Subgroups — #18073201

Order 4864.

498 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	#14563381	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊(C₁₉⋊C₈)	C₂	1	2	yes	maximal
4	#14561476	C₂	1	2	yes	maximal
5	C₁₉×#2295	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₁₉⋊C₈	—	2	1	no	—
8	C₂×C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
9	#11926299	—	2	2	no	—
10	#11926299	—	2	1	no	—
11	Q₈⋊(C₁₉⋊C₈)	C₄	1	8	yes	complemented
12	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
13	C₁₉×#391	C₄	1	2	yes	—
14	C₁₉×#389	C₄	1	2	yes	—
15	C₁₉×#387	—	2	2	no	—
16	C₄×C₄⋊C₄×C₁₉	—	2	1	no	—
17	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
18	C₁₉×#376	—	2	1	no	—
19	C₄³×C₁₉	C₂²	1	1	yes	characteristic
20	C₂²×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
21	C₄×Q₈×C₁₉	—	2	2	no	—
22	C₄×Q₈×C₁₉	C₂×C₄	1	4	yes	—
23	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
24	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
25	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
26	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₉	—	2	1	no	—
28	C₂×C₄²×C₁₉	—	2	1	no	—
29	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₉	D₄	1	2	yes	—
32	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
34	C₂²×C₁₉⋊C₈	—	2	2	no	—
35	C₂²×C₁₉⋊C₈	—	2	2	no	—
36	C₁₉×#95	—	2	2	no	—
37	C₁₉×#95	—	2	2	no	—
38	C₁₉×#95	—	2	2	no	—
39	C₁₉×#95	—	2	1	no	—
40	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
41	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	4	yes	—
42	C₄×C₁₉⋊C₈	—	2	2	no	—
43	C₄×C₁₉⋊C₈	—	4	2	no	—
44	C₄×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
45	C₂×Q₈×C₁₉	—	2	1	no	—
46	C₂×Q₈×C₁₉	—	2	4	no	—
47	C₂×Q₈×C₁₉	C₄²	1	2	yes	—
48	C₂²×C₄×C₁₉	—	2	1	no	—
49	C₂²×C₄×C₁₉	—	2	1	no	—
50	C₂²×C₄×C₁₉	—	2	2	no	—
51	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
56	C₄⋊C₄×C₁₉	—	2	1	no	—
57	C₄⋊C₄×C₁₉	—	2	4	no	—
58	C₄⋊C₄×C₁₉	—	2	4	no	—
59	C₄⋊C₄×C₁₉	—	4	4	no	—
60	C₄⋊C₄×C₁₉	C₄²	1	2	yes	—
61	C₄²×C₁₉	—	2	4	no	—
62	C₄²×C₁₉	—	2	4	no	—
63	C₄²×C₁₉	—	2	2	no	—
64	C₄²×C₁₉	—	4	2	no	—
65	C₄²×C₁₉	—	4	2	no	—
66	C₄²×C₁₉	SD₁₆	1	2	yes	—
67	C₄²×C₁₉	Q₁₆	1	2	yes	—
68	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
69	C₄²×C₁₉	#41	1	2	yes	—
70	C₂×C₁₉⋊C₈	—	2	4	no	—
71	C₂×C₁₉⋊C₈	—	2	4	no	—
72	C₂×C₁₉⋊C₈	—	4	4	no	—
73	C₂×C₁₉⋊C₈	—	4	2	no	—
74	C₂×C₁₉⋊C₈	—	4	2	no	—
75	#8182	—	19	1	no	maximal, Sylow
76	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	Q₈×C₁₉	—	2	4	no	—
78	Q₈×C₁₉	—	4	2	no	—
79	C₂×C₄×C₁₉	—	2	2	no	—
80	C₂×C₄×C₁₉	—	2	2	no	—
81	C₂×C₄×C₁₉	—	2	2	no	—
82	C₂×C₄×C₁₉	—	2	2	no	—
83	C₂×C₄×C₁₉	—	2	2	no	—
84	C₂×C₄×C₁₉	—	2	2	no	—
85	C₂×C₄×C₁₉	—	2	2	no	—
86	C₂×C₄×C₁₉	—	2	2	no	—
87	C₂×C₄×C₁₉	—	4	2	no	—
88	C₂×C₄×C₁₉	—	4	2	no	—
89	C₂×C₄×C₁₉	—	4	2	no	—
90	C₂×C₄×C₁₉	—	4	2	no	—
91	C₂×C₄×C₁₉	—	4	2	no	—
92	C₂×C₄×C₁₉	#103	1	4	yes	—
93	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
94	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
95	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
96	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
97	C₂×C₄×C₁₉	#131	1	4	yes	—
98	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
99	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
100	C₂×C₄×C₁₉	#135	1	2	yes	—
101	C₁₉⋊C₈	—	4	4	no	—
102	C₁₉⋊C₈	—	4	4	no	—
103	#2295	D₁₉	1	1	yes	characteristic
104	C₄×C₄⋊C₈	—	19	1	no	—
105	#1744	—	19	1	no	—
106	C₂×Q₈⋊C₈	—	19	2	no	—
107	#1279	—	19	2	no	—
108	C₂²×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
109	C₂²×C₁₉	#408	1	2	yes	—
110	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
111	C₂²×C₁₉	#416	1	1	yes	characteristic
112	C₂²×C₁₉	#421	1	2	yes	—
113	C₄×C₁₉	—	2	4	no	—
114	C₄×C₁₉	—	2	2	no	—
115	C₄×C₁₉	—	4	2	no	—
116	C₄×C₁₉	—	4	2	no	—
117	C₄×C₁₉	—	4	2	no	—
118	C₄×C₁₉	—	8	2	no	—
119	C₄×C₁₉	#405	1	2	yes	—
120	C₄×C₁₉	#433	1	2	yes	—
121	C₄×C₁₉	C₄×SD₁₆	1	2	yes	—
122	C₄×C₁₉	C₄×Q₁₆	1	2	yes	—
