Subgroups — #18091190

Order 4864.

722 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926587	C₂	1	2	yes	maximal
2	C₂²×#10597275	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926312	C₂	1	2	yes	maximal
4	C₂×#11926309	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	C₂²	1	2	yes	—
7	C₂²×C₄⋊Dic₁₉	C₂²	1	2	yes	—
8	#11926587	—	2	4	no	—
9	#11926587	—	2	4	no	—
10	C₂×#10597275	—	2	4	no	—
11	C₂×#10597275	—	2	2	no	—
12	C₂×#10597264	—	2	2	no	—
13	C₂×#10597265	—	2	2	no	—
14	#11926312	—	2	4	no	—
15	#11926309	C₄	1	4	yes	complemented
16	C₂²×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
18	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
19	C₁₉×#336	—	2	2	no	—
20	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
21	C₂³×Dic₁₉	—	2	2	no	—
22	C₂×M₁₆×C₁₉	—	2	1	no	—
23	C₂×M₁₆×C₁₉	—	2	2	no	—
24	C₂×M₁₆×C₁₉	—	2	2	no	—
25	C₂×M₁₆×C₁₉	Q₈	1	2	yes	—
26	C₂²×C₈×C₁₉	—	2	2	no	—
27	C₂²×C₈×C₁₉	—	2	2	no	—
28	C₂²×C₈×C₁₉	—	2	1	no	—
29	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₁₉	D₄	1	2	yes	—
31	C₂²×C₈×C₁₉	Q₈	1	2	yes	—
32	C₁₉×#117	—	2	2	no	—
33	C₁₉×#117	—	2	2	no	—
34	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
37	C₂×C₄²×C₁₉	—	2	1	no	—
38	C₂×C₄⋊Dic₁₉	—	2	2	no	—
39	C₂×C₄⋊Dic₁₉	—	2	2	no	—
40	C₂×C₄⋊Dic₁₉	—	2	2	no	—
41	C₂×C₄⋊Dic₁₉	—	2	2	no	—
42	C₂×C₄⋊Dic₁₉	—	2	2	no	—
43	C₂×C₄⋊Dic₁₉	D₄	1	4	yes	—
44	C₂×#64520	—	2	2	no	—
45	C₂×#64520	—	2	2	no	—
46	C₂×#64520	—	2	2	no	—
47	C₂×#64520	C₂×C₄	1	4	yes	—
48	C₂×#64520	D₄	1	4	yes	—
49	C₂×#64520	Q₈	1	4	yes	—
50	C₂²×C₁₉⋊C₈	—	2	2	no	—
51	#10597275	—	2	8	no	—
52	#10597265	—	2	8	no	—
53	#10597264	—	2	8	no	—
54	C₂⁴×C₁₉	#41	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	—	2	1	no	—
56	C₂²×C₄×C₁₉	—	2	1	no	—
57	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	#41	1	2	yes	—
67	C₂²×Dic₁₉	—	2	4	no	—
68	C₂²×Dic₁₉	—	4	2	no	—
69	C₂²×Dic₁₉	—	4	2	no	—
70	C₂²×Dic₁₉	—	8	4	no	—
71	M₁₆×C₁₉	—	2	4	no	—
72	M₁₆×C₁₉	—	4	2	no	—
73	C₂×C₈×C₁₉	—	2	4	no	—
74	C₂×C₈×C₁₉	—	2	2	no	—
75	C₂×C₈×C₁₉	—	2	4	no	—
76	C₂×C₈×C₁₉	—	2	2	no	—
77	C₂×C₈×C₁₉	—	2	2	no	—
78	C₂×C₈×C₁₉	—	4	1	no	—
79	C₂×C₈×C₁₉	—	4	1	no	—
80	C₂×C₈×C₁₉	C₄⋊C₄	1	4	yes	—
81	C₄⋊C₄×C₁₉	—	2	2	no	—
82	C₄⋊C₄×C₁₉	—	2	2	no	—
83	C₂²⋊C₄×C₁₉	—	4	2	no	—
84	C₄²×C₁₉	—	4	1	no	—
85	C₄²×C₁₉	—	4	1	no	—
86	C₄⋊Dic₁₉	—	2	4	no	—
87	C₄⋊Dic₁₉	—	2	4	no	—
88	C₄⋊Dic₁₉	—	4	2	no	—
89	C₄⋊Dic₁₉	—	4	2	no	—
