Subgroups — #18088457

Order 4864.

714 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10597279	C₂	1	2	yes	maximal
2	C₂×#11926500	C₂	1	1	yes	maximal, characteristic
3	#14563374	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₉⋊#338	C₂	1	2	yes	maximal
5	C₁₉×#1742	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	C₂²	1	2	yes	—
7	C₂²×#64520	C₂²	1	1	yes	characteristic
8	C₂³×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×#10597279	—	2	4	no	—
10	C₂×#10597279	—	2	4	no	—
11	C₂×#10597279	C₄	1	8	yes	complemented
12	#11926500	—	2	4	no	—
13	C₂×#10597260	C₄	1	2	yes	—
14	C₂×C₄×C₁₉⋊C₈	C₄	1	2	yes	—
15	#11926313	—	2	2	no	—
16	C₁₉⋊#338	—	2	4	no	—
17	C₂²×M₁₆×C₁₉	C₂²	1	2	yes	—
18	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
19	C₁₉×#343	—	2	2	no	—
20	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
21	C₂×M₁₆×C₁₉	—	2	2	no	—
22	C₂×M₁₆×C₁₉	—	2	2	no	—
23	C₂×M₁₆×C₁₉	—	2	2	no	—
24	C₂×M₁₆×C₁₉	—	2	2	no	—
25	C₂×M₁₆×C₁₉	—	2	2	no	—
26	C₂×M₁₆×C₁₉	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₁₉	—	2	2	no	—
28	C₁₉×#117	—	2	2	no	—
29	C₁₉×#117	—	2	2	no	—
30	C₂×C₄⋊C₄×C₁₉	D₄	1	2	yes	—
31	C₂×C₂²⋊C₄×C₁₉	Q₈	1	2	yes	—
32	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
33	C₂×#64520	—	2	2	no	—
34	C₂×#64520	—	2	2	no	—
35	C₂×#64520	—	2	2	no	—
36	C₂×#64520	—	2	2	no	—
37	C₂×#64520	—	2	2	no	—
38	C₂×#64520	—	2	2	no	—
39	C₂×#64520	—	2	2	no	—
40	C₂×#64520	C₂×C₄	1	4	yes	—
41	C₂×#64520	D₄	1	2	yes	—
42	C₂×#64520	Q₈	1	2	yes	—
43	C₂²×C₁₉⋊C₈	—	2	2	no	—
44	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
45	C₂²×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
46	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
49	C₂²×C₁₉⋊C₈	D₄	1	4	yes	—
50	C₂²×C₁₉⋊C₈	Q₈	1	2	yes	—
51	#10597279	—	2	16	no	—
52	#10597260	—	2	4	no	—
53	C₄×C₁₉⋊C₈	—	2	4	no	—
54	C₂⁴×C₁₉	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
56	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	#41	1	2	yes	—
65	C₂²×C₄×C₁₉	#41	1	2	yes	—
66	M₁₆×C₁₉	—	2	4	no	—
67	M₁₆×C₁₉	—	2	4	no	—
68	M₁₆×C₁₉	—	4	2	no	—
69	M₁₆×C₁₉	—	4	2	no	—
70	C₂×C₈×C₁₉	—	2	4	no	—
71	C₂×C₈×C₁₉	—	4	2	no	—
72	C₂×C₈×C₁₉	—	4	2	no	—
73	C₄⋊C₄×C₁₉	—	2	4	no	—
74	C₂²⋊C₄×C₁₉	—	2	4	no	—
75	C₄²×C₁₉	—	2	2	no	—
76	C₄²×C₁₉	—	2	2	no	—
77	#64520	—	2	4	no	—
78	#64520	—	2	4	no	—
79	#64520	—	2	4	no	—
80	#64520	—	2	4	no	—
81	#64520	—	4	2	no	—
82	#64520	—	4	2	no	—
83	C₂×C₁₉⋊C₈	—	2	2	no	—
84	C₂×C₁₉⋊C₈	—	2	2	no	—
85	C₂×C₁₉⋊C₈	—	2	4	no	—
86	C₂×C₁₉⋊C₈	—	4	2	no	—
87	C₂×C₁₉⋊C₈	—	4	2	no	—
88	C₂×C₁₉⋊C₈	C₄²	1	2	yes	—
89	C₂×C₁₉⋊C₈	C₄²	1	2	yes	—
90	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	2	yes	—
91	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	2	yes	—
92	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	8	yes	—
93	C₂×C₁₉⋊C₈	C₄⋊C₄	1	4	yes	—
94	C₂×C₁₉⋊C₈	C₄⋊C₄	1	2	yes	—
95	C₂×C₁₉⋊C₈	C₄⋊C₄	1	2	yes	—
96	C₂×C₁₉⋊C₈	C₄⋊C₄	1	8	yes	—
97	#10867	—	19	1	no	maximal, Sylow
98	C₂³×C₁₉	—	4	2	no	—
99	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂³×C₁₉	C₄×Q₈	1	2	yes	—
101	C₂³×C₁₉	C₂²⋊Q₈	1	2	yes	—
102	C₂³×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
