Subgroups — #18073380

Order 4864.

462 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926674	C₂	1	1	yes	maximal, characteristic
2	#14563502	C₂	1	1	yes	maximal, characteristic
3	C₄×#10597273	C₂	1	1	yes	maximal, characteristic
4	C₂×#11926276	C₂	1	1	yes	maximal, characteristic
5	#14561475	C₂	1	1	yes	maximal, characteristic
6	#14561470	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₉×#322	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Dic₃₈	C₂²	1	1	yes	characteristic
9	#11926674	—	2	2	no	—
10	#11926674	C₄	1	4	yes	complemented
11	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×Dic₁₉⋊C₈	—	2	1	no	—
14	C₂×#10597273	C₄	1	2	yes	—
15	C₂×#10597272	—	2	1	no	—
16	C₂×#10597272	—	2	1	no	—
17	C₄×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
18	#11926307	C₄	1	2	yes	—
19	#11926298	—	2	1	no	—
20	#11926276	C₄	1	4	yes	complemented
21	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
24	C₁₉×#322	—	2	2	no	—
25	C₂²×Dic₃₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₉	—	2	1	no	—
27	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
31	C₂×C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊Dic₁₉	C₂×C₄	1	1	yes	characteristic
33	C₂×Dic₁₉⋊C₄	—	2	1	no	—
34	C₂×C₄×Dic₁₉	—	2	1	no	—
35	C₂×C₄×Dic₁₉	—	2	1	no	—
36	C₂²×C₁₉⋊C₈	—	2	1	no	—
37	C₂²×C₁₉⋊C₈	—	2	1	no	—
38	C₄×Dic₃₈	—	2	1	no	—
39	C₄×Dic₃₈	—	2	1	no	—
40	C₄×Dic₃₈	C₂×C₄	1	2	yes	—
41	C₄×Dic₃₈	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₁₉	—	2	1	no	—
43	C₄×C₈×C₁₉	—	2	1	no	—
44	C₄×C₈×C₁₉	—	2	1	no	—
45	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₁₉	D₄	1	2	yes	—
48	C₁₉⋊#95	—	2	1	no	—
49	#10597273	—	2	4	no	—
50	#10597272	—	2	4	no	—
51	#10597272	—	4	2	no	—
52	Dic₁₉⋊C₈	—	2	4	no	—
53	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
54	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
55	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
56	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
57	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
60	C₂²×Dic₁₉	—	2	1	no	—
61	C₂²×Dic₁₉	—	2	1	no	—
62	C₂²×Dic₁₉	—	2	2	no	—
63	C₂×Dic₃₈	—	2	2	no	—
64	C₂×Dic₃₈	—	2	2	no	—
65	C₂×Dic₃₈	—	2	1	no	—
66	C₂×Dic₃₈	C₄²	1	2	yes	—
67	C₂×C₈×C₁₉	—	2	2	no	—
68	C₂×C₈×C₁₉	—	2	2	no	—
69	C₂×C₈×C₁₉	—	2	1	no	—
70	C₂×C₈×C₁₉	—	2	1	no	—
71	C₂×C₈×C₁₉	—	2	1	no	—
72	C₂×C₈×C₁₉	—	2	1	no	—
73	C₂×C₈×C₁₉	—	2	2	no	—
74	C₂×C₈×C₁₉	—	4	2	no	—
75	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
76	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
77	C₂×C₈×C₁₉	M₁₆	1	4	yes	—
78	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₁₉	#41	1	1	yes	characteristic
82	C₄⋊Dic₁₉	—	2	1	no	—
83	C₄⋊Dic₁₉	—	2	2	no	—
84	C₄⋊Dic₁₉	—	2	2	no	—
85	C₄⋊Dic₁₉	C₄²	1	2	yes	—
86	Dic₁₉⋊C₄	—	2	2	no	—
87	Dic₁₉⋊C₄	—	2	2	no	—
88	C₄×Dic₁₉	—	2	2	no	—
89	C₄×Dic₁₉	—	2	2	no	—
90	C₄×Dic₁₉	—	4	1	no	—
91	C₄×Dic₁₉	—	4	1	no	—
92	C₂×C₁₉⋊C₈	—	2	2	no	—
93	C₂×C₁₉⋊C₈	—	2	2	no	—
94	C₂×C₁₉⋊C₈	—	4	1	no	—
95	C₂×C₁₉⋊C₈	—	4	1	no	—
96	C₂×C₁₉⋊C₈	—	4	1	no	—
97	C₂×C₁₉⋊C₈	—	4	1	no	—
98	#8209	—	19	1	no	maximal, Sylow
99	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₉	—	2	2	no	—
101	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
