Subgroups — #18079057

Order 4864.

480 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14574765	C₂	1	1	yes	maximal, characteristic
2	#14572378	C₂	1	1	yes	maximal, characteristic
3	#14564623	C₂	1	1	yes	maximal, characteristic, complemented
4	#14562313	C₂	1	1	yes	maximal, characteristic
5	C₂×#11926289	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561484	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1301	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10597347	—	2	1	no	—
9	#11927094	C₂²	1	1	yes	characteristic
10	C₂×#10597315	C₂²	1	1	yes	characteristic
11	#11926942	—	2	1	no	—
12	#11926891	—	2	1	no	—
13	#11926863	C₄	1	4	yes	—
14	#11926835	—	2	2	no	—
15	#11926621	—	2	1	no	—
16	#11926614	—	2	2	no	—
17	#11926609	—	2	1	no	—
18	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
19	#11926510	—	2	1	no	—
20	C₄×#64520	C₂²	1	1	yes	characteristic
21	#11926312	—	2	1	no	—
22	#11926312	—	2	1	no	—
23	#11926291	—	2	1	no	—
24	#11926289	—	2	1	no	—
25	#11926289	—	2	1	no	—
26	#11926287	—	2	1	no	—
27	C₁₉×#431	C₂²	1	1	yes	characteristic
28	C₁₉×#423	C₂²	1	1	yes	characteristic
29	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×#64541	—	2	1	no	—
31	C₂²×Dic₃₈	—	2	1	no	—
32	C₂×M₁₆×C₁₉	—	2	1	no	—
33	C₂²×C₈×C₁₉	—	2	1	no	—
34	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
35	#10597376	—	2	1	no	—
36	#10597376	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊Dic₁₉	—	2	1	no	—
38	C₂×C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
39	C₂×#64520	D₄	1	1	yes	characteristic
40	C₂×#64520	D₄	1	1	yes	characteristic
41	C₂²×C₁₉⋊C₈	—	2	1	no	—
42	#10597347	—	4	2	no	—
43	#10597347	—	4	2	no	—
44	C₂×#64515	—	2	2	no	—
45	C₂×C₈×D₁₉	—	2	2	no	—
46	#10597321	—	2	1	no	—
47	#10597320	D₄	1	1	yes	characteristic
48	C₄×D₇₆	—	2	1	no	—
49	C₄×D₇₆	D₄	1	2	yes	—
50	#10597315	—	2	1	no	—
51	#10597315	—	2	1	no	—
52	#10597315	—	2	1	no	—
53	#10597315	—	2	1	no	—
54	#10597315	D₄	1	1	yes	characteristic
55	C₁₉×C₄⋊C₈	—	2	1	no	—
56	C₁₉×C₄⋊C₈	—	2	1	no	—
57	C₁₉×C₄⋊C₈	—	2	1	no	—
58	C₁₉×C₄⋊C₈	—	2	1	no	—
59	C₁₉×C₄⋊C₈	D₄	1	1	yes	characteristic
60	C₁₉×C₄⋊C₈	D₄	1	1	yes	characteristic
61	C₁₉×C₂²⋊C₈	C₂×C₄	1	2	yes	—
62	C₄×C₈×C₁₉	—	2	1	no	—
63	#10597278	—	2	4	no	—
64	(C₂×D₁₉)⋊C₈	—	2	2	no	—
65	(C₂×D₁₉)⋊C₈	—	2	2	no	—
66	#10597274	—	2	4	no	—
67	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
68	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
69	#10597260	—	2	1	no	—
70	C₄×C₁₉⋊C₈	—	2	1	no	—
71	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
74	C₂×C₂²⋊D₁₉	—	2	2	no	—
75	C₂²×Dic₁₉	—	2	2	no	—
76	#64541	—	4	1	no	—
77	#64541	—	4	1	no	—
78	#64541	—	4	1	no	—
79	#64541	—	4	1	no	—
80	C₂×D₇₆	—	2	1	no	—
81	C₂×D₇₆	—	2	1	no	—
82	C₂×D₇₆	C₂²⋊C₄	1	2	yes	—
83	C₂×C₄×D₁₉	—	2	2	no	—
84	C₂×Dic₃₈	—	2	1	no	—
