Subgroups — #18077026

Order 4864.

426 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14572381	C₂	1	1	yes	maximal, characteristic
2	#14562407	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562285	C₂	1	1	yes	maximal, characteristic
4	#14562085	C₂	1	1	yes	maximal, characteristic
5	#14561473	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561469	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1950	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10597353	—	2	1	no	—
9	#11927094	C₂²	1	1	yes	characteristic
10	#11926623	—	2	1	no	—
11	#11926617	—	2	2	no	—
12	C₈×D₇₆	—	2	1	no	—
13	#11926597	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
15	#11926512	—	2	1	no	—
16	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
17	#11926294	—	2	1	no	—
18	#11926294	—	2	1	no	—
19	#11926291	C₄	1	1	yes	characteristic
20	#11926291	C₄	1	1	yes	characteristic
21	#11926287	C₄	1	1	yes	characteristic
22	#11926287	C₄	1	1	yes	characteristic
23	#11926271	—	2	2	no	—
24	C₂×C₁₉×#108	—	2	1	no	—
25	C₁₉×#423	C₂²	1	1	yes	characteristic
26	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
28	C₁₉×#341	—	2	1	no	—
29	C₂×#64541	—	2	1	no	—
30	C₂×M₁₆×C₁₉	—	2	1	no	—
31	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
32	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₉	—	2	1	no	—
34	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
35	#10597376	—	2	1	no	—
36	#10597376	Q₈	1	2	yes	—
37	C₂²×C₁₉⋊C₈	—	2	1	no	—
38	C₂²×C₁₉⋊C₈	—	2	1	no	—
39	#10597353	—	4	2	no	—
40	#10597353	—	4	2	no	—
41	C₂×#64515	—	2	2	no	—
42	C₂×C₈×D₁₉	—	2	2	no	—
43	#10597321	—	2	1	no	—
44	#10597320	C₂×C₄	1	1	yes	characteristic
45	C₄×D₇₆	—	2	1	no	—
46	C₄×D₇₆	D₄	1	2	yes	—
47	#10597315	C₂×C₄	1	1	yes	characteristic
48	C₁₉×#108	—	4	2	no	—
49	C₁₉×C₄⋊C₈	—	2	1	no	—
50	C₁₉×C₄⋊C₈	—	2	1	no	—
51	C₁₉×C₄⋊C₈	—	2	1	no	—
52	C₁₉×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₁₉×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
54	C₁₉×#97	—	2	1	no	—
55	C₄×C₈×C₁₉	—	2	1	no	—
56	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
57	(C₂×D₁₉)⋊C₈	—	2	2	no	—
58	(C₂×D₁₉)⋊C₈	—	2	2	no	—
59	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
60	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	1	yes	characteristic
61	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	1	yes	characteristic
62	C₄⋊(C₁₉⋊C₈)	D₄	1	1	yes	characteristic
63	C₄⋊(C₁₉⋊C₈)	D₄	1	1	yes	characteristic
64	C₄×C₁₉⋊C₈	—	2	1	no	—
65	C₄×C₁₉⋊C₈	—	2	1	no	—
66	C₄×C₁₉⋊C₈	—	2	1	no	—
67	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
70	C₂×C₂²⋊D₁₉	—	2	2	no	—
71	#64541	—	4	1	no	—
