Subgroups — #18075450

Order 4864.

486 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563673	C₂	1	2	yes	maximal, complemented
2	#14561973	C₂	1	2	yes	maximal
3	#14561512	C₂	1	2	yes	maximal, complemented
4	C₂×C₈×D₄×C₁₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#10597276	—	2	2	no	—
6	#11926508	C₂²	1	4	yes	—
7	#11926309	—	2	2	no	—
8	#11926270	—	2	2	no	—
9	#11926268	—	2	2	no	—
10	C₂³×C₈×C₁₉	C₄	1	1	yes	characteristic
11	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
13	C₈×D₄×C₁₉	—	2	2	no	—
14	C₈×D₄×C₁₉	—	2	2	no	—
15	C₂×C₁₉×C₄⋊C₈	C₄	1	1	yes	characteristic
16	C₂×C₁₉×C₂²⋊C₈	C₄	1	1	yes	characteristic
17	C₂×C₁₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₉	C₄	1	1	yes	characteristic
19	C₂²×D₄×C₁₉	Q₈	1	1	yes	characteristic
20	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
22	C₂²×C₈×C₁₉	—	2	1	no	—
23	C₂²×C₈×C₁₉	—	2	1	no	—
24	C₂²×C₈×C₁₉	—	4	2	no	—
25	C₂²×C₈×C₁₉	—	4	2	no	—
26	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₉	D₄	1	2	yes	—
31	C₂²×C₈×C₁₉	D₄	1	2	yes	—
32	C₄×D₄×C₁₉	—	2	2	no	—
33	C₄×D₄×C₁₉	—	2	2	no	—
34	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₁₉	D₄	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₁₉	Q₈	1	1	yes	characteristic
37	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
38	C₂×#64520	—	2	4	no	—
39	C₁₉×C₄⋊C₈	—	2	2	no	—
40	C₁₉×C₂²⋊C₈	—	2	2	no	—
41	C₁₉×C₂²⋊C₈	—	2	2	no	—
42	C₄×C₈×C₁₉	—	2	1	no	—
43	C₄×C₈×C₁₉	—	2	1	no	—
44	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
45	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
46	#10597276	—	4	4	no	—
47	C₂⁴×C₁₉	C₄⋊C₄	1	1	yes	characteristic
48	C₂⁴×C₁₉	C₂×Q₈	1	1	yes	characteristic
49	C₂×D₄×C₁₉	—	2	2	no	—
50	C₂×D₄×C₁₉	—	2	2	no	—
51	C₂×D₄×C₁₉	—	2	2	no	—
52	C₂²×C₄×C₁₉	—	2	1	no	—
53	C₂²×C₄×C₁₉	—	2	1	no	—
54	C₂²×C₄×C₁₉	—	4	1	no	—
55	C₂²×C₄×C₁₉	—	4	1	no	—
56	C₂²×C₄×C₁₉	—	4	1	no	—
57	C₂²×C₄×C₁₉	—	4	1	no	—
58	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₁₉	#41	1	2	yes	—
65	C₂×C₈×C₁₉	—	2	2	no	—
66	C₂×C₈×C₁₉	—	2	4	no	—
67	C₂×C₈×C₁₉	—	2	1	no	—
68	C₂×C₈×C₁₉	—	2	1	no	—
69	C₂×C₈×C₁₉	—	4	1	no	—
70	C₂×C₈×C₁₉	—	4	2	no	—
71	C₂×C₈×C₁₉	—	4	2	no	—
72	C₂×C₈×C₁₉	—	4	2	no	—
73	C₂×C₈×C₁₉	—	4	1	no	—
74	C₂×C₈×C₁₉	—	4	1	no	—
75	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
77	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₄⋊C₄×C₁₉	—	2	2	no	—
79	C₂²⋊C₄×C₁₉	—	2	2	no	—
80	C₂²⋊C₄×C₁₉	—	2	2	no	—
81	C₄²×C₁₉	—	2	1	no	—
82	C₄²×C₁₉	—	2	1	no	—
83	#64520	—	4	4	no	—
84	#64520	—	4	4	no	—
85	C₂×C₁₉⋊C₈	—	4	4	no	—
86	#8612	—	19	1	no	maximal, Sylow
87	C₂³×C₁₉	—	2	1	no	—
88	C₂³×C₁₉	—	2	1	no	—
89	C₂³×C₁₉	—	2	1	no	—
90	C₂³×C₁₉	—	2	1	no	—
91	C₂³×C₁₉	—	4	1	no	—
92	C₂³×C₁₉	—	4	1	no	—
93	C₂³×C₁₉	—	4	1	no	—
94	C₂³×C₁₉	—	4	1	no	—
95	C₂³×C₁₉	#108	1	2	yes	complemented
96	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂³×C₁₉	C₂²⋊Q₈	1	2	yes	—
98	D₄×C₁₉	—	2	8	no	—
99	C₂×C₄×C₁₉	—	2	1	no	—
100	C₂×C₄×C₁₉	—	2	1	no	—
101	C₂×C₄×C₁₉	—	2	2	no	—
