Subgroups — #18088455

Order 4864.

746 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926580	C₂	1	2	yes	maximal
2	C₂×#11926502	C₂	1	1	yes	maximal, characteristic
3	#14563367	C₂	1	2	yes	maximal
4	C₂×#11926301	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₉×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×#10597372	C₂²	1	2	yes	—
8	C₂×#10597372	C₂²	1	1	yes	characteristic
9	#11926580	—	2	4	no	—
10	#11926580	C₄	1	8	yes	complemented
11	C₂×#10597264	—	2	2	no	—
12	C₂×#10597265	—	2	2	no	—
13	#11926502	—	2	2	no	—
14	#11926502	—	2	2	no	—
15	C₂×#10597272	C₄	1	2	yes	complemented
16	C₂×C₈×Dic₁₉	C₄	1	2	yes	complemented
17	#11926310	—	2	4	no	—
18	#11926301	—	2	2	no	—
19	C₂²×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
20	C₂×C₁₉×#117	C₂²	1	2	yes	—
21	C₁₉×H(3,ℤ/4)	—	2	2	no	—
22	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
23	C₂×M₁₆×C₁₉	—	2	2	no	—
24	C₂×M₁₆×C₁₉	—	2	2	no	—
25	C₂×M₁₆×C₁₉	D₄	1	2	yes	—
26	C₂×M₁₆×C₁₉	Q₈	1	2	yes	—
27	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
28	C₁₉×#117	—	2	2	no	—
29	C₁₉×#117	—	2	2	no	—
30	C₁₉×#117	—	2	2	no	—
31	C₁₉×#117	C₂×C₄	1	4	yes	—
32	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
33	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
34	C₂×C₄²×C₁₉	—	2	2	no	—
35	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
36	C₂×C₂²⋊Dic₁₉	D₄	1	2	yes	—
37	#10597372	—	2	2	no	—
38	#10597372	—	2	2	no	—
39	#10597372	—	2	2	no	—
40	#10597372	—	2	2	no	—
41	#10597372	—	2	2	no	—
42	#10597372	C₂×C₄	1	4	yes	—
43	C₂×C₄⋊Dic₁₉	—	2	2	no	—
44	C₂×C₄⋊Dic₁₉	Q₈	1	2	yes	—
45	C₂×C₄×Dic₁₉	—	2	2	no	—
46	C₂×C₄×Dic₁₉	C₂×C₄	1	2	yes	—
47	C₂²×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
48	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
51	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
52	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
53	C₂²×C₁₉⋊C₈	Q₈	1	2	yes	—
54	#10597272	—	2	4	no	—
55	C₈×Dic₁₉	—	2	2	no	—
56	C₈×Dic₁₉	—	2	2	no	—
57	#10597265	—	2	8	no	—
58	#10597264	—	2	8	no	—
59	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	—	4	2	no	—
61	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	#41	1	2	yes	—
70	C₂²×C₄×C₁₉	#41	1	2	yes	—
71	C₂²×Dic₁₉	—	4	2	no	—
72	C₂²×Dic₁₉	C₄⋊C₄	1	4	yes	—
73	M₁₆×C₁₉	—	2	4	no	—
74	M₁₆×C₁₉	—	2	4	no	—
75	C₂×C₈×C₁₉	—	2	2	no	—
76	C₂×C₈×C₁₉	—	2	2	no	—
77	C₂×C₈×C₁₉	C₄⋊C₄	1	4	yes	—
78	C₄⋊C₄×C₁₉	—	2	4	no	—
79	C₄⋊C₄×C₁₉	—	4	2	no	—
80	C₂²⋊C₄×C₁₉	—	2	4	no	—
81	C₂²⋊C₄×C₁₉	—	4	2	no	—
82	C₄²×C₁₉	—	2	4	no	—
83	C₄²×C₁₉	—	4	2	no	—
84	C₂²⋊Dic₁₉	—	2	4	no	—
85	C₂²⋊Dic₁₉	—	2	4	no	—
