Subgroups — #14575061

Order 2432.

334 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927156	C₂	1	1	yes	maximal, characteristic, complemented
2	#11926851	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926844	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926836	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926826	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926800	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926788	C₂	1	1	yes	maximal, characteristic
8	#11926783	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926771	C₂	1	1	yes	maximal, characteristic
10	#11926763	C₂	1	1	yes	maximal, characteristic, complemented
11	#11926763	C₂	1	1	yes	maximal, characteristic, complemented
12	#11926754	C₂	1	1	yes	maximal, characteristic, complemented
13	#11926749	C₂	1	1	yes	maximal, characteristic, complemented
14	#11926571	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₉×#542	C₂	1	1	yes	maximal, characteristic
16	C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₁₉×#126	—	2	1	no	—
18	C₁₉×#124	—	2	1	no	—
19	C₁₉×#124	C₂²	1	1	yes	characteristic
20	C₁₉×#123	—	2	1	no	—
21	C₁₉×C₂²⋊Q₈	—	2	1	no	—
22	C₁₉×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
23	C₁₉×#121	C₂²	1	1	yes	characteristic
24	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
26	C₁₉×#117	C₂²	1	1	yes	characteristic
27	C₂×#64528	C₂²	1	1	yes	characteristic
28	C₂×#64528	C₂²	1	1	yes	characteristic
29	C₂×#64526	C₂²	1	1	yes	characteristic
30	C₂×#64526	C₂²	1	1	yes	characteristic
31	#10597370	C₂²	1	1	yes	characteristic
32	#10597370	C₂²	1	1	yes	characteristic
33	C₂×#64520	C₂²	1	1	yes	characteristic
34	C₂²×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
35	#10597335	—	2	1	no	—
36	#10597335	—	2	1	no	—
37	#10597325	—	2	1	no	—
38	#10597325	—	2	1	no	—
39	#10597316	C₂²	1	1	yes	characteristic
40	#10597315	C₂²	1	1	yes	characteristic
41	C₄×Dic₃₈	C₂²	1	1	yes	characteristic
42	C₄×Dic₃₈	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	#10597267	C₂²	1	1	yes	characteristic
48	#10597267	C₂²	1	1	yes	characteristic
49	#10597267	C₂²	1	1	yes	characteristic
50	#10597267	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	C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
58	C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₉	—	2	1	no	—
60	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₁₉	—	2	1	no	—
62	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	—	2	1	no	—
64	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
65	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
66	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
67	C₄⋊C₄×C₁₉	—	2	1	no	—
68	C₄⋊C₄×C₁₉	—	2	1	no	—
69	C₄⋊C₄×C₁₉	—	2	1	no	—
70	C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
71	C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
72	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
74	C₂²⋊C₄×C₁₉	—	2	1	no	—
75	C₂²⋊C₄×C₁₉	—	2	1	no	—
76	C₂²⋊C₄×C₁₉	—	2	1	no	—
77	C₂²⋊C₄×C₁₉	—	2	1	no	—
78	C₂²⋊C₄×C₁₉	—	2	1	no	—
79	C₂²⋊C₄×C₁₉	D₄	1	1	yes	characteristic
80	C₂²⋊C₄×C₁₉	D₄	1	1	yes	characteristic
81	C₄²×C₁₉	—	2	1	no	—
82	C₄²×C₁₉	C₂³	1	1	yes	characteristic
83	C₄²×C₁₉	C₂³	1	1	yes	characteristic
84	C₂²⋊Dic₁₉	—	2	1	no	—
85	C₂²⋊Dic₁₉	—	2	1	no	—
86	#64528	—	4	1	no	—
87	#64528	—	4	1	no	—
88	#64526	—	4	1	no	—
89	#64526	—	4	1	no	—
90	C₄⋊Dic₁₉	—	2	1	no	—
91	C₄⋊Dic₁₉	—	2	1	no	—
