Subgroups — #18073869

Order 4864.

1502 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	#14563545	C₂	1	2	yes	maximal
2	C₂×#11926270	C₂	1	4	yes	maximal, complemented
3	C₂³×C₄⋊C₄×C₁₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
5	C₂×C₂²⋊(C₁₉⋊C₈)	C₂²	1	4	yes	—
6	#11926270	—	2	8	no	—
7	C₂⁴×C₄×C₁₉	C₄	1	1	yes	characteristic
8	C₂⁴×C₄×C₁₉	C₄	1	1	yes	characteristic
9	C₂⁴×C₄×C₁₉	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₁₉	—	2	4	no	—
11	C₂²×C₄⋊C₄×C₁₉	—	2	8	no	—
12	C₂²×C₄⋊C₄×C₁₉	C₄	1	2	yes	—
13	C₂²×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
14	C₂⁵×C₁₉	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₄×C₁₉	—	2	2	no	—
16	C₂³×C₄×C₁₉	—	2	2	no	—
17	C₂³×C₄×C₁₉	—	2	4	no	—
18	C₂³×C₄×C₁₉	—	2	4	no	—
19	C₂³×C₄×C₁₉	—	2	2	no	—
20	C₂³×C₄×C₁₉	—	2	4	no	—
21	C₂³×C₄×C₁₉	—	2	4	no	—
22	C₂³×C₄×C₁₉	—	4	4	no	—
23	C₂³×C₄×C₁₉	—	4	4	no	—
24	C₂³×C₄×C₁₉	C₈	1	4	yes	complemented
25	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
28	C₂³×C₄×C₁₉	D₄	1	4	yes	—
29	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₁₉	—	2	32	no	—
31	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
32	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
33	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
34	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
35	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
36	C₂×C₄⋊C₄×C₁₉	D₄	1	8	yes	—
37	C₂²×C₁₉⋊C₈	—	2	4	no	—
38	C₂²⋊(C₁₉⋊C₈)	—	2	8	no	—
39	C₂²⋊(C₁₉⋊C₈)	—	4	8	no	—
40	C₂⁴×C₁₉	—	2	4	no	—
41	C₂⁴×C₁₉	—	2	4	no	—
42	C₂⁴×C₁₉	—	2	2	no	—
43	C₂⁴×C₁₉	—	2	2	no	—
44	C₂⁴×C₁₉	C₂²⋊C₄	1	2	yes	—
45	C₂⁴×C₁₉	C₂×C₈	1	2	yes	—
46	C₂⁴×C₁₉	M₁₆	1	2	yes	—
47	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₉	—	2	8	no	—
49	C₂²×C₄×C₁₉	—	2	16	no	—
50	C₂²×C₄×C₁₉	—	2	8	no	—
51	C₂²×C₄×C₁₉	—	2	8	no	—
52	C₂²×C₄×C₁₉	—	2	2	no	—
53	C₂²×C₄×C₁₉	—	2	2	no	—
54	C₂²×C₄×C₁₉	—	2	2	no	—
55	C₂²×C₄×C₁₉	—	2	2	no	—
56	C₂²×C₄×C₁₉	—	2	16	no	—
57	C₂²×C₄×C₁₉	—	2	8	no	—
58	C₂²×C₄×C₁₉	—	2	8	no	—
59	C₂²×C₄×C₁₉	—	2	2	no	—
60	C₂²×C₄×C₁₉	—	2	2	no	—
61	C₂²×C₄×C₁₉	—	2	2	no	—
62	C₂²×C₄×C₁₉	—	2	8	no	—
63	C₂²×C₄×C₁₉	—	4	2	no	—
64	C₂²×C₄×C₁₉	—	4	4	no	—
65	C₂²×C₄×C₁₉	—	4	2	no	—
66	C₂²×C₄×C₁₉	—	4	16	no	—
67	C₂²×C₄×C₁₉	—	4	16	no	—
68	C₂²×C₄×C₁₉	—	4	4	no	—
69	C₂²×C₄×C₁₉	—	4	4	no	—
70	C₂²×C₄×C₁₉	—	4	4	no	—
71	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
72	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
