Subgroups — #18073565

Order 4864.

444 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14572347	C₂	1	1	yes	maximal, characteristic
2	#14562297	C₂	1	1	yes	maximal, characteristic
3	#14561922	C₂	1	1	yes	maximal, characteristic, complemented
4	M₁₆×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#14561525	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561525	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1525	C₂	1	1	yes	maximal, characteristic
8	C₂×#10597394	C₄	1	2	yes	—
9	#11926622	—	2	1	no	—
10	#11926622	—	2	1	no	—
11	D₄×C₁₉⋊C₈	—	2	1	no	—
12	D₄×C₁₉⋊C₈	—	2	1	no	—
13	#11926597	C₂²	1	1	yes	characteristic
14	#11926597	C₂²	1	1	yes	characteristic
15	#11926568	C₂²	1	1	yes	characteristic
16	#11926511	C₄	1	2	yes	—
17	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
18	#11926293	C₄	1	2	yes	—
19	#11926288	C₄	1	2	yes	—
20	#11926266	—	2	1	no	—
21	#11926266	—	2	1	no	—
22	#11926265	—	2	1	no	—
23	C₈×C₁₉⋊C₈	—	2	1	no	—
24	C₁₉×#532	C₂²	1	1	yes	characteristic
25	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
26	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
27	C₁₉×#329	C₄	1	4	yes	—
28	C₂×#41×C₁₉	C₂×C₄	1	2	yes	—
29	C₂×M₁₆×C₁₉	—	2	1	no	—
30	C₂×M₁₆×C₁₉	—	2	1	no	—
31	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
32	C₁₉×#127	C₂×C₄	1	1	yes	characteristic
33	C₁₉×#124	C₂×C₄	1	2	yes	—
34	C₁₉×#121	—	2	1	no	—
35	C₁₉×#121	—	2	1	no	—
36	C₄×D₄×C₁₉	—	2	1	no	—
37	C₄×D₄×C₁₉	—	2	1	no	—
38	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
39	#10597394	—	4	4	no	—
40	C₂×#64520	—	2	2	no	—
41	C₂×#64520	C₂×C₄	1	2	yes	—
42	C₂²×C₁₉⋊C₈	—	2	2	no	—
43	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₁₉×#97	C₂×C₄	1	2	yes	—
46	C₁₉×#97	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₁₉	—	2	1	no	—
48	C₄×C₈×C₁₉	—	2	1	no	—
49	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
50	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
51	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
52	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
53	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
54	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
55	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
56	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
57	C₄×C₁₉⋊C₈	—	2	1	no	—
58	C₄×C₁₉⋊C₈	—	2	1	no	—
59	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
60	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
61	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
62	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
63	#41×C₁₉	—	4	2	no	—
64	#41×C₁₉	—	4	2	no	—
65	C₂×Q₈×C₁₉	C₄²	1	2	yes	—
66	C₂×D₄×C₁₉	—	2	2	no	—
67	C₂×D₄×C₁₉	—	2	1	no	—
68	C₂×D₄×C₁₉	—	2	1	no	—
69	C₂×D₄×C₁₉	C₄²	1	2	yes	—
70	C₂²×C₄×C₁₉	—	2	2	no	—
71	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
73	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
74	M₁₆×C₁₉	—	2	4	no	—
75	M₁₆×C₁₉	—	4	2	no	—
76	C₂×C₈×C₁₉	—	2	2	no	—
77	C₂×C₈×C₁₉	—	2	2	no	—
78	C₄⋊C₄×C₁₉	—	2	1	no	—
