Subgroups — #14286183

Order 1920.

360 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#4830	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#6660	C₂	1	1	yes	maximal, characteristic
3	#11746994	C₂	1	1	yes	maximal, characteristic
4	#11746304	C₂	1	1	yes	maximal, characteristic
5	C₃×#64700	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745842	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃⋊C₈×#168	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×#688	—	3	1	no	maximal, Hall
9	C₄×D₆₀	C₂²	1	1	yes	characteristic
10	C₄×C₅×Dic₆	C₂²	1	1	yes	characteristic
11	C₂×#6660	C₄	1	1	yes	characteristic
12	C₂×#6648	—	2	1	no	—
13	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
15	C₅×Dic₃⋊C₈	—	2	1	no	—
16	C₅×C₈×Dic₃	—	2	1	no	—
17	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
19	C₃×C₄×#168	C₂²	1	1	yes	characteristic
20	#90985	C₄	1	1	yes	characteristic
21	#90973	C₄	1	1	yes	characteristic
22	#90969	C₄	1	1	yes	characteristic
23	#90957	—	2	1	no	—
24	#68516	—	5	1	no	maximal, Hall
25	C₅×C₈×Q₈	—	3	1	no	—
26	C₄×#688	—	3	1	no	—
27	C₈×D₂₀	—	3	1	no	—
28	Q₈⋊#168	—	3	1	no	—
29	#64703	—	3	1	no	—
30	#64700	D₃	1	1	yes	characteristic, complemented
31	C₈×#168	—	3	1	no	—
32	C₂×D₆₀	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×D₁₅	—	2	1	no	—
34	C₂×C₅×Dic₆	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×C₅×C₈	—	2	1	no	—
36	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
37	#6714	—	2	1	no	—
38	C₄⋊Dic₁₅	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃⋊#168	—	2	1	no	—
40	C₅×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
41	C₅×Dic₃⋊C₄	—	2	1	no	—
42	C₄×C₅×Dic₃	—	2	1	no	—
43	C₂×C₅×C₃⋊C₈	D₄	1	1	yes	characteristic
44	C₂×C₅×C₃⋊C₈	D₄	1	1	yes	characteristic
45	C₂×C₃×#168	—	2	1	no	—
46	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
47	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
48	#6660	C₈	1	4	yes	complemented
49	#6648	—	4	2	no	—
50	#4834	—	5	1	no	—
51	#4830	D₅	1	1	yes	characteristic, complemented
52	C₄×#204	—	5	1	no	—
53	C₃×#334	—	5	1	no	—
54	#4484	—	5	1	no	—
55	#4483	—	5	1	no	—
56	C₈×C₃⋊C₈	—	5	1	no	—
57	C₄×C₅×Q₈	—	3	1	no	—
58	C₂×#688	—	3	1	no	—
59	C₂×C₈×D₅	—	6	1	no	—
60	C₄×D₂₀	—	3	1	no	—
61	C₅×C₄⋊C₈	—	6	1	no	—
62	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
63	C₄×C₅×C₈	—	3	1	no	—
64	C₄×C₅×C₈	—	6	1	no	—
65	#3967	—	3	1	no	—
66	#3952	—	6	1	no	—
67	#3941	—	3	1	no	—
68	#3940	—	3	1	no	—
69	C₄⋊#168	C₂×D₃	1	1	yes	characteristic
70	C₄×#168	—	3	1	no	—
71	C₄×#168	C₂×D₃	1	1	yes	characteristic
72	C₈×SD₁₆	—	15	1	no	Sylow
73	C₂²×D₁₅	—	2	1	no	—
