Subgroups — #14574722

Order 2432.

408 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927163	C₂	1	1	yes	maximal, characteristic, complemented
2	#11927112	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926934	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926903	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926891	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926768	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926761	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926742	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926737	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926705	C₂	1	1	yes	maximal, characteristic, complemented
11	#11926693	C₂	1	1	yes	maximal, characteristic, complemented
12	#11926688	C₂	1	1	yes	maximal, characteristic, complemented
13	#11926676	C₂	1	1	yes	maximal, characteristic, complemented
14	#11926603	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₉×#495	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#64545	C₂²	1	1	yes	characteristic
17	C₂×D₄×D₁₉	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic, complemented
19	C₂×SD₁₆×C₁₉	C₂²	1	1	yes	characteristic
20	C₂×SD₁₆×C₁₉	C₂²	1	1	yes	characteristic, complemented
21	C₂×D₈×C₁₉	C₂²	1	1	yes	characteristic
22	C₁₉×#124	C₂²	1	1	yes	characteristic
23	C₁₉×#124	C₂²	1	1	yes	characteristic, complemented
24	#10597393	C₂²	1	1	yes	characteristic
25	#10597392	C₂²	1	1	yes	characteristic
26	Q₈×Dic₁₉	C₂²	1	1	yes	characteristic, complemented
27	C₂×#64527	C₂²	1	1	yes	characteristic
28	C₂×#64527	C₂²	1	1	yes	characteristic, complemented
29	#10597386	C₂²	1	1	yes	characteristic
30	#10597384	C₂²	1	1	yes	characteristic
31	D₄×Dic₁₉	C₂²	1	1	yes	characteristic, complemented
32	C₂×#64525	C₂²	1	1	yes	characteristic
33	C₂×#64525	C₂²	1	1	yes	characteristic, complemented
34	#10597332	—	2	1	no	—
35	#10597331	—	2	1	no	—
36	#10597330	—	2	1	no	—
37	#10597330	—	2	1	no	—
38	#10597328	—	2	1	no	—
39	#10597328	—	2	1	no	—
40	#10597325	—	2	1	no	—
41	#10597324	—	2	1	no	—
42	#10597320	C₂²	1	1	yes	characteristic
43	#10597318	C₂²	1	1	yes	characteristic
44	#10597315	C₂²	1	1	yes	characteristic, complemented
45	C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic, complemented
46	#10597292	C₂²	1	1	yes	characteristic, complemented
47	#10597292	C₂²	1	1	yes	characteristic, complemented
48	#10597289	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	#10597278	C₂²	1	1	yes	characteristic, complemented
53	#10597278	C₂²	1	1	yes	characteristic, complemented
54	(C₂×D₁₉)⋊C₈	C₂²	1	1	yes	characteristic
55	(C₂×D₁₉)⋊C₈	C₂²	1	1	yes	characteristic
