Subgroups — #18091214

Order 4864.

790 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14572225	C₂	1	1	yes	maximal, characteristic
2	#14566118	C₂	1	2	yes	maximal
3	#14563346	C₂	1	2	yes	maximal
4	C₂×#11926301	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1728	C₂	1	1	yes	maximal, characteristic
6	C₂×#10597372	C₂²	1	2	yes	—
7	C₂×#10597372	C₂²	1	2	yes	—
8	#11926586	—	2	4	no	—
9	#11926582	—	2	2	no	—
10	#11926442	—	2	4	no	—
11	#11926417	—	2	4	no	—
12	C₄×Dic₁₉⋊C₄	—	2	4	no	—
13	#11926305	—	2	4	no	—
14	#11926301	C₄	1	4	yes	complemented
15	C₂²×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
16	C₂²×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
18	C₁₉×#336	—	2	2	no	—
19	C₂³×C₄×C₁₉	D₄	1	2	yes	—
20	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
21	C₂×M₁₆×C₁₉	—	2	2	no	—
22	C₂×M₁₆×C₁₉	—	2	2	no	—
23	C₂×M₁₆×C₁₉	—	2	1	no	—
24	C₂×M₁₆×C₁₉	—	2	2	no	—
25	C₂×M₁₆×C₁₉	—	2	2	no	—
26	C₂×M₁₆×C₁₉	C₂×C₄	1	2	yes	—
27	C₂×M₁₆×C₁₉	Q₈	1	2	yes	—
28	C₂×M₁₆×C₁₉	Q₈	1	2	yes	—
29	C₂²×C₈×C₁₉	—	2	1	no	—
30	C₂²×C₈×C₁₉	D₄	1	2	yes	—
31	C₂×C₄²×C₁₉	—	2	2	no	—
32	C₂×C₄²×C₁₉	—	2	2	no	—
33	C₂×C₄²×C₁₉	—	2	1	no	—
34	C₂×C₄²×C₁₉	—	2	1	no	—
35	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
36	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
37	#10597372	—	2	2	no	—
38	#10597372	—	2	2	no	—
39	#10597372	—	2	2	no	—
40	#10597372	—	2	2	no	—
41	#10597372	—	2	2	no	—
42	#10597372	—	2	2	no	—
43	#10597372	C₂×C₄	1	4	yes	—
44	#10597372	Q₈	1	4	yes	—
45	C₂×C₄⋊Dic₁₉	—	2	2	no	—
46	C₂×C₄⋊Dic₁₉	D₄	1	4	yes	—
47	C₂×Dic₁₉⋊C₄	—	2	8	no	—
48	C₂×C₄×Dic₁₉	—	2	2	no	—
49	C₂×C₄×Dic₁₉	D₄	1	4	yes	—
50	C₁₉⋊#95	—	2	8	no	—
51	#10597275	—	2	4	no	—
52	#10597274	—	2	4	no	—
53	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	—	2	4	no	—
55	C₂²×C₄×C₁₉	—	2	2	no	—
56	C₂²×C₄×C₁₉	—	4	4	no	—
57	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
64	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	#41	1	2	yes	—
67	C₂²×Dic₁₉	—	2	4	no	—
68	C₂²×Dic₁₉	—	2	4	no	—
69	M₁₆×C₁₉	—	2	4	no	—
70	M₁₆×C₁₉	—	2	4	no	—
71	M₁₆×C₁₉	—	2	4	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	2	no	—
76	C₂×C₈×C₁₉	—	4	1	no	—
77	C₂×C₈×C₁₉	—	4	1	no	—
78	C₂×C₈×C₁₉	C₄⋊C₄	1	4	yes	complemented
79	C₄²×C₁₉	—	4	1	no	—
80	C₄²×C₁₉	—	4	2	no	—
81	C₄²×C₁₉	—	4	1	no	—
82	C₂²⋊Dic₁₉	—	2	4	no	—
83	C₂²⋊Dic₁₉	—	2	4	no	—
84	C₂²⋊Dic₁₉	—	4	2	no	—
85	C₂²⋊Dic₁₉	—	4	2	no	—
86	C₄⋊Dic₁₉	—	2	4	no	—
87	C₄⋊Dic₁₉	—	2	4	no	—
88	C₄⋊Dic₁₉	—	2	4	no	—
89	C₄⋊Dic₁₉	—	2	4	no	—
90	Dic₁₉⋊C₄	—	4	16	no	—
91	C₄×Dic₁₉	—	2	4	no	—
92	C₄×Dic₁₉	—	4	2	no	—
93	C₄×Dic₁₉	—	4	2	no	—
94	C₄×Dic₁₉	C₄⋊C₄	1	8	yes	complemented
95	#11335	—	19	1	no	maximal, Sylow
96	C₂³×C₁₉	—	4	2	no	—
97	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
98	C₂³×C₁₉	#120	1	2	yes	—
99	C₂³×C₁₉	#121	1	1	yes	characteristic
100	C₂³×C₁₉	C₂²⋊Q₈	1	2	yes	—
101	C₂³×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
102	C₂×C₄×C₁₉	—	2	8	no	—
103	C₂×C₄×C₁₉	—	2	4	no	—
104	C₂×C₄×C₁₉	—	4	4	no	—
105	C₂×C₄×C₁₉	—	4	2	no	—
106	C₂×C₄×C₁₉	—	4	2	no	—
107	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
