Subgroups — #14286458

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	D₄×C₃⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₄×#686	C₂	1	1	yes	maximal, characteristic
3	#11746996	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×D₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	#11745864	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745835	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745831	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	C₂²×C₃⋊#168	—	2	1	no	—
11	C₂×C₃×#686	C₄	1	1	yes	characteristic
12	C₃×C₄×D₂₀	C₂²	1	1	yes	characteristic
13	C₂×D₅×C₃⋊C₈	—	2	1	no	—
14	#91114	—	2	1	no	—
15	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
16	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
17	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
18	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₃×#3964	C₄	1	1	yes	characteristic
20	C₃×#3941	C₄	1	1	yes	characteristic
21	C₃×#3939	C₄	1	1	yes	characteristic
22	C₃×C₄×#168	C₂²	1	1	yes	characteristic
23	#90955	—	2	1	no	—
24	#70878	—	5	1	no	maximal, Hall
25	C₄×#686	D₃	1	1	yes	characteristic
26	D₄×#168	—	3	1	no	—
27	#65123	—	3	1	no	—
28	C₅×D₄⋊C₈	—	3	1	no	—
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×C₃×D₂₀	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×C₄×D₅	—	2	1	no	—
36	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
37	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
38	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
39	C₂×C₃⋊#168	—	2	1	no	—
40	C₂×C₃⋊#168	—	4	2	no	—
41	C₂×C₃⋊#168	D₄	1	1	yes	characteristic
42	C₂×C₃⋊#168	D₄	1	1	yes	characteristic
43	C₂×C₅×C₃⋊C₈	—	2	1	no	—
44	C₂×C₅×C₃⋊C₈	—	2	1	no	—
45	C₃×#686	C₈	1	4	yes	complemented
46	C₃×#685	—	2	1	no	—
47	C₃×C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
48	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
49	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
50	D₅×C₃⋊C₈	—	4	2	no	—
51	C₃×C₄×D₈	—	5	1	no	—
52	D₄×C₃⋊C₈	—	5	1	no	—
53	D₄×C₃⋊C₈	—	5	1	no	—
54	D₄⋊(C₃⋊C₈)	—	5	1	no	—
55	D₄⋊(C₃⋊C₈)	D₅	1	1	yes	characteristic, complemented
56	#4457	—	5	1	no	—
57	#4455	—	5	1	no	—
58	C₄×C₅×D₄	C₂×D₃	1	1	yes	characteristic
59	C₂×#686	Dic₃	1	1	yes	characteristic
60	C₂²×#168	—	6	1	no	—
61	C₂×C₈×D₅	—	6	1	no	—
62	C₄×D₂₀	C₂×D₃	1	1	yes	characteristic
63	C₅×C₄⋊C₈	—	3	1	no	—
64	C₄×C₅×C₈	—	3	1	no	—
65	#3964	Dic₃	1	1	yes	characteristic
66	C₂²⋊#168	—	6	1	no	—
67	#3952	—	6	1	no	—
68	#3941	Dic₃	1	1	yes	characteristic
69	#3939	Dic₃	1	1	yes	characteristic
70	C₄⋊#168	—	3	1	no	—