85	C₂×Dic₃₈	—	4	1	no	—
86	C₂×Dic₃₈	—	4	1	no	—
87	C₂×Dic₃₈	—	8	1	no	—
88	C₂×Dic₃₈	—	8	1	no	—
89	M₁₆×C₁₉	—	4	1	no	—
90	M₁₆×C₁₉	—	4	1	no	—
91	C₂×C₈×C₁₉	—	2	2	no	—
92	C₂×C₈×C₁₉	—	2	1	no	—
93	C₂×C₈×C₁₉	—	2	1	no	—
94	C₂×C₈×C₁₉	—	4	1	no	—
95	C₂×C₈×C₁₉	—	4	1	no	—
96	C₂×C₈×C₁₉	—	4	1	no	—
97	C₄²×C₁₉	C₂×D₄	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	#41	1	1	yes	characteristic
101	(C₂×D₁₉)⋊C₄	—	2	2	no	—
102	(C₂×D₁₉)⋊C₄	—	2	2	no	—
103	C₄⋊Dic₁₉	—	2	1	no	—
104	C₄⋊Dic₁₉	—	2	2	no	—
105	C₄⋊Dic₁₉	—	2	2	no	—
106	C₄⋊Dic₁₉	—	4	1	no	—
107	C₄⋊Dic₁₉	—	4	1	no	—
108	C₄⋊Dic₁₉	C₂²⋊C₄	1	2	yes	—
109	#64520	—	2	1	no	—
110	#64520	—	2	1	no	—
111	#64520	—	2	1	no	—
112	#64520	SD₁₆	1	2	yes	—
113	C₂×C₁₉⋊C₈	—	2	1	no	—
114	C₂×C₁₉⋊C₈	—	2	1	no	—
115	C₂×C₁₉⋊C₈	—	4	1	no	—
116	C₂×C₁₉⋊C₈	—	8	1	no	—
117	#64515	—	4	4	no	—
118	C₈×D₁₉	—	4	4	no	—
119	#9269	—	19	1	no	maximal, Sylow
120	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
121	C₂²×D₁₉	—	2	2	no	—
122	C₂×C₄×C₁₉	—	2	1	no	—
123	C₂×C₄×C₁₉	—	2	1	no	—
124	C₂×C₄×C₁₉	—	4	1	no	—
125	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
128	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
129	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
130	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
131	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
132	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
133	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
135	C₂²⋊D₁₉	—	4	2	no	—
136	C₂²⋊D₁₉	—	4	2	no	—
137	C₂×Dic₁₉	—	2	2	no	—
138	C₂×Dic₁₉	—	4	2	no	—
139	C₂×Dic₁₉	—	8	2	no	—
140	D₇₆	—	4	2	no	—
141	D₇₆	—	4	1	no	—
142	D₇₆	—	4	1	no	—
143	D₇₆	—	4	1	no	—
144	D₇₆	—	4	1	no	—
145	C₄×D₁₉	—	4	2	no	—
146	C₄×D₁₉	—	4	2	no	—
147	Dic₃₈	—	4	1	no	—
148	Dic₃₈	—	4	1	no	—
149	Dic₃₈	—	8	2	no	—
150	Dic₃₈	—	8	1	no	—
151	Dic₃₈	—	8	1	no	—
152	C₈×C₁₉	—	4	2	no	—
153	C₈×C₁₉	—	4	1	no	—
154	C₈×C₁₉	—	4	1	no	—
155	C₈×C₁₉	—	8	1	no	—
156	C₁₉⋊C₈	—	2	2	no	—
157	C₁₉⋊C₈	—	4	1	no	—
158	C₁₉⋊C₈	—	8	1	no	—
159	#3227	—	19	1	no	—
160	#2935	—	19	1	no	—
161	#1997	—	19	1	no	—
162	#1547	—	19	1	no	—
163	C₂×#332	—	19	1	no	—
164	#1301	D₁₉	1	1	yes	characteristic, complemented
165	#1283	—	19	1	no	—
166	C₂²×C₁₉	—	2	1	no	—
167	C₂²×C₁₉	—	2	1	no	—
168	C₂²×C₁₉	#386	1	1	yes	characteristic
169	C₂²×C₁₉	C₄×SD₁₆	1	1	yes	characteristic
170	C₂²×C₁₉	#440	1	1	yes	characteristic
171	C₂×D₁₉	—	4	2	no	—
172	C₂×D₁₉	—	4	2	no	—
173	C₂×D₁₉	—	4	2	no	—
174	C₄×C₁₉	—	2	1	no	—
175	C₄×C₁₉	—	2	1	no	—
176	C₄×C₁₉	—	2	1	no	—
177	C₄×C₁₉	#450	1	1	yes	characteristic
178	C₄×C₁₉	#452	1	1	yes	characteristic
179	C₄×C₁₉	#461	1	1	yes	characteristic
180	C₄×C₁₉	#463	1	1	yes	characteristic
181	C₄×C₁₉	#471	1	1	yes	characteristic
182	C₄×C₁₉	#472	1	1	yes	characteristic
183	Dic₁₉	—	4	2	no	—
184	Dic₁₉	—	8	2	no	—
