Subgroups — #18076858

Order 4864.

392 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14572301	C₂	1	1	yes	maximal, characteristic
2	#14562411	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562390	C₂	1	1	yes	maximal, characteristic, complemented
4	#14562017	C₂	1	1	yes	maximal, characteristic
5	#14561538	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561476	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1806	C₂	1	1	yes	maximal, characteristic
8	#11926605	—	2	2	no	—
9	#11926601	—	2	1	no	—
10	#11926597	—	2	1	no	—
11	#11926597	C₂²	1	1	yes	characteristic
12	#11926568	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
14	#11926511	—	2	1	no	—
15	#11926500	—	2	1	no	—
16	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
17	#11926296	—	2	1	no	—
18	#11926296	—	2	1	no	—
19	#11926294	—	2	2	no	—
20	Q₈⋊(C₁₉⋊C₈)	C₄	1	1	yes	characteristic, complemented
21	Q₈⋊(C₁₉⋊C₈)	C₄	1	1	yes	characteristic, complemented
22	D₄⋊(C₁₉⋊C₈)	C₄	1	1	yes	characteristic, complemented
23	D₄⋊(C₁₉⋊C₈)	C₄	1	1	yes	characteristic, complemented
24	C₄×#41×C₁₉	C₂²	1	1	yes	characteristic
25	C₁₉×#423	C₂²	1	1	yes	characteristic
26	C₂×C₁₉×#104	—	2	1	no	—
27	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
28	C₁₉×#339	—	2	1	no	—
29	C₂×#41×C₁₉	D₄	1	1	yes	characteristic
30	C₂×M₁₆×C₁₉	—	2	1	no	—
31	C₄×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
32	C₄×Q₈×C₁₉	Q₈	1	1	yes	characteristic
33	C₄×D₄×C₁₉	—	2	1	no	—
34	C₄×D₄×C₁₉	—	2	1	no	—
35	C₄×D₄×C₁₉	C₂×C₄	1	1	yes	characteristic
36	C₄×D₄×C₁₉	Q₈	1	1	yes	characteristic
37	C₁₉×#117	—	2	1	no	—
38	C₁₉×#117	D₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
40	C₂×C₄²×C₁₉	—	2	1	no	—
41	C₂×C₄²×C₁₉	—	2	1	no	—
42	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
43	C₂×#64520	—	2	1	no	—
44	C₂²×C₁₉⋊C₈	—	2	1	no	—
45	C₂²×C₁₉⋊C₈	—	2	1	no	—
46	C₁₉×C₄⋊C₈	—	2	1	no	—
47	C₁₉×C₄⋊C₈	—	2	1	no	—
48	C₁₉×#104	—	4	2	no	—
49	C₁₉×#95	—	2	1	no	—
50	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
51	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
52	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
53	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
54	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
55	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	1	yes	characteristic
56	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	1	yes	characteristic
57	C₄⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
58	C₄⋊(C₁₉⋊C₈)	D₄	1	1	yes	characteristic
59	C₄⋊(C₁₉⋊C₈)	D₄	1	1	yes	characteristic
60	#10597260	—	2	2	no	—
61	C₄×C₁₉⋊C₈	—	2	2	no	—
62	C₄×C₁₉⋊C₈	—	2	1	no	—
63	C₄×C₁₉⋊C₈	—	2	1	no	—
64	C₄×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
65	C₄×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
66	#41×C₁₉	—	2	1	no	—
67	#41×C₁₉	—	2	1	no	—