104	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
108	C₂×C₄×C₁₉	#131	1	2	yes	—
109	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
110	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
111	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
112	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
113	C₂×Dic₁₉	—	2	2	no	—
114	C₂×Dic₁₉	—	2	2	no	—
115	C₂×Dic₁₉	—	4	2	no	—
116	C₂×Dic₁₉	—	4	2	no	—
117	C₂×Dic₁₉	—	4	2	no	—
118	C₂×Dic₁₉	—	4	2	no	—
119	C₂×Dic₁₉	—	4	2	no	—
120	Dic₃₈	—	2	2	no	—
121	Dic₃₈	—	2	2	no	—
122	Dic₃₈	—	4	2	no	—
123	C₈×C₁₉	—	2	2	no	—
124	C₈×C₁₉	—	2	2	no	—
125	C₈×C₁₉	—	2	4	no	—
126	C₈×C₁₉	—	4	2	no	—
127	C₁₉⋊C₈	—	4	2	no	—
128	C₁₉⋊C₈	—	4	2	no	—
129	C₂×#446	—	19	1	no	—
130	#1770	—	19	1	no	—
131	C₄×#103	—	19	1	no	—
132	C₂×Q₈⋊C₈	—	19	1	no	—
133	C₂×#322	D₁₉	1	1	yes	characteristic
134	#1279	—	19	1	no	—
135	#1277	—	19	1	no	—
136	C₂²×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
137	C₂²×C₁₉	#436	1	2	yes	—
138	C₂²×C₁₉	#440	1	1	yes	characteristic
139	C₂²×C₁₉	#441	1	1	yes	characteristic
140	C₂²×C₁₉	#444	1	2	yes	—
141	C₄×C₁₉	—	2	1	no	—
142	C₄×C₁₉	—	2	1	no	—
143	C₄×C₁₉	C₄×M₁₆	1	1	yes	characteristic
144	C₄×C₁₉	#405	1	1	yes	characteristic
145	C₄×C₁₉	#417	1	1	yes	characteristic
146	C₄×C₁₉	#419	1	1	yes	characteristic
147	C₄×C₁₉	#435	1	2	yes	—
148	C₄×C₁₉	#440	1	1	yes	characteristic
149	C₄×C₁₉	#441	1	1	yes	characteristic
150	Dic₁₉	—	4	2	no	—
151	Dic₁₉	—	4	2	no	—
152	Dic₁₉	—	8	2	no	—
153	C₂×C₄×Q₈	—	19	1	no	—
154	#446	—	19	4	no	—
155	#446	—	38	2	no	—
156	C₂×C₄⋊C₈	—	19	1	no	—
157	C₂×C₄⋊C₈	—	19	1	no	—
158	C₂×C₄⋊C₈	—	38	1	no	—
159	C₂×#103	—	19	2	no	—
160	C₂×#97	—	38	1	no	—
161	C₂×#97	—	38	1	no	—
162	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
163	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
164	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
165	C₄×C₄⋊C₄	—	19	1	no	—
166	#340	—	19	2	no	—
167	#336	—	38	1	no	—
168	Q₈⋊C₈	—	19	4	no	—
169	#322	—	2	2	no	—
170	C₂×C₁₉	#1592	1	2	yes	—
171	C₂×C₁₉	#1593	1	2	yes	—
172	C₂×C₁₉	#1753	1	1	yes	characteristic
173	C₂×C₁₉	#1763	1	1	yes	characteristic
174	C₂×C₁₉	#1765	1	1	yes	characteristic
175	C₂²×Q₈	—	19	1	no	—
176	C₂²×C₈	—	2	1	no	—
177	C₂²×C₈	—	38	1	no	—
178	C₂²×C₈	—	38	1	no	—
179	C₂²×C₈	C₄×D₁₉	1	2	yes	—
180	C₂²×C₈	D₇₆	1	1	yes	characteristic
181	C₂²×C₈	D₇₆	1	1	yes	characteristic
182	C₄×Q₈	—	19	2	no	—
183	C₄×Q₈	—	19	2	no	—
184	C₄×Q₈	—	38	1	no	—
185	C₄×Q₈	—	38	1	no	—
186	C₂×C₄⋊C₄	—	19	1	no	—
187	C₂×C₄⋊C₄	—	19	2	no	—
188	C₂×C₄⋊C₄	—	38	1	no	—
189	C₂×C₄²	—	38	1	no	—
190	C₂×C₄²	—	38	1	no	—
191	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
192	C₄⋊C₈	—	19	2	no	—
193	C₄⋊C₈	—	19	2	no	—
194	C₄⋊C₈	—	38	4	no	—
195	C₄⋊C₈	—	38	1	no	—
196	C₄⋊C₈	—	38	1	no	—
197	#103	—	38	4	no	—
198	#97	—	38	4	no	—
199	#97	—	76	2	no	—
200	C₄×C₈	—	2	1	no	—
201	C₄×C₈	—	2	1	no	—
202	C₄×C₈	—	2	1	no	—
203	C₄×C₈	C₄×D₁₉	1	2	yes	—
204	C₄×C₈	C₄×D₁₉	1	2	yes	—
205	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
206	#95	—	38	1	no	—
207	C₁₉	#8209	1	1	yes	characteristic, complemented, Sylow
208	C₂×Q₈	—	19	2	no	—
209	C₂×Q₈	—	38	1	no	—
210	C₂×Q₈	—	38	2	no	—