56	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic, complemented
57	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic, complemented
58	#10597260	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₁₉	—	2	1	no	—
60	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
61	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
63	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₁₉	—	2	1	no	—
65	C₂×C₂²⋊D₁₉	—	2	1	no	—
66	C₂×C₂²⋊D₁₉	—	2	1	no	—
67	C₂×C₂²⋊D₁₉	—	2	1	no	—
68	C₂²×Dic₁₉	—	2	1	no	—
69	#64545	—	4	1	no	—
70	#64545	—	4	1	no	—
71	D₄×D₁₉	—	4	1	no	—
72	D₄×D₁₉	—	4	1	no	—
73	C₂×D₇₆	—	2	1	no	—
74	C₂×D₇₆	—	2	1	no	—
75	C₂×D₇₆	—	2	1	no	—
76	C₂×D₇₆	C₂³	1	1	yes	characteristic
77	C₂×D₇₆	C₂³	1	1	yes	characteristic
78	C₂×C₄×D₁₉	—	2	1	no	—
79	C₂×C₄×D₁₉	C₂³	1	1	yes	characteristic
80	C₂×Dic₃₈	—	2	1	no	—
81	Q₁₆×C₁₉	—	4	1	no	—
82	SD₁₆×C₁₉	—	4	1	no	—
83	SD₁₆×C₁₉	—	4	1	no	—
84	D₈×C₁₉	—	4	1	no	—
85	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
86	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
87	C₂²⋊C₄×C₁₉	—	2	1	no	—
88	C₂²⋊C₄×C₁₉	—	2	1	no	—
89	C₂²⋊C₄×C₁₉	—	2	1	no	—
90	C₂²⋊C₄×C₁₉	—	2	1	no	—
91	C₄²×C₁₉	C₂³	1	1	yes	characteristic
92	C₂²⋊Dic₁₉	—	2	1	no	—
93	C₂²⋊Dic₁₉	—	2	1	no	—
94	#64527	—	4	1	no	—
95	#64527	—	4	1	no	—
96	#64525	—	4	1	no	—
97	#64525	—	4	1	no	—
98	(C₂×D₁₉)⋊C₄	—	2	1	no	—
99	(C₂×D₁₉)⋊C₄	—	2	1	no	—
100	(C₂×D₁₉)⋊C₄	—	2	1	no	—
101	(C₂×D₁₉)⋊C₄	D₄	1	1	yes	characteristic
102	(C₂×D₁₉)⋊C₄	D₄	1	1	yes	characteristic
103	C₄⋊Dic₁₉	—	2	1	no	—
104	C₄⋊Dic₁₉	—	2	1	no	—
105	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
106	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
107	Dic₁₉⋊C₄	—	2	1	no	—
108	Dic₁₉⋊C₄	D₄	1	1	yes	characteristic
109	Dic₁₉⋊C₄	D₄	1	1	yes	characteristic
110	C₄×Dic₁₉	—	2	1	no	—
111	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
112	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
113	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
114	C₂³×C₁₉	—	2	1	no	—
115	C₂³×C₁₉	—	2	1	no	—
116	C₂²×D₁₉	—	2	1	no	—
117	C₂²×D₁₉	—	2	1	no	—
118	C₂²×D₁₉	—	4	2	no	—
119	C₂²×D₁₉	—	8	1	no	—
120	C₂²×D₁₉	—	8	1	no	—
121	C₂²×D₁₉	C₂×D₄	1	1	yes	characteristic
122	Q₈×C₁₉	—	2	1	no	—
123	Q₈×C₁₉	—	2	1	no	—
124	Q₈×C₁₉	—	2	1	no	—
125	Q₈×C₁₉	—	2	1	no	—
126	D₄×C₁₉	—	2	1	no	—
127	D₄×C₁₉	—	2	1	no	—
128	D₄×C₁₉	—	2	1	no	—
129	D₄×C₁₉	—	2	1	no	—
130	C₂×C₄×C₁₉	—	2	1	no	—
131	C₂×C₄×C₁₉	—	2	1	no	—
132	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
135	C₂²⋊D₁₉	—	4	2	no	—
136	C₂²⋊D₁₉	—	8	1	no	—
137	C₂²⋊D₁₉	—	8	1	no	—
138	C₂²⋊D₁₉	—	8	1	no	—
139	C₂×Dic₁₉	—	2	1	no	—
140	C₂×Dic₁₉	—	2	1	no	—
141	C₂×Dic₁₉	—	8	1	no	—
142	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
