Subgroups — #18069683

Order 4864.

402 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊(C₁₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	#14563664	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉×#340	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₉⋊C₁₆	—	2	1	no	—
5	C₂×#10597275	C₄	1	2	yes	complemented
6	#11926404	—	2	1	no	—
7	C₂²⋊(C₁₉⋊C₁₆)	—	2	1	no	—
8	C₂²⋊(C₁₉⋊C₁₆)	C₄	1	2	yes	complemented
9	#11926307	—	2	1	no	—
10	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊C₄×C₁₉	—	2	1	no	—
12	C₁₉×#340	—	2	1	no	—
13	C₁₉×#340	C₄	1	2	yes	complemented
14	C₂³×C₄×C₁₉	—	4	1	no	—
15	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₉	—	4	2	no	—
17	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₉	Q₈	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
24	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
25	C₂×C₄⋊C₄×C₁₉	—	4	2	no	—
26	C₂×C₄⋊C₄×C₁₉	—	4	1	no	—
27	C₂×C₂²⋊Dic₁₉	—	4	1	no	—
28	C₂×C₄⋊Dic₁₉	—	2	2	no	—
29	C₁₉⋊#95	—	4	1	no	—
30	#10597275	—	4	2	no	—
31	C₂×C₁₉⋊C₁₆	—	2	2	no	—
32	C₂×C₁₉⋊C₁₆	—	4	2	no	—
33	C₂⁴×C₁₉	D₈	1	1	yes	characteristic
34	C₂²×C₄×C₁₉	—	2	2	no	—
35	C₂²×C₄×C₁₉	—	2	2	no	—
36	C₂²×C₄×C₁₉	—	4	4	no	—
37	C₂²×C₄×C₁₉	—	4	2	no	—
38	C₂²×C₄×C₁₉	—	8	2	no	—
39	C₂²×C₄×C₁₉	—	8	1	no	—
40	C₂²×C₄×C₁₉	—	8	1	no	—
41	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
46	C₂²×C₄×C₁₉	Q₁₆	1	1	yes	characteristic
47	C₂²×Dic₁₉	—	4	2	no	—
48	C₂×C₈×C₁₉	—	2	2	no	—
49	C₂×C₈×C₁₉	—	2	2	no	—
50	C₂×C₈×C₁₉	—	4	1	no	—
51	C₂×C₈×C₁₉	—	4	1	no	—
52	C₂×C₈×C₁₉	—	4	1	no	—
53	C₂×C₈×C₁₉	—	4	1	no	—
54	C₂×C₈×C₁₉	C₄²	1	1	yes	characteristic
55	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
58	C₄⋊C₄×C₁₉	—	2	4	no	—
59	C₄⋊C₄×C₁₉	—	4	4	no	—
60	C₄⋊C₄×C₁₉	—	4	2	no	—
61	C₂²⋊Dic₁₉	—	8	1	no	—
62	C₂²⋊Dic₁₉	—	8	1	no	—
63	C₄⋊Dic₁₉	—	4	2	no	—
64	C₄⋊Dic₁₉	—	4	2	no	—
65	C₁₉⋊C₁₆	—	4	2	no	—
66	#7258	—	19	1	no	maximal, Sylow
67	C₂³×C₁₉	—	2	2	no	—
68	C₂³×C₁₉	—	2	2	no	—
69	C₂³×C₁₉	#102	1	1	yes	characteristic
70	C₂³×C₁₉	#102	1	1	yes	characteristic
71	C₂³×C₁₉	#107	1	1	yes	characteristic
72	C₂³×C₁₉	D₁₆	1	1	yes	characteristic
73	C₂³×C₁₉	SD₃₂	1	1	yes	characteristic
74	C₂³×C₁₉	SD₃₂	1	1	yes	characteristic
75	C₂³×C₁₉	Q₃₂	1	1	yes	characteristic
76	C₂×C₄×C₁₉	—	2	2	no	—
77	C₂×C₄×C₁₉	—	2	2	no	—
78	C₂×C₄×C₁₉	—	2	2	no	—
79	C₂×C₄×C₁₉	—	2	1	no	—
80	C₂×C₄×C₁₉	—	2	1	no	—
81	C₂×C₄×C₁₉	—	2	1	no	—
82	C₂×C₄×C₁₉	—	2	1	no	—
83	C₂×C₄×C₁₉	—	4	4	no	—
84	C₂×C₄×C₁₉	—	8	4	no	—
85	C₂×C₄×C₁₉	—	8	4	no	—
86	C₂×C₄×C₁₉	—	8	2	no	—
87	C₂×C₄×C₁₉	—	8	2	no	—
88	C₂×C₄×C₁₉	#95	1	1	yes	characteristic
89	C₂×C₄×C₁₉	#99	1	2	yes	—
90	C₂×C₄×C₁₉	#103	1	1	yes	characteristic
91	C₂×C₄×C₁₉	#103	1	1	yes	characteristic
92	C₂×C₄×C₁₉	#104	1	2	yes	—
93	C₂×C₄×C₁₉	#106	1	1	yes	characteristic
94	C₂×Dic₁₉	—	8	2	no	—
