Subgroups — #18088235

Order 4864.

674 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×Dic₁₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926316	C₂	1	2	yes	maximal
3	C₂×#11926305	C₂	1	2	yes	maximal
4	C₂×#11926301	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₉×#338	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	C₂²	1	1	yes	characteristic
7	C₂×#10597372	C₂²	1	2	yes	—
8	C₂²×C₄×Dic₁₉	C₂²	1	1	yes	characteristic
9	M₁₆×Dic₁₉	—	2	4	no	—
10	C₂×#10597272	C₄	1	2	yes	complemented
11	C₂×C₈×Dic₁₉	C₄	1	2	yes	complemented
12	#11926316	—	2	4	no	—
13	#11926305	—	2	4	no	—
14	#11926301	C₄	1	4	yes	complemented
15	C₂²×M₁₆×C₁₉	C₂²	1	2	yes	—
16	C₂²×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#338	C₄	1	4	yes	complemented
18	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
19	C₂³×Dic₁₉	—	2	1	no	—
20	C₂×M₁₆×C₁₉	—	2	2	no	—
21	C₂×M₁₆×C₁₉	—	2	2	no	—
22	C₂×M₁₆×C₁₉	—	2	2	no	—
23	C₂×M₁₆×C₁₉	—	2	2	no	—
24	C₂×M₁₆×C₁₉	—	2	2	no	—
25	C₂×M₁₆×C₁₉	—	2	2	no	—
26	C₂×M₁₆×C₁₉	—	2	2	no	—
27	C₂×M₁₆×C₁₉	C₂×C₄	1	4	yes	—
28	C₂×M₁₆×C₁₉	C₂×C₄	1	2	yes	—
29	C₂×M₁₆×C₁₉	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₁₉	—	2	2	no	—
31	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
32	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
33	#10597372	—	2	2	no	—
34	#10597372	—	2	2	no	—
35	#10597372	—	2	2	no	—
36	#10597372	C₂×C₄	1	4	yes	—
37	C₂×C₄⋊Dic₁₉	—	2	2	no	—
38	C₂×C₄×Dic₁₉	—	2	2	no	—
39	C₂×C₄×Dic₁₉	—	2	1	no	—
40	C₂×C₄×Dic₁₉	—	2	2	no	—
41	C₂×C₄×Dic₁₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×Dic₁₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×Dic₁₉	D₄	1	4	yes	—
44	C₂×#64520	—	2	2	no	—
45	C₂×#64520	—	2	2	no	—
46	C₂×#64520	D₄	1	2	yes	—
47	C₂×#64520	Q₈	1	2	yes	—
48	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
49	#10597272	—	2	4	no	—
50	C₈×Dic₁₉	—	2	4	no	—
51	C₂⁴×C₁₉	#41	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	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	#41	1	2	yes	—
59	C₂²×C₄×C₁₉	#41	1	2	yes	—
60	C₂²×C₄×C₁₉	#41	1	2	yes	—
61	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
62	C₂²×Dic₁₉	—	2	1	no	—
63	C₂²×Dic₁₉	—	2	1	no	—
64	C₂²×Dic₁₉	—	2	4	no	—
65	C₂²×Dic₁₉	—	4	4	no	—
66	C₂²×Dic₁₉	—	4	2	no	—
67	M₁₆×C₁₉	—	2	4	no	—
68	M₁₆×C₁₉	—	2	4	no	—
69	M₁₆×C₁₉	—	2	4	no	—
70	M₁₆×C₁₉	—	2	4	no	—
71	M₁₆×C₁₉	—	4	2	no	—
72	M₁₆×C₁₉	—	4	2	no	—
73	C₂×C₈×C₁₉	—	2	2	no	—
74	C₂×C₈×C₁₉	—	2	2	no	—
