Subgroups — #18082404

Order 4864.

626 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926918	C₂	1	1	yes	maximal, characteristic
2	#14564597	C₂	1	1	yes	maximal, characteristic, complemented
3	#14564441	C₂	1	1	yes	maximal, characteristic
4	#14564440	C₂	1	1	yes	maximal, characteristic
5	#14563794	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11926266	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1990	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#64527	—	2	2	no	—
9	C₂×#10597320	C₂²	1	1	yes	characteristic
10	#11926918	—	2	2	no	—
11	#11926918	C₄	1	4	yes	complemented
12	C₂×#10597264	—	2	2	no	—
13	C₂×#10597278	—	2	2	no	—
14	C₂×#10597266	—	2	2	no	—
15	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
17	#11926414	C₂²	1	1	yes	characteristic
18	#11926306	—	2	2	no	—
19	#11926266	—	2	2	no	—
20	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
22	C₁₉×#384	C₂²	1	1	yes	characteristic
23	C₁₉×#340	—	2	2	no	—
24	C₂²×Q₈×C₁₉	—	2	1	no	—
25	C₂²×D₇₆	—	2	1	no	—
26	C₂²×D₇₆	—	2	1	no	—
27	C₂²×D₇₆	—	2	1	no	—
28	C₂²×C₈×C₁₉	D₄	1	2	yes	—
29	C₁₉×C₄⋊Q₈	—	2	1	no	—
30	C₁₉×C₄⋊Q₈	—	2	1	no	—
31	C₁₉×C₄⋊Q₈	—	2	1	no	—
32	C₁₉×C₄⋊Q₈	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
34	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
35	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
36	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
37	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
38	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
39	C₂×#64527	—	2	8	no	—
40	C₂×#64527	—	4	4	no	—
41	C₂×#64527	—	4	4	no	—
42	C₂×C₄⋊Dic₁₉	—	2	1	no	—
43	C₂×C₄⋊Dic₁₉	—	2	1	no	—
44	C₂×C₄⋊Dic₁₉	—	2	1	no	—
45	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
46	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
47	#10597320	—	2	1	no	—
48	#10597320	—	2	1	no	—
49	#10597320	—	2	1	no	—
50	#10597320	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₁₉	—	2	1	no	—
52	C₄×C₈×C₁₉	—	2	1	no	—
53	#10597278	—	4	4	no	—
54	#10597266	—	4	4	no	—
55	#10597264	—	4	4	no	—
56	C₄×C₁₉⋊C₈	—	2	1	no	—
57	C₄×C₁₉⋊C₈	—	2	1	no	—
58	C₄×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
59	C₄×C₁₉⋊C₈	D₄	1	2	yes	—
60	C₂³×D₁₉	—	4	1	no	—
61	C₂×Q₈×C₁₉	—	2	2	no	—
62	C₂×Q₈×C₁₉	—	4	1	no	—
63	C₂×Q₈×C₁₉	—	4	2	no	—
64	C₂×Q₈×C₁₉	—	4	2	no	—
65	C₂²×C₄×C₁₉	—	4	1	no	—
66	C₂²×C₄×C₁₉	—	4	1	no	—
67	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×Dic₁₉	—	4	1	no	—
71	C₂×D₇₆	—	2	2	no	—
72	C₂×D₇₆	—	2	2	no	—
73	C₂×D₇₆	—	2	2	no	—
74	C₂×D₇₆	—	4	2	no	—
75	C₂×D₇₆	—	4	2	no	—
76	C₂×D₇₆	—	4	2	no	—
77	C₂×D₇₆	—	4	2	no	—
78	C₂×D₇₆	—	4	2	no	—
79	C₂×D₇₆	—	4	2	no	—
80	C₂×D₇₆	—	4	1	no	—
81	C₂×D₇₆	—	4	1	no	—
82	C₂×D₇₆	—	4	1	no	—
83	C₂×C₈×C₁₉	—	2	2	no	—
84	C₂×C₈×C₁₉	—	2	2	no	—
