Subgroups — #18088439

Order 4864.

982 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10597275	C₂	1	2	yes	maximal
2	C₂×#11926307	C₂	1	2	yes	maximal
3	#14561865	C₂	1	2	yes	maximal
4	C₂²×C₄×C₈×C₁₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₁₉	C₂²	1	4	yes	—
6	C₂×#10597275	—	2	4	no	—
7	C₂×#10597275	—	2	4	no	—
8	C₂×#10597275	C₄	1	8	yes	complemented
9	C₂×C₁₉⋊#95	—	2	4	no	—
10	#11926307	—	2	4	no	—
11	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₉	—	2	4	no	—
15	C₂×C₄×C₈×C₁₉	C₄	1	4	yes	complemented
16	C₂³×C₄×C₁₉	D₄	1	2	yes	—
17	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
18	C₂³×Dic₁₉	—	2	4	no	—
19	C₂²×C₈×C₁₉	—	2	2	no	—
20	C₂²×C₈×C₁₉	—	2	2	no	—
21	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₉	D₄	1	2	yes	—
25	C₂²×C₈×C₁₉	D₄	1	2	yes	—
26	C₂²×C₈×C₁₉	D₄	1	4	yes	—
27	C₂²×C₈×C₁₉	Q₈	1	2	yes	—
28	C₂²×C₈×C₁₉	Q₈	1	2	yes	—
29	C₂×C₄²×C₁₉	—	2	2	no	—
30	C₂×C₄²×C₁₉	—	2	2	no	—
31	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₉	Q₈	1	2	yes	—
33	C₂×C₄⋊Dic₁₉	—	2	4	no	—
34	C₂×C₄⋊Dic₁₉	—	2	4	no	—
35	C₂×C₄⋊Dic₁₉	—	2	4	no	—
36	C₂×C₄⋊Dic₁₉	—	2	4	no	—
37	C₂×C₄⋊Dic₁₉	—	2	4	no	—
38	C₂×C₄⋊Dic₁₉	C₂×C₄	1	8	yes	—
39	C₄×C₈×C₁₉	—	2	8	no	—
40	C₁₉⋊#95	—	4	8	no	—
41	#10597275	—	2	16	no	—
42	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₉	—	2	4	no	—
44	C₂²×C₄×C₁₉	—	2	4	no	—
45	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄×C₁₉	D₈	1	4	yes	—
47	C₂²×C₄×C₁₉	Q₁₆	1	4	yes	—
48	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	#41	1	2	yes	—
57	C₂²×C₄×C₁₉	#41	1	2	yes	—
58	C₂²×Dic₁₉	—	2	8	no	—
59	C₂²×Dic₁₉	—	4	4	no	—
60	C₂²×Dic₁₉	—	4	4	no	—
61	C₂²×Dic₁₉	—	4	8	no	—
62	C₂²×Dic₁₉	—	4	8	no	—
63	C₂×C₈×C₁₉	—	2	2	no	—
64	C₂×C₈×C₁₉	—	2	2	no	—
65	C₂×C₈×C₁₉	—	2	4	no	—
66	C₂×C₈×C₁₉	—	2	4	no	—
67	C₂×C₈×C₁₉	C₄²	1	4	yes	—
68	C₂×C₈×C₁₉	C₂²⋊C₄	1	8	yes	—
69	C₂×C₈×C₁₉	C₂²⋊C₄	1	4	yes	—
70	C₂×C₈×C₁₉	C₄⋊C₄	1	8	yes	—
71	C₂×C₈×C₁₉	C₄⋊C₄	1	4	yes	—
72	C₂×C₈×C₁₉	C₄⋊C₄	1	4	yes	—
73	C₄²×C₁₉	—	2	2	no	—
74	C₄²×C₁₉	—	2	2	no	—
75	C₄²×C₁₉	—	2	4	no	—
76	C₄⋊Dic₁₉	—	2	8	no	—
77	C₄⋊Dic₁₉	—	2	8	no	—
78	C₄⋊Dic₁₉	—	4	4	no	—
79	C₄⋊Dic₁₉	—	4	4	no	—
80	#10862	—	19	1	no	maximal, Sylow
81	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂³×C₁₉	C₄×D₄	1	2	yes	—
83	C₂³×C₁₉	#121	1	2	yes	—
84	C₂³×C₁₉	#127	1	1	yes	characteristic
85	C₂³×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
86	C₂³×C₁₉	C₂×D₈	1	2	yes	—
87	C₂³×C₁₉	C₂×Q₁₆	1	2	yes	—
88	C₂³×C₁₉	#135	1	4	yes	—
89	C₂×C₄×C₁₉	—	2	4	no	—
