Subgroups — #18073017

Order 4864.

642 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂²⋊(C₁₉⋊C₈)	C₂	1	2	yes	maximal
2	C₂×#11926265	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₈×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉⋊#1273	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×M₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂²⋊(C₁₉⋊C₈)	C₄	1	4	yes	—
7	C₂×C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
9	#11926299	C₄	1	4	yes	—
10	#11926265	—	2	2	no	—
11	C₈×C₁₉⋊C₈	—	2	2	no	—
12	C₂²×M₁₆×C₁₉	C₄	1	2	yes	—
13	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
14	C₄×M₁₆×C₁₉	—	2	2	no	—
15	C₄×M₁₆×C₁₉	—	2	2	no	—
16	C₂×C₁₉×#97	C₄	1	2	yes	—
17	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
19	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
21	C₂×M₁₆×C₁₉	—	2	4	no	—
22	C₂×M₁₆×C₁₉	—	2	4	no	—
23	C₂×M₁₆×C₁₉	C₈	1	8	yes	complemented
24	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₁₉	—	2	1	no	—
27	C₂×C₄²×C₁₉	—	2	1	no	—
28	C₂×C₄²×C₁₉	—	2	2	no	—
29	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
32	C₂²×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
33	C₂²×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
34	C₁₉×#97	—	2	4	no	—
35	C₄×C₈×C₁₉	—	2	1	no	—
36	C₄×C₈×C₁₉	—	2	1	no	—
37	C₄×C₈×C₁₉	D₄	1	2	yes	—
38	C₄×C₈×C₁₉	D₄	1	2	yes	—
39	C₂²⋊(C₁₉⋊C₈)	—	2	8	no	—
40	C₄×C₁₉⋊C₈	—	2	2	no	—
41	C₄×C₁₉⋊C₈	—	2	2	no	—
42	C₄×C₁₉⋊C₈	—	2	2	no	—
43	C₄×C₁₉⋊C₈	—	2	2	no	—
44	C₂⁴×C₁₉	C₄²	1	1	yes	characteristic
45	C₂²×C₄×C₁₉	—	2	1	no	—
46	C₂²×C₄×C₁₉	—	2	1	no	—
47	C₂²×C₄×C₁₉	—	2	2	no	—
48	C₂²×C₄×C₁₉	—	2	2	no	—
49	C₂²×C₄×C₁₉	—	2	2	no	—
50	C₂²×C₄×C₁₉	—	2	4	no	—
51	C₂²×C₄×C₁₉	—	2	4	no	—
52	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
53	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	C₂×C₈	1	4	yes	—
55	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
57	M₁₆×C₁₉	—	2	16	no	—
58	C₂×C₈×C₁₉	—	2	2	no	—
59	C₂×C₈×C₁₉	—	2	2	no	—
60	C₂×C₈×C₁₉	C₂²⋊C₄	1	4	yes	—
61	C₂×C₈×C₁₉	C₂×C₈	1	4	yes	—
62	C₂×C₈×C₁₉	C₂×C₈	1	4	yes	—
63	C₂×C₈×C₁₉	M₁₆	1	4	yes	—
64	C₄²×C₁₉	—	2	1	no	—
65	C₄²×C₁₉	—	2	1	no	—
66	C₄²×C₁₉	—	2	2	no	—
67	C₄²×C₁₉	—	2	2	no	—
68	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₁₉	#41	1	2	yes	—
71	C₂×C₁₉⋊C₈	—	2	4	no	—
72	C₂×C₁₉⋊C₈	—	2	4	no	—
73	C₂×C₁₉⋊C₈	—	2	4	no	—
74	C₂×C₁₉⋊C₈	—	2	4	no	—
75	C₂×C₁₉⋊C₈	—	2	4	no	—
76	C₂×C₁₉⋊C₈	M₁₆	1	8	yes	complemented
77	#8146	—	19	1	no	maximal, Sylow
78	C₂³×C₁₉	—	2	1	no	—
79	C₂³×C₁₉	—	2	2	no	—
80	C₂³×C₁₉	—	2	1	no	—
81	C₂³×C₁₉	—	2	2	no	—
82	C₂³×C₁₉	C₄×C₈	1	2	yes	—
83	C₂³×C₁₉	C₂×C₄²	1	1	yes	characteristic
84	C₂×C₄×C₁₉	—	2	1	no	—
85	C₂×C₄×C₁₉	—	2	1	no	—
86	C₂×C₄×C₁₉	—	2	4	no	—
87	C₂×C₄×C₁₉	—	2	4	no	—
88	C₂×C₄×C₁₉	—	2	2	no	—
89	C₂×C₄×C₁₉	—	2	2	no	—
90	C₂×C₄×C₁₉	—	2	2	no	—
