Subgroups — #18090623

Order 4864.

682 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10597274	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926570	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926312	C₂	1	2	yes	maximal
4	#14563355	C₂	1	2	yes	maximal
5	C₂×C₁₉×#341	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	C₂²	1	1	yes	characteristic
7	C₂×#10597372	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊Dic₁₉	C₂²	1	2	yes	—
9	C₂×#10597274	—	2	4	no	—
10	C₂×#10597274	—	2	2	no	—
11	#11926570	—	2	2	no	—
12	#11926570	C₄	1	4	yes	complemented
13	C₂×Dic₁₉⋊C₈	—	2	2	no	—
14	#11926312	—	2	4	no	—
15	#11926307	—	2	4	no	—
16	C₂²×M₁₆×C₁₉	C₂²	1	2	yes	—
17	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
18	C₁₉×#341	—	2	2	no	—
19	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
20	C₂³×Dic₁₉	—	2	2	no	—
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	Q₈	1	4	yes	—
27	C₂²×C₈×C₁₉	—	2	2	no	—
28	C₂²×C₈×C₁₉	—	2	2	no	—
29	C₂²×C₈×C₁₉	—	2	2	no	—
30	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₁₉	D₄	1	2	yes	—
32	C₂²×C₈×C₁₉	D₄	1	2	yes	—
33	C₂×C₂²⋊Dic₁₉	—	2	1	no	—
34	#10597372	—	2	1	no	—
35	#10597372	—	2	1	no	—
36	#10597372	C₂×C₄	1	2	yes	—
37	#10597372	Q₈	1	2	yes	—
38	C₂×C₄⋊Dic₁₉	—	2	2	no	—
39	C₂×C₄⋊Dic₁₉	—	2	2	no	—
40	C₂×C₄⋊Dic₁₉	—	2	2	no	—
41	C₂×C₄⋊Dic₁₉	—	2	2	no	—
42	C₂×C₄⋊Dic₁₉	—	2	2	no	—
43	C₂×C₄⋊Dic₁₉	D₄	1	4	yes	—
44	C₂×C₄⋊Dic₁₉	D₄	1	2	yes	—
45	C₂×C₄×Dic₁₉	—	2	1	no	—
46	C₂×#64520	—	2	1	no	—
47	C₂×#64520	—	2	2	no	—
48	C₂×#64520	—	2	2	no	—
49	C₂×#64520	C₂×C₄	1	2	yes	—
50	C₂²×C₁₉⋊C₈	—	2	1	no	—
51	#10597274	—	2	8	no	—
52	Dic₁₉⋊C₈	—	2	8	no	—
53	C₂⁴×C₁₉	#41	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	C₂×D₄	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×Q₈	1	2	yes	—
60	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	#41	1	2	yes	—
62	C₂²×C₄×C₁₉	#41	1	2	yes	—
63	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
64	C₂²×Dic₁₉	—	2	2	no	—
65	C₂²×Dic₁₉	—	2	4	no	—
66	C₂²×Dic₁₉	—	4	2	no	—
67	C₂²×Dic₁₉	—	4	2	no	—
68	C₂²×Dic₁₉	—	8	4	no	—
69	M₁₆×C₁₉	—	2	4	no	—
70	M₁₆×C₁₉	—	2	4	no	—
71	M₁₆×C₁₉	—	4	2	no	—
72	M₁₆×C₁₉	—	4	2	no	—
73	C₂×C₈×C₁₉	—	2	4	no	—
74	C₂×C₈×C₁₉	—	2	2	no	—
75	C₂×C₈×C₁₉	—	2	4	no	—
76	C₂×C₈×C₁₉	—	2	2	no	—
77	C₂×C₈×C₁₉	—	2	4	no	—
78	C₂×C₈×C₁₉	—	4	2	no	—
79	C₂×C₈×C₁₉	—	4	2	no	—
80	C₂×C₈×C₁₉	C₂²⋊C₄	1	4	yes	—
81	C₂²⋊Dic₁₉	—	2	2	no	—
82	C₂²⋊Dic₁₉	—	2	2	no	—
83	C₄⋊Dic₁₉	—	2	2	no	—
84	C₄⋊Dic₁₉	—	2	4	no	—
85	C₄⋊Dic₁₉	—	2	4	no	—
86	C₄⋊Dic₁₉	—	4	2	no	—
87	C₄⋊Dic₁₉	—	4	2	no	—
88	C₄⋊Dic₁₉	C₄⋊C₄	1	4	yes	—
89	C₄×Dic₁₉	—	2	2	no	—
90	C₄×Dic₁₉	—	2	2	no	—
91	#64520	—	2	4	no	—
92	#64520	—	4	2	no	—
93	C₂×C₁₉⋊C₈	—	2	2	no	—
94	C₂×C₁₉⋊C₈	—	4	1	no	—
95	C₂×C₁₉⋊C₈	—	4	1	no	—
96	#11238	—	19	1	no	maximal, Sylow
97	C₂³×C₁₉	—	2	4	no	—
98	C₂³×C₁₉	#117	1	1	yes	characteristic
99	C₂³×C₁₉	#121	1	1	yes	characteristic
100	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
101	C₂³×C₁₉	#123	1	2	yes	—
102	C₂³×C₁₉	#125	1	2	yes	—
103	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
