Subgroups — #18077765

Order 4864.

470 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14574195	C₂	1	1	yes	maximal, characteristic
2	#14572584	C₂	1	1	yes	maximal, characteristic, complemented
3	#14564553	C₂	1	1	yes	maximal, characteristic
4	#14562035	C₂	1	1	yes	maximal, characteristic
5	#14561476	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561471	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1503	C₂	1	1	yes	maximal, characteristic, complemented
8	#11927094	C₂²	1	1	yes	characteristic
9	#11926940	C₄	1	1	yes	characteristic, complemented
10	#11926918	C₄	1	1	yes	characteristic, complemented
11	#11926894	C₄	1	1	yes	characteristic, complemented
12	#11926892	C₄	1	1	yes	characteristic, complemented
13	#11926775	—	2	2	no	—
14	#11926675	—	2	1	no	—
15	#11926675	—	2	2	no	—
16	Q₈×C₁₉⋊C₈	—	2	1	no	—
17	C₄×#64527	—	2	2	no	—
18	#11926571	—	2	1	no	—
19	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
20	C₂×#10597262	—	2	1	no	—
21	C₂×#10597262	—	2	1	no	—
22	#11926500	—	2	1	no	—
23	C₄×#64520	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
25	#11926273	—	2	1	no	—
26	#11926273	—	2	1	no	—
27	C₁₉×#533	C₂²	1	1	yes	characteristic
28	C₁₉×#431	C₂²	1	1	yes	characteristic
29	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
30	C₁₉×#328	—	2	2	no	—
31	C₂×#64541	—	2	1	no	—
32	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
33	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
34	C₁₉×#125	—	2	1	no	—
35	C₁₉×C₂²⋊Q₈	—	2	1	no	—
36	C₁₉×C₂²⋊Q₈	D₄	1	2	yes	—
37	C₄×Q₈×C₁₉	—	2	1	no	—
38	C₄×Q₈×C₁₉	D₄	1	2	yes	—
39	C₁₉×#117	—	2	1	no	—
40	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
41	C₂×C₄²×C₁₉	—	2	1	no	—
42	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
43	C₂×#64527	—	2	2	no	—
44	#10597376	—	2	1	no	—
45	#10597376	—	2	1	no	—
46	C₂×#64520	D₄	1	1	yes	characteristic
47	C₂×#64520	D₄	1	1	yes	characteristic
48	C₂²×C₁₉⋊C₈	—	2	1	no	—
49	C₂²×C₁₉⋊C₈	—	2	1	no	—
50	#10597321	—	2	1	no	—
51	#10597320	C₂×C₄	1	1	yes	characteristic
52	C₄×D₇₆	—	2	1	no	—
53	C₄×D₇₆	—	2	1	no	—
54	#10597315	C₂×C₄	1	1	yes	characteristic
55	C₁₉×C₄⋊C₈	—	2	1	no	—
56	C₁₉×C₂²⋊C₈	—	2	1	no	—
57	C₄×C₈×C₁₉	—	2	1	no	—
58	C₁₉×#95	—	2	1	no	—
59	#10597292	—	2	2	no	—
60	#10597266	—	2	2	no	—
61	#10597265	—	2	2	no	—
62	#10597262	—	4	2	no	—
63	#10597262	—	4	2	no	—
64	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
65	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