74	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
77	C₂×Dic₁₅	—	2	1	no	—
78	D₆₀	—	2	1	no	—
79	D₆₀	C₂×C₈	1	2	yes	—
80	C₄×D₁₅	—	4	1	no	—
81	C₄×D₁₅	—	4	1	no	—
82	C₂×C₅×Dic₃	—	2	1	no	—
83	C₂×C₅×Dic₃	—	2	1	no	—
84	C₅×Dic₆	—	2	1	no	—
85	C₅×Dic₆	C₂×C₈	1	2	yes	complemented
86	C₃×C₅×C₈	—	4	1	no	—
87	C₃⋊#168	—	2	2	no	—
88	C₃×#168	—	2	1	no	—
89	C₃×#168	—	4	1	no	—
90	C₃×#168	C₂×C₈	1	2	yes	complemented
91	C₅×C₃⋊C₈	—	2	1	no	—
92	C₅×C₃⋊C₈	SD₁₆	1	2	yes	complemented
93	C₂×#204	—	5	1	no	—
94	C₂×D₃×C₈	—	10	1	no	—
95	C₄×D₁₂	—	5	1	no	—
96	C₄×Dic₆	C₂×D₅	1	1	yes	characteristic
97	C₃×C₄⋊C₈	—	5	1	no	—
98	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
99	C₃×C₄×C₈	—	5	1	no	—
100	#821	—	5	1	no	—
101	(C₂×D₃)⋊C₈	—	10	1	no	—
102	#817	—	5	1	no	—
103	#816	—	5	1	no	—
104	Dic₃⋊C₈	—	2	1	no	—
105	C₈×Dic₃	—	2	1	no	—
106	C₄×C₃⋊C₈	—	5	1	no	—
107	C₄×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
108	C₂×C₅×Q₈	—	3	1	no	—
109	C₂×D₂₀	—	3	1	no	—
110	C₂×C₄×D₅	—	6	1	no	—
111	C₂×C₅×C₈	—	2	1	no	—
112	C₂×C₅×C₈	—	3	1	no	—
113	C₂×C₅×C₈	—	3	1	no	—
114	C₅×C₄⋊C₄	—	3	1	no	—
115	C₅×C₄⋊C₄	—	6	1	no	—
116	C₄²×C₅	—	6	1	no	—
117	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
118	#688	—	3	4	no	—
119	#685	—	6	1	no	—
120	C₄⋊Dic₅	—	3	1	no	—
121	C₂×#168	—	2	1	no	—
122	C₂×#168	—	6	1	no	—
123	C₂×#168	C₄×D₃	1	1	yes	characteristic
124	C₂×#168	C₄×D₃	1	1	yes	characteristic
125	C₈×D₅	—	12	2	no	—
126	C₈×Q₈	—	3	1	no	—
127	C₄×SD₁₆	—	15	1	no	—
128	C₈×D₄	—	15	1	no	—
129	#334	—	5	1	no	—
130	Q₈⋊C₈	—	15	1	no	—
131	D₄⋊C₈	—	15	1	no	—
132	C₈²	—	15	1	no	—
133	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
134	C₂×D₁₅	—	2	2	no	—
135	C₂×D₁₅	—	4	1	no	—
136	C₂×D₁₅	—	4	1	no	—
137	C₃×C₄×C₅	—	2	1	no	—
138	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
139	C₃×C₄×C₅	#131	1	1	yes	characteristic
140	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
141	C₃×C₄×C₅	#135	1	1	yes	characteristic
142	Dic₁₅	—	4	1	no	—
143	C₅×Dic₃	—	2	2	no	—
144	C₅×Dic₃	—	4	1	no	—
145	C₂×D₁₂	—	5	1	no	—
146	C₂×C₄×D₃	—	10	1	no	—
147	C₂×Dic₆	C₄×D₅	1	1	yes	characteristic
148	C₂×C₃×C₈	—	2	1	no	—
149	C₂×C₃×C₈	—	5	1	no	—
150	C₂×C₃×C₈	—	5	1	no	—
151	C₂×C₃×C₈	—	10	1	no	—
152	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
153	(C₂×D₃)⋊C₄	—	10	1	no	—
154	C₄⋊Dic₃	—	5	1	no	—
155	C₄⋊Dic₃	C₄×D₅	1	1	yes	characteristic
156	Dic₃⋊C₄	—	2	1	no	—
157	C₄×Dic₃	—	2	1	no	—
158	C₂×C₃⋊C₈	—	10	1	no	—
159	C₂×C₃⋊C₈	C₂²⋊D₅	1	1	yes	characteristic
160	C₂×C₃⋊C₈	C₂²⋊D₅	1	1	yes	characteristic
161	#204	—	5	4	no	—
162	D₃×C₈	—	20	2	no	—
163	C₂²×D₅	—	6	1	no	—
164	C₅×Q₈	—	3	2	no	—
