Subgroups — #18081630

Order 4864.

610 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563787	C₂	1	1	yes	maximal, characteristic, complemented
2	#14563488	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11926290	C₂	1	2	yes	maximal, complemented
4	C₂×#11926288	C₂	1	2	yes	maximal
5	C₁₉×#2858	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
7	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
9	C₂×#10597260	—	2	1	no	—
10	#11926299	—	2	1	no	—
11	#11926290	—	2	4	no	—
12	#11926288	—	2	4	no	—
13	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₁₉×#127	C₂²	1	1	yes	characteristic
15	C₂×C₁₉×#124	—	2	1	no	—
16	C₂×C₁₉×#124	—	2	1	no	—
17	C₁₉×#390	—	2	2	no	—
18	C₁₉×#390	C₄	1	4	yes	—
19	C₄³×C₁₉	C₂²	1	1	yes	characteristic
20	C₂²×Q₈×C₁₉	—	2	1	no	—
21	C₂²×D₄×C₁₉	—	2	1	no	—
22	C₂²×D₄×C₁₉	—	2	1	no	—
23	C₂²×D₄×C₁₉	C₂×C₄	1	2	yes	—
24	C₁₉×C₄⋊Q₈	—	2	2	no	—
25	C₁₉×C₄⋊Q₈	—	2	2	no	—
26	C₁₉×#127	—	2	2	no	—
27	C₁₉×#127	—	2	2	no	—
28	C₁₉×#124	—	4	4	no	—
29	C₁₉×#124	—	4	2	no	—
30	C₁₉×#124	—	4	2	no	—
31	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
32	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
33	C₂×C₂²⋊C₄×C₁₉	—	2	4	no	—
34	C₂×C₂²⋊C₄×C₁₉	—	2	4	no	—
35	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₉	D₄	1	2	yes	—
38	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
40	C₂²×C₁₉⋊C₈	—	2	2	no	—
41	C₂²×C₁₉⋊C₈	—	2	2	no	—
42	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
43	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
44	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
45	C₄⋊(C₁₉⋊C₈)	—	4	4	no	—
46	C₄⋊(C₁₉⋊C₈)	D₄	1	4	yes	—
47	#10597260	—	4	2	no	—
48	C₂⁴×C₁₉	—	2	2	no	—
49	C₂×Q₈×C₁₉	—	4	2	no	—
50	C₂×Q₈×C₁₉	—	4	1	no	—
51	C₂×Q₈×C₁₉	—	4	1	no	—
52	C₂×Q₈×C₁₉	—	4	2	no	—
53	C₂×D₄×C₁₉	—	2	4	no	—
54	C₂×D₄×C₁₉	—	4	4	no	—
55	C₂×D₄×C₁₉	—	4	2	no	—
56	C₂×D₄×C₁₉	—	4	1	no	—
57	C₂×D₄×C₁₉	—	4	1	no	—
58	C₂×D₄×C₁₉	—	4	1	no	—
59	C₂×D₄×C₁₉	—	4	1	no	—
60	C₂×D₄×C₁₉	—	4	2	no	—
61	C₂×D₄×C₁₉	—	4	2	no	—
62	C₂×D₄×C₁₉	—	4	2	no	—
63	C₂²×C₄×C₁₉	—	2	2	no	—
64	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
69	C₄⋊C₄×C₁₉	—	2	4	no	—
70	C₄⋊C₄×C₁₉	—	4	1	no	—
71	C₄⋊C₄×C₁₉	—	4	1	no	—
72	C₂²⋊C₄×C₁₉	—	4	8	no	—
73	C₂²⋊C₄×C₁₉	—	4	8	no	—
74	C₄²×C₁₉	—	2	2	no	—
75	C₄²×C₁₉	—	2	2	no	—
76	C₄²×C₁₉	—	2	2	no	—
77	C₄²×C₁₉	—	4	2	no	—
78	C₄²×C₁₉	D₈	1	2	yes	—
79	C₄²×C₁₉	SD₁₆	1	2	yes	—
80	C₄²×C₁₉	C₂×D₄	1	2	yes	—
81	C₄²×C₁₉	#41	1	2	yes	—
82	C₂×C₁₉⋊C₈	—	2	4	no	—
83	C₂×C₁₉⋊C₈	—	4	2	no	—
84	C₂×C₁₉⋊C₈	—	4	2	no	—
85	C₂×C₁₉⋊C₈	—	4	2	no	—
86	C₂×C₁₉⋊C₈	—	4	4	no	—
87	#9704	—	19	1	no	maximal, Sylow
88	C₂³×C₁₉	—	4	2	no	—
89	C₂³×C₁₉	—	4	4	no	—
90	C₂³×C₁₉	—	4	2	no	—
91	C₂³×C₁₉	—	4	4	no	—
92	C₂³×C₁₉	—	4	2	no	—
93	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	Q₈×C₁₉	—	8	2	no	—
95	Q₈×C₁₉	—	8	2	no	—
96	D₄×C₁₉	—	8	4	no	—
97	D₄×C₁₉	—	8	2	no	—
98	D₄×C₁₉	—	8	2	no	—
99	D₄×C₁₉	—	8	2	no	—
100	D₄×C₁₉	—	8	1	no	—
101	D₄×C₁₉	—	8	1	no	—
