Subgroups — #18078192

Order 4864.

644 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	C₂²×C₄⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	#14571987	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926289	C₂	1	2	yes	maximal, complemented
4	#14561519	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1797	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₁₉⋊C₈	—	2	1	no	—
7	C₂×#10597315	C₂²	1	1	yes	characteristic
8	C₂×#10597370	—	2	1	no	—
9	C₂×C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
10	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
11	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
12	#11926450	—	2	1	no	—
13	#11926442	—	2	1	no	—
14	#11926442	C₄	1	2	yes	—
15	#11926414	C₂²	1	1	yes	characteristic
16	#11926404	—	2	1	no	—
17	#11926404	C₄	1	2	yes	—
18	#11926289	—	2	4	no	—
19	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
20	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₁₉×C₂²⋊C₈	—	2	1	no	—
22	C₁₉×#336	—	2	1	no	—
23	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
26	C₂²×Dic₃₈	—	2	1	no	—
27	C₂²×C₈×C₁₉	—	2	2	no	—
28	C₂×C₄²×C₁₉	—	2	2	no	—
29	C₂×C₄²×C₁₉	—	2	2	no	—
30	C₂×C₄²×C₁₉	C₂×C₄	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×C₂²⋊Dic₁₉	—	2	2	no	—
35	C₂×C₄⋊Dic₁₉	—	2	1	no	—
36	C₂×C₄⋊Dic₁₉	—	2	1	no	—
37	C₂×C₄⋊Dic₁₉	—	2	1	no	—
38	C₂×C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
39	C₂×C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
40	#10597370	—	4	2	no	—
41	#10597370	—	4	2	no	—
42	C₂×Dic₁₉⋊C₄	—	2	2	no	—
43	C₂²×C₁₉⋊C₈	—	2	4	no	—
44	C₂²×C₁₉⋊C₈	—	2	2	no	—
45	C₂²×C₁₉⋊C₈	—	4	2	no	—
46	C₂²×C₁₉⋊C₈	—	4	2	no	—
47	#10597315	—	2	2	no	—
48	#10597315	—	2	2	no	—
49	C₁₉×C₄⋊C₈	—	2	2	no	—
50	C₁₉×C₄⋊C₈	—	2	2	no	—
51	C₁₉×C₂²⋊C₈	—	4	2	no	—
52	C₁₉⋊#95	—	2	2	no	—
53	C₁₉⋊#95	—	2	2	no	—
54	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
55	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
56	C₄⋊(C₁₉⋊C₈)	D₄	1	4	yes	—
57	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	—	2	1	no	—
59	C₂²×C₄×C₁₉	—	2	2	no	—
60	C₂²×C₄×C₁₉	—	2	1	no	—
61	C₂²×C₄×C₁₉	—	2	2	no	—
62	C₂²×C₄×C₁₉	—	2	1	no	—
63	C₂²×C₄×C₁₉	—	2	1	no	—
64	C₂²×C₄×C₁₉	—	2	2	no	—
65	C₂²×C₄×C₁₉	—	2	2	no	—
66	C₂²×C₄×C₁₉	—	4	1	no	—
67	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	D₈	1	2	yes	—
71	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
72	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
73	C₂²×C₄×C₁₉	Q₁₆	1	2	yes	—
74	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×Dic₁₉	—	2	2	no	—
78	C₂²×Dic₁₉	—	2	2	no	—
79	C₂×Dic₃₈	—	4	1	no	—
80	C₂×Dic₃₈	—	4	1	no	—
81	C₂×Dic₃₈	—	4	1	no	—
82	C₂×Dic₃₈	—	4	1	no	—
83	C₂×Dic₃₈	—	4	1	no	—
84	C₂×Dic₃₈	—	4	1	no	—
85	C₂×C₈×C₁₉	—	4	2	no	—
86	C₂×C₈×C₁₉	—	4	2	no	—
87	C₂×C₈×C₁₉	—	4	2	no	—
88	C₄²×C₁₉	—	2	2	no	—
89	C₄²×C₁₉	—	2	1	no	—