143	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
144	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
145	D₇₆	—	2	1	no	—
146	D₇₆	—	2	1	no	—
147	D₇₆	—	2	1	no	—
148	D₇₆	—	2	1	no	—
149	D₇₆	—	4	2	no	—
150	D₇₆	—	8	1	no	—
151	D₇₆	—	8	1	no	—
152	C₄×D₁₉	—	2	1	no	—
153	C₄×D₁₉	—	2	1	no	—
154	C₄×D₁₉	—	4	2	no	—
155	Dic₃₈	—	8	1	no	—
156	C₈×C₁₉	—	2	1	no	—
157	C₈×C₁₉	—	2	1	no	—
158	C₁₉⋊C₈	—	2	1	no	—
159	C₁₉⋊C₈	—	2	1	no	—
160	#3224	—	19	1	no	maximal, Sylow
161	C₂²×C₁₉	—	4	1	no	—
162	C₂²×C₁₉	—	4	1	no	—
163	C₂²×C₁₉	—	4	1	no	—
164	C₂²×C₁₉	—	4	1	no	—
165	C₂²×C₁₉	—	4	1	no	—
166	C₂²×C₁₉	—	4	1	no	—
167	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
168	C₂×D₁₉	—	2	1	no	—
169	C₂×D₁₉	—	2	1	no	—
170	C₂×D₁₉	—	2	1	no	—
171	C₂×D₁₉	—	4	1	no	—
172	C₂×D₁₉	—	4	1	no	—
173	C₂×D₁₉	—	4	1	no	—
174	C₂×D₁₉	—	4	1	no	—
175	C₂×D₁₉	—	4	1	no	—
176	C₂×D₁₉	—	4	1	no	—
177	C₂×D₁₉	—	8	4	no	—
178	C₄×C₁₉	—	2	1	no	—
179	C₄×C₁₉	—	2	1	no	—
180	C₄×C₁₉	—	4	1	no	—
181	C₄×C₁₉	—	4	1	no	—
182	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
183	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
184	Dic₁₉	—	2	1	no	—
185	Dic₁₉	—	2	1	no	—
186	Dic₁₉	—	2	1	no	—
187	Dic₁₉	—	4	1	no	—
188	Dic₁₉	—	4	1	no	—
189	#542	—	19	1	no	—
190	#535	—	19	1	no	—
191	#496	—	19	1	no	—
192	#495	D₁₉	1	1	yes	characteristic, complemented
193	#489	—	19	1	no	—
194	#486	—	19	1	no	—
195	#464	—	19	1	no	—
196	#463	—	19	1	no	—
197	#460	—	19	1	no	—
198	#459	—	19	1	no	—
199	#452	—	19	1	no	—
200	#450	—	19	1	no	—
201	#449	—	19	1	no	—
202	#447	—	19	1	no	—
203	#433	—	19	1	no	—
204	C₂×C₁₉	—	8	1	no	—
205	C₂×C₁₉	—	8	1	no	—
206	C₂×C₁₉	#535	1	1	yes	characteristic
207	C₂×C₁₉	#576	1	1	yes	characteristic
208	C₂×C₁₉	#577	1	1	yes	characteristic
209	D₁₉	—	4	1	no	—
210	D₁₉	—	8	1	no	—
211	D₁₉	—	8	1	no	—
212	C₂×#41	—	19	1	no	—
213	C₂²×D₄	—	19	1	no	—
214	C₂×Q₁₆	C₂×D₁₉	1	1	yes	characteristic, complemented
215	C₂×SD₁₆	—	19	1	no	—
216	C₂×SD₁₆	—	19	1	no	—
217	C₂×SD₁₆	C₂×D₁₉	1	1	yes	characteristic
218	C₂×SD₁₆	C₂×D₁₉	1	1	yes	characteristic, complemented
219	C₂×D₈	—	19	1	no	—
220	C₂×D₈	—	19	1	no	—
221	C₂×D₈	C₂×D₁₉	1	1	yes	characteristic
222	C₄⋊Q₈	—	19	1	no	—
223	#127	—	19	1	no	—
224	#127	—	19	1	no	—
225	#126	—	38	1	no	—
226	#124	—	19	1	no	—
227	#124	—	38	1	no	—
228	#124	C₂×D₁₉	1	1	yes	characteristic
229	#124	C₂×D₁₉	1	1	yes	characteristic, complemented
230	#123	—	38	1	no	—
231	C₂²⋊Q₈	—	19	1	no	—
232	C₂²⋊Q₈	—	38	1	no	—
233	#121	—	19	1	no	—
234	#121	—	38	1	no	—
235	#121	—	38	1	no	—
236	#120	—	38	1	no	—
237	#120	—	38	1	no	—
238	C₄×Q₈	—	19	1	no	—
239	C₄×D₄	—	19	1	no	—
240	#117	—	19	1	no	—
241	C₄⋊C₈	—	19	1	no	—
