Subgroups — #14575489

Order 2432.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927246	C₂	1	1	yes	maximal, characteristic, complemented
2	#11927188	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926897	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926880	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926866	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926842	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926777	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926771	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926757	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926686	C₂	1	1	yes	maximal, characteristic, complemented
11	D₈×Dic₁₉	C₂	1	1	yes	maximal, characteristic, complemented
12	(C₂²⋊D₁₉)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
13	#11926537	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#10597289	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₉×#447	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×D₄×C₁₉	C₂²	1	1	yes	characteristic
17	C₂×#64543	C₂²	1	1	yes	characteristic
18	C₂×D₈×C₁₉	C₂²	1	1	yes	characteristic
19	C₂×D₈×C₁₉	C₂²	1	1	yes	characteristic
20	C₁₉×#121	C₂²	1	1	yes	characteristic
21	#10597399	—	2	1	no	—
22	C₂×C₂²⋊Dic₁₉	—	2	1	no	—
23	#10597385	—	2	1	no	—
24	#10597385	—	2	1	no	—
25	#10597384	C₂²	1	1	yes	characteristic
26	#10597384	C₂²	1	1	yes	characteristic, complemented
27	#10597382	C₂²	1	1	yes	characteristic
28	#10597381	—	2	1	no	—
29	D₄×Dic₁₉	—	2	1	no	—
30	D₄×Dic₁₉	C₂²	1	1	yes	characteristic
31	D₄×Dic₁₉	C₂²	1	1	yes	characteristic, complemented
32	C₂×#64526	C₂²	1	1	yes	characteristic, complemented
33	C₄×C₂²⋊D₁₉	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
35	#10597370	C₂²	1	1	yes	characteristic
36	C₂²×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
37	C₂×C₈×D₁₉	C₂²	1	1	yes	characteristic, complemented
38	#10597338	C₂²	1	1	yes	characteristic, complemented
39	#10597336	C₂²	1	1	yes	characteristic, complemented
40	#10597332	—	2	1	no	—
41	#10597325	—	2	1	no	—
42	C₁₉×#102	C₂²	1	1	yes	characteristic
43	C₁₉×#102	C₂²	1	1	yes	characteristic
44	C₁₉×C₂²⋊C₈	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	#10597289	C₂²	1	1	yes	characteristic
49	#10597289	C₂²	1	1	yes	characteristic, complemented
50	(C₂×D₁₉)⋊C₈	C₂²	1	1	yes	characteristic, complemented
51	#10597275	C₂²	1	1	yes	characteristic, complemented
52	#10597275	C₂²	1	1	yes	characteristic, complemented
53	#10597273	C₂²	1	1	yes	characteristic, complemented
54	#10597273	C₂²	1	1	yes	characteristic, complemented
