Subgroups — #18075635

Order 4864.

434 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926674	C₂	1	1	yes	maximal, characteristic
2	#14563637	C₂	1	2	yes	maximal
3	C₂×#11926295	C₂	1	1	yes	maximal, characteristic
4	#14561518	C₂	1	2	yes	maximal
5	C₁₉×#1845	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₃₈	C₂²	1	1	yes	characteristic
7	#11926674	—	2	2	no	—
8	#11926674	C₄	1	4	yes	complemented
9	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
10	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₉⋊C₈	—	2	1	no	—
12	C₂×#10597267	—	2	2	no	—
13	C₂×#10597272	—	2	1	no	—
14	#11926306	—	2	2	no	—
15	#11926295	—	2	2	no	—
16	C₂×C₁₉×#107	—	2	1	no	—
17	C₂×C₁₉×#107	—	2	1	no	—
18	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
19	C₁₉×#384	C₂²	1	2	yes	—
20	C₂²×Dic₃₈	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
24	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
25	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
26	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
27	C₂×C₄⋊Dic₁₉	D₄	1	1	yes	characteristic
28	C₂×Dic₁₉⋊C₄	—	2	1	no	—
29	C₂×C₄×Dic₁₉	—	2	1	no	—
30	C₂²×C₁₉⋊C₈	—	2	1	no	—
31	C₂²×C₁₉⋊C₈	—	2	2	no	—
32	C₄×Dic₃₈	—	2	1	no	—
33	C₄×Dic₃₈	—	2	1	no	—
34	C₄×Dic₃₈	C₂×C₄	1	2	yes	—
35	C₄×Dic₃₈	Q₈	1	2	yes	—
36	C₁₉×#107	—	4	2	no	—
37	C₁₉×#107	—	4	2	no	—
38	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₁₉	D₄	1	2	yes	—
40	#10597272	—	2	4	no	—
41	Dic₁₉⋊C₈	—	2	4	no	—
42	#10597267	—	4	4	no	—
43	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
44	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
45	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
46	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
47	C₂²×C₄×C₁₉	—	4	2	no	—
48	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
51	C₂²×Dic₁₉	—	2	1	no	—
52	C₂²×Dic₁₉	—	2	1	no	—
53	C₂×Dic₃₈	—	2	2	no	—
54	C₂×Dic₃₈	—	2	2	no	—
55	C₂×Dic₃₈	—	2	1	no	—
56	C₂×Dic₃₈	C₄⋊C₄	1	2	yes	—
57	C₂×C₈×C₁₉	—	2	1	no	—
58	C₂×C₈×C₁₉	—	2	1	no	—
59	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
62	C₂×C₈×C₁₉	Q₁₆	1	2	yes	—
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	—	4	2	no	—
67	C₄⋊C₄×C₁₉	—	8	2	no	—
68	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
71	C₄²×C₁₉	#41	1	1	yes	characteristic
72	C₄⋊Dic₁₉	—	2	1	no	—
73	C₄⋊Dic₁₉	C₄⋊C₄	1	2	yes	—
74	Dic₁₉⋊C₄	—	2	2	no	—
75	Dic₁₉⋊C₄	—	2	2	no	—
76	C₄×Dic₁₉	—	2	2	no	—
77	C₄×Dic₁₉	—	2	2	no	—
78	C₂×C₁₉⋊C₈	—	2	2	no	—
79	C₂×C₁₉⋊C₈	—	4	1	no	—
80	C₂×C₁₉⋊C₈	—	4	1	no	—
81	C₂×C₁₉⋊C₈	—	4	2	no	—
82	C₂×C₁₉⋊C₈	—	4	2	no	—
83	C₂×C₁₉⋊C₈	—	4	2	no	—
84	#8653	—	19	1	no	maximal, Sylow
85	C₂³×C₁₉	#121	1	1	yes	characteristic
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	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
94	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
95	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
97	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
98	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
