Subgroups — #18088367

Order 4864.

814 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×#11926568	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926308	C₂	1	4	yes	maximal
4	C₂×C₁₉×#423	C₂	1	1	yes	maximal, characteristic
5	C₂²×#64520	C₂²	1	2	yes	—
6	C₂³×C₁₉⋊C₈	C₂²	1	2	yes	—
7	#11926568	—	2	2	no	—
8	#11926568	C₄	1	4	yes	—
9	C₂×C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
10	C₂×C₄×C₁₉⋊C₈	—	2	2	no	—
11	C₂×C₄×C₁₉⋊C₈	—	2	2	no	—
12	C₂×C₄×C₁₉⋊C₈	C₄	1	2	yes	—
13	C₂×C₄×C₁₉⋊C₈	C₄	1	2	yes	—
14	#11926308	—	2	8	no	—
15	C₂²×M₁₆×C₁₉	C₂²	1	2	yes	—
16	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#423	—	2	2	no	—
18	C₁₉×#423	C₄	1	4	yes	—
19	C₂×C₁₉×C₄⋊C₈	—	2	2	no	—
20	C₂³×C₄×C₁₉	Q₈	1	2	yes	—
21	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₁₉	—	2	2	no	—
23	C₂×M₁₆×C₁₉	—	2	4	no	—
24	C₂×M₁₆×C₁₉	—	2	4	no	—
25	C₂×M₁₆×C₁₉	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₁₉	—	2	2	no	—
27	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₉	D₄	1	2	yes	—
32	C₂×C₄²×C₁₉	D₄	1	2	yes	—
33	C₂×C₄²×C₁₉	D₄	1	2	yes	—
34	C₂×#64520	—	2	2	no	—
35	C₂×#64520	—	2	4	no	—
36	C₂×#64520	—	2	4	no	—
37	C₂×#64520	C₂×C₄	1	4	yes	—
38	C₂²×C₁₉⋊C₈	—	2	2	no	—
39	C₂²×C₁₉⋊C₈	—	2	2	no	—
40	C₂²×C₁₉⋊C₈	—	2	2	no	—
41	C₂²×C₁₉⋊C₈	—	2	2	no	—
42	C₂²×C₁₉⋊C₈	—	2	2	no	—
43	C₂²×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
44	C₂²×C₁₉⋊C₈	D₄	1	4	yes	—
45	C₂²×C₁₉⋊C₈	Q₈	1	4	yes	—
46	C₁₉×C₄⋊C₈	—	2	8	no	—
47	C₄⋊(C₁₉⋊C₈)	—	2	8	no	—
48	C₄×C₁₉⋊C₈	—	2	4	no	—
49	C₄×C₁₉⋊C₈	—	2	4	no	—
50	C₂⁴×C₁₉	C₂×Q₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₉	—	2	8	no	—
52	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
54	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
56	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
64	C₂²×C₄×C₁₉	#41	1	2	yes	—
65	C₂²×C₄×C₁₉	#41	1	2	yes	—
66	M₁₆×C₁₉	—	2	8	no	—
67	M₁₆×C₁₉	—	4	4	no	—
68	C₂×C₈×C₁₉	—	2	4	no	—
69	C₂×C₈×C₁₉	—	4	2	no	—
70	C₂×C₈×C₁₉	—	4	2	no	—
71	C₄²×C₁₉	C₄²	1	2	yes	—
72	C₄²×C₁₉	C₄²	1	2	yes	—
73	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
74	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
75	C₄²×C₁₉	C₂²⋊C₄	1	4	yes	—
76	C₄²×C₁₉	C₂²⋊C₄	1	4	yes	—
77	#64520	—	2	8	no	—
78	#64520	—	4	4	no	—
79	C₂×C₁₉⋊C₈	—	2	4	no	—
80	C₂×C₁₉⋊C₈	—	2	4	no	—
81	C₂×C₁₉⋊C₈	—	2	4	no	—
82	C₂×C₁₉⋊C₈	—	2	4	no	—
83	C₂×C₁₉⋊C₈	—	2	4	no	—
84	C₂×C₁₉⋊C₈	—	2	4	no	—
85	C₂×C₁₉⋊C₈	—	2	4	no	—
86	C₂×C₁₉⋊C₈	—	4	2	no	—
87	C₂×C₁₉⋊C₈	—	4	2	no	—
88	C₂×C₁₉⋊C₈	C₄⋊C₄	1	8	yes	—
89	#10844	—	19	1	no	maximal, Sylow
90	C₂³×C₁₉	—	2	4	no	—
91	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂³×C₁₉	#125	1	2	yes	—
95	C₂³×C₁₉	C₄⋊Q₈	1	2	yes	—
96	C₂×C₄×C₁₉	—	2	4	no	—
97	C₂×C₄×C₁₉	—	2	8	no	—
98	C₂×C₄×C₁₉	—	2	8	no	—
99	C₂×C₄×C₁₉	#95	1	4	yes	—
100	C₂×C₄×C₁₉	#95	1	4	yes	—
101	C₂×C₄×C₁₉	#108	1	8	yes	—
102	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₄×C₁₉	#117	1	2	yes	—
108	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
111	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
112	C₂×C₄×C₁₉	#121	1	2	yes	—
