Subgroups — #14286432

Order 1920.

384 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#6715	C₂	1	1	yes	maximal, characteristic
2	D₄×C₃⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
3	#11750935	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#11746301	C₂	1	1	yes	maximal, characteristic, complemented
6	#11746280	C₂	1	1	yes	maximal, characteristic, complemented
7	#11746275	C₂	1	1	yes	maximal, characteristic, complemented
8	#10598944	—	3	1	no	maximal, Hall
9	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
10	C₂×#6715	C₄	1	1	yes	characteristic
11	C₂²×C₃⋊#168	—	2	1	no	—
12	C₂×C₈×D₁₅	—	2	1	no	—
13	C₄×D₆₀	C₂²	1	1	yes	characteristic
14	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	#91116	C₄	1	1	yes	characteristic
17	#91114	—	2	1	no	—
18	#91104	—	2	1	no	—
19	#91093	C₄	1	1	yes	characteristic
20	#91091	C₄	1	1	yes	characteristic
21	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
22	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
23	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
24	#68151	—	5	1	no	maximal, Hall
25	C₄×#686	—	3	1	no	—
26	D₄×#168	—	3	1	no	—
27	#65123	—	3	1	no	—
28	C₅×D₄⋊C₈	D₃	1	1	yes	characteristic, complemented
29	#64700	—	3	1	no	—
30	#64679	—	3	1	no	—
31	#64674	—	3	1	no	—
32	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₃×C₄×C₅	—	2	1	no	—
34	C₂×D₆₀	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×D₁₅	—	2	1	no	—
36	C₂×C₃×C₅×C₈	—	2	1	no	—
37	C₂×C₃×C₅×C₈	—	2	1	no	—
38	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
39	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
40	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
41	#6715	C₈	1	4	yes	complemented
42	#6714	—	2	1	no	—
43	C₄⋊Dic₁₅	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃⋊#168	—	2	1	no	—
45	C₂×C₃⋊#168	—	4	2	no	—
46	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
47	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
48	C₂×C₃⋊#168	D₄	1	1	yes	characteristic
49	C₂×C₃⋊#168	D₄	1	1	yes	characteristic
50	C₈×D₁₅	—	4	2	no	—
51	C₄×#213	—	5	1	no	—
52	D₄×C₃⋊C₈	—	5	1	no	—
53	#4834	—	5	1	no	—
54	C₃×D₄⋊C₈	D₅	1	1	yes	characteristic, complemented
55	#4480	—	5	1	no	—
56	#4459	—	5	1	no	—
57	#4454	—	5	1	no	—
58	C₄×C₅×D₄	C₂×D₃	1	1	yes	characteristic
59	C₂×#686	—	3	1	no	—
60	C₂²×#168	—	6	1	no	—
61	C₂×C₈×D₅	—	6	1	no	—
62	C₄×D₂₀	—	3	1	no	—
63	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
64	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
65	#3964	—	3	1	no	—
66	C₂²⋊#168	—	6	1	no	—
67	#3952	—	6	1	no	—
68	#3941	—	3	1	no	—
69	#3939	—	3	1	no	—
70	C₄⋊#168	—	3	1	no	—
71	C₄×#168	—	3	1	no	—
72	C₄×#168	—	3	1	no	—
73	#1580	—	15	1	no	Sylow
74	C₂³×C₃×C₅	—	2	1	no	—
75	C₂²×D₁₅	—	2	1	no	—
76	C₃×C₅×D₄	—	2	1	no	—
77	C₃×C₅×D₄	C₂×C₈	1	2	yes	complemented