86	C₂²⋊Dic₁₉	—	4	2	no	—
87	C₄⋊Dic₁₉	—	2	4	no	—
88	C₄⋊Dic₁₉	—	2	4	no	—
89	C₄⋊Dic₁₉	—	4	2	no	—
90	C₄×Dic₁₉	—	2	2	no	—
91	C₄×Dic₁₉	—	2	2	no	—
92	C₄×Dic₁₉	—	2	4	no	—
93	C₄×Dic₁₉	—	4	2	no	—
94	C₂×C₁₉⋊C₈	C₄²	1	2	yes	—
95	C₂×C₁₉⋊C₈	C₄²	1	2	yes	—
96	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	4	yes	—
97	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	4	yes	—
98	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	2	yes	—
101	C₂×C₁₉⋊C₈	C₄⋊C₄	1	2	yes	—
102	C₂×C₁₉⋊C₈	C₄⋊C₄	1	2	yes	—
103	C₂×C₁₉⋊C₈	C₄⋊C₄	1	4	yes	—
104	C₂×C₁₉⋊C₈	C₄⋊C₄	1	4	yes	—
105	#10866	—	19	1	no	maximal, Sylow
106	C₂³×C₁₉	—	4	2	no	—
107	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂³×C₁₉	C₄×D₄	1	2	yes	—
109	C₂³×C₁₉	#121	1	2	yes	—
110	C₂³×C₁₉	#127	1	1	yes	characteristic
111	C₂³×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
112	C₂×C₄×C₁₉	—	4	4	no	—
113	C₂×C₄×C₁₉	—	8	4	no	—
114	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
115	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
117	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
121	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
124	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
125	C₂×C₄×C₁₉	#121	1	2	yes	—
126	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
127	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
128	C₂×C₄×C₁₉	#124	1	2	yes	—
129	C₂×C₄×C₁₉	#125	1	2	yes	—
130	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
131	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
132	C₂×Dic₁₉	—	2	4	no	—
133	C₂×Dic₁₉	—	4	4	no	—
134	C₂×Dic₁₉	—	4	4	no	—
135	C₂×Dic₁₉	—	8	4	no	—
136	C₈×C₁₉	—	2	4	no	—
137	C₁₉⋊C₈	#95	1	2	yes	—
138	C₁₉⋊C₈	#95	1	4	yes	—
139	C₁₉⋊C₈	#95	1	2	yes	—
140	C₂×#427	—	19	2	no	—
141	C₂×#405	—	19	1	no	—
142	#1740	—	19	2	no	—
143	C₂×H(3,ℤ/4)	—	19	1	no	—
144	C₂×H(3,ℤ/4)	D₁₉	1	1	yes	characteristic
145	C₂²×C₁₉	—	2	2	no	—
146	C₂²×C₁₉	—	2	4	no	—
147	C₂²×C₁₉	—	4	2	no	—
148	C₂²×C₁₉	—	4	1	no	—
149	C₂²×C₁₉	—	4	1	no	—
150	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
151	C₂²×C₁₉	#380	1	1	yes	characteristic
152	C₂²×C₁₉	#384	1	1	yes	characteristic
153	C₂²×C₁₉	#389	1	2	yes	—
154	C₂²×C₁₉	#390	1	2	yes	—
155	C₄×C₁₉	—	8	4	no	—
156	C₄×C₁₉	C₂×#95	1	1	yes	characteristic
157	C₄×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
158	C₄×C₁₉	#380	1	1	yes	characteristic
159	C₄×C₁₉	#382	1	1	yes	characteristic
160	C₄×C₁₉	#389	1	2	yes	—
161	C₄×C₁₉	#391	1	2	yes	—
162	Dic₁₉	—	4	4	no	—
163	Dic₁₉	—	8	4	no	—
164	C₂²×M₁₆	C₂×D₁₉	1	1	yes	characteristic
165	C₂³×C₈	—	19	1	no	—
166	C₂×#117	—	19	1	no	—
167	C₂×#117	—	19	2	no	—
