Subgroups — #18090807

Order 4864.

642 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926568	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926308	C₂	1	2	yes	maximal
3	C₂×#11926308	C₂	1	2	yes	maximal
4	C₂×#11926299	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1728	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	C₂²	1	2	yes	—
7	C₂³×C₁₉⋊C₈	C₂²	1	2	yes	—
8	#11926568	—	2	4	no	—
9	C₂×C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
10	#11926308	—	2	4	no	—
11	#11926308	—	2	4	no	—
12	#11926299	C₄	1	2	yes	—
13	#11926299	C₄	1	2	yes	—
14	C₂²×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
15	C₂²×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#336	—	2	2	no	—
18	C₂³×C₄×C₁₉	Q₈	1	2	yes	—
19	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₁₉	—	2	1	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	—	2	2	no	—
25	C₂×M₁₆×C₁₉	—	2	2	no	—
26	C₂×M₁₆×C₁₉	D₄	1	2	yes	—
27	C₂×M₁₆×C₁₉	Q₈	1	2	yes	—
28	C₂²×C₈×C₁₉	—	2	1	no	—
29	C₂²×C₈×C₁₉	—	2	1	no	—
30	C₂×C₄²×C₁₉	—	2	2	no	—
31	C₂×C₄²×C₁₉	—	2	2	no	—
32	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₉	D₄	1	2	yes	—
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	D₄	1	4	yes	—
41	C₂²×C₁₉⋊C₈	—	2	2	no	—
42	C₂²×C₁₉⋊C₈	—	2	2	no	—
43	C₂²×C₁₉⋊C₈	—	2	2	no	—
44	C₂²×C₁₉⋊C₈	—	2	2	no	—
45	C₂²×C₁₉⋊C₈	—	2	2	no	—
46	C₂²×C₁₉⋊C₈	—	2	2	no	—
47	C₂²×C₁₉⋊C₈	—	2	2	no	—
48	C₂²×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
49	C₄⋊(C₁₉⋊C₈)	—	2	8	no	—
50	C₂⁴×C₁₉	C₂×Q₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₉	—	2	4	no	—
52	C₂²×C₄×C₁₉	—	2	4	no	—
53	C₂²×C₄×C₁₉	—	4	2	no	—
54	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
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	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	2	no	—
71	C₂×C₈×C₁₉	—	2	2	no	—
72	C₂×C₈×C₁₉	—	4	1	no	—
73	C₂×C₈×C₁₉	—	4	1	no	—
74	C₂×C₈×C₁₉	—	4	1	no	—
75	C₂×C₈×C₁₉	—	4	1	no	—
76	C₄²×C₁₉	—	2	2	no	—
77	C₄²×C₁₉	—	2	2	no	—
78	C₄²×C₁₉	—	2	2	no	—
79	C₄²×C₁₉	—	2	2	no	—
80	#64520	—	2	4	no	—
81	#64520	—	2	4	no	—
82	#64520	—	4	2	no	—
83	#64520	—	4	2	no	—
84	C₂×C₁₉⋊C₈	—	2	4	no	—
85	C₂×C₁₉⋊C₈	—	2	4	no	—
86	C₂×C₁₉⋊C₈	—	2	4	no	—
87	C₂×C₁₉⋊C₈	—	2	4	no	—
88	C₂×C₁₉⋊C₈	—	4	2	no	—
89	C₂×C₁₉⋊C₈	—	4	2	no	—
90	C₂×C₁₉⋊C₈	—	4	2	no	—
91	C₂×C₁₉⋊C₈	—	4	2	no	—
92	C₂×C₁₉⋊C₈	—	4	2	no	—
93	C₂×C₁₉⋊C₈	—	4	2	no	—
94	#11269	—	19	1	no	maximal, Sylow
95	C₂³×C₁₉	—	2	2	no	—
96	C₂³×C₁₉	—	2	2	no	—
97	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂³×C₁₉	C₂²⋊Q₈	1	2	yes	—
99	C₂³×C₁₉	#125	1	1	yes	characteristic
100	C₂³×C₁₉	#125	1	2	yes	—
101	C₂³×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
102	C₂×C₄×C₁₉	—	2	2	no	—
103	C₂×C₄×C₁₉	—	2	2	no	—
104	C₂×C₄×C₁₉	—	2	2	no	—
105	C₂×C₄×C₁₉	—	2	2	no	—
106	C₂×C₄×C₁₉	—	2	2	no	—
107	C₂×C₄×C₁₉	—	4	8	no	—
108	C₂×C₄×C₁₉	#108	1	4	yes	—
109	C₂×C₄×C₁₉	#117	1	2	yes	—
110	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
114	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
115	C₂×C₄×C₁₉	#121	1	2	yes	—
