Subgroups — #18075620

Order 4864.

510 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926623	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×#1844	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#64515	—	2	1	no	—
7	C₂×C₄×D₇₆	C₂²	1	1	yes	characteristic
8	#11926623	—	2	2	no	—
9	#11926623	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×#106	—	2	1	no	—
17	C₂×C₁₉×#106	—	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×#64515	—	2	4	no	—
33	C₂×#64515	—	4	2	no	—
34	C₂×#64515	—	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×#106	—	4	2	no	—
40	C₁₉×#106	—	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	2	no	—
56	C₂×D₇₆	—	2	2	no	—
57	C₂×D₇₆	—	2	1	no	—
58	C₂×D₇₆	C₄⋊C₄	1	2	yes	—
59	C₂×C₄×D₁₉	—	2	2	no	—
60	C₂×C₄×D₁₉	—	2	2	no	—
61	C₂×C₄×D₁₉	—	4	1	no	—
62	C₂×C₄×D₁₉	—	4	1	no	—
63	C₂×C₄×D₁₉	—	4	1	no	—
64	C₂×C₄×D₁₉	—	4	1	no	—
65	C₂×C₈×C₁₉	—	2	1	no	—
66	C₂×C₈×C₁₉	—	2	1	no	—
67	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
68	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
69	C₂×C₈×C₁₉	D₈	1	2	yes	—
70	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
71	C₄⋊C₄×C₁₉	—	4	2	no	—
72	C₄⋊C₄×C₁₉	—	4	2	no	—
73	C₄⋊C₄×C₁₉	—	4	2	no	—
74	C₄⋊C₄×C₁₉	—	4	2	no	—
75	C₄⋊C₄×C₁₉	—	8	2	no	—
76	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
79	C₄²×C₁₉	#41	1	1	yes	characteristic
80	(C₂×D₁₉)⋊C₄	—	2	2	no	—
81	(C₂×D₁₉)⋊C₄	—	2	2	no	—
82	C₄⋊Dic₁₉	—	2	1	no	—
83	C₄⋊Dic₁₉	C₄⋊C₄	1	2	yes	—
84	C₂×C₁₉⋊C₈	—	2	2	no	—
85	C₂×C₁₉⋊C₈	—	4	1	no	—
86	C₂×C₁₉⋊C₈	—	4	1	no	—
87	C₂×C₁₉⋊C₈	—	4	2	no	—
88	C₂×C₁₉⋊C₈	—	4	2	no	—
89	C₂×C₁₉⋊C₈	—	4	2	no	—
90	#64515	—	4	8	no	—
91	#8650	—	19	1	no	maximal, Sylow
92	C₂³×C₁₉	#121	1	1	yes	characteristic
93	C₂²×D₁₉	—	2	2	no	—
94	C₂²×D₁₉	—	4	1	no	—
95	C₂²×D₁₉	—	4	1	no	—
96	C₂²×D₁₉	—	4	2	no	—
97	C₂²×D₁₉	—	4	2	no	—
98	C₂×C₄×C₁₉	—	2	2	no	—
99	C₂×C₄×C₁₉	—	8	2	no	—
100	C₂×C₄×C₁₉	—	8	2	no	—
101	C₂×C₄×C₁₉	—	8	2	no	—
102	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
105	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
106	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
109	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
111	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
112	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
113	C₂×C₄×C₁₉	#137	1	2	yes	—
114	C₂×Dic₁₉	—	2	2	no	—
115	C₂×Dic₁₉	—	4	1	no	—
116	C₂×Dic₁₉	—	4	1	no	—
117	D₇₆	—	2	4	no	—
118	D₇₆	—	4	1	no	—
119	D₇₆	—	4	1	no	—
120	C₄×D₁₉	—	4	2	no	—
121	C₄×D₁₉	—	4	2	no	—
122	C₄×D₁₉	—	4	2	no	—
123	C₄×D₁₉	—	4	2	no	—
124	C₈×C₁₉	—	2	2	no	—
125	C₈×C₁₉	#102	1	4	yes	complemented
126	C₁₉⋊C₈	—	4	2	no	—
127	C₁₉⋊C₈	—	8	2	no	—
128	C₂×#436	—	19	1	no	—
129	#1844	D₁₉	1	1	yes	characteristic, complemented
130	#1802	—	19	2	no	—
131	C₂×#335	—	19	1	no	—
132	#1296	—	19	2	no	—
133	C₂²×C₁₉	#380	1	1	yes	characteristic
