Subgroups — C₃×#10600674 #14312635

Order 1920.

368 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#405	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3944	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×#672	C₂	1	1	yes	maximal, characteristic
4	C₃×#64681	C₂	1	2	yes	maximal, complemented
5	C₃×C₄×#672	C₂	1	2	yes	maximal, complemented
6	#10600674	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×M₁₆	C₄	1	1	yes	characteristic
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₃×C₅×#117	C₄	1	1	yes	characteristic
10	C₃×C₅×#97	C₄	1	2	yes	—
11	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
12	C₃×#3944	C₄	1	4	yes	complemented
13	C₂×C₃×#672	C₄	1	4	yes	complemented
14	C₂×C₃×#672	C₂²	1	1	yes	characteristic
15	C₂×C₃×#672	C₂²	1	1	yes	characteristic
16	C₂×C₃×#672	C₂²	1	2	yes	—
17	C₃×#2108	—	5	1	no	maximal, Hall
18	C₅×#405	C₂×C₃	1	1	yes	characteristic
19	C₂×#3944	C₂×C₃	1	1	yes	characteristic
20	C₂²×#672	C₂×C₃	1	1	yes	characteristic
21	#64681	C₂×C₃	1	2	yes	complemented
22	C₄×#672	C₂×C₃	1	2	yes	complemented
23	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
24	C₃×C₅×M₁₆	C₈	1	4	yes	—
25	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
27	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
30	C₃×C₅×C₄⋊C₄	C₈	1	2	yes	—
31	C₃×C₅×C₂²⋊C₄	C₈	1	2	yes	—
32	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
33	C₃×#672	C₂×C₄	1	2	yes	—
34	C₃×#672	C₂×C₄	1	2	yes	—
35	C₃×#672	C₂×C₄	1	4	yes	—
36	C₂×C₃×M₃₂	—	5	1	no	—
37	C₂²×C₃×C₁₆	—	5	1	no	—
38	C₃×#405	D₅	1	1	yes	characteristic
39	C₃×#346	—	5	2	no	—
40	C₃×C₄×C₁₆	—	5	2	no	—
41	C₂×C₅×M₁₆	C₃×C₄	1	1	yes	characteristic
42	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
43	C₅×#117	C₃×C₄	1	1	yes	characteristic
44	C₅×#97	C₃×C₄	1	2	yes	—
45	C₄×C₅×C₈	C₂²×C₃	1	2	yes	—
46	#3944	C₃×C₄	1	4	yes	complemented
47	C₂×#672	C₃×C₄	1	4	yes	complemented
48	C₂×#672	C₂²×C₃	1	1	yes	characteristic
49	C₂×#672	C₂²×C₃	1	1	yes	characteristic
50	C₂×#672	C₂²×C₃	1	2	yes	—
51	#2108	—	5	1	no	Sylow
52	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
54	C₂×C₃×C₄×C₅	C₂×C₈	1	4	yes	—
55	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
57	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₃×C₅×C₈	C₄²	1	2	yes	—
59	C₃×C₅×C₈	C₂×C₈	1	4	yes	—
60	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
61	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₃×M₁₆	Dic₅	1	1	yes	characteristic
63	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
64	C₃×#117	Dic₅	1	1	yes	characteristic
65	C₃×M₃₂	—	5	4	no	—
66	C₂×C₃×C₁₆	—	5	4	no	—
67	C₂×C₃×C₁₆	—	5	1	no	—
68	C₂×C₃×C₁₆	—	5	1	no	—
69	C₂×C₃×C₁₆	—	5	2	no	—
70	C₃×#97	Dic₅	1	2	yes	—
71	C₃×C₄×C₈	C₂×D₅	1	2	yes	—
72	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
73	C₅×M₁₆	C₃×C₈	1	4	yes	—
74	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
