Subgroups — #18078809

Order 4864.

604 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14573666	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×Dic₁₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#14564462	C₂	1	1	yes	maximal, characteristic
4	#14562131	C₂	1	1	yes	maximal, characteristic
5	#14561536	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561467	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1487	C₂	1	1	yes	maximal, characteristic, complemented
8	#11927094	C₂²	1	1	yes	characteristic
9	C₂×Q₈×Dic₁₉	—	2	1	no	—
10	#11926862	—	2	1	no	—
11	#11926829	C₄	1	2	yes	complemented
12	#11926744	C₄	1	2	yes	complemented
13	#11926683	—	2	1	no	—
14	C₄×#64527	—	2	2	no	—
15	#11926597	C₂²	1	1	yes	characteristic
16	C₂×#10597282	—	2	1	no	—
17	C₂×#10597282	—	2	1	no	—
18	C₄⋊C₄×Dic₁₉	—	2	2	no	—
19	C₄⋊C₄×Dic₁₉	—	2	1	no	—
20	C₄×C₄⋊Dic₁₉	—	2	1	no	—
21	C₄×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
22	C₄²×Dic₁₉	—	2	1	no	—
23	C₄²×Dic₁₉	C₂²	1	1	yes	characteristic
24	#11926279	—	2	2	no	—
25	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
26	C₁₉×#431	C₂²	1	1	yes	characteristic
27	C₁₉×#423	C₂²	1	1	yes	characteristic
28	C₁₉×Q₈⋊C₈	—	2	2	no	—
29	C₂²×Q₈×C₁₉	D₄	1	1	yes	characteristic
30	C₂×#64541	—	2	1	no	—
31	C₂×M₁₆×C₁₉	—	2	1	no	—
32	C₄×Q₈×C₁₉	—	2	1	no	—
33	C₄×Q₈×C₁₉	—	2	1	no	—
34	C₄×Q₈×C₁₉	C₂×C₄	1	2	yes	—
35	C₄×Q₈×C₁₉	D₄	1	2	yes	—
36	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
37	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₉	—	2	1	no	—
39	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
40	Q₈×Dic₁₉	—	2	8	no	—
41	C₂×#64527	—	2	2	no	—
42	#10597376	—	2	1	no	—
43	#10597376	—	2	1	no	—
44	C₂×C₄⋊Dic₁₉	—	2	2	no	—
45	C₂×C₄⋊Dic₁₉	—	2	1	no	—
46	C₂×C₄×Dic₁₉	—	2	2	no	—
47	C₂×C₄×Dic₁₉	—	2	1	no	—
48	C₂×C₄×Dic₁₉	—	2	2	no	—
49	C₂×C₄×Dic₁₉	—	2	1	no	—
50	C₂×C₄×Dic₁₉	—	2	1	no	—
51	C₂×C₄×Dic₁₉	D₄	1	1	yes	characteristic
52	C₂×C₄×Dic₁₉	D₄	1	1	yes	characteristic
53	#10597321	—	2	1	no	—
54	#10597320	C₂×C₄	1	1	yes	characteristic
55	C₄×D₇₆	—	2	1	no	—
56	C₄×D₇₆	—	2	1	no	—
57	#10597315	C₂×C₄	1	1	yes	characteristic
58	C₁₉×C₄⋊C₈	—	2	1	no	—
59	C₁₉×C₄⋊C₈	—	2	1	no	—
60	C₁₉×C₄⋊C₈	—	2	1	no	—
61	C₁₉×C₂²⋊C₈	—	2	1	no	—
62	C₄×C₈×C₁₉	—	2	1	no	—
63	#10597292	—	2	2	no	—
64	C₁₉⋊#95	—	2	2	no	—
65	C₁₉⋊#95	D₄	1	2	yes	—
66	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
67	#10597282	—	4	2	no	—
68	#10597282	—	4	2	no	—
