Subgroups — #18081786

Order 4864.

654 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14564563	C₂	1	2	yes	maximal, complemented
2	C₈×C₄⋊Dic₁₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11926297	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11926288	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₉×#493	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
8	C₂×#10597289	—	2	4	no	—
9	C₂×C₈×Dic₁₉	—	2	1	no	—
10	C₄×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
11	#11926298	—	2	1	no	—
12	#11926297	—	2	2	no	—
13	#11926288	—	2	4	no	—
14	C₂²×D₈×C₁₉	—	2	1	no	—
15	C₂²×D₈×C₁₉	—	2	1	no	—
16	C₂×C₁₉×#127	C₂²	1	2	yes	—
17	C₁₉×#493	—	2	2	no	—
18	C₁₉×#493	C₄	1	4	yes	complemented
19	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
20	C₂²×D₄×C₁₉	—	2	2	no	—
21	C₂²×D₄×C₁₉	—	2	2	no	—
22	C₂²×D₄×C₁₉	—	2	2	no	—
23	C₂×D₈×C₁₉	—	2	4	no	—
24	C₂×D₈×C₁₉	—	2	4	no	—
25	C₂×D₈×C₁₉	—	4	2	no	—
26	C₂×D₈×C₁₉	—	4	2	no	—
27	C₂×D₈×C₁₉	—	4	2	no	—
28	C₂×D₈×C₁₉	—	4	2	no	—
29	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
31	C₁₉×#127	—	2	2	no	—
32	C₁₉×#127	—	2	2	no	—
33	C₁₉×#127	—	2	2	no	—
34	C₁₉×#127	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
36	C₂×C₄⋊Dic₁₉	D₄	1	2	yes	—
37	C₂×C₄×Dic₁₉	—	2	1	no	—
38	C₂²×C₁₉⋊C₈	—	2	1	no	—
39	C₂²×C₁₉⋊C₈	—	2	2	no	—
40	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₁₉	D₄	1	2	yes	—
42	#10597289	—	4	8	no	—
43	C₁₉⋊#95	—	2	1	no	—
44	C₈×Dic₁₉	—	4	2	no	—
45	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
46	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
47	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
48	C₄⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
49	C₂⁴×C₁₉	—	4	2	no	—
50	C₂×D₄×C₁₉	—	2	4	no	—
51	C₂×D₄×C₁₉	—	2	4	no	—
52	C₂×D₄×C₁₉	—	2	4	no	—
53	C₂×D₄×C₁₉	—	4	2	no	—
54	C₂×D₄×C₁₉	—	4	2	no	—
55	C₂×D₄×C₁₉	—	4	2	no	—
56	C₂×D₄×C₁₉	—	4	2	no	—
57	C₂×D₄×C₁₉	—	4	4	no	—
58	C₂×D₄×C₁₉	—	4	2	no	—
59	C₂×D₄×C₁₉	—	4	4	no	—
60	C₂×D₄×C₁₉	—	4	2	no	—
61	C₂×D₄×C₁₉	—	4	2	no	—
62	C₂×D₄×C₁₉	—	8	4	no	—
63	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
66	C₂²×Dic₁₉	—	2	2	no	—
67	D₈×C₁₉	—	4	8	no	—
68	D₈×C₁₉	—	4	8	no	—
69	C₂×C₈×C₁₉	—	2	1	no	—
70	C₂×C₈×C₁₉	—	2	1	no	—
71	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₁₉	D₈	1	2	yes	—
74	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
75	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₁₉	#41	1	1	yes	characteristic
79	C₄⋊Dic₁₉	—	2	2	no	—
80	C₄⋊Dic₁₉	D₈	1	4	yes	complemented
81	C₄×Dic₁₉	—	4	1	no	—
82	C₄×Dic₁₉	—	4	1	no	—
83	C₂×C₁₉⋊C₈	—	2	2	no	—
84	C₂×C₁₉⋊C₈	—	4	1	no	—
85	C₂×C₁₉⋊C₈	—	4	1	no	—
86	C₂×C₁₉⋊C₈	—	4	2	no	—
87	C₂×C₁₉⋊C₈	—	4	4	no	—
88	#9734	—	19	1	no	maximal, Sylow
89	C₂³×C₁₉	—	4	4	no	—
90	C₂³×C₁₉	—	8	2	no	—
91	C₂³×C₁₉	—	8	2	no	—
92	C₂³×C₁₉	—	8	4	no	—
93	C₂³×C₁₉	—	8	4	no	—
94	C₂³×C₁₉	#121	1	1	yes	characteristic
95	D₄×C₁₉	—	4	4	no	—
96	D₄×C₁₉	—	4	4	no	—
97	D₄×C₁₉	—	4	8	no	—
98	D₄×C₁₉	—	8	4	no	—
99	D₄×C₁₉	—	8	4	no	—
100	D₄×C₁₉	—	8	2	no	—
101	D₄×C₁₉	—	8	2	no	—
102	D₄×C₁₉	—	8	2	no	—
103	D₄×C₁₉	—	8	2	no	—
104	C₂×C₄×C₁₉	—	2	2	no	—
105	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
109	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
