Subgroups — #14573641

Order 2432.

448 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927098	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₄×D₁₉	C₂	1	1	yes	maximal, characteristic
3	#11926876	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926873	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926833	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926832	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926748	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926747	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926696	C₂	1	1	yes	maximal, characteristic
10	#11926693	C₂	1	1	yes	maximal, characteristic
11	C₄×#64516	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#64527	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#64526	C₂	1	1	yes	maximal, characteristic, complemented
14	#11926595	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×SD₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₄×D₁₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₁₉	—	2	1	no	—
18	C₂×SD₁₆×C₁₉	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic, complemented
20	C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
21	#10597392	C₂²	1	1	yes	characteristic
22	#10597390	C₂²	1	1	yes	characteristic, complemented
23	C₂×#64527	C₂²	1	1	yes	characteristic
24	D₄×Dic₁₉	C₂²	1	1	yes	characteristic
25	C₂×#64526	C₂²	1	1	yes	characteristic
26	C₄×C₂²⋊D₁₉	—	2	1	no	—
27	C₂×#64516	C₂²	1	1	yes	characteristic
28	#10597343	C₂²	1	1	yes	characteristic
29	#10597342	—	2	1	no	—
30	#10597339	C₂²	1	1	yes	characteristic
31	C₄⋊C₄×D₁₉	C₂²	1	1	yes	characteristic
32	#10597335	—	2	1	no	—
33	#10597334	C₂²	1	1	yes	characteristic, complemented
34	(C₂²⋊D₁₉)⋊C₄	—	2	1	no	—
35	C₂²⋊C₄×D₁₉	—	2	1	no	—
36	C₄×D₇₆	C₂²	1	1	yes	characteristic
37	#10597318	—	2	1	no	—
38	C₄²×D₁₉	C₂²	1	1	yes	characteristic
39	C₄×Dic₃₈	—	2	1	no	—
40	C₄×Dic₃₈	C₂²	1	1	yes	characteristic, complemented
41	C₁₉×#106	C₂²	1	1	yes	characteristic, complemented
42	C₁₉×#103	C₂²	1	1	yes	characteristic, complemented
43	C₁₉×#102	C₂²	1	1	yes	characteristic
44	C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic, complemented
45	#10597292	C₂²	1	1	yes	characteristic, complemented
46	#10597289	C₂²	1	1	yes	characteristic
47	#10597278	C₂²	1	1	yes	characteristic
48	(C₂×D₁₉)⋊C₈	C₂²	1	1	yes	characteristic
49	(C₂×D₁₉)⋊C₈	C₂²	1	1	yes	characteristic
50	#10597274	C₂²	1	1	yes	characteristic, complemented
51	#10597273	C₂²	1	1	yes	characteristic, complemented
52	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic, complemented
53	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic, complemented
54	#10597267	C₂²	1	1	yes	characteristic, complemented
55	#10597266	C₂²	1	1	yes	characteristic
56	#10597265	C₂²	1	1	yes	characteristic, complemented
57	#10597265	C₂²	1	1	yes	characteristic, complemented
58	C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₁₉	—	2	1	no	—
60	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	—	2	1	no	—
63	C₂×C₂²⋊D₁₉	—	2	1	no	—
