Subgroups — #18088447

Order 4864.

806 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10597279	C₂	1	2	yes	maximal
2	C₂²×C₄×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926308	C₂	1	2	yes	maximal
4	#14563329	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1728	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	C₂²	1	2	yes	—
7	C₂³×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂³×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×#10597279	—	2	4	no	—
10	C₂×#10597279	—	2	4	no	—
11	C₂×#10597279	C₄	1	8	yes	complemented
12	C₂×C₄×C₁₉⋊C₈	—	2	4	no	—
13	C₂×C₄×C₁₉⋊C₈	C₄	1	4	yes	—
14	#11926308	—	2	4	no	—
15	#11926299	—	2	2	no	—
16	C₂²×M₁₆×C₁₉	C₂²	1	2	yes	—
17	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
18	C₁₉×#336	—	2	2	no	—
19	C₂³×C₄×C₁₉	Q₈	1	2	yes	—
20	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
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	—	2	2	no	—
25	C₂×M₁₆×C₁₉	—	2	2	no	—
26	C₂×M₁₆×C₁₉	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₁₉	—	2	2	no	—
28	C₂×C₄²×C₁₉	—	2	2	no	—
29	C₂×C₄²×C₁₉	—	2	2	no	—
30	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₉	D₄	1	2	yes	—
32	C₂×#64520	—	2	2	no	—
33	C₂×#64520	—	2	2	no	—
34	C₂×#64520	—	2	2	no	—
35	C₂×#64520	—	2	2	no	—
36	C₂×#64520	—	2	2	no	—
37	C₂×#64520	C₂×C₄	1	4	yes	—
38	C₂²×C₁₉⋊C₈	—	2	2	no	—
39	C₂²×C₁₉⋊C₈	—	2	2	no	—
40	C₂²×C₁₉⋊C₈	—	2	2	no	—
41	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
42	C₂²×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
43	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
44	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
45	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
46	C₂²×C₁₉⋊C₈	D₄	1	4	yes	—
47	C₂²×C₁₉⋊C₈	Q₈	1	2	yes	—
48	C₂²×C₁₉⋊C₈	Q₈	1	2	yes	—
49	#10597279	—	2	16	no	—
50	C₄×C₁₉⋊C₈	—	2	8	no	—
51	C₂⁴×C₁₉	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₁₉	—	2	4	no	—
53	C₂²×C₄×C₁₉	—	2	8	no	—
54	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
55	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	#41	1	2	yes	—
64	C₂²×C₄×C₁₉	#41	1	2	yes	—
65	M₁₆×C₁₉	—	2	4	no	—
66	M₁₆×C₁₉	—	2	4	no	—
67	M₁₆×C₁₉	—	4	2	no	—
68	M₁₆×C₁₉	—	4	2	no	—
69	C₂×C₈×C₁₉	—	2	4	no	—
70	C₂×C₈×C₁₉	—	4	2	no	—
71	C₂×C₈×C₁₉	—	4	2	no	—
72	C₄²×C₁₉	—	2	2	no	—
73	C₄²×C₁₉	—	2	2	no	—
74	C₄²×C₁₉	—	2	4	no	—
75	#64520	—	2	4	no	—
76	#64520	—	2	4	no	—
77	#64520	—	4	2	no	—
78	#64520	—	4	2	no	—
79	C₂×C₁₉⋊C₈	—	2	2	no	—
80	C₂×C₁₉⋊C₈	—	2	2	no	—
81	C₂×C₁₉⋊C₈	—	2	4	no	—
82	C₂×C₁₉⋊C₈	—	2	4	no	—
83	C₂×C₁₉⋊C₈	—	2	4	no	—
84	C₂×C₁₉⋊C₈	—	4	2	no	—
85	C₂×C₁₉⋊C₈	—	4	2	no	—
86	C₂×C₁₉⋊C₈	C₄²	1	4	yes	—
87	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	4	yes	—
88	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	8	yes	—
89	C₂×C₁₉⋊C₈	C₄⋊C₄	1	4	yes	—
90	C₂×C₁₉⋊C₈	C₄⋊C₄	1	4	yes	—
91	C₂×C₁₉⋊C₈	C₄⋊C₄	1	8	yes	—
92	#10864	—	19	1	no	maximal, Sylow
93	C₂³×C₁₉	—	2	4	no	—
94	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂³×C₁₉	C₄×Q₈	1	2	yes	—
96	C₂³×C₁₉	C₂²⋊Q₈	1	2	yes	—
97	C₂³×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
98	C₂³×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₁₉	—	2	16	no	—
