Subgroups — #18081490

Order 4864.

568 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#10597315	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926276	C₂	1	4	yes	maximal
4	C₄×C₁₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×#10597315	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₃₈	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₁₉⋊C₈	—	2	1	no	—
9	#11926450	C₄	1	2	yes	—
10	#11926442	—	2	1	no	—
11	#11926442	—	2	1	no	—
12	#11926414	C₄	1	1	yes	characteristic
13	C₄×C₄⋊Dic₁₉	—	2	1	no	—
14	C₄×C₄⋊Dic₁₉	—	2	1	no	—
15	#11926299	—	2	1	no	—
16	#11926276	—	2	8	no	—
17	C₂×C₁₉×C₄⋊C₈	—	2	1	no	—
18	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
19	C₄³×C₁₉	C₂²	1	1	yes	characteristic
20	C₁₉×#336	—	2	1	no	—
21	C₂²×Dic₃₈	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₉	—	2	2	no	—
23	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊Dic₁₉	—	2	1	no	—
31	C₂×C₄⋊Dic₁₉	—	2	1	no	—
32	C₂×C₄⋊Dic₁₉	—	2	1	no	—
33	C₂×C₄⋊Dic₁₉	—	2	1	no	—
34	C₂×C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
35	C₂×Dic₁₉⋊C₄	—	2	2	no	—
36	C₂×C₄×Dic₁₉	—	2	2	no	—
37	C₂²×C₁₉⋊C₈	—	2	2	no	—
38	#10597315	—	2	2	no	—
39	#10597315	C₈	1	4	yes	complemented
40	C₄×Dic₃₈	—	2	4	no	—
41	C₄×Dic₃₈	—	2	4	no	—
42	C₁₉×C₄⋊C₈	—	2	4	no	—
43	C₄×C₈×C₁₉	—	2	4	no	—
44	C₁₉⋊#95	—	2	2	no	—
45	C₁₉⋊#95	—	2	2	no	—
46	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
47	C₄⋊(C₁₉⋊C₈)	D₄	1	4	yes	—
48	C₄×C₁₉⋊C₈	—	4	2	no	—
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	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×Dic₁₉	—	2	2	no	—
57	C₂²×Dic₁₉	—	2	2	no	—
58	C₂×Dic₃₈	—	2	4	no	—
59	C₂×Dic₃₈	—	2	4	no	—
60	C₂×Dic₃₈	—	2	2	no	—
61	C₂×Dic₃₈	C₂×C₈	1	4	yes	—
62	C₂×C₈×C₁₉	—	2	4	no	—
63	C₂×C₈×C₁₉	—	4	2	no	—
64	C₂×C₈×C₁₉	—	4	2	no	—
65	C₄²×C₁₉	—	2	2	no	—
66	C₄²×C₁₉	—	2	2	no	—
67	C₄²×C₁₉	—	2	2	no	—
68	C₄²×C₁₉	—	4	1	no	—
69	C₄²×C₁₉	C₂×C₈	1	2	yes	—
70	C₄²×C₁₉	M₁₆	1	2	yes	—
71	C₄²×C₁₉	SD₁₆	1	2	yes	—
72	C₄²×C₁₉	Q₁₆	1	2	yes	—
73	C₄²×C₁₉	C₂×D₄	1	2	yes	—
74	C₄²×C₁₉	#41	1	2	yes	—
75	C₄⋊Dic₁₉	—	2	4	no	—
76	C₄⋊Dic₁₉	—	2	4	no	—
77	C₄⋊Dic₁₉	—	2	4	no	—
78	C₄⋊Dic₁₉	—	4	2	no	—
79	C₄⋊Dic₁₉	—	4	2	no	—
80	Dic₁₉⋊C₄	—	4	2	no	—
81	Dic₁₉⋊C₄	—	4	2	no	—
82	C₄×Dic₁₉	—	4	4	no	—
83	C₂×C₁₉⋊C₈	—	2	4	no	—
84	C₂×C₁₉⋊C₈	—	4	2	no	—
85	C₂×C₁₉⋊C₈	—	4	2	no	—
86	#9673	—	19	1	no	maximal, Sylow
87	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	1	no	—
95	C₂×C₄×C₁₉	—	2	1	no	—
96	C₂×C₄×C₁₉	—	2	1	no	—
97	C₂×C₄×C₁₉	C₂²⋊C₈	1	2	yes	—
98	C₂×C₄×C₁₉	C₂²⋊C₈	1	2	yes	—
99	C₂×C₄×C₁₉	#103	1	2	yes	—
100	C₂×C₄×C₁₉	#103	1	2	yes	—
101	C₂×C₄×C₁₉	#104	1	2	yes	—
102	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₁₉	#121	1	2	yes	—
104	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
105	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
106	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
108	C₂×C₄×C₁₉	#131	1	2	yes	—
109	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