111	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
112	C₂×C₄×C₁₉	#121	1	2	yes	—
113	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
118	C₂×C₄×C₁₉	#123	1	2	yes	—
119	C₂×C₄×C₁₉	#125	1	1	yes	characteristic
120	C₂×C₄×C₁₉	#125	1	2	yes	—
121	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
124	C₂×C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
125	C₂×Dic₁₉	—	2	8	no	—
126	C₂×Dic₁₉	—	4	8	no	—
127	C₂×Dic₁₉	—	4	8	no	—
128	C₂×Dic₁₉	—	4	4	no	—
129	C₈×C₁₉	—	2	4	no	—
130	C₈×C₁₉	—	4	2	no	—
131	#2911	—	19	1	no	—
132	#2296	—	19	2	no	—
133	#1733	—	19	2	no	—
134	C₂×H(3,ℤ/4)	—	19	1	no	—
135	#1728	D₁₉	1	1	yes	characteristic
136	C₂²×C₁₉	—	2	4	no	—
137	C₂²×C₁₉	—	2	2	no	—
138	C₂²×C₁₉	—	4	1	no	—
139	C₂²×C₁₉	—	4	2	no	—
140	C₂²×C₁₉	—	4	1	no	—
141	C₂²×C₁₉	#385	1	2	yes	—
142	C₂²×C₁₉	#389	1	1	yes	characteristic
143	C₂²×C₁₉	#392	1	1	yes	characteristic
144	C₂²×C₁₉	#395	1	2	yes	—
145	C₂²×C₁₉	#398	1	1	yes	characteristic
146	C₄×C₁₉	—	4	4	no	—
147	C₄×C₁₉	#384	1	1	yes	characteristic
148	C₄×C₁₉	#385	1	1	yes	characteristic
149	C₄×C₁₉	#389	1	2	yes	—
150	C₄×C₁₉	#395	1	1	yes	characteristic
151	C₄×C₁₉	#398	1	2	yes	—
152	C₄×C₁₉	#399	1	1	yes	characteristic
153	Dic₁₉	—	4	8	no	—
154	Dic₁₉	—	8	4	no	—
155	C₂²×M₁₆	C₂×D₁₉	1	1	yes	characteristic
156	C₂²×M₁₆	C₂×D₁₉	1	1	yes	characteristic
157	C₂×#117	—	19	2	no	—
158	C₂×#117	—	19	2	no	—
159	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
160	#428	—	38	4	no	—
161	#427	—	38	2	no	—
162	#389	—	38	4	no	—
163	#385	—	38	4	no	—
164	C₄×C₄⋊C₄	—	38	4	no	—
165	#339	—	38	4	no	—
166	H(3,ℤ/4)	—	19	4	no	—
167	#336	—	2	2	no	—
168	C₂×C₁₉	—	2	2	no	—
169	C₂×C₁₉	—	2	1	no	—
170	C₂×C₁₉	—	2	2	no	—
171	C₂×C₁₉	—	2	1	no	—
172	C₂×C₁₉	#1444	1	1	yes	characteristic
173	C₂×C₁₉	#2010	1	2	yes	—
174	C₂³×C₄	D₇₆	1	2	yes	—
175	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	1	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	Dic₃₈	1	2	yes	—
182	C₂×M₁₆	Dic₃₈	1	2	yes	—
183	C₂×M₁₆	C₄×D₁₉	1	2	yes	—
184	C₂²×C₈	—	2	1	no	—
185	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
186	#117	—	19	4	no	—
187	#117	—	19	4	no	—
188	#117	—	38	2	no	—
189	#117	—	38	2	no	—
190	#117	—	38	2	no	—
191	#117	—	38	2	no	—
192	#117	—	38	2	no	—
193	#117	—	38	2	no	—
194	C₂×C₄⋊C₄	—	19	4	no	—
195	C₂×C₄⋊C₄	—	38	2	no	—
196	C₂×C₄⋊C₄	—	38	8	no	—
197	C₂×C₂²⋊C₄	—	38	2	no	—
198	C₂×C₂²⋊C₄	—	38	2	no	—
199	C₂×C₄²	—	2	1	no	—
200	C₂×C₄²	—	2	2	no	—
201	C₂×C₄²	—	2	2	no	—
202	C₂×C₄²	—	2	1	no	—
203	C₂×C₄²	—	19	4	no	—
204	C₂×C₄²	—	38	2	no	—
205	#107	—	38	4	no	—
206	#106	—	38	4	no	—
207	#95	—	38	8	no	—
208	C₁₉	#11335	1	1	yes	characteristic, complemented, Sylow
209	C₂⁴	C₂×D₇₆	1	1	yes	characteristic
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	4	4	no	—
213	C₂²×C₄	—	38	4	no	—
214	C₂²×C₄	—	38	4	no	—
215	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
216	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
218	C₂²×C₄	#64541	1	1	yes	characteristic
219	C₂²×C₄	#64541	1	1	yes	characteristic
220	C₂²×C₄	D₄×D₁₉	1	2	yes	—
221	C₂²×C₄	D₄×D₁₉	1	2	yes	—
222	C₂²×C₄	#64543	1	2	yes	—
223	C₂²×C₄	Q₈×D₁₉	1	2	yes	—
