Subgroups — #18088181

Order 4864.

650 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	C₂×#11926500	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926316	C₂	1	2	yes	maximal
3	#14563368	C₂	1	1	yes	maximal, characteristic
4	#14563333	C₂	1	2	yes	maximal
5	C₁₉×#1742	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	C₂²	1	1	yes	characteristic
7	C₂³×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×#10597372	C₂²	1	2	yes	—
9	#11926500	—	2	4	no	—
10	C₂×#10597260	C₄	1	2	yes	complemented
11	C₂×C₄×C₁₉⋊C₈	C₄	1	2	yes	complemented
12	#11926316	—	2	4	no	—
13	#11926311	C₄	1	4	yes	—
14	#11926298	—	2	4	no	—
15	C₂²×M₁₆×C₁₉	C₂²	1	2	yes	—
16	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
17	C₁₉×#343	C₄	1	4	yes	complemented
18	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
19	C₂×M₁₆×C₁₉	—	2	2	no	—
20	C₂×M₁₆×C₁₉	—	2	2	no	—
21	C₂×M₁₆×C₁₉	—	2	2	no	—
22	C₂×M₁₆×C₁₉	—	2	2	no	—
23	C₂×M₁₆×C₁₉	—	2	2	no	—
24	C₂×M₁₆×C₁₉	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₁₉	—	2	2	no	—
26	C₁₉×#117	—	2	2	no	—
27	C₁₉×#117	—	2	2	no	—
28	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
29	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
31	C₂×C₂²⋊Dic₁₉	C₂×C₄	1	4	yes	—
32	#10597372	—	2	2	no	—
33	#10597372	—	2	2	no	—
34	#10597372	—	2	2	no	—
35	#10597372	—	2	2	no	—
36	C₂×C₄⋊Dic₁₉	—	2	2	no	—
37	C₂×C₄×Dic₁₉	—	2	2	no	—
38	C₂×#64520	—	2	2	no	—
39	C₂×#64520	—	2	2	no	—
40	C₂×#64520	D₄	1	2	yes	—
41	C₂×#64520	Q₈	1	2	yes	—
42	C₂²×C₁₉⋊C₈	—	2	2	no	—
43	C₂²×C₁₉⋊C₈	—	2	1	no	—
44	C₂²×C₁₉⋊C₈	—	2	1	no	—
45	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
48	C₂²×C₁₉⋊C₈	D₄	1	4	yes	—
49	#10597260	—	2	4	no	—
50	C₄×C₁₉⋊C₈	—	2	4	no	—
51	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₉	C₄²	1	4	yes	—
53	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	#41	1	2	yes	—
59	C₂²×C₄×C₁₉	#41	1	2	yes	—
60	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	#41	1	2	yes	—
62	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
63	C₂²×Dic₁₉	—	2	4	no	—
64	M₁₆×C₁₉	—	2	4	no	—
65	M₁₆×C₁₉	—	2	4	no	—
66	M₁₆×C₁₉	—	4	2	no	—
67	M₁₆×C₁₉	—	4	2	no	—
68	C₂×C₈×C₁₉	—	2	4	no	—
69	C₂×C₈×C₁₉	—	4	2	no	—
70	C₂×C₈×C₁₉	—	4	2	no	—
71	C₄⋊C₄×C₁₉	—	2	4	no	—
72	C₂²⋊C₄×C₁₉	—	2	4	no	—
73	C₄²×C₁₉	—	2	2	no	—
74	C₄²×C₁₉	—	2	2	no	—
75	C₂²⋊Dic₁₉	—	2	4	no	—
76	C₂²⋊Dic₁₉	—	2	4	no	—
77	C₄⋊Dic₁₉	—	4	2	no	—
78	C₄⋊Dic₁₉	—	4	2	no	—
79	C₄×Dic₁₉	—	4	2	no	—
80	C₄×Dic₁₉	—	4	2	no	—
81	#64520	—	2	4	no	—
82	#64520	—	2	4	no	—
83	C₂×C₁₉⋊C₈	—	2	2	no	—
84	C₂×C₁₉⋊C₈	—	2	1	no	—
85	C₂×C₁₉⋊C₈	—	2	1	no	—
86	C₂×C₁₉⋊C₈	—	2	1	no	—
87	C₂×C₁₉⋊C₈	—	2	1	no	—
88	C₂×C₁₉⋊C₈	—	2	2	no	—
89	C₂×C₁₉⋊C₈	—	2	4	no	—
90	C₂×C₁₉⋊C₈	—	2	4	no	—
91	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	2	yes	—
92	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	2	yes	—
93	C₂×C₁₉⋊C₈	C₄⋊C₄	1	4	yes	—
94	#10803	—	19	1	no	maximal, Sylow
95	C₂³×C₁₉	—	4	2	no	—
96	C₂³×C₁₉	C₂×C₄²	1	1	yes	characteristic
97	C₂³×C₁₉	#117	1	2	yes	—
98	C₂³×C₁₉	C₄×D₄	1	2	yes	—
99	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
100	C₂³×C₁₉	C₄×Q₈	1	1	yes	characteristic
101	C₂×C₄×C₁₉	—	2	4	no	—
102	C₂×C₄×C₁₉	—	4	4	no	—
103	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