66	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
67	C₄⋊(C₁₉⋊C₈)	D₄	1	1	yes	characteristic
68	C₄⋊(C₁₉⋊C₈)	D₄	1	1	yes	characteristic
69	#10597260	—	2	2	no	—
70	#10597260	—	2	1	no	—
71	C₄×C₁₉⋊C₈	—	2	2	no	—
72	C₄×C₁₉⋊C₈	—	2	1	no	—
73	C₄×C₁₉⋊C₈	—	2	1	no	—
74	C₄×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
75	C₄×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
76	C₂×Q₈×C₁₉	—	2	1	no	—
77	C₂×Q₈×C₁₉	C₂²⋊C₄	1	2	yes	—
78	C₂²×C₄×C₁₉	—	4	1	no	—
79	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
82	C₂×C₂²⋊D₁₉	—	4	1	no	—
83	#64541	—	4	1	no	—
84	#64541	—	4	1	no	—
85	#64541	—	4	1	no	—
86	#64541	—	4	1	no	—
87	C₂×D₇₆	—	2	1	no	—
88	C₂×D₇₆	—	2	1	no	—
89	C₂×D₇₆	—	2	1	no	—
90	C₂×C₄×D₁₉	—	4	1	no	—
91	C₂×Dic₃₈	—	2	1	no	—
92	C₂×C₈×C₁₉	—	4	1	no	—
93	C₄⋊C₄×C₁₉	—	2	1	no	—
94	C₄⋊C₄×C₁₉	—	2	1	no	—
95	C₄⋊C₄×C₁₉	—	2	1	no	—
96	C₄⋊C₄×C₁₉	—	2	2	no	—
97	C₄⋊C₄×C₁₉	—	2	2	no	—
98	C₄⋊C₄×C₁₉	—	2	2	no	—
99	C₄⋊C₄×C₁₉	—	4	1	no	—
100	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
101	C₂²⋊C₄×C₁₉	—	2	2	no	—
102	C₄²×C₁₉	—	2	2	no	—
103	C₄²×C₁₉	—	4	1	no	—
104	C₄²×C₁₉	—	4	1	no	—
105	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₁₉	#41	1	1	yes	characteristic
109	#64527	—	4	4	no	—
110	(C₂×D₁₉)⋊C₄	—	4	1	no	—
111	(C₂×D₁₉)⋊C₄	—	4	1	no	—
112	C₄⋊Dic₁₉	—	2	1	no	—
113	C₄⋊Dic₁₉	—	2	1	no	—
114	#64520	—	2	1	no	—
115	#64520	—	2	1	no	—
116	#64520	—	2	1	no	—
117	#64520	SD₁₆	1	2	yes	complemented
118	C₂×C₁₉⋊C₈	—	2	1	no	—
119	C₂×C₁₉⋊C₈	—	2	1	no	—
120	C₂×C₁₉⋊C₈	—	2	1	no	—
121	C₂×C₁₉⋊C₈	—	2	1	no	—
122	C₂×C₁₉⋊C₈	—	2	1	no	—
123	C₂×C₁₉⋊C₈	—	2	1	no	—
124	C₂×C₁₉⋊C₈	—	4	2	no	—
125	C₂×C₁₉⋊C₈	—	4	1	no	—
126	C₂×C₁₉⋊C₈	—	4	1	no	—
127	#9054	—	19	1	no	maximal, Sylow
128	C₂³×C₁₉	#121	1	1	yes	characteristic
129	C₂²×D₁₉	—	4	1	no	—
130	Q₈×C₁₉	—	4	2	no	—
131	Q₈×C₁₉	—	4	1	no	—
132	Q₈×C₁₉	—	4	1	no	—
133	C₂×C₄×C₁₉	—	2	1	no	—
134	C₂×C₄×C₁₉	—	2	1	no	—
135	C₂×C₄×C₁₉	—	2	2	no	—
136	C₂×C₄×C₁₉	—	2	2	no	—
137	C₂×C₄×C₁₉	—	4	1	no	—
138	C₂×C₄×C₁₉	—	4	2	no	—
139	C₂×C₄×C₁₉	—	8	1	no	—
140	C₂×C₄×C₁₉	—	8	1	no	—
141	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
144	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
145	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
146	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
147	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
148	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
149	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