90	#64520	—	2	4	no	—
91	#64520	—	2	4	no	—
92	#64520	—	2	4	no	—
93	#64520	C₄⋊C₄	1	8	yes	—
94	C₂×C₁₉⋊C₈	—	2	4	no	—
95	C₂×C₁₉⋊C₈	—	2	4	no	—
96	C₂×C₁₉⋊C₈	—	2	4	no	—
97	#11331	—	19	1	no	maximal, Sylow
98	C₂³×C₁₉	—	2	2	no	—
99	C₂³×C₁₉	—	2	2	no	—
100	C₂³×C₁₉	C₄×Q₈	1	1	yes	characteristic
101	C₂³×C₁₉	#121	1	1	yes	characteristic
102	C₂³×C₁₉	#121	1	2	yes	—
103	C₂³×C₁₉	#125	1	1	yes	characteristic
104	C₂³×C₁₉	#125	1	2	yes	—
105	C₂×C₄×C₁₉	—	4	1	no	—
106	C₂×C₄×C₁₉	—	4	1	no	—
107	C₂×C₄×C₁₉	—	4	2	no	—
108	C₂×C₄×C₁₉	—	4	2	no	—
109	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
114	C₂×C₄×C₁₉	#120	1	2	yes	—
115	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
117	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
118	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
121	C₂×C₄×C₁₉	#123	1	2	yes	—
122	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
126	C₂×C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
127	C₂×Dic₁₉	—	4	4	no	—
128	C₂×Dic₁₉	—	8	4	no	—
129	C₂×Dic₁₉	—	8	4	no	—
130	C₂×Dic₁₉	—	8	4	no	—
131	C₈×C₁₉	—	2	4	no	—
132	C₈×C₁₉	—	4	2	no	—
133	C₁₉⋊C₈	—	2	8	no	—
134	C₂×#428	—	19	2	no	—
135	C₂²×#107	—	19	1	no	—
136	C₂×#343	—	19	2	no	—
137	C₂×#341	—	19	1	no	—
138	#1729	D₁₉	1	1	yes	characteristic
139	C₂²×C₁₉	—	2	4	no	—
140	C₂²×C₁₉	—	2	2	no	—
141	C₂²×C₁₉	—	2	2	no	—
142	C₂²×C₁₉	—	2	2	no	—
143	C₂²×C₁₉	—	2	2	no	—
144	C₂²×C₁₉	#389	1	2	yes	—
145	C₂²×C₁₉	#389	1	1	yes	characteristic
146	C₂²×C₁₉	#392	1	1	yes	characteristic
147	C₂²×C₁₉	#398	1	1	yes	characteristic
148	C₂²×C₁₉	#398	1	2	yes	—
149	C₂²×C₁₉	#472	1	2	yes	—
150	C₂²×C₁₉	#473	1	2	yes	—
151	C₄×C₁₉	—	8	1	no	—
152	C₄×C₁₉	—	8	1	no	—
153	C₄×C₁₉	#384	1	1	yes	characteristic
154	C₄×C₁₉	#385	1	1	yes	characteristic
155	C₄×C₁₉	#385	1	2	yes	—
156	C₄×C₁₉	#395	1	1	yes	characteristic
157	C₄×C₁₉	#395	1	2	yes	—
158	C₄×C₁₉	#399	1	1	yes	characteristic
159	Dic₁₉	—	8	4	no	—
160	C₂²×M₁₆	—	19	2	no	—
161	C₂²×M₁₆	C₂×D₁₉	1	1	yes	characteristic
162	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
163	C₂×#117	C₂×D₁₉	1	1	yes	characteristic
164	C₂²×C₄⋊C₄	—	19	2	no	—
165	#428	—	38	4	no	—
166	#428	—	38	4	no	—
167	C₂×#107	—	38	4	no	—
168	C₂×#107	—	38	2	no	—
169	C₂×#107	—	38	2	no	—
170	C₂×#106	—	38	2	no	—
171	#343	—	38	4	no	—
172	#341	—	19	4	no	—
173	#336	—	2	2	no	—
174	C₂×C₁₉	—	2	2	no	—
175	C₂×C₁₉	—	2	2	no	—
176	C₂×C₁₉	#1444	1	1	yes	characteristic