123	C₂×C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
124	C₂×C₄⋊C₈	—	19	2	no	—
125	C₂×C₄×C₈	—	19	2	no	—
126	C₂×C₄×C₈	—	38	1	no	—
127	#391	Dic₁₉	1	2	yes	—
128	#389	Dic₁₉	1	2	yes	—
129	#387	—	2	2	no	—
130	C₄×C₄⋊C₄	—	2	1	no	—
131	C₄×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
132	#376	—	2	1	no	—
133	C₄³	C₂×D₁₉	1	1	yes	characteristic
134	#336	—	38	2	no	—
135	#336	—	38	1	no	—
136	Q₈⋊C₈	—	19	8	no	—
137	C₂×C₁₉	#1481	1	2	yes	—
138	C₂×C₁₉	#1482	1	2	yes	—
139	C₂×C₁₉	#1750	1	1	yes	characteristic
140	C₂×C₁₉	#1759	1	1	yes	characteristic
141	C₂×C₁₉	C₄×#103	1	1	yes	characteristic
142	C₂²×Q₈	C₂×Dic₁₉	1	1	yes	characteristic
143	C₂²×C₈	—	38	2	no	—
144	C₂²×C₈	—	38	2	no	—
145	C₄×Q₈	—	2	2	no	—
146	C₄×Q₈	C₄×D₁₉	1	4	yes	—
147	C₂×C₄⋊C₄	—	2	1	no	—
148	C₂×C₄⋊C₄	—	2	2	no	—
149	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
150	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
151	C₂×C₄²	—	2	1	no	—
152	C₂×C₄²	—	2	1	no	—
153	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
154	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
155	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
156	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
157	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
158	C₄⋊C₈	—	19	4	no	—
159	C₄⋊C₈	—	38	2	no	—
160	C₄×C₈	—	19	4	no	—
161	C₄×C₈	—	38	2	no	—
162	C₄×C₈	—	76	2	no	—
163	#95	—	2	2	no	—
164	#95	—	2	2	no	—
165	#95	—	2	2	no	—
166	#95	—	2	1	no	—
167	C₁₉	#8182	1	1	yes	characteristic, complemented, Sylow
168	C₂×Q₈	—	2	1	no	—
169	C₂×Q₈	—	2	4	no	—
170	C₂×Q₈	C₄×Dic₁₉	1	2	yes	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
175	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
176	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
178	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
179	C₂×C₈	—	38	4	no	—
180	C₂×C₈	—	38	4	no	—
181	C₂×C₈	—	76	4	no	—
182	C₂×C₈	—	76	2	no	—
183	C₂×C₈	—	76	2	no	—
184	C₄⋊C₄	—	2	1	no	—
185	C₄⋊C₄	—	2	4	no	—
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	4	4	no	—
188	C₄⋊C₄	C₄×Dic₁₉	1	2	yes	—
189	C₄²	—	2	4	no	—
190	C₄²	—	2	2	no	—
191	C₄²	—	2	4	no	—
192	C₄²	—	4	2	no	—
193	C₄²	—	4	2	no	—
194	C₄²	#64527	1	2	yes	—
195	C₄²	#64528	1	2	yes	—
196	C₄²	C₂×C₄×D₁₉	1	2	yes	—
197	C₄²	#64541	1	2	yes	—
198	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
199	Q₈	—	2	4	no	—
200	Q₈	—	4	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	#10597292	1	4	yes	—
215	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
216	C₂×C₄	#10597372	1	1	yes	characteristic
217	C₂×C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
218	C₂×C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
219	C₂×C₄	C₂×#64527	1	1	yes	characteristic
220	C₂×C₄	C₂×#64528	1	1	yes	characteristic
221	C₂×C₄	#10597394	1	4	yes	—
222	C₂×C₄	#10597396	1	2	yes	—
223	C₈	—	76	4	no	—
224	C₈	—	76	4	no	—
225	C₂²	C₄×C₂²⋊Dic₁₉	1	1	yes	characteristic
226	C₂²	#11926511	1	2	yes	—
227	C₂²	C₂×#10597292	1	1	yes	characteristic
228	C₂²	#11926540	1	1	yes	characteristic
229	C₂²	#11926561	1	2	yes	—
230	C₄	—	2	2	no	—
231	C₄	—	2	4	no	—
232	C₄	—	4	2	no	—
233	C₄	—	4	2	no	—
234	C₄	—	4	2	no	—
235	C₄	—	8	2	no	—
236	C₄	#11926500	1	2	yes	—
237	C₄	#11926605	1	2	yes	—
238	C₄	C₄×#64527	1	2	yes	—
239	C₄	C₄×#64528	1	2	yes	—
240	C₂	#14562038	1	2	yes	—
241	C₂	#14562043	1	2	yes	—
242	C₂	#14563401	1	1	yes	characteristic
243	C₂	#14563445	1	1	yes	characteristic
244	C₂	C₄×#10597292	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#2295	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2295	D₁₉
hypocenter	C₁₉	#8182
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #2295

Complements

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

N	H ≅ G/N	type
11. Q₈⋊(C₁₉⋊C₈)	C₄	semidirect
167. C₁₉	#8182	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊(C₁₉⋊C₈)	2
#14563381	2
C₂×Q₈⋊(C₁₉⋊C₈)	2
#14561476	2
C₁₉×#2295	2
#8182	19