Subgroups — #18081781

Order 4864.

518 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14564588	C₂	1	2	yes	maximal
2	C₈×C₄⋊Dic₁₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926297	C₂	1	1	yes	maximal, characteristic
4	C₂×#11926290	C₂	1	2	yes	maximal
5	C₂×C₁₉×#494	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×C₈×Dic₁₉	—	2	1	no	—
9	C₄×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
10	#11926306	—	2	4	no	—
11	#11926298	—	2	1	no	—
12	#11926297	—	2	2	no	—
13	#11926290	—	2	4	no	—
14	C₂²×Q₁₆×C₁₉	—	2	1	no	—
15	C₂²×Q₁₆×C₁₉	—	2	1	no	—
16	C₂×C₁₉×C₄⋊Q₈	C₂²	1	2	yes	—
17	C₁₉×#494	—	2	2	no	—
18	C₁₉×#494	C₄	1	4	yes	complemented
19	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
20	C₂²×Q₈×C₁₉	—	2	2	no	—
21	C₂×Q₁₆×C₁₉	—	2	4	no	—
22	C₂×Q₁₆×C₁₉	—	2	4	no	—
23	C₂×Q₁₆×C₁₉	—	4	2	no	—
24	C₂×Q₁₆×C₁₉	—	4	2	no	—
25	C₂×Q₁₆×C₁₉	—	4	2	no	—
26	C₂×Q₁₆×C₁₉	—	4	2	no	—
27	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
29	C₁₉×C₄⋊Q₈	—	2	2	no	—
30	C₁₉×C₄⋊Q₈	—	2	2	no	—
31	C₁₉×C₄⋊Q₈	—	2	2	no	—
32	C₁₉×C₄⋊Q₈	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
34	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
35	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
36	C₂×C₄⋊Dic₁₉	D₄	1	2	yes	—
37	C₂×C₄×Dic₁₉	—	2	1	no	—
38	C₂²×C₁₉⋊C₈	—	2	1	no	—
39	C₂²×C₁₉⋊C₈	—	2	2	no	—
40	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₁₉	D₄	1	2	yes	—
42	C₁₉⋊#95	—	2	1	no	—
43	C₈×Dic₁₉	—	4	2	no	—
44	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
45	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
46	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
47	C₄⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
48	C₂×Q₈×C₁₉	—	2	4	no	—
49	C₂×Q₈×C₁₉	—	4	2	no	—
50	C₂×Q₈×C₁₉	—	4	2	no	—
51	C₂×Q₈×C₁₉	—	4	2	no	—
52	C₂×Q₈×C₁₉	—	4	2	no	—
53	C₂×Q₈×C₁₉	—	4	2	no	—
54	C₂²×C₄×C₁₉	—	4	2	no	—
55	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
58	C₂²×Dic₁₉	—	2	2	no	—
59	Q₁₆×C₁₉	—	4	8	no	—
60	Q₁₆×C₁₉	—	4	8	no	—
61	C₂×C₈×C₁₉	—	2	1	no	—
62	C₂×C₈×C₁₉	—	2	1	no	—
63	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈×C₁₉	D₈	1	2	yes	—
66	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
67	C₄⋊C₄×C₁₉	—	2	4	no	—
68	C₄⋊C₄×C₁₉	—	2	4	no	—
69	C₄⋊C₄×C₁₉	—	4	2	no	—
70	C₄⋊C₄×C₁₉	—	4	2	no	—
71	C₄²×C₁₉	C₂²×C₄	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	C₄⋊Dic₁₉	—	2	2	no	—
76	C₄⋊Dic₁₉	Q₁₆	1	4	yes	complemented
77	C₄×Dic₁₉	—	4	1	no	—
78	C₄×Dic₁₉	—	4	1	no	—
79	C₂×C₁₉⋊C₈	—	2	2	no	—
80	C₂×C₁₉⋊C₈	—	4	1	no	—
81	C₂×C₁₉⋊C₈	—	4	1	no	—
82	C₂×C₁₉⋊C₈	—	4	2	no	—
83	C₂×C₁₉⋊C₈	—	4	4	no	—
84	#9733	—	19	1	no	maximal, Sylow
85	C₂³×C₁₉	#121	1	1	yes	characteristic
86	Q₈×C₁₉	—	4	4	no	—
87	Q₈×C₁₉	—	4	4	no	—
88	Q₈×C₁₉	—	8	2	no	—
89	Q₈×C₁₉	—	8	2	no	—
90	C₂×C₄×C₁₉	—	2	2	no	—
91	C₂×C₄×C₁₉	—	4	4	no	—
92	C₂×C₄×C₁₉	—	8	2	no	—
