Subgroups — #14575600

Order 2432.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927194	C₂	1	1	yes	maximal, characteristic, complemented
2	#11927188	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926902	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926875	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926861	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926832	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926794	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926771	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926757	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926686	C₂	1	1	yes	maximal, characteristic, complemented
11	#11926656	C₂	1	1	yes	maximal, characteristic, complemented
12	#11926612	C₂	1	1	yes	maximal, characteristic, complemented
13	#11926542	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#10597289	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₉×#447	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×D₄×C₁₉	C₂²	1	1	yes	characteristic
17	C₂×#64543	—	2	1	no	—
18	C₂×#64543	C₂²	1	1	yes	characteristic
19	C₂×D₈×C₁₉	C₂²	1	1	yes	characteristic
20	C₂×D₈×C₁₉	C₂²	1	1	yes	characteristic
21	C₁₉×#121	C₂²	1	1	yes	characteristic
22	#10597399	—	2	1	no	—
23	C₂×C₂²⋊Dic₁₉	—	2	1	no	—
24	#10597385	—	2	1	no	—
25	#10597384	C₂²	1	1	yes	characteristic
26	#10597384	C₂²	1	1	yes	characteristic, complemented
27	#10597382	C₂²	1	1	yes	characteristic
28	#10597381	—	2	1	no	—
29	#10597381	—	2	1	no	—
30	D₄×Dic₁₉	C₂²	1	1	yes	characteristic
31	D₄×Dic₁₉	C₂²	1	1	yes	characteristic, complemented
32	C₂×#64526	C₂²	1	1	yes	characteristic, complemented
33	C₄×C₂²⋊D₁₉	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
35	#10597370	C₂²	1	1	yes	characteristic
36	C₂²×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
37	C₂×#64515	C₂²	1	1	yes	characteristic, complemented
38	C₄⋊C₄×D₁₉	C₂²	1	1	yes	characteristic, complemented
39	#10597334	C₂²	1	1	yes	characteristic, complemented
40	#10597329	—	2	1	no	—
41	#10597324	—	2	1	no	—
42	C₁₉×#102	C₂²	1	1	yes	characteristic
43	C₁₉×#102	C₂²	1	1	yes	characteristic
44	C₁₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
45	#10597289	C₂²	1	1	yes	characteristic
46	#10597289	C₂²	1	1	yes	characteristic
47	#10597289	C₂²	1	1	yes	characteristic
48	#10597289	C₂²	1	1	yes	characteristic
49	#10597289	C₂²	1	1	yes	characteristic, complemented
50	(C₂×D₁₉)⋊C₈	C₂²	1	1	yes	characteristic, complemented
51	#10597274	C₂²	1	1	yes	characteristic, complemented
52	#10597274	C₂²	1	1	yes	characteristic, complemented
53	#10597273	C₂²	1	1	yes	characteristic, complemented
54	#10597273	C₂²	1	1	yes	characteristic, complemented
55	#10597272	C₂²	1	1	yes	characteristic, complemented
56	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic, complemented
57	#10597267	C₂²	1	1	yes	characteristic, complemented
58	#10597265	C₂²	1	1	yes	characteristic, complemented
59	C₂⁴×C₁₉	—	2	1	no	—
60	C₂×D₄×C₁₉	—	2	2	no	—
61	C₂×D₄×C₁₉	—	2	1	no	—
62	C₂×D₄×C₁₉	—	2	1	no	—
63	C₂×D₄×C₁₉	—	4	1	no	—
64	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
65	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
66	C₂×D₄×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	C₂×C₂²⋊D₁₉	D₄	1	1	yes	characteristic
72	C₂²×Dic₁₉	—	2	1	no	—
73	C₂²×Dic₁₉	—	2	1	no	—
74	C₂²×Dic₁₉	—	2	1	no	—
75	C₂²×Dic₁₉	—	2	1	no	—
76	#64543	—	2	2	no	—
77	#64543	—	4	1	no	—
78	#64543	—	4	4	no	—
79	C₂×C₄×D₁₉	—	2	1	no	—
80	C₂×C₄×D₁₉	C₂³	1	1	yes	characteristic
81	C₂×Dic₃₈	—	2	1	no	—
82	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
83	D₈×C₁₉	—	2	2	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	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