107	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
108	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
109	C₂×C₄×C₁₉	#121	1	2	yes	—
110	C₂×C₄×C₁₉	#121	1	2	yes	—
111	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
113	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
116	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
117	C₂×C₄×C₁₉	#125	1	2	yes	—
118	C₂×C₄×C₁₉	#126	1	2	yes	—
119	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
121	C₂×Dic₁₉	—	2	4	no	—
122	C₂×Dic₁₉	—	4	4	no	—
123	C₂×Dic₁₉	—	4	4	no	—
124	C₂×Dic₁₉	—	8	4	no	—
125	C₂×Dic₁₉	—	8	4	no	—
126	C₂×Dic₁₉	—	8	4	no	—
127	C₈×C₁₉	—	2	4	no	—
128	C₈×C₁₉	—	4	4	no	—
129	C₁₉⋊C₈	—	4	2	no	—
130	C₂²×#106	—	19	1	no	—
131	C₂×#424	—	19	1	no	—
132	C₂×#343	—	19	2	no	—
133	C₂×#341	D₁₉	1	1	yes	characteristic
134	#1736	—	19	2	no	—
135	C₂²×C₁₉	—	2	2	no	—
136	C₂²×C₁₉	—	2	2	no	—
137	C₂²×C₁₉	—	2	4	no	—
138	C₂²×C₁₉	—	2	2	no	—
139	C₂²×C₁₉	—	2	2	no	—
140	C₂²×C₁₉	#380	1	1	yes	characteristic
141	C₂²×C₁₉	#387	1	2	yes	—
142	C₂²×C₁₉	#394	1	1	yes	characteristic
143	C₂²×C₁₉	#398	1	1	yes	characteristic
144	C₂²×C₁₉	#398	1	2	yes	—
145	C₂²×C₁₉	#471	1	4	yes	—
146	C₄×C₁₉	#386	1	1	yes	characteristic
147	C₄×C₁₉	#389	1	2	yes	—
148	C₄×C₁₉	#391	1	1	yes	characteristic
149	C₄×C₁₉	#395	1	1	yes	characteristic
150	C₄×C₁₉	#396	1	2	yes	—
151	C₄×C₁₉	#400	1	1	yes	characteristic
152	Dic₁₉	—	4	4	no	—
153	Dic₁₉	—	8	4	no	—
154	C₂²×M₁₆	—	19	1	no	—
155	C₂²×M₁₆	C₂×D₁₉	1	2	yes	—
156	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
157	C₂×#117	—	19	1	no	—
158	C₂²×C₄⋊C₄	—	19	2	no	—
159	C₂×#106	—	38	4	no	—
160	C₂×#106	—	38	2	no	—
161	#424	—	19	4	no	—
162	#424	—	38	2	no	—
163	C₂×C₄⋊C₈	—	38	2	no	—
164	#343	—	38	4	no	—
165	#341	—	2	2	no	—
166	#340	—	38	4	no	—
167	C₂×C₁₉	—	2	2	no	—
168	C₂×C₁₉	—	2	2	no	—
169	C₂×C₁₉	#1441	1	1	yes	characteristic
170	C₂×C₁₉	#1815	1	2	yes	—
171	C₂×C₁₉	#2046	1	2	yes	—
172	C₂×C₁₉	#2090	1	2	yes	—
173	C₂³×C₄	—	38	2	no	—
174	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	19	2	no	—
181	C₂×M₁₆	—	38	1	no	—
182	C₂×M₁₆	—	38	2	no	—
183	C₂×M₁₆	—	38	2	no	—
184	C₂×M₁₆	Dic₃₈	1	4	yes	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	38	1	no	—
189	C₂²×C₈	C₄×D₁₉	1	2	yes	—
190	C₂²×C₈	D₇₆	1	2	yes	—
191	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
192	#117	—	19	2	no	—
193	#117	—	19	2	no	—
194	#117	—	38	1	no	—
195	#117	—	38	1	no	—
196	C₂×C₄⋊C₄	—	19	2	no	—
197	C₂×C₄⋊C₄	—	19	4	no	—
198	C₂×C₄⋊C₄	—	38	2	no	—
199	C₂×C₄⋊C₄	—	38	2	no	—
200	C₂×C₄⋊C₄	—	38	2	no	—
201	C₂×C₄⋊C₄	—	38	2	no	—
202	C₂×C₄⋊C₄	—	38	2	no	—
203	C₂×C₂²⋊C₄	—	38	1	no	—
204	C₂×C₄²	—	38	1	no	—
205	#106	—	38	8	no	—
206	C₄⋊C₈	—	38	8	no	—
207	C₁₉	#11238	1	1	yes	characteristic, complemented, Sylow
208	C₂⁴	#64545	1	1	yes	characteristic
209	C₂²×C₄	—	38	2	no	—
210	C₂²×C₄	—	38	4	no	—
211	C₂²×C₄	—	76	2	no	—
212	C₂²×C₄	—	76	2	no	—
213	C₂²×C₄	—	152	4	no	—
214	C₂²×C₄	C₂×Dic₃₈	1	2	yes	—
215	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
217	C₂²×C₄	#64541	1	2	yes	—
218	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
