Subgroups — #18081082

Order 4864.

398 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14564354	C₂	1	2	yes	maximal
2	C₂×#11926297	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#14561476	C₂	1	2	yes	maximal
5	C₄×C₁₉×#107	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
7	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
10	#11926299	—	2	2	no	—
11	#11926297	—	2	2	no	—
12	C₈×C₁₉⋊C₈	—	2	2	no	—
13	C₂×C₁₉×#107	C₄	1	1	yes	characteristic
14	C₂×C₁₉×#107	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
17	C₁₉×#340	C₄	1	2	yes	—
18	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
21	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₁₉	—	2	2	no	—
23	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
24	C₂²×C₁₉⋊C₈	—	2	1	no	—
25	C₂²×C₁₉⋊C₈	—	2	2	no	—
26	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
27	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
28	C₁₉×#107	—	2	2	no	—
29	C₁₉×#107	C₈	1	4	yes	complemented
30	C₄×C₈×C₁₉	D₄	1	2	yes	—
31	C₄×C₈×C₁₉	Q₈	1	2	yes	—
32	C₁₉×#95	—	2	2	no	—
33	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
34	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
35	C₄⋊(C₁₉⋊C₈)	—	4	4	no	—
36	C₄×C₁₉⋊C₈	—	2	1	no	—
37	C₄×C₁₉⋊C₈	—	2	1	no	—
38	C₄×C₁₉⋊C₈	—	2	1	no	—
39	C₄×C₁₉⋊C₈	Q₈	1	2	yes	—
40	C₂²×C₄×C₁₉	—	2	2	no	—
41	C₂²×C₄×C₁₉	—	2	2	no	—
42	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
45	C₂×C₈×C₁₉	—	2	1	no	—
46	C₂×C₈×C₁₉	—	2	1	no	—
47	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
49	C₂×C₈×C₁₉	C₂×C₈	1	2	yes	—
50	C₂×C₈×C₁₉	M₁₆	1	2	yes	—
51	C₄⋊C₄×C₁₉	—	2	2	no	—
52	C₄⋊C₄×C₁₉	—	2	2	no	—
53	C₄⋊C₄×C₁₉	—	2	2	no	—
54	C₄⋊C₄×C₁₉	C₂×C₈	1	4	yes	—
55	C₄²×C₁₉	—	4	2	no	—
56	C₄²×C₁₉	—	4	2	no	—
57	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
60	C₄²×C₁₉	#41	1	1	yes	characteristic
61	C₂×C₁₉⋊C₈	—	2	2	no	—
62	C₂×C₁₉⋊C₈	—	2	2	no	—
63	C₂×C₁₉⋊C₈	—	2	1	no	—
64	C₂×C₁₉⋊C₈	—	2	1	no	—
65	C₂×C₁₉⋊C₈	—	2	2	no	—
66	C₂×C₁₉⋊C₈	—	2	1	no	—
67	C₂×C₁₉⋊C₈	—	2	1	no	—
68	C₂×C₁₉⋊C₈	—	4	2	no	—
69	C₂×C₁₉⋊C₈	—	4	2	no	—
70	C₂×C₁₉⋊C₈	—	4	2	no	—
71	C₂×C₁₉⋊C₈	D₈	1	2	yes	—
72	C₂×C₁₉⋊C₈	Q₁₆	1	2	yes	—
73	#9594	—	19	1	no	maximal, Sylow
74	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
75	C₂×C₄×C₁₉	—	2	1	no	—
76	C₂×C₄×C₁₉	—	2	1	no	—
77	C₂×C₄×C₁₉	—	2	4	no	—
78	C₂×C₄×C₁₉	—	4	2	no	—
79	C₂×C₄×C₁₉	—	4	2	no	—
80	C₂×C₄×C₁₉	—	4	2	no	—
81	C₂×C₄×C₁₉	—	4	2	no	—
82	C₂×C₄×C₁₉	—	4	2	no	—
83	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
85	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
86	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₁₉	#125	1	1	yes	characteristic
88	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
89	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
91	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
92	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
93	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
