Subgroups — #14575469

Order 2432.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927232	C₂	1	1	yes	maximal, characteristic, complemented
2	#11927187	C₂	1	1	yes	maximal, characteristic
3	#11926873	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926867	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926841	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926784	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926772	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926763	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926709	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926696	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#64516	C₂	1	1	yes	maximal, characteristic, complemented
12	C₈×D₇₆	C₂	1	1	yes	maximal, characteristic, complemented
13	#11926539	C₂	1	1	yes	maximal, characteristic, complemented
14	#102×D₁₉	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₉×#488	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	C₁₉×#125	C₂²	1	1	yes	characteristic, complemented
19	C₁₉×#124	C₂²	1	1	yes	characteristic
20	#10597392	C₂²	1	1	yes	characteristic
21	C₂×#64528	C₂²	1	1	yes	characteristic, complemented
22	#10597384	C₂²	1	1	yes	characteristic
23	C₂×#64526	C₂²	1	1	yes	characteristic
24	C₂×#64516	C₂²	1	1	yes	characteristic
25	C₂×C₈×D₁₉	C₂²	1	1	yes	characteristic
26	C₂×C₈×D₁₉	C₂²	1	1	yes	characteristic, complemented
27	#10597344	—	2	1	no	—
28	#10597343	C₂²	1	1	yes	characteristic
29	#10597343	C₂²	1	1	yes	characteristic, complemented
30	#10597340	—	2	1	no	—
31	#10597338	—	2	1	no	—
32	#10597338	C₂²	1	1	yes	characteristic, complemented
33	C₄⋊C₄×D₁₉	C₂²	1	1	yes	characteristic
34	#10597336	—	2	1	no	—
35	#10597330	—	2	1	no	—
36	#10597329	—	2	1	no	—
37	#10597328	—	2	1	no	—
38	C₂²⋊C₄×D₁₉	—	2	1	no	—
39	C₄×D₇₆	C₂²	1	1	yes	characteristic
40	C₄×D₇₆	C₂²	1	1	yes	characteristic
41	C₄×Dic₃₈	C₂²	1	1	yes	characteristic, complemented
42	C₁₉×#103	C₂²	1	1	yes	characteristic, complemented
43	C₁₉×#103	C₂²	1	1	yes	characteristic, complemented
44	C₁₉×#102	C₂²	1	1	yes	characteristic
45	C₁₉×#102	C₂²	1	1	yes	characteristic
46	C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic, complemented
47	#10597292	C₂²	1	1	yes	characteristic, complemented
48	#10597289	C₂²	1	1	yes	characteristic
49	#10597278	C₂²	1	1	yes	characteristic
50	(C₂×D₁₉)⋊C₈	C₂²	1	1	yes	characteristic
51	(C₂×D₁₉)⋊C₈	C₂²	1	1	yes	characteristic, complemented
52	#10597274	C₂²	1	1	yes	characteristic, complemented
53	#10597273	C₂²	1	1	yes	characteristic, complemented
54	#10597266	C₂²	1	1	yes	characteristic
55	#10597266	C₂²	1	1	yes	characteristic
56	#10597265	C₂²	1	1	yes	characteristic, complemented
57	#10597264	C₂²	1	1	yes	characteristic, complemented
58	C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₁₉	—	2	1	no	—
60	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₉	—	2	1	no	—
63	C₂×C₂²⋊D₁₉	—	2	1	no	—
64	#64545	—	2	2	no	—
65	#64545	—	4	1	no	—
66	D₄×D₁₉	—	2	2	no	—
67	D₄×D₁₉	—	4	1	no	—
68	C₂×D₇₆	—	2	1	no	—
69	C₂×D₇₆	D₄	1	1	yes	characteristic
70	C₂×D₇₆	C₂³	1	1	yes	characteristic
71	C₂×C₄×D₁₉	—	2	1	no	—
72	C₂×C₄×D₁₉	—	2	1	no	—
73	C₂×C₄×D₁₉	—	2	1	no	—