211	C₂×Q₈	—	38	2	no	—
212	C₂²×C₄	—	38	1	no	—
213	C₂²×C₄	—	38	1	no	—
214	C₂²×C₄	—	38	2	no	—
215	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
217	C₂²×C₄	#64541	1	1	yes	characteristic
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	38	2	no	—
227	C₂×C₈	—	38	2	no	—
228	C₂×C₈	—	76	1	no	—
229	C₂×C₈	—	76	1	no	—
230	C₂×C₈	—	76	1	no	—
231	C₂×C₈	—	76	1	no	—
232	C₂×C₈	#64515	1	4	yes	—
233	C₂×C₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
234	C₂×C₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
235	C₄⋊C₄	—	19	2	no	—
236	C₄⋊C₄	—	38	2	no	—
237	C₄⋊C₄	—	38	2	no	—
238	C₄⋊C₄	—	38	1	no	—
239	C₄⋊C₄	—	38	2	no	—
240	C₄⋊C₄	—	38	2	no	—
241	C₄²	—	38	2	no	—
242	C₄²	—	38	2	no	—
243	C₄²	—	76	1	no	—
244	C₄²	—	76	1	no	—
245	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
246	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
247	C₄²	#64541	1	1	yes	characteristic
248	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
249	C₂³	C₄×D₇₆	1	1	yes	characteristic
250	Q₈	—	38	2	no	—
251	Q₈	—	38	2	no	—
252	Q₈	—	76	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	38	2	no	—
255	C₂×C₄	—	38	2	no	—
256	C₂×C₄	—	76	2	no	—
257	C₂×C₄	—	76	2	no	—
258	C₂×C₄	—	76	2	no	—
259	C₂×C₄	—	76	2	no	—
260	C₂×C₄	—	76	2	no	—
261	C₂×C₄	C₄²×D₁₉	1	1	yes	characteristic
262	C₂×C₄	#10597318	1	1	yes	characteristic
263	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
264	C₂×C₄	C₂×#64515	1	2	yes	—
265	C₂×C₄	#10597347	1	2	yes	—
266	C₂×C₄	#10597354	1	1	yes	characteristic
267	C₂×C₄	#10597355	1	1	yes	characteristic
268	C₂×C₄	#10597355	1	1	yes	characteristic
269	C₂×C₄	#10597355	1	1	yes	characteristic
270	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
271	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
272	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
273	C₈	—	2	2	no	—
274	C₈	—	2	2	no	—
275	C₈	—	2	4	no	—
276	C₈	—	4	2	no	—
277	C₈	—	76	2	no	—
278	C₈	—	76	2	no	—
279	C₂²	C₄×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
280	C₂²	#11926623	1	2	yes	—
281	C₂²	#11926646	1	1	yes	characteristic
282	C₂²	#11926652	1	1	yes	characteristic
283	C₂²	#11926667	1	2	yes	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	76	2	no	—
287	C₄	—	76	2	no	—
288	C₄	—	152	2	no	—
289	C₄	C₄×#64515	1	1	yes	characteristic
290	C₄	#11926498	1	1	yes	characteristic
291	C₄	#11926546	1	1	yes	characteristic
292	C₄	#11926554	1	1	yes	characteristic
293	C₄	#11926615	1	2	yes	—
294	C₄	#11926646	1	1	yes	characteristic
295	C₄	#11926652	1	1	yes	characteristic
296	C₂	#14562539	1	2	yes	—
297	C₂	#14562545	1	2	yes	—
298	C₂	#14563418	1	1	yes	characteristic
299	C₂	#14563466	1	1	yes	characteristic
300	C₂	#14563479	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×C₁₉×#322	C₂
socle, prime radical	C₂³×C₁₉	C₄×D₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#322	D₁₉
hypocenter	C₁₉	#8209
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
10. #11926674	C₄	semidirect
20. #11926276	C₄	semidirect
207. C₁₉	#8209	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926674	2
#14563502	2
C₄×#10597273	2
C₂×#11926276	2
#14561475	2
#14561470	2
C₂×C₁₉×#322	2
#8209	19