69	#10597266	—	2	2	no	—
70	#10597265	—	2	2	no	—
71	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
72	C₄×C₁₉⋊C₈	—	2	1	no	—
73	C₂×Q₈×C₁₉	—	2	4	no	—
74	C₂×Q₈×C₁₉	—	2	1	no	—
75	C₂×Q₈×C₁₉	C₂²⋊C₄	1	2	yes	—
76	C₂²×C₄×C₁₉	—	2	1	no	—
77	C₂²×C₄×C₁₉	—	2	1	no	—
78	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₂²⋊D₁₉	—	4	1	no	—
82	C₂²×Dic₁₉	—	2	1	no	—
83	C₂²×Dic₁₉	—	2	1	no	—
84	C₂²×Dic₁₉	—	2	2	no	—
85	#64541	—	4	1	no	—
86	#64541	—	4	1	no	—
87	#64541	—	4	1	no	—
88	#64541	—	4	1	no	—
89	C₂×D₇₆	—	2	1	no	—
90	C₂×D₇₆	—	2	1	no	—
91	C₂×D₇₆	—	2	1	no	—
92	C₂×C₄×D₁₉	—	4	1	no	—
93	C₂×Dic₃₈	—	2	1	no	—
94	M₁₆×C₁₉	—	4	1	no	—
95	M₁₆×C₁₉	—	4	1	no	—
96	C₂×C₈×C₁₉	—	4	1	no	—
97	C₂×C₈×C₁₉	—	4	1	no	—
98	C₄⋊C₄×C₁₉	—	2	2	no	—
99	C₄⋊C₄×C₁₉	—	2	2	no	—
100	C₄⋊C₄×C₁₉	—	2	1	no	—
101	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
102	C₄²×C₁₉	—	2	2	no	—
103	C₄²×C₁₉	—	2	2	no	—
104	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₁₉	#41	1	1	yes	characteristic
107	C₄²×C₁₉	#41	1	1	yes	characteristic
108	#64527	—	4	4	no	—
109	(C₂×D₁₉)⋊C₄	—	4	1	no	—
110	(C₂×D₁₉)⋊C₄	—	4	1	no	—
111	C₄⋊Dic₁₉	—	2	8	no	—
112	C₄⋊Dic₁₉	—	2	4	no	—
113	C₄⋊Dic₁₉	—	2	1	no	—
114	C₄⋊Dic₁₉	—	2	1	no	—
115	C₄×Dic₁₉	—	2	8	no	—
116	C₄×Dic₁₉	—	2	4	no	—
117	C₄×Dic₁₉	—	2	1	no	—
118	C₄×Dic₁₉	—	2	1	no	—
119	C₄×Dic₁₉	—	2	1	no	—
120	C₄×Dic₁₉	—	4	2	no	—
121	C₄×Dic₁₉	—	4	4	no	—
122	C₄×Dic₁₉	—	4	1	no	—
123	C₄×Dic₁₉	—	4	1	no	—
124	C₄×Dic₁₉	SD₁₆	1	2	yes	complemented
125	C₂×C₁₉⋊C₈	—	2	2	no	—
126	#9234	—	19	1	no	maximal, Sylow
127	C₂³×C₁₉	#120	1	1	yes	characteristic
128	C₂²×D₁₉	—	4	1	no	—
129	Q₈×C₁₉	—	2	4	no	—
130	Q₈×C₁₉	—	2	2	no	—
131	Q₈×C₁₉	#104	1	4	yes	complemented
132	C₂×C₄×C₁₉	—	2	4	no	—
133	C₂×C₄×C₁₉	—	2	2	no	—
134	C₂×C₄×C₁₉	—	2	2	no	—
135	C₂×C₄×C₁₉	—	2	2	no	—
136	C₂×C₄×C₁₉	—	4	1	no	—
137	C₂×C₄×C₁₉	—	4	2	no	—
138	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
142	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
143	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
144	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
145	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
146	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
147	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
148	C₂²⋊D₁₉	—	8	1	no	—
149	C₂²⋊D₁₉	—	8	1	no	—
150	C₂×Dic₁₉	—	2	2	no	—
151	C₂×Dic₁₉	—	2	2	no	—
152	C₂×Dic₁₉	—	2	2	no	—
153	C₂×Dic₁₉	—	2	2	no	—
154	C₂×Dic₁₉	—	2	4	no	—
155	C₂×Dic₁₉	—	4	1	no	—
