Subgroups — #14573625

Order 2432.

398 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927123	C₂	1	1	yes	maximal, characteristic
2	#11927025	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926888	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926886	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926876	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926874	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926866	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926865	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926861	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926583	C₂	1	1	yes	maximal, characteristic, complemented
11	#106×D₁₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#11926542	C₂	1	1	yes	maximal, characteristic, complemented
13	#11926537	C₂	1	1	yes	maximal, characteristic, complemented
14	#11926509	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₉×#481	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#64543	C₂²	1	1	yes	characteristic
17	C₂×#64541	C₂²	1	1	yes	characteristic
18	C₁₉×#121	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
20	#10597385	—	2	1	no	—
21	#10597384	—	2	1	no	—
22	#10597384	C₂²	1	1	yes	characteristic
23	C₄×C₂²⋊D₁₉	—	2	1	no	—
24	C₄×C₂²⋊D₁₉	—	2	1	no	—
25	#10597372	C₂²	1	1	yes	characteristic
26	#10597370	C₂²	1	1	yes	characteristic
27	C₂×#64515	C₂²	1	1	yes	characteristic
28	C₂×C₈×D₁₉	C₂²	1	1	yes	characteristic
29	#10597343	C₂²	1	1	yes	characteristic
30	#10597339	—	2	1	no	—
31	#10597338	C₂²	1	1	yes	characteristic
32	#10597338	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×D₁₉	C₂²	1	1	yes	characteristic
34	C₄⋊C₄×D₁₉	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₁₉	C₂²	1	1	yes	characteristic
36	#10597333	—	2	1	no	—
37	#10597330	—	2	1	no	—
38	#10597324	—	2	1	no	—
39	C₁₉×#106	C₂²	1	1	yes	characteristic
40	C₁₉×#106	C₂²	1	1	yes	characteristic
41	C₁₉×#102	C₂²	1	1	yes	characteristic
42	C₁₉×#102	C₂²	1	1	yes	characteristic
43	C₁₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
44	#10597289	C₂²	1	1	yes	characteristic
45	#10597289	C₂²	1	1	yes	characteristic
46	C₂²⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
47	(C₂×D₁₉)⋊C₈	C₂²	1	1	yes	characteristic
48	(C₂×D₁₉)⋊C₈	C₂²	1	1	yes	characteristic
49	#10597275	C₂²	1	1	yes	characteristic
50	#10597274	C₂²	1	1	yes	characteristic
51	#10597273	C₂²	1	1	yes	characteristic
52	#10597273	C₂²	1	1	yes	characteristic
53	#10597267	C₂²	1	1	yes	characteristic
54	#10597267	C₂²	1	1	yes	characteristic
55	#10597265	C₂²	1	1	yes	characteristic
56	#10597265	C₂²	1	1	yes	characteristic
57	#10597265	C₂²	1	1	yes	characteristic
58	#10597264	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₁₉	—	2	1	no	—
60	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	—	2	1	no	—
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	D₄	1	1	yes	characteristic
