Subgroups — #14567319

Order 2432.

508 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10597390	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#10597323	C₂	1	1	yes	maximal, characteristic
3	#11926467	C₂	1	1	yes	maximal, characteristic
4	#11926467	C₂	1	1	yes	maximal, characteristic
5	#11926456	C₂	1	1	yes	maximal, characteristic
6	#11926451	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926448	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926441	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926441	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926429	C₂	1	1	yes	maximal, characteristic, complemented
11	#11926426	C₂	1	1	yes	maximal, characteristic, complemented
12	#11926408	C₂	1	1	yes	maximal, characteristic, complemented
13	#11926402	C₂	1	1	yes	maximal, characteristic
14	C₂²⋊C₄×Dic₁₉	C₂	1	1	yes	maximal, characteristic
15	C₁₉×#393	C₂	1	1	yes	maximal, characteristic
16	C₂²×Q₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₂³×Dic₁₉	C₂²	1	1	yes	characteristic
18	C₂²×Dic₃₈	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×C₂²⋊Dic₁₉	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊Dic₁₉	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊Dic₁₉	C₂²	1	1	yes	characteristic
25	#10597390	—	2	4	no	—
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×Dic₁₉⋊C₄	C₂²	1	1	yes	characteristic
35	C₂×Dic₁₉⋊C₄	C₂²	1	1	yes	characteristic
36	C₂×C₄×Dic₁₉	C₂²	1	1	yes	characteristic
37	C₂×C₄×Dic₁₉	C₂²	1	1	yes	characteristic
38	C₂×C₄×Dic₁₉	C₂²	1	1	yes	characteristic
39	C₂×C₄×Dic₁₉	C₂²	1	1	yes	characteristic
40	#10597323	—	2	4	no	—
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	C₁₉⋊#95	C₂²	1	1	yes	characteristic
52	C₁₉⋊#95	C₂²	1	1	yes	characteristic
53	C₂⁴×C₁₉	C₂³	1	1	yes	characteristic
54	C₂×Q₈×C₁₉	—	2	2	no	—
55	C₂×Q₈×C₁₉	—	2	2	no	—
56	C₂×Q₈×C₁₉	—	4	1	no	—
57	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
63	C₂²×Dic₁₉	—	2	2	no	—
64	C₂²×Dic₁₉	—	2	2	no	—
65	C₂²×Dic₁₉	Q₈	1	4	yes	complemented
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	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
70	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
71	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
72	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
73	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
74	C₂×Dic₃₈	—	2	2	no	—
75	C₂×Dic₃₈	—	2	2	no	—
76	C₂×Dic₃₈	—	4	1	no	—
77	C₂²⋊C₄×C₁₉	—	2	2	no	—
78	C₂²⋊C₄×C₁₉	—	2	2	no	—
79	C₂²⋊C₄×C₁₉	—	4	1	no	—
80	C₂²⋊Dic₁₉	—	2	2	no	—
81	C₂²⋊Dic₁₉	—	2	2	no	—
82	C₂²⋊Dic₁₉	—	4	1	no	—
83	C₄⋊Dic₁₉	—	2	2	no	—