68	#41×C₁₉	—	2	1	no	—
69	#41×C₁₉	—	2	1	no	—
70	C₂×Q₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
71	C₂×D₄×C₁₉	—	2	1	no	—
72	C₂×D₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₉	—	2	1	no	—
74	C₂²×C₄×C₁₉	—	2	1	no	—
75	C₂²×C₄×C₁₉	—	2	1	no	—
76	C₂²×C₄×C₁₉	—	4	1	no	—
77	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
79	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
80	M₁₆×C₁₉	—	4	1	no	—
81	M₁₆×C₁₉	—	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	2	no	—
86	C₄⋊C₄×C₁₉	—	4	1	no	—
87	C₄⋊C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
88	C₄⋊C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
89	C₂²⋊C₄×C₁₉	—	2	1	no	—
90	C₂²⋊C₄×C₁₉	—	2	1	no	—
91	C₂²⋊C₄×C₁₉	—	2	1	no	—
92	C₄²×C₁₉	—	2	1	no	—
93	C₄²×C₁₉	—	2	1	no	—
94	C₄²×C₁₉	—	2	1	no	—
95	C₄²×C₁₉	—	4	1	no	—
96	C₄²×C₁₉	—	4	1	no	—
97	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
101	#64520	—	4	1	no	—
102	#64520	—	4	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	—	2	1	no	—
107	C₂×C₁₉⋊C₈	—	2	1	no	—
108	C₂×C₁₉⋊C₈	—	2	2	no	—
109	C₂×C₁₉⋊C₈	—	2	1	no	—
110	C₂×C₁₉⋊C₈	—	4	2	no	—
111	C₂×C₁₉⋊C₈	—	4	1	no	—
112	C₂×C₁₉⋊C₈	—	4	1	no	—
113	#8905	—	19	1	no	maximal, Sylow
114	C₂³×C₁₉	—	2	1	no	—
115	C₂³×C₁₉	#123	1	1	yes	characteristic
116	Q₈×C₁₉	—	2	1	no	—
117	Q₈×C₁₉	—	2	1	no	—
118	D₄×C₁₉	—	2	1	no	—
119	D₄×C₁₉	—	2	1	no	—
120	D₄×C₁₉	—	4	1	no	—
121	D₄×C₁₉	—	4	1	no	—
122	C₂×C₄×C₁₉	—	2	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	—	4	1	no	—
128	C₂×C₄×C₁₉	—	4	1	no	—
129	C₂×C₄×C₁₉	—	4	1	no	—
130	C₂×C₄×C₁₉	—	4	1	no	—
131	C₂×C₄×C₁₉	—	4	1	no	—
132	C₂×C₄×C₁₉	—	4	2	no	—
133	C₂×C₄×C₁₉	—	8	1	no	—
134	C₂×C₄×C₁₉	—	8	1	no	—
135	C₂×C₄×C₁₉	—	8	1	no	—
136	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₉	#120	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×C₄×C₁₉	#135	1	1	yes	characteristic
143	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
144	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
146	C₈×C₁₉	—	8	1	no	—
147	C₁₉⋊C₈	—	2	2	no	—
148	C₁₉⋊C₈	—	4	1	no	—
149	C₁₉⋊C₈	—	4	1	no	—
150	C₁₉⋊C₈	—	4	1	no	—
151	C₁₉⋊C₈	—	4	1	no	—
152	C₁₉⋊C₈	—	4	1	no	—
153	C₁₉⋊C₈	—	4	2	no	—
154	#2925	—	19	1	no	—
155	#1806	D₁₉	1	1	yes	characteristic
156	#1571	—	19	1	no	—
157	#1566	—	19	1	no	—
158	#1476	—	19	1	no	—
159	#1303	—	19	1	no	—
160	#1279	—	19	1	no	—
161	C₂²×C₁₉	—	2	1	no	—
162	C₂²×C₁₉	—	2	1	no	—
163	C₂²×C₁₉	—	4	1	no	—
164	C₂²×C₁₉	—	4	1	no	—
165	C₂²×C₁₉	—	4	1	no	—
166	C₂²×C₁₉	#385	1	1	yes	characteristic
167	C₂²×C₁₉	#482	1	1	yes	characteristic
168	C₂²×C₁₉	#485	1	1	yes	characteristic
169	C₄×C₁₉	—	2	1	no	—
170	C₄×C₁₉	—	2	1	no	—
171	C₄×C₁₉	—	2	1	no	—
172	C₄×C₁₉	—	4	1	no	—
173	C₄×C₁₉	—	4	1	no	—
174	C₄×C₁₉	—	4	1	no	—
175	C₄×C₁₉	—	8	2	no	—
176	C₄×C₁₉	#427	1	1	yes	characteristic
177	C₄×C₁₉	#428	1	1	yes	characteristic
178	C₄×C₁₉	#443	1	1	yes	characteristic
