Subgroups — #18087240

Order 4864.

418 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14564429	C₂	1	1	yes	maximal, characteristic
2	#14563632	C₂	1	2	yes	maximal
3	#14561518	C₂	1	2	yes	maximal, complemented
4	#14561470	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1845	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₇₆	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
9	C₂×#10597266	—	2	2	no	—
10	C₂×(C₂×D₁₉)⋊C₈	—	2	1	no	—
11	C₂×(C₂×D₁₉)⋊C₈	—	2	1	no	—
12	#11926306	—	2	2	no	—
13	C₂×C₁₉×#107	—	2	1	no	—
14	C₂×C₁₉×#107	—	2	1	no	—
15	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
16	C₁₉×#384	C₂²	1	2	yes	—
17	C₂²×D₇₆	D₄	1	1	yes	characteristic
18	C₂²×C₄×D₁₉	—	2	1	no	—
19	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
22	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
23	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
24	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
25	C₂×(C₂×D₁₉)⋊C₄	—	2	1	no	—
26	C₂×C₄⋊Dic₁₉	D₄	1	1	yes	characteristic
27	C₂²×C₁₉⋊C₈	—	2	2	no	—
28	C₂²×C₁₉⋊C₈	—	2	1	no	—
29	C₄×D₇₆	—	2	2	no	—
30	C₄×D₇₆	—	2	2	no	—
31	C₁₉×#107	—	4	2	no	—
32	C₁₉×#107	—	4	2	no	—
33	C₄×C₈×C₁₉	—	2	2	no	—
34	(C₂×D₁₉)⋊C₈	—	4	2	no	—
35	(C₂×D₁₉)⋊C₈	—	4	2	no	—
36	#10597266	—	4	4	no	—
37	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
38	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
39	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
40	C₂³×D₁₉	—	2	1	no	—
41	C₂²×C₄×C₁₉	—	4	2	no	—
42	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
45	C₂²×Dic₁₉	—	2	1	no	—
46	C₂×D₇₆	—	2	2	no	—
47	C₂×D₇₆	—	2	2	no	—
48	C₂×D₇₆	—	2	1	no	—
49	C₂×D₇₆	D₈	1	2	yes	—
50	C₂×C₄×D₁₉	—	4	1	no	—
51	C₂×C₄×D₁₉	—	4	1	no	—
52	C₂×C₄×D₁₉	—	4	1	no	—
53	C₂×C₄×D₁₉	—	4	1	no	—
54	C₂×C₄×D₁₉	—	4	1	no	—
55	C₂×C₄×D₁₉	—	4	1	no	—
56	C₂×C₈×C₁₉	—	2	1	no	—
57	C₂×C₈×C₁₉	—	2	1	no	—
58	C₂×C₈×C₁₉	—	2	1	no	—
59	C₂×C₈×C₁₉	—	2	1	no	—
60	C₂×C₈×C₁₉	D₈	1	2	yes	—
61	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
62	C₄⋊C₄×C₁₉	—	4	2	no	—
63	C₄⋊C₄×C₁₉	—	4	2	no	—
64	C₄⋊C₄×C₁₉	—	4	2	no	—
65	C₄⋊C₄×C₁₉	—	4	2	no	—
66	C₄⋊C₄×C₁₉	—	8	2	no	—
67	C₄²×C₁₉	#41	1	2	yes	—
68	C₄²×C₁₉	#41	1	2	yes	—
69	(C₂×D₁₉)⋊C₄	—	4	2	no	—
70	C₄⋊Dic₁₉	—	2	2	no	—
71	C₂×C₁₉⋊C₈	—	4	2	no	—
72	C₂×C₁₉⋊C₈	—	4	1	no	—
73	C₂×C₁₉⋊C₈	—	4	2	no	—
74	C₂×C₁₉⋊C₈	—	4	1	no	—
75	C₂×C₁₉⋊C₈	—	4	1	no	—
76	C₂×C₁₉⋊C₈	—	4	2	no	—
77	#10615	—	19	1	no	maximal, Sylow
78	C₂³×C₁₉	#121	1	1	yes	characteristic
79	C₂²×D₁₉	—	2	2	no	—
80	C₂²×D₁₉	—	4	1	no	—
81	C₂²×D₁₉	—	4	1	no	—
82	C₂²×D₁₉	—	4	1	no	—
83	C₂²×D₁₉	—	4	1	no	—
84	C₂²×D₁₉	—	4	1	no	—
85	C₂²×D₁₉	—	4	1	no	—
86	C₂×C₄×C₁₉	—	2	1	no	—
87	C₂×C₄×C₁₉	—	2	1	no	—
88	C₂×C₄×C₁₉	—	8	2	no	—
89	C₂×C₄×C₁₉	—	8	2	no	—
90	C₂×C₄×C₁₉	—	8	2	no	—
91	C₂×C₄×C₁₉	#123	1	2	yes	—
92	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
93	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
94	C₂×C₄×C₁₉	#126	1	2	yes	—
95	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
98	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