66	C₂×C₂²⋊D₁₉	D₄	1	1	yes	characteristic
67	C₂²×Dic₁₉	—	2	1	no	—
68	#64543	—	2	2	no	—
69	#64543	—	2	2	no	—
70	#64541	—	2	2	no	—
71	#64541	—	2	2	no	—
72	C₂×D₇₆	D₄	1	1	yes	characteristic
73	C₂×C₄×D₁₉	—	2	1	no	—
74	C₂×C₄×D₁₉	—	2	1	no	—
75	C₂×C₄×D₁₉	—	2	1	no	—
76	C₂×C₄×D₁₉	C₂³	1	1	yes	characteristic
77	C₂×C₄×D₁₉	C₂³	1	1	yes	characteristic
78	C₂×Dic₃₈	D₄	1	1	yes	characteristic
79	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
80	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
81	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₁₉	—	2	1	no	—
83	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
84	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
85	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
86	C₂²⋊C₄×C₁₉	—	2	1	no	—
87	C₂²⋊Dic₁₉	—	2	1	no	—
88	C₂²⋊Dic₁₉	—	2	1	no	—
89	C₂²⋊Dic₁₉	—	2	1	no	—
90	(C₂×D₁₉)⋊C₄	—	2	1	no	—
91	(C₂×D₁₉)⋊C₄	—	2	1	no	—
92	(C₂×D₁₉)⋊C₄	—	2	1	no	—
93	C₄⋊Dic₁₉	—	2	1	no	—
94	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
95	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
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	C₄×Dic₁₉	—	2	1	no	—
102	C₄×Dic₁₉	—	2	1	no	—
103	C₄×Dic₁₉	—	2	1	no	—
104	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
105	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
106	#64515	—	2	2	no	—
107	C₈×D₁₉	—	2	2	no	—
108	C₂³×C₁₉	—	2	1	no	—
109	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
110	C₂²×D₁₉	C₂×D₄	1	1	yes	characteristic
111	C₂²×D₁₉	C₂×D₄	1	1	yes	characteristic
112	D₄×C₁₉	—	2	1	no	—
113	D₄×C₁₉	—	2	1	no	—
114	D₄×C₁₉	—	8	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	—	2	1	no	—
119	C₂×C₄×C₁₉	—	2	1	no	—
120	C₂×C₄×C₁₉	—	8	1	no	—
121	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
123	C₂²⋊D₁₉	—	2	2	no	—
124	C₂²⋊D₁₉	—	2	2	no	—
125	C₂²⋊D₁₉	—	4	2	no	—
126	C₂²⋊D₁₉	—	8	1	no	—
127	C₂×Dic₁₉	—	2	1	no	—
128	C₂×Dic₁₉	—	2	1	no	—
129	C₂×Dic₁₉	—	2	1	no	—
130	C₂×Dic₁₉	—	2	1	no	—
131	C₂×Dic₁₉	—	4	2	no	—
132	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
133	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
134	D₇₆	—	2	2	no	—
135	C₄×D₁₉	—	2	1	no	—
136	C₄×D₁₉	—	2	1	no	—
137	C₄×D₁₉	—	8	1	no	—
138	C₄×D₁₉	—	8	1	no	—
139	C₄×D₁₉	—	8	1	no	—
140	C₄×D₁₉	#41	1	2	yes	—
141	C₄×D₁₉	#41	1	2	yes	—
142	Dic₃₈	—	2	2	no	—
143	Dic₃₈	—	2	1	no	—
144	Dic₃₈	—	2	1	no	—
145	C₈×C₁₉	—	2	1	no	—
146	C₈×C₁₉	—	2	1	no	—
147	C₁₉⋊C₈	—	2	1	no	—
148	C₁₉⋊C₈	—	2	1	no	—
149	#3099	—	19	1	no	maximal, Sylow
150	C₂²×C₁₉	—	2	1	no	—
151	C₂²×C₁₉	—	2	1	no	—
152	C₂²×C₁₉	—	2	1	no	—
153	C₂²×C₁₉	—	4	1	no	—
154	C₂²×C₁₉	—	4	1	no	—
155	C₂²×C₁₉	—	4	1	no	—
156	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
157	C₂×D₁₉	—	2	1	no	—
158	C₂×D₁₉	—	2	1	no	—
159	C₂×D₁₉	—	2	1	no	—
160	C₂×D₁₉	—	2	1	no	—
161	C₂×D₁₉	—	2	1	no	—
162	C₂×D₁₉	#123	1	2	yes	—
163	C₄×C₁₉	—	2	1	no	—
164	C₄×C₁₉	—	4	1	no	—
165	C₄×C₁₉	—	4	1	no	—
166	C₄×C₁₉	—	4	1	no	—
