Subgroups — #18087775

Order 4864.

1374 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14566018	C₂	1	2	yes	maximal
2	C₂×#11926301	C₂	1	4	yes	maximal, complemented
3	C₂³×M₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#10597372	C₂²	1	4	yes	—
5	#11926407	—	2	2	no	—
6	#11926404	—	2	4	no	—
7	#11926400	—	2	2	no	—
8	C₄×C₂²⋊Dic₁₉	C₄	1	8	yes	complemented
9	#11926301	—	2	8	no	—
10	C₂⁴×C₄×C₁₉	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₁₉	—	2	4	no	—
12	C₂²×M₁₆×C₁₉	—	2	8	no	—
13	C₂²×M₁₆×C₁₉	C₄	1	2	yes	complemented
14	C₂²×M₁₆×C₁₉	C₂²	1	2	yes	—
15	C₂³×C₈×C₁₉	C₄	1	2	yes	complemented
16	C₂⁵×C₁₉	D₄	1	1	yes	characteristic
17	C₂³×C₄×C₁₉	—	2	4	no	—
18	C₂³×C₄×C₁₉	—	2	4	no	—
19	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₉	C₂×C₄	1	4	yes	—
22	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₉	D₄	1	4	yes	—
24	C₂³×C₄×C₁₉	Q₈	1	1	yes	characteristic
25	C₂³×C₄×C₁₉	Q₈	1	1	yes	characteristic
26	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
27	C₂×M₁₆×C₁₉	—	2	32	no	—
28	C₂×M₁₆×C₁₉	—	2	4	no	—
29	C₂×M₁₆×C₁₉	—	2	4	no	—
30	C₂×M₁₆×C₁₉	—	2	4	no	—
31	C₂×M₁₆×C₁₉	—	2	8	no	—
32	C₂×M₁₆×C₁₉	D₄	1	8	yes	—
33	C₂²×C₈×C₁₉	—	2	4	no	—
34	C₂²×C₈×C₁₉	—	2	8	no	—
35	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
36	C₂×C₂²⋊Dic₁₉	—	2	8	no	—
37	C₂×C₂²⋊Dic₁₉	—	2	4	no	—
38	#10597372	—	2	8	no	—
39	#10597372	—	2	8	no	—
40	C₂×C₄⋊Dic₁₉	—	2	4	no	—
41	C₂×C₄×Dic₁₉	C₂×C₄	1	8	yes	—
42	C₁₉⋊#95	—	2	8	no	—
43	C₂⁴×C₁₉	—	2	4	no	—
44	C₂⁴×C₁₉	—	4	4	no	—
45	C₂⁴×C₁₉	C₂²⋊C₄	1	4	yes	—
46	C₂⁴×C₁₉	C₄⋊C₄	1	1	yes	characteristic
47	C₂⁴×C₁₉	C₄⋊C₄	1	1	yes	characteristic
48	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	—	2	4	no	—
50	C₂²×C₄×C₁₉	—	2	2	no	—
51	C₂²×C₄×C₁₉	—	2	2	no	—
52	C₂²×C₄×C₁₉	—	2	2	no	—
53	C₂²×C₄×C₁₉	—	2	2	no	—
54	C₂²×C₄×C₁₉	—	2	16	no	—
55	C₂²×C₄×C₁₉	—	2	8	no	—
56	C₂²×C₄×C₁₉	—	2	4	no	—
57	C₂²×C₄×C₁₉	—	2	4	no	—
58	C₂²×C₄×C₁₉	—	2	8	no	—
59	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
60	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
61	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
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	4	yes	—
65	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
66	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
72	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
73	C₂²×C₄×C₁₉	#41	1	4	yes	—
74	C₂²×Dic₁₉	—	2	8	no	—