134	C₂²×C₁₉	#459	1	1	yes	characteristic
135	C₂²×C₁₉	#461	1	1	yes	characteristic
136	C₂²×C₁₉	#468	1	2	yes	—
137	C₂²×C₁₉	#471	1	2	yes	—
138	C₂×D₁₉	—	4	2	no	—
139	C₂×D₁₉	—	4	2	no	—
140	C₂×D₁₉	—	4	2	no	—
141	C₂×D₁₉	—	8	2	no	—
142	C₂×D₁₉	—	8	2	no	—
143	C₄×C₁₉	—	2	1	no	—
144	C₄×C₁₉	—	2	1	no	—
145	C₄×C₁₉	—	16	2	no	—
146	C₄×C₁₉	C₂×#102	1	1	yes	characteristic
147	C₄×C₁₉	#417	1	1	yes	characteristic
148	C₄×C₁₉	#428	1	2	yes	—
149	C₄×C₁₉	#469	1	1	yes	characteristic
150	C₄×C₁₉	#470	1	1	yes	characteristic
151	C₄×C₁₉	#474	1	1	yes	characteristic
152	C₄×C₁₉	#476	1	1	yes	characteristic
153	Dic₁₉	—	4	2	no	—
154	C₂²×M₁₆	—	38	1	no	—
155	C₂×C₄×D₄	—	19	1	no	—
156	#436	—	19	4	no	—
157	#436	—	38	2	no	—
158	C₂×#106	—	2	1	no	—
159	C₂×#106	—	2	1	no	—
160	C₂×C₄⋊C₈	—	19	1	no	—
161	C₂×C₄⋊C₈	—	19	2	no	—
162	C₂×#102	—	38	2	no	—
163	C₂×C₂²⋊C₈	—	38	1	no	—
164	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
165	#384	C₂×D₁₉	1	2	yes	—
166	#340	—	38	2	no	—
167	#335	—	38	2	no	—
168	C₂×C₁₉	#1685	1	2	yes	—
169	C₂×C₁₉	#1691	1	2	yes	—
170	C₂×C₁₉	#1802	1	1	yes	characteristic
171	C₂×C₁₉	#1809	1	1	yes	characteristic
172	C₂×C₁₉	#1815	1	1	yes	characteristic
173	D₁₉	—	8	2	no	—
174	C₂²×D₄	—	19	1	no	—
175	C₂³×C₄	—	38	1	no	—
176	C₂×M₁₆	—	38	4	no	—
177	C₂×M₁₆	—	76	2	no	—
178	C₂×M₁₆	—	76	2	no	—
179	C₂²×C₈	—	38	1	no	—
180	C₂²×C₈	—	38	2	no	—
181	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
182	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
183	C₄×D₄	—	19	2	no	—
184	C₄×D₄	—	19	2	no	—
185	C₄×D₄	—	38	1	no	—
186	C₄×D₄	—	38	1	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	19	1	no	—
191	C₂×C₂²⋊C₄	—	38	1	no	—
192	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
193	#106	—	4	2	no	—
194	#106	—	4	2	no	—
195	C₄⋊C₈	—	19	2	no	—
196	C₄⋊C₈	—	38	1	no	—
197	C₄⋊C₈	—	38	2	no	—
198	C₄⋊C₈	—	38	2	no	—
199	#102	—	76	4	no	—
200	C₂²⋊C₈	—	38	4	no	—
201	C₄×C₈	C₄×D₁₉	1	2	yes	—
202	C₄×C₈	D₇₆	1	2	yes	—
203	C₁₉	#8650	1	1	yes	characteristic, complemented, Sylow
204	C₂⁴	—	38	1	no	—
205	C₂×D₄	—	19	2	no	—
206	C₂×D₄	—	38	1	no	—
207	C₂×D₄	—	38	2	no	—
208	C₂×D₄	—	38	2	no	—
209	C₂²×C₄	—	4	2	no	—
210	C₂²×C₄	—	38	1	no	—
211	C₂²×C₄	—	38	2	no	—
212	C₂²×C₄	—	38	2	no	—
213	C₂²×C₄	—	76	1	no	—
214	C₂²×C₄	—	76	1	no	—
215	C₂²×C₄	—	76	1	no	—
216	C₂²×C₄	—	76	1	no	—
217	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
218	C₂²×C₄	#64543	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
220	M₁₆	—	76	8	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	38	2	no	—
224	C₂×C₈	—	76	1	no	—
225	C₂×C₈	—	76	1	no	—
226	C₂×C₈	—	76	2	no	—
227	C₂×C₈	—	76	2	no	—
228	C₂×C₈	—	76	2	no	—
229	C₂×C₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
230	C₂×C₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
231	C₂×C₈	#64525	1	2	yes	—
232	C₂×C₈	#64527	1	2	yes	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	8	2	no	—