95	C₂×Dic₁₉	—	8	2	no	—
96	C₂×Dic₁₉	—	8	2	no	—
97	C₈×C₁₉	—	2	1	no	—
98	C₈×C₁₉	—	2	1	no	—
99	C₈×C₁₉	—	2	1	no	—
100	C₈×C₁₉	—	2	1	no	—
101	C₂×C₂²⋊C₁₆	—	19	1	no	—
102	#1808	—	19	1	no	—
103	C₂×#340	D₁₉	1	1	yes	characteristic
104	C₂²×C₁₉	—	2	1	no	—
105	C₂²×C₁₉	—	2	1	no	—
106	C₂²×C₁₉	—	2	2	no	—
107	C₂²×C₁₉	—	2	2	no	—
108	C₂²×C₁₉	—	2	2	no	—
109	C₂²×C₁₉	—	2	2	no	—
110	C₂²×C₁₉	—	2	1	no	—
111	C₂²×C₁₉	—	2	1	no	—
112	C₂²×C₁₉	#327	1	2	yes	—
113	C₂²×C₁₉	#340	1	1	yes	characteristic
114	C₂²×C₁₉	#357	1	1	yes	characteristic
115	C₂²×C₁₉	#357	1	1	yes	characteristic
116	C₂²×C₁₉	#358	1	1	yes	characteristic
117	C₂²×C₁₉	#358	1	1	yes	characteristic
118	C₂²×C₁₉	#360	1	2	yes	—
119	C₂²×C₁₉	#366	1	1	yes	characteristic
120	C₂²×C₁₉	#367	1	1	yes	characteristic
121	C₄×C₁₉	—	2	1	no	—
122	C₄×C₁₉	—	2	1	no	—
123	C₄×C₁₉	—	8	4	no	—
124	C₄×C₁₉	#328	1	2	yes	—
125	C₄×C₁₉	#339	1	1	yes	characteristic
126	C₄×C₁₉	#342	1	1	yes	characteristic
127	Dic₁₉	—	16	2	no	—
128	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
129	C₂²×C₄⋊C₄	—	2	1	no	—
130	C₂²×C₁₆	—	38	1	no	—
131	C₂×#107	—	19	2	no	—
132	#380	—	38	1	no	—
133	C₂²⋊C₁₆	—	19	2	no	—
134	C₂²⋊C₁₆	—	38	1	no	—
135	#340	—	2	1	no	—
136	#340	—	38	1	no	—
137	#340	Dic₁₉	1	2	yes	complemented
138	C₂×C₁₉	—	2	1	no	—
139	C₂×C₁₉	—	2	1	no	—
140	C₂×C₁₉	—	2	1	no	—
141	C₂×C₁₉	—	2	1	no	—
142	C₂×C₁₉	#1294	1	1	yes	characteristic
143	C₂×C₁₉	#1347	1	2	yes	—
144	C₂×C₁₉	#1348	1	2	yes	—
145	C₂×C₁₉	#1380	1	1	yes	characteristic
146	C₂×C₁₉	#1386	1	1	yes	characteristic
147	C₂³×C₄	—	4	1	no	—
148	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
149	C₂²×C₈	—	4	2	no	—
150	C₂²×C₈	Dic₃₈	1	1	yes	characteristic
151	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
152	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
153	C₂²×C₈	D₇₆	1	1	yes	characteristic
154	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
155	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
156	C₂×C₄⋊C₄	—	2	4	no	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	—	4	2	no	—
159	C₂×C₄⋊C₄	—	4	1	no	—
160	C₂×C₄⋊C₄	—	38	2	no	—
161	C₂×C₂²⋊C₄	—	76	1	no	—
162	C₂×C₁₆	—	38	2	no	—
163	C₂×C₁₆	—	76	2	no	—
164	#107	—	76	2	no	—
165	#95	—	76	1	no	—
166	C₁₉	#7258	1	1	yes	characteristic, complemented, Sylow
167	C₂⁴	#64525	1	1	yes	characteristic
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	4	4	no	—
171	C₂²×C₄	—	4	2	no	—
172	C₂²×C₄	—	8	2	no	—
173	C₂²×C₄	—	8	1	no	—
174	C₂²×C₄	—	8	1	no	—
175	C₂²×C₄	—	76	2	no	—
176	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
177	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
178	C₂²×C₄	#64526	1	1	yes	characteristic
179	C₂²×C₄	#64527	1	1	yes	characteristic
180	C₂²×C₄	#64528	1	1	yes	characteristic
181	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	4	1	no	—