79	C₄⋊C₄×C₁₉	—	2	1	no	—
80	C₂²⋊C₄×C₁₉	—	2	2	no	—
81	C₂²⋊C₄×C₁₉	—	2	2	no	—
82	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
86	#64520	—	4	4	no	—
87	#64520	—	4	2	no	—
88	C₂×C₁₉⋊C₈	—	2	2	no	—
89	C₂×C₁₉⋊C₈	—	2	2	no	—
90	C₂×C₁₉⋊C₈	—	2	2	no	—
91	C₂×C₁₉⋊C₈	—	2	2	no	—
92	C₂×C₁₉⋊C₈	—	4	1	no	—
93	C₂×C₁₉⋊C₈	—	4	4	no	—
94	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
98	#8245	—	19	1	no	maximal, Sylow
99	C₂³×C₁₉	—	2	2	no	—
100	C₂³×C₁₉	#117	1	1	yes	characteristic
101	Q₈×C₁₉	—	4	2	no	—
102	D₄×C₁₉	—	4	2	no	—
103	D₄×C₁₉	—	4	4	no	—
104	D₄×C₁₉	—	4	2	no	—
105	D₄×C₁₉	—	4	2	no	—
106	C₂×C₄×C₁₉	—	2	1	no	—
107	C₂×C₄×C₁₉	—	2	1	no	—
108	C₂×C₄×C₁₉	—	2	2	no	—
109	C₂×C₄×C₁₉	—	2	2	no	—
110	C₂×C₄×C₁₉	—	2	2	no	—
111	C₂×C₄×C₁₉	—	4	2	no	—
112	C₂×C₄×C₁₉	—	4	2	no	—
113	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
119	C₈×C₁₉	—	2	4	no	—
120	C₈×C₁₉	—	4	2	no	—
121	C₁₉⋊C₈	—	2	1	no	—
122	C₁₉⋊C₈	—	2	1	no	—
123	C₁₉⋊C₈	—	2	1	no	—
124	C₁₉⋊C₈	—	4	2	no	—
125	C₁₉⋊C₈	—	4	2	no	—
126	C₁₉⋊C₈	—	4	2	no	—
127	C₁₉⋊C₈	#100	1	2	yes	—
128	#2931	—	19	1	no	—
129	#1543	—	19	1	no	—
130	#1525	D₁₉	1	1	yes	characteristic
131	#1451	—	19	1	no	—
132	C₈×M₁₆	—	19	1	no	—
133	#1299	—	19	1	no	—
134	#1299	—	19	1	no	—
135	C₂²×C₁₉	—	2	1	no	—
136	C₂²×C₁₉	—	2	1	no	—
137	C₂²×C₁₉	—	2	1	no	—
138	C₂²×C₁₉	—	4	2	no	—
139	C₂²×C₁₉	—	4	2	no	—
140	C₂²×C₁₉	—	4	2	no	—
141	C₂²×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
142	C₄×C₁₉	—	2	1	no	—
143	C₄×C₁₉	—	2	1	no	—
144	C₄×C₁₉	—	2	1	no	—
145	C₄×C₁₉	—	4	2	no	—
146	C₄×C₁₉	C₂×#100	1	1	yes	characteristic
147	C₄×C₁₉	#413	1	1	yes	characteristic
148	C₄×C₁₉	#443	1	2	yes	—
149	C₄×C₁₉	#444	1	2	yes	—
150	C₂×#131	—	19	2	no	—
151	#532	C₂×D₁₉	1	1	yes	characteristic
152	#436	—	38	1	no	—
153	#436	—	38	1	no	—
154	C₈×D₄	—	38	1	no	—
155	C₈×D₄	—	38	1	no	—
156	#431	—	19	1	no	—
157	#431	—	19	1	no	—
158	#423	—	19	1	no	—
159	#408	—	19	2	no	—
160	C₄×M₁₆	C₂×D₁₉	1	1	yes	characteristic
161	C₄×M₁₆	C₂×D₁₉	1	1	yes	characteristic
162	C₂×C₄×C₈	—	19	1	no	—
163	#333	—	19	2	no	—
164	#331	—	19	2	no	—
165	#329	Dic₁₉	1	4	yes	—
166	#322	—	38	1	no	—
167	#322	—	38	1	no	—
168	#322	—	38	1	no	—
169	C₈²	—	38	1	no	—
170	C₂×C₁₉	—	4	1	no	—
171	C₂×C₁₉	—	8	2	no	—
172	C₂×C₁₉	C₄×#100	1	1	yes	characteristic
173	C₂×C₁₉	#1764	1	1	yes	characteristic
174	C₂×C₁₉	#1765	1	1	yes	characteristic
175	C₂×#41	C₄×D₁₉	1	2	yes	—
176	#131	—	76	4	no	—
177	C₂×M₁₆	—	2	1	no	—
178	C₂×M₁₆	—	2	1	no	—
179	C₂×M₁₆	—	19	2	no	—
180	C₂×M₁₆	—	38	2	no	—
181	C₂²×C₈	—	19	1	no	—
182	C₂²×C₈	—	19	1	no	—
183	C₂²×C₈	—	38	2	no	—
184	C₄⋊Q₈	C₄×D₁₉	1	1	yes	characteristic
185	#127	C₄×D₁₉	1	1	yes	characteristic
186	#124	C₂×Dic₁₉	1	2	yes	—
187	#121	—	2	1	no	—
188	#121	—	2	1	no	—
189	C₄×D₄	—	2	1	no	—
190	C₄×D₄	—	2	1	no	—
191	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
192	C₄⋊C₈	—	19	2	no	—
193	C₄⋊C₈	—	19	2	no	—
194	C₄⋊C₈	—	38	1	no	—
195	C₄⋊C₈	—	38	1	no	—
196	C₂²⋊C₈	—	38	2	no	—
197	C₂²⋊C₈	—	38	2	no	—
198	C₂²⋊C₈	—	38	1	no	—
199	C₂²⋊C₈	—	38	1	no	—
200	#97	C₂×Dic₁₉	1	2	yes	—
