Subgroups — #18073024

Order 4864.

618 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#64515	C₂	1	1	yes	maximal, characteristic
2	C₄×(C₂×D₁₉)⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×(C₂×D₁₉)⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×#11926266	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉⋊#1273	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉⋊#1273	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₉×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#64515	C₄	1	2	yes	—
9	C₂×C₄²×D₁₉	C₂²	1	1	yes	characteristic
10	C₂×(C₂×D₁₉)⋊C₈	C₄	1	2	yes	—
11	C₂×(C₂×D₁₉)⋊C₈	C₄	1	2	yes	—
12	C₄×#64515	—	2	2	no	—
13	C₄×#64515	—	2	2	no	—
14	C₂×#10597272	C₄	1	2	yes	—
15	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
18	#11926298	C₄	1	2	yes	—
19	#11926298	C₄	1	2	yes	—
20	#11926266	—	2	2	no	—
21	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
24	C₁₉×#322	—	2	2	no	—
25	C₂²×C₄×D₁₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×D₁₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×D₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
32	C₂×C₄×Dic₁₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×Dic₁₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×Dic₁₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
36	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
37	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
38	C₂×#64515	—	2	4	no	—
39	C₂×#64515	—	2	4	no	—
40	C₂×#64515	—	2	4	no	—
41	C₄²×D₁₉	—	2	1	no	—
42	C₄²×D₁₉	—	2	1	no	—
43	C₄²×D₁₉	—	2	1	no	—
44	C₄²×D₁₉	—	2	1	no	—
45	C₄×C₈×C₁₉	—	2	1	no	—
46	C₄×C₈×C₁₉	—	2	1	no	—
47	C₄×C₈×C₁₉	—	2	1	no	—
48	C₄×C₈×C₁₉	—	2	1	no	—
49	C₄×C₈×C₁₉	D₄	1	2	yes	—
50	C₄×C₈×C₁₉	D₄	1	2	yes	—
51	(C₂×D₁₉)⋊C₈	—	2	4	no	—
52	(C₂×D₁₉)⋊C₈	—	2	4	no	—
53	#10597272	—	2	4	no	—
54	C₄×C₁₉⋊C₈	—	2	1	no	—
55	C₄×C₁₉⋊C₈	—	2	1	no	—
56	C₄×C₁₉⋊C₈	—	2	1	no	—
57	C₄×C₁₉⋊C₈	—	2	1	no	—
58	C₄×C₁₉⋊C₈	—	2	1	no	—
59	C₄×C₁₉⋊C₈	—	2	1	no	—
60	C₂³×D₁₉	C₄²	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
64	C₂²×Dic₁₉	C₄²	1	1	yes	characteristic
65	C₂²×Dic₁₉	C₄²	1	1	yes	characteristic
66	C₂²×Dic₁₉	C₄²	1	1	yes	characteristic
67	C₂×C₄×D₁₉	—	2	1	no	—
68	C₂×C₄×D₁₉	—	2	1	no	—
69	C₂×C₄×D₁₉	—	2	2	no	—
70	C₂×C₄×D₁₉	—	2	1	no	—
71	C₂×C₄×D₁₉	—	2	1	no	—
72	C₂×C₄×D₁₉	—	2	1	no	—
73	C₂×C₄×D₁₉	—	2	1	no	—
74	C₂×C₄×D₁₉	—	2	2	no	—
75	C₂×C₄×D₁₉	—	2	2	no	—
76	C₂×C₄×D₁₉	—	2	2	no	—
77	C₂×C₄×D₁₉	—	2	2	no	—
78	C₂×C₄×D₁₉	M₁₆	1	4	yes	—
79	C₂×C₈×C₁₉	—	2	2	no	—
80	C₂×C₈×C₁₉	—	2	2	no	—
81	C₂×C₈×C₁₉	—	2	2	no	—
82	C₂×C₈×C₁₉	—	2	2	no	—
83	C₂×C₈×C₁₉	—	2	2	no	—
84	C₂×C₈×C₁₉	—	2	2	no	—
85	C₂×C₈×C₁₉	—	2	2	no	—
86	C₂×C₈×C₁₉	C₂²⋊C₄	1	4	yes	—
87	C₂×C₈×C₁₉	M₁₆	1	4	yes	complemented
