Subgroups — #14574643

Order 2432.

390 conjugacy classes of subgroups.

The subgroup lattice has 372 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	#11927090	C₂	1	1	yes	maximal, characteristic
3	#11926937	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926935	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926905	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926902	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926753	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926752	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926743	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926742	C₂	1	1	yes	maximal, characteristic, complemented
11	#11926708	C₂	1	1	yes	maximal, characteristic, complemented
12	#11926706	C₂	1	1	yes	maximal, characteristic, complemented
13	#11926704	C₂	1	1	yes	maximal, characteristic, complemented
14	#11926598	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₉×#452	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#41×C₁₉	C₂²	1	1	yes	characteristic
17	C₂×#64541	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic
19	C₂×SD₁₆×C₁₉	C₂²	1	1	yes	characteristic
20	C₁₉×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
21	#10597393	C₂²	1	1	yes	characteristic
22	Q₈×Dic₁₉	C₂²	1	1	yes	characteristic
23	#10597390	C₂²	1	1	yes	characteristic
24	#10597390	C₂²	1	1	yes	characteristic
25	C₂×#64528	C₂²	1	1	yes	characteristic
26	C₂×#64528	C₂²	1	1	yes	characteristic
27	C₂×#64527	C₂²	1	1	yes	characteristic
28	#10597386	C₂²	1	1	yes	characteristic
29	#10597385	—	2	1	no	—
30	#10597385	—	2	1	no	—
31	#10597382	C₂²	1	1	yes	characteristic
32	C₂×#64526	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₁₉	—	2	1	no	—
34	C₄×C₂²⋊D₁₉	—	2	1	no	—
35	#10597370	C₂²	1	1	yes	characteristic
36	C₂×C₄×Dic₁₉	C₂²	1	1	yes	characteristic
37	C₂×Dic₇₆	C₂²	1	1	yes	characteristic
38	C₂×#64516	C₂²	1	1	yes	characteristic
39	#10597335	—	2	1	no	—
40	#10597334	C₂²	1	1	yes	characteristic
41	#10597333	—	2	1	no	—
42	#10597333	C₂²	1	1	yes	characteristic
43	#10597324	—	2	1	no	—
44	#10597323	—	2	1	no	—
45	C₁₉×#103	C₂²	1	1	yes	characteristic
46	C₁₉×#102	C₂²	1	1	yes	characteristic
47	C₁₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
48	#10597292	C₂²	1	1	yes	characteristic
49	#10597289	C₂²	1	1	yes	characteristic
50	C₂²⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
51	#10597278	C₂²	1	1	yes	characteristic
52	#10597273	C₂²	1	1	yes	characteristic
53	#10597272	C₂²	1	1	yes	characteristic
54	#10597272	C₂²	1	1	yes	characteristic
55	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
56	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
57	#10597267	C₂²	1	1	yes	characteristic
58	#10597266	C₂²	1	1	yes	characteristic
59	#41×C₁₉	—	4	1	no	—
60	#41×C₁₉	—	4	1	no	—
61	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
62	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
63	C₂×D₄×C₁₉	—	2	1	no	—
64	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	—	2	1	no	—
