Subgroups — #18073675

Order 4864.

494 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄×#10597260	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926266	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₁₉⋊#1273	C₂	1	1	yes	maximal, characteristic
6	C₁₉⋊#1273	C₂	1	1	yes	maximal, characteristic
7	C₄×C₁₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊(C₁₉⋊C₈)	C₄	1	2	yes	—
9	C₂×#10597260	C₄	1	2	yes	—
10	C₂×#10597260	C₄	1	2	yes	—
11	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
15	#11926299	C₄	1	2	yes	—
16	#11926266	—	2	2	no	—
17	C₈×C₁₉⋊C₈	—	2	2	no	—
18	C₂×C₁₉×C₄⋊C₈	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
21	C₄³×C₁₉	C₂²	1	1	yes	characteristic
22	C₁₉×#336	C₄	1	2	yes	—
23	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
32	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
33	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
34	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
35	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
36	C₁₉×C₄⋊C₈	—	2	4	no	—
37	C₄×C₈×C₁₉	—	2	1	no	—
38	C₄×C₈×C₁₉	—	2	1	no	—
39	C₄×C₈×C₁₉	—	2	1	no	—
40	C₄×C₈×C₁₉	—	2	1	no	—
41	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
42	#10597260	—	2	4	no	—
43	#10597260	C₈	1	8	yes	complemented
44	C₄×C₁₉⋊C₈	—	2	1	no	—
45	C₄×C₁₉⋊C₈	—	2	1	no	—
46	C₄×C₁₉⋊C₈	—	2	1	no	—
47	C₄×C₁₉⋊C₈	—	2	1	no	—
48	C₄×C₁₉⋊C₈	—	2	1	no	—
49	C₄×C₁₉⋊C₈	—	2	1	no	—
50	C₄×C₁₉⋊C₈	D₄	1	2	yes	—
51	C₄×C₁₉⋊C₈	Q₈	1	2	yes	—
52	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
53	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₁₉	—	2	2	no	—
60	C₂×C₈×C₁₉	—	2	2	no	—
61	C₂×C₈×C₁₉	—	2	2	no	—
62	C₂×C₈×C₁₉	—	2	2	no	—
63	C₂×C₈×C₁₉	—	2	2	no	—
64	C₂×C₈×C₁₉	M₁₆	1	4	yes	—
65	C₄²×C₁₉	—	2	2	no	—
66	C₄²×C₁₉	—	2	2	no	—
67	C₄²×C₁₉	—	2	2	no	—
68	C₄²×C₁₉	—	2	2	no	—
69	C₄²×C₁₉	—	2	2	no	—
70	C₄²×C₁₉	C₂×C₈	1	4	yes	—
71	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
73	C₄²×C₁₉	#41	1	1	yes	characteristic
74	C₄²×C₁₉	#41	1	1	yes	characteristic
75	C₂×C₁₉⋊C₈	—	2	2	no	—
76	C₂×C₁₉⋊C₈	—	2	2	no	—
77	C₂×C₁₉⋊C₈	—	2	2	no	—
78	C₂×C₁₉⋊C₈	—	2	2	no	—
79	C₂×C₁₉⋊C₈	—	2	2	no	—
80	C₂×C₁₉⋊C₈	—	2	2	no	—
81	C₂×C₁₉⋊C₈	—	2	2	no	—
82	C₂×C₁₉⋊C₈	C₄⋊C₄	1	4	yes	—
83	C₂×C₁₉⋊C₈	C₂×C₈	1	4	yes	—
84	C₂×C₁₉⋊C₈	C₂×C₈	1	4	yes	—
85	C₂×C₁₉⋊C₈	M₁₆	1	4	yes	—
86	C₂×C₁₉⋊C₈	M₁₆	1	4	yes	—
87	#8262	—	19	1	no	maximal, Sylow
88	C₂³×C₁₉	C₂×C₄²	1	1	yes	characteristic
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	—	2	1	no	—
95	C₂×C₄×C₁₉	—	2	1	no	—
96	C₂×C₄×C₁₉	—	2	1	no	—
97	C₂×C₄×C₁₉	—	2	1	no	—
98	C₂×C₄×C₁₉	—	2	1	no	—
99	C₂×C₄×C₁₉	C₄×C₈	1	2	yes	—
100	C₂×C₄×C₁₉	C₄×C₈	1	2	yes	—
101	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
103	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
107	C₂×C₄×C₁₉	C₂²×C₈	1	2	yes	—
108	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
109	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
