Subgroups — #14575618

Order 2432.

394 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	#11927244	C₂	1	1	yes	maximal, characteristic, complemented
2	#11927197	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926912	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926884	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926873	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926845	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926787	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926770	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926688	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926688	C₂	1	1	yes	maximal, characteristic, complemented
11	#11926654	C₂	1	1	yes	maximal, characteristic, complemented
12	#11926614	C₂	1	1	yes	maximal, characteristic, complemented
13	#11926544	C₂	1	1	yes	maximal, characteristic, complemented
14	#11926539	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₉×#461	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#64545	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×C₄⋊Q₈	C₂²	1	1	yes	characteristic
21	C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
22	#10597392	C₂²	1	1	yes	characteristic
23	#10597392	C₂²	1	1	yes	characteristic
24	#10597384	C₂²	1	1	yes	characteristic
25	#10597384	C₂²	1	1	yes	characteristic
26	C₂×#64525	C₂²	1	1	yes	characteristic
27	#10597375	—	2	1	no	—
28	C₄×C₂²⋊D₁₉	—	2	1	no	—
29	C₂×#64515	C₂²	1	1	yes	characteristic
30	C₂×C₈×D₁₉	C₂²	1	1	yes	characteristic
31	#10597341	—	2	1	no	—
32	#10597340	—	2	1	no	—
33	#10597340	—	2	1	no	—
34	#10597338	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₁₉	—	2	1	no	—
36	C₄⋊C₄×D₁₉	C₂²	1	1	yes	characteristic
37	#10597330	—	2	1	no	—
38	#10597325	—	2	1	no	—
39	C₄×D₇₆	C₂²	1	1	yes	characteristic
40	C₄×D₇₆	C₂²	1	1	yes	characteristic
41	#10597316	C₂²	1	1	yes	characteristic
42	C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₁₉×#103	C₂²	1	1	yes	characteristic
44	C₁₉×#103	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	(C₂×D₁₉)⋊C₈	C₂²	1	1	yes	characteristic
54	#10597275	C₂²	1	1	yes	characteristic
55	#10597274	C₂²	1	1	yes	characteristic
56	#10597266	C₂²	1	1	yes	characteristic
57	#10597265	C₂²	1	1	yes	characteristic
58	#10597260	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
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	C₂×C₂²⋊D₁₉	—	2	1	no	—
65	#64545	—	2	2	no	—
66	#64545	—	2	2	no	—
67	#64545	—	4	1	no	—
68	#64545	—	4	1	no	—
69	C₂×D₇₆	—	2	1	no	—
70	C₂×D₇₆	—	2	1	no	—
71	C₂×D₇₆	D₄	1	1	yes	characteristic
72	C₂×D₇₆	C₂³	1	1	yes	characteristic
73	C₂×C₄×D₁₉	—	2	1	no	—
74	C₂×C₄×D₁₉	—	2	1	no	—
75	C₂×C₄×D₁₉	—	2	1	no	—
76	C₂×C₄×D₁₉	—	2	1	no	—
77	C₂×C₄×D₁₉	C₂³	1	1	yes	characteristic
78	C₂×C₄×D₁₉	C₂³	1	1	yes	characteristic
79	SD₁₆×C₁₉	—	4	1	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	—	2	1	no	—
84	C₄⋊C₄×C₁₉	—	2	1	no	—
85	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
86	C₂²⋊C₄×C₁₉	—	2	1	no	—
87	C₄²×C₁₉	C₂³	1	1	yes	characteristic
88	C₂²⋊Dic₁₉	—	2	1	no	—
89	C₂²⋊Dic₁₉	—	2	1	no	—
90	#64525	—	2	2	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	—	2	1	no	—
95	(C₂×D₁₉)⋊C₄	D₄	1	1	yes	characteristic
96	(C₂×D₁₉)⋊C₄	D₄	1	1	yes	characteristic
97	C₄⋊Dic₁₉	D₄	1	1	yes	characteristic
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	—	2	1	no	—