84	C₄⋊Dic₁₉	—	4	1	no	—
85	Dic₁₉⋊C₄	—	2	2	no	—
86	Dic₁₉⋊C₄	—	2	2	no	—
87	Dic₁₉⋊C₄	—	2	2	no	—
88	Dic₁₉⋊C₄	—	2	2	no	—
89	Dic₁₉⋊C₄	—	2	2	no	—
90	Dic₁₉⋊C₄	—	2	2	no	—
91	Dic₁₉⋊C₄	—	4	1	no	—
92	Dic₁₉⋊C₄	D₄	1	4	yes	—
93	C₄×Dic₁₉	—	2	2	no	—
94	C₄×Dic₁₉	—	2	2	no	—
95	C₄×Dic₁₉	—	2	2	no	—
96	C₄×Dic₁₉	—	2	2	no	—
97	C₂³×C₁₉	—	2	1	no	—
98	C₂³×C₁₉	—	2	1	no	—
99	C₂³×C₁₉	—	2	1	no	—
100	C₂³×C₁₉	—	2	1	no	—
101	C₂³×C₁₉	—	2	1	no	—
102	C₂³×C₁₉	—	2	1	no	—
103	C₂³×C₁₉	C₂×Q₈	1	2	yes	—
104	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
105	Q₈×C₁₉	—	4	4	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	—	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	—	2	1	no	—
121	C₂×C₄×C₁₉	—	2	1	no	—
122	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
123	C₂×C₄×C₁₉	#41	1	2	yes	—
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	8	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	—	2	1	no	—
138	C₂×Dic₁₉	—	2	1	no	—
139	C₂×Dic₁₉	—	2	1	no	—
140	C₂×Dic₁₉	—	2	1	no	—
141	C₂×Dic₁₉	—	2	1	no	—
142	C₂×Dic₁₉	—	2	1	no	—
143	C₂×Dic₁₉	C₂×D₄	1	2	yes	—
144	C₂×Dic₁₉	C₂×D₄	1	2	yes	—
145	C₂×Dic₁₉	C₂×Q₈	1	2	yes	—
146	C₂×Dic₁₉	C₂×Q₈	1	2	yes	—
147	C₂×Dic₁₉	#41	1	2	yes	—
148	C₂×Dic₁₉	#41	1	2	yes	—
149	Dic₃₈	—	4	4	no	—
150	#2439	—	19	1	no	maximal, Sylow
151	C₂²×C₁₉	—	2	2	no	—
152	C₂²×C₁₉	—	2	2	no	—
153	C₂²×C₁₉	—	2	2	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	C₂²×C₁₉	C₂²×Q₈	1	1	yes	characteristic
160	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
161	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
162	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
163	C₂²×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
164	C₂²×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
165	C₄×C₁₉	—	2	2	no	—
166	C₄×C₁₉	—	2	2	no	—
167	C₄×C₁₉	—	4	1	no	—
168	C₄×C₁₉	—	4	1	no	—
169	C₄×C₁₉	—	4	1	no	—
170	C₄×C₁₉	—	4	1	no	—
171	Dic₁₉	—	2	2	no	—
172	Dic₁₉	—	2	2	no	—
173	Dic₁₉	—	2	2	no	—
174	Dic₁₉	—	4	1	no	—
175	Dic₁₉	—	4	1	no	—
176	Dic₁₉	—	4	1	no	—
177	Dic₁₉	—	4	1	no	—
178	Dic₁₉	C₂²⋊Q₈	1	4	yes	—
179	C₂×C₄⋊Q₈	—	19	1	no	—
180	C₂×#117	—	19	1	no	—
181	#396	—	19	1	no	—
182	#396	—	19	1	no	—
183	#393	—	19	1	no	—
184	#393	D₁₉	1	1	yes	characteristic
185	#391	—	19	1	no	—
186	#391	—	19	1	no	—
187	#389	—	19	1	no	—
188	#389	—	19	1	no	—
189	#387	—	19	1	no	—
190	#387	—	19	1	no	—
191	#382	—	19	1	no	—
192	#380	—	19	1	no	—
193	C₄×C₂²⋊C₄	—	19	1	no	—
194	C₂×C₁₉	—	4	2	no	—
195	C₂×C₁₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
196	C₂×C₁₉	#533	1	1	yes	characteristic
197	C₂×C₁₉	#544	1	1	yes	characteristic
198	C₂×C₁₉	#546	1	1	yes	characteristic
