Subgroups — #18081567

Order 4864.

570 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14572031	C₂	1	1	yes	maximal, characteristic
2	#14563786	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×Dic₁₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#11926293	C₂	1	1	yes	maximal, characteristic
5	C₂×#11926292	C₂	1	1	yes	maximal, characteristic
6	C₂×#11926292	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₉×#333	C₂	1	1	yes	maximal, characteristic
8	C₂×#10597315	C₂²	1	1	yes	characteristic
9	C₂×#10597390	C₄	1	4	yes	—
10	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×Dic₁₉⋊C₈	—	2	2	no	—
13	C₂×C₈×Dic₁₉	—	2	1	no	—
14	#11926413	—	2	1	no	—
15	#11926413	—	2	2	no	—
16	#11926413	—	2	1	no	—
17	C₄²×Dic₁₉	C₂²	1	1	yes	characteristic
18	#11926298	—	2	1	no	—
19	#11926293	—	2	2	no	—
20	#11926292	—	2	2	no	—
21	#11926292	—	2	2	no	—
22	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
23	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₁₉×#333	—	2	2	no	—
26	C₂²×Q₈×C₁₉	C₂×C₄	1	2	yes	—
27	C₂²×Dic₃₈	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₁₉	—	2	1	no	—
29	C₂²×C₈×C₁₉	—	2	1	no	—
30	C₁₉×C₄⋊Q₈	—	2	1	no	—
31	C₁₉×C₄⋊Q₈	—	2	1	no	—
32	C₁₉×C₄⋊Q₈	—	2	1	no	—
33	C₁₉×C₄⋊Q₈	—	2	1	no	—
34	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
35	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
36	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
37	#10597390	—	4	8	no	—
38	C₂×C₄⋊Dic₁₉	—	2	1	no	—
39	C₂×C₄⋊Dic₁₉	—	2	1	no	—
40	C₂×Dic₁₉⋊C₄	—	2	4	no	—
41	C₂×Dic₁₉⋊C₄	—	2	4	no	—
42	C₂×C₄×Dic₁₉	C₂×C₄	1	2	yes	—
43	C₂×C₄×Dic₁₉	D₄	1	1	yes	characteristic
44	C₂×C₄×Dic₁₉	D₄	1	2	yes	—
45	C₂×C₄×Dic₁₉	D₄	1	1	yes	characteristic
46	C₂²×C₁₉⋊C₈	—	2	1	no	—
47	C₂²×C₁₉⋊C₈	—	2	1	no	—
48	#10597315	—	2	1	no	—
49	#10597315	—	2	1	no	—
50	#10597315	—	2	1	no	—
51	#10597315	—	2	1	no	—
52	C₁₉×C₄⋊C₈	—	2	1	no	—
53	C₁₉×C₄⋊C₈	—	2	1	no	—
54	C₁₉×C₄⋊C₈	—	2	1	no	—
55	C₁₉×C₄⋊C₈	D₄	1	2	yes	—
56	Dic₁₉⋊C₈	—	4	4	no	—
57	C₈×Dic₁₉	—	4	2	no	—
58	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
59	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
60	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
61	C₄⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
62	C₂×Q₈×C₁₉	—	2	2	no	—
63	C₂×Q₈×C₁₉	—	2	2	no	—
64	C₂×Q₈×C₁₉	—	4	4	no	—
65	C₂²×C₄×C₁₉	—	2	2	no	—
66	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×Dic₁₉	—	2	2	no	—
70	C₂²×Dic₁₉	C₂²⋊C₄	1	2	yes	—
71	C₂²×Dic₁₉	C₂²⋊C₄	1	2	yes	—
72	C₂×Dic₃₈	—	2	2	no	—
73	C₂×Dic₃₈	—	2	2	no	—
