Subgroups — #14573779

Order 2432.

430 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927089	C₂	1	1	yes	maximal, characteristic, complemented
2	#41×Dic₁₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926869	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926865	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926827	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926822	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926742	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926737	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926689	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926656	C₂	1	1	yes	maximal, characteristic, complemented
11	#11926653	C₂	1	1	yes	maximal, characteristic, complemented
12	#11926647	C₂	1	1	yes	maximal, characteristic, complemented
13	#11926645	C₂	1	1	yes	maximal, characteristic, complemented
14	#11926598	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₉×#449	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#41×C₁₉	C₂²	1	1	yes	characteristic
17	C₂×#64543	—	2	1	no	—
18	C₂×SD₁₆×C₁₉	C₂²	1	1	yes	characteristic
19	C₂×D₈×C₁₉	C₂²	1	1	yes	characteristic
20	C₁₉×#121	C₂²	1	1	yes	characteristic
21	Q₈×Dic₁₉	C₂²	1	1	yes	characteristic
22	Q₈×Dic₁₉	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	—	2	1	no	—
27	D₄×Dic₁₉	—	2	1	no	—
28	D₄×Dic₁₉	C₂²	1	1	yes	characteristic
29	D₄×Dic₁₉	C₂²	1	1	yes	characteristic
30	D₄×Dic₁₉	C₂²	1	1	yes	characteristic
31	C₂×#64525	C₂²	1	1	yes	characteristic
32	#10597376	C₂²	1	1	yes	characteristic
33	#10597372	—	2	1	no	—
34	#10597372	C₂²	1	1	yes	characteristic
35	C₂×C₄×Dic₁₉	—	2	1	no	—
36	C₂×C₄×Dic₁₉	C₂²	1	1	yes	characteristic
37	#10597339	C₂²	1	1	yes	characteristic
38	#10597334	C₂²	1	1	yes	characteristic
39	#10597331	—	2	1	no	—
40	(C₂²⋊D₁₉)⋊C₄	—	2	1	no	—
41	C₁₉×#103	C₂²	1	1	yes	characteristic
42	C₁₉×#102	C₂²	1	1	yes	characteristic
43	C₁₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
44	#10597292	C₂²	1	1	yes	characteristic
45	#10597289	C₂²	1	1	yes	characteristic
46	#10597289	C₂²	1	1	yes	characteristic
47	#10597289	C₂²	1	1	yes	characteristic
48	C₂²⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
49	#10597278	C₂²	1	1	yes	characteristic
50	#10597278	C₂²	1	1	yes	characteristic
51	#10597275	C₂²	1	1	yes	characteristic
52	#10597274	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	#10597265	C₂²	1	1	yes	characteristic
58	#10597264	C₂²	1	1	yes	characteristic
59	#41×C₁₉	—	2	2	no	—
60	#41×C₁₉	—	2	2	no	—
61	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₁₉	—	2	1	no	—
63	C₂×D₄×C₁₉	—	2	1	no	—
64	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
65	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	—	2	1	no	—
67	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
68	C₂×C₂²⋊D₁₉	—	2	1	no	—
69	C₂×C₂²⋊D₁₉	—	2	1	no	—
70	C₂×C₂²⋊D₁₉	—	2	1	no	—
71	C₂²×Dic₁₉	—	2	1	no	—
72	C₂²×Dic₁₉	—	2	1	no	—
73	C₂²×Dic₁₉	—	2	1	no	—
74	C₂²×Dic₁₉	D₄	1	1	yes	characteristic
75	C₂²×Dic₁₉	D₄	1	1	yes	characteristic
76	#64543	—	4	4	no	—
77	C₂×D₇₆	C₂³	1	1	yes	characteristic
78	C₂×C₄×D₁₉	—	2	1	no	—
79	C₂×Dic₃₈	—	2	1	no	—
80	SD₁₆×C₁₉	—	2	2	no	—
81	D₈×C₁₉	—	2	2	no	—
82	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
83	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
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	C₂²⋊Dic₁₉	—	2	1	no	—
90	#64527	—	2	2	no	—
91	#64525	—	2	2	no	—
92	(C₂×D₁₉)⋊C₄	—	2	1	no	—
93	(C₂×D₁₉)⋊C₄	—	2	1	no	—
94	C₄⋊Dic₁₉	—	2	1	no	—
95	C₄⋊Dic₁₉	—	2	1	no	—
96	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