94	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
95	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
96	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
97	C₈×C₁₉	—	2	2	no	—
98	C₈×C₁₉	C₄⋊C₈	1	4	yes	complemented
99	C₁₉⋊C₈	—	2	4	no	—
100	C₁₉⋊C₈	—	2	2	no	—
101	C₁₉⋊C₈	—	2	2	no	—
102	C₁₉⋊C₈	—	8	2	no	—
103	#1939	—	19	2	no	—
104	C₄×#107	D₁₉	1	1	yes	characteristic
105	C₂×#335	—	19	1	no	—
106	C₂×C₈²	—	19	1	no	—
107	#1279	—	19	2	no	—
108	C₂²×C₁₉	#389	1	1	yes	characteristic
109	C₂²×C₁₉	C₈×D₄	1	1	yes	characteristic
110	C₂²×C₁₉	#436	1	1	yes	characteristic
111	C₂²×C₁₉	C₄×D₈	1	1	yes	characteristic
112	C₂²×C₁₉	C₄×Q₁₆	1	1	yes	characteristic
113	C₂²×C₁₉	#443	1	1	yes	characteristic
114	C₂²×C₁₉	#443	1	1	yes	characteristic
115	C₄×C₁₉	—	2	1	no	—
116	C₄×C₁₉	—	2	1	no	—
117	C₄×C₁₉	—	4	4	no	—
118	C₄×C₁₉	—	8	2	no	—
119	C₄×C₁₉	C₂×C₄⋊C₈	1	1	yes	characteristic
120	C₄×C₁₉	#424	1	1	yes	characteristic
121	C₄×C₁₉	C₈×Q₈	1	1	yes	characteristic
122	C₄×C₁₉	#446	1	1	yes	characteristic
123	C₄×C₁₉	#466	1	1	yes	characteristic
124	C₄×C₁₉	#467	1	1	yes	characteristic
125	C₄×C₁₉	#499	1	1	yes	characteristic
126	C₄×C₁₉	#500	1	1	yes	characteristic
127	C₂×#107	Dic₁₉	1	1	yes	characteristic
128	C₂×#107	Dic₁₉	1	1	yes	characteristic
129	C₂×C₄⋊C₈	—	19	2	no	—
130	C₂×C₄⋊C₈	—	38	2	no	—
131	C₂×C₄×C₈	—	19	1	no	—
132	C₂×C₄×C₈	—	19	1	no	—
133	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
134	C₄×C₄⋊C₄	C₂×D₁₉	1	2	yes	—
135	#340	Dic₁₉	1	2	yes	—
136	#336	—	38	2	no	—
137	#335	—	38	2	no	—
138	C₈²	—	38	2	no	—
139	C₂×C₁₉	C₈×D₈	1	1	yes	characteristic
140	C₂×C₁₉	C₈×Q₁₆	1	1	yes	characteristic
141	C₂×C₁₉	#1589	1	1	yes	characteristic
142	C₂×C₁₉	#1591	1	1	yes	characteristic
143	C₂×C₁₉	#1939	1	1	yes	characteristic
144	C₂×C₁₉	#1942	1	1	yes	characteristic
145	C₂×C₁₉	#1945	1	1	yes	characteristic
146	C₂²×C₈	—	19	1	no	—
147	C₂²×C₈	—	19	1	no	—
148	C₂²×C₈	—	38	1	no	—
149	C₂²×C₈	—	38	2	no	—
150	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
151	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
152	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
153	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
154	C₂×C₄²	—	2	2	no	—
155	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
156	#107	—	2	2	no	—
157	#107	C₁₉⋊C₈	1	4	yes	complemented
158	C₄⋊C₈	—	38	2	no	—
159	C₄⋊C₈	—	38	2	no	—
160	C₄⋊C₈	—	76	4	no	—
161	C₄×C₈	—	19	2	no	—
162	C₄×C₈	—	38	1	no	—
163	C₄×C₈	—	38	1	no	—
164	C₄×C₈	—	38	1	no	—
165	C₄×C₈	Dic₃₈	1	2	yes	—
166	C₄×C₈	D₇₆	1	2	yes	—
167	#95	—	2	2	no	—
168	C₁₉	#9594	1	1	yes	characteristic, complemented, Sylow
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
172	C₂²×C₄	#64543	1	1	yes	characteristic
173	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	19	2	no	—
177	C₂×C₈	—	19	2	no	—
178	C₂×C₈	—	38	2	no	—
179	C₂×C₈	—	38	2	no	—
180	C₂×C₈	—	38	1	no	—
181	C₂×C₈	—	38	1	no	—
182	C₂×C₈	—	38	2	no	—
183	C₂×C₈	—	38	1	no	—
184	C₂×C₈	—	38	1	no	—