242	C₄⋊C₈	—	19	1	no	—
243	#103	—	19	1	no	—
244	#103	—	19	1	no	—
245	#102	—	19	1	no	—
246	#102	—	19	1	no	—
247	#102	—	19	1	no	—
248	#102	—	19	1	no	—
249	#102	—	19	1	no	—
250	#102	—	19	1	no	—
251	C₂²⋊C₈	—	19	1	no	—
252	C₂²⋊C₈	—	19	1	no	—
253	#97	—	19	1	no	—
254	C₄×C₈	C₂×D₁₉	1	1	yes	characteristic, complemented
255	C₁₉	#3224	1	1	yes	characteristic, complemented, Sylow
256	C₂⁴	—	38	1	no	—
257	#41	—	76	1	no	—
258	#41	—	76	1	no	—
259	C₂×Q₈	—	38	1	no	—
260	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
261	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
262	C₂×D₄	—	19	1	no	—
263	C₂×D₄	—	19	1	no	—
264	C₂×D₄	—	38	1	no	—
265	C₂×D₄	—	38	1	no	—
266	C₂×D₄	—	38	1	no	—
267	C₂×D₄	—	38	1	no	—
268	C₂×D₄	—	38	1	no	—
269	C₂×D₄	—	38	1	no	—
270	C₂×D₄	—	38	1	no	—
271	C₂×D₄	—	76	1	no	—
272	C₂×D₄	—	76	1	no	—
273	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
274	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
275	C₂²×C₄	—	19	1	no	—
276	C₂²×C₄	—	38	1	no	—
277	C₂²×C₄	—	38	1	no	—
278	Q₁₆	—	4	1	no	—
279	SD₁₆	—	4	1	no	—
280	SD₁₆	—	4	1	no	—
281	SD₁₆	—	76	1	no	—
282	SD₁₆	—	76	1	no	—
283	D₈	—	4	1	no	—
284	D₈	—	76	1	no	—
285	D₈	—	76	1	no	—
286	C₂×C₈	—	19	1	no	—
287	C₂×C₈	—	19	1	no	—
288	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
289	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
290	C₄⋊C₄	—	19	1	no	—
291	C₄⋊C₄	—	19	1	no	—
292	C₄⋊C₄	—	19	1	no	—
293	C₄⋊C₄	—	19	1	no	—
294	C₄⋊C₄	—	38	1	no	—
295	C₄⋊C₄	—	38	1	no	—
296	C₄⋊C₄	—	38	1	no	—
297	C₂²⋊C₄	—	2	1	no	—
298	C₂²⋊C₄	—	2	1	no	—
299	C₂²⋊C₄	—	2	1	no	—
300	C₂²⋊C₄	—	2	1	no	—
301	C₂²⋊C₄	—	19	1	no	—
302	C₂²⋊C₄	—	19	1	no	—
303	C₂²⋊C₄	—	38	1	no	—
304	C₂²⋊C₄	—	38	1	no	—
305	C₂²⋊C₄	—	38	1	no	—
306	C₂²⋊C₄	—	38	1	no	—
307	C₂²⋊C₄	—	38	1	no	—
308	C₄²	—	19	1	no	—
309	C₄²	—	38	1	no	—
310	C₄²	C₂²×D₁₉	1	1	yes	characteristic
311	C₂³	—	2	1	no	—
312	C₂³	—	2	1	no	—
313	C₂³	—	19	1	no	—
314	C₂³	—	38	1	no	—
315	C₂³	—	38	1	no	—
316	C₂³	—	76	2	no	—
317	C₂³	—	152	1	no	—
318	C₂³	—	152	1	no	—
319	Q₈	—	2	1	no	—
320	Q₈	—	2	1	no	—
321	Q₈	—	2	1	no	—
322	Q₈	—	2	1	no	—
323	Q₈	—	152	1	no	—
324	D₄	—	2	1	no	—
325	D₄	—	2	1	no	—
326	D₄	—	2	1	no	—
327	D₄	—	2	1	no	—
328	D₄	—	38	1	no	—
329	D₄	—	38	1	no	—
330	D₄	—	38	1	no	—
331	D₄	—	38	1	no	—
332	D₄	—	76	2	no	—
333	D₄	—	76	2	no	—
334	D₄	—	152	1	no	—
335	D₄	—	152	1	no	—
336	D₄	—	152	1	no	—
337	D₄	—	152	1	no	—
338	D₄	—	152	1	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	19	1	no	—
342	C₂×C₄	—	19	1	no	—
343	C₂×C₄	—	19	1	no	—
344	C₂×C₄	—	38	1	no	—
345	C₂×C₄	—	38	1	no	—
346	C₂×C₄	—	38	1	no	—
347	C₂×C₄	—	38	1	no	—
348	C₂×C₄	—	76	2	no	—
349	C₂×C₄	—	152	1	no	—
