Subgroups — #18090607

Order 4864.

608 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926593	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926314	C₂	1	2	yes	maximal
3	C₂×C₁₉⋊#338	C₂	1	2	yes	maximal
4	C₂×C₁₉⋊#338	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#2905	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	C₂²	1	2	yes	—
7	C₂²×#64520	C₂²	1	2	yes	—
8	#11926593	—	2	4	no	—
9	C₂×#10597276	—	2	2	no	—
10	#11926314	—	2	4	no	—
11	C₁₉⋊#338	—	2	4	no	—
12	C₁₉⋊#338	—	2	2	no	—
13	C₂²×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
14	C₂²×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
15	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
16	C₁₉×#424	—	2	2	no	—
17	C₁₉×#424	C₄	1	4	yes	—
18	C₁₉×#423	—	2	2	no	—
19	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₁₉	—	2	2	no	—
21	C₂×M₁₆×C₁₉	—	2	2	no	—
22	C₂×M₁₆×C₁₉	—	2	2	no	—
23	C₂×M₁₆×C₁₉	—	2	1	no	—
24	C₂×M₁₆×C₁₉	—	2	1	no	—
25	C₂×M₁₆×C₁₉	—	2	2	no	—
26	C₂×M₁₆×C₁₉	C₂×C₄	1	2	yes	—
27	C₂×M₁₆×C₁₉	D₄	1	2	yes	—
28	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₁₉	D₄	1	2	yes	—
30	C₁₉×#117	—	2	1	no	—
31	C₁₉×#117	—	2	1	no	—
32	C₁₉×#117	C₂×C₄	1	2	yes	—
33	C₁₉×#117	D₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
35	C₂×C₂²⋊C₄×C₁₉	Q₈	1	2	yes	—
36	C₂×C₄²×C₁₉	—	2	1	no	—
37	C₂×#64520	—	2	2	no	—
38	C₂×#64520	—	2	2	no	—
39	C₂×#64520	—	2	2	no	—
40	C₂×#64520	—	2	2	no	—
41	C₂×#64520	—	2	2	no	—
42	C₂×#64520	—	2	2	no	—
43	C₂×#64520	—	2	2	no	—
44	C₂×#64520	—	2	2	no	—
45	C₂×#64520	—	2	2	no	—
46	C₂×#64520	—	2	2	no	—
47	C₂×#64520	—	2	2	no	—
48	C₂×#64520	D₄	1	4	yes	—
49	C₂²×C₁₉⋊C₈	—	2	2	no	—
50	C₂²×C₁₉⋊C₈	Q₈	1	4	yes	—
51	C₁₉×C₄⋊C₈	—	2	8	no	—
52	#10597276	—	2	8	no	—
53	C₂⁴×C₁₉	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	—	2	2	no	—
55	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
61	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	#41	1	2	yes	—
64	C₂²×C₄×C₁₉	#41	1	2	yes	—
65	M₁₆×C₁₉	—	2	4	no	—
66	M₁₆×C₁₉	—	2	4	no	—
67	M₁₆×C₁₉	—	4	1	no	—
68	M₁₆×C₁₉	—	4	2	no	—
69	M₁₆×C₁₉	—	4	1	no	—
70	C₂×C₈×C₁₉	—	2	2	no	—
71	C₂×C₈×C₁₉	—	2	2	no	—
72	C₂×C₈×C₁₉	—	2	2	no	—
73	C₂×C₈×C₁₉	—	2	4	no	—
74	C₂×C₈×C₁₉	C₂²⋊C₄	1	4	yes	—
75	C₄⋊C₄×C₁₉	—	2	2	no	—
76	C₄⋊C₄×C₁₉	—	2	2	no	—
77	C₂²⋊C₄×C₁₉	—	2	2	no	—
78	C₂²⋊C₄×C₁₉	C₄⋊C₄	1	4	yes	—
79	C₄²×C₁₉	—	2	2	no	—
80	C₄²×C₁₉	—	2	2	no	—
81	#64520	—	2	4	no	—
82	#64520	—	2	4	no	—
83	#64520	—	4	2	no	—
84	#64520	—	4	2	no	—
85	#64520	—	4	2	no	—
86	#64520	—	4	2	no	—
87	#64520	—	4	2	no	—
88	#64520	—	4	2	no	—
89	C₂×C₁₉⋊C₈	—	2	4	no	—
90	C₂×C₁₉⋊C₈	—	2	4	no	—
91	C₂×C₁₉⋊C₈	—	2	4	no	—
92	C₂×C₁₉⋊C₈	—	2	4	no	—
93	C₂×C₁₉⋊C₈	—	4	2	no	—
94	C₂×C₁₉⋊C₈	—	4	2	no	—
95	#11235	—	19	1	no	maximal, Sylow
96	C₂³×C₁₉	—	8	1	no	—
97	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂³×C₁₉	#125	1	2	yes	—
101	C₂³×C₁₉	C₄⋊Q₈	1	2	yes	—
102	C₂×C₄×C₁₉	—	2	4	no	—
103	C₂×C₄×C₁₉	—	8	2	no	—
