Subgroups — #18069097

Order 4864.

560 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂²⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₉⋊#95	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1797	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄×Dic₁₉	—	2	1	no	—
5	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
6	C₂×C₄×C₁₉⋊C₈	—	2	1	no	—
7	C₂×C₁₉⋊#95	—	2	1	no	—
8	C₂×C₁₉⋊#95	C₄	1	2	yes	—
9	#11926299	C₄	1	2	yes	complemented
10	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
11	C₂×C₁₉×C₄⋊C₈	C₄	1	2	yes	complemented
12	C₂×C₁₉×C₂²⋊C₈	—	2	1	no	—
13	C₁₉×#336	—	2	1	no	—
14	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
16	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
17	C₂³×Dic₁₉	—	2	2	no	—
18	C₂²×C₈×C₁₉	—	2	2	no	—
19	C₂×C₄²×C₁₉	—	4	2	no	—
20	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₉	Q₈	1	1	yes	characteristic
24	C₂×C₄×Dic₁₉	—	2	4	no	—
25	C₂×C₄×Dic₁₉	—	4	2	no	—
26	C₂×C₄×Dic₁₉	—	4	1	no	—
27	C₂×C₄×Dic₁₉	—	4	1	no	—
28	C₂²×C₁₉⋊C₈	—	2	2	no	—
29	C₁₉×C₄⋊C₈	—	4	2	no	—
30	C₁₉×C₂²⋊C₈	—	4	1	no	—
31	C₁₉×C₂²⋊C₈	—	4	1	no	—
32	C₁₉⋊#95	—	2	4	no	—
33	C₁₉⋊#95	—	2	2	no	—
34	C₁₉⋊#95	C₈	1	4	yes	complemented
35	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
36	C₄×C₁₉⋊C₈	—	4	2	no	—
37	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₉	—	2	2	no	—
39	C₂²×C₄×C₁₉	—	2	2	no	—
40	C₂²×C₄×C₁₉	—	2	2	no	—
41	C₂²×C₄×C₁₉	—	2	2	no	—
42	C₂²×C₄×C₁₉	—	4	1	no	—
43	C₂²×C₄×C₁₉	—	4	1	no	—
44	C₂²×C₄×C₁₉	—	4	1	no	—
45	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
46	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₉	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₁₉	M₁₆	1	2	yes	—
53	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	Q₁₆	1	1	yes	characteristic
57	C₂²×Dic₁₉	—	2	4	no	—
58	C₂²×Dic₁₉	—	2	4	no	—
59	C₂²×Dic₁₉	—	2	4	no	—
60	C₂²×Dic₁₉	—	4	2	no	—
61	C₂²×Dic₁₉	—	4	2	no	—
62	C₂²×Dic₁₉	—	4	2	no	—
63	C₂²×Dic₁₉	—	4	2	no	—
64	C₂×C₈×C₁₉	—	4	2	no	—
65	C₂×C₈×C₁₉	—	4	2	no	—
66	C₂×C₈×C₁₉	—	4	2	no	—
67	C₄²×C₁₉	—	2	2	no	—
68	C₄²×C₁₉	—	2	2	no	—
69	C₄²×C₁₉	—	4	1	no	—
70	C₄²×C₁₉	—	4	1	no	—
71	C₄×Dic₁₉	—	4	4	no	—
72	C₄×Dic₁₉	—	4	4	no	—
73	C₂×C₁₉⋊C₈	—	2	4	no	—
74	C₂×C₁₉⋊C₈	—	4	2	no	—
75	C₂×C₁₉⋊C₈	—	4	2	no	—
76	#7028	—	19	1	no	maximal, Sylow
77	C₂³×C₁₉	—	2	1	no	—
78	C₂³×C₁₉	—	2	1	no	—
79	C₂³×C₁₉	—	2	1	no	—
80	C₂³×C₁₉	—	2	1	no	—
81	C₂³×C₁₉	#95	1	1	yes	characteristic
82	C₂³×C₁₉	C₂²⋊C₈	1	2	yes	—
83	C₂³×C₁₉	#102	1	1	yes	characteristic
84	C₂³×C₁₉	#103	1	1	yes	characteristic
85	C₂³×C₁₉	#104	1	2	yes	—
86	C₂×C₄×C₁₉	—	2	1	no	—
87	C₂×C₄×C₁₉	—	2	1	no	—
88	C₂×C₄×C₁₉	—	2	1	no	—
89	C₂×C₄×C₁₉	—	2	1	no	—
90	C₂×C₄×C₁₉	—	2	1	no	—
91	C₂×C₄×C₁₉	—	2	1	no	—
92	C₂×C₄×C₁₉	—	2	1	no	—
93	C₂×C₄×C₁₉	—	2	1	no	—
94	C₂×C₄×C₁₉	—	2	2	no	—
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	1	no	—
99	C₂×C₄×C₁₉	—	2	1	no	—
100	C₂×C₄×C₁₉	—	2	1	no	—
101	C₂×C₄×C₁₉	—	4	2	no	—
102	C₂×C₄×C₁₉	—	4	2	no	—
103	C₂×C₄×C₁₉	—	4	4	no	—
104	C₂×C₄×C₁₉	C₄×C₈	1	1	yes	characteristic
105	C₂×C₄×C₁₉	#97	1	1	yes	characteristic
106	C₂×C₄×C₁₉	#99	1	2	yes	—