111	C₂×C₄×C₁₉	#137	1	2	yes	—
112	C₂×Dic₁₉	—	2	4	no	—
113	C₂×Dic₁₉	—	4	2	no	—
114	C₂×Dic₁₉	—	4	2	no	—
115	C₂×Dic₁₉	—	4	2	no	—
116	C₂×Dic₁₉	—	4	2	no	—
117	C₂×Dic₁₉	—	4	2	no	—
118	Dic₃₈	—	2	8	no	—
119	Dic₃₈	—	4	4	no	—
120	C₈×C₁₉	—	4	4	no	—
121	C₁₉⋊C₈	—	4	4	no	—
122	C₄×C₄⋊Q₈	—	19	1	no	—
123	C₄×C₄⋊C₈	—	19	1	no	—
124	C₄×C₄⋊C₈	D₁₉	1	1	yes	characteristic
125	C₂×Q₈⋊C₈	—	19	4	no	—
126	C₂²×C₁₉	#390	1	1	yes	characteristic
127	C₂²×C₁₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
128	C₂²×C₁₉	#408	1	1	yes	characteristic
129	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
130	C₂²×C₁₉	#419	1	1	yes	characteristic
131	C₂²×C₁₉	C₂×#104	1	1	yes	characteristic
132	C₂²×C₁₉	#421	1	1	yes	characteristic
133	C₄×C₁₉	—	2	1	no	—
134	C₄×C₁₉	—	2	1	no	—
135	C₄×C₁₉	—	2	2	no	—
136	C₄×C₁₉	—	2	2	no	—
137	C₄×C₁₉	—	4	1	no	—
138	C₄×C₁₉	Q₈⋊C₈	1	4	yes	complemented
139	C₄×C₁₉	C₈×D₄	1	2	yes	—
140	C₄×C₁₉	#436	1	2	yes	—
141	C₄×C₁₉	#461	1	2	yes	—
142	C₄×C₁₉	#462	1	2	yes	—
143	Dic₁₉	—	4	4	no	—
144	Dic₁₉	—	8	2	no	—
145	C₂×C₄⋊Q₈	—	19	1	no	—
146	C₂×C₄×Q₈	—	19	2	no	—
147	C₂×C₄⋊C₈	—	2	1	no	—
148	C₂×C₄⋊C₈	—	19	2	no	—
149	C₂×C₄×C₈	—	38	1	no	—
150	C₂×C₄×C₈	C₂×D₁₉	1	2	yes	—
151	#391	—	19	2	no	—
152	#389	—	38	1	no	—
153	#389	—	38	1	no	—
154	#384	—	19	1	no	—
155	C₄×C₄⋊C₄	—	38	1	no	—
156	C₄×C₄⋊C₄	—	38	1	no	—
157	C₄³	C₂×D₁₉	1	1	yes	characteristic
158	#336	—	2	1	no	—
159	#336	—	38	1	no	—
160	Q₈⋊C₈	—	38	8	no	—
161	C₂×C₁₉	C₂×Q₈⋊C₈	1	1	yes	characteristic
162	C₂×C₁₉	#1479	1	1	yes	characteristic
163	C₂×C₁₉	#1481	1	1	yes	characteristic
164	C₂×C₁₉	#1485	1	1	yes	characteristic
165	C₂×C₁₉	#1954	1	1	yes	characteristic
166	C₂×C₁₉	#1980	1	1	yes	characteristic
167	C₂×C₁₉	#1981	1	1	yes	characteristic
168	C₂²×Q₈	—	19	2	no	—
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	—	38	2	no	—
171	C₄⋊Q₈	—	19	4	no	—
172	C₄⋊Q₈	—	38	2	no	—
173	C₄×Q₈	—	38	4	no	—
174	C₄×Q₈	—	38	4	no	—
175	C₂×C₄⋊C₄	—	19	2	no	—
176	C₂×C₄⋊C₄	—	38	1	no	—
177	C₂×C₄⋊C₄	—	38	1	no	—
178	C₂×C₄⋊C₄	—	38	1	no	—
179	C₂×C₄⋊C₄	—	38	1	no	—
180	C₂×C₄⋊C₄	—	38	2	no	—
181	C₂×C₄²	—	38	2	no	—
182	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
183	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
184	C₂×C₄²	D₇₆	1	1	yes	characteristic
185	C₂×C₄²	D₇₆	1	1	yes	characteristic
186	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
187	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
188	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
189	C₄⋊C₈	—	2	4	no	—
190	C₄⋊C₈	—	19	4	no	—
191	C₄⋊C₈	—	38	2	no	—
192	C₄×C₈	—	2	4	no	—
193	C₄×C₈	—	76	2	no	—
194	#95	—	38	2	no	—
195	#95	—	38	2	no	—
196	C₁₉	#9673	1	1	yes	characteristic, complemented, Sylow
197	C₂×Q₈	—	19	4	no	—
198	C₂×Q₈	—	38	2	no	—
199	C₂×Q₈	—	38	4	no	—
200	C₂×Q₈	—	38	4	no	—
201	C₂²×C₄	—	38	2	no	—
202	C₂²×C₄	—	38	2	no	—
203	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
204	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