97	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
98	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
99	Dic₁₉⋊C₄	—	2	1	no	—
100	C₄×Dic₁₉	—	2	1	no	—
101	C₄×Dic₁₉	—	2	1	no	—
102	C₄×Dic₁₉	—	2	1	no	—
103	C₄×Dic₁₉	D₄	1	1	yes	characteristic
104	C₄×Dic₁₉	D₄	1	1	yes	characteristic
105	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
106	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
107	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
108	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
109	C₂³×C₁₉	—	2	1	no	—
110	C₂³×C₁₉	—	2	1	no	—
111	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
112	C₂²×D₁₉	—	2	1	no	—
113	Q₈×C₁₉	—	2	1	no	—
114	Q₈×C₁₉	#41	1	2	yes	complemented
115	D₄×C₁₉	—	2	4	no	—
116	D₄×C₁₉	—	2	1	no	—
117	D₄×C₁₉	—	2	1	no	—
118	D₄×C₁₉	—	2	1	no	—
119	D₄×C₁₉	—	4	2	no	—
120	D₄×C₁₉	#41	1	2	yes	complemented
121	C₂×C₄×C₁₉	—	2	4	no	—
122	C₂×C₄×C₁₉	—	2	1	no	—
123	C₂×C₄×C₁₉	—	2	1	no	—
124	C₂×C₄×C₁₉	—	2	1	no	—
125	C₂×C₄×C₁₉	—	2	1	no	—
126	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
128	C₂²⋊D₁₉	—	4	2	no	—
129	C₂²⋊D₁₉	—	4	2	no	—
130	C₂²⋊D₁₉	—	4	2	no	—
131	C₂×Dic₁₉	—	2	2	no	—
132	C₂×Dic₁₉	—	2	1	no	—
133	C₂×Dic₁₉	—	2	1	no	—
134	C₂×Dic₁₉	—	2	1	no	—
135	C₂×Dic₁₉	—	4	1	no	—
136	C₂×Dic₁₉	—	4	2	no	—
137	C₂×Dic₁₉	—	4	2	no	—
138	C₂×Dic₁₉	—	4	2	no	—
139	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
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	D₇₆	—	2	1	no	—
144	D₇₆	—	2	1	no	—
145	C₄×D₁₉	—	4	2	no	—
146	Dic₃₈	—	4	2	no	—
147	C₈×C₁₉	—	2	1	no	—
148	C₈×C₁₉	—	2	1	no	—
149	C₁₉⋊C₈	—	2	1	no	—
150	C₁₉⋊C₈	—	2	1	no	—
151	#3115	—	19	1	no	maximal, Sylow
152	C₂²×C₁₉	—	2	1	no	—
153	C₂²×C₁₉	—	2	2	no	—
154	C₂²×C₁₉	—	2	1	no	—
155	C₂²×C₁₉	—	2	1	no	—
156	C₂²×C₁₉	—	4	1	no	—
157	C₂²×C₁₉	—	4	1	no	—
158	C₂²×C₁₉	—	4	1	no	—
159	C₂²×C₁₉	—	4	1	no	—
160	C₂²×C₁₉	—	4	1	no	—
161	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
162	C₂×D₁₉	—	4	1	no	—
163	C₂×D₁₉	—	4	1	no	—
164	C₂×D₁₉	—	4	1	no	—
165	C₄×C₁₉	—	2	1	no	—
166	C₄×C₁₉	—	2	2	no	—
167	C₄×C₁₉	—	4	1	no	—
168	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
169	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
170	Dic₁₉	—	2	1	no	—
171	Dic₁₉	—	2	1	no	—
172	Dic₁₉	—	2	1	no	—
173	Dic₁₉	—	4	1	no	—
174	Dic₁₉	—	4	1	no	—
175	Dic₁₉	—	4	1	no	—
176	Dic₁₉	AGL(1,ℤ/8)	1	2	yes	—
177	#532	—	19	1	no	—
178	C₄×#41	—	19	1	no	—
179	#482	—	19	1	no	—
180	#482	—	19	1	no	—
181	#475	—	19	1	no	—
182	#474	—	19	1	no	—
183	#460	—	19	1	no	—
184	#459	—	19	1	no	—
185	#449	—	19	1	no	—
186	#449	D₁₉	1	1	yes	characteristic, complemented
187	#442	—	19	1	no	—
188	#442	—	19	1	no	—
189	#440	—	19	1	no	—
190	#440	—	19	1	no	—
191	#432	—	19	1	no	—
192	C₂×C₁₉	—	4	1	no	—
193	C₂×C₁₉	—	4	2	no	—
194	C₂×C₁₉	—	8	1	no	—
195	C₂×C₁₉	#525	1	1	yes	characteristic
196	C₂×C₁₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
197	C₂×C₁₉	#575	1	1	yes	characteristic
198	D₁₉	—	8	1	no	—
199	C₂×#41	—	38	1	no	—
200	C₂×#41	C₂×D₁₉	1	1	yes	characteristic
201	C₂×SD₁₆	—	19	1	no	—
202	C₂×SD₁₆	C₂×D₁₉	1	1	yes	characteristic
203	C₂×D₈	—	19	1	no	—
204	C₂×D₈	C₂×D₁₉	1	1	yes	characteristic
205	C₄⋊Q₈	—	19	1	no	—
206	#127	—	19	1	no	—
207	#124	—	38	1	no	—
208	#121	—	19	1	no	—
209	#121	—	38	1	no	—
210	#121	C₂×D₁₉	1	1	yes	characteristic
211	C₄×Q₈	—	19	1	no	—
212	C₄×Q₈	—	19	1	no	—
213	C₄×D₄	—	19	1	no	—
214	C₄×D₄	—	19	1	no	—
215	C₄×D₄	—	19	1	no	—
216	C₄×D₄	—	19	1	no	—
217	C₄×D₄	—	38	1	no	—
218	C₄×D₄	—	38	1	no	—
219	C₄×D₄	—	38	1	no	—
220	#117	—	19	1	no	—
221	#117	—	38	1	no	—
222	C₂×C₄²	—	19	1	no	—
223	C₂×C₄²	—	38	1	no	—
224	#107	—	19	1	no	—
225	#107	—	19	1	no	—
