Subgroups — #18073124

Order 4864.

418 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#10597293	C₂	1	1	yes	maximal, characteristic
2	#14561959	C₂	1	1	yes	maximal, characteristic
3	#14561924	C₂	1	1	yes	maximal, characteristic
4	C₈×#64520	C₂	1	1	yes	maximal, characteristic
5	#14561485	C₂	1	1	yes	maximal, characteristic
6	#14561485	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₉×#446	C₂	1	1	yes	maximal, characteristic
8	#11926597	C₂²	1	1	yes	characteristic
9	#11926597	C₂²	1	1	yes	characteristic
10	C₂×#10597293	C₄	1	2	yes	—
11	C₄×#64520	C₂²	1	1	yes	characteristic
12	C₄×#64520	C₂²	1	1	yes	characteristic
13	#11926313	C₄	1	2	yes	—
14	#11926270	C₄	1	4	yes	—
15	#11926266	—	2	1	no	—
16	#11926266	—	2	1	no	—
17	#11926265	—	2	1	no	—
18	C₈×C₁₉⋊C₈	—	2	1	no	—
19	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
20	C₁₉×#446	—	2	1	no	—
21	C₁₉×#446	—	2	1	no	—
22	C₁₉×#446	—	2	1	no	—
23	C₁₉×#446	—	2	1	no	—
24	C₂×C₁₉×C₄⋊C₈	C₄	1	2	yes	—
25	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₁₉×#97	C₄	1	2	yes	—
27	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
28	C₂²×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
32	C₄×Q₈×C₁₉	—	2	1	no	—
33	C₄×Q₈×C₁₉	—	2	1	no	—
34	C₄×Q₈×C₁₉	—	2	1	no	—
35	C₄×Q₈×C₁₉	—	2	1	no	—
36	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
40	C₂×#64520	C₂×C₄	1	2	yes	—
41	C₂×#64520	C₂×C₄	1	2	yes	—
42	C₁₉×C₄⋊C₈	—	2	2	no	—
43	C₁₉×C₄⋊C₈	—	2	2	no	—
44	C₁₉×C₄⋊C₈	—	2	1	no	—
45	C₁₉×C₄⋊C₈	—	2	1	no	—
46	C₁₉×#97	—	2	2	no	—
47	C₁₉×#97	—	2	2	no	—
48	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
49	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
50	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
51	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
52	#10597293	—	2	4	no	—
53	C₂²⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
54	C₂²⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
55	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
56	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
57	#10597260	—	2	1	no	—
58	#10597260	—	2	1	no	—
59	C₄×C₁₉⋊C₈	—	2	1	no	—
60	C₄×C₁₉⋊C₈	—	2	1	no	—
61	C₄×C₁₉⋊C₈	—	2	1	no	—
62	C₄×C₁₉⋊C₈	—	2	1	no	—
63	C₂×Q₈×C₁₉	—	2	1	no	—
64	C₂×Q₈×C₁₉	—	2	1	no	—
65	C₂×Q₈×C₁₉	—	2	2	no	—
66	C₂×Q₈×C₁₉	—	2	2	no	—
67	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
68	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
69	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₁₉	—	2	2	no	—
73	C₂×C₈×C₁₉	—	2	2	no	—
74	C₂×C₈×C₁₉	—	4	1	no	—
75	C₂×C₈×C₁₉	—	4	2	no	—
76	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
80	C₄⋊C₄×C₁₉	—	2	1	no	—
81	C₄⋊C₄×C₁₉	—	2	1	no	—
82	C₄⋊C₄×C₁₉	—	2	2	no	—
83	C₄⋊C₄×C₁₉	—	2	2	no	—
84	C₄²×C₁₉	—	2	2	no	—
85	C₄²×C₁₉	—	2	2	no	—
86	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₁₉	#41	1	1	yes	characteristic
89	C₄²×C₁₉	#41	1	1	yes	characteristic
90	#64520	—	2	2	no	—
91	#64520	—	2	2	no	—
92	#64520	—	2	2	no	—
93	#64520	M₁₆	1	4	yes	—
94	C₂×C₁₉⋊C₈	—	2	2	no	—
95	C₂×C₁₉⋊C₈	—	2	2	no	—
96	C₂×C₁₉⋊C₈	—	2	2	no	—
97	C₂×C₁₉⋊C₈	—	2	2	no	—
98	#8167	—	19	1	no	maximal, Sylow