102	C₂×C₄×C₁₉	—	2	2	no	—
103	C₂×C₄×C₁₉	—	2	2	no	—
104	C₂×C₄×C₁₉	—	2	2	no	—
105	C₂×C₄×C₁₉	—	2	2	no	—
106	C₂×C₄×C₁₉	—	2	2	no	—
107	C₂×C₄×C₁₉	—	2	2	no	—
108	C₂×C₄×C₁₉	—	4	2	no	—
109	C₂×C₄×C₁₉	—	4	2	no	—
110	C₂×C₄×C₁₉	—	4	2	no	—
111	C₂×C₄×C₁₉	#102	1	4	yes	—
112	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₁₉	#121	1	2	yes	—
114	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
115	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
118	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
119	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
120	C₂×C₄×C₁₉	#137	1	2	yes	—
121	C₁₉⋊C₈	—	4	4	no	—
122	C₁₉⋊C₈	—	8	2	no	—
123	#2858	D₁₉	1	1	yes	characteristic
124	#1842	—	19	1	no	—
125	#1767	—	19	1	no	—
126	C₂×#332	—	19	2	no	—
127	C₂×#331	—	19	2	no	—
128	C₂²×C₁₉	—	8	4	no	—
129	C₂²×C₁₉	—	8	4	no	—
130	C₂²×C₁₉	—	8	2	no	—
131	C₂²×C₁₉	—	8	2	no	—
132	C₂²×C₁₉	—	8	2	no	—
133	C₂²×C₁₉	#390	1	1	yes	characteristic
134	C₂²×C₁₉	#413	1	2	yes	—
135	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
136	C₂²×C₁₉	#417	1	2	yes	—
137	C₂²×C₁₉	#418	1	1	yes	characteristic
138	C₄×C₁₉	—	2	4	no	—
139	C₄×C₁₉	—	2	2	no	—
140	C₄×C₁₉	—	4	2	no	—
141	C₄×C₁₉	—	8	2	no	—
142	C₄×C₁₉	#459	1	2	yes	—
143	C₄×C₁₉	#460	1	2	yes	—
144	C₄×C₁₉	#489	1	2	yes	—
145	C₄×C₁₉	#496	1	1	yes	characteristic
146	C₄×C₁₉	#497	1	1	yes	characteristic
147	C₂×C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
148	C₂×#127	C₂×D₁₉	1	1	yes	characteristic
149	C₂×#124	—	2	1	no	—
150	C₂×#124	—	2	1	no	—
151	C₂×C₄⋊C₈	—	19	2	no	—
152	C₂×C₄⋊C₈	—	19	2	no	—
153	C₂×C₄⋊C₈	—	38	2	no	—
154	C₂×#97	—	38	1	no	—
155	#390	—	2	2	no	—
156	#390	Dic₁₉	1	4	yes	—
157	C₄³	C₂×D₁₉	1	1	yes	characteristic
158	#336	—	38	1	no	—
159	#332	—	38	4	no	—
160	#331	—	38	4	no	—
161	C₂×C₁₉	—	16	2	no	—
162	C₂×C₁₉	#1547	1	2	yes	—
163	C₂×C₁₉	#1557	1	2	yes	—
164	C₂×C₁₉	#1959	1	1	yes	characteristic
165	C₂×C₁₉	#1967	1	1	yes	characteristic
166	C₂×C₁₉	#1982	1	1	yes	characteristic
167	C₂²×Q₈	—	2	1	no	—
168	C₂²×D₄	—	2	1	no	—
169	C₂²×D₄	—	2	1	no	—
170	C₂²×D₄	C₂×Dic₁₉	1	2	yes	—
171	C₂²×C₈	—	38	2	no	—
172	C₂²×C₈	—	38	2	no	—
173	C₄⋊Q₈	—	2	2	no	—
174	C₄⋊Q₈	—	2	2	no	—
175	#127	—	2	2	no	—
176	#127	—	2	2	no	—
177	#124	—	4	4	no	—
178	#124	—	4	2	no	—
179	#124	—	4	2	no	—
180	C₂×C₄⋊C₄	—	2	1	no	—
181	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
182	C₂×C₂²⋊C₄	—	2	4	no	—
183	C₂×C₂²⋊C₄	—	2	4	no	—
184	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
185	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
186	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
187	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
188	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
189	C₄⋊C₈	—	19	4	no	—
190	C₄⋊C₈	—	38	2	no	—
191	C₄⋊C₈	—	38	2	no	—
192	C₄⋊C₈	—	38	2	no	—
193	C₄⋊C₈	—	76	4	no	—
194	#97	—	76	2	no	—
195	C₁₉	#9704	1	1	yes	characteristic, complemented, Sylow
196	C₂⁴	—	2	2	no	—
197	C₂×Q₈	—	4	1	no	—
198	C₂×Q₈	—	4	2	no	—