226	#106	—	19	1	no	—
227	#106	—	19	1	no	—
228	C₄⋊C₈	—	19	1	no	—
229	C₄⋊C₈	—	19	1	no	—
230	#103	—	19	1	no	—
231	#103	C₂×D₁₉	1	1	yes	characteristic
232	#102	—	19	1	no	—
233	#102	—	19	1	no	—
234	#102	—	19	1	no	—
235	#102	—	19	1	no	—
236	#102	—	19	1	no	—
237	#102	C₂×D₁₉	1	1	yes	characteristic
238	C₂²⋊C₈	—	19	1	no	—
239	C₂²⋊C₈	C₂×D₁₉	1	1	yes	characteristic
240	#97	—	19	1	no	—
241	#97	—	19	1	no	—
242	C₁₉	#3115	1	1	yes	characteristic, complemented, Sylow
243	#41	—	2	2	no	—
244	#41	—	2	2	no	—
245	#41	—	76	4	no	—
246	C₂×Q₈	—	38	1	no	—
247	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
248	C₂×D₄	—	2	1	no	—
249	C₂×D₄	—	2	1	no	—
250	C₂×D₄	—	19	1	no	—
251	C₂×D₄	—	38	1	no	—
252	C₂×D₄	—	38	1	no	—
253	C₂×D₄	—	38	1	no	—
254	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
255	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
256	C₂²×C₄	—	2	1	no	—
257	C₂²×C₄	—	19	1	no	—
258	C₂²×C₄	—	19	1	no	—
259	C₂²×C₄	—	38	1	no	—
260	C₂²×C₄	—	38	1	no	—
261	C₂²×C₄	—	38	1	no	—
262	C₂²×C₄	—	38	1	no	—
263	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
264	SD₁₆	—	2	2	no	—
265	SD₁₆	—	38	2	no	—
266	D₈	—	2	2	no	—
267	D₈	—	38	2	no	—
268	C₂×C₈	—	19	1	no	—
269	C₂×C₈	—	19	1	no	—
270	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
271	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
272	C₄⋊C₄	—	19	1	no	—
273	C₄⋊C₄	—	19	1	no	—
274	C₄⋊C₄	—	19	1	no	—
275	C₄⋊C₄	—	38	1	no	—
276	C₄⋊C₄	—	38	1	no	—
277	C₄⋊C₄	—	38	1	no	—
278	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
279	C₂²⋊C₄	—	2	1	no	—
280	C₂²⋊C₄	—	38	1	no	—
281	C₂²⋊C₄	—	38	1	no	—
282	C₂²⋊C₄	—	38	1	no	—
283	C₂²⋊C₄	—	38	1	no	—
284	C₂²⋊C₄	—	38	1	no	—
285	C₂²⋊C₄	—	38	1	no	—
286	C₄²	—	19	1	no	—
287	C₄²	—	19	1	no	—
288	C₄²	—	19	1	no	—
289	C₄²	—	19	1	no	—
290	C₄²	—	38	1	no	—
291	C₄²	—	38	1	no	—
292	C₄²	—	38	1	no	—
293	C₂³	—	2	1	no	—
294	C₂³	—	2	1	no	—
295	C₂³	—	38	1	no	—
296	C₂³	D₄×D₁₉	1	1	yes	characteristic
297	Q₈	—	2	1	no	—
298	Q₈	—	76	2	no	—
299	Q₈	#64543	1	2	yes	complemented
300	D₄	—	2	4	no	—
301	D₄	—	2	1	no	—
302	D₄	—	2	1	no	—
303	D₄	—	2	1	no	—
304	D₄	—	4	2	no	—
305	D₄	—	38	1	no	—
306	D₄	—	38	1	no	—
307	D₄	—	76	2	no	—
308	D₄	—	76	2	no	—
309	D₄	—	76	2	no	—
310	D₄	#64543	1	2	yes	complemented
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	19	1	no	—
317	C₂×C₄	—	19	1	no	—
318	C₂×C₄	—	19	1	no	—
319	C₂×C₄	—	19	1	no	—
320	C₂×C₄	—	38	2	no	—
321	C₂×C₄	—	38	1	no	—
322	C₂×C₄	—	38	1	no	—
323	C₂×C₄	—	38	1	no	—
324	C₂×C₄	—	76	2	no	—
325	C₂×C₄	—	76	2	no	—
326	C₂×C₄	—	76	1	no	—
327	C₂×C₄	—	76	2	no	—
328	C₂×C₄	—	76	2	no	—
329	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
330	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
331	C₈	—	2	1	no	—
332	C₈	—	2	1	no	—
333	C₈	—	38	1	no	—
334	C₈	—	38	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	4	1	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	76	1	no	—
345	C₂²	—	76	1	no	—
346	C₂²	—	76	1	no	—
347	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	2	no	—
350	C₄	—	4	1	no	—
351	C₄	—	19	2	no	—
352	C₄	—	38	1	no	—
353	C₄	—	38	1	no	—
354	C₄	—	38	1	no	—
355	C₄	—	76	1	no	—
356	C₄	—	76	1	no	—
357	C₄	—	76	1	no	—
358	C₄	C₂×#64543	1	1	yes	characteristic
359	C₄	C₂×#64543	1	1	yes	characteristic
360	C₂	—	4	1	no	—
361	C₂	—	4	2	no	—
362	C₂	—	8	1	no	—
363	C₂	—	152	1	no	—
364	C₂	#11927050	1	1	yes	characteristic
365	C₂	AGL(1,ℤ/8)×D₁₉	1	1	yes	characteristic
366	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×#449	C₂
socle	C₂²×C₁₉	C₂²×D₄
hypercenter	#449	D₁₉
hypocenter	C₁₉	#3115
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #449