104	Dic₁₉⋊C₄	—	2	1	no	—
105	Dic₁₉⋊C₄	—	2	1	no	—
106	C₄×Dic₁₉	—	2	1	no	—
107	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
108	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
109	#64515	—	4	1	no	—
110	C₈×D₁₉	—	4	1	no	—
111	C₂³×C₁₉	—	2	1	no	—
112	C₂²×D₁₉	—	2	1	no	—
113	C₂²×D₁₉	C₂×D₄	1	1	yes	characteristic
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	—	2	1	no	—
119	D₄×C₁₉	—	2	1	no	—
120	D₄×C₁₉	—	2	1	no	—
121	C₂×C₄×C₁₉	—	2	1	no	—
122	C₂×C₄×C₁₉	—	2	1	no	—
123	C₂×C₄×C₁₉	—	2	1	no	—
124	C₂×C₄×C₁₉	—	8	1	no	—
125	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
128	C₂²⋊D₁₉	—	4	2	no	—
129	C₂²⋊D₁₉	—	8	1	no	—
130	C₂×Dic₁₉	—	2	1	no	—
131	C₂×Dic₁₉	—	2	1	no	—
132	C₂×Dic₁₉	—	2	1	no	—
133	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
135	D₇₆	—	2	1	no	—
136	D₇₆	—	4	1	no	—
137	D₇₆	—	4	2	no	—
138	D₇₆	—	4	2	no	—
139	D₇₆	#41	1	2	yes	—
140	C₄×D₁₉	—	2	1	no	—
141	C₄×D₁₉	—	2	1	no	—
142	C₄×D₁₉	—	2	1	no	—
143	C₄×D₁₉	—	2	1	no	—
144	C₄×D₁₉	—	4	2	no	—
145	C₄×D₁₉	—	4	2	no	—
146	C₄×D₁₉	—	4	2	no	—
147	C₄×D₁₉	—	8	1	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	#3307	—	19	1	no	maximal, Sylow
153	C₂²×C₁₉	—	4	1	no	—
154	C₂²×C₁₉	—	4	1	no	—
155	C₂²×C₁₉	—	4	1	no	—
156	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
157	C₂×D₁₉	—	2	1	no	—
158	C₂×D₁₉	—	2	1	no	—
159	C₂×D₁₉	—	2	2	no	—
160	C₂×D₁₉	—	2	1	no	—
161	C₂×D₁₉	—	2	1	no	—
162	C₂×D₁₉	—	2	1	no	—
163	C₂×D₁₉	—	2	1	no	—
164	C₂×D₁₉	—	4	1	no	—
165	C₂×D₁₉	—	4	1	no	—
166	C₄×C₁₉	—	2	1	no	—
167	C₄×C₁₉	—	2	1	no	—
168	C₄×C₁₉	—	4	1	no	—
169	C₄×C₁₉	—	4	1	no	—
170	C₄×C₁₉	—	4	1	no	—
171	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
172	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
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	#547	—	19	1	no	—
180	#491	—	19	1	no	—
181	#485	—	19	1	no	—
182	#483	—	19	1	no	—
183	#478	—	19	1	no	—
184	#468	—	19	1	no	—
185	#465	—	19	1	no	—
186	#461	D₁₉	1	1	yes	characteristic, complemented
187	#449	—	19	1	no	—
188	#449	—	19	1	no	—
189	#442	—	19	1	no	—
190	#435	—	19	1	no	—
191	#417	—	19	1	no	—
192	#416	—	19	1	no	—
193	C₂×C₁₉	—	8	1	no	—
194	C₂×C₁₉	#546	1	1	yes	characteristic
195	C₂×C₁₉	#575	1	1	yes	characteristic
196	C₂×C₁₉	#577	1	1	yes	characteristic
197	D₁₉	—	4	1	no	—
198	D₁₉	—	4	1	no	—
199	D₁₉	—	4	2	no	—
200	C₂×#41	—	19	1	no	—
201	C₂×#41	—	19	1	no	—
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	C₂×M₁₆	—	19	1	no	—
206	C₂²×C₈	—	19	1	no	—
207	C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
208	#126	—	38	1	no	—
209	#125	—	19	1	no	—
210	#123	—	38	1	no	—
211	#123	—	38	1	no	—
212	#123	—	38	1	no	—
213	C₂²⋊Q₈	—	19	1	no	—
214	C₂²⋊Q₈	—	19	1	no	—
215	#121	—	19	1	no	—
216	#121	—	19	1	no	—
217	#121	—	38	1	no	—
218	#121	—	38	1	no	—
219	C₄×D₄	—	19	1	no	—
220	C₄×D₄	—	19	1	no	—
221	C₄×D₄	—	38	1	no	—
222	C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
223	#117	—	19	1	no	—
224	C₂×C₄⋊C₄	—	19	1	no	—
225	C₂×C₄⋊C₄	—	38	1	no	—
226	#107	—	19	1	no	—
227	#106	—	19	1	no	—
228	#106	—	19	1	no	—
229	C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
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	—	19	1	no	—
242	#97	—	19	1	no	—
243	C₁₉	#3307	1	1	yes	characteristic, complemented, Sylow
244	#41	—	38	2	no	—
245	#41	—	38	2	no	—
246	#41	—	76	1	no	—
247	#41	—	76	1	no	—
248	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
