Subgroups — #18075374

Order 4864.

598 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563615	C₂	1	2	yes	maximal
2	C₂×#11926295	C₂	1	2	yes	maximal, complemented
3	#14561470	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄×C₈×C₁₉	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	4	yes	—
6	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
7	#11926299	—	2	2	no	—
8	#11926295	—	2	4	no	—
9	C₂³×C₈×C₁₉	C₄	1	1	yes	characteristic
10	C₂³×C₈×C₁₉	C₄	1	1	yes	characteristic
11	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₉	—	2	4	no	—
13	C₂×C₄×C₈×C₁₉	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₉	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
19	C₂³×C₄×C₁₉	Q₈	1	1	yes	characteristic
20	C₂²×C₈×C₁₉	—	2	2	no	—
21	C₂²×C₈×C₁₉	—	2	2	no	—
22	C₂²×C₈×C₁₉	—	2	2	no	—
23	C₂²×C₈×C₁₉	—	2	2	no	—
24	C₂²×C₈×C₁₉	—	2	2	no	—
25	C₂²×C₈×C₁₉	C₈	1	4	yes	complemented
26	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₉	—	2	2	no	—
31	C₂×C₄²×C₁₉	—	2	2	no	—
32	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₉	Q₈	1	1	yes	characteristic
35	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
36	C₂²×C₁₉⋊C₈	—	2	4	no	—
37	C₄×C₈×C₁₉	—	2	2	no	—
38	C₄×C₈×C₁₉	—	2	2	no	—
39	C₄×C₈×C₁₉	—	2	2	no	—
40	C₄×C₈×C₁₉	D₄	1	4	yes	—
41	C₂²⋊(C₁₉⋊C₈)	—	4	4	no	—
42	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
43	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
44	C₂⁴×C₁₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₉	—	2	2	no	—
46	C₂²×C₄×C₁₉	—	2	2	no	—
47	C₂²×C₄×C₁₉	—	2	1	no	—
48	C₂²×C₄×C₁₉	—	2	1	no	—
49	C₂²×C₄×C₁₉	—	2	2	no	—
50	C₂²×C₄×C₁₉	—	2	2	no	—
51	C₂²×C₄×C₁₉	—	2	1	no	—
52	C₂²×C₄×C₁₉	—	2	1	no	—
53	C₂²×C₄×C₁₉	—	2	2	no	—
54	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂×C₈	1	2	yes	—
58	C₂²×C₄×C₁₉	M₁₆	1	2	yes	—
59	C₂²×C₄×C₁₉	SD₁₆	1	4	yes	—
60	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₈×C₁₉	—	2	8	no	—
64	C₂×C₈×C₁₉	—	2	1	no	—
65	C₂×C₈×C₁₉	—	2	1	no	—
66	C₂×C₈×C₁₉	—	2	1	no	—
67	C₂×C₈×C₁₉	—	2	1	no	—
68	C₂×C₈×C₁₉	—	2	1	no	—
69	C₂×C₈×C₁₉	—	2	1	no	—
70	C₂×C₈×C₁₉	—	2	8	no	—
71	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₁₉	C₂×C₈	1	2	yes	—
74	C₂×C₈×C₁₉	C₂×C₈	1	2	yes	—
75	C₂×C₈×C₁₉	M₁₆	1	2	yes	—
76	C₂×C₈×C₁₉	M₁₆	1	2	yes	—
77	C₄²×C₁₉	—	2	1	no	—
78	C₄²×C₁₉	—	2	1	no	—
79	C₄²×C₁₉	—	2	2	no	—
80	C₄²×C₁₉	—	2	2	no	—
81	C₄²×C₁₉	C₂×D₄	1	2	yes	—
82	C₄²×C₁₉	#41	1	2	yes	—
83	C₂×C₁₉⋊C₈	—	4	4	no	—
84	C₂×C₁₉⋊C₈	—	4	4	no	—
85	C₂×C₁₉⋊C₈	—	4	4	no	—
86	#8595	—	19	1	no	maximal, Sylow
87	C₂³×C₁₉	—	2	1	no	—
88	C₂³×C₁₉	—	2	1	no	—
89	C₂³×C₁₉	—	2	1	no	—
90	C₂³×C₁₉	—	2	1	no	—
91	C₂³×C₁₉	C₄⋊C₈	1	2	yes	—
92	C₂³×C₁₉	#106	1	2	yes	—
93	C₂³×C₁₉	#108	1	2	yes	—
94	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₉	—	2	8	no	—
96	C₂×C₄×C₁₉	—	2	1	no	—
97	C₂×C₄×C₁₉	—	2	1	no	—
98	C₂×C₄×C₁₉	—	2	1	no	—
99	C₂×C₄×C₁₉	—	2	1	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	2	no	—
105	C₂×C₄×C₁₉	—	2	2	no	—
106	C₂×C₄×C₁₉	—	2	2	no	—
107	C₂×C₄×C₁₉	—	2	2	no	—
108	C₂×C₄×C₁₉	—	2	8	no	—
109	C₂×C₄×C₁₉	C₄⋊C₈	1	2	yes	—
110	C₂×C₄×C₁₉	#106	1	2	yes	—
111	C₂×C₄×C₁₉	#108	1	2	yes	—
112	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₉	#117	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	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