90	C₂×C₄×C₁₉	—	2	4	no	—
91	C₂×C₄×C₁₉	—	2	8	no	—
92	C₂×C₄×C₁₉	—	2	8	no	—
93	C₂×C₄×C₁₉	#107	1	8	yes	complemented
94	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
95	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
97	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
101	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
102	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
104	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
105	C₂×C₄×C₁₉	#121	1	2	yes	—
106	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
107	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₁₉	#124	1	2	yes	—
109	C₂×C₄×C₁₉	#125	1	2	yes	—
110	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
111	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
112	C₂×Dic₁₉	—	4	8	no	—
113	C₂×Dic₁₉	—	4	16	no	—
114	C₂×Dic₁₉	—	8	8	no	—
115	C₂×Dic₁₉	—	8	4	no	—
116	C₂×Dic₁₉	—	8	4	no	—
117	C₈×C₁₉	—	2	4	no	—
118	C₈×C₁₉	#95	1	8	yes	complemented
119	C₂²×#107	—	19	2	no	—
120	C₂²×C₄×C₈	D₁₉	1	1	yes	characteristic
121	C₂×#340	—	19	2	no	—
122	C₄⋊#95	—	19	2	no	—
123	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
124	C₂²×C₁₉	#380	1	1	yes	characteristic
125	C₂²×C₁₉	#384	1	1	yes	characteristic
126	C₂²×C₁₉	#389	1	2	yes	—
127	C₂²×C₁₉	#390	1	2	yes	—
128	C₂²×C₁₉	C₂×#107	1	2	yes	—
129	C₂²×C₁₉	#427	1	2	yes	—
130	C₂²×C₁₉	C₄×D₈	1	4	yes	—
131	C₂²×C₁₉	C₄×Q₁₆	1	4	yes	—
132	C₂²×C₁₉	#466	1	4	yes	—
133	C₂²×C₁₉	#467	1	4	yes	—
134	C₂²×C₁₉	#493	1	1	yes	characteristic
135	C₂²×C₁₉	#494	1	1	yes	characteristic
136	C₂²×C₁₉	#495	1	2	yes	—
137	C₂²×C₁₉	#499	1	2	yes	—
138	C₂²×C₁₉	#500	1	2	yes	—
139	C₄×C₁₉	—	2	8	no	—
140	C₄×C₁₉	C₂×#95	1	1	yes	characteristic
141	C₄×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
142	C₄×C₁₉	#380	1	1	yes	characteristic
143	C₄×C₁₉	#382	1	1	yes	characteristic
144	C₄×C₁₉	#389	1	2	yes	—
145	C₄×C₁₉	#391	1	2	yes	—
146	Dic₁₉	—	8	8	no	—
147	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
148	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
149	C₂²×C₄⋊C₄	—	19	4	no	—
150	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
151	C₂×#107	—	19	8	no	—
152	C₂×#107	—	38	4	no	—
153	C₂×#107	—	38	4	no	—
154	C₂×C₄×C₈	—	2	4	no	—
155	C₂×C₄×C₈	Dic₁₉	1	4	yes	complemented
156	C₂×#95	—	38	4	no	—
157	#340	—	38	4	no	—
158	C₂×C₁₉	C₄⋊#95	1	1	yes	characteristic
159	C₂×C₁₉	C₄×#107	1	2	yes	—
160	C₂×C₁₉	#1808	1	2	yes	—
161	C₂×C₁₉	#1845	1	1	yes	characteristic
