Subgroups — #18085372

Order 4864.

734 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926940	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11926916	C₂	1	1	yes	maximal, characteristic
3	#14564597	C₂	1	1	yes	maximal, characteristic, complemented
4	#14564596	C₂	1	1	yes	maximal, characteristic, complemented
5	#14564442	C₂	1	1	yes	maximal, characteristic
6	C₁₉⋊#1273	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₉×#492	C₂	1	1	yes	maximal, characteristic
8	C₂²×#64526	—	2	2	no	—
9	C₂×#10597315	C₂²	1	1	yes	characteristic
10	#11926940	—	2	2	no	—
11	#11926940	—	2	2	no	—
12	#11926916	—	2	2	no	—
13	#11926916	—	2	2	no	—
14	C₂×#10597265	—	2	2	no	—
15	C₂×#10597289	—	2	2	no	—
16	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
18	#11926414	C₂²	1	1	yes	characteristic
19	#11926307	—	2	2	no	—
20	#11926306	—	2	2	no	—
21	C₂²×SD₁₆×C₁₉	—	2	2	no	—
22	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
23	C₂×C₁₉×#127	C₂²	1	1	yes	characteristic
24	C₁₉×#492	—	2	2	no	—
25	C₁₉×#492	—	2	2	no	—
26	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
27	C₂²×Q₈×C₁₉	—	2	1	no	—
28	C₂²×D₄×C₁₉	—	2	1	no	—
29	C₂²×D₄×C₁₉	—	2	1	no	—
30	C₂²×D₄×C₁₉	—	2	1	no	—
31	C₂²×Dic₃₈	—	2	1	no	—
32	C₂×SD₁₆×C₁₉	—	2	8	no	—
33	C₂×SD₁₆×C₁₉	—	4	4	no	—
34	C₂×SD₁₆×C₁₉	—	4	4	no	—
35	C₂²×C₈×C₁₉	D₄	1	2	yes	—
36	C₁₉×C₄⋊Q₈	—	2	1	no	—
37	C₁₉×C₄⋊Q₈	—	2	1	no	—
38	C₁₉×C₄⋊Q₈	—	2	1	no	—
39	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
40	C₁₉×#127	—	2	1	no	—
41	C₁₉×#127	—	2	1	no	—
42	C₁₉×#127	—	2	1	no	—
43	C₁₉×#127	D₄	1	2	yes	—
44	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
45	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
46	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
47	C₂×#64526	—	2	8	no	—
48	C₂×#64526	—	4	4	no	—
49	C₂×#64526	—	4	4	no	—
50	C₂×C₄⋊Dic₁₉	—	2	1	no	—
51	C₂×C₄⋊Dic₁₉	—	2	1	no	—
52	C₂×C₄⋊Dic₁₉	—	2	1	no	—
53	C₂×C₄⋊Dic₁₉	—	2	1	no	—
54	C₂×C₄⋊Dic₁₉	—	2	1	no	—
55	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
56	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
57	#10597315	—	2	1	no	—
58	#10597315	—	2	1	no	—
59	#10597315	—	2	1	no	—
60	#10597315	D₄	1	2	yes	—
61	C₄×C₈×C₁₉	—	2	1	no	—
62	C₄×C₈×C₁₉	—	2	1	no	—
63	#10597289	—	4	4	no	—
64	#10597265	—	2	8	no	—
65	C₄×C₁₉⋊C₈	—	2	1	no	—
66	C₄×C₁₉⋊C₈	—	2	1	no	—
67	C₄×C₁₉⋊C₈	—	2	1	no	—
68	C₄×C₁₉⋊C₈	—	2	1	no	—
69	C₂⁴×C₁₉	—	4	1	no	—
70	C₂×Q₈×C₁₉	—	2	2	no	—
71	C₂×Q₈×C₁₉	—	4	1	no	—