205	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
206	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	4	2	no	—
212	C₂×C₈	—	4	2	no	—
213	C₂×C₈	—	38	4	no	—
214	C₂×C₈	—	76	2	no	—
215	C₂×C₈	—	76	2	no	—
216	C₄⋊C₄	—	38	4	no	—
217	C₄⋊C₄	—	38	4	no	—
218	C₄⋊C₄	—	38	4	no	—
219	C₄⋊C₄	—	76	2	no	—
220	C₄⋊C₄	—	76	2	no	—
221	C₄⋊C₄	—	76	2	no	—
222	C₄⋊C₄	—	76	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	2	2	no	—
225	C₄²	—	2	2	no	—
226	C₄²	—	4	1	no	—
227	C₄²	—	76	4	no	—
228	C₄²	C₈×D₁₉	1	2	yes	—
229	C₄²	#64515	1	2	yes	—
230	C₄²	#64516	1	2	yes	—
231	C₄²	Dic₇₆	1	2	yes	—
232	C₄²	D₄×D₁₉	1	2	yes	—
233	C₄²	#64545	1	2	yes	—
234	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
235	Q₈	—	38	8	no	—
236	Q₈	—	76	4	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	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	38	4	no	—
247	C₂×C₄	—	76	2	no	—
248	C₂×C₄	—	76	2	no	—
249	C₂×C₄	—	76	2	no	—
250	C₂×C₄	—	76	2	no	—
251	C₂×C₄	—	76	2	no	—
252	C₂×C₄	#10597262	1	2	yes	—
253	C₂×C₄	#10597273	1	2	yes	—
254	C₂×C₄	#10597273	1	2	yes	—
255	C₂×C₄	(C₂×D₁₉)⋊C₈	1	2	yes	—
256	C₂×C₄	(C₂×D₁₉)⋊C₈	1	2	yes	—
257	C₂×C₄	#10597339	1	2	yes	—
258	C₂×C₄	#10597341	1	2	yes	—
259	C₂×C₄	C₂×C₈×D₁₉	1	1	yes	characteristic
260	C₂×C₄	C₂×#64515	1	1	yes	characteristic
261	C₂×C₄	C₂×#64516	1	1	yes	characteristic
262	C₂×C₄	C₂×Dic₇₆	1	1	yes	characteristic
263	C₂×C₄	#10597353	1	2	yes	—
264	C₂×C₄	#10597355	1	2	yes	—
265	C₂×C₄	#10597386	1	1	yes	characteristic
266	C₂×C₄	#10597393	1	1	yes	characteristic
267	C₈	—	4	4	no	—
268	C₈	—	76	4	no	—
269	C₂²	#11926445	1	1	yes	characteristic
270	C₂²	C₂×(C₂×D₁₉)⋊C₈	1	1	yes	characteristic
271	C₂²	#11926512	1	1	yes	characteristic
272	C₂²	C₂×#10597273	1	1	yes	characteristic
273	C₂²	#11926554	1	1	yes	characteristic
274	C₂²	C₂×#10597262	1	1	yes	characteristic
275	C₂²	#11926562	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	4	1	no	—
281	C₄	—	76	4	no	—
282	C₄	—	152	2	no	—
283	C₄	#11926276	1	4	yes	complemented
284	C₄	#11926609	1	2	yes	—
285	C₄	#11926621	1	2	yes	—
286	C₄	#11926750	1	2	yes	—
287	C₄	#11926755	1	2	yes	—
288	C₂	C₂×#11926276	1	1	yes	characteristic
289	C₂	#14562031	1	1	yes	characteristic
290	C₂	#14562039	1	1	yes	characteristic
291	C₂	#14562057	1	1	yes	characteristic
292	C₂	#14564428	1	1	yes	characteristic
293	C₂	#14564553	1	1	yes	characteristic
294	C₂	#14564557	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₇₆
commutator	C₂×C₄×C₁₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₄×C₁₉×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₄×C₄⋊C₈	D₁₉
hypocenter	C₁₉	#9673
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
39. #10597315	C₈	semidirect
138. C₄×C₁₉	Q₈⋊C₈	semidirect
196. C₁₉	#9673	semidirect
283. C₄	#11926276	semidirect

Maximal subgroups

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

subgroup	index
C₄×#10597315	2
C₄×C₄⋊(C₁₉⋊C₈)	2
C₂×#11926276	2
C₄×C₁₉×C₄⋊C₈	2
#9673	19