74	C₂×C₄×D₁₉	C₂³	1	1	yes	characteristic
75	C₂×C₄×D₁₉	C₂³	1	1	yes	characteristic
76	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
78	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₁₉	—	2	1	no	—
80	C₄⋊C₄×C₁₉	—	2	1	no	—
81	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
83	C₂²⋊C₄×C₁₉	—	2	1	no	—
84	C₂²⋊C₄×C₁₉	—	2	1	no	—
85	C₄²×C₁₉	C₂³	1	1	yes	characteristic
86	C₂²⋊Dic₁₉	—	2	1	no	—
87	#64528	—	4	1	no	—
88	#64526	—	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	(C₂×D₁₉)⋊C₄	D₄	1	1	yes	characteristic
96	C₄⋊Dic₁₉	—	2	1	no	—
97	C₄⋊Dic₁₉	—	2	1	no	—
98	C₄⋊Dic₁₉	D₄	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	C₄×Dic₁₉	—	2	1	no	—
105	C₄×Dic₁₉	—	2	1	no	—
106	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
107	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
108	#64516	—	2	2	no	—
109	C₈×D₁₉	—	4	1	no	—
110	C₈×D₁₉	—	4	1	no	—
111	C₂³×C₁₉	—	2	1	no	—
112	C₂²×D₁₉	—	2	1	no	—
113	C₂²×D₁₉	—	4	2	no	—
114	C₂²×D₁₉	—	4	2	no	—
115	C₂²×D₁₉	—	8	1	no	—
116	C₂²×D₁₉	C₂×D₄	1	1	yes	characteristic
117	C₂²×D₁₉	C₂×D₄	1	1	yes	characteristic
118	Q₈×C₁₉	—	2	1	no	—
119	Q₈×C₁₉	—	2	1	no	—
120	D₄×C₁₉	—	2	1	no	—
121	D₄×C₁₉	—	2	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	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	2	no	—
136	D₇₆	—	2	1	no	—
137	D₇₆	—	4	2	no	—
138	D₇₆	#41	1	2	yes	—
139	C₄×D₁₉	—	2	1	no	—
140	C₄×D₁₉	—	2	1	no	—
141	C₄×D₁₉	—	2	1	no	—
142	C₄×D₁₉	—	2	1	no	—
143	C₄×D₁₉	—	4	2	no	—
144	C₄×D₁₉	—	4	2	no	—
145	C₄×D₁₉	—	8	1	no	—
146	Dic₃₈	—	2	1	no	—
147	Dic₃₈	—	2	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	#3296	—	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	1	no	—
160	C₂×D₁₉	—	2	2	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	—	8	4	no	—
166	C₂×D₁₉	#135	1	2	yes	—
167	C₄×C₁₉	—	2	1	no	—
168	C₄×C₁₉	—	2	1	no	—
169	C₄×C₁₉	—	4	1	no	—
170	C₄×C₁₉	—	4	1	no	—
171	C₄×C₁₉	—	4	1	no	—
172	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
173	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
174	Dic₁₉	—	2	1	no	—
175	Dic₁₉	—	2	1	no	—
176	Dic₁₉	—	4	1	no	—
177	Dic₁₉	—	4	1	no	—
178	Dic₁₉	—	4	1	no	—
179	#552	—	19	1	no	—
180	#546	—	19	1	no	—
181	#488	D₁₉	1	1	yes	characteristic, complemented
182	#483	—	19	1	no	—
183	#482	—	19	1	no	—
184	#478	—	19	1	no	—
185	#467	—	19	1	no	—
186	#465	—	19	1	no	—
187	#464	—	19	1	no	—
188	#452	—	19	1	no	—
189	#450	—	19	1	no	—
190	C₄×SD₁₆	—	19	1	no	—
191	C₈×D₄	—	19	1	no	—
192	#416	—	19	1	no	—
193	C₂×#102	—	19	1	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	—	2	2	no	—
199	D₁₉	—	4	1	no	—
200	D₁₉	—	4	2	no	—
201	C₂×#41	—	19	1	no	—
202	C₂²×D₄	—	19	1	no	—
203	C₂×Q₁₆	—	19	1	no	—
204	C₂×SD₁₆	—	19	1	no	—
205	C₂×SD₁₆	—	19	1	no	—
206	C₂²×C₈	—	19	1	no	—
207	C₂²×C₈	—	19	1	no	—
208	#126	—	38	1	no	—
209	#125	—	38	1	no	—
210	#125	C₂×D₁₉	1	1	yes	characteristic, complemented
211	#124	C₂×D₁₉	1	1	yes	characteristic
212	#123	—	38	1	no	—
213	#123	—	38	1	no	—
214	C₂²⋊Q₈	—	19	1	no	—
215	C₂²⋊Q₈	—	19	1	no	—
216	C₂²⋊Q₈	—	19	1	no	—
217	#121	—	19	1	no	—
218	#121	—	38	1	no	—
219	#120	—	38	1	no	—
220	C₄×Q₈	—	19	1	no	—
221	C₄×D₄	—	19	1	no	—
222	C₄×D₄	—	19	1	no	—
223	#117	—	19	1	no	—
224	#117	—	38	1	no	—
