Subgroups — #18070908

Order 4864.

442 conjugacy classes of subgroups.

The subgroup lattice has 276 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	C₄×C₄⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#14561461	C₂	1	4	yes	maximal
4	C₁₉×#2862	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
6	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
7	C₂×C₄×C₁₉⋊C₈	—	2	1	no	—
8	#11926299	—	2	1	no	—
9	#11926299	C₂²	1	2	yes	—
10	#11926299	C₂²	1	2	yes	—
11	C₂×C₁₉×C₄⋊Q₈	C₄	1	1	yes	characteristic
12	C₂×C₁₉×C₄⋊Q₈	C₄	1	1	yes	characteristic
13	C₁₉×#391	—	2	1	no	—
14	C₁₉×#391	—	2	1	no	—
15	C₁₉×#391	C₄	1	2	yes	—
16	C₁₉×#391	C₂²	1	2	yes	—
17	C₁₉×#382	—	2	1	no	—
18	C₁₉×#382	—	2	1	no	—
19	C₄³×C₁₉	C₂²	1	1	yes	characteristic
20	C₂²×Q₈×C₁₉	—	2	1	no	—
21	C₂²×Q₈×C₁₉	C₂×C₄	1	2	yes	—
22	C₁₉×C₄⋊Q₈	—	4	2	no	—
23	C₁₉×C₄⋊Q₈	—	4	2	no	—
24	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
25	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
26	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
27	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	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	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₉	D₄	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	C₂³	1	1	yes	characteristic
35	C₂²×C₁₉⋊C₈	—	2	2	no	—
36	C₂²×C₁₉⋊C₈	—	2	2	no	—
37	C₁₉×#95	—	2	2	no	—
38	C₁₉×#95	—	2	2	no	—
39	C₁₉×#95	—	2	2	no	—
40	C₁₉×#95	—	2	2	no	—
41	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
42	C₄⋊(C₁₉⋊C₈)	—	4	2	no	—
43	C₄×C₁₉⋊C₈	—	4	2	no	—
44	C₂×Q₈×C₁₉	—	2	4	no	—
45	C₂×Q₈×C₁₉	—	4	2	no	—
46	C₂×Q₈×C₁₉	—	4	2	no	—
47	C₂×Q₈×C₁₉	—	4	2	no	—
48	C₂×Q₈×C₁₉	—	4	2	no	—
49	C₂×Q₈×C₁₉	—	4	2	no	—
50	C₂²×C₄×C₁₉	—	2	2	no	—
51	C₂²×C₄×C₁₉	—	2	2	no	—
52	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
59	C₄⋊C₄×C₁₉	—	4	2	no	—
60	C₄⋊C₄×C₁₉	—	4	2	no	—
61	C₄⋊C₄×C₁₉	—	4	2	no	—
62	C₄⋊C₄×C₁₉	—	4	2	no	—
63	C₄²×C₁₉	—	2	2	no	—
64	C₄²×C₁₉	—	2	2	no	—
65	C₄²×C₁₉	—	2	2	no	—
66	C₄²×C₁₉	—	4	1	no	—
67	C₄²×C₁₉	—	4	1	no	—
68	C₄²×C₁₉	SD₁₆	1	2	yes	—
69	C₄²×C₁₉	SD₁₆	1	2	yes	—
70	C₄²×C₁₉	Q₁₆	1	2	yes	—
71	C₄²×C₁₉	Q₁₆	1	2	yes	—
72	C₂×C₁₉⋊C₈	—	2	4	no	—
73	C₂×C₁₉⋊C₈	—	4	2	no	—
74	C₂×C₁₉⋊C₈	—	4	2	no	—
75	C₂×C₁₉⋊C₈	—	4	2	no	—
76	C₂×C₁₉⋊C₈	—	4	2	no	—
77	C₂×C₁₉⋊C₈	—	4	2	no	—
78	#7662	—	19	1	no	maximal, Sylow
79	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	Q₈×C₁₉	—	8	4	no	—
81	Q₈×C₁₉	—	8	4	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	—	2	1	no	—
90	C₂×C₄×C₁₉	—	2	1	no	—
91	C₂×C₄×C₁₉	—	2	1	no	—
92	C₂×C₄×C₁₉	—	2	1	no	—
93	C₂×C₄×C₁₉	—	2	1	no	—
94	C₂×C₄×C₁₉	—	4	2	no	—
95	C₂×C₄×C₁₉	—	4	2	no	—