55	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic, complemented
56	C₈×Dic₁₉	C₂²	1	1	yes	characteristic, complemented
57	#10597267	C₂²	1	1	yes	characteristic, complemented
58	#10597265	C₂²	1	1	yes	characteristic, complemented
59	C₂⁴×C₁₉	—	2	1	no	—
60	C₂×D₄×C₁₉	—	2	2	no	—
61	C₂×D₄×C₁₉	—	2	1	no	—
62	C₂×D₄×C₁₉	—	2	1	no	—
63	C₂×D₄×C₁₉	—	4	1	no	—
64	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
65	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
66	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
68	C₂×C₂²⋊D₁₉	—	2	1	no	—
69	C₂×C₂²⋊D₁₉	—	2	1	no	—
70	C₂×C₂²⋊D₁₉	—	2	1	no	—
71	C₂×C₂²⋊D₁₉	D₄	1	1	yes	characteristic
72	C₂²×Dic₁₉	—	2	1	no	—
73	C₂²×Dic₁₉	—	2	1	no	—
74	C₂²×Dic₁₉	—	2	1	no	—
75	C₂²×Dic₁₉	—	2	1	no	—
76	#64543	—	2	2	no	—
77	#64543	—	4	1	no	—
78	C₂×C₄×D₁₉	C₂³	1	1	yes	characteristic
79	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
80	D₈×C₁₉	—	2	2	no	—
81	D₈×C₁₉	—	4	1	no	—
82	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
83	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
84	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
85	C₂²⋊C₄×C₁₉	—	2	1	no	—
86	C₂²⋊Dic₁₉	—	2	1	no	—
87	C₂²⋊Dic₁₉	—	2	1	no	—
88	C₂²⋊Dic₁₉	—	2	1	no	—
89	C₂²⋊Dic₁₉	—	2	1	no	—
90	C₂²⋊Dic₁₉	—	2	1	no	—
91	C₂²⋊Dic₁₉	—	2	1	no	—
92	C₂²⋊Dic₁₉	—	2	1	no	—
93	C₂²⋊Dic₁₉	D₄	1	1	yes	characteristic
94	#64526	—	4	1	no	—
95	(C₂×D₁₉)⋊C₄	—	2	1	no	—
96	(C₂×D₁₉)⋊C₄	D₄	1	1	yes	characteristic
97	C₄⋊Dic₁₉	—	2	1	no	—
98	C₄⋊Dic₁₉	—	2	1	no	—
99	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
100	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
101	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
102	Dic₁₉⋊C₄	—	2	1	no	—
103	Dic₁₉⋊C₄	—	2	1	no	—
104	Dic₁₉⋊C₄	D₄	1	1	yes	characteristic
105	C₄×Dic₁₉	—	2	1	no	—
106	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
107	C₂×C₁₉⋊C₈	—	4	1	no	—
108	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
109	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
110	C₈×D₁₉	—	4	1	no	—
111	C₂³×C₁₉	—	2	1	no	—
112	C₂³×C₁₉	—	2	1	no	—
113	C₂³×C₁₉	—	2	1	no	—
114	C₂³×C₁₉	—	4	2	no	—
115	C₂³×C₁₉	—	4	2	no	—
116	C₂³×C₁₉	—	8	1	no	—
117	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
118	C₂²×D₁₉	C₂×D₄	1	1	yes	characteristic
119	D₄×C₁₉	—	2	1	no	—
120	D₄×C₁₉	—	2	1	no	—
121	D₄×C₁₉	—	2	1	no	—
122	D₄×C₁₉	—	2	1	no	—
123	D₄×C₁₉	—	2	1	no	—
124	D₄×C₁₉	—	4	2	no	—
125	D₄×C₁₉	—	4	2	no	—
126	D₄×C₁₉	#41	1	2	yes	—
127	C₂×C₄×C₁₉	—	2	1	no	—
128	C₂×C₄×C₁₉	—	2	1	no	—
129	C₂×C₄×C₁₉	—	2	1	no	—
130	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
132	C₂²⋊D₁₉	—	2	2	no	—
133	C₂²⋊D₁₉	—	4	2	no	—
