Subgroups — #14575550

Order 2432.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927262	C₂	1	1	yes	maximal, characteristic, complemented
2	#11927196	C₂	1	1	yes	maximal, characteristic
3	#11926893	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926888	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926823	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926786	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926783	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926753	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926709	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926708	C₂	1	1	yes	maximal, characteristic, complemented
11	#11926631	C₂	1	1	yes	maximal, characteristic, complemented
12	(C₂²⋊D₁₉)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
13	#11926539	C₂	1	1	yes	maximal, characteristic, complemented
14	#11926538	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₉×#485	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#64545	C₂²	1	1	yes	characteristic
17	C₂×#64541	C₂²	1	1	yes	characteristic
18	C₁₉×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
19	C₁₉×#117	C₂²	1	1	yes	characteristic
20	#10597390	C₂²	1	1	yes	characteristic
21	C₂×#64528	C₂²	1	1	yes	characteristic
22	C₄×C₂²⋊D₁₉	C₂²	1	1	yes	characteristic
23	#10597370	C₂²	1	1	yes	characteristic
24	C₂²×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×Dic₇₆	C₂²	1	1	yes	characteristic
26	C₂×#64516	C₂²	1	1	yes	characteristic
27	C₂×C₈×D₁₉	C₂²	1	1	yes	characteristic
28	#10597343	C₂²	1	1	yes	characteristic
29	#10597343	C₂²	1	1	yes	characteristic
30	#10597340	—	2	1	no	—
31	#10597339	C₂²	1	1	yes	characteristic
32	#10597338	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×D₁₉	—	2	1	no	—
34	#10597334	C₂²	1	1	yes	characteristic
35	#10597333	C₂²	1	1	yes	characteristic
36	#10597331	—	2	1	no	—
37	#10597330	—	2	1	no	—
38	#10597329	—	2	1	no	—
39	#10597325	—	2	1	no	—
40	#10597322	—	2	1	no	—
41	C₄×Dic₃₈	—	2	1	no	—
42	C₁₉×#107	C₂²	1	1	yes	characteristic
43	C₁₉×#106	C₂²	1	1	yes	characteristic
44	C₁₉×#103	C₂²	1	1	yes	characteristic
45	C₁₉×#103	C₂²	1	1	yes	characteristic
46	C₁₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
47	#10597292	C₂²	1	1	yes	characteristic
48	#10597278	C₂²	1	1	yes	characteristic
49	(C₂×D₁₉)⋊C₈	C₂²	1	1	yes	characteristic
50	#10597273	C₂²	1	1	yes	characteristic
51	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
52	C₈×Dic₁₉	C₂²	1	1	yes	characteristic
53	#10597267	C₂²	1	1	yes	characteristic
54	#10597267	C₂²	1	1	yes	characteristic
55	#10597267	C₂²	1	1	yes	characteristic
56	#10597266	C₂²	1	1	yes	characteristic
57	#10597266	C₂²	1	1	yes	characteristic
58	#10597264	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₉	—	2	1	no	—
62	C₂×C₂²⋊D₁₉	D₄	1	1	yes	characteristic
