Subgroups — #14574519

Order 2432.

390 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927122	C₂	1	1	yes	maximal, characteristic
2	#11927115	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926886	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926880	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926875	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926869	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926802	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926790	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926785	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926773	C₂	1	1	yes	maximal, characteristic, complemented
11	#11926708	C₂	1	1	yes	maximal, characteristic, complemented
12	#11926708	C₂	1	1	yes	maximal, characteristic, complemented
13	#11926702	C₂	1	1	yes	maximal, characteristic, complemented
14	#11926685	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₉×#408	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#41×C₁₉	C₂²	1	1	yes	characteristic
17	C₂×#64541	C₂²	1	1	yes	characteristic
18	C₂²×Dic₃₈	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₁₉	C₂²	1	1	yes	characteristic
21	#10597392	—	2	1	no	—
22	#10597390	—	2	1	no	—
23	#10597385	—	2	1	no	—
24	#10597385	—	2	1	no	—
25	#10597384	—	2	1	no	—
26	#10597382	—	2	1	no	—
27	#10597381	—	2	1	no	—
28	#10597381	—	2	1	no	—
29	#10597376	C₂²	1	1	yes	characteristic
30	#10597376	C₂²	1	1	yes	characteristic
31	#10597372	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
33	#10597370	C₂²	1	1	yes	characteristic
34	#10597370	C₂²	1	1	yes	characteristic
35	#10597370	C₂²	1	1	yes	characteristic
36	#10597370	C₂²	1	1	yes	characteristic
37	#10597370	C₂²	1	1	yes	characteristic
38	#10597370	C₂²	1	1	yes	characteristic
39	C₂×Dic₇₆	C₂²	1	1	yes	characteristic
40	C₂×Dic₇₆	C₂²	1	1	yes	characteristic
41	C₂×#64516	C₂²	1	1	yes	characteristic
42	C₂×#64516	C₂²	1	1	yes	characteristic
43	C₁₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
44	C₁₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
45	C₁₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
46	C₁₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
47	#10597278	C₂²	1	1	yes	characteristic
48	#10597278	C₂²	1	1	yes	characteristic
49	#10597275	C₂²	1	1	yes	characteristic
50	#10597275	C₂²	1	1	yes	characteristic
51	#10597274	C₂²	1	1	yes	characteristic
52	#10597274	C₂²	1	1	yes	characteristic
53	#10597273	C₂²	1	1	yes	characteristic
54	#10597273	C₂²	1	1	yes	characteristic
55	#10597273	C₂²	1	1	yes	characteristic
56	#10597273	C₂²	1	1	yes	characteristic
57	#10597273	C₂²	1	1	yes	characteristic
58	#10597273	C₂²	1	1	yes	characteristic
59	#41×C₁₉	—	4	1	no	—
60	#41×C₁₉	—	4	1	no	—
61	C₂×Q₈×C₁₉	D₄	1	1	yes	characteristic
62	C₂×D₄×C₁₉	D₄	1	1	yes	characteristic
63	C₂×D₄×C₁₉	D₄	1	1	yes	characteristic