85	C₂×C₈×C₁₉	—	2	2	no	—
86	C₄⋊C₄×C₁₉	—	2	2	no	—
87	C₄⋊C₄×C₁₉	—	2	2	no	—
88	C₄⋊C₄×C₁₉	—	4	1	no	—
89	C₄⋊C₄×C₁₉	—	4	1	no	—
90	C₄⋊C₄×C₁₉	—	4	2	no	—
91	C₄⋊C₄×C₁₉	—	4	2	no	—
92	C₄⋊C₄×C₁₉	—	4	2	no	—
93	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₁₉	#41	1	1	yes	characteristic
96	C₄²×C₁₉	#41	1	1	yes	characteristic
97	#64527	—	4	16	no	—
98	C₄⋊Dic₁₉	—	4	2	no	—
99	C₄⋊Dic₁₉	—	4	2	no	—
100	C₄⋊Dic₁₉	—	4	2	no	—
101	C₂×C₁₉⋊C₈	—	2	2	no	—
102	C₂×C₁₉⋊C₈	—	2	2	no	—
103	C₂×C₁₉⋊C₈	—	2	2	no	—
104	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	4	yes	—
105	C₂×C₁₉⋊C₈	D₈	1	4	yes	—
106	C₂×C₁₉⋊C₈	SD₁₆	1	4	yes	—
107	#9842	—	19	1	no	maximal, Sylow
108	C₂³×C₁₉	#127	1	1	yes	characteristic
109	C₂²×D₁₉	—	4	2	no	—
110	C₂²×D₁₉	—	8	1	no	—
111	C₂²×D₁₉	—	8	1	no	—
112	C₂²×D₁₉	—	8	1	no	—
113	C₂²×D₁₉	—	8	1	no	—
114	C₂²×D₁₉	—	8	1	no	—
115	C₂²×D₁₉	—	8	1	no	—
116	Q₈×C₁₉	—	4	4	no	—
117	Q₈×C₁₉	—	8	2	no	—
118	C₂×C₄×C₁₉	—	4	2	no	—
119	C₂×C₄×C₁₉	—	8	1	no	—
120	C₂×C₄×C₁₉	—	8	1	no	—
121	C₂×C₄×C₁₉	—	8	1	no	—
122	C₂×C₄×C₁₉	—	8	1	no	—
123	C₂×C₄×C₁₉	—	8	1	no	—
124	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
128	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
129	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
130	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
131	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
132	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
133	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
134	C₂×C₄×C₁₉	#137	1	2	yes	—
135	C₂×C₄×C₁₉	#137	1	2	yes	—
136	C₂×Dic₁₉	—	8	1	no	—
137	C₂×Dic₁₉	—	8	1	no	—
138	C₂×Dic₁₉	—	8	1	no	—
139	D₇₆	—	4	4	no	—
140	D₇₆	—	4	4	no	—
141	D₇₆	—	4	4	no	—
142	D₇₆	—	8	2	no	—
143	D₇₆	—	8	2	no	—
144	D₇₆	—	8	2	no	—
145	C₈×C₁₉	—	4	2	no	—
146	C₁₉⋊C₈	—	4	2	no	—
147	C₁₉⋊C₈	#102	1	8	yes	complemented
148	C₂×#492	—	19	1	no	—
149	#1990	—	19	1	no	—
150	#1990	D₁₉	1	1	yes	characteristic, complemented
151	#1957	—	19	1	no	—
152	#1957	—	19	1	no	—
153	#1845	—	19	1	no	—
154	C₂×#322	—	19	1	no	—
155	C₂²×C₁₉	#390	1	1	yes	characteristic
156	C₂²×C₁₉	#492	1	1	yes	characteristic
157	C₂²×C₁₉	#493	1	1	yes	characteristic
158	C₂²×C₁₉	#496	1	1	yes	characteristic
159	C₂²×C₁₉	#496	1	1	yes	characteristic
160	C₂²×C₁₉	#497	1	1	yes	characteristic
161	C₂²×C₁₉	#497	1	1	yes	characteristic
162	C₂×D₁₉	—	8	2	no	—
163	C₂×D₁₉	—	8	2	no	—
164	C₂×D₁₉	—	8	2	no	—
165	C₂×D₁₉	—	16	1	no	—
166	C₂×D₁₉	—	16	1	no	—
167	C₂×D₁₉	—	16	1	no	—
168	C₂×D₁₉	—	16	1	no	—
169	C₄×C₁₉	—	8	2	no	—
170	C₄×C₁₉	—	16	1	no	—
171	C₄×C₁₉	C₂×#102	1	2	yes	—
172	C₄×C₁₉	#440	1	2	yes	—
173	C₄×C₁₉	#440	1	2	yes	—
174	C₄×C₁₉	#468	1	2	yes	—
175	C₄×C₁₉	#468	1	2	yes	—
176	C₄×C₁₉	#486	1	2	yes	—
177	Dic₁₉	—	16	1	no	—
178	C₂²×SD₁₆	—	38	2	no	—
179	C₂×C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