71	C₄×#168	—	3	1	no	—
72	C₄×#168	C₂×D₃	1	1	yes	characteristic
73	#1580	—	15	1	no	Sylow
74	C₂³×C₃×C₅	—	2	1	no	—
75	C₂²×C₃×D₅	—	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×C₃×Dic₅	—	2	1	no	—
85	C₃×D₂₀	—	2	1	no	—
86	C₃×D₂₀	C₂×C₈	1	2	yes	complemented
87	C₃×C₄×D₅	—	4	1	no	—
88	C₃×C₄×D₅	—	4	1	no	—
89	C₃⋊#168	—	2	1	no	—
90	C₃⋊#168	—	2	1	no	—
91	C₃⋊#168	—	4	1	no	—
92	C₃×#168	—	2	1	no	—
93	C₃×#168	C₂×C₈	1	2	yes	complemented
94	C₅×C₃⋊C₈	—	4	1	no	—
95	C₅×C₃⋊C₈	—	4	1	no	—
96	C₂×C₃×D₈	—	5	1	no	—
97	C₃×C₄×D₄	—	5	1	no	—
98	C₃×C₄×D₄	C₂×D₅	1	1	yes	characteristic
99	C₂²×C₃⋊C₈	—	10	1	no	—
100	C₂²×C₃⋊C₈	—	10	1	no	—
101	C₃×#107	—	5	1	no	—
102	C₃×#102	—	5	1	no	—
103	C₃×#102	—	5	1	no	—
104	C₃×C₄×C₈	—	5	1	no	—
105	#830	—	10	1	no	—
106	#830	—	10	1	no	—
107	C₄⋊(C₃⋊C₈)	—	5	1	no	—
108	C₄⋊(C₃⋊C₈)	C₂×D₅	1	1	yes	characteristic
109	C₄×C₃⋊C₈	—	5	1	no	—
110	C₄×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
111	C₂×C₅×D₄	C₂×Dic₃	1	1	yes	characteristic
112	C₂²×C₄×C₅	—	2	1	no	—
113	C₂×D₂₀	C₂×Dic₃	1	1	yes	characteristic
114	C₂×C₄×D₅	—	2	1	no	—
115	C₂×C₅×C₈	—	6	1	no	—
116	C₂×C₅×C₈	—	6	1	no	—
117	C₅×C₄⋊C₄	C₂×Dic₃	1	1	yes	characteristic
118	C₅×C₂²⋊C₄	—	2	1	no	—
119	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
120	#686	C₃⋊C₈	1	4	yes	complemented
121	#685	—	2	1	no	—
122	C₄⋊Dic₅	C₂×Dic₃	1	1	yes	characteristic
123	C₂×#168	—	3	1	no	—
124	C₂×#168	—	3	1	no	—
125	C₂×#168	—	6	1	no	—
126	C₂×#168	—	12	2	no	—
127	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
128	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
129	C₈×D₅	—	12	2	no	—
130	C₄×D₈	—	5	1	no	—
131	C₈×D₄	—	15	1	no	—
132	C₈×D₄	—	15	1	no	—
133	#334	—	15	1	no	—
134	D₄⋊C₈	—	3	1	no	—
135	D₄⋊C₈	—	15	1	no	—
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×C₃×D₅	—	2	2	no	—
142	C₂×C₃×D₅	—	4	1	no	—
143	C₂×C₃×D₅	—	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	C₃×Dic₅	—	4	1	no	—
151	C₂×C₃×D₄	—	5	1	no	—
152	C₂×C₃×D₄	C₄×D₅	1	1	yes	characteristic
153	C₂²×C₃×C₄	—	2	1	no	—
154	C₂²×C₃×C₄	—	10	1	no	—
155	C₃×D₈	—	5	4	no	—
156	C₂×C₃×C₈	—	5	1	no	—
157	C₂×C₃×C₈	—	5	1	no	—
158	C₃×C₄⋊C₄	—	5	1	no	—
159	C₃×C₄⋊C₄	C₄×D₅	1	1	yes	characteristic
160	C₃×C₂²⋊C₄	—	2	1	no	—
161	C₃×C₂²⋊C₄	—	10	1	no	—
162	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
163	C₂×C₃⋊C₈	—	2	1	no	—
164	C₂×C₃⋊C₈	—	2	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	C₂×C₃⋊C₈	—	20	2	no	—
170	C₂³×C₅	—	2	1	no	—
171	C₂²×D₅	—	2	1	no	—
172	C₅×D₄	—	2	1	no	—
173	C₅×D₄	C₂×C₃⋊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	D₃×D₄	1	1	yes	characteristic