116	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
117	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
119	C₂×C₄×C₁₉	#123	1	2	yes	—
120	C₂×C₄×C₁₉	#123	1	2	yes	—
121	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
122	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
123	C₂×C₄×C₁₉	#124	1	2	yes	—
124	C₂×C₄×C₁₉	#125	1	2	yes	—
125	C₂×C₄×C₁₉	#126	1	2	yes	—
126	C₂×C₄×C₁₉	#126	1	2	yes	—
127	C₈×C₁₉	—	4	2	no	—
128	C₈×C₁₉	—	4	2	no	—
129	C₁₉⋊C₈	—	4	4	no	—
130	C₁₉⋊C₈	—	4	4	no	—
131	C₂×#423	—	19	1	no	—
132	C₂×#341	—	19	2	no	—
133	C₂×#341	—	19	2	no	—
134	#1728	D₁₉	1	1	yes	characteristic
135	C₂×#336	—	19	1	no	—
136	C₂²×C₁₉	—	2	2	no	—
137	C₂²×C₁₉	—	2	2	no	—
138	C₂²×C₁₉	—	2	4	no	—
139	C₂²×C₁₉	—	2	2	no	—
140	C₂²×C₁₉	—	2	2	no	—
141	C₂²×C₁₉	#382	1	1	yes	characteristic
142	C₂²×C₁₉	#385	1	1	yes	characteristic
143	C₂²×C₁₉	#389	1	1	yes	characteristic
144	C₂²×C₁₉	#398	1	2	yes	—
145	C₂²×C₁₉	#400	1	2	yes	—
146	C₂²×C₁₉	C₂×#108	1	2	yes	—
147	C₂²×C₁₉	#430	1	2	yes	—
148	C₄×C₁₉	—	2	4	no	—
149	C₄×C₁₉	—	4	2	no	—
150	C₄×C₁₉	#387	1	2	yes	—
151	C₄×C₁₉	#388	1	2	yes	—
152	C₄×C₁₉	#393	1	1	yes	characteristic
153	C₄×C₁₉	#394	1	1	yes	characteristic
154	C₄×C₁₉	#397	1	1	yes	characteristic
155	C₄×C₁₉	#400	1	1	yes	characteristic
156	C₂²×M₁₆	—	19	2	no	—
157	C₂²×M₁₆	C₂×D₁₉	1	1	yes	characteristic
158	C₂²×M₁₆	C₂×D₁₉	1	1	yes	characteristic
159	C₂³×C₈	—	19	2	no	—
160	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
161	#423	—	38	4	no	—
162	C₂×C₄⋊C₈	—	38	2	no	—
163	#341	—	38	4	no	—
164	#341	—	38	4	no	—
165	#336	—	2	2	no	—
166	#336	—	19	2	no	—
167	#336	—	19	2	no	—
168	C₂×C₁₉	—	2	2	no	—
169	C₂×C₁₉	—	2	2	no	—
170	C₂×C₁₉	#1442	1	1	yes	characteristic
171	C₂×C₁₉	#1837	1	1	yes	characteristic
172	C₂×C₁₉	#1838	1	1	yes	characteristic
173	C₂×C₁₉	#1855	1	2	yes	—
174	C₂×C₁₉	#1950	1	2	yes	—
175	C₂³×C₄	Dic₃₈	1	2	yes	—
176	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
177	C₂×M₁₆	—	2	1	no	—
178	C₂×M₁₆	—	2	1	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	19	4	no	—
184	C₂×M₁₆	—	38	2	no	—
185	C₂×M₁₆	—	38	2	no	—
186	C₂×M₁₆	—	38	2	no	—
187	C₂×M₁₆	—	38	2	no	—
188	C₂×M₁₆	—	38	2	no	—
189	C₂×M₁₆	Dic₃₈	1	2	yes	—
190	C₂×M₁₆	D₇₆	1	2	yes	—
191	C₂²×C₈	—	2	1	no	—
192	C₂²×C₈	—	2	1	no	—
193	C₂²×C₈	—	19	4	no	—
194	C₂²×C₈	—	38	2	no	—
195	C₂²×C₈	—	38	2	no	—
196	C₂²×C₈	—	38	2	no	—
197	C₂²×C₈	—	38	2	no	—
198	C₂²×C₈	—	38	2	no	—
199	C₂²×C₈	—	38	2	no	—
200	C₂²×C₈	—	38	2	no	—
201	C₂×C₄²	—	2	2	no	—
202	C₂×C₄²	—	2	2	no	—
203	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
204	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
205	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
206	C₄⋊C₈	—	38	8	no	—
207	C₁₉	#11269	1	1	yes	characteristic, complemented, Sylow
208	C₂⁴	C₂×Dic₃₈	1	1	yes	characteristic
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	4	2	no	—
212	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
213	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
214	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