78	C₂×C₃×C₄×C₅	—	2	1	no	—
79	C₂×C₃×C₄×C₅	—	4	1	no	—
80	C₂×C₃×C₄×C₅	—	4	1	no	—
81	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
84	C₂×Dic₁₅	—	2	1	no	—
85	D₆₀	—	2	1	no	—
86	D₆₀	C₂×C₈	1	2	yes	complemented
87	C₄×D₁₅	—	4	1	no	—
88	C₄×D₁₅	—	4	1	no	—
89	C₃×C₅×C₈	—	4	1	no	—
90	C₃×C₅×C₈	—	4	1	no	—
91	C₃⋊#168	—	2	1	no	—
92	C₃⋊#168	—	2	1	no	—
93	C₃⋊#168	—	2	1	no	—
94	C₃⋊#168	—	4	1	no	—
95	C₃⋊#168	C₂×C₈	1	2	yes	complemented
96	C₃×C₄×D₄	C₂×D₅	1	1	yes	characteristic
97	C₂×#213	—	5	1	no	—
98	C₂²×C₃⋊C₈	—	10	1	no	—
99	C₂×D₃×C₈	—	10	1	no	—
100	C₄×D₁₂	—	5	1	no	—
101	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
102	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
103	#832	—	5	1	no	—
104	#830	—	10	1	no	—
105	(C₂×D₃)⋊C₈	—	10	1	no	—
106	#809	—	5	1	no	—
107	#807	—	5	1	no	—
108	C₄⋊(C₃⋊C₈)	—	5	1	no	—
109	C₄×C₃⋊C₈	—	5	1	no	—
110	C₄×C₃⋊C₈	—	5	1	no	—
111	C₂×C₅×D₄	C₄×D₃	1	1	yes	characteristic
112	C₂²×C₄×C₅	—	2	1	no	—
113	C₂×D₂₀	—	3	1	no	—
114	C₂×C₄×D₅	—	6	1	no	—
115	C₂×C₅×C₈	—	2	1	no	—
116	C₂×C₅×C₈	—	2	1	no	—
117	C₅×C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
118	C₅×C₂²⋊C₄	—	2	1	no	—
119	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
120	#686	—	3	4	no	—
121	#685	—	6	1	no	—
122	C₄⋊Dic₅	—	3	1	no	—
123	C₂×#168	—	3	1	no	—
124	C₂×#168	—	3	1	no	—
125	C₂×#168	—	3	1	no	—
126	C₂×#168	—	3	1	no	—
127	C₂×#168	—	6	1	no	—
128	C₂×#168	—	12	2	no	—
129	C₈×D₅	—	12	2	no	—
130	C₄×D₈	—	15	1	no	—
131	C₈×D₄	—	15	1	no	—
132	C₈×D₄	—	15	1	no	—
133	#334	—	15	1	no	—
134	D₄⋊C₈	—	15	1	no	—
135	D₄⋊C₈	D₁₅	1	1	yes	characteristic, complemented
136	#322	—	15	1	no	—
137	C₂²×C₃×C₅	—	2	2	no	—
138	C₂²×C₃×C₅	—	4	1	no	—
139	C₂²×C₃×C₅	—	4	1	no	—
140	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
141	C₂×D₁₅	—	2	2	no	—
142	C₂×D₁₅	—	4	1	no	—
143	C₂×D₁₅	—	4	1	no	—
144	C₃×C₄×C₅	—	2	1	no	—
145	C₃×C₄×C₅	—	4	1	no	—
146	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
147	C₃×C₄×C₅	#131	1	1	yes	characteristic
148	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
150	Dic₁₅	—	4	1	no	—
151	C₂×C₃×D₄	C₄×D₅	1	1	yes	characteristic
152	C₂²×C₃×C₄	—	2	1	no	—
153	C₂×D₁₂	—	5	1	no	—
154	C₂×C₄×D₃	—	10	1	no	—
155	C₂×C₃×C₈	—	2	1	no	—
156	C₂×C₃×C₈	—	2	1	no	—
157	C₃×C₄⋊C₄	C₄×D₅	1	1	yes	characteristic
158	C₃×C₂²⋊C₄	—	2	1	no	—
159	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
160	#213	—	5	4	no	—
161	(C₂×D₃)⋊C₄	—	10	1	no	—
162	C₄⋊Dic₃	—	5	1	no	—
163	C₂×C₃⋊C₈	—	5	1	no	—
164	C₂×C₃⋊C₈	—	5	1	no	—
165	C₂×C₃⋊C₈	—	5	1	no	—
166	C₂×C₃⋊C₈	—	5	1	no	—
167	C₂×C₃⋊C₈	—	10	1	no	—
168	C₂×C₃⋊C₈	—	20	2	no	—
169	D₃×C₈	—	20	2	no	—
170	C₂³×C₅	—	2	1	no	—
171	C₂²×D₅	—	6	1	no	—
172	C₅×D₄	—	2	1	no	—
173	C₅×D₄	D₃×C₈	1	2	yes	complemented
174	C₂×C₄×C₅	—	2	1	no	—
175	C₂×C₄×C₅	—	4	1	no	—
176	C₂×C₄×C₅	—	4	1	no	—
177	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