185	C₂×#131	—	38	1	no	—
186	#532	—	19	1	no	—
187	C₂×C₄⋊Q₈	—	19	1	no	—
188	#498	—	38	1	no	—
189	#486	—	38	1	no	—
190	#481	—	19	4	no	—
191	#476	—	38	2	no	—
192	#436	—	38	1	no	—
193	#435	—	38	2	no	—
194	C₈×D₄	—	38	1	no	—
195	#431	C₂×D₁₉	1	1	yes	characteristic
196	#423	C₂×D₁₉	1	1	yes	characteristic
197	C₂×C₄⋊C₈	—	19	1	no	—
198	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
199	#408	—	38	1	no	—
200	C₄×M₁₆	—	19	1	no	—
201	#343	—	38	1	no	—
202	#343	—	38	1	no	—
203	#332	—	38	1	no	—
204	#332	—	38	1	no	—
205	#332	—	38	1	no	—
206	#331	—	38	1	no	—
207	C₂×C₁₉	—	2	1	no	—
208	C₂×C₁₉	—	2	1	no	—
209	C₂×C₁₉	#1877	1	1	yes	characteristic
210	C₂×C₁₉	#1907	1	1	yes	characteristic
211	C₂×C₁₉	#1915	1	1	yes	characteristic
212	D₁₉	—	8	2	no	—
213	C₂×#41	—	38	1	no	—
214	C₂²×Q₈	—	38	1	no	—
215	#131	—	76	2	no	—
216	#131	—	76	2	no	—
217	C₂×M₁₆	—	2	1	no	—
218	C₂×M₁₆	—	19	1	no	—
219	C₂×M₁₆	—	19	1	no	—
220	C₂×M₁₆	—	38	2	no	—
221	C₂²×C₈	—	2	1	no	—
222	C₂²×C₈	—	38	2	no	—
223	C₂²×C₈	—	38	1	no	—
224	C₄⋊Q₈	—	19	1	no	—
225	C₄⋊Q₈	—	38	1	no	—
226	C₄⋊Q₈	—	38	1	no	—
227	C₄⋊Q₈	—	38	1	no	—
228	C₄⋊Q₈	—	38	1	no	—
229	#127	—	19	1	no	—
230	#124	—	38	1	no	—
231	#121	—	19	2	no	—
232	#121	—	38	1	no	—
233	C₄×D₄	—	19	2	no	—
234	C₄×D₄	—	38	1	no	—
235	C₂×C₄⋊C₄	—	19	2	no	—
236	C₂×C₄⋊C₄	—	38	1	no	—
237	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
238	#106	—	38	4	no	—
239	C₄⋊C₈	—	2	1	no	—
240	C₄⋊C₈	—	2	1	no	—
241	C₄⋊C₈	—	2	1	no	—
242	C₄⋊C₈	—	2	1	no	—
243	C₄⋊C₈	—	38	1	no	—
244	C₄⋊C₈	—	38	1	no	—
245	C₄⋊C₈	D₇₆	1	1	yes	characteristic
246	C₄⋊C₈	D₇₆	1	1	yes	characteristic
247	#102	—	38	4	no	—
248	C₂²⋊C₈	—	38	2	no	—
249	C₂²⋊C₈	—	38	2	no	—
250	C₂²⋊C₈	C₄×D₁₉	1	2	yes	—
251	#97	—	38	1	no	—
252	C₄×C₈	—	2	1	no	—
253	C₄×C₈	—	38	1	no	—
254	C₁₉	#9269	1	1	yes	characteristic, complemented, Sylow
255	#41	—	76	1	no	—
256	#41	—	76	1	no	—
257	#41	—	76	1	no	—
258	#41	—	76	1	no	—
259	C₂×Q₈	—	38	1	no	—
260	C₂×Q₈	—	76	1	no	—
261	C₂×Q₈	—	76	1	no	—
262	C₂×Q₈	—	152	1	no	—
263	C₂×Q₈	—	152	1	no	—
264	C₂×D₄	—	19	2	no	—
265	C₂×D₄	—	38	2	no	—
266	C₂×D₄	—	38	1	no	—
267	C₂×D₄	—	38	1	no	—
268	C₂²×C₄	—	38	2	no	—
269	C₂²×C₄	—	38	2	no	—
270	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
271	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
272	C₂²×C₄	#64545	1	1	yes	characteristic
273	M₁₆	—	4	1	no	—
274	M₁₆	—	4	1	no	—
275	M₁₆	—	19	2	no	—
276	M₁₆	—	38	1	no	—
277	M₁₆	—	38	1	no	—
278	M₁₆	—	38	1	no	—
279	M₁₆	—	76	4	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	4	1	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	4	1	no	—
286	C₂×C₈	—	38	1	no	—
287	C₂×C₈	—	38	1	no	—
288	C₂×C₈	—	76	4	no	—
289	C₂×C₈	—	76	1	no	—
290	C₂×C₈	—	152	1	no	—
291	C₄⋊C₄	—	19	2	no	—
292	C₄⋊C₄	—	38	1	no	—
293	C₄⋊C₄	—	38	2	no	—