72	#64541	—	4	1	no	—
73	#64541	—	4	1	no	—
74	#64541	—	4	1	no	—
75	C₂×D₇₆	—	2	1	no	—
76	C₂×D₇₆	—	2	1	no	—
77	C₂×D₇₆	C₄⋊C₄	1	2	yes	—
78	C₂×C₄×D₁₉	—	2	2	no	—
79	C₂×Dic₃₈	—	2	1	no	—
80	M₁₆×C₁₉	—	2	1	no	—
81	M₁₆×C₁₉	—	2	1	no	—
82	M₁₆×C₁₉	—	2	1	no	—
83	M₁₆×C₁₉	—	4	1	no	—
84	M₁₆×C₁₉	—	4	1	no	—
85	M₁₆×C₁₉	SD₁₆	1	2	yes	—
86	C₂×C₈×C₁₉	—	2	1	no	—
87	C₂×C₈×C₁₉	—	2	1	no	—
88	C₂×C₈×C₁₉	—	2	1	no	—
89	C₂×C₈×C₁₉	—	2	1	no	—
90	C₂×C₈×C₁₉	—	4	1	no	—
91	C₂×C₈×C₁₉	—	4	1	no	—
92	C₂×C₈×C₁₉	—	4	1	no	—
93	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₁₉	#41	1	1	yes	characteristic
97	(C₂×D₁₉)⋊C₄	—	2	2	no	—
98	(C₂×D₁₉)⋊C₄	—	2	2	no	—
99	C₄⋊Dic₁₉	—	2	1	no	—
100	C₄⋊Dic₁₉	C₄⋊C₄	1	2	yes	—
101	C₂×C₁₉⋊C₈	—	2	2	no	—
102	C₂×C₁₉⋊C₈	—	2	1	no	—
103	C₂×C₁₉⋊C₈	—	2	1	no	—
104	C₂×C₁₉⋊C₈	—	2	1	no	—
105	C₂×C₁₉⋊C₈	—	2	1	no	—
106	C₂×C₁₉⋊C₈	—	4	1	no	—
107	C₂×C₁₉⋊C₈	—	4	1	no	—
108	C₂×C₁₉⋊C₈	—	4	1	no	—
109	C₂×C₁₉⋊C₈	—	4	1	no	—
110	#64515	—	4	4	no	—
111	C₈×D₁₉	—	4	4	no	—
112	#8932	—	19	1	no	maximal, Sylow
113	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂²×D₁₉	—	2	2	no	—
115	C₂×C₄×C₁₉	—	2	1	no	—
116	C₂×C₄×C₁₉	—	2	1	no	—
117	C₂×C₄×C₁₉	—	4	1	no	—
118	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
126	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₂²⋊D₁₉	—	4	2	no	—
129	C₂²⋊D₁₉	—	4	2	no	—
130	C₂×Dic₁₉	—	2	2	no	—
131	D₇₆	—	4	2	no	—
132	D₇₆	—	4	1	no	—
133	D₇₆	—	4	1	no	—
134	D₇₆	—	4	1	no	—
135	D₇₆	—	4	1	no	—
136	C₄×D₁₉	—	4	2	no	—
137	C₄×D₁₉	—	4	2	no	—
138	Dic₃₈	—	4	1	no	—
139	Dic₃₈	—	4	1	no	—
140	C₈×C₁₉	—	2	2	no	—
141	C₈×C₁₉	—	4	1	no	—
142	C₈×C₁₉	—	4	1	no	—
143	C₈×C₁₉	—	4	1	no	—
144	C₈×C₁₉	—	8	1	no	—
145	C₁₉⋊C₈	—	4	2	no	—
146	C₁₉⋊C₈	—	4	1	no	—
147	C₁₉⋊C₈	—	4	1	no	—
148	C₁₉⋊C₈	—	4	1	no	—
149	C₁₉⋊C₈	—	4	1	no	—
150	#2935	—	19	1	no	—
151	#1950	D₁₉	1	1	yes	characteristic, complemented
152	#1570	—	19	1	no	—
153	#1540	—	19	1	no	—
154	#1492	—	19	1	no	—
155	#1278	—	19	1	no	—
156	#1277	—	19	1	no	—
157	C₂²×C₁₉	—	2	1	no	—
158	C₂²×C₁₉	—	2	1	no	—
159	C₂²×C₁₉	#385	1	1	yes	characteristic
160	C₂²×C₁₉	#476	1	1	yes	characteristic
161	C₂²×C₁₉	#478	1	1	yes	characteristic
162	C₂×D₁₉	—	4	2	no	—
163	C₂×D₁₉	—	4	2	no	—
164	C₂×D₁₉	—	4	2	no	—
165	C₄×C₁₉	—	2	1	no	—
166	C₄×C₁₉	—	2	1	no	—
167	C₄×C₁₉	—	2	1	no	—
168	C₄×C₁₉	C₄×SD₁₆	1	1	yes	characteristic
169	C₄×C₁₉	#442	1	1	yes	characteristic
170	C₄×C₁₉	#443	1	1	yes	characteristic
171	C₄×C₁₉	#444	1	1	yes	characteristic
172	C₄×C₁₉	#449	1	1	yes	characteristic
173	C₄×C₁₉	#450	1	1	yes	characteristic
174	Dic₁₉	—	4	2	no	—
175	C₂×#131	—	38	1	no	—
176	#532	—	19	1	no	—
177	#436	—	38	1	no	—
178	#435	—	38	2	no	—
179	C₈×D₄	—	38	1	no	—