162	C₂×C₁₉	#1846	1	1	yes	characteristic
163	C₂×C₁₉	#1942	1	4	yes	—
164	C₂×C₁₉	#1969	1	2	yes	—
165	C₂×C₁₉	#1970	1	2	yes	—
166	C₂³×C₄	—	38	4	no	—
167	C₂³×C₄	D₇₆	1	2	yes	—
168	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	Dic₃₈	1	2	yes	—
172	C₂²×C₈	Dic₃₈	1	2	yes	—
173	C₂²×C₈	C₄×D₁₉	1	4	yes	—
174	C₂²×C₈	D₇₆	1	2	yes	—
175	C₂²×C₈	D₇₆	1	2	yes	—
176	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
177	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
178	C₂²×C₈	C₂²⋊D₁₉	1	4	yes	—
179	C₂×C₄⋊C₄	—	19	8	no	—
180	C₂×C₄⋊C₄	—	38	4	no	—
181	C₂×C₄⋊C₄	—	38	4	no	—
182	C₂×C₄⋊C₄	—	38	4	no	—
183	C₂×C₄⋊C₄	—	38	4	no	—
184	C₂×C₄⋊C₄	—	38	4	no	—
185	C₂×C₄²	—	2	2	no	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	Dic₃₈	1	2	yes	—
188	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
189	#107	—	38	16	no	—
190	C₄×C₈	—	2	8	no	—
191	#95	—	76	8	no	—
192	C₁₉	#10862	1	1	yes	characteristic, complemented, Sylow
193	C₂⁴	C₂×D₇₆	1	1	yes	characteristic
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	38	8	no	—
197	C₂²×C₄	—	76	4	no	—
198	C₂²×C₄	—	76	4	no	—
199	C₂²×C₄	—	76	8	no	—
200	C₂²×C₄	—	76	8	no	—
201	C₂²×C₄	D₁₅₂	1	4	yes	—
202	C₂²×C₄	Dic₇₆	1	4	yes	—
203	C₂²×C₄	C₄⋊Dic₁₉	1	4	yes	—
204	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
205	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
206	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
208	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
210	C₂²×C₄	#64541	1	2	yes	—
211	C₂²×C₄	#64541	1	2	yes	—
212	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	C₄×Dic₁₉	1	4	yes	—
219	C₂×C₈	Dic₁₉⋊C₄	1	8	yes	—
220	C₂×C₈	C₄⋊Dic₁₉	1	4	yes	—
221	C₂×C₈	C₄⋊Dic₁₉	1	4	yes	—
222	C₂×C₈	(C₂×D₁₉)⋊C₄	1	8	yes	—
223	C₂×C₈	C₂²⋊Dic₁₉	1	4	yes	—
224	C₄⋊C₄	—	38	8	no	—
225	C₄⋊C₄	—	38	8	no	—
226	C₄⋊C₄	—	76	4	no	—
227	C₄⋊C₄	—	76	4	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	4	no	—
230	C₄²	—	2	2	no	—
231	C₂³	#10597315	1	1	yes	characteristic
232	C₂³	C₄×D₇₆	1	2	yes	—
233	C₂³	#10597320	1	1	yes	characteristic
234	C₂³	C₂×D₁₅₂	1	2	yes	—
235	C₂³	#10597350	1	4	yes	—
236	C₂³	C₂×Dic₇₆	1	2	yes	—
237	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
238	C₂³	#10597376	1	2	yes	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	8	no	—
243	C₂×C₄	—	76	16	no	—
244	C₂×C₄	—	76	8	no	—
245	C₂×C₄	—	152	4	no	—
246	C₂×C₄	—	152	4	no	—
247	C₂×C₄	—	152	8	no	—
248	C₂×C₄	#10597275	1	8	yes	complemented
249	C₂×C₄	C₄×Dic₃₈	1	2	yes	—
250	C₂×C₄	C₄×Dic₃₈	1	2	yes	—