107	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
108	C₂×C₄×C₁₉	#103	1	1	yes	characteristic
109	C₂×C₄×C₁₉	#104	1	2	yes	—
110	C₂×C₄×C₁₉	#104	1	2	yes	—
111	C₂×C₄×C₁₉	C₄⋊C₈	1	2	yes	—
112	C₂×C₄×C₁₉	#106	1	1	yes	characteristic
113	C₂×C₄×C₁₉	#107	1	1	yes	characteristic
114	C₂×Dic₁₉	—	2	8	no	—
115	C₂×Dic₁₉	—	4	4	no	—
116	C₂×Dic₁₉	—	4	4	no	—
117	C₂×Dic₁₉	—	4	4	no	—
118	C₂×Dic₁₉	—	4	4	no	—
119	C₂×Dic₁₉	—	4	4	no	—
120	C₂×Dic₁₉	—	4	4	no	—
121	C₈×C₁₉	—	8	2	no	—
122	C₁₉⋊C₈	—	4	4	no	—
123	#1797	D₁₉	1	1	yes	characteristic
124	C₄×C₂²⋊C₈	—	19	1	no	—
125	C₄×#95	—	19	1	no	—
126	C₂²×C₁₉	—	2	2	no	—
127	C₂²×C₁₉	—	2	1	no	—
128	C₂²×C₁₉	—	2	1	no	—
129	C₂²×C₁₉	—	2	2	no	—
130	C₂²×C₁₉	—	2	2	no	—
131	C₂²×C₁₉	—	2	1	no	—
132	C₂²×C₁₉	—	2	1	no	—
133	C₂²×C₁₉	—	2	2	no	—
134	C₂²×C₁₉	D₄⋊C₈	1	2	yes	—
135	C₂²×C₁₉	Q₈⋊C₈	1	2	yes	—
136	C₂²×C₁₉	#336	1	1	yes	characteristic
137	C₂²×C₁₉	#339	1	1	yes	characteristic
138	C₂²×C₁₉	#339	1	1	yes	characteristic
139	C₂²×C₁₉	#339	1	1	yes	characteristic
140	C₂²×C₁₉	#340	1	1	yes	characteristic
141	C₂²×C₁₉	#342	1	1	yes	characteristic
142	C₂²×C₁₉	#344	1	1	yes	characteristic
143	C₄×C₁₉	—	2	1	no	—
144	C₄×C₁₉	—	2	1	no	—
145	C₄×C₁₉	—	2	2	no	—
146	C₄×C₁₉	—	2	1	no	—
147	C₄×C₁₉	—	2	1	no	—
148	C₄×C₁₉	—	2	2	no	—
149	C₄×C₁₉	—	4	1	no	—
150	C₄×C₁₉	#327	1	2	yes	—
151	C₄×C₁₉	#328	1	2	yes	—
152	Dic₁₉	—	4	8	no	—
153	C₂²×C₄²	—	38	1	no	—
154	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
155	C₂×C₄⋊C₈	Dic₁₉	1	2	yes	complemented
156	C₂×C₂²⋊C₈	—	2	1	no	—
157	C₂×C₂²⋊C₈	—	38	1	no	—
158	C₂×C₄×C₈	—	38	1	no	—
159	C₂×#95	—	19	2	no	—
160	C₂×#95	—	38	1	no	—
161	#336	—	2	1	no	—
162	#336	—	19	2	no	—
163	C₂×C₁₉	—	2	2	no	—
164	C₂×C₁₉	—	2	2	no	—
165	C₂×C₁₉	#1276	1	1	yes	characteristic
166	C₂×C₁₉	#1279	1	1	yes	characteristic
167	C₂×C₁₉	#1282	1	1	yes	characteristic
168	C₂×C₁₉	#1294	1	1	yes	characteristic
169	C₂×C₁₉	#1297	1	1	yes	characteristic
170	C₂×C₁₉	#1300	1	1	yes	characteristic
171	C₂×C₁₉	#1303	1	1	yes	characteristic
172	C₂³×C₄	—	38	2	no	—
173	C₂³×C₄	C₄×D₁₉	1	1	yes	characteristic
174	C₂³×C₄	D₇₆	1	1	yes	characteristic
175	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	38	2	no	—
178	C₂×C₄²	—	4	2	no	—
179	C₂×C₄²	—	38	4	no	—
180	C₂×C₄²	—	76	2	no	—
181	C₂×C₄²	—	76	1	no	—
182	C₂×C₄²	—	76	1	no	—
183	C₂×C₄²	Dic₃₈	1	1	yes	characteristic
184	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
185	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
186	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
187	C₄⋊C₈	—	4	2	no	—
188	C₂²⋊C₈	—	4	1	no	—
189	C₂²⋊C₈	—	4	1	no	—
190	C₂²⋊C₈	—	38	4	no	—
191	C₄×C₈	—	76	2	no	—
192	#95	—	19	4	no	—
193	#95	—	38	4	no	—
194	#95	—	38	2	no	—
195	C₁₉	#7028	1	1	yes	characteristic, complemented, Sylow
196	C₂⁴	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	4	1	no	—
202	C₂²×C₄	—	4	1	no	—
203	C₂²×C₄	—	4	1	no	—
204	C₂²×C₄	—	38	4	no	—
205	C₂²×C₄	—	38	4	no	—
206	C₂²×C₄	—	38	4	no	—
207	C₂²×C₄	—	76	2	no	—
208	C₂²×C₄	—	76	2	no	—
209	C₂²×C₄	—	76	2	no	—
210	C₂²×C₄	—	76	2	no	—
211	C₂²×C₄	C₈×D₁₉	1	2	yes	—
212	C₂²×C₄	#64515	1	2	yes	—