134	C₂²⋊D₁₉	—	8	1	no	—
135	C₂²⋊D₁₉	—	8	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	—	4	2	no	—
141	C₂×Dic₁₉	—	8	1	no	—
142	C₂×Dic₁₉	—	8	1	no	—
143	C₂×Dic₁₉	—	8	1	no	—
144	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
145	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
146	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
147	C₄×D₁₉	—	2	1	no	—
148	C₄×D₁₉	—	2	1	no	—
149	Dic₃₈	—	2	1	no	—
150	Dic₃₈	—	2	1	no	—
151	C₈×C₁₉	—	2	1	no	—
152	C₈×C₁₉	—	2	1	no	—
153	C₁₉⋊C₈	—	2	1	no	—
154	C₁₉⋊C₈	—	2	1	no	—
155	#3297	—	19	1	no	maximal, Sylow
156	C₂²×C₁₉	—	2	1	no	—
157	C₂²×C₁₉	—	2	1	no	—
158	C₂²×C₁₉	—	2	2	no	—
159	C₂²×C₁₉	—	4	1	no	—
160	C₂²×C₁₉	—	4	1	no	—
161	C₂²×C₁₉	—	4	1	no	—
162	C₂²×C₁₉	—	4	1	no	—
163	C₂²×C₁₉	—	4	1	no	—
164	C₂²×C₁₉	—	4	1	no	—
165	C₂²×C₁₉	—	4	1	no	—
166	C₂²×C₁₉	—	4	1	no	—
167	C₂²×C₁₉	—	8	4	no	—
168	C₂²×C₁₉	#135	1	2	yes	—
169	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
170	C₂×D₁₉	—	2	1	no	—
171	C₂×D₁₉	—	2	1	no	—
172	C₂×D₁₉	—	2	1	no	—
173	C₄×C₁₉	—	2	1	no	—
174	C₄×C₁₉	—	4	1	no	—
175	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
176	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
177	Dic₁₉	—	2	1	no	—
178	Dic₁₉	—	2	1	no	—
179	Dic₁₉	—	2	1	no	—
180	Dic₁₉	—	4	1	no	—
181	Dic₁₉	—	4	1	no	—
182	Dic₁₉	—	4	1	no	—
183	Dic₁₉	—	4	1	no	—
184	#553	—	19	1	no	—
185	#546	—	19	1	no	—
186	#488	—	19	1	no	—
187	#484	—	19	1	no	—
188	#482	—	19	1	no	—
189	#478	—	19	1	no	—
190	#466	—	19	1	no	—
191	#465	—	19	1	no	—
192	#463	—	19	1	no	—
193	#449	—	19	1	no	—
194	#447	D₁₉	1	1	yes	characteristic, complemented
195	C₄×D₈	—	19	1	no	—
196	C₈×D₄	—	19	1	no	—
197	#416	—	19	1	no	—
198	C₂×#102	—	19	1	no	—
199	C₂×C₁₉	—	2	2	no	—
200	C₂×C₁₉	—	4	2	no	—
201	C₂×C₁₉	—	8	1	no	—
202	C₂×C₁₉	—	8	1	no	—
203	C₂×C₁₉	#546	1	1	yes	characteristic
204	C₂×C₁₉	C₂×#135	1	1	yes	characteristic
205	C₂×C₁₉	#576	1	1	yes	characteristic
206	D₁₉	—	4	1	no	—
207	C₂×#41	—	19	1	no	—
208	C₂²×D₄	C₂×D₁₉	1	1	yes	characteristic
209	C₂×SD₁₆	—	19	1	no	—
210	C₂×D₈	C₂×D₁₉	1	1	yes	characteristic
211	C₂×D₈	C₂×D₁₉	1	1	yes	characteristic
212	C₂²×C₈	—	19	1	no	—
213	C₂²×C₈	—	19	1	no	—
214	#126	—	38	1	no	—
215	#125	—	19	1	no	—
216	#124	—	19	1	no	—
217	#123	—	38	1	no	—
218	#123	—	38	1	no	—
219	C₂²⋊Q₈	—	19	1	no	—
220	#121	—	19	1	no	—
221	#121	—	19	1	no	—
222	#121	—	38	1	no	—
223	#121	—	38	1	no	—
224	#121	C₂×D₁₉	1	1	yes	characteristic
225	#120	—	38	1	no	—
226	C₄×D₄	—	19	1	no	—
227	C₄×D₄	—	19	1	no	—
228	C₄×D₄	—	19	1	no	—
229	C₄×D₄	—	38	1	no	—
230	#117	—	19	1	no	—
231	C₂×C₄⋊C₄	—	19	1	no	—
232	C₂×C₂²⋊C₄	—	38	1	no	—