104	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
106	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
108	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
109	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
110	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
111	C₂×C₄×C₁₉	#121	1	2	yes	—
112	C₂×C₄×C₁₉	#121	1	2	yes	—
113	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
117	C₂×C₄×C₁₉	#123	1	2	yes	—
118	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
119	C₂×C₄×C₁₉	#125	1	2	yes	—
120	C₂×C₄×C₁₉	#126	1	2	yes	—
121	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
122	C₈×C₁₉	—	2	4	no	—
123	C₈×C₁₉	—	4	2	no	—
124	C₁₉⋊C₈	—	4	4	no	—
125	C₁₉⋊C₈	—	4	4	no	—
126	C₂×#430	—	19	1	no	—
127	#2905	D₁₉	1	1	yes	characteristic
128	C₂×#344	—	19	2	no	—
129	C₂×#338	—	19	1	no	—
130	C₂×#338	—	19	2	no	—
131	C₂²×C₁₉	—	4	1	no	—
132	C₂²×C₁₉	—	4	1	no	—
133	C₂²×C₁₉	—	4	1	no	—
134	C₂²×C₁₉	—	4	2	no	—
135	C₂²×C₁₉	—	4	2	no	—
136	C₂²×C₁₉	#380	1	1	yes	characteristic
137	C₂²×C₁₉	#389	1	2	yes	—
138	C₂²×C₁₉	#395	1	1	yes	characteristic
139	C₂²×C₁₉	#398	1	2	yes	—
140	C₂²×C₁₉	#400	1	1	yes	characteristic
141	C₄×C₁₉	—	4	4	no	—
142	C₄×C₁₉	#386	1	1	yes	characteristic
143	C₄×C₁₉	#387	1	2	yes	—
144	C₄×C₁₉	#391	1	1	yes	characteristic
145	C₄×C₁₉	#394	1	1	yes	characteristic
146	C₄×C₁₉	#396	1	2	yes	—
147	C₄×C₁₉	#398	1	1	yes	characteristic
148	C₂²×M₁₆	—	19	2	no	—
149	C₂²×M₁₆	—	19	2	no	—
150	C₂²×M₁₆	C₂×D₁₉	1	1	yes	characteristic
151	C₂²×M₁₆	C₂×D₁₉	1	1	yes	characteristic
152	C₂×#117	C₂×D₁₉	1	1	yes	characteristic
153	#430	—	38	4	no	—
154	C₂×#108	—	38	2	no	—
155	#424	—	2	2	no	—
156	#424	Dic₁₉	1	4	yes	—
157	#423	—	2	2	no	—
158	#344	—	38	4	no	—
159	#338	—	38	4	no	—
160	#338	—	38	2	no	—
161	C₂×C₁₉	—	2	1	no	—
162	C₂×C₁₉	—	2	2	no	—
163	C₂×C₁₉	—	2	1	no	—
164	C₂×C₁₉	—	2	2	no	—
165	C₂×C₁₉	—	2	1	no	—
166	C₂×C₁₉	#1441	1	1	yes	characteristic
167	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	1	no	—
173	C₂×M₁₆	—	2	1	no	—
174	C₂×M₁₆	—	19	4	no	—
175	C₂×M₁₆	—	38	2	no	—
176	C₂×M₁₆	—	38	2	no	—
177	C₂×M₁₆	—	38	2	no	—
178	C₂×M₁₆	—	38	2	no	—
179	C₂×M₁₆	—	38	2	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	C₂×Dic₁₉	1	2	yes	—
187	C₂×M₁₆	C₂²⋊D₁₉	1	2	yes	—
188	C₂²×C₈	—	19	4	no	—
189	C₂²×C₈	—	38	2	no	—
190	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
191	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
192	#117	—	2	1	no	—
193	#117	—	2	1	no	—
194	#117	D₇₆	1	2	yes	—
195	#117	C₂×Dic₁₉	1	2	yes	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₂²⋊C₄	Dic₃₈	1	2	yes	—
198	C₂×C₄²	—	2	1	no	—
199	#108	—	38	8	no	—
200	C₄⋊C₈	—	2	8	no	—
201	C₁₉	#11235	1	1	yes	characteristic, complemented, Sylow
202	C₂⁴	C₂×Dic₃₈	1	1	yes	characteristic
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
205	C₂²×C₄	#64541	1	1	yes	characteristic
206	C₂²×C₄	#64541	1	1	yes	characteristic
207	C₂²×C₄	D₄×D₁₉	1	2	yes	—
208	C₂²×C₄	#64543	1	2	yes	—
