Subgroups — #18072087

Order 4864.

734 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	#14566033	C₂	1	1	yes	maximal, characteristic
3	#14561475	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×M₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₁₉⋊C₈	—	2	1	no	—
6	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
7	C₂×C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
8	C₂×C₄⋊(C₁₉⋊C₈)	C₄	1	4	yes	complemented
9	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
10	#11926412	—	2	1	no	—
11	#11926410	C₄	1	2	yes	—
12	#11926407	—	2	1	no	—
13	#11926404	C₄	1	2	yes	—
14	C₄×C₄⋊Dic₁₉	C₂²	1	2	yes	—
15	C₄×C₂²⋊Dic₁₉	—	2	1	no	—
16	#11926390	—	2	1	no	—
17	C₂²×M₁₆×C₁₉	—	2	1	no	—
18	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
19	C₄×M₁₆×C₁₉	—	2	2	no	—
20	C₄×M₁₆×C₁₉	C₄	1	4	yes	complemented
21	C₂×C₁₉×#97	—	2	1	no	—
22	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
23	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
26	C₂×M₁₆×C₁₉	—	2	8	no	—
27	C₂×M₁₆×C₁₉	—	2	4	no	—
28	C₂²×C₈×C₁₉	—	2	2	no	—
29	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₉	D₄	1	2	yes	—
35	C₂×C₄²×C₁₉	Q₈	1	2	yes	—
36	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
37	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
38	C₂×C₄⋊Dic₁₉	C₂×C₄	1	4	yes	—
39	C₂×C₄×Dic₁₉	—	2	2	no	—
40	C₂²×C₁₉⋊C₈	—	2	4	no	—
41	C₂²×C₁₉⋊C₈	—	2	2	no	—
42	C₂²×C₁₉⋊C₈	—	4	4	no	—
43	C₁₉×#97	—	2	4	no	—
44	C₄×C₈×C₁₉	—	2	2	no	—
45	C₄×C₈×C₁₉	C₂×C₄	1	4	yes	—
46	C₁₉⋊#95	—	2	2	no	—
47	C₁₉⋊#95	—	2	2	no	—
48	C₁₉⋊#95	—	2	2	no	—
49	C₁₉⋊#95	—	2	2	no	—
50	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
51	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
52	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	4	yes	—
53	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	—	2	4	no	—
55	C₂²×C₄×C₁₉	—	2	4	no	—
56	C₂²×C₄×C₁₉	—	2	2	no	—
57	C₂²×C₄×C₁₉	—	2	2	no	—
58	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
59	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×Dic₁₉	—	2	4	no	—
71	M₁₆×C₁₉	—	2	16	no	—
72	C₂×C₈×C₁₉	—	2	8	no	—
73	C₂×C₈×C₁₉	—	2	4	no	—
74	C₄²×C₁₉	C₄²	1	2	yes	—
75	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
76	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
77	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
78	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
79	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
80	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
84	C₂²⋊Dic₁₉	—	4	4	no	—
85	C₄⋊Dic₁₉	—	2	8	no	—
86	C₄×Dic₁₉	—	4	2	no	—
87	C₄×Dic₁₉	—	4	2	no	—
88	C₂×C₁₉⋊C₈	—	2	4	no	—
89	C₂×C₁₉⋊C₈	—	4	8	no	—
90	C₂×C₁₉⋊C₈	—	4	4	no	—
91	#7895	—	19	1	no	maximal, Sylow
92	C₂³×C₁₉	—	2	2	no	—
93	C₂³×C₁₉	—	2	2	no	—
94	C₂³×C₁₉	#95	1	2	yes	—
95	C₂³×C₁₉	#104	1	4	yes	—
96	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₉	—	2	4	no	—
98	C₂×C₄×C₁₉	—	2	4	no	—
99	C₂×C₄×C₁₉	—	2	2	no	—
100	C₂×C₄×C₁₉	—	2	2	no	—
101	C₂×C₄×C₁₉	—	2	2	no	—
102	C₂×C₄×C₁₉	—	2	2	no	—
103	C₂×C₄×C₁₉	—	2	4	no	—
104	C₂×C₄×C₁₉	—	2	2	no	—