88	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₁₉	#41	1	1	yes	characteristic
91	C₄²×C₁₉	#41	1	1	yes	characteristic
92	C₄×Dic₁₉	—	2	1	no	—
93	C₄×Dic₁₉	—	2	1	no	—
94	C₄×Dic₁₉	—	2	1	no	—
95	C₄×Dic₁₉	—	2	1	no	—
96	C₄×Dic₁₉	—	2	1	no	—
97	C₄×Dic₁₉	—	2	1	no	—
98	C₂×C₁₉⋊C₈	—	2	2	no	—
99	C₂×C₁₉⋊C₈	—	2	2	no	—
100	C₂×C₁₉⋊C₈	—	2	2	no	—
101	C₂×C₁₉⋊C₈	—	2	2	no	—
102	C₂×C₁₉⋊C₈	—	2	2	no	—
103	C₂×C₁₉⋊C₈	—	2	2	no	—
104	C₂×C₁₉⋊C₈	—	2	2	no	—
105	C₂×C₁₉⋊C₈	M₁₆	1	4	yes	—
106	C₂×C₁₉⋊C₈	M₁₆	1	4	yes	complemented
107	#64515	—	2	16	no	—
108	#8147	—	19	1	no	maximal, Sylow
109	C₂³×C₁₉	C₂×C₄²	1	1	yes	characteristic
110	C₂²×D₁₉	—	2	1	no	—
111	C₂²×D₁₉	—	2	1	no	—
112	C₂²×D₁₉	—	2	1	no	—
113	C₂²×D₁₉	—	2	1	no	—
114	C₂²×D₁₉	—	2	1	no	—
115	C₂²×D₁₉	—	2	1	no	—
116	C₂²×D₁₉	#97	1	2	yes	—
117	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
119	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
124	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
125	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
126	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
127	C₂×C₄×C₁₉	#131	1	2	yes	—
128	C₂×C₄×C₁₉	#131	1	2	yes	—
129	C₂×Dic₁₉	—	2	1	no	—
130	C₂×Dic₁₉	—	2	1	no	—
131	C₂×Dic₁₉	—	2	1	no	—
132	C₂×Dic₁₉	—	2	1	no	—
133	C₂×Dic₁₉	—	2	1	no	—
134	C₂×Dic₁₉	—	2	1	no	—
135	C₂×Dic₁₉	—	2	1	no	—
136	C₂×Dic₁₉	—	2	1	no	—
137	C₂×Dic₁₉	#97	1	2	yes	—
138	C₄×D₁₉	—	2	4	no	—
139	C₄×D₁₉	—	2	4	no	—
140	C₄×D₁₉	—	4	2	no	—
141	C₄×D₁₉	—	4	2	no	—
142	C₄×D₁₉	—	4	2	no	—
143	C₄×D₁₉	—	4	2	no	—
144	C₈×C₁₉	—	2	4	no	—
145	C₈×C₁₉	—	2	4	no	—
146	C₈×C₁₉	—	4	2	no	—
147	C₁₉⋊C₈	—	2	4	no	—
148	C₁₉⋊C₈	—	2	4	no	—
149	C₁₉⋊C₈	—	4	2	no	—
150	C₂×C₄×M₁₆	—	19	1	no	—
151	C₄×C₂²⋊C₈	—	19	1	no	—
152	C₄×C₂²⋊C₈	—	19	1	no	—
153	C₂×#322	—	19	1	no	—
154	C₂×#322	D₁₉	1	1	yes	characteristic, complemented
155	#1273	—	19	1	no	—
156	#1273	—	19	1	no	—
157	C₂²×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
158	C₂²×C₁₉	C₂×#97	1	1	yes	characteristic
159	C₂²×C₁₉	C₄×M₁₆	1	1	yes	characteristic
160	C₂²×C₁₉	C₄×M₁₆	1	1	yes	characteristic
161	C₂²×C₁₉	C₄×M₁₆	1	1	yes	characteristic
162	C₂²×C₁₉	#405	1	1	yes	characteristic
163	C₂²×C₁₉	#405	1	1	yes	characteristic
164	C₂×D₁₉	—	2	2	no	—
165	C₂×D₁₉	—	2	2	no	—
166	C₂×D₁₉	—	2	2	no	—
167	C₂×D₁₉	—	4	1	no	—
168	C₂×D₁₉	—	4	1	no	—
169	C₂×D₁₉	—	4	1	no	—
170	C₂×D₁₉	—	4	1	no	—
171	C₄×C₁₉	#407	1	2	yes	—
172	C₄×C₁₉	#432	1	2	yes	—
173	C₄×C₁₉	#435	1	2	yes	—
174	C₄×C₁₉	#435	1	2	yes	—
175	C₄×C₁₉	#436	1	2	yes	—
176	C₄×C₁₉	#436	1	2	yes	—
177	Dic₁₉	—	2	2	no	—
178	Dic₁₉	—	4	1	no	—
179	Dic₁₉	—	4	1	no	—
180	C₂²×M₁₆	—	19	2	no	—
181	C₂²×C₄²	—	19	1	no	—
182	C₂×C₂²⋊C₈	—	19	2	no	—
183	C₂×C₂²⋊C₈	—	19	2	no	—
184	C₄×M₁₆	—	38	2	no	—
185	C₄×M₁₆	—	38	2	no	—
186	C₂×#97	—	19	2	no	—
187	C₂×C₄×C₈	—	19	1	no	—
188	C₂×C₄×C₈	—	19	1	no	—
189	C₂×C₄×C₈	—	19	1	no	—
