Subgroups — #14574348

Order 2432.

402 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927096	C₂	1	1	yes	maximal, characteristic, complemented
2	#11927089	C₂	1	1	yes	maximal, characteristic
3	#11926948	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926915	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926893	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926889	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926792	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926790	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926788	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926647	C₂	1	1	yes	maximal, characteristic, complemented
11	#11926645	C₂	1	1	yes	maximal, characteristic, complemented
12	#11926643	C₂	1	1	yes	maximal, characteristic, complemented
13	#11926641	C₂	1	1	yes	maximal, characteristic, complemented
14	M₁₆×Dic₁₉	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₉×#468	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#64543	—	2	1	no	—
17	C₂×SD₁₆×C₁₉	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
19	C₁₉×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
20	C₁₉×#121	C₂²	1	1	yes	characteristic
21	Q₈×Dic₁₉	C₂²	1	1	yes	characteristic
22	#10597390	C₂²	1	1	yes	characteristic
23	C₂×#64527	C₂²	1	1	yes	characteristic
24	#10597386	C₂²	1	1	yes	characteristic
25	#10597385	—	2	1	no	—
26	D₄×Dic₁₉	C₂²	1	1	yes	characteristic
27	C₂×#64526	C₂²	1	1	yes	characteristic
28	#10597376	C₂²	1	1	yes	characteristic
29	#10597370	C₂²	1	1	yes	characteristic
30	C₂×C₄×Dic₁₉	C₂²	1	1	yes	characteristic
31	C₂×#64520	C₂²	1	1	yes	characteristic
32	C₂×#64516	C₂²	1	1	yes	characteristic
33	#10597339	C₂²	1	1	yes	characteristic
34	#10597335	—	2	1	no	—
35	#10597334	C₂²	1	1	yes	characteristic
36	#10597334	C₂²	1	1	yes	characteristic
37	#10597333	—	2	1	no	—
38	#10597333	C₂²	1	1	yes	characteristic
39	#10597331	—	2	1	no	—
40	(C₂²⋊D₁₉)⋊C₄	—	2	1	no	—
41	#10597324	—	2	1	no	—
42	#10597323	—	2	1	no	—
43	C₁₉×#107	C₂²	1	1	yes	characteristic
44	C₁₉×#103	C₂²	1	1	yes	characteristic
45	C₁₉×#102	C₂²	1	1	yes	characteristic
46	#10597292	C₂²	1	1	yes	characteristic
47	#10597289	C₂²	1	1	yes	characteristic
48	#10597278	C₂²	1	1	yes	characteristic
49	#10597275	C₂²	1	1	yes	characteristic
50	#10597273	C₂²	1	1	yes	characteristic
51	#10597272	C₂²	1	1	yes	characteristic
52	#10597272	C₂²	1	1	yes	characteristic
53	C₈×Dic₁₉	C₂²	1	1	yes	characteristic
54	C₈×Dic₁₉	C₂²	1	1	yes	characteristic
55	#10597267	C₂²	1	1	yes	characteristic
56	#10597266	C₂²	1	1	yes	characteristic
57	#10597264	C₂²	1	1	yes	characteristic
58	#10597264	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₁₉	—	2	1	no	—
61	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₉	—	2	1	no	—
64	C₂×C₂²⋊D₁₉	—	2	1	no	—
65	C₂×C₂²⋊D₁₉	—	2	1	no	—
66	C₂²×Dic₁₉	—	2	1	no	—
67	C₂²×Dic₁₉	D₄	1	1	yes	characteristic