209	C₂²×C₄	Q₈×D₁₉	1	2	yes	—
210	C₂²×C₄	#64545	1	2	yes	—
211	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
214	M₁₆	—	2	4	no	—
215	M₁₆	—	2	4	no	—
216	M₁₆	—	4	1	no	—
217	M₁₆	—	4	1	no	—
218	M₁₆	—	4	2	no	—
219	M₁₆	—	38	4	no	—
220	M₁₆	—	38	4	no	—
221	M₁₆	—	76	2	no	—
222	M₁₆	—	76	2	no	—
223	M₁₆	—	76	2	no	—
224	M₁₆	—	76	2	no	—
225	M₁₆	—	76	2	no	—
226	M₁₆	—	76	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	38	4	no	—
232	C₂×C₈	—	38	4	no	—
233	C₂×C₈	—	38	4	no	—
234	C₂×C₈	—	38	4	no	—
235	C₂×C₈	—	76	2	no	—
236	C₂×C₈	—	76	2	no	—
237	C₂×C₈	C₂²⋊Dic₁₉	1	4	yes	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₂²⋊C₄	C₄⋊Dic₁₉	1	4	yes	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₂³	—	8	1	no	—
245	C₂³	#10597334	1	2	yes	—
246	C₂³	#10597336	1	2	yes	—
247	C₂³	#10597370	1	1	yes	characteristic
248	C₂³	#10597370	1	1	yes	characteristic
249	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	8	2	no	—
252	C₂×C₄	#10597335	1	2	yes	—
253	C₂×C₄	#10597340	1	2	yes	—
254	C₂×C₄	#10597341	1	2	yes	—
255	C₂×C₄	#10597342	1	2	yes	—
256	C₂×C₄	#10597343	1	2	yes	—
257	C₂×C₄	#10597344	1	2	yes	—
258	C₂×C₄	#10597372	1	1	yes	characteristic
259	C₂×C₄	#10597376	1	1	yes	characteristic
260	C₂×C₄	#10597376	1	1	yes	characteristic
261	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
262	C₂×C₄	#10597381	1	1	yes	characteristic
263	C₂×C₄	#10597383	1	1	yes	characteristic
264	C₂×C₄	#10597385	1	2	yes	—
265	C₂×C₄	#10597390	1	1	yes	characteristic
266	C₂×C₄	Q₈×Dic₁₉	1	2	yes	—
267	C₂×C₄	#10597392	1	2	yes	—
268	C₂×C₄	#10597393	1	1	yes	characteristic
269	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
270	C₈	—	2	4	no	—
271	C₈	—	4	2	no	—
272	C₈	—	76	4	no	—
273	C₈	—	76	4	no	—
274	C₂²	—	4	1	no	—
275	C₂²	—	4	2	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	4	1	no	—
278	C₂²	—	4	1	no	—
279	C₂²	#11926404	1	1	yes	characteristic
280	C₂²	#11926443	1	2	yes	—
281	C₂²	#11926464	1	1	yes	characteristic
282	C₂²	#11926477	1	2	yes	—
283	C₂²	#11926483	1	1	yes	characteristic
284	C₄	—	4	4	no	—
285	C₄	#11926424	1	1	yes	characteristic
286	C₄	#11926430	1	2	yes	—
287	C₄	#11926451	1	1	yes	characteristic
288	C₄	#11926461	1	1	yes	characteristic
289	C₄	#11926466	1	2	yes	—
290	C₄	#11926474	1	1	yes	characteristic
291	C₂	—	2	1	no	—
292	C₂	—	2	2	no	—
293	C₂	—	2	1	no	—
294	C₂	—	2	2	no	—
295	C₂	—	2	1	no	—
296	C₂	#14561888	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#11926451
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₁₉×#2905	C₂
socle	C₂×C₁₉	#1441
prime radical	C₂⁴×C₁₉	C₂×Q₈
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	#2905	D₁₉
hypocenter	C₁₉	#11235
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ #2905

Complements

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

N	H ≅ G/N	type
201. C₁₉	#11235	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926593	2
C₂×#11926314	2
C₂×C₁₉⋊#338	2
C₁₉×#2905	2
#11235	19