73	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
74	C₂²×C₄×C₁₉	C₂×C₈	1	4	yes	—
75	C₂²×C₄×C₁₉	M₁₆	1	4	yes	—
76	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
78	C₄⋊C₄×C₁₉	—	2	16	no	—
79	C₄⋊C₄×C₁₉	—	2	16	no	—
80	C₂×C₁₉⋊C₈	—	4	8	no	—
81	C₂×C₁₉⋊C₈	—	4	4	no	—
82	#8299	—	19	1	no	maximal, Sylow
83	C₂³×C₁₉	—	2	8	no	—
84	C₂³×C₁₉	—	2	16	no	—
85	C₂³×C₁₉	—	2	8	no	—
86	C₂³×C₁₉	—	2	8	no	—
87	C₂³×C₁₉	—	2	8	no	—
88	C₂³×C₁₉	—	2	2	no	—
89	C₂³×C₁₉	—	2	4	no	—
90	C₂³×C₁₉	—	2	2	no	—
91	C₂³×C₁₉	—	2	4	no	—
92	C₂³×C₁₉	—	2	2	no	—
93	C₂³×C₁₉	—	2	2	no	—
94	C₂³×C₁₉	—	2	2	no	—
95	C₂³×C₁₉	—	2	2	no	—
96	C₂³×C₁₉	C₂²⋊C₈	1	4	yes	—
97	C₂³×C₁₉	#99	1	4	yes	—
98	C₂³×C₁₉	#101	1	4	yes	—
99	C₂³×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
100	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂³×C₁₉	C₂²×C₈	1	1	yes	characteristic
102	C₂³×C₁₉	C₂×M₁₆	1	1	yes	characteristic
103	C₂³×C₁₉	C₂×M₁₆	1	2	yes	—
104	C₂×C₄×C₁₉	—	2	32	no	—
105	C₂×C₄×C₁₉	—	2	4	no	—
106	C₂×C₄×C₁₉	—	2	4	no	—
107	C₂×C₄×C₁₉	—	2	4	no	—
108	C₂×C₄×C₁₉	—	2	4	no	—
109	C₂×C₄×C₁₉	—	2	4	no	—
110	C₂×C₄×C₁₉	—	2	4	no	—
111	C₂×C₄×C₁₉	—	2	4	no	—
112	C₂×C₄×C₁₉	—	2	2	no	—
113	C₂×C₄×C₁₉	—	2	4	no	—
114	C₂×C₄×C₁₉	—	2	4	no	—
115	C₂×C₄×C₁₉	—	4	4	no	—
116	C₂×C₄×C₁₉	—	4	16	no	—
117	C₂×C₄×C₁₉	—	4	4	no	—
118	C₂×C₄×C₁₉	—	4	16	no	—
119	C₂×C₄×C₁₉	—	4	4	no	—
120	C₂×C₄×C₁₉	—	4	4	no	—
121	C₂×C₄×C₁₉	—	4	4	no	—
122	C₂×C₄×C₁₉	C₂²⋊C₈	1	8	yes	complemented
123	C₂×C₄×C₁₉	#120	1	4	yes	—
124	C₁₉⋊C₈	—	8	4	no	—
125	C₂³×C₄⋊C₄	D₁₉	1	1	yes	characteristic
126	#1779	—	19	2	no	—
127	C₂×#324	—	19	4	no	—
128	C₂²×C₁₉	—	2	16	no	—
129	C₂²×C₁₉	—	2	16	no	—
130	C₂²×C₁₉	—	2	4	no	—
131	C₂²×C₁₉	—	2	2	no	—
132	C₂²×C₁₉	—	2	2	no	—
133	C₂²×C₁₉	—	2	4	no	—
134	C₂²×C₁₉	—	2	4	no	—
135	C₂²×C₁₉	—	2	4	no	—
136	C₂²×C₁₉	—	2	4	no	—
137	C₂²×C₁₉	—	2	4	no	—
138	C₂²×C₁₉	—	2	2	no	—
139	C₂²×C₁₉	—	2	4	no	—
140	C₂²×C₁₉	—	2	2	no	—
141	C₂²×C₁₉	#324	1	8	yes	complemented
142	C₂²×C₁₉	#379	1	1	yes	characteristic
143	C₂²×C₁₉	C₂×C₂²⋊C₈	1	2	yes	—
144	C₂²×C₁₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
145	C₂²×C₁₉	#407	1	2	yes	—
146	C₂²×C₁₉	#407	1	1	yes	characteristic
147	C₂²×C₁₉	C₂×#99	1	2	yes	—
148	C₂²×C₁₉	C₂×#101	1	2	yes	—
149	C₄×C₁₉	—	2	8	no	—
150	C₄×C₁₉	—	2	4	no	—
151	C₄×C₁₉	—	4	2	no	—
152	C₄×C₁₉	—	4	4	no	—
153	C₂⁴×C₄	Dic₁₉	1	1	yes	characteristic
154	C₂⁴×C₄	Dic₁₉	1	1	yes	characteristic