156	C₂×Dic₁₉	—	4	2	no	—
157	C₂×Dic₁₉	—	4	1	no	—
158	C₂×Dic₁₉	—	4	2	no	—
159	C₂×Dic₁₉	—	4	1	no	—
160	D₇₆	—	4	2	no	—
161	D₇₆	—	4	1	no	—
162	D₇₆	—	4	1	no	—
163	D₇₆	—	8	1	no	—
164	C₄×D₁₉	—	8	1	no	—
165	C₄×D₁₉	—	8	1	no	—
166	Dic₃₈	—	4	1	no	—
167	Dic₃₈	—	4	1	no	—
168	C₈×C₁₉	—	8	1	no	—
169	C₈×C₁₉	—	8	1	no	—
170	C₁₉⋊C₈	—	4	2	no	—
171	#3103	—	19	1	no	—
172	C₄²×Q₈	—	19	1	no	—
173	#1962	—	19	1	no	—
174	#1502	—	19	1	no	—
175	#1487	D₁₉	1	1	yes	characteristic, complemented
176	#1303	—	19	1	no	—
177	#1276	—	19	1	no	—
178	C₂²×C₁₉	—	2	1	no	—
179	C₂²×C₁₉	—	2	1	no	—
180	C₂²×C₁₉	#386	1	1	yes	characteristic
181	C₂²×C₁₉	#448	1	1	yes	characteristic
182	C₂²×C₁₉	#449	1	1	yes	characteristic
183	C₂×D₁₉	—	8	1	no	—
184	C₂×D₁₉	—	8	1	no	—
185	C₂×D₁₉	—	8	1	no	—
186	C₄×C₁₉	—	2	4	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	C₂×#104	1	2	yes	—
192	C₄×C₁₉	C₄×SD₁₆	1	1	yes	characteristic
193	C₄×C₁₉	#440	1	1	yes	characteristic
194	C₄×C₁₉	#460	1	1	yes	characteristic
195	C₄×C₁₉	#464	1	1	yes	characteristic
196	Dic₁₉	—	2	4	no	—
197	Dic₁₉	—	4	2	no	—
198	Dic₁₉	—	4	4	no	—
199	Dic₁₉	—	8	1	no	—
200	#532	—	19	1	no	—
201	C₂×C₄×Q₈	—	38	1	no	—
202	C₂×C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
203	#481	—	38	1	no	—
204	#475	—	19	2	no	—
205	#460	—	19	2	no	—
206	#448	—	38	1	no	—
207	C₄×SD₁₆	—	38	2	no	—
208	#431	—	19	1	no	—
209	#431	C₂×D₁₉	1	1	yes	characteristic
210	#423	C₂×D₁₉	1	1	yes	characteristic
211	C₂×#104	—	38	1	no	—
212	C₂×#104	—	38	1	no	—
213	C₄×C₄⋊C₄	—	19	1	no	—
214	C₄×C₄⋊C₄	—	38	2	no	—
215	C₄×C₄⋊C₄	—	38	1	no	—
216	C₄×C₄⋊C₄	—	38	1	no	—
217	C₄³	—	19	1	no	—
218	C₄³	—	38	1	no	—
219	#327	—	38	2	no	—
220	Q₈⋊C₈	—	2	2	no	—
221	C₂×C₁₉	—	2	1	no	—
222	C₂×C₁₉	—	2	1	no	—
223	C₂×C₁₉	#1880	1	1	yes	characteristic
224	C₂×C₁₉	#1896	1	1	yes	characteristic
225	C₂×C₁₉	#1906	1	1	yes	characteristic
226	D₁₉	—	16	1	no	—
227	C₂×#41	—	38	1	no	—
228	C₂²×Q₈	D₇₆	1	1	yes	characteristic
229	C₂×SD₁₆	—	38	2	no	—
230	C₂×M₁₆	—	2	1	no	—
231	C₄⋊Q₈	—	19	1	no	—
232	#127	—	19	1	no	—
233	#124	—	38	1	no	—
234	#121	—	38	1	no	—
235	#121	—	38	1	no	—
236	C₄×Q₈	—	2	1	no	—
237	C₄×Q₈	—	2	1	no	—
238	C₄×Q₈	—	38	8	no	—
239	C₄×Q₈	C₄×D₁₉	1	2	yes	—
240	C₄×Q₈	C₂²⋊D₁₉	1	2	yes	—
241	C₄×D₄	—	38	1	no	—
242	C₄×D₄	—	38	1	no	—
243	C₂×C₄⋊C₄	—	2	1	no	—
244	C₂×C₄⋊C₄	—	38	2	no	—
245	C₂×C₄⋊C₄	—	38	1	no	—
246	C₂×C₄⋊C₄	D₇₆	1	1	yes	characteristic
247	C₂×C₄²	—	2	1	no	—
248	C₂×C₄²	—	19	1	no	—
249	C₂×C₄²	—	19	1	no	—
250	C₂×C₄²	—	38	2	no	—
