Subgroups — #14576422

Order 2432.

384 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927276	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Dic₃₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926905	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926903	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926889	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926835	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926832	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926827	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926761	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926737	C₂	1	1	yes	maximal, characteristic
11	#11926673	C₂	1	1	yes	maximal, characteristic, complemented
12	#11926656	C₂	1	1	yes	maximal, characteristic, complemented
13	#11926654	C₂	1	1	yes	maximal, characteristic, complemented
14	SD₁₆×Dic₁₉	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₉×#463	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Dic₃₈	—	2	1	no	—
17	C₂×SD₁₆×C₁₉	C₂²	1	1	yes	characteristic
18	C₂×D₈×C₁₉	C₂²	1	1	yes	characteristic
19	C₁₉×#124	C₂²	1	1	yes	characteristic
20	C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
21	#10597393	C₂²	1	1	yes	characteristic
22	Q₈×Dic₁₉	C₂²	1	1	yes	characteristic
23	#10597386	C₂²	1	1	yes	characteristic
24	D₄×Dic₁₉	C₂²	1	1	yes	characteristic
25	D₄×Dic₁₉	C₂²	1	1	yes	characteristic
26	D₄×Dic₁₉	C₂²	1	1	yes	characteristic
27	C₂×#64525	C₂²	1	1	yes	characteristic
28	#10597370	—	2	1	no	—
29	#10597334	C₂²	1	1	yes	characteristic
30	#10597334	C₂²	1	1	yes	characteristic
31	#10597332	—	2	1	no	—
32	#10597331	—	2	1	no	—
33	#10597329	—	2	1	no	—
34	(C₂²⋊D₁₉)⋊C₄	—	2	1	no	—
35	#10597324	—	2	1	no	—
36	#10597324	—	2	1	no	—
37	C₄×D₇₆	C₂²	1	1	yes	characteristic
38	#10597315	C₂²	1	1	yes	characteristic
39	C₄×Dic₃₈	C₂²	1	1	yes	characteristic
40	C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
41	C₁₉×#103	C₂²	1	1	yes	characteristic
42	C₁₉×#102	C₂²	1	1	yes	characteristic
43	#10597292	C₂²	1	1	yes	characteristic
44	#10597292	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	#10597278	C₂²	1	1	yes	characteristic
49	#10597278	C₂²	1	1	yes	characteristic
50	#10597274	C₂²	1	1	yes	characteristic
51	#10597274	C₂²	1	1	yes	characteristic
52	#10597272	C₂²	1	1	yes	characteristic
53	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
54	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
55	C₈×Dic₁₉	C₂²	1	1	yes	characteristic
56	#10597266	C₂²	1	1	yes	characteristic
57	#10597265	C₂²	1	1	yes	characteristic
58	#10597260	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	—	2	1	no	—
63	C₂×C₂²⋊D₁₉	—	2	1	no	—
64	C₂×C₂²⋊D₁₉	—	2	1	no	—
65	C₂²×Dic₁₉	—	2	1	no	—
66	C₂²×Dic₁₉	—	2	1	no	—
67	C₂²×Dic₁₉	—	2	1	no	—
