Subgroups — C₂×C₃×D₃×#168 #13086484

Order 1440.

326 conjugacy classes of subgroups.

The subgroup lattice has 194 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×D₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₃⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃²×#168	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×D₃×#168	C₂	1	4	yes	maximal, direct product factor
5	C₂²×C₃×#168	—	3	1	no	maximal
6	C₂×D₃×#168	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃×C₅×D₃	C₄	1	1	yes	characteristic
8	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₅×Dic₃	C₄	1	1	yes	characteristic
10	C₃×C₄×C₅×D₃	C₄	1	2	yes	—
11	C₃×C₄×C₅×D₃	C₂²	1	2	yes	—
12	C₃×C₃⋊#168	C₂²	1	2	yes	complemented
13	C₃²×#168	C₂²	1	2	yes	complemented
14	C₂×C₃×D₃×C₈	—	5	1	no	maximal, Hall
15	C₂²×C₃×C₄×C₅	—	3	1	no	—
16	C₂×C₄×C₅×D₃	C₂×C₃	1	1	yes	characteristic
17	C₂×C₃⋊#168	C₂×C₃	1	1	yes	characteristic, complemented
18	C₂×C₃×#168	—	2	1	no	—
19	C₂×C₃×#168	—	3	1	no	—
20	C₂×C₃×#168	—	3	4	no	—
21	C₂×C₃×#168	D₃	1	1	yes	characteristic, direct product factor
22	C₂×C₃×#168	C₂×C₃	1	1	yes	characteristic, complemented
23	D₃×#168	C₂×C₃	1	4	yes	direct product factor
24	C₂²×C₃²×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₅×D₃	C₈	1	4	yes	complemented
26	C₂×C₃×C₅×D₃	C₂×C₄	1	2	yes	—
27	C₃²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
28	C₃²×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₃×C₅×Dic₃	C₂×C₄	1	2	yes	—
30	C₂²×#168	—	3	1	no	Hall
31	C₂×C₃×C₄×D₃	D₅	1	1	yes	characteristic
32	C₂×C₃²×C₈	—	5	1	no	—
33	C₂×C₃×C₃⋊C₈	—	5	1	no	—
34	C₃×D₃×C₈	—	5	4	no	—
35	C₂³×C₃×C₅	—	3	1	no	—
36	C₂²×C₅×D₃	C₃×C₄	1	1	yes	characteristic
37	C₂×C₃×C₄×C₅	—	2	1	no	—
38	C₂×C₃×C₄×C₅	—	3	1	no	—
39	C₂×C₃×C₄×C₅	—	3	2	no	—
40	C₂×C₃×C₄×C₅	—	3	2	no	—
41	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
42	C₂×C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
43	C₂×C₅×Dic₃	C₃×C₄	1	1	yes	characteristic
44	C₄×C₅×D₃	C₃×C₄	1	2	yes	—
45	C₄×C₅×D₃	C₂²×C₃	1	2	yes	—
46	C₃⋊#168	C₂²×C₃	1	2	yes	complemented
47	C₃×#168	—	2	2	no	—
48	C₃×#168	—	3	2	no	—
49	C₃×#168	C₂×D₃	1	2	yes	direct product factor
50	C₃×#168	C₂²×C₃	1	2	yes	complemented
51	C₂²×C₃×C₈	—	15	1	no	—
52	C₂×D₃×C₈	—	5	1	no	—