249	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
250	C₂×D₄	—	19	1	no	—
251	C₂×D₄	—	19	1	no	—
252	C₂×D₄	—	38	1	no	—
253	C₂×D₄	—	38	1	no	—
254	C₂×D₄	—	38	1	no	—
255	C₂×D₄	—	38	1	no	—
256	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
257	C₂²×C₄	—	2	1	no	—
258	C₂²×C₄	—	19	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	—	38	1	no	—
264	SD₁₆	—	4	1	no	—
265	SD₁₆	—	4	1	no	—
266	D₈	—	38	2	no	—
267	M₁₆	—	76	1	no	—
268	C₂×C₈	—	19	1	no	—
269	C₂×C₈	—	19	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	—	2	1	no	—
274	C₄⋊C₄	—	2	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	—	38	1	no	—
282	C₄⋊C₄	—	38	1	no	—
283	C₄⋊C₄	—	38	1	no	—
284	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
285	C₂²⋊C₄	—	2	1	no	—
286	C₂²⋊C₄	—	19	1	no	—
287	C₂²⋊C₄	—	19	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	C₂²⋊C₄	—	38	1	no	—
294	C₄²	—	38	1	no	—
295	C₄²	C₂²×D₁₉	1	1	yes	characteristic
296	C₂³	—	2	1	no	—
297	C₂³	—	19	1	no	—
298	C₂³	—	19	1	no	—
299	C₂³	—	38	1	no	—
300	Q₈	—	2	1	no	—
301	Q₈	—	2	1	no	—
302	Q₈	—	2	1	no	—
303	Q₈	—	2	1	no	—
304	D₄	—	2	1	no	—
305	D₄	—	2	1	no	—
306	D₄	—	19	2	no	—
307	D₄	—	38	1	no	—
308	D₄	—	76	1	no	—
309	D₄	—	76	2	no	—
310	D₄	—	76	2	no	—
311	D₄	—	76	2	no	—
312	D₄	—	152	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	8	1	no	—
317	C₂×C₄	—	19	1	no	—
318	C₂×C₄	—	19	1	no	—
319	C₂×C₄	—	38	1	no	—
320	C₂×C₄	—	38	1	no	—
321	C₂×C₄	—	38	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	—	76	2	no	—
327	C₂×C₄	—	76	2	no	—
328	C₂×C₄	—	76	2	no	—
329	C₂×C₄	—	152	1	no	—
330	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
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	—	4	1	no	—
338	C₂²	—	4	1	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	38	1	no	—
341	C₂²	—	38	1	no	—
342	C₂²	—	38	1	no	—
343	C₂²	—	38	1	no	—
344	C₂²	—	38	1	no	—
345	C₂²	—	38	1	no	—
346	C₂²	—	38	2	no	—
347	C₂²	—	76	1	no	—
348	C₂²	—	76	1	no	—
349	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	4	1	no	—
353	C₄	—	4	1	no	—
354	C₄	—	4	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	#10597430	1	1	yes	characteristic
361	C₄	#41×D₁₉	1	1	yes	characteristic
362	C₂	—	8	1	no	—
363	C₂	—	76	1	no	—
364	C₂	—	76	1	no	—
365	C₂	—	76	2	no	—
366	C₂	#11927191	1	1	yes	characteristic
367	C₂	#11927343	1	1	yes	characteristic
368	C₂	#11927350	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×#461	C₂
socle	C₂²×C₁₉	C₂²×D₄
hypercenter	#461	D₁₉
hypocenter	C₁₉	#3307
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #461

Complements

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

N	H ≅ G/N	type
1. #11927244	C₂	semidirect
2. #11927197	C₂	semidirect
3. #11926912	C₂	semidirect
4. #11926884	C₂	semidirect
5. #11926873	C₂	semidirect
6. #11926845	C₂	semidirect
7. #11926787	C₂	semidirect
8. #11926770	C₂	semidirect
9. #11926688	C₂	semidirect
10. #11926688	C₂	semidirect
11. #11926654	C₂	semidirect
12. #11926614	C₂	semidirect
13. #11926544	C₂	semidirect
14. #11926539	C₂	semidirect
15. C₁₉×#461	C₂	semidirect
186. #461	D₁₉	semidirect
243. C₁₉	#3307	semidirect

Maximal subgroups

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

subgroup	index
#11927244	2
#11927197	2
#11926912	2
#11926884	2
#11926873	2
#11926845	2
#11926787	2
#11926770	2
#11926688	2
#11926654	2
#11926614	2
#11926544	2
#11926539	2
C₁₉×#461	2
#3307	19