Subgroups — #14573601

Order 2432.

386 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927117	C₂	1	1	yes	maximal, characteristic
2	#11927024	C₂	1	1	yes	maximal, characteristic
3	#11926888	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926885	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926883	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926881	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926877	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926875	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926873	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926585	C₂	1	1	yes	maximal, characteristic, complemented
11	#11926579	C₂	1	1	yes	maximal, characteristic, complemented
12	#11926551	C₂	1	1	yes	maximal, characteristic, complemented
13	#103×D₁₉	C₂	1	1	yes	maximal, characteristic, complemented
14	#11926509	C₂	1	1	yes	maximal, characteristic
15	C₁₉×#485	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₁₉	C₂²	1	1	yes	characteristic
17	C₂×#64541	C₂²	1	1	yes	characteristic
18	C₁₉×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
19	C₁₉×#117	C₂²	1	1	yes	characteristic
20	#10597392	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
22	#10597370	C₂²	1	1	yes	characteristic
23	C₂×#64515	C₂²	1	1	yes	characteristic
24	C₂×C₈×D₁₉	C₂²	1	1	yes	characteristic
25	#10597343	C₂²	1	1	yes	characteristic
26	#10597340	—	2	1	no	—
27	#10597338	C₂²	1	1	yes	characteristic
28	#10597338	C₂²	1	1	yes	characteristic
29	#10597338	C₂²	1	1	yes	characteristic
30	C₄⋊C₄×D₁₉	C₂²	1	1	yes	characteristic
31	C₄⋊C₄×D₁₉	C₂²	1	1	yes	characteristic
32	#10597334	—	2	1	no	—
33	#10597331	—	2	1	no	—
34	#10597329	—	2	1	no	—
35	(C₂²⋊D₁₉)⋊C₄	—	2	1	no	—
36	(C₂²⋊D₁₉)⋊C₄	—	2	1	no	—
37	C₄×D₇₆	—	2	1	no	—
38	C₄×Dic₃₈	—	2	1	no	—
39	C₁₉×#107	C₂²	1	1	yes	characteristic
40	C₁₉×#106	C₂²	1	1	yes	characteristic
41	C₁₉×#103	C₂²	1	1	yes	characteristic
42	C₁₉×#103	C₂²	1	1	yes	characteristic
43	C₁₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
44	#10597292	C₂²	1	1	yes	characteristic
45	#10597292	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	#10597274	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	#10597264	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	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₉	D₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₉	D₄	1	1	yes	characteristic
63	C₂²×Dic₁₉	—	2	1	no	—
64	Q₈×D₁₉	—	2	2	no	—
65	Q₈×D₁₉	—	2	2	no	—
66	#64541	—	2	2	no	—
67	#64541	—	2	2	no	—
68	C₂×D₇₆	D₄	1	1	yes	characteristic