90	C₄²×C₁₉	—	2	2	no	—
91	C₄²×C₁₉	—	2	1	no	—
92	C₄²×C₁₉	C₂×D₄	1	2	yes	—
93	C₄²×C₁₉	#41	1	2	yes	—
94	C₂²⋊Dic₁₉	—	4	4	no	—
95	C₄⋊Dic₁₉	—	2	4	no	—
96	C₄⋊Dic₁₉	—	4	2	no	—
97	C₄⋊Dic₁₉	—	4	1	no	—
98	C₄⋊Dic₁₉	—	4	1	no	—
99	C₄⋊Dic₁₉	—	4	2	no	—
100	C₄⋊Dic₁₉	—	4	2	no	—
101	Dic₁₉⋊C₄	—	4	4	no	—
102	C₂×C₁₉⋊C₈	—	2	4	no	—
103	C₂×C₁₉⋊C₈	—	4	8	no	—
104	C₂×C₁₉⋊C₈	—	4	2	no	—
105	C₂×C₁₉⋊C₈	—	4	2	no	—
106	#9131	—	19	1	no	maximal, Sylow
107	C₂³×C₁₉	—	2	1	no	—
108	C₂³×C₁₉	—	2	1	no	—
109	C₂³×C₁₉	—	2	1	no	—
110	C₂³×C₁₉	—	2	1	no	—
111	C₂³×C₁₉	#101	1	2	yes	—
112	C₂³×C₁₉	#102	1	2	yes	—
113	C₂³×C₁₉	#103	1	2	yes	—
114	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	2	no	—
122	C₂×C₄×C₁₉	—	2	2	no	—
123	C₂×C₄×C₁₉	—	2	4	no	—
124	C₂×C₄×C₁₉	—	2	4	no	—
125	C₂×C₄×C₁₉	—	2	2	no	—
126	C₂×C₄×C₁₉	—	2	2	no	—
127	C₂×C₄×C₁₉	—	2	1	no	—
128	C₂×C₄×C₁₉	—	2	1	no	—
129	C₂×C₄×C₁₉	—	2	1	no	—
130	C₂×C₄×C₁₉	—	4	4	no	—
131	C₂×C₄×C₁₉	#102	1	2	yes	—
132	C₂×C₄×C₁₉	#103	1	2	yes	—
133	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
135	C₂×C₄×C₁₉	#121	1	2	yes	—
136	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
137	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
139	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
140	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
141	C₂×C₄×C₁₉	#135	1	2	yes	—
142	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
144	C₂×Dic₁₉	—	4	2	no	—
145	C₂×Dic₁₉	—	4	2	no	—
146	C₂×Dic₁₉	—	4	2	no	—
147	C₂×Dic₁₉	—	4	2	no	—
148	C₂×Dic₁₉	—	4	2	no	—
149	C₂×Dic₁₉	—	4	2	no	—
150	Dic₃₈	—	8	1	no	—
151	Dic₃₈	—	8	1	no	—
152	Dic₃₈	—	8	1	no	—
153	Dic₃₈	—	8	1	no	—
154	C₈×C₁₉	—	8	2	no	—
155	C₁₉⋊C₈	—	4	4	no	—
156	C₂²×C₄⋊C₈	—	19	1	no	—
157	#2857	—	19	1	no	—
158	#1797	D₁₉	1	1	yes	characteristic
159	C₂×#332	—	19	2	no	—
160	#1296	—	19	2	no	—
161	C₂²×C₁₉	—	2	2	no	—
162	C₂²×C₁₉	—	2	2	no	—
163	C₂²×C₁₉	—	2	2	no	—
164	C₂²×C₁₉	—	2	2	no	—
165	C₂²×C₁₉	—	2	2	no	—
166	C₂²×C₁₉	—	2	2	no	—
167	C₂²×C₁₉	#332	1	4	yes	complemented
168	C₂²×C₁₉	#386	1	1	yes	characteristic
169	C₂²×C₁₉	C₂×#101	1	1	yes	characteristic
170	C₂²×C₁₉	#413	1	1	yes	characteristic
171	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
172	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
173	C₂²×C₁₉	#416	1	1	yes	characteristic
174	C₂²×C₁₉	#417	1	1	yes	characteristic
175	C₄×C₁₉	—	2	2	no	—
176	C₄×C₁₉	—	2	2	no	—
177	C₄×C₁₉	—	2	2	no	—
178	C₄×C₁₉	—	4	1	no	—
179	C₄×C₁₉	#447	1	1	yes	characteristic
180	C₄×C₁₉	#448	1	1	yes	characteristic
181	C₄×C₁₉	#465	1	2	yes	—
182	C₄×C₁₉	#467	1	2	yes	—
183	C₄×C₁₉	#480	1	1	yes	characteristic
184	C₄×C₁₉	#481	1	1	yes	characteristic
185	Dic₁₉	—	8	2	no	—
186	Dic₁₉	—	8	2	no	—
187	C₂³×C₈	—	38	1	no	—
188	C₂×C₄⋊Q₈	—	19	1	no	—
189	C₂×C₂²⋊Q₈	—	38	1	no	—
190	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
191	C₂×C₄⋊C₈	—	19	2	no	—
