Subgroups — #18075342

Order 4864.

664 conjugacy classes of subgroups.

The subgroup lattice has 380 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	#14563752	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11926287	C₂	1	2	yes	maximal
4	#14561519	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1842	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10597320	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₇₆	—	2	1	no	—
8	C₂×C₄×D₇₆	—	2	1	no	—
9	C₂×C₄×D₇₆	C₄	1	2	yes	—
10	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
11	#11926446	—	2	1	no	—
12	#11926446	—	2	1	no	—
13	#11926446	C₄	1	2	yes	—
14	#11926414	C₂²	1	1	yes	characteristic
15	#11926299	—	2	1	no	—
16	#11926299	—	2	1	no	—
17	#11926287	—	2	4	no	—
18	C₂×C₁₉×C₄⋊C₈	—	2	1	no	—
19	C₂×C₁₉×C₄⋊C₈	—	2	1	no	—
20	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
21	C₄³×C₁₉	C₂²	1	1	yes	characteristic
22	C₂²×D₇₆	—	2	1	no	—
23	C₂²×D₇₆	—	2	1	no	—
24	C₂²×D₇₆	C₂×C₄	1	2	yes	—
25	C₂²×C₄×D₁₉	—	2	2	no	—
26	C₂²×C₈×C₁₉	—	2	2	no	—
27	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₉	C₂×C₄	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	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
34	C₂×(C₂×D₁₉)⋊C₄	—	2	2	no	—
35	C₂×(C₂×D₁₉)⋊C₄	—	2	2	no	—
36	C₂×(C₂×D₁₉)⋊C₄	—	2	2	no	—
37	C₂×C₄⋊Dic₁₉	—	2	1	no	—
38	C₂×C₄⋊Dic₁₉	—	2	1	no	—
39	C₂×C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
40	C₂²×C₁₉⋊C₈	—	2	2	no	—
41	#10597320	—	2	1	no	—
42	#10597320	—	2	1	no	—
43	#10597320	D₄	1	2	yes	—
44	#10597320	Q₈	1	2	yes	—
45	C₄×D₇₆	—	4	4	no	—
46	C₄×D₇₆	—	4	2	no	—
47	C₄×D₇₆	—	4	2	no	—
48	C₄×D₇₆	—	4	4	no	—
49	C₁₉×C₄⋊C₈	—	2	2	no	—
50	C₁₉×C₄⋊C₈	—	2	2	no	—
51	C₁₉×C₄⋊C₈	—	4	2	no	—
52	C₁₉×C₄⋊C₈	—	4	2	no	—
53	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
54	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
55	C₂³×D₁₉	—	2	2	no	—
56	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
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×D₄	1	1	yes	characteristic
63	C₂²×Dic₁₉	—	2	2	no	—
64	C₂×D₇₆	—	2	2	no	—
65	C₂×D₇₆	—	2	2	no	—
66	C₂×D₇₆	—	2	2	no	—
67	C₂×D₇₆	—	2	2	no	—
68	C₂×D₇₆	—	2	2	no	—
69	C₂×D₇₆	—	4	2	no	—
70	C₂×D₇₆	—	4	2	no	—
71	C₂×D₇₆	—	4	1	no	—
72	C₂×D₇₆	—	4	1	no	—
73	C₂×D₇₆	—	4	2	no	—
74	C₂×D₇₆	—	4	1	no	—
75	C₂×D₇₆	—	4	2	no	—
76	C₂×D₇₆	—	4	1	no	—
77	C₂×D₇₆	C₄⋊C₄	1	4	yes	—
78	C₂×C₄×D₁₉	—	4	4	no	—
79	C₂×C₄×D₁₉	—	4	4	no	—
80	C₂×C₄×D₁₉	—	4	4	no	—
81	C₂×C₈×C₁₉	—	4	2	no	—
82	C₂×C₈×C₁₉	—	4	2	no	—
83	C₂×C₈×C₁₉	—	4	2	no	—
84	C₄²×C₁₉	—	2	2	no	—
85	C₄²×C₁₉	—	2	2	no	—
86	C₄²×C₁₉	—	2	1	no	—
87	C₄²×C₁₉	—	2	2	no	—
88	C₄²×C₁₉	—	2	1	no	—
89	C₄²×C₁₉	—	4	1	no	—
90	C₄²×C₁₉	D₈	1	2	yes	—
91	C₄²×C₁₉	SD₁₆	1	2	yes	—
92	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
94	C₄²×C₁₉	#41	1	1	yes	characteristic
95	C₄²×C₁₉	#41	1	1	yes	characteristic
96	(C₂×D₁₉)⋊C₄	—	4	4	no	—
97	(C₂×D₁₉)⋊C₄	—	4	4	no	—
