Subgroups — C₃×#10604742 #14312865

Order 1920.

456 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#4053	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3998	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×#674	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#64887	C₂	1	4	yes	maximal, complemented
5	#10604742	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×#709	C₄	1	1	yes	characteristic
7	C₂×C₃×C₅×M₁₆	C₄	1	1	yes	characteristic
8	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
9	C₃×#4053	—	2	2	no	—
10	C₃×#4053	—	2	2	no	—
11	C₂×C₃×#676	C₄	1	2	yes	—
12	C₂×C₃×C₈×D₅	C₂²	1	2	yes	—
13	C₃×#3998	—	2	2	no	—
14	C₂×C₃×#674	—	2	2	no	—
15	C₂×C₃×#674	C₂²	1	2	yes	—
16	C₂×C₃×#674	C₂²	1	2	yes	—
17	C₃×#2113	—	5	1	no	maximal, Hall
18	C₂×#4053	C₂×C₃	1	1	yes	characteristic
19	C₂×#3998	C₂×C₃	1	1	yes	characteristic, complemented
20	C₂²×#674	C₂×C₃	1	1	yes	characteristic, complemented
21	#64887	C₂×C₃	1	4	yes	complemented
22	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₂²⋊D₅	C₈	1	2	yes	—
24	C₃×#709	—	2	2	no	—
25	C₃×#709	—	2	2	no	—
26	C₂×C₃×D₂₀	C₈	1	1	yes	characteristic
27	C₂×C₃×C₄×D₅	C₂×C₄	1	2	yes	—
28	C₂×C₃×Dic₁₀	C₈	1	1	yes	characteristic
29	C₃×C₅×M₁₆	—	2	2	no	—
30	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
31	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×#168	D₄	1	2	yes	—
33	C₂×C₃×#168	D₄	1	2	yes	—
34	C₂×C₃×#168	C₂³	1	1	yes	characteristic
35	C₃×#676	—	2	4	no	—
36	C₃×C₈×D₅	—	2	2	no	—
37	C₃×C₈×D₅	—	2	2	no	—
38	C₃×#674	—	2	2	no	—
39	C₃×#674	—	2	2	no	—
40	C₂×C₃×#131	—	5	1	no	—
41	C₂×C₃×M₃₂	—	5	1	no	—
42	C₂²×C₃×C₁₆	—	5	1	no	—
43	C₃×C₂²⋊C₁₆	—	5	4	no	—
44	C₂×#709	C₃×C₄	1	1	yes	characteristic
45	C₂×C₅×M₁₆	C₃×C₄	1	1	yes	characteristic
46	C₂²×#168	C₂²×C₃	1	1	yes	characteristic
47	#4053	—	2	2	no	—
48	#4053	—	2	2	no	—
49	C₂×#676	C₃×C₄	1	2	yes	—
50	C₂×C₈×D₅	C₂²×C₃	1	2	yes	—
51	#3998	—	2	2	no	—
52	C₂×#674	—	2	2	no	—
53	C₂×#674	C₂²×C₃	1	2	yes	—
54	C₂×#674	C₂²×C₃	1	2	yes	—
55	#2113	—	5	1	no	Sylow
56	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₃×D₅	C₂×C₈	1	2	yes	—
58	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
59	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
61	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₃×C₂²⋊D₅	—	2	4	no	—
63	C₂×C₃×Dic₅	C₂×C₈	1	2	yes	—
64	C₃×D₂₀	—	2	2	no	—
65	C₃×C₄×D₅	—	2	2	no	—
66	C₃×C₄×D₅	—	2	2	no	—
67	C₃×Dic₁₀	—	2	2	no	—
68	C₃×C₅×C₈	—	2	2	no	—
69	C₃×#168	C₂²⋊C₄	1	2	yes	—
70	C₃×#168	C₂×D₄	1	1	yes	characteristic
71	C₃×#168	C₂×D₄	1	1	yes	characteristic
72	C₂×C₃×#41	—	5	1	no	—
73	C₃×#131	—	10	2	no	—
74	C₃×#131	—	10	2	no	—
75	C₂×C₃×M₁₆	—	5	2	no	—
76	C₂×C₃×M₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic
77	C₂²×C₃×C₈	—	5	1	no	—
78	C₂²×C₃×C₈	—	5	2	no	—
79	C₃×M₃₂	—	10	2	no	—
80	C₂×C₃×C₁₆	—	5	2	no	—
81	C₂×C₃×C₁₆	—	5	2	no	—
82	C₂×C₃×C₁₆	—	10	2	no	—
83	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
84	C₂×C₂²⋊D₅	C₃×C₈	1	2	yes	—
85	#709	—	2	2	no	—
86	#709	—	2	2	no	—
