Subgroups — #18090622

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×#11926312	C₂	1	2	yes	maximal
3	C₂×#11926309	C₂	1	1	yes	maximal, characteristic
4	#14563356	C₂	1	2	yes	maximal
5	C₂×C₁₉×#424	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	C₂²	1	2	yes	—
7	C₂²×C₄⋊Dic₁₉	C₂²	1	2	yes	—
8	C₂×#10597274	—	2	4	no	—
9	C₂×#10597274	—	2	2	no	—
10	#11926312	—	2	4	no	—
11	#11926309	—	2	2	no	—
12	#11926306	—	2	4	no	—
13	C₂²×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
14	C₂³×C₈×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	complemented
18	C₂×C₁₉×C₄⋊C₈	—	2	2	no	—
19	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
20	C₂³×Dic₁₉	—	2	2	no	—
21	C₂×M₁₆×C₁₉	—	2	1	no	—
22	C₂×M₁₆×C₁₉	—	2	2	no	—
23	C₂×M₁₆×C₁₉	—	2	2	no	—
24	C₂×M₁₆×C₁₉	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₉	—	2	2	no	—
26	C₂²×C₈×C₁₉	—	2	2	no	—
27	C₂²×C₈×C₁₉	—	2	1	no	—
28	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₁₉	D₄	1	2	yes	—
30	C₂²×C₈×C₁₉	D₄	1	2	yes	—
31	C₁₉×#117	—	2	1	no	—
32	C₁₉×#117	—	2	1	no	—
33	C₁₉×#117	C₂×C₄	1	2	yes	—
34	C₁₉×#117	Q₈	1	2	yes	—
35	C₂×C₄⋊C₄×C₁₉	D₄	1	2	yes	—
36	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
37	C₂×C₄²×C₁₉	—	2	1	no	—
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×#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	—	2	2	no	—
49	C₂×#64520	Q₈	1	4	yes	—
50	C₂²×C₁₉⋊C₈	—	2	2	no	—
51	C₁₉×C₄⋊C₈	—	2	8	no	—
52	#10597274	—	2	8	no	—
53	C₂⁴×C₁₉	#41	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	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
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	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	—	4	2	no	—
71	C₂×C₈×C₁₉	—	2	4	no	—
72	C₂×C₈×C₁₉	—	2	2	no	—
73	C₂×C₈×C₁₉	—	2	4	no	—
74	C₂×C₈×C₁₉	—	2	2	no	—
75	C₂×C₈×C₁₉	—	2	2	no	—
76	C₂×C₈×C₁₉	—	4	1	no	—
77	C₂×C₈×C₁₉	—	4	1	no	—
78	C₂×C₈×C₁₉	C₂²⋊C₄	1	4	yes	—
79	C₄⋊C₄×C₁₉	—	2	2	no	—
80	C₄⋊C₄×C₁₉	C₄⋊C₄	1	4	yes	—
81	C₂²⋊C₄×C₁₉	—	2	2	no	—
82	C₂²⋊C₄×C₁₉	—	2	2	no	—
83	C₄²×C₁₉	—	2	2	no	—
84	C₄²×C₁₉	—	2	2	no	—
85	C₄⋊Dic₁₉	—	2	4	no	—
86	C₄⋊Dic₁₉	—	2	4	no	—
87	C₄⋊Dic₁₉	—	4	2	no	—
88	C₄⋊Dic₁₉	—	4	2	no	—
89	#64520	—	2	4	no	—
90	#64520	—	2	4	no	—
91	#64520	—	4	2	no	—
92	#64520	—	4	2	no	—
93	C₂×C₁₉⋊C₈	—	2	4	no	—
94	C₂×C₁₉⋊C₈	—	4	2	no	—
95	C₂×C₁₉⋊C₈	—	4	2	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	—	2	4	no	—
104	C₂×C₄×C₁₉	—	4	4	no	—
105	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
109	C₂×C₄×C₁₉	#120	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	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
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	C₂×C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
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	2	no	—
129	C₁₉⋊C₈	—	4	4	no	—
130	C₂²×#106	—	19	1	no	—
131	C₂×#424	D₁₉	1	1	yes	characteristic
132	C₂×#343	—	19	2	no	—
133	C₂×#341	—	19	1	no	—
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	—	4	4	no	—
147	C₄×C₁₉	#386	1	1	yes	characteristic
148	C₄×C₁₉	#389	1	2	yes	—
149	C₄×C₁₉	#391	1	1	yes	characteristic
150	C₄×C₁₉	#395	1	1	yes	characteristic
151	C₄×C₁₉	#396	1	2	yes	—
152	C₄×C₁₉	#400	1	1	yes	characteristic
153	Dic₁₉	—	8	4	no	—
154	C₂²×M₁₆	—	19	2	no	—
155	C₂²×M₁₆	C₂×D₁₉	1	1	yes	characteristic
156	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
157	C₂×#117	C₂×D₁₉	1	1	yes	characteristic
158	C₂²×C₄⋊C₄	—	19	2	no	—
159	C₂×#106	—	38	4	no	—
160	C₂×#106	—	38	2	no	—
161	#424	—	2	2	no	—
162	#424	Dic₁₉	1	4	yes	complemented
163	C₂×C₄⋊C₈	—	2	2	no	—
164	#343	—	38	4	no	—
165	#341	—	38	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	1	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	19	4	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	C₂×Dic₁₉	1	2	yes	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	2	1	no	—
