Subgroups — #18082695

Order 4864.

506 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	#14563752	C₂	1	1	yes	maximal, characteristic
2	#14563488	C₂	1	1	yes	maximal, characteristic
3	#14561518	C₂	1	4	yes	maximal
4	C₁₉×#2868	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
6	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×#10597260	—	2	1	no	—
8	#11926299	—	2	1	no	—
9	#11926299	—	2	1	no	—
10	#11926299	—	2	1	no	—
11	C₂×C₁₉×C₄⋊Q₈	—	2	1	no	—
12	C₂×C₁₉×C₄⋊Q₈	—	2	1	no	—
13	C₂×C₁₉×C₄⋊Q₈	C₄	1	2	yes	—
14	C₁₉×#384	—	2	1	no	—
15	C₁₉×#384	—	2	1	no	—
16	C₁₉×#384	C₄	1	2	yes	—
17	C₁₉×#384	C₂²	1	2	yes	—
18	C₄³×C₁₉	C₂²	1	1	yes	characteristic
19	C₂²×Q₈×C₁₉	—	2	2	no	—
20	C₁₉×C₄⋊Q₈	—	4	4	no	—
21	C₁₉×C₄⋊Q₈	—	4	4	no	—
22	C₁₉×C₄⋊Q₈	—	4	2	no	—
23	C₁₉×C₄⋊Q₈	—	4	2	no	—
24	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
25	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
26	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
27	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
28	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
29	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
37	C₂²×C₁₉⋊C₈	—	2	2	no	—
38	C₂²×C₁₉⋊C₈	—	2	2	no	—
39	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
40	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
41	C₄⋊(C₁₉⋊C₈)	Q₈	1	4	yes	complemented
42	#10597260	—	4	2	no	—
43	C₂×Q₈×C₁₉	—	4	4	no	—
44	C₂×Q₈×C₁₉	—	4	4	no	—
45	C₂×Q₈×C₁₉	—	4	4	no	—
46	C₂²×C₄×C₁₉	—	2	4	no	—
47	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
54	C₄⋊C₄×C₁₉	—	4	4	no	—
55	C₄⋊C₄×C₁₉	—	4	4	no	—
56	C₄⋊C₄×C₁₉	—	4	4	no	—
57	C₄⋊C₄×C₁₉	—	4	4	no	—
58	C₄⋊C₄×C₁₉	—	4	2	no	—
59	C₄⋊C₄×C₁₉	—	4	2	no	—
60	C₄⋊C₄×C₁₉	—	4	4	no	—
61	C₄²×C₁₉	—	2	2	no	—
62	C₄²×C₁₉	—	2	2	no	—
63	C₄²×C₁₉	—	2	2	no	—
64	C₄²×C₁₉	—	2	2	no	—
65	C₄²×C₁₉	—	2	2	no	—
66	C₄²×C₁₉	D₈	1	2	yes	—
67	C₄²×C₁₉	SD₁₆	1	2	yes	—
68	C₄²×C₁₉	C₂×Q₈	1	2	yes	—
69	C₄²×C₁₉	#41	1	2	yes	—
70	C₂×C₁₉⋊C₈	—	2	4	no	—
71	C₂×C₁₉⋊C₈	—	4	2	no	—
72	C₂×C₁₉⋊C₈	—	4	2	no	—
73	C₂×C₁₉⋊C₈	—	4	2	no	—
74	C₂×C₁₉⋊C₈	—	4	2	no	—
75	C₂×C₁₉⋊C₈	—	4	2	no	—
76	#9905	—	19	1	no	maximal, Sylow
77	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	Q₈×C₁₉	—	8	8	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	—	2	1	no	—
84	C₂×C₄×C₁₉	—	2	1	no	—
85	C₂×C₄×C₁₉	—	2	1	no	—
86	C₂×C₄×C₁₉	—	2	1	no	—
87	C₂×C₄×C₁₉	—	2	1	no	—
88	C₂×C₄×C₁₉	—	2	1	no	—
89	C₂×C₄×C₁₉	—	4	4	no	—
90	C₂×C₄×C₁₉	—	4	4	no	—
91	C₂×C₄×C₁₉	—	4	4	no	—
92	C₂×C₄×C₁₉	#100	1	2	yes	—
93	C₂×C₄×C₁₉	#101	1	2	yes	—
94	C₂×C₄×C₁₉	#102	1	2	yes	—
95	C₂×C₄×C₁₉	#102	1	2	yes	—
96	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
97	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
98	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
99	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
100	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
101	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
102	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
103	C₂×C₄×C₁₉	#137	1	2	yes	—
104	C₂×C₄×C₁₉	#137	1	2	yes	—
105	C₁₉⋊C₈	—	4	4	no	—
106	C₁₉⋊C₈	—	8	2	no	—
107	#2868	D₁₉	1	1	yes	characteristic
108	#1831	—	19	1	no	—
109	#1767	—	19	1	no	—
110	#1296	—	19	4	no	—
111	C₂²×C₁₉	#391	1	1	yes	characteristic
112	C₂²×C₁₉	C₂×#100	1	1	yes	characteristic
113	C₂²×C₁₉	C₂×#101	1	1	yes	characteristic
114	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
115	C₂²×C₁₉	#418	1	1	yes	characteristic
116	C₂²×C₁₉	#419	1	1	yes	characteristic
117	C₂²×C₁₉	#419	1	1	yes	characteristic
118	C₄×C₁₉	—	2	2	no	—
119	C₄×C₁₉	—	2	2	no	—
120	C₄×C₁₉	—	2	2	no	—
121	C₄×C₁₉	—	2	2	no	—
122	C₄×C₁₉	—	2	2	no	—
123	C₄×C₁₉	—	8	4	no	—
124	C₄×C₁₉	#474	1	2	yes	—
125	C₄×C₁₉	#475	1	2	yes	—
126	C₄×C₁₉	#489	1	2	yes	—
127	C₄×C₁₉	#501	1	2	yes	—
128	C₂×C₄⋊Q₈	—	2	1	no	—
129	C₂×C₄⋊Q₈	—	2	1	no	—
130	C₂×C₄⋊Q₈	Dic₁₉	1	2	yes	—
131	C₂×C₄⋊C₈	—	19	2	no	—
132	C₂×C₄⋊C₈	—	19	2	no	—
133	C₂×#97	—	38	1	no	—
134	#384	—	2	1	no	—
135	#384	—	2	1	no	—
136	#384	Dic₁₉	1	2	yes	—
137	#384	C₂×D₁₉	1	2	yes	—
138	C₄³	C₂×D₁₉	1	1	yes	characteristic