180	#431	—	19	1	no	—
181	C₂×#108	—	2	1	no	—
182	#423	C₂×D₁₉	1	1	yes	characteristic
183	C₂×C₄⋊C₈	—	19	1	no	—
184	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
185	#408	—	38	1	no	—
186	C₄×M₁₆	C₂×D₁₉	1	1	yes	characteristic
187	C₂×C₄×C₈	—	19	1	no	—
188	#341	—	2	1	no	—
189	#334	—	38	1	no	—
190	#334	—	38	1	no	—
191	#332	—	19	1	no	—
192	#332	—	19	1	no	—
193	#331	—	19	1	no	—
194	#331	—	19	1	no	—
195	D₄⋊C₈	—	38	2	no	—
196	C₂×C₁₉	—	2	1	no	—
197	C₂×C₁₉	—	2	1	no	—
198	C₂×C₁₉	#1858	1	1	yes	characteristic
199	C₂×C₁₉	#1864	1	1	yes	characteristic
200	C₂×C₁₉	#1866	1	1	yes	characteristic
201	D₁₉	—	8	2	no	—
202	C₂×#41	—	38	1	no	—
203	#131	—	76	2	no	—
204	#131	—	76	2	no	—
205	C₂×M₁₆	—	2	1	no	—
206	C₂×M₁₆	—	38	2	no	—
207	C₂×M₁₆	C₂²⋊D₁₉	1	1	yes	characteristic
208	C₂×M₁₆	C₂²⋊D₁₉	1	1	yes	characteristic
209	C₂²×C₈	—	2	1	no	—
210	C₂²×C₈	—	38	2	no	—
211	C₂²×C₈	—	38	1	no	—
212	C₂²×C₈	—	38	1	no	—
213	C₄⋊Q₈	—	19	1	no	—
214	#127	—	19	1	no	—
215	#124	—	38	1	no	—
216	#121	—	19	2	no	—
217	#121	—	38	1	no	—
218	C₄×D₄	—	19	2	no	—
219	C₄×D₄	—	38	1	no	—
220	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
221	#108	—	4	2	no	—
222	C₄⋊C₈	—	2	1	no	—
223	C₄⋊C₈	—	2	1	no	—
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	19	1	no	—
226	C₄⋊C₈	—	19	1	no	—
227	C₄⋊C₈	—	19	1	no	—
228	C₄⋊C₈	—	19	1	no	—
229	C₄⋊C₈	—	38	1	no	—
230	C₄⋊C₈	C₄×D₁₉	1	1	yes	characteristic
231	C₄⋊C₈	C₄×D₁₉	1	1	yes	characteristic
232	C₂²⋊C₈	—	38	2	no	—
233	C₂²⋊C₈	—	38	2	no	—
234	C₂²⋊C₈	—	38	1	no	—
235	#97	—	2	1	no	—
236	C₄×C₈	—	2	1	no	—
237	C₄×C₈	—	38	1	no	—
238	C₄×C₈	—	38	1	no	—
239	C₄×C₈	—	38	1	no	—
240	C₁₉	#8932	1	1	yes	characteristic, complemented, Sylow
241	#41	—	76	1	no	—
242	#41	—	76	1	no	—
243	#41	—	76	1	no	—
244	#41	—	76	1	no	—
245	C₂×Q₈	—	38	1	no	—
246	C₂×D₄	—	19	2	no	—
247	C₂×D₄	—	38	2	no	—
248	C₂×D₄	—	38	1	no	—
249	C₂×D₄	—	38	1	no	—
250	C₂²×C₄	—	38	2	no	—
251	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
252	C₂²×C₄	#64545	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
254	M₁₆	—	2	1	no	—
255	M₁₆	—	2	1	no	—
256	M₁₆	—	2	1	no	—
257	M₁₆	—	4	1	no	—
258	M₁₆	—	4	1	no	—
259	M₁₆	—	76	4	no	—
260	M₁₆	#64527	1	2	yes	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	4	1	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	4	1	no	—
268	C₂×C₈	—	38	2	no	—
269	C₂×C₈	—	38	1	no	—
270	C₂×C₈	—	38	1	no	—
271	C₂×C₈	—	38	1	no	—
272	C₂×C₈	—	38	1	no	—
273	C₂×C₈	—	76	4	no	—
274	C₂×C₈	—	76	1	no	—
275	C₂×C₈	—	76	1	no	—
276	C₂×C₈	—	76	1	no	—
277	C₂×C₈	—	76	1	no	—
278	C₄⋊C₄	—	19	2	no	—
279	C₄⋊C₄	—	38	1	no	—
280	C₂²⋊C₄	—	38	2	no	—
281	C₂²⋊C₄	—	38	2	no	—