165	C₅×Q₈	—	6	1	no	—
166	C₂×C₄×C₅	—	6	1	no	—
167	C₂×C₄×C₅	—	6	1	no	—
168	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
169	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
170	C₂×C₄×C₅	#239	1	1	yes	characteristic
171	C₂×Dic₅	—	6	1	no	—
172	D₂₀	—	3	2	no	—
173	D₂₀	—	6	1	no	—
174	C₄×D₅	—	12	1	no	—
175	C₄×D₅	—	12	1	no	—
176	C₅×C₈	—	3	2	no	—
177	C₅×C₈	—	4	1	no	—
178	C₅×C₈	—	6	1	no	—
179	#168	—	2	1	no	—
180	#168	—	4	1	no	—
181	#168	—	6	2	no	—
182	#168	D₃×C₈	1	2	yes	complemented
183	C₂×SD₁₆	—	15	1	no	—
184	C₂²×C₈	—	30	1	no	—
185	C₄×Q₈	—	3	1	no	—
186	C₄×D₄	—	15	1	no	—
187	#106	—	15	1	no	—
188	C₄⋊C₈	—	5	1	no	—
189	C₄⋊C₈	—	6	1	no	—
190	C₄⋊C₈	D₃×D₅	1	1	yes	characteristic
191	#103	—	15	1	no	—
192	#102	—	15	1	no	—
193	C₂²⋊C₈	—	30	1	no	—
194	C₄×C₈	—	3	1	no	—
195	C₄×C₈	—	5	1	no	—
196	C₄×C₈	—	6	1	no	—
197	C₄×C₈	—	15	1	no	—
198	C₂×C₃×C₅	C₈×D₄	1	1	yes	characteristic
199	C₂×C₃×C₅	C₄×SD₁₆	1	1	yes	characteristic
200	C₂×C₃×C₅	#443	1	1	yes	characteristic
201	D₁₅	—	4	2	no	—
202	C₂²×D₃	—	10	1	no	—
203	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
204	C₂×C₃×C₄	#709	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
206	C₂×Dic₃	—	2	1	no	—
207	C₂×Dic₃	—	2	1	no	—
208	C₂×Dic₃	—	10	1	no	—
209	D₁₂	—	5	2	no	—
210	D₁₂	—	10	1	no	—
211	C₄×D₃	—	20	1	no	—
212	C₄×D₃	—	20	1	no	—
213	Dic₆	—	2	1	no	—
214	Dic₆	C₈×D₅	1	2	yes	complemented
215	C₃×C₈	—	4	1	no	—
216	C₃×C₈	—	5	2	no	—
217	C₃×C₈	—	10	1	no	—
218	C₃×C₈	—	20	1	no	—
219	C₃⋊C₈	—	2	1	no	—
220	C₃⋊C₈	—	10	2	no	—
221	C₃⋊C₈	#688	1	2	yes	complemented
222	C₂²×C₅	#893	1	1	yes	characteristic
223	C₂×D₅	—	6	2	no	—
224	C₂×D₅	—	12	1	no	—
225	C₂×D₅	—	12	1	no	—
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	—	6	2	no	—
228	C₄×C₅	—	12	1	no	—
229	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
230	C₄×C₅	#907	1	1	yes	characteristic
231	C₄×C₅	D₃×SD₁₆	1	1	yes	characteristic
232	C₄×C₅	#916	1	1	yes	characteristic
233	Dic₅	—	12	1	no	—
234	C₂×Q₈	—	3	1	no	—
235	C₂×D₄	—	15	1	no	—
236	C₂²×C₄	—	30	1	no	—
237	SD₁₆	—	15	4	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	3	1	no	—
240	C₂×C₈	—	3	1	no	—
241	C₂×C₈	—	5	1	no	—
242	C₂×C₈	—	5	1	no	—
243	C₂×C₈	—	10	1	no	—
244	C₂×C₈	—	30	1	no	—
245	C₂×C₈	—	60	2	no	—
246	C₄⋊C₄	—	3	1	no	—
247	C₄⋊C₄	—	6	1	no	—
248	C₄⋊C₄	—	15	1	no	—
249	C₂²⋊C₄	—	30	1	no	—
250	C₄²	—	6	1	no	—
251	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
252	C₃×C₅	C₈×SD₁₆	1	1	yes	characteristic, complemented, Hall
253	C₂²×C₃	C₄×C₂²⋊D₅	1	1	yes	characteristic
254	C₂×D₃	—	10	2	no	—
255	C₂×D₃	—	20	1	no	—
256	C₂×D₃	—	20	1	no	—
257	C₃×C₄	—	2	1	no	—
