Subgroups — #14569084

Order 2432.

436 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11926486	C₂	1	1	yes	maximal, characteristic, complemented
2	#11926480	C₂	1	1	yes	maximal, characteristic
3	#11926477	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926456	C₂	1	1	yes	maximal, characteristic
5	#11926448	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926448	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926440	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926437	C₂	1	1	yes	maximal, characteristic
9	#11926431	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926428	C₂	1	1	yes	maximal, characteristic, complemented
11	#11926416	C₂	1	1	yes	maximal, characteristic
12	#11926416	C₂	1	1	yes	maximal, characteristic
13	C₄×Dic₁₉⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²⋊C₄×Dic₁₉	C₂	1	1	yes	maximal, characteristic
15	C₁₉×#388	C₂	1	1	yes	maximal, characteristic
16	C₂³×Dic₁₉	C₂²	1	1	yes	characteristic
17	C₂²×Dic₃₈	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊Dic₁₉	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊Dic₁₉	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊Dic₁₉	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
28	C₂×Dic₁₉⋊C₄	C₂²	1	1	yes	characteristic
29	C₂×Dic₁₉⋊C₄	C₂²	1	1	yes	characteristic
30	C₂×Dic₁₉⋊C₄	C₂²	1	1	yes	characteristic
31	C₂×Dic₁₉⋊C₄	C₂²	1	1	yes	characteristic
32	C₂×Dic₁₉⋊C₄	C₂²	1	1	yes	characteristic
33	C₂×Dic₁₉⋊C₄	C₂²	1	1	yes	characteristic
34	C₂×C₄×Dic₁₉	C₂²	1	1	yes	characteristic
35	C₂×C₄×Dic₁₉	C₂²	1	1	yes	characteristic
36	C₂×C₄×Dic₁₉	C₂²	1	1	yes	characteristic
37	C₁₉×#95	C₂²	1	1	yes	characteristic
38	C₁₉×#95	C₂²	1	1	yes	characteristic
39	C₁₉⋊#95	C₂²	1	1	yes	characteristic
40	C₁₉⋊#95	C₂²	1	1	yes	characteristic
41	C₁₉⋊#95	C₂²	1	1	yes	characteristic
42	C₁₉⋊#95	C₂²	1	1	yes	characteristic
43	C₁₉⋊#95	C₂²	1	1	yes	characteristic
44	C₁₉⋊#95	C₂²	1	1	yes	characteristic
45	C₁₉⋊#95	C₂²	1	1	yes	characteristic
46	C₁₉⋊#95	C₂²	1	1	yes	characteristic
47	C₁₉⋊#95	C₂²	1	1	yes	characteristic
48	C₁₉⋊#95	C₂²	1	1	yes	characteristic
49	C₁₉⋊#95	C₂²	1	1	yes	characteristic
50	C₁₉⋊#95	C₂²	1	1	yes	characteristic
51	C₂⁴×C₁₉	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
58	C₂²×Dic₁₉	—	2	2	no	—
59	C₂²×Dic₁₉	—	4	1	no	—
60	C₂²×Dic₁₉	—	4	1	no	—
61	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
62	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
63	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
64	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
65	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
66	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
67	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