282	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
283	C₄²	#64541	1	1	yes	characteristic
284	C₄²	D₄×D₁₉	1	1	yes	characteristic
285	C₄²	D₄×D₁₉	1	1	yes	characteristic
286	C₂³	—	38	2	no	—
287	C₂³	#10597392	1	1	yes	characteristic
288	Q₈	—	76	1	no	—
289	Q₈	—	76	1	no	—
290	D₄	—	76	2	no	—
291	D₄	—	76	2	no	—
292	D₄	—	76	2	no	—
293	D₄	—	76	1	no	—
294	D₄	—	76	1	no	—
295	D₄	—	76	1	no	—
296	D₄	—	76	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	38	2	no	—
301	C₂×C₄	—	76	2	no	—
302	C₂×C₄	—	76	2	no	—
303	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
304	C₂×C₄	#10597339	1	1	yes	characteristic
305	C₂×C₄	#10597340	1	1	yes	characteristic
306	C₂×C₄	#10597342	1	1	yes	characteristic
307	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
308	C₂×C₄	#10597378	1	1	yes	characteristic
309	C₂×C₄	#10597383	1	1	yes	characteristic
310	C₂×C₄	C₂×#64527	1	1	yes	characteristic
311	C₂×C₄	#10597395	1	1	yes	characteristic
312	C₂×C₄	#10597396	1	1	yes	characteristic
313	C₈	—	2	2	no	—
314	C₈	—	4	1	no	—
315	C₈	—	4	1	no	—
316	C₈	—	4	1	no	—
317	C₈	—	8	1	no	—
318	C₈	—	76	2	no	—
319	C₈	—	76	1	no	—
320	C₈	—	76	1	no	—
321	C₈	—	76	1	no	—
322	C₈	—	76	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	76	2	no	—
326	C₂²	—	76	2	no	—
327	C₂²	—	76	2	no	—
328	C₂²	#11926415	1	1	yes	characteristic
329	C₂²	#11926831	1	1	yes	characteristic
330	C₂²	#11926841	1	1	yes	characteristic
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	76	2	no	—
335	C₄	C₄×#64527	1	1	yes	characteristic
336	C₄	#11926654	1	1	yes	characteristic
337	C₄	#11926660	1	1	yes	characteristic
338	C₄	#11926665	1	1	yes	characteristic
339	C₄	#11926687	1	1	yes	characteristic
340	C₄	#11926694	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	152	2	no	—
344	C₂	#14563853	1	1	yes	characteristic
345	C₂	#14563895	1	1	yes	characteristic
346	C₂	#14563907	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597383
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1950	C₂
socle	C₂²×C₁₉	#385
prime radical	#11927094	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1950	D₁₉
hypocenter	C₁₉	#8932
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1950

Complements

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

N	H ≅ G/N	type
2. #14562407	C₂	semidirect
5. #14561473	C₂	semidirect
6. #14561469	C₂	semidirect
7. C₁₉×#1950	C₂	semidirect
151. #1950	D₁₉	semidirect
240. C₁₉	#8932	semidirect

Maximal subgroups

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

subgroup	index
#14572381	2
#14562407	2
#14562285	2
#14562085	2
#14561473	2
#14561469	2
C₁₉×#1950	2
#8932	19