75	C₂×C₈×C₁₉	—	2	4	no	—
76	C₂×C₈×C₁₉	—	4	2	no	—
77	C₂×C₈×C₁₉	—	4	2	no	—
78	C₂×C₈×C₁₉	C₄²	1	4	yes	—
79	C₂²⋊Dic₁₉	—	2	4	no	—
80	C₂²⋊Dic₁₉	—	4	2	no	—
81	C₄⋊Dic₁₉	—	2	4	no	—
82	C₄⋊Dic₁₉	—	4	2	no	—
83	C₄×Dic₁₉	—	2	1	no	—
84	C₄×Dic₁₉	—	2	1	no	—
85	C₄×Dic₁₉	—	2	4	no	—
86	C₄×Dic₁₉	—	2	4	no	—
87	C₄×Dic₁₉	—	4	2	no	—
88	C₄×Dic₁₉	C₂²⋊C₄	1	2	yes	—
89	C₄×Dic₁₉	C₂²⋊C₄	1	2	yes	—
90	#64520	—	2	4	no	—
91	#64520	—	2	4	no	—
92	C₂×C₁₉⋊C₈	—	2	2	no	—
93	C₂×C₁₉⋊C₈	—	2	2	no	—
94	C₂×C₁₉⋊C₈	C₄⋊C₄	1	4	yes	—
95	#10813	—	19	1	no	maximal, Sylow
96	C₂³×C₁₉	—	4	2	no	—
97	C₂³×C₁₉	#117	1	1	yes	characteristic
98	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
99	C₂³×C₁₉	C₄×Q₈	1	1	yes	characteristic
100	C₂³×C₁₉	#123	1	2	yes	—
101	C₂³×C₁₉	#126	1	2	yes	—
102	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
106	C₂×C₄×C₁₉	#117	1	2	yes	—
107	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
108	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
109	C₂×C₄×C₁₉	#117	1	2	yes	—
110	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₁₉	#121	1	2	yes	—
115	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₁₉	#123	1	2	yes	—
117	C₂×C₄×C₁₉	#124	1	2	yes	—
118	C₂×C₄×C₁₉	#125	1	2	yes	—
119	C₂×C₄×C₁₉	#126	1	2	yes	—
120	C₂×Dic₁₉	—	2	4	no	—
121	C₂×Dic₁₉	—	2	2	no	—
122	C₂×Dic₁₉	—	2	2	no	—
123	C₂×Dic₁₉	—	4	8	no	—
124	C₂×Dic₁₉	—	4	2	no	—
125	C₂×Dic₁₉	—	4	4	no	—
126	C₂×Dic₁₉	—	8	4	no	—
127	C₈×C₁₉	—	2	4	no	—
128	C₈×C₁₉	—	4	4	no	—
129	C₁₉⋊C₈	—	2	4	no	—
130	C₂×C₄×M₁₆	—	19	1	no	—
131	C₂×#344	—	19	2	no	—
132	C₂×#339	—	19	2	no	—
133	C₂×#338	D₁₉	1	1	yes	characteristic
134	C₂×H(3,ℤ/4)	—	19	1	no	—
135	C₂²×C₁₉	—	2	2	no	—
136	C₂²×C₁₉	—	2	4	no	—
137	C₂²×C₁₉	—	4	2	no	—
138	C₂²×C₁₉	—	4	1	no	—
139	C₂²×C₁₉	—	4	1	no	—
140	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
141	C₂²×C₁₉	#381	1	1	yes	characteristic
142	C₂²×C₁₉	#383	1	1	yes	characteristic
143	C₂²×C₁₉	#387	1	2	yes	—
144	C₂²×C₁₉	#388	1	2	yes	—
145	C₄×C₁₉	#376	1	1	yes	characteristic
146	C₄×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
147	C₄×C₁₉	#380	1	1	yes	characteristic
148	C₄×C₁₉	#382	1	1	yes	characteristic
149	C₄×C₁₉	#387	1	2	yes	—
150	C₄×C₁₉	#388	1	2	yes	—
151	Dic₁₉	—	2	4	no	—
152	Dic₁₉	—	4	2	no	—
153	Dic₁₉	—	8	4	no	—
154	C₂²×M₁₆	—	19	1	no	—
155	C₂²×M₁₆	C₂×D₁₉	1	2	yes	—
156	C₂²×M₁₆	C₂×D₁₉	1	1	yes	characteristic
157	C₂×#117	—	19	2	no	—
158	C₂²×C₄²	—	19	1	no	—
159	C₄×M₁₆	—	38	4	no	—