102	C₂×C₄×C₁₉	—	2	1	no	—
103	C₂×C₄×C₁₉	—	2	2	no	—
104	C₂×C₄×C₁₉	—	2	1	no	—
105	C₂×C₄×C₁₉	—	2	1	no	—
106	C₂×C₄×C₁₉	—	2	1	no	—
107	C₂×C₄×C₁₉	—	4	1	no	—
108	C₂×C₄×C₁₉	—	4	2	no	—
109	C₂×C₄×C₁₉	—	4	2	no	—
110	C₂×C₄×C₁₉	—	4	1	no	—
111	C₂×C₄×C₁₉	—	4	1	no	—
112	C₂×C₄×C₁₉	—	4	1	no	—
113	C₂×C₄×C₁₉	—	4	1	no	—
114	C₂×C₄×C₁₉	#108	1	2	yes	complemented
115	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
117	C₂×C₄×C₁₉	#121	1	2	yes	—
118	C₈×C₁₉	—	2	1	no	—
119	C₈×C₁₉	—	2	2	no	—
120	C₈×C₁₉	—	4	1	no	—
121	C₈×C₁₉	—	4	1	no	—
122	C₈×C₁₉	#99	1	2	yes	—
123	C₁₉⋊C₈	—	8	4	no	—
124	C₂×C₈×D₄	D₁₉	1	1	yes	characteristic
125	#1810	—	19	2	no	—
126	#1465	—	19	2	no	—
127	#1293	—	19	2	no	—
128	C₂²×C₁₉	—	2	1	no	—
129	C₂²×C₁₉	—	2	1	no	—
130	C₂²×C₁₉	—	2	2	no	—
131	C₂²×C₁₉	—	2	2	no	—
132	C₂²×C₁₉	—	2	2	no	—
133	C₂²×C₁₉	—	2	2	no	—
134	C₂²×C₁₉	—	4	2	no	—
135	C₂²×C₁₉	—	4	2	no	—
136	C₂²×C₁₉	—	4	2	no	—
137	C₂²×C₁₉	—	4	2	no	—
138	C₂²×C₁₉	—	4	1	no	—
139	C₂²×C₁₉	—	4	1	no	—
140	C₂²×C₁₉	#380	1	1	yes	characteristic
141	C₂²×C₁₉	C₂×#108	1	1	yes	characteristic
142	C₂²×C₁₉	C₂×#108	1	1	yes	characteristic
143	C₄×C₁₉	—	2	2	no	—
144	C₄×C₁₉	—	2	1	no	—
145	C₄×C₁₉	—	2	2	no	—
146	C₄×C₁₉	—	4	1	no	—
147	C₄×C₁₉	—	4	1	no	—
148	C₄×C₁₉	C₂×#99	1	1	yes	characteristic
149	C₄×C₁₉	#410	1	1	yes	characteristic
150	C₂³×C₈	Dic₁₉	1	1	yes	characteristic
151	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
152	C₂×C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
153	C₈×D₄	—	2	2	no	—
154	C₈×D₄	—	2	2	no	—
155	C₂×#108	—	38	2	no	—
156	C₂×C₄⋊C₈	Dic₁₉	1	1	yes	characteristic
157	#407	—	19	4	no	—
158	C₂×C₂²⋊C₈	Dic₁₉	1	1	yes	characteristic
159	C₂×C₂²⋊C₈	C₂×D₁₉	1	1	yes	characteristic
160	C₂×C₄×C₈	Dic₁₉	1	1	yes	characteristic
161	#341	—	38	2	no	—
162	#324	—	38	2	no	—
163	#323	—	38	2	no	—
164	C₂×C₁₉	—	2	1	no	—
165	C₂×C₁₉	—	2	1	no	—
166	C₂×C₁₉	—	4	2	no	—
167	C₂×C₁₉	—	4	2	no	—
168	C₂×C₁₉	#1799	1	1	yes	characteristic
169	C₂×C₁₉	#1810	1	1	yes	characteristic
170	C₂×C₁₉	#1813	1	1	yes	characteristic
171	C₂²×D₄	Dic₃₈	1	1	yes	characteristic
172	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
173	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
174	C₂×M₁₆	—	38	4	no	—
175	C₂²×C₈	—	2	1	no	—
176	C₂²×C₈	—	2	1	no	—
177	C₂²×C₈	—	4	2	no	—
178	C₂²×C₈	—	4	2	no	—
179	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
180	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
181	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
182	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
183	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
184	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
185	C₄×D₄	—	2	2	no	—
186	C₄×D₄	—	2	2	no	—
187	C₂×C₄⋊C₄	D₇₆	1	1	yes	characteristic
188	C₂×C₂²⋊C₄	Dic₃₈	1	1	yes	characteristic
189	C₂×C₂²⋊C₄	D₇₆	1	1	yes	characteristic
190	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
191	#108	—	76	4	no	—
192	C₄⋊C₈	—	2	2	no	—
193	C₂²⋊C₈	—	2	2	no	—
194	C₂²⋊C₈	—	2	2	no	—
