Subgroups — #14573831

Order 2432.

396 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927148	C₂	1	1	yes	maximal, characteristic, complemented
2	#11927023	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926861	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926856	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926852	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926844	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926741	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926686	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926686	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926656	C₂	1	1	yes	maximal, characteristic, complemented
11	#11926641	C₂	1	1	yes	maximal, characteristic, complemented
12	#11926635	C₂	1	1	yes	maximal, characteristic, complemented
13	D₈×Dic₁₉	C₂	1	1	yes	maximal, characteristic, complemented
14	#11926604	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₉×#459	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#64543	C₂²	1	1	yes	characteristic
17	C₂×D₈×C₁₉	C₂²	1	1	yes	characteristic
18	C₂×D₈×C₁₉	C₂²	1	1	yes	characteristic
19	C₁₉×#127	C₂²	1	1	yes	characteristic
20	C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
21	#10597385	—	2	1	no	—
22	#10597385	—	2	1	no	—
23	#10597384	C₂²	1	1	yes	characteristic
24	#10597384	C₂²	1	1	yes	characteristic
25	#10597381	—	2	1	no	—
26	#10597381	—	2	1	no	—
27	D₄×Dic₁₉	C₂²	1	1	yes	characteristic
28	D₄×Dic₁₉	C₂²	1	1	yes	characteristic
29	D₄×Dic₁₉	C₂²	1	1	yes	characteristic
30	C₂×#64526	C₂²	1	1	yes	characteristic
31	C₂×#64526	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊D₁₉	—	2	1	no	—
33	C₂×Dic₁₉⋊C₄	—	2	1	no	—
34	C₄⋊C₄×D₁₉	C₂²	1	1	yes	characteristic
35	#10597333	C₂²	1	1	yes	characteristic
36	(C₂²⋊D₁₉)⋊C₄	—	2	1	no	—
37	#10597323	—	2	1	no	—
38	#10597318	C₂²	1	1	yes	characteristic
39	#10597316	C₂²	1	1	yes	characteristic
40	C₄×Dic₃₈	C₂²	1	1	yes	characteristic
41	C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
42	C₁₉×#102	C₂²	1	1	yes	characteristic
43	C₁₉×#102	C₂²	1	1	yes	characteristic
44	#10597289	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	(C₂×D₁₉)⋊C₈	C₂²	1	1	yes	characteristic
49	(C₂×D₁₉)⋊C₈	C₂²	1	1	yes	characteristic
50	#10597275	C₂²	1	1	yes	characteristic
51	#10597274	C₂²	1	1	yes	characteristic
52	#10597273	C₂²	1	1	yes	characteristic
53	#10597273	C₂²	1	1	yes	characteristic
54	#10597272	C₂²	1	1	yes	characteristic
55	C₈×Dic₁₉	C₂²	1	1	yes	characteristic
56	#10597265	C₂²	1	1	yes	characteristic
57	#10597264	C₂²	1	1	yes	characteristic
58	C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₁₉	—	2	1	no	—
60	C₂×D₄×C₁₉	—	2	1	no	—
61	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
63	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	—	2	1	no	—
65	C₂×C₂²⋊D₁₉	—	2	1	no	—
66	C₂×C₂²⋊D₁₉	—	2	1	no	—
67	C₂×C₂²⋊D₁₉	—	2	1	no	—
68	C₂²×Dic₁₉	—	2	1	no	—
