Subgroups — #18090981

Order 4864.

578 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14572199	C₂	1	1	yes	maximal, characteristic
2	#14563374	C₂	1	1	yes	maximal, characteristic
3	#14563361	C₂	1	2	yes	maximal
4	C₂×#11926309	C₂	1	2	yes	maximal
5	C₁₉×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	C₂²	1	2	yes	—
7	C₂²×#64520	C₂²	1	2	yes	—
8	#11926571	—	2	4	no	—
9	#11926568	—	2	2	no	—
10	#11926313	C₄	1	4	yes	—
11	#11926309	—	2	4	no	—
12	#11926308	—	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×#336	—	2	2	no	—
17	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
18	C₂×M₁₆×C₁₉	—	2	1	no	—
19	C₂×M₁₆×C₁₉	—	2	2	no	—
20	C₂×M₁₆×C₁₉	—	2	2	no	—
21	C₂×M₁₆×C₁₉	Q₈	1	2	yes	—
22	C₂²×C₈×C₁₉	—	2	1	no	—
23	C₂²×C₈×C₁₉	—	2	1	no	—
24	C₂²×C₈×C₁₉	—	2	1	no	—
25	C₂²×C₈×C₁₉	—	2	2	no	—
26	C₂²×C₈×C₁₉	—	2	2	no	—
27	C₂²×C₈×C₁₉	D₄	1	2	yes	—
28	C₁₉×#117	—	2	2	no	—
29	C₁₉×#117	—	2	2	no	—
30	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
31	C₂×C₂²⋊C₄×C₁₉	Q₈	1	2	yes	—
32	C₂×C₄²×C₁₉	D₄	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	—	2	2	no	—
41	C₂×#64520	—	2	2	no	—
42	C₂×#64520	—	2	2	no	—
43	C₂×#64520	C₂×C₄	1	4	yes	—
44	C₂×#64520	D₄	1	4	yes	—
45	C₂²×C₁₉⋊C₈	—	2	2	no	—
46	C₂²×C₁₉⋊C₈	—	2	2	no	—
47	C₄⋊(C₁₉⋊C₈)	—	2	8	no	—
48	C₂⁴×C₁₉	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
50	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	#41	1	2	yes	—
58	C₂²×C₄×C₁₉	#41	1	2	yes	—
59	C₂²×C₄×C₁₉	#41	1	2	yes	—
60	M₁₆×C₁₉	—	2	4	no	—
61	M₁₆×C₁₉	—	4	2	no	—
62	C₂×C₈×C₁₉	—	2	2	no	—
63	C₂×C₈×C₁₉	—	2	2	no	—
64	C₂×C₈×C₁₉	—	2	4	no	—
65	C₂×C₈×C₁₉	—	4	1	no	—
66	C₂×C₈×C₁₉	—	4	1	no	—
67	C₂×C₈×C₁₉	—	4	1	no	—
68	C₂×C₈×C₁₉	—	4	1	no	—
69	C₂×C₈×C₁₉	—	4	2	no	—
70	C₄⋊C₄×C₁₉	—	2	4	no	—
71	C₂²⋊C₄×C₁₉	—	2	4	no	—
72	C₄²×C₁₉	—	2	2	no	—
73	C₄²×C₁₉	—	2	2	no	—
74	#64520	—	2	4	no	—
75	#64520	—	2	4	no	—
76	#64520	—	2	4	no	—
77	#64520	—	2	4	no	—
78	#64520	—	4	2	no	—
79	#64520	—	4	2	no	—
80	#64520	—	4	2	no	—
81	#64520	—	4	2	no	—
82	C₂×C₁₉⋊C₈	—	2	4	no	—
83	C₂×C₁₉⋊C₈	—	2	4	no	—
84	C₂×C₁₉⋊C₈	—	4	2	no	—
85	C₂×C₁₉⋊C₈	—	4	2	no	—
86	C₂×C₁₉⋊C₈	—	4	2	no	—
87	C₂×C₁₉⋊C₈	—	4	2	no	—
88	#11297	—	19	1	no	maximal, Sylow
89	C₂³×C₁₉	—	4	2	no	—
90	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂³×C₁₉	C₂²⋊Q₈	1	2	yes	—
92	C₂³×C₁₉	#125	1	1	yes	characteristic
93	C₂³×C₁₉	#125	1	2	yes	—
94	C₂³×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₁₉	—	2	4	no	—
96	C₂×C₄×C₁₉	—	4	4	no	—
97	C₂×C₄×C₁₉	#117	1	2	yes	—
98	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
100	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
101	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
102	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
103	C₂×C₄×C₁₉	#121	1	2	yes	—
104	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
105	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
