Subgroups — #18073285

Order 4864.

460 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14572602	C₂	1	1	yes	maximal, characteristic
2	C₄×#10597294	C₂	1	1	yes	maximal, characteristic
3	#14563380	C₂	1	1	yes	maximal, characteristic, complemented
4	#14561925	C₂	1	1	yes	maximal, characteristic, complemented
5	#14561472	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561467	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1476	C₂	1	1	yes	maximal, characteristic
8	#11926675	C₄	1	2	yes	complemented
9	#11926622	C₄	1	2	yes	complemented
10	#11926616	—	2	2	no	—
11	#11926597	—	2	1	no	—
12	#11926597	C₂²	1	1	yes	characteristic
13	C₂×#10597294	C₄	1	2	yes	—
14	C₄×#64520	—	2	1	no	—
15	C₄×#64520	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₄²×Dic₁₉	C₂²	1	1	yes	characteristic
18	#11926303	C₄	1	2	yes	—
19	#11926298	—	2	2	no	—
20	#11926266	—	2	1	no	—
21	#11926265	—	2	1	no	—
22	C₄×#41×C₁₉	C₂²	1	1	yes	characteristic
23	C₁₉×#431	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
25	C₁₉×Q₈⋊C₈	C₄	1	2	yes	complemented
26	C₁₉×D₄⋊C₈	C₄	1	2	yes	complemented
27	C₂×#41×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₉	—	2	1	no	—
29	C₄×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
30	C₄×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₄×D₄×C₁₉	—	2	1	no	—
32	C₄×D₄×C₁₉	—	2	1	no	—
33	C₄×D₄×C₁₉	C₂×C₄	1	1	yes	characteristic
34	C₄×D₄×C₁₉	C₂×C₄	1	1	yes	characteristic
35	C₁₉×#117	—	2	1	no	—
36	C₁₉×#117	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₉	—	2	1	no	—
38	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
39	C₂×C₄×Dic₁₉	—	2	1	no	—
40	C₂×C₄×Dic₁₉	—	2	1	no	—
41	C₂×C₄×Dic₁₉	C₂×C₄	1	2	yes	—
42	C₂×#64520	—	2	2	no	—
43	C₂×#64520	C₂×C₄	1	2	yes	—
44	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
45	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
46	C₁₉×C₂²⋊C₈	—	2	1	no	—
47	C₄×C₈×C₁₉	—	2	1	no	—
48	C₄×C₈×C₁₉	—	2	1	no	—
49	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
50	#10597294	—	2	4	no	—
51	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
52	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
53	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
54	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
55	#10597260	—	2	1	no	—
56	#10597260	—	2	2	no	—
57	C₄×C₁₉⋊C₈	—	2	1	no	—
58	C₄×C₁₉⋊C₈	—	2	1	no	—
59	C₄×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
60	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
61	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
62	#41×C₁₉	—	2	2	no	—
63	#41×C₁₉	—	2	2	no	—
64	C₂×Q₈×C₁₉	C₄²	1	1	yes	characteristic
65	C₂×D₄×C₁₉	—	2	1	no	—
66	C₂×D₄×C₁₉	C₄²	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	—	2	1	no	—
68	C₂²×C₄×C₁₉	—	2	1	no	—
69	C₂²×C₄×C₁₉	—	2	1	no	—
70	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
73	C₂²×Dic₁₉	—	2	2	no	—
74	C₂×C₈×C₁₉	—	2	2	no	—
75	C₂×C₈×C₁₉	—	2	2	no	—
76	C₂×C₈×C₁₉	—	4	2	no	—
77	C₄⋊C₄×C₁₉	—	2	1	no	—
78	C₄⋊C₄×C₁₉	—	2	1	no	—
79	C₄⋊C₄×C₁₉	C₄²	1	1	yes	characteristic
80	C₄⋊C₄×C₁₉	C₄²	1	1	yes	characteristic
81	C₂²⋊C₄×C₁₉	—	2	1	no	—
82	C₂²⋊C₄×C₁₉	—	2	1	no	—
83	C₂²⋊C₄×C₁₉	—	2	1	no	—
84	C₄²×C₁₉	—	2	1	no	—
85	C₄²×C₁₉	—	2	1	no	—
86	C₄²×C₁₉	—	2	1	no	—
87	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₁₉	#41	1	1	yes	characteristic
91	C₄×Dic₁₉	—	2	2	no	—
92	C₄×Dic₁₉	—	4	2	no	—
93	C₄×Dic₁₉	—	4	2	no	—
94	C₄×Dic₁₉	M₁₆	1	4	yes	complemented