68	C₂×D₇₆	C₂³	1	1	yes	characteristic
69	C₂×C₄×D₁₉	—	2	1	no	—
70	C₂×Dic₃₈	—	2	1	no	—
71	C₂×Dic₃₈	—	2	1	no	—
72	C₂×Dic₃₈	—	2	1	no	—
73	C₂×Dic₃₈	—	4	4	no	—
74	C₂×Dic₃₈	D₄	1	1	yes	characteristic
75	SD₁₆×C₁₉	—	2	2	no	—
76	D₈×C₁₉	—	2	2	no	—
77	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
78	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
80	C₂²⋊C₄×C₁₉	—	2	1	no	—
81	C₂²⋊C₄×C₁₉	—	2	1	no	—
82	C₂²⋊C₄×C₁₉	—	2	1	no	—
83	C₄²×C₁₉	C₂³	1	1	yes	characteristic
84	C₂²⋊Dic₁₉	—	2	1	no	—
85	C₂²⋊Dic₁₉	—	2	1	no	—
86	C₂²⋊Dic₁₉	—	2	1	no	—
87	#64525	—	2	2	no	—
88	(C₂×D₁₉)⋊C₄	—	2	1	no	—
89	(C₂×D₁₉)⋊C₄	—	2	1	no	—
90	(C₂×D₁₉)⋊C₄	—	2	1	no	—
91	C₄⋊Dic₁₉	—	2	1	no	—
92	C₄⋊Dic₁₉	—	2	1	no	—
93	C₄⋊Dic₁₉	D₄	1	1	yes	characteristic
94	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
95	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
96	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
97	Dic₁₉⋊C₄	—	2	1	no	—
98	Dic₁₉⋊C₄	—	2	1	no	—
99	Dic₁₉⋊C₄	D₄	1	1	yes	characteristic
100	Dic₁₉⋊C₄	D₄	1	1	yes	characteristic
101	C₄×Dic₁₉	—	2	1	no	—
102	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
103	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
104	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
105	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
106	C₂³×C₁₉	—	2	1	no	—
107	C₂³×C₁₉	—	2	1	no	—
108	C₂²×D₁₉	—	2	1	no	—
109	Q₈×C₁₉	—	2	1	no	—
110	Q₈×C₁₉	—	2	1	no	—
111	D₄×C₁₉	—	2	1	no	—
112	D₄×C₁₉	—	2	1	no	—
113	D₄×C₁₉	—	2	1	no	—
114	D₄×C₁₉	#41	1	2	yes	—
115	C₂×C₄×C₁₉	—	2	1	no	—
116	C₂×C₄×C₁₉	—	2	1	no	—
117	C₂×C₄×C₁₉	—	4	2	no	—
118	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
121	C₂²⋊D₁₉	—	4	2	no	—
122	C₂²⋊D₁₉	—	8	1	no	—
123	C₂×Dic₁₉	—	2	1	no	—
124	C₂×Dic₁₉	—	2	1	no	—
125	C₂×Dic₁₉	—	2	1	no	—
126	C₂×Dic₁₉	—	4	2	no	—
127	C₂×Dic₁₉	—	4	2	no	—
128	C₂×Dic₁₉	—	8	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	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
133	D₇₆	—	2	1	no	—
134	D₇₆	—	2	1	no	—
135	C₄×D₁₉	—	8	1	no	—
136	Dic₃₈	—	2	2	no	—
137	Dic₃₈	—	4	2	no	—
138	Dic₃₈	—	4	2	no	—
139	Dic₃₈	—	4	2	no	—
140	C₈×C₁₉	—	2	1	no	—
141	C₈×C₁₉	—	2	1	no	—
142	C₁₉⋊C₈	—	2	1	no	—
143	C₁₉⋊C₈	—	2	1	no	—
144	#3373	—	19	1	no	maximal, Sylow
145	C₂²×C₁₉	—	2	2	no	—
146	C₂²×C₁₉	—	4	1	no	—
147	C₂²×C₁₉	—	4	1	no	—
148	C₂²×C₁₉	—	4	1	no	—
149	C₂²×C₁₉	—	4	1	no	—
150	C₂²×C₁₉	—	4	1	no	—
151	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
152	C₂×D₁₉	—	4	1	no	—
153	C₂×D₁₉	—	4	1	no	—
154	C₂×D₁₉	—	4	1	no	—
155	C₄×C₁₉	—	2	1	no	—
156	C₄×C₁₉	—	2	1	no	—
157	C₄×C₁₉	—	4	1	no	—
158	C₄×C₁₉	—	4	1	no	—
159	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
160	C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
161	Dic₁₉	—	2	1	no	—
162	Dic₁₉	—	2	1	no	—
