Subgroups — #14575541

Order 2432.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 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	#11927205	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926900	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926883	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926863	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926849	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926778	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926770	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926761	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926688	C₂	1	1	yes	maximal, characteristic, complemented
11	SD₁₆×Dic₁₉	C₂	1	1	yes	maximal, characteristic, complemented
12	(C₂²⋊D₁₉)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
13	#11926539	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#10597292	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₉×#448	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Q₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₂×#64545	C₂²	1	1	yes	characteristic
18	C₂×SD₁₆×C₁₉	C₂²	1	1	yes	characteristic
19	C₂×SD₁₆×C₁₉	C₂²	1	1	yes	characteristic
20	C₁₉×#121	C₂²	1	1	yes	characteristic
21	#10597393	—	2	1	no	—
22	#10597393	C₂²	1	1	yes	characteristic
23	#10597392	—	2	1	no	—
24	#10597392	C₂²	1	1	yes	characteristic
25	Q₈×Dic₁₉	C₂²	1	1	yes	characteristic
26	#10597385	—	2	1	no	—
27	#10597384	C₂²	1	1	yes	characteristic
28	D₄×Dic₁₉	—	2	1	no	—
29	D₄×Dic₁₉	C₂²	1	1	yes	characteristic
30	C₂×#64525	C₂²	1	1	yes	characteristic
31	#10597376	—	2	1	no	—
32	#10597376	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₁₉	—	2	1	no	—
34	C₄×C₂²⋊D₁₉	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
36	C₂²×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
37	C₂×C₈×D₁₉	C₂²	1	1	yes	characteristic
38	#10597338	C₂²	1	1	yes	characteristic
39	#10597336	C₂²	1	1	yes	characteristic
40	#10597332	—	2	1	no	—
41	#10597325	—	2	1	no	—
42	C₁₉×#103	C₂²	1	1	yes	characteristic
43	C₁₉×#103	C₂²	1	1	yes	characteristic
44	C₁₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
45	#10597292	C₂²	1	1	yes	characteristic
46	#10597292	C₂²	1	1	yes	characteristic
47	#10597292	C₂²	1	1	yes	characteristic
48	#10597292	C₂²	1	1	yes	characteristic
49	#10597289	C₂²	1	1	yes	characteristic
50	#10597278	C₂²	1	1	yes	characteristic
51	#10597278	C₂²	1	1	yes	characteristic
52	(C₂×D₁₉)⋊C₈	C₂²	1	1	yes	characteristic
53	#10597274	C₂²	1	1	yes	characteristic
54	#10597274	C₂²	1	1	yes	characteristic
55	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
56	C₈×Dic₁₉	C₂²	1	1	yes	characteristic
57	#10597266	C₂²	1	1	yes	characteristic
58	#10597264	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₉	—	2	1	no	—
60	C₂×Q₈×C₁₉	—	2	2	no	—