177	C₂×C₁₉	#1952	1	2	yes	—
178	C₂×C₁₉	#2018	1	1	yes	characteristic
179	C₂×C₁₉	#2020	1	1	yes	characteristic
180	C₂×C₁₉	#2089	1	2	yes	—
181	C₂³×C₄	—	38	2	no	—
182	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
183	C₂×M₁₆	—	2	1	no	—
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	19	4	no	—
187	C₂×M₁₆	—	19	4	no	—
188	C₂×M₁₆	—	19	4	no	—
189	C₂×M₁₆	—	38	2	no	—
190	C₂×M₁₆	—	38	2	no	—
191	C₂×M₁₆	—	38	2	no	—
192	C₂×M₁₆	Dic₃₈	1	2	yes	—
193	C₂²×C₈	—	2	2	no	—
194	C₂²×C₈	—	2	2	no	—
195	C₂²×C₈	—	2	1	no	—
196	C₂²×C₈	—	38	2	no	—
197	C₂²×C₈	Dic₃₈	1	2	yes	—
198	C₂²×C₈	C₄×D₁₉	1	2	yes	—
199	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
200	#117	—	2	2	no	—
201	#117	—	2	2	no	—
202	C₂×C₄⋊C₄	—	19	4	no	—
203	C₂×C₄⋊C₄	—	38	2	no	—
204	C₂×C₄⋊C₄	—	38	2	no	—
205	C₂×C₄⋊C₄	—	38	2	no	—
206	C₂×C₄⋊C₄	—	38	2	no	—
207	C₂×C₄⋊C₄	—	38	2	no	—
208	C₂×C₄⋊C₄	D₇₆	1	1	yes	characteristic
209	C₂×C₄⋊C₄	D₇₆	1	1	yes	characteristic
210	C₂×C₂²⋊C₄	—	2	1	no	—
211	C₂×C₄²	—	2	1	no	—
212	#107	—	38	8	no	—
213	#107	—	38	8	no	—
214	#106	—	38	8	no	—
215	C₁₉	#11331	1	1	yes	characteristic, complemented, Sylow
216	C₂⁴	#64541	1	1	yes	characteristic
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	38	4	no	—
220	C₂²×C₄	—	76	2	no	—
221	C₂²×C₄	—	76	2	no	—
222	C₂²×C₄	—	152	4	no	—
223	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
224	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
226	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
227	C₂²×C₄	#64541	1	1	yes	characteristic
228	C₂²×C₄	D₄×D₁₉	1	2	yes	—
229	C₂²×C₄	D₄×D₁₉	1	2	yes	—
230	C₂²×C₄	#64543	1	2	yes	—
231	C₂²×C₄	Q₈×D₁₉	1	2	yes	—
232	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
233	M₁₆	—	2	4	no	—
234	M₁₆	—	4	2	no	—
235	M₁₆	—	19	8	no	—
236	M₁₆	—	38	4	no	—
237	M₁₆	—	38	4	no	—
238	M₁₆	—	38	4	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	4	1	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	—	38	4	no	—
247	C₂×C₈	—	38	4	no	—
248	C₂×C₈	—	38	4	no	—
249	C₂×C₈	Dic₁₉⋊C₄	1	4	yes	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	38	4	no	—
253	C₄⋊C₄	—	38	4	no	—
254	C₄⋊C₄	—	76	2	no	—
255	C₄⋊C₄	—	76	2	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₄²	—	4	1	no	—
258	C₄²	—	4	1	no	—
259	C₂³	—	2	2	no	—
260	C₂³	—	2	2	no	—
261	C₂³	C₄×Dic₃₈	1	1	yes	characteristic
262	C₂³	#10597316	1	1	yes	characteristic
