Subgroups — #14573876

Order 2432.

418 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927148	C₂	1	1	yes	maximal, characteristic, complemented
2	#11927016	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926867	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926866	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926845	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926844	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926736	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926676	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926676	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926656	C₂	1	1	yes	maximal, characteristic, complemented
11	#11926653	C₂	1	1	yes	maximal, characteristic, complemented
12	D₈×Dic₁₉	C₂	1	1	yes	maximal, characteristic, complemented
13	#11926624	C₂	1	1	yes	maximal, characteristic, complemented
14	#11926604	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₉×#459	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₄×D₁₉	C₂²	1	1	yes	characteristic
17	C₂×D₈×C₁₉	C₂²	1	1	yes	characteristic
18	C₂×D₈×C₁₉	C₂²	1	1	yes	characteristic
19	C₁₉×#127	C₂²	1	1	yes	characteristic, complemented
20	C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
21	#10597385	—	2	1	no	—
22	#10597385	—	2	1	no	—
23	#10597384	C₂²	1	1	yes	characteristic
24	#10597384	C₂²	1	1	yes	characteristic
25	#10597381	—	2	1	no	—
26	#10597381	—	2	1	no	—
27	D₄×Dic₁₉	C₂²	1	1	yes	characteristic
28	D₄×Dic₁₉	C₂²	1	1	yes	characteristic, complemented
29	D₄×Dic₁₉	C₂²	1	1	yes	characteristic, complemented
30	C₂×#64525	C₂²	1	1	yes	characteristic
31	C₂×#64525	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊D₁₉	—	2	1	no	—
33	C₂×(C₂×D₁₉)⋊C₄	—	2	1	no	—
34	#10597339	C₂²	1	1	yes	characteristic
35	#10597338	C₂²	1	1	yes	characteristic
36	(C₂²⋊D₁₉)⋊C₄	—	2	1	no	—
37	C₂²⋊C₄×D₁₉	—	2	1	no	—
38	C₄×D₇₆	C₂²	1	1	yes	characteristic
39	#10597318	C₂²	1	1	yes	characteristic
40	#10597316	C₂²	1	1	yes	characteristic, complemented
41	C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
42	C₁₉×#102	C₂²	1	1	yes	characteristic, complemented
43	C₁₉×#102	C₂²	1	1	yes	characteristic, complemented
44	#10597289	C₂²	1	1	yes	characteristic
45	#10597289	C₂²	1	1	yes	characteristic
46	#10597289	C₂²	1	1	yes	characteristic, complemented
47	#10597289	C₂²	1	1	yes	characteristic, complemented
48	#10597278	C₂²	1	1	yes	characteristic
49	#10597278	C₂²	1	1	yes	characteristic
50	(C₂×D₁₉)⋊C₈	C₂²	1	1	yes	characteristic
51	(C₂×D₁₉)⋊C₈	C₂²	1	1	yes	characteristic