99	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
100	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
101	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
102	C₂×C₄×C₁₉	#137	1	2	yes	—
103	C₂×Dic₁₉	—	2	2	no	—
104	C₂×Dic₁₉	—	2	2	no	—
105	C₂×Dic₁₉	—	4	1	no	—
106	C₂×Dic₁₉	—	4	1	no	—
107	C₂×Dic₁₉	—	4	1	no	—
108	C₂×Dic₁₉	—	4	1	no	—
109	Dic₃₈	—	2	4	no	—
110	Dic₃₈	—	4	2	no	—
111	C₈×C₁₉	—	2	2	no	—
112	C₈×C₁₉	#103	1	4	yes	complemented
113	C₁₉⋊C₈	—	4	2	no	—
114	C₁₉⋊C₈	—	8	2	no	—
115	C₂×#446	—	19	1	no	—
116	#1845	D₁₉	1	1	yes	characteristic
117	#1803	—	19	2	no	—
118	C₂×#334	—	19	1	no	—
119	#1296	—	19	2	no	—
120	C₂²×C₁₉	#380	1	1	yes	characteristic
121	C₂²×C₁₉	#460	1	1	yes	characteristic
122	C₂²×C₁₉	#462	1	1	yes	characteristic
123	C₂²×C₁₉	#469	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	C₄×C₁₉	—	2	1	no	—
128	C₄×C₁₉	—	2	1	no	—
129	C₄×C₁₉	—	16	2	no	—
130	C₄×C₁₉	C₂×#103	1	1	yes	characteristic
131	C₄×C₁₉	#417	1	1	yes	characteristic
132	C₄×C₁₉	#428	1	2	yes	—
133	C₄×C₁₉	#468	1	2	yes	—
134	C₄×C₁₉	#475	1	1	yes	characteristic
135	C₄×C₁₉	#477	1	1	yes	characteristic
136	Dic₁₉	—	4	2	no	—
137	Dic₁₉	—	4	2	no	—
138	C₂×C₄×Q₈	—	19	1	no	—
139	#446	—	19	4	no	—
140	#446	—	38	2	no	—
141	C₂×#107	—	2	1	no	—
142	C₂×#107	—	2	1	no	—
143	C₂×C₄⋊C₈	—	19	1	no	—
144	C₂×C₄⋊C₈	—	19	2	no	—
145	C₂×C₄⋊C₈	—	38	1	no	—
146	C₂×#103	—	38	2	no	—
147	C₂×#97	—	38	1	no	—
148	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
149	#384	C₂×D₁₉	1	2	yes	—
150	#340	—	38	2	no	—
151	#334	—	38	2	no	—
152	C₂×C₁₉	#1688	1	1	yes	characteristic
153	C₂×C₁₉	#1690	1	1	yes	characteristic
154	C₂×C₁₉	#1694	1	1	yes	characteristic
155	C₂×C₁₉	#1696	1	1	yes	characteristic
156	C₂×C₁₉	#1803	1	1	yes	characteristic
157	C₂×C₁₉	#1809	1	1	yes	characteristic
158	C₂×C₁₉	#1816	1	1	yes	characteristic
159	C₂²×Q₈	—	19	1	no	—
160	C₂²×C₈	—	38	1	no	—
161	C₂²×C₈	—	38	2	no	—
162	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
163	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
164	C₄×Q₈	—	19	2	no	—
165	C₄×Q₈	—	19	2	no	—
166	C₄×Q₈	—	38	1	no	—
167	C₄×Q₈	—	38	1	no	—
168	C₂×C₄⋊C₄	—	2	2	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	19	1	no	—
172	C₂×C₄⋊C₄	—	38	1	no	—
173	C₂×C₄²	—	38	1	no	—
174	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
175	#107	—	4	2	no	—
176	#107	—	4	2	no	—
177	C₄⋊C₈	—	19	2	no	—
178	C₄⋊C₈	—	38	4	no	—
179	C₄⋊C₈	—	38	1	no	—
180	C₄⋊C₈	—	38	2	no	—
181	C₄⋊C₈	—	38	2	no	—
182	#103	—	76	4	no	—
183	#97	—	38	4	no	—
184	C₄×C₈	C₄×D₁₉	1	2	yes	—
185	C₄×C₈	D₇₆	1	2	yes	—
186	C₁₉	#8653	1	1	yes	characteristic, complemented, Sylow
187	C₂×Q₈	—	19	2	no	—
188	C₂×Q₈	—	38	1	no	—
189	C₂×Q₈	—	38	2	no	—
190	C₂×Q₈	—	38	2	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	—	38	1	no	—
193	C₂²×C₄	—	38	1	no	—
194	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
195	C₂²×C₄	#64543	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	38	2	no	—
200	C₂×C₈	—	76	1	no	—
201	C₂×C₈	—	76	1	no	—