263	C₂³	#10597330	1	2	yes	—
264	C₂³	#10597335	1	2	yes	—
265	C₂³	#10597376	1	1	yes	characteristic
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	76	4	no	—
271	C₂×C₄	—	152	4	no	—
272	C₂×C₄	—	152	4	no	—
273	C₂×C₄	—	152	4	no	—
274	C₂×C₄	#10597315	1	1	yes	characteristic
275	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
276	C₂×C₄	#10597324	1	2	yes	—
277	C₂×C₄	#10597324	1	2	yes	—
278	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
279	C₂×C₄	#10597328	1	2	yes	—
280	C₂×C₄	#10597329	1	2	yes	—
281	C₂×C₄	#10597334	1	2	yes	—
282	C₂×C₄	#10597334	1	2	yes	—
283	C₂×C₄	C₄⋊C₄×D₁₉	1	2	yes	—
284	C₂×C₄	#10597342	1	2	yes	—
285	C₂×C₄	#10597343	1	2	yes	—
286	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
287	C₂×C₄	#10597370	1	1	yes	characteristic
288	C₂×C₄	#10597381	1	1	yes	characteristic
289	C₂×C₄	#10597383	1	1	yes	characteristic
290	C₂×C₄	#10597386	1	1	yes	characteristic
291	C₂×C₄	#10597390	1	1	yes	characteristic
292	C₈	—	2	4	no	—
293	C₈	—	4	2	no	—
294	C₈	—	38	8	no	—
295	C₂²	—	2	4	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	#11926441	1	2	yes	—
301	C₂²	#11926442	1	1	yes	characteristic
302	C₂²	#11926453	1	1	yes	characteristic
303	C₂²	#11926474	1	1	yes	characteristic
304	C₂²	#11926478	1	2	yes	—
305	C₂²	#11926814	1	2	yes	—
306	C₂²	#11926819	1	2	yes	—
307	C₄	—	8	1	no	—
308	C₄	—	8	1	no	—
309	C₄	—	152	4	no	—
310	C₄	#11926413	1	1	yes	characteristic
311	C₄	#11926416	1	1	yes	characteristic
312	C₄	#11926418	1	2	yes	—
313	C₄	#11926464	1	1	yes	characteristic
314	C₄	#11926465	1	2	yes	—
315	C₄	#11926480	1	1	yes	characteristic
316	C₂	—	2	2	no	—
317	C₂	—	2	2	no	—
318	C₂	#14561899	1	1	yes	characteristic
319	C₂	#14564420	1	2	yes	—
320	C₂	#14564744	1	1	yes	characteristic
321	C₂	#14564756	1	1	yes	characteristic
322	C₂	#14565140	1	2	yes	—

	subgroup	G/H
center	C₂³	#10597376
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₁₉×#1729	C₂
socle	C₂³×C₁₉	#121
prime radical	C₂⁴×C₁₉	#41
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	#1729	D₁₉
hypocenter	C₁₉	#11331
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #1729

Complements

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

N	H ≅ G/N	type
15. #11926309	C₄	semidirect
215. C₁₉	#11331	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926587	2
C₂²×#10597275	2
C₂×#11926312	2
C₂×#11926309	2
C₁₉×#1729	2
#11331	19