Subgroups — #18075590

Order 4864.

542 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₇₆	C₂	1	1	yes	maximal, characteristic
2	#14563632	C₂	1	2	yes	maximal
3	C₂×#11926297	C₂	1	1	yes	maximal, characteristic, complemented
4	#14561518	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1845	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₁₉	—	2	1	no	—
7	C₂×C₄×D₇₆	C₂²	1	1	yes	characteristic
8	C₈×D₇₆	—	2	2	no	—
9	C₈×D₇₆	C₄	1	4	yes	complemented
10	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×#10597266	—	2	2	no	—
13	C₂×(C₂×D₁₉)⋊C₈	—	2	1	no	—
14	#11926306	—	2	2	no	—
15	#11926297	—	2	2	no	—
16	C₂×C₁₉×#107	—	2	1	no	—
17	C₂×C₁₉×#107	—	2	1	no	—
18	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
19	C₁₉×#384	C₂²	1	2	yes	—
20	C₂²×D₇₆	D₄	1	1	yes	characteristic
21	C₂²×C₄×D₁₉	—	2	1	no	—
22	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
25	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
26	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
27	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
28	C₂×(C₂×D₁₉)⋊C₄	—	2	1	no	—
29	C₂×C₄⋊Dic₁₉	D₄	1	1	yes	characteristic
30	C₂²×C₁₉⋊C₈	—	2	1	no	—
31	C₂²×C₁₉⋊C₈	—	2	2	no	—
32	C₂×C₈×D₁₉	—	2	4	no	—
33	C₂×C₈×D₁₉	—	4	2	no	—
34	C₂×C₈×D₁₉	—	4	2	no	—
35	C₄×D₇₆	—	2	1	no	—
36	C₄×D₇₆	—	2	1	no	—
37	C₄×D₇₆	C₂×C₄	1	2	yes	—
38	C₄×D₇₆	Q₈	1	2	yes	—
39	C₁₉×#107	—	4	2	no	—
40	C₁₉×#107	—	4	2	no	—
41	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₁₉	D₄	1	2	yes	—
43	(C₂×D₁₉)⋊C₈	—	2	4	no	—
44	#10597266	—	4	4	no	—
45	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
46	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
47	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
48	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
49	C₂³×D₁₉	—	2	1	no	—
50	C₂²×C₄×C₁₉	—	4	2	no	—
51	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
54	C₂²×Dic₁₉	—	2	1	no	—
55	C₂×D₇₆	—	2	1	no	—
56	C₂×D₇₆	—	2	1	no	—
57	C₂×D₇₆	—	2	1	no	—
58	C₂×D₇₆	C₄⋊C₄	1	2	yes	—
59	C₂×D₇₆	D₈	1	2	yes	—
60	C₂×D₇₆	Q₁₆	1	2	yes	—
61	C₂×C₄×D₁₉	—	2	2	no	—
62	C₂×C₄×D₁₉	—	2	2	no	—
63	C₂×C₄×D₁₉	—	4	1	no	—
64	C₂×C₄×D₁₉	—	4	1	no	—
65	C₂×C₄×D₁₉	—	4	1	no	—
66	C₂×C₄×D₁₉	—	4	1	no	—
67	C₂×C₈×C₁₉	—	2	1	no	—
68	C₂×C₈×C₁₉	—	2	1	no	—
69	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
70	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
71	C₂×C₈×C₁₉	D₈	1	2	yes	—
72	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
73	C₄⋊C₄×C₁₉	—	4	2	no	—
74	C₄⋊C₄×C₁₉	—	4	2	no	—
75	C₄⋊C₄×C₁₉	—	4	2	no	—
76	C₄⋊C₄×C₁₉	—	4	2	no	—
77	C₄⋊C₄×C₁₉	—	8	2	no	—
78	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
81	C₄²×C₁₉	#41	1	1	yes	characteristic
82	(C₂×D₁₉)⋊C₄	—	2	2	no	—
83	(C₂×D₁₉)⋊C₄	—	2	2	no	—