98	(C₂×D₁₉)⋊C₄	—	4	4	no	—
99	C₄⋊Dic₁₉	—	4	1	no	—
100	C₄⋊Dic₁₉	—	4	1	no	—
101	C₄⋊Dic₁₉	—	4	2	no	—
102	C₄⋊Dic₁₉	—	4	2	no	—
103	C₂×C₁₉⋊C₈	—	4	2	no	—
104	C₂×C₁₉⋊C₈	—	4	2	no	—
105	C₂×C₁₉⋊C₈	—	4	2	no	—
106	#8588	—	19	1	no	maximal, Sylow
107	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂²×D₁₉	—	2	4	no	—
109	C₂²×D₁₉	—	4	2	no	—
110	C₂²×D₁₉	—	4	2	no	—
111	C₂²×D₁₉	—	4	2	no	—
112	C₂²×D₁₉	—	4	2	no	—
113	C₂²×D₁₉	—	4	2	no	—
114	C₂²×D₁₉	—	4	2	no	—
115	C₂×C₄×C₁₉	—	2	1	no	—
116	C₂×C₄×C₁₉	—	2	1	no	—
117	C₂×C₄×C₁₉	—	2	1	no	—
118	C₂×C₄×C₁₉	—	2	1	no	—
119	C₂×C₄×C₁₉	—	2	1	no	—
120	C₂×C₄×C₁₉	—	2	1	no	—
121	C₂×C₄×C₁₉	—	2	1	no	—
122	C₂×C₄×C₁₉	—	2	1	no	—
123	C₂×C₄×C₁₉	—	2	1	no	—
124	C₂×C₄×C₁₉	—	2	1	no	—
125	C₂×C₄×C₁₉	—	2	1	no	—
126	C₂×C₄×C₁₉	#100	1	2	yes	—
127	C₂×C₄×C₁₉	#102	1	2	yes	—
128	C₂×C₄×C₁₉	#102	1	2	yes	—
129	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
131	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
132	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
133	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
134	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
135	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
137	C₂×C₄×C₁₉	#135	1	2	yes	—
138	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
140	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
141	C₂×Dic₁₉	—	4	2	no	—
142	C₂×Dic₁₉	—	4	2	no	—
143	C₂×Dic₁₉	—	4	2	no	—
144	D₇₆	—	4	4	no	—
145	D₇₆	—	4	4	no	—
146	D₇₆	—	4	2	no	—
147	D₇₆	—	4	2	no	—
148	D₇₆	—	4	4	no	—
149	D₇₆	—	4	2	no	—
150	D₇₆	—	4	2	no	—
151	D₇₆	—	4	4	no	—
152	C₄×D₁₉	—	8	8	no	—
153	C₈×C₁₉	—	8	2	no	—
154	C₁₉⋊C₈	—	8	2	no	—
155	C₄×#127	—	19	1	no	—
156	#1842	D₁₉	1	1	yes	characteristic, complemented
157	#1831	—	19	1	no	—
158	C₂×#331	—	19	2	no	—
159	#1296	—	19	2	no	—
160	C₂²×C₁₉	#380	1	1	yes	characteristic
161	C₂²×C₁₉	C₂×#100	1	1	yes	characteristic
162	C₂²×C₁₉	#413	1	1	yes	characteristic
163	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
164	C₂²×C₁₉	#416	1	1	yes	characteristic
165	C₂²×C₁₉	#417	1	1	yes	characteristic
166	C₂²×C₁₉	#418	1	1	yes	characteristic
167	C₂×D₁₉	—	4	4	no	—
168	C₂×D₁₉	—	4	4	no	—
169	C₂×D₁₉	—	4	4	no	—
170	C₂×D₁₉	—	8	2	no	—
171	C₂×D₁₉	—	8	2	no	—
172	C₂×D₁₉	—	8	2	no	—
173	C₂×D₁₉	—	8	2	no	—
174	C₄×C₁₉	—	2	2	no	—
175	C₄×C₁₉	—	2	1	no	—
176	C₄×C₁₉	—	2	1	no	—
177	C₄×C₁₉	—	2	2	no	—
178	C₄×C₁₉	—	2	2	no	—
179	C₄×C₁₉	—	4	1	no	—
180	C₄×C₁₉	#459	1	1	yes	characteristic
181	C₄×C₁₉	#461	1	1	yes	characteristic
182	C₄×C₁₉	#463	1	2	yes	—
183	C₄×C₁₉	#474	1	1	yes	characteristic
184	C₄×C₁₉	#476	1	1	yes	characteristic
185	C₄×C₁₉	#478	1	2	yes	—
186	Dic₁₉	—	8	2	no	—
187	C₂×#127	—	19	1	no	—
188	C₂×C₄×D₄	—	19	2	no	—
189	C₂×C₄×D₄	—	38	1	no	—
190	C₂×C₄×D₄	—	38	1	no	—
191	C₂×C₄⋊C₈	—	2	1	no	—
192	C₂×C₄⋊C₈	—	2	1	no	—
193	C₂×C₄⋊C₈	—	19	2	no	—
194	C₂×C₄⋊C₈	C₂×D₁₉	1	2	yes	—
195	#390	—	19	2	no	—
196	#390	—	38	1	no	—
197	#390	—	38	1	no	—
