Subgroups — #18090547

Order 4864.

706 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14572199	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926314	C₂	1	2	yes	maximal
3	#14563374	C₂	1	1	yes	maximal, characteristic
4	#14563329	C₂	1	2	yes	maximal
5	C₁₉×#2296	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	C₂²	1	2	yes	—
7	C₂²×#64520	C₂²	1	2	yes	—
8	#11926571	—	2	4	no	—
9	#11926568	—	2	2	no	—
10	#11926314	—	2	4	no	—
11	#11926313	—	2	2	no	—
12	#11926299	—	2	4	no	—
13	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
14	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
16	C₁₉×#389	—	2	2	no	—
17	C₁₉×#385	C₄	1	4	yes	—
18	C₄×C₄⋊C₄×C₁₉	—	2	2	no	—
19	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
21	C₁₉×#117	—	2	2	no	—
22	C₁₉×#117	—	2	1	no	—
23	C₁₉×#117	—	2	1	no	—
24	C₁₉×#117	—	2	2	no	—
25	C₁₉×#117	D₄	1	2	yes	—
26	C₁₉×#117	Q₈	1	2	yes	—
27	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
28	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
29	C₂×C₄⋊C₄×C₁₉	D₄	1	2	yes	—
30	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
31	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₉	—	2	1	no	—
33	C₂×C₄²×C₁₉	—	2	2	no	—
34	C₂×C₄²×C₁₉	—	2	1	no	—
35	C₂×C₄²×C₁₉	—	2	1	no	—
36	C₂×C₄²×C₁₉	—	2	2	no	—
37	C₂×C₄²×C₁₉	D₄	1	2	yes	—
38	C₂×#64520	—	2	2	no	—
39	C₂×#64520	—	2	2	no	—
40	C₂×#64520	—	2	2	no	—
41	C₂×#64520	—	2	2	no	—
42	C₂×#64520	—	2	2	no	—
43	C₂×#64520	—	2	2	no	—
44	C₂×#64520	—	2	2	no	—
45	C₂×#64520	—	2	2	no	—
46	C₂×#64520	—	2	2	no	—
47	C₂×#64520	—	2	2	no	—
48	C₂×#64520	—	2	2	no	—
49	C₂×#64520	Q₈	1	4	yes	—
50	C₂²×C₁₉⋊C₈	—	2	2	no	—
51	C₂²×C₁₉⋊C₈	D₄	1	4	yes	—
52	C₁₉×#95	—	2	4	no	—
53	C₄⋊(C₁₉⋊C₈)	—	2	8	no	—
54	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	—	2	2	no	—
56	C₂²×C₄×C₁₉	—	2	2	no	—
57	C₂²×C₄×C₁₉	—	2	4	no	—
58	C₂²×C₄×C₁₉	—	4	4	no	—
59	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
60	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
66	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	#41	1	2	yes	—
69	C₂²×C₄×C₁₉	#41	1	2	yes	—
70	C₄⋊C₄×C₁₉	—	2	2	no	—
71	C₄⋊C₄×C₁₉	—	2	4	no	—
72	C₄⋊C₄×C₁₉	—	2	2	no	—
73	C₄⋊C₄×C₁₉	—	4	8	no	—
74	C₂²⋊C₄×C₁₉	—	2	2	no	—
75	C₂²⋊C₄×C₁₉	—	2	4	no	—
76	C₂²⋊C₄×C₁₉	C₄⋊C₄	1	4	yes	—
77	C₄²×C₁₉	—	2	2	no	—
78	C₄²×C₁₉	—	2	2	no	—
79	C₄²×C₁₉	—	2	2	no	—
80	C₄²×C₁₉	—	2	2	no	—
81	C₄²×C₁₉	—	4	1	no	—
82	C₄²×C₁₉	—	4	2	no	—
83	C₄²×C₁₉	—	4	1	no	—
84	#64520	—	2	4	no	—
85	#64520	—	2	4	no	—
86	#64520	—	4	2	no	—
87	#64520	—	4	2	no	—
88	#64520	—	4	2	no	—
89	#64520	—	4	2	no	—
90	#64520	—	4	2	no	—
91	#64520	—	4	2	no	—
92	C₂×C₁₉⋊C₈	—	2	4	no	—
93	C₂×C₁₉⋊C₈	—	2	4	no	—
94	C₂×C₁₉⋊C₈	—	2	4	no	—
95	C₂×C₁₉⋊C₈	—	2	4	no	—
96	C₂×C₁₉⋊C₈	—	4	2	no	—
97	C₂×C₁₉⋊C₈	—	4	2	no	—
98	#11223	—	19	1	no	maximal, Sylow
99	C₂³×C₁₉	—	4	2	no	—
100	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂³×C₁₉	#117	1	1	yes	characteristic
103	C₂³×C₁₉	C₄×D₄	1	2	yes	—
104	C₂³×C₁₉	C₄×Q₈	1	2	yes	—
105	C₂×C₄×C₁₉	—	2	4	no	—
106	C₂×C₄×C₁₉	—	2	8	no	—
107	C₂×C₄×C₁₉	—	2	4	no	—
108	C₂×C₄×C₁₉	—	2	4	no	—
109	C₂×C₄×C₁₉	—	4	4	no	—
110	C₂×C₄×C₁₉	—	4	2	no	—
111	C₂×C₄×C₁₉	—	4	4	no	—
112	C₂×C₄×C₁₉	—	4	2	no	—
113	C₂×C₄×C₁₉	—	4	4	no	—