91	C₂×C₄×C₁₉	—	4	4	no	—
92	C₂×C₄×C₁₉	—	4	4	no	—
93	C₂×C₄×C₁₉	C₄×C₈	1	2	yes	—
94	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
96	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
97	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
98	C₂×C₄×C₁₉	C₂²×C₈	1	2	yes	—
99	C₂×C₄×C₁₉	C₂×M₁₆	1	4	yes	—
100	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
101	C₂×C₄×C₁₉	#131	1	4	yes	—
102	C₈×C₁₉	—	2	4	no	—
103	C₈×C₁₉	C₂²⋊C₈	1	8	yes	complemented
104	C₁₉⋊C₈	—	2	8	no	—
105	C₁₉⋊C₈	—	4	4	no	—
106	C₂×C₄×M₁₆	D₁₉	1	1	yes	characteristic
107	C₄×C₂²⋊C₈	—	19	2	no	—
108	C₂×#322	—	19	1	no	—
109	C₂×C₈²	—	19	1	no	—
110	#1273	—	19	2	no	—
111	C₂²×C₁₉	—	2	4	no	—
112	C₂²×C₁₉	—	2	2	no	—
113	C₂²×C₁₉	—	4	1	no	—
114	C₂²×C₁₉	—	4	1	no	—
115	C₂²×C₁₉	—	4	2	no	—
116	C₂²×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
117	C₂²×C₁₉	C₂×C₄×C₈	1	1	yes	characteristic
118	C₂²×C₁₉	C₄×M₁₆	1	2	yes	—
119	C₂²×C₁₉	C₄×M₁₆	1	1	yes	characteristic
120	C₂²×C₁₉	#405	1	2	yes	—
121	C₄×C₁₉	—	2	2	no	—
122	C₄×C₁₉	—	4	2	no	—
123	C₄×C₁₉	C₂×C₂²⋊C₈	1	2	yes	—
124	C₄×C₁₉	#431	1	2	yes	—
125	C₄×C₁₉	#435	1	4	yes	—
126	C₄×C₁₉	#436	1	4	yes	—
127	C₂²×M₁₆	Dic₁₉	1	2	yes	—
128	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
129	C₂×C₂²⋊C₈	—	19	4	no	—
130	C₄×M₁₆	—	2	2	no	—
131	C₄×M₁₆	—	2	2	no	—
132	C₂×#97	Dic₁₉	1	2	yes	—
133	C₂×C₄×C₈	—	19	2	no	—
134	C₂×C₄×C₈	—	19	2	no	—
135	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
136	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
137	#336	—	19	4	no	—
138	#322	—	38	2	no	—
139	C₈²	—	38	2	no	—
140	C₂×C₁₉	—	4	2	no	—
141	C₂×C₁₉	C₈×M₁₆	1	2	yes	—
142	C₂×C₁₉	#1453	1	1	yes	characteristic
143	C₂×C₁₉	#1455	1	1	yes	characteristic
144	C₂×C₁₉	C₄×C₂²⋊C₈	1	1	yes	characteristic
145	C₂×C₁₉	#1753	1	2	yes	—
146	C₂³×C₄	C₄×D₁₉	1	2	yes	—
147	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
148	C₂×M₁₆	—	2	4	no	—
149	C₂×M₁₆	—	2	4	no	—
150	C₂×M₁₆	C₁₉⋊C₈	1	8	yes	complemented
151	C₂²×C₈	—	19	4	no	—
152	C₂²×C₈	—	19	4	no	—
153	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
154	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
155	C₂×C₄²	—	2	2	no	—
156	C₂×C₄²	—	2	1	no	—
157	C₂×C₄²	—	2	1	no	—
158	C₂×C₄²	C₄×D₁₉	1	2	yes	—
159	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
160	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
161	C₂²⋊C₈	—	38	8	no	—
162	#97	—	2	4	no	—
163	C₄×C₈	—	2	1	no	—
164	C₄×C₈	—	2	1	no	—
165	C₄×C₈	—	38	2	no	—
166	C₄×C₈	—	38	2	no	—
167	C₄×C₈	—	38	2	no	—
168	C₄×C₈	—	38	2	no	—
169	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
170	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
171	C₁₉	#8146	1	1	yes	characteristic, complemented, Sylow
172	C₂⁴	C₄×Dic₁₉	1	1	yes	characteristic
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	C₂×C₁₉⋊C₈	1	4	yes	—