72	C₂×Q₈×C₁₉	—	4	2	no	—
73	C₂×Q₈×C₁₉	—	4	2	no	—
74	C₂×D₄×C₁₉	—	2	2	no	—
75	C₂×D₄×C₁₉	—	2	2	no	—
76	C₂×D₄×C₁₉	—	2	2	no	—
77	C₂×D₄×C₁₉	—	4	1	no	—
78	C₂×D₄×C₁₉	—	4	1	no	—
79	C₂×D₄×C₁₉	—	4	1	no	—
80	C₂×D₄×C₁₉	—	4	2	no	—
81	C₂×D₄×C₁₉	—	4	2	no	—
82	C₂×D₄×C₁₉	—	4	2	no	—
83	C₂×D₄×C₁₉	—	4	2	no	—
84	C₂×D₄×C₁₉	—	4	2	no	—
85	C₂×D₄×C₁₉	—	4	2	no	—
86	C₂²×C₄×C₁₉	—	4	1	no	—
87	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
90	C₂²×Dic₁₉	—	4	1	no	—
91	C₂²×Dic₁₉	—	4	1	no	—
92	C₂×Dic₃₈	—	2	2	no	—
93	C₂×Dic₃₈	—	4	2	no	—
94	C₂×Dic₃₈	—	4	2	no	—
95	C₂×Dic₃₈	—	4	1	no	—
96	SD₁₆×C₁₉	—	4	16	no	—
97	C₂×C₈×C₁₉	—	2	2	no	—
98	C₂×C₈×C₁₉	—	2	2	no	—
99	C₂×C₈×C₁₉	SD₁₆	1	4	yes	complemented
100	C₄⋊C₄×C₁₉	—	2	2	no	—
101	C₄⋊C₄×C₁₉	—	2	2	no	—
102	C₄⋊C₄×C₁₉	—	4	1	no	—
103	C₄⋊C₄×C₁₉	—	4	1	no	—
104	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₁₉	#41	1	1	yes	characteristic
108	#64526	—	4	16	no	—
109	C₄⋊Dic₁₉	—	2	2	no	—
110	C₄⋊Dic₁₉	—	2	2	no	—
111	C₄⋊Dic₁₉	—	4	1	no	—
112	C₄⋊Dic₁₉	—	4	1	no	—
113	C₄⋊Dic₁₉	—	4	2	no	—
114	C₄⋊Dic₁₉	—	4	2	no	—
115	C₄⋊Dic₁₉	—	4	2	no	—
116	C₂×C₁₉⋊C₈	—	2	2	no	—
117	C₂×C₁₉⋊C₈	—	2	2	no	—
118	C₂×C₁₉⋊C₈	—	2	2	no	—
119	C₂×C₁₉⋊C₈	—	2	2	no	—
120	C₂×C₁₉⋊C₈	SD₁₆	1	4	yes	complemented
121	C₂×C₁₉⋊C₈	SD₁₆	1	4	yes	complemented
122	#10315	—	19	1	no	maximal, Sylow
123	C₂³×C₁₉	—	4	2	no	—
124	C₂³×C₁₉	—	8	1	no	—
125	C₂³×C₁₉	—	8	1	no	—
126	C₂³×C₁₉	—	8	1	no	—
127	C₂³×C₁₉	—	8	1	no	—
128	C₂³×C₁₉	—	8	1	no	—
129	C₂³×C₁₉	—	8	1	no	—
130	C₂³×C₁₉	#127	1	1	yes	characteristic
131	Q₈×C₁₉	—	4	4	no	—
132	Q₈×C₁₉	—	8	2	no	—
133	D₄×C₁₉	—	4	4	no	—
134	D₄×C₁₉	—	4	4	no	—
135	D₄×C₁₉	—	4	4	no	—
136	D₄×C₁₉	—	8	2	no	—
137	D₄×C₁₉	—	8	2	no	—
138	D₄×C₁₉	—	8	2	no	—
139	C₂×C₄×C₁₉	—	4	2	no	—
140	C₂×C₄×C₁₉	—	8	1	no	—
141	C₂×C₄×C₁₉	—	8	1	no	—
142	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
143	C₂×C₄×C₁₉	#120	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	#121	1	1	yes	characteristic
147	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
148	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
149	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
150	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
151	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
152	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
153	C₂×C₄×C₁₉	#137	1	2	yes	—
154	C₂×Dic₁₉	—	4	2	no	—
155	C₂×Dic₁₉	—	8	1	no	—
156	C₂×Dic₁₉	—	8	1	no	—
157	C₂×Dic₁₉	—	8	1	no	—
