Subgroups — #18082066

Order 4864.

510 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14564621	C₂	1	2	yes	maximal
2	#14563660	C₂	1	1	yes	maximal, characteristic
3	#14563411	C₂	1	1	yes	maximal, characteristic
4	#14562207	C₂	1	2	yes	maximal
5	C₂×C₁₉×#467	C₂	1	1	yes	maximal, characteristic
6	C₂×#10597274	—	2	1	no	—
7	C₂×C₂²⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
8	C₂×#10597272	—	2	1	no	—
9	#11926404	C₂²	1	2	yes	—
10	C₄×C₂²⋊Dic₁₉	C₂²	1	1	yes	characteristic
11	#11926311	—	2	4	no	—
12	#11926307	—	2	1	no	—
13	#11926298	—	2	1	no	—
14	#11926281	—	2	4	no	—
15	C₂²×Q₁₆×C₁₉	C₄	1	1	yes	characteristic, complemented
16	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₂×C₁₉×C₂²⋊Q₈	C₂²	1	2	yes	—
18	C₁₉×#467	—	2	1	no	—
19	C₁₉×#467	—	2	1	no	—
20	C₁₉×#467	—	2	1	no	—
21	C₁₉×#467	—	2	1	no	—
22	C₂×C₁₉×#107	C₄	1	1	yes	characteristic, complemented
23	C₂×C₁₉×#103	C₄	1	2	yes	complemented
24	C₂²×Q₈×C₁₉	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
26	C₂×Q₁₆×C₁₉	—	2	1	no	—
27	C₂×Q₁₆×C₁₉	—	2	1	no	—
28	C₂×Q₁₆×C₁₉	—	4	2	no	—
29	C₂²×C₈×C₁₉	—	4	2	no	—
30	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
36	C₁₉×C₂²⋊Q₈	—	2	2	no	—
37	C₁₉×C₂²⋊Q₈	—	2	2	no	—
38	C₁₉×C₂²⋊Q₈	—	2	2	no	—
39	C₁₉×C₂²⋊Q₈	—	2	2	no	—
40	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
41	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
42	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
43	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
44	C₂×C₂²⋊Dic₁₉	D₄	1	2	yes	—
45	C₂×C₄⋊Dic₁₉	—	2	2	no	—
46	C₂×C₄×Dic₁₉	—	2	1	no	—
47	C₂²×C₁₉⋊C₈	—	2	1	no	—
48	C₁₉×#107	—	2	1	no	—
49	C₁₉×#107	—	2	1	no	—
50	C₁₉×#103	—	2	2	no	—
51	C₁₉×#103	—	2	2	no	—
52	C₁₉⋊#95	—	2	1	no	—
53	C₁₉⋊#95	—	2	2	no	—
54	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
55	C₂²⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
56	#10597274	—	4	2	no	—
57	#10597272	—	4	2	no	—
58	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
59	C₂×Q₈×C₁₉	—	2	2	no	—
60	C₂×Q₈×C₁₉	—	2	2	no	—
61	C₂×Q₈×C₁₉	—	4	2	no	—
62	C₂×Q₈×C₁₉	—	4	2	no	—
63	C₂²×C₄×C₁₉	—	2	2	no	—
64	C₂²×C₄×C₁₉	—	2	2	no	—
65	C₂²×C₄×C₁₉	—	4	1	no	—
66	C₂²×C₄×C₁₉	—	4	1	no	—
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×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
73	C₂²×Dic₁₉	—	2	2	no	—
74	C₂²×Dic₁₉	—	4	2	no	—
75	Q₁₆×C₁₉	—	4	2	no	—
76	Q₁₆×C₁₉	—	4	2	no	—
77	C₂×C₈×C₁₉	—	2	2	no	—
78	C₂×C₈×C₁₉	—	2	2	no	—
79	C₂×C₈×C₁₉	—	2	2	no	—
80	C₂×C₈×C₁₉	—	4	1	no	—
81	C₂×C₈×C₁₉	—	4	1	no	—
82	C₂×C₈×C₁₉	—	4	1	no	—
83	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
87	C₄⋊C₄×C₁₉	—	2	2	no	—
88	C₄⋊C₄×C₁₉	—	2	2	no	—
89	C₄⋊C₄×C₁₉	—	4	2	no	—
90	C₄⋊C₄×C₁₉	—	4	2	no	—
91	C₂²⋊C₄×C₁₉	—	4	2	no	—
92	C₂²⋊C₄×C₁₉	—	4	2	no	—
93	C₂²⋊Dic₁₉	—	2	4	no	—
94	C₂²⋊Dic₁₉	—	4	4	no	—
95	C₄⋊Dic₁₉	—	4	2	no	—
96	C₄⋊Dic₁₉	—	4	2	no	—
97	C₄×Dic₁₉	—	4	1	no	—
98	C₄×Dic₁₉	—	4	1	no	—
99	C₂×C₁₉⋊C₈	—	2	2	no	—
100	C₂×C₁₉⋊C₈	—	4	1	no	—
101	C₂×C₁₉⋊C₈	—	4	1	no	—
102	#9784	—	19	1	no	maximal, Sylow
103	C₂³×C₁₉	—	4	1	no	—
104	C₂³×C₁₉	—	4	1	no	—
105	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
106	C₂³×C₁₉	#121	1	1	yes	characteristic
107	C₂³×C₁₉	#127	1	1	yes	characteristic
108	C₂³×C₁₉	#137	1	4	yes	—
109	Q₈×C₁₉	—	4	2	no	—
110	Q₈×C₁₉	—	4	2	no	—