96	C₂×C₄×C₁₉	—	4	2	no	—
97	C₂×C₄×C₁₉	—	4	2	no	—
98	C₂×C₄×C₁₉	—	4	2	no	—
99	C₂×C₄×C₁₉	—	4	2	no	—
100	C₂×C₄×C₁₉	#100	1	2	yes	—
101	C₂×C₄×C₁₉	#103	1	2	yes	—
102	C₂×C₄×C₁₉	#103	1	2	yes	—
103	C₂×C₄×C₁₉	#103	1	2	yes	—
104	C₂×C₄×C₁₉	#103	1	2	yes	—
105	C₂×C₄×C₁₉	#104	1	2	yes	—
106	C₂×C₄×C₁₉	#131	1	2	yes	—
107	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
110	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
111	C₁₉⋊C₈	—	4	4	no	—
112	C₁₉⋊C₈	—	8	2	no	—
113	#2862	D₁₉	1	1	yes	characteristic
114	#1831	—	19	1	no	—
115	C₄×C₄⋊C₈	—	19	1	no	—
116	#1272	—	19	4	no	—
117	C₂²×C₁₉	#408	1	1	yes	characteristic
118	C₂²×C₁₉	#410	1	1	yes	characteristic
119	C₂²×C₁₉	C₂×#100	1	1	yes	characteristic
120	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
121	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
122	C₂²×C₁₉	C₂×#104	1	1	yes	characteristic
123	C₂²×C₁₉	#421	1	1	yes	characteristic
124	C₄×C₁₉	—	2	2	no	—
125	C₄×C₁₉	—	2	2	no	—
126	C₄×C₁₉	—	2	2	no	—
127	C₄×C₁₉	—	2	1	no	—
128	C₄×C₁₉	—	2	1	no	—
129	C₄×C₁₉	—	2	1	no	—
130	C₄×C₁₉	—	2	1	no	—
131	C₄×C₁₉	—	2	2	no	—
132	C₄×C₁₉	—	8	2	no	—
133	C₄×C₁₉	—	8	2	no	—
134	C₄×C₁₉	#333	1	4	yes	—
135	C₂×C₄⋊Q₈	Dic₁₉	1	1	yes	characteristic
136	C₂×C₄⋊Q₈	Dic₁₉	1	1	yes	characteristic
137	C₂×C₄⋊C₈	—	38	1	no	—
138	C₂×C₄⋊C₈	—	38	1	no	—
139	C₂×C₄×C₈	—	38	1	no	—
140	#391	—	2	1	no	—
141	#391	—	2	1	no	—
142	#391	Dic₁₉	1	2	yes	—
143	#391	C₂×D₁₉	1	2	yes	—
144	#382	—	2	1	no	—
145	#382	—	2	1	no	—
146	C₄³	C₂×D₁₉	1	1	yes	characteristic
147	#336	—	19	2	no	—
148	#336	—	19	2	no	—
149	#336	—	38	1	no	—
150	C₂×C₁₉	#1469	1	1	yes	characteristic
151	C₂×C₁₉	#1481	1	1	yes	characteristic
152	C₂×C₁₉	#1493	1	1	yes	characteristic
153	C₂×C₁₉	#1503	1	1	yes	characteristic
154	C₂×C₁₉	#1518	1	1	yes	characteristic
155	C₂×C₁₉	#1527	1	1	yes	characteristic
156	C₂×C₁₉	C₂×#333	1	1	yes	characteristic
157	C₂²×Q₈	—	2	1	no	—
158	C₂²×Q₈	C₂×Dic₁₉	1	2	yes	—
159	C₂²×C₈	—	38	2	no	—
160	C₂²×C₈	—	38	2	no	—
161	C₄⋊Q₈	—	4	2	no	—
162	C₄⋊Q₈	—	4	2	no	—
163	C₂×C₄⋊C₄	—	2	1	no	—
164	C₂×C₄⋊C₄	—	2	1	no	—
165	C₂×C₄⋊C₄	—	2	1	no	—
166	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
167	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
168	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
169	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
170	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
171	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
172	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
173	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
174	C₄⋊C₈	—	38	4	no	—
175	C₄⋊C₈	—	76	2	no	—
176	C₄×C₈	—	76	2	no	—
177	#95	—	2	2	no	—