178	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
179	C₂×C₄×C₅	#237	1	1	yes	characteristic
180	C₂×Dic₅	—	6	1	no	—
181	D₂₀	—	3	2	no	—
182	D₂₀	—	6	1	no	—
183	C₄×D₅	—	12	1	no	—
184	C₄×D₅	—	12	1	no	—
185	C₅×C₈	—	4	1	no	—
186	C₅×C₈	—	4	1	no	—
187	#168	—	3	2	no	—
188	#168	—	6	1	no	—
189	#168	—	6	1	no	—
190	#168	—	6	1	no	—
191	#168	—	12	1	no	—
192	C₂×D₈	—	15	1	no	—
193	C₂²×C₈	—	30	1	no	—
194	C₂²×C₈	—	30	1	no	—
195	C₄×D₄	—	15	1	no	—
196	C₄×D₄	C₂×D₁₅	1	1	yes	characteristic
197	#107	—	15	1	no	—
198	C₄⋊C₈	—	15	1	no	—
199	C₄⋊C₈	C₂×D₁₅	1	1	yes	characteristic
200	#102	—	15	1	no	—
201	#102	—	15	1	no	—
202	C₂²⋊C₈	—	30	1	no	—
203	C₂²⋊C₈	—	30	1	no	—
204	C₄×C₈	—	15	1	no	—
205	C₄×C₈	—	15	1	no	—
206	C₄×C₈	C₂×D₁₅	1	1	yes	characteristic
207	C₂×C₃×C₅	—	4	2	no	—
208	C₂×C₃×C₅	C₈×D₄	1	1	yes	characteristic
209	C₂×C₃×C₅	#442	1	1	yes	characteristic
210	C₂×C₃×C₅	#444	1	1	yes	characteristic
211	D₁₅	—	4	2	no	—
212	C₂³×C₃	—	2	1	no	—
213	C₂²×D₃	—	10	1	no	—
214	C₃×D₄	—	2	1	no	—
215	C₃×D₄	C₈×D₅	1	2	yes	complemented
216	C₂×C₃×C₄	—	2	1	no	—
217	C₂×C₃×C₄	—	4	1	no	—
218	C₂×C₃×C₄	—	4	1	no	—
219	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
220	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
221	C₂×C₃×C₄	#711	1	1	yes	characteristic
222	C₂×Dic₃	—	10	1	no	—
223	D₁₂	—	5	2	no	—
224	D₁₂	—	10	1	no	—
225	C₄×D₃	—	20	1	no	—
226	C₄×D₃	—	20	1	no	—
227	C₃×C₈	—	4	1	no	—
228	C₃×C₈	—	4	1	no	—
229	C₃⋊C₈	—	5	2	no	—
230	C₃⋊C₈	—	10	1	no	—
231	C₃⋊C₈	—	10	1	no	—
232	C₃⋊C₈	—	10	1	no	—
233	C₃⋊C₈	—	20	1	no	—
234	C₂²×C₅	—	2	2	no	—
235	C₂²×C₅	—	4	1	no	—
236	C₂²×C₅	—	4	1	no	—
237	C₂²×C₅	#67⋊C₄	1	1	yes	characteristic
238	C₂×D₅	—	6	2	no	—
239	C₂×D₅	—	12	1	no	—
240	C₂×D₅	—	12	1	no	—
241	C₄×C₅	—	2	1	no	—
242	C₄×C₅	—	4	1	no	—
243	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
244	C₄×C₅	#907	1	1	yes	characteristic
245	C₄×C₅	#911	1	1	yes	characteristic
246	C₄×C₅	#914	1	1	yes	characteristic
247	Dic₅	—	12	1	no	—
248	C₂×D₄	—	15	1	no	—
249	C₂×D₄	C₄×D₁₅	1	1	yes	characteristic
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	—	30	1	no	—
252	D₈	—	15	4	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	15	1	no	—
256	C₂×C₈	—	15	1	no	—
257	C₂×C₈	—	15	1	no	—
258	C₂×C₈	—	15	1	no	—
259	C₂×C₈	—	30	1	no	—
260	C₂×C₈	—	60	2	no	—
261	C₂×C₈	—	60	2	no	—
262	C₄⋊C₄	—	15	1	no	—
263	C₄⋊C₄	C₄×D₁₅	1	1	yes	characteristic
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	30	1	no	—
266	C₄²	C₂²×D₁₅	1	1	yes	characteristic
267	C₃×C₅	#1580	1	1	yes	characteristic, complemented, Hall
268	C₂²×C₃	—	2	2	no	—
269	C₂²×C₃	—	4	1	no	—
270	C₂²×C₃	—	4	1	no	—
271	C₂²×C₃	#4027	1	1	yes	characteristic
272	C₂×D₃	—	10	2	no	—
273	C₂×D₃	—	20	1	no	—
274	C₂×D₃	—	20	1	no	—
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	4	1	no	—
