Subgroups — #18081398

Order 4864.

502 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14564572	C₂	1	2	yes	maximal
2	C₂×#11926297	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊(C₁₉⋊C₈)	C₂	1	2	yes	maximal
4	C₂×C₈×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Q₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
9	#11926299	—	2	4	no	—
10	#11926297	—	2	2	no	—
11	Q₈⋊(C₁₉⋊C₈)	—	2	4	no	—
12	C₈×C₁₉⋊C₈	—	2	2	no	—
13	C₂²×Q₁₆×C₁₉	C₄	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₁₉	C₂²	1	2	yes	—
15	C₄×Q₁₆×C₁₉	—	2	2	no	—
16	C₄×Q₁₆×C₁₉	—	2	2	no	—
17	C₂×C₁₉×#107	C₄	1	1	yes	characteristic
18	C₂×C₁₉×#103	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
20	C₂²×Q₈×C₁₉	C₂×C₄	1	2	yes	—
21	C₂×Q₁₆×C₁₉	—	2	2	no	—
22	C₂×Q₁₆×C₁₉	—	2	2	no	—
23	C₂×Q₁₆×C₁₉	C₈	1	4	yes	complemented
24	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
26	C₄×Q₈×C₁₉	—	2	2	no	—
27	C₄×Q₈×C₁₉	—	2	2	no	—
28	C₄×Q₈×C₁₉	—	2	2	no	—
29	C₄×Q₈×C₁₉	—	2	2	no	—
30	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
31	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₉	—	2	2	no	—
33	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
34	C₂²×C₁₉⋊C₈	—	2	1	no	—
35	C₂²×C₁₉⋊C₈	—	2	2	no	—
36	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
37	C₁₉×#107	—	2	2	no	—
38	C₁₉×#103	—	2	4	no	—
39	C₄×C₈×C₁₉	D₄	1	2	yes	—
40	C₄×C₈×C₁₉	D₄	1	2	yes	—
41	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
42	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
43	C₄×C₁₉⋊C₈	—	2	1	no	—
44	C₄×C₁₉⋊C₈	—	2	1	no	—
45	C₄×C₁₉⋊C₈	—	2	1	no	—
46	C₄×C₁₉⋊C₈	D₄	1	2	yes	—
47	C₂×Q₈×C₁₉	—	2	2	no	—
48	C₂×Q₈×C₁₉	—	2	2	no	—
49	C₂×Q₈×C₁₉	—	2	2	no	—
50	C₂×Q₈×C₁₉	—	2	2	no	—
51	C₂×Q₈×C₁₉	—	2	2	no	—
52	C₂×Q₈×C₁₉	C₂×C₈	1	4	yes	—
53	C₂²×C₄×C₁₉	—	2	2	no	—
54	C₂²×C₄×C₁₉	—	2	2	no	—
55	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
58	Q₁₆×C₁₉	—	2	8	no	—
59	C₂×C₈×C₁₉	—	2	1	no	—
60	C₂×C₈×C₁₉	—	2	1	no	—
61	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
62	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₁₉	C₂×C₈	1	2	yes	—
64	C₂×C₈×C₁₉	M₁₆	1	2	yes	—
65	C₄⋊C₄×C₁₉	—	2	2	no	—
66	C₄⋊C₄×C₁₉	—	2	2	no	—
67	C₄⋊C₄×C₁₉	—	4	2	no	—
68	C₄⋊C₄×C₁₉	—	4	2	no	—
69	C₄²×C₁₉	—	4	2	no	—
70	C₄²×C₁₉	—	4	2	no	—
71	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₁₉	#41	1	1	yes	characteristic
75	C₂×C₁₉⋊C₈	—	2	4	no	—
76	C₂×C₁₉⋊C₈	—	2	2	no	—
77	C₂×C₁₉⋊C₈	—	2	2	no	—
78	C₂×C₁₉⋊C₈	—	2	2	no	—
79	C₂×C₁₉⋊C₈	—	4	2	no	—
80	C₂×C₁₉⋊C₈	—	4	4	no	—
81	C₂×C₁₉⋊C₈	Q₁₆	1	4	yes	complemented
82	#9656	—	19	1	no	maximal, Sylow
83	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
84	Q₈×C₁₉	—	2	4	no	—
85	Q₈×C₁₉	—	2	4	no	—
86	Q₈×C₁₉	—	4	2	no	—
87	Q₈×C₁₉	—	4	2	no	—
88	C₂×C₄×C₁₉	—	2	2	no	—
89	C₂×C₄×C₁₉	—	2	4	no	—
90	C₂×C₄×C₁₉	—	4	2	no	—
91	C₂×C₄×C₁₉	—	4	2	no	—
92	C₂×C₄×C₁₉	—	4	2	no	—
93	C₂×C₄×C₁₉	—	4	4	no	—
94	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