95	#64520	—	2	2	no	—
96	#64520	—	2	2	no	—
97	#64520	—	4	4	no	—
98	C₂×C₁₉⋊C₈	—	2	2	no	—
99	C₂×C₁₉⋊C₈	—	2	2	no	—
100	C₂×C₁₉⋊C₈	—	4	2	no	—
101	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	2	yes	—
102	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	2	yes	—
103	#8195	—	19	1	no	maximal, Sylow
104	C₂³×C₁₉	—	2	1	no	—
105	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
106	Q₈×C₁₉	—	2	1	no	—
107	Q₈×C₁₉	—	2	1	no	—
108	D₄×C₁₉	—	2	1	no	—
109	D₄×C₁₉	—	2	1	no	—
110	D₄×C₁₉	—	4	2	no	—
111	C₂×C₄×C₁₉	—	2	2	no	—
112	C₂×C₄×C₁₉	—	2	2	no	—
113	C₂×C₄×C₁₉	—	2	1	no	—
114	C₂×C₄×C₁₉	—	2	1	no	—
115	C₂×C₄×C₁₉	—	2	1	no	—
116	C₂×C₄×C₁₉	—	4	2	no	—
117	C₂×C₄×C₁₉	—	4	2	no	—
118	C₂×C₄×C₁₉	—	4	2	no	—
119	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
126	C₂×C₄×C₁₉	#131	1	2	yes	—
127	C₂×Dic₁₉	—	2	4	no	—
128	C₂×Dic₁₉	—	4	2	no	—
129	C₂×Dic₁₉	—	4	2	no	—
130	C₈×C₁₉	—	4	2	no	—
131	C₈×C₁₉	—	4	2	no	—
132	C₁₉⋊C₈	—	2	2	no	—
133	C₁₉⋊C₈	—	2	2	no	—
134	C₁₉⋊C₈	—	4	2	no	—
135	C₁₉⋊C₈	—	4	2	no	—
136	#2965	—	19	1	no	—
137	C₄×#104	—	19	1	no	—
138	#1744	—	19	1	no	—
139	#1476	D₁₉	1	1	yes	characteristic
140	#1451	—	19	1	no	—
141	#1278	—	19	1	no	—
142	#1276	—	19	1	no	—
143	C₂²×C₁₉	—	2	1	no	—
144	C₂²×C₁₉	—	2	1	no	—
145	C₂²×C₁₉	—	4	1	no	—
146	C₂²×C₁₉	—	4	1	no	—
147	C₂²×C₁₉	—	4	1	no	—
148	C₂²×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
149	C₂²×C₁₉	#435	1	2	yes	—
150	C₄×C₁₉	—	2	1	no	—
151	C₄×C₁₉	—	2	1	no	—
152	C₄×C₁₉	—	4	1	no	—
153	C₄×C₁₉	—	4	1	no	—
154	C₄×C₁₉	—	4	1	no	—
155	C₄×C₁₉	C₄×M₁₆	1	1	yes	characteristic
156	C₄×C₁₉	#405	1	1	yes	characteristic
157	C₄×C₁₉	#421	1	2	yes	—
158	C₄×C₁₉	#436	1	2	yes	—
159	C₄×C₁₉	#444	1	2	yes	—
160	Dic₁₉	—	4	4	no	—
161	C₄×#41	C₂×D₁₉	1	1	yes	characteristic
162	#446	—	19	2	no	—
163	#436	—	19	2	no	—
164	#435	—	38	2	no	—
165	#431	—	19	1	no	—
166	#431	—	38	1	no	—
167	#431	C₂×D₁₉	1	1	yes	characteristic
168	C₂×#104	—	19	2	no	—
169	C₄×M₁₆	—	19	1	no	—
170	C₄×M₁₆	—	38	1	no	—
171	C₂×C₄×C₈	—	19	1	no	—
172	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
173	C₄³	—	19	1	no	—
174	#339	—	19	2	no	—
175	#336	—	38	2	no	—
176	Q₈⋊C₈	Dic₁₉	1	2	yes	complemented
177	D₄⋊C₈	Dic₁₉	1	2	yes	complemented
178	#322	—	38	1	no	—
179	#322	—	38	1	no	—
180	C₂×C₁₉	—	2	2	no	—
181	C₂×C₁₉	—	8	1	no	—
182	C₂×C₁₉	#1753	1	1	yes	characteristic
183	C₂×C₁₉	#1759	1	1	yes	characteristic
184	C₂×C₁₉	#1765	1	1	yes	characteristic
185	C₂×#41	C₄×D₁₉	1	1	yes	characteristic
186	C₂×M₁₆	—	19	2	no	—
187	C₂×M₁₆	—	38	2	no	—
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	—	19	2	no	—
190	C₄×Q₈	C₄×D₁₉	1	1	yes	characteristic
191	C₄×Q₈	C₂×Dic₁₉	1	1	yes	characteristic
192	C₄×D₄	—	2	1	no	—
193	C₄×D₄	—	2	1	no	—
194	C₄×D₄	C₄×D₁₉	1	1	yes	characteristic
195	C₄×D₄	C₂×Dic₁₉	1	1	yes	characteristic
196	#117	—	2	1	no	—
197	#117	C₄×D₁₉	1	1	yes	characteristic
198	C₂×C₄²	—	2	1	no	—
199	C₂×C₄²	—	19	2	no	—
200	C₂×C₄²	—	38	1	no	—
201	C₂×C₄²	—	38	1	no	—
202	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
203	C₄⋊C₈	—	19	2	no	—
204	C₄⋊C₈	—	38	2	no	—
205	C₄⋊C₈	C₂×Dic₁₉	1	2	yes	—
206	#104	—	38	4	no	—
207	C₂²⋊C₈	—	2	1	no	—
208	C₂²⋊C₈	—	38	2	no	—
209	C₂²⋊C₈	—	38	1	no	—
210	#97	—	38	2	no	—
211	#97	—	38	1	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	—	2	1	no	—
214	C₄×C₈	—	19	2	no	—
215	C₄×C₈	—	19	1	no	—
216	C₄×C₈	—	19	1	no	—
217	C₄×C₈	—	38	1	no	—
218	C₄×C₈	—	38	1	no	—
219	C₄×C₈	C₂×Dic₁₉	1	2	yes	—
