Subgroups — #18072009

Order 4864.

662 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926600	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926566	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#64515	C₂	1	1	yes	maximal, characteristic
4	#14561470	C₂	1	1	yes	maximal, characteristic, complemented
5	#14561469	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561469	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1277	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#64515	—	2	1	no	—
9	C₂²×#64515	C₄	1	2	yes	—
10	C₂×C₄²×D₁₉	C₂²	1	1	yes	characteristic
11	#11926600	—	2	2	no	—
12	#11926600	C₄	1	4	yes	—
13	#11926566	—	2	2	no	—
14	#11926566	C₄	1	4	yes	—
15	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₉⋊C₈	—	2	1	no	—
18	C₂×(C₂×D₁₉)⋊C₈	—	2	1	no	—
19	C₄×#64515	—	2	2	no	—
20	C₄×#64515	—	2	2	no	—
21	C₂×#10597272	—	2	1	no	—
22	C₂×#10597272	C₄	1	2	yes	—
23	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
27	C₂²×C₄×D₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×D₁₉	D₄	1	1	yes	characteristic
29	C₂²×C₄×D₁₉	Q₈	1	1	yes	characteristic
30	C₂²×C₈×C₁₉	—	2	1	no	—
31	C₂²×C₈×C₁₉	—	2	1	no	—
32	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
34	C₂×C₄×Dic₁₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×Dic₁₉	D₄	1	1	yes	characteristic
36	C₂×C₄×Dic₁₉	Q₈	1	1	yes	characteristic
37	C₂²×C₁₉⋊C₈	—	2	1	no	—
38	C₂²×C₁₉⋊C₈	—	2	1	no	—
39	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
40	C₂×#64515	—	2	4	no	—
41	C₂×#64515	—	2	4	no	—
42	C₂×#64515	—	2	4	no	—
43	C₂×#64515	—	2	4	no	—
44	C₂×#64515	—	4	2	no	—
45	C₂×#64515	—	4	2	no	—
46	C₄²×D₁₉	C₂×C₄	1	2	yes	—
47	C₄²×D₁₉	C₂×C₄	1	2	yes	—
48	C₄²×D₁₉	D₄	1	2	yes	—
49	C₄²×D₁₉	D₄	1	2	yes	—
50	C₁₉×C₄⋊C₈	—	2	1	no	—
51	C₁₉×C₄⋊C₈	—	2	1	no	—
52	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
53	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
54	C₄×C₈×C₁₉	—	2	1	no	—
55	C₄×C₈×C₁₉	—	2	1	no	—
56	(C₂×D₁₉)⋊C₈	—	2	4	no	—
57	#10597272	—	2	4	no	—
58	#10597272	—	2	4	no	—
59	Dic₁₉⋊C₈	—	2	4	no	—
60	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
61	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
62	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
63	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
64	C₄×C₁₉⋊C₈	—	2	1	no	—
65	C₄×C₁₉⋊C₈	—	2	1	no	—
66	C₂³×D₁₉	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
70	C₂²×Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
72	C₂²×Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×D₁₉	—	2	2	no	—
74	C₂×C₄×D₁₉	—	2	2	no	—
75	C₂×C₄×D₁₉	—	2	1	no	—
76	C₂×C₄×D₁₉	—	2	1	no	—
77	C₂×C₄×D₁₉	—	2	1	no	—
78	C₂×C₄×D₁₉	—	2	1	no	—
79	C₂×C₄×D₁₉	—	4	2	no	—
80	C₂×C₄×D₁₉	C₄²	1	2	yes	—
81	C₂×C₄×D₁₉	C₂²⋊C₄	1	2	yes	—
82	C₂×C₄×D₁₉	C₂²⋊C₄	1	2	yes	—
83	C₂×C₄×D₁₉	C₂²⋊C₄	1	2	yes	—
84	C₂×C₄×D₁₉	C₄⋊C₄	1	2	yes	—
