Subgroups — #18077055

Order 4864.

356 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14572737	C₂	1	1	yes	maximal, characteristic
2	#14564613	C₂	1	1	yes	maximal, characteristic
3	#14562424	C₂	1	1	yes	maximal, characteristic
4	#14562424	C₂	1	1	yes	maximal, characteristic
5	#14561467	C₂	1	1	yes	maximal, characteristic
6	#14561467	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1544	C₂	1	1	yes	maximal, characteristic
8	#11926675	—	2	1	no	—
9	#11926675	—	2	1	no	—
10	Q₈×C₁₉⋊C₈	—	2	1	no	—
11	Q₈×C₁₉⋊C₈	—	2	1	no	—
12	#11926605	—	2	2	no	—
13	#11926597	C₂²	1	1	yes	characteristic
14	#11926597	C₂²	1	1	yes	characteristic
15	#11926568	C₂²	1	1	yes	characteristic
16	C₄²×Dic₁₉	C₂²	1	1	yes	characteristic
17	#11926303	—	2	2	no	—
18	#11926297	—	2	1	no	—
19	#11926297	—	2	1	no	—
20	#11926295	—	2	1	no	—
21	#11926295	—	2	1	no	—
22	C₁₉×#533	C₂²	1	1	yes	characteristic
23	C₁₉×#431	C₂²	1	1	yes	characteristic
24	C₁₉×#431	C₂²	1	1	yes	characteristic
25	C₁₉×#332	C₄	1	1	yes	characteristic, complemented
26	C₁₉×#332	C₄	1	1	yes	characteristic, complemented
27	C₁₉×#332	C₄	1	1	yes	characteristic, complemented
28	C₁₉×#332	C₄	1	1	yes	characteristic, complemented
29	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
30	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
31	C₁₉×#125	Q₈	1	2	yes	—
32	C₁₉×C₂²⋊Q₈	—	2	1	no	—
33	C₁₉×C₂²⋊Q₈	—	2	1	no	—
34	C₄×Q₈×C₁₉	—	2	1	no	—
35	C₄×Q₈×C₁₉	—	2	1	no	—
36	C₁₉×#117	D₄	1	2	yes	—
37	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
38	C₂×C₄×Dic₁₉	—	2	1	no	—
39	C₂×C₄×Dic₁₉	—	2	1	no	—
40	C₂×C₄×Dic₁₉	—	2	1	no	—
41	C₂×#64520	—	2	1	no	—
42	C₁₉×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
43	C₁₉×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
44	C₁₉×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₁₉×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
46	C₁₉×C₂²⋊C₈	—	2	1	no	—
47	C₁₉×C₂²⋊C₈	—	2	1	no	—
48	C₄×C₈×C₁₉	—	2	1	no	—
49	C₄×C₈×C₁₉	—	2	1	no	—
50	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
51	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
52	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
53	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
54	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
55	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
56	C₄⋊(C₁₉⋊C₈)	D₄	1	1	yes	characteristic
57	C₄⋊(C₁₉⋊C₈)	D₄	1	1	yes	characteristic
58	C₄⋊(C₁₉⋊C₈)	D₄	1	1	yes	characteristic
59	C₄⋊(C₁₉⋊C₈)	D₄	1	1	yes	characteristic
60	#10597260	—	2	2	no	—
61	C₄×C₁₉⋊C₈	—	2	2	no	—
62	C₄×C₁₉⋊C₈	—	2	1	no	—
63	C₄×C₁₉⋊C₈	—	2	1	no	—
64	C₂×Q₈×C₁₉	—	2	1	no	—
65	C₂×Q₈×C₁₉	—	2	1	no	—
66	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
69	C₂²×Dic₁₉	—	4	1	no	—
70	C₂×C₈×C₁₉	—	2	1	no	—
71	C₂×C₈×C₁₉	—	2	1	no	—
72	C₂×C₈×C₁₉	—	2	1	no	—
73	C₂×C₈×C₁₉	—	2	1	no	—
74	C₄⋊C₄×C₁₉	—	2	1	no	—
75	C₄⋊C₄×C₁₉	—	2	1	no	—
76	C₄⋊C₄×C₁₉	—	2	2	no	—
77	C₄⋊C₄×C₁₉	—	2	2	no	—
78	C₄⋊C₄×C₁₉	C₄⋊C₄	1	2	yes	—
79	C₄⋊C₄×C₁₉	C₄⋊C₄	1	2	yes	—
80	C₂²⋊C₄×C₁₉	—	2	2	no	—
81	C₄²×C₁₉	—	2	2	no	—
82	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
86	C₄×Dic₁₉	—	2	2	no	—
87	C₄×Dic₁₉	—	2	2	no	—
88	C₄×Dic₁₉	—	2	2	no	—
