Subgroups — #18071920

Order 4864.

798 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926597	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₁₉⋊#1273	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×M₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₁₉⋊C₈	C₄	1	2	yes	—
6	#11926597	—	2	2	no	—
7	#11926597	—	2	2	no	—
8	C₂×C₄⋊(C₁₉⋊C₈)	C₄	1	2	yes	—
9	C₂×C₂²⋊(C₁₉⋊C₈)	C₄	1	2	yes	—
10	C₂×C₄×C₁₉⋊C₈	—	2	2	no	—
11	C₂×C₄×C₁₉⋊C₈	—	2	2	no	—
12	C₂×C₄×C₁₉⋊C₈	C₄	1	2	yes	—
13	C₂×C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
14	C₂×C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
15	C₂²×M₁₆×C₁₉	C₄	1	2	yes	—
16	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
17	C₄×M₁₆×C₁₉	—	2	2	no	—
18	C₄×M₁₆×C₁₉	—	2	2	no	—
19	C₂×C₁₉×#97	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
21	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₂×M₁₆×C₁₉	—	2	4	no	—
25	C₂×M₁₆×C₁₉	—	2	4	no	—
26	C₂×M₁₆×C₁₉	C₈	1	8	yes	complemented
27	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₉	D₄	1	2	yes	—
32	C₂×C₄²×C₁₉	D₄	1	2	yes	—
33	C₂×C₄²×C₁₉	D₄	1	2	yes	—
34	C₂×C₄²×C₁₉	Q₈	1	2	yes	—
35	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
36	C₂²×C₁₉⋊C₈	—	2	4	no	—
37	C₂²×C₁₉⋊C₈	—	2	8	no	—
38	C₂²×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
39	C₂²×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
40	C₁₉×#97	—	2	4	no	—
41	C₄×C₈×C₁₉	—	2	2	no	—
42	C₄×C₈×C₁₉	—	2	2	no	—
43	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
44	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
45	C₄×C₁₉⋊C₈	—	2	4	no	—
46	C₄×C₁₉⋊C₈	—	2	2	no	—
47	C₄×C₁₉⋊C₈	—	2	2	no	—
48	C₄×C₁₉⋊C₈	—	2	2	no	—
49	C₄×C₁₉⋊C₈	—	2	2	no	—
50	C₂⁴×C₁₉	C₄²	1	1	yes	characteristic
51	C₂²×C₄×C₁₉	—	2	4	no	—
52	C₂²×C₄×C₁₉	—	2	4	no	—
53	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₁₉	C₂×C₈	1	4	yes	—
63	C₂²×C₄×C₁₉	M₁₆	1	4	yes	—
64	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
67	M₁₆×C₁₉	—	2	16	no	—
68	C₂×C₈×C₁₉	—	2	4	no	—
69	C₂×C₈×C₁₉	—	2	4	no	—
70	C₂×C₈×C₁₉	C₂×C₈	1	8	yes	complemented
71	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
72	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
73	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
74	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
75	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
76	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
77	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
81	C₂×C₁₉⋊C₈	—	2	16	no	—
82	C₂×C₁₉⋊C₈	—	2	4	no	—
83	C₂×C₁₉⋊C₈	—	2	4	no	—
84	C₂×C₁₉⋊C₈	—	2	4	no	—
85	C₂×C₁₉⋊C₈	—	2	8	no	—
86	C₂×C₁₉⋊C₈	M₁₆	1	8	yes	complemented
87	#7848	—	19	1	no	maximal, Sylow
88	C₂³×C₁₉	—	2	2	no	—
89	C₂³×C₁₉	—	2	2	no	—
90	C₂³×C₁₉	#95	1	2	yes	—
91	C₂³×C₁₉	C₄×C₈	1	2	yes	—
92	C₂³×C₁₉	#97	1	2	yes	—
93	C₂³×C₁₉	C₂×C₄²	1	1	yes	characteristic
94	C₂×C₄×C₁₉	—	2	2	no	—
95	C₂×C₄×C₁₉	—	2	2	no	—
96	C₂×C₄×C₁₉	—	2	4	no	—
97	C₂×C₄×C₁₉	—	2	4	no	—
98	C₂×C₄×C₁₉	—	2	4	no	—
99	C₂×C₄×C₁₉	—	2	4	no	—
100	C₂×C₄×C₁₉	#95	1	2	yes	—
101	C₂×C₄×C₁₉	#95	1	2	yes	—
102	C₂×C₄×C₁₉	#95	1	2	yes	—
103	C₂×C₄×C₁₉	C₄×C₈	1	2	yes	—
104	C₂×C₄×C₁₉	#97	1	2	yes	—
105	C₂×C₄×C₁₉	C₂²⋊C₈	1	8	yes	—
106	C₂×C₄×C₁₉	C₄⋊C₈	1	8	yes	—