105	C₂×C₄×C₁₉	—	2	2	no	—
106	C₂×C₄×C₁₉	—	2	4	no	—
107	C₂×C₄×C₁₉	—	2	4	no	—
108	C₂×C₄×C₁₉	#95	1	2	yes	—
109	C₂×C₄×C₁₉	#95	1	2	yes	—
110	C₂×C₄×C₁₉	#95	1	2	yes	—
111	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₉	#131	1	4	yes	—
118	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
119	C₂×C₄×C₁₉	#137	1	2	yes	—
120	C₂×Dic₁₉	—	4	4	no	—
121	C₂×Dic₁₉	—	4	4	no	—
122	C₂×Dic₁₉	—	4	4	no	—
123	C₈×C₁₉	—	2	8	no	—
124	C₁₉⋊C₈	—	4	4	no	—
125	C₂²×C₄⋊C₈	—	19	1	no	—
126	C₂×C₄×M₁₆	D₁₉	1	1	yes	characteristic
127	#2283	—	19	1	no	—
128	#1279	—	19	4	no	—
129	C₂²×C₁₉	—	2	2	no	—
130	C₂²×C₁₉	—	2	2	no	—
131	C₂²×C₁₉	—	2	8	no	—
132	C₂²×C₁₉	#339	1	4	yes	—
133	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
134	C₂²×C₁₉	#408	1	2	yes	—
135	C₂²×C₁₉	#417	1	2	yes	—
136	C₂²×C₁₉	C₂×#104	1	2	yes	—
137	C₄×C₁₉	—	2	2	no	—
138	C₄×C₁₉	—	2	1	no	—
139	C₄×C₁₉	—	2	2	no	—
140	C₄×C₁₉	—	2	1	no	—
141	C₄×C₁₉	—	2	2	no	—
142	C₄×C₁₉	#405	1	2	yes	—
143	C₄×C₁₉	#418	1	1	yes	characteristic
144	C₄×C₁₉	#419	1	1	yes	characteristic
145	C₄×C₁₉	#424	1	2	yes	—
146	C₄×C₁₉	#428	1	2	yes	—
147	Dic₁₉	—	8	4	no	—
148	C₂²×M₁₆	—	2	1	no	—
149	C₂³×C₈	—	38	1	no	—
150	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
151	C₂×C₄⋊C₈	—	19	4	no	—
152	C₂×C₄⋊C₈	—	19	2	no	—
153	C₂×C₄⋊C₈	—	38	1	no	—
154	C₂×C₄⋊C₈	—	38	2	no	—
155	C₄×M₁₆	—	2	2	no	—
156	C₄×M₁₆	Dic₁₉	1	4	yes	complemented
157	C₂×#97	—	2	1	no	—
158	C₂×C₄×C₈	C₂×D₁₉	1	2	yes	—
159	#383	—	38	1	no	—
160	#382	—	19	2	no	—
161	#381	—	38	1	no	—
162	#380	—	19	2	no	—
163	C₄×C₄⋊C₄	—	19	2	no	—
164	C₄×C₂²⋊C₄	—	38	1	no	—
165	#376	—	38	1	no	—
166	C₂×C₁₉	—	2	4	no	—
167	C₂×C₁₉	#1483	1	4	yes	—
168	C₂×C₁₉	#1729	1	1	yes	characteristic
169	C₂×C₁₉	C₂×#339	1	1	yes	characteristic
170	C₂×C₁₉	#1736	1	1	yes	characteristic
171	C₂³×C₄	C₄×D₁₉	1	1	yes	characteristic
172	C₂³×C₄	D₇₆	1	1	yes	characteristic
173	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
174	C₂×M₁₆	—	2	8	no	—
175	C₂×M₁₆	—	2	4	no	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	38	4	no	—
178	C₂²×C₈	—	38	2	no	—
179	C₂²×C₈	—	76	4	no	—
180	C₂×C₄⋊C₄	—	19	4	no	—
181	C₂×C₂²⋊C₄	—	38	2	no	—
182	C₂×C₄²	—	38	2	no	—
183	C₂×C₄²	Dic₃₈	1	2	yes	—
184	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
185	C₂×C₄²	C₄×D₁₉	1	2	yes	—
186	C₂×C₄²	D₇₆	1	1	yes	characteristic
187	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
188	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
189	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
190	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
191	C₄⋊C₈	—	19	4	no	—
192	C₄⋊C₈	—	38	4	no	—
193	C₄⋊C₈	—	38	2	no	—
194	#97	—	2	4	no	—
195	C₄×C₈	—	2	2	no	—
196	C₄×C₈	C₂×Dic₁₉	1	4	yes	—
197	#95	—	38	2	no	—
198	#95	—	38	2	no	—
199	#95	—	38	2	no	—
200	#95	—	38	2	no	—
201	C₁₉	#7895	1	1	yes	characteristic, complemented, Sylow
202	C₂⁴	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	38	4	no	—
208	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
209	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
210	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