178	#95	—	2	2	no	—
179	#95	—	2	2	no	—
180	#95	—	2	2	no	—
181	C₁₉	#7662	1	1	yes	characteristic, complemented, Sylow
182	C₂×Q₈	—	2	4	no	—
183	C₂×Q₈	—	4	2	no	—
184	C₂×Q₈	—	4	2	no	—
185	C₂×Q₈	—	4	2	no	—
186	C₂×Q₈	—	4	2	no	—
187	C₂×Q₈	—	4	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
191	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
192	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
193	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
194	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
197	C₂×C₈	—	38	4	no	—
198	C₂×C₈	—	76	2	no	—
199	C₂×C₈	—	76	2	no	—
200	C₂×C₈	—	76	2	no	—
201	C₂×C₈	—	76	2	no	—
202	C₂×C₈	—	76	2	no	—
203	C₄⋊C₄	—	4	2	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	4	1	no	—
211	C₄²	—	4	1	no	—
212	C₄²	#64527	1	2	yes	—
213	C₄²	#64527	1	2	yes	—
214	C₄²	#64528	1	2	yes	—
215	C₄²	#64528	1	2	yes	—
216	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
217	Q₈	—	8	4	no	—
218	Q₈	—	8	4	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	#10597290	1	2	yes	—
238	C₂×C₄	#10597292	1	2	yes	—
239	C₂×C₄	#10597292	1	2	yes	—
240	C₂×C₄	#10597292	1	2	yes	—
241	C₂×C₄	#10597292	1	2	yes	—
242	C₂×C₄	#10597294	1	2	yes	—
243	C₂×C₄	C₂×#64527	1	1	yes	characteristic
244	C₂×C₄	C₂×#64527	1	1	yes	characteristic
245	C₂×C₄	C₂×#64528	1	1	yes	characteristic
246	C₂×C₄	C₂×#64528	1	1	yes	characteristic
247	C₂×C₄	#10597394	1	2	yes	—
248	C₈	—	76	4	no	—
249	C₈	—	152	2	no	—
250	C₂²	#11926511	1	1	yes	characteristic
251	C₂²	#11926518	1	1	yes	characteristic
252	C₂²	C₂×#10597290	1	1	yes	characteristic
253	C₂²	C₂×#10597292	1	1	yes	characteristic
254	C₂²	C₂×#10597292	1	1	yes	characteristic
255	C₂²	C₂×#10597294	1	1	yes	characteristic
256	C₂²	#11926561	1	1	yes	characteristic
257	C₄	—	2	2	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	8	2	no	—
266	C₄	—	8	2	no	—
267	C₄	#11926293	1	4	yes	—
268	C₂	#14561987	1	1	yes	characteristic
269	C₂	#14562038	1	1	yes	characteristic
270	C₂	#14562090	1	1	yes	characteristic
271	C₂	#14562134	1	1	yes	characteristic
272	C₂	#14562199	1	1	yes	characteristic
273	C₂	#14562233	1	1	yes	characteristic
274	C₂	C₂×#11926293	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×#2862	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2862	D₁₉
hypocenter	C₁₉	#7662
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2862

Complements

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

N	H ≅ G/N	type
181. C₁₉	#7662	semidirect

Maximal subgroups

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

subgroup	index
#14563752	2
C₄×C₄⋊(C₁₉⋊C₈)	2
#14561461	2
C₁₉×#2862	2
#7662	19