Subgroups — #18072102

Order 4864.

638 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926597	C₂	1	1	yes	maximal, characteristic
2	#14566033	C₂	1	1	yes	maximal, characteristic
3	#14561474	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₉×#423	C₂	1	1	yes	maximal, characteristic
5	#11926597	—	2	2	no	—
6	#11926597	C₄	1	4	yes	complemented
7	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
8	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
9	C₂×C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
10	#11926412	—	2	1	no	—
11	#11926410	C₄	1	2	yes	—
12	#11926407	—	2	1	no	—
13	#11926404	C₄	1	2	yes	—
14	C₄×C₄⋊Dic₁₉	C₂²	1	2	yes	—
15	C₄×C₂²⋊Dic₁₉	—	2	1	no	—
16	#11926390	—	2	1	no	—
17	C₂²×M₁₆×C₁₉	—	2	1	no	—
18	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
19	C₁₉×#423	—	2	2	no	—
20	C₁₉×#423	C₄	1	4	yes	complemented
21	C₂×C₁₉×C₄⋊C₈	—	2	1	no	—
22	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
23	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
26	C₂×M₁₆×C₁₉	—	2	4	no	—
27	C₂×M₁₆×C₁₉	—	4	4	no	—
28	C₂²×C₈×C₁₉	—	2	2	no	—
29	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₉	D₄	1	2	yes	—
35	C₂×C₄²×C₁₉	Q₈	1	2	yes	—
36	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
37	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
38	C₂×C₄⋊Dic₁₉	C₂×C₄	1	4	yes	—
39	C₂×C₄×Dic₁₉	—	2	2	no	—
40	C₂²×C₁₉⋊C₈	—	2	2	no	—
41	C₁₉×C₄⋊C₈	—	2	4	no	—
42	C₁₉×C₄⋊C₈	—	2	2	no	—
43	C₁₉×C₄⋊C₈	C₂×C₄	1	4	yes	—
44	C₁₉⋊#95	—	2	2	no	—
45	C₁₉⋊#95	—	2	2	no	—
46	C₁₉⋊#95	—	2	2	no	—
47	C₁₉⋊#95	—	2	2	no	—
48	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
49	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
50	C₄×C₁₉⋊C₈	—	2	2	no	—
51	C₄×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
52	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₉	—	2	4	no	—
54	C₂²×C₄×C₁₉	—	2	2	no	—
55	C₂²×C₄×C₁₉	—	2	2	no	—
56	C₂²×C₄×C₁₉	—	4	2	no	—
57	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
58	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₁₉	C₄⋊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	C₂²×Dic₁₉	—	2	4	no	—
70	M₁₆×C₁₉	—	4	8	no	—
71	C₂×C₈×C₁₉	—	2	4	no	—
72	C₂×C₈×C₁₉	—	4	4	no	—
73	C₄²×C₁₉	C₄²	1	2	yes	—
74	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
75	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
76	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
77	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
78	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
79	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
83	C₂²⋊Dic₁₉	—	4	4	no	—
84	C₄⋊Dic₁₉	—	2	8	no	—
85	C₄×Dic₁₉	—	4	2	no	—
86	C₄×Dic₁₉	—	4	2	no	—
87	C₂×C₁₉⋊C₈	—	2	4	no	—
88	C₂×C₁₉⋊C₈	—	4	4	no	—
89	#7899	—	19	1	no	maximal, Sylow
90	C₂³×C₁₉	—	2	2	no	—
91	C₂³×C₁₉	—	2	2	no	—
92	C₂³×C₁₉	—	2	2	no	—
93	C₂³×C₁₉	#95	1	2	yes	—
94	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₉	—	2	2	no	—
96	C₂×C₄×C₁₉	—	2	2	no	—
97	C₂×C₄×C₁₉	—	2	2	no	—
98	C₂×C₄×C₁₉	—	2	2	no	—
