅×C₈	Q₈	1	2	yes	—
22	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
23	C₅×#217	—	4	2	no	—
24	C₅×C₄⋊Dic₃	—	2	4	no	—
25	C₄×C₅×Dic₃	—	4	2	no	—
26	C₃×C₅×C₁₆	C₂×C₄	1	2	yes	—
27	C₃×C₅×C₁₆	C₂×C₄	1	2	yes	—
28	C₂²×C₃×C₁₆	C₂×C₅	1	1	yes	characteristic
29	#5106	C₂×C₅	1	2	yes	—
30	#4505	C₂×C₅	1	2	yes	complemented
31	#4504	C₂×C₅	1	2	yes	complemented
32	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
33	C₅×#117	—	6	2	no	—
34	C₂×C₅×C₁₆	Dic₃	1	4	yes	complemented
35	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
36	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
37	C₅×#107	—	3	4	no	—
38	C₅×#106	—	6	2	no	—
39	#2158	—	3	1	no	Sylow
40	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
41	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
42	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
43	C₂×C₃×C₄×C₅	Q₁₆	1	2	yes	—
44	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
45	C₂×C₅×Dic₃	—	4	4	no	—
46	C₃×C₅×C₈	C₄⋊C₄	1	2	yes	—
47	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
48	C₃×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
50	#926	—	2	2	no	—
51	C₂×C₃×C₁₆	C₄×C₅	1	4	yes	complemented
52	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
53	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
54	#818	C₂²×C₅	1	4	yes	—
55	#817	—	2	2	no	—
56	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
57	C₂×C₅×C₈	Dic₆	1	2	yes	—
58	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
59	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
60	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
61	C₅×C₄⋊C₄	—	6	4	no	—
62	C₅×C₂²⋊C₄	—	12	2	no	—
63	C₄²×C₅	—	12	2	no	—
64	C₅×C₁₆	C₂×Dic₃	1	2	yes	—
65	C₅×C₁₆	C₂×Dic₃	1	2	yes	—
66	C₂²×C₁₆	C₅×D₃	1	1	yes	characteristic
67	#427	—	3	2	no	—
68	#367	—	3	2	no	—
69	#366	—	3	2	no	—
70	C₂²×C₃×C₅	—	2	2	no	—
71	C₂²×C₃×C₅	#107	1	2	yes	—
72	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
73	C₃×C₄×C₅	#107	1	2	yes	—
74	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
76	C₅×Dic₃	—	8	4	no	—
77	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
78	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
79	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
80	C₂×C₃×C₈	C₅×Q₈	1	2	yes	—
81	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
82	#217	—	4	2	no	—
83	C₄⋊Dic₃	—	2	4	no	—
84	C₄×Dic₃	—	4	2	no	—
85	C₃×C₁₆	C₂×C₄×C₅	1	2	yes	—
86	C₃×C₁₆	C₂×C₄×C₅	1	2	yes	—
87	C₂³×C₅	D₂₄	1	1	yes	characteristic
88	C₂×C₄×C₅	—	12	4	no	—
89	C₂×C₄×C₅	D₂₄	1	1	yes	characteristic
90	C₂×C₄×C₅	Dic₁₂	1	2	yes	—
91	C₂×C₄×C₅	C₄⋊Dic₃	1	2	yes	—
92	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
93	C₅×C₈	C₄⋊Dic₃	1	2	yes	—
94	C₅×C₈	C₂×Dic₆	1	1	yes	characteristic
95	C₅×C₈	C₂²×Dic₃	1	1	yes	characteristic
96	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
97	#117	—	6	2	no	—
98	C₂×C₁₆	C₅×Dic₃	1	4	yes	complemented
99	C₂×C₁₆	C₂×C₅×D₃	1	1	yes	characteristic
100	C₂×C₁₆	C₂×C₅×D₃	1	1	yes	characteristic
101	#107	—	3	4	no	—
102	#106	—	6	2	no	—
103	C₂×C₃×C₅	—	2	2	no	—
104	C₂×C₃×C₅	C₂×#107	1	1	yes	characteristic
105	C₂×C₃×C₅	#508	1	2	yes	—
106	C₂³×C₃	C₅×D₈	1	1	yes	characteristic
107	C₂×C₃×C₄	C₅×C₄⋊C₄	1	2	yes	—
