Subgroups — C₃×#10600584 #14298894

Order 1920.

376 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₃×#65213	C₂	1	2	yes	maximal
3	C₃×#64772	C₂	1	4	yes	maximal
4	#10600584	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
6	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₂×C₃×#681	C₂²	1	4	yes	—
8	C₃×C₄×C₅×C₈	C₄	1	2	yes	—
9	C₃×C₄×C₅×C₈	C₄	1	2	yes	—
10	C₃×C₄⋊#168	—	2	4	no	—
11	C₃×#1837	—	5	1	no	maximal, Hall
12	C₂×C₄×C₅×C₈	C₂×C₃	1	1	yes	characteristic
13	#65213	C₂×C₃	1	2	yes	—
14	#64772	C₂×C₃	1	4	yes	—
15	C₂²×C₃×C₄×C₅	Q₈	1	2	yes	—
16	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
17	C₂×C₃×C₅×C₈	—	2	4	no	—
18	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	—
19	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
20	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
21	C₃×C₄²×C₅	D₄	1	2	yes	—
22	C₃×#681	—	2	8	no	—
23	C₂×C₃×#168	—	2	4	no	—
24	C₃×#423	—	5	2	no	—
25	C₂×C₃×C₄×C₈	D₅	1	1	yes	characteristic
26	C₃×#341	—	5	4	no	—
27	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
28	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
29	C₂×#681	C₂²×C₃	1	4	yes	—
30	C₄×C₅×C₈	C₃×C₄	1	2	yes	—
31	C₄×C₅×C₈	C₃×C₄	1	2	yes	—
32	C₄⋊#168	—	2	4	no	—
33	#1837	—	5	1	no	Sylow
34	C₂³×C₃×C₅	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₃×C₄×C₅	—	2	4	no	—
36	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
37	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
38	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
40	C₂×C₃×C₄×C₅	#41	1	4	yes	—
41	C₃×C₅×C₈	—	2	4	no	—
42	C₃×#168	—	4	4	no	—
43	C₂×C₃×M₁₆	—	5	4	no	—
44	C₂²×C₃×C₈	C₂×D₅	1	2	yes	—
45	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
46	C₃×C₄⋊C₈	—	10	4	no	—
47	C₃×C₄×C₈	Dic₅	1	2	yes	—
48	C₃×C₄×C₈	Dic₅	1	2	yes	—
49	C₂²×C₄×C₅	C₃×Q₈	1	2	yes	—
50	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
51	C₂×C₅×C₈	—	2	4	no	—
52	C₂×C₅×C₈	C₂×C₃×C₄	1	4	yes	—
53	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
54	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
55	C₄²×C₅	C₃×D₄	1	2	yes	—
56	#681	—	2	8	no	—
57	C₂×#168	—	2	4	no	—
58	#423	—	5	2	no	—
59	C₂×C₄×C₈	C₃×D₅	1	1	yes	characteristic
60	#341	—	5	4	no	—
61	C₂²×C₃×C₅	—	2	2	no	—
62	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₃×C₅	#125	1	2	yes	—
64	C₃×C₄×C₅	—	2	2	no	—
65	C₃×C₄×C₅	#108	1	4	yes	—
66	C₃×C₄×C₅	#117	1	2	yes	—
67	C₃×C₄×C₅	#124	1	2	yes	—
68	C₂²×C₃×C₄	Dic₁₀	1	2	yes	—
69	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
70	C₃×M₁₆	—	10	8	no	—
71	C₂×C₃×C₈	—	2	4	no	—
72	C₂×C₃×C₈	—	10	4	no	—
73	C₂×C₃×C₈	C₂×Dic₅	1	4	yes	—
74	C₃×C₄²	D₂₀	1	2	yes	—
75	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
76	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
77	C₂³×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
78	C₂×C₄×C₅	—	2	4	no	—
79	C₂×C₄×C₅	C₃×C₄⋊C₄	1	2	yes	—
80	C₂×C₄×C₅	C₃×C₄⋊C₄	1	2	yes	—
81	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
82	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
83	C₂×C₄×C₅	C₃×#41	1	4	yes	—
84	C₅×C₈	—	2	4	no	—