53	C₂×C₃²×C₅	C₂×C₈	1	2	yes	complemented
54	C₂×C₃²×C₅	C₂²×C₄	1	1	yes	characteristic
55	C₃×C₅×D₃	C₂×C₈	1	4	yes	complemented
56	C₂²×C₄×C₅	—	3	1	no	—
57	C₂×#168	—	3	1	no	—
58	C₂×#168	—	3	4	no	—
59	C₂×#168	C₃×D₃	1	1	yes	characteristic, direct product factor
60	C₂²×C₃×D₃	Dic₅	1	1	yes	characteristic
61	C₂×C₃²×C₄	C₂×D₅	1	1	yes	characteristic
62	C₂×C₃×Dic₃	Dic₅	1	1	yes	characteristic
63	C₃×C₄×D₃	Dic₅	1	2	yes	—
64	C₃×C₄×D₃	C₂×D₅	1	2	yes	—
65	C₃²×C₈	—	5	2	no	—
66	C₃×C₃⋊C₈	—	5	2	no	—
67	C₂²×C₃×C₅	—	2	1	no	—
68	C₂²×C₃×C₅	—	3	4	no	—
69	C₂²×C₃×C₅	—	3	2	no	—
70	C₂²×C₃×C₅	C₄×D₃	1	1	yes	characteristic
71	C₂²×C₃×C₅	C₂×C₃×C₄	1	1	yes	characteristic
72	C₂×C₅×D₃	C₃×C₈	1	4	yes	complemented
73	C₂×C₅×D₃	C₂×C₃×C₄	1	2	yes	—
74	C₃×C₄×C₅	—	2	1	no	—
75	C₃×C₄×C₅	—	2	1	no	—
76	C₃×C₄×C₅	—	3	2	no	—
77	C₃×C₄×C₅	C₄×D₃	1	1	yes	characteristic
78	C₃×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
79	C₃×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
80	C₃×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
81	C₅×Dic₃	C₂×C₃×C₄	1	2	yes	—
82	C₂²×C₃×C₄	—	3	1	no	—
83	C₂×C₄×D₃	C₃×D₅	1	1	yes	characteristic
84	C₂×C₃×C₈	—	5	1	no	—
85	C₂×C₃×C₈	—	5	1	no	—
86	C₂×C₃×C₈	—	10	1	no	—
87	C₂×C₃×C₈	—	15	4	no	—
88	C₂×C₃×C₈	—	15	1	no	—
89	C₂×C₃⋊C₈	—	5	1	no	—
90	D₃×C₈	—	5	4	no	—
91	C₃²×C₅	C₂²×C₈	1	1	yes	characteristic, complemented, Hall
92	C₂³×C₅	—	3	1	no	—
93	C₂×C₄×C₅	—	3	1	no	—
94	C₂×C₄×C₅	—	3	2	no	—
95	C₂×C₄×C₅	—	3	2	no	—
96	C₂×C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
97	#168	—	3	2	no	—
98	#168	C₂×C₃×D₃	1	2	yes	direct product factor
99	C₂²×C₃²	C₂×Dic₅	1	1	yes	characteristic
100	C₂×C₃×D₃	#168	1	4	yes	direct product factor
101	C₂×C₃×D₃	C₂×Dic₅	1	2	yes	—
102	C₃²×C₄	C₂×Dic₅	1	1	yes	characteristic
103	C₃²×C₄	C₂²×D₅	1	1	yes	characteristic
104	C₃×Dic₃	C₂×Dic₅	1	2	yes	—
105	C₂²×C₈	—	15	1	no	Sylow
106	C₂×C₃×C₅	—	2	2	no	—
107	C₂×C₃×C₅	—	2	1	no	—
108	C₂×C₃×C₅	—	3	4	no	—
109	C₂×C₃×C₅	D₃×C₈	1	2	yes	complemented
110	C₂×C₃×C₅	C₂×C₃×C₈	1	2	yes	complemented
111	C₂×C₃×C₅	C₂×C₄×D₃	1	1	yes	characteristic
112	C₂×C₃×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
113	C₅×D₃	C₂×C₃×C₈	1	4	yes	complemented
114	C₂³×C₃	—	3	1	no	—
115	C₂²×D₃	C₃×Dic₅	1	1	yes	characteristic