100	C₂×C₄×C₁₉	—	2	4	no	—
101	C₂×C₄×C₁₉	—	2	4	no	—
102	C₂×C₄×C₁₉	#108	1	8	yes	complemented
103	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
106	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
110	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
111	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
114	C₂×C₄×C₁₉	#121	1	2	yes	—
115	C₂×C₄×C₁₉	#121	1	2	yes	—
116	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
117	C₂×C₄×C₁₉	#124	1	2	yes	—
118	C₂×C₄×C₁₉	#125	1	2	yes	—
119	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
121	C₈×C₁₉	—	4	4	no	—
122	C₁₉⋊C₈	—	2	4	no	—
123	C₁₉⋊C₈	—	4	4	no	—
124	C₁₉⋊C₈	#95	1	8	yes	—
125	C₂²×#108	—	19	2	no	—
126	C₂²×C₄×C₈	—	19	1	no	—
127	C₂×#341	—	19	2	no	—
128	#1728	—	19	1	no	—
129	#1728	D₁₉	1	1	yes	characteristic
130	C₂²×C₁₉	—	2	4	no	—
131	C₂²×C₁₉	—	2	2	no	—
132	C₂²×C₁₉	—	2	2	no	—
133	C₂²×C₁₉	—	2	4	no	—
134	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
135	C₂²×C₁₉	#380	1	1	yes	characteristic
136	C₂²×C₁₉	#384	1	1	yes	characteristic
137	C₂²×C₁₉	#389	1	2	yes	—
138	C₂²×C₁₉	#391	1	2	yes	—
139	C₂²×C₁₉	C₂×#108	1	4	yes	—
140	C₄×C₁₉	—	2	8	no	—
141	C₄×C₁₉	C₂×#95	1	1	yes	characteristic
142	C₄×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
143	C₄×C₁₉	#380	1	1	yes	characteristic
144	C₄×C₁₉	#382	1	1	yes	characteristic
145	C₄×C₁₉	#389	1	2	yes	—
146	C₄×C₁₉	#390	1	2	yes	—
147	C₂²×M₁₆	—	19	2	no	—
148	C₂²×M₁₆	C₂×D₁₉	1	2	yes	—
149	C₂³×C₈	—	19	1	no	—
150	C₂³×C₈	—	19	1	no	—
151	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
152	C₂×#108	—	19	8	no	—
153	C₂×#108	—	38	4	no	—
154	C₂×#108	—	38	4	no	—
155	C₂×C₄×C₈	—	19	4	no	—
156	C₂×C₄×C₈	—	38	4	no	—
157	#341	—	38	4	no	—
158	#336	—	2	2	no	—
159	#336	—	38	2	no	—
160	C₂×C₁₉	—	2	2	no	—
161	C₂×C₁₉	—	2	2	no	—
162	C₂×C₁₉	C₄⋊#95	1	1	yes	characteristic
163	C₂×C₁₉	C₄×#108	1	2	yes	—
164	C₂×C₁₉	#1810	1	2	yes	—
165	C₂×C₁₉	#1848	1	2	yes	—
166	C₂³×C₄	Dic₃₈	1	2	yes	—
167	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
168	C₂×M₁₆	—	2	2	no	—
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	—	19	4	no	—
174	C₂×M₁₆	—	38	2	no	—
175	C₂×M₁₆	—	38	2	no	—
176	C₂×M₁₆	—	38	2	no	—
177	C₂×M₁₆	—	38	2	no	—
178	C₂×M₁₆	—	38	2	no	—
179	C₂×M₁₆	C₄×D₁₉	1	4	yes	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	19	2	no	—
182	C₂²×C₈	—	19	4	no	—
183	C₂²×C₈	—	19	4	no	—
184	C₂²×C₈	—	19	2	no	—
185	C₂²×C₈	—	19	2	no	—
186	C₂²×C₈	—	19	2	no	—
187	C₂²×C₈	—	19	2	no	—
188	C₂²×C₈	—	19	2	no	—
189	C₂²×C₈	—	38	2	no	—
190	C₂²×C₈	—	38	2	no	—
191	C₂²×C₈	—	38	2	no	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	C₄×D₁₉	1	2	yes	—
195	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
196	#108	—	38	16	no	—
197	C₄×C₈	—	38	8	no	—
198	C₁₉	#10864	1	1	yes	characteristic, complemented, Sylow
199	C₂⁴	C₂×Dic₃₈	1	1	yes	characteristic
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	8	no	—
202	C₂²×C₄	Dic₁₉⋊C₄	1	4	yes	—
203	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
205	C₂²×C₄	#64541	1	2	yes	—
206	C₂²×C₄	D₄×D₁₉	1	2	yes	—
207	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
208	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