185	C₂×C₈	—	4	1	no	—
186	C₂×C₈	—	4	1	no	—
187	C₂×C₈	—	4	1	no	—
188	C₂×C₈	C₄×Dic₁₉	1	1	yes	characteristic
189	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
190	C₂×C₈	C₄⋊Dic₁₉	1	1	yes	characteristic
191	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	4	4	no	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄⋊C₄	—	76	2	no	—
196	C₄⋊C₄	—	76	2	no	—
197	C₂²⋊C₄	—	152	1	no	—
198	C₂²⋊C₄	—	152	1	no	—
199	C₁₆	—	76	2	no	—
200	C₂³	—	2	2	no	—
201	C₂³	—	2	2	no	—
202	C₂³	#10597264	1	1	yes	characteristic
203	C₂³	#10597266	1	1	yes	characteristic
204	C₂³	#10597283	1	1	yes	characteristic
205	C₂³	#10597284	1	1	yes	characteristic
206	C₂³	#10597285	1	1	yes	characteristic
207	C₂³	#10597286	1	1	yes	characteristic
208	C₂³	#10597289	1	1	yes	characteristic
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	8	4	no	—
218	C₂×C₄	—	8	4	no	—
219	C₂×C₄	—	8	2	no	—
220	C₂×C₄	—	8	2	no	—
221	C₂×C₄	—	152	2	no	—
222	C₂×C₄	—	152	2	no	—
223	C₂×C₄	—	152	2	no	—
224	C₂×C₄	#10597262	1	2	yes	—
225	C₂×C₄	#10597263	1	2	yes	—
226	C₂×C₄	#10597265	1	1	yes	characteristic
227	C₂×C₄	#10597267	1	1	yes	characteristic
228	C₂×C₄	C₁₉⋊#95	1	1	yes	characteristic
229	C₂×C₄	#10597292	1	1	yes	characteristic
230	C₈	—	2	1	no	—
231	C₈	—	2	1	no	—
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	2	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	1	no	—
242	C₂²	#11926277	1	2	yes	—
243	C₂²	#11926306	1	1	yes	characteristic
244	C₂²	#11926344	1	1	yes	characteristic
245	C₂²	#11926345	1	1	yes	characteristic
246	C₂²	#11926347	1	1	yes	characteristic
247	C₂²	#11926348	1	1	yes	characteristic
248	C₂²	#11926353	1	2	yes	—
249	C₂²	#11926369	1	1	yes	characteristic
250	C₂²	#11926371	1	1	yes	characteristic
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	8	4	no	—
254	C₄	—	304	2	no	—
255	C₄	#11926280	1	2	yes	—
256	C₄	#11926304	1	1	yes	characteristic
257	C₄	#11926310	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	#14561514	1	1	yes	characteristic
263	C₂	#14561643	1	2	yes	—
264	C₂	#14561646	1	2	yes	—
265	C₂	#14561726	1	1	yes	characteristic
266	C₂	#14561742	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597266
commutator	C₂×C₈×C₁₉	C₄²
Frattini	C₂³×C₈	C₂×D₁₉
Fitting	C₂×C₁₉×#340	C₂
socle	C₂³×C₁₉	#102
prime radical	C₂⁴×C₁₉	D₈
cosocle	C₂³×C₈×C₁₉	C₂²
hypercenter	C₂×#340	D₁₉
hypocenter	C₁₉	#7258
LCS	G ▷ C₂×C₈×C₁₉ ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#340

Complements

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

N	H ≅ G/N	type
5. C₂×#10597275	C₄	semidirect
8. C₂²⋊(C₁₉⋊C₁₆)	C₄	semidirect
13. C₁₉×#340	C₄	semidirect
137. #340	Dic₁₉	semidirect
166. C₁₉	#7258	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊(C₁₉⋊C₁₆)	2
#14563664	2
C₂×C₁₉×#340	2
#7258	19