151	C₂²⋊D₁₉	—	8	1	no	—
152	C₂²⋊D₁₉	—	8	1	no	—
153	C₂×Dic₁₉	—	4	1	no	—
154	D₇₆	—	4	2	no	—
155	D₇₆	—	4	1	no	—
156	D₇₆	—	4	1	no	—
157	D₇₆	—	4	1	no	—
158	D₇₆	—	4	1	no	—
159	C₄×D₁₉	—	8	1	no	—
160	C₄×D₁₉	—	8	1	no	—
161	Dic₃₈	—	4	1	no	—
162	Dic₃₈	—	4	1	no	—
163	C₈×C₁₉	—	8	1	no	—
164	C₁₉⋊C₈	—	2	2	no	—
165	C₁₉⋊C₈	—	2	2	no	—
166	C₁₉⋊C₈	—	4	1	no	—
167	C₁₉⋊C₈	—	4	1	no	—
168	C₁₉⋊C₈	—	4	1	no	—
169	C₁₉⋊C₈	—	4	1	no	—
170	#3167	—	19	1	no	—
171	#2962	—	19	1	no	—
172	#1980	—	19	1	no	—
173	#1503	D₁₉	1	1	yes	characteristic, complemented
174	#1480	—	19	1	no	—
175	#1279	—	19	1	no	—
176	#1278	—	19	1	no	—
177	C₂²×C₁₉	—	2	1	no	—
178	C₂²×C₁₉	—	2	1	no	—
179	C₂²×C₁₉	#386	1	1	yes	characteristic
180	C₂²×C₁₉	#460	1	1	yes	characteristic
181	C₂²×C₁₉	#464	1	1	yes	characteristic
182	C₂×D₁₉	—	8	1	no	—
183	C₂×D₁₉	—	8	1	no	—
184	C₂×D₁₉	—	8	1	no	—
185	C₄×C₁₉	—	2	1	no	—
186	C₄×C₁₉	—	2	1	no	—
187	C₄×C₁₉	—	2	1	no	—
188	C₄×C₁₉	—	4	2	no	—
189	C₄×C₁₉	—	4	2	no	—
190	C₄×C₁₉	—	8	2	no	—
191	C₄×C₁₉	C₄×SD₁₆	1	1	yes	characteristic
192	C₄×C₁₉	#440	1	1	yes	characteristic
193	C₄×C₁₉	#443	1	1	yes	characteristic
194	C₄×C₁₉	#444	1	1	yes	characteristic
195	C₄×C₁₉	#448	1	1	yes	characteristic
196	C₄×C₁₉	#449	1	1	yes	characteristic
197	Dic₁₉	—	8	1	no	—
198	#533	C₂×D₁₉	1	1	yes	characteristic
199	#532	—	19	1	no	—
200	#498	—	19	1	no	—
201	#492	—	19	1	no	—
202	#487	—	19	1	no	—
203	#486	—	19	1	no	—
204	#465	—	38	2	no	—
205	#446	—	38	1	no	—
206	#446	—	38	2	no	—
207	C₈×Q₈	—	38	1	no	—
208	C₄×SD₁₆	—	38	2	no	—
209	#431	C₂×D₁₉	1	1	yes	characteristic
210	#424	—	38	1	no	—
211	C₂×C₄⋊C₈	—	19	1	no	—
212	C₂×#104	—	38	1	no	—
213	C₂×#104	—	38	1	no	—
214	#405	—	38	1	no	—
215	C₄×M₁₆	—	19	1	no	—
216	C₂×C₄×C₈	—	19	1	no	—
217	C₄×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
218	#328	—	2	2	no	—
219	D₄⋊C₈	—	38	1	no	—
220	D₄⋊C₈	—	38	1	no	—
221	C₂×C₁₉	—	2	1	no	—
222	C₂×C₁₉	—	2	1	no	—
223	C₂×C₁₉	#1876	1	1	yes	characteristic
224	C₂×C₁₉	#1880	1	1	yes	characteristic
225	C₂×C₁₉	#1884	1	1	yes	characteristic
226	D₁₉	—	16	1	no	—
227	C₂×#41	—	38	1	no	—
228	C₂×SD₁₆	—	38	2	no	—
229	C₂×M₁₆	—	19	1	no	—
230	C₂×M₁₆	—	19	1	no	—
231	C₂²×C₈	—	38	1	no	—
232	C₂²×C₈	—	38	1	no	—
233	C₄⋊Q₈	—	19	1	no	—
234	C₄⋊Q₈	C₄×D₁₉	1	1	yes	characteristic
235	C₄⋊Q₈	C₄×D₁₉	1	1	yes	characteristic
236	#127	—	19	1	no	—
237	#125	—	2	1	no	—
238	#124	—	38	1	no	—
239	C₂²⋊Q₈	—	2	1	no	—
240	C₂²⋊Q₈	C₂²⋊D₁₉	1	2	yes	—
241	#121	—	38	1	no	—
242	#121	—	38	1	no	—