225	C₂×C₄⋊C₄	—	19	1	no	—
226	C₂×C₂²⋊C₄	—	38	1	no	—
227	#107	—	19	1	no	—
228	#106	—	19	1	no	—
229	#106	—	19	1	no	—
230	C₄⋊C₈	—	19	1	no	—
231	#103	—	19	1	no	—
232	#103	—	19	1	no	—
233	#103	C₂×D₁₉	1	1	yes	characteristic, complemented
234	#103	C₂×D₁₉	1	1	yes	characteristic, complemented
235	#102	—	19	1	no	—
236	#102	—	19	1	no	—
237	#102	—	19	1	no	—
238	#102	—	19	1	no	—
239	#102	C₂×D₁₉	1	1	yes	characteristic
240	#102	C₂×D₁₉	1	1	yes	characteristic
241	C₂²⋊C₈	—	19	1	no	—
242	C₂²⋊C₈	—	19	1	no	—
243	C₄×C₈	C₂×D₁₉	1	1	yes	characteristic, complemented
244	C₁₉	#3296	1	1	yes	characteristic, complemented, Sylow
245	C₂⁴	—	38	1	no	—
246	#41	—	38	2	no	—
247	#41	—	76	1	no	—
248	C₂×Q₈	—	19	1	no	—
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	2	no	—
255	C₂×D₄	—	38	1	no	—
256	C₂×D₄	—	76	1	no	—
257	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
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	Q₁₆	—	76	1	no	—
264	SD₁₆	—	38	2	no	—
265	SD₁₆	—	76	1	no	—
266	C₂×C₈	—	19	1	no	—
267	C₂×C₈	—	19	1	no	—
268	C₂×C₈	—	76	1	no	—
269	C₂×C₈	—	76	1	no	—
270	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
271	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	2	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	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
284	C₂²⋊C₄	—	2	1	no	—
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	—	38	1	no	—
296	C₄²	C₂²×D₁₉	1	1	yes	characteristic
297	C₂³	—	2	1	no	—
298	C₂³	—	19	1	no	—
299	C₂³	—	19	1	no	—
300	C₂³	—	38	1	no	—
301	C₂³	—	76	2	no	—
302	C₂³	—	76	2	no	—
303	C₂³	—	152	1	no	—
304	Q₈	—	2	1	no	—
305	Q₈	—	2	1	no	—
306	Q₈	—	38	1	no	—
307	Q₈	—	38	1	no	—
308	D₄	—	2	1	no	—
309	D₄	—	2	1	no	—
310	D₄	—	19	2	no	—
311	D₄	—	38	2	no	—
312	D₄	—	38	1	no	—
313	D₄	—	76	2	no	—
314	D₄	—	76	2	no	—
315	D₄	—	152	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	19	1	no	—
320	C₂×C₄	—	19	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	—	38	1	no	—
327	C₂×C₄	—	38	1	no	—
328	C₂×C₄	—	76	2	no	—
329	C₂×C₄	—	76	2	no	—
330	C₂×C₄	—	152	1	no	—
331	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
332	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
333	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	38	1	no	—
337	C₈	—	38	1	no	—
338	C₂²	—	4	1	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	19	2	no	—
342	C₂²	—	38	1	no	—
343	C₂²	—	38	1	no	—
344	C₂²	—	38	1	no	—
345	C₂²	—	38	1	no	—
346	C₂²	—	38	1	no	—
347	C₂²	—	38	2	no	—
348	C₂²	—	76	1	no	—
349	C₂²	—	76	1	no	—
350	C₂²	—	152	4	no	—
351	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	4	1	no	—
356	C₄	—	4	1	no	—
357	C₄	—	38	1	no	—
358	C₄	—	38	1	no	—
359	C₄	—	76	1	no	—
360	C₄	—	76	1	no	—
361	C₄	—	76	1	no	—
362	C₄	#41×D₁₉	1	1	yes	characteristic
363	C₄	#10597435	1	1	yes	characteristic
364	C₂	—	8	1	no	—
365	C₂	—	38	2	no	—
366	C₂	—	76	1	no	—
367	C₂	—	76	2	no	—
368	C₂	#11927187	1	1	yes	characteristic
369	C₂	#135×D₁₉	1	1	yes	characteristic
370	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×#488	C₂
socle	C₂²×C₁₉	C₂²×D₄
prime radical	#11927187	C₂
hypercenter	#488	D₁₉
hypocenter	C₁₉	#3296
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #488