107	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₉	C₂²×C₈	1	2	yes	—
114	C₂×C₄×C₁₉	C₂×M₁₆	1	4	yes	—
115	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
116	C₂×C₄×C₁₉	#131	1	4	yes	—
117	C₈×C₁₉	—	2	8	no	—
118	C₁₉⋊C₈	—	2	8	no	—
119	C₁₉⋊C₈	—	4	4	no	—
120	C₂×#431	—	19	1	no	—
121	C₂×C₄×M₁₆	D₁₉	1	1	yes	characteristic
122	C₂²×C₄×C₈	—	19	1	no	—
123	#1273	—	19	4	no	—
124	C₂²×C₁₉	—	2	4	no	—
125	C₂²×C₁₉	—	2	4	no	—
126	C₂²×C₁₉	—	2	2	no	—
127	C₂²×C₁₉	—	2	2	no	—
128	C₂²×C₁₉	#336	1	4	yes	—
129	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
130	C₂²×C₁₉	C₂×C₄×C₈	1	1	yes	characteristic
131	C₂²×C₁₉	C₂×#97	1	1	yes	characteristic
132	C₂²×C₁₉	C₄×M₁₆	1	2	yes	—
133	C₂²×C₁₉	#405	1	2	yes	—
134	C₄×C₁₉	—	2	2	no	—
135	C₄×C₁₉	—	2	2	no	—
136	C₄×C₁₉	#336	1	4	yes	—
137	C₄×C₁₉	C₂×C₂²⋊C₈	1	2	yes	—
138	C₄×C₁₉	#407	1	2	yes	—
139	C₄×C₁₉	#408	1	2	yes	—
140	C₄×C₁₉	C₂×C₄⋊C₈	1	2	yes	—
141	C₄×C₁₉	#423	1	2	yes	—
142	C₄×C₁₉	#424	1	2	yes	—
143	C₂²×M₁₆	Dic₁₉	1	2	yes	—
144	C₂³×C₈	—	19	2	no	—
145	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
146	#431	—	38	2	no	—
147	#431	—	38	2	no	—
148	C₂×C₄⋊C₈	—	19	2	no	—
149	C₂×C₂²⋊C₈	—	19	2	no	—
150	C₄×M₁₆	—	2	2	no	—
151	C₄×M₁₆	—	2	2	no	—
152	C₂×#97	Dic₁₉	1	2	yes	—
153	C₂×C₄×C₈	—	19	2	no	—
154	C₂×C₄×C₈	—	19	2	no	—
155	C₂×C₄×C₈	—	19	2	no	—
156	C₂×C₄×C₈	—	38	2	no	—
157	C₂×C₄×C₈	—	38	2	no	—
158	C₂×C₄×C₈	C₂×D₁₉	1	2	yes	—
159	C₂×C₁₉	—	2	4	no	—
160	C₂×C₁₉	C₈×M₁₆	1	2	yes	—
161	C₂×C₁₉	#1451	1	2	yes	—
162	C₂×C₁₉	C₂×#336	1	1	yes	characteristic
163	C₂×C₁₉	#1728	1	1	yes	characteristic
164	C₂×C₁₉	#1729	1	1	yes	characteristic
165	C₂³×C₄	C₄×D₁₉	1	1	yes	characteristic
166	C₂³×C₄	C₄×D₁₉	1	1	yes	characteristic
167	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
168	C₂×M₁₆	—	2	4	no	—
169	C₂×M₁₆	—	2	4	no	—
170	C₂×M₁₆	C₁₉⋊C₈	1	8	yes	complemented
171	C₂²×C₈	—	19	4	no	—
172	C₂²×C₈	—	19	4	no	—
173	C₂²×C₈	—	38	8	no	—
174	C₂²×C₈	—	38	4	no	—
175	C₂²×C₈	C₂×Dic₁₉	1	4	yes	—
176	C₂×C₄²	Dic₃₈	1	2	yes	—
177	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
178	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
179	C₂×C₄²	D₇₆	1	2	yes	—
180	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
181	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
182	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
183	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
184	C₄⋊C₈	—	38	4	no	—
185	C₂²⋊C₈	—	38	4	no	—
186	#97	—	2	4	no	—
187	C₄×C₈	—	2	2	no	—
188	C₄×C₈	—	2	2	no	—
189	C₄×C₈	—	38	4	no	—
190	C₄×C₈	—	38	2	no	—
191	C₄×C₈	—	38	2	no	—
192	C₄×C₈	—	38	2	no	—
193	C₄×C₈	—	38	2	no	—
194	C₁₉	#7848	1	1	yes	characteristic, complemented, Sylow
195	C₂⁴	C₄×Dic₁₉	1	1	yes	characteristic
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	C₂×C₁₉⋊C₈	1	4	yes	—
199	C₂²×C₄	#64520	1	4	yes	—
200	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
201	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
202	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
203	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
204	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
205	C₂²×C₄	C₄⋊Dic₁₉	1	2	yes	—
206	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
207	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
