Subgroups — C₃×#10598304 #14278755

Order 1920.

264 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₃×#65390	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#3944	C₂	1	1	yes	maximal, characteristic
4	#10598304	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₃×#4072	C₄	1	1	yes	characteristic
7	C₂×C₃×#681	C₄	1	1	yes	characteristic
8	C₃×C₅×M₃₂	—	2	1	no	—
9	C₃×C₅×M₃₂	—	2	1	no	—
10	C₂×C₃×C₅×C₁₆	C₄	1	2	yes	complemented
11	C₃×#3947	—	2	1	no	—
12	C₃×C₈×Dic₅	—	2	1	no	—
13	C₃×#3944	—	2	1	no	—
14	C₃×#3944	C₄	1	2	yes	complemented
15	C₂×C₃×#672	—	2	1	no	—
16	C₃×#1378	—	5	1	no	maximal, Hall
17	C₂×C₅×M₃₂	C₂×C₃	1	1	yes	characteristic
18	#65390	C₂×C₃	1	1	yes	characteristic
19	C₂×#3944	C₂×C₃	1	1	yes	characteristic
20	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
24	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
25	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
26	C₂×C₃×C₅×C₈	Q₈	1	1	yes	characteristic
27	C₃×C₂²⋊Dic₅	C₈	1	2	yes	—
28	C₃×C₄⋊Dic₅	—	2	1	no	—
29	C₃×C₄×Dic₅	—	2	1	no	—
30	C₃×#681	—	2	1	no	—
31	C₃×#681	C₈	1	2	yes	—
32	C₂×C₃×#168	—	2	1	no	—
33	C₃×C₅×C₁₆	—	2	2	no	—
34	C₃×#672	—	2	2	no	—
35	C₂×C₃×M₃₂	—	5	1	no	—
36	C₂×C₃×M₃₂	D₅	1	1	yes	characteristic
37	C₃×#405	—	5	1	no	—
38	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
39	#4072	C₃×C₄	1	1	yes	characteristic
40	C₂×#681	C₃×C₄	1	1	yes	characteristic
41	C₅×M₃₂	—	2	1	no	—
42	C₅×M₃₂	—	2	1	no	—
43	C₂×C₅×C₁₆	C₃×C₄	1	2	yes	complemented
44	#3947	—	2	1	no	—
45	C₈×Dic₅	—	2	1	no	—
46	#3944	—	2	1	no	—
47	#3944	C₃×C₄	1	2	yes	complemented
48	C₂×#672	—	2	1	no	—
49	#1378	—	5	1	no	Sylow
50	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	M₁₆	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	M₁₆	1	1	yes	characteristic
57	C₂×C₃×Dic₅	—	2	2	no	—
58	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
59	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
60	C₃×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
61	C₃×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
62	C₃×#168	—	2	2	no	—
63	C₂×C₃×M₁₆	—	5	1	no	—
64	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
65	C₃×#117	—	5	1	no	—
66	C₃×M₃₂	—	2	1	no	—
67	C₃×M₃₂	—	2	1	no	—
68	C₃×M₃₂	—	5	2	no	—
69	C₃×M₃₂	—	10	1	no	—
70	C₂×C₃×C₁₆	—	10	1	no	—
71	C₂×C₃×C₁₆	Dic₅	1	2	yes	complemented
72	C₃×#97	—	10	1	no	—
73	C₃×C₄×C₈	—	10	1	no	—
74	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
75	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
76	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
77	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
78	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
79	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
80	C₂×C₅×C₈	C₃×Q₈	1	1	yes	characteristic
81	C₂²⋊Dic₅	C₃×C₈	1	2	yes	—
82	C₄⋊Dic₅	—	2	1	no	—
83	C₄×Dic₅	—	2	1	no	—
84	#681	—	2	1	no	—
85	#681	C₃×C₈	1	2	yes	—
86	C₂×#168	—	2	1	no	—
87	C₅×C₁₆	—	2	2	no	—
88	#672	—	2	2	no	—
