Subgroups — C₅×#69806 #14294929

Order 1920.

480 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#97	C₂	1	1	yes	maximal, characteristic
2	C₅×#4932	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₅×#4550	C₂	1	4	yes	maximal
5	C₅×#1746	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₅×Dic₃	C₂²	1	2	yes	—
9	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
10	C₃×C₅×#97	C₄	1	4	yes	complemented
11	C₅×C₃⋊#95	C₄	1	4	yes	—
12	C₄×C₅×C₃⋊C₈	C₄	1	4	yes	complemented
13	#69806	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂×C₅×#97	D₃	1	1	yes	characteristic
15	C₂×C₄×C₅×C₈	—	3	1	no	—
16	C₅×#376	—	3	1	no	—
17	C₅×#336	—	3	4	no	—
18	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
19	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
20	C₂²×C₅×Dic₃	C₂×C₄	1	4	yes	—
21	C₂×C₃×C₅×C₈	—	2	4	no	—
22	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	—
23	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
24	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
25	C₄×C₅×Dic₃	—	2	4	no	—
26	C₂×C₅×C₃⋊C₈	—	2	4	no	—
27	C₂×C₅×C₃⋊C₈	C₂×C₄	1	4	yes	—
28	C₂×C₃×#97	C₂×C₅	1	1	yes	characteristic
29	#4932	C₂×C₅	1	1	yes	characteristic
30	C₂×C₄×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic
31	#4550	C₂×C₅	1	4	yes	—
32	C₂²×C₅×C₈	—	3	2	no	—
33	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
34	C₂×C₄²×C₅	—	3	2	no	—
35	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
36	C₅×#97	Dic₃	1	4	yes	complemented
37	C₄×C₅×C₈	—	3	4	no	—
38	C₅×#95	—	3	4	no	—
39	#1746	—	3	1	no	Sylow
40	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₃×C₄×C₅	—	2	4	no	—
42	C₂×C₃×C₄×C₅	C₄²	1	4	yes	—
43	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₃×C₄×C₅	#41	1	4	yes	—
46	C₂×C₅×Dic₃	—	2	4	no	—
47	C₂×C₅×Dic₃	—	2	8	no	—
48	C₃×C₅×C₈	—	2	4	no	—
49	C₅×C₃⋊C₈	—	2	4	no	—
50	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
51	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
52	C₂×C₄×Dic₃	C₂²×C₅	1	2	yes	—
53	C₂²×C₃⋊C₈	C₂²×C₅	1	2	yes	—
54	C₃×#97	C₄×C₅	1	4	yes	complemented
55	C₃⋊#95	C₄×C₅	1	4	yes	—
56	C₄×C₃⋊C₈	C₄×C₅	1	4	yes	complemented
57	C₂²×C₄×C₅	—	3	4	no	—
58	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
59	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
60	C₂×C₅×C₈	—	2	4	no	—
61	C₂×C₅×C₈	—	3	4	no	—
62	C₂×C₅×C₈	—	6	4	no	—
63	C₂×C₅×C₈	C₂×Dic₃	1	4	yes	—
64	C₄²×C₅	—	6	4	no	—
65	C₄²×C₅	C₄×D₃	1	2	yes	—
66	C₄²×C₅	C₂×Dic₃	1	2	yes	—
67	C₂×#97	C₅×D₃	1	1	yes	characteristic
68	C₂×C₄×C₈	—	3	1	no	—
69	#376	—	3	1	no	—
70	#336	—	3	4	no	—
71	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
72	C₂²×C₃×C₅	#117	1	2	yes	—
73	C₂²×C₃×C₅	#131	1	4	yes	—
74	C₃×C₄×C₅	—	2	4	no	—
75	C₃×C₄×C₅	#117	1	2	yes	—
76	C₃×C₄×C₅	#117	1	2	yes	—
77	C₅×Dic₃	—	4	4	no	—
78	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
79	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
80	C₂²×Dic₃	C₂×C₄×C₅	1	4	yes	—
81	C₂×C₃×C₈	—	2	4	no	—
82	C₂×C₃×C₈	C₂×C₄×C₅	1	4	yes	—
83	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
84	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
85	C₄×Dic₃	—	2	4	no	—
86	C₂×C₃⋊C₈	—	2	4	no	—
87	C₂×C₃⋊C₈	C₂×C₄×C₅	1	4	yes	—
88	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
89	C₂×C₄×C₅	—	2	4	no	—
90	C₂×C₄×C₅	—	6	4	no	—
91	C₂×C₄×C₅	—	6	8	no	—
92	C₂×C₄×C₅	C₄×Dic₃	1	4	yes	—
93	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