192	C₂×C₄⋊C₈	—	38	4	no	—
193	C₂×C₄⋊C₈	—	38	1	no	—
194	C₂×C₄⋊C₈	C₂×D₁₉	1	2	yes	—
195	C₂×C₂²⋊C₈	—	2	1	no	—
196	#391	—	38	1	no	—
197	#389	—	19	2	no	—
198	#389	—	38	1	no	—
199	#384	—	19	1	no	—
200	#380	—	19	2	no	—
201	#380	—	38	1	no	—
202	#336	—	2	1	no	—
203	#332	—	38	4	no	—
204	C₂×C₁₉	—	2	4	no	—
205	C₂×C₁₉	C₂×#332	1	1	yes	characteristic
206	C₂×C₁₉	#1540	1	1	yes	characteristic
207	C₂×C₁₉	#1550	1	1	yes	characteristic
208	C₂×C₁₉	#1552	1	1	yes	characteristic
209	C₂×C₁₉	#1874	1	1	yes	characteristic
210	C₂×C₁₉	#1892	1	1	yes	characteristic
211	C₂×C₁₉	#1901	1	1	yes	characteristic
212	C₂²×Q₈	—	38	1	no	—
213	C₂³×C₄	C₄×D₁₉	1	1	yes	characteristic
214	C₂³×C₄	D₇₆	1	1	yes	characteristic
215	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
216	C₂²×C₈	—	2	2	no	—
217	C₂²×C₈	—	38	4	no	—
218	C₂²×C₈	—	38	2	no	—
219	C₂²×C₈	—	76	2	no	—
220	C₂²×C₈	—	76	2	no	—
221	C₄⋊Q₈	—	38	2	no	—
222	C₄⋊Q₈	—	38	2	no	—
223	C₂²⋊Q₈	—	76	2	no	—
224	C₂²⋊Q₈	—	76	2	no	—
225	C₂×C₄⋊C₄	—	19	2	no	—
226	C₂×C₄⋊C₄	—	19	2	no	—
227	C₂×C₄⋊C₄	—	38	2	no	—
228	C₂×C₄⋊C₄	—	38	1	no	—
229	C₂×C₄⋊C₄	—	38	1	no	—
230	C₂×C₄⋊C₄	—	38	1	no	—
231	C₂×C₂²⋊C₄	—	38	2	no	—
232	C₂×C₄²	—	2	2	no	—
233	C₂×C₄²	—	2	2	no	—
234	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
235	C₂×C₄²	D₇₆	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	—	19	4	no	—
241	C₄⋊C₈	—	38	4	no	—
242	C₄⋊C₈	—	38	2	no	—
243	C₂²⋊C₈	—	4	2	no	—
244	#95	—	38	2	no	—
245	#95	—	38	2	no	—
246	C₁₉	#9131	1	1	yes	characteristic, complemented, Sylow
247	C₂⁴	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
248	C₂×Q₈	—	76	1	no	—
249	C₂×Q₈	—	76	1	no	—
250	C₂×Q₈	—	76	1	no	—
251	C₂×Q₈	—	76	1	no	—
252	C₂×Q₈	—	76	1	no	—
253	C₂×Q₈	—	76	1	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	1	no	—
257	C₂²×C₄	—	2	1	no	—
258	C₂²×C₄	—	2	1	no	—
259	C₂²×C₄	—	2	1	no	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	38	2	no	—
264	C₂²×C₄	—	38	2	no	—
265	C₂²×C₄	#64516	1	2	yes	—
266	C₂²×C₄	D₁₅₂	1	2	yes	—
267	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
268	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
269	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
270	C₂²×C₄	#64526	1	2	yes	—
271	C₂²×C₄	#64528	1	2	yes	—
272	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
273	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
274	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
275	C₂×C₈	—	4	2	no	—
276	C₂×C₈	—	4	2	no	—
277	C₂×C₈	—	4	2	no	—
278	C₂×C₈	—	38	4	no	—
279	C₂×C₈	—	76	8	no	—
280	C₂×C₈	—	76	2	no	—
281	C₂×C₈	—	76	2	no	—
282	C₄⋊C₄	—	38	4	no	—
283	C₄⋊C₄	—	76	2	no	—
284	C₄⋊C₄	—	76	2	no	—
285	C₄⋊C₄	—	76	2	no	—
286	C₄⋊C₄	—	76	4	no	—
287	C₄⋊C₄	—	76	1	no	—
288	C₄⋊C₄	—	76	1	no	—
289	C₂²⋊C₄	—	76	4	no	—
290	C₄²	—	2	2	no	—
291	C₄²	—	2	2	no	—
292	C₄²	—	2	1	no	—
293	C₄²	—	2	1	no	—
294	C₄²	D₄×D₁₉	1	2	yes	—