198	#384	—	19	1	no	—
199	C₄³	C₂×D₁₉	1	1	yes	characteristic
200	#336	—	38	1	no	—
201	#336	—	38	1	no	—
202	#331	—	38	4	no	—
203	C₂×C₁₉	#1540	1	1	yes	characteristic
204	C₂×C₁₉	#1545	1	1	yes	characteristic
205	C₂×C₁₉	#1551	1	1	yes	characteristic
206	C₂×C₁₉	#1557	1	1	yes	characteristic
207	C₂×C₁₉	#1800	1	1	yes	characteristic
208	C₂×C₁₉	#1802	1	1	yes	characteristic
209	C₂×C₁₉	#1804	1	1	yes	characteristic
210	D₁₉	—	8	4	no	—
211	C₂²×D₄	—	19	2	no	—
212	C₂²×D₄	—	38	1	no	—
213	C₂²×D₄	—	38	1	no	—
214	C₂³×C₄	—	38	2	no	—
215	C₂²×C₈	—	2	2	no	—
216	C₂²×C₈	—	38	2	no	—
217	#127	—	19	2	no	—
218	#127	—	19	2	no	—
219	#127	—	38	1	no	—
220	#127	—	38	1	no	—
221	C₄×D₄	—	76	4	no	—
222	C₄×D₄	—	76	4	no	—
223	C₄×D₄	—	76	2	no	—
224	C₄×D₄	—	76	2	no	—
225	C₂×C₄⋊C₄	—	19	2	no	—
226	C₂×C₄⋊C₄	—	38	1	no	—
227	C₂×C₄⋊C₄	—	38	1	no	—
228	C₂×C₂²⋊C₄	—	38	2	no	—
229	C₂×C₂²⋊C₄	—	38	2	no	—
230	C₂×C₂²⋊C₄	—	38	2	no	—
231	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
232	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
233	C₂×C₄²	D₇₆	1	1	yes	characteristic
234	C₂×C₄²	D₇₆	1	1	yes	characteristic
235	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
236	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
237	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
238	C₄⋊C₈	—	2	2	no	—
239	C₄⋊C₈	—	2	2	no	—
240	C₄⋊C₈	—	4	2	no	—
241	C₄⋊C₈	—	4	2	no	—
242	C₄⋊C₈	—	38	2	no	—
243	C₄⋊C₈	—	38	2	no	—
244	C₁₉	#8588	1	1	yes	characteristic, complemented, Sylow
245	C₂⁴	—	38	2	no	—
246	C₂×D₄	—	19	4	no	—
247	C₂×D₄	—	38	2	no	—
248	C₂×D₄	—	38	2	no	—
249	C₂×D₄	—	38	2	no	—
250	C₂×D₄	—	38	2	no	—
251	C₂×D₄	—	38	2	no	—
252	C₂×D₄	—	76	2	no	—
253	C₂×D₄	—	76	2	no	—
254	C₂×D₄	—	76	1	no	—
255	C₂×D₄	—	76	1	no	—
256	C₂×D₄	—	76	1	no	—
257	C₂×D₄	—	76	1	no	—
258	C₂×D₄	—	76	2	no	—
259	C₂×D₄	—	76	2	no	—
260	C₂²×C₄	—	38	2	no	—
261	C₂²×C₄	—	76	4	no	—
262	C₂²×C₄	—	76	4	no	—
263	C₂²×C₄	—	76	4	no	—
264	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
265	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
266	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
267	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
268	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
269	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
270	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
271	C₂×C₈	—	4	2	no	—
272	C₂×C₈	—	4	2	no	—
273	C₂×C₈	—	4	2	no	—
274	C₂×C₈	—	76	2	no	—
275	C₂×C₈	—	76	2	no	—
276	C₂×C₈	—	76	2	no	—
277	C₄⋊C₄	—	76	2	no	—
278	C₄⋊C₄	—	76	1	no	—
279	C₄⋊C₄	—	76	1	no	—
280	C₄⋊C₄	—	76	2	no	—
281	C₂²⋊C₄	—	76	4	no	—
282	C₂²⋊C₄	—	76	4	no	—
283	C₂²⋊C₄	—	76	4	no	—
284	C₄²	—	2	1	no	—
285	C₄²	—	2	1	no	—
286	C₄²	—	2	2	no	—
287	C₄²	—	2	2	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	4	1	no	—
290	C₄²	#64525	1	2	yes	—
291	C₄²	#64527	1	2	yes	—
292	C₄²	D₄×D₁₉	1	1	yes	characteristic
293	C₄²	#64543	1	1	yes	characteristic
