Subgroups — #18074302

Order 4864.

782 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563627	C₂	1	1	yes	maximal, characteristic
2	#14562320	C₂	1	2	yes	maximal, complemented
3	#14562064	C₂	1	2	yes	maximal, complemented
4	#14561956	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₉×#554	C₂	1	1	yes	maximal, characteristic
6	#11926597	C₂²	1	2	yes	—
7	#11926568	C₂²	1	2	yes	—
8	C₂×#10597290	—	2	1	no	—
9	#11926313	—	2	1	no	—
10	#11926290	—	2	4	no	—
11	Q₈⋊(C₁₉⋊C₈)	—	2	4	no	—
12	#11926268	—	2	2	no	—
13	C₁₉×#554	—	2	4	no	—
14	C₁₉×#554	—	2	4	no	—
15	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₁₉×#125	—	2	1	no	—
17	C₂×C₁₉×C₂²⋊Q₈	—	2	2	no	—
18	C₂×C₁₉×C₂²⋊Q₈	C₄	1	2	yes	—
19	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₁₉	—	2	1	no	—
21	C₂×C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₁₉	C₄	1	2	yes	—
23	C₂²×Q₈×C₁₉	—	2	1	no	—
24	C₂²×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×D₄×C₁₉	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₉	—	2	2	no	—
27	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
28	C₁₉×C₄⋊Q₈	—	2	2	no	—
29	C₁₉×C₄⋊Q₈	D₄	1	4	yes	—
30	C₁₉×#125	—	2	8	no	—
31	C₁₉×C₂²⋊Q₈	—	2	16	no	—
32	C₁₉×C₂²⋊Q₈	—	2	4	no	—
33	C₁₉×C₂²⋊Q₈	—	2	4	no	—
34	C₄×Q₈×C₁₉	—	2	2	no	—
35	C₄×Q₈×C₁₉	D₄	1	4	yes	—
36	C₄×D₄×C₁₉	—	2	8	no	—
37	C₄×D₄×C₁₉	—	2	2	no	—
38	C₄×D₄×C₁₉	—	2	2	no	—
39	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
40	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
41	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
42	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
43	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
44	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
45	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
48	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
49	C₂×C₂²⋊C₄×C₁₉	D₄	1	2	yes	—
50	C₂×C₄²×C₁₉	—	2	1	no	—
51	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
52	C₂×#64520	—	2	2	no	—
53	#10597290	—	4	2	no	—
54	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
55	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
56	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
57	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
58	C₄×C₁₉⋊C₈	—	2	2	no	—
59	C₂⁴×C₁₉	C₂²⋊C₄	1	2	yes	—
60	C₂×Q₈×C₁₉	—	2	1	no	—
61	C₂×Q₈×C₁₉	—	2	4	no	—
62	C₂×Q₈×C₁₉	—	4	2	no	—
63	C₂×Q₈×C₁₉	—	8	2	no	—
64	C₂×Q₈×C₁₉	C₂²⋊C₄	1	2	yes	—
65	C₂×D₄×C₁₉	—	2	2	no	—
66	C₂×D₄×C₁₉	—	2	2	no	—
67	C₂×D₄×C₁₉	SD₁₆	1	2	yes	—
68	C₂×D₄×C₁₉	Q₁₆	1	2	yes	—
69	C₂²×C₄×C₁₉	—	2	1	no	—
70	C₂²×C₄×C₁₉	—	2	1	no	—
71	C₂²×C₄×C₁₉	—	2	2	no	—
72	C₂²×C₄×C₁₉	—	2	8	no	—
73	C₂²×C₄×C₁₉	—	2	2	no	—
74	C₂²×C₄×C₁₉	—	2	2	no	—
75	C₂²×C₄×C₁₉	—	4	4	no	—
76	C₂²×C₄×C₁₉	—	8	4	no	—
77	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
78	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
81	C₄⋊C₄×C₁₉	—	2	8	no	—
82	C₄⋊C₄×C₁₉	—	2	8	no	—
83	C₄⋊C₄×C₁₉	—	2	1	no	—
84	C₄⋊C₄×C₁₉	—	2	2	no	—
85	C₄⋊C₄×C₁₉	—	2	4	no	—
86	C₄⋊C₄×C₁₉	—	2	4	no	—
87	C₄⋊C₄×C₁₉	—	2	2	no	—
88	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
89	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
90	C₂²⋊C₄×C₁₉	—	2	8	no	—