179	C₄×C₁₉	#444	1	1	yes	characteristic
180	C₄×C₁₉	#449	1	1	yes	characteristic
181	C₄×C₁₉	#452	1	1	yes	characteristic
182	C₄×#41	C₂×D₁₉	1	1	yes	characteristic
183	#433	—	38	2	no	—
184	#432	—	38	1	no	—
185	#431	—	19	1	no	—
186	#431	—	38	1	no	—
187	#423	—	19	1	no	—
188	#423	C₂×D₁₉	1	1	yes	characteristic
189	C₂×C₄⋊C₈	—	19	1	no	—
190	C₂×#104	—	2	1	no	—
191	#408	—	38	1	no	—
192	#405	—	38	1	no	—
193	C₂×C₄×C₈	—	19	1	no	—
194	C₄×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
195	#339	—	2	1	no	—
196	#335	—	38	1	no	—
197	#335	—	38	1	no	—
198	#334	—	38	2	no	—
199	Q₈⋊C₈	—	19	1	no	—
200	Q₈⋊C₈	—	19	1	no	—
201	D₄⋊C₈	—	19	1	no	—
202	D₄⋊C₈	—	19	1	no	—
203	C₂×C₁₉	—	2	1	no	—
204	C₂×C₁₉	—	2	1	no	—
205	C₂×C₁₉	—	8	1	no	—
206	C₂×C₁₉	#1854	1	1	yes	characteristic
207	C₂×C₁₉	#1861	1	1	yes	characteristic
208	C₂×C₁₉	#1866	1	1	yes	characteristic
209	C₂×#41	C₂²⋊D₁₉	1	1	yes	characteristic
210	C₂×M₁₆	—	2	1	no	—
211	C₂×M₁₆	—	38	1	no	—
212	C₂²×C₈	—	38	1	no	—
213	C₂²×C₈	—	38	1	no	—
214	C₄×Q₈	Dic₃₈	1	1	yes	characteristic
215	C₄×Q₈	C₄×D₁₉	1	1	yes	characteristic
216	C₄×D₄	—	2	1	no	—
217	C₄×D₄	—	2	1	no	—
218	C₄×D₄	Dic₃₈	1	1	yes	characteristic
219	C₄×D₄	C₄×D₁₉	1	1	yes	characteristic
220	#117	—	2	1	no	—
221	#117	C₂²⋊D₁₉	1	1	yes	characteristic
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄²	—	2	1	no	—
224	C₂×C₄²	—	2	1	no	—
225	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
226	C₄⋊C₈	—	2	1	no	—
227	C₄⋊C₈	—	2	1	no	—
228	C₄⋊C₈	—	19	1	no	—
229	C₄⋊C₈	—	19	1	no	—
230	C₄⋊C₈	—	19	2	no	—
231	C₄⋊C₈	—	19	1	no	—
232	C₄⋊C₈	—	19	1	no	—
233	C₄⋊C₈	—	38	1	no	—
234	C₄⋊C₈	—	38	1	no	—
235	#104	—	4	2	no	—
236	C₂²⋊C₈	—	38	2	no	—
237	C₂²⋊C₈	—	38	2	no	—
238	C₂²⋊C₈	—	38	1	no	—
239	#97	—	38	2	no	—
240	C₄×C₈	—	19	1	no	—
241	C₄×C₈	—	19	1	no	—
242	C₄×C₈	—	38	2	no	—
243	C₄×C₈	—	38	1	no	—
244	C₄×C₈	—	38	1	no	—
245	#95	—	2	1	no	—
246	C₁₉	#8905	1	1	yes	characteristic, complemented, Sylow
247	#41	—	2	1	no	—
248	#41	—	2	1	no	—
249	#41	—	2	1	no	—
250	#41	—	2	1	no	—
251	C₂×Q₈	Dic₁₉⋊C₄	1	1	yes	characteristic
252	C₂×D₄	—	2	1	no	—
253	C₂×D₄	Dic₁₉⋊C₄	1	1	yes	characteristic
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	2	1	no	—
256	C₂²×C₄	—	2	1	no	—
257	C₂²×C₄	—	4	1	no	—
258	C₂²×C₄	#64543	1	1	yes	characteristic
259	C₂²×C₄	#64543	1	1	yes	characteristic
260	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
261	M₁₆	—	4	1	no	—
262	M₁₆	—	4	1	no	—
263	M₁₆	—	76	1	no	—
264	M₁₆	—	76	1	no	—
265	C₂×C₈	—	4	1	no	—
266	C₂×C₈	—	38	1	no	—
267	C₂×C₈	—	38	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	2	no	—
274	C₂×C₈	—	76	1	no	—
275	C₂×C₈	—	76	1	no	—
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
281	C₄⋊C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
282	C₂²⋊C₄	—	2	1	no	—
283	C₂²⋊C₄	—	2	1	no	—
284	C₂²⋊C₄	—	2	1	no	—