294	C₄²	Q₈×D₁₉	1	1	yes	characteristic
295	C₄²	#64545	1	1	yes	characteristic
296	C₂³	—	38	4	no	—
297	C₂³	—	76	2	no	—
298	C₂³	—	76	2	no	—
299	C₂³	—	76	2	no	—
300	C₂³	—	76	2	no	—
301	C₂³	—	76	2	no	—
302	C₂³	—	76	2	no	—
303	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
304	D₄	—	76	4	no	—
305	D₄	—	76	4	no	—
306	D₄	—	76	2	no	—
307	D₄	—	76	2	no	—
308	D₄	—	76	4	no	—
309	D₄	—	76	2	no	—
310	D₄	—	76	2	no	—
311	D₄	—	76	4	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	76	2	no	—
324	C₂×C₄	—	76	2	no	—
325	C₂×C₄	—	76	2	no	—
326	C₂×C₄	—	152	8	no	—
327	C₂×C₄	#10597266	1	2	yes	—
328	C₂×C₄	#10597266	1	2	yes	—
329	C₂×C₄	#10597280	1	2	yes	—
330	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
331	C₂×C₄	#10597338	1	1	yes	characteristic
332	C₂×C₄	#10597341	1	1	yes	characteristic
333	C₂×C₄	#10597343	1	1	yes	characteristic
334	C₂×C₄	#10597350	1	2	yes	—
335	C₂×C₄	#10597354	1	2	yes	—
336	C₂×C₄	C₂×#64525	1	1	yes	characteristic
337	C₂×C₄	#10597378	1	1	yes	characteristic
338	C₂×C₄	#10597384	1	1	yes	characteristic
339	C₂×C₄	C₂×#64527	1	1	yes	characteristic
340	C₂×C₄	#10597388	1	1	yes	characteristic
341	C₂×C₄	#10597392	1	1	yes	characteristic
342	C₈	—	8	2	no	—
343	C₈	—	152	2	no	—
344	C₂²	—	76	4	no	—
345	C₂²	—	76	4	no	—
346	C₂²	—	76	4	no	—
347	C₂²	—	152	2	no	—
348	C₂²	—	152	2	no	—
349	C₂²	—	152	2	no	—
350	C₂²	—	152	2	no	—
351	C₂²	#11926401	1	1	yes	characteristic
352	C₂²	C₂×#10597280	1	1	yes	characteristic
353	C₂²	#11926526	1	1	yes	characteristic
354	C₂²	C₂×#10597266	1	1	yes	characteristic
355	C₂²	#11926541	1	1	yes	characteristic
356	C₂²	#11926543	1	1	yes	characteristic
357	C₂²	#11926550	1	1	yes	characteristic
358	C₄	—	2	2	no	—
359	C₄	—	2	2	no	—
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	2	2	no	—
363	C₄	—	4	1	no	—
364	C₄	—	152	2	no	—
365	C₄	#11926736	1	1	yes	characteristic
366	C₄	#11926746	1	1	yes	characteristic
367	C₄	#11926760	1	2	yes	—
368	C₄	#11926821	1	1	yes	characteristic
369	C₄	#11926831	1	1	yes	characteristic
370	C₄	#11926845	1	2	yes	—
371	C₂	—	152	4	no	—
372	C₂	#14562285	1	1	yes	characteristic
373	C₂	#14562304	1	1	yes	characteristic
374	C₂	#14562328	1	1	yes	characteristic
375	C₂	#14562355	1	1	yes	characteristic
376	C₂	#14563624	1	1	yes	characteristic
377	C₂	#14563632	1	1	yes	characteristic
378	C₂	#14563646	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₁₉)⋊C₄
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1842	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂×#10597320	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1842	D₁₉
hypocenter	C₁₉	#8588
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
2. #14563752	C₂	semidirect
4. #14561519	C₂	semidirect
5. C₁₉×#1842	C₂	semidirect
156. #1842	D₁₉	semidirect
244. C₁₉	#8588	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
#14563752	2
C₂×#11926287	2
#14561519	2
C₁₉×#1842	2
#8588	19