69	C₂²×Dic₁₉	—	2	1	no	—
70	C₂²×Dic₁₉	—	2	1	no	—
71	C₂²×Dic₁₉	—	2	1	no	—
72	#64543	—	2	2	no	—
73	#64543	—	2	2	no	—
74	C₂×C₄×D₁₉	—	2	1	no	—
75	C₂×C₄×D₁₉	C₂³	1	1	yes	characteristic
76	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
77	D₈×C₁₉	—	2	2	no	—
78	D₈×C₁₉	—	2	2	no	—
79	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
80	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
82	C₂²⋊C₄×C₁₉	—	2	1	no	—
83	C₄²×C₁₉	C₂³	1	1	yes	characteristic
84	C₂²⋊Dic₁₉	—	2	1	no	—
85	C₂²⋊Dic₁₉	—	2	1	no	—
86	C₂²⋊Dic₁₉	—	2	1	no	—
87	C₂²⋊Dic₁₉	—	2	1	no	—
88	C₂²⋊Dic₁₉	—	2	1	no	—
89	#64526	—	2	2	no	—
90	#64526	—	2	2	no	—
91	(C₂×D₁₉)⋊C₄	D₄	1	1	yes	characteristic
92	(C₂×D₁₉)⋊C₄	D₄	1	1	yes	characteristic
93	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
94	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
95	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
96	Dic₁₉⋊C₄	—	2	1	no	—
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	D₄	1	1	yes	characteristic
102	Dic₁₉⋊C₄	D₄	1	1	yes	characteristic
103	C₄×Dic₁₉	—	2	1	no	—
104	C₄×Dic₁₉	—	2	1	no	—
105	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
106	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
107	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
108	C₂³×C₁₉	—	2	1	no	—
109	C₂³×C₁₉	—	2	1	no	—
110	C₂³×C₁₉	—	2	1	no	—
111	C₂²×D₁₉	C₂×D₄	1	1	yes	characteristic
112	D₄×C₁₉	—	2	1	no	—
113	D₄×C₁₉	—	2	1	no	—
114	D₄×C₁₉	—	2	1	no	—
115	D₄×C₁₉	—	2	1	no	—
116	D₄×C₁₉	—	2	1	no	—
117	D₄×C₁₉	—	8	1	no	—
118	D₄×C₁₉	—	8	1	no	—
119	D₄×C₁₉	#41	1	2	yes	—
120	C₂×C₄×C₁₉	—	2	1	no	—
121	C₂×C₄×C₁₉	—	8	1	no	—
122	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
125	C₂²⋊D₁₉	—	2	4	no	—
126	C₂²⋊D₁₉	—	8	1	no	—
127	C₂²⋊D₁₉	—	8	1	no	—
128	C₂×Dic₁₉	—	2	4	no	—
129	C₂×Dic₁₉	—	2	1	no	—
130	C₂×Dic₁₉	—	2	1	no	—
131	C₂×Dic₁₉	—	2	1	no	—
132	C₂×Dic₁₉	—	2	1	no	—
133	C₂×Dic₁₉	—	8	1	no	—
134	C₂×Dic₁₉	—	8	1	no	—
135	C₂×Dic₁₉	—	8	1	no	—
136	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
137	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
138	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
139	C₄×D₁₉	—	2	1	no	—
140	C₄×D₁₉	—	2	1	no	—
141	C₄×D₁₉	—	8	1	no	—
142	Dic₃₈	—	2	1	no	—
143	Dic₃₈	#41	1	2	yes	—
144	C₈×C₁₉	—	2	1	no	—
145	C₈×C₁₉	—	2	1	no	—
146	C₁₉⋊C₈	—	2	1	no	—
147	C₁₉⋊C₈	—	2	1	no	—
148	#3119	—	19	1	no	maximal, Sylow
149	C₂²×C₁₉	—	2	2	no	—
150	C₂²×C₁₉	—	4	1	no	—
151	C₂²×C₁₉	—	4	1	no	—
152	C₂²×C₁₉	—	4	1	no	—
153	C₂²×C₁₉	—	4	1	no	—
154	C₂²×C₁₉	—	4	1	no	—
155	C₂²×C₁₉	—	4	1	no	—
156	C₂²×C₁₉	—	4	1	no	—
157	C₂²×C₁₉	—	4	1	no	—
158	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
159	C₂×D₁₉	—	2	1	no	—
160	C₂×D₁₉	—	2	1	no	—
161	C₂×D₁₉	—	2	1	no	—
162	C₄×C₁₉	—	2	1	no	—
163	C₄×C₁₉	—	2	1	no	—
164	C₄×C₁₉	—	4	1	no	—
165	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
166	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