350	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
351	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
352	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
353	C₈	—	2	1	no	—
354	C₈	—	2	1	no	—
355	C₈	—	38	1	no	—
356	C₈	—	38	1	no	—
357	C₂²	—	4	1	no	—
358	C₂²	—	4	1	no	—
359	C₂²	—	4	1	no	—
360	C₂²	—	4	1	no	—
361	C₂²	—	4	1	no	—
362	C₂²	—	4	1	no	—
363	C₂²	—	38	1	no	—
364	C₂²	—	38	1	no	—
365	C₂²	—	38	1	no	—
366	C₂²	—	76	1	no	—
367	C₂²	—	76	1	no	—
368	C₂²	—	76	1	no	—
369	C₂²	—	76	1	no	—
370	C₂²	—	76	1	no	—
371	C₂²	—	76	1	no	—
372	C₂²	—	152	4	no	—
373	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	4	1	no	—
377	C₄	—	4	1	no	—
378	C₄	—	38	1	no	—
379	C₄	—	38	1	no	—
380	C₄	—	38	1	no	—
381	C₄	—	76	1	no	—
382	C₄	—	76	1	no	—
383	C₄	#10597430	1	1	yes	characteristic
384	C₄	#10597430	1	1	yes	characteristic
385	C₂	—	8	1	no	—
386	C₂	—	8	1	no	—
387	C₂	—	76	1	no	—
388	C₂	—	152	1	no	—
389	C₂	—	152	1	no	—
390	C₂	#11927107	1	1	yes	characteristic
391	C₂	#11927346	1	1	yes	characteristic
392	C₂	#11927351	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×#495	C₂
socle	C₂²×C₁₉	C₂²×D₄
hypercenter	#495	D₁₉
hypocenter	C₁₉	#3224
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #495

Complements

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

N	H ≅ G/N	type
1. #11927163	C₂	semidirect
2. #11927112	C₂	semidirect
3. #11926934	C₂	semidirect
4. #11926903	C₂	semidirect
5. #11926891	C₂	semidirect
6. #11926768	C₂	semidirect
7. #11926761	C₂	semidirect
8. #11926742	C₂	semidirect
9. #11926737	C₂	semidirect
10. #11926705	C₂	semidirect
11. #11926693	C₂	semidirect
12. #11926688	C₂	semidirect
13. #11926676	C₂	semidirect
14. #11926603	C₂	semidirect
15. C₁₉×#495	C₂	semidirect
18. C₂×Q₁₆×C₁₉	C₂²	semidirect
20. C₂×SD₁₆×C₁₉	C₂²	semidirect
23. C₁₉×#124	C₂²	semidirect
26. Q₈×Dic₁₉	C₂²	semidirect
28. C₂×#64527	C₂²	semidirect
31. D₄×Dic₁₉	C₂²	semidirect
33. C₂×#64525	C₂²	semidirect
44. #10597315	C₂²	semidirect
45. C₄×C₈×C₁₉	C₂²	semidirect
46. #10597292	C₂²	semidirect
47. #10597292	C₂²	semidirect
52. #10597278	C₂²	semidirect
53. #10597278	C₂²	semidirect
56. Dic₁₉⋊C₈	C₂²	semidirect
57. Dic₁₉⋊C₈	C₂²	semidirect
58. #10597260	C₂²	semidirect
192. #495	D₁₉	semidirect
214. C₂×Q₁₆	C₂×D₁₉	semidirect
218. C₂×SD₁₆	C₂×D₁₉	semidirect
229. #124	C₂×D₁₉	semidirect
254. C₄×C₈	C₂×D₁₉	semidirect
255. C₁₉	#3224	semidirect

Maximal subgroups

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

subgroup	index
#11927163	2
#11927112	2
#11926934	2
#11926903	2
#11926891	2
#11926768	2
#11926761	2
#11926742	2
#11926737	2
#11926705	2
#11926693	2
#11926688	2
#11926676	2
#11926603	2
C₁₉×#495	2
#3224	19