Subgroups — #14280458

Order 1920.

424 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#4685	C₂	1	1	yes	maximal, characteristic
2	C₄×D₃×#168	C₂	1	1	yes	maximal, characteristic
3	#11746876	C₂	1	1	yes	maximal, characteristic
4	C₈×C₃⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64674	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	#10601533	—	3	1	no	maximal, Hall
9	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
10	C₂×D₃×#168	C₄	1	2	yes	—
11	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
13	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
14	C₅×(C₂×D₃)⋊C₈	C₄	1	2	yes	—
15	C₅×Dic₃⋊C₈	C₄	1	2	yes	—
16	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₃×C₄×#168	C₂²	1	1	yes	characteristic
18	C₃×C₄×#168	C₂²	1	1	yes	characteristic
19	Dic₃⋊#168	C₄	1	2	yes	—
20	(C₂×D₃)⋊#168	C₄	1	2	yes	—
21	Dic₃×#168	C₄	1	2	yes	—
22	#68068	—	5	1	no	maximal, Hall
23	C₅×#431	—	3	1	no	—
24	#65214	—	3	1	no	—
25	C₂×C₄×#168	—	3	1	no	—
26	#64674	—	3	1	no	—
27	#64674	D₃	1	1	yes	characteristic, complemented
28	C₈×#168	—	3	1	no	—
29	C₈×#168	—	3	1	no	—
30	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
34	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
35	C₂×C₃⋊#168	C₂×C₄	1	2	yes	—
36	C₂×C₃⋊#168	C₂×C₄	1	2	yes	—
37	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
38	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
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	D₃×#168	C₈	1	8	yes	complemented
44	#4685	—	5	1	no	—
45	#4685	D₅	1	1	yes	characteristic
46	C₄×D₃×C₈	—	5	1	no	—
47	C₃×#322	—	5	1	no	—
48	#4455	—	5	1	no	—
49	C₈×C₃⋊C₈	—	5	1	no	—
50	C₈×C₃⋊C₈	—	5	1	no	—
51	C₂×C₄²×C₅	—	3	1	no	—
52	C₂²×#168	—	3	2	no	—
53	C₅×C₄⋊C₈	—	3	2	no	—
54	C₅×C₂²⋊C₈	—	3	2	no	—
55	C₄×C₅×C₈	—	3	1	no	—
56	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
57	C₂²⋊#168	—	3	2	no	—
58	C₄⋊#168	—	3	2	no	—
59	C₄×#168	—	3	1	no	—
60	C₄×#168	—	3	1	no	—
61	C₄×#168	—	3	2	no	—
62	C₄×#168	C₂×D₃	1	1	yes	characteristic
63	C₄×#168	C₂×D₃	1	1	yes	characteristic
64	#1450	—	15	1	no	Sylow
65	C₂²×C₅×D₃	C₄²	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₅×Dic₃	C₄²	1	1	yes	characteristic
70	C₂×C₅×Dic₃	C₄²	1	1	yes	characteristic
71	C₂×C₅×Dic₃	C₄²	1	1	yes	characteristic
72	C₄×C₅×D₃	—	2	2	no	—
73	C₄×C₅×D₃	—	2	2	no	—
74	C₄×C₅×D₃	C₂×C₈	1	4	yes	—
75	C₃×C₅×C₈	—	2	2	no	—
76	C₃⋊#168	—	2	2	no	—
77	C₃⋊#168	C₂×C₈	1	4	yes	—
78	C₃×#168	—	2	2	no	—
79	C₃×#168	C₂×C₈	1	4	yes	—
80	C₅×C₃⋊C₈	—	2	2	no	—
81	C₂×D₃×C₈	—	5	2	no	—
82	C₄²×D₃	C₂×D₅	1	1	yes	characteristic
83	C₃×C₄×C₈	—	5	1	no	—
84	C₃×C₄×C₈	—	5	1	no	—
85	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
86	(C₂×D₃)⋊C₈	—	5	2	no	—
87	(C₂×D₃)⋊C₈	Dic₅	1	2	yes	—
88	Dic₃⋊C₈	—	5	2	no	—
89	Dic₃⋊C₈	Dic₅	1	2	yes	—
90	C₈×Dic₃	—	5	2	no	—
91	C₄×C₃⋊C₈	—	5	1	no	—
92	C₄×C₃⋊C₈	—	5	1	no	—