118	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
119	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
121	C₂×C₄×C₁₉	#135	1	2	yes	—
122	C₈×C₁₉	—	2	2	no	—
123	C₈×C₁₉	—	2	2	no	—
124	C₈×C₁₉	—	2	2	no	—
125	C₈×C₁₉	C₂²⋊C₈	1	4	yes	complemented
126	C₁₉⋊C₈	—	8	4	no	—
127	C₂²×C₄×C₈	D₁₉	1	1	yes	characteristic
128	#1797	—	19	2	no	—
129	C₂×#334	—	19	2	no	—
130	#1277	—	19	2	no	—
131	C₂²×C₁₉	—	2	2	no	—
132	C₂²×C₁₉	—	2	2	no	—
133	C₂²×C₁₉	—	2	2	no	—
134	C₂²×C₁₉	—	2	2	no	—
135	C₂²×C₁₉	—	2	2	no	—
136	C₂²×C₁₉	—	2	2	no	—
137	C₂²×C₁₉	#334	1	4	yes	complemented
138	C₂²×C₁₉	#380	1	1	yes	characteristic
139	C₂²×C₁₉	C₂×C₄⋊C₈	1	1	yes	characteristic
140	C₂²×C₁₉	#423	1	1	yes	characteristic
141	C₂²×C₁₉	C₂×#106	1	1	yes	characteristic
142	C₂²×C₁₉	#427	1	1	yes	characteristic
143	C₂²×C₁₉	C₂×#108	1	1	yes	characteristic
144	C₂²×C₁₉	C₂×#108	1	1	yes	characteristic
145	C₄×C₁₉	—	2	2	no	—
146	C₄×C₁₉	—	2	2	no	—
147	C₄×C₁₉	—	2	2	no	—
148	C₄×C₁₉	—	2	2	no	—
149	C₄×C₁₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
150	C₄×C₁₉	#407	1	1	yes	characteristic
151	C₄×C₁₉	#431	1	1	yes	characteristic
152	C₄×C₁₉	#432	1	1	yes	characteristic
153	C₄×C₁₉	#465	1	4	yes	—
154	C₂³×C₈	Dic₁₉	1	1	yes	characteristic
155	C₂³×C₈	Dic₁₉	1	1	yes	characteristic
156	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
157	C₂×C₄⋊C₈	—	19	4	no	—
158	C₂×C₂²⋊C₈	—	38	2	no	—
159	C₂×C₄×C₈	—	2	4	no	—
160	C₂×C₄×C₈	Dic₁₉	1	1	yes	characteristic
161	C₂×C₄×C₈	Dic₁₉	1	1	yes	characteristic
162	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
163	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
164	#336	—	38	2	no	—
165	#334	—	38	4	no	—
166	C₂×C₁₉	—	2	4	no	—
167	C₂×C₁₉	C₂×#334	1	1	yes	characteristic
168	C₂×C₁₉	#1564	1	1	yes	characteristic
169	C₂×C₁₉	#1566	1	1	yes	characteristic
170	C₂×C₁₉	#1568	1	1	yes	characteristic
171	C₂×C₁₉	#1797	1	1	yes	characteristic
172	C₂×C₁₉	#1807	1	1	yes	characteristic
173	C₂×C₁₉	#1810	1	1	yes	characteristic
174	C₂³×C₄	Dic₃₈	1	1	yes	characteristic
175	C₂³×C₄	D₇₆	1	1	yes	characteristic
176	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	38	4	no	—
183	C₂²×C₈	C₁₉⋊C₈	1	4	yes	complemented
184	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
185	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
186	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
187	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	—	2	2	no	—
190	C₂×C₄²	Dic₃₈	1	1	yes	characteristic
191	C₂×C₄²	D₇₆	1	1	yes	characteristic
192	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
193	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
194	C₄⋊C₈	—	38	4	no	—
195	C₄⋊C₈	—	38	4	no	—
196	C₂²⋊C₈	—	76	4	no	—
197	C₄×C₈	—	2	2	no	—
198	C₄×C₈	—	2	2	no	—
199	C₄×C₈	—	2	2	no	—
200	C₄×C₈	C₂²⋊D₁₉	1	4	yes	—
201	C₁₉	#8595	1	1	yes	characteristic, complemented, Sylow
202	C₂⁴	C₄⋊Dic₁₉	1	1	yes	characteristic
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	#64516	1	4	yes	—
213	C₂²×C₄	C₂×C₁₉⋊C₈	1	2	yes	—
214	C₂²×C₄	#64520	1	2	yes	—
215	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
216	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
217	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