107	C₂×C₄×C₁₉	#123	1	2	yes	—
108	C₂×C₄×C₁₉	#123	1	2	yes	—
109	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
110	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
111	C₂×C₄×C₁₉	#124	1	2	yes	—
112	C₂×C₄×C₁₉	#125	1	2	yes	—
113	C₂×C₄×C₁₉	#126	1	2	yes	—
114	C₂×C₄×C₁₉	#126	1	2	yes	—
115	C₈×C₁₉	—	4	2	no	—
116	C₈×C₁₉	—	4	2	no	—
117	C₁₉⋊C₈	—	4	4	no	—
118	C₁₉⋊C₈	—	4	4	no	—
119	#2905	—	19	1	no	—
120	#1742	—	19	1	no	—
121	#1738	—	19	2	no	—
122	C₂×#341	—	19	2	no	—
123	#1729	D₁₉	1	1	yes	characteristic
124	C₂²×C₁₉	—	2	2	no	—
125	C₂²×C₁₉	—	2	4	no	—
126	C₂²×C₁₉	—	4	1	no	—
127	C₂²×C₁₉	—	4	2	no	—
128	C₂²×C₁₉	—	4	1	no	—
129	C₂²×C₁₉	#382	1	1	yes	characteristic
130	C₂²×C₁₉	#385	1	1	yes	characteristic
131	C₂²×C₁₉	#389	1	1	yes	characteristic
132	C₂²×C₁₉	#398	1	2	yes	—
133	C₂²×C₁₉	#400	1	2	yes	—
134	C₄×C₁₉	—	4	4	no	—
135	C₄×C₁₉	#387	1	2	yes	—
136	C₄×C₁₉	#388	1	2	yes	—
137	C₄×C₁₉	#393	1	1	yes	characteristic
138	C₄×C₁₉	#394	1	1	yes	characteristic
139	C₄×C₁₉	#397	1	1	yes	characteristic
140	C₄×C₁₉	#400	1	1	yes	characteristic
141	C₂²×M₁₆	—	19	2	no	—
142	C₂²×M₁₆	—	19	2	no	—
143	C₂²×M₁₆	C₂×D₁₉	1	1	yes	characteristic
144	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
145	C₂×#117	C₂×D₁₉	1	1	yes	characteristic
146	#424	—	38	4	no	—
147	#423	—	38	2	no	—
148	#343	—	19	4	no	—
149	#341	—	38	4	no	—
150	#341	—	38	4	no	—
151	#336	—	2	2	no	—
152	C₂×C₁₉	—	2	1	no	—
153	C₂×C₁₉	—	2	1	no	—
154	C₂×C₁₉	—	2	2	no	—
155	C₂×C₁₉	—	2	2	no	—
156	C₂×C₁₉	#1442	1	1	yes	characteristic
157	C₂×C₁₉	#1858	1	2	yes	—
158	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
159	C₂×M₁₆	—	2	1	no	—
160	C₂×M₁₆	—	2	2	no	—
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	—	19	4	no	—
163	C₂×M₁₆	—	19	4	no	—
164	C₂×M₁₆	—	38	2	no	—
165	C₂×M₁₆	—	38	2	no	—
166	C₂×M₁₆	—	38	2	no	—
167	C₂×M₁₆	—	38	2	no	—
168	C₂×M₁₆	—	38	2	no	—
169	C₂×M₁₆	—	38	2	no	—
170	C₂×M₁₆	—	38	2	no	—
171	C₂×M₁₆	—	38	2	no	—
172	C₂×M₁₆	—	38	2	no	—
173	C₂×M₁₆	—	38	2	no	—
174	C₂×M₁₆	Dic₃₈	1	2	yes	—
175	C₂²×C₈	—	2	1	no	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	2	1	no	—
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	—	38	2	no	—
181	C₂²×C₈	—	38	2	no	—
182	C₂²×C₈	D₇₆	1	2	yes	—
183	#117	—	2	2	no	—
184	#117	—	2	2	no	—
185	C₂×C₄⋊C₄	C₄×D₁₉	1	2	yes	—
186	C₂×C₂²⋊C₄	Dic₃₈	1	2	yes	—
187	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
188	C₄⋊C₈	—	38	8	no	—
189	C₁₉	#11297	1	1	yes	characteristic, complemented, Sylow
190	C₂⁴	C₂×Dic₃₈	1	1	yes	characteristic
191	C₂²×C₄	Dic₁₉⋊C₄	1	4	yes	—
192	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
195	C₂²×C₄	#64541	1	1	yes	characteristic
196	C₂²×C₄	#64541	1	1	yes	characteristic
197	C₂²×C₄	D₄×D₁₉	1	2	yes	—
198	C₂²×C₄	#64543	1	2	yes	—
199	C₂²×C₄	#64543	1	2	yes	—
200	C₂²×C₄	#64545	1	2	yes	—
201	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
