Subgroups — #18075588

Order 4864.

454 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563792	C₂	1	1	yes	maximal, characteristic
2	#14563640	C₂	1	2	yes	maximal
3	C₂×#11926295	C₂	1	1	yes	maximal, characteristic
4	#14561519	C₂	1	2	yes	maximal
5	C₂×C₈×Q₈×C₁₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#10597274	—	2	1	no	—
7	C₂×#10597274	—	2	1	no	—
8	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
9	C₂×#10597292	—	2	2	no	—
10	#11926414	C₂²	1	2	yes	—
11	#11926306	—	2	2	no	—
12	#11926295	—	2	2	no	—
13	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
14	C₈×Q₈×C₁₉	—	2	2	no	—
15	C₈×Q₈×C₁₉	C₄	1	4	yes	complemented
16	C₂×C₁₉×C₄⋊C₈	—	2	1	no	—
17	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₉	—	2	1	no	—
19	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
20	C₂²×Q₈×C₁₉	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₉	—	2	1	no	—
22	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
24	C₄×Q₈×C₁₉	—	2	1	no	—
25	C₄×Q₈×C₁₉	—	2	1	no	—
26	C₄×Q₈×C₁₉	C₂×C₄	1	2	yes	—
27	C₄×Q₈×C₁₉	Q₈	1	2	yes	—
28	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₉	—	2	1	no	—
31	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊Dic₁₉	—	2	2	no	—
33	C₂×C₄⋊Dic₁₉	—	2	2	no	—
34	C₂×C₄⋊Dic₁₉	—	2	2	no	—
35	C₂²×C₁₉⋊C₈	—	2	2	no	—
36	C₁₉×C₄⋊C₈	—	2	4	no	—
37	C₁₉×C₄⋊C₈	—	2	1	no	—
38	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₁₉	—	2	4	no	—
40	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₁₉	D₄	1	2	yes	—
42	#10597292	—	4	4	no	—
43	#10597274	—	4	2	no	—
44	#10597274	—	4	2	no	—
45	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
46	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
47	C₂×Q₈×C₁₉	—	2	1	no	—
48	C₂×Q₈×C₁₉	—	2	2	no	—
49	C₂×Q₈×C₁₉	C₄⋊C₄	1	2	yes	—
50	C₂×Q₈×C₁₉	SD₁₆	1	4	yes	—
51	C₂²×C₄×C₁₉	—	2	1	no	—
52	C₂²×C₄×C₁₉	—	2	1	no	—
53	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
56	C₂²×Dic₁₉	—	4	2	no	—
57	C₂×C₈×C₁₉	—	2	1	no	—
58	C₂×C₈×C₁₉	—	2	1	no	—
59	C₂×C₈×C₁₉	—	2	2	no	—
60	C₂×C₈×C₁₉	—	4	1	no	—
61	C₂×C₈×C₁₉	—	4	1	no	—
62	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
65	C₂×C₈×C₁₉	Q₁₆	1	2	yes	—
66	C₄⋊C₄×C₁₉	—	2	2	no	—
67	C₄⋊C₄×C₁₉	—	2	1	no	—
68	C₄⋊C₄×C₁₉	—	2	2	no	—
69	C₄⋊C₄×C₁₉	C₄⋊C₄	1	2	yes	—
70	C₄²×C₁₉	—	2	2	no	—
71	C₄²×C₁₉	—	2	2	no	—
72	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
75	C₄²×C₁₉	#41	1	1	yes	characteristic
76	C₄⋊Dic₁₉	—	4	2	no	—
77	C₄⋊Dic₁₉	—	4	2	no	—
78	C₄⋊Dic₁₉	—	4	2	no	—
79	C₄⋊Dic₁₉	—	4	2	no	—
80	C₄⋊Dic₁₉	—	8	2	no	—
81	C₂×C₁₉⋊C₈	—	4	2	no	—
82	C₂×C₁₉⋊C₈	—	4	2	no	—
83	C₂×C₁₉⋊C₈	—	4	2	no	—
84	#8643	—	19	1	no	maximal, Sylow
85	C₂³×C₁₉	#121	1	1	yes	characteristic
86	Q₈×C₁₉	—	2	2	no	—
87	Q₈×C₁₉	—	2	4	no	—
88	Q₈×C₁₉	#106	1	4	yes	complemented
89	C₂×C₄×C₁₉	—	2	2	no	—
90	C₂×C₄×C₁₉	—	2	4	no	—
91	C₂×C₄×C₁₉	—	2	2	no	—
92	C₂×C₄×C₁₉	—	2	2	no	—
93	C₂×C₄×C₁₉	—	4	2	no	—
94	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
97	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
