Subgroups — #18081227

Order 4864.

370 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14564399	C₂	1	2	yes	maximal
2	#14563795	C₂	1	1	yes	maximal, characteristic
3	#14563502	C₂	1	1	yes	maximal, characteristic
4	#14561519	C₂	1	2	yes	maximal
5	C₂×C₁₉×#334	C₂	1	1	yes	maximal, characteristic
6	C₂×#10597275	—	2	1	no	—
7	C₂×#10597275	—	2	1	no	—
8	C₂×#10597274	—	2	2	no	—
9	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×#10597272	—	2	1	no	—
11	#11926414	C₂²	1	2	yes	—
12	C₄×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
13	#11926307	—	2	2	no	—
14	#11926298	—	2	1	no	—
15	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#334	C₄	1	4	yes	complemented
18	C₂²×C₈×C₁₉	—	2	2	no	—
19	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₉	—	2	2	no	—
23	C₂×C₄⋊Dic₁₉	—	2	2	no	—
24	C₂×C₄⋊Dic₁₉	—	2	2	no	—
25	C₂×C₄⋊Dic₁₉	D₄	1	1	yes	characteristic
26	C₂×C₄⋊Dic₁₉	D₄	1	1	yes	characteristic
27	C₂×C₄×Dic₁₉	—	2	1	no	—
28	C₂²×C₁₉⋊C₈	—	2	1	no	—
29	C₁₉×C₄⋊C₈	—	2	2	no	—
30	C₁₉×C₄⋊C₈	C₂×C₄	1	4	yes	—
31	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₁₉	Q₈	1	2	yes	—
33	C₁₉⋊#95	—	2	1	no	—
34	#10597275	—	4	2	no	—
35	#10597275	—	4	2	no	—
36	#10597274	—	4	4	no	—
37	#10597272	—	4	2	no	—
38	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
39	C₄⋊(C₁₉⋊C₈)	Q₈	1	2	yes	—
40	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
43	C₂²×Dic₁₉	—	2	1	no	—
44	C₂²×Dic₁₉	—	2	1	no	—
45	C₂²×Dic₁₉	—	4	2	no	—
46	C₂×C₈×C₁₉	—	2	1	no	—
47	C₂×C₈×C₁₉	—	2	1	no	—
48	C₂×C₈×C₁₉	—	2	4	no	—
49	C₂×C₈×C₁₉	—	4	2	no	—
50	C₂×C₈×C₁₉	—	4	2	no	—
51	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
52	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
53	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
54	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
55	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
57	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₁₉	#41	1	1	yes	characteristic
59	C₄⋊Dic₁₉	—	2	2	no	—
60	C₄⋊Dic₁₉	—	2	2	no	—
61	C₄⋊Dic₁₉	—	4	2	no	—
62	C₄⋊Dic₁₉	—	4	2	no	—
63	C₄⋊Dic₁₉	—	4	2	no	—
64	C₄⋊Dic₁₉	—	4	2	no	—
65	C₄⋊Dic₁₉	—	8	2	no	—
66	C₄×Dic₁₉	—	4	1	no	—
67	C₄×Dic₁₉	—	4	1	no	—
68	C₂×C₁₉⋊C₈	—	2	2	no	—
69	C₂×C₁₉⋊C₈	—	4	1	no	—
70	C₂×C₁₉⋊C₈	—	4	1	no	—
71	#9621	—	19	1	no	maximal, Sylow
72	C₂³×C₁₉	#121	1	1	yes	characteristic
73	C₂×C₄×C₁₉	—	2	1	no	—
74	C₂×C₄×C₁₉	—	2	1	no	—
75	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
77	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
78	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₁₉	#125	1	1	yes	characteristic
80	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
82	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
83	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
84	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
85	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