69	C₂×C₄×D₁₉	—	2	1	no	—
70	C₂×C₄×D₁₉	—	2	1	no	—
71	C₂×C₄×D₁₉	—	2	1	no	—
72	C₂×C₄×D₁₉	C₂³	1	1	yes	characteristic
73	C₂×C₄×D₁₉	C₂³	1	1	yes	characteristic
74	C₂×Dic₃₈	—	2	1	no	—
75	C₂×Dic₃₈	D₄	1	1	yes	characteristic
76	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
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	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
83	C₂²⋊C₄×C₁₉	—	2	1	no	—
84	C₂²⋊C₄×C₁₉	—	2	1	no	—
85	C₄²×C₁₉	—	2	1	no	—
86	C₂²⋊Dic₁₉	—	2	1	no	—
87	(C₂×D₁₉)⋊C₄	—	2	1	no	—
88	(C₂×D₁₉)⋊C₄	—	2	1	no	—
89	(C₂×D₁₉)⋊C₄	—	2	1	no	—
90	(C₂×D₁₉)⋊C₄	—	2	1	no	—
91	(C₂×D₁₉)⋊C₄	—	2	1	no	—
92	C₄⋊Dic₁₉	—	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	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₁₉	C₂×D₄	1	1	yes	characteristic
110	C₂²×D₁₉	C₂×D₄	1	1	yes	characteristic
111	Q₈×C₁₉	—	2	1	no	—
112	Q₈×C₁₉	—	2	1	no	—
113	C₂×C₄×C₁₉	—	2	1	no	—
114	C₂×C₄×C₁₉	—	2	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	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
121	C₂²⋊D₁₉	—	2	2	no	—
122	C₂²⋊D₁₉	—	2	2	no	—
123	C₂×Dic₁₉	—	2	1	no	—
124	C₂×Dic₁₉	—	2	1	no	—
125	C₂×Dic₁₉	—	2	1	no	—
126	C₂×Dic₁₉	—	2	1	no	—
127	C₂×Dic₁₉	—	8	1	no	—
128	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
130	D₇₆	—	2	2	no	—
131	C₄×D₁₉	—	2	1	no	—
132	C₄×D₁₉	—	2	1	no	—
133	C₄×D₁₉	—	4	2	no	—
134	C₄×D₁₉	—	8	1	no	—
135	C₄×D₁₉	—	8	1	no	—
136	C₄×D₁₉	#41	1	2	yes	—
137	C₄×D₁₉	#41	1	2	yes	—
138	Dic₃₈	—	2	2	no	—
139	Dic₃₈	—	2	1	no	—
140	Dic₃₈	—	2	1	no	—
141	Dic₃₈	—	4	2	no	—
142	C₈×C₁₉	—	2	1	no	—
143	C₈×C₁₉	—	2	1	no	—
144	C₁₉⋊C₈	—	2	1	no	—
145	C₁₉⋊C₈	—	2	1	no	—
146	#3097	—	19	1	no	maximal, Sylow
147	C₂²×C₁₉	—	2	1	no	—
148	C₂²×C₁₉	—	2	1	no	—
149	C₂²×C₁₉	—	2	1	no	—
150	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
151	C₂×D₁₉	—	2	1	no	—
152	C₂×D₁₉	—	2	1	no	—
153	C₂×D₁₉	—	2	1	no	—
154	C₂×D₁₉	—	2	1	no	—
155	C₂×D₁₉	—	2	1	no	—
156	C₂×D₁₉	#123	1	2	yes	—
157	C₄×C₁₉	—	2	1	no	—
158	C₄×C₁₉	—	4	1	no	—
159	C₄×C₁₉	—	4	1	no	—
160	C₄×C₁₉	—	4	1	no	—
161	C₄×C₁₉	—	4	1	no	—
162	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
163	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
164	Dic₁₉	—	2	1	no	—
165	Dic₁₉	—	4	1	no	—
166	Dic₁₉	—	4	1	no	—
167	Dic₁₉	—	4	1	no	—
168	Dic₁₉	—	4	1	no	—
169	Dic₁₉	#123	1	2	yes	—
170	#536	—	19	1	no	—
171	#520	—	19	1	no	—
172	#485	—	19	1	no	—
173	#485	—	19	1	no	—
174	#485	—	19	1	no	—
175	#485	D₁₉	1	1	yes	characteristic, complemented
176	#484	—	19	1	no	—
177	#484	—	19	1	no	—
178	#483	—	19	1	no	—
179	#483	—	19	1	no	—
180	#428	—	19	1	no	—