295	C₄²	#64543	1	2	yes	—
296	C₂³	—	2	1	no	—
297	C₂³	—	2	1	no	—
298	C₂³	—	2	1	no	—
299	C₂³	—	2	1	no	—
300	C₂³	#10597267	1	2	yes	—
301	C₂³	#10597278	1	2	yes	—
302	C₂³	#10597281	1	2	yes	—
303	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
304	Q₈	—	152	1	no	—
305	Q₈	—	152	1	no	—
306	Q₈	—	152	1	no	—
307	Q₈	—	152	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	4	no	—
319	C₂×C₄	—	2	4	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	4	4	no	—
324	C₂×C₄	—	76	2	no	—
325	C₂×C₄	—	76	2	no	—
326	C₂×C₄	—	76	2	no	—
327	C₂×C₄	—	76	2	no	—
328	C₂×C₄	—	76	2	no	—
329	C₂×C₄	—	76	2	no	—
330	C₂×C₄	#10597267	1	2	yes	—
331	C₂×C₄	#10597278	1	2	yes	—
332	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
333	C₂×C₄	#10597328	1	1	yes	characteristic
334	C₂×C₄	#10597332	1	1	yes	characteristic
335	C₂×C₄	C₂×#64516	1	1	yes	characteristic
336	C₂×C₄	C₂×D₁₅₂	1	1	yes	characteristic
337	C₂×C₄	#10597354	1	1	yes	characteristic
338	C₂×C₄	#10597355	1	1	yes	characteristic
339	C₂×C₄	C₂×#64526	1	1	yes	characteristic
340	C₂×C₄	#10597385	1	2	yes	—
341	C₂×C₄	C₂×#64528	1	1	yes	characteristic
342	C₂×C₄	#10597396	1	2	yes	—
343	C₈	—	8	2	no	—
344	C₈	—	76	4	no	—
345	C₂²	—	2	2	no	—
346	C₂²	—	2	2	no	—
347	C₂²	—	2	2	no	—
348	C₂²	—	2	2	no	—
349	C₂²	—	2	2	no	—
350	C₂²	—	2	2	no	—
351	C₂²	#11926289	1	4	yes	complemented
352	C₂²	#11926421	1	1	yes	characteristic
353	C₂²	C₂×#10597281	1	1	yes	characteristic
354	C₂²	#11926526	1	1	yes	characteristic
355	C₂²	C₂×#10597278	1	1	yes	characteristic
356	C₂²	C₂×#10597267	1	1	yes	characteristic
357	C₂²	#11926538	1	1	yes	characteristic
358	C₂²	#11926546	1	1	yes	characteristic
359	C₄	—	2	2	no	—
360	C₄	—	2	2	no	—
361	C₄	—	2	2	no	—
362	C₄	—	4	1	no	—
363	C₄	—	152	2	no	—
364	C₄	—	152	2	no	—
365	C₄	#11926680	1	1	yes	characteristic
366	C₄	#11926685	1	1	yes	characteristic
367	C₄	#11926771	1	2	yes	—
368	C₄	#11926783	1	2	yes	—
369	C₄	#11926858	1	1	yes	characteristic
370	C₄	#11926863	1	1	yes	characteristic
371	C₂	—	2	4	no	—
372	C₂	C₂×#11926289	1	1	yes	characteristic
373	C₂	#14562282	1	1	yes	characteristic
374	C₂	#14562326	1	1	yes	characteristic
375	C₂	#14562336	1	1	yes	characteristic
376	C₂	#14563950	1	1	yes	characteristic
377	C₂	#14564056	1	1	yes	characteristic
378	C₂	#14564101	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×#1797	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1797	D₁₉
hypocenter	C₁₉	#9131
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1797

Complements

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

N	H ≅ G/N	type
3. C₂×#11926289	C₂	semidirect
4. #14561519	C₂	semidirect
167. C₂²×C₁₉	#332	semidirect
246. C₁₉	#9131	semidirect
351. C₂²	#11926289	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊(C₁₉⋊C₈)	2
#14571987	2
C₂×#11926289	2
#14561519	2
C₁₉×#1797	2
#9131	19