93	C₄×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
94	C₂²×C₄×C₅	—	3	1	no	—
95	C₂²×C₄×C₅	—	3	1	no	—
96	C₂²×C₄×C₅	—	3	1	no	—
97	C₂×C₅×C₈	—	3	2	no	—
98	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
99	C₄²×C₅	—	3	1	no	—
100	C₄²×C₅	—	3	1	no	—
101	C₄²×C₅	—	3	1	no	—
102	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
103	C₂×#168	—	3	2	no	—
104	C₂×#168	—	3	8	no	—
105	C₂×#168	—	3	2	no	—
106	C₂×#168	C₄×D₃	1	2	yes	—
107	C₂×#168	C₄×D₃	1	2	yes	—
108	#431	—	3	1	no	—
109	#431	—	15	1	no	—
110	C₂×C₄×C₈	—	15	1	no	—
111	#322	—	5	1	no	—
112	#322	—	15	1	no	—
113	C₈²	—	15	1	no	—
114	C₈²	—	15	1	no	—
115	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₅×D₃	—	2	1	no	—
117	C₂×C₅×D₃	—	2	1	no	—
118	C₂×C₅×D₃	C₄×C₈	1	2	yes	—
119	C₃×C₄×C₅	C₂²×C₈	1	2	yes	—
120	C₃×C₄×C₅	#131	1	2	yes	—
121	C₃×C₄×C₅	#131	1	2	yes	—
122	C₅×Dic₃	—	2	1	no	—
123	C₅×Dic₃	—	2	1	no	—
124	C₅×Dic₃	C₄×C₈	1	2	yes	—
125	C₂×C₄×D₃	C₄×D₅	1	1	yes	characteristic
126	C₂×C₄×D₃	C₄×D₅	1	1	yes	characteristic
127	C₂×C₄×D₃	C₂×Dic₅	1	1	yes	characteristic
128	C₂×C₃×C₈	—	5	2	no	—
129	C₂×C₃×C₈	—	5	2	no	—
130	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
131	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
132	C₄×Dic₃	C₄×D₅	1	1	yes	characteristic
133	C₄×Dic₃	C₄×D₅	1	1	yes	characteristic
134	C₄×Dic₃	C₂×Dic₅	1	1	yes	characteristic
135	C₂×C₃⋊C₈	—	5	2	no	—
136	C₂×C₃⋊C₈	—	5	2	no	—
137	C₂×C₃⋊C₈	C₂×Dic₅	1	2	yes	—
138	D₃×C₈	—	5	8	no	—
139	C₂³×C₅	—	3	1	no	—
140	C₂×C₄×C₅	—	3	1	no	—
141	C₂×C₄×C₅	—	3	4	no	—
142	C₂×C₄×C₅	—	3	1	no	—
143	C₂×C₄×C₅	—	3	1	no	—
144	C₂×C₄×C₅	—	6	2	no	—
145	C₂×C₄×C₅	—	6	2	no	—
146	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
148	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
149	C₅×C₈	—	2	2	no	—
150	C₅×C₈	—	6	2	no	—
151	#168	—	2	2	no	—
152	#168	—	3	4	no	—
153	#168	—	6	2	no	—
154	#168	D₃×C₈	1	4	yes	—
155	C₂²×C₈	—	15	2	no	—
156	C₂×C₄²	—	3	1	no	—
157	C₄⋊C₈	—	3	2	no	—
158	C₄⋊C₈	—	15	2	no	—
159	C₂²⋊C₈	—	3	2	no	—
160	C₂²⋊C₈	—	15	2	no	—
161	C₄×C₈	—	3	1	no	—
162	C₄×C₈	—	5	1	no	—
163	C₄×C₈	—	5	1	no	—
164	C₄×C₈	—	15	2	no	—
165	C₄×C₈	—	15	1	no	—
166	C₄×C₈	—	15	1	no	—
167	C₄×C₈	D₃×D₅	1	1	yes	characteristic
168	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
169	C₂×C₃×C₅	#405	1	1	yes	characteristic
170	C₂×C₃×C₅	#405	1	1	yes	characteristic
171	C₅×D₃	—	2	2	no	—
172	C₂²×D₃	C₄×Dic₅	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
176	C₂×Dic₃	C₄×Dic₅	1	1	yes	characteristic
177	C₂×Dic₃	C₄×Dic₅	1	1	yes	characteristic
178	C₂×Dic₃	C₄×Dic₅	1	1	yes	characteristic
179	C₄×D₃	—	2	2	no	—
180	C₄×D₃	—	2	2	no	—
181	C₄×D₃	C₈×D₅	1	4	yes	—
182	C₃×C₈	—	2	2	no	—
183	C₃×C₈	—	5	4	no	—
184	C₃×C₈	—	10	2	no	—
185	C₃⋊C₈	—	2	2	no	—
186	C₃⋊C₈	—	5	4	no	—
187	C₃⋊C₈	—	10	2	no	—
188	C₂²×C₅	—	3	2	no	—
189	C₂²×C₅	—	6	1	no	—
190	C₂²×C₅	—	6	1	no	—