190	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
191	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
192	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
193	#336	—	19	2	no	—
194	#336	—	19	2	no	—
195	#322	—	2	2	no	—
196	#322	—	38	2	no	—
197	C₂×C₁₉	#1451	1	1	yes	characteristic
198	C₂×C₁₉	#1451	1	1	yes	characteristic
199	C₂×C₁₉	#1453	1	1	yes	characteristic
200	C₂×C₁₉	#1455	1	1	yes	characteristic
201	C₂×C₁₉	#1749	1	1	yes	characteristic
202	C₂×C₁₉	#1753	1	1	yes	characteristic
203	C₂×C₁₉	#1753	1	1	yes	characteristic
204	D₁₉	—	4	2	no	—
205	C₂³×C₄	—	19	1	no	—
206	C₂³×C₄	—	19	1	no	—
207	C₂³×C₄	—	19	1	no	—
208	C₂×M₁₆	—	38	4	no	—
209	C₂×M₁₆	—	38	4	no	—
210	C₂×M₁₆	—	38	4	no	—
211	C₂²×C₈	—	19	2	no	—
212	C₂²×C₈	—	19	2	no	—
213	C₂²×C₈	—	19	2	no	—
214	C₂²×C₈	C₄×D₁₉	1	2	yes	—
215	C₂²×C₈	C₄×D₁₉	1	2	yes	—
216	C₂²×C₈	C₄×D₁₉	1	2	yes	—
217	C₂×C₄²	—	19	1	no	—
218	C₂×C₄²	—	19	1	no	—
219	C₂×C₄²	—	19	1	no	—
220	C₂×C₄²	—	38	1	no	—
221	C₂×C₄²	—	38	1	no	—
222	C₂×C₄²	—	38	1	no	—
223	C₂×C₄²	—	38	1	no	—
224	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
225	C₂²⋊C₈	—	38	4	no	—
226	C₂²⋊C₈	—	38	4	no	—
227	#97	—	38	4	no	—
228	C₄×C₈	—	2	1	no	—
229	C₄×C₈	—	2	1	no	—
230	C₄×C₈	—	2	1	no	—
231	C₄×C₈	—	2	1	no	—
232	C₄×C₈	—	38	1	no	—
233	C₄×C₈	—	38	1	no	—
234	C₄×C₈	—	38	1	no	—
235	C₄×C₈	—	38	1	no	—
236	C₄×C₈	—	38	1	no	—
237	C₄×C₈	—	38	1	no	—
238	C₄×C₈	D₇₆	1	2	yes	—
239	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
240	C₁₉	#8147	1	1	yes	characteristic, complemented, Sylow
241	C₂⁴	—	19	1	no	—
242	C₂²×C₄	—	19	1	no	—
243	C₂²×C₄	—	19	4	no	—
244	C₂²×C₄	—	19	1	no	—
245	C₂²×C₄	—	19	1	no	—
246	C₂²×C₄	—	38	1	no	—
247	C₂²×C₄	—	38	2	no	—
248	C₂²×C₄	—	38	1	no	—
249	C₂²×C₄	—	38	1	no	—
250	C₂²×C₄	—	38	1	no	—
251	C₂²×C₄	—	38	2	no	—
252	C₂²×C₄	—	38	2	no	—
253	C₂²×C₄	—	38	2	no	—
254	C₂²×C₄	—	38	2	no	—
255	C₂²×C₄	—	38	1	no	—
256	C₂²×C₄	—	38	1	no	—
257	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
258	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
259	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
260	M₁₆	—	38	16	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	19	4	no	—
269	C₂×C₈	—	19	4	no	—
270	C₂×C₈	—	38	2	no	—
271	C₂×C₈	—	38	2	no	—
272	C₂×C₈	—	38	2	no	—
273	C₂×C₈	—	38	2	no	—
274	C₂×C₈	—	38	2	no	—
275	C₂×C₈	—	38	2	no	—
276	C₂×C₈	—	38	2	no	—
277	C₂×C₈	#64515	1	4	yes	complemented
278	C₂×C₈	(C₂×D₁₉)⋊C₄	1	4	yes	—
279	C₄²	—	38	1	no	—
280	C₄²	—	38	1	no	—
281	C₄²	—	38	1	no	—
282	C₄²	—	38	1	no	—
283	C₄²	—	38	1	no	—
284	C₄²	—	38	1	no	—
285	C₄²	C₂×D₇₆	1	1	yes	characteristic
286	C₄²	#64541	1	1	yes	characteristic
287	C₄²	#64541	1	1	yes	characteristic
288	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
289	C₂³	—	19	2	no	—
290	C₂³	—	38	1	no	—