52	#10597275	C₂²	1	1	yes	characteristic, complemented
53	#10597274	C₂²	1	1	yes	characteristic, complemented
54	#10597272	C₂²	1	1	yes	characteristic, complemented
55	C₈×Dic₁₉	C₂²	1	1	yes	characteristic, complemented
56	#10597265	C₂²	1	1	yes	characteristic, complemented
57	#10597264	C₂²	1	1	yes	characteristic, complemented
58	C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₁₉	—	2	1	no	—
60	C₂×D₄×C₁₉	—	2	1	no	—
61	C₂×D₄×C₁₉	—	2	1	no	—
62	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
63	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
64	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	—	2	1	no	—
66	C₂×C₂²⋊D₁₉	—	2	1	no	—
67	C₂×C₂²⋊D₁₉	—	2	1	no	—
68	C₂×C₂²⋊D₁₉	—	2	1	no	—
69	C₂²×Dic₁₉	—	2	1	no	—
70	C₂²×Dic₁₉	—	2	1	no	—
71	C₂²×Dic₁₉	—	2	1	no	—
72	D₄×D₁₉	—	2	2	no	—
73	D₄×D₁₉	—	2	2	no	—
74	C₂×D₇₆	C₂³	1	1	yes	characteristic
75	C₂×C₄×D₁₉	—	2	1	no	—
76	C₂×C₄×D₁₉	—	2	1	no	—
77	C₂×C₄×D₁₉	C₂³	1	1	yes	characteristic
78	D₈×C₁₉	—	2	2	no	—
79	D₈×C₁₉	—	2	2	no	—
80	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
81	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
83	C₂²⋊C₄×C₁₉	—	2	1	no	—
84	C₄²×C₁₉	C₂³	1	1	yes	characteristic
85	C₂²⋊Dic₁₉	—	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	#64525	—	2	2	no	—
91	#64525	—	2	2	no	—
92	(C₂×D₁₉)⋊C₄	—	2	1	no	—
93	(C₂×D₁₉)⋊C₄	—	2	1	no	—
94	(C₂×D₁₉)⋊C₄	—	2	1	no	—
95	(C₂×D₁₉)⋊C₄	D₄	1	1	yes	characteristic
96	(C₂×D₁₉)⋊C₄	D₄	1	1	yes	characteristic
97	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
98	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
99	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
100	Dic₁₉⋊C₄	—	2	1	no	—
101	Dic₁₉⋊C₄	—	2	1	no	—
102	Dic₁₉⋊C₄	D₄	1	1	yes	characteristic
103	Dic₁₉⋊C₄	D₄	1	1	yes	characteristic
104	C₄×Dic₁₉	—	2	1	no	—
105	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
106	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
107	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
108	C₂³×C₁₉	—	2	1	no	—
109	C₂³×C₁₉	—	2	1	no	—
110	C₂³×C₁₉	—	2	1	no	—
111	C₂²×D₁₉	—	2	1	no	—
112	C₂²×D₁₉	—	2	4	no	—
113	C₂²×D₁₉	—	8	1	no	—
114	C₂²×D₁₉	—	8	1	no	—
115	C₂²×D₁₉	C₂×D₄	1	1	yes	characteristic
116	D₄×C₁₉	—	2	1	no	—
117	D₄×C₁₉	—	2	1	no	—
118	D₄×C₁₉	—	2	1	no	—
119	D₄×C₁₉	—	2	1	no	—
120	D₄×C₁₉	—	2	1	no	—
121	D₄×C₁₉	—	8	1	no	—
122	D₄×C₁₉	—	8	1	no	—
123	D₄×C₁₉	#41	1	2	yes	—
124	C₂×C₄×C₁₉	—	2	1	no	—
125	C₂×C₄×C₁₉	—	8	1	no	—
126	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
129	C₂²⋊D₁₉	—	2	4	no	—
130	C₂²⋊D₁₉	—	8	1	no	—
131	C₂²⋊D₁₉	—	8	1	no	—
132	C₂×Dic₁₉	—	2	1	no	—
