Subgroups — #18075664

Order 4864.

482 conjugacy classes of subgroups.

The subgroup lattice has 308 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	#14563807	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562061	C₂	1	2	yes	maximal
4	#14561531	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1564	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	1	no	—
9	C₈×D₇₆	—	2	1	no	—
10	C₈×D₇₆	C₄	1	2	yes	—
11	C₈×D₇₆	C₄	1	2	yes	—
12	C₂×#10597276	—	2	1	no	—
13	C₂×#10597276	—	2	1	no	—
14	#11926568	C₂²	1	2	yes	—
15	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×(C₂×D₁₉)⋊C₈	—	2	1	no	—
17	#11926273	—	2	4	no	—
18	C₁₉×#431	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
20	C₁₉×#335	—	2	1	no	—
21	C₁₉×#334	—	2	1	no	—
22	C₂²×D₇₆	Q₈	1	1	yes	characteristic
23	C₂²×C₄×D₁₉	—	2	1	no	—
24	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
27	C₂×(C₂×D₁₉)⋊C₄	—	2	1	no	—
28	C₂×C₄⋊Dic₁₉	Q₈	1	1	yes	characteristic
29	C₂×#64520	—	2	2	no	—
30	C₂²×C₁₉⋊C₈	—	2	1	no	—
31	C₂×C₈×D₁₉	—	2	2	no	—
32	C₂×C₈×D₁₉	—	2	2	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	D₄	1	2	yes	—
39	C₁₉×C₄⋊C₈	—	2	2	no	—
40	C₁₉×C₂²⋊C₈	—	2	2	no	—
41	C₄×C₈×C₁₉	—	2	2	no	—
42	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
45	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
46	(C₂×D₁₉)⋊C₈	—	2	2	no	—
47	(C₂×D₁₉)⋊C₈	—	2	2	no	—
48	#10597276	—	4	2	no	—
49	#10597276	—	4	2	no	—
50	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
51	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
52	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
53	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
54	C₂³×D₁₉	—	2	1	no	—
55	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
58	C₂²×Dic₁₉	—	2	1	no	—
59	C₂×D₇₆	—	2	1	no	—
60	C₂×D₇₆	—	2	4	no	—
61	C₂×D₇₆	C₄⋊C₄	1	2	yes	—
62	C₂×C₄×D₁₉	—	2	2	no	—
63	C₂×C₄×D₁₉	—	2	2	no	—
64	C₂×C₄×D₁₉	—	4	1	no	—
65	C₂×C₄×D₁₉	—	4	1	no	—
66	C₂×C₄×D₁₉	—	4	1	no	—
67	C₂×C₄×D₁₉	—	4	1	no	—
68	C₂×C₈×C₁₉	—	2	1	no	—
69	C₂×C₈×C₁₉	—	2	1	no	—
70	C₂×C₈×C₁₉	—	4	2	no	—
71	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₉	D₈	1	2	yes	—
76	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
77	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₁₉	#41	1	1	yes	characteristic
81	(C₂×D₁₉)⋊C₄	—	2	2	no	—
82	(C₂×D₁₉)⋊C₄	—	2	2	no	—
83	C₄⋊Dic₁₉	—	2	1	no	—
84	C₄⋊Dic₁₉	C₄⋊C₄	1	2	yes	—
85	#64520	—	4	2	no	—
86	#64520	—	4	2	no	—
87	C₂×C₁₉⋊C₈	—	2	2	no	—
88	C₂×C₁₉⋊C₈	—	4	1	no	—
89	C₂×C₁₉⋊C₈	—	4	1	no	—
90	C₂×C₁₉⋊C₈	—	4	2	no	—
91	C₈×D₁₉	—	4	4	no	—
92	C₈×D₁₉	—	4	4	no	—
93	#8658	—	19	1	no	maximal, Sylow
94	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂²×D₁₉	—	2	4	no	—
96	C₂²×D₁₉	—	2	2	no	—
97	C₂²×D₁₉	—	4	2	no	—
98	C₂²×D₁₉	—	4	1	no	—
99	C₂²×D₁₉	—	4	1	no	—
100	C₂×C₄×C₁₉	—	2	2	no	—
101	C₂×C₄×C₁₉	—	4	1	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	#121	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	#137	1	1	yes	characteristic
112	C₂×Dic₁₉	—	2	2	no	—
113	C₂×Dic₁₉	—	4	1	no	—
114	C₂×Dic₁₉	—	4	1	no	—
115	D₇₆	—	2	4	no	—
116	D₇₆	—	2	2	no	—