68	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
69	C₂×Dic₃₈	—	2	2	no	—
70	C₂×Dic₃₈	—	2	2	no	—
71	C₂×Dic₃₈	—	4	1	no	—
72	C₄⋊C₄×C₁₉	—	4	1	no	—
73	C₂²⋊C₄×C₁₉	—	4	1	no	—
74	C₂²⋊C₄×C₁₉	—	4	1	no	—
75	C₂²⋊C₄×C₁₉	Q₈	1	4	yes	complemented
76	C₄²×C₁₉	—	4	1	no	—
77	C₂²⋊Dic₁₉	—	2	2	no	—
78	C₂²⋊Dic₁₉	—	2	2	no	—
79	C₂²⋊Dic₁₉	—	2	2	no	—
80	C₄⋊Dic₁₉	—	2	2	no	—
81	C₄⋊Dic₁₉	—	4	1	no	—
82	Dic₁₉⋊C₄	—	2	2	no	—
83	Dic₁₉⋊C₄	—	2	2	no	—
84	Dic₁₉⋊C₄	—	2	2	no	—
85	Dic₁₉⋊C₄	—	2	2	no	—
86	Dic₁₉⋊C₄	—	4	1	no	—
87	Dic₁₉⋊C₄	—	4	1	no	—
88	C₄×Dic₁₉	—	2	2	no	—
89	C₄×Dic₁₉	—	4	1	no	—
90	C₄×Dic₁₉	—	4	1	no	—
91	C₂³×C₁₉	—	2	1	no	—
92	C₂³×C₁₉	—	2	1	no	—
93	C₂³×C₁₉	—	2	1	no	—
94	C₂³×C₁₉	—	2	1	no	—
95	C₂³×C₁₉	—	2	1	no	—
96	C₂³×C₁₉	—	2	1	no	—
97	C₂³×C₁₉	C₂×Q₈	1	2	yes	—
98	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
99	C₂×C₄×C₁₉	—	2	1	no	—
100	C₂×C₄×C₁₉	—	2	1	no	—
101	C₂×C₄×C₁₉	—	2	1	no	—
102	C₂×C₄×C₁₉	—	2	1	no	—
103	C₂×C₄×C₁₉	—	2	1	no	—
104	C₂×C₄×C₁₉	—	2	1	no	—
105	C₂×C₄×C₁₉	—	2	1	no	—
106	C₂×C₄×C₁₉	—	2	1	no	—
107	C₂×C₄×C₁₉	—	2	1	no	—
108	C₂×C₄×C₁₉	—	2	1	no	—
109	C₂×C₄×C₁₉	—	2	1	no	—
110	C₂×C₄×C₁₉	—	2	1	no	—
111	C₂×C₄×C₁₉	—	2	1	no	—
112	C₂×C₄×C₁₉	—	2	1	no	—
113	C₂×C₄×C₁₉	—	2	1	no	—
114	C₂×C₄×C₁₉	—	2	1	no	—
115	C₂×C₄×C₁₉	C₂×Q₈	1	2	yes	—
116	C₂×C₄×C₁₉	C₂×Q₈	1	2	yes	—
117	C₂×Dic₁₉	—	2	1	no	—
118	C₂×Dic₁₉	—	2	1	no	—
119	C₂×Dic₁₉	—	2	1	no	—
120	C₂×Dic₁₉	—	2	1	no	—
121	C₂×Dic₁₉	—	2	1	no	—
122	C₂×Dic₁₉	—	2	1	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	—	2	1	no	—
128	C₂×Dic₁₉	—	2	1	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	—	2	1	no	—
134	C₂×Dic₁₉	—	2	1	no	—
135	C₂×Dic₁₉	—	2	1	no	—
136	C₂×Dic₁₉	—	2	1	no	—
137	C₂×Dic₁₉	—	4	4	no	—
138	C₂×Dic₁₉	#41	1	2	yes	—
139	C₂×Dic₁₉	#41	1	2	yes	—
140	C₂×Dic₁₉	#41	1	2	yes	—
141	C₂×Dic₁₉	#41	1	2	yes	—
142	Dic₃₈	—	4	4	no	—
143	#2589	—	19	1	no	maximal, Sylow
144	C₂²×C₁₉	—	2	2	no	—
145	C₂²×C₁₉	—	2	2	no	—
146	C₂²×C₁₉	—	2	2	no	—
147	C₂²×C₁₉	—	4	1	no	—
148	C₂²×C₁₉	—	4	1	no	—
149	C₂²×C₁₉	—	4	1	no	—
150	C₂²×C₁₉	—	4	1	no	—
151	C₂²×C₁₉	C₂²×Q₈	1	1	yes	characteristic
152	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
153	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
154	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
155	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
156	C₂²×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
157	C₂²×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
158	C₄×C₁₉	—	2	2	no	—
159	C₄×C₁₉	—	2	2	no	—
160	C₄×C₁₉	—	4	1	no	—
161	C₄×C₁₉	—	4	1	no	—
162	C₄×C₁₉	—	4	1	no	—
163	C₄×C₁₉	—	4	1	no	—
164	Dic₁₉	—	2	2	no	—
165	Dic₁₉	—	2	2	no	—
166	Dic₁₉	—	2	2	no	—
167	Dic₁₉	—	2	2	no	—
168	Dic₁₉	—	4	1	no	—
169	Dic₁₉	—	4	1	no	—
170	Dic₁₉	—	4	1	no	—
171	Dic₁₉	—	4	1	no	—