87	C₂×D₂₀	C₃×C₈	1	1	yes	characteristic
88	C₂×C₄×D₅	C₂×C₃×C₄	1	2	yes	—
89	C₂×Dic₁₀	C₃×C₈	1	1	yes	characteristic
90	C₅×M₁₆	—	2	2	no	—
91	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
92	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
93	C₂×#168	C₃×D₄	1	2	yes	—
94	C₂×#168	C₃×D₄	1	2	yes	—
95	C₂×#168	C₂³×C₃	1	1	yes	characteristic
96	#676	—	2	4	no	—
97	C₈×D₅	—	2	2	no	—
98	C₈×D₅	—	2	2	no	—
99	#674	—	2	2	no	—
100	#674	—	2	2	no	—
101	C₂×#131	—	5	1	no	—
102	C₂×M₃₂	—	5	1	no	—
103	C₂²×C₁₆	—	5	1	no	—
104	C₂²⋊C₁₆	—	5	4	no	—
105	C₂²×C₃×C₅	—	2	2	no	—
106	C₂²×C₃×C₅	C₂²⋊C₈	1	2	yes	—
107	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₃×D₅	—	2	2	no	—
109	C₂×C₃×D₅	—	2	4	no	—
110	C₃×C₄×C₅	C₂²⋊C₈	1	2	yes	—
111	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
113	C₃×Dic₅	—	2	2	no	—
114	C₃×#41	—	10	2	no	—
115	C₃×#41	—	10	2	no	—
116	C₂×C₃×Q₈	—	5	1	no	—
117	C₂×C₃×D₄	—	5	2	no	—
118	C₂×C₃×D₄	—	5	1	no	—
119	C₂²×C₃×C₄	—	5	2	no	—
120	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
121	C₃×M₁₆	—	2	2	no	—
122	C₃×M₁₆	—	10	4	no	—
123	C₂×C₃×C₈	—	5	2	no	—
124	C₂×C₃×C₈	—	5	2	no	—
125	C₂×C₃×C₈	—	5	1	no	—
126	C₂×C₃×C₈	—	5	1	no	—
127	C₂×C₃×C₈	—	10	2	no	—
128	C₂×C₃×C₈	—	10	2	no	—
129	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
130	C₃×C₁₆	—	10	2	no	—
131	C₃×C₁₆	—	10	2	no	—
132	C₂³×C₅	C₂×C₃×C₈	1	1	yes	characteristic
133	C₂²×D₅	C₂×C₃×C₈	1	2	yes	—
134	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
135	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
136	C₂×C₄×C₅	C₃×M₁₆	1	2	yes	—
137	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
138	C₂²⋊D₅	—	2	4	no	—
139	C₂×Dic₅	C₂×C₃×C₈	1	2	yes	—
140	D₂₀	—	2	2	no	—
141	C₄×D₅	—	2	2	no	—
142	C₄×D₅	—	2	2	no	—
143	Dic₁₀	—	2	2	no	—
144	C₅×C₈	—	2	2	no	—
145	#168	C₃×C₂²⋊C₄	1	2	yes	—
146	#168	C₂×C₃×D₄	1	1	yes	characteristic
147	#168	C₂×C₃×D₄	1	1	yes	characteristic
148	C₂×#41	—	5	1	no	—
149	#131	—	10	2	no	—
150	#131	—	10	2	no	—
151	C₂×M₁₆	—	5	2	no	—
152	C₂×M₁₆	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
153	C₂²×C₈	—	5	1	no	—
154	C₂²×C₈	—	5	2	no	—
155	M₃₂	—	10	2	no	—
156	C₂×C₁₆	—	5	2	no	—
157	C₂×C₁₆	—	5	2	no	—
158	C₂×C₁₆	—	10	2	no	—
159	C₂×C₃×C₅	—	2	2	no	—
160	C₂×C₃×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
161	C₂×C₃×C₅	#504	1	2	yes	—
162	C₃×D₅	—	4	2	no	—
163	C₂³×C₃	—	5	2	no	—
164	C₂³×C₃	D₅⋊C₈	1	1	yes	characteristic
165	C₃×Q₈	—	10	2	no	—
166	C₃×D₄	—	10	4	no	—
167	C₃×D₄	—	10	2	no	—
168	C₂×C₃×C₄	—	5	2	no	—
169	C₂×C₃×C₄	—	10	2	no	—
170	C₂×C₃×C₄	—	10	2	no	—
171	C₂×C₃×C₄	D₅⋊C₈	1	1	yes	characteristic
172	C₂×C₃×C₄	#700	1	2	yes	—
173	C₂×C₃×C₄	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
174	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
175	C₃×C₈	—	2	2	no	—
176	C₃×C₈	—	5	2	no	—
177	C₃×C₈	—	5	1	no	—
178	C₃×C₈	—	5	1	no	—
179	C₂²×C₅	—	2	2	no	—
180	C₂²×C₅	C₃×C₂²⋊C₈	1	2	yes	—
181	C₂²×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
182	C₂×D₅	—	2	4	no	—
183	C₂×D₅	—	2	2	no	—
184	C₄×C₅	C₃×C₂²⋊C₈	1	2	yes	—
