Subgroups — #18088341

Order 4864.

850 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14566042	C₂	1	1	yes	maximal, characteristic
2	C₄×#10597372	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926301	C₂	1	4	yes	maximal
4	C₂×C₁₉×#423	C₂	1	1	yes	maximal, characteristic
5	C₂×#10597372	C₂²	1	2	yes	—
6	C₂×#10597372	C₂²	1	2	yes	—
7	#11926414	—	2	1	no	—
8	#11926410	—	2	1	no	—
9	#11926404	—	2	2	no	—
10	C₄×C₄⋊Dic₁₉	—	2	2	no	—
11	C₄×C₄⋊Dic₁₉	C₄	1	4	yes	complemented
12	C₄×C₂²⋊Dic₁₉	—	2	2	no	—
13	#11926390	C₄	1	2	yes	complemented
14	C₄²×Dic₁₉	C₄	1	2	yes	complemented
15	#11926301	—	2	8	no	—
16	C₂²×M₁₆×C₁₉	C₂²	1	2	yes	—
17	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
18	C₁₉×#423	—	2	2	no	—
19	C₁₉×#423	—	2	1	no	—
20	C₁₉×#423	—	2	1	no	—
21	C₂×C₁₉×C₄⋊C₈	C₄	1	4	yes	complemented
22	C₂³×C₄×C₁₉	D₄	1	2	yes	—
23	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
24	C₂×M₁₆×C₁₉	—	2	4	no	—
25	C₂×M₁₆×C₁₉	—	2	2	no	—
26	C₂×M₁₆×C₁₉	—	2	2	no	—
27	C₂×M₁₆×C₁₉	—	2	4	no	—
28	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₁₉	D₄	1	2	yes	—
34	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₉	Q₈	1	1	yes	characteristic
37	C₂×C₄²×C₁₉	Q₈	1	1	yes	characteristic
38	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
39	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
40	#10597372	—	2	2	no	—
41	#10597372	—	2	2	no	—
42	#10597372	—	2	4	no	—
43	#10597372	—	2	4	no	—
44	#10597372	D₄	1	4	yes	—
45	#10597372	Q₈	1	4	yes	—
46	C₂×C₄⋊Dic₁₉	—	2	2	no	—
47	C₂×C₄⋊Dic₁₉	—	2	4	no	—
48	C₂×C₄⋊Dic₁₉	—	2	2	no	—
49	C₂×C₄×Dic₁₉	—	2	4	no	—
50	C₂×C₄×Dic₁₉	C₂×C₄	1	4	yes	—
51	C₂×C₄×Dic₁₉	C₂×C₄	1	4	yes	—
52	C₁₉×C₄⋊C₈	—	2	4	no	—
53	C₁₉×C₄⋊C₈	—	2	4	no	—
54	C₁₉⋊#95	—	2	4	no	—
55	C₁₉⋊#95	—	2	4	no	—
56	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	—	4	4	no	—
58	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₁₉	#41	1	2	yes	—
73	C₂²×C₄×C₁₉	#41	1	2	yes	—
74	C₂²×Dic₁₉	—	2	4	no	—
75	C₂²×Dic₁₉	—	2	4	no	—
76	M₁₆×C₁₉	—	4	4	no	—
77	M₁₆×C₁₉	—	4	2	no	—
78	M₁₆×C₁₉	—	4	2	no	—
79	C₂×C₈×C₁₉	—	2	4	no	—
80	C₂×C₈×C₁₉	—	2	8	no	—
81	C₄²×C₁₉	C₄²	1	2	yes	—
82	C₄²×C₁₉	C₄²	1	2	yes	—
83	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
84	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
85	C₄²×C₁₉	C₄⋊C₄	1	1	yes	characteristic
86	C₄²×C₁₉	C₄⋊C₄	1	1	yes	characteristic
87	C₄²×C₁₉	C₄⋊C₄	1	1	yes	characteristic
88	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
89	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
90	C₄²×C₁₉	C₄⋊C₄	1	1	yes	characteristic
91	C₂²⋊Dic₁₉	—	2	4	no	—
92	C₂²⋊Dic₁₉	—	2	4	no	—
93	C₂²⋊Dic₁₉	—	4	4	no	—
94	C₄⋊Dic₁₉	—	2	4	no	—
95	C₄⋊Dic₁₉	—	2	4	no	—