219	C₂²×C₄	D₄×D₁₉	1	2	yes	—
220	C₂²×C₄	#64543	1	1	yes	characteristic
221	C₂²×C₄	#64543	1	2	yes	—
222	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
224	M₁₆	—	2	4	no	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	4	2	no	—
227	M₁₆	—	4	2	no	—
228	M₁₆	—	38	4	no	—
229	M₁₆	—	76	2	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	38	2	no	—
238	C₂×C₈	—	76	1	no	—
239	C₂×C₈	—	76	1	no	—
240	C₂×C₈	(C₂×D₁₉)⋊C₄	1	4	yes	—
241	C₄⋊C₄	—	19	4	no	—
242	C₄⋊C₄	—	38	2	no	—
243	C₄⋊C₄	—	38	4	no	—
244	C₄⋊C₄	—	38	4	no	—
245	C₄⋊C₄	—	76	2	no	—
246	C₄⋊C₄	—	76	2	no	—
247	C₂²⋊C₄	—	38	2	no	—
248	C₂²⋊C₄	—	38	2	no	—
249	C₄²	—	38	2	no	—
250	C₄²	—	38	2	no	—
251	C₂³	—	2	4	no	—
252	C₂³	#10597336	1	2	yes	—
253	C₂³	#10597338	1	1	yes	characteristic
254	C₂³	#10597340	1	2	yes	—
255	C₂³	#10597341	1	1	yes	characteristic
256	C₂³	#10597392	1	1	yes	characteristic
257	C₂×C₄	—	38	4	no	—
258	C₂×C₄	—	76	4	no	—
259	C₂×C₄	—	76	4	no	—
260	C₂×C₄	—	152	4	no	—
261	C₂×C₄	—	152	4	no	—
262	C₂×C₄	—	152	4	no	—
263	C₂×C₄	C₄×Dic₃₈	1	2	yes	—
264	C₂×C₄	#10597315	1	1	yes	characteristic
265	C₂×C₄	#10597321	1	1	yes	characteristic
266	C₂×C₄	#10597324	1	2	yes	—
267	C₂×C₄	#10597324	1	2	yes	—
268	C₂×C₄	#10597325	1	2	yes	—
269	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
270	C₂×C₄	#10597328	1	1	yes	characteristic
271	C₂×C₄	#10597330	1	2	yes	—
272	C₂×C₄	#10597332	1	1	yes	characteristic
273	C₂×C₄	#10597334	1	2	yes	—
274	C₂×C₄	#10597335	1	2	yes	—
275	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
276	C₂×C₄	#10597343	1	1	yes	characteristic
277	C₂×C₄	#10597370	1	2	yes	—
278	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
279	C₂×C₄	#10597384	1	1	yes	characteristic
280	C₂×C₄	#10597385	1	2	yes	—
281	C₈	—	2	4	no	—
282	C₈	—	4	4	no	—
283	C₈	—	76	2	no	—
284	C₂²	—	2	4	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	#11926401	1	1	yes	characteristic
290	C₂²	#11926431	1	2	yes	—
291	C₂²	#11926462	1	1	yes	characteristic
292	C₂²	#11926476	1	1	yes	characteristic
293	C₂²	#11926477	1	2	yes	—
294	C₂²	#11926808	1	4	yes	—
295	C₄	—	76	4	no	—
296	C₄	—	152	4	no	—
297	C₄	#11926421	1	1	yes	characteristic
298	C₄	#11926441	1	2	yes	—
299	C₄	#11926450	1	1	yes	characteristic
300	C₄	#11926465	1	1	yes	characteristic
301	C₄	#11926467	1	2	yes	—
302	C₄	#11926484	1	1	yes	characteristic
303	C₂	—	2	2	no	—
304	C₂	—	2	2	no	—
305	C₂	#14561889	1	1	yes	characteristic
306	C₂	#14563691	1	2	yes	—
307	C₂	#14564899	1	2	yes	—
308	C₂	#14565147	1	2	yes	—

	subgroup	G/H
center	C₂³	#10597341
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₁₉×#341	C₂
socle	C₂³×C₁₉	#121
prime radical	C₂⁴×C₁₉	#41
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×#341	D₁₉
hypocenter	C₁₉	#11238
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#341

Complements

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

N	H ≅ G/N	type
12. #11926570	C₄	semidirect
207. C₁₉	#11238	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#10597274	2
C₂×#11926570	2
C₂×#11926312	2
#14563355	2
C₂×C₁₉×#341	2
#11238	19