113	C₂×C₄×C₁₉	#121	1	2	yes	—
114	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
117	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
118	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
119	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
120	C₈×C₁₉	—	4	4	no	—
121	C₁₉⋊C₈	—	2	8	no	—
122	C₁₉⋊C₈	—	4	4	no	—
123	C₂×#423	—	19	1	no	—
124	C₂×#423	D₁₉	1	1	yes	characteristic
125	C₂²×C₄×C₈	—	19	1	no	—
126	C₂×#341	—	19	4	no	—
127	C₂²×C₁₉	—	2	4	no	—
128	C₂²×C₁₉	—	2	4	no	—
129	C₂²×C₁₉	—	2	2	no	—
130	C₂²×C₁₉	—	2	2	no	—
131	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
132	C₂²×C₁₉	#382	1	1	yes	characteristic
133	C₂²×C₁₉	#384	1	1	yes	characteristic
134	C₂²×C₁₉	#389	1	2	yes	—
135	C₂²×C₁₉	#389	1	2	yes	—
136	C₂²×C₁₉	C₂×#108	1	4	yes	—
137	C₄×C₁₉	—	2	4	no	—
138	C₄×C₁₉	#341	1	8	yes	complemented
139	C₄×C₁₉	C₄×C₄⋊C₄	1	2	yes	—
140	C₄×C₁₉	#380	1	2	yes	—
141	C₄×C₁₉	#390	1	1	yes	characteristic
142	C₄×C₁₉	#390	1	1	yes	characteristic
143	C₄×C₁₉	#391	1	1	yes	characteristic
144	C₄×C₁₉	#391	1	1	yes	characteristic
145	C₂²×M₁₆	—	19	2	no	—
146	C₂²×M₁₆	C₂×D₁₉	1	2	yes	—
147	C₂³×C₈	—	19	2	no	—
148	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
149	#423	—	2	2	no	—
150	#423	—	19	4	no	—
151	#423	—	38	2	no	—
152	#423	Dic₁₉	1	4	yes	—
153	C₂×C₄⋊C₈	—	2	2	no	—
154	C₂×C₄⋊C₈	—	38	2	no	—
155	C₂×C₄×C₈	—	19	2	no	—
156	C₂×C₄×C₈	—	19	2	no	—
157	C₂×C₄×C₈	—	38	2	no	—
158	C₂×C₄×C₈	—	38	2	no	—
159	#341	—	38	8	no	—
160	C₂×C₁₉	—	2	2	no	—
161	C₂×C₁₉	—	2	2	no	—
162	C₂×C₁₉	C₄⋊#95	1	1	yes	characteristic
163	C₂×C₁₉	C₂×#341	1	2	yes	—
164	C₂×C₁₉	#1837	1	2	yes	—
165	C₂×C₁₉	#1848	1	2	yes	—
166	C₂³×C₄	Dic₃₈	1	2	yes	—
167	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	4	no	—
170	C₂×M₁₆	—	2	4	no	—
171	C₂×M₁₆	—	19	4	no	—
172	C₂×M₁₆	—	38	2	no	—
173	C₂×M₁₆	—	38	4	no	—
174	C₂×M₁₆	—	38	4	no	—
175	C₂×M₁₆	C₂×Dic₁₉	1	4	yes	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	19	4	no	—
178	C₂²×C₈	—	19	4	no	—
179	C₂²×C₈	—	19	4	no	—
180	C₂²×C₈	—	38	2	no	—
181	C₂²×C₈	—	38	2	no	—
182	C₂²×C₈	—	38	2	no	—
183	C₂²×C₈	—	38	2	no	—
184	C₂²×C₈	—	38	2	no	—
185	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
186	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
187	C₂×C₄²	C₄×D₁₉	1	2	yes	—
188	C₂×C₄²	D₇₆	1	2	yes	—
189	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
190	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
191	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
192	C₄⋊C₈	—	2	8	no	—
193	C₄⋊C₈	—	38	8	no	—
194	C₄×C₈	—	38	4	no	—
195	C₄×C₈	—	38	4	no	—
196	C₁₉	#10844	1	1	yes	characteristic, complemented, Sylow
197	C₂⁴	C₂×Dic₃₈	1	1	yes	characteristic
198	C₂²×C₄	—	2	8	no	—
199	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
200	C₂²×C₄	Dic₁₉⋊C₄	1	4	yes	—
201	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
202	C₂²×C₄	C₄⋊Dic₁₉	1	4	yes	—
203	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
206	C₂²×C₄	D₄×D₁₉	1	2	yes	—
207	C₂²×C₄	#64543	1	2	yes	—
208	C₂²×C₄	Q₈×D₁₉	1	2	yes	—
209	C₂²×C₄	#64545	1	2	yes	—