181	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
182	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
184	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
185	M₁₆	—	2	16	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	19	8	no	—
189	C₂×C₈	—	38	4	no	—
190	C₂×C₈	—	38	4	no	—
191	C₂×C₈	—	38	4	no	—
192	C₂×C₈	—	38	4	no	—
193	C₂×C₈	—	38	4	no	—
194	C₂×C₈	C₂×C₁₉⋊C₈	1	4	yes	—
195	C₂×C₈	C₂×C₁₉⋊C₈	1	4	yes	—
196	C₂×C₈	#64520	1	4	yes	—
197	C₂×C₈	C₂²⋊Dic₁₉	1	4	yes	—
198	C₄²	—	2	1	no	—
199	C₄²	—	2	1	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	#64541	1	2	yes	—
203	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
204	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
205	C₂³	—	2	1	no	—
206	C₂³	—	2	2	no	—
207	C₂³	—	2	1	no	—
208	C₂³	—	2	2	no	—
209	C₂³	C₄×C₁₉⋊C₈	1	2	yes	—
210	C₂³	C₂×C₄×Dic₁₉	1	1	yes	characteristic
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	C₄×C₁₉⋊C₈	1	2	yes	—
221	C₂×C₄	M₁₆×D₁₉	1	4	yes	—
222	C₂×C₄	#10597353	1	4	yes	—
223	C₂×C₄	C₂²×C₁₉⋊C₈	1	2	yes	—
224	C₂×C₄	C₂×#64520	1	2	yes	—
225	C₂×C₄	#10597372	1	1	yes	characteristic
226	C₂×C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
227	C₂×C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
228	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
229	C₈	—	2	4	no	—
230	C₈	—	38	8	no	—
231	C₈	—	76	4	no	—
232	C₈	C₂²⋊(C₁₉⋊C₈)	1	8	yes	complemented
233	C₂²	—	2	2	no	—
234	C₂²	—	2	4	no	—
235	C₂²	—	4	2	no	—
236	C₂²	—	4	1	no	—
237	C₂²	—	4	1	no	—
238	C₂²	C₄×C₂²⋊Dic₁₉	1	1	yes	characteristic
239	C₂²	C₂×C₄×C₁₉⋊C₈	1	1	yes	characteristic
240	C₂²	M₁₆×Dic₁₉	1	2	yes	—
241	C₂²	C₄×#64520	1	1	yes	characteristic
242	C₂²	#11926502	1	2	yes	—
243	C₄	—	2	2	no	—
244	C₄	—	4	2	no	—
245	C₄	C₂×C₂²⋊(C₁₉⋊C₈)	1	2	yes	—
246	C₄	#11926597	1	2	yes	—
247	C₄	#11926613	1	4	yes	—
248	C₄	#11926620	1	4	yes	—
249	C₂	—	4	2	no	—
250	C₂	M₁₆×C₁₉⋊C₈	1	2	yes	—
251	C₂	#14561932	1	1	yes	characteristic
252	C₂	#14561938	1	1	yes	characteristic
253	C₂	C₄×C₂²⋊(C₁₉⋊C₈)	1	1	yes	characteristic
254	C₂	#14563416	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₁₉
commutator	C₂²×C₁₉	C₂×C₄×C₈
Fitting	C₂×C₄×M₁₆×C₁₉	C₂
socle	C₂³×C₁₉	C₂×C₄²
prime radical	C₂⁴×C₁₉	C₄²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×C₄×M₁₆	D₁₉
hypocenter	C₁₉	#8146
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#11926265	C₂	semidirect
3. C₂×C₈×C₁₉⋊C₈	C₂	semidirect
4. C₁₉⋊#1273	C₂	semidirect
23. C₂×M₁₆×C₁₉	C₈	semidirect
76. C₂×C₁₉⋊C₈	M₁₆	semidirect
103. C₈×C₁₉	C₂²⋊C₈	semidirect
150. C₂×M₁₆	C₁₉⋊C₈	semidirect
171. C₁₉	#8146	semidirect
232. C₈	C₂²⋊(C₁₉⋊C₈)	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₂²⋊(C₁₉⋊C₈)	2
C₂×#11926265	2
C₂×C₈×C₁₉⋊C₈	2
C₁₉⋊#1273	2
C₂×C₄×M₁₆×C₁₉	2
#8146	19