233	#107	—	19	1	no	—
234	#107	—	19	1	no	—
235	#106	—	19	1	no	—
236	C₄⋊C₈	—	19	1	no	—
237	#103	—	19	1	no	—
238	#103	—	19	1	no	—
239	#103	—	19	1	no	—
240	#102	—	19	1	no	—
241	#102	—	19	1	no	—
242	#102	—	19	1	no	—
243	#102	—	19	1	no	—
244	#102	—	19	1	no	—
245	#102	C₂×D₁₉	1	1	yes	characteristic
246	#102	C₂×D₁₉	1	1	yes	characteristic
247	C₂²⋊C₈	—	19	1	no	—
248	C₂²⋊C₈	C₂×D₁₉	1	1	yes	characteristic
249	C₄×C₈	—	19	1	no	—
250	C₁₉	#3297	1	1	yes	characteristic, complemented, Sylow
251	C₂⁴	—	2	1	no	—
252	#41	—	38	2	no	—
253	#41	—	76	1	no	—
254	C₂×Q₈	—	19	1	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	2	1	no	—
258	C₂×D₄	—	4	1	no	—
259	C₂×D₄	—	19	1	no	—
260	C₂×D₄	—	38	1	no	—
261	C₂×D₄	—	38	1	no	—
262	C₂×D₄	—	38	1	no	—
263	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
264	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
265	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
266	C₂²×C₄	—	19	1	no	—
267	C₂²×C₄	—	38	1	no	—
268	C₂²×C₄	—	38	1	no	—
269	C₂²×C₄	—	38	1	no	—
270	C₂²×C₄	—	38	1	no	—
271	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
272	SD₁₆	—	76	1	no	—
273	D₈	—	2	2	no	—
274	D₈	—	4	1	no	—
275	C₂×C₈	—	19	1	no	—
276	C₂×C₈	—	19	1	no	—
277	C₂×C₈	—	76	1	no	—
278	C₂×C₈	—	76	1	no	—
279	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
280	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
281	C₄⋊C₄	—	19	1	no	—
282	C₄⋊C₄	—	19	1	no	—
283	C₄⋊C₄	—	19	1	no	—
284	C₄⋊C₄	—	19	1	no	—
285	C₄⋊C₄	—	38	1	no	—
286	C₄⋊C₄	—	38	1	no	—
287	C₄⋊C₄	—	38	1	no	—
288	C₄⋊C₄	—	38	1	no	—
289	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
290	C₂²⋊C₄	—	2	1	no	—
291	C₂²⋊C₄	—	19	1	no	—
292	C₂²⋊C₄	—	19	1	no	—
293	C₂²⋊C₄	—	38	1	no	—
294	C₂²⋊C₄	—	38	1	no	—
295	C₂²⋊C₄	—	38	1	no	—
296	C₂²⋊C₄	—	38	1	no	—
297	C₂²⋊C₄	—	38	1	no	—
298	C₂²⋊C₄	—	38	1	no	—
299	C₂²⋊C₄	—	38	1	no	—
300	C₂²⋊C₄	—	38	1	no	—
301	C₄²	—	19	1	no	—
302	C₄²	—	38	1	no	—
303	C₂³	—	2	1	no	—
304	C₂³	—	2	1	no	—
305	C₂³	—	2	1	no	—
306	C₂³	—	4	2	no	—
307	C₂³	—	4	2	no	—
308	C₂³	—	8	1	no	—
309	C₂³	—	19	1	no	—
310	C₂³	D₄×D₁₉	1	1	yes	characteristic
311	Q₈	—	38	1	no	—
312	Q₈	—	38	1	no	—
313	D₄	—	2	1	no	—
314	D₄	—	2	1	no	—
315	D₄	—	2	1	no	—
316	D₄	—	2	1	no	—
317	D₄	—	2	1	no	—
318	D₄	—	4	2	no	—
319	D₄	—	4	2	no	—
320	D₄	—	38	2	no	—
321	D₄	—	76	2	no	—
322	D₄	—	152	1	no	—
323	D₄	—	152	1	no	—
324	D₄	#64543	1	2	yes	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	19	1	no	—
329	C₂×C₄	—	19	1	no	—
330	C₂×C₄	—	19	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	—	38	1	no	—
336	C₂×C₄	—	38	1	no	—
337	C₂×C₄	—	76	2	no	—
338	C₂×C₄	—	152	1	no	—