180	C₂×#127	—	19	1	no	—
181	#492	—	19	4	no	—
182	#492	—	38	2	no	—
183	C₂×#107	—	38	2	no	—
184	C₂×#102	—	38	2	no	—
185	C₂×#102	—	38	2	no	—
186	C₂×C₄×C₈	—	19	1	no	—
187	C₂×C₄×C₈	—	19	1	no	—
188	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
189	#384	—	19	1	no	—
190	#384	C₂×D₁₉	1	1	yes	characteristic
191	#340	—	2	2	no	—
192	#340	—	38	2	no	—
193	#322	—	38	2	no	—
194	C₂×C₁₉	#1714	1	1	yes	characteristic
195	C₂×C₁₉	#1714	1	1	yes	characteristic
196	C₂×C₁₉	#1721	1	1	yes	characteristic
197	C₂×C₁₉	#1722	1	1	yes	characteristic
198	C₂×C₁₉	#1957	1	1	yes	characteristic
199	C₂×C₁₉	#1973	1	1	yes	characteristic
200	C₂×C₁₉	#1973	1	1	yes	characteristic
201	D₁₉	—	16	2	no	—
202	C₂²×Q₈	—	2	1	no	—
203	C₂²×D₄	—	38	1	no	—
204	C₂²×D₄	—	38	1	no	—
205	C₂²×D₄	—	38	1	no	—
206	C₂×SD₁₆	—	38	8	no	—
207	C₂×SD₁₆	—	76	4	no	—
208	C₂×SD₁₆	—	76	4	no	—
209	C₂²×C₈	—	19	2	no	—
210	C₂²×C₈	—	19	2	no	—
211	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	C₄×D₁₉	1	2	yes	—
216	#127	—	19	2	no	—
217	#127	—	38	1	no	—
218	#127	—	38	1	no	—
219	#127	—	38	1	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄⋊C₄	—	2	1	no	—
225	C₂×C₄⋊C₄	—	38	1	no	—
226	C₂×C₄⋊C₄	—	38	1	no	—
227	C₂×C₄⋊C₄	—	38	1	no	—
228	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
229	#107	—	76	4	no	—
230	#102	—	76	4	no	—
231	#102	—	76	4	no	—
232	C₄×C₈	—	2	1	no	—
233	C₄×C₈	—	2	1	no	—
234	C₄×C₈	—	19	2	no	—
235	C₄×C₈	—	19	2	no	—
236	C₄×C₈	—	38	1	no	—
237	C₄×C₈	—	38	1	no	—
238	C₁₉	#9842	1	1	yes	characteristic, complemented, Sylow
239	C₂⁴	—	76	1	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×Q₈	—	4	2	no	—
243	C₂×Q₈	—	4	2	no	—
244	C₂×D₄	—	38	2	no	—
245	C₂×D₄	—	38	2	no	—
246	C₂×D₄	—	38	2	no	—
247	C₂×D₄	—	76	1	no	—
248	C₂×D₄	—	76	2	no	—
249	C₂×D₄	—	76	2	no	—
250	C₂×D₄	—	76	1	no	—
251	C₂×D₄	—	76	1	no	—
252	C₂×D₄	—	76	2	no	—
253	C₂×D₄	—	76	2	no	—
254	C₂×D₄	—	76	2	no	—
255	C₂×D₄	—	76	2	no	—
256	C₂²×C₄	—	4	1	no	—
257	C₂²×C₄	—	4	1	no	—
258	C₂²×C₄	—	76	1	no	—
259	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
260	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
261	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
262	SD₁₆	—	76	16	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	19	4	no	—
267	C₂×C₈	—	19	4	no	—
268	C₂×C₈	—	19	4	no	—
269	C₂×C₈	—	38	2	no	—
270	C₂×C₈	—	38	2	no	—
271	C₂×C₈	—	38	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	76	2	no	—
280	C₄⋊C₄	—	76	2	no	—
281	C₄⋊C₄	—	76	2	no	—
282	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
283	C₄²	#64541	1	1	yes	characteristic
284	C₄²	D₄×D₁₉	1	1	yes	characteristic
285	C₄²	#64543	1	1	yes	characteristic
286	C₂³	—	76	2	no	—
287	C₂³	—	152	1	no	—
288	C₂³	—	152	1	no	—
289	C₂³	—	152	1	no	—
