Subgroups — #18081193

Order 4864.

382 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14564399	C₂	1	2	yes	maximal
2	#14563795	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊Dic₁₉	C₂	1	1	yes	maximal, characteristic
4	#14561519	C₂	1	2	yes	maximal
5	C₂×C₁₉×#335	C₂	1	1	yes	maximal, characteristic
6	C₂×#10597275	—	2	1	no	—
7	C₂×#10597275	—	2	1	no	—
8	C₂×#10597274	—	2	2	no	—
9	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₁₉	—	2	1	no	—
11	#11926414	C₂²	1	2	yes	—
12	C₄×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
13	#11926307	—	2	2	no	—
14	#11926298	—	2	1	no	—
15	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#335	C₄	1	4	yes	complemented
18	C₂²×C₈×C₁₉	—	2	2	no	—
19	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₉	—	2	2	no	—
23	C₂×C₄⋊Dic₁₉	—	2	2	no	—
24	C₂×C₄⋊Dic₁₉	—	2	2	no	—
25	C₂×C₄⋊Dic₁₉	D₄	1	1	yes	characteristic
26	C₂×C₄⋊Dic₁₉	D₄	1	1	yes	characteristic
27	C₂×C₄×Dic₁₉	—	2	1	no	—
28	C₂²×C₁₉⋊C₈	—	2	1	no	—
29	C₁₉×C₄⋊C₈	—	2	2	no	—
30	C₁₉×C₄⋊C₈	C₂×C₄	1	4	yes	—
31	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₁₉	Q₈	1	2	yes	—
33	C₁₉⋊#95	—	2	1	no	—
34	#10597275	—	4	2	no	—
35	#10597275	—	4	2	no	—
36	#10597274	—	4	4	no	—
37	C₈×Dic₁₉	—	4	2	no	—
38	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
39	C₄⋊(C₁₉⋊C₈)	Q₈	1	2	yes	—
40	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
43	C₂²×Dic₁₉	—	2	1	no	—
44	C₂²×Dic₁₉	—	2	1	no	—
45	C₂²×Dic₁₉	—	4	2	no	—
46	C₂×C₈×C₁₉	—	2	1	no	—
47	C₂×C₈×C₁₉	—	2	1	no	—
48	C₂×C₈×C₁₉	—	2	4	no	—
49	C₂×C₈×C₁₉	—	4	2	no	—
50	C₂×C₈×C₁₉	—	4	2	no	—
51	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
52	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
53	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
54	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
55	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
57	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₁₉	#41	1	1	yes	characteristic
59	C₄⋊Dic₁₉	—	2	1	no	—
60	C₄⋊Dic₁₉	—	2	1	no	—
61	C₄⋊Dic₁₉	—	4	2	no	—
62	C₄⋊Dic₁₉	—	4	2	no	—
63	C₄⋊Dic₁₉	—	4	2	no	—
64	C₄⋊Dic₁₉	—	4	2	no	—
65	C₄⋊Dic₁₉	—	8	2	no	—
66	C₄⋊Dic₁₉	D₈	1	2	yes	—
67	C₄⋊Dic₁₉	Q₁₆	1	2	yes	—
68	C₄×Dic₁₉	—	4	1	no	—
69	C₄×Dic₁₉	—	4	1	no	—
70	C₂×C₁₉⋊C₈	—	2	2	no	—
71	C₂×C₁₉⋊C₈	—	4	1	no	—
72	C₂×C₁₉⋊C₈	—	4	1	no	—
73	#9615	—	19	1	no	maximal, Sylow
74	C₂³×C₁₉	#121	1	1	yes	characteristic
75	C₂×C₄×C₁₉	—	2	1	no	—
76	C₂×C₄×C₁₉	—	2	1	no	—
77	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
80	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₁₉	#125	1	1	yes	characteristic
82	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
85	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
86	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
87	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
88	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
89	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
90	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
91	C₂×Dic₁₉	—	2	2	no	—