93	C₂×C₄×C₁₉	—	8	2	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	C₄×D₄	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	C₂×D₈	1	1	yes	characteristic
101	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
102	C₂×C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
103	C₂×C₄×C₁₉	#135	1	2	yes	—
104	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
105	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
106	C₂×Dic₁₉	—	2	4	no	—
107	C₂×Dic₁₉	—	4	2	no	—
108	C₂×Dic₁₉	—	4	2	no	—
109	C₈×C₁₉	—	2	2	no	—
110	C₈×C₁₉	#102	1	4	yes	—
111	C₁₉⋊C₈	—	4	2	no	—
112	C₁₉⋊C₈	—	8	2	no	—
113	C₂×#494	D₁₉	1	1	yes	characteristic
114	#1988	—	19	2	no	—
115	C₈×C₄⋊C₄	—	19	1	no	—
116	C₂×#335	—	19	1	no	—
117	C₂×#332	—	19	2	no	—
118	C₂²×C₁₉	#390	1	1	yes	characteristic
119	C₂²×C₁₉	#459	1	1	yes	characteristic
120	C₂²×C₁₉	#460	1	1	yes	characteristic
121	C₂²×C₁₉	#467	1	2	yes	—
122	C₂²×C₁₉	#473	1	2	yes	—
123	C₄×C₁₉	—	2	1	no	—
124	C₄×C₁₉	—	2	1	no	—
125	C₄×C₁₉	—	8	4	no	—
126	C₄×C₁₉	C₂×#102	1	1	yes	characteristic
127	C₄×C₁₉	#418	1	1	yes	characteristic
128	C₄×C₁₉	C₄×Q₁₆	1	2	yes	—
129	C₄×C₁₉	#459	1	1	yes	characteristic
130	C₄×C₁₉	#460	1	1	yes	characteristic
131	C₄×C₁₉	#494	1	1	yes	characteristic
132	C₄×C₁₉	#495	1	1	yes	characteristic
133	Dic₁₉	—	4	4	no	—
134	C₂²×Q₁₆	—	2	1	no	—
135	C₂²×Q₁₆	—	2	1	no	—
136	C₂×C₄⋊Q₈	C₂×D₁₉	1	2	yes	—
137	#494	—	2	2	no	—
138	#494	Dic₁₉	1	4	yes	complemented
139	C₂×C₄⋊C₈	—	19	1	no	—
140	C₂×C₄⋊C₈	—	19	2	no	—
141	C₂×C₄×C₈	—	38	1	no	—
142	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
143	C₄×C₄⋊C₄	—	19	1	no	—
144	#340	—	38	4	no	—
145	#336	—	38	1	no	—
146	#335	—	38	2	no	—
147	#332	—	38	4	no	—
148	C₂×C₁₉	#1669	1	2	yes	—
149	C₂×C₁₉	#1670	1	2	yes	—
150	C₂×C₁₉	#1969	1	1	yes	characteristic
151	C₂×C₁₉	#1979	1	1	yes	characteristic
152	C₂×C₁₉	#1982	1	1	yes	characteristic
153	C₂²×Q₈	—	2	2	no	—
154	C₂×Q₁₆	—	2	4	no	—
155	C₂×Q₁₆	—	2	4	no	—
156	C₂×Q₁₆	—	4	2	no	—
157	C₂×Q₁₆	—	4	2	no	—
158	C₂×Q₁₆	—	4	2	no	—
159	C₂×Q₁₆	—	4	2	no	—
160	C₂²×C₈	—	38	1	no	—
161	C₂²×C₈	—	38	2	no	—
162	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
163	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
164	C₄⋊Q₈	—	2	2	no	—
165	C₄⋊Q₈	—	2	2	no	—
166	C₄⋊Q₈	—	2	2	no	—
167	C₄⋊Q₈	C₂×Dic₁₉	1	4	yes	—
168	C₂×C₄⋊C₄	—	2	2	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	19	2	no	—
171	C₂×C₄²	—	38	1	no	—
172	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
173	C₄⋊C₈	—	19	2	no	—
174	C₄⋊C₈	—	38	2	no	—
175	C₄⋊C₈	—	38	1	no	—
176	C₄⋊C₈	—	38	2	no	—
177	C₄×C₈	—	76	2	no	—
178	C₄×C₈	C₂×Dic₁₉	1	2	yes	—
179	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
180	#95	—	38	1	no	—
181	C₁₉	#9733	1	1	yes	characteristic, complemented, Sylow
182	C₂×Q₈	—	2	4	no	—
183	C₂×Q₈	—	4	2	no	—
184	C₂×Q₈	—	4	2	no	—
185	C₂×Q₈	—	4	2	no	—
186	C₂×Q₈	—	4	2	no	—
187	C₂×Q₈	—	4	2	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	—	38	2	no	—
190	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