160	C₂×#97	—	19	2	no	—
161	C₂×C₄×C₈	—	19	2	no	—
162	#344	—	38	4	no	—
163	#339	—	38	4	no	—
164	#338	Dic₁₉	1	4	yes	complemented
165	H(3,ℤ/4)	—	19	4	no	—
166	C₂×C₁₉	—	2	1	no	—
167	C₂×C₁₉	—	2	2	no	—
168	C₂×C₁₉	—	2	1	no	—
169	C₂×C₁₉	—	2	2	no	—
170	C₂×C₁₉	#1434	1	1	yes	characteristic
171	C₂×C₁₉	#1841	1	2	yes	—
172	C₂³×C₄	—	38	1	no	—
173	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	19	2	no	—
182	C₂×M₁₆	—	19	2	no	—
183	C₂×M₁₆	—	38	2	no	—
184	C₂×M₁₆	—	38	2	no	—
185	C₂×M₁₆	C₄×D₁₉	1	2	yes	—
186	C₂×M₁₆	C₂×Dic₁₉	1	4	yes	—
187	C₂×M₁₆	C₂×Dic₁₉	1	2	yes	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	—	19	2	no	—
190	C₂²×C₈	C₄×D₁₉	1	2	yes	—
191	#117	—	19	4	no	—
192	#117	—	38	2	no	—
193	#117	—	38	2	no	—
194	#117	—	38	2	no	—
195	C₂×C₄⋊C₄	—	38	2	no	—
196	C₂×C₂²⋊C₄	—	38	2	no	—
197	C₂×C₄²	—	19	1	no	—
198	C₂×C₄²	—	19	1	no	—
199	C₂×C₄²	—	19	4	no	—
200	C₂×C₄²	—	38	2	no	—
201	C₂×C₄²	—	38	1	no	—
202	C₂×C₄²	—	38	2	no	—
203	#97	—	38	4	no	—
204	C₄×C₈	—	38	4	no	—
205	C₁₉	#10813	1	1	yes	characteristic, complemented, Sylow
206	C₂⁴	#64545	1	1	yes	characteristic
207	C₂²×C₄	—	38	1	no	—
208	C₂²×C₄	—	38	1	no	—
209	C₂²×C₄	—	38	4	no	—
210	C₂²×C₄	—	76	4	no	—
211	C₂²×C₄	—	76	2	no	—
212	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
214	C₂²×C₄	#64541	1	2	yes	—
215	C₂²×C₄	#64541	1	2	yes	—
216	C₂²×C₄	D₄×D₁₉	1	2	yes	—
217	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
218	C₂²×C₄	#64543	1	2	yes	—
219	C₂²×C₄	#64543	1	1	yes	characteristic
220	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
221	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
222	M₁₆	—	2	4	no	—
223	M₁₆	—	2	4	no	—
224	M₁₆	—	2	4	no	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	4	2	no	—
227	M₁₆	—	4	2	no	—
228	M₁₆	—	38	4	no	—
229	M₁₆	—	38	4	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	—	4	2	no	—
235	C₂×C₈	—	19	4	no	—
236	C₂×C₈	—	38	2	no	—
237	C₂×C₈	—	38	2	no	—
238	C₂×C₈	C₄×Dic₁₉	1	4	yes	—
239	C₄⋊C₄	—	38	4	no	—
240	C₄⋊C₄	—	76	2	no	—
241	C₂²⋊C₄	—	38	4	no	—
242	C₂²⋊C₄	—	76	2	no	—
243	C₄²	—	19	2	no	—
244	C₄²	—	19	2	no	—
245	C₄²	—	38	1	no	—
246	C₄²	—	38	1	no	—
247	C₄²	—	38	4	no	—
248	C₄²	—	38	4	no	—
249	C₄²	—	76	2	no	—
250	C₂³	—	4	2	no	—
251	C₂³	#10597338	1	1	yes	characteristic
252	C₂³	#10597339	1	1	yes	characteristic
253	C₂³	#10597340	1	2	yes	—
254	C₂³	#10597344	1	2	yes	—
255	C₂³	Q₈×Dic₁₉	1	1	yes	characteristic
256	C₂×C₄	—	38	4	no	—