92	C₂×Dic₁₉	—	2	2	no	—
93	C₂×Dic₁₉	—	4	1	no	—
94	C₂×Dic₁₉	—	4	1	no	—
95	C₂×Dic₁₉	—	4	1	no	—
96	C₂×Dic₁₉	—	4	1	no	—
97	C₂×Dic₁₉	—	8	2	no	—
98	C₂×Dic₁₉	—	8	2	no	—
99	C₂×Dic₁₉	—	8	2	no	—
100	C₈×C₁₉	—	2	2	no	—
101	C₈×C₁₉	—	4	4	no	—
102	C₈×C₁₉	#106	1	4	yes	complemented
103	C₁₉⋊C₈	—	4	2	no	—
104	#1948	—	19	2	no	—
105	#1845	—	19	1	no	—
106	C₈×C₄⋊C₄	—	19	1	no	—
107	C₂×#335	D₁₉	1	1	yes	characteristic
108	#1296	—	19	2	no	—
109	C₂²×C₁₉	#389	1	1	yes	characteristic
110	C₂²×C₁₉	#460	1	1	yes	characteristic
111	C₂²×C₁₉	#461	1	1	yes	characteristic
112	C₂²×C₁₉	#466	1	1	yes	characteristic
113	C₂²×C₁₉	#467	1	1	yes	characteristic
114	C₂²×C₁₉	#472	1	1	yes	characteristic
115	C₂²×C₁₉	#473	1	1	yes	characteristic
116	C₄×C₁₉	—	2	1	no	—
117	C₄×C₁₉	—	2	1	no	—
118	C₄×C₁₉	C₂×#106	1	1	yes	characteristic
119	C₄×C₁₉	#428	1	1	yes	characteristic
120	C₄×C₁₉	C₄×D₈	1	1	yes	characteristic
121	C₄×C₁₉	C₄×Q₁₆	1	1	yes	characteristic
122	C₄×C₁₉	#475	1	1	yes	characteristic
123	C₄×C₁₉	#476	1	1	yes	characteristic
124	C₄×C₁₉	#499	1	1	yes	characteristic
125	C₄×C₁₉	#500	1	1	yes	characteristic
126	Dic₁₉	—	4	2	no	—
127	Dic₁₉	—	4	2	no	—
128	Dic₁₉	—	16	2	no	—
129	C₂×#107	—	38	1	no	—
130	C₂×#107	—	38	1	no	—
131	C₂×#106	—	38	2	no	—
132	C₂×C₄⋊C₈	—	19	1	no	—
133	C₂×C₄⋊C₈	C₂×D₁₉	1	2	yes	—
134	C₂×C₄×C₈	—	38	1	no	—
135	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
136	#384	—	19	2	no	—
137	C₄×C₄⋊C₄	—	19	1	no	—
138	#340	—	38	2	no	—
139	#336	—	38	1	no	—
140	#335	Dic₁₉	1	4	yes	complemented
141	C₂×C₁₉	#1668	1	1	yes	characteristic
142	C₂×C₁₉	#1670	1	1	yes	characteristic
143	C₂×C₁₉	#1673	1	1	yes	characteristic
144	C₂×C₁₉	#1675	1	1	yes	characteristic
145	C₂×C₁₉	#1942	1	1	yes	characteristic
146	C₂×C₁₉	#1948	1	1	yes	characteristic
147	C₂×C₁₉	#1952	1	1	yes	characteristic
148	C₂²×C₈	—	2	2	no	—
149	C₂²×C₈	—	38	1	no	—
150	C₂²×C₈	D₇₆	1	1	yes	characteristic
151	C₂²×C₈	D₇₆	1	1	yes	characteristic
152	C₂×C₄⋊C₄	—	19	1	no	—
153	C₂×C₄⋊C₄	—	19	1	no	—
154	C₂×C₄⋊C₄	—	38	2	no	—
155	C₂×C₄⋊C₄	—	38	2	no	—
156	C₂×C₄⋊C₄	—	38	2	no	—
157	C₂×C₄²	—	38	1	no	—
158	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
159	#107	—	76	2	no	—
160	#107	—	76	2	no	—
161	#106	—	76	4	no	—
162	C₄⋊C₈	—	2	2	no	—
163	C₄⋊C₈	—	19	2	no	—
164	C₄⋊C₈	—	38	1	no	—
165	C₄⋊C₈	C₂×Dic₁₉	1	4	yes	—
166	C₄×C₈	—	76	2	no	—
167	C₄×C₈	Dic₃₈	1	2	yes	—
168	C₄×C₈	C₂×Dic₁₉	1	2	yes	—
169	#95	—	38	1	no	—
170	C₁₉	#9615	1	1	yes	characteristic, complemented, Sylow
171	C₂²×C₄	—	38	1	no	—
172	C₂²×C₄	—	38	1	no	—
173	C₂²×C₄	—	76	2	no	—
174	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
175	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
176	C₂²×C₄	#64545	1	1	yes	characteristic
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	4	2	no	—
181	C₂×C₈	—	4	2	no	—
182	C₂×C₈	—	38	2	no	—
183	C₂×C₈	—	76	1	no	—
184	C₂×C₈	—	76	1	no	—
185	C₂×C₈	#64516	1	2	yes	—
186	C₂×C₈	#64516	1	2	yes	—
187	C₂×C₈	C₄⋊Dic₁₉	1	2	yes	—
188	C₂×C₈	C₄⋊Dic₁₉	1	2	yes	—
189	C₄⋊C₄	—	19	2	no	—
190	C₄⋊C₄	—	19	2	no	—
191	C₄⋊C₄	—	38	1	no	—