84	C₄⋊Dic₁₉	—	2	1	no	—
85	C₄⋊Dic₁₉	C₄⋊C₄	1	2	yes	—
86	C₂×C₁₉⋊C₈	—	2	2	no	—
87	C₂×C₁₉⋊C₈	—	4	1	no	—
88	C₂×C₁₉⋊C₈	—	4	1	no	—
89	C₂×C₁₉⋊C₈	—	4	2	no	—
90	C₂×C₁₉⋊C₈	—	4	2	no	—
91	C₂×C₁₉⋊C₈	—	4	2	no	—
92	C₈×D₁₉	—	4	8	no	—
93	#8644	—	19	1	no	maximal, Sylow
94	C₂³×C₁₉	#121	1	1	yes	characteristic
95	C₂²×D₁₉	—	2	2	no	—
96	C₂²×D₁₉	—	2	2	no	—
97	C₂²×D₁₉	—	2	2	no	—
98	C₂²×D₁₉	—	4	1	no	—
99	C₂²×D₁₉	—	4	1	no	—
100	C₂²×D₁₉	—	4	1	no	—
101	C₂²×D₁₉	—	4	1	no	—
102	C₂×C₄×C₁₉	—	2	1	no	—
103	C₂×C₄×C₁₉	—	2	1	no	—
104	C₂×C₄×C₁₉	—	8	2	no	—
105	C₂×C₄×C₁₉	—	8	2	no	—
106	C₂×C₄×C₁₉	—	8	2	no	—
107	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
110	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
111	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
112	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
117	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
118	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
119	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
121	C₂×Dic₁₉	—	2	2	no	—
122	C₂×Dic₁₉	—	4	1	no	—
123	C₂×Dic₁₉	—	4	1	no	—
124	D₇₆	—	2	2	no	—
125	D₇₆	—	2	2	no	—
126	D₇₆	—	2	2	no	—
127	D₇₆	#107	1	4	yes	complemented
128	C₄×D₁₉	—	4	2	no	—
129	C₄×D₁₉	—	4	2	no	—
130	C₄×D₁₉	—	4	2	no	—
131	C₄×D₁₉	—	4	2	no	—
132	C₈×C₁₉	—	2	2	no	—
133	C₈×C₁₉	#102	1	4	yes	complemented
134	C₁₉⋊C₈	—	4	2	no	—
135	C₁₉⋊C₈	—	8	2	no	—
136	C₂×C₈×D₄	—	19	1	no	—
137	#1845	D₁₉	1	1	yes	characteristic, complemented
138	#1802	—	19	2	no	—
139	C₂×#335	—	19	1	no	—
140	#1296	—	19	2	no	—
141	C₂²×C₁₉	#380	1	1	yes	characteristic
142	C₂²×C₁₉	#459	1	1	yes	characteristic
143	C₂²×C₁₉	#461	1	1	yes	characteristic
144	C₂²×C₁₉	#466	1	1	yes	characteristic
145	C₂²×C₁₉	#467	1	1	yes	characteristic
146	C₂²×C₁₉	#472	1	1	yes	characteristic
147	C₂²×C₁₉	#473	1	1	yes	characteristic
148	C₂×D₁₉	—	2	4	no	—
149	C₂×D₁₉	—	4	2	no	—
150	C₂×D₁₉	—	4	2	no	—
151	C₂×D₁₉	—	4	2	no	—
152	C₂×D₁₉	—	4	2	no	—
153	C₂×D₁₉	—	4	2	no	—
154	C₂×D₁₉	—	4	2	no	—
155	C₄×C₁₉	—	2	1	no	—
156	C₄×C₁₉	—	2	1	no	—
157	C₄×C₁₉	—	16	2	no	—
158	C₄×C₁₉	C₂×#102	1	1	yes	characteristic
159	C₄×C₁₉	#417	1	1	yes	characteristic
160	C₄×C₁₉	C₂×#107	1	1	yes	characteristic
161	C₄×C₁₉	#427	1	1	yes	characteristic
162	C₄×C₁₉	#466	1	1	yes	characteristic
163	C₄×C₁₉	#467	1	1	yes	characteristic
164	C₄×C₁₉	#474	1	1	yes	characteristic
165	C₄×C₁₉	#476	1	1	yes	characteristic
166	Dic₁₉	—	4	2	no	—
167	C₂³×C₈	—	38	1	no	—
168	C₂×C₄×D₄	—	19	1	no	—
169	C₈×D₄	—	19	4	no	—
170	C₈×D₄	—	38	2	no	—
171	C₂×#107	—	2	1	no	—
172	C₂×#107	—	2	1	no	—
173	C₂×C₄⋊C₈	—	19	1	no	—
174	C₂×C₄⋊C₈	—	19	2	no	—
175	C₂×#102	—	38	2	no	—
176	C₂×C₂²⋊C₈	—	38	1	no	—
177	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