139	#336	—	38	1	no	—
140	#336	—	38	1	no	—
141	#336	—	38	1	no	—
142	C₂×C₁₉	#1545	1	1	yes	characteristic
143	C₂×C₁₉	#1546	1	1	yes	characteristic
144	C₂×C₁₉	#1556	1	1	yes	characteristic
145	C₂×C₁₉	#1558	1	1	yes	characteristic
146	C₂×C₁₉	#1988	1	1	yes	characteristic
147	C₂×C₁₉	#1993	1	1	yes	characteristic
148	C₂×C₁₉	#1997	1	1	yes	characteristic
149	C₂²×Q₈	—	2	2	no	—
150	C₂²×C₈	—	38	2	no	—
151	C₂²×C₈	—	38	2	no	—
152	C₄⋊Q₈	—	4	4	no	—
153	C₄⋊Q₈	—	4	4	no	—
154	C₄⋊Q₈	—	4	2	no	—
155	C₄⋊Q₈	—	4	2	no	—
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	—	2	2	no	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	C₂×Dic₁₉	1	4	yes	—
162	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
163	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
164	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
165	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
166	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
167	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
168	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
169	C₄⋊C₈	—	19	4	no	—
170	C₄⋊C₈	—	38	2	no	—
171	C₄⋊C₈	—	38	4	no	—
172	#97	—	76	2	no	—
173	C₁₉	#9905	1	1	yes	characteristic, complemented, Sylow
174	C₂×Q₈	—	4	4	no	—
175	C₂×Q₈	—	4	4	no	—
176	C₂×Q₈	—	4	4	no	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
179	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
180	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
181	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
182	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
185	C₂×C₈	—	38	4	no	—
186	C₂×C₈	—	76	2	no	—
187	C₂×C₈	—	76	2	no	—
188	C₂×C₈	—	76	2	no	—
189	C₂×C₈	—	76	2	no	—
190	C₂×C₈	—	76	2	no	—
191	C₄⋊C₄	—	4	4	no	—
192	C₄⋊C₄	—	4	4	no	—
193	C₄⋊C₄	—	4	2	no	—
194	C₄⋊C₄	—	4	4	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	4	no	—
197	C₄⋊C₄	—	4	4	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	#64525	1	2	yes	—
204	C₄²	#64526	1	2	yes	—
205	C₄²	Q₈×D₁₉	1	2	yes	—
206	C₄²	#64545	1	2	yes	—
207	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
208	Q₈	—	8	8	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	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	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	#10597289	1	2	yes	—
223	C₂×C₄	#10597289	1	2	yes	—
224	C₂×C₄	#10597290	1	2	yes	—
225	C₂×C₄	#10597293	1	2	yes	—
226	C₂×C₄	C₂×#64525	1	1	yes	characteristic
227	C₂×C₄	#10597378	1	2	yes	—
228	C₂×C₄	C₂×#64526	1	1	yes	characteristic
229	C₂×C₄	#10597388	1	2	yes	—
230	C₂×C₄	#10597388	1	2	yes	—
231	C₂×C₄	#10597390	1	1	yes	characteristic
232	C₂×C₄	Q₈×Dic₁₉	1	2	yes	—
233	C₂×C₄	#10597392	1	2	yes	—
234	C₂×C₄	#10597393	1	1	yes	characteristic
235	C₈	—	76	4	no	—
236	C₈	—	152	2	no	—
237	C₂²	#11926451	1	1	yes	characteristic
238	C₂²	C₂×#10597290	1	1	yes	characteristic
239	C₂²	C₂×#10597293	1	1	yes	characteristic
240	C₂²	C₂×#10597289	1	1	yes	characteristic
241	C₂²	#11926549	1	1	yes	characteristic
242	C₂²	#11926553	1	1	yes	characteristic
243	C₂²	#11926553	1	1	yes	characteristic
244	C₄	—	2	2	no	—
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	8	4	no	—
250	C₄	#11926821	1	2	yes	—
251	C₄	#11926826	1	2	yes	—
252	C₄	#11926906	1	2	yes	—
253	C₄	#11926950	1	2	yes	—
254	C₂	#14562303	1	1	yes	characteristic
255	C₂	#14562307	1	1	yes	characteristic
256	C₂	#14562352	1	1	yes	characteristic
257	C₂	#14562361	1	1	yes	characteristic
258	C₂	#14564588	1	1	yes	characteristic
259	C₂	#14564609	1	1	yes	characteristic
260	C₂	#14564625	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#2868	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2868	D₁₉
hypocenter	C₁₉	#9905
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

Complements

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

N	H ≅ G/N	type
41. C₄⋊(C₁₉⋊C₈)	Q₈	semidirect
173. C₁₉	#9905	semidirect

Maximal subgroups

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

subgroup	index
#14563752	2
#14563488	2
#14561518	2
C₁₉×#2868	2
#9905	19