Subgroups — C₃×#10604369 #14312759

Order 1920.

320 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#3998	C₂	1	2	yes	maximal
2	C₃×#65286	C₂	1	1	yes	maximal, characteristic
3	C₃×#64860	C₂	1	4	yes	maximal, complemented
4	#10604369	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₃×C₅×#117	C₄	1	1	yes	characteristic
6	C₂×C₃×#681	C₄	1	1	yes	characteristic
7	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
8	C₃×#3998	C₄	1	4	yes	complemented
9	C₃×#3998	C₂²	1	4	yes	—
10	C₂×C₃×#674	C₄	1	4	yes	complemented
11	C₃×#3935	C₂²	1	2	yes	—
12	C₃×C₄×#168	C₄	1	2	yes	—
13	C₃×#2110	—	5	1	no	maximal, Hall
14	C₂×#3998	C₂×C₃	1	2	yes	—
15	#65286	C₂×C₃	1	1	yes	characteristic
16	#64860	C₂×C₃	1	4	yes	complemented
17	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
18	C₃×C₅×C₄⋊C₄	—	2	1	no	—
19	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
20	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
21	C₃×#681	—	2	2	no	—
22	C₂×C₃×#168	C₂×C₄	1	2	yes	—
23	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×#168	C₂×C₄	1	2	yes	—
25	C₂×C₃×#168	C₂×C₄	1	2	yes	—
26	C₂×C₃×#168	C₂³	1	1	yes	characteristic
27	C₃×#674	C₂×C₄	1	8	yes	—
28	C₂×C₃×M₃₂	—	5	2	no	—
29	C₃×#405	—	5	1	no	—
30	C₃×#347	—	5	4	no	—
31	C₅×#117	C₃×C₄	1	1	yes	characteristic
32	C₂×#681	C₃×C₄	1	1	yes	characteristic
33	C₂²×#168	C₂²×C₃	1	1	yes	characteristic
34	#3998	C₃×C₄	1	4	yes	complemented
35	#3998	C₂²×C₃	1	4	yes	—
36	C₂×#674	C₃×C₄	1	4	yes	complemented
37	#3935	C₂²×C₃	1	2	yes	—
38	C₄×#168	C₃×C₄	1	2	yes	—
39	#2110	—	5	1	no	Sylow
40	C₂³×C₃×C₅	M₁₆	1	1	yes	characteristic
41	C₂×C₃×C₄×C₅	—	2	2	no	—
42	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
43	C₂×C₃×C₄×C₅	M₁₆	1	1	yes	characteristic
44	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
45	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
46	C₃×#168	—	2	2	no	—
47	C₃×#168	C₄²	1	1	yes	characteristic
48	C₃×#168	C₄²	1	1	yes	characteristic
49	C₃×#168	C₂²×C₄	1	2	yes	—
50	C₂×C₃×M₁₆	—	5	1	no	—
51	C₂²×C₃×C₈	—	5	1	no	—
52	C₃×#117	AGL(1,𝔽₅)	1	1	yes	characteristic
53	C₃×M₃₂	—	5	4	no	—
54	C₃×M₃₂	—	5	4	no	—
55	C₂×C₃×C₁₆	—	5	4	no	—
56	C₃×#97	—	5	2	no	—
57	C₃×C₄×C₈	—	5	2	no	—
58	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
59	C₅×C₄⋊C₄	—	2	1	no	—
60	C₅×C₂²⋊C₄	—	2	1	no	—
61	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
62	#681	—	2	2	no	—
63	C₂×#168	C₂×C₃×C₄	1	2	yes	—
64	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
65	C₂×#168	C₂×C₃×C₄	1	2	yes	—
66	C₂×#168	C₂×C₃×C₄	1	2	yes	—
67	C₂×#168	C₂³×C₃	1	1	yes	characteristic
68	#674	C₂×C₃×C₄	1	8	yes	—
69	C₂×M₃₂	—	5	2	no	—
70	#405	—	5	1	no	—
71	#347	—	5	4	no	—
72	C₂²×C₃×C₅	—	4	1	no	—
73	C₂²×C₃×C₅	#97	1	2	yes	—
74	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
75	C₃×C₄×C₅	—	4	2	no	—
76	C₃×C₄×C₅	#97	1	2	yes	—
77	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
78	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
79	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
80	C₃×M₁₆	—	10	2	no	—
81	C₂×C₃×C₈	—	5	2	no	—
82	C₂×C₃×C₈	—	5	2	no	—
83	C₂×C₃×C₈	—	5	1	no	—
84	C₂×C₃×C₈	—	5	1	no	—
85	C₂×C₃×C₈	—	5	2	no	—
86	C₃×C₄⋊C₄	—	2	1	no	—
87	C₃×C₂²⋊C₄	—	2	1	no	—
88	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	2	yes	—
89	C₃×C₁₆	—	5	8	no	—
90	C₂³×C₅	C₃×M₁₆	1	1	yes	characteristic
91	C₂×C₄×C₅	—	2	2	no	—
92	C₂×C₄×C₅	C₃×C₄²	1	2	yes	—
93	C₂×C₄×C₅	C₃×M₁₆	1	2	yes	—
94	C₂×C₄×C₅	C₃×M₁₆	1	1	yes	characteristic
95	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
96	#168	—	2	2	no	—
97	#168	C₃×C₄²	1	1	yes	characteristic