96	C₄⋊Dic₁₉	—	4	8	no	—
97	C₄⋊Dic₁₉	—	4	2	no	—
98	C₄⋊Dic₁₉	—	4	2	no	—
99	C₄×Dic₁₉	—	2	4	no	—
100	C₄×Dic₁₉	—	2	8	no	—
101	C₄×Dic₁₉	—	4	8	no	—
102	C₄×Dic₁₉	C₄⋊C₄	1	8	yes	complemented
103	#10836	—	19	1	no	maximal, Sylow
104	C₂³×C₁₉	—	4	2	no	—
105	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂³×C₁₉	#121	1	2	yes	—
109	C₂³×C₁₉	C₂²⋊Q₈	1	2	yes	—
110	C₂×C₄×C₁₉	—	2	4	no	—
111	C₂×C₄×C₁₉	—	2	4	no	—
112	C₂×C₄×C₁₉	—	2	4	no	—
113	C₂×C₄×C₁₉	—	4	2	no	—
114	C₂×C₄×C₁₉	—	4	2	no	—
115	C₂×C₄×C₁₉	—	4	2	no	—
116	C₂×C₄×C₁₉	#95	1	2	yes	—
117	C₂×C₄×C₁₉	#95	1	2	yes	—
118	C₂×C₄×C₁₉	#95	1	2	yes	—
119	C₂×C₄×C₁₉	#95	1	2	yes	—
120	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₉	#117	1	2	yes	—
126	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
129	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
130	C₂×C₄×C₁₉	#121	1	2	yes	—
131	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
132	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
133	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
134	C₂×C₄×C₁₉	#125	1	2	yes	—
135	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
136	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
137	C₂×C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
138	C₂×Dic₁₉	—	2	8	no	—
139	C₂×Dic₁₉	—	4	8	no	—
140	C₂×Dic₁₉	—	4	8	no	—
141	C₂×Dic₁₉	—	4	4	no	—
142	C₈×C₁₉	—	4	4	no	—
143	C₂×#423	D₁₉	1	1	yes	characteristic
144	#2285	—	19	1	no	—
145	C₄×#117	—	19	1	no	—
146	C₂×H(3,ℤ/4)	—	19	4	no	—
147	C₂²×C₁₉	—	2	2	no	—
148	C₂²×C₁₉	—	2	2	no	—
149	C₂²×C₁₉	—	2	2	no	—
150	C₂²×C₁₉	—	4	2	no	—
151	C₂²×C₁₉	—	4	2	no	—
152	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
153	C₂²×C₁₉	#380	1	2	yes	—
154	C₂²×C₁₉	#389	1	2	yes	—
155	C₂²×C₁₉	#390	1	1	yes	characteristic
156	C₂²×C₁₉	#391	1	1	yes	characteristic
157	C₄×C₁₉	—	2	2	no	—
158	C₄×C₁₉	—	2	2	no	—
159	C₄×C₁₉	—	2	2	no	—
160	C₄×C₁₉	H(3,ℤ/4)	1	4	yes	complemented
161	C₄×C₁₉	C₄×C₄⋊C₄	1	2	yes	—
162	C₄×C₁₉	#382	1	1	yes	characteristic
163	C₄×C₁₉	#384	1	1	yes	characteristic
164	C₄×C₁₉	#389	1	2	yes	—
165	C₄×C₁₉	#390	1	1	yes	characteristic
166	C₄×C₁₉	#391	1	1	yes	characteristic
167	Dic₁₉	—	4	8	no	—
168	Dic₁₉	—	8	4	no	—
169	C₂²×M₁₆	C₂×D₁₉	1	2	yes	—
170	C₂×#117	—	19	2	no	—
171	C₂×#117	—	19	2	no	—
172	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
173	#423	—	2	2	no	—
174	#423	—	2	1	no	—
175	#423	—	2	1	no	—
176	C₂×C₄⋊C₈	Dic₁₉	1	4	yes	complemented
177	#384	—	38	1	no	—
178	#382	—	38	1	no	—
179	#380	—	38	2	no	—
180	C₄×C₄⋊C₄	—	19	4	no	—
181	C₄×C₄⋊C₄	—	38	2	no	—
182	C₄×C₂²⋊C₄	—	38	2	no	—
183	#376	—	19	2	no	—
184	C₄³	—	19	2	no	—
185	H(3,ℤ/4)	—	38	8	no	—
186	C₂×C₁₉	—	2	2	no	—