211	C₂²×C₄	C₄⋊Dic₁₉	1	2	yes	—
212	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
213	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
214	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
215	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
216	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
217	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
220	M₁₆	—	2	16	no	—
221	C₂×C₈	—	2	8	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	38	4	no	—
224	C₂×C₈	—	76	8	no	—
225	C₂×C₈	—	76	4	no	—
226	C₄⋊C₄	—	38	8	no	—
227	C₂²⋊C₄	—	76	4	no	—
228	C₄²	—	76	2	no	—
229	C₄²	—	76	2	no	—
230	C₄²	C₄×Dic₁₉	1	2	yes	—
231	C₄²	Dic₁₉⋊C₄	1	2	yes	—
232	C₄²	Dic₁₉⋊C₄	1	2	yes	—
233	C₄²	C₄⋊Dic₁₉	1	2	yes	—
234	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
235	C₄²	C₂²⋊Dic₁₉	1	2	yes	—
236	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
237	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
238	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
239	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
240	C₂³	—	2	2	no	—
241	C₂³	—	2	2	no	—
242	C₂³	#10597262	1	4	yes	—
243	C₂³	C₁₉⋊#95	1	2	yes	—
244	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	76	4	no	—
257	C₂×C₄	—	76	4	no	—
258	C₂×C₄	—	76	4	no	—
259	C₂×C₄	C₁₉⋊#95	1	2	yes	—
260	C₂×C₄	C₁₉⋊#95	1	2	yes	—
261	C₂×C₄	C₁₉⋊#95	1	2	yes	—
262	C₂×C₄	#10597353	1	4	yes	—
263	C₂×C₄	#10597354	1	2	yes	—
264	C₂×C₄	#10597355	1	2	yes	—
265	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
266	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
267	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
268	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
269	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
270	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
271	C₈	—	2	8	no	—
272	C₈	—	76	4	no	—
273	C₂²	—	2	8	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	#11926304	1	4	yes	—
277	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
278	C₂²	#11926512	1	2	yes	—
279	C₂²	#11926546	1	2	yes	—
280	C₂²	C₂×#10597262	1	2	yes	—
281	C₄	—	2	2	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	2	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	2	no	—
286	C₄	—	152	4	no	—
287	C₄	#11926502	1	2	yes	—
288	C₄	#11926550	1	1	yes	characteristic
289	C₄	#11926554	1	1	yes	characteristic
290	C₄	#11926572	1	2	yes	—
291	C₄	#11926586	1	2	yes	—
292	C₂	—	2	4	no	—
293	C₂	#14562049	1	4	yes	—
294	C₂	#14563333	1	1	yes	characteristic
295	C₂	C₂×#11926304	1	1	yes	characteristic
296	C₂	#14563355	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₇₆
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₄×M₁₆×C₁₉	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×C₄×M₁₆	D₁₉
hypocenter	C₁₉	#7895
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #14561475	C₂	semidirect
8. C₂×C₄⋊(C₁₉⋊C₈)	C₄	semidirect
20. C₄×M₁₆×C₁₉	C₄	semidirect
156. C₄×M₁₆	Dic₁₉	semidirect
201. C₁₉	#7895	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊(C₁₉⋊C₈)	2
#14566033	2
#14561475	2
C₂×C₄×M₁₆×C₁₉	2
#7895	19