Subgroups — #18075905

Order 4864.

496 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14566033	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#14561474	C₂	1	2	yes	maximal, complemented
4	#14561474	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1797	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
7	C₂×C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
8	#11926412	—	2	1	no	—
9	#11926410	C₄	1	1	yes	characteristic
10	#11926410	C₄	1	1	yes	characteristic
11	#11926407	—	2	1	no	—
12	#11926404	C₄	1	1	yes	characteristic
13	#11926404	C₄	1	1	yes	characteristic
14	C₄×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
16	C₄×C₂²⋊Dic₁₉	—	2	1	no	—
17	#11926390	—	2	1	no	—
18	#11926299	—	2	1	no	—
19	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
20	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₁₉×C₂²⋊C₈	—	2	1	no	—
22	C₁₉×#336	—	2	1	no	—
23	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₉	—	2	2	no	—
27	C₂×C₄²×C₁₉	—	2	4	no	—
28	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
32	C₂×C₂²⋊Dic₁₉	—	2	1	no	—
33	C₂×C₂²⋊Dic₁₉	—	2	1	no	—
34	C₂×C₄⋊Dic₁₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊Dic₁₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄⋊Dic₁₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄⋊Dic₁₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×Dic₁₉	—	2	1	no	—
39	C₂×C₄×Dic₁₉	—	2	1	no	—
40	C₂²×C₁₉⋊C₈	—	2	2	no	—
41	C₁₉×C₄⋊C₈	—	2	4	no	—
42	C₁₉×C₂²⋊C₈	—	4	2	no	—
43	C₁₉⋊#95	—	2	1	no	—
44	C₁₉⋊#95	—	2	1	no	—
45	C₁₉⋊#95	—	2	1	no	—
46	C₁₉⋊#95	—	2	1	no	—
47	C₁₉⋊#95	—	2	1	no	—
48	C₁₉⋊#95	—	2	1	no	—
49	C₁₉⋊#95	—	2	1	no	—
50	C₁₉⋊#95	—	2	1	no	—
51	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
52	C₄×C₁₉⋊C₈	—	2	4	no	—
53	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	—	2	2	no	—
55	C₂²×C₄×C₁₉	—	2	2	no	—
56	C₂²×C₄×C₁₉	—	2	2	no	—
57	C₂²×C₄×C₁₉	—	2	2	no	—
58	C₂²×C₄×C₁₉	—	2	2	no	—
59	C₂²×C₄×C₁₉	—	2	2	no	—
60	C₂²×C₄×C₁₉	—	2	2	no	—
61	C₂²×C₄×C₁₉	—	2	1	no	—
62	C₂²×C₄×C₁₉	—	2	1	no	—
63	C₂²×C₄×C₁₉	—	4	1	no	—
64	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×Dic₁₉	—	2	1	no	—
71	C₂²×Dic₁₉	—	2	1	no	—
72	C₂²×Dic₁₉	—	2	1	no	—
73	C₂²×Dic₁₉	—	2	1	no	—
74	C₂×C₈×C₁₉	—	4	2	no	—
75	C₂×C₈×C₁₉	—	4	2	no	—
76	C₂×C₈×C₁₉	—	4	2	no	—
77	C₄²×C₁₉	—	2	2	no	—
78	C₄²×C₁₉	—	2	2	no	—
