Subgroups — #14573834

Order 2432.

396 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927149	C₂	1	1	yes	maximal, characteristic, complemented
2	#11926862	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926857	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926851	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926844	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926744	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926739	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926689	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926689	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926654	C₂	1	1	yes	maximal, characteristic, complemented
11	#11926642	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#64527	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#64525	C₂	1	1	yes	maximal, characteristic, complemented
14	#11926605	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₉×#520	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#41×C₁₉	C₂²	1	1	yes	characteristic
17	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₁₉	—	2	1	no	—
20	C₄×D₄×C₁₉	—	2	1	no	—
21	C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
22	C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
23	C₁₉×#117	—	2	1	no	—
24	C₁₉×#117	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
26	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
27	C₂×#64527	C₂²	1	1	yes	characteristic
28	C₂×#64527	C₂²	1	1	yes	characteristic
29	C₂×#64525	C₂²	1	1	yes	characteristic
30	C₂×#64525	C₂²	1	1	yes	characteristic
31	#10597376	C₂²	1	1	yes	characteristic
32	#10597376	C₂²	1	1	yes	characteristic
33	#10597340	—	2	1	no	—
34	#10597340	—	2	1	no	—
35	#10597330	—	2	1	no	—
36	#10597330	—	2	1	no	—
37	#10597320	C₂²	1	1	yes	characteristic
38	C₄×D₇₆	C₂²	1	1	yes	characteristic
39	C₄×D₇₆	C₂²	1	1	yes	characteristic
40	#10597316	C₂²	1	1	yes	characteristic
41	#10597292	C₂²	1	1	yes	characteristic
42	#10597292	C₂²	1	1	yes	characteristic
43	#10597289	C₂²	1	1	yes	characteristic
44	#10597289	C₂²	1	1	yes	characteristic
45	C₂²⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
46	C₂²⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
47	#10597266	C₂²	1	1	yes	characteristic
48	#10597266	C₂²	1	1	yes	characteristic
49	#10597266	C₂²	1	1	yes	characteristic
50	#10597266	C₂²	1	1	yes	characteristic
51	#10597265	C₂²	1	1	yes	characteristic
52	#10597265	C₂²	1	1	yes	characteristic
53	#10597264	C₂²	1	1	yes	characteristic
54	#10597264	C₂²	1	1	yes	characteristic
55	C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
56	C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
57	#10597260	C₂²	1	1	yes	characteristic
58	C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
59	#41×C₁₉	—	2	2	no	—
60	#41×C₁₉	—	2	2	no	—
61	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₁₉	—	2	1	no	—
63	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	—	2	1	no	—