168	C₂×#117	C₂×D₁₉	1	2	yes	—
169	#427	—	19	8	no	—
170	#427	—	38	4	no	—
171	C₂×#107	—	38	2	no	—
172	C₂×#106	—	38	2	no	—
173	#405	—	38	2	no	—
174	#405	—	38	2	no	—
175	C₂×#97	—	19	2	no	—
176	C₂×C₄×C₈	—	19	2	no	—
177	#342	—	38	4	no	—
178	H(3,ℤ/4)	—	2	2	no	—
179	H(3,ℤ/4)	—	38	2	no	—
180	C₂×C₁₉	—	2	2	no	—
181	C₂×C₁₉	—	2	1	no	—
182	C₂×C₁₉	—	2	1	no	—
183	C₂×C₁₉	—	2	2	no	—
184	C₂×C₁₉	C₄⋊#95	1	1	yes	characteristic
185	C₂×C₁₉	#1812	1	2	yes	—
186	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	Dic₃₈	1	2	yes	—
190	C₂×M₁₆	C₂²⋊D₁₉	1	2	yes	—
191	C₂²×C₈	—	19	1	no	—
192	C₂²×C₈	—	19	1	no	—
193	C₂²×C₈	—	19	4	no	—
194	C₂²×C₈	—	19	2	no	—
195	C₂²×C₈	—	19	2	no	—
196	C₂²×C₈	—	19	2	no	—
197	C₂²×C₈	—	19	2	no	—
198	C₂²×C₈	C₄×D₁₉	1	2	yes	—
199	#117	—	2	2	no	—
200	#117	—	2	2	no	—
201	#117	—	2	2	no	—
202	#117	—	19	4	no	—
203	#117	—	38	2	no	—
204	#117	—	38	2	no	—
205	#117	—	38	2	no	—
206	#117	—	38	2	no	—
207	#117	—	38	2	no	—
208	#117	C₄×D₁₉	1	4	yes	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	—	19	2	no	—
211	C₂×C₄⋊C₄	—	38	2	no	—
212	C₂×C₂²⋊C₄	—	2	2	no	—
213	C₂×C₂²⋊C₄	—	19	2	no	—
214	C₂×C₂²⋊C₄	—	38	2	no	—
215	C₂×C₄²	—	2	2	no	—
216	C₂×C₄²	—	19	2	no	—
217	C₂×C₄²	—	38	2	no	—
218	#107	—	38	8	no	—
219	#106	—	38	8	no	—
220	#97	—	38	4	no	—
221	C₄×C₈	—	38	2	no	—
222	C₄×C₈	—	38	2	no	—
223	C₁₉	#10866	1	1	yes	characteristic, complemented, Sylow
224	C₂⁴	D₄×D₁₉	1	1	yes	characteristic
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	—	19	4	no	—
227	C₂²×C₄	—	76	2	no	—
228	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
230	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
231	C₂²×C₄	#64541	1	2	yes	—
232	C₂²×C₄	D₄×D₁₉	1	2	yes	—
233	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
234	C₂²×C₄	#64543	1	2	yes	—
235	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
236	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
238	M₁₆	—	2	4	no	—
239	M₁₆	—	2	4	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	19	2	no	—
243	C₂×C₈	—	19	2	no	—
244	C₂×C₈	—	19	1	no	—
245	C₂×C₈	—	19	1	no	—
246	C₂×C₈	—	19	2	no	—
247	C₂×C₈	—	19	4	no	—
248	C₂×C₈	—	19	4	no	—
249	C₂×C₈	—	19	2	no	—
250	C₂×C₈	—	19	2	no	—
251	C₂×C₈	—	19	4	no	—
252	C₂×C₈	—	19	4	no	—
253	C₂×C₈	Dic₁₉⋊C₄	1	4	yes	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	38	4	no	—
257	C₄⋊C₄	—	38	4	no	—
258	C₄⋊C₄	—	76	2	no	—
259	C₂²⋊C₄	—	2	4	no	—
260	C₂²⋊C₄	—	4	2	no	—
261	C₂²⋊C₄	—	38	4	no	—
262	C₂²⋊C₄	—	38	4	no	—
263	C₂²⋊C₄	—	76	2	no	—
264	C₄²	—	2	4	no	—
265	C₄²	—	4	2	no	—
266	C₄²	—	38	2	no	—
267	C₄²	—	38	2	no	—
268	C₄²	—	38	4	no	—
