Subgroups — #18078821

Order 4864.

550 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14573639	C₂	1	1	yes	maximal, characteristic
2	#14566099	C₂	1	1	yes	maximal, characteristic
3	#14562134	C₂	1	1	yes	maximal, characteristic, complemented
4	#14562059	C₂	1	1	yes	maximal, characteristic
5	#14561536	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561468	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1994	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₇₆	C₂²	1	1	yes	characteristic
9	C₂×#10597318	—	2	1	no	—
10	#11926872	—	2	1	no	—
11	#11926831	C₄	1	2	yes	complemented
12	#11926746	C₄	1	2	yes	complemented
13	#11926694	—	2	1	no	—
14	C₄×#64527	—	2	2	no	—
15	#11926597	C₂²	1	1	yes	characteristic
16	#11926597	C₂²	1	1	yes	characteristic
17	#11926446	—	2	1	no	—
18	#11926442	—	2	1	no	—
19	#11926435	—	2	2	no	—
20	C₄×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
21	C₄×(C₂×D₁₉)⋊C₄	—	2	1	no	—
22	#11926281	—	2	2	no	—
23	#11926271	—	2	2	no	—
24	C₁₉×#533	C₂²	1	1	yes	characteristic
25	C₁₉×#423	C₂²	1	1	yes	characteristic
26	C₄³×C₁₉	C₂²	1	1	yes	characteristic
27	C₁₉×#339	—	2	1	no	—
28	C₁₉×#339	—	2	1	no	—
29	C₂²×D₇₆	D₄	1	1	yes	characteristic
30	C₂²×C₄×D₁₉	—	2	1	no	—
31	C₂×M₁₆×C₁₉	—	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	—	2	1	no	—
37	C₄×Q₈×C₁₉	—	2	1	no	—
38	C₄×Q₈×C₁₉	—	2	1	no	—
39	C₁₉×#117	—	2	1	no	—
40	C₂×C₄²×C₁₉	—	2	1	no	—
41	C₂×C₄²×C₁₉	—	2	1	no	—
42	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
45	C₂×#64527	—	2	2	no	—
46	C₂×(C₂×D₁₉)⋊C₄	—	2	1	no	—
47	C₂×(C₂×D₁₉)⋊C₄	—	2	2	no	—
48	C₂×(C₂×D₁₉)⋊C₄	—	2	2	no	—
49	C₂×C₄⋊Dic₁₉	—	2	1	no	—
50	C₂×C₄⋊Dic₁₉	D₄	1	1	yes	characteristic
51	C₂×Dic₁₉⋊C₄	—	2	2	no	—
52	C₂×C₄×Dic₁₉	—	2	1	no	—
53	C₄×D₇₆	—	2	1	no	—
54	C₄×D₇₆	—	2	1	no	—
55	C₄×D₇₆	C₂×C₄	1	2	yes	—
56	C₄×D₇₆	D₄	1	2	yes	—
57	#10597318	—	4	2	no	—
58	#10597318	—	4	2	no	—
59	C₁₉×C₄⋊C₈	—	2	1	no	—
60	C₁₉×C₄⋊C₈	—	2	1	no	—
61	#10597292	—	2	2	no	—
62	C₁₉⋊#95	—	2	2	no	—
63	C₁₉⋊#95	D₄	1	2	yes	—
64	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
65	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
66	#10597266	—	2	2	no	—
67	#10597265	—	2	2	no	—
68	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
69	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
70	C₄×C₁₉⋊C₈	—	2	1	no	—
71	C₄×C₁₉⋊C₈	—	2	1	no	—
72	C₂³×D₁₉	—	2	1	no	—
73	C₂×Q₈×C₁₉	—	2	1	no	—