63	#64545	—	2	2	no	—
64	#64545	—	4	1	no	—
65	#64541	—	2	2	no	—
66	#64541	—	4	1	no	—
67	C₂×D₇₆	—	2	1	no	—
68	C₂×D₇₆	C₂³	1	1	yes	characteristic
69	C₂×C₄×D₁₉	—	2	1	no	—
70	C₂×C₄×D₁₉	—	2	1	no	—
71	C₂×C₄×D₁₉	—	2	1	no	—
72	C₂×C₄×D₁₉	C₂³	1	1	yes	characteristic
73	C₂×Dic₃₈	—	2	1	no	—
74	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
75	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
76	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₁₉	—	2	1	no	—
79	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
82	C₂²⋊C₄×C₁₉	—	2	1	no	—
83	C₂²⋊C₄×C₁₉	—	2	1	no	—
84	C₄²×C₁₉	—	2	1	no	—
85	C₂²⋊Dic₁₉	—	2	1	no	—
86	C₂²⋊Dic₁₉	D₄	1	1	yes	characteristic
87	#64528	—	4	1	no	—
88	(C₂×D₁₉)⋊C₄	—	2	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	D₄	1	1	yes	characteristic
93	C₄⋊Dic₁₉	—	2	1	no	—
94	C₄⋊Dic₁₉	—	2	1	no	—
95	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
96	Dic₁₉⋊C₄	—	2	1	no	—
97	Dic₁₉⋊C₄	—	2	1	no	—
98	Dic₁₉⋊C₄	—	2	1	no	—
99	Dic₁₉⋊C₄	—	2	1	no	—
100	Dic₁₉⋊C₄	—	2	1	no	—
101	Dic₁₉⋊C₄	D₄	1	1	yes	characteristic
102	C₄×Dic₁₉	—	2	1	no	—
103	C₄×Dic₁₉	—	2	1	no	—
104	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
105	C₂×C₁₉⋊C₈	—	4	1	no	—
106	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
107	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
108	Dic₇₆	—	4	1	no	—
109	#64516	—	2	2	no	—
110	C₈×D₁₉	—	4	1	no	—
111	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
112	C₂²×D₁₉	—	2	1	no	—
113	C₂²×D₁₉	C₂×D₄	1	1	yes	characteristic
114	Q₈×C₁₉	—	2	1	no	—
115	Q₈×C₁₉	—	2	1	no	—
116	C₂×C₄×C₁₉	—	2	1	no	—
117	C₂×C₄×C₁₉	—	2	1	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	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
124	C₂²⋊D₁₉	—	2	2	no	—
125	C₂²⋊D₁₉	—	4	2	no	—
126	C₂×Dic₁₉	—	2	1	no	—
127	C₂×Dic₁₉	—	2	1	no	—
128	C₂×Dic₁₉	—	2	1	no	—
129	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
132	D₇₆	—	2	1	no	—
133	D₇₆	—	4	2	no	—
134	D₇₆	#41	1	2	yes	—
135	C₄×D₁₉	—	2	1	no	—
136	C₄×D₁₉	—	2	1	no	—
137	C₄×D₁₉	—	4	2	no	—
138	C₄×D₁₉	—	4	2	no	—
139	C₄×D₁₉	—	4	2	no	—
140	Dic₃₈	—	2	1	no	—
141	Dic₃₈	—	2	1	no	—
142	Dic₃₈	—	2	1	no	—
143	Dic₃₈	—	2	1	no	—
144	Dic₃₈	—	4	2	no	—
145	C₈×C₁₉	—	2	1	no	—
146	C₈×C₁₉	—	2	1	no	—
147	C₁₉⋊C₈	—	2	1	no	—
148	C₁₉⋊C₈	—	2	1	no	—
149	#3302	—	19	1	no	maximal, Sylow
150	C₂²×C₁₉	—	2	1	no	—
151	C₂²×C₁₉	—	2	1	no	—
152	C₂²×C₁₉	—	2	1	no	—
153	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
154	C₂×D₁₉	—	2	1	no	—
155	C₂×D₁₉	—	2	2	no	—
156	C₂×D₁₉	—	2	1	no	—
157	C₂×D₁₉	—	2	1	no	—
158	C₂×D₁₉	—	4	1	no	—
159	C₂×D₁₉	—	4	1	no	—
160	C₄×C₁₉	—	2	1	no	—
161	C₄×C₁₉	—	4	1	no	—
162	C₄×C₁₉	—	4	1	no	—
163	C₄×C₁₉	—	4	1	no	—
164	C₄×C₁₉	—	4	1	no	—
165	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
166	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