75	M₁₆×C₁₉	—	2	16	no	—
76	M₁₆×C₁₉	—	2	16	no	—
77	C₂×C₈×C₁₉	—	2	16	no	—
78	C₂×C₈×C₁₉	—	2	4	no	—
79	C₂×C₈×C₁₉	—	2	4	no	—
80	C₂×C₈×C₁₉	C₂²⋊C₄	1	8	yes	complemented
81	C₂²⋊Dic₁₉	—	4	16	no	—
82	C₂²⋊Dic₁₉	—	4	8	no	—
83	C₄⋊Dic₁₉	—	4	4	no	—
84	C₄⋊Dic₁₉	—	4	4	no	—
85	C₄×Dic₁₉	—	2	16	no	—
86	#10717	—	19	1	no	maximal, Sylow
87	C₂³×C₁₉	—	2	4	no	—
88	C₂³×C₁₉	—	2	2	no	—
89	C₂³×C₁₉	—	2	8	no	—
90	C₂³×C₁₉	—	2	2	no	—
91	C₂³×C₁₉	—	2	8	no	—
92	C₂³×C₁₉	—	4	4	no	—
93	C₂³×C₁₉	—	4	2	no	—
94	C₂³×C₁₉	—	4	2	no	—
95	C₂³×C₁₉	—	4	16	no	—
96	C₂³×C₁₉	#95	1	2	yes	—
97	C₂³×C₁₉	#95	1	2	yes	—
98	C₂³×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
99	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂³×C₁₉	#121	1	4	yes	—
101	C₂×C₄×C₁₉	—	2	2	no	—
102	C₂×C₄×C₁₉	—	2	2	no	—
103	C₂×C₄×C₁₉	—	2	8	no	—
104	C₂×C₄×C₁₉	—	2	4	no	—
105	C₂×C₄×C₁₉	—	2	4	no	—
106	C₂×C₄×C₁₉	—	2	2	no	—
107	C₂×C₄×C₁₉	—	2	2	no	—
108	C₂×C₄×C₁₉	—	2	2	no	—
109	C₂×C₄×C₁₉	—	2	2	no	—
110	C₂×C₄×C₁₉	—	2	8	no	—
111	C₂×C₄×C₁₉	—	2	8	no	—
112	C₂×C₄×C₁₉	#95	1	2	yes	—
113	C₂×C₄×C₁₉	#95	1	2	yes	—
114	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
115	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
116	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₁₉	#117	1	2	yes	—
119	C₂×C₄×C₁₉	C₄×D₄	1	8	yes	—
120	C₂×C₄×C₁₉	#120	1	4	yes	—
121	C₂×C₄×C₁₉	C₂²⋊Q₈	1	4	yes	—
122	C₂×C₄×C₁₉	#123	1	4	yes	—
123	C₂×Dic₁₉	—	4	16	no	—
124	C₂×Dic₁₉	—	4	8	no	—
125	C₈×C₁₉	—	2	8	no	—
126	C₂³×M₁₆	D₁₉	1	1	yes	characteristic
127	#2280	—	19	2	no	—
128	C₂×H(3,ℤ/4)	—	19	4	no	—
129	C₂²×C₁₉	—	2	8	no	—
130	C₂²×C₁₉	—	2	2	no	—
131	C₂²×C₁₉	—	2	4	no	—
132	C₂²×C₁₉	—	2	2	no	—
133	C₂²×C₁₉	—	2	2	no	—
134	C₂²×C₁₉	—	2	2	no	—
135	C₂²×C₁₉	—	2	4	no	—
136	C₂²×C₁₉	—	4	4	no	—
137	C₂²×C₁₉	—	4	16	no	—
138	C₂²×C₁₉	—	4	4	no	—
139	C₂²×C₁₉	H(3,ℤ/4)	1	4	yes	complemented
140	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
141	C₂²×C₁₉	#380	1	2	yes	—
142	C₂²×C₁₉	#386	1	4	yes	—
143	C₄×C₁₉	—	2	2	no	—
144	C₄×C₁₉	—	2	2	no	—
145	C₄×C₁₉	C₄×C₂²⋊C₄	1	2	yes	—
146	C₄×C₁₉	#379	1	1	yes	characteristic
147	C₄×C₁₉	#381	1	1	yes	characteristic
148	C₄×C₁₉	#385	1	4	yes	—
149	Dic₁₉	—	8	8	no	—
150	C₂⁴×C₄	C₂×D₁₉	1	1	yes	characteristic
151	C₂²×M₁₆	—	2	4	no	—
152	C₂²×M₁₆	—	2	8	no	—
153	C₂²×M₁₆	Dic₁₉	1	2	yes	complemented
154	C₂²×M₁₆	C₂×D₁₉	1	2	yes	—
155	C₂³×C₈	Dic₁₉	1	2	yes	complemented
156	C₂×#117	—	19	4	no	—
157	#381	—	38	2	no	—