195	C₂²⋊C₈	—	38	4	no	—
196	C₂²⋊C₈	—	38	4	no	—
197	C₄×C₈	—	2	1	no	—
198	C₄×C₈	—	2	1	no	—
199	C₁₉	#8612	1	1	yes	characteristic, complemented, Sylow
200	C₂⁴	C₄⋊Dic₁₉	1	1	yes	characteristic
201	C₂⁴	C₂×Dic₃₈	1	1	yes	characteristic
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	2	2	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	4	1	no	—
208	C₂²×C₄	—	4	1	no	—
209	C₂²×C₄	—	4	1	no	—
210	C₂²×C₄	—	4	1	no	—
211	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
212	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
213	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
215	C₂²×C₄	#64541	1	2	yes	—
216	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
218	M₁₆	—	76	4	no	—
219	M₁₆	—	76	4	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	76	4	no	—
231	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
232	C₂×C₈	C₂²⋊Dic₁₉	1	2	yes	—
233	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
234	C₄⋊C₄	—	2	2	no	—
235	C₂²⋊C₄	—	2	2	no	—
236	C₂²⋊C₄	—	2	2	no	—
237	C₄²	—	2	1	no	—
238	C₄²	—	2	1	no	—
239	C₂³	—	2	1	no	—
240	C₂³	—	2	1	no	—
241	C₂³	—	2	1	no	—
242	C₂³	—	2	1	no	—
243	C₂³	—	4	1	no	—
244	C₂³	—	4	1	no	—
245	C₂³	—	4	1	no	—
246	C₂³	—	4	1	no	—
247	C₂³	#10597276	1	2	yes	complemented
248	C₂³	#10597370	1	2	yes	—
249	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
250	D₄	—	2	8	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	#10597276	1	2	yes	complemented
267	C₂×C₄	#10597372	1	1	yes	characteristic
268	C₂×C₄	#10597376	1	2	yes	—
269	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
270	C₈	—	2	1	no	—
271	C₈	—	2	2	no	—
272	C₈	—	4	1	no	—
273	C₈	—	4	1	no	—
274	C₈	—	152	4	no	—
275	C₈	#10597291	1	2	yes	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	—	4	1	no	—
287	C₂²	—	4	1	no	—
288	C₂²	#11926404	1	1	yes	characteristic
289	C₂²	C₂×#10597276	1	1	yes	characteristic
290	C₂²	C₂×#10597276	1	1	yes	characteristic
291	C₄	—	2	2	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	2	no	—
294	C₄	—	4	1	no	—
295	C₄	—	4	1	no	—
296	C₄	C₂×#10597291	1	1	yes	characteristic
297	C₄	#11926518	1	1	yes	characteristic
298	C₂	—	2	1	no	—
299	C₂	—	2	1	no	—
300	C₂	—	4	2	no	—
301	C₂	—	4	2	no	—
302	C₂	#14563623	1	1	yes	characteristic
303	C₂	#14563673	1	1	yes	characteristic
304	C₂	#14563685	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926404
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₈×D₄×C₁₉	C₂
socle	C₂²×C₁₉	#380
prime radical	C₂²×D₄×C₁₉	Q₈
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×C₈×D₄	D₁₉
hypocenter	C₁₉	#8612
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #14563673	C₂	semidirect
3. #14561512	C₂	semidirect
95. C₂³×C₁₉	#108	semidirect
114. C₂×C₄×C₁₉	#108	semidirect
199. C₁₉	#8612	semidirect
247. C₂³	#10597276	semidirect
266. C₂×C₄	#10597276	semidirect

Maximal subgroups

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

subgroup	index
#14563673	2
#14561973	2
#14561512	2
C₂×C₈×D₄×C₁₉	2
#8612	19