68	C₂²×Dic₁₉	D₄	1	1	yes	characteristic
69	#64543	—	4	4	no	—
70	C₂×D₇₆	C₂³	1	1	yes	characteristic
71	C₂×C₄×D₁₉	—	2	1	no	—
72	C₂×Dic₃₈	—	2	1	no	—
73	C₂×Dic₃₈	—	2	1	no	—
74	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
75	SD₁₆×C₁₉	—	2	2	no	—
76	M₁₆×C₁₉	—	2	2	no	—
77	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
78	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₁₉	—	2	1	no	—
80	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
82	C₂²⋊C₄×C₁₉	—	2	1	no	—
83	C₂²⋊C₄×C₁₉	—	2	1	no	—
84	C₂²⋊Dic₁₉	—	2	1	no	—
85	C₂²⋊Dic₁₉	—	2	1	no	—
86	#64527	—	2	2	no	—
87	#64526	—	2	2	no	—
88	(C₂×D₁₉)⋊C₄	—	2	1	no	—
89	(C₂×D₁₉)⋊C₄	—	2	1	no	—
90	C₄⋊Dic₁₉	—	2	1	no	—
91	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
92	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
93	Dic₁₉⋊C₄	—	2	1	no	—
94	Dic₁₉⋊C₄	—	2	1	no	—
95	Dic₁₉⋊C₄	—	2	1	no	—
96	Dic₁₉⋊C₄	—	2	1	no	—
97	Dic₁₉⋊C₄	—	2	1	no	—
98	Dic₁₉⋊C₄	—	2	1	no	—
99	C₄×Dic₁₉	—	2	1	no	—
100	C₄×Dic₁₉	—	2	1	no	—
101	C₄×Dic₁₉	D₄	1	1	yes	characteristic
102	C₄×Dic₁₉	D₄	1	1	yes	characteristic
103	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
104	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
105	#64520	—	2	2	no	—
106	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
107	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
108	#64516	—	2	2	no	—
109	C₂³×C₁₉	—	2	1	no	—
110	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
111	C₂²×D₁₉	—	2	1	no	—
112	Q₈×C₁₉	—	2	1	no	—
113	Q₈×C₁₉	—	2	1	no	—
114	D₄×C₁₉	—	2	1	no	—
115	D₄×C₁₉	—	2	1	no	—
116	D₄×C₁₉	—	4	2	no	—
117	C₂×C₄×C₁₉	—	2	1	no	—
118	C₂×C₄×C₁₉	—	2	1	no	—
119	C₂×C₄×C₁₉	—	2	1	no	—
120	C₂×C₄×C₁₉	—	2	1	no	—
121	C₂×C₄×C₁₉	—	2	1	no	—
122	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
124	C₂²⋊D₁₉	—	4	2	no	—
125	C₂²⋊D₁₉	—	4	2	no	—
126	C₂²⋊D₁₉	—	4	2	no	—
127	C₂×Dic₁₉	—	2	2	no	—
128	C₂×Dic₁₉	—	2	2	no	—
129	C₂×Dic₁₉	—	2	1	no	—
130	C₂×Dic₁₉	—	2	1	no	—
131	C₂×Dic₁₉	—	2	1	no	—
132	C₂×Dic₁₉	—	4	2	no	—
133	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
137	D₇₆	—	2	1	no	—
138	D₇₆	—	2	1	no	—
139	C₄×D₁₉	—	4	2	no	—
140	Dic₃₈	—	2	1	no	—
141	Dic₃₈	—	2	1	no	—
142	Dic₃₈	—	4	2	no	—
143	Dic₃₈	—	4	2	no	—
144	C₈×C₁₉	—	2	1	no	—
145	C₈×C₁₉	#41	1	2	yes	complemented
146	C₁₉⋊C₈	—	2	1	no	—
147	C₁₉⋊C₈	#41	1	2	yes	complemented
148	#3182	—	19	1	no	maximal, Sylow
149	C₂²×C₁₉	—	2	1	no	—
150	C₂²×C₁₉	—	2	1	no	—
151	C₂²×C₁₉	—	2	1	no	—
152	C₂²×C₁₉	—	4	1	no	—
153	C₂²×C₁₉	—	4	1	no	—
154	C₂²×C₁₉	—	4	1	no	—
155	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
156	C₂×D₁₉	—	4	1	no	—
157	C₂×D₁₉	—	4	1	no	—
158	C₂×D₁₉	—	4	1	no	—
159	C₄×C₁₉	—	2	1	no	—
160	C₄×C₁₉	—	4	1	no	—
161	C₄×C₁₉	—	4	1	no	—
162	C₄×C₁₉	—	4	1	no	—
163	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
164	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
165	Dic₁₉	—	2	1	no	—
166	Dic₁₉	—	2	1	no	—
167	Dic₁₉	—	4	1	no	—