64	C₂×D₄×C₁₉	D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
66	C₂²×C₄×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	C₂²×Dic₁₉	—	2	1	no	—
72	C₂²×Dic₁₉	—	2	1	no	—
73	#64541	—	4	1	no	—
74	#64541	—	4	1	no	—
75	C₂×D₇₆	C₂³	1	1	yes	characteristic
76	C₂×C₄×D₁₉	—	2	1	no	—
77	C₂×Dic₃₈	—	2	1	no	—
78	C₂×Dic₃₈	—	4	1	no	—
79	C₂×Dic₃₈	—	4	1	no	—
80	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
81	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
82	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
83	M₁₆×C₁₉	—	4	1	no	—
84	C₂×C₈×C₁₉	—	4	1	no	—
85	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
86	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
87	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
88	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
94	C₂²⋊Dic₁₉	—	2	1	no	—
95	C₂²⋊Dic₁₉	—	2	1	no	—
96	(C₂×D₁₉)⋊C₄	—	2	1	no	—
97	(C₂×D₁₉)⋊C₄	—	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	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
103	Dic₁₉⋊C₄	—	2	1	no	—
104	Dic₁₉⋊C₄	—	2	1	no	—
105	Dic₁₉⋊C₄	—	2	1	no	—
106	Dic₁₉⋊C₄	—	2	1	no	—
107	Dic₁₉⋊C₄	—	2	1	no	—
108	Dic₁₉⋊C₄	—	2	1	no	—
109	C₄×Dic₁₉	—	2	1	no	—
110	Dic₇₆	—	4	1	no	—
111	Dic₇₆	—	4	1	no	—
112	#64516	—	4	1	no	—
113	#64516	—	4	1	no	—
114	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
115	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
116	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
117	C₂²×D₁₉	—	2	1	no	—
118	Q₈×C₁₉	—	4	1	no	—
119	D₄×C₁₉	—	4	1	no	—
120	D₄×C₁₉	—	4	1	no	—
121	D₄×C₁₉	—	4	1	no	—
122	C₂×C₄×C₁₉	—	2	1	no	—
123	C₂×C₄×C₁₉	—	2	1	no	—
124	C₂×C₄×C₁₉	—	2	1	no	—
125	C₂×C₄×C₁₉	—	2	1	no	—
126	C₂×C₄×C₁₉	—	2	1	no	—
127	C₂×C₄×C₁₉	—	2	1	no	—
128	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
132	C₂²⋊D₁₉	—	4	2	no	—
133	C₂²⋊D₁₉	—	8	1	no	—
134	C₂²⋊D₁₉	—	8	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	—	4	2	no	—
143	C₂×Dic₁₉	—	8	1	no	—
144	D₇₆	—	2	1	no	—
145	D₇₆	—	2	1	no	—
146	C₄×D₁₉	—	4	2	no	—
147	Dic₃₈	—	2	1	no	—
148	Dic₃₈	—	2	1	no	—
149	Dic₃₈	—	2	1	no	—
150	Dic₃₈	—	2	1	no	—
151	Dic₃₈	—	2	1	no	—
152	Dic₃₈	—	2	1	no	—
153	Dic₃₈	—	4	2	no	—
154	C₈×C₁₉	—	2	1	no	—
155	C₈×C₁₉	—	2	1	no	—
156	C₈×C₁₉	—	2	1	no	—
157	C₈×C₁₉	—	2	1	no	—
158	#3201	—	19	1	no	maximal, Sylow
159	C₂²×C₁₉	—	2	1	no	—
160	C₂²×C₁₉	—	2	1	no	—
161	C₂²×C₁₉	—	2	1	no	—
162	C₂²×C₁₉	—	2	1	no	—
163	C₂²×C₁₉	—	2	1	no	—
164	C₂²×C₁₉	—	2	1	no	—
165	C₂²×C₁₉	—	2	1	no	—
166	C₂²×C₁₉	—	2	1	no	—
167	C₂²×C₁₉	—	2	1	no	—
168	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
169	C₂×D₁₉	—	4	1	no	—
170	C₂×D₁₉	—	4	1	no	—
171	C₂×D₁₉	—	4	1	no	—
172	C₄×C₁₉	—	2	1	no	—
173	C₄×C₁₉	—	2	1	no	—
174	C₄×C₁₉	—	2	1	no	—
175	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
176	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
177	Dic₁₉	—	4	1	no	—
178	Dic₁₉	—	4	1	no	—
179	Dic₁₉	—	4	1	no	—
180	Dic₁₉	—	4	1	no	—
181	Dic₁₉	—	4	1	no	—