191	C₂²×C₄	#64543	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
193	Q₁₆	—	4	8	no	—
194	Q₁₆	—	4	8	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	38	2	no	—
198	C₂×C₈	—	76	1	no	—
199	C₂×C₈	—	76	1	no	—
200	C₂×C₈	—	76	2	no	—
201	C₂×C₈	—	76	4	no	—
202	C₂×C₈	#64525	1	2	yes	—
203	C₂×C₈	#64526	1	2	yes	—
204	C₂×C₈	C₂²⋊Dic₁₉	1	2	yes	—
205	C₂×C₈	C₂²⋊Dic₁₉	1	2	yes	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	19	4	no	—
211	C₄⋊C₄	—	38	2	no	—
212	C₄²	—	76	1	no	—
213	C₄²	—	76	1	no	—
214	C₄²	D₄×D₁₉	1	1	yes	characteristic
215	C₄²	#64543	1	1	yes	characteristic
216	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
217	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
218	C₂³	#10597384	1	1	yes	characteristic
219	Q₈	—	4	4	no	—
220	Q₈	—	4	4	no	—
221	Q₈	—	8	2	no	—
222	Q₈	—	8	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	8	2	no	—
226	C₂×C₄	—	8	2	no	—
227	C₂×C₄	—	38	4	no	—
228	C₂×C₄	—	76	2	no	—
229	C₂×C₄	—	76	2	no	—
230	C₂×C₄	Q₁₆×D₁₉	1	2	yes	—
231	C₂×C₄	#10597365	1	2	yes	—
232	C₂×C₄	C₂×#64525	1	1	yes	characteristic
233	C₂×C₄	#10597378	1	1	yes	characteristic
234	C₂×C₄	#10597378	1	1	yes	characteristic
235	C₂×C₄	C₂×#64526	1	1	yes	characteristic
236	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
237	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
238	C₂×C₄	#10597382	1	1	yes	characteristic
239	C₂×C₄	#10597384	1	1	yes	characteristic
240	C₂×C₄	#10597386	1	1	yes	characteristic
241	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
242	C₈	—	2	2	no	—
243	C₈	—	76	2	no	—
244	C₈	—	152	2	no	—
245	C₈	#10597289	1	4	yes	—
246	C₂²	#11926447	1	1	yes	characteristic
247	C₂²	#11926736	1	1	yes	characteristic
248	C₂²	#11926741	1	1	yes	characteristic
249	C₂²	#11926782	1	2	yes	—
250	C₂²	#11926816	1	2	yes	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	8	4	no	—
254	C₄	—	76	4	no	—
255	C₄	C₂×#10597289	1	1	yes	characteristic
256	C₄	#11926549	1	1	yes	characteristic
257	C₄	Q₁₆×Dic₁₉	1	2	yes	—
258	C₄	#11926736	1	1	yes	characteristic
259	C₄	#11926741	1	1	yes	characteristic
260	C₄	#11926922	1	1	yes	characteristic
261	C₄	#11926929	1	1	yes	characteristic
262	C₂	#14562997	1	2	yes	—
263	C₂	#14563002	1	2	yes	—
264	C₂	#14564497	1	1	yes	characteristic
265	C₂	#14564551	1	1	yes	characteristic
266	C₂	#14564563	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597384
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#494	C₂
socle, prime radical	C₂³×C₁₉	#121
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#494	D₁₉
hypocenter	C₁₉	#9733
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
18. C₁₉×#494	C₄	semidirect
76. C₄⋊Dic₁₉	Q₁₆	semidirect
138. #494	Dic₁₉	semidirect
181. C₁₉	#9733	semidirect

Maximal subgroups

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

subgroup	index
#14564588	2
C₈×C₄⋊Dic₁₉	2
C₂×#11926297	2
C₂×#11926290	2
C₂×C₁₉×#494	2
#9733	19