103	C₂³×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₁₉	—	2	4	no	—
105	C₂×C₄×C₁₉	—	4	4	no	—
106	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
109	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
113	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
114	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
117	C₂×C₄×C₁₉	#121	1	2	yes	—
118	C₂×C₄×C₁₉	#121	1	2	yes	—
119	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
120	C₂×C₄×C₁₉	#124	1	2	yes	—
121	C₂×C₄×C₁₉	#125	1	2	yes	—
122	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
124	C₈×C₁₉	—	4	4	no	—
125	C₁₉⋊C₈	—	2	4	no	—
126	C₁₉⋊C₈	—	4	4	no	—
127	C₁₉⋊C₈	#95	1	4	yes	—
128	C₁₉⋊C₈	#95	1	4	yes	—
129	C₂²×#108	—	19	2	no	—
130	C₂×#405	—	19	1	no	—
131	#1742	—	19	1	no	—
132	#1742	D₁₉	1	1	yes	characteristic
133	C₂×#338	—	19	2	no	—
134	C₂²×C₁₉	—	2	2	no	—
135	C₂²×C₁₉	—	2	4	no	—
136	C₂²×C₁₉	—	4	2	no	—
137	C₂²×C₁₉	—	4	1	no	—
138	C₂²×C₁₉	—	4	1	no	—
139	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
140	C₂²×C₁₉	#380	1	1	yes	characteristic
141	C₂²×C₁₉	#384	1	1	yes	characteristic
142	C₂²×C₁₉	#389	1	2	yes	—
143	C₂²×C₁₉	#391	1	2	yes	—
144	C₄×C₁₉	—	4	4	no	—
145	C₄×C₁₉	C₂×#95	1	1	yes	characteristic
146	C₄×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
147	C₄×C₁₉	#380	1	1	yes	characteristic
148	C₄×C₁₉	#382	1	1	yes	characteristic
149	C₄×C₁₉	#389	1	2	yes	—
150	C₄×C₁₉	#390	1	2	yes	—
151	C₂²×M₁₆	—	19	1	no	—
152	C₂²×M₁₆	—	19	2	no	—
153	C₂²×M₁₆	C₂×D₁₉	1	2	yes	—
154	C₂³×C₈	—	19	1	no	—
155	C₂×#117	C₂×D₁₉	1	1	yes	characteristic
156	C₂×#108	—	19	8	no	—
157	C₂×#108	—	38	4	no	—
158	C₂×#108	—	38	4	no	—
159	#405	—	38	4	no	—
160	C₂×#97	—	19	2	no	—
161	C₂×C₄×C₈	—	19	2	no	—
162	#343	—	2	2	no	—
163	#343	—	38	2	no	—
164	#338	—	38	4	no	—
165	C₂×C₁₉	—	2	1	no	—
166	C₂×C₁₉	—	2	1	no	—
167	C₂×C₁₉	—	2	2	no	—
168	C₂×C₁₉	—	2	2	no	—
169	C₂×C₁₉	C₄⋊#95	1	1	yes	characteristic
170	C₂×C₁₉	#1813	1	2	yes	—
171	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	19	2	no	—
178	C₂×M₁₆	—	19	2	no	—
179	C₂×M₁₆	—	19	4	no	—
180	C₂×M₁₆	—	38	2	no	—
181	C₂×M₁₆	—	38	2	no	—
182	C₂×M₁₆	—	38	2	no	—
183	C₂×M₁₆	—	38	2	no	—
184	C₂×M₁₆	—	38	2	no	—
185	C₂×M₁₆	—	38	2	no	—
186	C₂×M₁₆	—	38	2	no	—
187	C₂×M₁₆	C₄×D₁₉	1	4	yes	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	—	19	1	no	—
190	C₂²×C₈	—	19	1	no	—
191	C₂²×C₈	—	19	4	no	—
192	C₂²×C₈	—	19	4	no	—
193	C₂²×C₈	—	19	2	no	—
194	C₂²×C₈	—	19	2	no	—
195	C₂²×C₈	—	19	2	no	—
196	C₂²×C₈	—	38	2	no	—
197	#117	—	2	2	no	—
198	#117	—	2	2	no	—
199	C₂×C₄⋊C₄	C₂²⋊D₁₉	1	2	yes	—
200	C₂×C₂²⋊C₄	Dic₃₈	1	2	yes	—
201	C₂×C₄²	C₄×D₁₉	1	2	yes	—
202	#108	—	38	16	no	—
203	#97	—	38	4	no	—
204	C₄×C₈	—	38	4	no	—
205	C₁₉	#10867	1	1	yes	characteristic, complemented, Sylow
206	C₂⁴	C₂×Dic₃₈	1	1	yes	characteristic
207	C₂²×C₄	Dic₁₉⋊C₄	1	4	yes	—
208	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
210	C₂²×C₄	#64541	1	2	yes	—
211	C₂²×C₄	D₄×D₁₉	1	2	yes	—
212	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
213	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
214	C₂²×C₄	#64543	1	2	yes	—