158	C₂×Dic₁₉	—	8	1	no	—
159	C₂×Dic₁₉	—	8	1	no	—
160	Dic₃₈	—	4	4	no	—
161	Dic₃₈	—	8	2	no	—
162	C₈×C₁₉	—	2	4	no	—
163	C₁₉⋊C₈	—	2	4	no	—
164	C₁₉⋊C₈	—	2	4	no	—
165	C₂×#498	—	19	1	no	—
166	C₂×#492	—	19	1	no	—
167	C₂×#492	D₁₉	1	1	yes	characteristic
168	#1990	—	19	1	no	—
169	#1990	—	19	1	no	—
170	#1957	—	19	1	no	—
171	#1273	—	19	1	no	—
172	C₂²×C₁₉	—	8	2	no	—
173	C₂²×C₁₉	—	8	2	no	—
174	C₂²×C₁₉	—	8	2	no	—
175	C₂²×C₁₉	—	16	1	no	—
176	C₂²×C₁₉	—	16	1	no	—
177	C₂²×C₁₉	—	16	1	no	—
178	C₂²×C₁₉	—	16	1	no	—
179	C₂²×C₁₉	#392	1	1	yes	characteristic
180	C₂²×C₁₉	#492	1	1	yes	characteristic
181	C₂²×C₁₉	#492	1	1	yes	characteristic
182	C₂²×C₁₉	#492	1	1	yes	characteristic
183	C₂²×C₁₉	#496	1	1	yes	characteristic
184	C₂²×C₁₉	#496	1	1	yes	characteristic
185	C₂²×C₁₉	#497	1	1	yes	characteristic
186	C₄×C₁₉	—	8	2	no	—
187	C₄×C₁₉	#448	1	2	yes	—
188	C₄×C₁₉	#450	1	2	yes	—
189	C₄×C₁₉	#450	1	2	yes	—
190	C₄×C₁₉	#460	1	2	yes	—
191	C₄×C₁₉	#460	1	2	yes	—
192	C₄×C₁₉	#461	1	2	yes	—
193	Dic₁₉	—	8	2	no	—
194	Dic₁₉	—	16	1	no	—
195	C₂²×SD₁₆	—	2	2	no	—
196	C₂²×SD₁₆	—	38	2	no	—
197	C₂×C₄⋊Q₈	—	19	1	no	—
198	C₂×C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
199	C₂×#127	C₂×D₁₉	1	1	yes	characteristic
200	#498	—	38	2	no	—
201	#498	—	38	2	no	—
202	#492	—	2	2	no	—
203	#492	—	2	2	no	—
204	#492	—	38	2	no	—
205	#492	—	38	2	no	—
206	C₂×#106	—	38	2	no	—
207	C₂×#102	—	38	2	no	—
208	C₂×C₄×C₈	—	19	1	no	—
209	C₂×C₄×C₈	—	19	1	no	—
210	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
211	#384	—	19	1	no	—
212	#340	—	38	2	no	—
213	#340	—	38	2	no	—
214	C₂×C₁₉	—	16	2	no	—
215	C₂×C₁₉	#1705	1	1	yes	characteristic
216	C₂×C₁₉	#1714	1	1	yes	characteristic
217	C₂×C₁₉	#1714	1	1	yes	characteristic
218	C₂×C₁₉	#1715	1	1	yes	characteristic
219	C₂×C₁₉	#2013	1	1	yes	characteristic
220	C₂×C₁₉	#2014	1	1	yes	characteristic
221	C₂×C₁₉	#2014	1	1	yes	characteristic
222	C₂²×Q₈	—	2	1	no	—
223	C₂²×Q₈	—	38	1	no	—
224	C₂²×D₄	—	2	1	no	—
225	C₂²×D₄	—	2	1	no	—
226	C₂²×D₄	—	2	1	no	—
227	C₂×SD₁₆	—	2	8	no	—
228	C₂×SD₁₆	—	4	4	no	—
229	C₂×SD₁₆	—	4	4	no	—
230	C₂×SD₁₆	—	38	8	no	—
231	C₂×SD₁₆	—	76	4	no	—
232	C₂×SD₁₆	—	76	4	no	—
233	C₂²×C₈	—	19	2	no	—
234	C₂²×C₈	—	19	2	no	—
235	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
236	C₄⋊Q₈	—	2	1	no	—
237	C₄⋊Q₈	—	2	1	no	—
238	C₄⋊Q₈	—	2	1	no	—
239	C₄⋊Q₈	—	19	2	no	—
240	C₄⋊Q₈	—	38	1	no	—
241	C₄⋊Q₈	—	38	1	no	—
242	C₄⋊Q₈	—	38	1	no	—
243	C₄⋊Q₈	C₂²⋊D₁₉	1	2	yes	—