217	C₂²×C₄	#64541	1	1	yes	characteristic
218	C₂²×C₄	#64541	1	1	yes	characteristic
219	C₂²×C₄	D₄×D₁₉	1	2	yes	—
220	C₂²×C₄	#64543	1	2	yes	—
221	C₂²×C₄	#64543	1	2	yes	—
222	C₂²×C₄	#64545	1	2	yes	—
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	—	38	4	no	—
230	M₁₆	—	76	2	no	—
231	M₁₆	—	76	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	4	1	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	—	4	1	no	—
238	C₂×C₈	—	38	4	no	—
239	C₂×C₈	—	38	4	no	—
240	C₂×C₈	—	38	4	no	—
241	C₂×C₈	—	38	4	no	—
242	C₂×C₈	—	76	2	no	—
243	C₂×C₈	—	76	2	no	—
244	C₂×C₈	—	76	2	no	—
245	C₂×C₈	—	76	2	no	—
246	C₂×C₈	—	76	2	no	—
247	C₂×C₈	—	76	2	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₂³	—	2	2	no	—
253	C₂³	—	2	2	no	—
254	C₂³	#10597315	1	1	yes	characteristic
255	C₂³	#10597316	1	1	yes	characteristic
256	C₂³	#10597324	1	2	yes	—
257	C₂³	#10597336	1	2	yes	—
258	C₂³	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	4	8	no	—
265	C₂×C₄	#10597279	1	4	yes	—
266	C₂×C₄	C₄×Dic₃₈	1	1	yes	characteristic
267	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
268	C₂×C₄	#10597324	1	2	yes	—
269	C₂×C₄	#10597325	1	2	yes	—
270	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
271	C₂×C₄	#10597331	1	2	yes	—
272	C₂×C₄	#10597332	1	2	yes	—
273	C₂×C₄	#10597336	1	2	yes	—
274	C₂×C₄	#10597338	1	2	yes	—
275	C₂×C₄	#10597340	1	2	yes	—
276	C₂×C₄	#10597341	1	2	yes	—
277	C₂×C₄	#10597344	1	2	yes	—
278	C₂×C₄	#10597370	1	1	yes	characteristic
279	C₂×C₄	#10597376	1	1	yes	characteristic
280	C₂×C₄	#10597381	1	1	yes	characteristic
281	C₂×C₄	#10597382	1	1	yes	characteristic
282	C₂×C₄	#10597383	1	1	yes	characteristic
283	C₂×C₄	#10597393	1	1	yes	characteristic
284	C₈	—	4	2	no	—
285	C₈	—	4	2	no	—
286	C₈	—	76	4	no	—
287	C₈	—	76	4	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	4	no	—
293	C₂²	#11926409	1	1	yes	characteristic
294	C₂²	#11926416	1	1	yes	characteristic
295	C₂²	#11926442	1	1	yes	characteristic
296	C₂²	#11926476	1	2	yes	—
297	C₂²	#11926486	1	2	yes	—
298	C₂²	C₂×#10597279	1	2	yes	—
299	C₂²	#11926592	1	2	yes	—
300	C₄	—	2	4	no	—
301	C₄	—	4	2	no	—
302	C₄	#11926431	1	2	yes	—
303	C₄	#11926436	1	2	yes	—
304	C₄	#11926456	1	1	yes	characteristic
305	C₄	#11926461	1	1	yes	characteristic
306	C₄	#11926469	1	1	yes	characteristic
307	C₄	#11926483	1	1	yes	characteristic
308	C₂	—	2	2	no	—
309	C₂	—	2	2	no	—
310	C₂	#14561891	1	1	yes	characteristic
311	C₂	#14563772	1	1	yes	characteristic
312	C₂	#14563775	1	1	yes	characteristic
313	C₂	#14563839	1	2	yes	—
314	C₂	#14564410	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₁₉×#1728	C₂
socle	C₂³×C₁₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₉	C₂×Q₈
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	#1728	D₁₉
hypocenter	C₁₉	#11269
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #1728

Complements

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

N	H ≅ G/N	type
207. C₁₉	#11269	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926568	2
C₂×#11926308	2
C₂×#11926299	2
C₁₉×#1728	2
#11269	19