108	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
109	C₂×C₃×C₄	C₅×Q₁₆	1	2	yes	—
110	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
111	C₂×Dic₃	—	4	4	no	—
112	C₃×C₈	C₅×C₄⋊C₄	1	2	yes	—
113	C₃×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
114	C₃×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
115	C₂²×C₅	—	2	2	no	—
116	C₂²×C₅	#818	1	2	yes	—
117	C₂²×C₅	C₂×D₂₄	1	1	yes	characteristic
118	C₄×C₅	—	24	4	no	—
119	C₄×C₅	#818	1	2	yes	—
120	C₄×C₅	C₂×Dic₁₂	1	1	yes	characteristic
121	C₄×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
122	C₂²×C₄	C₅×D₁₂	1	1	yes	characteristic
123	C₂×C₈	C₅×Dic₆	1	2	yes	—
124	C₂×C₈	C₅×D₁₂	1	1	yes	characteristic
125	C₂×C₈	C₂×C₅×Dic₃	1	2	yes	—
126	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
127	C₄⋊C₄	—	6	4	no	—
128	C₂²⋊C₄	—	12	2	no	—
129	C₄²	—	12	2	no	—
130	C₁₆	C₂×C₅×Dic₃	1	2	yes	—
131	C₁₆	C₂×C₅×Dic₃	1	2	yes	—
132	C₃×C₅	#2158	1	1	yes	characteristic, complemented, Hall
133	C₂²×C₃	—	2	2	no	—
134	C₂²×C₃	C₅×#107	1	2	yes	—
135	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
136	C₃×C₄	C₅×#107	1	2	yes	—
137	C₃×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
138	C₃×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
139	Dic₃	—	8	4	no	—
140	C₂×C₅	—	2	2	no	—
141	C₂×C₅	#4904	1	2	yes	—
142	C₂×C₅	C₂×#818	1	1	yes	characteristic
143	C₂³	C₅×D₂₄	1	1	yes	characteristic
144	C₂×C₄	—	12	4	no	—
145	C₂×C₄	C₅×D₂₄	1	1	yes	characteristic
146	C₂×C₄	C₅×Dic₁₂	1	2	yes	—
147	C₂×C₄	C₅×C₄⋊Dic₃	1	2	yes	—
148	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
149	C₈	C₅×C₄⋊Dic₃	1	2	yes	—
150	C₈	C₂×C₅×Dic₆	1	1	yes	characteristic
151	C₈	C₂²×C₅×Dic₃	1	1	yes	characteristic
152	C₂×C₃	—	2	2	no	—
153	C₂×C₃	C₂×C₅×#107	1	1	yes	characteristic
154	C₂×C₃	C₅×#508	1	2	yes	—
155	C₅	#71404	1	1	yes	characteristic, direct product factor, Sylow
156	C₂²	—	2	2	no	—
157	C₂²	C₅×#818	1	2	yes	—
158	C₂²	C₂×C₅×D₂₄	1	1	yes	characteristic
159	C₄	—	24	4	no	—
160	C₄	C₅×#818	1	2	yes	—
161	C₄	C₂×C₅×Dic₁₂	1	1	yes	characteristic
162	C₄	C₂×C₅×C₄⋊Dic₃	1	1	yes	characteristic
163	C₃	C₅×#2158	1	1	yes	characteristic, complemented, Sylow
164	C₂	—	2	2	no	—
165	C₂	C₅×#4904	1	2	yes	—
166	C₂	C₂×C₅×#818	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	D₂₄
commutator	C₃×C₈	C₂²×C₄×C₅
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₂²×C₃×C₅×C₁₆	C₂
socle	C₂²×C₃×C₅	C₂×D₈
prime radical	C₂³×C₃×C₅	D₈
cosocle	C₂×C₃×C₈	C₂³×C₅
hypercenter	C₂²×C₅×C₁₆	D₃
hypocenter	C₃	C₅×#2158
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂²×C₅×C₈ ◁ C₂²×C₅×C₁₆

Complements

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

N	H ≅ G/N	type
3. C₅×#4505	C₂	semidirect
4. C₅×#4504	C₂	semidirect
8. C₂×C₃×C₅×C₁₆	C₄	semidirect
13. #71404	C₅	direct (N × H)
30. #4505	C₂×C₅	semidirect
31. #4504	C₂×C₅	semidirect
34. C₂×C₅×C₁₆	Dic₃	semidirect
51. C₂×C₃×C₁₆	C₄×C₅	semidirect
98. C₂×C₁₆	C₅×Dic₃	semidirect
132. C₃×C₅	#2158	semidirect
155. C₅	#71404	direct (N × H)
163. C₃	C₅×#2158	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×#5106	2
C₅×#4505	2
C₅×#4504	2
C₅×#2158	3
#71404	5

Subgroups — C5×#71404 #14314593

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71404 #14314593