218	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
220	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
221	C₂×C₈	—	2	8	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	8	no	—
229	C₂×C₈	—	76	4	no	—
230	C₂×C₈	—	76	4	no	—
231	C₂×C₈	—	76	4	no	—
232	C₂×C₈	C₂×C₁₉⋊C₈	1	2	yes	—
233	C₂×C₈	C₂×C₁₉⋊C₈	1	2	yes	—
234	C₂×C₈	#64520	1	2	yes	—
235	C₂×C₈	#64520	1	2	yes	—
236	C₂×C₈	C₂²⋊Dic₁₉	1	2	yes	—
237	C₂×C₈	C₂²⋊Dic₁₉	1	2	yes	—
238	C₄²	—	2	2	no	—
239	C₄²	—	2	1	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	1	no	—
242	C₄²	#64541	1	2	yes	—
243	C₄²	C₂×C₂²⋊D₁₉	1	2	yes	—
244	C₂³	—	2	1	no	—
245	C₂³	—	2	1	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	C₄⋊(C₁₉⋊C₈)	1	2	yes	—
249	C₂³	#10597274	1	2	yes	—
250	C₂³	#10597276	1	2	yes	—
251	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
252	C₂×C₄	—	2	8	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	8	no	—
266	C₂×C₄	C₄⋊(C₁₉⋊C₈)	1	2	yes	—
267	C₂×C₄	#10597274	1	2	yes	—
268	C₂×C₄	#10597276	1	2	yes	—
269	C₂×C₄	C₂×#64516	1	2	yes	—
270	C₂×C₄	#10597350	1	2	yes	—
271	C₂×C₄	C₂²×C₁₉⋊C₈	1	1	yes	characteristic
272	C₂×C₄	C₂×#64520	1	1	yes	characteristic
273	C₂×C₄	C₂×#64520	1	1	yes	characteristic
274	C₂×C₄	C₂×#64520	1	1	yes	characteristic
275	C₂×C₄	#10597370	1	2	yes	—
276	C₂×C₄	#10597372	1	1	yes	characteristic
277	C₂×C₄	#10597376	1	2	yes	—
278	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
279	C₈	—	2	2	no	—
280	C₈	—	2	2	no	—
281	C₈	—	2	2	no	—
282	C₈	—	152	4	no	—
283	C₈	C₂²⋊(C₁₉⋊C₈)	1	4	yes	complemented
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	#11926295	1	4	yes	complemented
291	C₂²	#11926404	1	1	yes	characteristic
292	C₂²	C₂×C₄⋊(C₁₉⋊C₈)	1	1	yes	characteristic
293	C₂²	#11926568	1	1	yes	characteristic
294	C₂²	C₂×#10597274	1	1	yes	characteristic
295	C₂²	#11926582	1	1	yes	characteristic
296	C₂²	C₂×#10597276	1	1	yes	characteristic
297	C₂²	C₂×#10597276	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	2	2	no	—
302	C₄	C₂×C₂²⋊(C₁₉⋊C₈)	1	1	yes	characteristic
303	C₄	#11926508	1	1	yes	characteristic
304	C₄	#11926597	1	1	yes	characteristic
305	C₄	#11926601	1	1	yes	characteristic
306	C₄	#11926773	1	4	yes	—
307	C₂	—	2	4	no	—
308	C₂	C₂×#11926295	1	1	yes	characteristic
309	C₂	#14562383	1	1	yes	characteristic
310	C₂	#14562391	1	1	yes	characteristic
311	C₂	#14562398	1	1	yes	characteristic
312	C₂	#14563615	1	1	yes	characteristic
313	C₂	#14563660	1	1	yes	characteristic
314	C₂	#14563673	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₇₆
commutator	C₂×C₄×C₁₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂²×C₄×C₈×C₁₉	C₂
socle	C₂³×C₁₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₉	C₄⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂²×C₄×C₈	D₁₉
hypocenter	C₁₉	#8595
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#11926295	C₂	semidirect
3. #14561470	C₂	semidirect
25. C₂²×C₈×C₁₉	C₈	semidirect
125. C₈×C₁₉	C₂²⋊C₈	semidirect
137. C₂²×C₁₉	#334	semidirect
183. C₂²×C₈	C₁₉⋊C₈	semidirect
201. C₁₉	#8595	semidirect
283. C₈	C₂²⋊(C₁₉⋊C₈)	semidirect
290. C₂²	#11926295	semidirect

Maximal subgroups

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

subgroup	index
#14563615	2
C₂×#11926295	2
#14561470	2
C₂²×C₄×C₈×C₁₉	2
#8595	19