Subgroups — C₃×#10605007 #14312863

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×#4094	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×C₂²⋊#674	C₂	1	4	yes	maximal, complemented
5	#10605007	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×C₅×#41	C₄	1	1	yes	characteristic
7	C₃×#4094	—	2	2	no	—
8	C₃×#4094	—	2	2	no	—
9	C₂×C₃×#681	C₄	1	1	yes	characteristic
10	C₂×C₃×#681	C₄	1	2	yes	—
11	C₂²×C₃×#168	C₂²	1	2	yes	—
12	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
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×#4094	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	C₂²⋊#674	C₂×C₃	1	4	yes	complemented
22	C₃×C₅×#41	—	2	2	no	—
23	C₃×C₅×#41	—	2	2	no	—
24	C₂×C₃×C₅×Q₈	C₈	1	1	yes	characteristic
25	C₂×C₃×C₅×D₄	C₈	1	2	yes	—
26	C₂×C₃×C₅×D₄	C₈	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
28	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
29	C₃×#681	—	2	2	no	—
30	C₃×#681	—	2	4	no	—
31	C₂×C₃×#168	—	2	2	no	—
32	C₂×C₃×#168	—	2	2	no	—
33	C₂×C₃×#168	C₂×C₄	1	2	yes	—
34	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×#168	D₄	1	2	yes	—
36	C₂×C₃×#168	D₄	1	2	yes	—
37	C₂×C₃×#168	C₂³	1	1	yes	characteristic
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×C₅×#41	C₃×C₄	1	1	yes	characteristic
45	#4094	—	2	2	no	—
46	#4094	—	2	2	no	—
47	C₂×#681	C₃×C₄	1	1	yes	characteristic
48	C₂×#681	C₃×C₄	1	2	yes	—
49	C₂²×#168	C₂²×C₃	1	2	yes	—
50	C₂²×#168	C₂²×C₃	1	1	yes	characteristic
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	2	yes	—
57	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
58	C₃×C₅×Q₈	—	2	2	no	—
59	C₃×C₅×D₄	—	2	4	no	—
60	C₃×C₅×D₄	—	2	2	no	—
61	C₂×C₃×C₄×C₅	—	2	2	no	—
62	C₂×C₃×C₄×C₅	—	2	2	no	—
63	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
64	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
65	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
67	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
68	C₃×#168	—	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	AGL(1,𝔽₅)	1	1	yes	characteristic
73	C₃×#131	—	10	2	no	—
74	C₃×#131	—	10	2	no	—
75	C₂×C₃×M₁₆	—	5	2	no	—
76	C₂×C₃×M₁₆	—	5	1	no	—
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	C₅×#41	—	2	2	no	—
84	C₅×#41	—	2	2	no	—
85	C₂×C₅×Q₈	C₃×C₈	1	1	yes	characteristic
86	C₂×C₅×D₄	C₃×C₈	1	2	yes	—
87	C₂×C₅×D₄	C₃×C₈	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
89	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
90	#681	—	2	2	no	—
91	#681	—	2	4	no	—
92	C₂×#168	—	2	2	no	—
93	C₂×#168	—	2	2	no	—
94	C₂×#168	C₂×C₃×C₄	1	2	yes	—
95	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
96	C₂×#168	C₃×D₄	1	2	yes	—
97	C₂×#168	C₃×D₄	1	2	yes	—
98	C₂×#168	C₂³×C₃	1	1	yes	characteristic
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	—	2	2	no	—
107	C₂²×C₃×C₅	—	2	4	no	—
108	C₂²×C₃×C₅	C₂²⋊C₈	1	2	yes	—
109	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
110	C₃×C₄×C₅	—	2	2	no	—
111	C₃×C₄×C₅	C₂²⋊C₈	1	2	yes	—
112	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
114	C₃×#41	—	2	2	no	—
115	C₃×#41	—	2	2	no	—
116	C₂×C₃×Q₈	#170	1	1	yes	characteristic
117	C₂×C₃×D₄	#170	1	2	yes	—
118	C₂×C₃×D₄	#170	1	1	yes	characteristic
119	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
120	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
121	C₃×M₁₆	—	10	4	no	—
122	C₃×M₁₆	—	10	2	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	—	5	2	no	—
128	C₂×C₃×C₈	—	10	2	no	—
129	C₂×C₃×C₈	—	10	2	no	—
130	C₃×C₁₆	—	10	2	no	—
131	C₃×C₁₆	—	10	2	no	—
132	C₂³×C₅	C₂×C₃×C₈	1	2	yes	—
133	C₂³×C₅	C₂×C₃×C₈	1	1	yes	characteristic
134	C₅×Q₈	—	2	2	no	—
135	C₅×D₄	—	2	4	no	—
136	C₅×D₄	—	2	2	no	—
137	C₂×C₄×C₅	—	2	2	no	—
138	C₂×C₄×C₅	—	2	2	no	—
139	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
140	C₂×C₄×C₅	C₂×C₃×C₈	1	2	yes	—
141	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
