Subgroups — #18070596

Order 4864.

646 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#10597320	C₂	1	1	yes	maximal, characteristic
2	#14563381	C₂	1	1	yes	maximal, characteristic, complemented
3	#14561455	C₂	1	4	yes	maximal, complemented
4	C₁₉×#1831	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#10597320	C₄	1	1	yes	characteristic
6	C₂×C₄×D₇₆	—	2	1	no	—
7	C₂×C₄×D₇₆	—	2	1	no	—
8	C₂×C₄×D₇₆	C₄	1	2	yes	—
9	C₂×C₄×C₁₉⋊C₈	—	2	1	no	—
10	#11926446	—	2	1	no	—
11	#11926446	—	2	1	no	—
12	#11926446	C₂²	1	2	yes	—
13	#11926414	C₄	1	1	yes	characteristic
14	#11926299	—	2	1	no	—
15	#11926299	C₂²	1	2	yes	—
16	C₂×C₁₉×C₄⋊C₈	—	2	1	no	—
17	C₄³×C₁₉	C₂²	1	1	yes	characteristic
18	C₁₉×#336	—	2	1	no	—
19	C₁₉×#336	C₂²	1	2	yes	—
20	C₂²×D₇₆	—	2	1	no	—
21	C₂²×D₇₆	—	2	1	no	—
22	C₂²×D₇₆	C₂×C₄	1	2	yes	—
23	C₂²×C₄×D₁₉	—	2	2	no	—
24	C₂²×C₈×C₁₉	—	2	2	no	—
25	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
32	C₂×(C₂×D₁₉)⋊C₄	—	2	2	no	—
33	C₂×(C₂×D₁₉)⋊C₄	—	2	2	no	—
34	C₂×(C₂×D₁₉)⋊C₄	—	2	2	no	—
35	C₂×C₄⋊Dic₁₉	—	2	1	no	—
36	C₂×C₄⋊Dic₁₉	—	2	1	no	—
37	C₂×C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
38	C₂²×C₁₉⋊C₈	—	2	2	no	—
39	#10597320	—	2	2	no	—
40	#10597320	C₈	1	4	yes	complemented
41	C₄×D₇₆	—	4	4	no	—
42	C₄×D₇₆	—	4	2	no	—
43	C₄×D₇₆	—	4	2	no	—
44	C₄×D₇₆	—	4	4	no	—
45	C₁₉×C₄⋊C₈	—	4	2	no	—
46	C₄×C₁₉⋊C₈	—	2	4	no	—
47	C₂³×D₁₉	—	2	2	no	—
48	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×Dic₁₉	—	2	2	no	—
56	C₂×D₇₆	—	2	4	no	—
57	C₂×D₇₆	—	2	4	no	—
58	C₂×D₇₆	—	2	2	no	—
59	C₂×D₇₆	—	4	2	no	—
60	C₂×D₇₆	—	4	2	no	—
61	C₂×D₇₆	—	4	2	no	—
62	C₂×D₇₆	—	4	2	no	—
63	C₂×D₇₆	—	4	2	no	—
64	C₂×D₇₆	—	4	2	no	—
65	C₂×D₇₆	C₂×C₈	1	4	yes	—
66	C₂×C₄×D₁₉	—	4	4	no	—
67	C₂×C₄×D₁₉	—	4	4	no	—
68	C₂×C₄×D₁₉	—	4	4	no	—
69	C₂×C₈×C₁₉	—	4	2	no	—
70	C₂×C₈×C₁₉	—	4	2	no	—
71	C₂×C₈×C₁₉	—	4	2	no	—
72	C₄²×C₁₉	—	2	2	no	—
73	C₄²×C₁₉	—	2	2	no	—
74	C₄²×C₁₉	—	2	2	no	—
75	C₄²×C₁₉	—	4	1	no	—
76	C₄²×C₁₉	—	4	1	no	—
77	C₄²×C₁₉	C₂×C₈	1	2	yes	—
78	C₄²×C₁₉	M₁₆	1	2	yes	—
79	C₄²×C₁₉	D₈	1	2	yes	—
80	C₄²×C₁₉	SD₁₆	1	2	yes	—
81	(C₂×D₁₉)⋊C₄	—	4	4	no	—
82	(C₂×D₁₉)⋊C₄	—	4	2	no	—
83	(C₂×D₁₉)⋊C₄	—	4	2	no	—
84	(C₂×D₁₉)⋊C₄	—	4	4	no	—
85	C₄⋊Dic₁₉	—	4	2	no	—
86	C₄⋊Dic₁₉	—	4	2	no	—
87	C₄⋊Dic₁₉	—	4	2	no	—
88	C₂×C₁₉⋊C₈	—	2	4	no	—
89	C₂×C₁₉⋊C₈	—	4	2	no	—
90	C₂×C₁₉⋊C₈	—	4	2	no	—
91	#7587	—	19	1	no	maximal, Sylow