79	C₄²×C₁₉	—	2	2	no	—
80	C₄²×C₁₉	#41	1	4	yes	—
81	C₂²⋊Dic₁₉	—	4	2	no	—
82	C₂²⋊Dic₁₉	—	4	2	no	—
83	C₄⋊Dic₁₉	—	2	2	no	—
84	C₄⋊Dic₁₉	—	2	2	no	—
85	C₄⋊Dic₁₉	—	2	2	no	—
86	C₄⋊Dic₁₉	—	2	2	no	—
87	C₄×Dic₁₉	—	4	2	no	—
88	C₄×Dic₁₉	—	4	1	no	—
89	C₄×Dic₁₉	—	4	1	no	—
90	C₂×C₁₉⋊C₈	—	2	4	no	—
91	C₂×C₁₉⋊C₈	—	4	2	no	—
92	C₂×C₁₉⋊C₈	—	4	2	no	—
93	#8711	—	19	1	no	maximal, Sylow
94	C₂³×C₁₉	—	2	1	no	—
95	C₂³×C₁₉	—	2	1	no	—
96	C₂³×C₁₉	—	2	1	no	—
97	C₂³×C₁₉	—	2	1	no	—
98	C₂³×C₁₉	—	2	1	no	—
99	C₂³×C₁₉	—	2	1	no	—
100	C₂³×C₁₉	#104	1	2	yes	—
101	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₉	—	2	1	no	—
103	C₂×C₄×C₁₉	—	2	1	no	—
104	C₂×C₄×C₁₉	—	2	1	no	—
105	C₂×C₄×C₁₉	—	2	1	no	—
106	C₂×C₄×C₁₉	—	2	1	no	—
107	C₂×C₄×C₁₉	—	2	1	no	—
108	C₂×C₄×C₁₉	—	2	1	no	—
109	C₂×C₄×C₁₉	—	2	1	no	—
110	C₂×C₄×C₁₉	—	2	1	no	—
111	C₂×C₄×C₁₉	—	2	1	no	—
112	C₂×C₄×C₁₉	—	2	1	no	—
113	C₂×C₄×C₁₉	—	4	4	no	—
114	C₂×C₄×C₁₉	—	4	4	no	—
115	C₂×C₄×C₁₉	—	4	2	no	—
116	C₂×C₄×C₁₉	—	4	2	no	—
117	C₂×C₄×C₁₉	#117	1	2	yes	—
118	C₂×C₄×C₁₉	#123	1	2	yes	—
119	C₂×C₄×C₁₉	#123	1	2	yes	—
120	C₂×C₄×C₁₉	#131	1	2	yes	—
121	C₂×C₄×C₁₉	#131	1	2	yes	—
122	C₂×C₄×C₁₉	#135	1	2	yes	—
123	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
125	C₂×Dic₁₉	—	4	1	no	—
126	C₂×Dic₁₉	—	4	1	no	—
127	C₂×Dic₁₉	—	4	1	no	—
128	C₂×Dic₁₉	—	4	1	no	—
129	C₂×Dic₁₉	—	4	1	no	—
130	C₂×Dic₁₉	—	4	1	no	—
131	C₂×Dic₁₉	—	4	1	no	—
132	C₂×Dic₁₉	—	4	1	no	—
133	C₂×Dic₁₉	—	4	1	no	—
134	C₂×Dic₁₉	—	4	1	no	—
135	C₂×Dic₁₉	—	4	1	no	—
136	C₂×Dic₁₉	—	4	1	no	—
137	C₈×C₁₉	—	8	2	no	—
138	C₁₉⋊C₈	—	4	4	no	—
139	#2283	—	19	1	no	—
140	#1797	D₁₉	1	1	yes	characteristic
141	C₄×C₂²⋊C₈	—	19	1	no	—
142	#1279	—	19	2	no	—
143	#1279	—	19	2	no	—
144	C₂²×C₁₉	—	2	2	no	—
145	C₂²×C₁₉	—	2	2	no	—
146	C₂²×C₁₉	—	2	2	no	—
147	C₂²×C₁₉	—	4	1	no	—
148	C₂²×C₁₉	—	4	1	no	—
149	C₂²×C₁₉	—	4	1	no	—
150	C₂²×C₁₉	—	4	1	no	—
151	C₂²×C₁₉	#381	1	1	yes	characteristic
152	C₂²×C₁₉	#408	1	1	yes	characteristic
153	C₂²×C₁₉	#408	1	1	yes	characteristic
154	C₂²×C₁₉	#416	1	1	yes	characteristic
155	C₂²×C₁₉	#417	1	1	yes	characteristic
156	C₂²×C₁₉	C₂×#104	1	1	yes	characteristic
157	C₂²×C₁₉	#421	1	1	yes	characteristic
158	C₄×C₁₉	—	2	2	no	—
159	C₄×C₁₉	—	4	1	no	—
160	C₄×C₁₉	—	4	1	no	—
161	C₄×C₁₉	—	4	1	no	—
162	C₄×C₁₉	#433	1	4	yes	—
163	C₄×C₁₉	#482	1	2	yes	—
164	C₄×C₁₉	#485	1	2	yes	—