209	C₂²×C₄	#64543	1	2	yes	—
210	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
211	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
213	M₁₆	—	2	4	no	—
214	M₁₆	—	2	4	no	—
215	M₁₆	—	4	2	no	—
216	M₁₆	—	4	2	no	—
217	M₁₆	—	38	4	no	—
218	M₁₆	—	38	4	no	—
219	M₁₆	—	76	2	no	—
220	M₁₆	—	76	2	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	19	4	no	—
225	C₂×C₈	—	19	4	no	—
226	C₂×C₈	—	19	8	no	—
227	C₂×C₈	—	19	4	no	—
228	C₂×C₈	—	19	8	no	—
229	C₂×C₈	—	19	4	no	—
230	C₂×C₈	—	38	2	no	—
231	C₂×C₈	—	38	2	no	—
232	C₂×C₈	—	38	4	no	—
233	C₂×C₈	—	38	4	no	—
234	C₂×C₈	—	38	4	no	—
235	C₂×C₈	—	76	2	no	—
236	C₂×C₈	—	76	2	no	—
237	C₄²	—	2	4	no	—
238	C₄²	—	2	2	no	—
239	C₄²	—	2	2	no	—
240	C₂³	—	2	4	no	—
241	C₂³	C₄×Dic₃₈	1	2	yes	—
242	C₂³	#10597324	1	2	yes	—
243	C₂³	#10597334	1	1	yes	characteristic
244	C₂³	#10597334	1	1	yes	characteristic
245	C₂³	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
246	C₂×C₄	—	2	16	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	#10597279	1	8	yes	complemented
250	C₂×C₄	C₄²×D₁₉	1	1	yes	characteristic
251	C₂×C₄	#10597318	1	1	yes	characteristic
252	C₂×C₄	#10597323	1	1	yes	characteristic
253	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
254	C₂×C₄	C₂²⋊C₄×D₁₉	1	2	yes	—
255	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
256	C₂×C₄	#10597330	1	2	yes	—
257	C₂×C₄	#10597331	1	2	yes	—
258	C₂×C₄	#10597333	1	2	yes	—
259	C₂×C₄	#10597335	1	2	yes	—
260	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
261	C₂×C₄	C₄⋊C₄×D₁₉	1	2	yes	—
262	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
263	C₂×C₄	#10597342	1	2	yes	—
264	C₂×C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
265	C₂×C₄	#10597385	1	2	yes	—
266	C₂×C₄	#10597386	1	1	yes	characteristic
267	C₂×C₄	#10597390	1	1	yes	characteristic
268	C₈	—	4	4	no	—
269	C₈	—	19	8	no	—
270	C₈	—	38	4	no	—
271	C₈	—	76	4	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	C₄×Dic₁₉⋊C₄	1	1	yes	characteristic
277	C₂²	#11926402	1	1	yes	characteristic
278	C₂²	#11926413	1	1	yes	characteristic
279	C₂²	#11926441	1	2	yes	—
280	C₂²	#11926448	1	2	yes	—
281	C₂²	C₂×#10597279	1	4	yes	—
282	C₄	—	2	8	no	—
283	C₄	#95×D₁₉	1	1	yes	characteristic
284	C₄	#11926395	1	1	yes	characteristic
285	C₄	#11926403	1	1	yes	characteristic
286	C₄	#11926408	1	1	yes	characteristic
287	C₄	#11926439	1	2	yes	—
288	C₄	#11926444	1	2	yes	—
289	C₂	—	2	2	no	—
290	C₂	—	2	2	no	—
291	C₂	Dic₁₉⋊#95	1	1	yes	characteristic
292	C₂	C₄×#10597279	1	2	yes	—
293	C₂	#14563671	1	2	yes	—
294	C₂	#14563806	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₁₉×#1728	C₂
socle	C₂³×C₁₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₉	C₂×Q₈
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	#1728	D₁₉
hypocenter	C₁₉	#10864
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #1728

Complements

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

N	H ≅ G/N	type
11. C₂×#10597279	C₄	semidirect
102. C₂×C₄×C₁₉	#108	semidirect
198. C₁₉	#10864	semidirect
249. C₂×C₄	#10597279	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#10597279	2
C₂²×C₄×C₁₉⋊C₈	2
C₂×#11926308	2
#14563329	2
C₁₉×#1728	2
#10864	19