188	C₂²×C₈	—	38	2	no	—
189	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
190	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
191	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
192	#117	—	2	1	no	—
193	#117	—	2	1	no	—
194	#117	Dic₃₈	1	2	yes	—
195	#117	C₂×Dic₁₉	1	2	yes	—
196	C₂×C₄⋊C₄	—	19	4	no	—
197	C₂×C₄⋊C₄	—	38	2	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	D₇₆	1	2	yes	—
203	C₂×C₂²⋊C₄	—	2	1	no	—
204	C₂×C₄²	—	2	1	no	—
205	#106	—	38	8	no	—
206	C₄⋊C₈	—	2	8	no	—
207	C₁₉	#11238	1	1	yes	characteristic, complemented, Sylow
208	C₂⁴	#64541	1	1	yes	characteristic
209	C₂²×C₄	—	2	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	1	yes	characteristic
215	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
216	C₂²×C₄	#64541	1	1	yes	characteristic
217	C₂²×C₄	D₄×D₁₉	1	2	yes	—
218	C₂²×C₄	#64543	1	2	yes	—
219	C₂²×C₄	Q₈×D₁₉	1	2	yes	—
220	C₂²×C₄	#64545	1	2	yes	—
221	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
224	M₁₆	—	2	4	no	—
225	M₁₆	—	4	2	no	—
226	M₁₆	—	38	4	no	—
227	M₁₆	—	38	4	no	—
228	M₁₆	—	76	2	no	—
229	M₁₆	—	76	2	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	—	38	4	no	—
238	C₂×C₈	—	76	2	no	—
239	C₂×C₈	—	76	2	no	—
240	C₂×C₈	C₂²⋊Dic₁₉	1	4	yes	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	38	4	no	—
243	C₄⋊C₄	—	38	4	no	—
244	C₄⋊C₄	—	76	2	no	—
245	C₄⋊C₄	—	76	2	no	—
246	C₄⋊C₄	C₄⋊Dic₁₉	1	4	yes	—
247	C₂²⋊C₄	—	2	2	no	—
248	C₂²⋊C₄	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₂³	—	2	4	no	—
252	C₂³	#10597335	1	2	yes	—
253	C₂³	#10597340	1	2	yes	—
254	C₂³	#10597370	1	1	yes	characteristic
255	C₂³	#10597372	1	1	yes	characteristic
256	C₂³	#10597376	1	1	yes	characteristic
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	4	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	#10597334	1	2	yes	—
264	C₂×C₄	#10597336	1	2	yes	—
265	C₂×C₄	#10597341	1	2	yes	—
266	C₂×C₄	#10597342	1	2	yes	—
267	C₂×C₄	#10597343	1	2	yes	—
268	C₂×C₄	#10597344	1	2	yes	—
269	C₂×C₄	#10597370	1	1	yes	characteristic
270	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
271	C₂×C₄	#10597376	1	1	yes	characteristic
272	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
273	C₂×C₄	#10597381	1	1	yes	characteristic
274	C₂×C₄	#10597383	1	1	yes	characteristic
275	C₂×C₄	#10597385	1	2	yes	—
276	C₂×C₄	#10597390	1	1	yes	characteristic
277	C₂×C₄	Q₈×Dic₁₉	1	2	yes	—
278	C₂×C₄	#10597392	1	2	yes	—
279	C₂×C₄	#10597393	1	1	yes	characteristic
280	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
281	C₈	—	2	4	no	—
282	C₈	—	4	2	no	—
283	C₈	—	76	4	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	#11926404	1	1	yes	characteristic
290	C₂²	#11926430	1	2	yes	—
291	C₂²	#11926461	1	1	yes	characteristic
292	C₂²	#11926474	1	1	yes	characteristic
293	C₂²	#11926477	1	2	yes	—
294	C₂²	#11926807	1	4	yes	—
295	C₄	—	4	4	no	—
296	C₄	—	152	4	no	—
297	C₄	#11926424	1	1	yes	characteristic
298	C₄	#11926443	1	2	yes	—
299	C₄	#11926451	1	1	yes	characteristic
300	C₄	#11926464	1	1	yes	characteristic
301	C₄	#11926466	1	2	yes	—
302	C₄	#11926483	1	1	yes	characteristic
303	C₂	—	2	2	no	—
304	C₂	—	2	2	no	—
305	C₂	#14561888	1	1	yes	characteristic
306	C₂	#14563694	1	2	yes	—
307	C₂	#14564898	1	2	yes	—
308	C₂	#14565145	1	2	yes	—

	subgroup	G/H
center	C₂³	#10597376
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₁₉×#424	C₂
socle	C₂³×C₁₉	#121
prime radical	C₂⁴×C₁₉	#41
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×#424	D₁₉
hypocenter	C₁₉	#11238
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#424

Complements

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

N	H ≅ G/N	type
17. C₁₉×#424	C₄	semidirect
162. #424	Dic₁₉	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×#11926312	2
C₂×#11926309	2
#14563356	2
C₂×C₁₉×#424	2
#11238	19