65	C₂²×C₄×C₁₉	—	2	1	no	—
66	C₂²×C₄×C₁₉	—	2	1	no	—
67	C₂²×C₄×C₁₉	—	2	1	no	—
68	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
69	C₂×C₂²⋊D₁₉	—	2	1	no	—
70	C₂×C₂²⋊D₁₉	—	2	1	no	—
71	C₂×D₇₆	—	2	1	no	—
72	C₂×D₇₆	—	2	1	no	—
73	C₂×D₇₆	C₂³	1	1	yes	characteristic
74	C₂×D₇₆	C₂³	1	1	yes	characteristic
75	C₂×C₄×D₁₉	—	2	1	no	—
76	C₂×C₄×D₁₉	—	2	1	no	—
77	C₄⋊C₄×C₁₉	—	2	1	no	—
78	C₄⋊C₄×C₁₉	—	2	1	no	—
79	C₄⋊C₄×C₁₉	—	2	1	no	—
80	C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
81	C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
82	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
84	C₂²⋊C₄×C₁₉	—	2	1	no	—
85	C₂²⋊C₄×C₁₉	—	2	1	no	—
86	C₂²⋊C₄×C₁₉	D₄	1	1	yes	characteristic
87	C₂²⋊C₄×C₁₉	D₄	1	1	yes	characteristic
88	C₄²×C₁₉	—	2	1	no	—
89	C₄²×C₁₉	—	2	1	no	—
90	C₄²×C₁₉	C₂³	1	1	yes	characteristic
91	C₄²×C₁₉	C₂³	1	1	yes	characteristic
92	#64527	—	2	2	no	—
93	#64527	—	2	2	no	—
94	#64525	—	2	2	no	—
95	#64525	—	2	2	no	—
96	(C₂×D₁₉)⋊C₄	—	2	1	no	—
97	(C₂×D₁₉)⋊C₄	—	2	1	no	—
98	(C₂×D₁₉)⋊C₄	—	2	1	no	—
99	(C₂×D₁₉)⋊C₄	—	2	1	no	—
100	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
101	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
102	Dic₁₉⋊C₄	—	2	1	no	—
103	Dic₁₉⋊C₄	—	2	1	no	—
104	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
105	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
106	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
107	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
108	C₂³×C₁₉	—	2	1	no	—
109	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
110	C₂²×D₁₉	—	2	1	no	—
111	C₂²×D₁₉	—	2	1	no	—
112	Q₈×C₁₉	—	2	1	no	—
113	Q₈×C₁₉	#41	1	2	yes	—
114	D₄×C₁₉	—	2	4	no	—
115	D₄×C₁₉	—	2	1	no	—
116	D₄×C₁₉	#41	1	2	yes	—
117	C₂×C₄×C₁₉	—	2	4	no	—
118	C₂×C₄×C₁₉	—	2	1	no	—
119	C₂×C₄×C₁₉	—	2	1	no	—
120	C₂×C₄×C₁₉	—	2	1	no	—
121	C₂×C₄×C₁₉	—	2	1	no	—
122	C₂×C₄×C₁₉	—	2	1	no	—
123	C₂×C₄×C₁₉	—	8	1	no	—
124	C₂×C₄×C₁₉	—	8	1	no	—
125	C₂×C₄×C₁₉	—	8	1	no	—
126	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
132	C₂²⋊D₁₉	—	8	1	no	—
133	C₂²⋊D₁₉	—	8	1	no	—
134	C₂×Dic₁₉	—	2	1	no	—
135	C₂×Dic₁₉	—	2	1	no	—
136	D₇₆	—	2	1	no	—
137	D₇₆	—	2	1	no	—
138	D₇₆	—	2	1	no	—
139	D₇₆	—	2	1	no	—
140	D₇₆	—	8	1	no	—
141	D₇₆	—	8	1	no	—
142	C₄×D₁₉	—	8	1	no	—
143	C₄×D₁₉	—	8	1	no	—
144	C₁₉⋊C₈	—	2	1	no	—
145	C₁₉⋊C₈	—	2	1	no	—
146	C₁₉⋊C₈	—	2	1	no	—
147	C₁₉⋊C₈	—	2	1	no	—
148	#3119	—	19	1	no	maximal, Sylow
149	C₂²×C₁₉	—	2	1	no	—
150	C₂²×C₁₉	—	2	2	no	—
151	C₂²×C₁₉	—	2	1	no	—
152	C₂²×C₁₉	—	2	1	no	—
153	C₂²×C₁₉	—	4	1	no	—
154	C₂²×C₁₉	—	4	1	no	—
155	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
156	C₂×D₁₉	—	4	1	no	—
157	C₂×D₁₉	—	4	1	no	—
158	C₂×D₁₉	—	4	1	no	—
159	C₂×D₁₉	—	4	1	no	—
160	C₂×D₁₉	—	4	1	no	—
161	C₂×D₁₉	—	4	1	no	—
162	C₄×C₁₉	—	2	1	no	—
163	C₄×C₁₉	—	2	2	no	—
164	C₄×C₁₉	—	2	1	no	—
165	C₄×C₁₉	—	2	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	C₂×#41	1	1	yes	characteristic
171	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
172	Dic₁₉	—	4	1	no	—