88	C₂²⋊C₄×C₁₉	—	2	1	no	—
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	C₂²⋊Dic₁₉	D₄	1	1	yes	characteristic
97	#64526	—	4	1	no	—
98	(C₂×D₁₉)⋊C₄	D₄	1	1	yes	characteristic
99	C₄⋊Dic₁₉	—	2	1	no	—
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	D₄	1	1	yes	characteristic
107	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
108	C₂×C₁₉⋊C₈	—	4	1	no	—
109	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
110	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
111	#64515	—	4	1	no	—
112	C₂³×C₁₉	—	2	1	no	—
113	C₂³×C₁₉	—	2	1	no	—
114	C₂³×C₁₉	—	2	1	no	—
115	C₂³×C₁₉	—	4	2	no	—
116	C₂³×C₁₉	—	4	2	no	—
117	C₂³×C₁₉	—	8	1	no	—
118	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
119	C₂²×D₁₉	C₂×D₄	1	1	yes	characteristic
120	D₄×C₁₉	—	2	1	no	—
121	D₄×C₁₉	—	2	1	no	—
122	D₄×C₁₉	—	2	1	no	—
123	D₄×C₁₉	—	2	1	no	—
124	D₄×C₁₉	—	2	1	no	—
125	D₄×C₁₉	—	4	2	no	—
126	D₄×C₁₉	—	4	2	no	—
127	D₄×C₁₉	#41	1	2	yes	complemented
128	C₂×C₄×C₁₉	—	2	1	no	—
129	C₂×C₄×C₁₉	—	2	1	no	—
130	C₂×C₄×C₁₉	—	2	1	no	—
131	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
133	C₂²⋊D₁₉	—	2	2	no	—
134	C₂²⋊D₁₉	—	4	2	no	—
135	C₂²⋊D₁₉	—	4	2	no	—
136	C₂²⋊D₁₉	—	8	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	—	4	2	no	—
142	C₂×Dic₁₉	—	4	2	no	—
143	C₂×Dic₁₉	—	4	2	no	—
144	C₂×Dic₁₉	—	8	1	no	—
145	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
146	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
147	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
148	C₄×D₁₉	—	2	1	no	—
149	C₄×D₁₉	—	2	1	no	—
150	C₄×D₁₉	—	4	2	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	C₁₉⋊C₈	—	2	1	no	—
157	C₁₉⋊C₈	—	2	1	no	—
158	#3306	—	19	1	no	maximal, Sylow
159	C₂²×C₁₉	—	2	1	no	—
160	C₂²×C₁₉	—	2	1	no	—
161	C₂²×C₁₉	—	2	2	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	—	4	1	no	—
168	C₂²×C₁₉	—	4	1	no	—
169	C₂²×C₁₉	—	4	1	no	—
170	C₂²×C₁₉	—	8	4	no	—
171	C₂²×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
172	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
173	C₂×D₁₉	—	2	1	no	—
174	C₂×D₁₉	—	2	1	no	—
175	C₂×D₁₉	—	2	1	no	—
176	C₄×C₁₉	—	2	1	no	—
177	C₄×C₁₉	—	4	1	no	—
178	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
179	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
180	Dic₁₉	—	2	1	no	—
181	Dic₁₉	—	2	1	no	—
182	Dic₁₉	—	2	1	no	—
183	Dic₁₉	—	4	1	no	—
184	Dic₁₉	—	4	1	no	—
185	Dic₁₉	—	4	1	no	—
186	Dic₁₉	—	4	1	no	—
187	#547	—	19	1	no	—
188	#546	—	19	1	no	—
189	#489	—	19	1	no	—
190	#483	—	19	1	no	—
191	#481	—	19	1	no	—
192	#476	—	19	1	no	—
193	#469	—	19	1	no	—
194	#465	—	19	1	no	—
195	#463	—	19	1	no	—
196	#449	—	19	1	no	—
197	#447	D₁₉	1	1	yes	characteristic, complemented
198	#442	—	19	1	no	—
199	#435	—	19	1	no	—
200	#417	—	19	1	no	—
201	C₂×#102	—	19	1	no	—
202	C₂×C₁₉	—	2	2	no	—
203	C₂×C₁₉	—	4	2	no	—
204	C₂×C₁₉	—	8	1	no	—
205	C₂×C₁₉	—	8	1	no	—
206	C₂×C₁₉	#546	1	1	yes	characteristic
207	C₂×C₁₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
208	C₂×C₁₉	#577	1	1	yes	characteristic
209	D₁₉	—	4	1	no	—
210	C₂×#41	—	19	1	no	—
211	C₂×#41	—	38	1	no	—
212	C₂²×D₄	C₂×D₁₉	1	1	yes	characteristic
213	C₂×SD₁₆	—	19	1	no	—
214	C₂×D₈	C₂×D₁₉	1	1	yes	characteristic
215	C₂×D₈	C₂×D₁₉	1	1	yes	characteristic
216	C₂×M₁₆	—	19	1	no	—
217	C₂²×C₈	—	19	1	no	—
218	C₄⋊Q₈	—	19	1	no	—
219	#124	—	19	1	no	—
220	#123	—	38	1	no	—
221	#123	—	38	1	no	—
222	#123	—	38	1	no	—
223	C₂²⋊Q₈	—	19	1	no	—
224	C₂²⋊Q₈	—	38	1	no	—
225	#121	—	19	1	no	—
226	#121	—	19	1	no	—
227	#121	—	38	1	no	—
228	#121	C₂×D₁₉	1	1	yes	characteristic
229	#120	—	38	1	no	—
230	C₄×D₄	—	19	1	no	—
231	C₄×D₄	—	19	1	no	—
232	C₄×D₄	—	19	1	no	—
233	C₂×C₄⋊C₄	—	19	1	no	—
234	C₂×C₄⋊C₄	—	19	1	no	—
235	C₂×C₂²⋊C₄	—	38	1	no	—
236	#106	—	19	1	no	—
237	#106	—	19	1	no	—