158	#380	—	38	4	no	—
159	#379	—	38	2	no	—
160	C₄×C₂²⋊C₄	—	19	8	no	—
161	H(3,ℤ/4)	—	38	8	no	—
162	C₂×C₁₉	—	2	4	no	—
163	C₂×C₁₉	—	2	4	no	—
164	C₂×C₁₉	—	2	2	no	—
165	C₂×C₁₉	—	4	2	no	—
166	C₂×C₁₉	C₂²⋊#95	1	1	yes	characteristic
167	C₂×C₁₉	C₂×H(3,ℤ/4)	1	2	yes	—
168	C₂⁵	D₇₆	1	1	yes	characteristic
169	C₂³×C₄	—	2	4	no	—
170	C₂³×C₄	—	2	4	no	—
171	C₂³×C₄	Dic₃₈	1	1	yes	characteristic
172	C₂³×C₄	Dic₃₈	1	1	yes	characteristic
173	C₂³×C₄	C₄×D₁₉	1	4	yes	—
174	C₂³×C₄	D₇₆	1	1	yes	characteristic
175	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
176	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
177	C₂³×C₄	C₂²⋊D₁₉	1	4	yes	—
178	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
179	C₂×M₁₆	—	2	32	no	—
180	C₂×M₁₆	—	2	4	no	—
181	C₂×M₁₆	—	2	4	no	—
182	C₂×M₁₆	—	2	4	no	—
183	C₂×M₁₆	—	2	8	no	—
184	C₂×M₁₆	C₂²⋊D₁₉	1	8	yes	—
185	C₂²×C₈	—	2	4	no	—
186	C₂²×C₈	—	2	8	no	—
187	C₂²×C₈	C₂×Dic₁₉	1	4	yes	—
188	#117	—	38	8	no	—
189	#117	—	38	8	no	—
190	C₂×C₄⋊C₄	—	38	4	no	—
191	C₂×C₂²⋊C₄	—	38	8	no	—
192	C₂×C₂²⋊C₄	—	38	4	no	—
193	C₂×C₄²	—	19	8	no	—
194	#95	—	38	8	no	—
195	C₁₉	#10717	1	1	yes	characteristic, complemented, Sylow
196	C₂⁴	—	2	4	no	—
197	C₂⁴	—	4	4	no	—
198	C₂⁴	C₄⋊Dic₁₉	1	1	yes	characteristic
199	C₂⁴	C₄⋊Dic₁₉	1	1	yes	characteristic
200	C₂⁴	(C₂×D₁₉)⋊C₄	1	4	yes	—
201	C₂⁴	C₂×D₇₆	1	1	yes	characteristic
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	8	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	16	no	—
211	C₂²×C₄	—	2	8	no	—
212	C₂²×C₄	—	38	8	no	—
213	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
214	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
215	C₂²×C₄	Dic₁₉⋊C₄	1	4	yes	—
216	C₂²×C₄	Dic₁₉⋊C₄	1	4	yes	—
217	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
218	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
219	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	4	yes	—
220	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
221	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
222	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
224	C₂²×C₄	#64541	1	4	yes	—
225	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
227	C₂²×C₄	C₂×C₂²⋊D₁₉	1	4	yes	—
228	M₁₆	—	2	16	no	—
229	M₁₆	—	2	16	no	—
230	C₂×C₈	—	2	16	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	C₂²⋊Dic₁₉	1	8	yes	complemented
234	C₄⋊C₄	—	76	4	no	—
235	C₄⋊C₄	—	76	4	no	—
236	C₂²⋊C₄	—	76	16	no	—