85	C₂×C₄×D₁₉	C₄⋊C₄	1	2	yes	—
86	C₂×C₈×C₁₉	—	2	2	no	—
87	C₂×C₈×C₁₉	—	2	2	no	—
88	C₂×C₈×C₁₉	—	2	2	no	—
89	C₂×C₈×C₁₉	—	2	2	no	—
90	C₂×C₈×C₁₉	—	4	1	no	—
91	C₂×C₈×C₁₉	—	4	1	no	—
92	C₂×C₈×C₁₉	—	4	1	no	—
93	C₂×C₈×C₁₉	—	4	1	no	—
94	C₂×C₈×C₁₉	M₁₆	1	4	yes	complemented
95	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
99	C₄×Dic₁₉	C₄²	1	2	yes	—
100	C₄×Dic₁₉	C₂²⋊C₄	1	2	yes	—
101	C₄×Dic₁₉	C₂²⋊C₄	1	2	yes	—
102	C₄×Dic₁₉	C₂²⋊C₄	1	2	yes	—
103	C₄×Dic₁₉	C₄⋊C₄	1	2	yes	—
104	C₄×Dic₁₉	C₄⋊C₄	1	2	yes	—
105	C₂×C₁₉⋊C₈	—	2	2	no	—
106	C₂×C₁₉⋊C₈	—	2	2	no	—
107	C₂×C₁₉⋊C₈	—	2	2	no	—
108	C₂×C₁₉⋊C₈	—	2	2	no	—
109	C₂×C₁₉⋊C₈	—	4	1	no	—
110	C₂×C₁₉⋊C₈	—	4	1	no	—
111	C₂×C₁₉⋊C₈	—	4	1	no	—
112	C₂×C₁₉⋊C₈	—	4	1	no	—
113	C₂×C₁₉⋊C₈	M₁₆	1	4	yes	complemented
114	#64515	—	2	16	no	—
115	#64515	—	4	8	no	—
116	#7876	—	19	1	no	maximal, Sylow
117	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂²×D₁₉	—	2	2	no	—
119	C₂²×D₁₉	—	2	1	no	—
120	C₂²×D₁₉	—	2	1	no	—
121	C₂²×D₁₉	—	2	2	no	—
122	C₂²×D₁₉	#95	1	2	yes	—
123	C₂×C₄×C₁₉	—	2	2	no	—
124	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
131	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
132	C₂×C₄×C₁₉	AGL(1,ℤ/8)	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	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
136	C₂×Dic₁₉	—	2	2	no	—
137	C₂×Dic₁₉	—	2	1	no	—
138	C₂×Dic₁₉	—	2	1	no	—
139	C₂×Dic₁₉	—	2	2	no	—
140	C₂×Dic₁₉	#95	1	2	yes	—
141	C₂×Dic₁₉	#95	1	2	yes	—
142	C₂×Dic₁₉	#95	1	2	yes	—
143	C₄×D₁₉	—	2	2	no	—
144	C₄×D₁₉	—	2	2	no	—
145	C₄×D₁₉	—	2	4	no	—
146	C₄×D₁₉	—	2	2	no	—
147	C₄×D₁₉	—	2	2	no	—
148	C₄×D₁₉	—	2	2	no	—
149	C₄×D₁₉	—	2	2	no	—
150	C₄×D₁₉	—	4	2	no	—
151	C₄×D₁₉	—	4	2	no	—
152	C₈×C₁₉	—	2	4	no	—
153	C₈×C₁₉	—	4	2	no	—
154	C₈×C₁₉	—	4	2	no	—
155	C₁₉⋊C₈	—	2	4	no	—
156	C₁₉⋊C₈	—	4	2	no	—
157	C₁₉⋊C₈	—	4	2	no	—
158	C₂×#432	—	19	1	no	—
159	C₂×#423	—	19	1	no	—
160	C₂×C₄×M₁₆	—	19	1	no	—
161	#1277	—	19	1	no	—
162	#1277	—	19	1	no	—
163	#1277	—	19	1	no	—
164	#1277	D₁₉	1	1	yes	characteristic, complemented
165	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
166	C₂²×C₁₉	#423	1	2	yes	—
167	C₂²×C₁₉	#428	1	1	yes	characteristic
168	C₂²×C₁₉	#428	1	1	yes	characteristic
169	C₂²×C₁₉	#430	1	2	yes	—
170	C₂×D₁₉	—	2	2	no	—
171	C₂×D₁₉	—	2	2	no	—
172	C₂×D₁₉	—	2	2	no	—
173	C₂×D₁₉	—	4	2	no	—
174	C₂×D₁₉	—	4	2	no	—
175	C₄×C₁₉	—	2	1	no	—
176	C₄×C₁₉	—	2	1	no	—
177	C₄×C₁₉	C₄×M₁₆	1	1	yes	characteristic
178	C₄×C₁₉	C₄×M₁₆	1	1	yes	characteristic
179	C₄×C₁₉	#407	1	2	yes	—
180	C₄×C₁₉	#417	1	1	yes	characteristic
181	C₄×C₁₉	#417	1	1	yes	characteristic
182	C₄×C₁₉	#418	1	1	yes	characteristic
183	C₄×C₁₉	#419	1	1	yes	characteristic
184	Dic₁₉	—	2	2	no	—
185	Dic₁₉	—	2	2	no	—
186	Dic₁₉	—	2	2	no	—
187	C₂²×M₁₆	—	19	2	no	—
188	C₂²×M₁₆	—	38	1	no	—
189	C₂²×C₄²	—	19	1	no	—
190	#432	—	19	4	no	—
191	#432	—	38	2	no	—
192	#423	—	19	4	no	—