182	Dic₁₉	—	4	1	no	—
183	Dic₁₉	—	4	1	no	—
184	#537	—	19	1	no	—
185	#536	—	19	1	no	—
186	#485	—	19	1	no	—
187	#484	—	19	1	no	—
188	#483	—	19	1	no	—
189	#482	—	19	1	no	—
190	#470	—	19	1	no	—
191	#468	—	19	1	no	—
192	#467	—	19	1	no	—
193	#465	—	19	1	no	—
194	#452	—	19	1	no	—
195	#452	—	19	1	no	—
196	#451	—	19	1	no	—
197	#448	—	19	1	no	—
198	#408	D₁₉	1	1	yes	characteristic, complemented
199	C₂×C₁₉	—	4	1	no	—
200	C₂×C₁₉	—	4	1	no	—
201	C₂×C₁₉	—	4	1	no	—
202	C₂×C₁₉	#534	1	1	yes	characteristic
203	C₂×C₁₉	#577	1	1	yes	characteristic
204	C₂×C₁₉	#578	1	1	yes	characteristic
205	D₁₉	—	8	1	no	—
206	C₂×#41	—	19	1	no	—
207	C₂×#41	C₂×D₁₉	1	1	yes	characteristic
208	C₂²×Q₈	—	19	1	no	—
209	C₂×Q₁₆	—	19	1	no	—
210	C₂×Q₁₆	—	19	1	no	—
211	C₂×SD₁₆	—	19	1	no	—
212	C₂×SD₁₆	—	19	1	no	—
213	C₂×M₁₆	C₂×D₁₉	1	1	yes	characteristic
214	C₂²×C₈	C₂×D₁₉	1	1	yes	characteristic
215	C₄⋊Q₈	—	38	1	no	—
216	#124	—	38	1	no	—
217	#123	—	38	1	no	—
218	#123	—	38	1	no	—
219	C₂²⋊Q₈	—	19	1	no	—
220	C₂²⋊Q₈	—	19	1	no	—
221	C₂²⋊Q₈	—	19	1	no	—
222	C₂²⋊Q₈	—	19	1	no	—
223	C₂²⋊Q₈	—	19	1	no	—
224	C₂²⋊Q₈	—	19	1	no	—
225	C₂²⋊Q₈	—	38	1	no	—
226	#121	—	19	1	no	—
227	#121	—	19	1	no	—
228	#121	—	38	1	no	—
229	#121	—	38	1	no	—
230	#121	—	38	1	no	—
231	#117	—	19	1	no	—
232	C₂×C₄⋊C₄	—	19	1	no	—
233	#107	—	19	1	no	—
234	#107	—	19	1	no	—
235	#106	—	19	1	no	—
236	#106	—	19	1	no	—
237	#103	—	19	1	no	—
238	#103	—	19	1	no	—
239	#103	—	19	1	no	—
240	#103	—	19	1	no	—
241	#103	—	19	1	no	—
242	#103	—	19	1	no	—
243	#102	—	19	1	no	—
244	#102	—	19	1	no	—
245	C₂²⋊C₈	C₂×D₁₉	1	1	yes	characteristic
246	C₂²⋊C₈	C₂×D₁₉	1	1	yes	characteristic
247	C₂²⋊C₈	C₂×D₁₉	1	1	yes	characteristic
248	C₂²⋊C₈	C₂×D₁₉	1	1	yes	characteristic
249	C₁₉	#3201	1	1	yes	characteristic, complemented, Sylow
250	#41	—	4	1	no	—
251	#41	—	4	1	no	—
252	#41	—	76	1	no	—
253	#41	—	76	1	no	—
254	C₂×Q₈	—	19	1	no	—
255	C₂×Q₈	—	19	1	no	—
256	C₂×Q₈	—	19	1	no	—
257	C₂×Q₈	—	38	1	no	—
258	C₂×Q₈	—	76	1	no	—
259	C₂×Q₈	—	76	1	no	—
260	C₂×Q₈	D₇₆	1	1	yes	characteristic
261	C₂×D₄	—	19	1	no	—
262	C₂×D₄	—	38	1	no	—
263	C₂×D₄	—	38	1	no	—
264	C₂×D₄	—	38	1	no	—
265	C₂×D₄	D₇₆	1	1	yes	characteristic
266	C₂×D₄	D₇₆	1	1	yes	characteristic
267	C₂×D₄	D₇₆	1	1	yes	characteristic
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	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
273	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
274	Q₁₆	—	76	1	no	—
275	Q₁₆	—	76	1	no	—
276	SD₁₆	—	76	1	no	—
277	SD₁₆	—	76	1	no	—
278	M₁₆	—	4	1	no	—
279	C₂×C₈	—	4	1	no	—
280	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
281	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
282	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
283	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
284	C₄⋊C₄	—	19	1	no	—
285	C₄⋊C₄	—	19	1	no	—
286	C₄⋊C₄	—	19	1	no	—
287	C₄⋊C₄	—	19	1	no	—
288	C₄⋊C₄	—	38	1	no	—
