Subgroups — C₃×#10604414 #14315138

Order 1920.

324 conjugacy classes of subgroups.

The subgroup lattice has 216 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, complemented
2	C₃×D₄×#168	C₂	1	1	yes	maximal, characteristic
3	C₃×C₂²⋊#674	C₂	1	2	yes	maximal, complemented
4	C₃×C₄⋊#674	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₄×#674	C₂	1	1	yes	maximal, characteristic, complemented
6	#10604414	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×C₄×C₅×D₄	C₄	1	1	yes	characteristic
8	C₂²×C₃×#168	C₂²	1	2	yes	—
9	C₃×#3998	—	2	4	no	—
10	C₂×C₃×#674	C₂²	1	1	yes	characteristic
11	C₂×C₃×#674	C₂²	1	2	yes	—
12	C₂×C₃×#674	C₂²	1	1	yes	characteristic
13	C₃×C₂²⋊#168	C₄	1	2	yes	—
14	C₃×C₄⋊#168	C₄	1	1	yes	characteristic
15	C₃×C₄×#168	C₂²	1	1	yes	characteristic
16	C₃×#2169	—	5	1	no	maximal, Hall
17	C₂×#3998	C₂×C₃	1	2	yes	complemented
18	D₄×#168	C₂×C₃	1	1	yes	characteristic
19	C₂²⋊#674	C₂×C₃	1	2	yes	complemented
20	C₄⋊#674	C₂×C₃	1	1	yes	characteristic, complemented
21	C₄×#674	C₂×C₃	1	1	yes	characteristic, complemented
22	C₂×C₃×C₅×D₄	C₈	1	1	yes	characteristic
23	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
24	C₃×C₅×C₄⋊C₄	C₈	1	1	yes	characteristic
25	C₃×C₅×C₂²⋊C₄	C₈	1	2	yes	—
26	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×#168	—	2	2	no	—
28	C₂×C₃×#168	—	2	2	no	—
29	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×#168	C₂×C₄	1	2	yes	—
31	C₂×C₃×#168	C₂³	1	1	yes	characteristic
32	C₃×#674	—	2	2	no	—
33	C₃×#674	—	2	1	no	—
34	C₃×#674	D₄	1	2	yes	complemented
35	C₂×C₃×M₃₂	—	5	2	no	—
36	C₃×C₈×D₄	—	5	1	no	—
37	C₃×C₄⋊C₁₆	—	5	1	no	—
38	C₃×C₂²⋊C₁₆	—	5	2	no	—
39	C₃×C₄×C₁₆	—	5	1	no	—
40	C₄×C₅×D₄	C₃×C₄	1	1	yes	characteristic
41	C₂²×#168	C₂²×C₃	1	2	yes	—
42	#3998	—	2	4	no	—
43	C₂×#674	C₂²×C₃	1	1	yes	characteristic
44	C₂×#674	C₂²×C₃	1	2	yes	—
45	C₂×#674	C₂²×C₃	1	1	yes	characteristic
46	C₂²⋊#168	C₃×C₄	1	2	yes	—
47	C₄⋊#168	C₃×C₄	1	1	yes	characteristic
48	C₄×#168	C₂²×C₃	1	1	yes	characteristic
49	#2169	—	5	1	no	Sylow
50	C₂³×C₃×C₅	C₂×C₈	1	2	yes	—
51	C₃×C₅×D₄	—	2	2	no	—
52	C₂×C₃×C₄×C₅	—	2	2	no	—
53	C₂×C₃×C₄×C₅	—	2	2	no	—
54	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
56	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₃×#168	—	2	2	no	—
59	C₃×#168	—	2	1	no	—
60	C₃×#168	C₂×D₄	1	1	yes	characteristic
61	C₃×#168	#41	1	1	yes	characteristic
62	C₂²×C₃×C₈	—	5	2	no	—
63	C₃×C₄×D₄	AGL(1,𝔽₅)	1	1	yes	characteristic
64	C₃×M₃₂	—	10	4	no	—
65	C₂×C₃×C₁₆	—	5	1	no	—
66	C₂×C₃×C₁₆	—	5	2	no	—
67	C₂×C₃×C₁₆	—	5	1	no	—
68	C₃×C₄⋊C₈	—	5	1	no	—
69	C₃×C₂²⋊C₈	—	5	2	no	—
70	C₃×C₄×C₈	—	5	1	no	—
71	C₂×C₅×D₄	C₃×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
73	C₅×C₄⋊C₄	C₃×C₈	1	1	yes	characteristic
74	C₅×C₂²⋊C₄	C₃×C₈	1	2	yes	—
75	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
76	C₂×#168	—	2	2	no	—