172	#400	—	19	1	no	—
173	#399	—	19	1	no	—
174	#398	—	19	1	no	—
175	#393	—	19	1	no	—
176	#391	—	19	1	no	—
177	#391	—	19	1	no	—
178	#389	—	19	1	no	—
179	#388	—	19	1	no	—
180	#388	D₁₉	1	1	yes	characteristic
181	#387	—	19	1	no	—
182	#387	—	19	1	no	—
183	#385	—	19	1	no	—
184	#385	—	19	1	no	—
185	C₄×C₄⋊C₄	—	19	1	no	—
186	C₄×C₂²⋊C₄	—	19	1	no	—
187	C₂×C₁₉	—	4	2	no	—
188	C₂×C₁₉	#529	1	1	yes	characteristic
189	C₂×C₁₉	#533	1	1	yes	characteristic
190	C₂×C₁₉	#538	1	1	yes	characteristic
191	C₂×C₁₉	#551	1	1	yes	characteristic
192	C₂×C₁₉	#554	1	1	yes	characteristic
193	C₂×C₁₉	#554	1	1	yes	characteristic
194	C₂×C₁₉	#559	1	1	yes	characteristic
195	C₂²×Q₈	—	19	1	no	—
196	C₂³×C₄	—	19	1	no	—
197	C₂×C₄⋊C₄	—	19	1	no	—
198	C₂×C₄⋊C₄	—	19	1	no	—
199	C₂×C₄⋊C₄	—	19	1	no	—
200	C₂×C₄⋊C₄	—	19	1	no	—
201	C₂×C₄⋊C₄	—	19	1	no	—
202	C₂×C₄⋊C₄	—	19	1	no	—
203	C₂×C₄⋊C₄	—	19	1	no	—
204	C₂×C₄⋊C₄	—	19	1	no	—
205	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
206	C₂×C₂²⋊C₄	—	19	1	no	—
207	C₂×C₂²⋊C₄	—	19	1	no	—
208	C₂×C₂²⋊C₄	—	19	1	no	—
209	C₂×C₂²⋊C₄	C₂×D₁₉	1	1	yes	characteristic
210	C₂×C₂²⋊C₄	C₂×D₁₉	1	1	yes	characteristic
211	C₂×C₂²⋊C₄	C₂×D₁₉	1	1	yes	characteristic
212	C₂×C₄²	—	19	1	no	—
213	C₂×C₄²	—	19	1	no	—
214	C₂×C₄²	—	19	1	no	—
215	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
216	#95	—	19	1	no	—
217	#95	—	19	1	no	—
218	#95	—	19	1	no	—
219	#95	—	19	1	no	—
220	#95	—	19	1	no	—
221	#95	—	19	1	no	—
222	#95	—	19	1	no	—
223	#95	—	19	1	no	—
224	#95	—	19	1	no	—
225	#95	—	19	1	no	—
226	#95	—	19	1	no	—
227	#95	—	19	1	no	—
228	#95	C₂×D₁₉	1	1	yes	characteristic
229	#95	C₂×D₁₉	1	1	yes	characteristic
230	C₁₉	#2589	1	1	yes	characteristic, complemented, Sylow
231	C₂⁴	C₂²×D₁₉	1	1	yes	characteristic
232	C₂×Q₈	—	38	2	no	—
233	C₂×Q₈	—	38	2	no	—
234	C₂×Q₈	—	76	1	no	—
235	C₂²×C₄	—	19	1	no	—
236	C₂²×C₄	—	19	1	no	—
237	C₂²×C₄	—	19	1	no	—
238	C₂²×C₄	—	19	1	no	—
239	C₂²×C₄	—	19	1	no	—
240	C₂²×C₄	—	19	1	no	—
241	C₂²×C₄	—	19	1	no	—
242	C₂²×C₄	—	19	1	no	—
243	C₂²×C₄	—	38	2	no	—
244	C₂²×C₄	—	76	1	no	—
245	C₂²×C₄	—	76	1	no	—
246	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
247	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
248	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
249	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
250	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
251	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
252	C₄⋊C₄	—	4	1	no	—
253	C₄⋊C₄	—	38	2	no	—
254	C₄⋊C₄	—	38	2	no	—
255	C₄⋊C₄	—	38	2	no	—
256	C₄⋊C₄	—	38	2	no	—
257	C₄⋊C₄	—	38	2	no	—
258	C₄⋊C₄	—	76	1	no	—
259	C₄⋊C₄	—	76	1	no	—
260	C₄⋊C₄	—	76	1	no	—
261	C₂²⋊C₄	—	4	1	no	—
262	C₂²⋊C₄	—	4	1	no	—
263	C₂²⋊C₄	—	38	2	no	—
264	C₂²⋊C₄	—	38	2	no	—