97	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
98	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
99	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
100	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
101	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
102	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
103	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
104	C₂×C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
105	C₂×C₄×C₁₉	#135	1	2	yes	—
106	C₈×C₁₉	—	2	2	no	—
107	C₈×C₁₉	C₂²⋊C₈	1	4	yes	—
108	C₁₉⋊C₈	—	2	4	no	—
109	C₁₉⋊C₈	—	2	4	no	—
110	C₁₉⋊C₈	—	8	2	no	—
111	C₂×C₄×Q₁₆	D₁₉	1	1	yes	characteristic
112	#1984	—	19	2	no	—
113	C₂×#335	—	19	1	no	—
114	C₂×Q₈⋊C₈	—	19	2	no	—
115	C₂×C₈²	—	19	1	no	—
116	C₂²×C₁₉	#390	1	1	yes	characteristic
117	C₂²×C₁₉	C₈×D₄	1	1	yes	characteristic
118	C₂²×C₁₉	#436	1	1	yes	characteristic
119	C₂²×C₁₉	C₄×Q₁₆	1	2	yes	—
120	C₂²×C₁₉	#443	1	2	yes	—
121	C₄×C₁₉	—	2	1	no	—
122	C₄×C₁₉	—	2	1	no	—
123	C₄×C₁₉	—	4	4	no	—
124	C₄×C₁₉	—	8	2	no	—
125	C₄×C₁₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
126	C₄×C₁₉	#408	1	1	yes	characteristic
127	C₄×C₁₉	C₈×D₄	1	1	yes	characteristic
128	C₄×C₁₉	#436	1	1	yes	characteristic
129	C₄×C₁₉	#467	1	2	yes	—
130	C₄×C₁₉	#494	1	1	yes	characteristic
131	C₄×C₁₉	#495	1	1	yes	characteristic
132	C₂²×Q₁₆	Dic₁₉	1	1	yes	characteristic
133	C₂×C₄×Q₈	C₂×D₁₉	1	2	yes	—
134	C₄×Q₁₆	—	2	2	no	—
135	C₄×Q₁₆	—	2	2	no	—
136	C₂×#107	Dic₁₉	1	1	yes	characteristic
137	C₂×C₄⋊C₈	—	19	2	no	—
138	C₂×#103	Dic₁₉	1	2	yes	—
139	C₂×C₄×C₈	—	19	1	no	—
140	C₂×C₄×C₈	—	19	1	no	—
141	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
142	#336	—	38	4	no	—
143	#335	—	38	2	no	—
144	Q₈⋊C₈	—	38	4	no	—
145	C₈²	—	38	2	no	—
146	C₂×C₁₉	C₈×Q₁₆	1	2	yes	—
147	C₂×C₁₉	#1591	1	2	yes	—
148	C₂×C₁₉	#1954	1	1	yes	characteristic
149	C₂×C₁₉	#1969	1	1	yes	characteristic
150	C₂×C₁₉	#1974	1	1	yes	characteristic
151	C₂²×Q₈	C₂×Dic₁₉	1	2	yes	—
152	C₂×Q₁₆	—	2	2	no	—
153	C₂×Q₁₆	—	2	2	no	—
154	C₂×Q₁₆	C₁₉⋊C₈	1	4	yes	complemented
155	C₂²×C₈	—	19	2	no	—
156	C₂²×C₈	—	38	1	no	—
157	C₂²×C₈	—	38	2	no	—
158	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
159	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
160	C₄×Q₈	—	2	2	no	—
161	C₄×Q₈	—	2	2	no	—
162	C₄×Q₈	—	2	2	no	—
163	C₄×Q₈	—	2	2	no	—
164	C₂×C₄⋊C₄	—	2	2	no	—
165	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
166	C₂×C₄²	—	2	2	no	—
167	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
168	#107	—	2	2	no	—
169	C₄⋊C₈	—	38	2	no	—
170	C₄⋊C₈	—	38	2	no	—
171	#103	—	2	4	no	—
172	C₄×C₈	—	19	2	no	—
173	C₄×C₈	—	38	1	no	—
174	C₄×C₈	—	38	1	no	—
175	C₄×C₈	—	38	1	no	—
176	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
177	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
178	C₁₉	#9656	1	1	yes	characteristic, complemented, Sylow
179	C₂×Q₈	—	2	2	no	—
180	C₂×Q₈	—	2	2	no	—
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	—	2	2	no	—