114	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
115	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
116	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
117	C₂×C₄×C₁₉	#121	1	2	yes	—
118	C₂×C₄×C₁₉	#121	1	2	yes	—
119	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
120	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
121	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
122	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
124	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
125	C₂×C₄×C₁₉	#123	1	2	yes	—
126	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
127	C₂×C₄×C₁₉	#125	1	2	yes	—
128	C₂×C₄×C₁₉	#125	1	2	yes	—
129	C₂×C₄×C₁₉	#126	1	2	yes	—
130	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
131	C₂×C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
132	C₁₉⋊C₈	—	4	4	no	—
133	C₁₉⋊C₈	—	4	4	no	—
134	#2905	—	19	1	no	—
135	#2296	D₁₉	1	1	yes	characteristic
136	C₂×#344	—	19	2	no	—
137	#1742	—	19	1	no	—
138	#1728	—	19	2	no	—
139	C₂²×C₁₉	—	2	2	no	—
140	C₂²×C₁₉	—	2	4	no	—
141	C₂²×C₁₉	—	4	1	no	—
142	C₂²×C₁₉	—	4	1	no	—
143	C₂²×C₁₉	—	4	2	no	—
144	C₂²×C₁₉	#380	1	1	yes	characteristic
145	C₂²×C₁₉	#386	1	1	yes	characteristic
146	C₂²×C₁₉	#387	1	2	yes	—
147	C₂²×C₁₉	#389	1	2	yes	—
148	C₂²×C₁₉	#391	1	1	yes	characteristic
149	C₄×C₁₉	—	4	4	no	—
150	C₄×C₁₉	—	4	4	no	—
151	C₄×C₁₉	—	4	4	no	—
152	C₄×C₁₉	#394	1	1	yes	characteristic
153	C₄×C₁₉	#395	1	1	yes	characteristic
154	C₄×C₁₉	#396	1	2	yes	—
155	C₄×C₁₉	#398	1	1	yes	characteristic
156	C₄×C₁₉	#398	1	2	yes	—
157	C₄×C₁₉	#400	1	1	yes	characteristic
158	C₂²×M₁₆	—	19	2	no	—
159	C₂²×M₁₆	—	19	2	no	—
160	C₂×#117	C₂×D₁₉	1	1	yes	characteristic
161	C₂×#117	C₂×D₁₉	1	1	yes	characteristic
162	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
163	#424	—	38	4	no	—
164	#423	—	38	2	no	—
165	#389	—	2	2	no	—
166	#385	Dic₁₉	1	4	yes	—
167	C₄×C₄⋊C₄	—	2	2	no	—
168	#344	—	38	4	no	—
169	#343	—	38	2	no	—
170	#336	—	38	4	no	—
171	C₂×C₁₉	—	2	1	no	—
172	C₂×C₁₉	—	2	2	no	—
173	C₂×C₁₉	—	2	2	no	—
174	C₂×C₁₉	—	2	1	no	—
175	C₂×C₁₉	#1441	1	1	yes	characteristic
176	C₂×C₁₉	#1891	1	2	yes	—
177	C₂³×C₄	C₂×Dic₁₉	1	2	yes	—
178	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
179	C₂×M₁₆	—	19	4	no	—
180	C₂×M₁₆	—	38	2	no	—
181	C₂×M₁₆	—	38	2	no	—
182	C₂×M₁₆	—	38	2	no	—
183	C₂×M₁₆	—	38	2	no	—
184	C₂×M₁₆	—	38	2	no	—
185	C₂×M₁₆	—	38	2	no	—
186	C₂×M₁₆	—	38	2	no	—
187	C₂×M₁₆	—	38	2	no	—
188	C₂×M₁₆	—	38	2	no	—
189	C₂×M₁₆	—	38	2	no	—
190	C₂×M₁₆	—	38	2	no	—
191	C₂²×C₈	—	19	4	no	—
192	C₂²×C₈	—	38	2	no	—
193	#117	—	2	1	no	—
194	#117	—	2	1	no	—
195	#117	—	2	2	no	—
196	#117	—	2	2	no	—
197	#117	Dic₃₈	1	2	yes	—
198	#117	D₇₆	1	2	yes	—
199	C₂×C₄⋊C₄	—	2	1	no	—
200	C₂×C₄⋊C₄	—	2	4	no	—
201	C₂×C₄⋊C₄	C₂²⋊D₁₉	1	2	yes	—
202	C₂×C₂²⋊C₄	C₂×Dic₁₉	1	2	yes	—
203	C₂×C₂²⋊C₄	C₂×Dic₁₉	1	2	yes	—
204	C₂×C₄²	—	2	1	no	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	—	2	2	no	—
207	C₂×C₄²	—	2	1	no	—
208	C₂×C₄²	—	2	1	no	—
209	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
210	C₄⋊C₈	—	38	8	no	—
211	#95	—	2	4	no	—
212	C₁₉	#11223	1	1	yes	characteristic, complemented, Sylow
213	C₂⁴	C₂²×Dic₁₉	1	1	yes	characteristic