98	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
102	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
103	C₂×C₄×C₁₉	#135	1	2	yes	—
104	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
105	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
106	C₂×Dic₁₉	—	8	2	no	—
107	C₂×Dic₁₉	—	8	2	no	—
108	C₂×Dic₁₉	—	8	2	no	—
109	C₈×C₁₉	—	2	2	no	—
110	C₈×C₁₉	—	4	2	no	—
111	C₈×C₁₉	#103	1	4	yes	complemented
112	C₁₉⋊C₈	—	8	2	no	—
113	C₂×C₈×Q₈	D₁₉	1	1	yes	characteristic
114	#1844	—	19	1	no	—
115	#1803	—	19	2	no	—
116	C₂×#334	—	19	1	no	—
117	#1296	—	19	2	no	—
118	C₂²×C₁₉	#380	1	1	yes	characteristic
119	C₂²×C₁₉	#460	1	1	yes	characteristic
120	C₂²×C₁₉	#462	1	1	yes	characteristic
121	C₂²×C₁₉	#465	1	2	yes	—
122	C₂²×C₁₉	#471	1	2	yes	—
123	C₄×C₁₉	—	2	4	no	—
124	C₄×C₁₉	—	2	1	no	—
125	C₄×C₁₉	—	2	1	no	—
126	C₄×C₁₉	—	4	2	no	—
127	C₄×C₁₉	C₂×#103	1	1	yes	characteristic
128	C₄×C₁₉	#417	1	1	yes	characteristic
129	C₄×C₁₉	C₂×#106	1	1	yes	characteristic
130	C₄×C₁₉	#427	1	1	yes	characteristic
131	C₄×C₁₉	#465	1	2	yes	—
132	C₄×C₁₉	#475	1	1	yes	characteristic
133	C₄×C₁₉	#477	1	1	yes	characteristic
134	Dic₁₉	—	16	2	no	—
135	C₂×C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
136	C₈×Q₈	—	2	2	no	—
137	C₈×Q₈	Dic₁₉	1	4	yes	complemented
138	C₂×#106	—	38	1	no	—
139	C₂×#106	—	38	1	no	—
140	C₂×C₄⋊C₈	—	2	1	no	—
141	C₂×C₄⋊C₈	—	19	2	no	—
142	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
143	C₂×#103	—	38	2	no	—
144	C₂×C₄×C₈	—	2	1	no	—
145	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
146	#384	—	19	2	no	—
147	#340	—	38	2	no	—
148	#334	—	38	2	no	—
149	C₂×C₁₉	#1666	1	2	yes	—
150	C₂×C₁₉	#1672	1	2	yes	—
151	C₂×C₁₉	#1803	1	1	yes	characteristic
152	C₂×C₁₉	#1807	1	1	yes	characteristic
153	C₂×C₁₉	#1815	1	1	yes	characteristic
154	C₂²×Q₈	D₇₆	1	1	yes	characteristic
155	C₂²×C₈	—	2	1	no	—
156	C₂²×C₈	—	38	2	no	—
157	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
158	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
159	C₄×Q₈	—	2	1	no	—
160	C₄×Q₈	—	2	1	no	—
161	C₄×Q₈	Dic₃₈	1	2	yes	—
162	C₄×Q₈	C₂×Dic₁₉	1	2	yes	—
163	C₂×C₄⋊C₄	—	2	1	no	—
164	C₂×C₄⋊C₄	—	38	2	no	—
165	C₂×C₄⋊C₄	—	38	2	no	—
166	C₂×C₄⋊C₄	—	38	2	no	—
167	C₂×C₄⋊C₄	D₇₆	1	1	yes	characteristic
168	C₂×C₄²	—	2	1	no	—
169	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
170	#106	—	76	2	no	—
171	#106	—	76	2	no	—
172	C₄⋊C₈	—	2	4	no	—
173	C₄⋊C₈	—	2	1	no	—
174	C₄⋊C₈	—	38	2	no	—
175	C₄⋊C₈	—	38	2	no	—
176	C₄⋊C₈	C₂×Dic₁₉	1	2	yes	—
177	#103	—	76	4	no	—
178	C₄×C₈	—	2	4	no	—
179	C₄×C₈	C₂×Dic₁₉	1	2	yes	—
180	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
181	C₁₉	#8643	1	1	yes	characteristic, complemented, Sylow
182	C₂×Q₈	—	2	1	no	—
183	C₂×Q₈	—	2	2	no	—
184	C₂×Q₈	#64516	1	4	yes	—
185	C₂×Q₈	C₄⋊Dic₁₉	1	2	yes	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	76	2	no	—
189	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