243	C₄×Q₈	—	2	1	no	—
244	C₄×Q₈	D₇₆	1	2	yes	—
245	C₄×D₄	—	38	1	no	—
246	C₄×D₄	—	38	1	no	—
247	#117	—	2	1	no	—
248	C₂×C₄⋊C₄	—	2	1	no	—
249	C₂×C₄²	—	2	1	no	—
250	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
251	#106	—	38	2	no	—
252	C₄⋊C₈	—	2	1	no	—
253	C₄⋊C₈	—	19	1	no	—
254	C₄⋊C₈	—	19	1	no	—
255	C₄⋊C₈	—	38	2	no	—
256	C₄⋊C₈	—	38	2	no	—
257	C₄⋊C₈	—	38	1	no	—
258	#104	—	76	2	no	—
259	#104	—	76	2	no	—
260	#103	—	38	2	no	—
261	#102	—	38	2	no	—
262	C₂²⋊C₈	—	2	1	no	—
263	#97	—	38	2	no	—
264	#97	—	38	1	no	—
265	C₄×C₈	—	2	1	no	—
266	C₄×C₈	—	19	1	no	—
267	C₄×C₈	—	19	1	no	—
268	C₄×C₈	—	38	2	no	—
269	C₄×C₈	—	38	1	no	—
270	C₄×C₈	—	38	1	no	—
271	#95	—	2	1	no	—
272	C₁₉	#9054	1	1	yes	characteristic, complemented, Sylow
273	#41	—	76	1	no	—
274	#41	—	76	1	no	—
275	#41	—	76	1	no	—
276	#41	—	76	1	no	—
277	C₂×Q₈	—	2	1	no	—
278	C₂×Q₈	—	38	1	no	—
279	C₂×Q₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
280	C₂×D₄	—	38	1	no	—
281	C₂×D₄	—	38	1	no	—
282	C₂×D₄	—	38	1	no	—
283	C₂×D₄	—	76	1	no	—
284	C₂²×C₄	—	4	1	no	—
285	C₂²×C₄	—	76	1	no	—
286	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
287	C₂²×C₄	#64543	1	1	yes	characteristic
288	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
289	SD₁₆	—	76	4	no	—
290	M₁₆	—	19	2	no	—
291	M₁₆	—	38	1	no	—
292	M₁₆	—	38	1	no	—
293	M₁₆	—	38	1	no	—
294	C₂×C₈	—	4	1	no	—
295	C₂×C₈	—	38	1	no	—
296	C₂×C₈	—	38	1	no	—
297	C₂×C₈	—	38	1	no	—
298	C₂×C₈	—	38	1	no	—
299	C₂×C₈	—	38	1	no	—
300	C₂×C₈	—	38	1	no	—
301	C₂×C₈	—	76	2	no	—
302	C₂×C₈	—	76	1	no	—
303	C₂×C₈	—	76	1	no	—
304	C₄⋊C₄	—	2	2	no	—
305	C₄⋊C₄	—	2	1	no	—
306	C₄⋊C₄	—	2	1	no	—
307	C₄⋊C₄	—	2	1	no	—
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	—	4	1	no	—
311	C₄⋊C₄	—	38	1	no	—
312	C₄⋊C₄	—	38	1	no	—
313	C₄⋊C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
314	C₂²⋊C₄	—	2	2	no	—
315	C₂²⋊C₄	—	76	1	no	—
316	C₂²⋊C₄	—	76	1	no	—
317	C₄²	—	2	2	no	—
318	C₄²	—	4	1	no	—
319	C₄²	—	4	1	no	—
320	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
321	C₄²	C₂×D₇₆	1	1	yes	characteristic
322	C₄²	D₄×D₁₉	1	1	yes	characteristic
323	C₄²	#64543	1	1	yes	characteristic
324	C₂³	—	76	1	no	—
325	C₂³	#10597385	1	1	yes	characteristic
326	Q₈	—	4	2	no	—
327	Q₈	—	4	1	no	—
328	Q₈	—	4	1	no	—
329	Q₈	—	76	1	no	—
330	Q₈	—	76	1	no	—
331	D₄	—	76	2	no	—
332	D₄	—	76	1	no	—
333	D₄	—	76	1	no	—
334	D₄	—	76	1	no	—
335	D₄	—	76	1	no	—
336	D₄	—	152	1	no	—
337	D₄	—	152	1	no	—
