Subgroups — #18076388

Order 4864.

520 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926568	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926294	C₂	1	2	yes	maximal, complemented
4	#14561469	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1797	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	—	2	1	no	—
7	#11926568	—	2	4	no	—
8	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
9	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
10	C₂×C₂²⋊(C₁₉⋊C₈)	C₄	1	2	yes	—
11	C₂×C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
12	#11926299	C₄	1	1	yes	characteristic
13	#11926299	C₄	1	1	yes	characteristic
14	#11926294	—	2	4	no	—
15	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
16	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₁₉×C₂²⋊C₈	—	2	1	no	—
19	C₁₉×#336	—	2	1	no	—
20	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₉	Q₈	1	1	yes	characteristic
23	C₂²×C₈×C₁₉	—	2	1	no	—
24	C₂²×C₈×C₁₉	—	2	1	no	—
25	C₂×C₄²×C₁₉	—	2	2	no	—
26	C₂×C₄²×C₁₉	—	2	2	no	—
27	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₉	Q₈	1	1	yes	characteristic
30	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
31	C₂×#64520	—	2	4	no	—
32	C₂×#64520	—	4	2	no	—
33	C₂×#64520	—	4	2	no	—
34	C₂²×C₁₉⋊C₈	—	2	2	no	—
35	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
36	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
37	C₁₉×C₄⋊C₈	—	2	2	no	—
38	C₁₉×C₄⋊C₈	—	2	2	no	—
39	C₁₉×C₂²⋊C₈	—	4	2	no	—
40	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
41	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
42	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
43	C₄⋊(C₁₉⋊C₈)	D₄	1	4	yes	—
44	C₄×C₁₉⋊C₈	—	2	2	no	—
45	C₄×C₁₉⋊C₈	—	2	2	no	—
46	C₂⁴×C₁₉	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₉	—	2	1	no	—
48	C₂²×C₄×C₁₉	—	2	1	no	—
49	C₂²×C₄×C₁₉	—	2	1	no	—
50	C₂²×C₄×C₁₉	—	2	1	no	—
51	C₂²×C₄×C₁₉	—	2	2	no	—
52	C₂²×C₄×C₁₉	—	2	2	no	—
53	C₂²×C₄×C₁₉	—	2	2	no	—
54	C₂²×C₄×C₁₉	—	2	2	no	—
55	C₂²×C₄×C₁₉	—	2	1	no	—
56	C₂²×C₄×C₁₉	—	2	1	no	—
57	C₂²×C₄×C₁₉	—	4	1	no	—
58	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
62	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
63	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₈×C₁₉	—	4	1	no	—
67	C₂×C₈×C₁₉	—	4	1	no	—
68	C₂×C₈×C₁₉	—	4	1	no	—
69	C₂×C₈×C₁₉	—	4	1	no	—
70	C₂×C₈×C₁₉	—	4	1	no	—
71	C₂×C₈×C₁₉	—	4	1	no	—
72	C₄²×C₁₉	—	2	2	no	—
73	C₄²×C₁₉	—	2	1	no	—
74	C₄²×C₁₉	—	2	2	no	—
75	C₄²×C₁₉	—	2	1	no	—
76	C₄²×C₁₉	C₂×D₄	1	2	yes	—
77	C₄²×C₁₉	#41	1	2	yes	—
78	#64520	—	4	8	no	—
79	C₂×C₁₉⋊C₈	—	2	2	no	—
80	C₂×C₁₉⋊C₈	—	2	2	no	—
81	C₂×C₁₉⋊C₈	—	2	2	no	—
82	C₂×C₁₉⋊C₈	—	2	4	no	—
83	C₂×C₁₉⋊C₈	—	2	2	no	—
84	C₂×C₁₉⋊C₈	—	2	2	no	—
85	C₂×C₁₉⋊C₈	—	4	2	no	—
86	C₂×C₁₉⋊C₈	—	4	2	no	—
87	C₂×C₁₉⋊C₈	M₁₆	1	4	yes	complemented
88	#8833	—	19	1	no	maximal, Sylow
89	C₂³×C₁₉	—	2	1	no	—
90	C₂³×C₁₉	—	2	1	no	—
91	C₂³×C₁₉	—	2	1	no	—
92	C₂³×C₁₉	—	2	1	no	—
93	C₂³×C₁₉	—	2	1	no	—
94	C₂³×C₁₉	—	2	1	no	—
95	C₂³×C₁₉	#106	1	2	yes	—
96	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₉	—	2	1	no	—
98	C₂×C₄×C₁₉	—	2	1	no	—
99	C₂×C₄×C₁₉	—	2	1	no	—
100	C₂×C₄×C₁₉	—	2	1	no	—
101	C₂×C₄×C₁₉	—	2	1	no	—
102	C₂×C₄×C₁₉	—	2	1	no	—
103	C₂×C₄×C₁₉	—	2	2	no	—
104	C₂×C₄×C₁₉	—	2	2	no	—
105	C₂×C₄×C₁₉	—	2	2	no	—
106	C₂×C₄×C₁₉	—	2	2	no	—