99	C₂×C₄×C₁₉	—	2	4	no	—
100	C₂×C₄×C₁₉	—	2	2	no	—
101	C₂×C₄×C₁₉	—	2	2	no	—
102	C₂×C₄×C₁₉	—	2	4	no	—
103	C₂×C₄×C₁₉	—	2	4	no	—
104	C₂×C₄×C₁₉	—	4	2	no	—
105	C₂×C₄×C₁₉	—	4	2	no	—
106	C₂×C₄×C₁₉	#95	1	2	yes	—
107	C₂×C₄×C₁₉	#95	1	2	yes	—
108	C₂×C₄×C₁₉	#95	1	2	yes	—
109	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₉	#131	1	4	yes	—
116	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
117	C₂×C₄×C₁₉	#137	1	2	yes	—
118	C₂×Dic₁₉	—	4	4	no	—
119	C₂×Dic₁₉	—	4	4	no	—
120	C₂×Dic₁₉	—	4	4	no	—
121	C₈×C₁₉	—	4	4	no	—
122	C₁₉⋊C₈	—	4	4	no	—
123	C₂×#431	—	19	1	no	—
124	C₂×#423	D₁₉	1	1	yes	characteristic
125	#2283	—	19	1	no	—
126	#1279	—	19	4	no	—
127	C₂²×C₁₉	—	2	2	no	—
128	C₂²×C₁₉	—	2	2	no	—
129	C₂²×C₁₉	—	2	2	no	—
130	C₂²×C₁₉	—	4	4	no	—
131	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
132	C₂²×C₁₉	#408	1	2	yes	—
133	C₂²×C₁₉	#417	1	2	yes	—
134	C₂²×C₁₉	#421	1	2	yes	—
135	C₄×C₁₉	—	2	2	no	—
136	C₄×C₁₉	—	2	1	no	—
137	C₄×C₁₉	—	2	2	no	—
138	C₄×C₁₉	—	2	1	no	—
139	C₄×C₁₉	—	2	2	no	—
140	C₄×C₁₉	#405	1	2	yes	—
141	C₄×C₁₉	#418	1	1	yes	characteristic
142	C₄×C₁₉	#419	1	1	yes	characteristic
143	C₄×C₁₉	#424	1	2	yes	—
144	C₄×C₁₉	#428	1	2	yes	—
145	Dic₁₉	—	8	4	no	—
146	C₂²×M₁₆	—	2	1	no	—
147	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
148	#431	—	19	4	no	—
149	#431	—	38	2	no	—
150	#423	—	2	2	no	—
151	#423	Dic₁₉	1	4	yes	complemented
152	C₂×C₄⋊C₈	—	2	1	no	—
153	C₂×C₄⋊C₈	—	38	1	no	—
154	C₂×C₄⋊C₈	C₂×D₁₉	1	2	yes	—
155	C₂×C₂²⋊C₈	—	38	1	no	—
156	C₂×C₄×C₈	—	19	2	no	—
157	#383	—	38	1	no	—
158	#382	—	19	2	no	—
159	#381	—	38	1	no	—
160	#380	—	19	2	no	—
161	C₄×C₄⋊C₄	—	19	2	no	—
162	C₄×C₂²⋊C₄	—	38	1	no	—
163	#376	—	38	1	no	—
164	C₂×C₁₉	—	4	2	no	—
165	C₂×C₁₉	#1495	1	4	yes	—
166	C₂×C₁₉	#1729	1	1	yes	characteristic
167	C₂×C₁₉	#1733	1	1	yes	characteristic
168	C₂×C₁₉	#1736	1	1	yes	characteristic
169	C₂³×C₄	C₄×D₁₉	1	1	yes	characteristic
170	C₂³×C₄	D₇₆	1	1	yes	characteristic
171	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
172	C₂×M₁₆	—	2	4	no	—
173	C₂×M₁₆	—	4	4	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	38	2	no	—
176	C₂×C₄⋊C₄	—	19	4	no	—
177	C₂×C₂²⋊C₄	—	38	2	no	—
178	C₂×C₄²	—	38	2	no	—
179	C₂×C₄²	Dic₃₈	1	2	yes	—
180	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
181	C₂×C₄²	C₄×D₁₉	1	2	yes	—
182	C₂×C₄²	D₇₆	1	1	yes	characteristic
183	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
184	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
185	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
186	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
187	C₄⋊C₈	—	2	4	no	—
188	C₄⋊C₈	—	2	2	no	—
189	C₄⋊C₈	—	38	4	no	—
190	C₄⋊C₈	C₂×Dic₁₉	1	4	yes	—
191	C₂²⋊C₈	—	38	4	no	—
192	C₄×C₈	—	19	4	no	—
193	C₄×C₈	—	38	2	no	—
194	#95	—	38	2	no	—
195	#95	—	38	2	no	—
196	#95	—	38	2	no	—
197	#95	—	38	2	no	—
198	C₁₉	#7899	1	1	yes	characteristic, complemented, Sylow
199	C₂⁴	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	—	38	4	no	—
205	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