89	C₂×M₃₂	—	5	1	no	—
90	C₂×M₃₂	C₃×D₅	1	1	yes	characteristic
91	#405	—	5	1	no	—
92	C₂²×C₃×C₅	—	4	1	no	—
93	C₂²×C₃×C₅	C₄×C₈	1	1	yes	characteristic
94	C₂²×C₃×C₅	C₂²⋊C₈	1	1	yes	characteristic
95	C₂²×C₃×C₅	C₄⋊C₈	1	1	yes	characteristic
96	C₃×C₄×C₅	#95	1	1	yes	characteristic
97	C₃×C₄×C₅	#97	1	1	yes	characteristic
98	C₃×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
99	C₃×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
100	C₃×Dic₅	—	4	2	no	—
101	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
102	C₃×M₁₆	—	5	2	no	—
103	C₃×M₁₆	—	10	1	no	—
104	C₂×C₃×C₈	—	10	1	no	—
105	C₂×C₃×C₈	Dic₁₀	1	1	yes	characteristic
106	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
107	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
108	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
109	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
110	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
111	C₃×C₄⋊C₄	—	10	1	no	—
112	C₃×C₂²⋊C₄	—	5	2	no	—
113	C₃×C₄²	—	10	1	no	—
114	C₃×C₁₆	—	2	2	no	—
115	C₃×C₁₆	—	10	2	no	—
116	C₂³×C₅	C₂×C₃×C₈	1	1	yes	characteristic
117	C₂×C₄×C₅	C₃×C₄²	1	1	yes	characteristic
118	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
121	C₂×C₄×C₅	C₃×M₁₆	1	1	yes	characteristic
122	C₂×C₄×C₅	C₃×M₁₆	1	1	yes	characteristic
123	C₂×Dic₅	—	2	2	no	—
124	C₅×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
125	C₅×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
126	C₅×C₈	C₃×C₄⋊C₄	1	1	yes	characteristic
127	C₅×C₈	C₃×C₄⋊C₄	1	1	yes	characteristic
128	#168	—	2	2	no	—
129	C₂×M₁₆	—	5	1	no	—
130	C₂²×C₈	C₂×C₃×D₅	1	1	yes	characteristic
131	#117	—	5	1	no	—
132	M₃₂	—	2	1	no	—
133	M₃₂	—	2	1	no	—
134	M₃₂	—	5	2	no	—
135	M₃₂	—	10	1	no	—
136	C₂×C₁₆	—	10	1	no	—
137	C₂×C₁₆	C₃×Dic₅	1	2	yes	complemented
138	#97	—	10	1	no	—
139	C₄×C₈	—	10	1	no	—
140	C₂×C₃×C₅	—	2	1	no	—
141	C₂×C₃×C₅	—	2	1	no	—
142	C₂×C₃×C₅	—	2	1	no	—
143	C₂×C₃×C₅	#336	1	1	yes	characteristic
144	C₂³×C₃	C₈×D₅	1	1	yes	characteristic
145	C₂×C₃×C₄	—	10	2	no	—
146	C₂×C₃×C₄	C₈×D₅	1	1	yes	characteristic
147	C₂×C₃×C₄	#676	1	1	yes	characteristic
148	C₂×C₃×C₄	#676	1	1	yes	characteristic
149	C₂×C₃×C₄	C₄×Dic₅	1	1	yes	characteristic
150	C₂×C₃×C₄	#683	1	1	yes	characteristic
151	C₂×C₃×C₄	#685	1	1	yes	characteristic
152	C₃×C₈	—	10	2	no	—
153	C₃×C₈	#683	1	1	yes	characteristic
154	C₃×C₈	C₄⋊Dic₅	1	1	yes	characteristic
155	C₃×C₈	#685	1	1	yes	characteristic
156	C₃×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
157	C₂²×C₅	—	4	1	no	—
158	C₂²×C₅	C₃×C₄×C₈	1	1	yes	characteristic
159	C₂²×C₅	C₃×C₂²⋊C₈	1	1	yes	characteristic
160	C₂²×C₅	C₃×C₄⋊C₈	1	1	yes	characteristic
161	C₄×C₅	C₃×#95	1	1	yes	characteristic
162	C₄×C₅	C₃×#97	1	1	yes	characteristic
163	C₄×C₅	C₃×C₂²⋊C₈	1	1	yes	characteristic
164	C₄×C₅	C₃×C₄⋊C₈	1	1	yes	characteristic
165	Dic₅	—	4	2	no	—
166	C₂²×C₄	C₃×C₄×D₅	1	1	yes	characteristic
167	M₁₆	—	5	2	no	—
168	M₁₆	—	10	1	no	—
169	C₂×C₈	—	10	1	no	—
170	C₂×C₈	C₃×Dic₁₀	1	1	yes	characteristic