155	C₂⁴×C₄	C₂×D₁₉	1	1	yes	characteristic
156	C₂²×C₄⋊C₄	—	2	4	no	—
157	C₂²×C₄⋊C₄	—	2	8	no	—
158	C₂²×C₄⋊C₄	Dic₁₉	1	2	yes	—
159	C₂²×C₄⋊C₄	C₂×D₁₉	1	2	yes	—
160	C₂×C₂²⋊C₈	—	19	4	no	—
161	C₂×C₂²⋊C₈	—	38	4	no	—
162	#324	—	38	8	no	—
163	C₂×C₁₉	—	2	8	no	—
164	C₂×C₁₉	—	2	2	no	—
165	C₂×C₁₉	—	2	2	no	—
166	C₂×C₁₉	C₂×#324	1	2	yes	—
167	C₂×C₁₉	#1463	1	2	yes	—
168	C₂×C₁₉	#1779	1	1	yes	characteristic
169	C₂×C₁₉	#1781	1	1	yes	characteristic
170	C₂×C₁₉	#1784	1	1	yes	characteristic
171	C₂⁵	C₂×Dic₁₉	1	1	yes	characteristic
172	C₂³×C₄	—	2	2	no	—
173	C₂³×C₄	—	2	2	no	—
174	C₂³×C₄	—	2	4	no	—
175	C₂³×C₄	—	2	4	no	—
176	C₂³×C₄	—	2	2	no	—
177	C₂³×C₄	—	2	4	no	—
178	C₂³×C₄	—	2	4	no	—
179	C₂³×C₄	—	4	4	no	—
180	C₂³×C₄	—	4	4	no	—
181	C₂³×C₄	C₁₉⋊C₈	1	4	yes	complemented
182	C₂³×C₄	C₂×Dic₁₉	1	2	yes	—
183	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
184	C₂³×C₄	C₂×Dic₁₉	1	2	yes	—
185	C₂³×C₄	C₂²⋊D₁₉	1	4	yes	—
186	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
187	C₂²×C₈	—	38	4	no	—
188	C₂×C₄⋊C₄	—	2	32	no	—
189	C₂×C₄⋊C₄	—	2	4	no	—
190	C₂×C₄⋊C₄	—	2	4	no	—
191	C₂×C₄⋊C₄	—	2	4	no	—
192	C₂×C₄⋊C₄	—	2	4	no	—
193	C₂×C₄⋊C₄	—	2	4	no	—
194	C₂×C₄⋊C₄	C₂²⋊D₁₉	1	8	yes	—
195	C₂²⋊C₈	—	38	8	no	—
196	C₂²⋊C₈	—	76	8	no	—
197	C₁₉	#8299	1	1	yes	characteristic, complemented, Sylow
198	C₂⁴	—	2	2	no	—
199	C₂⁴	—	2	4	no	—
200	C₂⁴	—	2	4	no	—
201	C₂⁴	—	2	2	no	—
202	C₂⁴	C₂×C₁₉⋊C₈	1	2	yes	—
203	C₂⁴	#64520	1	2	yes	—
204	C₂⁴	C₂²⋊Dic₁₉	1	2	yes	—
205	C₂⁴	C₂²×Dic₁₉	1	1	yes	characteristic
206	C₂²×C₄	—	2	8	no	—
207	C₂²×C₄	—	2	8	no	—
208	C₂²×C₄	—	2	16	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	8	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	16	no	—
215	C₂²×C₄	—	2	8	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	8	no	—
220	C₂²×C₄	—	2	8	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	—	4	4	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	—	4	16	no	—
225	C₂²×C₄	—	4	16	no	—
226	C₂²×C₄	—	4	4	no	—
227	C₂²×C₄	—	4	4	no	—
228	C₂²×C₄	—	4	4	no	—
229	C₂²×C₄	C₂×C₁₉⋊C₈	1	4	yes	—
230	C₂²×C₄	#64520	1	4	yes	—
231	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
232	C₂²×C₄	C₂²⋊Dic₁₉	1	4	yes	—
233	C₂²×C₄	C₂²⋊Dic₁₉	1	4	yes	—
234	C₂²×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
235	C₂²×C₄	C₂×C₂²⋊D₁₉	1	4	yes	—
236	C₂×C₈	—	76	8	no	—
237	C₂×C₈	—	76	4	no	—
238	C₄⋊C₄	—	2	16	no	—
239	C₄⋊C₄	—	2	16	no	—
240	C₂³	—	2	8	no	—