110	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
111	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
113	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
114	C₂×C₄×C₁₉	#135	1	2	yes	—
115	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₂×Dic₁₉	—	2	4	no	—
118	C₂×Dic₁₉	—	4	2	no	—
119	C₂×Dic₁₉	—	4	2	no	—
120	C₈×C₁₉	—	2	2	no	—
121	C₈×C₁₉	#102	1	4	yes	complemented
122	C₁₉⋊C₈	—	4	2	no	—
123	C₁₉⋊C₈	—	8	2	no	—
124	C₂×#493	D₁₉	1	1	yes	characteristic
125	#1982	—	19	2	no	—
126	C₈×C₄⋊C₄	—	19	1	no	—
127	C₂×#335	—	19	1	no	—
128	C₂×#331	—	19	2	no	—
129	C₂²×C₁₉	—	8	4	no	—
130	C₂²×C₁₉	—	8	4	no	—
131	C₂²×C₁₉	—	8	4	no	—
132	C₂²×C₁₉	—	16	4	no	—
133	C₂²×C₁₉	—	16	4	no	—
134	C₂²×C₁₉	#390	1	1	yes	characteristic
135	C₂²×C₁₉	#459	1	1	yes	characteristic
136	C₂²×C₁₉	#460	1	1	yes	characteristic
137	C₂²×C₁₉	#466	1	2	yes	—
138	C₂²×C₁₉	#472	1	2	yes	—
139	C₄×C₁₉	—	2	1	no	—
140	C₄×C₁₉	—	2	1	no	—
141	C₄×C₁₉	C₂×#102	1	1	yes	characteristic
142	C₄×C₁₉	#418	1	1	yes	characteristic
143	C₄×C₁₉	C₄×D₈	1	2	yes	—
144	C₄×C₁₉	#459	1	1	yes	characteristic
145	C₄×C₁₉	#460	1	1	yes	characteristic
146	C₄×C₁₉	#493	1	1	yes	characteristic
147	C₄×C₁₉	#495	1	1	yes	characteristic
148	Dic₁₉	—	4	4	no	—
149	C₂²×D₈	—	2	1	no	—
150	C₂²×D₈	—	2	1	no	—
151	C₂×#127	C₂×D₁₉	1	2	yes	—
152	#493	—	2	2	no	—
153	#493	Dic₁₉	1	4	yes	complemented
154	C₂×C₄⋊C₈	—	19	1	no	—
155	C₂×C₄⋊C₈	—	19	2	no	—
156	C₂×#102	—	38	4	no	—
157	C₂×C₄×C₈	—	38	1	no	—
158	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
159	C₄×C₄⋊C₄	—	19	1	no	—
160	#336	—	38	1	no	—
161	#335	—	38	2	no	—
162	#331	—	38	4	no	—
163	C₂×C₁₉	—	16	4	no	—
164	C₂×C₁₉	#1667	1	2	yes	—
165	C₂×C₁₉	#1668	1	2	yes	—
166	C₂×C₁₉	#1970	1	1	yes	characteristic
167	C₂×C₁₉	#1978	1	1	yes	characteristic
168	C₂×C₁₉	#1982	1	1	yes	characteristic
169	C₂²×D₄	—	2	2	no	—
170	C₂²×D₄	—	2	2	no	—
171	C₂²×D₄	—	2	2	no	—
172	C₂×D₈	—	2	4	no	—
173	C₂×D₈	—	2	4	no	—
174	C₂×D₈	—	4	2	no	—
175	C₂×D₈	—	4	2	no	—
176	C₂×D₈	—	4	2	no	—
177	C₂×D₈	—	4	2	no	—
178	C₂²×C₈	—	38	1	no	—
179	C₂²×C₈	—	38	2	no	—
180	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
181	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
182	#127	—	2	2	no	—
183	#127	—	2	2	no	—
184	#127	—	2	2	no	—
185	#127	C₂×Dic₁₉	1	4	yes	—
186	C₂×C₄⋊C₄	—	19	2	no	—
187	C₂×C₄²	—	38	1	no	—
188	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
189	C₄⋊C₈	—	19	2	no	—
190	C₄⋊C₈	—	38	2	no	—
191	C₄⋊C₈	—	38	1	no	—
192	C₄⋊C₈	—	38	2	no	—
193	#102	—	76	8	no	—
194	C₄×C₈	—	76	2	no	—
195	C₄×C₈	C₂×Dic₁₉	1	2	yes	—
196	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
197	#95	—	38	1	no	—
198	C₁₉	#9734	1	1	yes	characteristic, complemented, Sylow
199	C₂⁴	—	4	2	no	—
200	C₂×D₄	—	2	4	no	—
201	C₂×D₄	—	2	4	no	—
202	C₂×D₄	—	2	4	no	—
203	C₂×D₄	—	4	2	no	—
204	C₂×D₄	—	4	2	no	—
205	C₂×D₄	—	4	2	no	—
206	C₂×D₄	—	4	2	no	—
207	C₂×D₄	—	4	2	no	—
208	C₂×D₄	—	4	2	no	—
209	C₂×D₄	—	4	2	no	—
210	C₂×D₄	—	4	4	no	—
211	C₂×D₄	—	4	4	no	—
212	C₂×D₄	—	8	4	no	—
213	C₂²×C₄	—	38	2	no	—
214	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
215	C₂²×C₄	#64543	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
217	D₈	—	4	8	no	—