201	#97	C₂×Dic₁₉	1	2	yes	—
202	C₄×C₈	—	2	1	no	—
203	C₄×C₈	—	2	1	no	—
204	C₄×C₈	—	19	1	no	—
205	C₄×C₈	—	19	1	no	—
206	C₄×C₈	—	19	1	no	—
207	C₄×C₈	—	19	1	no	—
208	C₄×C₈	—	38	1	no	—
209	C₄×C₈	—	38	1	no	—
210	C₁₉	#8245	1	1	yes	characteristic, complemented, Sylow
211	#41	—	4	2	no	—
212	#41	—	4	2	no	—
213	C₂×Q₈	C₄×Dic₁₉	1	2	yes	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	2	1	no	—
216	C₂×D₄	—	2	1	no	—
217	C₂×D₄	C₄×Dic₁₉	1	2	yes	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
220	C₂²×C₄	#64543	1	1	yes	characteristic
221	C₂²×C₄	#64543	1	1	yes	characteristic
222	M₁₆	—	2	4	no	—
223	M₁₆	—	4	2	no	—
224	M₁₆	—	76	4	no	—
225	M₁₆	—	76	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	19	1	no	—
229	C₂×C₈	—	19	1	no	—
230	C₂×C₈	—	19	1	no	—
231	C₂×C₈	—	19	1	no	—
232	C₂×C₈	—	38	2	no	—
233	C₂×C₈	—	38	2	no	—
234	C₂×C₈	—	38	2	no	—
235	C₂×C₈	—	38	2	no	—
236	C₂×C₈	—	76	1	no	—
237	C₂×C₈	—	76	4	no	—
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	—	2	1	no	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₂²⋊C₄	—	2	2	no	—
242	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
243	C₄²	D₄×D₁₉	1	1	yes	characteristic
244	C₄²	D₄×D₁₉	1	1	yes	characteristic
245	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
246	C₂³	—	2	2	no	—
247	C₂³	#10597323	1	1	yes	characteristic
248	Q₈	—	4	2	no	—
249	D₄	—	4	2	no	—
250	D₄	—	4	4	no	—
251	D₄	—	4	2	no	—
252	D₄	—	4	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
261	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
262	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
263	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
264	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
265	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
266	C₈	—	2	4	no	—
267	C₈	—	4	2	no	—
268	C₈	—	19	2	no	—
269	C₈	—	38	1	no	—
270	C₈	—	38	1	no	—
271	C₈	—	38	1	no	—
272	C₈	—	76	2	no	—
273	C₈	—	76	2	no	—
274	C₈	—	76	2	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	C₂²⋊C₄×Dic₁₉	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	4	2	no	—
286	C₄	#100×D₁₉	1	1	yes	characteristic
287	C₄	#11926525	1	1	yes	characteristic
288	C₄	#11926658	1	2	yes	—
289	C₄	#11926663	1	2	yes	—
290	C₂	—	4	1	no	—
291	C₂	—	8	2	no	—
292	C₂	#100×Dic₁₉	1	1	yes	characteristic
293	C₂	#14563473	1	1	yes	characteristic
294	C₂	#14563480	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1525	C₂
socle	C₂²×C₁₉	C₄×C₂²⋊C₄
prime radical	C₁₉×#532	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1525	D₁₉
hypocenter	C₁₉	#8245
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1525

Complements

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

N	H ≅ G/N	type
3. #14561922	C₂	semidirect
4. M₁₆×C₁₉⋊C₈	C₂	semidirect
5. #14561525	C₂	semidirect
6. #14561525	C₂	semidirect
210. C₁₉	#8245	semidirect

Maximal subgroups

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

subgroup	index
#14572347	2
#14562297	2
#14561922	2
M₁₆×C₁₉⋊C₈	2
#14561525	2
C₁₉×#1525	2
#8245	19