74	C₂×Dic₃₈	—	4	4	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	—	4	1	no	—
79	C₂×C₈×C₁₉	—	4	2	no	—
80	C₄⋊C₄×C₁₉	—	4	1	no	—
81	C₄⋊C₄×C₁₉	—	4	1	no	—
82	C₄⋊C₄×C₁₉	—	4	1	no	—
83	C₄⋊C₄×C₁₉	—	4	1	no	—
84	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₁₉	#41	1	1	yes	characteristic
87	C₄²×C₁₉	#41	1	1	yes	characteristic
88	C₄⋊Dic₁₉	—	4	1	no	—
89	C₄⋊Dic₁₉	—	4	1	no	—
90	C₄⋊Dic₁₉	—	4	1	no	—
91	C₄⋊Dic₁₉	—	4	1	no	—
92	Dic₁₉⋊C₄	—	4	8	no	—
93	Dic₁₉⋊C₄	—	4	8	no	—
94	C₄×Dic₁₉	—	2	1	no	—
95	C₄×Dic₁₉	—	2	1	no	—
96	C₄×Dic₁₉	—	2	4	no	—
97	C₄×Dic₁₉	SD₁₆	1	2	yes	—
98	C₄×Dic₁₉	SD₁₆	1	4	yes	—
99	C₄×Dic₁₉	Q₁₆	1	4	yes	—
100	C₄×Dic₁₉	Q₁₆	1	2	yes	—
101	C₂×C₁₉⋊C₈	—	2	2	no	—
102	C₂×C₁₉⋊C₈	—	4	1	no	—
103	C₂×C₁₉⋊C₈	—	4	1	no	—
104	C₂×C₁₉⋊C₈	—	4	1	no	—
105	C₂×C₁₉⋊C₈	—	4	2	no	—
106	#9691	—	19	1	no	maximal, Sylow
107	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	Q₈×C₁₉	—	8	2	no	—
109	Q₈×C₁₉	—	8	2	no	—
110	C₂×C₄×C₁₉	—	2	2	no	—
111	C₂×C₄×C₁₉	—	4	2	no	—
112	C₂×C₄×C₁₉	—	4	4	no	—
113	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₉	C₄×D₄	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	#124	1	1	yes	characteristic
118	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
119	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
121	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
123	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
124	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
125	C₂×Dic₁₉	—	2	2	no	—
126	C₂×Dic₁₉	—	2	2	no	—
127	C₂×Dic₁₉	—	4	2	no	—
128	C₂×Dic₁₉	—	4	4	no	—
129	C₂×Dic₁₉	#100	1	4	yes	—
130	C₂×Dic₁₉	#103	1	4	yes	—
131	C₂×Dic₁₉	#103	1	4	yes	—
132	C₂×Dic₁₉	#103	1	4	yes	—
133	Dic₃₈	—	8	2	no	—
134	Dic₃₈	—	8	2	no	—
135	C₈×C₁₉	—	4	2	no	—
136	C₈×C₁₉	—	8	1	no	—
137	C₁₉⋊C₈	—	4	2	no	—
138	C₁₉⋊C₈	—	8	1	no	—
139	#2868	—	19	1	no	—
140	#1842	—	19	1	no	—
141	C₄×C₄⋊C₈	—	19	1	no	—
142	C₂×#333	—	19	1	no	—
143	C₂×#333	—	19	1	no	—
144	C₂×#333	—	19	1	no	—
145	C₂×#333	D₁₉	1	1	yes	characteristic
146	C₂²×C₁₉	#390	1	1	yes	characteristic
147	C₂²×C₁₉	C₂×#100	1	2	yes	—
148	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
149	C₂²×C₁₉	C₂×#103	1	2	yes	—
150	C₂²×C₁₉	#419	1	1	yes	characteristic
151	C₄×C₁₉	—	2	1	no	—
152	C₄×C₁₉	—	2	1	no	—
153	C₄×C₁₉	—	8	2	no	—
154	C₄×C₁₉	#461	1	1	yes	characteristic
155	C₄×C₁₉	#461	1	1	yes	characteristic
156	C₄×C₁₉	#462	1	1	yes	characteristic
157	C₄×C₁₉	#462	1	1	yes	characteristic
158	C₄×C₁₉	#487	1	2	yes	—
159	C₄×C₁₉	#492	1	1	yes	characteristic
160	C₄×C₁₉	#494	1	1	yes	characteristic
161	Dic₁₉	—	2	4	no	—
162	Dic₁₉	—	8	2	no	—
163	Dic₁₉	#333	1	8	yes	complemented