Complements

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

N	H ≅ G/N	type
1. #11927232	C₂	semidirect
3. #11926873	C₂	semidirect
4. #11926867	C₂	semidirect
5. #11926841	C₂	semidirect
6. #11926784	C₂	semidirect
7. #11926772	C₂	semidirect
8. #11926763	C₂	semidirect
9. #11926709	C₂	semidirect
10. #11926696	C₂	semidirect
11. C₄×#64516	C₂	semidirect
12. C₈×D₇₆	C₂	semidirect
13. #11926539	C₂	semidirect
14. #102×D₁₉	C₂	semidirect
15. C₁₉×#488	C₂	semidirect
18. C₁₉×#125	C₂²	semidirect
21. C₂×#64528	C₂²	semidirect
26. C₂×C₈×D₁₉	C₂²	semidirect
29. #10597343	C₂²	semidirect
32. #10597338	C₂²	semidirect
41. C₄×Dic₃₈	C₂²	semidirect
42. C₁₉×#103	C₂²	semidirect
43. C₁₉×#103	C₂²	semidirect
46. C₄×C₈×C₁₉	C₂²	semidirect
47. #10597292	C₂²	semidirect
51. (C₂×D₁₉)⋊C₈	C₂²	semidirect
52. #10597274	C₂²	semidirect
53. #10597273	C₂²	semidirect
56. #10597265	C₂²	semidirect
57. #10597264	C₂²	semidirect
58. C₄⋊(C₁₉⋊C₈)	C₂²	semidirect
181. #488	D₁₉	semidirect
210. #125	C₂×D₁₉	semidirect
233. #103	C₂×D₁₉	semidirect
234. #103	C₂×D₁₉	semidirect
243. C₄×C₈	C₂×D₁₉	semidirect
244. C₁₉	#3296	semidirect

Maximal subgroups

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

subgroup	index
#11927232	2
#11927187	2
#11926873	2
#11926867	2
#11926841	2
#11926784	2
#11926772	2
#11926763	2
#11926709	2
#11926696	2
C₄×#64516	2
C₈×D₇₆	2
#11926539	2
#102×D₁₉	2
C₁₉×#488	2
#3296	19