202	C₂×C₈	—	76	2	no	—
203	C₂×C₈	—	76	2	no	—
204	C₂×C₈	—	76	2	no	—
205	C₂×C₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
206	C₂×C₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
207	C₂×C₈	#64526	1	2	yes	—
208	C₂×C₈	#64528	1	2	yes	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₄⋊C₄	—	8	2	no	—
214	C₄⋊C₄	—	19	2	no	—
215	C₄⋊C₄	—	38	2	no	—
216	C₄⋊C₄	—	38	1	no	—
217	C₄⋊C₄	—	38	2	no	—
218	C₄²	—	38	2	no	—
219	C₄²	—	38	2	no	—
220	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
221	C₄²	C₂×D₇₆	1	1	yes	characteristic
222	C₄²	Q₈×D₁₉	1	1	yes	characteristic
223	C₄²	#64545	1	1	yes	characteristic
224	C₂³	#10597384	1	1	yes	characteristic
225	Q₈	—	38	4	no	—
226	Q₈	—	76	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	—	8	2	no	—
232	C₂×C₄	—	38	2	no	—
233	C₂×C₄	—	38	2	no	—
234	C₂×C₄	—	76	1	no	—
235	C₂×C₄	—	76	1	no	—
236	C₂×C₄	—	76	1	no	—
237	C₂×C₄	—	76	1	no	—
238	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
239	C₂×C₄	#10597338	1	1	yes	characteristic
240	C₂×C₄	#10597341	1	1	yes	characteristic
241	C₂×C₄	#10597343	1	1	yes	characteristic
242	C₂×C₄	#10597357	1	2	yes	—
243	C₂×C₄	#10597364	1	2	yes	—
244	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
245	C₂×C₄	#10597378	1	1	yes	characteristic
246	C₂×C₄	C₂×#64526	1	1	yes	characteristic
247	C₂×C₄	#10597388	1	1	yes	characteristic
248	C₂×C₄	C₂×#64528	1	1	yes	characteristic
249	C₂×C₄	#10597392	1	1	yes	characteristic
250	C₈	—	2	2	no	—
251	C₈	—	76	2	no	—
252	C₈	—	152	2	no	—
253	C₈	#10597267	1	4	yes	complemented
254	C₂²	#11926401	1	1	yes	characteristic
255	C₂²	#11926741	1	1	yes	characteristic
256	C₂²	#11926751	1	1	yes	characteristic
257	C₂²	#11926794	1	1	yes	characteristic
258	C₂²	#11926799	1	1	yes	characteristic
259	C₂²	#11926811	1	1	yes	characteristic
260	C₂²	#11926816	1	1	yes	characteristic
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	16	2	no	—
264	C₄	—	76	2	no	—
265	C₄	—	76	2	no	—
266	C₄	C₂×#10597267	1	1	yes	characteristic
267	C₄	#11926543	1	1	yes	characteristic
268	C₄	#11926585	1	2	yes	—
269	C₄	#11926791	1	2	yes	—
270	C₄	#11926826	1	1	yes	characteristic
271	C₄	#11926836	1	1	yes	characteristic
272	C₂	#14563116	1	1	yes	characteristic
273	C₂	#14563126	1	1	yes	characteristic
274	C₂	#14563150	1	1	yes	characteristic
275	C₂	#14563160	1	1	yes	characteristic
276	C₂	#14563637	1	1	yes	characteristic
277	C₂	#14563669	1	1	yes	characteristic
278	C₂	#14563700	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597384
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1845	C₂
socle, prime radical	C₂³×C₁₉	#121
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1845	D₁₉
hypocenter	C₁₉	#8653
LCS	G ▷ 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
8. #11926674	C₄	semidirect
112. C₈×C₁₉	#103	semidirect
186. C₁₉	#8653	semidirect
253. C₈	#10597267	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926674	2
#14563637	2
C₂×#11926295	2
#14561518	2
C₁₉×#1845	2
#8653	19