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	4	4	no	—
218	C₂²×C₄	C₂²⋊Dic₁₉	1	4	yes	—
219	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
221	C₂²×C₄	#64541	1	1	yes	characteristic
222	C₂²×C₄	#64541	1	1	yes	characteristic
223	C₂²×C₄	D₄×D₁₉	1	2	yes	—
224	C₂²×C₄	#64543	1	2	yes	—
225	C₂²×C₄	Q₈×D₁₉	1	2	yes	—
226	C₂²×C₄	#64545	1	2	yes	—
227	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
229	M₁₆	—	38	4	no	—
230	M₁₆	—	38	4	no	—
231	M₁₆	—	76	2	no	—
232	M₁₆	—	76	2	no	—
233	M₁₆	—	76	2	no	—
234	M₁₆	—	76	2	no	—
235	M₁₆	—	76	2	no	—
236	M₁₆	—	76	2	no	—
237	C₂×C₈	—	38	4	no	—
238	C₂×C₈	—	38	4	no	—
239	C₂×C₈	—	38	4	no	—
240	C₂×C₈	—	38	4	no	—
241	C₂×C₈	—	76	2	no	—
242	C₂×C₈	—	76	2	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	4	8	no	—
247	C₂²⋊C₄	—	2	4	no	—
248	C₂²⋊C₄	—	2	2	no	—
249	C₂²⋊C₄	C₄⋊Dic₁₉	1	4	yes	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	4	1	no	—
255	C₄²	—	4	1	no	—
256	C₄²	—	4	2	no	—
257	C₂³	—	4	2	no	—
258	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
259	C₂³	#10597372	1	1	yes	characteristic
260	C₂³	D₄×Dic₁₉	1	2	yes	—
261	C₂³	Q₈×Dic₁₉	1	2	yes	—
262	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	#10597334	1	2	yes	—
273	C₂×C₄	#10597335	1	2	yes	—
274	C₂×C₄	#10597336	1	2	yes	—
275	C₂×C₄	#10597340	1	2	yes	—
276	C₂×C₄	#10597341	1	2	yes	—
277	C₂×C₄	#10597342	1	2	yes	—
278	C₂×C₄	#10597343	1	2	yes	—
279	C₂×C₄	#10597344	1	2	yes	—
280	C₂×C₄	#10597370	1	1	yes	characteristic
281	C₂×C₄	#10597370	1	1	yes	characteristic
282	C₂×C₄	#10597376	1	1	yes	characteristic
283	C₂×C₄	#10597376	1	1	yes	characteristic
284	C₂×C₄	#10597381	1	1	yes	characteristic
285	C₂×C₄	#10597383	1	1	yes	characteristic
286	C₂×C₄	#10597385	1	2	yes	—
287	C₂×C₄	#10597390	1	1	yes	characteristic
288	C₂×C₄	#10597392	1	2	yes	—
289	C₂×C₄	#10597393	1	1	yes	characteristic
290	C₈	—	76	4	no	—
291	C₈	—	76	4	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	4	no	—
294	C₂²	—	4	2	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	#11926404	1	1	yes	characteristic
298	C₂²	#11926424	1	1	yes	characteristic
299	C₂²	#11926430	1	2	yes	—
300	C₂²	#11926443	1	2	yes	—
301	C₂²	#11926451	1	1	yes	characteristic
302	C₄	—	4	4	no	—
303	C₄	—	4	4	no	—
304	C₄	—	4	4	no	—
305	C₄	#11926461	1	1	yes	characteristic
306	C₄	#11926464	1	1	yes	characteristic
307	C₄	#11926466	1	2	yes	—
308	C₄	#11926474	1	1	yes	characteristic
309	C₄	#11926477	1	2	yes	—
310	C₄	#11926483	1	1	yes	characteristic
311	C₂	—	2	1	no	—
312	C₂	—	2	2	no	—
313	C₂	—	2	2	no	—
314	C₂	—	2	1	no	—
315	C₂	#14561888	1	1	yes	characteristic
316	C₂	#14564054	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#10597383
commutator, prime radical	C₂⁴×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₁₉×#2296	C₂
socle	C₂²×C₁₉	#386
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	#2296	D₁₉
hypocenter	C₁₉	#11223
LCS	G ▷ C₂⁴×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #2296

Complements

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

N	H ≅ G/N	type
212. C₁₉	#11223	semidirect

Maximal subgroups

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

subgroup	index
#14572199	2
C₂×#11926314	2
#14563374	2
#14563329	2
C₁₉×#2296	2
#11223	19