238	C₄⋊C₄	—	19	2	no	—
239	C₄⋊C₄	—	38	1	no	—
240	C₂²⋊C₄	—	38	2	no	—
241	C₂²⋊C₄	—	38	2	no	—
242	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
243	C₄²	C₂×D₇₆	1	1	yes	characteristic
244	C₄²	Q₈×D₁₉	1	1	yes	characteristic
245	C₄²	#64545	1	1	yes	characteristic
246	C₂³	—	38	2	no	—
247	C₂³	—	76	2	no	—
248	C₂³	—	76	2	no	—
249	C₂³	—	76	1	no	—
250	C₂³	—	76	1	no	—
251	C₂³	#10597384	1	1	yes	characteristic
252	D₄	—	38	4	no	—
253	D₄	—	76	1	no	—
254	D₄	—	76	1	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	8	2	no	—
257	C₂×C₄	—	8	2	no	—
258	C₂×C₄	—	8	2	no	—
259	C₂×C₄	—	38	2	no	—
260	C₂×C₄	—	76	1	no	—
261	C₂×C₄	—	76	1	no	—
262	C₂×C₄	—	76	2	no	—
263	C₂×C₄	—	76	2	no	—
264	C₂×C₄	—	76	2	no	—
265	C₂×C₄	—	76	2	no	—
266	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
267	C₂×C₄	#10597338	1	1	yes	characteristic
268	C₂×C₄	#10597341	1	1	yes	characteristic
269	C₂×C₄	#10597343	1	1	yes	characteristic
270	C₂×C₄	#10597360	1	2	yes	—
271	C₂×C₄	#10597361	1	2	yes	—
272	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
273	C₂×C₄	C₂×#64525	1	1	yes	characteristic
274	C₂×C₄	#10597378	1	1	yes	characteristic
275	C₂×C₄	C₂×#64527	1	1	yes	characteristic
276	C₂×C₄	#10597388	1	1	yes	characteristic
277	C₂×C₄	#10597392	1	1	yes	characteristic
278	C₈	—	2	2	no	—
279	C₈	—	76	2	no	—
280	C₈	—	152	2	no	—
281	C₈	#10597266	1	4	yes	complemented
282	C₂²	—	76	2	no	—
283	C₂²	—	76	2	no	—
284	C₂²	—	76	2	no	—
285	C₂²	—	152	2	no	—
286	C₂²	—	152	2	no	—
287	C₂²	#11926401	1	1	yes	characteristic
288	C₂²	#11926736	1	1	yes	characteristic
289	C₂²	#11926746	1	1	yes	characteristic
290	C₂²	#11926787	1	2	yes	—
291	C₂²	#11926804	1	2	yes	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	16	2	no	—
295	C₄	—	76	2	no	—
296	C₄	C₂×#10597266	1	1	yes	characteristic
297	C₄	#11926543	1	1	yes	characteristic
298	C₄	#11926583	1	2	yes	—
299	C₄	#11926798	1	1	yes	characteristic
300	C₄	#11926803	1	1	yes	characteristic
301	C₄	#11926821	1	1	yes	characteristic
302	C₄	#11926831	1	1	yes	characteristic
303	C₂	—	152	2	no	—
304	C₂	#14563097	1	2	yes	—
305	C₂	#14563131	1	2	yes	—
306	C₂	#14563632	1	1	yes	characteristic
307	C₂	#14563665	1	1	yes	characteristic
308	C₂	#14563695	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597384
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1844	C₂
socle	C₂³×C₁₉	#121
prime radical	C₂²×D₇₆	D₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1844	D₁₉
hypocenter	C₁₉	#8650
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

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×#1844	C₂	semidirect
9. #11926623	C₄	semidirect
125. C₈×C₁₉	#102	semidirect
129. #1844	D₁₉	semidirect
203. C₁₉	#8650	semidirect
281. 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×#11926623	2
#14563632	2
C₂×#11926297	2
#14561518	2
C₁₉×#1844	2
#8650	19