168	Dic₁₉	—	4	1	no	—
169	Dic₁₉	—	4	1	no	—
170	Dic₁₉	AGL(1,ℤ/8)	1	2	yes	—
171	Dic₁₉	#137	1	2	yes	—
172	#533	—	19	1	no	—
173	#532	—	19	1	no	—
174	#500	—	19	1	no	—
175	#492	—	19	1	no	—
176	#487	—	19	1	no	—
177	#486	—	19	1	no	—
178	#468	—	19	1	no	—
179	#468	—	19	1	no	—
180	#468	—	19	1	no	—
181	#468	D₁₉	1	1	yes	characteristic, complemented
182	#440	—	19	1	no	—
183	#440	—	19	1	no	—
184	#440	—	19	1	no	—
185	#440	—	19	1	no	—
186	C₄×M₁₆	—	19	1	no	—
187	C₂×C₁₉	—	4	1	no	—
188	C₂×C₁₉	—	8	1	no	—
189	C₂×C₁₉	#532	1	1	yes	characteristic
190	C₂×C₁₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
191	C₂×C₁₉	C₂×#137	1	1	yes	characteristic
192	D₁₉	—	8	1	no	—
193	C₂×#41	—	38	1	no	—
194	C₂×SD₁₆	—	19	1	no	—
195	C₂×SD₁₆	—	19	1	no	—
196	C₂×SD₁₆	—	19	1	no	—
197	C₂×SD₁₆	C₂×D₁₉	1	1	yes	characteristic
198	C₂×M₁₆	—	19	1	no	—
199	C₂×M₁₆	C₂×D₁₉	1	1	yes	characteristic
200	C₄⋊Q₈	—	19	1	no	—
201	C₄⋊Q₈	—	19	1	no	—
202	C₄⋊Q₈	—	19	1	no	—
203	#127	—	19	1	no	—
204	#125	—	38	1	no	—
205	#124	—	38	1	no	—
206	C₂²⋊Q₈	—	19	1	no	—
207	C₂²⋊Q₈	—	38	1	no	—
208	C₂²⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
209	#121	—	19	1	no	—
210	#121	—	38	1	no	—
211	#121	C₂×D₁₉	1	1	yes	characteristic
212	C₄×Q₈	—	19	1	no	—
213	C₄×Q₈	—	19	1	no	—
214	C₄×Q₈	—	38	1	no	—
215	C₄×D₄	—	19	1	no	—
216	C₄×D₄	—	19	1	no	—
217	C₄×D₄	—	38	1	no	—
218	#117	—	38	1	no	—
219	C₂×C₄²	—	19	1	no	—
220	#107	—	19	1	no	—
221	#107	—	19	1	no	—
222	#107	—	19	1	no	—
223	#107	C₂×D₁₉	1	1	yes	characteristic
224	#103	—	19	1	no	—
225	#103	—	19	1	no	—
226	#103	—	19	1	no	—
227	#103	C₂×D₁₉	1	1	yes	characteristic
228	#102	—	19	1	no	—
229	#102	—	19	1	no	—
230	#102	—	19	1	no	—
231	#102	C₂×D₁₉	1	1	yes	characteristic
232	#97	—	19	1	no	—
233	#97	—	19	1	no	—
234	C₄×C₈	—	19	1	no	—
235	C₄×C₈	—	19	1	no	—
236	C₁₉	#3182	1	1	yes	characteristic, complemented, Sylow
237	#41	—	76	4	no	—
238	C₂×Q₈	—	19	1	no	—
239	C₂×Q₈	—	38	1	no	—
240	C₂×Q₈	—	38	1	no	—
241	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
242	C₂×D₄	—	2	1	no	—
243	C₂×D₄	—	19	1	no	—
244	C₂×D₄	—	38	1	no	—
245	C₂×D₄	—	38	1	no	—
246	C₂×D₄	—	38	1	no	—
247	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
248	C₂²×C₄	—	19	1	no	—
249	C₂²×C₄	—	19	1	no	—
250	C₂²×C₄	—	38	1	no	—
251	C₂²×C₄	—	38	1	no	—
252	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
253	SD₁₆	—	2	2	no	—
254	SD₁₆	—	38	2	no	—
255	SD₁₆	—	38	2	no	—
256	SD₁₆	—	38	2	no	—
257	M₁₆	—	2	2	no	—
258	M₁₆	—	38	2	no	—
259	C₂×C₈	—	19	1	no	—
260	C₂×C₈	—	19	1	no	—
261	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
262	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	19	1	no	—
265	C₄⋊C₄	—	19	1	no	—
266	C₄⋊C₄	—	38	1	no	—
267	C₄⋊C₄	—	38	1	no	—
268	C₄⋊C₄	—	38	1	no	—
269	C₄⋊C₄	—	38	1	no	—
270	C₄⋊C₄	—	38	1	no	—
271	C₄⋊C₄	—	38	1	no	—
272	C₄⋊C₄	—	38	1	no	—
273	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