107	C₂×C₄×C₁₉	—	2	1	no	—
108	C₂×C₄×C₁₉	—	2	1	no	—
109	C₂×C₄×C₁₉	—	2	1	no	—
110	C₂×C₄×C₁₉	—	4	4	no	—
111	C₂×C₄×C₁₉	—	4	2	no	—
112	C₂×C₄×C₁₉	—	4	2	no	—
113	C₂×C₄×C₁₉	#106	1	2	yes	—
114	C₂×C₄×C₁₉	#108	1	2	yes	—
115	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
117	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
118	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
119	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
120	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
121	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
126	C₈×C₁₉	—	8	1	no	—
127	C₈×C₁₉	—	8	1	no	—
128	C₁₉⋊C₈	—	2	4	no	—
129	C₁₉⋊C₈	—	4	2	no	—
130	C₁₉⋊C₈	—	4	4	no	—
131	C₂×#423	—	19	1	no	—
132	#1797	D₁₉	1	1	yes	characteristic
133	C₄×C₂²⋊C₈	—	19	1	no	—
134	C₂×#334	—	19	2	no	—
135	#1277	—	19	2	no	—
136	C₂²×C₁₉	—	2	2	no	—
137	C₂²×C₁₉	—	2	2	no	—
138	C₂²×C₁₉	—	2	2	no	—
139	C₂²×C₁₉	—	4	1	no	—
140	C₂²×C₁₉	—	4	1	no	—
141	C₂²×C₁₉	—	4	1	no	—
142	C₂²×C₁₉	—	4	1	no	—
143	C₂²×C₁₉	#385	1	1	yes	characteristic
144	C₂²×C₁₉	#423	1	1	yes	characteristic
145	C₂²×C₁₉	#424	1	1	yes	characteristic
146	C₂²×C₁₉	C₂×#106	1	1	yes	characteristic
147	C₂²×C₁₉	#428	1	1	yes	characteristic
148	C₂²×C₁₉	C₂×#108	1	1	yes	characteristic
149	C₂²×C₁₉	#430	1	1	yes	characteristic
150	C₄×C₁₉	—	2	2	no	—
151	C₄×C₁₉	—	2	2	no	—
152	C₄×C₁₉	—	2	2	no	—
153	C₄×C₁₉	—	4	1	no	—
154	C₄×C₁₉	#435	1	2	yes	—
155	C₄×C₁₉	#436	1	2	yes	—
156	C₄×C₁₉	#448	1	1	yes	characteristic
157	C₄×C₁₉	#450	1	1	yes	characteristic
158	C₄×C₁₉	#481	1	1	yes	characteristic
159	C₄×C₁₉	#483	1	1	yes	characteristic
160	C₂²×M₁₆	—	38	1	no	—
161	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
162	#423	—	38	4	no	—
163	C₂×C₄⋊C₈	—	19	2	no	—
164	C₂×C₄⋊C₈	—	38	1	no	—
165	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
166	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
167	C₂×C₂²⋊C₈	—	2	1	no	—
168	C₂×C₂²⋊C₈	—	19	2	no	—
169	C₂×C₄×C₈	—	19	2	no	—
170	#336	—	2	1	no	—
171	#336	—	19	1	no	—
172	#336	—	19	1	no	—
173	#334	—	38	4	no	—
174	C₂×C₁₉	—	4	2	no	—
175	C₂×C₁₉	#1566	1	1	yes	characteristic
176	C₂×C₁₉	#1569	1	1	yes	characteristic
177	C₂×C₁₉	#1570	1	1	yes	characteristic
178	C₂×C₁₉	#1574	1	1	yes	characteristic
179	C₂×C₁₉	#1851	1	1	yes	characteristic
180	C₂×C₁₉	#1853	1	1	yes	characteristic
181	C₂×C₁₉	#1855	1	1	yes	characteristic
182	C₂³×C₄	Dic₃₈	1	1	yes	characteristic
183	C₂³×C₄	C₄×D₁₉	1	1	yes	characteristic
184	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
185	C₂×M₁₆	—	38	4	no	—
186	C₂×M₁₆	—	76	2	no	—
187	C₂×M₁₆	—	76	2	no	—
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	—	19	2	no	—
191	C₂²×C₈	—	19	2	no	—
192	C₂²×C₈	—	38	2	no	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	Dic₃₈	1	1	yes	characteristic
196	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
197	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
198	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
199	C₄⋊C₈	—	2	2	no	—
200	C₄⋊C₈	—	2	2	no	—
201	C₄⋊C₈	—	19	4	no	—
202	C₄⋊C₈	—	38	4	no	—
203	C₄⋊C₈	—	38	2	no	—
204	C₂²⋊C₈	—	4	2	no	—
205	C₂²⋊C₈	—	38	4	no	—
206	C₄×C₈	—	38	2	no	—
207	C₄×C₈	—	38	2	no	—
208	C₁₉	#8833	1	1	yes	characteristic, complemented, Sylow
209	C₂⁴	Dic₁₉⋊C₄	1	1	yes	characteristic
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	4	1	no	—