215	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
216	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
218	M₁₆	—	2	4	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	4	2	no	—
221	M₁₆	—	4	2	no	—
222	M₁₆	—	38	4	no	—
223	M₁₆	—	38	4	no	—
224	M₁₆	—	38	4	no	—
225	M₁₆	—	38	4	no	—
226	M₁₆	—	76	2	no	—
227	M₁₆	—	76	2	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	19	2	no	—
232	C₂×C₈	—	19	2	no	—
233	C₂×C₈	—	19	2	no	—
234	C₂×C₈	—	19	2	no	—
235	C₂×C₈	—	19	8	no	—
236	C₂×C₈	—	19	2	no	—
237	C₂×C₈	—	19	2	no	—
238	C₂×C₈	—	19	8	no	—
239	C₂×C₈	—	19	4	no	—
240	C₂×C₈	—	38	2	no	—
241	C₂×C₈	—	38	2	no	—
242	C₂×C₈	—	38	4	no	—
243	C₂×C₈	—	76	2	no	—
244	C₂×C₈	—	76	2	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₂²⋊C₄	—	2	4	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	2	no	—
249	C₂³	—	4	2	no	—
250	C₂³	C₄×Dic₃₈	1	2	yes	—
251	C₂³	#10597324	1	2	yes	—
252	C₂³	#10597334	1	1	yes	characteristic
253	C₂³	#10597334	1	1	yes	characteristic
254	C₂³	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	C₄²×D₁₉	1	1	yes	characteristic
258	C₂×C₄	#10597318	1	1	yes	characteristic
259	C₂×C₄	#10597323	1	1	yes	characteristic
260	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
261	C₂×C₄	C₂²⋊C₄×D₁₉	1	2	yes	—
262	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
263	C₂×C₄	#10597330	1	2	yes	—
264	C₂×C₄	#10597331	1	2	yes	—
265	C₂×C₄	#10597333	1	2	yes	—
266	C₂×C₄	#10597335	1	2	yes	—
267	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
268	C₂×C₄	C₄⋊C₄×D₁₉	1	2	yes	—
269	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
270	C₂×C₄	#10597342	1	2	yes	—
271	C₂×C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
272	C₂×C₄	#10597385	1	2	yes	—
273	C₂×C₄	#10597386	1	1	yes	characteristic
274	C₂×C₄	#10597390	1	1	yes	characteristic
275	C₈	—	4	4	no	—
276	C₈	—	19	4	no	—
277	C₈	—	19	4	no	—
278	C₈	—	38	4	no	—
279	C₈	—	76	4	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	C₄×Dic₁₉⋊C₄	1	1	yes	characteristic
286	C₂²	#11926402	1	1	yes	characteristic
287	C₂²	#11926413	1	1	yes	characteristic
288	C₂²	#11926441	1	2	yes	—
289	C₂²	#11926448	1	2	yes	—
290	C₄	—	4	4	no	—
291	C₄	#95×D₁₉	1	1	yes	characteristic
292	C₄	#11926395	1	1	yes	characteristic
293	C₄	#11926403	1	1	yes	characteristic
294	C₄	#11926408	1	1	yes	characteristic
295	C₄	#11926439	1	2	yes	—
296	C₄	#11926444	1	2	yes	—
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	2	no	—
300	C₂	—	2	2	no	—
301	C₂	Dic₁₉⋊#95	1	1	yes	characteristic
302	C₂	#14563683	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#10597323
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₁₉×#1742	C₂
socle	C₂²×C₁₉	#380
prime radical	C₂⁴×C₁₉	C₂×Q₈
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	#1742	D₁₉
hypocenter	C₁₉	#10867
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1742

Complements

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

N	H ≅ G/N	type
11. C₂×#10597279	C₄	semidirect
205. C₁₉	#10867	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#10597279	2
C₂×#11926500	2
#14563374	2
C₂×C₁₉⋊#338	2
C₁₉×#1742	2
#10867	19