92	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂²×D₁₉	—	2	4	no	—
94	C₂²×D₁₉	—	4	2	no	—
95	C₂²×D₁₉	—	4	2	no	—
96	C₂²×D₁₉	—	4	2	no	—
97	C₂²×D₁₉	—	4	2	no	—
98	C₂²×D₁₉	—	4	2	no	—
99	C₂²×D₁₉	—	4	2	no	—
100	C₂×C₄×C₁₉	—	2	1	no	—
101	C₂×C₄×C₁₉	—	2	1	no	—
102	C₂×C₄×C₁₉	—	2	1	no	—
103	C₂×C₄×C₁₉	—	2	1	no	—
104	C₂×C₄×C₁₉	—	2	1	no	—
105	C₂×C₄×C₁₉	—	2	1	no	—
106	C₂×C₄×C₁₉	—	2	1	no	—
107	C₂×C₄×C₁₉	—	2	1	no	—
108	C₂×C₄×C₁₉	—	2	1	no	—
109	C₂×C₄×C₁₉	—	2	1	no	—
110	C₂×C₄×C₁₉	—	2	1	no	—
111	C₂×C₄×C₁₉	—	2	1	no	—
112	C₂×C₄×C₁₉	C₂²⋊C₈	1	2	yes	—
113	C₂×C₄×C₁₉	C₂²⋊C₈	1	2	yes	—
114	C₂×C₄×C₁₉	#100	1	2	yes	—
115	C₂×C₄×C₁₉	#102	1	2	yes	—
116	C₂×C₄×C₁₉	#102	1	2	yes	—
117	C₂×C₄×C₁₉	#104	1	2	yes	—
118	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
122	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
123	C₂×Dic₁₉	—	4	2	no	—
124	C₂×Dic₁₉	—	4	2	no	—
125	C₂×Dic₁₉	—	4	2	no	—
126	D₇₆	—	4	4	no	—
127	D₇₆	—	4	4	no	—
128	D₇₆	—	4	8	no	—
129	D₇₆	—	4	8	no	—
130	C₄×D₁₉	—	8	8	no	—
131	C₈×C₁₉	—	8	2	no	—
132	C₁₉⋊C₈	—	4	4	no	—
133	C₄×#127	—	19	1	no	—
134	#1831	D₁₉	1	1	yes	characteristic, complemented
135	#1744	—	19	1	no	—
136	#1270	—	19	4	no	—
137	C₂²×C₁₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
138	C₂²×C₁₉	#410	1	1	yes	characteristic
139	C₂²×C₁₉	C₂×#100	1	1	yes	characteristic
140	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
141	C₂²×C₁₉	#418	1	1	yes	characteristic
142	C₂²×C₁₉	C₂×#104	1	1	yes	characteristic
143	C₂²×C₁₉	#421	1	1	yes	characteristic
144	C₂×D₁₉	—	4	4	no	—
145	C₂×D₁₉	—	4	4	no	—
146	C₂×D₁₉	—	4	4	no	—
147	C₂×D₁₉	—	8	2	no	—
148	C₂×D₁₉	—	8	2	no	—
149	C₂×D₁₉	—	8	2	no	—
150	C₂×D₁₉	—	8	2	no	—
151	C₄×C₁₉	—	2	2	no	—
152	C₄×C₁₉	—	2	1	no	—
153	C₄×C₁₉	—	2	1	no	—
154	C₄×C₁₉	—	2	2	no	—
155	C₄×C₁₉	—	2	2	no	—
156	C₄×C₁₉	—	2	1	no	—
157	C₄×C₁₉	—	2	1	no	—
158	C₄×C₁₉	—	2	2	no	—
159	C₄×C₁₉	D₄⋊C₈	1	4	yes	—
160	Dic₁₉	—	8	2	no	—
161	C₂×#127	—	19	1	no	—
162	C₂×C₄×D₄	—	19	2	no	—
163	C₂×C₄×D₄	—	38	1	no	—
164	C₂×C₄×D₄	—	38	1	no	—
165	C₂×C₄⋊C₈	—	2	1	no	—
166	C₂×C₄×C₈	—	38	1	no	—
167	#390	—	19	2	no	—
168	#390	—	38	1	no	—
169	#390	—	38	1	no	—
170	#384	—	19	1	no	—
171	C₄³	C₂×D₁₉	1	1	yes	characteristic
172	#336	—	2	1	no	—
173	#336	—	19	2	no	—
174	#336	—	38	1	no	—
175	#336	C₂×D₁₉	1	2	yes	—
176	C₂×C₁₉	#1460	1	1	yes	characteristic
177	C₂×C₁₉	C₂×D₄⋊C₈	1	1	yes	characteristic
