Subgroups — #14280683

Order 1920.

356 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11751026	C₂	1	1	yes	maximal, characteristic
2	C₅×#4925	C₂	1	1	yes	maximal, characteristic
3	C₃×#65214	C₂	1	1	yes	maximal, characteristic
4	#11745833	C₂	1	1	yes	maximal, characteristic, complemented
5	#11745832	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745831	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃⋊C₈×#168	C₂	1	1	yes	maximal, characteristic, complemented
8	#10601537	—	3	1	no	maximal, Hall
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	#91114	C₄	1	2	yes	—
11	C₄⋊(C₃⋊#168)	C₄	1	2	yes	—
12	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
13	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
14	C₅×#830	C₄	1	2	yes	—
15	C₅×C₄⋊(C₃⋊C₈)	C₄	1	2	yes	—
16	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₃×C₂²⋊#168	C₄	1	2	yes	—
19	C₃×C₄⋊#168	C₄	1	2	yes	—
20	C₃×C₄×#168	C₂²	1	1	yes	characteristic
21	C₃×C₄×#168	C₂²	1	1	yes	characteristic
22	#70744	—	5	1	no	maximal, Hall
23	C₅×#431	—	3	1	no	—
24	#65214	—	3	1	no	—
25	#65214	D₃	1	1	yes	characteristic
26	#64675	—	3	1	no	—
27	#64674	—	3	1	no	—
28	#64674	—	3	1	no	—
29	C₈×#168	—	3	1	no	—
30	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
33	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
34	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
35	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
37	C₂×C₃⋊#168	C₂×C₄	1	2	yes	—
38	C₂×C₃⋊#168	C₂×C₄	1	2	yes	—
39	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
40	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
41	C₂×C₃×#168	C₂×C₄	1	2	yes	—
42	C₂×C₃×#168	C₂×C₄	1	2	yes	—
43	C₃×#431	—	5	1	no	—
44	#4925	—	5	1	no	—
45	#4925	D₅	1	1	yes	characteristic
46	#4455	—	5	1	no	—
47	#4454	—	5	1	no	—
48	#4454	—	5	1	no	—
49	C₈×C₃⋊C₈	—	5	1	no	—
50	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
51	C₅×C₄⋊C₈	—	3	2	no	—
52	C₅×C₂²⋊C₈	—	3	2	no	—
53	C₄×C₅×C₈	—	3	1	no	—
54	C₄×C₅×C₈	—	3	1	no	—
55	C₂²⋊#168	—	3	2	no	—
56	C₂²⋊#168	Dic₃	1	2	yes	—
57	C₄⋊#168	—	3	2	no	—
58	C₄⋊#168	Dic₃	1	2	yes	—
59	C₄×#168	—	3	1	no	—
60	C₄×#168	—	3	1	no	—
61	C₄×#168	C₂×D₃	1	1	yes	characteristic
62	C₄×#168	C₂×D₃	1	1	yes	characteristic
63	#1454	—	15	1	no	Sylow
64	C₂³×C₃×C₅	C₄²	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	—	2	2	no	—
66	C₂×C₃×C₄×C₅	—	2	2	no	—
67	C₂×C₃×C₄×C₅	—	2	2	no	—
68	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
71	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₃⋊#168	—	2	2	no	—
75	C₃⋊#168	—	2	2	no	—
76	C₃×#168	—	2	2	no	—
77	C₃×#168	—	2	2	no	—
78	C₅×C₃⋊C₈	—	2	2	no	—
79	C₅×C₃⋊C₈	—	2	2	no	—
80	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
81	C₃×C₄⋊C₈	—	5	2	no	—
82	C₃×C₂²⋊C₈	—	5	2	no	—
83	C₃×C₄×C₈	—	5	1	no	—
84	C₃×C₄×C₈	—	5	1	no	—
85	#830	—	5	2	no	—
86	#830	Dic₅	1	2	yes	—
87	C₄⋊(C₃⋊C₈)	—	5	2	no	—
88	C₄⋊(C₃⋊C₈)	Dic₅	1	2	yes	—
89	C₄×C₃⋊C₈	—	5	1	no	—
90	C₄×C₃⋊C₈	—	5	1	no	—
91	C₄×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