237	C₂²⋊C₄	—	76	8	no	—
238	C₄²	—	38	16	no	—
239	C₂³	—	2	4	no	—
240	C₂³	—	2	2	no	—
241	C₂³	—	2	8	no	—
242	C₂³	—	2	2	no	—
243	C₂³	—	2	8	no	—
244	C₂³	—	4	2	no	—
245	C₂³	—	4	4	no	—
246	C₂³	—	4	2	no	—
247	C₂³	—	4	16	no	—
248	C₂³	C₁₉⋊#95	1	2	yes	—
249	C₂³	C₁₉⋊#95	1	2	yes	—
250	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
251	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	2	yes	—
252	C₂³	#10597376	1	4	yes	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	8	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	76	16	no	—
265	C₂×C₄	—	76	8	no	—
266	C₂×C₄	C₁₉⋊#95	1	2	yes	—
267	C₂×C₄	C₁₉⋊#95	1	2	yes	—
268	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
269	C₂×C₄	C₂×Dic₁₉⋊C₄	1	2	yes	—
270	C₂×C₄	#10597370	1	4	yes	—
271	C₂×C₄	#10597372	1	2	yes	—
272	C₂×C₄	C₄×C₂²⋊D₁₉	1	8	yes	—
273	C₂×C₄	#10597375	1	4	yes	—
274	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	2	yes	—
275	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
276	C₂×C₄	#10597399	1	4	yes	—
277	C₈	—	2	8	no	—
278	C₂²	—	2	8	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	4	4	no	—
286	C₂²	—	4	16	no	—
287	C₂²	—	4	4	no	—
288	C₂²	#11926301	1	4	yes	complemented
289	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
290	C₂²	#11926404	1	2	yes	—
291	C₂²	#11926423	1	4	yes	—
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	152	8	no	—
295	C₄	C₄×C₂²⋊Dic₁₉	1	2	yes	—
296	C₄	#11926400	1	1	yes	characteristic
297	C₄	#11926407	1	1	yes	characteristic
298	C₄	#11926417	1	4	yes	—
299	C₂	—	2	4	no	—
300	C₂	—	2	4	no	—
301	C₂	—	2	2	no	—
302	C₂	—	4	2	no	—
303	C₂	#14561858	1	1	yes	characteristic
304	C₂	C₂×#11926301	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂³×M₁₆×C₁₉	C₂
socle	C₂²×C₁₉	C₂×#95
prime radical	C₂⁵×C₁₉	D₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂³×M₁₆	D₁₉
hypocenter	C₁₉	#10717
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#11926301	C₂	semidirect
8. C₄×C₂²⋊Dic₁₉	C₄	semidirect
13. C₂²×M₁₆×C₁₉	C₄	semidirect
15. C₂³×C₈×C₁₉	C₄	semidirect
80. C₂×C₈×C₁₉	C₂²⋊C₄	semidirect
139. C₂²×C₁₉	H(3,ℤ/4)	semidirect
153. C₂²×M₁₆	Dic₁₉	semidirect
155. C₂³×C₈	Dic₁₉	semidirect
195. C₁₉	#10717	semidirect
233. C₂×C₈	C₂²⋊Dic₁₉	semidirect
288. C₂²	#11926301	semidirect

Maximal subgroups

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

subgroup	index
#14566018	2
C₂×#11926301	2
C₂³×M₁₆×C₁₉	2
#10717	19