224	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
225	M₁₆	—	2	4	no	—
226	M₁₆	—	2	4	no	—
227	M₁₆	—	2	4	no	—
228	M₁₆	—	4	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	4	1	no	—
234	C₂×C₈	Dic₁₉⋊C₄	1	4	yes	complemented
235	C₄⋊C₄	—	38	4	no	—
236	C₄⋊C₄	—	38	4	no	—
237	C₄⋊C₄	—	38	4	no	—
238	C₄⋊C₄	—	38	4	no	—
239	C₄⋊C₄	—	76	16	no	—
240	C₂²⋊C₄	—	38	4	no	—
241	C₂²⋊C₄	—	38	4	no	—
242	C₂²⋊C₄	—	76	2	no	—
243	C₂²⋊C₄	—	76	2	no	—
244	C₄²	—	4	1	no	—
245	C₄²	—	4	1	no	—
246	C₄²	—	4	2	no	—
247	C₄²	—	19	8	no	—
248	C₄²	—	38	4	no	—
249	C₄²	—	76	2	no	—
250	C₄²	—	76	2	no	—
251	C₂³	—	4	2	no	—
252	C₂³	#10597315	1	1	yes	characteristic
253	C₂³	C₄×D₇₆	1	1	yes	characteristic
254	C₂³	#10597328	1	2	yes	—
255	C₂³	#10597343	1	2	yes	—
256	C₂³	#10597376	1	1	yes	characteristic
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	38	8	no	—
263	C₂×C₄	—	76	8	no	—
264	C₂×C₄	—	76	8	no	—
265	C₂×C₄	—	76	4	no	—
266	C₂×C₄	C₄×Dic₃₈	1	1	yes	characteristic
267	C₂×C₄	#10597316	1	1	yes	characteristic
268	C₂×C₄	#10597324	1	2	yes	—
269	C₂×C₄	#10597324	1	2	yes	—
270	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
271	C₂×C₄	#10597329	1	2	yes	—
272	C₂×C₄	#10597330	1	2	yes	—
273	C₂×C₄	#10597334	1	2	yes	—
274	C₂×C₄	#10597334	1	2	yes	—
275	C₂×C₄	#10597335	1	2	yes	—
276	C₂×C₄	C₄⋊C₄×D₁₉	1	2	yes	—
277	C₂×C₄	#10597342	1	2	yes	—
278	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
279	C₂×C₄	#10597370	1	1	yes	characteristic
280	C₂×C₄	#10597381	1	1	yes	characteristic
281	C₂×C₄	#10597383	1	1	yes	characteristic
282	C₂×C₄	#10597386	1	1	yes	characteristic
283	C₂×C₄	#10597390	1	1	yes	characteristic
284	C₈	—	2	4	no	—
285	C₈	—	4	2	no	—
286	C₂²	—	2	4	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	4	1	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	2	no	—
291	C₂²	#11926418	1	2	yes	—
292	C₂²	#11926442	1	1	yes	characteristic
293	C₂²	#11926453	1	1	yes	characteristic
294	C₂²	#11926465	1	2	yes	—
295	C₂²	#11926474	1	1	yes	characteristic
296	C₄	—	4	4	no	—
297	C₄	—	76	8	no	—
298	C₄	—	152	4	no	—
299	C₄	#11926413	1	1	yes	characteristic
300	C₄	#11926416	1	1	yes	characteristic
301	C₄	#11926441	1	2	yes	—
302	C₄	#11926464	1	1	yes	characteristic
303	C₄	#11926478	1	2	yes	—
304	C₄	#11926480	1	1	yes	characteristic
305	C₂	—	2	2	no	—
306	C₂	—	2	1	no	—
307	C₂	—	2	2	no	—
308	C₂	—	2	1	no	—
309	C₂	#14561899	1	1	yes	characteristic
310	C₂	#14564702	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#10597386
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₁₉×#1728	C₂
socle	C₂²×C₁₉	#392
prime radical	C₂⁴×C₁₉	C₂×D₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	#1728	D₁₉
hypocenter	C₁₉	#11335
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1728

Complements

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

N	H ≅ G/N	type
14. #11926301	C₄	semidirect
78. C₂×C₈×C₁₉	C₄⋊C₄	semidirect
94. C₄×Dic₁₉	C₄⋊C₄	semidirect
208. C₁₉	#11335	semidirect
234. C₂×C₈	Dic₁₉⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
#14572225	2
#14566118	2
#14563346	2
C₂×#11926301	2
C₁₉×#1728	2
#11335	19