85	#168	—	4	4	no	—
86	C₂×M₁₆	—	5	4	no	—
87	C₂²×C₈	C₂×C₃×D₅	1	2	yes	—
88	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
89	C₄⋊C₈	—	10	4	no	—
90	C₄×C₈	C₃×Dic₅	1	2	yes	—
91	C₄×C₈	C₃×Dic₅	1	2	yes	—
92	C₂×C₃×C₅	—	2	2	no	—
93	C₂×C₃×C₅	#382	1	1	yes	characteristic
94	C₂×C₃×C₅	C₂×#108	1	2	yes	—
95	C₂³×C₃	C₂×Dic₁₀	1	1	yes	characteristic
96	C₂×C₃×C₄	—	2	4	no	—
97	C₂×C₃×C₄	C₄⋊Dic₅	1	2	yes	—
98	C₂×C₃×C₄	C₄⋊Dic₅	1	2	yes	—
99	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
100	C₂×C₃×C₄	#709	1	4	yes	—
101	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
102	C₃×C₈	—	2	4	no	—
103	C₃×C₈	—	20	4	no	—
104	C₂²×C₅	—	2	2	no	—
105	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
106	C₂²×C₅	C₃×#125	1	2	yes	—
107	C₄×C₅	—	2	2	no	—
108	C₄×C₅	C₃×#108	1	4	yes	—
109	C₄×C₅	C₃×#117	1	2	yes	—
110	C₄×C₅	C₃×#124	1	2	yes	—
111	C₂²×C₄	C₃×Dic₁₀	1	2	yes	—
112	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
113	M₁₆	—	10	8	no	—
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	10	4	no	—
116	C₂×C₈	C₂×C₃×Dic₅	1	4	yes	—
117	C₄²	C₃×D₂₀	1	2	yes	—
118	C₄²	C₂×C₃×Dic₅	1	1	yes	characteristic
119	C₄²	C₂×C₃×Dic₅	1	1	yes	characteristic
120	C₃×C₅	#1837	1	1	yes	characteristic, complemented, Hall
121	C₂²×C₃	—	2	2	no	—
122	C₂²×C₃	#4016	1	2	yes	—
123	C₂²×C₃	C₂×C₄⋊Dic₅	1	1	yes	characteristic
124	C₃×C₄	—	2	2	no	—
125	C₃×C₄	#3951	1	4	yes	—
126	C₃×C₄	#4021	1	2	yes	—
127	C₃×C₄	#4072	1	2	yes	—
128	C₂×C₅	—	2	2	no	—
129	C₂×C₅	C₃×#382	1	1	yes	characteristic
130	C₂×C₅	C₂×C₃×#108	1	2	yes	—
131	C₂³	C₂×C₃×Dic₁₀	1	1	yes	characteristic
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	C₃×C₄⋊Dic₅	1	2	yes	—
134	C₂×C₄	C₃×C₄⋊Dic₅	1	2	yes	—
135	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
136	C₂×C₄	C₃×#709	1	4	yes	—
137	C₂×C₄	C₂²×C₃×Dic₅	1	1	yes	characteristic
138	C₈	—	2	4	no	—
139	C₈	—	20	4	no	—
140	C₂×C₃	—	2	2	no	—
141	C₂×C₃	#65225	1	1	yes	characteristic
142	C₂×C₃	C₂×#3951	1	2	yes	—
143	C₅	C₃×#1837	1	1	yes	characteristic, complemented, Sylow
144	C₂²	—	2	2	no	—
145	C₂²	C₃×#4016	1	2	yes	—
146	C₂²	C₂×C₃×C₄⋊Dic₅	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	C₃×#3951	1	4	yes	—
149	C₄	C₃×#4021	1	2	yes	—
150	C₄	C₃×#4072	1	2	yes	—
151	C₃	#10600584	1	1	yes	characteristic, direct product factor, Sylow
152	C₂	—	2	2	no	—
153	C₂	C₃×#65225	1	1	yes	characteristic
154	C₂	C₂×C₃×#3951	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₂₀
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₂×C₃×C₄×C₅×C₈	C₂
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂³×C₃×C₅	C₂×Q₈
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₂×C₃×C₄×C₈	D₅
hypocenter	C₅	C₃×#1837
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. #10600584	C₃	direct (N × H)
120. C₃×C₅	#1837	semidirect
143. C₅	C₃×#1837	semidirect
151. C₃	#10600584	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×C₄×C₅×C₈	2
C₃×#65213	2
C₃×#64772	2
#10600584	3
C₃×#1837	5

Subgroups — C3×#10600584 #14298894

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10600584 #14298894