241	C₂³	—	2	16	no	—
242	C₂³	—	2	8	no	—
243	C₂³	—	2	2	no	—
244	C₂³	—	2	4	no	—
245	C₂³	—	2	4	no	—
246	C₂³	—	2	2	no	—
247	C₂³	—	2	2	no	—
248	C₂³	—	2	2	no	—
249	C₂³	—	2	2	no	—
250	C₂³	—	2	2	no	—
251	C₂³	—	2	8	no	—
252	C₂³	—	2	8	no	—
253	C₂³	C₂²⋊(C₁₉⋊C₈)	1	4	yes	—
254	C₂³	#10597291	1	4	yes	—
255	C₂³	#10597293	1	4	yes	—
256	C₂³	C₂²×C₁₉⋊C₈	1	1	yes	characteristic
257	C₂³	C₂×#64520	1	1	yes	characteristic
258	C₂³	C₂×#64520	1	2	yes	—
259	C₂³	C₂×C₂²⋊Dic₁₉	1	2	yes	—
260	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
261	C₂×C₄	—	2	32	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	16	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	16	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	8	yes	complemented
280	C₂×C₄	#10597399	1	4	yes	—
281	C₈	—	152	4	no	—
282	C₂²	—	2	16	no	—
283	C₂²	—	2	16	no	—
284	C₂²	—	2	4	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	2	4	no	—
290	C₂²	—	2	4	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	4	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	4	no	—
295	C₂²	#11926270	1	8	yes	complemented
296	C₂²	#11926400	1	1	yes	characteristic
297	C₂²	C₂×C₂²⋊(C₁₉⋊C₈)	1	2	yes	—
298	C₂²	C₂×C₂²⋊(C₁₉⋊C₈)	1	1	yes	characteristic
299	C₂²	#11926508	1	2	yes	—
300	C₂²	#11926508	1	1	yes	characteristic
301	C₂²	C₂×#10597291	1	2	yes	—
302	C₂²	C₂×#10597293	1	2	yes	—
303	C₄	—	2	8	no	—
304	C₄	—	2	4	no	—
305	C₄	—	4	2	no	—
306	C₄	—	4	4	no	—
307	C₂	—	2	2	no	—
308	C₂	—	2	2	no	—
309	C₂	—	2	8	no	—
310	C₂	C₂×#11926270	1	2	yes	—
311	C₂	#14561965	1	2	yes	—
312	C₂	#14563545	1	1	yes	characteristic
313	C₂	#14563550	1	1	yes	characteristic
314	C₂	#14563559	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₉
commutator	C₂³×C₁₉	C₂²×C₈
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂³×C₄⋊C₄×C₁₉	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁵×C₁₉	C₂×C₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂³×C₄⋊C₄	D₁₉
hypocenter	C₁₉	#8299
LCS	G ▷ C₂³×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×#11926270	C₂	semidirect
24. C₂³×C₄×C₁₉	C₈	semidirect
122. C₂×C₄×C₁₉	C₂²⋊C₈	semidirect
141. C₂²×C₁₉	#324	semidirect
181. C₂³×C₄	C₁₉⋊C₈	semidirect
197. C₁₉	#8299	semidirect
279. C₂×C₄	C₂²⋊(C₁₉⋊C₈)	semidirect
295. C₂²	#11926270	semidirect

Maximal subgroups

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

subgroup	index
#14563545	2
C₂×#11926270	2
C₂³×C₄⋊C₄×C₁₉	2
#8299	19