178	C₂×C₁₉	#1478	1	1	yes	characteristic
179	C₂×C₁₉	#1505	1	1	yes	characteristic
180	C₂×C₁₉	#1517	1	1	yes	characteristic
181	C₂×C₁₉	#1530	1	1	yes	characteristic
182	C₂×C₁₉	#1545	1	1	yes	characteristic
183	D₁₉	—	8	4	no	—
184	C₂²×D₄	—	19	2	no	—
185	C₂²×D₄	—	38	1	no	—
186	C₂²×D₄	—	38	1	no	—
187	C₂³×C₄	—	38	2	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	—	38	2	no	—
190	#127	—	19	4	no	—
191	#127	—	38	2	no	—
192	C₄×D₄	—	76	4	no	—
193	C₄×D₄	—	76	4	no	—
194	C₄×D₄	—	76	2	no	—
195	C₄×D₄	—	76	2	no	—
196	C₂×C₄⋊C₄	—	19	2	no	—
197	C₂×C₄⋊C₄	—	38	1	no	—
198	C₂×C₄⋊C₄	—	38	1	no	—
199	C₂×C₂²⋊C₄	—	38	2	no	—
200	C₂×C₂²⋊C₄	—	38	2	no	—
201	C₂×C₂²⋊C₄	—	38	2	no	—
202	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
203	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
204	C₂×C₄²	D₇₆	1	1	yes	characteristic
205	C₂×C₄²	D₇₆	1	1	yes	characteristic
206	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
207	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
208	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
209	C₄⋊C₈	—	4	2	no	—
210	C₄×C₈	—	38	4	no	—
211	C₁₉	#7587	1	1	yes	characteristic, complemented, Sylow
212	C₂⁴	—	38	2	no	—
213	C₂×D₄	—	19	4	no	—
214	C₂×D₄	—	38	4	no	—
215	C₂×D₄	—	38	4	no	—
216	C₂×D₄	—	38	2	no	—
217	C₂×D₄	—	76	2	no	—
218	C₂×D₄	—	76	2	no	—
219	C₂×D₄	—	76	2	no	—
220	C₂×D₄	—	76	2	no	—
221	C₂×D₄	—	76	2	no	—
222	C₂×D₄	—	76	2	no	—
223	C₂²×C₄	—	38	2	no	—
224	C₂²×C₄	—	76	4	no	—
225	C₂²×C₄	—	76	4	no	—
226	C₂²×C₄	—	76	4	no	—
227	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
228	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
229	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
230	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
234	C₂×C₈	—	4	2	no	—
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	38	4	no	—
238	C₂×C₈	—	76	2	no	—
239	C₂×C₈	—	76	2	no	—
240	C₄⋊C₄	—	76	2	no	—
241	C₄⋊C₄	—	76	2	no	—
242	C₄⋊C₄	—	76	2	no	—
243	C₂²⋊C₄	—	76	2	no	—
244	C₂²⋊C₄	—	76	2	no	—
245	C₂²⋊C₄	—	76	4	no	—
246	C₂²⋊C₄	—	76	4	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	4	1	no	—
251	C₄²	—	4	1	no	—
252	C₄²	C₈×D₁₉	1	2	yes	—
253	C₄²	#64515	1	2	yes	—
254	C₄²	#64525	1	2	yes	—
255	C₄²	#64527	1	2	yes	—
256	C₂³	—	38	4	no	—
257	C₂³	—	76	2	no	—
258	C₂³	—	76	2	no	—
259	C₂³	—	76	2	no	—
260	C₂³	—	76	2	no	—
