Subgroups — #18075851

Order 4864.

384 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14572611	C₂	1	1	yes	maximal, characteristic
2	C₄×#10597265	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562396	C₂	1	1	yes	maximal, characteristic, complemented
4	#14562160	C₂	1	1	yes	maximal, characteristic
5	#14561491	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561490	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1509	C₂	1	1	yes	maximal, characteristic
8	#11926675	C₄	1	1	yes	characteristic, complemented
9	Q₈×C₁₉⋊C₈	C₄	1	1	yes	characteristic, complemented
10	#11926622	C₄	1	1	yes	characteristic, complemented
11	#11926616	—	2	1	no	—
12	D₄×C₁₉⋊C₈	—	2	1	no	—
13	D₄×C₁₉⋊C₈	C₄	1	1	yes	characteristic, complemented
14	#11926605	—	2	1	no	—
15	#11926601	C₂²	1	1	yes	characteristic
16	#11926601	C₂²	1	1	yes	characteristic
17	C₂×#10597265	—	2	1	no	—
18	#11926500	—	2	1	no	—
19	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
20	C₄×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
21	#11926306	—	2	1	no	—
22	#11926296	—	2	1	no	—
23	#11926294	—	2	1	no	—
24	#11926281	—	2	1	no	—
25	#11926278	—	2	1	no	—
26	C₁₉×#529	C₂²	1	1	yes	characteristic
27	C₁₉×#432	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
29	C₁₉×#328	—	2	1	no	—
30	C₁₉×#327	—	2	1	no	—
31	C₁₉×#126	—	2	1	no	—
32	C₁₉×#125	Q₈	1	1	yes	characteristic
33	C₁₉×#124	Q₈	1	1	yes	characteristic
34	C₁₉×#123	—	2	1	no	—
35	C₁₉×C₂²⋊Q₈	D₄	1	1	yes	characteristic
36	C₁₉×#121	D₄	1	1	yes	characteristic
37	C₄×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
38	C₄×D₄×C₁₉	—	2	1	no	—
39	C₄×D₄×C₁₉	C₂×C₄	1	1	yes	characteristic
40	C₁₉×#117	—	2	1	no	—
41	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
42	C₂×C₄²×C₁₉	—	2	1	no	—
43	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
44	C₂×C₄⋊Dic₁₉	—	2	1	no	—
45	C₂×C₄×Dic₁₉	—	2	1	no	—
46	C₂×#64520	—	2	1	no	—
47	C₂²×C₁₉⋊C₈	—	2	1	no	—
48	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
49	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
50	C₁₉×C₄⋊C₈	—	2	1	no	—
51	C₁₉×C₂²⋊C₈	—	2	1	no	—
52	C₁₉×#97	—	2	1	no	—
53	C₁₉×#95	—	2	1	no	—
54	C₁₉⋊#95	—	2	1	no	—
55	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
56	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
57	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
58	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
59	#10597265	—	2	4	no	—
60	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
61	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
62	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
63	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	1	yes	characteristic
64	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	1	yes	characteristic
65	#10597260	—	2	1	no	—
66	#10597260	—	2	1	no	—
67	#10597260	—	2	1	no	—