86	C₂×C₄×C₁₉	#137	1	2	yes	—
87	C₂×Dic₁₉	—	4	1	no	—
88	C₂×Dic₁₉	—	4	1	no	—
89	C₂×Dic₁₉	—	4	1	no	—
90	C₂×Dic₁₉	—	4	1	no	—
91	C₂×Dic₁₉	—	4	1	no	—
92	C₂×Dic₁₉	—	4	1	no	—
93	C₂×Dic₁₉	—	8	2	no	—
94	C₂×Dic₁₉	—	8	2	no	—
95	C₂×Dic₁₉	—	8	2	no	—
96	C₈×C₁₉	—	2	2	no	—
97	C₈×C₁₉	—	4	4	no	—
98	C₈×C₁₉	#106	1	4	yes	complemented
99	C₁₉⋊C₈	—	4	2	no	—
100	#1948	—	19	2	no	—
101	#1845	—	19	1	no	—
102	#1770	—	19	1	no	—
103	C₂×#334	D₁₉	1	1	yes	characteristic
104	#1296	—	19	2	no	—
105	C₂²×C₁₉	#389	1	1	yes	characteristic
106	C₂²×C₁₉	#460	1	1	yes	characteristic
107	C₂²×C₁₉	#461	1	1	yes	characteristic
108	C₂²×C₁₉	#469	1	1	yes	characteristic
109	C₂²×C₁₉	#470	1	1	yes	characteristic
110	C₂²×C₁₉	#472	1	1	yes	characteristic
111	C₂²×C₁₉	#473	1	1	yes	characteristic
112	C₄×C₁₉	—	2	1	no	—
113	C₄×C₁₉	—	2	1	no	—
114	C₄×C₁₉	C₂×#106	1	1	yes	characteristic
115	C₄×C₁₉	#428	1	1	yes	characteristic
116	C₄×C₁₉	#440	1	2	yes	—
117	C₄×C₁₉	#475	1	1	yes	characteristic
118	C₄×C₁₉	#476	1	1	yes	characteristic
119	C₄×C₁₉	#501	1	2	yes	—
120	Dic₁₉	—	8	1	no	—
121	Dic₁₉	—	8	1	no	—
122	Dic₁₉	—	16	2	no	—
123	C₂×#107	—	38	1	no	—
124	C₂×#107	—	38	1	no	—
125	C₂×#106	—	38	2	no	—
126	C₂×C₄⋊C₈	—	19	1	no	—
127	C₂×C₄⋊C₈	C₂×D₁₉	1	2	yes	—
128	C₂×#97	—	38	1	no	—
129	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
130	#384	—	19	2	no	—
131	C₄×C₄⋊C₄	—	19	1	no	—
132	#340	—	38	2	no	—
133	#336	—	38	1	no	—
134	#334	Dic₁₉	1	4	yes	complemented
135	C₂×C₁₉	#1688	1	1	yes	characteristic
136	C₂×C₁₉	#1690	1	1	yes	characteristic
137	C₂×C₁₉	#1693	1	1	yes	characteristic
138	C₂×C₁₉	#1695	1	1	yes	characteristic
139	C₂×C₁₉	#1944	1	1	yes	characteristic
140	C₂×C₁₉	#1948	1	1	yes	characteristic
141	C₂×C₁₉	#1952	1	1	yes	characteristic
142	C₂²×C₈	—	2	2	no	—
143	C₂²×C₈	—	38	1	no	—
144	C₂²×C₈	D₇₆	1	1	yes	characteristic
145	C₂²×C₈	D₇₆	1	1	yes	characteristic
146	C₂×C₄⋊C₄	—	19	1	no	—
147	C₂×C₄⋊C₄	—	19	1	no	—
148	C₂×C₄⋊C₄	—	38	2	no	—
149	C₂×C₄⋊C₄	—	38	2	no	—
150	C₂×C₄⋊C₄	—	38	2	no	—
151	C₂×C₄²	—	38	1	no	—
152	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
153	#107	—	76	2	no	—
154	#107	—	76	2	no	—
155	#106	—	76	4	no	—
156	C₄⋊C₈	—	2	2	no	—
157	C₄⋊C₈	—	19	2	no	—
158	C₄⋊C₈	—	38	1	no	—
159	C₄⋊C₈	C₂×Dic₁₉	1	4	yes	—
160	#97	—	76	2	no	—
161	C₄×C₈	Dic₃₈	1	2	yes	—
162	C₄×C₈	C₂×Dic₁₉	1	2	yes	—
163	#95	—	38	1	no	—
164	C₁₉	#9621	1	1	yes	characteristic, complemented, Sylow
165	C₂²×C₄	—	38	1	no	—
166	C₂²×C₄	—	38	1	no	—
167	C₂²×C₄	—	76	2	no	—
168	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
169	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
170	C₂²×C₄	#64545	1	1	yes	characteristic
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	4	2	no	—
175	C₂×C₈	—	4	2	no	—
176	C₂×C₈	—	38	2	no	—
177	C₂×C₈	—	76	1	no	—
178	C₂×C₈	—	76	1	no	—
179	C₂×C₈	#64516	1	2	yes	—
180	C₂×C₈	#64516	1	2	yes	—
181	C₂×C₈	C₄⋊Dic₁₉	1	2	yes	—