238	#106	—	19	1	no	—
239	C₄⋊C₈	—	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	#102	—	19	1	no	—
246	#102	—	19	1	no	—
247	#102	—	19	1	no	—
248	#102	C₂×D₁₉	1	1	yes	characteristic
249	#102	C₂×D₁₉	1	1	yes	characteristic
250	C₂²⋊C₈	—	19	1	no	—
251	C₂²⋊C₈	C₂×D₁₉	1	1	yes	characteristic
252	#97	—	19	1	no	—
253	C₁₉	#3306	1	1	yes	characteristic, complemented, Sylow
254	C₂⁴	—	2	1	no	—
255	#41	—	38	2	no	—
256	#41	—	76	1	no	—
257	#41	—	76	4	no	—
258	C₂×Q₈	—	19	1	no	—
259	C₂×Q₈	—	38	1	no	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	2	1	no	—
262	C₂×D₄	—	2	1	no	—
263	C₂×D₄	—	4	1	no	—
264	C₂×D₄	—	19	1	no	—
265	C₂×D₄	—	38	1	no	—
266	C₂×D₄	—	38	1	no	—
267	C₂×D₄	—	38	1	no	—
268	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
269	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
270	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
271	C₂²×C₄	—	19	1	no	—
272	C₂²×C₄	—	38	1	no	—
273	C₂²×C₄	—	38	1	no	—
274	C₂²×C₄	—	38	1	no	—
275	C₂²×C₄	—	38	1	no	—
276	C₂²×C₄	—	38	1	no	—
277	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
278	SD₁₆	—	76	1	no	—
279	D₈	—	2	2	no	—
280	D₈	—	4	1	no	—
281	M₁₆	—	76	1	no	—
282	C₂×C₈	—	19	1	no	—
283	C₂×C₈	—	19	1	no	—
284	C₂×C₈	—	76	1	no	—
285	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
286	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
287	C₄⋊C₄	—	19	1	no	—
288	C₄⋊C₄	—	19	1	no	—
289	C₄⋊C₄	—	19	1	no	—
290	C₄⋊C₄	—	19	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	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
296	C₂²⋊C₄	—	2	1	no	—
297	C₂²⋊C₄	—	19	1	no	—
298	C₂²⋊C₄	—	19	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	C₂²⋊C₄	—	38	1	no	—
305	C₂²⋊C₄	—	38	1	no	—
306	C₄²	—	19	1	no	—
307	C₂³	—	2	1	no	—
308	C₂³	—	2	1	no	—
309	C₂³	—	2	1	no	—
310	C₂³	—	4	2	no	—
311	C₂³	—	4	2	no	—
312	C₂³	—	8	1	no	—
313	C₂³	—	19	1	no	—
314	C₂³	D₄×D₁₉	1	1	yes	characteristic
315	Q₈	—	38	1	no	—
316	Q₈	—	38	1	no	—
317	Q₈	—	76	2	no	—
318	D₄	—	2	1	no	—
319	D₄	—	2	1	no	—
320	D₄	—	2	1	no	—
321	D₄	—	2	1	no	—
322	D₄	—	2	1	no	—
323	D₄	—	4	2	no	—
324	D₄	—	4	2	no	—
325	D₄	—	38	2	no	—
326	D₄	—	76	2	no	—
327	D₄	—	76	2	no	—
328	D₄	—	152	1	no	—
329	D₄	#64543	1	2	yes	complemented
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	19	1	no	—
334	C₂×C₄	—	19	1	no	—
335	C₂×C₄	—	19	1	no	—
336	C₂×C₄	—	38	1	no	—
337	C₂×C₄	—	38	1	no	—
338	C₂×C₄	—	38	1	no	—
339	C₂×C₄	—	38	1	no	—
340	C₂×C₄	—	38	1	no	—
341	C₂×C₄	—	38	1	no	—
342	C₂×C₄	—	76	2	no	—
343	C₂×C₄	—	76	2	no	—
344	C₂×C₄	—	76	2	no	—
345	C₂×C₄	—	76	2	no	—
346	C₂×C₄	—	152	1	no	—
347	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
348	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
349	C₈	—	2	1	no	—
350	C₈	—	2	1	no	—
351	C₈	—	38	1	no	—
352	C₈	—	38	1	no	—
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	2	2	no	—
356	C₂²	—	4	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	—	4	1	no	—
364	C₂²	—	8	4	no	—
365	C₂²	—	38	1	no	—
366	C₂²	—	38	1	no	—
367	C₂²	—	38	1	no	—
368	C₂²	#10597357	1	2	yes	—
369	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
370	C₄	—	2	1	no	—
371	C₄	—	4	1	no	—
372	C₄	—	38	1	no	—
373	C₄	—	38	1	no	—
374	C₄	—	38	1	no	—
375	C₄	—	76	1	no	—
376	C₄	—	76	1	no	—
377	C₄	—	76	1	no	—
378	C₄	—	76	1	no	—
379	C₄	C₂×#64543	1	1	yes	characteristic
380	C₄	#10597430	1	1	yes	characteristic
381	C₂	—	2	2	no	—
382	C₂	—	4	2	no	—
383	C₂	—	8	1	no	—
384	C₂	—	8	1	no	—
385	C₂	—	76	1	no	—
386	C₂	#11927181	1	1	yes	characteristic
387	C₂	C₂×#10597357	1	1	yes	characteristic
388	C₂	#11927354	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×#447	C₂
socle	C₂²×C₁₉	C₂²×D₄
hypercenter	#447	D₁₉
hypocenter	C₁₉	#3306
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #447