178	C₂×C₄×C₅	#237	1	1	yes	characteristic
179	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
180	C₂×Dic₅	—	2	1	no	—
181	D₂₀	—	2	1	no	—
182	D₂₀	C₂×C₃⋊C₈	1	2	yes	complemented
183	C₄×D₅	—	4	1	no	—
184	C₄×D₅	—	4	1	no	—
185	C₅×C₈	—	12	1	no	—
186	C₅×C₈	—	12	1	no	—
187	#168	—	2	1	no	—
188	#168	—	6	1	no	—
189	#168	—	6	1	no	—
190	#168	—	12	1	no	—
191	#168	C₂×C₃⋊C₈	1	2	yes	complemented
192	C₂×D₈	—	5	1	no	—
193	C₂²×C₈	—	30	1	no	—
194	C₂²×C₈	—	30	1	no	—
195	C₄×D₄	—	5	1	no	—
196	C₄×D₄	D₃×D₅	1	1	yes	characteristic
197	#107	—	5	1	no	—
198	C₄⋊C₈	—	3	1	no	—
199	C₄⋊C₈	—	15	1	no	—
200	#102	—	5	1	no	—
201	#102	—	5	1	no	—
202	C₂²⋊C₈	—	30	1	no	—
203	C₂²⋊C₈	—	30	1	no	—
204	C₄×C₈	—	3	1	no	—
205	C₄×C₈	—	5	1	no	—
206	C₄×C₈	—	15	1	no	—
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	C₃×D₅	—	4	2	no	—
212	C₂³×C₃	—	2	1	no	—
213	C₂³×C₃	—	10	1	no	—
214	C₃×D₄	—	2	1	no	—
215	C₃×D₄	—	5	2	no	—
216	C₃×D₄	—	10	1	no	—
217	C₃×D₄	C₈×D₅	1	2	yes	complemented
218	C₂×C₃×C₄	—	2	1	no	—
219	C₂×C₃×C₄	—	4	1	no	—
220	C₂×C₃×C₄	—	4	1	no	—
221	C₂×C₃×C₄	—	10	1	no	—
222	C₂×C₃×C₄	—	20	1	no	—
223	C₂×C₃×C₄	—	20	1	no	—
224	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
225	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
226	C₂×C₃×C₄	#711	1	1	yes	characteristic
227	C₃×C₈	—	5	2	no	—
228	C₃×C₈	—	10	1	no	—
229	C₃⋊C₈	—	4	1	no	—
230	C₃⋊C₈	—	4	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	D₄×Dic₃	1	1	yes	characteristic
238	C₂×D₅	—	2	2	no	—
239	C₂×D₅	—	4	1	no	—
240	C₂×D₅	—	4	1	no	—
241	C₄×C₅	—	2	1	no	—
242	C₄×C₅	—	4	1	no	—
243	C₄×C₅	#911	1	1	yes	characteristic
244	C₄×C₅	#911	1	1	yes	characteristic
245	C₄×C₅	C₂²×C₃⋊C₈	1	1	yes	characteristic
246	C₄×C₅	#948	1	1	yes	characteristic
247	Dic₅	—	4	1	no	—
248	C₂×D₄	—	5	1	no	—
249	C₂×D₄	D₅×Dic₃	1	1	yes	characteristic
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	—	10	1	no	—
252	D₈	—	5	4	no	—
253	C₂×C₈	—	5	1	no	—
254	C₂×C₈	—	5	1	no	—
255	C₂×C₈	—	6	1	no	—
256	C₂×C₈	—	6	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	—	5	1	no	—
263	C₄⋊C₄	D₅×Dic₃	1	1	yes	characteristic
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	10	1	no	—
266	C₄²	C₂×D₃×D₅	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	—	10	2	no	—
272	C₂²×C₃	—	20	1	no	—
273	C₂²×C₃	—	20	1	no	—
274	C₂²×C₃	#4027	1	1	yes	characteristic
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	4	1	no	—
277	C₃×C₄	—	20	1	no	—
278	C₃×C₄	C₂×C₈×D₅	1	1	yes	characteristic
279	C₃×C₄	#4053	1	1	yes	characteristic
280	C₃×C₄	#4057	1	1	yes	characteristic