199	C₂×C₁₉	#548	1	1	yes	characteristic
200	C₂×C₁₉	#555	1	1	yes	characteristic
201	C₂×C₁₉	#556	1	1	yes	characteristic
202	C₂²×Q₈	—	19	1	no	—
203	C₂²×Q₈	C₂×D₁₉	1	1	yes	characteristic
204	C₂³×C₄	—	19	1	no	—
205	C₄⋊Q₈	—	38	4	no	—
206	#117	—	38	4	no	—
207	C₂×C₄⋊C₄	—	19	1	no	—
208	C₂×C₄⋊C₄	—	19	1	no	—
209	C₂×C₄⋊C₄	—	19	1	no	—
210	C₂×C₄⋊C₄	—	19	1	no	—
211	C₂×C₄⋊C₄	—	19	1	no	—
212	C₂×C₄⋊C₄	—	19	1	no	—
213	C₂×C₄⋊C₄	—	19	1	no	—
214	C₂×C₄⋊C₄	—	19	1	no	—
215	C₂×C₄⋊C₄	—	19	1	no	—
216	C₂×C₄⋊C₄	—	19	1	no	—
217	C₂×C₂²⋊C₄	—	19	1	no	—
218	C₂×C₂²⋊C₄	—	19	1	no	—
219	C₂×C₂²⋊C₄	—	19	1	no	—
220	C₂×C₂²⋊C₄	C₂×D₁₉	1	1	yes	characteristic
221	C₂×C₂²⋊C₄	C₂×D₁₉	1	1	yes	characteristic
222	C₂×C₂²⋊C₄	C₂×D₁₉	1	1	yes	characteristic
223	C₂×C₄²	—	19	1	no	—
224	C₂×C₄²	—	19	1	no	—
225	C₂×C₄²	—	19	1	no	—
226	C₂×C₄²	—	19	1	no	—
227	#95	—	19	1	no	—
228	#95	—	19	1	no	—
229	#95	—	19	1	no	—
230	#95	—	19	1	no	—
231	#95	—	19	1	no	—
232	#95	—	19	1	no	—
233	#95	—	19	1	no	—
234	#95	—	19	1	no	—
235	#95	—	19	1	no	—
236	#95	C₂×D₁₉	1	1	yes	characteristic
237	#95	C₂×D₁₉	1	1	yes	characteristic
238	#95	C₂×D₁₉	1	1	yes	characteristic
239	C₁₉	#2439	1	1	yes	characteristic, complemented, Sylow
240	C₂⁴	C₂²×D₁₉	1	1	yes	characteristic
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	4	1	no	—
244	C₂×Q₈	—	38	2	no	—
245	C₂×Q₈	—	38	2	no	—
246	C₂×Q₈	—	76	1	no	—
247	C₂²×C₄	—	19	1	no	—
248	C₂²×C₄	—	19	4	no	—
249	C₂²×C₄	—	19	1	no	—
250	C₂²×C₄	—	19	1	no	—
251	C₂²×C₄	—	19	1	no	—
252	C₂²×C₄	—	19	1	no	—
253	C₂²×C₄	—	19	1	no	—
254	C₂²×C₄	—	19	1	no	—
255	C₂²×C₄	—	19	1	no	—
256	C₂²×C₄	—	38	2	no	—
257	C₂²×C₄	—	38	2	no	—
258	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
259	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
260	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
261	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
262	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
263	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
264	C₄⋊C₄	—	19	4	no	—
265	C₄⋊C₄	—	38	2	no	—
266	C₄⋊C₄	—	38	2	no	—
267	C₄⋊C₄	—	38	2	no	—
268	C₄⋊C₄	—	38	2	no	—
269	C₄⋊C₄	—	38	2	no	—
270	C₄⋊C₄	—	38	2	no	—
271	C₄⋊C₄	—	38	2	no	—
272	C₄⋊C₄	—	76	1	no	—
273	C₄⋊C₄	—	76	1	no	—
274	C₂²⋊C₄	—	2	2	no	—
275	C₂²⋊C₄	—	2	2	no	—
276	C₂²⋊C₄	—	4	1	no	—
277	C₂²⋊C₄	—	38	2	no	—
278	C₂²⋊C₄	—	38	2	no	—
279	C₂²⋊C₄	—	76	1	no	—
280	C₄²	—	38	2	no	—
281	C₄²	—	38	2	no	—
282	C₄²	—	38	2	no	—
283	C₄²	—	38	2	no	—
284	C₂³	—	2	1	no	—
285	C₂³	—	2	1	no	—
286	C₂³	—	2	1	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	—	2	1	no	—
290	C₂³	Q₈×D₁₉	1	2	yes	—
291	C₂³	C₂³×D₁₉	1	1	yes	characteristic
292	Q₈	—	4	4	no	—