89	C₄×Dic₁₉	—	4	1	no	—
90	C₄×Dic₁₉	—	4	1	no	—
91	C₄×Dic₁₉	—	4	1	no	—
92	#64520	—	4	2	no	—
93	C₂×C₁₉⋊C₈	—	2	1	no	—
94	C₂×C₁₉⋊C₈	—	2	1	no	—
95	C₂×C₁₉⋊C₈	—	2	1	no	—
96	C₂×C₁₉⋊C₈	—	2	1	no	—
97	C₂×C₁₉⋊C₈	—	4	1	no	—
98	#8936	—	19	1	no	maximal, Sylow
99	C₂³×C₁₉	#123	1	1	yes	characteristic
100	Q₈×C₁₉	—	4	1	no	—
101	Q₈×C₁₉	—	4	1	no	—
102	Q₈×C₁₉	—	4	1	no	—
103	Q₈×C₁₉	—	4	1	no	—
104	C₂×C₄×C₁₉	—	2	1	no	—
105	C₂×C₄×C₁₉	—	2	1	no	—
106	C₂×C₄×C₁₉	—	2	1	no	—
107	C₂×C₄×C₁₉	—	2	1	no	—
108	C₂×C₄×C₁₉	—	2	1	no	—
109	C₂×C₄×C₁₉	—	2	1	no	—
110	C₂×C₄×C₁₉	—	2	2	no	—
111	C₂×C₄×C₁₉	—	2	2	no	—
112	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×Dic₁₉	—	4	2	no	—
119	C₂×Dic₁₉	—	4	2	no	—
120	C₂×Dic₁₉	—	4	2	no	—
121	C₈×C₁₉	—	4	1	no	—
122	C₈×C₁₉	—	4	1	no	—
123	C₈×C₁₉	—	4	1	no	—
124	C₈×C₁₉	—	4	1	no	—
125	C₁₉⋊C₈	—	4	1	no	—
126	C₁₉⋊C₈	—	4	1	no	—
127	C₁₉⋊C₈	—	4	1	no	—
128	C₁₉⋊C₈	—	4	1	no	—
129	C₁₉⋊C₈	—	8	1	no	—
130	#2986	—	19	1	no	—
131	#1994	—	19	1	no	—
132	#1574	—	19	1	no	—
133	#1574	—	19	1	no	—
134	#1544	D₁₉	1	1	yes	characteristic
135	#1276	—	19	1	no	—
136	#1276	—	19	1	no	—
137	C₂²×C₁₉	—	2	1	no	—
138	C₂²×C₁₉	—	2	1	no	—
139	C₂²×C₁₉	—	2	1	no	—
140	C₂²×C₁₉	#385	1	1	yes	characteristic
141	C₄×C₁₉	—	2	1	no	—
142	C₄×C₁₉	—	2	1	no	—
143	C₄×C₁₉	—	2	1	no	—
144	C₄×C₁₉	—	4	2	no	—
145	C₄×C₁₉	—	4	2	no	—
146	C₄×C₁₉	#443	1	1	yes	characteristic
147	C₄×C₁₉	#443	1	1	yes	characteristic
148	C₄×C₁₉	#444	1	1	yes	characteristic
149	C₄×C₁₉	#444	1	1	yes	characteristic
150	C₄×C₁₉	#454	1	1	yes	characteristic
151	C₄×C₁₉	#456	1	1	yes	characteristic
152	Dic₁₉	—	4	4	no	—
153	#533	C₂×D₁₉	1	1	yes	characteristic
154	#446	—	38	1	no	—
155	#446	—	38	1	no	—
156	C₈×Q₈	—	38	1	no	—
157	C₈×Q₈	—	38	1	no	—
158	#433	—	38	2	no	—
159	#431	—	19	1	no	—
160	#431	—	19	1	no	—
161	#431	C₂×D₁₉	1	1	yes	characteristic
162	#431	C₂×D₁₉	1	1	yes	characteristic
163	#423	—	19	1	no	—
164	C₄³	—	19	1	no	—
165	#339	—	38	2	no	—
166	#335	—	38	1	no	—
167	#335	—	38	1	no	—
168	#334	—	38	1	no	—
169	#334	—	38	1	no	—
170	#332	Dic₁₉	1	1	yes	characteristic, complemented
171	#332	Dic₁₉	1	1	yes	characteristic, complemented
172	#332	Dic₁₉	1	1	yes	characteristic, complemented
173	#332	Dic₁₉	1	1	yes	characteristic, complemented
174	C₂×C₁₉	—	4	1	no	—
175	C₂×C₁₉	#1857	1	1	yes	characteristic
176	C₂×C₁₉	#1866	1	1	yes	characteristic
177	C₂×C₁₉	#1866	1	1	yes	characteristic
178	C₂×M₁₆	—	38	1	no	—
179	C₄⋊Q₈	C₂×Dic₁₉	1	1	yes	characteristic
180	C₄⋊Q₈	C₂×Dic₁₉	1	1	yes	characteristic
181	#125	Dic₃₈	1	2	yes	—
182	C₂²⋊Q₈	—	2	1	no	—
183	C₂²⋊Q₈	—	2	1	no	—
184	C₄×Q₈	—	2	1	no	—
185	C₄×Q₈	—	2	1	no	—
186	#117	D₇₆	1	2	yes	—
187	C₂×C₄²	—	38	1	no	—
188	C₂×C₄²	—	38	1	no	—
189	C₂×C₄²	—	38	1	no	—
190	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
191	C₄⋊C₈	—	19	1	no	—
192	C₄⋊C₈	—	19	1	no	—
193	C₄⋊C₈	—	19	1	no	—
194	C₄⋊C₈	—	19	1	no	—
195	C₄⋊C₈	—	38	2	no	—
196	C₄⋊C₈	—	38	2	no	—
197	C₄⋊C₈	—	38	1	no	—
198	C₄⋊C₈	—	38	1	no	—
199	C₄⋊C₈	C₂×Dic₁₉	1	1	yes	characteristic
200	C₄⋊C₈	C₂×Dic₁₉	1	1	yes	characteristic
201	C₄⋊C₈	C₂×Dic₁₉	1	1	yes	characteristic
202	C₄⋊C₈	C₂×Dic₁₉	1	1	yes	characteristic
203	C₂²⋊C₈	—	2	1	no	—
204	C₂²⋊C₈	—	2	1	no	—