133	C₂×Dic₁₉	—	2	1	no	—
134	C₂×Dic₁₉	—	2	1	no	—
135	C₂×Dic₁₉	—	8	1	no	—
136	C₂×Dic₁₉	—	8	1	no	—
137	C₂×Dic₁₉	—	8	1	no	—
138	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
139	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
140	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
141	D₇₆	—	2	1	no	—
142	D₇₆	#41	1	2	yes	—
143	C₄×D₁₉	—	2	1	no	—
144	C₄×D₁₉	—	2	1	no	—
145	C₄×D₁₉	—	8	1	no	—
146	C₄×D₁₉	—	8	1	no	—
147	C₈×C₁₉	—	2	1	no	—
148	C₈×C₁₉	—	2	1	no	—
149	C₁₉⋊C₈	—	2	1	no	—
150	C₁₉⋊C₈	—	2	1	no	—
151	#3122	—	19	1	no	maximal, Sylow
152	C₂²×C₁₉	—	2	2	no	—
153	C₂²×C₁₉	—	4	1	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	—	4	1	no	—
159	C₂²×C₁₉	—	4	1	no	—
160	C₂²×C₁₉	—	4	1	no	—
161	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
162	C₂×D₁₉	—	2	1	no	—
163	C₂×D₁₉	—	2	2	no	—
164	C₂×D₁₉	—	2	1	no	—
165	C₂×D₁₉	—	2	1	no	—
166	C₂×D₁₉	—	4	1	no	—
167	C₂×D₁₉	—	4	1	no	—
168	C₂×D₁₉	—	8	4	no	—
169	C₄×C₁₉	—	2	1	no	—
170	C₄×C₁₉	—	2	1	no	—
171	C₄×C₁₉	—	4	1	no	—
172	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
173	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
174	Dic₁₉	—	2	1	no	—
175	Dic₁₉	—	2	1	no	—
176	Dic₁₉	—	2	1	no	—
177	Dic₁₉	—	4	1	no	—
178	Dic₁₉	—	4	1	no	—
179	Dic₁₉	—	4	1	no	—
180	#540	—	19	1	no	—
181	#518	—	19	1	no	—
182	#482	—	19	1	no	—
183	#482	—	19	1	no	—
184	#478	—	19	1	no	—
185	#478	—	19	1	no	—
186	#459	—	19	1	no	—
187	#459	D₁₉	1	1	yes	characteristic, complemented
188	#447	—	19	1	no	—
189	#447	—	19	1	no	—
190	#442	—	19	1	no	—
191	#442	—	19	1	no	—
192	C₄×D₈	—	19	1	no	—
193	C₄×D₈	—	19	1	no	—
194	#433	—	19	1	no	—
195	C₂×C₁₉	—	4	2	no	—
196	C₂×C₁₉	—	8	1	no	—
197	C₂×C₁₉	—	8	1	no	—
198	C₂×C₁₉	#525	1	1	yes	characteristic
199	C₂×C₁₉	#576	1	1	yes	characteristic
200	C₂×C₁₉	#576	1	1	yes	characteristic
201	D₁₉	—	4	1	no	—
202	D₁₉	—	4	2	no	—
203	C₂²×D₄	—	19	1	no	—
204	C₂×D₈	—	19	1	no	—
205	C₂×D₈	—	19	1	no	—
206	C₂×D₈	C₂×D₁₉	1	1	yes	characteristic
207	C₂×D₈	C₂×D₁₉	1	1	yes	characteristic
208	#127	C₂×D₁₉	1	1	yes	characteristic, complemented
209	#125	—	19	1	no	—
210	#123	—	38	1	no	—
211	#123	—	38	1	no	—
212	#121	—	19	1	no	—
213	#121	—	19	1	no	—
214	#121	—	38	1	no	—
215	#121	—	38	1	no	—
216	C₄×D₄	—	19	1	no	—
217	C₄×D₄	—	19	1	no	—
218	C₄×D₄	—	19	1	no	—
219	C₄×D₄	—	19	1	no	—
220	C₄×D₄	—	19	1	no	—
221	C₄×D₄	—	38	1	no	—
222	C₄×D₄	—	38	1	no	—
223	C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
224	#117	—	19	1	no	—
225	#117	—	19	1	no	—
226	C₂×C₂²⋊C₄	—	38	1	no	—
227	C₂×C₂²⋊C₄	—	38	1	no	—
228	#107	—	19	1	no	—
229	#107	—	19	1	no	—
230	#106	—	19	1	no	—