178	#384	C₂×D₁₉	1	2	yes	—
179	#340	—	38	2	no	—
180	#335	—	38	2	no	—
181	C₂×C₁₉	#1667	1	1	yes	characteristic
182	C₂×C₁₉	#1669	1	1	yes	characteristic
183	C₂×C₁₉	#1673	1	1	yes	characteristic
184	C₂×C₁₉	#1675	1	1	yes	characteristic
185	C₂×C₁₉	#1802	1	1	yes	characteristic
186	C₂×C₁₉	#1808	1	1	yes	characteristic
187	C₂×C₁₉	#1816	1	1	yes	characteristic
188	D₁₉	—	4	4	no	—
189	C₂²×D₄	—	19	1	no	—
190	C₂³×C₄	—	38	1	no	—
191	C₂²×C₈	—	38	1	no	—
192	C₂²×C₈	—	38	4	no	—
193	C₂²×C₈	—	38	2	no	—
194	C₂²×C₈	—	76	2	no	—
195	C₂²×C₈	—	76	2	no	—
196	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
197	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
198	C₄×D₄	—	19	2	no	—
199	C₄×D₄	—	19	2	no	—
200	C₄×D₄	—	38	1	no	—
201	C₄×D₄	—	38	1	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	19	1	no	—
206	C₂×C₂²⋊C₄	—	38	1	no	—
207	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
208	#107	—	4	2	no	—
209	#107	—	4	2	no	—
210	C₄⋊C₈	—	19	2	no	—
211	C₄⋊C₈	—	38	1	no	—
212	C₄⋊C₈	—	38	2	no	—
213	C₄⋊C₈	—	38	2	no	—
214	#102	—	76	4	no	—
215	C₂²⋊C₈	—	38	4	no	—
216	C₄×C₈	C₄×D₁₉	1	2	yes	—
217	C₄×C₈	D₇₆	1	2	yes	—
218	C₁₉	#8644	1	1	yes	characteristic, complemented, Sylow
219	C₂⁴	—	38	1	no	—
220	C₂×D₄	—	19	2	no	—
221	C₂×D₄	—	19	2	no	—
222	C₂×D₄	—	19	2	no	—
223	C₂×D₄	—	38	1	no	—
224	C₂×D₄	—	38	1	no	—
225	C₂×D₄	—	38	1	no	—
226	C₂²×C₄	—	4	2	no	—
227	C₂²×C₄	—	38	1	no	—
228	C₂²×C₄	—	38	2	no	—
229	C₂²×C₄	—	38	2	no	—
230	C₂²×C₄	—	76	1	no	—
231	C₂²×C₄	—	76	1	no	—
232	C₂²×C₄	—	76	1	no	—
233	C₂²×C₄	—	76	1	no	—
234	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
235	C₂²×C₄	#64543	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	38	2	no	—
240	C₂×C₈	—	76	1	no	—
241	C₂×C₈	—	76	1	no	—
242	C₂×C₈	—	76	8	no	—
243	C₂×C₈	—	76	2	no	—
244	C₂×C₈	—	76	2	no	—
245	C₂×C₈	—	76	2	no	—
246	C₂×C₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
247	C₂×C₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
248	C₂×C₈	#64525	1	2	yes	—
249	C₂×C₈	#64527	1	2	yes	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	8	2	no	—
255	C₄⋊C₄	—	19	2	no	—
256	C₄⋊C₄	—	38	1	no	—
257	C₂²⋊C₄	—	38	2	no	—
258	C₂²⋊C₄	—	38	2	no	—
259	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
260	C₄²	C₂×D₇₆	1	1	yes	characteristic
261	C₄²	Q₈×D₁₉	1	1	yes	characteristic
262	C₄²	#64545	1	1	yes	characteristic
263	C₂³	—	38	2	no	—
264	C₂³	—	38	2	no	—
265	C₂³	—	38	2	no	—
266	C₂³	—	76	1	no	—
267	C₂³	—	76	1	no	—
268	C₂³	—	76	1	no	—
269	C₂³	—	76	1	no	—
270	C₂³	#10597384	1	1	yes	characteristic
271	D₄	—	19	4	no	—
272	D₄	—	38	2	no	—
273	D₄	—	38	2	no	—
274	D₄	—	38	2	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	8	2	no	—
278	C₂×C₄	—	8	2	no	—
279	C₂×C₄	—	8	2	no	—