98	#168	C₃×C₄²	1	1	yes	characteristic
99	#168	C₂²×C₃×C₄	1	2	yes	—
100	C₂×M₁₆	—	5	1	no	—
101	C₂²×C₈	—	5	1	no	—
102	#117	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
103	M₃₂	—	5	4	no	—
104	M₃₂	—	5	4	no	—
105	C₂×C₁₆	—	5	4	no	—
106	#97	—	5	2	no	—
107	C₄×C₈	—	5	2	no	—
108	C₂×C₃×C₅	—	2	1	no	—
109	C₂×C₃×C₅	—	2	2	no	—
110	C₂×C₃×C₅	C₂×#97	1	1	yes	characteristic
111	C₂³×C₃	#704	1	1	yes	characteristic
112	C₂×C₃×C₄	—	2	2	no	—
113	C₂×C₃×C₄	#700	1	2	yes	—
114	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	2	yes	—
115	C₂×C₃×C₄	#704	1	1	yes	characteristic
116	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
117	C₃×C₈	—	5	1	no	—
118	C₃×C₈	—	5	1	no	—
119	C₃×C₈	—	5	2	no	—
120	C₃×C₈	—	10	2	no	—
121	C₂²×C₅	—	4	1	no	—
122	C₂²×C₅	C₃×#97	1	2	yes	—
123	C₂²×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
124	C₄×C₅	—	4	2	no	—
125	C₄×C₅	C₃×#97	1	2	yes	—
126	C₄×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
127	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
128	C₂²×C₄	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
129	M₁₆	—	10	2	no	—
130	C₂×C₈	—	5	2	no	—
131	C₂×C₈	—	5	2	no	—
132	C₂×C₈	—	5	1	no	—
133	C₂×C₈	—	5	1	no	—
134	C₂×C₈	—	5	2	no	—
135	C₄⋊C₄	—	2	1	no	—
136	C₂²⋊C₄	—	2	1	no	—
137	C₄²	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
138	C₁₆	—	5	8	no	—
139	C₃×C₅	#2110	1	1	yes	characteristic, complemented, Hall
140	C₂²×C₃	—	4	1	no	—
141	C₂²×C₃	#4002	1	2	yes	—
142	C₂²×C₃	C₂×#704	1	1	yes	characteristic
143	C₃×C₄	—	4	2	no	—
144	C₃×C₄	#4002	1	2	yes	—
145	C₃×C₄	C₂×#700	1	1	yes	characteristic
146	C₃×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
147	C₂×C₅	—	2	1	no	—
148	C₂×C₅	—	2	2	no	—
149	C₂×C₅	C₂×C₃×#97	1	1	yes	characteristic
150	C₂³	C₃×#704	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	C₃×#700	1	2	yes	—
153	C₂×C₄	C₃×C₄×AGL(1,𝔽₅)	1	2	yes	—
154	C₂×C₄	C₃×#704	1	1	yes	characteristic
155	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
156	C₈	—	5	2	no	—
157	C₈	—	5	1	no	—
158	C₈	—	5	1	no	—
159	C₈	—	10	2	no	—
160	C₂×C₃	—	2	1	no	—
161	C₂×C₃	—	2	2	no	—
162	C₂×C₃	C₂×#4002	1	1	yes	characteristic
163	C₅	C₃×#2110	1	1	yes	characteristic, complemented, Sylow
164	C₂²	—	4	1	no	—
165	C₂²	C₃×#4002	1	2	yes	—
166	C₂²	C₂×C₃×#704	1	1	yes	characteristic
167	C₄	—	4	2	no	—
168	C₄	C₃×#4002	1	2	yes	—
169	C₄	C₂×C₃×#700	1	1	yes	characteristic
170	C₄	C₂×C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
171	C₃	#10604369	1	1	yes	characteristic, direct product factor, Sylow
172	C₂	—	2	1	no	—
173	C₂	—	2	2	no	—
174	C₂	C₂×C₃×#4002	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	C₂×C₄×AGL(1,𝔽₅)
commutator	C₄×C₅	C₂×C₃×C₄²
Frattini	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₅×#117	C₄
socle	C₂×C₃×C₅	C₂×#97
prime radical	C₂³×C₃×C₅	M₁₆
cosocle	C₂×#168	C₂³×C₃
hypercenter	C₃×#117	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#2110
LCS	G ▷ C₄×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₄ ◁ C₃×#117

Complements

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

N	H ≅ G/N	type
3. C₃×#64860	C₂	semidirect
4. #10604369	C₃	direct (N × H)
8. C₃×#3998	C₄	semidirect
10. C₂×C₃×#674	C₄	semidirect
16. #64860	C₂×C₃	semidirect
34. #3998	C₃×C₄	semidirect
36. C₂×#674	C₃×C₄	semidirect
139. C₃×C₅	#2110	semidirect
163. C₅	C₃×#2110	semidirect
171. C₃	#10604369	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#3998	2
C₃×#65286	2
C₃×#64860	2
#10604369	3
C₃×#2110	5

Subgroups — C3×#10604369 #14312759

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10604369 #14312759