210	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
213	M₁₆	—	2	8	no	—
214	M₁₆	—	4	4	no	—
215	M₁₆	—	38	8	no	—
216	M₁₆	—	76	4	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	19	8	no	—
221	C₂×C₈	—	38	4	no	—
222	C₂×C₈	—	38	4	no	—
223	C₂×C₈	—	38	4	no	—
224	C₂×C₈	—	38	4	no	—
225	C₂×C₈	—	38	4	no	—
226	C₂×C₈	—	38	4	no	—
227	C₂×C₈	—	38	4	no	—
228	C₂×C₈	—	76	2	no	—
229	C₂×C₈	—	76	2	no	—
230	C₄²	C₄×Dic₁₉	1	2	yes	—
231	C₄²	C₄×Dic₁₉	1	2	yes	—
232	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
233	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
234	C₄²	(C₂×D₁₉)⋊C₄	1	4	yes	—
235	C₄²	C₂²⋊Dic₁₉	1	4	yes	—
236	C₂³	—	2	4	no	—
237	C₂³	#10597334	1	2	yes	—
238	C₂³	#10597336	1	2	yes	—
239	C₂³	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
240	C₂³	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
241	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	#10597279	1	8	yes	—
246	C₂×C₄	C₁₉⋊#95	1	4	yes	—
247	C₂×C₄	C₁₉⋊#95	1	4	yes	—
248	C₂×C₄	#10597333	1	2	yes	—
249	C₂×C₄	C₄⋊C₄×D₁₉	1	2	yes	—
250	C₂×C₄	#10597338	1	2	yes	—
251	C₂×C₄	#10597339	1	2	yes	—
252	C₂×C₄	#10597341	1	2	yes	—
253	C₂×C₄	#10597343	1	2	yes	—
254	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
255	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
256	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
257	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
258	C₂×C₄	#10597382	1	1	yes	characteristic
259	C₂×C₄	#10597384	1	2	yes	—
260	C₂×C₄	#10597386	1	1	yes	characteristic
261	C₂×C₄	#10597390	1	1	yes	characteristic
262	C₂×C₄	Q₈×Dic₁₉	1	2	yes	—
263	C₂×C₄	#10597392	1	2	yes	—
264	C₂×C₄	#10597393	1	1	yes	characteristic
265	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
266	C₈	—	4	4	no	—
267	C₈	—	38	8	no	—
268	C₈	—	76	4	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	4	no	—
272	C₂²	—	2	4	no	—
273	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
274	C₂²	#11926409	1	1	yes	characteristic
275	C₂²	#11926413	1	1	yes	characteristic
276	C₂²	#11926440	1	2	yes	—
277	C₂²	#11926443	1	2	yes	—
278	C₂²	C₂×#10597279	1	4	yes	—
279	C₄	—	2	4	no	—
280	C₄	#11926308	1	8	yes	complemented
281	C₄	C₄⋊C₄×Dic₁₉	1	2	yes	—
282	C₄	#11926401	1	2	yes	—
283	C₄	#11926445	1	1	yes	characteristic
284	C₄	#11926447	1	1	yes	characteristic
285	C₄	#11926449	1	1	yes	characteristic
286	C₄	#11926451	1	1	yes	characteristic
287	C₂	—	2	2	no	—
288	C₂	—	2	2	no	—
289	C₂	#14561864	1	1	yes	characteristic
290	C₂	C₂×#11926308	1	2	yes	—
291	C₂	#14563772	1	2	yes	—
292	C₂	#14563806	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₁₉×#423	C₂
socle	C₂³×C₁₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₉	C₂×Q₈
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×#423	D₁₉
hypocenter	C₁₉	#10844
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
138. C₄×C₁₉	#341	semidirect
196. C₁₉	#10844	semidirect
280. C₄	#11926308	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926568	2
C₂²×C₄×C₁₉⋊C₈	2
C₂×#11926308	2
C₂×C₁₉×#423	2
#10844	19