92	C₄×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
93	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
94	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
95	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
96	C₂×C₅×C₈	—	3	2	no	—
97	C₂×C₅×C₈	—	3	2	no	—
98	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
99	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
100	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
101	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
102	C₂×#168	—	3	2	no	—
103	C₂×#168	—	3	2	no	—
104	C₂×#168	C₂×Dic₃	1	2	yes	—
105	C₂×#168	C₂×Dic₃	1	2	yes	—
106	#431	—	3	1	no	—
107	#431	—	5	1	no	—
108	#431	—	15	1	no	—
109	#322	—	15	1	no	—
110	#322	—	15	1	no	—
111	#322	—	15	1	no	—
112	C₈²	—	15	1	no	—
113	C₂²×C₃×C₅	—	2	1	no	—
114	C₂²×C₃×C₅	—	2	1	no	—
115	C₂²×C₃×C₅	—	2	1	no	—
116	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
117	C₃×C₄×C₅	—	2	1	no	—
118	C₃×C₄×C₅	—	2	1	no	—
119	C₃×C₄×C₅	—	2	1	no	—
120	C₃×C₄×C₅	#131	1	2	yes	—
121	C₃×C₄×C₅	#131	1	2	yes	—
122	C₃×C₄×C₅	#131	1	2	yes	—
123	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
124	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
125	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
126	C₂×C₃×C₈	—	5	2	no	—
127	C₂×C₃×C₈	—	5	2	no	—
128	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
129	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
130	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
131	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
132	C₂×C₃⋊C₈	—	5	2	no	—
133	C₂×C₃⋊C₈	—	5	2	no	—
134	C₂×C₃⋊C₈	C₂×Dic₅	1	2	yes	—
135	C₂×C₃⋊C₈	C₂×Dic₅	1	2	yes	—
136	C₂³×C₅	C₄×Dic₃	1	1	yes	characteristic
137	C₂×C₄×C₅	—	2	2	no	—
138	C₂×C₄×C₅	—	2	2	no	—
139	C₂×C₄×C₅	—	2	2	no	—
140	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
141	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
142	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
146	C₅×C₈	—	6	2	no	—
147	C₅×C₈	—	6	2	no	—
148	#168	—	2	2	no	—
149	#168	—	2	2	no	—
150	#168	—	6	2	no	—
151	#168	—	6	2	no	—
152	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
153	C₄⋊C₈	—	3	2	no	—
154	C₄⋊C₈	—	5	2	no	—
155	C₄⋊C₈	—	15	2	no	—
156	C₂²⋊C₈	—	3	2	no	—
157	C₂²⋊C₈	—	5	2	no	—
158	C₂²⋊C₈	—	15	2	no	—
159	C₄×C₈	—	3	1	no	—
160	C₄×C₈	—	3	1	no	—
161	C₄×C₈	—	5	1	no	—
162	C₄×C₈	—	5	1	no	—
163	C₄×C₈	—	15	1	no	—
164	C₄×C₈	—	15	1	no	—
165	C₂×C₃×C₅	—	4	1	no	—
166	C₂×C₃×C₅	#405	1	1	yes	characteristic
167	C₂×C₃×C₅	#405	1	1	yes	characteristic
168	C₂×C₃×C₅	#405	1	1	yes	characteristic
169	C₂³×C₃	C₄×Dic₅	1	1	yes	characteristic
170	C₂×C₃×C₄	—	2	2	no	—
171	C₂×C₃×C₄	—	2	2	no	—
172	C₂×C₃×C₄	—	2	2	no	—
173	C₂×C₃×C₄	C₄×Dic₅	1	1	yes	characteristic
174	C₂×C₃×C₄	C₄×Dic₅	1	1	yes	characteristic
175	C₂×C₃×C₄	C₄×Dic₅	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
179	C₃×C₈	—	10	2	no	—
180	C₃×C₈	—	10	2	no	—
181	C₃⋊C₈	—	2	2	no	—
182	C₃⋊C₈	—	2	2	no	—
183	C₃⋊C₈	—	10	2	no	—
184	C₃⋊C₈	—	10	2	no	—