77	C₂×#168	—	2	2	no	—
78	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
79	C₂×#168	C₂×C₃×C₄	1	2	yes	—
80	C₂×#168	C₂³×C₃	1	1	yes	characteristic
81	#674	—	2	2	no	—
82	#674	—	2	1	no	—
83	#674	C₃×D₄	1	2	yes	complemented
84	C₂×M₃₂	—	5	2	no	—
85	C₈×D₄	—	5	1	no	—
86	C₄⋊C₁₆	—	5	1	no	—
87	C₂²⋊C₁₆	—	5	2	no	—
88	C₄×C₁₆	—	5	1	no	—
89	C₂²×C₃×C₅	—	2	2	no	—
90	C₂²×C₃×C₅	—	2	2	no	—
91	C₂²×C₃×C₅	—	2	2	no	—
92	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
93	C₃×C₄×C₅	—	2	2	no	—
94	C₃×C₄×C₅	—	2	1	no	—
95	C₃×C₄×C₅	M₃₂	1	2	yes	complemented
96	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
97	C₃×C₄×C₅	#131	1	1	yes	characteristic
98	C₂×C₃×D₄	#170	1	1	yes	characteristic
99	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
100	C₂×C₃×C₈	—	5	1	no	—
101	C₂×C₃×C₈	—	5	1	no	—
102	C₂×C₃×C₈	—	5	2	no	—
103	C₂×C₃×C₈	—	10	2	no	—
104	C₂×C₃×C₈	—	10	2	no	—
105	C₃×C₄⋊C₄	#170	1	1	yes	characteristic
106	C₃×C₂²⋊C₄	#170	1	2	yes	—
107	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
108	C₃×C₁₆	—	5	2	no	—
109	C₃×C₁₆	—	10	2	no	—
110	C₃×C₁₆	—	10	1	no	—
111	C₂³×C₅	C₂×C₃×C₈	1	2	yes	—
112	C₅×D₄	—	2	2	no	—
113	C₂×C₄×C₅	—	2	2	no	—
114	C₂×C₄×C₅	—	2	2	no	—
115	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
116	C₂×C₄×C₅	C₂×C₃×C₈	1	2	yes	—
117	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
118	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
119	#168	—	2	2	no	—
120	#168	—	2	1	no	—
121	#168	C₂×C₃×D₄	1	1	yes	characteristic
122	#168	C₃×#41	1	1	yes	characteristic
123	C₂²×C₈	—	5	2	no	—
124	C₄×D₄	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
125	M₃₂	—	10	4	no	—
126	C₂×C₁₆	—	5	1	no	—
127	C₂×C₁₆	—	5	1	no	—
128	C₂×C₁₆	—	5	2	no	—
129	C₄⋊C₈	—	5	1	no	—
130	C₂²⋊C₈	—	5	2	no	—
131	C₄×C₈	—	5	1	no	—
132	C₂×C₃×C₅	—	4	2	no	—
133	C₂×C₃×C₅	C₈×D₄	1	1	yes	characteristic
134	C₂×C₃×C₅	C₂×M₃₂	1	1	yes	characteristic
135	C₂×C₃×C₅	#504	1	1	yes	characteristic
136	C₂³×C₃	C₂×#170	1	2	yes	—
137	C₃×D₄	—	2	2	no	—
138	C₂×C₃×C₄	—	2	2	no	—
139	C₂×C₃×C₄	—	2	2	no	—
140	C₂×C₃×C₄	C₂×#170	1	1	yes	characteristic
141	C₂×C₃×C₄	C₂×#170	1	2	yes	—
142	C₂×C₃×C₄	C₂×#170	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
144	C₃×C₈	—	5	1	no	—
145	C₃×C₈	—	5	1	no	—
146	C₃×C₈	—	10	1	no	—
147	C₃×C₈	—	10	2	no	—
148	C₂²×C₅	—	2	2	no	—
149	C₂²×C₅	—	2	2	no	—
150	C₂²×C₅	—	2	2	no	—
151	C₂²×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
152	C₄×C₅	—	2	1	no	—
153	C₄×C₅	—	2	2	no	—
154	C₄×C₅	C₃×M₃₂	1	2	yes	complemented
155	C₄×C₅	C₃×C₄×D₄	1	1	yes	characteristic
156	C₄×C₅	C₃×#131	1	1	yes	characteristic
157	C₂×D₄	C₃×#170	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
159	C₂×C₈	—	5	1	no	—
160	C₂×C₈	—	5	1	no	—
161	C₂×C₈	—	5	2	no	—
162	C₂×C₈	—	10	2	no	—
163	C₂×C₈	—	10	2	no	—
164	C₄⋊C₄	C₃×#170	1	1	yes	characteristic
165	C₂²⋊C₄	C₃×#170	1	2	yes	—