251	C₂×C₄²	—	38	1	no	—
252	C₂×C₄²	—	38	2	no	—
253	C₂×C₄²	—	38	1	no	—
254	C₂×C₄²	—	38	1	no	—
255	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
256	#106	—	38	2	no	—
257	C₄⋊C₈	—	2	1	no	—
258	C₄⋊C₈	—	2	1	no	—
259	C₄⋊C₈	—	2	1	no	—
260	C₄⋊C₈	—	19	2	no	—
261	#104	—	76	2	no	—
262	#104	—	76	2	no	—
263	#103	—	38	2	no	—
264	#102	—	38	2	no	—
265	C₂²⋊C₈	—	2	1	no	—
266	C₂²⋊C₈	—	38	1	no	—
267	C₄×C₈	—	2	1	no	—
268	C₄×C₈	—	38	1	no	—
269	#95	—	19	2	no	—
270	#95	—	38	2	no	—
271	C₁₉	#9234	1	1	yes	characteristic, complemented, Sylow
272	#41	—	76	1	no	—
273	#41	—	76	1	no	—
274	#41	—	76	1	no	—
275	#41	—	76	1	no	—
276	C₂×Q₈	—	2	4	no	—
277	C₂×Q₈	—	2	1	no	—
278	C₂×Q₈	—	38	1	no	—
279	C₂×Q₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
280	C₂×D₄	—	38	1	no	—
281	C₂×D₄	—	38	1	no	—
282	C₂×D₄	—	38	1	no	—
283	C₂×D₄	—	76	1	no	—
284	C₂²×C₄	—	2	1	no	—
285	C₂²×C₄	—	2	1	no	—
286	C₂²×C₄	—	38	1	no	—
287	C₂²×C₄	—	38	2	no	—
288	C₂²×C₄	—	38	1	no	—
289	C₂²×C₄	—	76	1	no	—
290	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
291	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
292	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
293	SD₁₆	—	76	4	no	—
294	M₁₆	—	4	1	no	—
295	M₁₆	—	4	1	no	—
296	C₂×C₈	—	4	1	no	—
297	C₂×C₈	—	4	1	no	—
298	C₂×C₈	—	38	2	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	2	1	no	—
302	C₄⋊C₄	—	38	8	no	—
303	C₄⋊C₄	—	38	4	no	—
304	C₄⋊C₄	—	38	1	no	—
305	C₄⋊C₄	—	38	1	no	—
306	C₄⋊C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
307	C₂²⋊C₄	—	76	1	no	—
308	C₂²⋊C₄	—	76	1	no	—
309	C₄²	—	2	2	no	—
310	C₄²	—	2	2	no	—
311	C₄²	—	19	2	no	—
312	C₄²	—	38	8	no	—
313	C₄²	—	38	4	no	—
314	C₄²	—	38	1	no	—
315	C₄²	—	38	1	no	—
316	C₄²	—	38	1	no	—
317	C₄²	—	76	2	no	—
318	C₄²	—	76	4	no	—
319	C₄²	—	76	1	no	—
320	C₄²	—	76	1	no	—
321	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
322	C₄²	#64543	1	1	yes	characteristic
323	C₄²	#64543	1	1	yes	characteristic
324	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
325	C₂³	—	76	1	no	—
326	C₂³	#10597328	1	1	yes	characteristic
327	Q₈	—	2	4	no	—
328	Q₈	—	2	2	no	—
329	Q₈	—	76	1	no	—
330	Q₈	—	76	1	no	—
331	Q₈	#10597282	1	4	yes	complemented
332	D₄	—	76	2	no	—
333	D₄	—	76	1	no	—
334	D₄	—	76	1	no	—
335	D₄	—	152	1	no	—
336	D₄	—	152	1	no	—
337	D₄	—	152	1	no	—
338	C₂×C₄	—	2	4	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	4	2	no	—
344	C₂×C₄	—	38	2	no	—
345	C₂×C₄	—	38	2	no	—