193	#423	—	38	2	no	—
194	C₂×C₄⋊C₈	—	19	1	no	—
195	C₂×C₄⋊C₈	—	19	1	no	—
196	C₂×C₄⋊C₈	—	38	1	no	—
197	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
198	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
199	C₂×C₂²⋊C₈	—	38	1	no	—
200	C₄×M₁₆	—	38	2	no	—
201	C₄×M₁₆	—	38	2	no	—
202	C₂×#97	—	19	2	no	—
203	C₂×#97	—	38	1	no	—
204	C₂×C₄×C₈	—	19	1	no	—
205	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
206	C₂×C₁₉	#1569	1	2	yes	—
207	C₂×C₁₉	#1569	1	2	yes	—
208	C₂×C₁₉	#1728	1	1	yes	characteristic
209	C₂×C₁₉	#1736	1	1	yes	characteristic
210	C₂×C₁₉	#1738	1	1	yes	characteristic
211	D₁₉	—	4	2	no	—
212	C₂³×C₄	—	19	1	no	—
213	C₂³×C₄	—	19	1	no	—
214	C₂³×C₄	—	19	1	no	—
215	C₂×M₁₆	—	38	4	no	—
216	C₂×M₁₆	—	38	4	no	—
217	C₂×M₁₆	—	38	4	no	—
218	C₂×M₁₆	—	38	4	no	—
219	C₂×M₁₆	—	76	2	no	—
220	C₂×M₁₆	—	76	2	no	—
221	C₂²×C₈	—	2	1	no	—
222	C₂²×C₈	—	2	1	no	—
223	C₂²×C₈	—	19	2	no	—
224	C₂²×C₈	—	38	1	no	—
225	C₂²×C₈	—	38	1	no	—
226	C₂²×C₈	C₄×D₁₉	1	2	yes	—
227	C₂×C₄²	—	19	1	no	—
228	C₂×C₄²	—	19	1	no	—
229	C₂×C₄²	—	19	1	no	—
230	C₂×C₄²	—	19	2	no	—
231	C₂×C₄²	—	19	2	no	—
232	C₂×C₄²	—	19	2	no	—
233	C₂×C₄²	—	19	2	no	—
234	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
235	C₄⋊C₈	—	2	1	no	—
236	C₄⋊C₈	—	2	1	no	—
237	C₄⋊C₈	—	19	2	no	—
238	C₄⋊C₈	—	19	2	no	—
239	C₄⋊C₈	—	38	4	no	—
240	C₄⋊C₈	—	38	1	no	—
241	C₄⋊C₈	—	38	1	no	—
242	C₄⋊C₈	C₄×D₁₉	1	2	yes	—
243	C₄⋊C₈	C₄×D₁₉	1	2	yes	—
244	C₂²⋊C₈	—	38	4	no	—
245	#97	—	38	4	no	—
246	#97	—	38	4	no	—
247	C₄×C₈	—	2	1	no	—
248	C₄×C₈	—	2	1	no	—
249	C₄×C₈	—	38	1	no	—
250	C₄×C₈	—	38	1	no	—
251	C₁₉	#7876	1	1	yes	characteristic, complemented, Sylow
252	C₂⁴	—	19	1	no	—
253	C₂²×C₄	—	19	1	no	—
254	C₂²×C₄	—	19	1	no	—
255	C₂²×C₄	—	19	1	no	—
256	C₂²×C₄	—	19	2	no	—
257	C₂²×C₄	—	19	2	no	—
258	C₂²×C₄	—	19	2	no	—
259	C₂²×C₄	—	19	2	no	—
260	C₂²×C₄	—	19	2	no	—
261	C₂²×C₄	—	19	2	no	—
262	C₂²×C₄	—	38	2	no	—
263	C₂²×C₄	—	38	2	no	—
264	C₂²×C₄	—	38	1	no	—
265	C₂²×C₄	—	38	1	no	—
266	C₂²×C₄	—	38	1	no	—
267	C₂²×C₄	—	38	1	no	—
268	C₂²×C₄	—	76	2	no	—
269	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
270	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
271	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
272	M₁₆	—	38	16	no	—
273	M₁₆	—	76	8	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	4	1	no	—
279	C₂×C₈	—	4	1	no	—
280	C₂×C₈	—	4	1	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	—	19	4	no	—
283	C₂×C₈	—	38	2	no	—
284	C₂×C₈	—	38	2	no	—
285	C₂×C₈	—	38	2	no	—
286	C₂×C₈	—	38	2	no	—
287	C₂×C₈	—	76	1	no	—
288	C₂×C₈	—	76	1	no	—
289	C₂×C₈	—	76	1	no	—
290	C₂×C₈	—	76	1	no	—
291	C₂×C₈	#64515	1	4	yes	complemented
292	C₄²	—	19	2	no	—
293	C₄²	—	19	2	no	—
294	C₄²	—	19	2	no	—
295	C₄²	—	19	2	no	—
296	C₄²	—	19	2	no	—
297	C₄²	—	19	2	no	—
298	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