181	#427	—	19	1	no	—
182	#419	—	19	1	no	—
183	C₂×#103	—	19	1	no	—
184	#408	—	19	1	no	—
185	C₂×C₁₉	—	4	1	no	—
186	C₂×C₁₉	C₂×#123	1	1	yes	characteristic
187	C₂×C₁₉	#577	1	1	yes	characteristic
188	C₂×C₁₉	#578	1	1	yes	characteristic
189	D₁₉	—	2	2	no	—
190	D₁₉	—	4	1	no	—
191	C₂×#41	—	19	1	no	—
192	C₂²×Q₈	—	19	1	no	—
193	C₂×M₁₆	—	19	1	no	—
194	C₂²×C₈	—	19	1	no	—
195	C₄⋊Q₈	—	38	1	no	—
196	#124	—	38	1	no	—
197	#123	—	38	1	no	—
198	#123	—	38	1	no	—
199	C₂²⋊Q₈	—	19	1	no	—
200	C₂²⋊Q₈	—	19	1	no	—
201	C₂²⋊Q₈	—	19	1	no	—
202	C₂²⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
203	C₄×Q₈	—	38	1	no	—
204	C₄×D₄	—	38	1	no	—
205	C₄×D₄	—	38	1	no	—
206	C₄×D₄	—	38	1	no	—
207	#117	—	19	1	no	—
208	#117	—	19	1	no	—
209	#117	—	19	1	no	—
210	#117	C₂×D₁₉	1	1	yes	characteristic
211	C₂×C₄⋊C₄	—	19	1	no	—
212	C₂×C₄⋊C₄	—	19	1	no	—
213	C₂×C₄⋊C₄	—	19	1	no	—
214	#107	—	19	1	no	—
215	#107	—	19	1	no	—
216	#107	—	19	1	no	—
217	#107	C₂×D₁₉	1	1	yes	characteristic
218	#106	—	19	1	no	—
219	#106	—	19	1	no	—
220	#106	—	19	1	no	—
221	#106	C₂×D₁₉	1	1	yes	characteristic
222	#103	—	19	1	no	—
223	#103	—	19	1	no	—
224	#103	—	19	1	no	—
225	#103	—	19	1	no	—
226	#103	—	19	1	no	—
227	#103	—	19	1	no	—
228	#103	C₂×D₁₉	1	1	yes	characteristic
229	#103	C₂×D₁₉	1	1	yes	characteristic
230	C₂²⋊C₈	—	19	1	no	—
231	C₂²⋊C₈	—	19	1	no	—
232	C₂²⋊C₈	—	19	1	no	—
233	C₂²⋊C₈	C₂×D₁₉	1	1	yes	characteristic
234	C₁₉	#3097	1	1	yes	characteristic, complemented, Sylow
235	#41	—	38	2	no	—
236	#41	—	38	2	no	—
237	C₂×Q₈	—	19	1	no	—
238	C₂×Q₈	—	19	1	no	—
239	C₂×Q₈	—	38	1	no	—
240	C₂×Q₈	—	38	2	no	—
241	C₂×Q₈	—	38	2	no	—
242	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
243	C₂×D₄	—	19	1	no	—
244	C₂×D₄	—	19	1	no	—
245	C₂×D₄	—	19	1	no	—
246	C₂²×C₄	—	19	1	no	—
247	C₂²×C₄	—	19	1	no	—
248	C₂²×C₄	—	38	1	no	—
249	C₂²×C₄	—	38	1	no	—
250	C₂²×C₄	—	38	1	no	—
251	C₂²×C₄	—	38	1	no	—
252	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
253	M₁₆	—	38	2	no	—
254	C₂×C₈	—	19	1	no	—
255	C₂×C₈	—	19	1	no	—
256	C₂×C₈	—	38	2	no	—
257	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
258	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	19	1	no	—
261	C₄⋊C₄	—	19	1	no	—
262	C₄⋊C₄	—	19	1	no	—
263	C₄⋊C₄	—	38	1	no	—
264	C₄⋊C₄	—	38	1	no	—
265	C₄⋊C₄	—	38	1	no	—
266	C₄⋊C₄	—	38	1	no	—
267	C₄⋊C₄	—	38	1	no	—
268	C₄⋊C₄	—	38	1	no	—
269	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
270	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
271	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
272	C₂²⋊C₄	—	2	1	no	—
273	C₂²⋊C₄	—	2	1	no	—