277	C₃×C₄	C₂×C₈×D₅	1	1	yes	characteristic
278	C₃×C₄	#4053	1	1	yes	characteristic
279	C₃×C₄	#4057	1	1	yes	characteristic
280	C₃×C₄	#4060	1	1	yes	characteristic
281	Dic₃	—	20	1	no	—
282	C₂×C₅	—	4	2	no	—
283	C₂×C₅	#67⋊C₈	1	1	yes	characteristic
284	C₂×C₅	#4785	1	1	yes	characteristic
285	C₂×C₅	#4823	1	1	yes	characteristic
286	D₅	—	12	2	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	30	1	no	—
289	D₄	—	2	1	no	—
290	D₄	—	15	2	no	—
291	D₄	—	30	1	no	—
292	D₄	C₈×D₁₅	1	2	yes	complemented
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	30	1	no	—
297	C₂×C₄	—	60	1	no	—
298	C₂×C₄	—	60	1	no	—
299	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
300	C₂×C₄	D₄×D₁₅	1	1	yes	characteristic
301	C₂×C₄	#6819	1	1	yes	characteristic
302	C₈	—	4	1	no	—
303	C₈	—	4	1	no	—
304	C₈	—	15	2	no	—
305	C₈	—	30	1	no	—
306	C₈	—	30	1	no	—
307	C₈	—	30	1	no	—
308	C₈	—	60	1	no	—
309	C₂×C₃	—	4	2	no	—
310	C₂×C₃	#65014	1	1	yes	characteristic
311	C₂×C₃	#65074	1	1	yes	characteristic
312	C₂×C₃	#65112	1	1	yes	characteristic
313	D₃	—	20	2	no	—
314	C₅	#68151	1	1	yes	characteristic, complemented, Sylow
315	C₂²	—	2	2	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	30	2	no	—
319	C₂²	—	60	1	no	—
320	C₂²	—	60	1	no	—
321	C₂²	#91770	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	60	1	no	—
325	C₄	C₂×C₈×D₁₅	1	1	yes	characteristic
326	C₄	#91796	1	1	yes	characteristic
327	C₄	#91800	1	1	yes	characteristic
328	C₄	#91803	1	1	yes	characteristic
329	C₃	#10598944	1	1	yes	characteristic, complemented, Sylow
330	C₂	—	4	2	no	—
331	C₂	—	60	2	no	—
332	C₂	#11750826	1	1	yes	characteristic
333	C₂	#11750886	1	1	yes	characteristic
334	C₂	#11750924	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₅
commutator	C₃×C₄×C₅	C₂²×C₈
Frattini	C₄²	C₂²×D₁₅
Fitting	C₃×C₅×D₄⋊C₈	C₂
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂×#6715	C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	D₄⋊C₈	D₁₅
hypocenter	C₃×C₅	#1580
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ D₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. D₄×C₃⋊#168	C₂	semidirect
3. #11750935	C₂	semidirect
4. C₃×C₅×D₄⋊C₈	C₂	semidirect
5. #11746301	C₂	semidirect
6. #11746280	C₂	semidirect
7. #11746275	C₂	semidirect
28. C₅×D₄⋊C₈	D₃	semidirect
41. #6715	C₈	semidirect
54. C₃×D₄⋊C₈	D₅	semidirect
77. C₃×C₅×D₄	C₂×C₈	semidirect
86. D₆₀	C₂×C₈	semidirect
95. C₃⋊#168	C₂×C₈	semidirect
135. D₄⋊C₈	D₁₅	semidirect
173. C₅×D₄	D₃×C₈	semidirect
215. C₃×D₄	C₈×D₅	semidirect
267. C₃×C₅	#1580	semidirect
292. D₄	C₈×D₁₅	semidirect
314. C₅	#68151	semidirect
329. C₃	#10598944	semidirect

Maximal subgroups

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

subgroup	index
C₄×#6715	2
D₄×C₃⋊#168	2
#11750935	2
C₃×C₅×D₄⋊C₈	2
#11746301	2
#11746280	2
#11746275	2
#10598944	3
#68151	5