231	#106	—	19	1	no	—
232	C₄⋊C₈	—	19	1	no	—
233	C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic, complemented
234	#102	—	19	1	no	—
235	#102	—	19	1	no	—
236	#102	—	19	1	no	—
237	#102	—	19	1	no	—
238	#102	—	19	1	no	—
239	#102	—	19	1	no	—
240	#102	C₂×D₁₉	1	1	yes	characteristic, complemented
241	#102	C₂×D₁₉	1	1	yes	characteristic, complemented
242	C₂²⋊C₈	—	19	1	no	—
243	C₂²⋊C₈	—	19	1	no	—
244	#97	—	19	1	no	—
245	C₄×C₈	—	19	1	no	—
246	C₁₉	#3122	1	1	yes	characteristic, complemented, Sylow
247	C₂⁴	—	38	1	no	—
248	C₂×D₄	—	2	1	no	—
249	C₂×D₄	—	2	1	no	—
250	C₂×D₄	—	19	1	no	—
251	C₂×D₄	—	38	1	no	—
252	C₂×D₄	—	38	2	no	—
253	C₂×D₄	—	38	2	no	—
254	C₂×D₄	—	38	1	no	—
255	C₂×D₄	—	38	1	no	—
256	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
257	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
258	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
259	C₂²×C₄	—	2	1	no	—
260	C₂²×C₄	—	19	1	no	—
261	C₂²×C₄	—	38	1	no	—
262	C₂²×C₄	—	38	1	no	—
263	C₂²×C₄	—	38	1	no	—
264	C₂²×C₄	—	38	1	no	—
265	C₂²×C₄	—	38	1	no	—
266	D₈	—	2	2	no	—
267	D₈	—	2	2	no	—
268	D₈	—	38	2	no	—
269	D₈	—	38	2	no	—
270	C₂×C₈	—	19	1	no	—
271	C₂×C₈	—	19	1	no	—
272	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
273	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
274	C₄⋊C₄	—	19	1	no	—
275	C₄⋊C₄	—	19	1	no	—
276	C₄⋊C₄	—	19	1	no	—
277	C₄⋊C₄	—	19	1	no	—
278	C₄⋊C₄	—	19	1	no	—
279	C₄⋊C₄	—	38	1	no	—
280	C₄⋊C₄	—	38	1	no	—
281	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
282	C₂²⋊C₄	—	2	1	no	—
283	C₂²⋊C₄	—	19	1	no	—
284	C₂²⋊C₄	—	19	1	no	—
285	C₂²⋊C₄	—	38	1	no	—
286	C₂²⋊C₄	—	38	1	no	—
287	C₂²⋊C₄	—	38	1	no	—
288	C₂²⋊C₄	—	38	1	no	—
289	C₂²⋊C₄	—	38	1	no	—
290	C₂²⋊C₄	—	38	1	no	—
291	C₂²⋊C₄	—	38	1	no	—
292	C₂²⋊C₄	—	38	1	no	—
293	C₄²	—	19	1	no	—
294	C₄²	—	38	1	no	—
295	C₄²	C₂²×D₁₉	1	1	yes	characteristic
296	C₂³	—	2	1	no	—
297	C₂³	—	2	1	no	—
298	C₂³	—	2	1	no	—
299	C₂³	—	19	1	no	—
300	C₂³	—	38	4	no	—
301	C₂³	—	38	1	no	—
302	C₂³	—	152	1	no	—
303	C₂³	—	152	1	no	—
304	D₄	—	2	1	no	—
305	D₄	—	2	1	no	—
306	D₄	—	2	1	no	—
307	D₄	—	2	1	no	—
308	D₄	—	2	1	no	—
309	D₄	—	8	1	no	—
310	D₄	—	8	1	no	—
311	D₄	—	19	2	no	—
312	D₄	—	38	4	no	—
313	D₄	—	38	1	no	—
314	D₄	—	152	1	no	—
315	D₄	—	152	1	no	—
316	D₄	#64543	1	2	yes	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	8	1	no	—
319	C₂×C₄	—	19	1	no	—
320	C₂×C₄	—	19	1	no	—
321	C₂×C₄	—	19	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	—	152	1	no	—
328	C₂×C₄	—	152	1	no	—
329	C₂×C₄	—	152	1	no	—
330	C₂×C₄	—	152	1	no	—
331	C₂×C₄	—	152	1	no	—