61	C₂×Q₈×C₁₉	—	4	1	no	—
62	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
63	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
64	C₂×D₄×C₁₉	—	2	1	no	—
65	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	—	2	1	no	—
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	D₄	1	1	yes	characteristic
71	C₂²×Dic₁₉	—	2	1	no	—
72	C₂²×Dic₁₉	—	2	1	no	—
73	#64545	—	2	2	no	—
74	#64545	—	4	1	no	—
75	C₂×D₇₆	—	2	1	no	—
76	C₂×D₇₆	C₂³	1	1	yes	characteristic
77	C₂×C₄×D₁₉	—	2	1	no	—
78	C₂×C₄×D₁₉	C₂³	1	1	yes	characteristic
79	SD₁₆×C₁₉	—	2	2	no	—
80	SD₁₆×C₁₉	—	4	1	no	—
81	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
82	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
84	C₂²⋊C₄×C₁₉	—	2	1	no	—
85	C₂²⋊Dic₁₉	—	2	1	no	—
86	C₂²⋊Dic₁₉	—	2	1	no	—
87	C₂²⋊Dic₁₉	D₄	1	1	yes	characteristic
88	#64525	—	4	1	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	(C₂×D₁₉)⋊C₄	—	2	1	no	—
94	(C₂×D₁₉)⋊C₄	D₄	1	1	yes	characteristic
95	C₄⋊Dic₁₉	—	2	1	no	—
96	C₄⋊Dic₁₉	—	2	1	no	—
97	C₄⋊Dic₁₉	—	2	1	no	—
98	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
99	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
100	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
101	Dic₁₉⋊C₄	—	2	1	no	—
102	Dic₁₉⋊C₄	—	2	1	no	—
103	Dic₁₉⋊C₄	D₄	1	1	yes	characteristic
104	C₄×Dic₁₉	—	2	1	no	—
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	C₂×D₄	1	1	yes	characteristic
113	C₂²×D₁₉	—	2	1	no	—
114	C₂²×D₁₉	C₂×D₄	1	1	yes	characteristic
115	Q₈×C₁₉	—	2	1	no	—
116	Q₈×C₁₉	—	2	1	no	—
117	Q₈×C₁₉	—	2	1	no	—
118	Q₈×C₁₉	—	4	2	no	—
119	Q₈×C₁₉	#41	1	2	yes	—
120	D₄×C₁₉	—	2	1	no	—
121	D₄×C₁₉	—	2	1	no	—
122	D₄×C₁₉	—	4	2	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	—	4	2	no	—
129	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
131	C₂²⋊D₁₉	—	2	2	no	—
132	C₂²⋊D₁₉	—	4	2	no	—
133	C₂²⋊D₁₉	—	8	1	no	—
134	C₂×Dic₁₉	—	2	1	no	—
135	C₂×Dic₁₉	—	2	1	no	—
136	C₂×Dic₁₉	—	2	1	no	—
137	C₂×Dic₁₉	—	8	1	no	—
138	C₂×Dic₁₉	—	8	1	no	—
139	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
140	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
141	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
142	D₇₆	—	2	1	no	—
143	D₇₆	—	2	1	no	—
144	D₇₆	—	4	2	no	—
145	C₄×D₁₉	—	2	1	no	—
146	C₄×D₁₉	—	2	1	no	—
147	C₄×D₁₉	—	4	2	no	—
148	C₈×C₁₉	—	2	1	no	—
149	C₈×C₁₉	—	2	1	no	—
150	C₁₉⋊C₈	—	2	1	no	—
151	C₁₉⋊C₈	—	2	1	no	—
152	#3301	—	19	1	no	maximal, Sylow
153	C₂²×C₁₉	—	2	1	no	—
154	C₂²×C₁₉	—	2	1	no	—
155	C₂²×C₁₉	—	4	1	no	—
156	C₂²×C₁₉	—	4	1	no	—
157	C₂²×C₁₉	—	4	1	no	—
158	C₂²×C₁₉	#135	1	2	yes	—
159	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
160	C₂×D₁₉	—	2	1	no	—
161	C₂×D₁₉	—	2	1	no	—
162	C₂×D₁₉	—	2	1	no	—
163	C₂×D₁₉	—	4	1	no	—
164	C₂×D₁₉	—	4	1	no	—
165	C₂×D₁₉	—	4	1	no	—
166	C₄×C₁₉	—	2	2	no	—
167	C₄×C₁₉	—	2	1	no	—
168	C₄×C₁₉	—	4	1	no	—
169	C₄×C₁₉	—	4	1	no	—