183	C₂×Q₈	—	2	2	no	—
184	C₂×Q₈	C₂×C₁₉⋊C₈	1	4	yes	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
188	C₂²×C₄	#64543	1	1	yes	characteristic
189	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
190	Q₁₆	—	2	8	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	19	4	no	—
194	C₂×C₈	—	38	4	no	—
195	C₂×C₈	—	38	2	no	—
196	C₂×C₈	—	38	2	no	—
197	C₂×C₈	—	38	2	no	—
198	C₂×C₈	—	76	2	no	—
199	C₂×C₈	—	76	4	no	—
200	C₂×C₈	C₂×C₁₉⋊C₈	1	2	yes	—
201	C₂×C₈	#64520	1	2	yes	—
202	C₂×C₈	C₂²⋊Dic₁₉	1	2	yes	—
203	C₂×C₈	C₂²⋊Dic₁₉	1	2	yes	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄²	—	4	2	no	—
209	C₄²	—	4	2	no	—
210	C₄²	D₄×D₁₉	1	1	yes	characteristic
211	C₄²	#64543	1	1	yes	characteristic
212	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
213	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
214	C₂³	D₄×Dic₁₉	1	1	yes	characteristic
215	Q₈	—	2	4	no	—
216	Q₈	—	2	4	no	—
217	Q₈	—	4	2	no	—
218	Q₈	—	4	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	Q₁₆×D₁₉	1	2	yes	—
226	C₂×C₄	#10597365	1	2	yes	—
227	C₂×C₄	C₂²×C₁₉⋊C₈	1	1	yes	characteristic
228	C₂×C₄	C₂×#64520	1	1	yes	characteristic
229	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
230	C₂×C₄	#10597382	1	1	yes	characteristic
231	C₂×C₄	#10597384	1	1	yes	characteristic
232	C₂×C₄	#10597384	1	1	yes	characteristic
233	C₂×C₄	#10597386	1	1	yes	characteristic
234	C₂×C₄	#10597394	1	1	yes	characteristic
235	C₂×C₄	#10597394	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	—	38	4	no	—
239	C₈	—	38	4	no	—
240	C₈	—	152	2	no	—
241	C₈	C₂²⋊(C₁₉⋊C₈)	1	4	yes	—
242	C₂²	#11926447	1	1	yes	characteristic
243	C₂²	D₄×C₁₉⋊C₈	1	1	yes	characteristic
244	C₂²	#11926622	1	1	yes	characteristic
245	C₂²	Q₁₆×Dic₁₉	1	2	yes	—
246	C₂²	#11926661	1	2	yes	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	4	4	no	—
250	C₄	—	8	2	no	—
251	C₄	C₂×C₂²⋊(C₁₉⋊C₈)	1	1	yes	characteristic
252	C₄	#11926511	1	1	yes	characteristic
253	C₄	D₄×C₁₉⋊C₈	1	1	yes	characteristic
254	C₄	#11926622	1	1	yes	characteristic
255	C₄	#11926782	1	2	yes	—
256	C₄	#11926922	1	1	yes	characteristic
257	C₄	#11926929	1	1	yes	characteristic
258	C₂	Q₁₆×C₁₉⋊C₈	1	2	yes	—
259	C₂	#14562532	1	2	yes	—
260	C₂	#14564430	1	1	yes	characteristic
261	C₂	#14564497	1	1	yes	characteristic
262	C₂	#14564524	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₉
commutator	C₂×C₄×C₁₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₄×Q₁₆×C₁₉	C₂
socle, prime radical	C₂³×C₁₉	C₄×D₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×C₄×Q₁₆	D₁₉
hypocenter	C₁₉	#9656
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
23. C₂×Q₁₆×C₁₉	C₈	semidirect
81. C₂×C₁₉⋊C₈	Q₁₆	semidirect
154. C₂×Q₁₆	C₁₉⋊C₈	semidirect
178. C₁₉	#9656	semidirect

Maximal subgroups

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

subgroup	index
#14564572	2
C₂×#11926297	2
C₂×Q₈⋊(C₁₉⋊C₈)	2
C₂×C₈×C₁₉⋊C₈	2
C₂×C₄×Q₁₆×C₁₉	2
#9656	19