206	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
207	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
208	C₂²×C₄	C₄⋊Dic₁₉	1	2	yes	—
209	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
210	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
211	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
212	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
213	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
214	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
217	M₁₆	—	4	8	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	4	4	no	—
220	C₂×C₈	—	38	4	no	—
221	C₂×C₈	—	76	4	no	—
222	C₄⋊C₄	—	38	8	no	—
223	C₂²⋊C₄	—	76	4	no	—
224	C₄²	—	76	2	no	—
225	C₄²	—	76	2	no	—
226	C₄²	C₄×Dic₁₉	1	2	yes	—
227	C₄²	Dic₁₉⋊C₄	1	2	yes	—
228	C₄²	Dic₁₉⋊C₄	1	2	yes	—
229	C₄²	C₄⋊Dic₁₉	1	2	yes	—
230	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
231	C₄²	C₂²⋊Dic₁₉	1	2	yes	—
232	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
233	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
234	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
235	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
236	C₂³	—	2	2	no	—
237	C₂³	—	2	2	no	—
238	C₂³	—	2	2	no	—
239	C₂³	C₁₉⋊#95	1	2	yes	—
240	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	76	4	no	—
253	C₂×C₄	—	76	4	no	—
254	C₂×C₄	—	76	4	no	—
255	C₂×C₄	C₁₉⋊#95	1	2	yes	—
256	C₂×C₄	C₁₉⋊#95	1	2	yes	—
257	C₂×C₄	C₁₉⋊#95	1	2	yes	—
258	C₂×C₄	#10597353	1	4	yes	—
259	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
260	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
261	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
262	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
263	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
264	C₂×C₄	#10597378	1	2	yes	—
265	C₂×C₄	#10597388	1	2	yes	—
266	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
267	C₈	—	4	4	no	—
268	C₈	—	76	4	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	4	4	no	—
273	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
274	C₂²	#11926512	1	2	yes	—
275	C₂²	#11926543	1	2	yes	—
276	C₂²	#11926560	1	2	yes	—
277	C₄	—	2	2	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	2	no	—
282	C₄	—	152	4	no	—
283	C₄	#11926502	1	2	yes	—
284	C₄	#11926549	1	1	yes	characteristic
285	C₄	#11926553	1	1	yes	characteristic
286	C₄	#11926572	1	2	yes	—
287	C₄	#11926584	1	2	yes	—
288	C₂	—	4	2	no	—
289	C₂	#14562098	1	4	yes	—
290	C₂	#14563333	1	1	yes	characteristic
291	C₂	#14563346	1	1	yes	characteristic
292	C₂	#14563354	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#423	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#423	D₁₉
hypocenter	C₁₉	#7899
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
3. #14561474	C₂	semidirect
6. #11926597	C₄	semidirect
20. C₁₉×#423	C₄	semidirect
151. #423	Dic₁₉	semidirect
198. C₁₉	#7899	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926597	2
#14566033	2
#14561474	2
C₂×C₁₉×#423	2
#7899	19