291	C₂³	—	38	1	no	—
292	C₂³	—	38	1	no	—
293	C₂³	—	38	1	no	—
294	C₂³	—	38	1	no	—
295	C₂³	—	38	1	no	—
296	C₂³	C₄²×D₁₉	1	1	yes	characteristic
297	C₂×C₄	—	19	2	no	—
298	C₂×C₄	—	38	1	no	—
299	C₂×C₄	—	38	1	no	—
300	C₂×C₄	—	38	4	no	—
301	C₂×C₄	—	38	4	no	—
302	C₂×C₄	—	38	1	no	—
303	C₂×C₄	—	38	1	no	—
304	C₂×C₄	—	38	1	no	—
305	C₂×C₄	—	38	1	no	—
306	C₂×C₄	—	38	1	no	—
307	C₂×C₄	—	38	1	no	—
308	C₂×C₄	—	76	2	no	—
309	C₂×C₄	—	76	2	no	—
310	C₂×C₄	—	76	2	no	—
311	C₂×C₄	—	76	2	no	—
312	C₂×C₄	#10597318	1	1	yes	characteristic
313	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
314	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
315	C₂×C₄	C₂×#64515	1	2	yes	—
316	C₂×C₄	#10597347	1	2	yes	—
317	C₂×C₄	M₁₆×D₁₉	1	2	yes	—
318	C₂×C₄	M₁₆×D₁₉	1	2	yes	—
319	C₂×C₄	M₁₆×D₁₉	1	2	yes	—
320	C₂×C₄	#10597353	1	2	yes	—
321	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
322	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
323	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
324	C₈	—	2	4	no	—
325	C₈	—	2	4	no	—
326	C₈	—	4	2	no	—
327	C₈	—	38	4	no	—
328	C₈	—	38	4	no	—
329	C₈	—	76	2	no	—
330	C₂²	—	38	2	no	—
331	C₂²	—	38	2	no	—
332	C₂²	—	38	2	no	—
333	C₂²	—	76	1	no	—
334	C₂²	—	76	1	no	—
335	C₂²	—	76	1	no	—
336	C₂²	—	76	1	no	—
337	C₂²	C₄×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
338	C₂²	#97×D₁₉	1	1	yes	characteristic
339	C₂²	C₄×#64515	1	1	yes	characteristic
340	C₂²	#11926496	1	1	yes	characteristic
341	C₂²	#11926496	1	1	yes	characteristic
342	C₂²	#11926498	1	1	yes	characteristic
343	C₂²	#11926501	1	1	yes	characteristic
344	C₄	—	38	2	no	—
345	C₄	—	76	1	no	—
346	C₄	—	76	1	no	—
347	C₄	#11926507	1	2	yes	—
348	C₄	#11926599	1	2	yes	—
349	C₄	#11926615	1	2	yes	—
350	C₄	#11926617	1	2	yes	—
351	C₄	#11926620	1	2	yes	—
352	C₄	#11926623	1	2	yes	—
353	C₂	—	76	2	no	—
354	C₂	#14561921	1	1	yes	characteristic
355	C₂	#14561923	1	1	yes	characteristic
356	C₂	#14561931	1	1	yes	characteristic
357	C₂	#14561939	1	1	yes	characteristic
358	C₂	#14563398	1	1	yes	characteristic
359	C₂	#14563418	1	1	yes	characteristic
360	C₂	#14563420	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₁₉
commutator, socle	C₂³×C₁₉	C₂×C₄²
Fitting	C₂×C₁₉×#322	C₂
prime radical	C₂³×D₁₉	C₄²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#322	D₁₉
hypocenter	C₁₉	#8147
LCS	G ▷ C₂³×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
4. C₂×#11926266	C₂	semidirect
5. C₁₉⋊#1273	C₂	semidirect
6. C₁₉⋊#1273	C₂	semidirect
7. C₂×C₁₉×#322	C₂	semidirect
87. C₂×C₈×C₁₉	M₁₆	semidirect
106. C₂×C₁₉⋊C₈	M₁₆	semidirect
154. C₂×#322	D₁₉	semidirect
240. C₁₉	#8147	semidirect
277. C₂×C₈	#64515	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#64515	2
C₄×(C₂×D₁₉)⋊C₈	2
C₂×#11926266	2
C₁₉⋊#1273	2
C₂×C₁₉×#322	2
#8147	19