289	C₄⋊C₄	—	38	1	no	—
290	C₄⋊C₄	—	38	1	no	—
291	C₄⋊C₄	—	38	1	no	—
292	C₄⋊C₄	—	38	1	no	—
293	C₄⋊C₄	—	38	1	no	—
294	C₄⋊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	C₂²⋊C₄	—	38	1	no	—
302	C₂²⋊C₄	—	38	1	no	—
303	C₂²⋊C₄	—	38	1	no	—
304	C₄²	—	38	1	no	—
305	C₂³	—	38	1	no	—
306	C₂³	C₂×D₇₆	1	1	yes	characteristic
307	C₂³	C₂×D₇₆	1	1	yes	characteristic
308	C₂³	C₂×D₇₆	1	1	yes	characteristic
309	Q₈	—	4	1	no	—
310	Q₈	—	38	1	no	—
311	Q₈	—	38	1	no	—
312	Q₈	—	38	1	no	—
313	Q₈	—	38	1	no	—
314	Q₈	—	38	1	no	—
315	Q₈	—	38	1	no	—
316	Q₈	—	76	2	no	—
317	D₄	—	4	1	no	—
318	D₄	—	4	1	no	—
319	D₄	—	4	1	no	—
320	D₄	—	38	1	no	—
321	D₄	—	38	1	no	—
322	D₄	—	76	2	no	—
323	D₄	—	152	1	no	—
324	D₄	—	152	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	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	—	38	1	no	—
338	C₂×C₄	—	76	2	no	—
339	C₂×C₄	—	76	2	no	—
340	C₂×C₄	—	152	1	no	—
341	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
342	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
343	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
344	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
345	C₈	—	2	1	no	—
346	C₈	—	2	1	no	—
347	C₈	—	2	1	no	—
348	C₈	—	2	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	76	1	no	—
359	C₂²	—	76	1	no	—
360	C₂²	—	76	1	no	—
361	C₂²	C₂²×D₇₆	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	76	1	no	—
366	C₄	—	76	1	no	—
367	C₄	—	76	1	no	—
368	C₄	—	76	1	no	—
369	C₄	—	76	1	no	—
370	C₄	—	76	1	no	—
371	C₄	—	76	1	no	—
372	C₄	#10597430	1	1	yes	characteristic
373	C₄	#10597430	1	1	yes	characteristic
374	C₂	—	4	1	no	—
375	C₂	—	4	1	no	—
376	C₂	—	4	1	no	—
377	C₂	—	152	1	no	—
378	C₂	#11927106	1	1	yes	characteristic
379	C₂	#11927356	1	1	yes	characteristic
380	C₂	#11927361	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₇₆
commutator, cosocle	C₂×C₄×C₁₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉
Fitting	C₁₉×#408	C₂
socle	C₂²×C₁₉	C₂²×D₄
prime radical	#11927122	C₂
hypercenter	#408	D₁₉
hypocenter	C₁₉	#3201
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ #408

Complements

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

N	H ≅ G/N	type
2. #11927115	C₂	semidirect
3. #11926886	C₂	semidirect
4. #11926880	C₂	semidirect
5. #11926875	C₂	semidirect
6. #11926869	C₂	semidirect
7. #11926802	C₂	semidirect
8. #11926790	C₂	semidirect
9. #11926785	C₂	semidirect
10. #11926773	C₂	semidirect
11. #11926708	C₂	semidirect
12. #11926708	C₂	semidirect
13. #11926702	C₂	semidirect
14. #11926685	C₂	semidirect
15. C₁₉×#408	C₂	semidirect
198. #408	D₁₉	semidirect
249. C₁₉	#3201	semidirect

Maximal subgroups

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

subgroup	index
#11927122	2
#11927115	2
#11926886	2
#11926880	2
#11926875	2
#11926869	2
#11926802	2
#11926790	2
#11926785	2
#11926773	2
#11926708	2
#11926702	2
#11926685	2
C₁₉×#408	2
#3201	19