294	C₄⋊C₄	—	38	2	no	—
295	C₄⋊C₄	—	76	1	no	—
296	C₄⋊C₄	—	76	1	no	—
297	C₂²⋊C₄	—	38	2	no	—
298	C₂²⋊C₄	—	38	2	no	—
299	C₄²	C₂×D₇₆	1	1	yes	characteristic
300	C₄²	#64541	1	1	yes	characteristic
301	C₄²	D₄×D₁₉	1	1	yes	characteristic
302	C₄²	D₄×D₁₉	1	1	yes	characteristic
303	C₂³	—	38	2	no	—
304	C₂³	#10597339	1	1	yes	characteristic
305	Q₈	—	76	1	no	—
306	Q₈	—	76	1	no	—
307	Q₈	—	152	2	no	—
308	Q₈	—	152	1	no	—
309	Q₈	—	152	1	no	—
310	D₄	—	76	2	no	—
311	D₄	—	76	2	no	—
312	D₄	—	76	2	no	—
313	D₄	—	76	1	no	—
314	D₄	—	76	1	no	—
315	D₄	—	76	1	no	—
316	D₄	—	76	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	38	2	no	—
321	C₂×C₄	—	76	2	no	—
322	C₂×C₄	—	76	2	no	—
323	C₂×C₄	—	76	2	no	—
324	C₂×C₄	—	152	2	no	—
325	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
326	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
327	C₂×C₄	#10597328	1	1	yes	characteristic
328	C₂×C₄	#10597330	1	1	yes	characteristic
329	C₂×C₄	#10597340	1	1	yes	characteristic
330	C₂×C₄	#10597341	1	1	yes	characteristic
331	C₂×C₄	#10597357	1	1	yes	characteristic
332	C₂×C₄	SD₁₆×D₁₉	1	1	yes	characteristic
333	C₂×C₄	#10597362	1	1	yes	characteristic
334	C₂×C₄	#10597364	1	1	yes	characteristic
335	C₈	—	4	2	no	—
336	C₈	—	4	1	no	—
337	C₈	—	4	1	no	—
338	C₈	—	8	1	no	—
339	C₈	—	38	2	no	—
340	C₈	—	76	1	no	—
341	C₈	—	152	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	76	2	no	—
345	C₂²	—	76	2	no	—
346	C₂²	—	76	2	no	—
347	C₂²	#11926422	1	1	yes	characteristic
348	C₂²	#11926631	1	1	yes	characteristic
349	C₂²	#11926643	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	76	2	no	—
354	C₄	—	152	2	no	—
355	C₄	#11926696	1	1	yes	characteristic
356	C₄	#11926709	1	1	yes	characteristic
357	C₄	#11926748	1	1	yes	characteristic
358	C₄	#11926758	1	1	yes	characteristic
359	C₄	#11926808	1	1	yes	characteristic
360	C₄	#11926815	1	1	yes	characteristic
361	C₂	—	2	1	no	—
362	C₂	—	2	1	no	—
363	C₂	—	152	2	no	—
364	C₂	#14563970	1	1	yes	characteristic
365	C₂	#14564142	1	1	yes	characteristic
366	C₂	#14564186	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926422
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1301	C₂
socle	C₂²×C₁₉	#386
prime radical	#11927094	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1301	D₁₉
hypocenter	C₁₉	#9269
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1301

Complements

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

N	H ≅ G/N	type
3. #14564623	C₂	semidirect
5. C₂×#11926289	C₂	semidirect
6. #14561484	C₂	semidirect
7. C₁₉×#1301	C₂	semidirect
164. #1301	D₁₉	semidirect
254. C₁₉	#9269	semidirect

Maximal subgroups

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

subgroup	index
#14574765	2
#14572378	2
#14564623	2
#14562313	2
C₂×#11926289	2
#14561484	2
C₁₉×#1301	2
#9269	19