205	C₂²⋊C₈	—	38	1	no	—
206	C₂²⋊C₈	—	38	1	no	—
207	#97	—	38	2	no	—
208	C₄×C₈	—	2	1	no	—
209	C₄×C₈	—	2	1	no	—
210	C₄×C₈	—	38	2	no	—
211	C₄×C₈	—	38	1	no	—
212	C₄×C₈	—	38	1	no	—
213	C₁₉	#8936	1	1	yes	characteristic, complemented, Sylow
214	C₂×Q₈	—	2	1	no	—
215	C₂×Q₈	—	2	1	no	—
216	C₂²×C₄	—	76	1	no	—
217	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
218	C₂²×C₄	#64543	1	1	yes	characteristic
219	C₂²×C₄	#64543	1	1	yes	characteristic
220	M₁₆	—	76	2	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	38	1	no	—
226	C₂×C₈	—	38	1	no	—
227	C₂×C₈	—	38	1	no	—
228	C₂×C₈	—	38	1	no	—
229	C₂×C₈	—	76	1	no	—
230	C₄⋊C₄	—	2	1	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	C₄⋊Dic₁₉	1	2	yes	—
235	C₄⋊C₄	C₄⋊Dic₁₉	1	2	yes	—
236	C₂²⋊C₄	—	2	2	no	—
237	C₄²	—	2	2	no	—
238	C₄²	—	38	2	no	—
239	C₄²	—	38	2	no	—
240	C₄²	—	38	2	no	—
241	C₄²	—	76	1	no	—
242	C₄²	—	76	1	no	—
243	C₄²	—	76	1	no	—
244	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
245	C₄²	D₄×D₁₉	1	1	yes	characteristic
246	C₄²	D₄×D₁₉	1	1	yes	characteristic
247	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
248	C₂³	#10597332	1	1	yes	characteristic
249	Q₈	—	4	1	no	—
250	Q₈	—	4	1	no	—
251	Q₈	—	4	1	no	—
252	Q₈	—	4	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	76	2	no	—
262	C₂×C₄	—	76	2	no	—
263	C₂×C₄	—	76	2	no	—
264	C₂×C₄	#10597324	1	1	yes	characteristic
265	C₂×C₄	#10597324	1	1	yes	characteristic
266	C₂×C₄	#10597328	1	1	yes	characteristic
267	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
268	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
269	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
270	C₈	—	4	1	no	—
271	C₈	—	4	1	no	—
272	C₈	—	4	1	no	—
273	C₈	—	4	1	no	—
274	C₈	—	76	1	no	—
275	C₈	—	76	1	no	—
276	C₈	—	76	1	no	—
277	C₈	—	76	1	no	—
278	C₈	—	152	1	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	#11926419	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	4	2	no	—
287	C₄	—	4	2	no	—
288	C₄	—	76	4	no	—
289	C₄	#11926661	1	1	yes	characteristic
290	C₄	#11926661	1	1	yes	characteristic
291	C₄	#11926666	1	1	yes	characteristic
292	C₄	#11926666	1	1	yes	characteristic
293	C₄	#11926719	1	1	yes	characteristic
294	C₄	#11926729	1	1	yes	characteristic
295	C₂	—	4	1	no	—
296	C₂	#14563852	1	1	yes	characteristic
297	C₂	#14563912	1	1	yes	characteristic
298	C₂	#14563912	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597328
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1544	C₂
socle	C₂²×C₁₉	#385
prime radical	C₂³×C₁₉	#123
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1544	D₁₉
hypocenter	C₁₉	#8936
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1544

Complements

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

N	H ≅ G/N	type
25. C₁₉×#332	C₄	semidirect
26. C₁₉×#332	C₄	semidirect
27. C₁₉×#332	C₄	semidirect
28. C₁₉×#332	C₄	semidirect
170. #332	Dic₁₉	semidirect
171. #332	Dic₁₉	semidirect
172. #332	Dic₁₉	semidirect
173. #332	Dic₁₉	semidirect
213. C₁₉	#8936	semidirect

Maximal subgroups

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

subgroup	index
#14572737	2
#14564613	2
#14562424	2
#14561467	2
C₁₉×#1544	2
#8936	19