75	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
76	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
77	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
78	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
79	C₅×C₄⋊C₄	C₃×C₈	1	2	yes	—
80	C₅×C₂²⋊C₄	C₃×C₈	1	2	yes	—
81	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
82	#672	C₂×C₃×C₄	1	2	yes	—
83	#672	C₂×C₃×C₄	1	2	yes	—
84	#672	C₂×C₃×C₄	1	4	yes	—
85	C₂×M₃₂	—	5	1	no	—
86	C₂²×C₁₆	—	5	1	no	—
87	#405	C₃×D₅	1	1	yes	characteristic
88	#346	—	5	2	no	—
89	C₄×C₁₆	—	5	2	no	—
90	C₂²×C₃×C₅	—	2	2	no	—
91	C₂²×C₃×C₅	C₄×C₈	1	2	yes	—
92	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
93	C₃×C₄×C₅	—	2	4	no	—
94	C₃×C₄×C₅	C₄×C₈	1	2	yes	—
95	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
96	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
97	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
98	C₃×M₁₆	#168	1	4	yes	—
99	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
100	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
101	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
102	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
103	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
104	C₃×C₄⋊C₄	#168	1	2	yes	—
105	C₃×C₂²⋊C₄	#168	1	2	yes	—
106	C₃×C₄²	C₂×Dic₅	1	2	yes	—
107	C₃×C₁₆	—	5	2	no	—
108	C₃×C₁₆	—	5	2	no	—
109	C₃×C₁₆	—	5	4	no	—
110	C₂³×C₅	C₂×C₃×C₈	1	1	yes	characteristic
111	C₂×C₄×C₅	C₃×C₄²	1	2	yes	—
112	C₂×C₄×C₅	C₂×C₃×C₈	1	4	yes	—
113	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂×C₃×C₈	1	2	yes	—
115	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
116	C₅×C₈	C₃×C₄²	1	2	yes	—
117	C₅×C₈	C₂×C₃×C₈	1	4	yes	—
118	C₅×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
119	C₅×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
120	C₂×M₁₆	C₃×Dic₅	1	1	yes	characteristic
121	C₂²×C₈	C₂×C₃×D₅	1	1	yes	characteristic
122	#117	C₃×Dic₅	1	1	yes	characteristic
123	M₃₂	—	5	4	no	—
124	C₂×C₁₆	—	5	4	no	—
125	C₂×C₁₆	—	5	1	no	—
126	C₂×C₁₆	—	5	2	no	—
127	C₂×C₁₆	—	5	1	no	—
128	#97	C₃×Dic₅	1	2	yes	—
129	C₄×C₈	C₂×C₃×D₅	1	2	yes	—
130	C₂×C₃×C₅	—	2	2	no	—
131	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
132	C₂×C₃×C₅	#504	1	2	yes	—
133	C₂³×C₃	C₂×#168	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂×#168	1	4	yes	—
135	C₂×C₃×C₄	C₂×#168	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂×#168	1	2	yes	—
137	C₂×C₃×C₄	C₄×Dic₅	1	2	yes	—
138	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
139	C₃×C₈	C₂×#168	1	4	yes	—
140	C₃×C₈	C₄×Dic₅	1	2	yes	—
141	C₃×C₈	C₂×C₄×D₅	1	1	yes	characteristic
142	C₃×C₈	C₂×C₄×D₅	1	1	yes	characteristic
143	C₂²×C₅	—	2	2	no	—
144	C₂²×C₅	C₃×C₄×C₈	1	2	yes	—
145	C₂²×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
146	C₄×C₅	—	2	4	no	—
147	C₄×C₅	C₃×C₄×C₈	1	2	yes	—
148	C₄×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