280	C₂×C₄	—	38	2	no	—
281	C₂×C₄	—	76	1	no	—
282	C₂×C₄	—	76	1	no	—
283	C₂×C₄	—	76	2	no	—
284	C₂×C₄	—	76	2	no	—
285	C₂×C₄	—	76	2	no	—
286	C₂×C₄	—	76	2	no	—
287	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
288	C₂×C₄	#10597338	1	1	yes	characteristic
289	C₂×C₄	#10597341	1	1	yes	characteristic
290	C₂×C₄	#10597343	1	1	yes	characteristic
291	C₂×C₄	D₈×D₁₉	1	1	yes	characteristic
292	C₂×C₄	#10597358	1	1	yes	characteristic
293	C₂×C₄	Q₁₆×D₁₉	1	1	yes	characteristic
294	C₂×C₄	#10597365	1	1	yes	characteristic
295	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
296	C₂×C₄	C₂×#64525	1	1	yes	characteristic
297	C₂×C₄	#10597378	1	1	yes	characteristic
298	C₂×C₄	C₂×#64527	1	1	yes	characteristic
299	C₂×C₄	#10597388	1	1	yes	characteristic
300	C₂×C₄	#10597392	1	1	yes	characteristic
301	C₈	—	2	2	no	—
302	C₈	—	76	2	no	—
303	C₈	—	152	2	no	—
304	C₈	#10597266	1	4	yes	complemented
305	C₂²	—	38	4	no	—
306	C₂²	—	76	2	no	—
307	C₂²	—	76	2	no	—
308	C₂²	—	76	2	no	—
309	C₂²	—	76	2	no	—
310	C₂²	—	76	2	no	—
311	C₂²	—	76	2	no	—
312	C₂²	#11926401	1	1	yes	characteristic
313	C₂²	#11926736	1	1	yes	characteristic
314	C₂²	#11926746	1	1	yes	characteristic
315	C₂²	#11926777	1	1	yes	characteristic
316	C₂²	#11926782	1	1	yes	characteristic
317	C₂²	#11926811	1	1	yes	characteristic
318	C₂²	#11926816	1	1	yes	characteristic
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	16	2	no	—
322	C₄	—	76	2	no	—
323	C₄	C₂×#10597266	1	1	yes	characteristic
324	C₄	#11926543	1	1	yes	characteristic
325	C₄	#107×D₁₉	1	1	yes	characteristic
326	C₄	#11926581	1	1	yes	characteristic
327	C₄	#11926781	1	1	yes	characteristic
328	C₄	#11926786	1	1	yes	characteristic
329	C₄	#11926821	1	1	yes	characteristic
330	C₄	#11926831	1	1	yes	characteristic
331	C₂	—	76	4	no	—
332	C₂	#14562987	1	1	yes	characteristic
333	C₂	#14562997	1	1	yes	characteristic
334	C₂	#14563021	1	1	yes	characteristic
335	C₂	#14563031	1	1	yes	characteristic
336	C₂	#14563632	1	1	yes	characteristic
337	C₂	#14563661	1	1	yes	characteristic
338	C₂	#14563700	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597384
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1845	C₂
socle	C₂³×C₁₉	#121
prime radical	C₂²×D₇₆	D₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1845	D₁₉
hypocenter	C₁₉	#8644
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
3. C₂×#11926297	C₂	semidirect
4. #14561518	C₂	semidirect
5. C₁₉×#1845	C₂	semidirect
9. C₈×D₇₆	C₄	semidirect
127. D₇₆	#107	semidirect
133. C₈×C₁₉	#102	semidirect
137. #1845	D₁₉	semidirect
218. C₁₉	#8644	semidirect
304. C₈	#10597266	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₇₆	2
#14563632	2
C₂×#11926297	2
#14561518	2
C₁₉×#1845	2
#8644	19