218	D₈	—	4	8	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	38	2	no	—
222	C₂×C₈	—	76	1	no	—
223	C₂×C₈	—	76	1	no	—
224	C₂×C₈	—	76	2	no	—
225	C₂×C₈	—	76	4	no	—
226	C₂×C₈	#64525	1	2	yes	—
227	C₂×C₈	#64526	1	2	yes	—
228	C₂×C₈	C₂²⋊Dic₁₉	1	2	yes	—
229	C₂×C₈	C₂²⋊Dic₁₉	1	2	yes	—
230	C₄⋊C₄	—	19	4	no	—
231	C₄⋊C₄	—	38	2	no	—
232	C₄²	—	76	1	no	—
233	C₄²	—	76	1	no	—
234	C₄²	D₄×D₁₉	1	1	yes	characteristic
235	C₄²	#64543	1	1	yes	characteristic
236	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
237	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
238	C₂³	—	4	4	no	—
239	C₂³	—	8	4	no	—
240	C₂³	—	8	2	no	—
241	C₂³	—	8	2	no	—
242	C₂³	—	8	4	no	—
243	C₂³	#10597384	1	1	yes	characteristic
244	D₄	—	4	4	no	—
245	D₄	—	4	4	no	—
246	D₄	—	4	8	no	—
247	D₄	—	8	4	no	—
248	D₄	—	8	4	no	—
249	D₄	—	8	2	no	—
250	D₄	—	8	2	no	—
251	D₄	—	8	2	no	—
252	D₄	—	8	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	38	4	no	—
255	C₂×C₄	—	76	2	no	—
256	C₂×C₄	—	76	2	no	—
257	C₂×C₄	D₈×D₁₉	1	2	yes	—
258	C₂×C₄	#10597358	1	2	yes	—
259	C₂×C₄	C₂×#64525	1	1	yes	characteristic
260	C₂×C₄	#10597378	1	1	yes	characteristic
261	C₂×C₄	#10597378	1	1	yes	characteristic
262	C₂×C₄	C₂×#64526	1	1	yes	characteristic
263	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
264	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
265	C₂×C₄	#10597382	1	1	yes	characteristic
266	C₂×C₄	#10597384	1	1	yes	characteristic
267	C₂×C₄	#10597386	1	1	yes	characteristic
268	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
269	C₈	—	2	2	no	—
270	C₈	—	76	2	no	—
271	C₈	—	152	2	no	—
272	C₈	#10597289	1	4	yes	complemented
273	C₂²	—	8	4	no	—
274	C₂²	—	8	4	no	—
275	C₂²	—	8	4	no	—
276	C₂²	—	16	4	no	—
277	C₂²	—	16	4	no	—
278	C₂²	#11926447	1	1	yes	characteristic
279	C₂²	#11926736	1	1	yes	characteristic
280	C₂²	#11926741	1	1	yes	characteristic
281	C₂²	#11926777	1	2	yes	—
282	C₂²	#11926811	1	2	yes	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	76	4	no	—
286	C₄	C₂×#10597289	1	1	yes	characteristic
287	C₄	#11926549	1	1	yes	characteristic
288	C₄	D₈×Dic₁₉	1	2	yes	—
289	C₄	#11926736	1	1	yes	characteristic
290	C₄	#11926741	1	1	yes	characteristic
291	C₄	#11926919	1	1	yes	characteristic
292	C₄	#11926927	1	1	yes	characteristic
293	C₂	—	16	4	no	—
294	C₂	#14562987	1	2	yes	—
295	C₂	#14562992	1	2	yes	—
296	C₂	#14564502	1	1	yes	characteristic
297	C₂	#14564546	1	1	yes	characteristic
298	C₂	#14564563	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597384
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting, prime radical	C₂×C₁₉×#493	C₂
socle	C₂³×C₁₉	#121
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#493	D₁₉
hypocenter	C₁₉	#9734
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. #14564563	C₂	semidirect
2. C₈×C₄⋊Dic₁₉	C₂	semidirect
3. C₂×#11926297	C₂	semidirect
4. C₂×#11926288	C₂	semidirect
18. C₁₉×#493	C₄	semidirect
80. C₄⋊Dic₁₉	D₈	semidirect
121. C₈×C₁₉	#102	semidirect
153. #493	Dic₁₉	semidirect
198. C₁₉	#9734	semidirect
272. C₈	#10597289	semidirect

Maximal subgroups

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

subgroup	index
#14564563	2
C₈×C₄⋊Dic₁₉	2
C₂×#11926297	2
C₂×#11926288	2
C₂×C₁₉×#493	2
#9734	19