166	C₄²	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
167	C₁₆	—	5	2	no	—
168	C₁₆	—	10	2	no	—
169	C₁₆	—	10	1	no	—
170	C₃×C₅	#2169	1	1	yes	characteristic, complemented, Hall
171	C₂²×C₃	—	2	2	no	—
172	C₂²×C₃	—	2	2	no	—
173	C₂²×C₃	—	2	2	no	—
174	C₂²×C₃	C₂²×#170	1	1	yes	characteristic
175	C₃×C₄	—	2	2	no	—
176	C₃×C₄	—	2	1	no	—
177	C₃×C₄	#3998	1	2	yes	complemented
178	C₃×C₄	#4131	1	1	yes	characteristic
179	C₃×C₄	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
180	C₂×C₅	—	4	2	no	—
181	C₂×C₅	C₃×C₈×D₄	1	1	yes	characteristic
182	C₂×C₅	C₂×C₃×M₃₂	1	1	yes	characteristic
183	C₂×C₅	C₃×#504	1	1	yes	characteristic
184	C₂³	C₂×C₃×#170	1	2	yes	—
185	D₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	C₂×C₃×#170	1	1	yes	characteristic
189	C₂×C₄	C₂×C₃×#170	1	2	yes	—
190	C₂×C₄	C₂×C₃×#170	1	1	yes	characteristic
191	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
192	C₈	—	5	1	no	—
193	C₈	—	5	1	no	—
194	C₈	—	10	1	no	—
195	C₈	—	10	2	no	—
196	C₂×C₃	—	4	2	no	—
197	C₂×C₃	C₂×#3998	1	1	yes	characteristic
198	C₂×C₃	D₄×#170	1	1	yes	characteristic
199	C₂×C₃	#65791	1	1	yes	characteristic
200	C₅	C₃×#2169	1	1	yes	characteristic, complemented, Sylow
201	C₂²	—	2	2	no	—
202	C₂²	—	2	2	no	—
203	C₂²	—	2	2	no	—
204	C₂²	C₂²×C₃×#170	1	1	yes	characteristic
205	C₄	—	2	2	no	—
206	C₄	—	2	1	no	—
207	C₄	C₃×#3998	1	2	yes	complemented
208	C₄	C₃×#4131	1	1	yes	characteristic
209	C₄	C₃×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
210	C₃	#10604414	1	1	yes	characteristic, direct product factor, Sylow
211	C₂	—	4	2	no	—
212	C₂	C₂×C₃×#3998	1	1	yes	characteristic
213	C₂	C₃×D₄×#170	1	1	yes	characteristic
214	C₂	C₃×#65791	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)
commutator	C₂²×C₅	C₂²×C₃×C₈
Frattini	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₄×C₅×D₄	C₄
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂×C₃×C₅×D₄	C₈
cosocle	C₂×#168	C₂³×C₃
hypercenter	C₃×C₄×D₄	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#2169
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#3998	C₂	semidirect
3. C₃×C₂²⋊#674	C₂	semidirect
4. C₃×C₄⋊#674	C₂	semidirect
5. C₃×C₄×#674	C₂	semidirect
6. #10604414	C₃	direct (N × H)
17. C₂×#3998	C₂×C₃	semidirect
19. C₂²⋊#674	C₂×C₃	semidirect
20. C₄⋊#674	C₂×C₃	semidirect
21. C₄×#674	C₂×C₃	semidirect
34. C₃×#674	D₄	semidirect
83. #674	C₃×D₄	semidirect
95. C₃×C₄×C₅	M₃₂	semidirect
154. C₄×C₅	C₃×M₃₂	semidirect
170. C₃×C₅	#2169	semidirect
177. C₃×C₄	#3998	semidirect
200. C₅	C₃×#2169	semidirect
207. C₄	C₃×#3998	semidirect
210. C₃	#10604414	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×D₄×#168	2
C₃×C₂²⋊#674	2
C₃×C₄⋊#674	2
C₃×C₄×#674	2
#10604414	3
C₃×#2169	5

Subgroups — C3×#10604414 #14315138

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10604414 #14315138