92	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
93	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
94	Dic₁₉⋊C₄	—	2	1	no	—
95	Dic₁₉⋊C₄	—	2	1	no	—
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	C₄×Dic₁₉	—	2	1	no	—
101	C₄×Dic₁₉	—	2	1	no	—
102	#64520	—	4	1	no	—
103	C₂×C₁₉⋊C₈	—	4	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	Q₈×C₁₉	—	2	1	no	—
111	Q₈×C₁₉	—	2	1	no	—
112	Q₈×C₁₉	—	8	1	no	—
113	D₄×C₁₉	—	2	1	no	—
114	D₄×C₁₉	—	2	1	no	—
115	D₄×C₁₉	—	8	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	—	8	1	no	—
122	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
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	C₂×Dic₁₉	—	2	1	no	—
129	C₂×Dic₁₉	—	2	1	no	—
130	C₂×Dic₁₉	—	2	1	no	—
131	C₂×Dic₁₉	—	2	1	no	—
132	Dic₃₈	—	2	1	no	—
133	Dic₃₈	—	2	1	no	—
134	Dic₃₈	—	2	1	no	—
135	Dic₃₈	—	2	1	no	—
136	C₁₉⋊C₈	—	2	1	no	—
137	C₁₉⋊C₈	—	2	1	no	—
138	C₁₉⋊C₈	—	2	1	no	—
139	C₁₉⋊C₈	—	2	1	no	—
140	#3258	—	19	1	no	maximal, Sylow
141	C₂²×C₁₉	—	2	1	no	—
142	C₂²×C₁₉	—	2	1	no	—
143	C₂²×C₁₉	—	2	1	no	—
144	C₂²×C₁₉	—	4	1	no	—
145	C₂²×C₁₉	—	4	1	no	—
146	C₂²×C₁₉	—	4	1	no	—
147	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
148	C₄×C₁₉	—	2	1	no	—
149	C₄×C₁₉	—	2	1	no	—
150	C₄×C₁₉	—	2	1	no	—
151	C₄×C₁₉	—	2	1	no	—
152	C₄×C₁₉	—	2	1	no	—
153	C₄×C₁₉	—	4	1	no	—
154	C₄×C₁₉	—	4	1	no	—
155	C₄×C₁₉	—	4	1	no	—
156	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
157	C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
158	Dic₁₉	—	4	1	no	—
159	Dic₁₉	—	4	1	no	—
160	Dic₁₉	—	4	1	no	—
161	Dic₁₉	—	4	1	no	—
162	#542	D₁₉	1	1	yes	characteristic
163	#541	—	19	1	no	—
164	#479	—	19	1	no	—
165	#478	—	19	1	no	—
166	#477	—	19	1	no	—
167	#475	—	19	1	no	—
168	#470	—	19	1	no	—
169	#468	—	19	1	no	—
170	#467	—	19	1	no	—
171	#465	—	19	1	no	—
172	#464	—	19	1	no	—
173	#464	—	19	1	no	—
174	#462	—	19	1	no	—
175	#461	—	19	1	no	—
176	#424	—	19	1	no	—
177	C₂×C₁₉	—	4	1	no	—
178	C₂×C₁₉	—	8	1	no	—
179	C₂×C₁₉	#537	1	1	yes	characteristic
180	C₂×C₁₉	#577	1	1	yes	characteristic
181	C₂×C₁₉	#578	1	1	yes	characteristic
182	C₂×Q₁₆	—	19	1	no	—
183	C₂×Q₁₆	—	19	1	no	—
184	C₂×SD₁₆	—	19	1	no	—
185	C₂×SD₁₆	—	19	1	no	—
186	C₂×M₁₆	—	19	1	no	—
187	C₂²×C₈	—	19	1	no	—
188	C₄⋊Q₈	—	19	1	no	—
189	C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
190	#126	—	2	1	no	—
191	#126	—	38	1	no	—
192	#126	—	38	1	no	—
193	#125	—	19	1	no	—
194	#125	—	38	1	no	—
195	#125	—	38	1	no	—
196	#124	—	2	1	no	—
197	#124	C₂×D₁₉	1	1	yes	characteristic
198	#123	—	2	1	no	—
199	C₂²⋊Q₈	—	2	1	no	—
200	C₂²⋊Q₈	—	19	1	no	—
201	C₂²⋊Q₈	—	19	1	no	—
202	C₂²⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
203	#121	C₂×D₁₉	1	1	yes	characteristic
204	C₄×Q₈	—	19	1	no	—
205	C₄×Q₈	—	19	1	no	—
206	C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
207	C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
208	#117	C₂×D₁₉	1	1	yes	characteristic
209	#107	—	19	1	no	—
210	#107	—	19	1	no	—
211	#106	—	19	1	no	—
212	#106	—	19	1	no	—
213	C₄⋊C₈	—	19	1	no	—
214	C₄⋊C₈	—	19	1	no	—
215	C₄⋊C₈	—	19	1	no	—
216	C₄⋊C₈	—	19	1	no	—
217	#103	—	19	1	no	—
218	#103	—	19	1	no	—
219	#103	—	19	1	no	—
220	#103	—	19	1	no	—
221	#103	—	19	1	no	—
222	#103	—	19	1	no	—
223	#102	—	19	1	no	—
224	#102	—	19	1	no	—
225	C₁₉	#3258	1	1	yes	characteristic, complemented, Sylow
226	C₂×Q₈	—	2	1	no	—
227	C₂×Q₈	—	19	1	no	—
228	C₂×Q₈	—	19	1	no	—
229	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
230	C₂×D₄	—	2	1	no	—