265	C₂²⋊C₄	—	38	2	no	—
266	C₂²⋊C₄	Dic₃₈	1	4	yes	complemented
267	C₄²	—	4	1	no	—
268	C₄²	—	38	2	no	—
269	C₄²	—	76	1	no	—
270	C₄²	—	76	1	no	—
271	C₂³	—	2	1	no	—
272	C₂³	—	2	1	no	—
273	C₂³	—	2	1	no	—
274	C₂³	—	2	1	no	—
275	C₂³	—	2	1	no	—
276	C₂³	—	2	1	no	—
277	C₂³	C₂×Dic₃₈	1	2	yes	—
278	C₂³	C₂³×D₁₉	1	1	yes	characteristic
279	Q₈	—	76	4	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	19	2	no	—
297	C₂×C₄	—	19	2	no	—
298	C₂×C₄	—	19	2	no	—
299	C₂×C₄	—	19	2	no	—
300	C₂×C₄	—	38	1	no	—
301	C₂×C₄	—	38	1	no	—
302	C₂×C₄	—	38	1	no	—
303	C₂×C₄	—	38	1	no	—
304	C₂×C₄	—	38	1	no	—
305	C₂×C₄	—	38	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	—	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	—	38	1	no	—
319	C₂×C₄	—	38	1	no	—
320	C₂×C₄	—	76	4	no	—
321	C₂×C₄	C₂×Dic₃₈	1	2	yes	—
322	C₂×C₄	C₂×Dic₃₈	1	2	yes	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	2	2	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	C₂²×Dic₃₈	1	1	yes	characteristic
331	C₂²	#10597430	1	1	yes	characteristic
332	C₂²	#10597430	1	1	yes	characteristic
333	C₂²	#41×D₁₉	1	1	yes	characteristic
334	C₂²	#41×D₁₉	1	1	yes	characteristic
335	C₂²	#41×D₁₉	1	1	yes	characteristic
336	C₂²	#41×D₁₉	1	1	yes	characteristic
337	C₄	—	2	2	no	—
338	C₄	—	2	2	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	4	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	38	2	no	—
344	C₄	—	38	2	no	—
345	C₄	—	38	2	no	—
346	C₄	—	38	2	no	—
347	C₄	—	76	1	no	—
348	C₄	—	76	1	no	—
349	C₄	—	76	1	no	—
350	C₄	—	76	1	no	—
351	C₂	—	4	2	no	—
352	C₂	#11927077	1	1	yes	characteristic
353	C₂	#11927097	1	1	yes	characteristic
354	C₂	#11927126	1	1	yes	characteristic
355	C₂	#11927224	1	1	yes	characteristic
356	C₂	#11927253	1	1	yes	characteristic
357	C₂	#11927253	1	1	yes	characteristic
358	C₂	#11927278	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉
commutator, socle, cosocle	C₂³×C₁₉	C₂⁴
Fitting	C₁₉×#388	C₂
prime radical	C₂⁴×C₁₉	C₂³
hypercenter	#388	D₁₉
hypocenter	C₁₉	#2589
LCS	G ▷ C₂³×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ #388

Complements

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

N	H ≅ G/N	type
1. #11926486	C₂	semidirect
3. #11926477	C₂	semidirect
5. #11926448	C₂	semidirect
6. #11926448	C₂	semidirect
7. #11926440	C₂	semidirect
9. #11926431	C₂	semidirect
10. #11926428	C₂	semidirect
13. C₄×Dic₁₉⋊C₄	C₂	semidirect
75. C₂²⋊C₄×C₁₉	Q₈	semidirect
230. C₁₉	#2589	semidirect
266. C₂²⋊C₄	Dic₃₈	semidirect

Maximal subgroups

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

subgroup	index
#11926486	2
#11926480	2
#11926477	2
#11926456	2
#11926448	2
#11926440	2
#11926437	2
#11926431	2
#11926428	2
#11926416	2
C₄×Dic₁₉⋊C₄	2
C₂²⋊C₄×Dic₁₉	2
C₁₉×#388	2
#2589	19