117	D₇₆	#108	1	4	yes	complemented
118	C₄×D₁₉	—	4	2	no	—
119	C₄×D₁₉	—	4	2	no	—
120	C₄×D₁₉	—	4	2	no	—
121	C₄×D₁₉	—	4	2	no	—
122	C₈×C₁₉	—	2	1	no	—
123	C₈×C₁₉	—	2	1	no	—
124	C₈×C₁₉	—	8	2	no	—
125	C₈×C₁₉	#102	1	2	yes	—
126	C₈×C₁₉	#102	1	2	yes	—
127	C₁₉⋊C₈	—	4	2	no	—
128	C₁₉⋊C₈	—	8	2	no	—
129	C₂×C₈×D₄	—	19	1	no	—
130	#1848	—	19	1	no	—
131	#1564	D₁₉	1	1	yes	characteristic, complemented
132	#1486	—	19	2	no	—
133	#1301	—	19	2	no	—
134	C₂²×C₁₉	—	2	1	no	—
135	C₂²×C₁₉	—	2	1	no	—
136	C₂²×C₁₉	#380	1	1	yes	characteristic
137	C₂²×C₁₉	#474	1	1	yes	characteristic
138	C₂²×C₁₉	#476	1	1	yes	characteristic
139	C₂×D₁₉	—	2	4	no	—
140	C₂×D₁₉	—	4	4	no	—
141	C₂×D₁₉	—	4	4	no	—
142	C₂×D₁₉	—	4	2	no	—
143	C₂×D₁₉	—	4	2	no	—
144	C₄×C₁₉	—	2	1	no	—
145	C₄×C₁₉	—	2	1	no	—
146	C₄×C₁₉	—	2	1	no	—
147	C₄×C₁₉	C₂×#102	1	1	yes	characteristic
148	C₄×C₁₉	#417	1	1	yes	characteristic
149	C₄×C₁₉	C₂×#108	1	2	yes	—
150	C₄×C₁₉	#459	1	1	yes	characteristic
151	C₄×C₁₉	#461	1	1	yes	characteristic
152	Dic₁₉	—	4	2	no	—
153	C₂³×C₈	—	38	1	no	—
154	C₂×C₄×D₄	—	19	1	no	—
155	C₈×D₄	—	19	2	no	—
156	C₈×D₄	—	19	2	no	—
157	C₈×D₄	—	38	1	no	—
158	C₈×D₄	—	38	1	no	—
159	#431	C₂×D₁₉	1	2	yes	—
160	C₂×#108	—	38	1	no	—
161	C₂×#108	—	38	1	no	—
162	#423	—	19	2	no	—
163	C₂×C₄⋊C₈	—	19	1	no	—
164	C₂×C₂²⋊C₈	—	38	1	no	—
165	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
166	#335	—	2	1	no	—
167	#334	—	2	1	no	—
168	D₄⋊C₈	—	38	4	no	—
169	C₂×C₁₉	—	2	1	no	—
170	C₂×C₁₉	—	2	1	no	—
171	C₂×C₁₉	#1802	1	1	yes	characteristic
172	C₂×C₁₉	#1810	1	1	yes	characteristic
173	C₂×C₁₉	#1814	1	1	yes	characteristic
174	D₁₉	—	4	4	no	—
175	C₂²×D₄	—	19	1	no	—
176	C₂³×C₄	—	38	1	no	—
177	C₂×M₁₆	—	38	2	no	—
178	C₂²×C₈	—	38	1	no	—
179	C₂²×C₈	—	38	2	no	—
180	C₂²×C₈	—	38	2	no	—
181	C₂²×C₈	—	76	2	no	—
182	C₂²×C₈	—	76	2	no	—
183	C₂²×C₈	D₇₆	1	1	yes	characteristic
184	C₂²×C₈	D₇₆	1	1	yes	characteristic
185	C₄×D₄	—	19	2	no	—
186	C₄×D₄	—	19	2	no	—
187	C₄×D₄	—	38	1	no	—
188	C₄×D₄	—	38	1	no	—
189	C₂×C₄⋊C₄	—	19	1	no	—
190	C₂×C₂²⋊C₄	—	38	1	no	—
191	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
192	#108	—	76	2	no	—
193	#108	—	76	2	no	—
194	C₄⋊C₈	—	2	2	no	—
195	C₄⋊C₈	—	19	2	no	—
196	C₄⋊C₈	—	38	2	no	—
197	C₄⋊C₈	—	38	2	no	—
198	C₄⋊C₈	—	38	1	no	—
199	C₂²⋊C₈	—	2	2	no	—
200	C₂²⋊C₈	—	38	2	no	—
201	C₂²⋊C₈	—	38	2	no	—
202	C₄×C₈	—	2	2	no	—
203	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
204	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
205	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
206	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
207	C₁₉	#8658	1	1	yes	characteristic, complemented, Sylow
208	C₂⁴	—	38	1	no	—
209	C₂×D₄	—	19	2	no	—
210	C₂×D₄	—	38	4	no	—
211	C₂×D₄	—	38	1	no	—
212	C₂²×C₄	—	38	1	no	—
213	C₂²×C₄	—	38	2	no	—
214	C₂²×C₄	—	38	2	no	—
215	C₂²×C₄	—	76	1	no	—
216	C₂²×C₄	—	76	1	no	—
217	C₂²×C₄	—	76	1	no	—
218	C₂²×C₄	—	76	1	no	—
219	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
220	C₂²×C₄	#64543	1	1	yes	characteristic
221	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
222	M₁₆	—	76	2	no	—