167	Dic₁₉	—	2	1	no	—
168	Dic₁₉	—	2	1	no	—
169	Dic₁₉	—	4	1	no	—
170	Dic₁₉	—	4	1	no	—
171	Dic₁₉	—	4	1	no	—
172	Dic₁₉	#135	1	2	yes	—
173	#556	—	19	1	no	—
174	#547	—	19	1	no	—
175	#487	—	19	1	no	—
176	#485	—	19	1	no	—
177	#485	D₁₉	1	1	yes	characteristic, complemented
178	#474	—	19	1	no	—
179	#467	—	19	1	no	—
180	#467	—	19	1	no	—
181	#462	—	19	1	no	—
182	#452	—	19	1	no	—
183	#452	—	19	1	no	—
184	C₄×SD₁₆	—	19	1	no	—
185	C₈×D₄	—	19	1	no	—
186	#416	—	19	1	no	—
187	#416	—	19	1	no	—
188	C₂×C₁₉	—	4	1	no	—
189	C₂×C₁₉	#546	1	1	yes	characteristic
190	C₂×C₁₉	C₂×#135	1	1	yes	characteristic
191	C₂×C₁₉	#578	1	1	yes	characteristic
192	D₁₉	—	4	1	no	—
193	D₁₉	—	4	2	no	—
194	C₂×#41	—	19	1	no	—
195	C₂×#41	—	19	1	no	—
196	C₂×Q₁₆	—	19	1	no	—
197	C₂×Q₁₆	—	19	1	no	—
198	C₂×SD₁₆	—	19	1	no	—
199	C₂²×C₈	—	19	1	no	—
200	C₂²×C₈	—	19	1	no	—
201	C₄⋊Q₈	—	19	1	no	—
202	C₄⋊Q₈	—	19	1	no	—
203	#126	—	38	1	no	—
204	#126	—	38	1	no	—
205	#124	—	38	1	no	—
206	#123	—	38	1	no	—
207	#123	—	38	1	no	—
208	C₂²⋊Q₈	—	19	1	no	—
209	C₂²⋊Q₈	—	19	1	no	—
210	C₂²⋊Q₈	—	19	1	no	—
211	C₂²⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
212	#121	—	38	1	no	—
213	C₄×Q₈	—	19	1	no	—
214	C₄×Q₈	—	38	1	no	—
215	C₄×D₄	—	19	1	no	—
216	C₄×D₄	—	19	1	no	—
217	#117	—	19	1	no	—
218	#117	C₂×D₁₉	1	1	yes	characteristic
219	C₂×C₄⋊C₄	—	38	1	no	—
220	#107	—	19	1	no	—
221	#107	C₂×D₁₉	1	1	yes	characteristic
222	#106	C₂×D₁₉	1	1	yes	characteristic
223	C₄⋊C₈	—	19	1	no	—
224	#103	—	19	1	no	—
225	#103	—	19	1	no	—
226	#103	—	19	1	no	—
227	#103	—	19	1	no	—
228	#103	—	19	1	no	—
229	#103	C₂×D₁₉	1	1	yes	characteristic
230	#103	C₂×D₁₉	1	1	yes	characteristic
231	#102	—	19	1	no	—
232	#102	—	19	1	no	—
233	#102	—	19	1	no	—
234	C₂²⋊C₈	—	19	1	no	—
235	C₂²⋊C₈	C₂×D₁₉	1	1	yes	characteristic
236	C₄×C₈	—	19	1	no	—
237	C₁₉	#3302	1	1	yes	characteristic, complemented, Sylow
238	#41	—	38	2	no	—
239	#41	—	38	2	no	—
240	#41	—	76	1	no	—
241	#41	—	76	1	no	—
242	C₂×Q₈	—	19	1	no	—
243	C₂×Q₈	—	19	1	no	—
244	C₂×Q₈	—	38	1	no	—
245	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
246	C₂×D₄	—	19	1	no	—
247	C₂×D₄	—	19	1	no	—
248	C₂×D₄	—	38	1	no	—
249	C₂×D₄	—	38	1	no	—
250	C₂²×C₄	—	19	1	no	—
251	C₂²×C₄	—	38	1	no	—
252	C₂²×C₄	—	38	1	no	—
253	C₂²×C₄	—	38	1	no	—
254	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
255	Q₁₆	—	76	1	no	—
256	Q₁₆	—	76	1	no	—
257	SD₁₆	—	38	2	no	—
258	C₂×C₈	—	19	1	no	—
259	C₂×C₈	—	19	1	no	—
260	C₂×C₈	—	76	1	no	—
261	C₂×C₈	—	76	1	no	—
262	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
263	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	19	1	no	—
266	C₄⋊C₄	—	19	1	no	—
267	C₄⋊C₄	—	38	1	no	—
268	C₄⋊C₄	—	38	1	no	—
269	C₄⋊C₄	—	38	1	no	—
270	C₄⋊C₄	—	38	1	no	—
271	C₄⋊C₄	—	38	1	no	—
272	C₄⋊C₄	—	38	1	no	—
273	C₄⋊C₄	—	38	1	no	—