192	C₄⋊C₄	—	38	1	no	—
193	C₄⋊C₄	—	76	2	no	—
194	C₄⋊C₄	—	76	2	no	—
195	C₄⋊C₄	—	76	2	no	—
196	C₄⋊C₄	—	76	2	no	—
197	C₄⋊C₄	—	152	2	no	—
198	C₄²	—	76	1	no	—
199	C₄²	—	76	1	no	—
200	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
201	C₄²	#64543	1	1	yes	characteristic
202	C₄²	Q₈×D₁₉	1	1	yes	characteristic
203	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
204	C₂³	#10597341	1	1	yes	characteristic
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	38	2	no	—
208	C₂×C₄	—	38	2	no	—
209	C₂×C₄	—	76	1	no	—
210	C₂×C₄	—	76	1	no	—
211	C₂×C₄	—	76	1	no	—
212	C₂×C₄	—	76	1	no	—
213	C₂×C₄	—	152	2	no	—
214	C₂×C₄	—	152	2	no	—
215	C₂×C₄	—	152	2	no	—
216	C₂×C₄	#10597334	1	1	yes	characteristic
217	C₂×C₄	#10597336	1	1	yes	characteristic
218	C₂×C₄	#10597343	1	1	yes	characteristic
219	C₂×C₄	C₂×#64516	1	1	yes	characteristic
220	C₂×C₄	C₂×#64516	1	1	yes	characteristic
221	C₂×C₄	#10597354	1	1	yes	characteristic
222	C₂×C₄	#10597355	1	1	yes	characteristic
223	C₂×C₄	D₈×D₁₉	1	1	yes	characteristic
224	C₂×C₄	#10597358	1	1	yes	characteristic
225	C₂×C₄	Q₁₆×D₁₉	1	1	yes	characteristic
226	C₂×C₄	#10597365	1	1	yes	characteristic
227	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
228	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
229	C₂×C₄	Q₈×Dic₁₉	1	1	yes	characteristic
230	C₈	—	2	2	no	—
231	C₈	—	4	4	no	—
232	C₈	—	76	2	no	—
233	C₈	#10597274	1	4	yes	complemented
234	C₂²	#11926443	1	1	yes	characteristic
235	C₂²	#11926745	1	1	yes	characteristic
236	C₂²	#11926750	1	1	yes	characteristic
237	C₂²	#11926781	1	1	yes	characteristic
238	C₂²	#11926786	1	1	yes	characteristic
239	C₂²	#11926815	1	1	yes	characteristic
240	C₂²	#11926820	1	1	yes	characteristic
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	304	2	no	—
244	C₄	—	76	2	no	—
245	C₄	—	76	2	no	—
246	C₄	C₂×#10597274	1	1	yes	characteristic
247	C₄	#11926586	1	1	yes	characteristic
248	C₄	D₈×Dic₁₉	1	1	yes	characteristic
249	C₄	Q₁₆×Dic₁₉	1	1	yes	characteristic
250	C₄	#11926830	1	1	yes	characteristic
251	C₄	#11926835	1	1	yes	characteristic
252	C₄	#11926945	1	1	yes	characteristic
253	C₄	#11926948	1	1	yes	characteristic
254	C₂	#14562996	1	1	yes	characteristic
255	C₂	#14563006	1	1	yes	characteristic
256	C₂	#14563025	1	1	yes	characteristic
257	C₂	#14563035	1	1	yes	characteristic
258	C₂	#14564369	1	1	yes	characteristic
259	C₂	#14564399	1	1	yes	characteristic
260	C₂	#14564421	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597341
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#335	C₂
socle, prime radical	C₂³×C₁₉	#121
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#335	D₁₉
hypocenter	C₁₉	#9615
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
17. C₁₉×#335	C₄	semidirect
102. C₈×C₁₉	#106	semidirect
140. #335	Dic₁₉	semidirect
170. C₁₉	#9615	semidirect
233. C₈	#10597274	semidirect

Maximal subgroups

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

subgroup	index
#14564399	2
#14563795	2
C₈×C₄⋊Dic₁₉	2
#14561519	2
C₂×C₁₉×#335	2
#9615	19