185	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
186	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
187	Dic₅	—	2	2	no	—
188	#41	—	10	2	no	—
189	#41	—	10	2	no	—
190	C₂×Q₈	—	5	1	no	—
191	C₂×D₄	—	5	2	no	—
192	C₂×D₄	—	5	1	no	—
193	C₂²×C₄	—	5	2	no	—
194	C₂²×C₄	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
195	M₁₆	—	2	2	no	—
196	M₁₆	—	10	4	no	—
197	C₂×C₈	—	5	2	no	—
198	C₂×C₈	—	5	2	no	—
199	C₂×C₈	—	5	1	no	—
200	C₂×C₈	—	5	1	no	—
201	C₂×C₈	—	10	2	no	—
202	C₂×C₈	—	10	2	no	—
203	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
204	C₁₆	—	10	2	no	—
205	C₁₆	—	10	2	no	—
206	C₃×C₅	#2113	1	1	yes	characteristic, complemented, Hall
207	C₂²×C₃	—	2	2	no	—
208	C₂²×C₃	—	10	4	no	—
209	C₂²×C₃	—	10	2	no	—
210	C₂²×C₃	#4003	1	2	yes	—
211	C₂²×C₃	C₂×D₅⋊C₈	1	1	yes	characteristic
212	C₃×C₄	—	10	2	no	—
213	C₃×C₄	#4003	1	2	yes	—
214	C₃×C₄	C₂×#700	1	1	yes	characteristic
215	C₃×C₄	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
216	C₂×C₅	—	2	2	no	—
217	C₂×C₅	C₂×C₃×C₂²⋊C₈	1	1	yes	characteristic
218	C₂×C₅	C₃×#504	1	2	yes	—
219	D₅	—	4	2	no	—
220	C₂³	—	5	2	no	—
221	C₂³	C₃×D₅⋊C₈	1	1	yes	characteristic
222	Q₈	—	10	2	no	—
223	D₄	—	10	4	no	—
224	D₄	—	10	2	no	—
225	C₂×C₄	—	5	2	no	—
226	C₂×C₄	—	10	2	no	—
227	C₂×C₄	—	10	2	no	—
228	C₂×C₄	C₃×D₅⋊C₈	1	1	yes	characteristic
229	C₂×C₄	C₃×#700	1	2	yes	—
230	C₂×C₄	C₃×C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
231	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
232	C₈	—	2	2	no	—
233	C₈	—	5	1	no	—
234	C₈	—	5	1	no	—
235	C₈	—	5	2	no	—
236	C₂×C₃	—	2	2	no	—
237	C₂×C₃	—	20	2	no	—
238	C₂×C₃	#65725	1	2	yes	—
239	C₂×C₃	C₂×#4003	1	1	yes	characteristic
240	C₅	C₃×#2113	1	1	yes	characteristic, complemented, Sylow
241	C₂²	—	2	2	no	—
242	C₂²	—	10	4	no	—
243	C₂²	—	10	2	no	—
244	C₂²	C₃×#4003	1	2	yes	—
245	C₂²	C₂×C₃×D₅⋊C₈	1	1	yes	characteristic
246	C₄	—	10	2	no	—
247	C₄	C₃×#4003	1	2	yes	—
248	C₄	C₂×C₃×#700	1	1	yes	characteristic
249	C₄	C₂×C₃×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
250	C₃	#10604742	1	1	yes	characteristic, direct product factor, Sylow
251	C₂	—	2	2	no	—
252	C₂	—	20	2	no	—
253	C₂	C₃×#65725	1	2	yes	—
254	C₂	C₂×C₃×#4003	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	C₂×C₃×C₅×M₁₆	C₄
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂×C₃×#709	C₄
cosocle	C₂×#168	C₂³×C₃
hypercenter	C₂×C₃×M₁₆	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#2113
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#3998	C₂	semidirect
3. C₂²×C₃×#674	C₂	semidirect
4. C₃×#64887	C₂	semidirect
5. #10604742	C₃	direct (N × H)
19. C₂×#3998	C₂×C₃	semidirect
20. C₂²×#674	C₂×C₃	semidirect
21. #64887	C₂×C₃	semidirect
206. C₃×C₅	#2113	semidirect
240. C₅	C₃×#2113	semidirect
250. C₃	#10604742	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×#4053	2
C₂×C₃×#3998	2
C₂²×C₃×#674	2
C₃×#64887	2
#10604742	3
C₃×#2113	5

Subgroups — C3×#10604742 #14312865

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10604742 #14312865