164	C₂×C₄⋊Q₈	—	19	4	no	—
165	C₂×C₄⋊Q₈	—	19	1	no	—
166	C₂×C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
167	C₂×C₄⋊C₈	—	19	1	no	—
168	C₂×C₄⋊C₈	—	19	1	no	—
169	C₂×C₄⋊C₈	—	38	2	no	—
170	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
171	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
172	C₂×C₄×C₈	—	38	1	no	—
173	#384	—	38	1	no	—
174	#384	—	38	2	no	—
175	#384	—	38	1	no	—
176	C₄³	—	19	1	no	—
177	#336	—	38	1	no	—
178	#333	—	2	2	no	—
179	#333	—	38	2	no	—
180	#333	—	38	2	no	—
181	#333	—	38	2	no	—
182	C₂×C₁₉	C₂×#333	1	2	yes	—
183	C₂×C₁₉	#1548	1	2	yes	—
184	C₂×C₁₉	#1956	1	1	yes	characteristic
185	C₂×C₁₉	#1965	1	1	yes	characteristic
186	C₂×C₁₉	#1981	1	1	yes	characteristic
187	C₂²×Q₈	—	19	2	no	—
188	C₂²×Q₈	C₄×D₁₉	1	2	yes	—
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	—	2	1	no	—
191	C₂²×C₈	—	38	1	no	—
192	C₂²×C₈	—	38	1	no	—
193	C₄⋊Q₈	—	2	1	no	—
194	C₄⋊Q₈	—	2	1	no	—
195	C₄⋊Q₈	—	2	1	no	—
196	C₄⋊Q₈	—	2	1	no	—
197	C₄⋊Q₈	—	38	1	no	—
198	C₄⋊Q₈	—	38	1	no	—
199	C₄⋊Q₈	—	38	1	no	—
200	C₄⋊Q₈	—	38	1	no	—
201	C₄⋊Q₈	—	76	8	no	—
202	C₂×C₄⋊C₄	—	2	1	no	—
203	C₂×C₄⋊C₄	—	2	1	no	—
204	C₂×C₄⋊C₄	—	38	4	no	—
205	C₂×C₄⋊C₄	—	38	4	no	—
206	C₂×C₄⋊C₄	—	38	1	no	—
207	C₂×C₄⋊C₄	—	38	1	no	—
208	C₂×C₄²	—	19	1	no	—
209	C₂×C₄²	—	19	2	no	—
210	C₂×C₄²	—	19	2	no	—
211	C₂×C₄²	—	19	1	no	—
212	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
213	C₄⋊C₈	—	2	1	no	—
214	C₄⋊C₈	—	2	1	no	—
215	C₄⋊C₈	—	2	1	no	—
216	C₄⋊C₈	—	19	2	no	—
217	C₄⋊C₈	—	38	1	no	—
218	C₄⋊C₈	—	38	1	no	—
219	C₄⋊C₈	—	38	1	no	—
220	C₄⋊C₈	—	76	4	no	—
221	C₄⋊C₈	C₂²⋊D₁₉	1	2	yes	—
222	C₄×C₈	—	76	2	no	—
223	C₁₉	#9691	1	1	yes	characteristic, complemented, Sylow
224	C₂×Q₈	—	2	2	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	—	4	4	no	—
227	C₂×Q₈	—	38	2	no	—
228	C₂×Q₈	—	38	2	no	—
229	C₂×Q₈	—	76	4	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	19	2	no	—
232	C₂²×C₄	—	19	2	no	—
233	C₂²×C₄	—	38	2	no	—
234	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
235	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
236	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	4	1	no	—
239	C₂×C₈	—	4	1	no	—
240	C₂×C₈	—	4	1	no	—
241	C₂×C₈	—	4	2	no	—
242	C₂×C₈	—	38	2	no	—
243	C₂×C₈	—	76	1	no	—
244	C₂×C₈	—	76	1	no	—
245	C₂×C₈	—	76	1	no	—
246	C₂×C₈	—	76	2	no	—
247	C₄⋊C₄	—	4	1	no	—
248	C₄⋊C₄	—	4	1	no	—
249	C₄⋊C₄	—	4	1	no	—
250	C₄⋊C₄	—	4	1	no	—
251	C₄⋊C₄	—	76	8	no	—
252	C₄⋊C₄	—	76	8	no	—
253	C₄⋊C₄	—	76	1	no	—
254	C₄⋊C₄	—	76	1	no	—
255	C₄⋊C₄	—	76	1	no	—
256	C₄⋊C₄	—	76	1	no	—
257	C₄²	—	19	2	no	—