94	C₂×C₄×C₅	#235	1	4	yes	—
95	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
96	C₅×C₈	—	2	4	no	—
97	C₅×C₈	—	6	4	no	—
98	C₂²×C₈	—	3	2	no	—
99	C₂²×C₈	C₂×C₅×D₃	1	2	yes	—
100	C₂×C₄²	—	3	2	no	—
101	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
102	#97	C₅×Dic₃	1	4	yes	complemented
103	C₄×C₈	—	3	4	no	—
104	#95	—	3	4	no	—
105	C₂×C₃×C₅	#376	1	1	yes	characteristic
106	C₂×C₃×C₅	#405	1	2	yes	—
107	C₂×C₃×C₅	#433	1	4	yes	—
108	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
109	C₂×C₃×C₄	—	2	4	no	—
110	C₂×C₃×C₄	C₄²×C₅	1	4	yes	—
111	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
112	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
113	C₂×C₃×C₄	C₅×#41	1	4	yes	—
114	C₂×Dic₃	—	2	4	no	—
115	C₂×Dic₃	—	2	8	no	—
116	C₃×C₈	—	2	4	no	—
117	C₃⋊C₈	—	2	4	no	—
118	C₂²×C₅	#872	1	2	yes	—
119	C₂²×C₅	#907	1	4	yes	—
120	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
121	C₄×C₅	—	2	4	no	—
122	C₄×C₅	—	12	4	no	—
123	C₄×C₅	#872	1	2	yes	—
124	C₄×C₅	#926	1	2	yes	—
125	C₂²×C₄	—	3	4	no	—
126	C₂²×C₄	C₄×C₅×D₃	1	2	yes	—
127	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	—	3	4	no	—
130	C₂×C₈	—	6	4	no	—
131	C₂×C₈	C₂×C₅×Dic₃	1	4	yes	—
132	C₄²	—	6	4	no	—
133	C₄²	C₄×C₅×D₃	1	2	yes	—
134	C₄²	C₂×C₅×Dic₃	1	2	yes	—
135	C₃×C₅	#1746	1	1	yes	characteristic, complemented, Hall
136	C₂²×C₃	C₂×C₄²×C₅	1	1	yes	characteristic
137	C₂²×C₃	C₅×#117	1	2	yes	—
138	C₂²×C₃	C₅×#131	1	4	yes	—
139	C₃×C₄	—	2	4	no	—
140	C₃×C₄	C₅×#117	1	2	yes	—
141	C₃×C₄	C₅×#117	1	2	yes	—
142	Dic₃	—	4	4	no	—
143	C₂×C₅	#4709	1	4	yes	—
144	C₂×C₅	#4932	1	1	yes	characteristic
145	C₂×C₅	#5122	1	2	yes	—
146	C₂³	C₂×C₄×C₅×D₃	1	1	yes	characteristic
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	6	4	no	—
149	C₂×C₄	—	6	8	no	—
150	C₂×C₄	C₄×C₅×Dic₃	1	4	yes	—
151	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
152	C₂×C₄	C₅×#235	1	4	yes	—
153	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
154	C₈	—	2	4	no	—
155	C₈	—	6	4	no	—
156	C₂×C₃	C₅×#376	1	1	yes	characteristic
157	C₂×C₃	C₅×#405	1	2	yes	—
158	C₂×C₃	C₅×#433	1	4	yes	—
159	C₅	#69806	1	1	yes	characteristic, direct product factor, Sylow
160	C₂²	C₅×#872	1	2	yes	—
161	C₂²	C₅×#907	1	4	yes	—
162	C₂²	C₂×C₄×C₅×Dic₃	1	1	yes	characteristic
163	C₄	—	2	4	no	—
164	C₄	—	12	4	no	—
165	C₄	C₅×#872	1	2	yes	—
166	C₄	C₅×#926	1	2	yes	—
167	C₃	C₅×#1746	1	1	yes	characteristic, complemented, Sylow
168	C₂	C₅×#4709	1	4	yes	—
169	C₂	C₅×#4932	1	1	yes	characteristic
170	C₂	C₅×#5122	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂²×D₃
commutator	C₂²×C₃	C₂×C₄²×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₂×C₃×C₅×#97	C₂
socle, prime radical	C₂³×C₃×C₅	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₂×C₅×#97	D₃
hypocenter	C₃	C₅×#1746
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#97

Complements

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

N	H ≅ G/N	type
10. C₃×C₅×#97	C₄	semidirect
12. C₄×C₅×C₃⋊C₈	C₄	semidirect
13. #69806	C₅	direct (N × H)
36. C₅×#97	Dic₃	semidirect
54. C₃×#97	C₄×C₅	semidirect
56. C₄×C₃⋊C₈	C₄×C₅	semidirect
102. #97	C₅×Dic₃	semidirect
135. C₃×C₅	#1746	semidirect
159. C₅	#69806	direct (N × H)
167. C₃	C₅×#1746	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×#97	2
C₅×#4932	2
C₂×C₄×C₅×C₃⋊C₈	2
C₅×#4550	2
C₅×#1746	3
#69806	5

Subgroups — C5×#69806 #14294929

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69806 #14294929