116	C₂×C₃×C₄	—	2	1	no	—
117	C₂×C₃×C₄	—	3	1	no	—
118	C₂×C₃×C₄	—	3	2	no	—
119	C₂×C₃×C₄	—	3	2	no	—
120	C₂×C₃×C₄	D₃×D₅	1	1	yes	characteristic
121	C₂×C₃×C₄	C₂×C₃×D₅	1	1	yes	characteristic
122	C₂×Dic₃	C₃×Dic₅	1	1	yes	characteristic
123	C₄×D₃	C₃×Dic₅	1	2	yes	—
124	C₄×D₃	C₂×C₃×D₅	1	2	yes	—
125	C₃×C₈	—	5	2	no	—
126	C₃×C₈	—	5	2	no	—
127	C₃×C₈	—	10	2	no	—
128	C₃×C₈	—	15	2	no	—
129	C₃⋊C₈	—	5	2	no	—
130	C₂²×C₅	—	3	4	no	—
131	C₂²×C₅	—	3	2	no	—
132	C₂²×C₅	C₃×C₄×D₃	1	1	yes	characteristic
133	C₄×C₅	—	3	2	no	—
134	C₄×C₅	C₃×C₄×D₃	1	1	yes	characteristic
135	C₄×C₅	C₂²×C₃×D₃	1	1	yes	characteristic
136	C₂×C₃²	C₂×#168	1	2	yes	complemented
137	C₂×C₃²	C₂²×Dic₅	1	1	yes	characteristic
138	C₃×D₃	C₂×#168	1	4	yes	direct product factor
139	C₂²×C₄	—	3	1	no	—
140	C₂×C₈	—	5	1	no	—
141	C₂×C₈	—	15	4	no	—
142	C₂×C₈	—	15	1	no	—
143	C₃×C₅	—	2	1	no	—
144	C₃×C₅	C₂×D₃×C₈	1	1	yes	characteristic, complemented
145	C₃×C₅	C₂²×C₃×C₈	1	1	yes	characteristic, complemented
146	C₂²×C₃	—	2	1	no	—
147	C₂²×C₃	—	3	2	no	—
148	C₂²×C₃	—	3	4	no	—
149	C₂²×C₃	D₃×Dic₅	1	1	yes	characteristic
150	C₂²×C₃	C₂×C₃×Dic₅	1	1	yes	characteristic
151	C₂×D₃	C₃×#168	1	4	yes	direct product factor
152	C₂×D₃	C₂×C₃×Dic₅	1	2	yes	—
153	C₃×C₄	—	2	1	no	—
154	C₃×C₄	—	2	1	no	—
155	C₃×C₄	—	3	2	no	—
156	C₃×C₄	D₃×Dic₅	1	1	yes	characteristic
157	C₃×C₄	C₂×C₃×Dic₅	1	1	yes	characteristic
158	C₃×C₄	C₂×D₃×D₅	1	1	yes	characteristic
159	C₃×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
160	Dic₃	C₂×C₃×Dic₅	1	2	yes	—
161	C₂×C₅	—	3	4	no	—
162	C₂×C₅	C₃×D₃×C₈	1	2	yes	complemented
163	C₂×C₅	C₂×C₃×C₄×D₃	1	1	yes	characteristic
164	C₃²	C₂²×#168	1	1	yes	characteristic, complemented, Sylow
165	C₂³	—	3	1	no	—
166	C₂×C₄	—	3	1	no	—
167	C₂×C₄	—	3	2	no	—
168	C₂×C₄	—	3	2	no	—
169	C₂×C₄	C₃×D₃×D₅	1	1	yes	characteristic
170	C₈	—	5	2	no	—
171	C₈	—	15	2	no	—
172	C₂×C₃	—	2	1	no	—
173	C₂×C₃	—	2	2	no	—
174	C₂×C₃	—	3	4	no	—
175	C₂×C₃	D₃×#168	1	2	yes	direct product factor
176	C₂×C₃	C₂×C₃×#168	1	2	yes	complemented
177	C₂×C₃	C₂×D₃×Dic₅	1	1	yes	characteristic
178	C₂×C₃	C₂²×C₃×Dic₅	1	1	yes	characteristic
179	D₃	C₂×C₃×#168	1	4	yes	direct product factor
180	C₅	C₂×C₃×D₃×C₈	1	1	yes	characteristic, complemented, Sylow
181	C₂²	—	3	2	no	—