281	C₃×C₄	#4060	1	1	yes	characteristic
282	C₂×C₅	—	4	2	no	—
283	C₂×C₅	D₄×C₃⋊C₈	1	1	yes	characteristic
284	C₂×C₅	#5152	1	1	yes	characteristic
285	C₂×C₅	#5197	1	1	yes	characteristic
286	D₅	—	4	2	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	10	1	no	—
289	D₄	—	2	1	no	—
290	D₄	—	5	2	no	—
291	D₄	—	10	1	no	—
292	D₄	D₅×C₃⋊C₈	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	—	10	1	no	—
297	C₂×C₄	—	20	1	no	—
298	C₂×C₄	—	20	1	no	—
299	C₂×C₄	C₂×D₅×Dic₃	1	1	yes	characteristic
300	C₂×C₄	#6780	1	1	yes	characteristic
301	C₂×C₄	#6790	1	1	yes	characteristic
302	C₈	—	5	2	no	—
303	C₈	—	10	1	no	—
304	C₈	—	12	1	no	—
305	C₈	—	12	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	—	20	2	no	—
311	C₂×C₃	#65014	1	1	yes	characteristic
312	C₂×C₃	#65074	1	1	yes	characteristic
313	C₂×C₃	#65112	1	1	yes	characteristic
314	C₅	#70878	1	1	yes	characteristic, complemented, Sylow
315	C₂²	—	2	2	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	10	2	no	—
319	C₂²	—	20	1	no	—
320	C₂²	—	20	1	no	—
321	C₂²	#91559	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	20	1	no	—
325	C₄	C₂×D₅×C₃⋊C₈	1	1	yes	characteristic
326	C₄	#91284	1	1	yes	characteristic
327	C₄	#91482	1	1	yes	characteristic
328	C₄	#91498	1	1	yes	characteristic
329	C₃	#10598944	1	1	yes	characteristic, complemented, Sylow
330	C₂	—	4	2	no	—
331	C₂	—	20	2	no	—
332	C₂	#11748493	1	1	yes	characteristic
333	C₂	#11748995	1	1	yes	characteristic
334	C₂	#11749441	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6790
commutator	C₃×C₄×C₅	C₂²×C₈
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₃×C₄×C₅×D₄	C₂²
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂×C₃×#686	C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄×D₄	D₃×D₅
hypocenter	C₃×C₅	#1580
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. D₄×C₃⋊#168	C₂	semidirect
3. #11746996	C₂	semidirect
4. C₅×D₄⋊(C₃⋊C₈)	C₂	semidirect
5. #11745864	C₂	semidirect
6. #11745835	C₂	semidirect
7. #11745831	C₂	semidirect
45. C₃×#686	C₈	semidirect
55. D₄⋊(C₃⋊C₈)	D₅	semidirect
77. C₃×C₅×D₄	C₂×C₈	semidirect
86. C₃×D₂₀	C₂×C₈	semidirect
93. C₃×#168	C₂×C₈	semidirect
120. #686	C₃⋊C₈	semidirect
173. C₅×D₄	C₂×C₃⋊C₈	semidirect
182. D₂₀	C₂×C₃⋊C₈	semidirect
191. #168	C₂×C₃⋊C₈	semidirect
217. C₃×D₄	C₈×D₅	semidirect
267. C₃×C₅	#1580	semidirect
292. D₄	D₅×C₃⋊C₈	semidirect
314. C₅	#70878	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
D₄×C₃⋊#168	2
C₃×C₄×#686	2
#11746996	2
C₅×D₄⋊(C₃⋊C₈)	2
#11745864	2
#11745835	2
#11745831	2
#10598944	3
#70878	5