Subgroups — #18075688

Order 4864.

374 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	#14563807	C₂	1	1	yes	maximal, characteristic
2	#14562383	C₂	1	1	yes	maximal, characteristic
3	#14562064	C₂	1	2	yes	maximal
4	#14561532	C₂	1	2	yes	maximal
5	C₂×C₁₉×#446	C₂	1	1	yes	maximal, characteristic
6	#11926597	C₂²	1	2	yes	—
7	C₂×#10597276	—	2	1	no	—
8	C₂×#10597276	—	2	1	no	—
9	#11926568	C₂²	1	2	yes	—
10	#11926297	—	2	1	no	—
11	#11926295	—	2	1	no	—
12	Q₈⋊(C₁₉⋊C₈)	—	2	4	no	—
13	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
14	C₁₉×#446	—	2	1	no	—
15	C₁₉×#446	—	2	1	no	—
16	C₁₉×#446	C₄	1	2	yes	—
17	C₁₉×#446	C₄	1	2	yes	—
18	C₂×C₁₉×C₄⋊C₈	—	2	1	no	—
19	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₁₉×#97	—	2	1	no	—
21	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
22	C₂²×Q₈×C₁₉	Q₈	1	1	yes	characteristic
23	C₂²×C₈×C₁₉	—	2	1	no	—
24	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
26	C₄×Q₈×C₁₉	—	2	1	no	—
27	C₄×Q₈×C₁₉	—	2	1	no	—
28	C₄×Q₈×C₁₉	C₂×C₄	1	2	yes	—
29	C₄×Q₈×C₁₉	D₄	1	2	yes	—
30	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
31	C₂×C₄⋊C₄×C₁₉	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₁₉	—	2	1	no	—
33	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
34	C₂×#64520	—	2	2	no	—
35	C₁₉×C₄⋊C₈	—	2	2	no	—
36	C₁₉×C₄⋊C₈	—	2	2	no	—
37	C₁₉×C₄⋊C₈	—	2	1	no	—
38	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
39	C₁₉×#97	—	2	2	no	—
40	C₁₉×#97	—	2	2	no	—
41	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
44	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
45	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
46	#10597276	—	4	2	no	—
47	#10597276	—	4	2	no	—
48	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
49	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
50	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
51	C₄×C₁₉⋊C₈	—	2	2	no	—
52	C₂×Q₈×C₁₉	—	2	1	no	—
53	C₂×Q₈×C₁₉	—	4	2	no	—
54	C₂×Q₈×C₁₉	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₁₉	—	2	1	no	—
56	C₂²×C₄×C₁₉	—	2	1	no	—
57	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
60	C₂×C₈×C₁₉	—	2	1	no	—
61	C₂×C₈×C₁₉	—	2	1	no	—
62	C₂×C₈×C₁₉	—	2	2	no	—
63	C₂×C₈×C₁₉	—	4	1	no	—
64	C₂×C₈×C₁₉	—	4	1	no	—
65	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
70	C₂×C₈×C₁₉	Q₁₆	1	2	yes	—
71	C₄⋊C₄×C₁₉	—	2	2	no	—
72	C₄⋊C₄×C₁₉	—	2	2	no	—
73	C₄⋊C₄×C₁₉	—	2	1	no	—
74	C₄⋊C₄×C₁₉	C₄⋊C₄	1	2	yes	—
75	C₄²×C₁₉	—	2	2	no	—
76	C₄²×C₁₉	—	2	2	no	—
77	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₁₉	#41	1	1	yes	characteristic
81	#64520	—	4	2	no	—
82	#64520	—	4	2	no	—
83	C₂×C₁₉⋊C₈	—	4	2	no	—
84	C₂×C₁₉⋊C₈	—	4	2	no	—
85	#8663	—	19	1	no	maximal, Sylow
86	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
87	Q₈×C₁₉	—	2	4	no	—
88	Q₈×C₁₉	—	4	2	no	—
89	C₂×C₄×C₁₉	—	2	2	no	—
90	C₂×C₄×C₁₉	—	2	2	no	—
91	C₂×C₄×C₁₉	—	2	2	no	—
92	C₂×C₄×C₁₉	—	4	1	no	—
93	C₂×C₄×C₁₉	—	4	2	no	—
94	C₂×C₄×C₁₉	—	4	2	no	—
95	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
98	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
99	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
100	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
102	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
103	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
104	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
105	C₈×C₁₉	—	2	1	no	—
106	C₈×C₁₉	—	2	1	no	—
107	C₈×C₁₉	—	4	2	no	—
108	C₈×C₁₉	#103	1	2	yes	—
109	C₈×C₁₉	#103	1	2	yes	—
110	C₁₉⋊C₈	—	8	2	no	—
111	C₁₉⋊C₈	—	8	2	no	—
112	C₂×#446	D₁₉	1	1	yes	characteristic
113	#1848	—	19	1	no	—
114	#1564	—	19	1	no	—
115	#1487	—	19	2	no	—
116	#1301	—	19	2	no	—
117	C₂²×C₁₉	—	2	1	no	—
118	C₂²×C₁₉	—	2	1	no	—
119	C₂²×C₁₉	#380	1	1	yes	characteristic
120	C₂²×C₁₉	#475	1	1	yes	characteristic
121	C₂²×C₁₉	#477	1	1	yes	characteristic
122	C₄×C₁₉	—	2	1	no	—
123	C₄×C₁₉	—	2	1	no	—
124	C₄×C₁₉	—	2	1	no	—
125	C₄×C₁₉	—	4	2	no	—
126	C₄×C₁₉	—	4	2	no	—
127	C₄×C₁₉	C₂×#103	1	1	yes	characteristic
128	C₄×C₁₉	#417	1	1	yes	characteristic
129	C₄×C₁₉	#430	1	2	yes	—
130	C₄×C₁₉	#460	1	1	yes	characteristic
131	C₄×C₁₉	#462	1	1	yes	characteristic
132	C₂×C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
133	#446	—	2	1	no	—
134	#446	—	2	1	no	—
135	#446	Dic₁₉	1	2	yes	—
136	#446	Dic₁₉	1	2	yes	—
137	#431	—	19	2	no	—