68	C₄×C₁₉⋊C₈	—	2	1	no	—
69	C₄×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
70	C₄×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
71	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
72	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
73	C₂×Q₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
74	C₂×D₄×C₁₉	—	2	1	no	—
75	C₂×D₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₉	—	2	1	no	—
77	C₂²×C₄×C₁₉	—	4	1	no	—
78	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
81	C₂²×Dic₁₉	—	4	1	no	—
82	C₂×C₈×C₁₉	—	4	1	no	—
83	C₄⋊C₄×C₁₉	—	2	1	no	—
84	C₄⋊C₄×C₁₉	—	2	1	no	—
85	C₄⋊C₄×C₁₉	—	2	1	no	—
86	C₄⋊C₄×C₁₉	—	2	1	no	—
87	C₄⋊C₄×C₁₉	—	2	2	no	—
88	C₄⋊C₄×C₁₉	—	4	1	no	—
89	C₄⋊C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
90	C₄⋊C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
91	C₂²⋊C₄×C₁₉	—	2	1	no	—
92	C₂²⋊C₄×C₁₉	—	2	1	no	—
93	C₂²⋊C₄×C₁₉	—	2	1	no	—
94	C₂²⋊C₄×C₁₉	—	2	1	no	—
95	C₂²⋊C₄×C₁₉	—	2	1	no	—
96	C₄²×C₁₉	—	2	1	no	—
97	C₄²×C₁₉	—	8	1	no	—
98	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
101	C₄²×C₁₉	#41	1	1	yes	characteristic
102	C₄⋊Dic₁₉	—	2	2	no	—
103	C₄⋊Dic₁₉	—	4	1	no	—
104	C₄×Dic₁₉	—	8	1	no	—
105	#64520	—	4	2	no	—
106	C₂×C₁₉⋊C₈	—	2	2	no	—
107	C₂×C₁₉⋊C₈	—	2	1	no	—
108	C₂×C₁₉⋊C₈	—	2	1	no	—
109	C₂×C₁₉⋊C₈	—	4	1	no	—
110	C₂×C₁₉⋊C₈	—	4	2	no	—
111	C₂×C₁₉⋊C₈	—	4	1	no	—
112	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
116	#8696	—	19	1	no	maximal, Sylow
117	C₂³×C₁₉	—	2	1	no	—
118	C₂³×C₁₉	#121	1	1	yes	characteristic
119	Q₈×C₁₉	—	4	1	no	—
120	D₄×C₁₉	—	4	1	no	—
121	D₄×C₁₉	—	4	1	no	—
122	D₄×C₁₉	—	4	1	no	—
123	C₂×C₄×C₁₉	—	2	1	no	—
124	C₂×C₄×C₁₉	—	2	1	no	—
125	C₂×C₄×C₁₉	—	2	1	no	—
126	C₂×C₄×C₁₉	—	2	1	no	—
127	C₂×C₄×C₁₉	—	2	1	no	—
128	C₂×C₄×C₁₉	—	2	1	no	—
129	C₂×C₄×C₁₉	—	2	1	no	—
130	C₂×C₄×C₁₉	—	4	1	no	—
131	C₂×C₄×C₁₉	—	4	1	no	—
132	C₂×C₄×C₁₉	—	4	1	no	—
133	C₂×C₄×C₁₉	—	4	2	no	—
134	C₂×C₄×C₁₉	—	8	1	no	—
135	C₂×C₄×C₁₉	—	8	1	no	—
136	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
139	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
140	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
141	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
142	C₂×Dic₁₉	—	4	2	no	—
143	C₂×Dic₁₉	—	8	1	no	—
144	C₂×Dic₁₉	—	8	1	no	—
145	C₈×C₁₉	—	8	1	no	—
146	C₁₉⋊C₈	—	2	1	no	—
147	C₁₉⋊C₈	—	2	1	no	—
148	C₁₉⋊C₈	—	2	1	no	—
149	C₁₉⋊C₈	—	4	1	no	—
150	C₁₉⋊C₈	—	4	1	no	—
151	C₁₉⋊C₈	—	8	1	no	—
152	C₁₉⋊C₈	#104	1	2	yes	—
153	#2966	—	19	1	no	—
154	C₄×#106	—	19	1	no	—
155	#1568	—	19	1	no	—
156	#1509	—	19	1	no	—
157	#1509	D₁₉	1	1	yes	characteristic
158	#1285	—	19	1	no	—
159	#1285	—	19	1	no	—
160	C₂²×C₁₉	—	2	1	no	—
161	C₂²×C₁₉	—	2	1	no	—
162	C₂²×C₁₉	—	4	1	no	—
163	C₂²×C₁₉	—	4	1	no	—
164	C₂²×C₁₉	—	4	1	no	—
165	C₂²×C₁₉	#380	1	1	yes	characteristic
166	C₂²×C₁₉	#471	1	1	yes	characteristic
167	C₂²×C₁₉	#471	1	1	yes	characteristic