274	C₂²⋊C₄	—	38	1	no	—
275	C₂²⋊C₄	—	38	1	no	—
276	C₂²⋊C₄	—	38	1	no	—
277	C₂²⋊C₄	—	38	1	no	—
278	C₂²⋊C₄	—	38	1	no	—
279	C₂²⋊C₄	—	38	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	38	1	no	—
282	C₄²	—	38	1	no	—
283	C₄²	—	38	1	no	—
284	C₂³	—	19	1	no	—
285	C₂³	—	19	1	no	—
286	C₂³	D₄×D₁₉	1	1	yes	characteristic
287	Q₈	—	2	1	no	—
288	Q₈	—	2	1	no	—
289	Q₈	—	38	2	no	—
290	Q₈	—	38	1	no	—
291	Q₈	—	38	1	no	—
292	Q₈	—	76	2	no	—
293	D₄	—	38	2	no	—
294	D₄	—	38	2	no	—
295	D₄	—	38	2	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	19	1	no	—
303	C₂×C₄	—	19	2	no	—
304	C₂×C₄	—	19	2	no	—
305	C₂×C₄	—	19	1	no	—
306	C₂×C₄	—	38	1	no	—
307	C₂×C₄	—	38	1	no	—
308	C₂×C₄	—	38	1	no	—
309	C₂×C₄	—	38	1	no	—
310	C₂×C₄	—	38	1	no	—
311	C₂×C₄	—	38	1	no	—
312	C₂×C₄	—	76	2	no	—
313	C₂×C₄	—	152	1	no	—
314	C₂×C₄	—	152	1	no	—
315	C₂×C₄	—	152	1	no	—
316	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
317	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₈	—	38	1	no	—
321	C₈	—	38	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	19	2	no	—
326	C₂²	—	38	1	no	—
327	C₂²	—	38	1	no	—
328	C₂²	—	38	1	no	—
329	C₂²	—	38	1	no	—
330	C₂²	—	38	1	no	—
331	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	19	2	no	—
338	C₄	—	38	1	no	—
339	C₄	—	76	1	no	—
340	C₄	—	76	1	no	—
341	C₄	—	76	1	no	—
342	C₄	—	76	1	no	—
343	C₄	#41×D₁₉	1	1	yes	characteristic
344	C₄	#41×D₁₉	1	1	yes	characteristic
345	C₂	—	4	1	no	—
346	C₂	—	38	2	no	—
347	C₂	—	76	1	no	—
348	C₂	#123×D₁₉	1	1	yes	characteristic
349	C₂	#11927351	1	1	yes	characteristic
350	C₂	#11927360	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×#485	C₂
socle	C₂²×C₁₉	C₂²×D₄
prime radical	C₂×#64541	C₂²
hypercenter	#485	D₁₉
hypocenter	C₁₉	#3097
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #485

Complements

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

N	H ≅ G/N	type
3. #11926888	C₂	semidirect
4. #11926885	C₂	semidirect
5. #11926883	C₂	semidirect
6. #11926881	C₂	semidirect
7. #11926877	C₂	semidirect
8. #11926875	C₂	semidirect
9. #11926873	C₂	semidirect
10. #11926585	C₂	semidirect
11. #11926579	C₂	semidirect
12. #11926551	C₂	semidirect
13. #103×D₁₉	C₂	semidirect
15. C₁₉×#485	C₂	semidirect
175. #485	D₁₉	semidirect
234. C₁₉	#3097	semidirect

Maximal subgroups

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

subgroup	index
#11927117	2
#11927024	2
#11926888	2
#11926885	2
#11926883	2
#11926881	2
#11926877	2
#11926875	2
#11926873	2
#11926585	2
#11926579	2
#11926551	2
#103×D₁₉	2
#11926509	2
C₁₉×#485	2
#3097	19