99	C₂³×C₁₉	C₂×C₄²	1	1	yes	characteristic
100	Q₈×C₁₉	—	4	2	no	—
101	Q₈×C₁₉	—	4	2	no	—
102	C₂×C₄×C₁₉	—	2	1	no	—
103	C₂×C₄×C₁₉	—	2	1	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	1	no	—
109	C₂×C₄×C₁₉	—	4	2	no	—
110	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
112	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
117	C₂×C₄×C₁₉	#131	1	2	yes	—
118	C₈×C₁₉	—	2	1	no	—
119	C₈×C₁₉	—	2	1	no	—
120	C₈×C₁₉	—	2	1	no	—
121	C₈×C₁₉	—	4	2	no	—
122	C₈×C₁₉	#101	1	2	yes	complemented
123	C₁₉⋊C₈	—	2	4	no	—
124	C₁₉⋊C₈	—	4	2	no	—
125	C₁₉⋊C₈	—	4	2	no	—
126	C₁₉⋊C₈	—	4	2	no	—
127	C₂×#446	D₁₉	1	1	yes	characteristic
128	C₄×#101	—	19	1	no	—
129	#1461	—	19	1	no	—
130	#1451	—	19	1	no	—
131	C₈×M₁₆	—	19	1	no	—
132	#1283	—	19	1	no	—
133	#1283	—	19	1	no	—
134	C₂²×C₁₉	—	2	1	no	—
135	C₂²×C₁₉	—	2	1	no	—
136	C₂²×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
137	C₂²×C₁₉	C₄×M₁₆	1	1	yes	characteristic
138	C₂²×C₁₉	#405	1	1	yes	characteristic
139	C₄×C₁₉	—	2	1	no	—
140	C₄×C₁₉	—	2	1	no	—
141	C₄×C₁₉	—	2	1	no	—
142	C₄×C₁₉	—	4	2	no	—
143	C₄×C₁₉	—	4	2	no	—
144	C₄×C₁₉	C₂×#101	1	1	yes	characteristic
145	C₄×C₁₉	#413	1	1	yes	characteristic
146	C₄×C₁₉	#435	1	2	yes	—
147	C₄×C₁₉	#436	1	2	yes	—
148	C₂×C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
149	#446	—	2	1	no	—
150	#446	—	2	1	no	—
151	#446	—	2	1	no	—
152	#446	—	2	1	no	—
153	#431	—	19	1	no	—
154	#431	—	19	1	no	—
155	C₂×C₄⋊C₈	Dic₁₉	1	2	yes	—
156	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
157	C₂×#101	—	19	2	no	—
158	C₄×M₁₆	—	19	1	no	—
159	C₄×M₁₆	—	19	1	no	—
160	C₂×#97	Dic₁₉	1	2	yes	—
161	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
162	#343	—	19	2	no	—
163	#324	—	19	4	no	—
164	#322	—	38	1	no	—
165	#322	—	38	1	no	—
166	#322	—	38	1	no	—
167	C₈²	—	38	1	no	—
168	C₂×C₁₉	—	2	1	no	—
169	C₂×C₁₉	—	2	1	no	—
170	C₂×C₁₉	#1753	1	1	yes	characteristic
171	C₂×C₁₉	C₄×#101	1	1	yes	characteristic
172	C₂×C₁₉	#1757	1	1	yes	characteristic
173	C₂²×Q₈	C₄×D₁₉	1	1	yes	characteristic
174	C₂×M₁₆	—	19	2	no	—
175	C₂×M₁₆	—	19	2	no	—
176	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
177	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
178	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
179	C₄×Q₈	—	2	1	no	—
180	C₄×Q₈	—	2	1	no	—
181	C₄×Q₈	—	2	1	no	—
182	C₄×Q₈	—	2	1	no	—
183	C₂×C₄⋊C₄	C₄×D₁₉	1	2	yes	—
184	C₂×C₄⋊C₄	C₄×D₁₉	1	1	yes	characteristic
185	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
186	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
187	C₄⋊C₈	—	2	2	no	—
188	C₄⋊C₈	—	2	2	no	—
189	C₄⋊C₈	—	2	1	no	—
190	C₄⋊C₈	—	2	1	no	—
191	C₄⋊C₈	—	38	1	no	—
192	C₄⋊C₈	—	38	1	no	—
193	#101	—	38	4	no	—
194	C₂²⋊C₈	—	19	2	no	—
195	C₂²⋊C₈	—	19	2	no	—
196	#97	—	2	2	no	—
197	#97	—	2	2	no	—
198	#97	—	38	1	no	—
199	#97	—	38	1	no	—
200	C₄×C₈	—	38	1	no	—
201	C₄×C₈	—	38	1	no	—
202	C₄×C₈	—	38	1	no	—
203	C₄×C₈	—	38	1	no	—
204	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
205	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
206	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
207	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