299	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
300	C₄²	D₄×D₁₉	1	1	yes	characteristic
301	C₄²	D₄×D₁₉	1	1	yes	characteristic
302	C₂³	—	19	2	no	—
303	C₂³	—	38	2	no	—
304	C₂³	—	38	1	no	—
305	C₂³	—	38	1	no	—
306	C₂³	—	38	2	no	—
307	C₂³	C₄⋊C₄×D₁₉	1	1	yes	characteristic
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	19	2	no	—
310	C₂×C₄	—	19	2	no	—
311	C₂×C₄	—	19	2	no	—
312	C₂×C₄	—	38	2	no	—
313	C₂×C₄	—	38	1	no	—
314	C₂×C₄	—	38	1	no	—
315	C₂×C₄	—	38	2	no	—
316	C₂×C₄	—	38	2	no	—
317	C₂×C₄	—	38	2	no	—
318	C₂×C₄	—	38	2	no	—
319	C₂×C₄	—	38	2	no	—
320	C₂×C₄	—	38	2	no	—
321	C₂×C₄	—	38	2	no	—
322	C₂×C₄	—	38	4	no	—
323	C₂×C₄	—	76	2	no	—
324	C₂×C₄	—	76	2	no	—
325	C₂×C₄	C₄²×D₁₉	1	1	yes	characteristic
326	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
327	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
328	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
329	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
330	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
331	C₂×C₄	C₂×#64515	1	2	yes	—
332	C₂×C₄	M₁₆×D₁₉	1	2	yes	—
333	C₂×C₄	#10597357	1	1	yes	characteristic
334	C₂×C₄	#10597360	1	1	yes	characteristic
335	C₂×C₄	#10597361	1	1	yes	characteristic
336	C₂×C₄	#10597364	1	1	yes	characteristic
337	C₈	—	2	4	no	—
338	C₈	—	4	2	no	—
339	C₈	—	4	2	no	—
340	C₈	—	38	4	no	—
341	C₈	—	76	2	no	—
342	C₈	—	76	2	no	—
343	C₂²	—	38	2	no	—
344	C₂²	—	38	2	no	—
345	C₂²	—	38	2	no	—
346	C₂²	—	76	2	no	—
347	C₂²	—	76	2	no	—
348	C₂²	#95×D₁₉	1	1	yes	characteristic
349	C₂²	#11926566	1	2	yes	—
350	C₂²	#11926583	1	1	yes	characteristic
351	C₂²	#11926585	1	1	yes	characteristic
352	C₂²	#11926591	1	2	yes	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	38	2	no	—
356	C₄	—	38	2	no	—
357	C₄	—	38	2	no	—
358	C₄	C₄×#64515	1	1	yes	characteristic
359	C₄	#11926496	1	1	yes	characteristic
360	C₄	#11926506	1	2	yes	—
361	C₄	#11926542	1	1	yes	characteristic
362	C₄	#11926544	1	1	yes	characteristic
363	C₄	#11926547	1	1	yes	characteristic
364	C₄	#11926551	1	1	yes	characteristic
365	C₂	—	76	2	no	—
366	C₂	#14562399	1	2	yes	—
367	C₂	#14562401	1	2	yes	—
368	C₂	#14563327	1	1	yes	characteristic
369	C₂	#14563353	1	1	yes	characteristic
370	C₂	#14563360	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1277	C₂
socle	C₂³×C₁₉	C₂×C₄⋊C₄
prime radical	C₂³×D₁₉	C₄⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1277	D₁₉
hypocenter	C₁₉	#7876
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
4. #14561470	C₂	semidirect
5. #14561469	C₂	semidirect
6. #14561469	C₂	semidirect
7. C₁₉×#1277	C₂	semidirect
94. C₂×C₈×C₁₉	M₁₆	semidirect
113. C₂×C₁₉⋊C₈	M₁₆	semidirect
164. #1277	D₁₉	semidirect
251. C₁₉	#7876	semidirect
291. C₂×C₈	#64515	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926600	2
C₂×#11926566	2
C₂×C₄×#64515	2
#14561470	2
#14561469	2
C₁₉×#1277	2
#7876	19