208	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
209	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
211	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
212	M₁₆	—	2	16	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	19	8	no	—
216	C₂×C₈	—	38	16	no	—
217	C₂×C₈	—	38	4	no	—
218	C₂×C₈	—	38	4	no	—
219	C₂×C₈	—	38	8	no	—
220	C₂×C₈	—	38	4	no	—
221	C₂×C₈	C₂×C₁₉⋊C₈	1	8	yes	complemented
222	C₄²	Dic₁₉⋊C₄	1	2	yes	—
223	C₄²	Dic₁₉⋊C₄	1	2	yes	—
224	C₄²	C₄⋊Dic₁₉	1	2	yes	—
225	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
226	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
227	C₄²	C₂²⋊Dic₁₉	1	2	yes	—
228	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
229	C₄²	C₂×D₇₆	1	1	yes	characteristic
230	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
231	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
232	C₂³	—	2	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	C₄×C₁₉⋊C₈	1	2	yes	—
235	C₂³	#10597260	1	2	yes	—
236	C₂³	C₁₉⋊#95	1	2	yes	—
237	C₂³	C₂×C₄×Dic₁₉	1	1	yes	characteristic
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	C₄×C₁₉⋊C₈	1	2	yes	—
245	C₂×C₄	#10597260	1	2	yes	—
246	C₂×C₄	C₄⋊(C₁₉⋊C₈)	1	8	yes	—
247	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	8	yes	—
248	C₂×C₄	C₁₉⋊#95	1	2	yes	—
249	C₂×C₄	C₁₉⋊#95	1	2	yes	—
250	C₂×C₄	C₁₉⋊#95	1	2	yes	—
251	C₂×C₄	M₁₆×D₁₉	1	4	yes	—
252	C₂×C₄	#10597353	1	4	yes	—
253	C₂×C₄	C₂²×C₁₉⋊C₈	1	2	yes	—
254	C₂×C₄	C₂×#64520	1	2	yes	—
255	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
256	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
258	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
259	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
260	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
261	C₈	—	2	8	no	—
262	C₈	—	38	8	no	—
263	C₈	—	76	4	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	4	no	—
267	C₂²	—	2	4	no	—
268	C₂²	#11926299	1	4	yes	—
269	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
270	C₂²	C₂×C₄×C₁₉⋊C₈	1	1	yes	characteristic
271	C₂²	C₂×#10597260	1	1	yes	characteristic
272	C₂²	M₁₆×Dic₁₉	1	2	yes	—
273	C₂²	#11926502	1	2	yes	—
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	#11926299	1	4	yes	—
277	C₄	C₂×C₂²⋊(C₁₉⋊C₈)	1	2	yes	—
278	C₄	#11926507	1	2	yes	—
279	C₄	#11926512	1	2	yes	—
280	C₄	C₂×C₄⋊(C₁₉⋊C₈)	1	2	yes	—
281	C₄	#11926567	1	2	yes	—
282	C₄	#11926572	1	2	yes	—
283	C₂	—	2	4	no	—
284	C₂	M₁₆×C₁₉⋊C₈	1	2	yes	—
285	C₂	#14561922	1	2	yes	—
286	C₂	C₂×#11926299	1	1	yes	characteristic
287	C₂	#14563328	1	1	yes	characteristic
288	C₂	#14563333	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₁₉
commutator	C₂²×C₁₉	C₂×C₄×C₈
Fitting	C₂×C₄×M₁₆×C₁₉	C₂
socle	C₂³×C₁₉	C₂×C₄²
prime radical	C₂⁴×C₁₉	C₄²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×C₄×M₁₆	D₁₉
hypocenter	C₁₉	#7848
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₁₉⋊#1273	C₂	semidirect
26. C₂×M₁₆×C₁₉	C₈	semidirect
70. C₂×C₈×C₁₉	C₂×C₈	semidirect
86. C₂×C₁₉⋊C₈	M₁₆	semidirect
170. C₂×M₁₆	C₁₉⋊C₈	semidirect
194. C₁₉	#7848	semidirect
221. C₂×C₈	C₂×C₁₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926597	2
C₂²×C₄×C₁₉⋊C₈	2
C₁₉⋊#1273	2
C₂×C₄×M₁₆×C₁₉	2
#7848	19