66	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₁₉	—	2	1	no	—
68	C₂×C₂²⋊D₁₉	—	2	1	no	—
69	C₂×C₂²⋊D₁₉	—	2	1	no	—
70	C₂²×Dic₁₉	D₄	1	1	yes	characteristic
71	C₂²×Dic₁₉	D₄	1	1	yes	characteristic
72	#64541	—	4	1	no	—
73	#64541	—	4	1	no	—
74	C₂×D₇₆	C₂³	1	1	yes	characteristic
75	C₂×C₄×D₁₉	—	2	1	no	—
76	C₂×Dic₃₈	—	2	1	no	—
77	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
78	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
79	Q₁₆×C₁₉	—	4	1	no	—
80	SD₁₆×C₁₉	—	4	1	no	—
81	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
82	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₁₉	—	2	1	no	—
84	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
85	C₂²⋊C₄×C₁₉	—	2	1	no	—
86	C₂²⋊Dic₁₉	—	2	1	no	—
87	C₂²⋊Dic₁₉	—	2	1	no	—
88	C₂²⋊Dic₁₉	—	2	1	no	—
89	#64528	—	4	1	no	—
90	#64528	—	4	1	no	—
91	#64527	—	4	1	no	—
92	#64526	—	4	1	no	—
93	(C₂×D₁₉)⋊C₄	—	2	1	no	—
94	(C₂×D₁₉)⋊C₄	—	2	1	no	—
95	C₄⋊Dic₁₉	—	2	1	no	—
96	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
97	Dic₁₉⋊C₄	—	2	1	no	—
98	Dic₁₉⋊C₄	—	2	1	no	—
99	Dic₁₉⋊C₄	—	2	1	no	—
100	Dic₁₉⋊C₄	—	2	1	no	—
101	Dic₁₉⋊C₄	—	2	1	no	—
102	Dic₁₉⋊C₄	—	2	1	no	—
103	Dic₁₉⋊C₄	—	2	1	no	—
104	C₄×Dic₁₉	—	2	1	no	—
105	C₄×Dic₁₉	—	2	1	no	—
106	C₄×Dic₁₉	D₄	1	1	yes	characteristic
107	C₄×Dic₁₉	D₄	1	1	yes	characteristic
108	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
109	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
110	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
111	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
112	Dic₇₆	—	4	1	no	—
113	#64516	—	4	1	no	—
114	C₂³×C₁₉	—	2	1	no	—
115	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
116	C₂²×D₁₉	—	2	1	no	—
117	Q₈×C₁₉	—	2	1	no	—
118	Q₈×C₁₉	—	2	1	no	—
119	Q₈×C₁₉	—	2	1	no	—
120	Q₈×C₁₉	—	2	1	no	—
121	D₄×C₁₉	—	2	1	no	—
122	D₄×C₁₉	—	2	1	no	—
123	D₄×C₁₉	—	4	2	no	—
124	C₂×C₄×C₁₉	—	2	1	no	—
125	C₂×C₄×C₁₉	—	2	1	no	—
126	C₂×C₄×C₁₉	—	2	1	no	—
127	C₂×C₄×C₁₉	—	2	1	no	—
128	C₂×C₄×C₁₉	—	2	1	no	—
129	C₂×C₄×C₁₉	—	4	2	no	—
130	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
132	C₂²⋊D₁₉	—	4	2	no	—
133	C₂²⋊D₁₉	—	4	2	no	—
134	C₂²⋊D₁₉	—	4	2	no	—
135	C₂×Dic₁₉	—	2	2	no	—
136	C₂×Dic₁₉	—	2	1	no	—
137	C₂×Dic₁₉	—	2	1	no	—
138	C₂×Dic₁₉	—	2	1	no	—
139	C₂×Dic₁₉	—	4	1	no	—
140	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
141	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
142	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
143	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
144	D₇₆	—	2	1	no	—
145	D₇₆	—	2	1	no	—
146	C₄×D₁₉	—	4	2	no	—
147	Dic₃₈	—	2	1	no	—
148	Dic₃₈	—	2	1	no	—
149	Dic₃₈	—	2	1	no	—
150	Dic₃₈	—	2	1	no	—
151	Dic₃₈	—	4	2	no	—
152	C₈×C₁₉	—	2	1	no	—
153	C₈×C₁₉	—	2	1	no	—
154	C₁₉⋊C₈	—	2	1	no	—
155	C₁₉⋊C₈	—	2	1	no	—
156	#3216	—	19	1	no	maximal, Sylow
157	C₂²×C₁₉	—	2	1	no	—
158	C₂²×C₁₉	—	2	1	no	—
159	C₂²×C₁₉	—	2	1	no	—
160	C₂²×C₁₉	—	4	1	no	—
161	C₂²×C₁₉	—	4	1	no	—
162	C₂²×C₁₉	—	4	1	no	—
163	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
164	C₂×D₁₉	—	4	1	no	—
165	C₂×D₁₉	—	4	1	no	—
166	C₂×D₁₉	—	4	1	no	—