74	C₂×Q₈×C₁₉	—	2	1	no	—
75	C₂²×C₄×C₁₉	—	2	1	no	—
76	C₂²×C₄×C₁₉	—	2	1	no	—
77	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×Dic₁₉	—	2	1	no	—
81	C₂²×Dic₁₉	—	2	2	no	—
82	C₂×D₇₆	—	2	1	no	—
83	C₂×D₇₆	—	4	2	no	—
84	C₂×D₇₆	C₂²⋊C₄	1	2	yes	—
85	C₂×C₄×D₁₉	—	2	2	no	—
86	C₂×C₄×D₁₉	—	2	2	no	—
87	C₂×C₄×D₁₉	—	4	1	no	—
88	C₂×C₄×D₁₉	—	4	1	no	—
89	C₂×C₄×D₁₉	—	4	1	no	—
90	C₂×C₄×D₁₉	—	4	1	no	—
91	M₁₆×C₁₉	—	4	1	no	—
92	M₁₆×C₁₉	—	4	1	no	—
93	C₂×C₈×C₁₉	—	4	1	no	—
94	C₄⋊C₄×C₁₉	—	2	1	no	—
95	C₄⋊C₄×C₁₉	—	2	1	no	—
96	C₄⋊C₄×C₁₉	—	2	1	no	—
97	C₄⋊C₄×C₁₉	—	2	1	no	—
98	C₄⋊C₄×C₁₉	—	4	1	no	—
99	C₄⋊C₄×C₁₉	—	4	1	no	—
100	C₂²⋊C₄×C₁₉	—	4	1	no	—
101	C₄²×C₁₉	—	2	1	no	—
102	C₄²×C₁₉	—	2	1	no	—
103	C₄²×C₁₉	—	2	1	no	—
104	C₄²×C₁₉	—	4	2	no	—
105	C₄²×C₁₉	—	4	2	no	—
106	C₄²×C₁₉	—	4	1	no	—
107	C₄²×C₁₉	SD₁₆	1	2	yes	complemented
108	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₁₉	#41	1	1	yes	characteristic
111	C₄²×C₁₉	#41	1	1	yes	characteristic
112	#64527	—	4	4	no	—
113	(C₂×D₁₉)⋊C₄	—	2	2	no	—
114	(C₂×D₁₉)⋊C₄	—	2	2	no	—
115	(C₂×D₁₉)⋊C₄	—	4	4	no	—
116	(C₂×D₁₉)⋊C₄	—	4	4	no	—
117	C₄⋊Dic₁₉	—	2	1	no	—
118	C₄⋊Dic₁₉	—	4	2	no	—
119	C₄⋊Dic₁₉	C₂²⋊C₄	1	2	yes	—
120	Dic₁₉⋊C₄	—	4	4	no	—
121	C₄×Dic₁₉	—	4	1	no	—
122	C₄×Dic₁₉	—	4	1	no	—
123	C₂×C₁₉⋊C₈	—	2	2	no	—
124	C₂×C₁₉⋊C₈	—	4	1	no	—
125	#9235	—	19	1	no	maximal, Sylow
126	C₂³×C₁₉	#120	1	1	yes	characteristic
127	C₂²×D₁₉	—	2	2	no	—
128	C₂²×D₁₉	—	4	2	no	—
129	C₂²×D₁₉	—	4	2	no	—
130	C₂²×D₁₉	—	4	1	no	—
131	C₂²×D₁₉	—	4	1	no	—
132	Q₈×C₁₉	—	4	2	no	—
133	Q₈×C₁₉	—	8	1	no	—
134	C₂×C₄×C₁₉	—	2	2	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	1	no	—
139	C₂×C₄×C₁₉	—	4	1	no	—
140	C₂×C₄×C₁₉	—	4	1	no	—
141	C₂×C₄×C₁₉	—	4	1	no	—
142	C₂×C₄×C₁₉	—	4	1	no	—
143	C₂×C₄×C₁₉	—	4	1	no	—
144	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
148	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
149	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
150	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
151	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
152	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
153	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
154	C₂×Dic₁₉	—	2	2	no	—
155	C₂×Dic₁₉	—	4	1	no	—
156	C₂×Dic₁₉	—	4	1	no	—
157	C₂×Dic₁₉	—	4	2	no	—
158	C₂×Dic₁₉	—	4	2	no	—
159	C₂×Dic₁₉	—	4	2	no	—
160	D₇₆	—	2	2	no	—
161	D₇₆	—	2	2	no	—
162	D₇₆	—	4	2	no	—