91	C₂²⋊C₄×C₁₉	—	2	4	no	—
92	C₄²×C₁₉	—	2	4	no	—
93	C₄²×C₁₉	C₂×D₄	1	2	yes	—
94	C₄²×C₁₉	C₂×D₄	1	2	yes	—
95	#64520	—	4	2	no	—
96	#64520	—	4	2	no	—
97	C₂×C₁₉⋊C₈	—	4	2	no	—
98	C₂×C₁₉⋊C₈	—	4	2	no	—
99	#8395	—	19	1	no	maximal, Sylow
100	C₂³×C₁₉	—	2	2	no	—
101	C₂³×C₁₉	—	2	2	no	—
102	C₂³×C₁₉	—	4	2	no	—
103	C₂³×C₁₉	—	4	2	no	—
104	C₂³×C₁₉	#103	1	4	yes	—
105	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	Q₈×C₁₉	—	2	4	no	—
107	Q₈×C₁₉	—	4	2	no	—
108	Q₈×C₁₉	—	8	4	no	—
109	D₄×C₁₉	—	2	4	no	—
110	D₄×C₁₉	—	2	4	no	—
111	C₂×C₄×C₁₉	—	2	2	no	—
112	C₂×C₄×C₁₉	—	2	2	no	—
113	C₂×C₄×C₁₉	—	2	2	no	—
114	C₂×C₄×C₁₉	—	2	2	no	—
115	C₂×C₄×C₁₉	—	2	4	no	—
116	C₂×C₄×C₁₉	—	4	1	no	—
117	C₂×C₄×C₁₉	—	4	4	no	—
118	C₂×C₄×C₁₉	—	4	2	no	—
119	C₂×C₄×C₁₉	—	4	4	no	—
120	C₂×C₄×C₁₉	—	4	2	no	—
121	C₂×C₄×C₁₉	—	4	4	no	—
122	C₂×C₄×C₁₉	—	4	2	no	—
123	C₂×C₄×C₁₉	—	8	8	no	—
124	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₉	#120	1	2	yes	—
127	C₂×C₄×C₁₉	#120	1	2	yes	—
128	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
130	C₂×C₄×C₁₉	#137	1	2	yes	—
131	C₁₉⋊C₈	—	8	2	no	—
132	C₁₉⋊C₈	—	8	2	no	—
133	C₂×#554	D₁₉	1	1	yes	characteristic
134	#1800	—	19	1	no	—
135	#1549	—	19	2	no	—
136	#1487	—	19	2	no	—
137	#1460	—	19	1	no	—
138	C₂²×C₁₉	—	2	1	no	—
139	C₂²×C₁₉	—	2	1	no	—
140	C₂²×C₁₉	—	2	4	no	—
141	C₂²×C₁₉	—	4	4	no	—
142	C₂²×C₁₉	—	4	4	no	—
143	C₂²×C₁₉	—	4	2	no	—
144	C₂²×C₁₉	—	4	2	no	—
145	C₂²×C₁₉	#379	1	1	yes	characteristic
146	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
147	C₂²×C₁₉	#419	1	1	yes	characteristic
148	C₄×C₁₉	—	2	2	no	—
149	C₄×C₁₉	—	2	1	no	—
150	C₄×C₁₉	—	4	2	no	—
151	C₄×C₁₉	—	4	4	no	—
152	C₄×C₁₉	—	4	2	no	—
153	C₄×C₁₉	—	4	2	no	—
154	C₄×C₁₉	#413	1	2	yes	—
155	C₄×C₁₉	#448	1	2	yes	—
156	C₄×C₁₉	#451	1	2	yes	—
157	#554	—	2	4	no	—
158	#554	—	2	4	no	—
159	C₂×C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
160	C₂×#125	—	2	1	no	—
161	C₂×C₂²⋊Q₈	—	2	2	no	—
162	C₂×C₂²⋊Q₈	Dic₁₉	1	2	yes	—
163	C₂×C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
164	C₂×C₄×D₄	—	2	1	no	—
165	C₂×C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
166	C₂²×C₄⋊C₄	Dic₁₉	1	2	yes	—
167	#431	—	19	2	no	—
168	#423	—	19	2	no	—
169	C₂×#100	—	38	1	no	—
170	#343	—	38	1	no	—
171	#332	—	38	4	no	—
172	Q₈⋊C₈	—	38	4	no	—
173	#323	—	38	2	no	—
174	C₂×C₁₉	—	2	1	no	—
175	C₂×C₁₉	—	2	1	no	—
176	C₂×C₁₉	—	4	4	no	—
177	C₂×C₁₉	#1785	1	1	yes	characteristic
178	C₂×C₁₉	#1787	1	1	yes	characteristic
179	C₂×C₁₉	#1795	1	1	yes	characteristic
180	C₂²×Q₈	—	2	1	no	—
181	C₂²×Q₈	C₂×Dic₁₉	1	1	yes	characteristic
182	C₂²×D₄	C₂²⋊D₁₉	1	1	yes	characteristic
183	C₂³×C₄	—	2	2	no	—
184	C₂³×C₄	C₂×Dic₁₉	1	2	yes	—
185	C₂×M₁₆	—	38	2	no	—
186	C₄⋊Q₈	—	2	2	no	—
187	C₄⋊Q₈	C₂²⋊D₁₉	1	4	yes	—
188	#125	—	2	8	no	—
189	C₂²⋊Q₈	—	2	16	no	—
190	C₂²⋊Q₈	—	2	4	no	—
191	C₂²⋊Q₈	—	2	4	no	—
192	C₄×Q₈	—	2	2	no	—
193	C₄×Q₈	C₂²⋊D₁₉	1	4	yes	—
194	C₄×D₄	—	2	8	no	—
195	C₄×D₄	—	2	2	no	—
196	C₄×D₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	1	no	—
200	C₂×C₄⋊C₄	—	2	1	no	—
201	C₂×C₄⋊C₄	—	2	4	no	—
202	C₂×C₄⋊C₄	—	2	4	no	—
203	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
204	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