165	Dic₁₉	—	8	1	no	—
166	Dic₁₉	—	8	1	no	—
167	Dic₁₉	—	8	1	no	—
168	Dic₁₉	—	8	1	no	—
169	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
170	C₂×C₄⋊C₈	C₂×D₁₉	1	2	yes	—
171	C₂×C₂²⋊C₈	—	2	1	no	—
172	C₂×C₂²⋊C₈	—	38	1	no	—
173	C₂×C₄×C₈	—	19	2	no	—
174	#383	—	38	1	no	—
175	#382	—	19	1	no	—
176	#382	—	19	1	no	—
177	#381	—	38	1	no	—
178	#380	—	19	1	no	—
179	#380	—	19	1	no	—
180	C₄×C₄⋊C₄	—	19	1	no	—
181	C₄×C₄⋊C₄	—	19	1	no	—
182	C₄×C₂²⋊C₄	—	38	1	no	—
183	#376	—	38	1	no	—
184	#336	—	2	1	no	—
185	#336	—	38	1	no	—
186	C₂×C₁₉	—	4	2	no	—
187	C₂×C₁₉	#1482	1	1	yes	characteristic
188	C₂×C₁₉	#1483	1	1	yes	characteristic
189	C₂×C₁₉	#1494	1	1	yes	characteristic
190	C₂×C₁₉	#1495	1	1	yes	characteristic
191	C₂×C₁₉	#1818	1	1	yes	characteristic
192	C₂×C₁₉	#1825	1	1	yes	characteristic
193	C₂×C₁₉	#1826	1	1	yes	characteristic
194	C₂³×C₄	C₄×D₁₉	1	1	yes	characteristic
195	C₂³×C₄	D₇₆	1	1	yes	characteristic
196	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
197	C₂²×C₈	—	2	2	no	—
198	C₂²×C₈	—	38	2	no	—
199	C₂×C₄⋊C₄	—	19	1	no	—
200	C₂×C₄⋊C₄	—	19	1	no	—
201	C₂×C₄⋊C₄	—	19	1	no	—
202	C₂×C₄⋊C₄	—	19	1	no	—
203	C₂×C₂²⋊C₄	—	38	1	no	—
204	C₂×C₂²⋊C₄	—	38	1	no	—
205	C₂×C₄²	—	2	4	no	—
206	C₂×C₄²	—	38	1	no	—
207	C₂×C₄²	—	38	1	no	—
208	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
209	C₂×C₄²	D₇₆	1	1	yes	characteristic
210	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
211	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
212	C₄⋊C₈	—	2	4	no	—
213	C₂²⋊C₈	—	4	2	no	—
214	C₂²⋊C₈	—	38	4	no	—
215	C₄×C₈	—	38	4	no	—
216	#95	—	38	1	no	—
217	#95	—	38	1	no	—
218	#95	—	38	1	no	—
219	#95	—	38	1	no	—
220	#95	—	38	1	no	—
221	#95	—	38	1	no	—
222	#95	—	38	1	no	—
223	#95	—	38	1	no	—
224	C₁₉	#8711	1	1	yes	characteristic, complemented, Sylow
225	C₂⁴	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	4	1	no	—
236	C₂²×C₄	—	38	1	no	—
237	C₂²×C₄	—	38	1	no	—
238	C₂²×C₄	—	38	1	no	—
239	C₂²×C₄	—	38	1	no	—
240	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
241	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
242	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	—	4	2	no	—
248	C₂×C₈	—	4	2	no	—
249	C₂×C₈	—	38	4	no	—
250	C₂×C₈	—	76	2	no	—
251	C₂×C₈	—	76	2	no	—
252	C₄⋊C₄	—	38	2	no	—
253	C₄⋊C₄	—	38	2	no	—
254	C₄⋊C₄	—	38	2	no	—
255	C₄⋊C₄	—	38	2	no	—