163	C₄×D₁₉	—	4	2	no	—
164	C₄×D₁₉	—	4	2	no	—
165	C₄×D₁₉	—	4	2	no	—
166	C₄×D₁₉	—	4	2	no	—
167	C₈×C₁₉	—	8	1	no	—
168	C₁₉⋊C₈	—	4	2	no	—
169	C₁₉⋊C₈	—	8	1	no	—
170	#3101	—	19	1	no	—
171	#2293	—	19	1	no	—
172	#1994	D₁₉	1	1	yes	characteristic, complemented
173	#1503	—	19	1	no	—
174	#1486	—	19	1	no	—
175	#1303	—	19	1	no	—
176	#1276	—	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	#450	1	1	yes	characteristic
181	C₂²×C₁₉	#452	1	1	yes	characteristic
182	C₂×D₁₉	—	4	2	no	—
183	C₂×D₁₉	—	4	2	no	—
184	C₂×D₁₉	—	4	2	no	—
185	C₂×D₁₉	—	8	2	no	—
186	C₂×D₁₉	—	8	2	no	—
187	C₄×C₁₉	—	2	1	no	—
188	C₄×C₁₉	—	2	1	no	—
189	C₄×C₁₉	—	2	1	no	—
190	C₄×C₁₉	—	4	2	no	—
191	C₄×C₁₉	—	4	2	no	—
192	C₄×C₁₉	—	8	1	no	—
193	C₄×C₁₉	—	8	1	no	—
194	C₄×C₁₉	#421	1	2	yes	—
195	C₄×C₁₉	C₄×SD₁₆	1	1	yes	characteristic
196	C₄×C₁₉	#440	1	1	yes	characteristic
197	C₄×C₁₉	#461	1	1	yes	characteristic
198	C₄×C₁₉	#463	1	1	yes	characteristic
199	Dic₁₉	—	4	2	no	—
200	Dic₁₉	—	8	2	no	—
201	#533	C₂×D₁₉	1	1	yes	characteristic
202	C₂×C₄×D₄	—	19	1	no	—
203	C₂×#117	—	38	1	no	—
204	#483	—	38	1	no	—
205	#476	—	19	2	no	—
206	#461	—	19	2	no	—
207	#450	—	38	1	no	—
208	C₄×SD₁₆	—	38	2	no	—
209	#431	—	19	1	no	—
210	#431	—	19	1	no	—
211	#423	C₂×D₁₉	1	1	yes	characteristic
212	#390	—	38	1	no	—
213	#389	—	38	1	no	—
214	#388	—	38	2	no	—
215	C₄×C₄⋊C₄	—	19	1	no	—
216	C₄×C₂²⋊C₄	—	38	1	no	—
217	C₄³	C₂×D₁₉	1	1	yes	characteristic
218	#339	—	2	1	no	—
219	#339	—	2	1	no	—
220	#328	—	38	2	no	—
221	D₄⋊C₈	—	38	2	no	—
222	C₂×C₁₉	—	2	1	no	—
223	C₂×C₁₉	—	2	1	no	—
224	C₂×C₁₉	#1877	1	1	yes	characteristic
225	C₂×C₁₉	#1897	1	1	yes	characteristic
226	C₂×C₁₉	#1906	1	1	yes	characteristic
227	D₁₉	—	8	2	no	—
228	C₂²×D₄	—	19	1	no	—
229	C₂³×C₄	—	38	1	no	—
230	C₂×SD₁₆	—	38	2	no	—
231	C₂×M₁₆	—	2	1	no	—
232	C₄⋊Q₈	C₄×D₁₉	1	1	yes	characteristic
233	C₄⋊Q₈	C₄×D₁₉	1	1	yes	characteristic
234	#125	—	2	1	no	—
235	C₂²⋊Q₈	—	2	1	no	—
236	C₂²⋊Q₈	—	2	1	no	—
237	C₄×Q₈	—	2	1	no	—
238	C₄×Q₈	—	2	1	no	—
239	C₄×D₄	—	19	2	no	—
240	C₄×D₄	—	19	2	no	—
241	C₄×D₄	—	38	1	no	—
242	C₄×D₄	—	38	1	no	—
243	#117	—	2	1	no	—
244	#117	—	76	2	no	—
245	#117	—	76	2	no	—
246	C₂×C₄⋊C₄	—	19	1	no	—
247	C₂×C₄⋊C₄	—	38	2	no	—
248	C₂×C₄⋊C₄	—	38	1	no	—
249	C₂×C₂²⋊C₄	—	38	1	no	—
250	C₂×C₂²⋊C₄	—	38	2	no	—
251	C₂×C₂²⋊C₄	—	38	2	no	—
252	C₂×C₄²	—	2	1	no	—
253	C₂×C₄²	—	2	1	no	—
254	C₂×C₄²	—	38	1	no	—