138	C₂×#108	—	38	1	no	—
139	C₂×#108	—	38	1	no	—
140	#423	—	19	2	no	—
141	C₂×C₄⋊C₈	—	2	1	no	—
142	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
143	C₂×#97	—	2	1	no	—
144	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
145	#335	—	38	1	no	—
146	#334	—	38	1	no	—
147	Q₈⋊C₈	—	38	4	no	—
148	C₂×C₁₉	—	2	1	no	—
149	C₂×C₁₉	—	2	1	no	—
150	C₂×C₁₉	#1803	1	1	yes	characteristic
151	C₂×C₁₉	#1811	1	1	yes	characteristic
152	C₂×C₁₉	#1814	1	1	yes	characteristic
153	C₂²×Q₈	Dic₃₈	1	1	yes	characteristic
154	C₂×M₁₆	—	38	2	no	—
155	C₂²×C₈	—	2	1	no	—
156	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
157	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
158	C₄×Q₈	—	2	1	no	—
159	C₄×Q₈	—	2	1	no	—
160	C₄×Q₈	D₇₆	1	2	yes	—
161	C₄×Q₈	C₂×Dic₁₉	1	2	yes	—
162	C₂×C₄⋊C₄	—	2	1	no	—
163	C₂×C₄⋊C₄	Dic₃₈	1	1	yes	characteristic
164	C₂×C₄²	—	2	1	no	—
165	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
166	#108	—	76	2	no	—
167	#108	—	76	2	no	—
168	C₄⋊C₈	—	2	2	no	—
169	C₄⋊C₈	—	2	2	no	—
170	C₄⋊C₈	—	2	1	no	—
171	C₄⋊C₈	—	38	2	no	—
172	C₄⋊C₈	—	38	2	no	—
173	C₄⋊C₈	—	38	2	no	—
174	C₄⋊C₈	C₂×Dic₁₉	1	2	yes	—
175	C₂²⋊C₈	—	38	2	no	—
176	#97	—	2	2	no	—
177	#97	—	2	2	no	—
178	C₄×C₈	—	38	2	no	—
179	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
180	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
181	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
182	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
183	C₁₉	#8663	1	1	yes	characteristic, complemented, Sylow
184	C₂×Q₈	—	2	1	no	—
185	C₂×Q₈	—	4	2	no	—
186	C₂×Q₈	C₄⋊Dic₁₉	1	2	yes	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
190	C₂²×C₄	#64541	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
192	M₁₆	—	76	2	no	—
193	M₁₆	—	76	2	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	—	4	1	no	—
199	C₂×C₈	—	76	2	no	—
200	C₂×C₈	—	76	2	no	—
201	C₂×C₈	#64527	1	2	yes	—
202	C₂×C₈	#64528	1	2	yes	—
203	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
204	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
205	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
206	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	C₄⋊Dic₁₉	1	2	yes	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	C₂×D₇₆	1	1	yes	characteristic
214	C₄²	#64541	1	1	yes	characteristic
215	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
216	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
217	C₂³	#10597370	1	1	yes	characteristic
218	Q₈	—	2	4	no	—
219	Q₈	—	4	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	4	1	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	#10597370	1	1	yes	characteristic
227	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
228	C₂×C₄	#10597372	1	1	yes	characteristic
229	C₂×C₄	#10597376	1	1	yes	characteristic
230	C₂×C₄	#10597376	1	1	yes	characteristic
231	C₂×C₄	C₂×#64527	1	1	yes	characteristic
232	C₂×C₄	C₂×#64528	1	1	yes	characteristic
233	C₂×C₄	#10597395	1	1	yes	characteristic
234	C₂×C₄	#10597397	1	1	yes	characteristic
235	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	4	2	no	—
239	C₈	—	152	2	no	—
240	C₈	—	152	2	no	—
241	C₈	#10597292	1	2	yes	—
242	C₈	#10597292	1	2	yes	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	#11926404	1	1	yes	characteristic
246	C₂²	#11926829	1	1	yes	characteristic
247	C₂²	#11926839	1	1	yes	characteristic
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	4	2	no	—
252	C₄	—	4	2	no	—
253	C₄	C₂×#10597292	1	1	yes	characteristic
254	C₄	#11926545	1	1	yes	characteristic
255	C₄	#11926593	1	2	yes	—
256	C₄	#11926744	1	1	yes	characteristic
257	C₄	#11926754	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	#14563640	1	1	yes	characteristic
261	C₂	#14563677	1	1	yes	characteristic
262	C₂	#14563689	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597376
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#446	C₂
socle	C₂²×C₁₉	#380
prime radical	C₂³×C₁₉	C₂²⋊Q₈
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#446	D₁₉
hypocenter	C₁₉	#8663
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
183. C₁₉	#8663	semidirect

Maximal subgroups

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

subgroup	index
#14563807	2
#14562383	2
#14562064	2
#14561532	2
C₂×C₁₉×#446	2
#8663	19