106	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
107	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
108	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
109	C₂×C₄×C₁₉	#117	1	2	yes	—
110	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₁₉	#121	1	2	yes	—
114	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₁₉	#123	1	2	yes	—
116	C₂×C₄×C₁₉	#123	1	2	yes	—
117	C₂×C₄×C₁₉	#124	1	2	yes	—
118	C₂×C₄×C₁₉	#125	1	2	yes	—
119	C₂×C₄×C₁₉	#126	1	2	yes	—
120	C₂×C₄×C₁₉	#126	1	2	yes	—
121	C₂×Dic₁₉	—	4	8	no	—
122	C₂×Dic₁₉	—	4	4	no	—
123	C₈×C₁₉	—	4	4	no	—
124	C₁₉⋊C₈	—	2	4	no	—
125	C₁₉⋊C₈	—	2	2	no	—
126	C₁₉⋊C₈	—	2	2	no	—
127	C₂×#405	—	19	1	no	—
128	C₂×#344	—	19	2	no	—
129	#1742	D₁₉	1	1	yes	characteristic
130	#1740	—	19	1	no	—
131	#1729	—	19	2	no	—
132	C₂²×C₁₉	—	2	4	no	—
133	C₂²×C₁₉	—	2	2	no	—
134	C₂²×C₁₉	—	4	2	no	—
135	C₂²×C₁₉	—	4	1	no	—
136	C₂²×C₁₉	—	4	1	no	—
137	C₂²×C₁₉	#376	1	1	yes	characteristic
138	C₂²×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
140	C₂²×C₁₉	#387	1	2	yes	—
141	C₂²×C₁₉	#388	1	2	yes	—
142	C₄×C₁₉	—	4	4	no	—
143	C₄×C₁₉	#380	1	1	yes	characteristic
144	C₄×C₁₉	#381	1	1	yes	characteristic
145	C₄×C₁₉	#382	1	1	yes	characteristic
146	C₄×C₁₉	#383	1	1	yes	characteristic
147	C₄×C₁₉	#387	1	2	yes	—
148	C₄×C₁₉	#388	1	2	yes	—
149	Dic₁₉	—	8	4	no	—
150	C₂²×M₁₆	—	19	1	no	—
151	C₂²×M₁₆	C₂×D₁₉	1	2	yes	—
152	C₂³×C₈	—	19	1	no	—
153	C₂×#117	—	19	2	no	—
154	C₂×#117	C₂×D₁₉	1	1	yes	characteristic
155	#405	—	38	4	no	—
156	C₂×#97	—	19	2	no	—
157	C₂×C₄×C₈	—	19	2	no	—
158	#344	—	38	4	no	—
159	#343	Dic₁₉	1	4	yes	complemented
160	#342	—	19	4	no	—
161	#336	—	38	4	no	—
162	C₂×C₁₉	—	2	2	no	—
163	C₂×C₁₉	—	2	1	no	—
164	C₂×C₁₉	—	2	1	no	—
165	C₂×C₁₉	—	2	2	no	—
166	C₂×C₁₉	#1434	1	1	yes	characteristic
167	C₂×C₁₉	#1757	1	2	yes	—
168	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	19	2	no	—
175	C₂×M₁₆	—	19	2	no	—
176	C₂×M₁₆	—	38	2	no	—
177	C₂×M₁₆	—	38	2	no	—
178	C₂×M₁₆	C₂×Dic₁₉	1	4	yes	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	19	1	no	—
181	C₂²×C₈	—	19	1	no	—
182	C₂²×C₈	—	19	2	no	—
183	C₂²×C₈	—	19	4	no	—
184	C₂²×C₈	—	38	2	no	—
185	C₂²×C₈	—	38	1	no	—
186	C₂²×C₈	—	38	1	no	—
187	#117	—	2	2	no	—
188	#117	—	2	2	no	—
189	#117	—	38	2	no	—
190	#117	—	38	2	no	—
191	#117	—	38	2	no	—
192	#117	—	38	2	no	—
193	C₂×C₄⋊C₄	—	38	2	no	—
194	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
195	C₂×C₂²⋊C₄	—	19	4	no	—
196	C₂×C₂²⋊C₄	C₄×D₁₉	1	2	yes	—
197	C₂×C₄²	—	38	2	no	—
198	C₂×C₄²	C₄×D₁₉	1	2	yes	—
199	#97	—	38	4	no	—
200	C₄×C₈	—	38	4	no	—
201	C₁₉	#10803	1	1	yes	characteristic, complemented, Sylow
202	C₂⁴	C₂×C₄×D₁₉	1	1	yes	characteristic
203	C₂²×C₄	—	38	4	no	—
204	C₂²×C₄	C₄×Dic₁₉	1	4	yes	—
205	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
206	C₂²×C₄	#64541	1	2	yes	—
207	C₂²×C₄	#64541	1	2	yes	—
208	C₂²×C₄	D₄×D₁₉	1	2	yes	—
209	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
210	C₂²×C₄	#64543	1	2	yes	—
211	C₂²×C₄	#64543	1	1	yes	characteristic
212	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
213	C₂²×C₄	#64545	1	1	yes	characteristic
214	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
215	M₁₆	—	2	4	no	—
216	M₁₆	—	2	4	no	—