111	Q₈×C₁₉	—	8	2	no	—
112	C₂×C₄×C₁₉	—	2	1	no	—
113	C₂×C₄×C₁₉	—	2	1	no	—
114	C₂×C₄×C₁₉	—	2	1	no	—
115	C₂×C₄×C₁₉	—	2	1	no	—
116	C₂×C₄×C₁₉	—	2	1	no	—
117	C₂×C₄×C₁₉	—	2	1	no	—
118	C₂×C₄×C₁₉	—	4	1	no	—
119	C₂×C₄×C₁₉	—	4	1	no	—
120	C₂×C₄×C₁₉	—	4	1	no	—
121	C₂×C₄×C₁₉	—	4	2	no	—
122	C₂×C₄×C₁₉	—	4	2	no	—
123	C₂×C₄×C₁₉	—	8	2	no	—
124	C₂×C₄×C₁₉	—	8	2	no	—
125	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	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	#124	1	1	yes	characteristic
129	C₂×Dic₁₉	—	4	2	no	—
130	C₂×Dic₁₉	—	4	2	no	—
131	C₂×Dic₁₉	—	4	2	no	—
132	C₂×Dic₁₉	—	8	2	no	—
133	C₂×Dic₁₉	—	8	2	no	—
134	C₂×Dic₁₉	—	8	2	no	—
135	C₈×C₁₉	—	2	1	no	—
136	C₈×C₁₉	—	2	1	no	—
137	C₈×C₁₉	—	2	1	no	—
138	C₈×C₁₉	#99	1	2	yes	—
139	C₁₉⋊C₈	—	4	2	no	—
140	C₂×#467	D₁₉	1	1	yes	characteristic
141	#1996	—	19	2	no	—
142	#1807	—	19	1	no	—
143	#1752	—	19	1	no	—
144	#1520	—	19	2	no	—
145	C₂²×C₁₉	—	2	1	no	—
146	C₂²×C₁₉	—	2	1	no	—
147	C₂²×C₁₉	—	2	1	no	—
148	C₂²×C₁₉	—	2	1	no	—
149	C₂²×C₁₉	—	2	1	no	—
150	C₂²×C₁₉	—	2	1	no	—
151	C₂²×C₁₉	—	4	4	no	—
152	C₂²×C₁₉	#390	1	1	yes	characteristic
153	C₂²×C₁₉	#441	1	2	yes	—
154	C₂²×C₁₉	#470	1	2	yes	—
155	C₂²×C₁₉	#497	1	2	yes	—
156	C₄×C₁₉	—	2	1	no	—
157	C₄×C₁₉	—	2	1	no	—
158	C₄×C₁₉	—	2	1	no	—
159	C₄×C₁₉	—	8	2	no	—
160	C₄×C₁₉	—	8	2	no	—
161	C₄×C₁₉	C₂×#99	1	1	yes	characteristic
162	C₄×C₁₉	#410	1	1	yes	characteristic
163	Dic₁₉	—	8	2	no	—
164	Dic₁₉	—	16	2	no	—
165	C₂²×Q₁₆	Dic₁₉	1	1	yes	characteristic, complemented
166	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
167	C₂×C₂²⋊Q₈	C₂×D₁₉	1	2	yes	—
168	#467	—	2	1	no	—
169	#467	—	2	1	no	—
170	#467	—	2	1	no	—
171	#467	—	2	1	no	—
172	C₂×#107	Dic₁₉	1	1	yes	characteristic, complemented
173	C₂×#106	—	38	1	no	—
174	C₂×#103	Dic₁₉	1	2	yes	complemented
175	C₂×C₂²⋊C₈	—	19	1	no	—
176	C₂×#97	—	38	1	no	—
177	#380	—	19	2	no	—
178	C₄×C₂²⋊C₄	—	19	1	no	—
179	#342	—	38	4	no	—
180	#340	—	38	1	no	—
181	#336	—	38	1	no	—
182	#328	—	38	4	no	—
183	C₂×C₁₉	—	2	2	no	—
184	C₂×C₁₉	—	2	2	no	—
185	C₂×C₁₉	—	2	2	no	—
186	C₂×C₁₉	#1964	1	1	yes	characteristic
187	C₂×C₁₉	#1971	1	1	yes	characteristic
188	C₂×C₁₉	#1976	1	1	yes	characteristic
189	C₂²×Q₈	C₂×Dic₁₉	1	2	yes	—
190	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
191	C₂×Q₁₆	—	2	1	no	—
192	C₂×Q₁₆	—	2	1	no	—
193	C₂×Q₁₆	—	4	2	no	—
194	C₂²×C₈	—	4	2	no	—
195	C₂²×C₈	—	38	1	no	—
196	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
197	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
198	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
199	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
200	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
201	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
202	C₂²⋊Q₈	—	2	2	no	—
203	C₂²⋊Q₈	—	2	2	no	—
204	C₂²⋊Q₈	—	2	2	no	—
205	C₂²⋊Q₈	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	38	2	no	—
208	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
209	C₂×C₂²⋊C₄	—	2	2	no	—
210	C₂×C₂²⋊C₄	—	19	2	no	—
211	C₂×C₂²⋊C₄	—	38	2	no	—
212	C₂×C₄²	—	38	1	no	—
213	#107	—	2	1	no	—
214	#107	—	2	1	no	—
215	#106	—	76	2	no	—
216	#103	—	2	2	no	—
217	#103	—	2	2	no	—
218	C₂²⋊C₈	—	19	2	no	—
219	C₂²⋊C₈	—	38	1	no	—
220	#97	—	76	2	no	—
221	#95	—	38	1	no	—
222	#95	—	38	2	no	—
