Subgroups — #14280440

Order 1920.

424 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#431	C₂	1	1	yes	maximal, characteristic
2	#11751026	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₃⋊#168	C₂	1	1	yes	maximal, characteristic
4	#11746275	C₂	1	2	yes	maximal, complemented
5	C₈×C₃⋊#168	C₂	1	2	yes	maximal, complemented
6	#10601533	—	3	1	no	maximal, Hall
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊#168	C₄	1	2	yes	—
9	C₃×C₅×C₄⋊C₈	C₄	1	2	yes	—
10	C₃×C₅×C₂²⋊C₈	C₄	1	2	yes	—
11	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
12	#91114	C₄	1	2	yes	—
13	C₄⋊(C₃⋊#168)	C₄	1	2	yes	—
14	C₄×C₃⋊#168	C₄	1	2	yes	—
15	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
16	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
17	C₄×C₃⋊#168	C₂²	1	2	yes	—
18	#70740	—	5	1	no	maximal, Hall
19	C₅×#431	D₃	1	1	yes	characteristic
20	#65214	—	3	1	no	—
21	C₂×C₄×#168	—	3	1	no	—
22	#64674	—	3	2	no	—
23	C₈×#168	—	3	2	no	—
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	—
28	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
29	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
30	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
31	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
32	C₂×C₃⋊#168	C₈	1	8	yes	complemented
33	C₂×C₃⋊#168	C₂×C₄	1	4	yes	—
34	C₂×C₃⋊#168	C₂×C₄	1	2	yes	—
35	C₂×C₃⋊#168	C₂×C₄	1	2	yes	—
36	C₃×#431	D₅	1	1	yes	characteristic
37	#4925	—	5	1	no	—
38	C₂×C₄×C₃⋊C₈	—	5	1	no	—
39	#4454	—	5	2	no	—
40	C₈×C₃⋊C₈	—	5	2	no	—
41	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
42	C₂²×#168	—	3	2	no	—
43	C₅×C₄⋊C₈	Dic₃	1	2	yes	—
44	C₅×C₂²⋊C₈	Dic₃	1	2	yes	—
45	C₄×C₅×C₈	C₂×D₃	1	2	yes	—
46	C₂²⋊#168	—	3	2	no	—
47	C₄⋊#168	—	3	2	no	—
48	C₄×#168	—	3	2	no	—
49	C₄×#168	—	3	1	no	—
50	C₄×#168	—	3	2	no	—
51	C₄×#168	—	3	1	no	—
52	#1450	—	15	1	no	Sylow
53	C₂³×C₃×C₅	C₄²	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	—	2	2	no	—
55	C₂×C₃×C₄×C₅	—	2	2	no	—
56	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₂×C₈	1	4	yes	—
60	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
63	C₃×C₅×C₈	—	2	4	no	—
64	C₃⋊#168	—	2	4	no	—
65	C₃⋊#168	C₂×C₈	1	4	yes	—
66	C₃⋊#168	C₂×C₈	1	4	yes	—
67	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
68	C₂²×C₃⋊C₈	—	5	2	no	—
69	C₃×C₄⋊C₈	Dic₅	1	2	yes	—
70	C₃×C₂²⋊C₈	Dic₅	1	2	yes	—
71	C₃×C₄×C₈	C₂×D₅	1	2	yes	—
72	#830	—	5	2	no	—
73	C₄⋊(C₃⋊C₈)	—	5	2	no	—
74	C₄×C₃⋊C₈	—	5	2	no	—
75	C₄×C₃⋊C₈	—	5	1	no	—
76	C₄×C₃⋊C₈	—	5	1	no	—
77	C₄×C₃⋊C₈	—	5	2	no	—
78	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
81	C₂×C₅×C₈	C₂×Dic₃	1	4	yes	—
82	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
83	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
84	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