191	C₂²×C₅	C₄²×D₃	1	1	yes	characteristic
192	C₄×C₅	—	3	2	no	—
193	C₄×C₅	—	6	1	no	—
194	C₄×C₅	—	6	1	no	—
195	C₄×C₅	C₂×D₃×C₈	1	2	yes	—
196	C₄×C₅	#907	1	2	yes	—
197	C₄×C₅	#907	1	2	yes	—
198	C₂²×C₄	—	3	1	no	—
199	C₂²×C₄	—	3	1	no	—
200	C₂²×C₄	—	3	1	no	—
201	C₂×C₈	—	3	2	no	—
202	C₂×C₈	—	5	2	no	—
203	C₂×C₈	—	5	2	no	—
204	C₂×C₈	—	15	8	no	—
205	C₂×C₈	—	15	2	no	—
206	C₂×C₈	—	15	2	no	—
207	C₂×C₈	D₃×Dic₅	1	2	yes	—
208	C₄²	—	3	1	no	—
209	C₄²	—	3	1	no	—
210	C₄²	—	3	1	no	—
211	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
212	C₃×C₅	#1450	1	1	yes	characteristic, complemented, Hall
213	C₂²×C₃	C₂×C₄×Dic₅	1	1	yes	characteristic
214	C₂×D₃	—	2	1	no	—
215	C₂×D₃	—	2	1	no	—
216	C₂×D₃	C₈×Dic₅	1	2	yes	—
217	C₃×C₄	C₂×C₈×D₅	1	2	yes	—
218	C₃×C₄	#4053	1	2	yes	—
219	C₃×C₄	#4094	1	2	yes	—
220	Dic₃	—	2	1	no	—
221	Dic₃	—	2	1	no	—
222	Dic₃	C₈×Dic₅	1	2	yes	—
223	C₂×C₅	—	6	2	no	—
224	C₂×C₅	C₄×D₃×C₈	1	1	yes	characteristic
225	C₂×C₅	#4708	1	1	yes	characteristic
226	C₂×C₅	#4708	1	1	yes	characteristic
227	C₂³	—	3	1	no	—
228	C₂×C₄	—	3	1	no	—
229	C₂×C₄	—	3	4	no	—
230	C₂×C₄	—	3	1	no	—
231	C₂×C₄	—	3	1	no	—
232	C₂×C₄	—	6	2	no	—
233	C₂×C₄	—	6	2	no	—
234	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
235	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
236	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	—	5	4	no	—
239	C₈	—	6	2	no	—
240	C₈	—	10	2	no	—
241	C₈	—	15	4	no	—
242	C₈	—	30	2	no	—
243	C₂×C₃	#65286	1	1	yes	characteristic
244	C₂×C₃	C₂×C₈×Dic₅	1	1	yes	characteristic
245	C₂×C₃	#65411	1	1	yes	characteristic
246	D₃	—	2	2	no	—
247	C₅	#68068	1	1	yes	characteristic, complemented, Sylow
248	C₂²	—	3	2	no	—
249	C₂²	—	6	1	no	—
250	C₂²	—	6	1	no	—
251	C₂²	C₄×D₃×Dic₅	1	1	yes	characteristic
252	C₄	—	3	2	no	—
253	C₄	—	6	1	no	—
254	C₄	—	6	1	no	—
255	C₄	D₃×C₈×D₅	1	2	yes	—
256	C₄	#91268	1	2	yes	—
257	C₄	#91288	1	2	yes	—
258	C₃	#10601533	1	1	yes	characteristic, complemented, Sylow
259	C₂	—	6	2	no	—
260	C₂	#11746979	1	1	yes	characteristic
261	C₂	D₃×C₈×Dic₅	1	1	yes	characteristic
262	C₂	#11748055	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂×D₃×D₅
commutator	C₂×C₃×C₅	C₂×C₄×C₈
Fitting	C₃×C₄×C₅×C₈	C₂²
socle	C₂²×C₃×C₅	C₂×C₄²
prime radical	C₂²×C₅×D₃	C₄²
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄×C₈	D₃×D₅
hypocenter	C₃×C₅	#1450
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
4. C₈×C₃⋊#168	C₂	semidirect
5. C₃×#64674	C₂	semidirect
6. #11745831	C₂	semidirect
7. C₃⋊C₈×#168	C₂	semidirect
27. #64674	D₃	semidirect
43. D₃×#168	C₈	semidirect
212. C₃×C₅	#1450	semidirect
247. C₅	#68068	semidirect
258. C₃	#10601533	semidirect

Maximal subgroups

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

subgroup	index
C₅×#4685	2
C₄×D₃×#168	2
#11746876	2
C₈×C₃⋊#168	2
C₃×#64674	2
#11745831	2
C₃⋊C₈×#168	2
#10601533	3
#68068	5