221	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
222	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
223	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
224	C₂²×C₄	#64526	1	2	yes	—
225	C₂²×C₄	#64527	1	2	yes	—
226	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
229	M₁₆	—	76	8	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	4	1	no	—
234	C₂×C₈	—	4	1	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	—	19	4	no	—
237	C₂×C₈	—	38	2	no	—
238	C₂×C₈	—	38	2	no	—
239	C₂×C₈	—	38	4	no	—
240	C₂×C₈	—	38	2	no	—
241	C₂×C₈	—	38	2	no	—
242	C₂×C₈	—	38	2	no	—
243	C₂×C₈	—	76	2	no	—
244	C₂×C₈	—	76	2	no	—
245	C₄²	—	2	1	no	—
246	C₄²	—	2	1	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	2	no	—
249	C₄²	D₄×D₁₉	1	2	yes	—
250	C₄²	#64543	1	2	yes	—
251	C₂³	—	2	1	no	—
252	C₂³	—	2	1	no	—
253	C₂³	—	2	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	—	2	1	no	—
256	C₂³	—	2	1	no	—
257	C₂³	#10597265	1	2	yes	—
258	C₂³	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	#10597265	1	2	yes	—
276	C₂×C₄	#10597279	1	2	yes	—
277	C₂×C₄	#10597324	1	2	yes	—
278	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
279	C₂×C₄	#10597347	1	1	yes	characteristic
280	C₂×C₄	#10597347	1	1	yes	characteristic
281	C₂×C₄	M₁₆×D₁₉	1	2	yes	—
282	C₂×C₄	C₂×#64526	1	1	yes	characteristic
283	C₂×C₄	#10597381	1	1	yes	characteristic
284	C₂×C₄	#10597383	1	1	yes	characteristic
285	C₂×C₄	C₂×#64527	1	1	yes	characteristic
286	C₂×C₄	#10597395	1	1	yes	characteristic
287	C₂×C₄	#10597397	1	1	yes	characteristic
288	C₈	—	8	1	no	—
289	C₈	—	8	1	no	—
290	C₈	—	38	4	no	—
291	C₈	—	76	2	no	—
292	C₈	—	76	4	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	4	1	no	—
297	C₂²	—	4	1	no	—
298	C₂²	—	4	1	no	—
299	C₂²	—	4	1	no	—
300	C₂²	#11926416	1	1	yes	characteristic
301	C₂²	#11926567	1	1	yes	characteristic
302	C₂²	#11926570	1	1	yes	characteristic
303	C₂²	C₂×#10597265	1	1	yes	characteristic
304	C₂²	#11926587	1	1	yes	characteristic
305	C₂²	C₂×#10597279	1	1	yes	characteristic
306	C₂²	#11926592	1	1	yes	characteristic
307	C₄	—	2	2	no	—
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	4	1	no	—
311	C₄	#11926618	1	2	yes	—
312	C₄	#11926619	1	2	yes	—
313	C₄	#11926682	1	1	yes	characteristic
314	C₄	#11926694	1	1	yes	characteristic
315	C₄	#11926860	1	1	yes	characteristic
316	C₄	#11926872	1	1	yes	characteristic
317	C₂	—	4	2	no	—
318	C₂	#14562390	1	1	yes	characteristic
319	C₂	#14562403	1	1	yes	characteristic
320	C₂	#14562405	1	1	yes	characteristic
321	C₂	#14562423	1	1	yes	characteristic
322	C₂	#14563817	1	1	yes	characteristic
323	C₂	#14563827	1	1	yes	characteristic
324	C₂	#14563839	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1797	C₂
socle	C₂³×C₁₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₉	C₄⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1797	D₁₉
hypocenter	C₁₉	#8833
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #1797

Complements

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

N	H ≅ G/N	type
3. C₂×#11926294	C₂	semidirect
4. #14561469	C₂	semidirect
87. C₂×C₁₉⋊C₈	M₁₆	semidirect
208. C₁₉	#8833	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926568	2
C₄×C₂²⋊(C₁₉⋊C₈)	2
C₂×#11926294	2
#14561469	2
C₁₉×#1797	2
#8833	19