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	—	19	1	no	—
280	C₂²⋊C₄	—	19	1	no	—
281	C₂²⋊C₄	—	38	1	no	—
282	C₂²⋊C₄	—	38	1	no	—
283	C₂²⋊C₄	—	38	1	no	—
284	C₂²⋊C₄	—	38	1	no	—
285	C₂²⋊C₄	—	38	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	—	19	1	no	—
288	C₄²	—	38	1	no	—
289	C₄²	—	38	1	no	—
290	C₂³	—	19	1	no	—
291	C₂³	—	38	1	no	—
292	C₂³	D₄×D₁₉	1	1	yes	characteristic
293	Q₈	—	2	1	no	—
294	Q₈	—	2	1	no	—
295	Q₈	—	38	1	no	—
296	Q₈	—	38	1	no	—
297	Q₈	—	38	1	no	—
298	Q₈	—	38	1	no	—
299	Q₈	—	76	2	no	—
300	D₄	—	19	2	no	—
301	D₄	—	38	2	no	—
302	D₄	—	38	1	no	—
303	D₄	—	76	2	no	—
304	D₄	—	76	2	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	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	—	19	1	no	—
312	C₂×C₄	—	19	1	no	—
313	C₂×C₄	—	19	1	no	—
314	C₂×C₄	—	38	1	no	—
315	C₂×C₄	—	38	1	no	—
316	C₂×C₄	—	38	1	no	—
317	C₂×C₄	—	38	1	no	—
318	C₂×C₄	—	38	1	no	—
319	C₂×C₄	—	76	2	no	—
320	C₂×C₄	—	76	2	no	—
321	C₂×C₄	—	76	2	no	—
322	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
323	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
324	C₈	—	2	1	no	—
325	C₈	—	2	1	no	—
326	C₈	—	38	1	no	—
327	C₈	—	38	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	38	1	no	—
332	C₂²	—	38	1	no	—
333	C₂²	—	38	1	no	—
334	C₂²	—	38	2	no	—
335	C₂²	—	76	1	no	—
336	C₂²	—	76	1	no	—
337	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	4	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	19	2	no	—
344	C₄	—	38	1	no	—
345	C₄	—	38	1	no	—
346	C₄	—	76	1	no	—
347	C₄	—	76	1	no	—
348	C₄	—	76	1	no	—
349	C₄	#41×D₁₉	1	1	yes	characteristic
350	C₄	#10597435	1	1	yes	characteristic
351	C₂	—	4	1	no	—
352	C₂	—	76	1	no	—
353	C₂	—	76	2	no	—
354	C₂	#11927186	1	1	yes	characteristic
355	C₂	#135×D₁₉	1	1	yes	characteristic
356	C₂	#11927360	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×#485	C₂
socle	C₂²×C₁₉	C₂²×D₄
prime radical	#11927196	C₂
hypercenter	#485	D₁₉
hypocenter	C₁₉	#3302
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #485

Complements

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

N	H ≅ G/N	type
1. #11927262	C₂	semidirect
3. #11926893	C₂	semidirect
4. #11926888	C₂	semidirect
5. #11926823	C₂	semidirect
6. #11926786	C₂	semidirect
7. #11926783	C₂	semidirect
8. #11926753	C₂	semidirect
9. #11926709	C₂	semidirect
10. #11926708	C₂	semidirect
11. #11926631	C₂	semidirect
12. (C₂²⋊D₁₉)⋊C₈	C₂	semidirect
13. #11926539	C₂	semidirect
14. #11926538	C₂	semidirect
15. C₁₉×#485	C₂	semidirect
177. #485	D₁₉	semidirect
237. C₁₉	#3302	semidirect

Maximal subgroups

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

subgroup	index
#11927262	2
#11927196	2
#11926893	2
#11926888	2
#11926823	2
#11926786	2
#11926783	2
#11926753	2
#11926709	2
#11926708	2
#11926631	2
(C₂²⋊D₁₉)⋊C₈	2
#11926539	2
#11926538	2
C₁₉×#485	2
#3302	19