185	C₂×C₈	—	76	2	no	—
186	C₂×C₈	—	76	2	no	—
187	C₂×C₈	—	76	2	no	—
188	C₂×C₈	C₂×C₁₉⋊C₈	1	2	yes	—
189	C₂×C₈	#64520	1	2	yes	—
190	C₂×C₈	C₄⋊Dic₁₉	1	2	yes	—
191	C₂×C₈	C₄⋊Dic₁₉	1	2	yes	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	C₂×C₁₉⋊C₈	1	4	yes	—
196	C₄²	—	4	2	no	—
197	C₄²	—	4	2	no	—
198	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
199	C₄²	C₂×D₇₆	1	1	yes	characteristic
200	C₄²	Q₈×D₁₉	1	1	yes	characteristic
201	C₄²	#64545	1	1	yes	characteristic
202	C₂³	D₄×Dic₁₉	1	1	yes	characteristic
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	#10597334	1	1	yes	characteristic
212	C₂×C₄	#10597336	1	1	yes	characteristic
213	C₂×C₄	#10597341	1	1	yes	characteristic
214	C₂×C₄	#10597343	1	1	yes	characteristic
215	C₂×C₄	D₈×D₁₉	1	1	yes	characteristic
216	C₂×C₄	#10597358	1	1	yes	characteristic
217	C₂×C₄	Q₁₆×D₁₉	1	1	yes	characteristic
218	C₂×C₄	#10597365	1	1	yes	characteristic
219	C₂×C₄	C₂²×C₁₉⋊C₈	1	1	yes	characteristic
220	C₂×C₄	C₂×#64520	1	1	yes	characteristic
221	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
222	C₂×C₄	Q₈×Dic₁₉	1	1	yes	characteristic
223	C₂×C₄	#10597394	1	1	yes	characteristic
224	C₂×C₄	#10597394	1	1	yes	characteristic
225	C₈	—	2	2	no	—
226	C₈	—	38	4	no	—
227	C₈	—	38	2	no	—
228	C₈	—	38	2	no	—
229	C₈	—	152	2	no	—
230	C₈	C₄⋊(C₁₉⋊C₈)	1	4	yes	complemented
231	C₂²	#11926443	1	1	yes	characteristic
232	C₂²	D₄×C₁₉⋊C₈	1	1	yes	characteristic
233	C₂²	#11926622	1	1	yes	characteristic
234	C₂²	D₈×Dic₁₉	1	1	yes	characteristic
235	C₂²	Q₁₆×Dic₁₉	1	1	yes	characteristic
236	C₂²	#11926661	1	1	yes	characteristic
237	C₂²	#11926661	1	1	yes	characteristic
238	C₄	—	2	1	no	—
239	C₄	—	2	1	no	—
240	C₄	—	4	4	no	—
241	C₄	—	8	2	no	—
242	C₄	C₂×C₄⋊(C₁₉⋊C₈)	1	1	yes	characteristic
243	C₄	#11926571	1	1	yes	characteristic
244	C₄	Q₈×C₁₉⋊C₈	1	1	yes	characteristic
245	C₄	#11926675	1	1	yes	characteristic
246	C₄	#11926781	1	1	yes	characteristic
247	C₄	#11926786	1	1	yes	characteristic
248	C₄	#11926945	1	1	yes	characteristic
249	C₄	#11926948	1	1	yes	characteristic
250	C₂	D₈×C₁₉⋊C₈	1	1	yes	characteristic
251	C₂	Q₁₆×C₁₉⋊C₈	1	1	yes	characteristic
252	C₂	#14562520	1	1	yes	characteristic
253	C₂	#14562532	1	1	yes	characteristic
254	C₂	#14564354	1	1	yes	characteristic
255	C₂	#14564369	1	1	yes	characteristic
256	C₂	#14564384	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₉
commutator	C₂×C₄×C₁₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₄×C₁₉×#107	C₂
socle, prime radical	C₂³×C₁₉	C₄×D₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₄×#107	D₁₉
hypocenter	C₁₉	#9594
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
29. C₁₉×#107	C₈	semidirect
98. C₈×C₁₉	C₄⋊C₈	semidirect
157. #107	C₁₉⋊C₈	semidirect
168. C₁₉	#9594	semidirect
230. C₈	C₄⋊(C₁₉⋊C₈)	semidirect

Maximal subgroups

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

subgroup	index
#14564354	2
C₂×#11926297	2
C₂×C₈×C₁₉⋊C₈	2
#14561476	2
C₄×C₁₉×#107	2
#9594	19