64	C₂²×Dic₁₉	—	2	1	no	—
65	D₄×D₁₉	—	2	2	no	—
66	D₄×D₁₉	—	2	2	no	—
67	C₂×D₇₆	C₂³	1	1	yes	characteristic
68	C₂×C₄×D₁₉	—	2	1	no	—
69	C₂×C₄×D₁₉	—	2	1	no	—
70	C₂×C₄×D₁₉	—	2	1	no	—
71	C₂×C₄×D₁₉	—	4	4	no	—
72	C₂×C₄×D₁₉	D₄	1	1	yes	characteristic
73	C₂×C₄×D₁₉	D₄	1	1	yes	characteristic
74	C₂×C₄×D₁₉	C₂³	1	1	yes	characteristic
75	C₂×Dic₃₈	—	2	1	no	—
76	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
77	SD₁₆×C₁₉	—	2	2	no	—
78	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
79	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₁₉	—	2	1	no	—
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	C₄²×C₁₉	—	2	1	no	—
85	C₄²×C₁₉	C₂³	1	1	yes	characteristic
86	C₂²⋊Dic₁₉	—	2	1	no	—
87	#64527	—	2	2	no	—
88	#64526	—	2	2	no	—
89	(C₂×D₁₉)⋊C₄	—	2	1	no	—
90	(C₂×D₁₉)⋊C₄	—	2	1	no	—
91	(C₂×D₁₉)⋊C₄	—	2	1	no	—
92	(C₂×D₁₉)⋊C₄	—	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	Dic₁₉⋊C₄	—	2	1	no	—
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	Dic₁₉⋊C₄	—	2	1	no	—
102	C₄×Dic₁₉	—	2	1	no	—
103	C₄×Dic₁₉	D₄	1	1	yes	characteristic
104	C₄×Dic₁₉	D₄	1	1	yes	characteristic
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	#64516	—	2	2	no	—
109	C₂³×C₁₉	—	2	1	no	—
110	C₂²×D₁₉	—	2	4	no	—
111	C₂²×D₁₉	—	2	1	no	—
112	C₂²×D₁₉	—	4	2	no	—
113	C₂²×D₁₉	—	4	2	no	—
114	C₂²×D₁₉	C₂×D₄	1	1	yes	characteristic
115	Q₈×C₁₉	—	2	1	no	—
116	Q₈×C₁₉	—	2	1	no	—
117	D₄×C₁₉	—	2	1	no	—
118	D₄×C₁₉	#41	1	2	yes	—
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	—	4	2	no	—
123	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
126	C₂²⋊D₁₉	—	2	4	no	—
127	C₂×Dic₁₉	—	2	1	no	—
128	C₂×Dic₁₉	—	2	1	no	—
129	C₂×Dic₁₉	—	2	1	no	—
130	C₂×Dic₁₉	—	4	2	no	—
131	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
133	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
134	D₇₆	—	2	1	no	—
135	D₇₆	#41	1	2	yes	—
136	C₄×D₁₉	—	2	1	no	—
137	C₄×D₁₉	—	2	1	no	—
138	C₄×D₁₉	—	4	1	no	—
139	C₄×D₁₉	—	4	2	no	—
140	C₄×D₁₉	—	4	2	no	—
141	C₄×D₁₉	—	4	2	no	—
142	C₄×D₁₉	SD₁₆	1	4	yes	complemented
143	Dic₃₈	—	2	1	no	—
144	Dic₃₈	—	2	1	no	—
145	Dic₃₈	—	4	2	no	—
146	C₈×C₁₉	—	2	1	no	—
147	C₈×C₁₉	—	2	1	no	—
148	C₁₉⋊C₈	—	2	1	no	—
149	C₁₉⋊C₈	—	2	1	no	—
150	#3101	—	19	1	no	maximal, Sylow
151	C₂²×C₁₉	—	2	2	no	—
152	C₂²×C₁₉	—	4	1	no	—
153	C₂²×C₁₉	—	4	1	no	—
154	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
155	C₂×D₁₉	—	2	1	no	—
156	C₂×D₁₉	—	2	1	no	—
157	C₂×D₁₉	—	2	2	no	—
158	C₂×D₁₉	—	4	8	no	—
159	C₂×D₁₉	—	4	1	no	—
160	C₂×D₁₉	—	4	1	no	—
161	C₂×D₁₉	C₂×SD₁₆	1	2	yes	—
162	C₄×C₁₉	—	2	1	no	—
163	C₄×C₁₉	—	4	1	no	—
164	C₄×C₁₉	—	4	1	no	—
165	C₄×C₁₉	—	4	1	no	—
166	C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
167	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
168	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
169	Dic₁₉	—	2	1	no	—
170	Dic₁₉	—	2	1	no	—
171	Dic₁₉	—	4	1	no	—
172	Dic₁₉	—	4	1	no	—
173	Dic₁₉	—	4	1	no	—
174	Dic₁₉	C₂×SD₁₆	1	2	yes	—
175	#533	—	19	1	no	—
176	C₂×C₄×D₄	—	19	1	no	—
177	#483	—	19	1	no	—
178	#483	—	19	1	no	—
179	#476	—	19	1	no	—
180	#476	—	19	1	no	—
181	#461	—	19	1	no	—