170	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
171	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
172	Dic₁₉	—	2	1	no	—
173	Dic₁₉	—	2	1	no	—
174	Dic₁₉	—	2	1	no	—
175	Dic₁₉	—	4	1	no	—
176	Dic₁₉	—	4	1	no	—
177	Dic₁₉	—	4	1	no	—
178	#553	—	19	1	no	—
179	#548	—	19	1	no	—
180	#488	—	19	1	no	—
181	#485	—	19	1	no	—
182	#481	—	19	1	no	—
183	#479	—	19	1	no	—
184	#466	—	19	1	no	—
185	#465	—	19	1	no	—
186	#463	—	19	1	no	—
187	#449	—	19	1	no	—
188	#448	D₁₉	1	1	yes	characteristic, complemented
189	C₄×SD₁₆	—	19	1	no	—
190	C₈×D₄	—	19	1	no	—
191	#416	—	19	1	no	—
192	C₂×#103	—	19	1	no	—
193	C₂×C₁₉	—	2	2	no	—
194	C₂×C₁₉	—	8	1	no	—
195	C₂×C₁₉	#546	1	1	yes	characteristic
196	C₂×C₁₉	C₂×#135	1	1	yes	characteristic
197	C₂×C₁₉	#577	1	1	yes	characteristic
198	D₁₉	—	4	1	no	—
199	D₁₉	—	8	1	no	—
200	C₂×#41	—	19	1	no	—
201	C₂²×Q₈	C₂×D₁₉	1	1	yes	characteristic
202	C₂×SD₁₆	C₂×D₁₉	1	1	yes	characteristic
203	C₂×SD₁₆	C₂×D₁₉	1	1	yes	characteristic
204	C₂×D₈	—	19	1	no	—
205	C₂²×C₈	—	19	1	no	—
206	C₂²×C₈	—	19	1	no	—
207	#126	—	38	1	no	—
208	#125	—	19	1	no	—
209	#124	—	19	1	no	—
210	#124	—	38	1	no	—
211	#123	—	38	1	no	—
212	C₂²⋊Q₈	—	19	1	no	—
213	C₂²⋊Q₈	—	38	1	no	—
214	#121	—	19	1	no	—
215	#121	—	19	1	no	—
216	#121	—	38	1	no	—
217	#121	—	38	1	no	—
218	#121	C₂×D₁₉	1	1	yes	characteristic
219	C₄×Q₈	—	19	1	no	—
220	C₄×D₄	—	19	1	no	—
221	C₄×D₄	—	19	1	no	—
222	C₄×D₄	—	38	1	no	—
223	C₄×D₄	—	38	1	no	—
224	#117	—	19	1	no	—
225	C₂×C₄⋊C₄	—	19	1	no	—
226	#107	—	19	1	no	—
227	#106	—	19	1	no	—
228	#106	—	19	1	no	—
229	C₄⋊C₈	—	19	1	no	—
230	#103	—	19	1	no	—
231	#103	—	19	1	no	—
232	#103	—	19	1	no	—
233	#103	—	19	1	no	—
234	#103	C₂×D₁₉	1	1	yes	characteristic
235	#103	C₂×D₁₉	1	1	yes	characteristic
236	#102	—	19	1	no	—
237	#102	—	19	1	no	—
238	#102	—	19	1	no	—
239	#102	—	19	1	no	—
240	C₂²⋊C₈	—	19	1	no	—
241	C₂²⋊C₈	C₂×D₁₉	1	1	yes	characteristic
242	C₄×C₈	—	19	1	no	—
243	C₁₉	#3301	1	1	yes	characteristic, complemented, Sylow
244	#41	—	38	2	no	—
245	#41	—	76	1	no	—
246	C₂×Q₈	—	2	1	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
250	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	—	19	1	no	—
253	C₂×D₄	—	19	1	no	—
254	C₂×D₄	—	38	1	no	—
255	C₂×D₄	—	38	1	no	—
256	C₂×D₄	—	38	1	no	—
257	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
258	C₂²×C₄	—	2	1	no	—
259	C₂²×C₄	—	19	1	no	—
260	C₂²×C₄	—	38	1	no	—
261	C₂²×C₄	—	38	1	no	—
262	C₂²×C₄	—	38	1	no	—
263	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
264	SD₁₆	—	2	2	no	—
265	SD₁₆	—	4	1	no	—
266	D₈	—	76	1	no	—
267	C₂×C₈	—	19	1	no	—
268	C₂×C₈	—	19	1	no	—
269	C₂×C₈	—	76	1	no	—
270	C₂×C₈	—	76	1	no	—
271	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
272	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
273	C₄⋊C₄	—	19	1	no	—
274	C₄⋊C₄	—	19	1	no	—