256	C₂²⋊C₄	—	76	2	no	—
257	C₂²⋊C₄	—	76	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	76	1	no	—
262	C₄²	—	76	1	no	—
263	C₄²	—	76	2	no	—
264	C₄²	#64543	1	4	yes	—
265	C₂³	—	2	1	no	—
266	C₂³	—	2	1	no	—
267	C₂³	—	2	1	no	—
268	C₂³	—	2	1	no	—
269	C₂³	—	2	1	no	—
270	C₂³	—	2	1	no	—
271	C₂³	#10597282	1	2	yes	—
272	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	76	1	no	—
289	C₂×C₄	—	76	1	no	—
290	C₂×C₄	—	76	1	no	—
291	C₂×C₄	—	76	1	no	—
292	C₂×C₄	—	76	1	no	—
293	C₂×C₄	—	76	1	no	—
294	C₂×C₄	—	76	1	no	—
295	C₂×C₄	—	76	1	no	—
296	C₂×C₄	—	76	1	no	—
297	C₂×C₄	—	76	1	no	—
298	C₂×C₄	—	76	1	no	—
299	C₂×C₄	—	76	1	no	—
300	C₂×C₄	#10597323	1	2	yes	—
301	C₂×C₄	#10597332	1	2	yes	—
302	C₂×C₄	#10597347	1	2	yes	—
303	C₂×C₄	#10597353	1	2	yes	—
304	C₂×C₄	#10597378	1	1	yes	characteristic
305	C₂×C₄	#10597381	1	2	yes	—
306	C₂×C₄	#10597388	1	1	yes	characteristic
307	C₂×C₄	#10597396	1	2	yes	—
308	C₈	—	8	2	no	—
309	C₈	—	76	4	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	#11926405	1	1	yes	characteristic
318	C₂²	#11926510	1	1	yes	characteristic
319	C₂²	#11926512	1	1	yes	characteristic
320	C₂²	#11926538	1	1	yes	characteristic
321	C₂²	#11926543	1	1	yes	characteristic
322	C₂²	C₂×#10597282	1	1	yes	characteristic
323	C₂²	#11926560	1	1	yes	characteristic
324	C₄	—	2	2	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	152	1	no	—
329	C₄	—	152	1	no	—
330	C₄	—	152	1	no	—
331	C₄	—	152	1	no	—
332	C₄	#11926602	1	4	yes	—
333	C₄	#11926865	1	2	yes	—
334	C₄	#11926882	1	2	yes	—
335	C₂	—	4	2	no	—
336	C₂	#14562041	1	1	yes	characteristic
337	C₂	#14562046	1	1	yes	characteristic
338	C₂	#14562093	1	1	yes	characteristic
339	C₂	#14562098	1	1	yes	characteristic
340	C₂	#14563704	1	1	yes	characteristic
341	C₂	#14563727	1	1	yes	characteristic
342	C₂	#14563732	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1797	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1797	D₁₉
hypocenter	C₁₉	#8711
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #1797

Complements

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

N	H ≅ G/N	type
3. #14561474	C₂	semidirect
4. #14561474	C₂	semidirect
224. C₁₉	#8711	semidirect

Maximal subgroups

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

subgroup	index
#14566033	2
C₄×C₂²⋊(C₁₉⋊C₈)	2
#14561474	2
C₁₉×#1797	2
#8711	19