338	C₂×C₄	—	2	1	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	4	2	no	—
344	C₂×C₄	—	8	1	no	—
345	C₂×C₄	—	8	1	no	—
346	C₂×C₄	—	76	1	no	—
347	C₂×C₄	—	152	1	no	—
348	C₂×C₄	—	152	1	no	—
349	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
350	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
351	C₂×C₄	#10597328	1	1	yes	characteristic
352	C₂×C₄	#10597332	1	1	yes	characteristic
353	C₂×C₄	SD₁₆×D₁₉	1	1	yes	characteristic
354	C₂×C₄	#10597360	1	1	yes	characteristic
355	C₂×C₄	#10597364	1	1	yes	characteristic
356	C₂×C₄	#10597365	1	1	yes	characteristic
357	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
358	C₂×C₄	#10597385	1	1	yes	characteristic
359	C₈	—	8	1	no	—
360	C₈	—	38	2	no	—
361	C₈	—	38	2	no	—
362	C₈	—	76	1	no	—
363	C₈	—	76	1	no	—
364	C₈	—	76	1	no	—
365	C₈	—	76	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	152	1	no	—
369	C₂²	—	152	1	no	—
370	C₂²	—	152	1	no	—
371	C₂²	#11926421	1	1	yes	characteristic
372	C₂²	#11926742	1	1	yes	characteristic
373	C₂²	#11926768	1	1	yes	characteristic
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	4	2	no	—
378	C₄	—	4	2	no	—
379	C₄	—	8	2	no	—
380	C₄	—	152	1	no	—
381	C₄	#11926629	1	1	yes	characteristic
382	C₄	#11926647	1	1	yes	characteristic
383	C₄	#11926658	1	1	yes	characteristic
384	C₄	#11926663	1	1	yes	characteristic
385	C₄	#11926684	1	1	yes	characteristic
386	C₄	#11926692	1	1	yes	characteristic
387	C₂	—	2	1	no	—
388	C₂	—	2	1	no	—
389	C₂	—	304	1	no	—
390	C₂	#14563960	1	1	yes	characteristic
391	C₂	#14563988	1	1	yes	characteristic
392	C₂	#14564016	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597328
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1503	C₂
socle	C₂²×C₁₉	#386
prime radical	#11927094	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1503	D₁₉
hypocenter	C₁₉	#9054
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1503

Complements

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

N	H ≅ G/N	type
2. #14572584	C₂	semidirect
5. #14561476	C₂	semidirect
6. #14561471	C₂	semidirect
7. C₁₉×#1503	C₂	semidirect
9. #11926940	C₄	semidirect
10. #11926918	C₄	semidirect
11. #11926894	C₄	semidirect
12. #11926892	C₄	semidirect
117. #64520	SD₁₆	semidirect
173. #1503	D₁₉	semidirect
272. C₁₉	#9054	semidirect

Maximal subgroups

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

subgroup	index
#14574195	2
#14572584	2
#14564553	2
#14562035	2
#14561476	2
#14561471	2
C₁₉×#1503	2
#9054	19