85	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
86	C₂×#168	—	3	8	no	—
87	C₂×#168	—	3	2	no	—
88	C₂×#168	—	3	2	no	—
89	C₂×#168	—	3	4	no	—
90	#431	—	15	1	no	—
91	#431	D₁₅	1	1	yes	characteristic
92	C₂×C₄×C₈	—	15	1	no	—
93	#322	—	15	2	no	—
94	C₈²	—	15	2	no	—
95	C₂²×C₃×C₅	—	2	1	no	—
96	C₂²×C₃×C₅	—	2	1	no	—
97	C₂²×C₃×C₅	C₄×C₈	1	2	yes	—
98	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
99	C₃×C₄×C₅	—	2	1	no	—
100	C₃×C₄×C₅	—	2	1	no	—
101	C₃×C₄×C₅	C₄×C₈	1	2	yes	—
102	C₃×C₄×C₅	C₂²×C₈	1	2	yes	—
103	C₃×C₄×C₅	#131	1	2	yes	—
104	C₃×C₄×C₅	#131	1	2	yes	—
105	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
106	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
107	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
108	C₂×C₃×C₈	C₂×Dic₅	1	4	yes	—
109	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
110	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
111	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
112	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
113	C₂×C₃⋊C₈	—	5	8	no	—
114	C₂×C₃⋊C₈	—	5	2	no	—
115	C₂×C₃⋊C₈	—	5	2	no	—
116	C₂×C₃⋊C₈	—	5	4	no	—
117	C₂³×C₅	C₄×Dic₃	1	1	yes	characteristic
118	C₂×C₄×C₅	—	2	2	no	—
119	C₂×C₄×C₅	—	2	2	no	—
120	C₂×C₄×C₅	D₃×C₈	1	4	yes	—
121	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
122	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
123	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
124	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
126	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
127	C₅×C₈	—	2	4	no	—
128	#168	—	3	4	no	—
129	#168	—	3	4	no	—
130	#168	—	6	4	no	—
131	C₂²×C₈	—	15	2	no	—
132	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
133	C₄⋊C₈	—	15	2	no	—
134	C₄⋊C₈	Dic₁₅	1	2	yes	—
135	C₂²⋊C₈	—	15	2	no	—
136	C₂²⋊C₈	Dic₁₅	1	2	yes	—
137	C₄×C₈	—	15	2	no	—
138	C₄×C₈	—	15	1	no	—
139	C₄×C₈	—	15	2	no	—
140	C₄×C₈	—	15	1	no	—
141	C₄×C₈	C₂×D₁₅	1	2	yes	—
142	C₂×C₃×C₅	—	2	2	no	—
143	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
144	C₂×C₃×C₅	#405	1	1	yes	characteristic
145	C₂×C₃×C₅	#405	1	1	yes	characteristic
146	C₂³×C₃	C₄×Dic₅	1	1	yes	characteristic
147	C₂×C₃×C₄	—	2	2	no	—
148	C₂×C₃×C₄	—	2	2	no	—
149	C₂×C₃×C₄	C₈×D₅	1	4	yes	—
150	C₂×C₃×C₄	C₄×Dic₅	1	1	yes	characteristic
151	C₂×C₃×C₄	C₄×Dic₅	1	1	yes	characteristic
152	C₂×C₃×C₄	C₄×Dic₅	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
156	C₃×C₈	—	2	4	no	—
157	C₃⋊C₈	—	5	4	no	—
158	C₃⋊C₈	—	5	4	no	—
159	C₃⋊C₈	—	10	4	no	—
160	C₂²×C₅	—	2	1	no	—
161	C₂²×C₅	—	2	1	no	—
162	C₂²×C₅	C₈×Dic₃	1	2	yes	—
163	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
164	C₄×C₅	—	2	1	no	—
165	C₄×C₅	—	2	1	no	—
166	C₄×C₅	C₈×Dic₃	1	2	yes	—
167	C₄×C₅	C₂×D₃×C₈	1	2	yes	—
168	C₄×C₅	#907	1	2	yes	—
169	C₄×C₅	#948	1	2	yes	—