208	C₁₉	#8167	1	1	yes	characteristic, complemented, Sylow
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	—	2	2	no	—
211	C₂×Q₈	—	2	1	no	—
212	C₂×Q₈	—	2	1	no	—
213	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
214	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
215	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
217	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
218	M₁₆	—	19	4	no	—
219	M₁₆	—	38	2	no	—
220	M₁₆	—	38	2	no	—
221	M₁₆	—	38	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	38	2	no	—
227	C₂×C₈	—	38	2	no	—
228	C₂×C₈	—	38	2	no	—
229	C₂×C₈	—	38	2	no	—
230	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
231	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
232	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
233	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄²	—	2	2	no	—
239	C₄²	—	2	2	no	—
240	C₄²	#64541	1	1	yes	characteristic
241	C₄²	#64541	1	1	yes	characteristic
242	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
243	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
244	C₂³	C₂×C₄×Dic₁₉	1	1	yes	characteristic
245	Q₈	—	4	2	no	—
246	Q₈	—	4	2	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	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	—	4	1	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	M₁₆×D₁₉	1	2	yes	—
256	C₂×C₄	#10597353	1	2	yes	—
257	C₂×C₄	#10597372	1	1	yes	characteristic
258	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
259	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
260	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
261	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
262	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
263	C₈	—	2	1	no	—
264	C₈	—	2	1	no	—
265	C₈	—	2	1	no	—
266	C₈	—	4	2	no	—
267	C₈	—	38	4	no	—
268	C₈	—	76	2	no	—
269	C₈	—	76	2	no	—
270	C₈	—	76	2	no	—
271	C₈	#10597293	1	2	yes	complemented
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	C₄×C₂²⋊Dic₁₉	1	1	yes	characteristic
275	C₂²	M₁₆×Dic₁₉	1	1	yes	characteristic
276	C₂²	#11926502	1	1	yes	characteristic
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	4	2	no	—
281	C₄	—	4	2	no	—
282	C₄	C₂×#10597293	1	1	yes	characteristic
283	C₄	#11926528	1	1	yes	characteristic
284	C₄	#11926613	1	2	yes	—
285	C₄	#11926620	1	2	yes	—
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	#14563416	1	1	yes	characteristic
289	C₂	C₄×#10597293	1	1	yes	characteristic
290	C₂	#14563435	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₁₉
commutator, prime radical	C₂³×C₁₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#446	C₂
socle	C₂²×C₁₉	C₄×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#446	D₁₉
hypocenter	C₁₉	#8167
LCS	G ▷ C₂³×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
122. C₈×C₁₉	#101	semidirect
208. C₁₉	#8167	semidirect
271. C₈	#10597293	semidirect

Maximal subgroups

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

subgroup	index
C₄×#10597293	2
#14561959	2
#14561924	2
C₈×#64520	2
#14561485	2
C₂×C₁₉×#446	2
#8167	19