223	C₁₉	#9784	1	1	yes	characteristic, complemented, Sylow
224	C₂⁴	D₄×D₁₉	1	1	yes	characteristic
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×Q₈	—	4	2	no	—
228	C₂×Q₈	—	4	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	4	1	no	—
232	C₂²×C₄	—	4	1	no	—
233	C₂²×C₄	—	38	2	no	—
234	C₂²×C₄	—	76	2	no	—
235	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
236	C₂²×C₄	#64543	1	1	yes	characteristic
237	C₂²×C₄	#64543	1	1	yes	characteristic
238	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
241	Q₁₆	—	4	2	no	—
242	Q₁₆	—	4	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	4	1	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	—	38	2	no	—
250	C₂×C₈	—	76	1	no	—
251	C₂×C₈	—	76	1	no	—
252	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
253	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
254	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
255	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	—	76	2	no	—
261	C₄⋊C₄	—	76	2	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₂²⋊C₄	—	38	4	no	—
265	C₂²⋊C₄	—	76	4	no	—
266	C₄²	—	76	1	no	—
267	C₄²	—	76	1	no	—
268	C₂³	—	4	1	no	—
269	C₂³	—	4	1	no	—
270	C₂³	#10597364	1	4	yes	—
271	C₂³	D₄×Dic₁₉	1	1	yes	characteristic
272	C₂³	#10597384	1	1	yes	characteristic
273	C₂³	#10597386	1	1	yes	characteristic
274	Q₈	—	4	2	no	—
275	Q₈	—	4	2	no	—
276	Q₈	—	8	2	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	8	2	no	—
289	C₂×C₄	—	8	2	no	—
290	C₂×C₄	—	76	2	no	—
291	C₂×C₄	—	76	2	no	—
292	C₂×C₄	—	76	2	no	—
293	C₂×C₄	—	152	2	no	—
294	C₂×C₄	—	152	2	no	—
295	C₂×C₄	—	152	2	no	—
296	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
297	C₂×C₄	#10597382	1	1	yes	characteristic
298	C₂×C₄	#10597384	1	1	yes	characteristic
299	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
300	C₈	—	2	1	no	—
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	76	2	no	—
304	C₈	#10597291	1	2	yes	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	4	4	no	—
312	C₂²	#11926447	1	1	yes	characteristic
313	C₂²	#11926651	1	2	yes	—
314	C₂²	#11926799	1	2	yes	—
315	C₂²	#11926937	1	2	yes	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	8	2	no	—
320	C₄	—	8	2	no	—
321	C₄	—	304	2	no	—
322	C₄	—	152	2	no	—
323	C₄	C₂×#10597291	1	1	yes	characteristic
324	C₄	#11926518	1	1	yes	characteristic
325	C₂	—	2	2	no	—
326	C₂	—	2	2	no	—
327	C₂	—	2	2	no	—
328	C₂	#14564472	1	1	yes	characteristic
329	C₂	#14564507	1	1	yes	characteristic
330	C₂	#14564534	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926447
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₁₉×#467	C₂
socle	C₂²×C₁₉	#390
prime radical	C₂⁴×C₁₉	C₂×D₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×#467	D₁₉
hypocenter	C₁₉	#9784
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#467

Complements

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

N	H ≅ G/N	type
15. C₂²×Q₁₆×C₁₉	C₄	semidirect
22. C₂×C₁₉×#107	C₄	semidirect
23. C₂×C₁₉×#103	C₄	semidirect
165. C₂²×Q₁₆	Dic₁₉	semidirect
172. C₂×#107	Dic₁₉	semidirect
174. C₂×#103	Dic₁₉	semidirect
223. C₁₉	#9784	semidirect

Maximal subgroups

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

subgroup	index
#14564621	2
#14563660	2
#14563411	2
#14562207	2
C₂×C₁₉×#467	2
#9784	19