182	#461	—	19	1	no	—
183	#450	—	19	1	no	—
184	#450	—	19	1	no	—
185	C₄×SD₁₆	—	19	1	no	—
186	C₄×SD₁₆	—	19	1	no	—
187	C₄×SD₁₆	—	19	1	no	—
188	C₄×SD₁₆	D₁₉	1	1	yes	characteristic, complemented
189	#431	—	19	1	no	—
190	C₂×C₁₉	—	4	2	no	—
191	C₂×C₁₉	#525	1	1	yes	characteristic
192	C₂×C₁₉	C₂²×SD₁₆	1	1	yes	characteristic
193	C₂×C₁₉	#575	1	1	yes	characteristic
194	D₁₉	—	2	2	no	—
195	D₁₉	—	4	2	no	—
196	C₂²×D₄	—	19	1	no	—
197	C₂³×C₄	—	38	1	no	—
198	C₂×SD₁₆	—	19	1	no	—
199	C₂×SD₁₆	—	19	1	no	—
200	C₂×SD₁₆	—	19	1	no	—
201	C₂×SD₁₆	C₂×D₁₉	1	1	yes	characteristic
202	C₄⋊Q₈	—	19	1	no	—
203	C₄⋊Q₈	—	19	1	no	—
204	#125	—	38	1	no	—
205	C₂²⋊Q₈	—	19	1	no	—
206	C₂²⋊Q₈	—	19	1	no	—
207	C₂²⋊Q₈	—	38	1	no	—
208	C₄×Q₈	—	19	1	no	—
209	C₄×Q₈	—	38	1	no	—
210	C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic, complemented
211	C₄×D₄	—	19	1	no	—
212	C₄×D₄	—	19	1	no	—
213	C₄×D₄	—	19	1	no	—
214	C₄×D₄	—	38	1	no	—
215	C₄×D₄	—	38	1	no	—
216	C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
217	#117	—	38	1	no	—
218	C₂×C₄⋊C₄	—	19	1	no	—
219	C₂×C₂²⋊C₄	—	38	1	no	—
220	C₂×C₄²	—	19	1	no	—
221	#106	—	19	1	no	—
222	#106	—	19	1	no	—
223	#106	—	19	1	no	—
224	#106	C₂×D₁₉	1	1	yes	characteristic, complemented
225	C₄⋊C₈	—	19	1	no	—
226	C₄⋊C₈	—	19	1	no	—
227	#103	—	19	1	no	—
228	#103	—	19	1	no	—
229	#103	—	19	1	no	—
230	#103	C₂×D₁₉	1	1	yes	characteristic, complemented
231	#102	—	19	1	no	—
232	#102	—	19	1	no	—
233	#102	—	19	1	no	—
234	#102	C₂×D₁₉	1	1	yes	characteristic
235	C₂²⋊C₈	—	19	1	no	—
236	C₂²⋊C₈	—	19	1	no	—
237	C₄×C₈	—	19	1	no	—
238	C₄×C₈	C₂×D₁₉	1	1	yes	characteristic, complemented
239	C₁₉	#3101	1	1	yes	characteristic, complemented, Sylow
240	C₂⁴	—	38	1	no	—
241	C₂×Q₈	—	19	1	no	—
242	C₂×Q₈	—	38	1	no	—
243	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
244	C₂×D₄	—	19	1	no	—
245	C₂×D₄	—	38	1	no	—
246	C₂×D₄	—	38	2	no	—
247	C₂×D₄	—	38	2	no	—
248	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	19	1	no	—
251	C₂²×C₄	—	19	1	no	—
252	C₂²×C₄	—	19	1	no	—
253	C₂²×C₄	—	38	1	no	—
254	C₂²×C₄	—	38	1	no	—
255	C₂²×C₄	—	38	1	no	—
256	C₂²×C₄	—	38	1	no	—
257	C₂²×C₄	—	76	4	no	—
258	SD₁₆	—	2	2	no	—
259	SD₁₆	—	38	2	no	—
260	SD₁₆	—	38	2	no	—
261	SD₁₆	—	38	2	no	—
262	C₂×C₈	—	19	1	no	—
263	C₂×C₈	—	19	1	no	—
264	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
265	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	19	1	no	—
268	C₄⋊C₄	—	19	1	no	—
269	C₄⋊C₄	—	38	1	no	—
270	C₄⋊C₄	—	38	1	no	—
271	C₄⋊C₄	—	38	1	no	—
272	C₄⋊C₄	—	38	1	no	—
273	C₄⋊C₄	—	38	1	no	—
274	C₄⋊C₄	—	38	1	no	—
275	C₄⋊C₄	—	38	1	no	—
276	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
277	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	38	1	no	—
280	C₂²⋊C₄	—	38	1	no	—
281	C₂²⋊C₄	—	38	1	no	—
282	C₂²⋊C₄	—	38	1	no	—
283	C₂²⋊C₄	—	38	1	no	—
284	C₄²	—	2	1	no	—
285	C₄²	—	19	1	no	—
286	C₄²	—	19	1	no	—
287	C₄²	—	19	1	no	—
288	C₄²	—	38	1	no	—
289	C₄²	C₂²×D₁₉	1	1	yes	characteristic
290	C₂³	—	2	1	no	—
291	C₂³	—	19	1	no	—
292	C₂³	—	38	4	no	—
293	C₂³	—	38	1	no	—
294	C₂³	—	76	2	no	—
295	C₂³	—	76	2	no	—