258	C₄²	—	19	4	no	—
259	C₄²	—	19	2	no	—
260	C₄²	—	19	4	no	—
261	C₄²	—	38	4	no	—
262	C₄²	—	38	1	no	—
263	C₄²	—	38	1	no	—
264	C₄²	#64541	1	1	yes	characteristic
265	C₄²	D₄×D₁₉	1	1	yes	characteristic
266	C₄²	#64543	1	1	yes	characteristic
267	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
268	C₂³	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
269	Q₈	—	8	2	no	—
270	Q₈	—	8	2	no	—
271	Q₈	—	152	2	no	—
272	Q₈	—	152	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	19	4	no	—
277	C₂×C₄	—	19	4	no	—
278	C₂×C₄	—	19	4	no	—
279	C₂×C₄	—	19	4	no	—
280	C₂×C₄	—	38	2	no	—
281	C₂×C₄	—	38	2	no	—
282	C₂×C₄	—	76	2	no	—
283	C₂×C₄	—	76	4	no	—
284	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
285	C₂×C₄	#10597330	1	1	yes	characteristic
286	C₂×C₄	#10597331	1	1	yes	characteristic
287	C₂×C₄	SD₁₆×D₁₉	1	2	yes	—
288	C₂×C₄	SD₁₆×D₁₉	1	1	yes	characteristic
289	C₂×C₄	#10597361	1	1	yes	characteristic
290	C₂×C₄	Q₁₆×D₁₉	1	2	yes	—
291	C₂×C₄	Q₁₆×D₁₉	1	1	yes	characteristic
292	C₂×C₄	#10597364	1	1	yes	characteristic
293	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
294	C₂×C₄	#10597385	1	1	yes	characteristic
295	C₂×C₄	#10597386	1	1	yes	characteristic
296	C₈	—	4	2	no	—
297	C₈	—	8	1	no	—
298	C₈	—	76	2	no	—
299	C₈	—	152	1	no	—
300	C₂²	#11926444	1	1	yes	characteristic
301	C₂²	#100×D₁₉	1	2	yes	—
302	C₂²	#103×D₁₉	1	2	yes	—
303	C₂²	#103×D₁₉	1	1	yes	characteristic
304	C₂²	#11926551	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	8	2	no	—
308	C₄	—	19	8	no	—
309	C₄	—	38	4	no	—
310	C₄	—	152	2	no	—
311	C₄	#11926747	1	1	yes	characteristic
312	C₄	#11926748	1	1	yes	characteristic
313	C₄	#11926752	1	1	yes	characteristic
314	C₄	#11926753	1	1	yes	characteristic
315	C₄	#11926893	1	2	yes	—
316	C₄	#11926915	1	1	yes	characteristic
317	C₄	#11926922	1	1	yes	characteristic
318	C₂	D₁₉×#333	1	2	yes	—
319	C₂	#14562314	1	2	yes	—
320	C₂	#14564435	1	1	yes	characteristic
321	C₂	#14564473	1	1	yes	characteristic
322	C₂	#14564556	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊C₄×D₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#333	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#333	D₁₉
hypocenter	C₁₉	#9691
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#333

Complements

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

N	H ≅ G/N	type
163. Dic₁₉	#333	semidirect
223. C₁₉	#9691	semidirect

Maximal subgroups

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

subgroup	index
#14572031	2
#14563786	2
C₄⋊C₈×Dic₁₉	2
C₂×#11926293	2
C₂×#11926292	2
C₂×C₁₉×#333	2
#9691	19