339	C₂×C₄	—	152	1	no	—
340	C₂×C₄	—	152	1	no	—
341	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
342	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
343	C₈	—	2	1	no	—
344	C₈	—	2	1	no	—
345	C₈	—	38	1	no	—
346	C₈	—	38	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	2	2	no	—
350	C₂²	—	4	1	no	—
351	C₂²	—	4	1	no	—
352	C₂²	—	4	1	no	—
353	C₂²	—	4	1	no	—
354	C₂²	—	4	1	no	—
355	C₂²	—	4	1	no	—
356	C₂²	—	4	1	no	—
357	C₂²	—	4	1	no	—
358	C₂²	—	8	4	no	—
359	C₂²	—	38	1	no	—
360	C₂²	—	38	1	no	—
361	C₂²	—	38	1	no	—
362	C₂²	#10597358	1	2	yes	—
363	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
364	C₄	—	2	1	no	—
365	C₄	—	4	1	no	—
366	C₄	—	38	1	no	—
367	C₄	—	38	1	no	—
368	C₄	—	38	1	no	—
369	C₄	—	76	1	no	—
370	C₄	—	76	1	no	—
371	C₄	—	76	1	no	—
372	C₄	—	76	1	no	—
373	C₄	C₂×#64543	1	1	yes	characteristic
374	C₄	#10597430	1	1	yes	characteristic
375	C₂	—	2	2	no	—
376	C₂	—	4	2	no	—
377	C₂	—	8	1	no	—
378	C₂	—	8	1	no	—
379	C₂	—	76	1	no	—
380	C₂	#11927181	1	1	yes	characteristic
381	C₂	C₂×#10597358	1	1	yes	characteristic
382	C₂	#11927348	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×#447	C₂
socle	C₂²×C₁₉	C₂²×D₄
hypercenter	#447	D₁₉
hypocenter	C₁₉	#3297
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #447

Complements

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

N	H ≅ G/N	type
1. #11927246	C₂	semidirect
2. #11927188	C₂	semidirect
3. #11926897	C₂	semidirect
4. #11926880	C₂	semidirect
5. #11926866	C₂	semidirect
6. #11926842	C₂	semidirect
7. #11926777	C₂	semidirect
8. #11926771	C₂	semidirect
9. #11926757	C₂	semidirect
10. #11926686	C₂	semidirect
11. D₈×Dic₁₉	C₂	semidirect
12. (C₂²⋊D₁₉)⋊C₈	C₂	semidirect
13. #11926537	C₂	semidirect
14. C₂×#10597289	C₂	semidirect
15. C₁₉×#447	C₂	semidirect
26. #10597384	C₂²	semidirect
31. D₄×Dic₁₉	C₂²	semidirect
32. C₂×#64526	C₂²	semidirect
37. C₂×C₈×D₁₉	C₂²	semidirect
38. #10597338	C₂²	semidirect
39. #10597336	C₂²	semidirect
49. #10597289	C₂²	semidirect
50. (C₂×D₁₉)⋊C₈	C₂²	semidirect
51. #10597275	C₂²	semidirect
52. #10597275	C₂²	semidirect
53. #10597273	C₂²	semidirect
54. #10597273	C₂²	semidirect
55. Dic₁₉⋊C₈	C₂²	semidirect
56. C₈×Dic₁₉	C₂²	semidirect
57. #10597267	C₂²	semidirect
58. #10597265	C₂²	semidirect
194. #447	D₁₉	semidirect
250. C₁₉	#3297	semidirect

Maximal subgroups

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

subgroup	index
#11927246	2
#11927188	2
#11926897	2
#11926880	2
#11926866	2
#11926842	2
#11926777	2
#11926771	2
#11926757	2
#11926686	2
D₈×Dic₁₉	2
(C₂²⋊D₁₉)⋊C₈	2
#11926537	2
C₂×#10597289	2
C₁₉×#447	2
#3297	19