Complements

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

N	H ≅ G/N	type
1. #11927089	C₂	semidirect
2. #41×Dic₁₉	C₂	semidirect
3. #11926869	C₂	semidirect
4. #11926865	C₂	semidirect
5. #11926827	C₂	semidirect
6. #11926822	C₂	semidirect
7. #11926742	C₂	semidirect
8. #11926737	C₂	semidirect
9. #11926689	C₂	semidirect
10. #11926656	C₂	semidirect
11. #11926653	C₂	semidirect
12. #11926647	C₂	semidirect
13. #11926645	C₂	semidirect
14. #11926598	C₂	semidirect
15. C₁₉×#449	C₂	semidirect
114. Q₈×C₁₉	#41	semidirect
120. D₄×C₁₉	#41	semidirect
186. #449	D₁₉	semidirect
242. C₁₉	#3115	semidirect
299. Q₈	#64543	semidirect
310. D₄	#64543	semidirect

Maximal subgroups

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

subgroup	index
#11927089	2
#41×Dic₁₉	2
#11926869	2
#11926865	2
#11926827	2
#11926822	2
#11926742	2
#11926737	2
#11926689	2
#11926656	2
#11926653	2
#11926647	2
#11926645	2
#11926598	2
C₁₉×#449	2
#3115	19