110	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
111	C₂×C₄×C₁₉	#131	1	2	yes	—
112	C₂×C₄×C₁₉	#131	1	2	yes	—
113	C₈×C₁₉	—	2	4	no	—
114	C₈×C₁₉	—	4	2	no	—
115	C₁₉⋊C₈	—	2	4	no	—
116	C₁₉⋊C₈	—	2	4	no	—
117	C₁₉⋊C₈	—	4	2	no	—
118	C₁₉⋊C₈	C₄⋊C₈	1	8	yes	complemented
119	C₄×C₄⋊C₈	—	19	1	no	—
120	C₄×C₄⋊C₈	D₁₉	1	1	yes	characteristic
121	C₄×#97	—	19	1	no	—
122	C₂×#322	—	19	1	no	—
123	C₂×C₈²	—	19	1	no	—
124	#1273	—	19	1	no	—
125	#1273	—	19	1	no	—
126	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
127	C₂²×C₁₉	C₂×C₄×C₈	1	1	yes	characteristic
128	C₂²×C₁₉	C₄×M₁₆	1	1	yes	characteristic
129	C₂²×C₁₉	C₄×M₁₆	1	1	yes	characteristic
130	C₂²×C₁₉	C₄×M₁₆	1	1	yes	characteristic
131	C₂²×C₁₉	#405	1	1	yes	characteristic
132	C₂²×C₁₉	#405	1	1	yes	characteristic
133	C₄×C₁₉	—	2	2	no	—
134	C₄×C₁₉	—	2	2	no	—
135	C₄×C₁₉	—	4	1	no	—
136	C₄×C₁₉	—	4	1	no	—
137	C₄×C₁₉	C₂×C₄⋊C₈	1	2	yes	—
138	C₄×C₁₉	#431	1	2	yes	—
139	C₄×C₁₉	#435	1	2	yes	—
140	C₄×C₁₉	#435	1	2	yes	—
141	C₄×C₁₉	#446	1	2	yes	—
142	C₄×C₁₉	#446	1	2	yes	—
143	C₂×C₄⋊C₈	—	19	2	no	—
144	C₂×C₄⋊C₈	Dic₁₉	1	2	yes	—
145	C₂×#97	—	19	2	no	—
146	C₂×#97	—	19	2	no	—
147	C₂×C₄×C₈	—	19	1	no	—
148	C₂×C₄×C₈	—	19	1	no	—
149	C₂×C₄×C₈	—	19	1	no	—
150	C₂×C₄×C₈	—	19	1	no	—
151	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
152	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
153	C₄³	C₂×D₁₉	1	1	yes	characteristic
154	#336	—	19	2	no	—
155	#336	Dic₁₉	1	2	yes	—
156	#322	—	38	2	no	—
157	C₈²	—	38	2	no	—
158	C₂×C₁₉	C₈×M₁₆	1	1	yes	characteristic
159	C₂×C₁₉	C₈×M₁₆	1	1	yes	characteristic
160	C₂×C₁₉	#1453	1	1	yes	characteristic
161	C₂×C₁₉	#1455	1	1	yes	characteristic
162	C₂×C₁₉	C₄×C₄⋊C₈	1	1	yes	characteristic
163	C₂×C₁₉	#1771	1	1	yes	characteristic
164	C₂×C₁₉	#1771	1	1	yes	characteristic
165	C₂²×C₈	—	19	2	no	—
166	C₂²×C₈	—	19	2	no	—
167	C₂²×C₈	—	19	2	no	—
168	C₂²×C₈	—	19	2	no	—
169	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
170	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
171	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
172	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
173	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
174	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
175	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
176	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
177	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
178	C₄⋊C₈	—	2	4	no	—
179	C₄⋊C₈	—	38	4	no	—
180	#97	—	19	8	no	—
181	#97	—	38	4	no	—
182	C₄×C₈	—	2	1	no	—
183	C₄×C₈	—	2	1	no	—
184	C₄×C₈	—	2	1	no	—
185	C₄×C₈	—	2	1	no	—
186	C₄×C₈	—	19	2	no	—
187	C₄×C₈	—	19	2	no	—
188	C₄×C₈	—	38	1	no	—
189	C₄×C₈	—	38	1	no	—
190	C₄×C₈	—	38	1	no	—
191	C₄×C₈	—	38	1	no	—
192	C₄×C₈	—	38	1	no	—
193	C₄×C₈	—	38	1	no	—
194	C₁₉	#8262	1	1	yes	characteristic, complemented, Sylow
195	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
196	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