244	#127	—	2	1	no	—
245	#127	—	2	1	no	—
246	#127	—	2	1	no	—
247	#127	C₂²⋊D₁₉	1	2	yes	—
248	C₂×C₄⋊C₄	—	2	1	no	—
249	C₂×C₄⋊C₄	—	2	1	no	—
250	C₂×C₄⋊C₄	—	38	1	no	—
251	C₂×C₄⋊C₄	—	38	1	no	—
252	C₂×C₄⋊C₄	—	38	1	no	—
253	C₂×C₄⋊C₄	—	38	1	no	—
254	C₂×C₄⋊C₄	—	38	1	no	—
255	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
256	#106	—	38	8	no	—
257	#102	—	76	4	no	—
258	C₄×C₈	—	2	1	no	—
259	C₄×C₈	—	2	1	no	—
260	C₄×C₈	—	38	1	no	—
261	C₄×C₈	—	38	1	no	—
262	C₄×C₈	—	38	1	no	—
263	C₄×C₈	—	38	1	no	—
264	C₁₉	#10315	1	1	yes	characteristic, complemented, Sylow
265	C₂⁴	—	4	1	no	—
266	C₂×Q₈	—	2	2	no	—
267	C₂×Q₈	—	4	1	no	—
268	C₂×Q₈	—	4	2	no	—
269	C₂×Q₈	—	4	2	no	—
270	C₂×Q₈	—	38	2	no	—
271	C₂×Q₈	—	76	1	no	—
272	C₂×Q₈	—	76	2	no	—
273	C₂×Q₈	—	76	2	no	—
274	C₂×D₄	—	2	2	no	—
275	C₂×D₄	—	2	2	no	—
276	C₂×D₄	—	2	2	no	—
277	C₂×D₄	—	4	2	no	—
278	C₂×D₄	—	4	2	no	—
279	C₂×D₄	—	4	1	no	—
280	C₂×D₄	—	4	1	no	—
281	C₂×D₄	—	4	1	no	—
282	C₂×D₄	—	4	2	no	—
283	C₂×D₄	—	4	2	no	—
284	C₂×D₄	—	4	2	no	—
285	C₂×D₄	—	4	2	no	—
286	C₂²×C₄	—	4	1	no	—
287	C₂²×C₄	—	76	1	no	—
288	C₂²×C₄	—	76	1	no	—
289	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
290	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
291	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
292	SD₁₆	—	4	16	no	—
293	SD₁₆	—	76	16	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	19	4	no	—
297	C₂×C₈	—	19	4	no	—
298	C₂×C₈	—	38	2	no	—
299	C₂×C₈	—	38	2	no	—
300	C₂×C₈	—	38	2	no	—
301	C₂×C₈	—	38	2	no	—
302	C₂×C₈	#64526	1	4	yes	complemented
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	—	2	2	no	—
305	C₄⋊C₄	—	4	1	no	—
306	C₄⋊C₄	—	4	1	no	—
307	C₄⋊C₄	—	38	2	no	—
308	C₄⋊C₄	—	38	2	no	—
309	C₄⋊C₄	—	76	1	no	—
310	C₄⋊C₄	—	76	1	no	—
311	C₄⋊C₄	—	76	2	no	—
312	C₄⋊C₄	—	76	2	no	—
313	C₄⋊C₄	—	76	2	no	—
314	C₄²	D₄×D₁₉	1	1	yes	characteristic
315	C₄²	#64543	1	1	yes	characteristic
316	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
317	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
318	C₂³	—	4	2	no	—
319	C₂³	—	8	1	no	—
320	C₂³	—	8	1	no	—
321	C₂³	—	8	1	no	—
322	C₂³	—	8	1	no	—
323	C₂³	—	8	1	no	—
324	C₂³	—	8	1	no	—
325	C₂³	#10597386	1	1	yes	characteristic
326	Q₈	—	4	4	no	—
327	Q₈	—	8	2	no	—
328	Q₈	—	76	4	no	—
329	Q₈	—	152	2	no	—
330	D₄	—	4	4	no	—
331	D₄	—	4	4	no	—
332	D₄	—	4	4	no	—
333	D₄	—	8	2	no	—
334	D₄	—	8	2	no	—
335	D₄	—	8	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	8	1	no	—