173	Dic₁₉	—	4	1	no	—
174	#540	—	19	1	no	—
175	#520	D₁₉	1	1	yes	characteristic, complemented
176	#481	—	19	1	no	—
177	#480	—	19	1	no	—
178	#479	—	19	1	no	—
179	#478	—	19	1	no	—
180	#460	—	19	1	no	—
181	#459	—	19	1	no	—
182	#449	—	19	1	no	—
183	#449	—	19	1	no	—
184	#442	—	19	1	no	—
185	#440	—	19	1	no	—
186	C₄×SD₁₆	—	19	1	no	—
187	C₄×D₈	—	19	1	no	—
188	#433	—	19	1	no	—
189	C₂×C₁₉	—	4	1	no	—
190	C₂×C₁₉	—	4	2	no	—
191	C₂×C₁₉	#525	1	1	yes	characteristic
192	C₂×C₁₉	#576	1	1	yes	characteristic
193	C₂×C₁₉	#577	1	1	yes	characteristic
194	D₁₉	—	8	1	no	—
195	D₁₉	—	8	1	no	—
196	C₂×#41	C₂×D₁₉	1	1	yes	characteristic
197	C₂×SD₁₆	—	19	1	no	—
198	C₂×SD₁₆	—	19	1	no	—
199	C₂×D₈	—	19	1	no	—
200	C₂×D₈	—	19	1	no	—
201	#127	—	19	1	no	—
202	#125	—	19	1	no	—
203	#123	—	38	1	no	—
204	#123	—	38	1	no	—
205	#121	—	19	1	no	—
206	#121	—	19	1	no	—
207	#121	—	38	1	no	—
208	#121	—	38	1	no	—
209	C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
210	C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
211	C₄×D₄	—	2	1	no	—
212	C₄×D₄	—	2	1	no	—
213	C₄×D₄	—	19	1	no	—
214	C₄×D₄	—	19	1	no	—
215	C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
216	C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
217	#117	—	2	1	no	—
218	#117	C₂×D₁₉	1	1	yes	characteristic
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
221	#107	—	19	1	no	—
222	#107	—	19	1	no	—
223	#106	—	19	1	no	—
224	#106	—	19	1	no	—
225	C₄⋊C₈	—	19	1	no	—
226	C₄⋊C₈	—	19	1	no	—
227	#103	—	19	1	no	—
228	#103	—	19	1	no	—
229	#102	—	19	1	no	—
230	#102	—	19	1	no	—
231	#102	—	19	1	no	—
232	#102	—	19	1	no	—
233	#102	—	19	1	no	—
234	#102	—	19	1	no	—
235	C₂²⋊C₈	—	19	1	no	—
236	C₂²⋊C₈	—	19	1	no	—
237	#97	—	19	1	no	—
238	C₄×C₈	—	19	1	no	—
239	C₁₉	#3119	1	1	yes	characteristic, complemented, Sylow
240	#41	—	2	2	no	—
241	#41	—	2	2	no	—
242	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
243	C₂×D₄	—	2	1	no	—
244	C₂×D₄	—	19	1	no	—
245	C₂×D₄	—	19	1	no	—
246	C₂×D₄	—	38	1	no	—
247	C₂×D₄	—	38	1	no	—
248	C₂×D₄	—	38	1	no	—
249	C₂×D₄	—	38	1	no	—
250	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
251	C₂²×C₄	—	2	1	no	—
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	38	1	no	—
256	C₂²×C₄	—	38	1	no	—
257	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
258	SD₁₆	—	38	2	no	—
259	SD₁₆	—	38	2	no	—
260	D₈	—	38	2	no	—
261	D₈	—	38	2	no	—
262	C₂×C₈	—	19	1	no	—
263	C₂×C₈	—	19	1	no	—
264	C₂×C₈	—	19	1	no	—
265	C₂×C₈	—	19	1	no	—
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	19	1	no	—
270	C₄⋊C₄	—	19	1	no	—
271	C₄⋊C₄	—	38	1	no	—
272	C₄⋊C₄	—	38	1	no	—
273	C₄⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
274	C₄⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
275	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	38	1	no	—
280	C₂²⋊C₄	—	38	1	no	—
281	C₂²⋊C₄	—	38	1	no	—
282	C₂²⋊C₄	—	38	1	no	—
283	C₂²⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
284	C₂²⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