205	C₂×C₄⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
206	C₂×C₂²⋊C₄	—	2	2	no	—
207	C₂×C₂²⋊C₄	C₂²⋊D₁₉	1	2	yes	—
208	C₂×C₄²	—	2	1	no	—
209	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
210	C₄⋊C₈	—	38	2	no	—
211	C₄⋊C₈	—	38	2	no	—
212	C₄⋊C₈	—	38	2	no	—
213	#100	—	76	2	no	—
214	C₂²⋊C₈	—	38	2	no	—
215	C₄×C₈	—	38	2	no	—
216	C₁₉	#8395	1	1	yes	characteristic, complemented, Sylow
217	C₂⁴	C₂²⋊Dic₁₉	1	2	yes	—
218	C₂×Q₈	—	2	1	no	—
219	C₂×Q₈	—	2	4	no	—
220	C₂×Q₈	—	4	2	no	—
221	C₂×Q₈	—	8	2	no	—
222	C₂×Q₈	C₂²⋊Dic₁₉	1	2	yes	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	#64527	1	2	yes	—
226	C₂×D₄	#64528	1	2	yes	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	8	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	4	4	no	—
234	C₂²×C₄	—	8	4	no	—
235	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
236	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
239	M₁₆	—	76	2	no	—
240	M₁₆	—	76	2	no	—
241	C₂×C₈	—	76	2	no	—
242	C₂×C₈	—	76	2	no	—
243	C₄⋊C₄	—	2	8	no	—
244	C₄⋊C₄	—	2	8	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	C₂²⋊Dic₁₉	1	2	yes	—
251	C₄⋊C₄	C₂²⋊Dic₁₉	1	4	yes	—
252	C₂²⋊C₄	—	2	8	no	—
253	C₂²⋊C₄	—	2	4	no	—
254	C₄²	—	2	4	no	—
255	C₄²	C₂×C₂²⋊D₁₉	1	2	yes	—
256	C₄²	C₂×C₂²⋊D₁₉	1	2	yes	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	4	2	no	—
260	C₂³	—	4	2	no	—
261	C₂³	#10597292	1	4	yes	—
262	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
263	Q₈	—	2	4	no	—
264	Q₈	—	4	2	no	—
265	Q₈	—	8	4	no	—
266	D₄	—	2	4	no	—
267	D₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	8	8	no	—
281	C₂×C₄	C₂×#64527	1	1	yes	characteristic
282	C₂×C₄	#10597388	1	2	yes	—
283	C₂×C₄	C₂×#64528	1	1	yes	characteristic
284	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
285	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
286	C₂×C₄	#10597399	1	2	yes	—
287	C₂×C₄	#10597399	1	2	yes	—
288	C₈	—	152	2	no	—
289	C₈	—	152	2	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	4	no	—
293	C₂²	—	4	4	no	—
294	C₂²	—	4	4	no	—
295	C₂²	—	4	2	no	—
296	C₂²	—	4	2	no	—
297	C₂²	#11926400	1	1	yes	characteristic
298	C₂²	C₂×#10597292	1	1	yes	characteristic
299	C₂²	#11926553	1	1	yes	characteristic
300	C₄	—	2	2	no	—
301	C₄	—	2	1	no	—
302	C₄	—	4	2	no	—
303	C₄	—	4	4	no	—
304	C₄	—	4	2	no	—
305	C₄	—	4	2	no	—
306	C₄	#11926528	1	2	yes	—
307	C₄	#11926683	1	2	yes	—
308	C₄	#11926700	1	2	yes	—
309	C₂	—	2	1	no	—
310	C₂	—	2	1	no	—
311	C₂	—	4	4	no	—
312	C₂	#14563563	1	1	yes	characteristic
313	C₂	#14563570	1	1	yes	characteristic
314	C₂	#14563605	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926400
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#554	C₂
socle	C₂²×C₁₉	#379
prime radical	C₂²×D₄×C₁₉	D₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#554	D₁₉
hypocenter	C₁₉	#8395
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#554

Complements

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

N	H ≅ G/N	type
2. #14562320	C₂	semidirect
3. #14562064	C₂	semidirect
216. C₁₉	#8395	semidirect

Maximal subgroups

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

subgroup	index
#14563627	2
#14562320	2
#14562064	2
#14561956	2
C₂×C₁₉×#554	2
#8395	19