142	C₂×C₄×C₅	C₃×M₁₆	1	2	yes	—
143	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
144	#168	—	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	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
149	#131	—	10	2	no	—
150	#131	—	10	2	no	—
151	C₂×M₁₆	—	5	2	no	—
152	C₂×M₁₆	—	5	1	no	—
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	—	4	2	no	—
161	C₂×C₃×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
162	C₂×C₃×C₅	#504	1	2	yes	—
163	C₂³×C₃	C₂×#170	1	2	yes	—
164	C₂³×C₃	C₂×#170	1	1	yes	characteristic
165	C₃×Q₈	—	2	2	no	—
166	C₃×D₄	—	2	4	no	—
167	C₃×D₄	—	2	2	no	—
168	C₂×C₃×C₄	—	2	2	no	—
169	C₂×C₃×C₄	—	2	2	no	—
170	C₂×C₃×C₄	C₂×#170	1	2	yes	—
171	C₂×C₃×C₄	C₂×#170	1	1	yes	characteristic
172	C₂×C₃×C₄	#704	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	—	5	2	no	—
176	C₃×C₈	—	5	1	no	—
177	C₃×C₈	—	5	1	no	—
178	C₃×C₈	—	10	2	no	—
179	C₂²×C₅	—	2	2	no	—
180	C₂²×C₅	—	2	4	no	—
181	C₂²×C₅	—	2	2	no	—
182	C₂²×C₅	C₃×C₂²⋊C₈	1	2	yes	—
183	C₂²×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
184	C₄×C₅	—	2	2	no	—
185	C₄×C₅	C₃×C₂²⋊C₈	1	2	yes	—
186	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
187	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
188	#41	—	2	2	no	—
189	#41	—	2	2	no	—
190	C₂×Q₈	C₃×#170	1	1	yes	characteristic
191	C₂×D₄	C₃×#170	1	2	yes	—
192	C₂×D₄	C₃×#170	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
194	C₂²×C₄	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
195	M₁₆	—	10	4	no	—
196	M₁₆	—	10	2	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	—	5	2	no	—
202	C₂×C₈	—	10	2	no	—
203	C₂×C₈	—	10	2	no	—
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	—	2	2	no	—
209	C₂²×C₃	—	2	4	no	—
210	C₂²×C₃	C₂²⋊#170	1	2	yes	—
211	C₂²×C₃	C₂²×#170	1	1	yes	characteristic
212	C₃×C₄	—	2	2	no	—
213	C₃×C₄	C₂²⋊#170	1	2	yes	—
214	C₃×C₄	C₂×#704	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	—	4	2	no	—
218	C₂×C₅	C₂×C₃×C₂²⋊C₈	1	1	yes	characteristic
219	C₂×C₅	C₃×#504	1	2	yes	—
220	C₂³	C₂×C₃×#170	1	2	yes	—
221	C₂³	C₂×C₃×#170	1	1	yes	characteristic
222	Q₈	—	2	2	no	—
223	D₄	—	2	4	no	—
224	D₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	C₂×C₃×#170	1	2	yes	—
228	C₂×C₄	C₂×C₃×#170	1	1	yes	characteristic
229	C₂×C₄	C₃×#704	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	—	5	1	no	—
233	C₈	—	5	1	no	—
234	C₈	—	5	2	no	—
235	C₈	—	10	2	no	—
236	C₂×C₃	—	2	2	no	—
237	C₂×C₃	—	4	2	no	—
238	C₂×C₃	#65791	1	2	yes	—
239	C₂×C₃	C₂×C₂²⋊#170	1	1	yes	characteristic
240	C₅	C₃×#2113	1	1	yes	characteristic, complemented, Sylow
241	C₂²	—	2	2	no	—
242	C₂²	—	2	4	no	—
243	C₂²	—	2	2	no	—
244	C₂²	C₃×C₂²⋊#170	1	2	yes	—
245	C₂²	C₂²×C₃×#170	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	C₃×C₂²⋊#170	1	2	yes	—
248	C₄	C₂×C₃×#704	1	1	yes	characteristic
249	C₄	C₂×C₃×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
250	C₃	#10605007	1	1	yes	characteristic, direct product factor, Sylow
251	C₂	—	2	2	no	—
252	C₂	—	4	2	no	—
253	C₂	C₃×#65791	1	2	yes	—
254	C₂	C₂×C₃×C₂²⋊#170	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, prime radical	C₂×C₃×C₅×#41	C₄
socle	C₂²×C₃×C₅	C₂²×C₈
cosocle	C₂×#168	C₂³×C₃
hypercenter	C₂×C₃×#41	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#2113
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×#41

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×C₂²⋊#674	C₂	semidirect
5. #10605007	C₃	direct (N × H)
19. C₂×#3998	C₂×C₃	semidirect
20. C₂²×#674	C₂×C₃	semidirect
21. C₂²⋊#674	C₂×C₃	semidirect
206. C₃×C₅	#2113	semidirect
240. C₅	C₃×#2113	semidirect
250. C₃	#10605007	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×#4094	2
C₂×C₃×#3998	2
C₂²×C₃×#674	2
C₃×C₂²⋊#674	2
#10605007	3
C₃×#2113	5

Subgroups — C3×#10605007 #14312863

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10605007 #14312863