285	C₄²	—	2	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	C₂²×D₁₉	1	1	yes	characteristic
288	C₄²	C₂²×D₁₉	1	1	yes	characteristic
289	C₂³	—	2	1	no	—
290	C₂³	—	38	1	no	—
291	C₂³	—	38	1	no	—
292	C₂³	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
293	Q₈	—	2	1	no	—
294	Q₈	#64541	1	2	yes	—
295	D₄	—	2	4	no	—
296	D₄	—	2	1	no	—
297	D₄	—	38	1	no	—
298	D₄	—	38	1	no	—
299	D₄	—	38	1	no	—
300	D₄	—	38	1	no	—
301	D₄	—	152	1	no	—
302	D₄	—	152	1	no	—
303	D₄	—	152	1	no	—
304	D₄	—	152	1	no	—
305	D₄	#64541	1	2	yes	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	8	1	no	—
313	C₂×C₄	—	8	1	no	—
314	C₂×C₄	—	8	1	no	—
315	C₂×C₄	—	38	1	no	—
316	C₂×C₄	—	38	1	no	—
317	C₂×C₄	—	152	1	no	—
318	C₂×C₄	—	152	1	no	—
319	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
320	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
321	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
322	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
323	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
324	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
325	C₈	—	38	1	no	—
326	C₈	—	38	1	no	—
327	C₈	—	38	1	no	—
328	C₈	—	38	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	4	1	no	—
334	C₂²	—	4	1	no	—
335	C₂²	—	76	1	no	—
336	C₂²	—	76	1	no	—
337	C₂²	—	76	1	no	—
338	C₂²	—	76	1	no	—
339	C₂²	—	76	1	no	—
340	C₂²	—	76	1	no	—
341	C₂²	C₂²×C₂²⋊D₁₉	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	2	2	no	—
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	4	1	no	—
349	C₄	—	4	1	no	—
350	C₄	—	76	1	no	—
351	C₄	—	76	1	no	—
352	C₄	C₂×#64541	1	1	yes	characteristic
353	C₄	C₂×#64541	1	1	yes	characteristic
354	C₂	—	4	1	no	—
355	C₂	—	4	2	no	—
356	C₂	—	152	1	no	—
357	C₂	—	152	1	no	—
358	C₂	#11927055	1	1	yes	characteristic
359	C₂	#11927347	1	1	yes	characteristic
360	C₂	#11927352	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₉
commutator, cosocle	C₂×C₄×C₁₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉
Fitting	C₁₉×#520	C₂
socle	C₂²×C₁₉	C₂²×D₄
hypercenter	#520	D₁₉
hypocenter	C₁₉	#3119
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #520

Complements

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

N	H ≅ G/N	type
1. #11927149	C₂	semidirect
2. #11926862	C₂	semidirect
3. #11926857	C₂	semidirect
4. #11926851	C₂	semidirect
5. #11926844	C₂	semidirect
6. #11926744	C₂	semidirect
7. #11926739	C₂	semidirect
8. #11926689	C₂	semidirect
9. #11926689	C₂	semidirect
10. #11926654	C₂	semidirect
11. #11926642	C₂	semidirect
12. C₄×#64527	C₂	semidirect
13. C₄×#64525	C₂	semidirect
14. #11926605	C₂	semidirect
15. C₁₉×#520	C₂	semidirect
175. #520	D₁₉	semidirect
239. C₁₉	#3119	semidirect

Maximal subgroups

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

subgroup	index
#11927149	2
#11926862	2
#11926857	2
#11926851	2
#11926844	2
#11926744	2
#11926739	2
#11926689	2
#11926654	2
#11926642	2
C₄×#64527	2
C₄×#64525	2
#11926605	2
C₁₉×#520	2
#3119	19