217	M₁₆	—	4	2	no	—
218	M₁₆	—	4	2	no	—
219	M₁₆	—	38	4	no	—
220	M₁₆	—	38	4	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	19	2	no	—
225	C₂×C₈	—	19	2	no	—
226	C₂×C₈	—	19	4	no	—
227	C₂×C₈	—	38	1	no	—
228	C₂×C₈	—	38	1	no	—
229	C₂×C₈	—	38	2	no	—
230	C₂×C₈	—	38	2	no	—
231	C₂×C₈	—	38	4	no	—
232	C₂×C₈	—	38	1	no	—
233	C₂×C₈	—	38	1	no	—
234	C₂×C₈	—	38	4	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	76	2	no	—
237	C₄⋊C₄	—	76	2	no	—
238	C₂²⋊C₄	—	2	4	no	—
239	C₂²⋊C₄	—	38	4	no	—
240	C₂²⋊C₄	—	38	4	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	76	2	no	—
244	C₄²	—	76	2	no	—
245	C₂³	—	4	2	no	—
246	C₂³	#10597318	1	2	yes	—
247	C₂³	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
248	C₂³	#10597333	1	1	yes	characteristic
249	C₂³	#10597339	1	1	yes	characteristic
250	C₂³	C₂×C₄×Dic₁₉	1	1	yes	characteristic
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	76	8	no	—
254	C₂×C₄	—	76	4	no	—
255	C₂×C₄	#10597318	1	1	yes	characteristic
256	C₂×C₄	#10597318	1	1	yes	characteristic
257	C₂×C₄	#10597323	1	1	yes	characteristic
258	C₂×C₄	#10597325	1	2	yes	—
259	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
260	C₂×C₄	#10597329	1	2	yes	—
261	C₂×C₄	#10597330	1	2	yes	—
262	C₂×C₄	#10597331	1	2	yes	—
263	C₂×C₄	#10597335	1	2	yes	—
264	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
265	C₂×C₄	#10597338	1	1	yes	characteristic
266	C₂×C₄	#10597340	1	2	yes	—
267	C₂×C₄	#10597342	1	2	yes	—
268	C₂×C₄	#10597344	1	2	yes	—
269	C₂×C₄	#10597372	1	2	yes	—
270	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
271	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
272	C₂×C₄	Q₈×Dic₁₉	1	1	yes	characteristic
273	C₈	—	4	4	no	—
274	C₈	—	38	4	no	—
275	C₈	—	38	2	no	—
276	C₈	—	38	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	4	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	1	no	—
281	C₂²	—	4	1	no	—
282	C₂²	#11926390	1	1	yes	characteristic
283	C₂²	C₂²⋊C₄×Dic₁₉	1	1	yes	characteristic
284	C₂²	C₄⋊C₄×Dic₁₉	1	1	yes	characteristic
285	C₂²	#11926426	1	2	yes	—
286	C₂²	#11926433	1	2	yes	—
287	C₄	—	4	4	no	—
288	C₄	—	152	4	no	—
289	C₄	#11926403	1	1	yes	characteristic
290	C₄	#11926406	1	1	yes	characteristic
291	C₄	#11926408	1	1	yes	characteristic
292	C₄	#11926411	1	1	yes	characteristic
293	C₄	#11926430	1	2	yes	—
294	C₄	#11926437	1	2	yes	—
295	C₂	—	2	2	no	—
296	C₂	—	2	1	no	—
297	C₂	—	2	1	no	—
298	C₂	—	2	2	no	—
299	C₂	#14561860	1	1	yes	characteristic
300	C₂	#14563438	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₁₉
commutator	C₂³×C₁₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₁₉×#1742	C₂
socle	C₂²×C₁₉	C₄×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂²×C₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	#1742	D₁₉
hypocenter	C₁₉	#10803
LCS	G ▷ C₂³×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1742

Complements

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

N	H ≅ G/N	type
10. C₂×#10597260	C₄	semidirect
11. C₂×C₄×C₁₉⋊C₈	C₄	semidirect
17. C₁₉×#343	C₄	semidirect
159. #343	Dic₁₉	semidirect
201. C₁₉	#10803	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926500	2
C₂×#11926316	2
#14563368	2
#14563333	2
C₁₉×#1742	2
#10803	19