290	C₂³	—	152	1	no	—
291	C₂³	—	152	1	no	—
292	C₂³	—	152	1	no	—
293	C₂³	#10597386	1	1	yes	characteristic
294	Q₈	—	4	4	no	—
295	Q₈	—	8	2	no	—
296	D₄	—	76	4	no	—
297	D₄	—	76	4	no	—
298	D₄	—	76	4	no	—
299	D₄	—	152	2	no	—
300	D₄	—	152	2	no	—
301	D₄	—	152	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	8	1	no	—
304	C₂×C₄	—	8	1	no	—
305	C₂×C₄	—	8	1	no	—
306	C₂×C₄	—	8	1	no	—
307	C₂×C₄	—	8	1	no	—
308	C₂×C₄	—	152	1	no	—
309	C₂×C₄	—	152	1	no	—
310	C₂×C₄	—	152	1	no	—
311	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
312	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
313	C₂×C₄	#10597330	1	1	yes	characteristic
314	C₂×C₄	#10597331	1	1	yes	characteristic
315	C₂×C₄	D₈×D₁₉	1	2	yes	—
316	C₂×C₄	SD₁₆×D₁₉	1	2	yes	—
317	C₂×C₄	#10597360	1	2	yes	—
318	C₂×C₄	#10597364	1	2	yes	—
319	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
320	C₂×C₄	#10597385	1	1	yes	characteristic
321	C₂×C₄	#10597388	1	2	yes	—
322	C₂×C₄	#10597395	1	2	yes	—
323	C₈	—	4	2	no	—
324	C₈	—	19	8	no	—
325	C₈	—	76	2	no	—
326	C₂²	—	304	1	no	—
327	C₂²	—	304	1	no	—
328	C₂²	—	304	1	no	—
329	C₂²	—	304	1	no	—
330	C₂²	—	152	2	no	—
331	C₂²	—	152	2	no	—
332	C₂²	—	152	2	no	—
333	C₂²	#11926444	1	1	yes	characteristic
334	C₂²	#11926915	1	1	yes	characteristic
335	C₂²	#11926919	1	1	yes	characteristic
336	C₂²	#11926930	1	1	yes	characteristic
337	C₂²	#11926934	1	1	yes	characteristic
338	C₂²	#11926936	1	1	yes	characteristic
339	C₂²	#11926937	1	1	yes	characteristic
340	C₄	—	8	2	no	—
341	C₄	—	16	1	no	—
342	C₄	—	304	1	no	—
343	C₄	#102×D₁₉	1	2	yes	—
344	C₄	#11926642	1	2	yes	—
345	C₄	#11926647	1	2	yes	—
346	C₄	#11926792	1	2	yes	—
347	C₄	#11926793	1	2	yes	—
348	C₄	#11926889	1	2	yes	—
349	C₂	—	304	2	no	—
350	C₂	#14563259	1	1	yes	characteristic
351	C₂	#14563260	1	1	yes	characteristic
352	C₂	#14563303	1	1	yes	characteristic
353	C₂	#14563305	1	1	yes	characteristic
354	C₂	#14564439	1	1	yes	characteristic
355	C₂	#14564513	1	1	yes	characteristic
356	C₂	#14564519	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597386
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1990	C₂
socle	C₂³×C₁₉	#127
prime radical	C₂×#10597320	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1990	D₁₉
hypocenter	C₁₉	#9842
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1990

Complements

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

N	H ≅ G/N	type
2. #14564597	C₂	semidirect
5. #14563794	C₂	semidirect
6. C₂×#11926266	C₂	semidirect
7. C₁₉×#1990	C₂	semidirect
11. #11926918	C₄	semidirect
147. C₁₉⋊C₈	#102	semidirect
150. #1990	D₁₉	semidirect
238. C₁₉	#9842	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926918	2
#14564597	2
#14564441	2
#14564440	2
#14563794	2
C₂×#11926266	2
C₁₉×#1990	2
#9842	19