269	C₄²	—	76	2	no	—
270	C₂³	—	4	2	no	—
271	C₂³	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
272	C₂³	#10597330	1	2	yes	—
273	C₂³	#10597334	1	1	yes	characteristic
274	C₂³	C₄⋊C₄×D₁₉	1	1	yes	characteristic
275	C₂³	#10597386	1	1	yes	characteristic
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	8	4	no	—
278	C₂×C₄	—	38	4	no	—
279	C₂×C₄	—	76	4	no	—
280	C₂×C₄	—	76	4	no	—
281	C₂×C₄	—	152	4	no	—
282	C₂×C₄	C₄×Dic₃₈	1	2	yes	—
283	C₂×C₄	C₄²×D₁₉	1	1	yes	characteristic
284	C₂×C₄	#10597318	1	1	yes	characteristic
285	C₂×C₄	#10597323	1	1	yes	characteristic
286	C₂×C₄	#10597324	1	2	yes	—
287	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
288	C₂×C₄	C₂²⋊C₄×D₁₉	1	2	yes	—
289	C₂×C₄	#10597331	1	2	yes	—
290	C₂×C₄	#10597333	1	2	yes	—
291	C₂×C₄	#10597334	1	1	yes	characteristic
292	C₂×C₄	#10597335	1	2	yes	—
293	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
294	C₂×C₄	C₄⋊C₄×D₁₉	1	2	yes	—
295	C₂×C₄	#10597342	1	2	yes	—
296	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
297	C₂×C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
298	C₂×C₄	#10597385	1	2	yes	—
299	C₂×C₄	#10597390	1	1	yes	characteristic
300	C₈	—	2	4	no	—
301	C₈	—	19	2	no	—
302	C₈	—	19	2	no	—
303	C₈	—	19	4	no	—
304	C₂²	—	2	4	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	2	no	—
308	C₂²	—	4	1	no	—
309	C₂²	#11926395	1	1	yes	characteristic
310	C₂²	#11926403	1	1	yes	characteristic
311	C₂²	#11926413	1	1	yes	characteristic
312	C₂²	#11926441	1	2	yes	—
313	C₂²	#11926444	1	2	yes	—
314	C₄	—	8	4	no	—
315	C₄	—	76	4	no	—
316	C₄	—	152	4	no	—
317	C₄	#95×D₁₉	1	1	yes	characteristic
318	C₄	C₄×Dic₁₉⋊C₄	1	1	yes	characteristic
319	C₄	#11926402	1	1	yes	characteristic
320	C₄	#11926408	1	1	yes	characteristic
321	C₄	#11926439	1	2	yes	—
322	C₄	#11926448	1	2	yes	—
323	C₂	—	2	2	no	—
324	C₂	—	2	1	no	—
325	C₂	—	2	1	no	—
326	C₂	—	2	2	no	—
327	C₂	Dic₁₉⋊#95	1	1	yes	characteristic
328	C₂	#14563680	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#10597318
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₁₉×H(3,ℤ/4)	C₂
socle	C₂²×C₁₉	#380
prime radical	C₂⁴×C₁₉	C₂×D₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×H(3,ℤ/4)	D₁₉
hypocenter	C₁₉	#10866
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×H(3,ℤ/4)

Complements

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

N	H ≅ G/N	type
10. #11926580	C₄	semidirect
15. C₂×#10597272	C₄	semidirect
16. C₂×C₈×Dic₁₉	C₄	semidirect
223. C₁₉	#10866	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926580	2
C₂×#11926502	2
#14563367	2
C₂×#11926301	2
C₂×C₁₉×H(3,ℤ/4)	2
#10866	19