99	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
100	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
101	C₂×C₄×C₁₉	#137	1	2	yes	—
102	C₂×Dic₁₉	—	4	1	no	—
103	C₂×Dic₁₉	—	4	1	no	—
104	C₂×Dic₁₉	—	4	1	no	—
105	D₇₆	—	2	4	no	—
106	D₇₆	—	4	2	no	—
107	C₄×D₁₉	—	8	1	no	—
108	C₄×D₁₉	—	8	1	no	—
109	C₄×D₁₉	—	8	1	no	—
110	C₄×D₁₉	—	8	1	no	—
111	C₈×C₁₉	—	2	2	no	—
112	C₈×C₁₉	—	2	2	no	—
113	C₁₉⋊C₈	—	8	2	no	—
114	C₁₉⋊C₈	—	8	1	no	—
115	#1954	—	19	1	no	—
116	#1845	D₁₉	1	1	yes	characteristic, complemented
117	#1802	—	19	2	no	—
118	#1296	—	19	2	no	—
119	#1277	—	19	1	no	—
120	C₂²×C₁₉	#397	1	1	yes	characteristic
121	C₂²×C₁₉	#459	1	1	yes	characteristic
122	C₂²×C₁₉	#461	1	1	yes	characteristic
123	C₂²×C₁₉	#466	1	1	yes	characteristic
124	C₂²×C₁₉	#470	1	1	yes	characteristic
125	C₂²×C₁₉	#472	1	1	yes	characteristic
126	C₂²×C₁₉	#473	1	1	yes	characteristic
127	C₂×D₁₉	—	4	2	no	—
128	C₂×D₁₉	—	4	2	no	—
129	C₂×D₁₉	—	4	2	no	—
130	C₂×D₁₉	—	8	1	no	—
131	C₂×D₁₉	—	8	1	no	—
132	C₂×D₁₉	—	8	1	no	—
133	C₂×D₁₉	—	8	1	no	—
134	C₄×C₁₉	—	2	2	no	—
135	C₄×C₁₉	—	16	2	no	—
136	C₄×C₁₉	#480	1	1	yes	characteristic
137	C₄×C₁₉	#480	1	2	yes	—
138	C₄×C₁₉	#481	1	1	yes	characteristic
139	C₄×C₁₉	#485	1	2	yes	—
140	C₄×C₁₉	#486	1	1	yes	characteristic
141	C₄×C₁₉	#489	1	1	yes	characteristic
142	Dic₁₉	—	8	1	no	—
143	C₂×C₄×D₄	—	19	1	no	—
144	C₂×#107	—	2	1	no	—
145	C₂×#107	—	2	1	no	—
146	C₂×C₄⋊C₈	—	19	2	no	—
147	C₂×C₄⋊C₈	—	19	1	no	—
148	C₂×#102	—	38	2	no	—
149	C₂×C₂²⋊C₈	—	38	1	no	—
150	C₂×C₂²⋊C₈	—	38	1	no	—
151	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
152	#384	C₂×D₁₉	1	2	yes	—
153	#340	—	38	2	no	—
154	C₂×C₁₉	#1667	1	1	yes	characteristic
155	C₂×C₁₉	#1673	1	1	yes	characteristic
156	C₂×C₁₉	#1689	1	1	yes	characteristic
157	C₂×C₁₉	#1695	1	1	yes	characteristic
158	C₂×C₁₉	#2071	1	1	yes	characteristic
159	C₂×C₁₉	#2075	1	1	yes	characteristic
160	C₂×C₁₉	#2078	1	1	yes	characteristic
161	D₁₉	—	8	2	no	—
162	C₂²×D₄	—	19	1	no	—
163	C₂³×C₄	—	38	1	no	—
164	C₂²×C₈	—	38	2	no	—
165	C₂²×C₈	—	38	1	no	—
166	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
167	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
168	C₄×D₄	—	38	2	no	—
169	C₄×D₄	—	38	2	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	—	19	1	no	—
174	C₂×C₂²⋊C₄	—	38	1	no	—
175	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
176	#107	—	4	2	no	—
177	#107	—	4	2	no	—
178	C₄⋊C₈	—	38	2	no	—
179	C₄⋊C₈	—	38	2	no	—
180	C₄⋊C₈	—	38	2	no	—
181	#102	—	76	4	no	—
182	C₂²⋊C₈	—	76	2	no	—
183	C₂²⋊C₈	—	76	2	no	—
184	C₄×C₈	—	2	2	no	—
185	C₁₉	#10615	1	1	yes	characteristic, complemented, Sylow
186	C₂⁴	—	38	1	no	—
187	C₂×D₄	—	19	2	no	—
188	C₂×D₄	—	38	2	no	—
189	C₂×D₄	—	38	1	no	—
190	C₂×D₄	—	38	2	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	—	38	1	no	—
193	C₂²×C₄	—	76	1	no	—
194	C₂²×C₄	—	76	1	no	—
195	C₂²×C₄	—	76	1	no	—
196	C₂²×C₄	—	76	1	no	—
197	C₂²×C₄	—	76	1	no	—
198	C₂²×C₄	—	76	1	no	—
199	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
200	C₂²×C₄	#64543	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	—	76	1	no	—