163	Dic₁₉	—	2	1	no	—
164	Dic₁₉	—	4	1	no	—
165	Dic₁₉	—	4	1	no	—
166	Dic₁₉	—	4	1	no	—
167	Dic₁₉	AGL(1,ℤ/8)	1	2	yes	—
168	#559	—	19	1	no	—
169	D₄×Q₈	—	19	1	no	—
170	#489	—	19	1	no	—
171	#489	—	19	1	no	—
172	#486	—	19	1	no	—
173	#476	—	19	1	no	—
174	#476	—	19	1	no	—
175	#475	—	19	1	no	—
176	#463	—	19	1	no	—
177	#463	D₁₉	1	1	yes	characteristic, complemented
178	#459	—	19	1	no	—
179	#446	—	19	1	no	—
180	#442	—	19	1	no	—
181	#442	—	19	1	no	—
182	C₄×SD₁₆	—	19	1	no	—
183	C₂×C₁₉	—	4	2	no	—
184	C₂×C₁₉	—	8	1	no	—
185	C₂×C₁₉	#548	1	1	yes	characteristic
186	C₂×C₁₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
187	C₂×C₁₉	#577	1	1	yes	characteristic
188	D₁₉	—	8	1	no	—
189	C₂²×Q₈	—	38	1	no	—
190	C₂×SD₁₆	C₂×D₁₉	1	1	yes	characteristic
191	C₂×D₈	—	19	1	no	—
192	C₂×D₈	C₂×D₁₉	1	1	yes	characteristic
193	C₄⋊Q₈	—	19	1	no	—
194	C₄⋊Q₈	—	19	1	no	—
195	C₄⋊Q₈	—	19	1	no	—
196	#127	—	19	1	no	—
197	#124	—	19	1	no	—
198	#124	—	38	1	no	—
199	#124	C₂×D₁₉	1	1	yes	characteristic
200	#123	—	38	1	no	—
201	#123	—	38	1	no	—
202	C₂²⋊Q₈	—	38	1	no	—
203	C₂²⋊Q₈	—	38	1	no	—
204	C₂²⋊Q₈	—	38	1	no	—
205	C₄×Q₈	—	19	1	no	—
206	C₄×Q₈	—	19	1	no	—
207	C₄×D₄	—	19	1	no	—
208	C₄×D₄	—	19	1	no	—
209	C₄×D₄	—	19	1	no	—
210	C₄×D₄	—	19	1	no	—
211	C₄×D₄	—	38	1	no	—
212	C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
213	#106	—	19	1	no	—
214	#106	—	19	1	no	—
215	#106	—	19	1	no	—
216	C₄⋊C₈	—	19	1	no	—
217	C₄⋊C₈	—	19	1	no	—
218	C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
219	#103	—	19	1	no	—
220	#103	—	19	1	no	—
221	#103	C₂×D₁₉	1	1	yes	characteristic
222	#102	—	19	1	no	—
223	#102	—	19	1	no	—
224	#102	—	19	1	no	—
225	#102	—	19	1	no	—
226	#102	—	19	1	no	—
227	#102	—	19	1	no	—
228	#102	C₂×D₁₉	1	1	yes	characteristic
229	#97	—	19	1	no	—
230	#97	—	19	1	no	—
231	C₄×C₈	—	19	1	no	—
232	C₁₉	#3373	1	1	yes	characteristic, complemented, Sylow
233	C₂×Q₈	—	19	1	no	—
234	C₂×Q₈	—	38	1	no	—
235	C₂×Q₈	—	38	1	no	—
236	C₂×Q₈	—	38	1	no	—
237	C₂×Q₈	—	76	4	no	—
238	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
239	C₂×D₄	—	19	1	no	—
240	C₂×D₄	—	38	1	no	—
241	C₂×D₄	—	38	1	no	—
242	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
243	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	38	1	no	—
246	C₂²×C₄	—	38	1	no	—
247	C₂²×C₄	—	38	1	no	—
248	C₂²×C₄	—	38	1	no	—
249	SD₁₆	—	2	2	no	—
250	D₈	—	2	2	no	—
251	D₈	—	38	2	no	—
252	C₂×C₈	—	19	1	no	—
253	C₂×C₈	—	19	1	no	—
254	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
255	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
256	C₄⋊C₄	—	19	1	no	—
257	C₄⋊C₄	—	19	1	no	—
258	C₄⋊C₄	—	19	1	no	—
259	C₄⋊C₄	—	19	1	no	—
260	C₄⋊C₄	—	19	1	no	—
261	C₄⋊C₄	—	19	1	no	—
262	C₄⋊C₄	—	38	1	no	—
263	C₄⋊C₄	—	38	1	no	—
264	C₄⋊C₄	—	38	1	no	—
265	C₄⋊C₄	—	38	1	no	—