285	C₄²	—	2	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	—	2	1	no	—
288	C₄²	—	4	1	no	—
289	C₄²	—	4	1	no	—
290	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
291	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
292	C₄²	D₄×D₁₉	1	1	yes	characteristic
293	C₄²	D₄×D₁₉	1	1	yes	characteristic
294	C₂³	—	2	1	no	—
295	C₂³	#10597381	1	1	yes	characteristic
296	Q₈	—	2	1	no	—
297	Q₈	—	2	1	no	—
298	D₄	—	2	1	no	—
299	D₄	—	2	1	no	—
300	D₄	—	4	1	no	—
301	D₄	—	4	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	—	4	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	8	1	no	—
314	C₂×C₄	—	8	1	no	—
315	C₂×C₄	—	8	1	no	—
316	C₂×C₄	#10597324	1	1	yes	characteristic
317	C₂×C₄	#10597324	1	1	yes	characteristic
318	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
319	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
320	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
321	C₂×C₄	#10597378	1	1	yes	characteristic
322	C₂×C₄	#10597383	1	1	yes	characteristic
323	C₂×C₄	#10597388	1	1	yes	characteristic
324	C₂×C₄	#10597396	1	1	yes	characteristic
325	C₂×C₄	#10597396	1	1	yes	characteristic
326	C₈	—	8	1	no	—
327	C₈	—	38	2	no	—
328	C₈	—	76	1	no	—
329	C₈	—	76	1	no	—
330	C₈	—	76	1	no	—
331	C₈	—	76	1	no	—
332	C₈	—	76	1	no	—
333	C₈	—	76	2	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	4	1	no	—
339	C₂²	#11926416	1	1	yes	characteristic
340	C₂²	#11926865	1	1	yes	characteristic
341	C₂²	#11926882	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	4	1	no	—
346	C₄	—	4	1	no	—
347	C₄	—	4	1	no	—
348	C₄	—	8	2	no	—
349	C₄	#11926580	1	1	yes	characteristic
350	C₄	#11926584	1	1	yes	characteristic
351	C₄	#11926658	1	1	yes	characteristic
352	C₄	#11926663	1	1	yes	characteristic
353	C₄	#11926687	1	1	yes	characteristic
354	C₄	#11926704	1	1	yes	characteristic
355	C₂	—	2	1	no	—
356	C₂	—	2	1	no	—
357	C₂	—	8	1	no	—
358	C₂	#14563832	1	1	yes	characteristic
359	C₂	#14563873	1	1	yes	characteristic
360	C₂	#14563908	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597383
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1806	C₂
socle	C₂²×C₁₉	#385
prime radical	C₂×#41×C₁₉	D₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1806	D₁₉
hypocenter	C₁₉	#8905
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1806

Complements

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

N	H ≅ G/N	type
2. #14562411	C₂	semidirect
3. #14562390	C₂	semidirect
5. #14561538	C₂	semidirect
6. #14561476	C₂	semidirect
20. Q₈⋊(C₁₉⋊C₈)	C₄	semidirect
21. Q₈⋊(C₁₉⋊C₈)	C₄	semidirect
22. D₄⋊(C₁₉⋊C₈)	C₄	semidirect
23. D₄⋊(C₁₉⋊C₈)	C₄	semidirect
246. C₁₉	#8905	semidirect

Maximal subgroups

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

subgroup	index
#14572301	2
#14562411	2
#14562390	2
#14562017	2
#14561538	2
#14561476	2
C₁₉×#1806	2
#8905	19