220	C₁₉	#8195	1	1	yes	characteristic, complemented, Sylow
221	#41	—	2	2	no	—
222	#41	—	2	2	no	—
223	C₂×Q₈	C₄×Dic₁₉	1	1	yes	characteristic
224	C₂×D₄	—	2	1	no	—
225	C₂×D₄	C₄×Dic₁₉	1	1	yes	characteristic
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	38	2	no	—
230	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
231	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
232	C₂²×C₄	#64543	1	1	yes	characteristic
233	M₁₆	—	38	2	no	—
234	M₁₆	—	38	2	no	—
235	M₁₆	—	76	4	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	19	2	no	—
240	C₂×C₈	—	19	2	no	—
241	C₂×C₈	—	38	2	no	—
242	C₂×C₈	—	38	2	no	—
243	C₂×C₈	—	76	2	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	C₄×Dic₁₉	1	1	yes	characteristic
247	C₄⋊C₄	C₄×Dic₁₉	1	1	yes	characteristic
248	C₂²⋊C₄	—	2	1	no	—
249	C₂²⋊C₄	—	2	1	no	—
250	C₂²⋊C₄	—	2	1	no	—
251	C₄²	—	2	1	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	19	4	no	—
255	C₄²	—	38	2	no	—
256	C₄²	—	76	2	no	—
257	C₄²	—	76	2	no	—
258	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
259	C₄²	D₄×D₁₉	1	1	yes	characteristic
260	C₄²	#64543	1	1	yes	characteristic
261	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
262	C₂³	—	2	1	no	—
263	C₂³	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
264	Q₈	—	2	1	no	—
265	Q₈	—	2	1	no	—
266	D₄	—	2	1	no	—
267	D₄	—	2	1	no	—
268	D₄	—	4	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	38	4	no	—
278	C₂×C₄	—	76	2	no	—
279	C₂×C₄	—	76	2	no	—
280	C₂×C₄	#10597323	1	1	yes	characteristic
281	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
282	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
283	C₂×C₄	#10597347	1	2	yes	—
284	C₂×C₄	M₁₆×D₁₉	1	2	yes	—
285	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
286	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
287	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
288	C₈	—	4	2	no	—
289	C₈	—	4	2	no	—
290	C₈	—	38	2	no	—
291	C₈	—	38	2	no	—
292	C₈	—	76	2	no	—
293	C₈	—	76	2	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	—	4	1	no	—
298	C₂²	—	4	1	no	—
299	C₂²	C₂²⋊C₄×Dic₁₉	1	1	yes	characteristic
300	C₂²	#11926618	1	2	yes	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	76	4	no	—
307	C₄	M₁₆×Dic₁₉	1	1	yes	characteristic
308	C₄	#11926499	1	1	yes	characteristic
309	C₄	#11926559	1	2	yes	—
310	C₄	#11926619	1	2	yes	—
311	C₄	#11926666	1	2	yes	—
312	C₂	—	2	2	no	—
313	C₂	—	8	1	no	—
314	C₂	#14563421	1	1	yes	characteristic
315	C₂	#14563447	1	1	yes	characteristic
316	C₂	#14563482	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1476	C₂
socle	C₂²×C₁₉	C₄×C₂²⋊C₄
prime radical	C₂×#41×C₁₉	C₂×C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1476	D₁₉
hypocenter	C₁₉	#8195
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1476

Complements

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

N	H ≅ G/N	type
3. #14563380	C₂	semidirect
4. #14561925	C₂	semidirect
5. #14561472	C₂	semidirect
6. #14561467	C₂	semidirect
8. #11926675	C₄	semidirect
9. #11926622	C₄	semidirect
25. C₁₉×Q₈⋊C₈	C₄	semidirect
26. C₁₉×D₄⋊C₈	C₄	semidirect
94. C₄×Dic₁₉	M₁₆	semidirect
176. Q₈⋊C₈	Dic₁₉	semidirect
177. D₄⋊C₈	Dic₁₉	semidirect
220. C₁₉	#8195	semidirect

Maximal subgroups

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

subgroup	index
#14572602	2
C₄×#10597294	2
#14563380	2
#14561925	2
#14561472	2
#14561467	2
C₁₉×#1476	2
#8195	19