274	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	2	1	no	—
277	C₂²⋊C₄	—	38	1	no	—
278	C₂²⋊C₄	—	38	1	no	—
279	C₂²⋊C₄	—	38	1	no	—
280	C₂²⋊C₄	—	38	1	no	—
281	C₄²	—	19	1	no	—
282	C₄²	—	19	1	no	—
283	C₄²	—	19	1	no	—
284	C₄²	—	19	1	no	—
285	C₄²	—	38	1	no	—
286	C₄²	—	38	1	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	38	1	no	—
289	C₂³	D₄×D₁₉	1	1	yes	characteristic
290	Q₈	—	2	1	no	—
291	Q₈	—	2	1	no	—
292	Q₈	—	38	1	no	—
293	Q₈	—	38	1	no	—
294	Q₈	—	76	2	no	—
295	Q₈	—	76	2	no	—
296	D₄	—	2	1	no	—
297	D₄	—	2	1	no	—
298	D₄	—	4	2	no	—
299	D₄	—	38	1	no	—
300	D₄	—	38	1	no	—
301	D₄	—	76	2	no	—
302	D₄	—	76	2	no	—
303	D₄	—	76	2	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	19	1	no	—
310	C₂×C₄	—	19	1	no	—
311	C₂×C₄	—	19	1	no	—
312	C₂×C₄	—	19	1	no	—
313	C₂×C₄	—	38	2	no	—
314	C₂×C₄	—	38	2	no	—
315	C₂×C₄	—	38	1	no	—
316	C₂×C₄	—	38	1	no	—
317	C₂×C₄	—	38	1	no	—
318	C₂×C₄	—	76	2	no	—
319	C₂×C₄	—	76	2	no	—
320	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
321	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
322	C₈	—	2	1	no	—
323	C₈	—	19	2	no	—
324	C₈	—	38	1	no	—
325	C₈	#64543	1	2	yes	complemented
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	76	1	no	—
333	C₂²	—	76	1	no	—
334	C₂²	—	76	1	no	—
335	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	19	2	no	—
341	C₄	—	19	2	no	—
342	C₄	—	38	1	no	—
343	C₄	—	38	1	no	—
344	C₄	—	76	1	no	—
345	C₄	—	76	1	no	—
346	C₄	—	76	1	no	—
347	C₄	C₂×#64543	1	1	yes	characteristic
348	C₄	#41×D₁₉	1	1	yes	characteristic
349	C₂	—	4	1	no	—
350	C₂	—	8	1	no	—
351	C₂	—	152	1	no	—
352	C₂	#11927089	1	1	yes	characteristic
353	C₂	AGL(1,ℤ/8)×D₁₉	1	1	yes	characteristic
354	C₂	#137×D₁₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₉
commutator, cosocle	C₂×C₄×C₁₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉
Fitting	C₁₉×#468	C₂
socle	C₂²×C₁₉	C₂²×D₄
prime radical	#11927089	C₂
hypercenter	#468	D₁₉
hypocenter	C₁₉	#3182
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #468

Complements

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

N	H ≅ G/N	type
1. #11927096	C₂	semidirect
3. #11926948	C₂	semidirect
4. #11926915	C₂	semidirect
5. #11926893	C₂	semidirect
6. #11926889	C₂	semidirect
7. #11926792	C₂	semidirect
8. #11926790	C₂	semidirect
9. #11926788	C₂	semidirect
10. #11926647	C₂	semidirect
11. #11926645	C₂	semidirect
12. #11926643	C₂	semidirect
13. #11926641	C₂	semidirect
14. M₁₆×Dic₁₉	C₂	semidirect
15. C₁₉×#468	C₂	semidirect
145. C₈×C₁₉	#41	semidirect
147. C₁₉⋊C₈	#41	semidirect
181. #468	D₁₉	semidirect
236. C₁₉	#3182	semidirect
325. C₈	#64543	semidirect

Maximal subgroups

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

subgroup	index
#11927096	2
#11927089	2
#11926948	2
#11926915	2
#11926893	2
#11926889	2
#11926792	2
#11926790	2
#11926788	2
#11926647	2
#11926645	2
#11926643	2
#11926641	2
M₁₆×Dic₁₉	2
C₁₉×#468	2
#3182	19