202	M₁₆	—	2	4	no	—
203	M₁₆	—	4	2	no	—
204	M₁₆	—	38	4	no	—
205	M₁₆	—	38	4	no	—
206	M₁₆	—	38	4	no	—
207	M₁₆	—	38	4	no	—
208	M₁₆	—	76	2	no	—
209	M₁₆	—	76	2	no	—
210	M₁₆	—	76	2	no	—
211	M₁₆	—	76	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	4	1	no	—
216	C₂×C₈	—	4	1	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	38	4	no	—
221	C₂×C₈	—	38	4	no	—
222	C₂×C₈	—	76	2	no	—
223	C₂×C₈	—	76	2	no	—
224	C₂×C₈	—	76	2	no	—
225	C₂×C₈	—	76	2	no	—
226	C₄⋊C₄	—	2	4	no	—
227	C₂²⋊C₄	—	2	4	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	2	no	—
230	C₂³	—	4	2	no	—
231	C₂³	#10597315	1	1	yes	characteristic
232	C₂³	#10597316	1	1	yes	characteristic
233	C₂³	#10597324	1	2	yes	—
234	C₂³	#10597336	1	2	yes	—
235	C₂³	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	C₄×Dic₃₈	1	1	yes	characteristic
239	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
240	C₂×C₄	#10597324	1	2	yes	—
241	C₂×C₄	#10597325	1	2	yes	—
242	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
243	C₂×C₄	#10597331	1	2	yes	—
244	C₂×C₄	#10597332	1	2	yes	—
245	C₂×C₄	#10597336	1	2	yes	—
246	C₂×C₄	#10597338	1	2	yes	—
247	C₂×C₄	#10597340	1	2	yes	—
248	C₂×C₄	#10597341	1	2	yes	—
249	C₂×C₄	#10597344	1	2	yes	—
250	C₂×C₄	#10597370	1	1	yes	characteristic
251	C₂×C₄	#10597376	1	1	yes	characteristic
252	C₂×C₄	#10597381	1	1	yes	characteristic
253	C₂×C₄	#10597382	1	1	yes	characteristic
254	C₂×C₄	#10597383	1	1	yes	characteristic
255	C₂×C₄	#10597393	1	1	yes	characteristic
256	C₈	—	4	2	no	—
257	C₈	—	4	2	no	—
258	C₈	—	76	4	no	—
259	C₈	—	76	4	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	4	no	—
262	C₂²	—	4	1	no	—
263	C₂²	—	4	1	no	—
264	C₂²	—	4	2	no	—
265	C₂²	#11926409	1	1	yes	characteristic
266	C₂²	#11926416	1	1	yes	characteristic
267	C₂²	#11926442	1	1	yes	characteristic
268	C₂²	#11926476	1	2	yes	—
269	C₂²	#11926486	1	2	yes	—
270	C₄	—	4	4	no	—
271	C₄	#11926431	1	2	yes	—
272	C₄	#11926436	1	2	yes	—
273	C₄	#11926456	1	1	yes	characteristic
274	C₄	#11926461	1	1	yes	characteristic
275	C₄	#11926469	1	1	yes	characteristic
276	C₄	#11926483	1	1	yes	characteristic
277	C₂	—	2	1	no	—
278	C₂	—	2	1	no	—
279	C₂	—	2	2	no	—
280	C₂	—	2	2	no	—
281	C₂	#14561891	1	1	yes	characteristic
282	C₂	#14563854	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#10597383
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₁₉×#1729	C₂
socle	C₂²×C₁₉	#385
prime radical	C₂⁴×C₁₉	C₂×Q₈
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	#1729	D₁₉
hypocenter	C₁₉	#11297
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1729

Complements

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

N	H ≅ G/N	type
189. C₁₉	#11297	semidirect

Maximal subgroups

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

subgroup	index
#14572199	2
#14563374	2
#14563361	2
C₂×#11926309	2
C₁₉×#1729	2
#11297	19