275	C₄⋊C₄	—	19	1	no	—
276	C₄⋊C₄	—	19	1	no	—
277	C₄⋊C₄	—	38	1	no	—
278	C₄⋊C₄	—	38	1	no	—
279	C₄⋊C₄	—	38	1	no	—
280	C₄⋊C₄	—	38	1	no	—
281	C₄⋊C₄	—	38	1	no	—
282	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
283	C₂²⋊C₄	—	2	1	no	—
284	C₂²⋊C₄	—	19	1	no	—
285	C₂²⋊C₄	—	19	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	—	38	1	no	—
296	C₂³	—	2	1	no	—
297	C₂³	—	19	1	no	—
298	C₂³	—	38	1	no	—
299	C₂³	D₄×D₁₉	1	1	yes	characteristic
300	Q₈	—	2	1	no	—
301	Q₈	—	2	1	no	—
302	Q₈	—	2	1	no	—
303	Q₈	—	4	2	no	—
304	Q₈	#64543	1	2	yes	—
305	D₄	—	2	1	no	—
306	D₄	—	2	1	no	—
307	D₄	—	4	2	no	—
308	D₄	—	38	2	no	—
309	D₄	—	38	1	no	—
310	D₄	—	38	1	no	—
311	D₄	—	76	2	no	—
312	D₄	—	76	2	no	—
313	D₄	—	152	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	19	1	no	—
321	C₂×C₄	—	19	1	no	—
322	C₂×C₄	—	19	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	—	76	2	no	—
329	C₂×C₄	—	152	1	no	—
330	C₂×C₄	—	152	1	no	—
331	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
332	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	38	1	no	—
336	C₈	—	38	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	4	1	no	—
342	C₂²	—	38	1	no	—
343	C₂²	—	38	1	no	—
344	C₂²	—	38	1	no	—
345	C₂²	—	76	1	no	—
346	C₂²	—	76	1	no	—
347	C₂²	—	76	1	no	—
348	C₂²	#10597362	1	2	yes	—
349	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	2	no	—
352	C₄	—	4	1	no	—
353	C₄	—	4	1	no	—
354	C₄	—	38	1	no	—
355	C₄	—	38	1	no	—
356	C₄	—	38	1	no	—
357	C₄	—	76	1	no	—
358	C₄	—	76	1	no	—
359	C₄	—	76	1	no	—
360	C₄	C₂×#64543	1	1	yes	characteristic
361	C₄	#10597430	1	1	yes	characteristic
362	C₂	—	2	2	no	—
363	C₂	—	8	1	no	—
364	C₂	—	76	1	no	—
365	C₂	—	152	1	no	—
366	C₂	#11927181	1	1	yes	characteristic
367	C₂	C₂×#10597362	1	1	yes	characteristic
368	C₂	#11927353	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×#448	C₂
socle	C₂²×C₁₉	C₂²×D₄
hypercenter	#448	D₁₉
hypocenter	C₁₉	#3301
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #448

Complements

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

N	H ≅ G/N	type
1. #11927246	C₂	semidirect
2. #11927205	C₂	semidirect
3. #11926900	C₂	semidirect
4. #11926883	C₂	semidirect
5. #11926863	C₂	semidirect
6. #11926849	C₂	semidirect
7. #11926778	C₂	semidirect
8. #11926770	C₂	semidirect
9. #11926761	C₂	semidirect
10. #11926688	C₂	semidirect
11. SD₁₆×Dic₁₉	C₂	semidirect
12. (C₂²⋊D₁₉)⋊C₈	C₂	semidirect
13. #11926539	C₂	semidirect
14. C₂×#10597292	C₂	semidirect
15. C₁₉×#448	C₂	semidirect
188. #448	D₁₉	semidirect
243. C₁₉	#3301	semidirect

Maximal subgroups

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

subgroup	index
#11927246	2
#11927205	2
#11926900	2
#11926883	2
#11926863	2
#11926849	2
#11926778	2
#11926770	2
#11926761	2
#11926688	2
SD₁₆×Dic₁₉	2
(C₂²⋊D₁₉)⋊C₈	2
#11926539	2
C₂×#10597292	2
C₁₉×#448	2
#3301	19