197	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
198	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
201	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	19	4	no	—
208	C₂×C₈	—	19	4	no	—
209	C₂×C₈	—	19	4	no	—
210	C₂×C₈	—	19	4	no	—
211	C₂×C₈	—	19	4	no	—
212	C₂×C₈	—	38	2	no	—
213	C₂×C₈	—	38	2	no	—
214	C₂×C₈	—	38	2	no	—
215	C₂×C₈	—	38	2	no	—
216	C₂×C₈	—	38	2	no	—
217	C₂×C₈	—	38	2	no	—
218	C₂×C₈	—	38	2	no	—
219	C₂×C₈	#64520	1	4	yes	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	2	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	2	2	no	—
225	C₄²	C₈×D₁₉	1	4	yes	—
226	C₄²	D₄×D₁₉	1	1	yes	characteristic
227	C₄²	#64543	1	1	yes	characteristic
228	C₄²	Q₈×D₁₉	1	1	yes	characteristic
229	C₄²	#64545	1	1	yes	characteristic
230	C₂³	C₂×C₄×Dic₁₉	1	1	yes	characteristic
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	C₈×Dic₁₉	1	2	yes	—
242	C₂×C₄	C₈×Dic₁₉	1	2	yes	—
243	C₂×C₄	#10597333	1	1	yes	characteristic
244	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
245	C₂×C₄	#10597338	1	1	yes	characteristic
246	C₂×C₄	#10597339	1	1	yes	characteristic
247	C₂×C₄	C₂×C₈×D₁₉	1	2	yes	—
248	C₂×C₄	#10597347	1	2	yes	—
249	C₂×C₄	M₁₆×D₁₉	1	2	yes	—
250	C₂×C₄	M₁₆×D₁₉	1	2	yes	—
251	C₂×C₄	C₂×#64520	1	2	yes	—
252	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
253	C₂×C₄	Q₈×Dic₁₉	1	1	yes	characteristic
254	C₂×C₄	#10597394	1	2	yes	—
255	C₈	—	2	4	no	—
256	C₈	—	4	2	no	—
257	C₈	—	19	8	no	—
258	C₈	—	38	4	no	—
259	C₈	—	38	4	no	—
260	C₈	—	76	2	no	—
261	C₂²	C₄⋊C₄×Dic₁₉	1	1	yes	characteristic
262	C₂²	C₂×C₈×Dic₁₉	1	1	yes	characteristic
263	C₂²	M₁₆×Dic₁₉	1	1	yes	characteristic
264	C₂²	M₁₆×Dic₁₉	1	1	yes	characteristic
265	C₂²	C₄×#64520	1	1	yes	characteristic
266	C₂²	#11926499	1	1	yes	characteristic
267	C₂²	#11926500	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	2	2	no	—
270	C₄	—	4	1	no	—
271	C₄	—	4	1	no	—
272	C₄	C₄⋊C₈×D₁₉	1	2	yes	—
273	C₄	#11926596	1	2	yes	—
274	C₄	#11926614	1	2	yes	—
275	C₄	#11926616	1	2	yes	—
276	C₄	#11926673	1	2	yes	—
277	C₄	#11926675	1	2	yes	—
278	C₂	C₈×#64520	1	1	yes	characteristic
279	C₂	#14561916	1	1	yes	characteristic
280	C₂	#14561930	1	1	yes	characteristic
281	C₂	#14561938	1	1	yes	characteristic
282	C₂	C₄⋊C₈×Dic₁₉	1	1	yes	characteristic
283	C₂	#14563510	1	1	yes	characteristic
284	C₂	#14563512	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₁₉
commutator	C₂²×C₁₉	C₂×C₄×C₈
Fitting	C₄×C₁₉×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₄²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₄×C₄⋊C₈	D₁₉
hypocenter	C₁₉	#8262
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄² ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
43. #10597260	C₈	semidirect
118. C₁₉⋊C₈	C₄⋊C₈	semidirect
194. C₁₉	#8262	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊(C₁₉⋊C₈)	2
C₄×#10597260	2
C₂×#11926266	2
C₂×C₈×C₁₉⋊C₈	2
C₁₉⋊#1273	2
C₄×C₁₉×C₄⋊C₈	2
#8262	19