293	Q₈	—	76	4	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	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	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	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	—	19	2	no	—
311	C₂×C₄	—	19	2	no	—
312	C₂×C₄	—	19	2	no	—
313	C₂×C₄	—	19	2	no	—
314	C₂×C₄	—	19	2	no	—
315	C₂×C₄	—	19	2	no	—
316	C₂×C₄	—	38	8	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	—	38	1	no	—
322	C₂×C₄	—	38	1	no	—
323	C₂×C₄	—	38	1	no	—
324	C₂×C₄	—	38	1	no	—
325	C₂×C₄	—	38	1	no	—
326	C₂×C₄	—	38	1	no	—
327	C₂×C₄	—	38	1	no	—
328	C₂×C₄	—	38	1	no	—
329	C₂×C₄	—	38	1	no	—
330	C₂×C₄	—	38	1	no	—
331	C₂×C₄	—	38	1	no	—
332	C₂×C₄	—	38	1	no	—
333	C₂×C₄	—	38	1	no	—
334	C₂×C₄	—	38	1	no	—
335	C₂×C₄	D₄×D₁₉	1	2	yes	—
336	C₂×C₄	#64543	1	2	yes	—
337	C₂²	—	2	2	no	—
338	C₂²	—	2	2	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	4	1	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
345	C₂²	C₂×#64543	1	1	yes	characteristic
346	C₂²	#10597430	1	1	yes	characteristic
347	C₂²	C₂×Q₈×D₁₉	1	1	yes	characteristic
348	C₂²	#41×D₁₉	1	1	yes	characteristic
349	C₂²	#41×D₁₉	1	1	yes	characteristic
350	C₂²	#10597436	1	1	yes	characteristic
351	C₄	—	2	2	no	—
352	C₄	—	2	2	no	—
353	C₄	—	4	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	4	1	no	—
356	C₄	—	4	1	no	—
357	C₄	—	19	4	no	—
358	C₄	—	38	2	no	—
359	C₄	—	38	2	no	—
360	C₄	—	38	2	no	—
361	C₄	—	76	1	no	—
362	C₄	—	76	1	no	—
363	C₄	—	76	1	no	—
364	C₄	—	76	1	no	—
365	C₂	—	4	2	no	—
366	C₂	C₂²⋊Q₈×D₁₉	1	1	yes	characteristic
367	C₂	#11927096	1	1	yes	characteristic
368	C₂	#11927173	1	1	yes	characteristic
369	C₂	#11927181	1	1	yes	characteristic
370	C₂	#11927202	1	1	yes	characteristic
371	C₂	#11927256	1	1	yes	characteristic
372	C₂	#11927263	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉
commutator, socle, cosocle	C₂³×C₁₉	C₂⁴
Fitting	C₁₉×#393	C₂
prime radical	C₂⁴×C₁₉	C₂³
hypercenter	#393	D₁₉
hypocenter	C₁₉	#2439
LCS	G ▷ C₂³×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ #393

Complements

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

N	H ≅ G/N	type
1. C₂×#10597390	C₂	semidirect
6. #11926451	C₂	semidirect
7. #11926448	C₂	semidirect
8. #11926441	C₂	semidirect
9. #11926441	C₂	semidirect
10. #11926429	C₂	semidirect
11. #11926426	C₂	semidirect
12. #11926408	C₂	semidirect
65. C₂²×Dic₁₉	Q₈	semidirect
239. C₁₉	#2439	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10597390	2
C₂×#10597323	2
#11926467	2
#11926456	2
#11926451	2
#11926448	2
#11926441	2
#11926429	2
#11926426	2
#11926408	2
#11926402	2
C₂²⋊C₄×Dic₁₉	2
C₁₉×#393	2
#2439	19