167	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
168	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
169	Dic₁₉	—	2	1	no	—
170	Dic₁₉	—	4	1	no	—
171	Dic₁₉	—	4	1	no	—
172	Dic₁₉	—	4	1	no	—
173	Dic₁₉	—	4	1	no	—
174	Dic₁₉	#123	1	2	yes	—
175	#537	—	19	1	no	—
176	#520	—	19	1	no	—
177	#485	—	19	1	no	—
178	#485	—	19	1	no	—
179	#483	—	19	1	no	—
180	#483	—	19	1	no	—
181	#482	—	19	1	no	—
182	#482	—	19	1	no	—
183	#481	—	19	1	no	—
184	#481	D₁₉	1	1	yes	characteristic, complemented
185	#428	—	19	1	no	—
186	C₂×#106	—	19	1	no	—
187	#417	—	19	1	no	—
188	#416	—	19	1	no	—
189	#408	—	19	1	no	—
190	C₂×C₁₉	—	4	1	no	—
191	C₂×C₁₉	—	8	1	no	—
192	C₂×C₁₉	C₂×#123	1	1	yes	characteristic
193	C₂×C₁₉	#577	1	1	yes	characteristic
194	C₂×C₁₉	#577	1	1	yes	characteristic
195	D₁₉	—	2	2	no	—
196	D₁₉	—	4	1	no	—
197	C₂×#41	—	19	1	no	—
198	C₂×#41	—	19	1	no	—
199	C₂×M₁₆	—	19	1	no	—
200	C₂²×C₈	—	19	1	no	—
201	C₂²⋊Q₈	—	19	1	no	—
202	C₂²⋊Q₈	—	19	1	no	—
203	C₂²⋊Q₈	—	38	1	no	—
204	#121	—	19	1	no	—
205	#121	—	38	1	no	—
206	#121	—	38	1	no	—
207	#121	—	38	1	no	—
208	#121	C₂×D₁₉	1	1	yes	characteristic
209	C₄×Q₈	—	38	1	no	—
210	C₄×D₄	—	38	1	no	—
211	C₄×D₄	—	38	1	no	—
212	C₄×D₄	—	38	1	no	—
213	#117	—	19	1	no	—
214	#117	—	19	1	no	—
215	#117	—	19	1	no	—
216	C₂×C₄⋊C₄	—	19	1	no	—
217	C₂×C₄⋊C₄	—	19	1	no	—
218	C₂×C₄⋊C₄	—	19	1	no	—
219	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
220	#107	—	19	1	no	—
221	#107	—	19	1	no	—
222	#106	—	19	1	no	—
223	#106	—	19	1	no	—
224	#106	—	19	1	no	—
225	#106	—	19	1	no	—
226	#106	C₂×D₁₉	1	1	yes	characteristic
227	#106	C₂×D₁₉	1	1	yes	characteristic
228	#103	—	19	1	no	—
229	#103	—	19	1	no	—
230	#103	—	19	1	no	—
231	#103	—	19	1	no	—
232	#102	—	19	1	no	—
233	#102	—	19	1	no	—
234	#102	C₂×D₁₉	1	1	yes	characteristic
235	#102	C₂×D₁₉	1	1	yes	characteristic
236	C₂²⋊C₈	—	19	1	no	—
237	C₂²⋊C₈	—	19	1	no	—
238	C₂²⋊C₈	—	19	1	no	—
239	C₂²⋊C₈	C₂×D₁₉	1	1	yes	characteristic
240	C₁₉	#3099	1	1	yes	characteristic, complemented, Sylow
241	#41	—	38	2	no	—
242	#41	—	38	2	no	—
243	#41	—	38	2	no	—
244	#41	—	38	2	no	—
245	C₂×Q₈	—	19	1	no	—
246	C₂×Q₈	—	19	1	no	—
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	—	19	1	no	—
249	C₂×D₄	—	19	1	no	—
250	C₂×D₄	—	19	1	no	—
251	C₂×D₄	—	38	1	no	—
252	C₂×D₄	—	38	1	no	—
253	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	19	1	no	—
256	C₂²×C₄	—	19	1	no	—
257	C₂²×C₄	—	38	1	no	—
258	C₂²×C₄	—	38	1	no	—
259	C₂²×C₄	—	38	1	no	—
260	C₂²×C₄	—	38	1	no	—
261	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
262	M₁₆	—	38	2	no	—
263	C₂×C₈	—	19	1	no	—
264	C₂×C₈	—	19	1	no	—
265	C₂×C₈	—	38	2	no	—
266	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
267	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	19	1	no	—
270	C₄⋊C₄	—	19	1	no	—
271	C₄⋊C₄	—	19	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	—	38	1	no	—
277	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