170	C₂²×C₄	C₄×D₁₅	1	1	yes	characteristic
171	C₂²×C₄	C₄×D₁₅	1	1	yes	characteristic
172	C₂²×C₄	C₂×Dic₁₅	1	1	yes	characteristic
173	C₂×C₈	—	15	8	no	—
174	C₂×C₈	—	15	2	no	—
175	C₂×C₈	—	15	2	no	—
176	C₂×C₈	—	15	4	no	—
177	C₂×C₈	C₂×Dic₁₅	1	4	yes	—
178	C₄²	C₄×D₁₅	1	1	yes	characteristic
179	C₄²	C₄×D₁₅	1	1	yes	characteristic
180	C₄²	C₂×Dic₁₅	1	1	yes	characteristic
181	C₄²	C₂²×D₁₅	1	1	yes	characteristic
182	C₃×C₅	#1450	1	1	yes	characteristic, complemented, Hall
183	C₂²×C₃	—	2	1	no	—
184	C₂²×C₃	—	2	1	no	—
185	C₂²×C₃	C₈×Dic₅	1	2	yes	—
186	C₂²×C₃	C₂×C₄×Dic₅	1	1	yes	characteristic
187	C₃×C₄	—	2	1	no	—
188	C₃×C₄	—	2	1	no	—
189	C₃×C₄	C₈×Dic₅	1	2	yes	—
190	C₃×C₄	C₂×C₈×D₅	1	2	yes	—
191	C₃×C₄	#4053	1	2	yes	—
192	C₃×C₄	#4094	1	2	yes	—
193	C₂×C₅	—	2	2	no	—
194	C₂×C₅	#4997	1	1	yes	characteristic
195	C₂×C₅	C₂×C₈×Dic₃	1	1	yes	characteristic
196	C₂×C₅	#5122	1	1	yes	characteristic
197	C₂³	C₄×Dic₁₅	1	1	yes	characteristic
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	C₈×D₁₅	1	4	yes	—
201	C₂×C₄	C₄×Dic₁₅	1	1	yes	characteristic
202	C₂×C₄	C₄×Dic₁₅	1	1	yes	characteristic
203	C₂×C₄	C₄×Dic₁₅	1	1	yes	characteristic
204	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
205	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
206	C₂×C₄	C₂²×Dic₁₅	1	1	yes	characteristic
207	C₈	—	2	4	no	—
208	C₈	—	15	4	no	—
209	C₈	—	15	4	no	—
210	C₈	—	30	4	no	—
211	C₂×C₃	—	2	2	no	—
212	C₂×C₃	#65286	1	1	yes	characteristic
213	C₂×C₃	C₂×C₈×Dic₅	1	1	yes	characteristic
214	C₂×C₃	#65411	1	1	yes	characteristic
215	C₅	#70740	1	1	yes	characteristic, complemented, Sylow
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	C₈×Dic₁₅	1	2	yes	—
219	C₂²	C₂×C₄×Dic₁₅	1	1	yes	characteristic
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	C₈×Dic₁₅	1	2	yes	—
223	C₄	C₂×C₈×D₁₅	1	2	yes	—
224	C₄	#91796	1	2	yes	—
225	C₄	#91837	1	2	yes	—
226	C₃	#10601533	1	1	yes	characteristic, complemented, Sylow
227	C₂	—	2	2	no	—
228	C₂	#11751098	1	1	yes	characteristic
229	C₂	C₂×C₈×Dic₁₅	1	1	yes	characteristic
230	C₂	#11751223	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₁₅
commutator	C₂×C₃×C₅	C₂×C₄×C₈
Fitting	C₃×C₅×#431	C₂
socle	C₂²×C₃×C₅	C₂×C₄²
prime radical	C₂³×C₃×C₅	C₄²
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	#431	D₁₅
hypocenter	C₃×C₅	#1450
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄² ◁ #431

Complements

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

N	H ≅ G/N	type
4. #11746275	C₂	semidirect
5. C₈×C₃⋊#168	C₂	semidirect
32. C₂×C₃⋊#168	C₈	semidirect
182. C₃×C₅	#1450	semidirect
215. C₅	#70740	semidirect
226. C₃	#10601533	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×C₅×#431	2
#11751026	2
C₂×C₄×C₃⋊#168	2
#11746275	2
C₈×C₃⋊#168	2
#10601533	3
#70740	5