207	C₂×C₈	—	76	1	no	—
208	C₂×C₈	—	76	1	no	—
209	C₂×C₈	—	76	2	no	—
210	C₂×C₈	—	76	2	no	—
211	C₂×C₈	—	76	2	no	—
212	C₂×C₈	#64525	1	2	yes	—
213	C₂×C₈	#64527	1	2	yes	—
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄⋊C₄	—	4	2	no	—
218	C₄⋊C₄	—	8	2	no	—
219	C₄⋊C₄	—	38	2	no	—
220	C₂²⋊C₄	—	76	2	no	—
221	C₄²	#64541	1	2	yes	—
222	C₄²	#64545	1	2	yes	—
223	C₂³	—	38	2	no	—
224	C₂³	—	76	1	no	—
225	C₂³	—	76	1	no	—
226	C₂³	—	76	1	no	—
227	C₂³	—	76	1	no	—
228	C₂³	—	76	1	no	—
229	C₂³	—	76	1	no	—
230	C₂³	#10597384	1	1	yes	characteristic
231	D₄	—	38	4	no	—
232	D₄	—	76	2	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	8	2	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	—	8	2	no	—
238	C₂×C₄	—	76	1	no	—
239	C₂×C₄	—	76	1	no	—
240	C₂×C₄	—	76	1	no	—
241	C₂×C₄	—	152	1	no	—
242	C₂×C₄	—	152	1	no	—
243	C₂×C₄	—	152	1	no	—
244	C₂×C₄	—	152	1	no	—
245	C₂×C₄	#10597340	1	2	yes	—
246	C₂×C₄	#10597344	1	2	yes	—
247	C₂×C₄	D₈×D₁₉	1	1	yes	characteristic
248	C₂×C₄	#10597358	1	1	yes	characteristic
249	C₂×C₄	#10597364	1	2	yes	—
250	C₂×C₄	#10597375	1	1	yes	characteristic
251	C₂×C₄	C₂×#64525	1	1	yes	characteristic
252	C₂×C₄	#10597378	1	1	yes	characteristic
253	C₂×C₄	C₂×#64527	1	1	yes	characteristic
254	C₂×C₄	#10597388	1	1	yes	characteristic
255	C₂×C₄	#10597393	1	1	yes	characteristic
256	C₈	—	2	2	no	—
257	C₈	—	2	2	no	—
258	C₈	—	152	2	no	—
259	C₈	—	152	1	no	—
260	C₂²	—	76	2	no	—
261	C₂²	—	76	2	no	—
262	C₂²	—	76	2	no	—
263	C₂²	—	152	1	no	—
264	C₂²	—	152	1	no	—
265	C₂²	—	152	1	no	—
266	C₂²	—	152	1	no	—
267	C₂²	#11926470	1	1	yes	characteristic
268	C₂²	#11926736	1	1	yes	characteristic
269	C₂²	#11926746	1	1	yes	characteristic
270	C₂²	#11926777	1	1	yes	characteristic
271	C₂²	#11926799	1	1	yes	characteristic
272	C₂²	#11926811	1	1	yes	characteristic
273	C₂²	#11926816	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	16	2	no	—
276	C₄	—	152	1	no	—
277	C₄	#11926857	1	1	yes	characteristic
278	C₄	#11926859	1	2	yes	—
279	C₄	#11926862	1	1	yes	characteristic
280	C₄	#11926887	1	2	yes	—
281	C₄	#11926892	1	1	yes	characteristic
282	C₄	#11926906	1	1	yes	characteristic
283	C₂	—	152	2	no	—
284	C₂	#14562987	1	1	yes	characteristic
285	C₂	#14563021	1	1	yes	characteristic
286	C₂	#14563121	1	1	yes	characteristic
287	C₂	#14563155	1	1	yes	characteristic
288	C₂	#14565039	1	1	yes	characteristic
289	C₂	#14565059	1	1	yes	characteristic
290	C₂	#14565076	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597384
commutator, cosocle	C₂×C₄²×C₁₉	C₂³
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1845	C₂
socle	C₂³×C₁₉	#121
prime radical	C₂²×D₇₆	D₄
hypercenter	#1845	D₁₉
hypocenter	C₁₉	#10615
LCS	G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
3. #14561518	C₂	semidirect
4. #14561470	C₂	semidirect
5. C₁₉×#1845	C₂	semidirect
116. #1845	D₁₉	semidirect
185. C₁₉	#10615	semidirect

Maximal subgroups

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

subgroup	index
#14564429	2
#14563632	2
#14561518	2
#14561470	2
C₁₉×#1845	2
#10615	19