213	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
214	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
215	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
216	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
217	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
218	C₂²×C₄	#64525	1	1	yes	characteristic
219	C₂²×C₄	#64526	1	1	yes	characteristic
220	C₂²×C₄	#64527	1	1	yes	characteristic
221	C₂²×C₄	#64528	1	1	yes	characteristic
222	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	38	4	no	—
227	C₂×C₈	—	76	2	no	—
228	C₂×C₈	—	76	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	4	1	no	—
232	C₄²	—	4	1	no	—
233	C₄²	—	76	4	no	—
234	C₄²	—	76	4	no	—
235	C₂³	—	2	1	no	—
236	C₂³	—	2	1	no	—
237	C₂³	—	2	1	no	—
238	C₂³	—	2	1	no	—
239	C₂³	#10597266	1	1	yes	characteristic
240	C₂³	#10597267	1	1	yes	characteristic
241	C₂³	(C₂×D₁₉)⋊C₈	1	2	yes	—
242	C₂³	#10597282	1	2	yes	—
243	C₂³	C₁₉⋊#95	1	1	yes	characteristic
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	38	8	no	—
263	C₂×C₄	—	76	4	no	—
264	C₂×C₄	—	76	4	no	—
265	C₂×C₄	—	76	4	no	—
266	C₂×C₄	—	76	4	no	—
267	C₂×C₄	—	76	4	no	—
268	C₂×C₄	—	76	4	no	—
269	C₂×C₄	#10597264	1	1	yes	characteristic
270	C₂×C₄	#10597265	1	1	yes	characteristic
271	C₂×C₄	C₈×Dic₁₉	1	1	yes	characteristic
272	C₂×C₄	Dic₁₉⋊C₈	1	2	yes	—
273	C₂×C₄	#10597272	1	1	yes	characteristic
274	C₂×C₄	#10597282	1	2	yes	—
275	C₂×C₄	#10597289	1	1	yes	characteristic
276	C₂×C₄	#10597291	1	2	yes	—
277	C₂×C₄	#10597292	1	1	yes	characteristic
278	C₂×C₄	#10597294	1	2	yes	—
279	C₈	—	8	2	no	—
280	C₈	—	76	4	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	#11926271	1	2	yes	—
290	C₂²	#11926274	1	2	yes	—
291	C₂²	#11926298	1	1	yes	characteristic
292	C₂²	#11926303	1	1	yes	characteristic
293	C₂²	#11926305	1	1	yes	characteristic
294	C₂²	#11926305	1	1	yes	characteristic
295	C₂²	#11926306	1	1	yes	characteristic
296	C₂²	#11926311	1	1	yes	characteristic
297	C₂²	#11926315	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	2	no	—
304	C₄	—	4	1	no	—
305	C₄	—	76	8	no	—
306	C₄	#11926278	1	2	yes	—
307	C₄	#11926281	1	2	yes	—
308	C₂	—	2	2	no	—
309	C₂	—	2	2	no	—
310	C₂	#14561467	1	1	yes	characteristic
311	C₂	#14561474	1	1	yes	characteristic
312	C₂	#14561482	1	1	yes	characteristic
313	C₂	#14561513	1	1	yes	characteristic
314	C₂	#14561520	1	1	yes	characteristic
315	C₂	#14561527	1	1	yes	characteristic
316	C₂	#14561536	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₁₉
commutator	C₂×C₄×C₁₉	C₄×C₈
Frattini	C₂²×C₄²	C₂×D₁₉
Fitting	C₁₉×#1797	C₂
socle	C₂³×C₁₉	#95
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
cosocle	C₂²×C₄²×C₁₉	C₂²
hypercenter	#1797	D₁₉
hypocenter	C₁₉	#7028
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #1797

Complements

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

N	H ≅ G/N	type
9. #11926299	C₄	semidirect
11. C₂×C₁₉×C₄⋊C₈	C₄	semidirect
34. C₁₉⋊#95	C₈	semidirect
155. C₂×C₄⋊C₈	Dic₁₉	semidirect
195. C₁₉	#7028	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₂²⋊(C₁₉⋊C₈)	2
C₄×C₁₉⋊#95	2
C₁₉×#1797	2
#7028	19