168	C₄×C₁₉	—	2	1	no	—
169	C₄×C₁₉	—	2	1	no	—
170	C₄×C₁₉	—	2	1	no	—
171	C₄×C₁₉	—	2	1	no	—
172	C₄×C₁₉	—	2	1	no	—
173	C₄×C₁₉	—	4	1	no	—
174	C₄×C₁₉	—	4	1	no	—
175	C₄×C₁₉	—	4	1	no	—
176	C₄×C₁₉	—	8	2	no	—
177	C₄×C₁₉	C₂×#104	1	1	yes	characteristic
178	C₄×C₁₉	#421	1	1	yes	characteristic
179	Dic₁₉	—	8	2	no	—
180	#529	C₂×D₁₉	1	1	yes	characteristic
181	#446	—	19	1	no	—
182	C₈×Q₈	—	19	1	no	—
183	#436	—	19	1	no	—
184	#435	—	38	1	no	—
185	C₈×D₄	—	19	1	no	—
186	C₈×D₄	—	38	1	no	—
187	#433	—	38	1	no	—
188	#432	—	19	1	no	—
189	#432	—	19	1	no	—
190	#432	C₂×D₁₉	1	1	yes	characteristic
191	C₂×#106	—	38	1	no	—
192	#405	—	38	1	no	—
193	C₂×C₄×C₈	—	19	1	no	—
194	C₄×C₄⋊C₄	—	19	1	no	—
195	C₄×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
196	#340	—	38	1	no	—
197	#335	—	38	1	no	—
198	#334	—	38	1	no	—
199	#328	—	2	1	no	—
200	#328	—	38	1	no	—
201	#327	—	2	1	no	—
202	#327	—	38	1	no	—
203	C₂×C₁₉	—	2	1	no	—
204	C₂×C₁₉	—	2	1	no	—
205	C₂×C₁₉	—	8	1	no	—
206	C₂×C₁₉	#1806	1	1	yes	characteristic
207	C₂×C₁₉	#1812	1	1	yes	characteristic
208	C₂×C₁₉	#1815	1	1	yes	characteristic
209	C₂×M₁₆	—	38	1	no	—
210	C₂²×C₈	—	19	1	no	—
211	C₂²×C₈	—	19	1	no	—
212	C₂²×C₈	—	38	1	no	—
213	#126	—	2	1	no	—
214	#125	Dic₃₈	1	1	yes	characteristic
215	#124	Dic₃₈	1	1	yes	characteristic
216	#123	—	2	1	no	—
217	C₂²⋊Q₈	C₂²⋊D₁₉	1	1	yes	characteristic
218	#121	C₂²⋊D₁₉	1	1	yes	characteristic
219	C₄×Q₈	C₄×D₁₉	1	1	yes	characteristic
220	C₄×D₄	—	2	1	no	—
221	C₄×D₄	C₄×D₁₉	1	1	yes	characteristic
222	#117	—	2	1	no	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄⋊C₄	—	38	1	no	—
225	C₂×C₄²	—	2	1	no	—
226	C₂×C₄²	—	38	1	no	—
227	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
228	#106	—	38	4	no	—
229	C₄⋊C₈	—	2	1	no	—
230	C₄⋊C₈	—	19	1	no	—
231	C₄⋊C₈	—	19	1	no	—
232	C₄⋊C₈	—	38	1	no	—
233	C₄⋊C₈	—	38	1	no	—
234	C₄⋊C₈	—	38	1	no	—
235	C₂²⋊C₈	—	2	1	no	—
236	C₂²⋊C₈	—	38	1	no	—
237	C₂²⋊C₈	—	38	1	no	—
238	C₂²⋊C₈	—	38	1	no	—
239	C₂²⋊C₈	—	38	1	no	—
240	#97	—	2	1	no	—
241	#97	—	38	1	no	—
242	#97	—	38	1	no	—
243	#97	—	38	1	no	—
244	C₄×C₈	—	19	1	no	—
245	C₄×C₈	—	19	1	no	—
246	C₄×C₈	—	19	1	no	—
247	C₄×C₈	—	19	1	no	—
248	C₄×C₈	—	38	1	no	—
249	#95	—	2	1	no	—
250	#95	—	38	1	no	—
251	C₁₉	#8696	1	1	yes	characteristic, complemented, Sylow
252	C₂×Q₈	Dic₁₉⋊C₄	1	1	yes	characteristic
253	C₂×D₄	—	2	1	no	—
254	C₂×D₄	Dic₁₉⋊C₄	1	1	yes	characteristic
255	C₂²×C₄	—	2	1	no	—
256	C₂²×C₄	—	4	1	no	—
257	C₂²×C₄	—	76	1	no	—
258	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
259	C₂²×C₄	#64543	1	1	yes	characteristic
260	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
261	M₁₆	—	76	2	no	—
262	C₂×C₈	—	4	1	no	—
263	C₂×C₈	—	19	1	no	—
264	C₂×C₈	—	19	1	no	—
265	C₂×C₈	—	19	1	no	—
266	C₂×C₈	—	19	1	no	—
267	C₂×C₈	—	38	2	no	—
268	C₂×C₈	—	38	1	no	—
269	C₂×C₈	—	38	1	no	—
270	C₂×C₈	—	76	1	no	—