190	C₂²×C₄	#64541	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	4	1	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	76	2	no	—
198	C₂×C₈	—	76	2	no	—
199	C₂×C₈	—	76	2	no	—
200	C₂×C₈	#64527	1	2	yes	—
201	C₂×C₈	#64528	1	2	yes	—
202	C₂×C₈	C₂²⋊Dic₁₉	1	2	yes	—
203	C₂×C₈	C₂²⋊Dic₁₉	1	2	yes	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	76	2	no	—
208	C₄⋊C₄	—	76	2	no	—
209	C₄⋊C₄	—	76	2	no	—
210	C₄⋊C₄	—	76	2	no	—
211	C₄⋊C₄	—	152	2	no	—
212	C₄⋊C₄	C₄⋊Dic₁₉	1	2	yes	—
213	C₄²	—	2	2	no	—
214	C₄²	—	2	2	no	—
215	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
216	C₄²	#64541	1	1	yes	characteristic
217	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
218	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
219	C₂³	#10597376	1	1	yes	characteristic
220	Q₈	—	2	2	no	—
221	Q₈	—	2	4	no	—
222	Q₈	#10597274	1	4	yes	complemented
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	152	2	no	—
229	C₂×C₄	—	152	2	no	—
230	C₂×C₄	—	152	2	no	—
231	C₂×C₄	C₂×#64516	1	2	yes	—
232	C₂×C₄	#10597350	1	2	yes	—
233	C₂×C₄	#10597370	1	1	yes	characteristic
234	C₂×C₄	#10597370	1	1	yes	characteristic
235	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
236	C₂×C₄	#10597372	1	1	yes	characteristic
237	C₂×C₄	#10597376	1	1	yes	characteristic
238	C₂×C₄	C₂×#64527	1	1	yes	characteristic
239	C₂×C₄	C₂×#64528	1	1	yes	characteristic
240	C₂×C₄	#10597395	1	1	yes	characteristic
241	C₂×C₄	#10597397	1	1	yes	characteristic
242	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	—	4	2	no	—
245	C₈	—	152	2	no	—
246	C₈	#10597292	1	4	yes	complemented
247	C₂²	#11926404	1	1	yes	characteristic
248	C₂²	#11926744	1	1	yes	characteristic
249	C₂²	#11926754	1	1	yes	characteristic
250	C₂²	#11926773	1	2	yes	—
251	C₂²	#11926807	1	2	yes	—
252	C₄	—	2	4	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	4	2	no	—
256	C₄	—	304	2	no	—
257	C₄	C₂×#10597292	1	1	yes	characteristic
258	C₄	#11926545	1	1	yes	characteristic
259	C₄	C₂×#10597274	1	1	yes	characteristic
260	C₄	#11926582	1	1	yes	characteristic
261	C₄	#11926773	1	2	yes	—
262	C₄	#11926829	1	1	yes	characteristic
263	C₄	#11926839	1	1	yes	characteristic
264	C₂	#14562983	1	2	yes	—
265	C₂	#14563017	1	2	yes	—
266	C₂	#14563640	1	1	yes	characteristic
267	C₂	#14563660	1	1	yes	characteristic
268	C₂	#14563694	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597376
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₈×Q₈×C₁₉	C₂
socle, prime radical	C₂³×C₁₉	#121
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×C₈×Q₈	D₁₉
hypocenter	C₁₉	#8643
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
15. C₈×Q₈×C₁₉	C₄	semidirect
88. Q₈×C₁₉	#106	semidirect
111. C₈×C₁₉	#103	semidirect
137. C₈×Q₈	Dic₁₉	semidirect
181. C₁₉	#8643	semidirect
222. Q₈	#10597274	semidirect
246. C₈	#10597292	semidirect

Maximal subgroups

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

subgroup	index
#14563792	2
#14563640	2
C₂×#11926295	2
#14561519	2
C₂×C₈×Q₈×C₁₉	2
#8643	19