185	C₂²×C₅	—	2	1	no	—
186	C₂²×C₅	—	2	1	no	—
187	C₂²×C₅	—	2	1	no	—
188	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
189	C₄×C₅	—	2	1	no	—
190	C₄×C₅	—	2	1	no	—
191	C₄×C₅	—	2	1	no	—
192	C₄×C₅	#901	1	2	yes	—
193	C₄×C₅	#907	1	2	yes	—
194	C₄×C₅	#948	1	2	yes	—
195	C₂²×C₄	D₅×Dic₃	1	1	yes	characteristic
196	C₂²×C₄	D₃×Dic₅	1	1	yes	characteristic
197	C₂²×C₄	#1200	1	1	yes	characteristic
198	C₂×C₈	—	3	2	no	—
199	C₂×C₈	—	3	2	no	—
200	C₂×C₈	—	5	2	no	—
201	C₂×C₈	—	5	2	no	—
202	C₂×C₈	—	15	2	no	—
203	C₂×C₈	—	15	2	no	—
204	C₄²	D₅×Dic₃	1	1	yes	characteristic
205	C₄²	D₃×Dic₅	1	1	yes	characteristic
206	C₄²	#1200	1	1	yes	characteristic
207	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
208	C₃×C₅	#1454	1	1	yes	characteristic, complemented, Hall
209	C₂²×C₃	—	2	1	no	—
210	C₂²×C₃	—	2	1	no	—
211	C₂²×C₃	—	2	1	no	—
212	C₂²×C₃	C₂×C₄×Dic₅	1	1	yes	characteristic
213	C₃×C₄	—	2	1	no	—
214	C₃×C₄	—	2	1	no	—
215	C₃×C₄	—	2	1	no	—
216	C₃×C₄	#4047	1	2	yes	—
217	C₃×C₄	#4053	1	2	yes	—
218	C₃×C₄	#4094	1	2	yes	—
219	C₂×C₅	—	4	1	no	—
220	C₂×C₅	#4997	1	1	yes	characteristic
221	C₂×C₅	#5101	1	1	yes	characteristic
222	C₂×C₅	#5122	1	1	yes	characteristic
223	C₂³	Dic₃×Dic₅	1	1	yes	characteristic
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	Dic₃×Dic₅	1	1	yes	characteristic
228	C₂×C₄	Dic₃×Dic₅	1	1	yes	characteristic
229	C₂×C₄	Dic₃×Dic₅	1	1	yes	characteristic
230	C₂×C₄	C₂×D₅×Dic₃	1	1	yes	characteristic
231	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
232	C₂×C₄	C₂×#1200	1	1	yes	characteristic
233	C₈	—	6	2	no	—
234	C₈	—	6	2	no	—
235	C₈	—	10	2	no	—
236	C₈	—	10	2	no	—
237	C₈	—	30	2	no	—
238	C₈	—	30	2	no	—
239	C₂×C₃	—	4	1	no	—
240	C₂×C₃	#65286	1	1	yes	characteristic
241	C₂×C₃	#65390	1	1	yes	characteristic
242	C₂×C₃	#65411	1	1	yes	characteristic
243	C₅	#70744	1	1	yes	characteristic, complemented, Sylow
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	C₂×Dic₃×Dic₅	1	1	yes	characteristic
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	#91284	1	2	yes	—
252	C₄	#91286	1	2	yes	—
253	C₄	#91291	1	2	yes	—
254	C₃	#10601537	1	1	yes	characteristic, complemented, Sylow
255	C₂	—	4	1	no	—
256	C₂	#11748320	1	1	yes	characteristic
257	C₂	#11748323	1	1	yes	characteristic
258	C₂	#11748326	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂×D₃×D₅
commutator, socle	C₂²×C₃×C₅	C₂×C₄²
Fitting	C₂×C₃×C₄²×C₅	C₂²
prime radical	C₂³×C₃×C₅	C₄²
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1454
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄² ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
4. #11745833	C₂	semidirect
5. #11745832	C₂	semidirect
6. #11745831	C₂	semidirect
7. C₃⋊C₈×#168	C₂	semidirect
208. C₃×C₅	#1454	semidirect
243. C₅	#70744	semidirect
254. C₃	#10601537	semidirect

Maximal subgroups

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

subgroup	index
#11751026	2
C₅×#4925	2
C₃×#65214	2
#11745833	2
#11745832	2
#11745831	2
C₃⋊C₈×#168	2
#10601537	3
#70744	5