167	C₄×C₁₉	—	2	1	no	—
168	C₄×C₁₉	—	4	1	no	—
169	C₄×C₁₉	—	4	1	no	—
170	C₄×C₁₉	—	4	1	no	—
171	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
172	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
173	Dic₁₉	—	2	1	no	—
174	Dic₁₉	—	2	1	no	—
175	Dic₁₉	—	2	1	no	—
176	Dic₁₉	—	4	1	no	—
177	Dic₁₉	—	4	1	no	—
178	Dic₁₉	—	4	1	no	—
179	Dic₁₉	#137	1	2	yes	—
180	#533	—	19	1	no	—
181	#532	—	19	1	no	—
182	#497	—	19	1	no	—
183	#497	—	19	1	no	—
184	#489	—	19	1	no	—
185	#489	—	19	1	no	—
186	#462	—	19	1	no	—
187	#462	—	19	1	no	—
188	#460	—	19	1	no	—
189	#460	—	19	1	no	—
190	#452	—	19	1	no	—
191	#452	—	19	1	no	—
192	#452	—	19	1	no	—
193	#452	D₁₉	1	1	yes	characteristic, complemented
194	#432	—	19	1	no	—
195	C₂×C₁₉	—	4	1	no	—
196	C₂×C₁₉	—	8	1	no	—
197	C₂×C₁₉	#535	1	1	yes	characteristic
198	C₂×C₁₉	C₂×#137	1	1	yes	characteristic
199	C₂×C₁₉	#575	1	1	yes	characteristic
200	D₁₉	—	8	1	no	—
201	C₂×#41	—	19	1	no	—
202	C₂×#41	C₂×D₁₉	1	1	yes	characteristic
203	C₂×Q₁₆	—	19	1	no	—
204	C₂×Q₁₆	—	19	1	no	—
205	C₂×Q₁₆	—	19	1	no	—
206	C₂×Q₁₆	C₂×D₁₉	1	1	yes	characteristic
207	C₂×SD₁₆	—	19	1	no	—
208	C₂×SD₁₆	—	19	1	no	—
209	C₂×SD₁₆	—	19	1	no	—
210	C₂×SD₁₆	C₂×D₁₉	1	1	yes	characteristic
211	C₄⋊Q₈	—	19	1	no	—
212	C₄⋊Q₈	—	19	1	no	—
213	C₄⋊Q₈	—	19	1	no	—
214	#127	—	19	1	no	—
215	#125	—	38	1	no	—
216	#124	—	19	1	no	—
217	#124	—	19	1	no	—
218	C₂²⋊Q₈	—	19	1	no	—
219	C₂²⋊Q₈	—	38	1	no	—
220	C₂²⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
221	#121	—	38	1	no	—
222	#121	—	38	1	no	—
223	C₄×Q₈	—	19	1	no	—
224	C₄×Q₈	—	19	1	no	—
225	C₄×Q₈	—	38	1	no	—
226	C₄×D₄	—	38	1	no	—
227	C₄×D₄	—	38	1	no	—
228	#117	—	38	1	no	—
229	C₂×C₄²	—	19	1	no	—
230	C₄⋊C₈	—	19	1	no	—
231	C₄⋊C₈	—	19	1	no	—
232	#103	—	19	1	no	—
233	#103	—	19	1	no	—
234	#103	—	19	1	no	—
235	#103	C₂×D₁₉	1	1	yes	characteristic
236	#102	—	19	1	no	—
237	#102	—	19	1	no	—
238	#102	—	19	1	no	—
239	#102	C₂×D₁₉	1	1	yes	characteristic
240	C₂²⋊C₈	—	19	1	no	—
241	C₂²⋊C₈	C₂×D₁₉	1	1	yes	characteristic
242	#97	—	19	1	no	—
243	#97	—	19	1	no	—
244	C₁₉	#3216	1	1	yes	characteristic, complemented, Sylow
245	#41	—	4	1	no	—
246	#41	—	4	1	no	—
247	#41	—	76	1	no	—
248	#41	—	76	1	no	—
249	C₂×Q₈	—	19	1	no	—
250	C₂×Q₈	—	19	1	no	—
251	C₂×Q₈	—	38	1	no	—
252	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
253	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
254	C₂×D₄	—	2	1	no	—
255	C₂×D₄	—	19	1	no	—
256	C₂×D₄	—	38	1	no	—
257	C₂×D₄	—	38	1	no	—
258	C₂×D₄	—	38	1	no	—
259	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
260	C₂²×C₄	—	2	1	no	—
261	C₂²×C₄	—	19	1	no	—
262	C₂²×C₄	—	19	1	no	—
263	C₂²×C₄	—	38	1	no	—
264	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
265	Q₁₆	—	4	1	no	—
266	Q₁₆	—	76	1	no	—
267	Q₁₆	—	76	1	no	—
268	Q₁₆	—	76	1	no	—
269	SD₁₆	—	4	1	no	—
270	SD₁₆	—	76	1	no	—
271	SD₁₆	—	76	1	no	—
272	SD₁₆	—	76	1	no	—
273	C₂×C₈	—	19	1	no	—
274	C₂×C₈	—	19	1	no	—
275	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
276	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	19	1	no	—
279	C₄⋊C₄	—	38	1	no	—