271	C₂×C₈	—	76	2	no	—
272	C₂×C₈	—	76	1	no	—
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	38	2	no	—
280	C₄⋊C₄	—	76	1	no	—
281	C₄⋊C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
282	C₄⋊C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
283	C₂²⋊C₄	—	2	1	no	—
284	C₂²⋊C₄	—	2	1	no	—
285	C₂²⋊C₄	—	2	1	no	—
286	C₂²⋊C₄	—	2	1	no	—
287	C₂²⋊C₄	—	2	1	no	—
288	C₄²	—	2	1	no	—
289	C₄²	—	8	1	no	—
290	C₄²	—	152	1	no	—
291	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
292	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
293	C₄²	D₄×D₁₉	1	1	yes	characteristic
294	C₄²	#64543	1	1	yes	characteristic
295	C₂³	—	2	1	no	—
296	C₂³	#10597385	1	1	yes	characteristic
297	Q₈	—	4	1	no	—
298	D₄	—	4	1	no	—
299	D₄	—	4	1	no	—
300	D₄	—	4	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	8	1	no	—
313	C₂×C₄	—	8	1	no	—
314	C₂×C₄	—	76	2	no	—
315	C₂×C₄	—	152	1	no	—
316	C₂×C₄	—	152	1	no	—
317	C₂×C₄	#10597323	1	1	yes	characteristic
318	C₂×C₄	#10597324	1	1	yes	characteristic
319	C₂×C₄	#10597324	1	1	yes	characteristic
320	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
321	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
322	C₂×C₄	#10597385	1	1	yes	characteristic
323	C₈	—	8	1	no	—
324	C₈	—	19	2	no	—
325	C₈	—	38	1	no	—
326	C₈	—	38	1	no	—
327	C₈	—	38	1	no	—
328	C₈	—	76	1	no	—
329	C₈	—	76	1	no	—
330	C₈	—	152	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	4	1	no	—
334	C₂²	—	4	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	#11926402	1	1	yes	characteristic
337	C₂²	#11926805	1	1	yes	characteristic
338	C₂²	#11926809	1	1	yes	characteristic
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	4	1	no	—
345	C₄	—	4	1	no	—
346	C₄	—	4	1	no	—
347	C₄	—	8	2	no	—
348	C₄	—	152	2	no	—
349	C₄	#104×D₁₉	1	1	yes	characteristic
350	C₄	#11926559	1	1	yes	characteristic
351	C₂	—	2	1	no	—
352	C₂	—	2	1	no	—
353	C₂	—	8	1	no	—
354	C₂	#14563653	1	1	yes	characteristic
355	C₂	#14563679	1	1	yes	characteristic
356	C₂	#14563692	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926402
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1509	C₂
socle	C₂²×C₁₉	#380
prime radical	C₁₉×#121	D₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1509	D₁₉
hypocenter	C₁₉	#8696
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1509

Complements

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

N	H ≅ G/N	type
2. C₄×#10597265	C₂	semidirect
3. #14562396	C₂	semidirect
5. #14561491	C₂	semidirect
6. #14561490	C₂	semidirect
8. #11926675	C₄	semidirect
9. Q₈×C₁₉⋊C₈	C₄	semidirect
10. #11926622	C₄	semidirect
13. D₄×C₁₉⋊C₈	C₄	semidirect
251. C₁₉	#8696	semidirect

Maximal subgroups

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

subgroup	index
#14572611	2
C₄×#10597265	2
#14562396	2
#14562160	2
#14561491	2
#14561490	2
C₁₉×#1509	2
#8696	19