261	C₂³	—	76	2	no	—
262	C₂³	—	76	2	no	—
263	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
264	D₄	—	76	4	no	—
265	D₄	—	76	4	no	—
266	D₄	—	76	8	no	—
267	D₄	—	76	8	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	76	2	no	—
281	C₂×C₄	—	76	2	no	—
282	C₂×C₄	—	76	2	no	—
283	C₂×C₄	—	152	8	no	—
284	C₂×C₄	#10597266	1	2	yes	—
285	C₂×C₄	#10597266	1	2	yes	—
286	C₂×C₄	(C₂×D₁₉)⋊C₈	1	2	yes	—
287	C₂×C₄	(C₂×D₁₉)⋊C₈	1	2	yes	—
288	C₂×C₄	#10597280	1	2	yes	—
289	C₂×C₄	#10597282	1	2	yes	—
290	C₂×C₄	C₂×C₈×D₁₉	1	1	yes	characteristic
291	C₂×C₄	C₂×#64515	1	1	yes	characteristic
292	C₂×C₄	#10597354	1	2	yes	—
293	C₂×C₄	C₂×#64525	1	1	yes	characteristic
294	C₂×C₄	C₂×#64527	1	1	yes	characteristic
295	C₈	—	8	2	no	—
296	C₈	—	76	4	no	—
297	C₂²	—	76	4	no	—
298	C₂²	—	76	4	no	—
299	C₂²	—	76	4	no	—
300	C₂²	—	152	2	no	—
301	C₂²	—	152	2	no	—
302	C₂²	—	152	2	no	—
303	C₂²	—	152	2	no	—
304	C₂²	C₂×(C₂×D₁₉)⋊C₈	1	1	yes	characteristic
305	C₂²	#11926517	1	1	yes	characteristic
306	C₂²	C₂×#10597280	1	1	yes	characteristic
307	C₂²	C₂×#10597266	1	1	yes	characteristic
308	C₂²	#11926550	1	1	yes	characteristic
309	C₂²	C₂×#10597282	1	1	yes	characteristic
310	C₂²	#11926562	1	1	yes	characteristic
311	C₄	—	2	2	no	—
312	C₄	—	2	2	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	2	no	—
318	C₄	—	2	2	no	—
319	C₄	—	152	2	no	—
320	C₄	#11926271	1	4	yes	—
321	C₂	—	152	4	no	—
322	C₂	#14561955	1	1	yes	characteristic
323	C₂	C₂×#11926271	1	1	yes	characteristic
324	C₂	#14562027	1	1	yes	characteristic
325	C₂	#14562144	1	1	yes	characteristic
326	C₂	#14562194	1	1	yes	characteristic
327	C₂	#14562247	1	1	yes	characteristic
328	C₂	#14562304	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₁₉)⋊C₄
commutator	C₂×C₄×C₁₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1831	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂×#10597320	C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1831	D₁₉
hypocenter	C₁₉	#7587
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
2. #14563381	C₂	semidirect
3. #14561455	C₂	semidirect
4. C₁₉×#1831	C₂	semidirect
40. #10597320	C₈	semidirect
134. #1831	D₁₉	semidirect
211. C₁₉	#7587	semidirect

Maximal subgroups

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

subgroup	index
C₄×#10597320	2
#14563381	2
#14561455	2
C₁₉×#1831	2
#7587	19