167	Dic₁₉	—	2	2	no	—
168	Dic₁₉	—	2	1	no	—
169	Dic₁₉	—	2	1	no	—
170	Dic₁₉	—	2	1	no	—
171	Dic₁₉	—	4	1	no	—
172	Dic₁₉	—	4	1	no	—
173	Dic₁₉	—	4	1	no	—
174	#540	—	19	1	no	—
175	#520	—	19	1	no	—
176	#481	—	19	1	no	—
177	#480	—	19	1	no	—
178	#479	—	19	1	no	—
179	#478	—	19	1	no	—
180	#460	—	19	1	no	—
181	#459	D₁₉	1	1	yes	characteristic, complemented
182	#449	—	19	1	no	—
183	#449	—	19	1	no	—
184	#442	—	19	1	no	—
185	#440	—	19	1	no	—
186	C₄×SD₁₆	—	19	1	no	—
187	C₄×D₈	—	19	1	no	—
188	#433	—	19	1	no	—
189	C₂×C₁₉	—	4	2	no	—
190	C₂×C₁₉	—	8	1	no	—
191	C₂×C₁₉	—	8	1	no	—
192	C₂×C₁₉	#525	1	1	yes	characteristic
193	C₂×C₁₉	#576	1	1	yes	characteristic
194	C₂×C₁₉	#577	1	1	yes	characteristic
195	D₁₉	—	4	1	no	—
196	C₂×#41	—	19	1	no	—
197	C₂×SD₁₆	—	19	1	no	—
198	C₂×SD₁₆	—	19	1	no	—
199	C₂×D₈	C₂×D₁₉	1	1	yes	characteristic
200	C₂×D₈	C₂×D₁₉	1	1	yes	characteristic
201	#127	C₂×D₁₉	1	1	yes	characteristic
202	#125	—	19	1	no	—
203	#123	—	38	1	no	—
204	#123	—	38	1	no	—
205	#121	—	19	1	no	—
206	#121	—	19	1	no	—
207	#121	—	38	1	no	—
208	#121	—	38	1	no	—
209	C₄×Q₈	—	19	1	no	—
210	C₄×Q₈	—	19	1	no	—
211	C₄×D₄	—	19	1	no	—
212	C₄×D₄	—	19	1	no	—
213	C₄×D₄	—	19	1	no	—
214	C₄×D₄	—	38	1	no	—
215	C₄×D₄	—	38	1	no	—
216	C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
217	#117	—	19	1	no	—
218	#117	—	38	1	no	—
219	C₂×C₄⋊C₄	—	19	1	no	—
220	C₂×C₄⋊C₄	—	38	1	no	—
221	#107	—	19	1	no	—
222	#107	—	19	1	no	—
223	#106	—	19	1	no	—
224	#106	—	19	1	no	—
225	C₄⋊C₈	—	19	1	no	—
226	C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
227	#103	—	19	1	no	—
228	#103	—	19	1	no	—
229	#102	—	19	1	no	—
230	#102	—	19	1	no	—
231	#102	—	19	1	no	—
232	#102	—	19	1	no	—
233	#102	C₂×D₁₉	1	1	yes	characteristic
234	#102	C₂×D₁₉	1	1	yes	characteristic
235	C₂²⋊C₈	—	19	1	no	—
236	C₂²⋊C₈	—	19	1	no	—
237	#97	—	19	1	no	—
238	C₄×C₈	—	19	1	no	—
239	C₁₉	#3119	1	1	yes	characteristic, complemented, Sylow
240	#41	—	38	2	no	—
241	#41	—	38	2	no	—
242	C₂×Q₈	—	19	1	no	—
243	C₂×D₄	—	2	1	no	—
244	C₂×D₄	—	2	1	no	—
245	C₂×D₄	—	38	1	no	—
246	C₂×D₄	—	38	1	no	—
247	C₂×D₄	—	38	1	no	—
248	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
249	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
250	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
251	C₂²×C₄	—	2	1	no	—
252	C₂²×C₄	—	19	1	no	—
253	C₂²×C₄	—	38	1	no	—
254	C₂²×C₄	—	38	1	no	—
255	C₂²×C₄	—	38	1	no	—
256	C₂²×C₄	—	38	1	no	—
257	C₂²×C₄	—	38	1	no	—
258	SD₁₆	—	38	2	no	—
259	SD₁₆	—	38	2	no	—
260	D₈	—	2	2	no	—
261	D₈	—	2	2	no	—
262	C₂×C₈	—	19	1	no	—
263	C₂×C₈	—	19	1	no	—
264	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
265	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
266	C₄⋊C₄	—	19	1	no	—
267	C₄⋊C₄	—	19	1	no	—
268	C₄⋊C₄	—	19	1	no	—
269	C₄⋊C₄	—	19	1	no	—
270	C₄⋊C₄	—	19	1	no	—
271	C₄⋊C₄	—	38	1	no	—
272	C₄⋊C₄	—	38	1	no	—
273	C₄⋊C₄	—	38	1	no	—
274	C₄⋊C₄	—	38	1	no	—
275	C₄⋊C₄	—	38	1	no	—