255	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
256	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
257	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
258	#106	—	38	2	no	—
259	C₄⋊C₈	—	2	1	no	—
260	C₄⋊C₈	—	2	1	no	—
261	C₄⋊C₈	—	19	2	no	—
262	C₄⋊C₈	—	38	1	no	—
263	#103	—	38	2	no	—
264	#102	—	38	2	no	—
265	C₂²⋊C₈	—	38	1	no	—
266	C₂²⋊C₈	—	38	1	no	—
267	C₄×C₈	—	38	1	no	—
268	C₄×C₈	—	38	1	no	—
269	#95	—	19	2	no	—
270	#95	—	38	2	no	—
271	C₁₉	#9235	1	1	yes	characteristic, complemented, Sylow
272	C₂⁴	—	38	1	no	—
273	C₂×Q₈	—	2	1	no	—
274	C₂×Q₈	—	2	1	no	—
275	C₂×D₄	—	19	2	no	—
276	C₂×D₄	—	38	1	no	—
277	C₂×D₄	—	76	2	no	—
278	C₂²×C₄	—	2	1	no	—
279	C₂²×C₄	—	2	1	no	—
280	C₂²×C₄	—	38	1	no	—
281	C₂²×C₄	—	38	2	no	—
282	C₂²×C₄	—	38	2	no	—
283	C₂²×C₄	—	38	2	no	—
284	C₂²×C₄	—	76	1	no	—
285	C₂²×C₄	—	76	1	no	—
286	C₂²×C₄	—	76	1	no	—
287	C₂²×C₄	—	76	1	no	—
288	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
289	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
290	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
291	SD₁₆	—	76	4	no	—
292	M₁₆	—	4	1	no	—
293	M₁₆	—	4	1	no	—
294	C₂×C₈	—	4	1	no	—
295	C₂×C₈	—	38	2	no	—
296	C₂×C₈	—	76	1	no	—
297	C₄⋊C₄	—	2	1	no	—
298	C₄⋊C₄	—	2	1	no	—
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	—	4	1	no	—
303	C₄⋊C₄	—	19	2	no	—
304	C₄⋊C₄	—	38	1	no	—
305	C₄⋊C₄	—	76	4	no	—
306	C₄⋊C₄	—	76	2	no	—
307	C₂²⋊C₄	—	4	1	no	—
308	C₂²⋊C₄	—	38	2	no	—
309	C₂²⋊C₄	—	38	2	no	—
310	C₂²⋊C₄	—	76	4	no	—
311	C₂²⋊C₄	—	76	4	no	—
312	C₄²	—	2	1	no	—
313	C₄²	—	2	1	no	—
314	C₄²	—	2	1	no	—
315	C₄²	—	4	2	no	—
316	C₄²	—	4	2	no	—
317	C₄²	—	4	1	no	—
318	C₄²	—	76	1	no	—
319	C₄²	—	76	1	no	—
320	C₄²	#64527	1	2	yes	complemented
321	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
322	C₄²	#64541	1	1	yes	characteristic
323	C₄²	D₄×D₁₉	1	1	yes	characteristic
324	C₄²	#64543	1	1	yes	characteristic
325	C₂³	—	38	2	no	—
326	C₂³	—	76	2	no	—
327	C₂³	—	76	2	no	—
328	C₂³	—	76	1	no	—
329	C₂³	—	76	1	no	—
330	C₂³	#10597383	1	1	yes	characteristic
331	Q₈	—	4	2	no	—
332	Q₈	—	8	1	no	—
333	D₄	—	38	2	no	—
334	D₄	—	38	2	no	—
335	D₄	—	76	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	4	1	no	—
340	C₂×C₄	—	4	1	no	—
341	C₂×C₄	—	4	1	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	4	1	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	38	2	no	—
347	C₂×C₄	—	76	1	no	—
348	C₂×C₄	—	76	1	no	—