199	C₂×Q₈	—	4	1	no	—
200	C₂×Q₈	—	4	2	no	—
201	C₂×D₄	—	2	4	no	—
202	C₂×D₄	—	4	4	no	—
203	C₂×D₄	—	4	1	no	—
204	C₂×D₄	—	4	2	no	—
205	C₂×D₄	—	4	1	no	—
206	C₂×D₄	—	4	1	no	—
207	C₂×D₄	—	4	2	no	—
208	C₂×D₄	—	4	1	no	—
209	C₂×D₄	—	4	2	no	—
210	C₂×D₄	—	4	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
213	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
214	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
217	C₂×C₈	—	38	4	no	—
218	C₂×C₈	—	76	2	no	—
219	C₂×C₈	—	76	2	no	—
220	C₂×C₈	—	76	2	no	—
221	C₂×C₈	—	76	4	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	4	1	no	—
224	C₄⋊C₄	—	4	1	no	—
225	C₂²⋊C₄	—	4	8	no	—
226	C₂²⋊C₄	—	4	8	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	4	2	no	—
231	C₄²	#64525	1	2	yes	—
232	C₄²	#64526	1	2	yes	—
233	C₄²	D₄×D₁₉	1	2	yes	—
234	C₄²	#64543	1	2	yes	—
235	C₂³	—	4	4	no	—
236	C₂³	—	4	2	no	—
237	C₂³	—	4	4	no	—
238	C₂³	—	4	2	no	—
239	C₂³	—	4	2	no	—
240	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
241	Q₈	—	8	2	no	—
242	Q₈	—	8	2	no	—
243	D₄	—	8	4	no	—
244	D₄	—	8	2	no	—
245	D₄	—	8	1	no	—
246	D₄	—	8	1	no	—
247	D₄	—	8	2	no	—
248	D₄	—	8	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	#10597289	1	4	yes	—
259	C₂×C₄	C₂×#64525	1	1	yes	characteristic
260	C₂×C₄	#10597378	1	2	yes	—
261	C₂×C₄	C₂×#64526	1	1	yes	characteristic
262	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
263	C₂×C₄	#10597382	1	1	yes	characteristic
264	C₂×C₄	#10597384	1	2	yes	—
265	C₂×C₄	#10597386	1	1	yes	characteristic
266	C₂×C₄	#10597395	1	2	yes	—
267	C₂×C₄	#10597397	1	2	yes	—
268	C₈	—	76	4	no	—
269	C₈	—	152	2	no	—
270	C₂²	—	8	4	no	—
271	C₂²	—	8	4	no	—
272	C₂²	—	8	2	no	—
273	C₂²	—	8	2	no	—
274	C₂²	—	8	2	no	—
275	C₂²	#11926447	1	1	yes	characteristic
276	C₂²	#11926528	1	2	yes	—
277	C₂²	C₂×#10597289	1	1	yes	characteristic
278	C₂²	#11926545	1	2	yes	—
279	C₂²	#11926549	1	1	yes	characteristic
280	C₄	—	2	4	no	—
281	C₄	—	2	2	no	—
282	C₄	—	4	2	no	—
283	C₄	—	8	2	no	—
284	C₄	#11926736	1	2	yes	—
285	C₄	#11926741	1	2	yes	—
286	C₄	#11926903	1	2	yes	—
287	C₄	#11926934	1	1	yes	characteristic
288	C₄	#11926939	1	1	yes	characteristic
289	C₂	—	16	2	no	—
290	C₂	#14562312	1	2	yes	—
291	C₂	#14562357	1	2	yes	—
292	C₂	#14564452	1	1	yes	characteristic
293	C₂	#14564485	1	1	yes	characteristic
294	C₂	#14564563	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#2858	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂×C₁₉×#127	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2858	D₁₉
hypocenter	C₁₉	#9704
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2858

Complements

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

N	H ≅ G/N	type
1. #14563787	C₂	semidirect
2. #14563488	C₂	semidirect
3. C₂×#11926290	C₂	semidirect
195. C₁₉	#9704	semidirect

Maximal subgroups

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

subgroup	index
#14563787	2
#14563488	2
C₂×#11926290	2
C₂×#11926288	2
C₁₉×#2858	2
#9704	19