278	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
279	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
280	C₂²⋊C₄	—	2	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	C₂²⋊C₄	—	38	1	no	—
287	C₄²	—	38	1	no	—
288	C₄²	—	38	1	no	—
289	C₄²	—	38	1	no	—
290	C₂³	—	2	1	no	—
291	C₂³	—	19	1	no	—
292	C₂³	—	19	1	no	—
293	C₂³	D₄×D₁₉	1	1	yes	characteristic
294	Q₈	—	38	2	no	—
295	Q₈	—	38	1	no	—
296	Q₈	—	38	1	no	—
297	D₄	—	2	1	no	—
298	D₄	—	2	1	no	—
299	D₄	—	8	1	no	—
300	D₄	—	38	2	no	—
301	D₄	—	38	2	no	—
302	D₄	—	38	2	no	—
303	D₄	—	76	2	no	—
304	D₄	—	152	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	—	2	1	no	—
310	C₂×C₄	—	8	1	no	—
311	C₂×C₄	—	19	1	no	—
312	C₂×C₄	—	19	2	no	—
313	C₂×C₄	—	19	2	no	—
314	C₂×C₄	—	19	1	no	—
315	C₂×C₄	—	38	1	no	—
316	C₂×C₄	—	38	1	no	—
317	C₂×C₄	—	38	1	no	—
318	C₂×C₄	—	38	1	no	—
319	C₂×C₄	—	38	1	no	—
320	C₂×C₄	—	38	1	no	—
321	C₂×C₄	—	76	2	no	—
322	C₂×C₄	—	152	1	no	—
323	C₂×C₄	—	152	1	no	—
324	C₂×C₄	—	152	1	no	—
325	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
326	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
327	C₈	—	2	1	no	—
328	C₈	—	2	1	no	—
329	C₈	—	38	1	no	—
330	C₈	—	38	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	4	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	19	2	no	—
338	C₂²	—	38	1	no	—
339	C₂²	—	38	1	no	—
340	C₂²	—	38	1	no	—
341	C₂²	—	38	1	no	—
342	C₂²	—	38	1	no	—
343	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	4	1	no	—
346	C₄	—	4	1	no	—
347	C₄	—	4	1	no	—
348	C₄	—	19	2	no	—
349	C₄	—	38	1	no	—
350	C₄	—	76	1	no	—
351	C₄	—	76	1	no	—
352	C₄	—	76	1	no	—
353	C₄	—	76	1	no	—
354	C₄	#41×D₁₉	1	1	yes	characteristic
355	C₄	#41×D₁₉	1	1	yes	characteristic
356	C₂	—	4	1	no	—
357	C₂	—	8	1	no	—
358	C₂	—	38	2	no	—
359	C₂	—	76	1	no	—
360	C₂	#123×D₁₉	1	1	yes	characteristic
361	C₂	#11927350	1	1	yes	characteristic
362	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×#481	C₂
socle	C₂²×C₁₉	C₂²×D₄
prime radical	#11927123	C₂
hypercenter	#481	D₁₉
hypocenter	C₁₉	#3099
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #481

Complements

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

N	H ≅ G/N	type
2. #11927025	C₂	semidirect
3. #11926888	C₂	semidirect
4. #11926886	C₂	semidirect
5. #11926876	C₂	semidirect
6. #11926874	C₂	semidirect
7. #11926866	C₂	semidirect
8. #11926865	C₂	semidirect
9. #11926861	C₂	semidirect
10. #11926583	C₂	semidirect
11. #106×D₁₉	C₂	semidirect
12. #11926542	C₂	semidirect
13. #11926537	C₂	semidirect
14. #11926509	C₂	semidirect
15. C₁₉×#481	C₂	semidirect
184. #481	D₁₉	semidirect
240. C₁₉	#3099	semidirect

Maximal subgroups

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

subgroup	index
#11927123	2
#11927025	2
#11926888	2
#11926886	2
#11926876	2
#11926874	2
#11926866	2
#11926865	2
#11926861	2
#11926583	2
#106×D₁₉	2
#11926542	2
#11926537	2
#11926509	2
C₁₉×#481	2
#3099	19