266	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	2	1	no	—
269	C₂²⋊C₄	—	2	1	no	—
270	C₂²⋊C₄	—	38	1	no	—
271	C₂²⋊C₄	—	38	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	C₄²	—	19	1	no	—
277	C₄²	—	19	1	no	—
278	C₄²	—	38	1	no	—
279	C₄²	C₂²×D₁₉	1	1	yes	characteristic
280	C₂³	—	2	1	no	—
281	C₂³	—	2	1	no	—
282	C₂³	—	38	1	no	—
283	Q₈	—	2	1	no	—
284	Q₈	—	2	1	no	—
285	Q₈	—	38	2	no	—
286	Q₈	—	76	2	no	—
287	Q₈	—	76	2	no	—
288	Q₈	—	76	2	no	—
289	D₄	—	2	1	no	—
290	D₄	—	2	1	no	—
291	D₄	—	2	1	no	—
292	D₄	—	38	1	no	—
293	D₄	—	38	1	no	—
294	D₄	—	76	2	no	—
295	D₄	—	152	1	no	—
296	D₄	#64543	1	2	yes	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	19	1	no	—
301	C₂×C₄	—	19	1	no	—
302	C₂×C₄	—	19	1	no	—
303	C₂×C₄	—	19	1	no	—
304	C₂×C₄	—	38	1	no	—
305	C₂×C₄	—	38	1	no	—
306	C₂×C₄	—	38	1	no	—
307	C₂×C₄	—	76	2	no	—
308	C₂×C₄	—	76	2	no	—
309	C₂×C₄	—	152	1	no	—
310	C₂×C₄	—	152	1	no	—
311	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
312	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
313	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
314	C₈	—	2	1	no	—
315	C₈	—	2	1	no	—
316	C₈	—	38	1	no	—
317	C₈	—	38	1	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	4	1	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	76	1	no	—
325	C₂²	—	76	1	no	—
326	C₂²	—	76	1	no	—
327	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	19	2	no	—
333	C₄	—	38	1	no	—
334	C₄	—	38	1	no	—
335	C₄	—	38	1	no	—
336	C₄	—	76	1	no	—
337	C₄	—	76	1	no	—
338	C₄	—	76	1	no	—
339	C₄	C₂×#64543	1	1	yes	characteristic
340	C₄	#10597436	1	1	yes	characteristic
341	C₂	—	4	2	no	—
342	C₂	—	8	1	no	—
343	C₂	—	152	1	no	—
344	C₂	#11927200	1	1	yes	characteristic
345	C₂	AGL(1,ℤ/8)×D₁₉	1	1	yes	characteristic
346	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×#463	C₂
socle	C₂²×C₁₉	C₂²×D₄
prime radical	#11926737	C₂
hypercenter	#463	D₁₉
hypocenter	C₁₉	#3373
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #463

Complements

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

N	H ≅ G/N	type
1. #11927276	C₂	semidirect
2. D₄×Dic₃₈	C₂	semidirect
3. #11926905	C₂	semidirect
4. #11926903	C₂	semidirect
5. #11926889	C₂	semidirect
6. #11926835	C₂	semidirect
7. #11926832	C₂	semidirect
8. #11926827	C₂	semidirect
9. #11926761	C₂	semidirect
11. #11926673	C₂	semidirect
12. #11926656	C₂	semidirect
13. #11926654	C₂	semidirect
14. SD₁₆×Dic₁₉	C₂	semidirect
15. C₁₉×#463	C₂	semidirect
177. #463	D₁₉	semidirect
232. C₁₉	#3373	semidirect

Maximal subgroups

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

subgroup	index
#11927276	2
D₄×Dic₃₈	2
#11926905	2
#11926903	2
#11926889	2
#11926835	2
#11926832	2
#11926827	2
#11926761	2
#11926737	2
#11926673	2
#11926656	2
#11926654	2
SD₁₆×Dic₁₉	2
C₁₉×#463	2
#3373	19