349	C₂×C₄	—	76	2	no	—
350	C₂×C₄	—	76	2	no	—
351	C₂×C₄	—	76	2	no	—
352	C₂×C₄	—	76	2	no	—
353	C₂×C₄	—	76	2	no	—
354	C₂×C₄	—	76	2	no	—
355	C₂×C₄	—	76	2	no	—
356	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
357	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
358	C₂×C₄	#10597329	1	1	yes	characteristic
359	C₂×C₄	#10597330	1	1	yes	characteristic
360	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
361	C₂×C₄	#10597384	1	1	yes	characteristic
362	C₂×C₄	C₂×#64527	1	1	yes	characteristic
363	C₂×C₄	#10597388	1	1	yes	characteristic
364	C₂×C₄	#10597395	1	1	yes	characteristic
365	C₂×C₄	#10597396	1	1	yes	characteristic
366	C₈	—	8	1	no	—
367	C₈	—	76	2	no	—
368	C₈	—	152	1	no	—
369	C₂²	—	2	1	no	—
370	C₂²	—	2	1	no	—
371	C₂²	—	76	2	no	—
372	C₂²	—	76	2	no	—
373	C₂²	—	76	2	no	—
374	C₂²	—	152	2	no	—
375	C₂²	—	152	2	no	—
376	C₂²	#11926420	1	1	yes	characteristic
377	C₂²	#11926694	1	1	yes	characteristic
378	C₂²	#11926704	1	1	yes	characteristic
379	C₄	—	2	1	no	—
380	C₄	—	2	1	no	—
381	C₄	—	2	1	no	—
382	C₄	—	4	2	no	—
383	C₄	—	4	2	no	—
384	C₄	—	8	1	no	—
385	C₄	—	8	1	no	—
386	C₄	—	76	2	no	—
387	C₄	—	152	2	no	—
388	C₄	#11926559	1	2	yes	—
389	C₄	C₄×#64527	1	1	yes	characteristic
390	C₄	#11926642	1	1	yes	characteristic
391	C₄	#11926746	1	1	yes	characteristic
392	C₄	#11926756	1	1	yes	characteristic
393	C₂	—	2	1	no	—
394	C₂	—	2	1	no	—
395	C₂	—	152	2	no	—
396	C₂	#14563972	1	1	yes	characteristic
397	C₂	#14564080	1	1	yes	characteristic
398	C₂	#14564139	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597384
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1994	C₂
socle	C₂²×C₁₉	#386
prime radical	C₂²×D₇₆	D₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1994	D₁₉
hypocenter	C₁₉	#9235
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1994

Complements

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

N	H ≅ G/N	type
3. #14562134	C₂	semidirect
5. #14561536	C₂	semidirect
6. #14561468	C₂	semidirect
7. C₁₉×#1994	C₂	semidirect
11. #11926831	C₄	semidirect
12. #11926746	C₄	semidirect
107. C₄²×C₁₉	SD₁₆	semidirect
172. #1994	D₁₉	semidirect
271. C₁₉	#9235	semidirect
320. C₄²	#64527	semidirect

Maximal subgroups

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

subgroup	index
#14573639	2
#14566099	2
#14562134	2
#14562059	2
#14561536	2
#14561468	2
C₁₉×#1994	2
#9235	19