182	C₂×C₈	C₄⋊Dic₁₉	1	2	yes	—
183	C₄⋊C₄	—	38	2	no	—
184	C₄⋊C₄	—	38	2	no	—
185	C₄⋊C₄	—	76	2	no	—
186	C₄⋊C₄	—	76	2	no	—
187	C₄⋊C₄	—	76	2	no	—
188	C₄⋊C₄	—	76	2	no	—
189	C₄⋊C₄	—	152	2	no	—
190	C₄²	—	76	1	no	—
191	C₄²	—	76	1	no	—
192	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
193	C₄²	#64543	1	1	yes	characteristic
194	C₄²	Q₈×D₁₉	1	1	yes	characteristic
195	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
196	C₂³	#10597341	1	1	yes	characteristic
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	76	1	no	—
200	C₂×C₄	—	76	1	no	—
201	C₂×C₄	—	76	1	no	—
202	C₂×C₄	—	76	1	no	—
203	C₂×C₄	—	76	1	no	—
204	C₂×C₄	—	76	1	no	—
205	C₂×C₄	—	152	2	no	—
206	C₂×C₄	—	152	2	no	—
207	C₂×C₄	—	152	2	no	—
208	C₂×C₄	#10597334	1	1	yes	characteristic
209	C₂×C₄	#10597336	1	1	yes	characteristic
210	C₂×C₄	#10597343	1	1	yes	characteristic
211	C₂×C₄	C₂×#64516	1	1	yes	characteristic
212	C₂×C₄	C₂×#64516	1	1	yes	characteristic
213	C₂×C₄	#10597354	1	1	yes	characteristic
214	C₂×C₄	#10597355	1	1	yes	characteristic
215	C₂×C₄	#10597360	1	2	yes	—
216	C₂×C₄	#10597361	1	2	yes	—
217	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
218	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
219	C₂×C₄	Q₈×Dic₁₉	1	1	yes	characteristic
220	C₈	—	2	2	no	—
221	C₈	—	4	4	no	—
222	C₈	—	76	2	no	—
223	C₈	#10597274	1	4	yes	complemented
224	C₂²	#11926443	1	1	yes	characteristic
225	C₂²	#11926745	1	1	yes	characteristic
226	C₂²	#11926750	1	1	yes	characteristic
227	C₂²	#11926798	1	1	yes	characteristic
228	C₂²	#11926803	1	1	yes	characteristic
229	C₂²	#11926815	1	1	yes	characteristic
230	C₂²	#11926820	1	1	yes	characteristic
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	304	2	no	—
234	C₄	—	152	1	no	—
235	C₄	—	152	1	no	—
236	C₄	C₂×#10597274	1	1	yes	characteristic
237	C₄	#11926586	1	1	yes	characteristic
238	C₄	#11926645	1	2	yes	—
239	C₄	#11926830	1	1	yes	characteristic
240	C₄	#11926835	1	1	yes	characteristic
241	C₄	#11926951	1	2	yes	—
242	C₂	#14563120	1	1	yes	characteristic
243	C₂	#14563130	1	1	yes	characteristic
244	C₂	#14563149	1	1	yes	characteristic
245	C₂	#14563159	1	1	yes	characteristic
246	C₂	#14564379	1	1	yes	characteristic
247	C₂	#14564399	1	1	yes	characteristic
248	C₂	#14564421	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597341
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#334	C₂
socle, prime radical	C₂³×C₁₉	#121
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#334	D₁₉
hypocenter	C₁₉	#9621
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
17. C₁₉×#334	C₄	semidirect
98. C₈×C₁₉	#106	semidirect
134. #334	Dic₁₉	semidirect
164. C₁₉	#9621	semidirect
223. C₈	#10597274	semidirect

Maximal subgroups

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

subgroup	index
#14564399	2
#14563795	2
#14563502	2
#14561519	2
C₂×C₁₉×#334	2
#9621	19