171	C₂×C₈	C₃×C₄×D₅	1	1	yes	characteristic
172	C₂×C₈	C₃×D₂₀	1	1	yes	characteristic
173	C₂×C₈	C₂×C₃×Dic₅	1	1	yes	characteristic
174	C₂×C₈	C₃×C₂²⋊D₅	1	1	yes	characteristic
175	C₂×C₈	C₃×C₂²⋊D₅	1	1	yes	characteristic
176	C₄⋊C₄	—	10	1	no	—
177	C₂²⋊C₄	—	5	2	no	—
178	C₄²	—	10	1	no	—
179	C₁₆	—	2	2	no	—
180	C₁₆	—	10	2	no	—
181	C₃×C₅	#1378	1	1	yes	characteristic, complemented, Hall
182	C₂²×C₃	—	4	1	no	—
183	C₂²×C₃	C₈×Dic₅	1	1	yes	characteristic
184	C₂²×C₃	#3946	1	1	yes	characteristic
185	C₂²×C₃	#3952	1	1	yes	characteristic
186	C₃×C₄	—	20	2	no	—
187	C₃×C₄	#3946	1	1	yes	characteristic
188	C₃×C₄	#3947	1	1	yes	characteristic
189	C₃×C₄	#3952	1	1	yes	characteristic
190	C₃×C₄	#3963	1	1	yes	characteristic
191	C₂×C₅	—	2	1	no	—
192	C₂×C₅	—	2	1	no	—
193	C₂×C₅	—	2	1	no	—
194	C₂×C₅	C₃×#336	1	1	yes	characteristic
195	C₂³	C₃×C₈×D₅	1	1	yes	characteristic
196	C₂×C₄	—	10	2	no	—
197	C₂×C₄	C₃×C₈×D₅	1	1	yes	characteristic
198	C₂×C₄	C₃×#676	1	1	yes	characteristic
199	C₂×C₄	C₃×#676	1	1	yes	characteristic
200	C₂×C₄	C₃×C₄×Dic₅	1	1	yes	characteristic
201	C₂×C₄	C₃×#683	1	1	yes	characteristic
202	C₂×C₄	C₃×#685	1	1	yes	characteristic
203	C₈	—	10	2	no	—
204	C₈	C₃×#683	1	1	yes	characteristic
205	C₈	C₃×C₄⋊Dic₅	1	1	yes	characteristic
206	C₈	C₃×#685	1	1	yes	characteristic
207	C₈	C₃×C₂²⋊Dic₅	1	1	yes	characteristic
208	C₂×C₃	—	2	1	no	—
209	C₂×C₃	—	2	1	no	—
210	C₂×C₃	—	2	1	no	—
211	C₂×C₃	#64770	1	1	yes	characteristic
212	C₅	C₃×#1378	1	1	yes	characteristic, complemented, Sylow
213	C₂²	—	4	1	no	—
214	C₂²	C₃×C₈×Dic₅	1	1	yes	characteristic
215	C₂²	C₃×#3946	1	1	yes	characteristic
216	C₂²	C₃×#3952	1	1	yes	characteristic
217	C₄	—	20	2	no	—
218	C₄	C₃×#3946	1	1	yes	characteristic
219	C₄	C₃×#3947	1	1	yes	characteristic
220	C₄	C₃×#3952	1	1	yes	characteristic
221	C₄	C₃×#3963	1	1	yes	characteristic
222	C₃	#10598304	1	1	yes	characteristic, direct product factor, Sylow
223	C₂	—	2	1	no	—
224	C₂	—	2	1	no	—
225	C₂	—	2	1	no	—
226	C₂	C₃×#64770	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₈	C₂²⋊Dic₅
commutator	C₂²×C₅	C₃×C₄×C₈
Frattini	C₂²×C₈	C₂×C₃×D₅
Fitting	C₂×C₃×C₅×M₃₂	C₂
socle	C₂×C₃×C₅	#336
prime radical	C₂³×C₃×C₅	C₂×C₈
cosocle	C₂²×C₅×C₈	C₂²×C₃
hypercenter	C₂×C₃×M₃₂	D₅
hypocenter	C₅	C₃×#1378
LCS	G ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₈ ◁ C₂²×C₃×C₈ ◁ C₂×C₃×M₃₂

Complements

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

N	H ≅ G/N	type
4. #10598304	C₃	direct (N × H)
10. C₂×C₃×C₅×C₁₆	C₄	semidirect
14. C₃×#3944	C₄	semidirect
43. C₂×C₅×C₁₆	C₃×C₄	semidirect
47. #3944	C₃×C₄	semidirect
71. C₂×C₃×C₁₆	Dic₅	semidirect
137. C₂×C₁₆	C₃×Dic₅	semidirect
181. C₃×C₅	#1378	semidirect
212. C₅	C₃×#1378	semidirect
222. C₃	#10598304	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×C₅×M₃₂	2
C₃×#65390	2
C₂×C₃×#3944	2
#10598304	3
C₃×#1378	5

Subgroups — C3×#10598304 #14278755

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10598304 #14278755