187	C₂×C₁₉	—	2	1	no	—
188	C₂×C₁₉	—	2	2	no	—
189	C₂×C₁₉	—	2	1	no	—
190	C₂×C₁₉	C₄⋊#95	1	1	yes	characteristic
191	C₂×C₁₉	C₂×H(3,ℤ/4)	1	2	yes	—
192	C₂³×C₄	D₇₆	1	2	yes	—
193	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
194	C₂×M₁₆	—	2	4	no	—
195	C₂×M₁₆	—	2	4	no	—
196	C₂×M₁₆	—	2	2	no	—
197	C₂×M₁₆	—	2	2	no	—
198	C₂²×C₈	C₂×Dic₁₉	1	4	yes	—
199	#117	—	19	4	no	—
200	#117	—	19	4	no	—
201	#117	—	38	2	no	—
202	#117	—	38	2	no	—
203	#117	—	38	4	no	—
204	#117	—	38	4	no	—
205	C₂×C₄⋊C₄	—	38	2	no	—
206	C₂×C₄⋊C₄	—	38	4	no	—
207	C₂×C₄⋊C₄	—	38	2	no	—
208	C₂×C₂²⋊C₄	—	38	2	no	—
209	C₂×C₂²⋊C₄	—	38	2	no	—
210	C₂×C₄²	—	19	4	no	—
211	C₂×C₄²	—	19	4	no	—
212	C₂×C₄²	—	38	4	no	—
213	C₂×C₄²	Dic₃₈	1	1	yes	characteristic
214	C₂×C₄²	Dic₃₈	1	1	yes	characteristic
215	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
216	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
217	C₂×C₄²	C₄×D₁₉	1	2	yes	—
218	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
219	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
220	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
221	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
222	C₄⋊C₈	—	2	4	no	—
223	C₄⋊C₈	—	2	4	no	—
224	#95	—	38	4	no	—
225	#95	—	38	4	no	—
226	C₁₉	#10836	1	1	yes	characteristic, complemented, Sylow
227	C₂⁴	C₂×D₇₆	1	1	yes	characteristic
228	C₂²×C₄	—	4	4	no	—
229	C₂²×C₄	—	38	4	no	—
230	C₂²×C₄	—	38	4	no	—
231	C₂²×C₄	C₄⋊Dic₁₉	1	2	yes	—
232	C₂²×C₄	C₄⋊Dic₁₉	1	2	yes	—
233	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
234	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
235	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
236	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
237	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
240	C₂²×C₄	D₄×D₁₉	1	2	yes	—
241	C₂²×C₄	#64543	1	2	yes	—
242	C₂²×C₄	Q₈×D₁₉	1	2	yes	—
243	C₂²×C₄	#64545	1	2	yes	—
244	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
247	M₁₆	—	4	2	no	—
248	M₁₆	—	4	4	no	—
249	M₁₆	—	4	2	no	—
250	C₂×C₈	—	2	8	no	—
251	C₂×C₈	—	2	4	no	—
252	C₄⋊C₄	—	38	4	no	—
253	C₄⋊C₄	—	38	4	no	—
254	C₄⋊C₄	—	76	8	no	—
255	C₄⋊C₄	—	76	2	no	—
256	C₄⋊C₄	—	76	2	no	—
257	C₂²⋊C₄	—	38	4	no	—
258	C₂²⋊C₄	—	38	4	no	—
259	C₂²⋊C₄	—	76	4	no	—
260	C₄²	—	19	8	no	—
261	C₄²	—	38	4	no	—
262	C₄²	—	38	8	no	—
263	C₄²	—	76	8	no	—
264	C₄²	C₄×Dic₁₉	1	2	yes	—
265	C₄²	C₄×Dic₁₉	1	2	yes	—
266	C₄²	Dic₁₉⋊C₄	1	1	yes	characteristic
267	C₄²	Dic₁₉⋊C₄	1	1	yes	characteristic
268	C₄²	Dic₁₉⋊C₄	1	1	yes	characteristic
269	C₄²	Dic₁₉⋊C₄	1	2	yes	—
270	C₄²	Dic₁₉⋊C₄	1	2	yes	—
271	C₄²	Dic₁₉⋊C₄	1	1	yes	characteristic
272	C₄²	C₂²⋊Dic₁₉	1	2	yes	—
273	C₄²	C₂²⋊Dic₁₉	1	2	yes	—
274	C₂³	—	4	2	no	—