149	C₄×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
150	C₂²×C₄	C₂×C₃×Dic₅	1	1	yes	characteristic
151	M₁₆	C₃×#168	1	4	yes	—
152	C₂×C₈	C₃×C₄×D₅	1	2	yes	—
153	C₂×C₈	C₃×C₄×D₅	1	2	yes	—
154	C₂×C₈	C₂×C₃×Dic₅	1	2	yes	—
155	C₂×C₈	C₂×C₃×Dic₅	1	1	yes	characteristic
156	C₂×C₈	C₂²×C₃×D₅	1	1	yes	characteristic
157	C₄⋊C₄	C₃×#168	1	2	yes	—
158	C₂²⋊C₄	C₃×#168	1	2	yes	—
159	C₄²	C₂×C₃×Dic₅	1	2	yes	—
160	C₁₆	—	5	2	no	—
161	C₁₆	—	5	2	no	—
162	C₁₆	—	5	4	no	—
163	C₃×C₅	#2108	1	1	yes	characteristic, complemented, Hall
164	C₂²×C₃	—	2	2	no	—
165	C₂²×C₃	C₄×#168	1	2	yes	—
166	C₂²×C₃	C₂²×#168	1	1	yes	characteristic
167	C₃×C₄	—	2	4	no	—
168	C₃×C₄	C₄×#168	1	2	yes	—
169	C₃×C₄	C₂²×#168	1	1	yes	characteristic
170	C₃×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
171	C₂×C₅	—	2	2	no	—
172	C₂×C₅	C₂×C₃×C₄×C₈	1	1	yes	characteristic
173	C₂×C₅	C₃×#504	1	2	yes	—
174	C₂³	C₂×C₃×#168	1	1	yes	characteristic
175	C₂×C₄	C₂×C₃×#168	1	4	yes	—
176	C₂×C₄	C₂×C₃×#168	1	1	yes	characteristic
177	C₂×C₄	C₂×C₃×#168	1	2	yes	—
178	C₂×C₄	C₃×C₄×Dic₅	1	2	yes	—
179	C₂×C₄	C₂²×C₃×Dic₅	1	1	yes	characteristic
180	C₈	C₂×C₃×#168	1	4	yes	—
181	C₈	C₃×C₄×Dic₅	1	2	yes	—
182	C₈	C₂×C₃×C₄×D₅	1	1	yes	characteristic
183	C₈	C₂×C₃×C₄×D₅	1	1	yes	characteristic
184	C₂×C₃	—	2	2	no	—
185	C₂×C₃	C₂×C₄×#168	1	1	yes	characteristic
186	C₂×C₃	#65429	1	2	yes	—
187	C₅	C₃×#2108	1	1	yes	characteristic, complemented, Sylow
188	C₂²	—	2	2	no	—
189	C₂²	C₃×C₄×#168	1	2	yes	—
190	C₂²	C₂²×C₃×#168	1	1	yes	characteristic
191	C₄	—	2	4	no	—
192	C₄	C₃×C₄×#168	1	2	yes	—
193	C₄	C₂²×C₃×#168	1	1	yes	characteristic
194	C₄	C₂×C₃×C₄×Dic₅	1	1	yes	characteristic
195	C₃	#10600674	1	1	yes	characteristic, direct product factor, Sylow
196	C₂	—	2	2	no	—
197	C₂	C₂×C₃×C₄×#168	1	1	yes	characteristic
198	C₂	C₃×#65429	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₈	C₂²×D₅
commutator	C₂×C₅	C₂×C₃×C₄×C₈
Frattini	C₂×C₈	C₂²×C₃×D₅
Fitting	C₃×C₅×#405	C₂
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂³×C₃×C₅	C₂×C₈
cosocle	C₂×C₅×C₈	C₂³×C₃
hypercenter	C₃×#405	D₅
hypocenter	C₅	C₃×#2108
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₈ ◁ C₃×#405

Complements

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

N	H ≅ G/N	type
4. C₃×#64681	C₂	semidirect
5. C₃×C₄×#672	C₂	semidirect
6. #10600674	C₃	direct (N × H)
12. C₃×#3944	C₄	semidirect
13. C₂×C₃×#672	C₄	semidirect
21. #64681	C₂×C₃	semidirect
22. C₄×#672	C₂×C₃	semidirect
46. #3944	C₃×C₄	semidirect
47. C₂×#672	C₃×C₄	semidirect
163. C₃×C₅	#2108	semidirect
187. C₅	C₃×#2108	semidirect
195. C₃	#10600674	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅×#405	2
C₂×C₃×#3944	2
C₂²×C₃×#672	2
C₃×#64681	2
C₃×C₄×#672	2
#10600674	3
C₃×#2108	5

Subgroups — C3×#10600674 #14312635

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10600674 #14312635