280	C₄⋊C₄	—	38	1	no	—
281	C₄⋊C₄	—	38	1	no	—
282	C₄⋊C₄	—	38	1	no	—
283	C₄⋊C₄	—	38	1	no	—
284	C₄⋊C₄	—	38	1	no	—
285	C₄⋊C₄	—	38	1	no	—
286	C₄⋊C₄	—	38	1	no	—
287	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
288	C₂²⋊C₄	—	2	1	no	—
289	C₂²⋊C₄	—	38	1	no	—
290	C₂²⋊C₄	—	38	1	no	—
291	C₂²⋊C₄	—	38	1	no	—
292	C₂²⋊C₄	—	38	1	no	—
293	C₂²⋊C₄	—	38	1	no	—
294	C₄²	—	19	1	no	—
295	C₄²	—	19	1	no	—
296	C₄²	—	19	1	no	—
297	C₄²	—	19	1	no	—
298	C₄²	—	38	1	no	—
299	C₄²	—	38	1	no	—
300	C₂³	—	2	1	no	—
301	C₂³	—	38	1	no	—
302	C₂³	D₄×D₁₉	1	1	yes	characteristic
303	Q₈	—	2	1	no	—
304	Q₈	—	2	1	no	—
305	Q₈	—	2	1	no	—
306	Q₈	—	2	1	no	—
307	Q₈	—	38	1	no	—
308	Q₈	—	38	1	no	—
309	Q₈	—	38	1	no	—
310	Q₈	—	38	1	no	—
311	Q₈	—	76	2	no	—
312	D₄	—	2	1	no	—
313	D₄	—	2	1	no	—
314	D₄	—	4	2	no	—
315	D₄	—	38	1	no	—
316	D₄	—	38	1	no	—
317	D₄	—	76	2	no	—
318	D₄	—	76	2	no	—
319	D₄	—	76	2	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	19	1	no	—
327	C₂×C₄	—	19	1	no	—
328	C₂×C₄	—	19	1	no	—
329	C₂×C₄	—	19	1	no	—
330	C₂×C₄	—	38	2	no	—
331	C₂×C₄	—	38	1	no	—
332	C₂×C₄	—	38	1	no	—
333	C₂×C₄	—	38	1	no	—
334	C₂×C₄	—	76	2	no	—
335	C₂×C₄	—	76	1	no	—
336	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
337	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
338	C₈	—	2	1	no	—
339	C₈	—	2	1	no	—
340	C₈	—	38	1	no	—
341	C₈	—	38	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	4	1	no	—
346	C₂²	—	4	1	no	—
347	C₂²	—	4	1	no	—
348	C₂²	—	76	1	no	—
349	C₂²	—	76	1	no	—
350	C₂²	—	76	1	no	—
351	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	4	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	4	1	no	—
356	C₄	—	19	2	no	—
357	C₄	—	38	1	no	—
358	C₄	—	38	1	no	—
359	C₄	—	38	1	no	—
360	C₄	—	76	1	no	—
361	C₄	—	76	1	no	—
362	C₄	—	76	1	no	—
363	C₄	#10597430	1	1	yes	characteristic
364	C₄	#10597430	1	1	yes	characteristic
365	C₂	—	4	1	no	—
366	C₂	—	8	1	no	—
367	C₂	—	152	1	no	—
368	C₂	#11927109	1	1	yes	characteristic
369	C₂	#137×D₁₉	1	1	yes	characteristic
370	C₂	#11927341	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×#452	C₂
socle	C₂²×C₁₉	C₂²×D₄
prime radical	#11927090	C₂
hypercenter	#452	D₁₉
hypocenter	C₁₉	#3216
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #452

Complements

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

N	H ≅ G/N	type
1. #11927096	C₂	semidirect
3. #11926937	C₂	semidirect
4. #11926935	C₂	semidirect
5. #11926905	C₂	semidirect
6. #11926902	C₂	semidirect
7. #11926753	C₂	semidirect
8. #11926752	C₂	semidirect
9. #11926743	C₂	semidirect
10. #11926742	C₂	semidirect
11. #11926708	C₂	semidirect
12. #11926706	C₂	semidirect
13. #11926704	C₂	semidirect
14. #11926598	C₂	semidirect
15. C₁₉×#452	C₂	semidirect
193. #452	D₁₉	semidirect
244. C₁₉	#3216	semidirect

Maximal subgroups

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

subgroup	index
#11927096	2
#11927090	2
#11926937	2
#11926935	2
#11926905	2
#11926902	2
#11926753	2
#11926752	2
#11926743	2
#11926742	2
#11926708	2
#11926706	2
#11926704	2
#11926598	2
C₁₉×#452	2
#3216	19