275	C₂³	#10597341	1	2	yes	—
276	C₂³	#10597343	1	2	yes	—
277	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
278	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
279	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	38	8	no	—
287	C₂×C₄	—	76	8	no	—
288	C₂×C₄	—	76	8	no	—
289	C₂×C₄	—	76	4	no	—
290	C₂×C₄	C₁₉⋊#95	1	2	yes	—
291	C₂×C₄	C₁₉⋊#95	1	2	yes	—
292	C₂×C₄	C₁₉⋊#95	1	2	yes	—
293	C₂×C₄	C₁₉⋊#95	1	2	yes	—
294	C₂×C₄	#10597333	1	2	yes	—
295	C₂×C₄	#10597334	1	2	yes	—
296	C₂×C₄	#10597336	1	2	yes	—
297	C₂×C₄	C₄⋊C₄×D₁₉	1	2	yes	—
298	C₂×C₄	#10597338	1	2	yes	—
299	C₂×C₄	#10597339	1	2	yes	—
300	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
301	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
302	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
303	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
304	C₂×C₄	#10597382	1	1	yes	characteristic
305	C₂×C₄	#10597384	1	2	yes	—
306	C₂×C₄	#10597386	1	1	yes	characteristic
307	C₂×C₄	#10597390	1	1	yes	characteristic
308	C₂×C₄	Q₈×Dic₁₉	1	2	yes	—
309	C₂×C₄	#10597392	1	2	yes	—
310	C₂×C₄	#10597393	1	1	yes	characteristic
311	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
312	C₈	—	4	4	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	4	2	no	—
318	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
319	C₂²	#11926401	1	2	yes	—
320	C₂²	#11926443	1	2	yes	—
321	C₂²	#11926445	1	1	yes	characteristic
322	C₂²	#11926449	1	1	yes	characteristic
323	C₄	—	2	2	no	—
324	C₄	—	2	2	no	—
325	C₄	—	2	2	no	—
326	C₄	—	76	8	no	—
327	C₄	—	152	4	no	—
328	C₄	#11926301	1	4	yes	complemented
329	C₄	C₄⋊C₄×Dic₁₉	1	2	yes	—
330	C₄	#11926409	1	1	yes	characteristic
331	C₄	#11926413	1	1	yes	characteristic
332	C₄	#11926440	1	2	yes	—
333	C₄	#11926447	1	1	yes	characteristic
334	C₄	#11926451	1	1	yes	characteristic
335	C₂	—	2	2	no	—
336	C₂	—	2	1	no	—
337	C₂	—	2	2	no	—
338	C₂	—	2	1	no	—
339	C₂	#14561864	1	1	yes	characteristic
340	C₂	C₂×#11926301	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₁₉×#423	C₂
socle	C₂²×C₁₉	C₂×#95
prime radical	C₂⁴×C₁₉	C₂×D₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×#423	D₁₉
hypocenter	C₁₉	#10836
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
11. C₄×C₄⋊Dic₁₉	C₄	semidirect
13. #11926390	C₄	semidirect
14. C₄²×Dic₁₉	C₄	semidirect
21. C₂×C₁₉×C₄⋊C₈	C₄	semidirect
102. C₄×Dic₁₉	C₄⋊C₄	semidirect
160. C₄×C₁₉	H(3,ℤ/4)	semidirect
176. C₂×C₄⋊C₈	Dic₁₉	semidirect
226. C₁₉	#10836	semidirect
328. C₄	#11926301	semidirect

Maximal subgroups

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

subgroup	index
#14566042	2
C₄×#10597372	2
C₂×#11926301	2
C₂×C₁₉×#423	2
#10836	19