332	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
333	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
334	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
335	C₈	—	2	1	no	—
336	C₈	—	2	1	no	—
337	C₈	—	38	1	no	—
338	C₈	—	38	1	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	—	4	1	no	—
345	C₂²	—	4	1	no	—
346	C₂²	—	4	1	no	—
347	C₂²	—	4	1	no	—
348	C₂²	—	38	1	no	—
349	C₂²	—	38	2	no	—
350	C₂²	—	38	1	no	—
351	C₂²	—	38	1	no	—
352	C₂²	—	76	1	no	—
353	C₂²	—	76	1	no	—
354	C₂²	—	152	4	no	—
355	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	4	1	no	—
359	C₄	—	38	1	no	—
360	C₄	—	38	1	no	—
361	C₄	—	38	1	no	—
362	C₄	—	76	1	no	—
363	C₄	—	76	1	no	—
364	C₄	—	76	1	no	—
365	C₄	C₂×#64543	1	1	yes	characteristic
366	C₄	#41×D₁₉	1	1	yes	characteristic
367	C₂	—	4	2	no	—
368	C₂	—	8	1	no	—
369	C₂	—	8	1	no	—
370	C₂	—	76	1	no	—
371	C₂	—	76	2	no	—
372	C₂	#11927054	1	1	yes	characteristic
373	C₂	#11927345	1	1	yes	characteristic
374	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×#459	C₂
socle	C₂²×C₁₉	C₂²×D₄
hypercenter	#459	D₁₉
hypocenter	C₁₉	#3122
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #459

Complements

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

N	H ≅ G/N	type
1. #11927148	C₂	semidirect
2. #11927016	C₂	semidirect
3. #11926867	C₂	semidirect
4. #11926866	C₂	semidirect
5. #11926845	C₂	semidirect
6. #11926844	C₂	semidirect
7. #11926736	C₂	semidirect
8. #11926676	C₂	semidirect
9. #11926676	C₂	semidirect
10. #11926656	C₂	semidirect
11. #11926653	C₂	semidirect
12. D₈×Dic₁₉	C₂	semidirect
13. #11926624	C₂	semidirect
14. #11926604	C₂	semidirect
15. C₁₉×#459	C₂	semidirect
19. C₁₉×#127	C₂²	semidirect
28. D₄×Dic₁₉	C₂²	semidirect
29. D₄×Dic₁₉	C₂²	semidirect
40. #10597316	C₂²	semidirect
41. C₁₉×C₄⋊C₈	C₂²	semidirect
42. C₁₉×#102	C₂²	semidirect
43. C₁₉×#102	C₂²	semidirect
46. #10597289	C₂²	semidirect
47. #10597289	C₂²	semidirect
52. #10597275	C₂²	semidirect
53. #10597274	C₂²	semidirect
54. #10597272	C₂²	semidirect
55. C₈×Dic₁₉	C₂²	semidirect
56. #10597265	C₂²	semidirect
57. #10597264	C₂²	semidirect
58. C₄⋊(C₁₉⋊C₈)	C₂²	semidirect
187. #459	D₁₉	semidirect
208. #127	C₂×D₁₉	semidirect
233. C₄⋊C₈	C₂×D₁₉	semidirect
240. #102	C₂×D₁₉	semidirect
241. #102	C₂×D₁₉	semidirect
246. C₁₉	#3122	semidirect

Maximal subgroups

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

subgroup	index
#11927148	2
#11927016	2
#11926867	2
#11926866	2
#11926845	2
#11926844	2
#11926736	2
#11926676	2
#11926656	2
#11926653	2
D₈×Dic₁₉	2
#11926624	2
#11926604	2
C₁₉×#459	2
#3122	19