258	C₃×C₄	C₂×C₈×D₅	1	1	yes	characteristic
259	C₃×C₄	#4047	1	1	yes	characteristic
260	C₃×C₄	C₂×#688	1	1	yes	characteristic
261	C₃×C₄	#4096	1	1	yes	characteristic
262	Dic₃	—	2	2	no	—
263	Dic₃	—	4	1	no	—
264	Dic₃	—	20	1	no	—
265	C₂×C₅	#4834	1	1	yes	characteristic
266	C₂×C₅	#4852	1	1	yes	characteristic
267	C₂×C₅	#4895	1	1	yes	characteristic
268	D₅	—	12	2	no	—
269	C₂³	—	30	1	no	—
270	Q₈	—	3	2	no	—
271	Q₈	—	6	1	no	—
272	D₄	—	15	2	no	—
273	D₄	—	30	1	no	—
274	C₂×C₄	—	6	1	no	—
275	C₂×C₄	—	6	1	no	—
276	C₂×C₄	—	30	1	no	—
277	C₂×C₄	—	60	1	no	—
278	C₂×C₄	—	60	1	no	—
279	C₂×C₄	#6769	1	1	yes	characteristic
280	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
281	C₂×C₄	D₃×C₂²⋊D₅	1	1	yes	characteristic
282	C₈	—	3	2	no	—
283	C₈	—	4	1	no	—
284	C₈	—	5	2	no	—
285	C₈	—	6	1	no	—
286	C₈	—	10	1	no	—
287	C₈	—	20	1	no	—
288	C₈	—	30	2	no	—
289	C₂×C₃	C₄×#688	1	1	yes	characteristic
290	C₂×C₃	C₈×C₂²⋊D₅	1	1	yes	characteristic
291	C₂×C₃	#65433	1	1	yes	characteristic
292	D₃	—	20	2	no	—
293	C₅	#68516	1	1	yes	characteristic, complemented, Sylow
294	C₂²	—	30	2	no	—
295	C₂²	—	60	1	no	—
296	C₂²	—	60	1	no	—
297	C₂²	#91446	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	6	2	no	—
300	C₄	—	12	1	no	—
301	C₄	—	60	1	no	—
302	C₄	D₃×C₈×D₅	1	1	yes	characteristic
303	C₄	#91267	1	1	yes	characteristic
304	C₄	#91495	1	1	yes	characteristic
305	C₄	D₃×#688	1	1	yes	characteristic
306	C₃	C₈×#688	1	1	yes	characteristic, complemented, Sylow
307	C₂	—	60	2	no	—
308	C₂	#11747232	1	1	yes	characteristic
309	C₂	#11747700	1	1	yes	characteristic
310	C₂	#11748208	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₃×C₂²⋊D₅
commutator	C₃×C₄×C₅	C₂²×C₈
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₃×C₅×C₄⋊C₈	C₂²
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂×D₆₀	C₂×C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄⋊C₈	D₃×D₅
hypocenter	C₃×C₅	C₈×SD₁₆
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₅×#4830	C₂	semidirect
5. C₃×#64700	C₂	semidirect
6. #11745842	C₂	semidirect
7. C₃⋊C₈×#168	C₂	semidirect
30. #64700	D₃	semidirect
48. #6660	C₈	semidirect
51. #4830	D₅	semidirect
85. C₅×Dic₆	C₂×C₈	semidirect
90. C₃×#168	C₂×C₈	semidirect
92. C₅×C₃⋊C₈	SD₁₆	semidirect
182. #168	D₃×C₈	semidirect
214. Dic₆	C₈×D₅	semidirect
221. C₃⋊C₈	#688	semidirect
252. C₃×C₅	C₈×SD₁₆	semidirect
293. C₅	#68516	semidirect
306. C₃	C₈×#688	semidirect

Maximal subgroups

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

subgroup	index
C₅×#4830	2
C₄×#6660	2
#11746994	2
#11746304	2
C₃×#64700	2
#11745842	2
C₃⋊C₈×#168	2
C₈×#688	3
#68516	5