251	C₂×C₄	#10597315	1	1	yes	characteristic
252	C₂×C₄	#10597315	1	1	yes	characteristic
253	C₂×C₄	#10597316	1	2	yes	—
254	C₂×C₄	C₄×D₇₆	1	2	yes	—
255	C₂×C₄	#10597321	1	2	yes	—
256	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
257	C₂×C₄	C₂×Dic₁₉⋊C₄	1	2	yes	—
258	C₂×C₄	#10597370	1	2	yes	—
259	C₂×C₄	#10597370	1	2	yes	—
260	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
261	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
262	C₂×C₄	#10597372	1	1	yes	characteristic
263	C₂×C₄	#10597372	1	1	yes	characteristic
264	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	2	yes	—
265	C₂×C₄	#10597376	1	2	yes	—
266	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
267	C₈	—	2	4	no	—
268	C₈	C₁₉⋊#95	1	8	yes	complemented
269	C₂²	C₄×C₄⋊Dic₁₉	1	1	yes	characteristic
270	C₂²	#11926404	1	1	yes	characteristic
271	C₂²	#11926414	1	1	yes	characteristic
272	C₂²	#11926442	1	2	yes	—
273	C₂²	#11926446	1	2	yes	—
274	C₂²	C₂×#10597275	1	2	yes	—
275	C₂²	#11926582	1	2	yes	—
276	C₂²	C₄×D₁₅₂	1	4	yes	—
277	C₂²	C₄×Dic₇₆	1	4	yes	—
278	C₂²	#11926780	1	4	yes	—
279	C₂²	#11926785	1	4	yes	—
280	C₂²	#11926921	1	1	yes	characteristic
281	C₂²	#11926924	1	1	yes	characteristic
282	C₂²	#11926928	1	2	yes	—
283	C₂²	#11926946	1	2	yes	—
284	C₂²	#11926949	1	2	yes	—
285	C₄	—	2	8	no	—
286	C₄	—	152	8	no	—
287	C₄	C₂×C₁₉⋊#95	1	1	yes	characteristic
288	C₄	C₄×C₄⋊Dic₁₉	1	1	yes	characteristic
289	C₄	#11926404	1	1	yes	characteristic
290	C₄	#11926410	1	1	yes	characteristic
291	C₄	#11926442	1	2	yes	—
292	C₄	#11926450	1	2	yes	—
293	C₂	#14561865	1	1	yes	characteristic
294	C₂	C₄×#10597275	1	2	yes	—
295	C₂	#14563664	1	2	yes	—
296	C₂	#14563795	1	1	yes	characteristic
297	C₂	#14563799	1	1	yes	characteristic
298	C₂	#14564368	1	4	yes	—
299	C₂	#14564496	1	2	yes	—
300	C₂	#14564501	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₇₆
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂²×C₄×C₈×C₁₉	C₂
socle, prime radical	C₂⁴×C₁₉	C₂×D₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂²×C₄×C₈	D₁₉
hypocenter	C₁₉	#10862
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
8. C₂×#10597275	C₄	semidirect
15. C₂×C₄×C₈×C₁₉	C₄	semidirect
93. C₂×C₄×C₁₉	#107	semidirect
118. C₈×C₁₉	#95	semidirect
155. C₂×C₄×C₈	Dic₁₉	semidirect
192. C₁₉	#10862	semidirect
248. C₂×C₄	#10597275	semidirect
268. C₈	C₁₉⋊#95	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#10597275	2
C₂×#11926307	2
#14561865	2
C₂²×C₄×C₈×C₁₉	2
#10862	19