346	C₂×C₄	—	38	2	no	—
347	C₂×C₄	—	38	4	no	—
348	C₂×C₄	—	38	2	no	—
349	C₂×C₄	—	76	2	no	—
350	C₂×C₄	—	76	1	no	—
351	C₂×C₄	—	76	1	no	—
352	C₂×C₄	—	76	2	no	—
353	C₂×C₄	—	76	1	no	—
354	C₂×C₄	—	152	1	no	—
355	C₂×C₄	—	152	1	no	—
356	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
357	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
358	C₂×C₄	#10597332	1	1	yes	characteristic
359	C₂×C₄	SD₁₆×D₁₉	1	1	yes	characteristic
360	C₂×C₄	#10597360	1	1	yes	characteristic
361	C₂×C₄	#10597364	1	1	yes	characteristic
362	C₂×C₄	#10597365	1	1	yes	characteristic
363	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
364	C₂×C₄	#10597385	1	1	yes	characteristic
365	C₂×C₄	#10597385	1	1	yes	characteristic
366	C₈	—	8	1	no	—
367	C₈	—	8	1	no	—
368	C₈	—	76	2	no	—
369	C₂²	—	2	1	no	—
370	C₂²	—	2	1	no	—
371	C₂²	—	152	1	no	—
372	C₂²	—	152	1	no	—
373	C₂²	—	152	1	no	—
374	C₂²	#11926421	1	1	yes	characteristic
375	C₂²	#11926684	1	1	yes	characteristic
376	C₂²	#11926692	1	1	yes	characteristic
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	2	4	no	—
380	C₄	—	2	1	no	—
381	C₄	—	4	2	no	—
382	C₄	—	38	4	no	—
383	C₄	—	76	2	no	—
384	C₄	—	76	4	no	—
385	C₄	—	152	1	no	—
386	C₄	C₂×#10597282	1	2	yes	—
387	C₄	#11926629	1	1	yes	characteristic
388	C₄	#11926647	1	1	yes	characteristic
389	C₄	#11926742	1	1	yes	characteristic
390	C₄	#11926768	1	1	yes	characteristic
391	C₂	—	2	1	no	—
392	C₂	—	2	1	no	—
393	C₂	—	304	1	no	—
394	C₂	#14563988	1	1	yes	characteristic
395	C₂	#14564076	1	1	yes	characteristic
396	C₂	#14564134	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597385
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1487	C₂
socle	C₂²×C₁₉	#386
prime radical	#11927094	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1487	D₁₉
hypocenter	C₁₉	#9234
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1487

Complements

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

N	H ≅ G/N	type
2. C₄×Q₈×Dic₁₉	C₂	semidirect
5. #14561536	C₂	semidirect
6. #14561467	C₂	semidirect
7. C₁₉×#1487	C₂	semidirect
11. #11926829	C₄	semidirect
12. #11926744	C₄	semidirect
124. C₄×Dic₁₉	SD₁₆	semidirect
131. Q₈×C₁₉	#104	semidirect
175. #1487	D₁₉	semidirect
271. C₁₉	#9234	semidirect
331. Q₈	#10597282	semidirect

Maximal subgroups

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

subgroup	index
#14573666	2
C₄×Q₈×Dic₁₉	2
#14564462	2
#14562131	2
#14561536	2
#14561467	2
C₁₉×#1487	2
#9234	19