231	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
234	Q₁₆	—	76	1	no	—
235	Q₁₆	—	76	1	no	—
236	SD₁₆	—	76	1	no	—
237	SD₁₆	—	76	1	no	—
238	M₁₆	—	76	1	no	—
239	C₂×C₈	—	19	1	no	—
240	C₂×C₈	—	19	1	no	—
241	C₂×C₈	—	19	1	no	—
242	C₂×C₈	—	19	1	no	—
243	C₂×C₈	—	76	1	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	19	1	no	—
248	C₄⋊C₄	—	19	1	no	—
249	C₄⋊C₄	—	38	1	no	—
250	C₄⋊C₄	—	38	1	no	—
251	C₄⋊C₄	—	38	1	no	—
252	C₄⋊C₄	—	38	1	no	—
253	C₄⋊C₄	—	38	1	no	—
254	C₄⋊C₄	—	38	1	no	—
255	C₄⋊C₄	—	38	1	no	—
256	C₄⋊C₄	—	38	1	no	—
257	C₄⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
258	C₄⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
259	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
260	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
261	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	2	1	no	—
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	38	1	no	—
267	C₂²⋊C₄	—	38	1	no	—
268	C₂²⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
269	C₂²⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
270	C₄²	—	2	1	no	—
271	C₄²	—	38	1	no	—
272	C₄²	—	38	1	no	—
273	C₄²	C₂²×D₁₉	1	1	yes	characteristic
274	C₄²	C₂²×D₁₉	1	1	yes	characteristic
275	C₂³	—	2	1	no	—
276	C₂³	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
277	Q₈	—	2	1	no	—
278	Q₈	—	2	1	no	—
279	Q₈	—	8	1	no	—
280	Q₈	—	38	1	no	—
281	Q₈	—	38	1	no	—
282	Q₈	—	38	1	no	—
283	Q₈	—	38	1	no	—
284	D₄	—	2	1	no	—
285	D₄	—	2	1	no	—
286	D₄	—	8	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	8	1	no	—
293	C₂×C₄	—	38	1	no	—
294	C₂×C₄	—	38	1	no	—
295	C₂×C₄	—	38	1	no	—
296	C₂×C₄	—	38	1	no	—
297	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
298	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
299	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
300	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
301	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
302	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
303	C₈	—	38	1	no	—
304	C₈	—	38	1	no	—
305	C₈	—	38	1	no	—
306	C₈	—	38	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	C₂²×C₂²⋊D₁₉	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	76	1	no	—
323	C₄	—	76	1	no	—
324	C₄	—	76	1	no	—
325	C₄	—	76	1	no	—
326	C₄	#10597435	1	1	yes	characteristic
327	C₄	#10597436	1	1	yes	characteristic
328	C₂	—	4	1	no	—
329	C₂	—	8	1	no	—
330	C₂	#11927122	1	1	yes	characteristic
331	C₂	#11927355	1	1	yes	characteristic
332	C₂	#11927359	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×#542	C₂
socle	C₂²×C₁₉	C₂²×D₄
prime radical	C₁₉×#121	C₂²
hypercenter	#542	D₁₉
hypocenter	C₁₉	#3258
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #542

Complements

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

N	H ≅ G/N	type
1. #11927156	C₂	semidirect
2. #11926851	C₂	semidirect
3. #11926844	C₂	semidirect
4. #11926836	C₂	semidirect
5. #11926826	C₂	semidirect
6. #11926800	C₂	semidirect
8. #11926783	C₂	semidirect
10. #11926763	C₂	semidirect
11. #11926763	C₂	semidirect
12. #11926754	C₂	semidirect
13. #11926749	C₂	semidirect
14. #11926571	C₂	semidirect
225. C₁₉	#3258	semidirect

Maximal subgroups

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

subgroup	index
#11927156	2
#11926851	2
#11926844	2
#11926836	2
#11926826	2
#11926800	2
#11926788	2
#11926783	2
#11926771	2
#11926763	2
#11926754	2
#11926749	2
#11926571	2
C₁₉×#542	2
#3258	19