182	C₂²	—	3	4	no	—
183	C₂²	C₃×D₃×Dic₅	1	1	yes	characteristic
184	C₄	—	3	2	no	—
185	C₄	C₃×D₃×Dic₅	1	1	yes	characteristic
186	C₄	C₂×C₃×D₃×D₅	1	1	yes	characteristic
187	C₃	—	2	1	no	—
188	C₃	C₂×D₃×#168	1	1	yes	characteristic, direct product factor
189	C₃	C₂²×C₃×#168	1	1	yes	characteristic, complemented
190	C₂	—	3	4	no	—
191	C₂	C₃×D₃×#168	1	2	yes	direct product factor
192	C₂	C₂×C₃×D₃×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃×C₄	D₃×D₅
commutator, hypocenter	C₃×C₅	C₂²×C₃×C₈
Frattini	C₄	C₂×C₃×D₃×D₅
Fitting	C₂×C₃²×C₄×C₅	C₂²
socle	C₂²×C₃²×C₅	C₂×C₄
prime radical	C₂²×C₃×C₅×D₃	C₄
cosocle	C₃×C₄×C₅	C₂³×C₃
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₃×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₃⋊#168	C₂	semidirect
3. C₂×C₃²×#168	C₂	semidirect
4. C₃×D₃×#168	C₂	direct (N × H)
6. C₂×D₃×#168	C₃	direct (N × H)
12. C₃×C₃⋊#168	C₂²	semidirect
13. C₃²×#168	C₂²	semidirect
17. C₂×C₃⋊#168	C₂×C₃	semidirect
21. C₂×C₃×#168	D₃	direct (N × H)
22. C₂×C₃×#168	C₂×C₃	semidirect
23. D₃×#168	C₂×C₃	direct (N × H)
25. C₂×C₃×C₅×D₃	C₈	semidirect
46. C₃⋊#168	C₂²×C₃	semidirect
49. C₃×#168	C₂×D₃	direct (N × H)
50. C₃×#168	C₂²×C₃	semidirect
53. C₂×C₃²×C₅	C₂×C₈	semidirect
55. C₃×C₅×D₃	C₂×C₈	semidirect
59. C₂×#168	C₃×D₃	direct (N × H)
72. C₂×C₅×D₃	C₃×C₈	semidirect
91. C₃²×C₅	C₂²×C₈	semidirect
98. #168	C₂×C₃×D₃	direct (N × H)
100. C₂×C₃×D₃	#168	direct (N × H)
109. C₂×C₃×C₅	D₃×C₈	semidirect
110. C₂×C₃×C₅	C₂×C₃×C₈	semidirect
113. C₅×D₃	C₂×C₃×C₈	semidirect
136. C₂×C₃²	C₂×#168	semidirect
138. C₃×D₃	C₂×#168	direct (N × H)
144. C₃×C₅	C₂×D₃×C₈	semidirect
145. C₃×C₅	C₂²×C₃×C₈	semidirect
151. C₂×D₃	C₃×#168	direct (N × H)
162. C₂×C₅	C₃×D₃×C₈	semidirect
164. C₃²	C₂²×#168	semidirect
175. C₂×C₃	D₃×#168	direct (N × H)
176. C₂×C₃	C₂×C₃×#168	semidirect
179. D₃	C₂×C₃×#168	direct (N × H)
180. C₅	C₂×C₃×D₃×C₈	semidirect
188. C₃	C₂×D₃×#168	direct (N × H)
189. C₃	C₂²×C₃×#168	semidirect
191. C₂	C₃×D₃×#168	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×D₃	2
C₂×C₃×C₃⋊#168	2
C₂×C₃²×#168	2
C₃×D₃×#168	2
C₂²×C₃×#168	3
C₂×D₃×#168	3
C₂×C₃×D₃×C₈	5

Subgroups — C2×C3×D3×#168 #13086484

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×D₃×#168 #13086484