Complements

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

N	H ≅ G/N	type
1. #11927194	C₂	semidirect
2. #11927188	C₂	semidirect
3. #11926902	C₂	semidirect
4. #11926875	C₂	semidirect
5. #11926861	C₂	semidirect
6. #11926832	C₂	semidirect
7. #11926794	C₂	semidirect
8. #11926771	C₂	semidirect
9. #11926757	C₂	semidirect
10. #11926686	C₂	semidirect
11. #11926656	C₂	semidirect
12. #11926612	C₂	semidirect
13. #11926542	C₂	semidirect
14. C₂×#10597289	C₂	semidirect
15. C₁₉×#447	C₂	semidirect
26. #10597384	C₂²	semidirect
31. D₄×Dic₁₉	C₂²	semidirect
32. C₂×#64526	C₂²	semidirect
37. C₂×#64515	C₂²	semidirect
38. C₄⋊C₄×D₁₉	C₂²	semidirect
39. #10597334	C₂²	semidirect
49. #10597289	C₂²	semidirect
50. (C₂×D₁₉)⋊C₈	C₂²	semidirect
51. #10597274	C₂²	semidirect
52. #10597274	C₂²	semidirect
53. #10597273	C₂²	semidirect
54. #10597273	C₂²	semidirect
55. #10597272	C₂²	semidirect
56. Dic₁₉⋊C₈	C₂²	semidirect
57. #10597267	C₂²	semidirect
58. #10597265	C₂²	semidirect
127. D₄×C₁₉	#41	semidirect
197. #447	D₁₉	semidirect
253. C₁₉	#3306	semidirect
329. D₄	#64543	semidirect

Maximal subgroups

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

subgroup	index
#11927194	2
#11927188	2
#11926902	2
#11926875	2
#11926861	2
#11926832	2
#11926794	2
#11926771	2
#11926757	2
#11926686	2
#11926656	2
#11926612	2
#11926542	2
C₂×#10597289	2
C₁₉×#447	2
#3306	19