296	Q₈	—	2	1	no	—
297	Q₈	—	2	1	no	—
298	Q₈	—	38	1	no	—
299	Q₈	—	38	1	no	—
300	Q₈	—	76	2	no	—
301	D₄	—	2	1	no	—
302	D₄	—	19	2	no	—
303	D₄	—	38	4	no	—
304	D₄	—	38	1	no	—
305	D₄	#64541	1	2	yes	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	19	1	no	—
311	C₂×C₄	—	19	4	no	—
312	C₂×C₄	—	19	1	no	—
313	C₂×C₄	—	19	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	—	76	1	no	—
320	C₂×C₄	—	76	2	no	—
321	C₂×C₄	—	76	2	no	—
322	C₂×C₄	—	76	2	no	—
323	C₂×C₄	—	76	2	no	—
324	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
325	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
326	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
327	C₈	—	2	1	no	—
328	C₈	—	2	1	no	—
329	C₈	—	38	1	no	—
330	C₈	—	38	1	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	4	1	no	—
334	C₂²	—	19	2	no	—
335	C₂²	—	38	1	no	—
336	C₂²	—	38	1	no	—
337	C₂²	—	38	2	no	—
338	C₂²	—	76	8	no	—
339	C₂²	—	76	1	no	—
340	C₂²	—	76	1	no	—
341	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	4	1	no	—
344	C₄	—	4	1	no	—
345	C₄	—	4	1	no	—
346	C₄	—	19	2	no	—
347	C₄	—	38	1	no	—
348	C₄	—	38	1	no	—
349	C₄	—	76	1	no	—
350	C₄	—	76	1	no	—
351	C₄	—	76	1	no	—
352	C₄	SD₁₆×D₁₉	1	2	yes	—
353	C₄	C₂×#64541	1	1	yes	characteristic
354	C₄	#41×D₁₉	1	1	yes	characteristic
355	C₂	—	4	2	no	—
356	C₂	—	38	2	no	—
357	C₂	—	76	2	no	—
358	C₂	#11927053	1	1	yes	characteristic
359	C₂	C₂×SD₁₆×D₁₉	1	1	yes	characteristic
360	C₂	#11927341	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₉
commutator, cosocle	C₂×C₄×C₁₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉
Fitting	C₄×SD₁₆×C₁₉	C₂
socle	C₂²×C₁₉	C₂²×D₄
prime radical	C₂×D₄×D₁₉	C₂²
hypercenter	C₄×SD₁₆	D₁₉
hypocenter	C₁₉	#3101
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #11927098	C₂	semidirect
3. #11926876	C₂	semidirect
4. #11926873	C₂	semidirect
5. #11926833	C₂	semidirect
6. #11926832	C₂	semidirect
7. #11926748	C₂	semidirect
8. #11926747	C₂	semidirect
11. C₄×#64516	C₂	semidirect
12. C₄×#64527	C₂	semidirect
13. C₄×#64526	C₂	semidirect
14. #11926595	C₂	semidirect
15. C₄×SD₁₆×C₁₉	C₂	semidirect
19. C₄×Q₈×C₁₉	C₂²	semidirect
22. #10597390	C₂²	semidirect
33. #10597334	C₂²	semidirect
40. C₄×Dic₃₈	C₂²	semidirect
41. C₁₉×#106	C₂²	semidirect
42. C₁₉×#103	C₂²	semidirect
44. C₄×C₈×C₁₉	C₂²	semidirect
45. #10597292	C₂²	semidirect
50. #10597274	C₂²	semidirect
51. #10597273	C₂²	semidirect
52. Dic₁₉⋊C₈	C₂²	semidirect
53. Dic₁₉⋊C₈	C₂²	semidirect
54. #10597267	C₂²	semidirect
56. #10597265	C₂²	semidirect
57. #10597265	C₂²	semidirect
58. C₄×C₁₉⋊C₈	C₂²	semidirect
142. C₄×D₁₉	SD₁₆	semidirect
188. C₄×SD₁₆	D₁₉	semidirect
210. C₄×Q₈	C₂×D₁₉	semidirect
224. #106	C₂×D₁₉	semidirect
230. #103	C₂×D₁₉	semidirect
238. C₄×C₈	C₂×D₁₉	semidirect
239. C₁₉	#3101	semidirect

Maximal subgroups

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

subgroup	index
#11927098	2
C₄×D₄×D₁₉	2
#11926876	2
#11926873	2
#11926833	2
#11926832	2
#11926748	2
#11926747	2
#11926696	2
#11926693	2
C₄×#64516	2
C₄×#64527	2
C₄×#64526	2
#11926595	2
C₄×SD₁₆×C₁₉	2
#3101	19