338	C₂×C₄	—	8	1	no	—
339	C₂×C₄	—	76	2	no	—
340	C₂×C₄	—	152	1	no	—
341	C₂×C₄	—	152	1	no	—
342	C₂×C₄	—	152	1	no	—
343	C₂×C₄	—	152	1	no	—
344	C₂×C₄	—	152	1	no	—
345	C₂×C₄	SD₁₆×D₁₉	1	2	yes	—
346	C₂×C₄	SD₁₆×D₁₉	1	2	yes	—
347	C₂×C₄	#10597360	1	2	yes	—
348	C₂×C₄	#10597361	1	2	yes	—
349	C₂×C₄	#10597378	1	2	yes	—
350	C₂×C₄	C₂×#64526	1	2	yes	—
351	C₂×C₄	#10597383	1	1	yes	characteristic
352	C₂×C₄	#10597383	1	1	yes	characteristic
353	C₂×C₄	#10597384	1	1	yes	characteristic
354	C₂×C₄	#10597385	1	1	yes	characteristic
355	C₂×C₄	#10597385	1	1	yes	characteristic
356	C₂×C₄	#10597399	1	1	yes	characteristic
357	C₈	—	2	4	no	—
358	C₈	—	38	4	no	—
359	C₈	—	38	4	no	—
360	C₂²	—	8	2	no	—
361	C₂²	—	8	2	no	—
362	C₂²	—	8	2	no	—
363	C₂²	—	16	1	no	—
364	C₂²	—	16	1	no	—
365	C₂²	—	16	1	no	—
366	C₂²	—	16	1	no	—
367	C₂²	#11926452	1	1	yes	characteristic
368	C₂²	#11926915	1	1	yes	characteristic
369	C₂²	#11926915	1	1	yes	characteristic
370	C₂²	#11926916	1	1	yes	characteristic
371	C₂²	#11926930	1	1	yes	characteristic
372	C₂²	#11926931	1	1	yes	characteristic
373	C₂²	#11926935	1	1	yes	characteristic
374	C₄	—	8	2	no	—
375	C₄	—	304	1	no	—
376	C₄	—	152	2	no	—
377	C₄	#11926681	1	2	yes	—
378	C₄	#11926693	1	2	yes	—
379	C₄	#11926695	1	2	yes	—
380	C₄	#11926741	1	2	yes	—
381	C₄	#11926742	1	2	yes	—
382	C₄	#11926747	1	2	yes	—
383	C₂	—	16	2	no	—
384	C₂	#14563221	1	1	yes	characteristic
385	C₂	#14563254	1	1	yes	characteristic
386	C₂	#14563255	1	1	yes	characteristic
387	C₂	#14563262	1	1	yes	characteristic
388	C₂	#14564716	1	1	yes	characteristic
389	C₂	#14564721	1	1	yes	characteristic
390	C₂	#14564723	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597386
commutator, cosocle	C₂×C₄²×C₁₉	C₂³
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#492	C₂
socle	C₂³×C₁₉	#127
prime radical	C₂×C₁₉×#127	C₂²
hypercenter	C₂×#492	D₁₉
hypocenter	C₁₉	#10315
LCS	G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#492

Complements

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

N	H ≅ G/N	type
1. C₂×#11926940	C₂	semidirect
3. #14564597	C₂	semidirect
4. #14564596	C₂	semidirect
6. C₁₉⋊#1273	C₂	semidirect
99. C₂×C₈×C₁₉	SD₁₆	semidirect
120. C₂×C₁₉⋊C₈	SD₁₆	semidirect
121. C₂×C₁₉⋊C₈	SD₁₆	semidirect
264. C₁₉	#10315	semidirect
302. C₂×C₈	#64526	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926940	2
C₂×#11926916	2
#14564597	2
#14564596	2
#14564442	2
C₁₉⋊#1273	2
C₂×C₁₉×#492	2
#10315	19