257	C₂×C₄	—	38	2	no	—
258	C₂×C₄	—	38	2	no	—
259	C₂×C₄	—	76	8	no	—
260	C₂×C₄	—	76	2	no	—
261	C₂×C₄	—	76	4	no	—
262	C₂×C₄	—	152	4	no	—
263	C₂×C₄	#10597318	1	1	yes	characteristic
264	C₂×C₄	#10597318	1	2	yes	—
265	C₂×C₄	#10597318	1	1	yes	characteristic
266	C₂×C₄	#10597323	1	1	yes	characteristic
267	C₂×C₄	#10597325	1	2	yes	—
268	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
269	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
270	C₂×C₄	#10597329	1	2	yes	—
271	C₂×C₄	#10597330	1	2	yes	—
272	C₂×C₄	#10597331	1	2	yes	—
273	C₂×C₄	#10597333	1	1	yes	characteristic
274	C₂×C₄	#10597335	1	2	yes	—
275	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
276	C₂×C₄	#10597342	1	2	yes	—
277	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
278	C₂×C₄	#10597372	1	2	yes	—
279	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
280	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
281	C₈	—	2	4	no	—
282	C₈	—	4	4	no	—
283	C₈	—	38	4	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	4	no	—
286	C₂²	—	4	2	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	1	no	—
289	C₂²	C₄⋊C₄×Dic₁₉	1	1	yes	characteristic
290	C₂²	#11926406	1	1	yes	characteristic
291	C₂²	#11926411	1	1	yes	characteristic
292	C₂²	#11926430	1	2	yes	—
293	C₂²	#11926433	1	2	yes	—
294	C₄	—	38	4	no	—
295	C₄	—	76	2	no	—
296	C₄	—	152	4	no	—
297	C₄	#11926390	1	1	yes	characteristic
298	C₄	C₂²⋊C₄×Dic₁₉	1	1	yes	characteristic
299	C₄	#11926403	1	1	yes	characteristic
300	C₄	#11926408	1	1	yes	characteristic
301	C₄	#11926426	1	2	yes	—
302	C₄	#11926437	1	2	yes	—
303	C₂	—	2	1	no	—
304	C₂	—	2	2	no	—
305	C₂	—	2	1	no	—
306	C₂	—	2	2	no	—
307	C₂	#14561860	1	1	yes	characteristic
308	C₂	#14563785	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#10597318
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₁₉×#338	C₂
socle	C₂²×C₁₉	#383
prime radical	C₂⁴×C₁₉	#41
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×#338	D₁₉
hypocenter	C₁₉	#10813
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#338

Complements

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

N	H ≅ G/N	type
10. C₂×#10597272	C₄	semidirect
11. C₂×C₈×Dic₁₉	C₄	semidirect
14. #11926301	C₄	semidirect
17. C₁₉×#338	C₄	semidirect
164. #338	Dic₁₉	semidirect
205. C₁₉	#10813	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×Dic₁₉	2
C₂×#11926316	2
C₂×#11926305	2
C₂×#11926301	2
C₂×C₁₉×#338	2
#10813	19