276	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	19	1	no	—
279	C₂²⋊C₄	—	19	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	C₄²	—	19	1	no	—
286	C₄²	—	38	1	no	—
287	C₄²	—	38	1	no	—
288	C₄²	C₂²×D₁₉	1	1	yes	characteristic
289	C₂³	—	2	1	no	—
290	C₂³	—	2	1	no	—
291	C₂³	—	2	1	no	—
292	C₂³	—	19	1	no	—
293	Q₈	—	19	2	no	—
294	Q₈	—	38	1	no	—
295	D₄	—	2	1	no	—
296	D₄	—	2	1	no	—
297	D₄	—	2	1	no	—
298	D₄	—	2	1	no	—
299	D₄	—	2	1	no	—
300	D₄	—	8	1	no	—
301	D₄	—	8	1	no	—
302	D₄	—	38	4	no	—
303	D₄	—	152	1	no	—
304	D₄	—	152	1	no	—
305	D₄	#64543	1	2	yes	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	8	1	no	—
308	C₂×C₄	—	19	1	no	—
309	C₂×C₄	—	19	1	no	—
310	C₂×C₄	—	19	1	no	—
311	C₂×C₄	—	38	4	no	—
312	C₂×C₄	—	38	1	no	—
313	C₂×C₄	—	38	1	no	—
314	C₂×C₄	—	38	1	no	—
315	C₂×C₄	—	38	1	no	—
316	C₂×C₄	—	38	1	no	—
317	C₂×C₄	—	38	1	no	—
318	C₂×C₄	—	152	1	no	—
319	C₂×C₄	—	152	1	no	—
320	C₂×C₄	—	152	1	no	—
321	C₂×C₄	—	152	1	no	—
322	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
323	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
324	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
325	C₈	—	2	1	no	—
326	C₈	—	2	1	no	—
327	C₈	—	38	1	no	—
328	C₈	—	38	1	no	—
329	C₂²	—	2	2	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	4	1	no	—
334	C₂²	—	4	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	38	1	no	—
339	C₂²	—	38	1	no	—
340	C₂²	—	38	1	no	—
341	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	4	1	no	—
345	C₄	—	38	1	no	—
346	C₄	—	38	2	no	—
347	C₄	—	38	1	no	—
348	C₄	—	38	1	no	—
349	C₄	—	76	1	no	—
350	C₄	—	76	1	no	—
351	C₄	—	76	1	no	—
352	C₄	C₂×#64543	1	1	yes	characteristic
353	C₄	#41×D₁₉	1	1	yes	characteristic
354	C₂	—	4	2	no	—
355	C₂	—	8	1	no	—
356	C₂	—	8	1	no	—
357	C₂	—	76	1	no	—
358	C₂	#11927054	1	1	yes	characteristic
359	C₂	#11927345	1	1	yes	characteristic
360	C₂	#11927354	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×#459	C₂
socle	C₂²×C₁₉	C₂²×D₄
hypercenter	#459	D₁₉
hypocenter	C₁₉	#3119
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #459

Complements

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

N	H ≅ G/N	type
1. #11927148	C₂	semidirect
2. #11927023	C₂	semidirect
3. #11926861	C₂	semidirect
4. #11926856	C₂	semidirect
5. #11926852	C₂	semidirect
6. #11926844	C₂	semidirect
7. #11926741	C₂	semidirect
8. #11926686	C₂	semidirect
9. #11926686	C₂	semidirect
10. #11926656	C₂	semidirect
11. #11926641	C₂	semidirect
12. #11926635	C₂	semidirect
13. D₈×Dic₁₉	C₂	semidirect
14. #11926604	C₂	semidirect
15. C₁₉×#459	C₂	semidirect
181. #459	D₁₉	semidirect
239. C₁₉	#3119	semidirect

Maximal subgroups

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

subgroup	index
#11927148	2
#11927023	2
#11926861	2
#11926856	2
#11926852	2
#11926844	2
#11926741	2
#11926686	2
#11926656	2
#11926641	2
#11926635	2
D₈×Dic₁₉	2
#11926604	2
C₁₉×#459	2
#3119	19