223	M₁₆	—	76	2	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	38	2	no	—
228	C₂×C₈	—	76	1	no	—
229	C₂×C₈	—	76	1	no	—
230	C₂×C₈	—	76	4	no	—
231	C₂×C₈	—	76	4	no	—
232	C₂×C₈	—	76	2	no	—
233	C₂×C₈	#64516	1	2	yes	—
234	C₂×C₈	D₁₅₂	1	2	yes	—
235	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
236	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
237	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
238	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
239	C₄⋊C₄	—	19	2	no	—
240	C₄⋊C₄	—	38	1	no	—
241	C₂²⋊C₄	—	38	2	no	—
242	C₂²⋊C₄	—	38	2	no	—
243	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
244	C₄²	D₄×D₁₉	1	1	yes	characteristic
245	C₄²	#64545	1	1	yes	characteristic
246	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
247	C₂³	—	38	4	no	—
248	C₂³	—	38	2	no	—
249	C₂³	—	76	2	no	—
250	C₂³	—	76	1	no	—
251	C₂³	—	76	1	no	—
252	C₂³	#10597343	1	1	yes	characteristic
253	D₄	—	19	4	no	—
254	D₄	—	38	4	no	—
255	D₄	—	38	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	38	2	no	—
259	C₂×C₄	—	76	1	no	—
260	C₂×C₄	—	76	1	no	—
261	C₂×C₄	—	76	2	no	—
262	C₂×C₄	—	76	2	no	—
263	C₂×C₄	—	76	2	no	—
264	C₂×C₄	—	76	2	no	—
265	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
266	C₂×C₄	#10597338	1	1	yes	characteristic
267	C₂×C₄	#10597341	1	1	yes	characteristic
268	C₂×C₄	C₂×#64516	1	1	yes	characteristic
269	C₂×C₄	C₂×D₁₅₂	1	1	yes	characteristic
270	C₂×C₄	#10597354	1	1	yes	characteristic
271	C₂×C₄	#10597355	1	1	yes	characteristic
272	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
273	C₂×C₄	#10597384	1	1	yes	characteristic
274	C₂×C₄	#10597392	1	1	yes	characteristic
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	8	2	no	—
278	C₈	—	76	2	no	—
279	C₈	—	152	2	no	—
280	C₈	#10597278	1	2	yes	—
281	C₈	#10597278	1	2	yes	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	38	4	no	—
285	C₂²	—	76	4	no	—
286	C₂²	—	76	4	no	—
287	C₂²	—	76	2	no	—
288	C₂²	—	76	2	no	—
289	C₂²	#11926401	1	1	yes	characteristic
290	C₂²	#11926825	1	1	yes	characteristic
291	C₂²	#11926835	1	1	yes	characteristic
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	76	2	no	—
296	C₄	C₂×#10597278	1	1	yes	characteristic
297	C₄	#11926546	1	1	yes	characteristic
298	C₄	#108×D₁₉	1	2	yes	—
299	C₄	#11926740	1	1	yes	characteristic
300	C₄	#11926750	1	1	yes	characteristic
301	C₂	—	2	1	no	—
302	C₂	—	2	1	no	—
303	C₂	—	76	4	no	—
304	C₂	#14563636	1	1	yes	characteristic
305	C₂	#14563670	1	1	yes	characteristic
306	C₂	#14563686	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597341
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1564	C₂
socle	C₂²×C₁₉	#380
prime radical	C₂²×D₇₆	Q₈
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1564	D₁₉
hypocenter	C₁₉	#8658
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
2. #14563807	C₂	semidirect
4. #14561531	C₂	semidirect
5. C₁₉×#1564	C₂	semidirect
117. D₇₆	#108	semidirect
131. #1564	D₁₉	semidirect
207. C₁₉	#8658	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
#14563807	2
#14562061	2
#14561531	2
C₁₉×#1564	2
#8658	19