Subgroups — #18072684

Order 4864.

874 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926508	C₂	1	2	yes	maximal
2	#14561522	C₂	1	4	yes	maximal, complemented
3	C₂×C₁₉×#407	C₂	1	1	yes	maximal, characteristic
4	C₂²×#64520	—	2	2	no	—
5	#11926508	—	2	2	no	—
6	#11926508	—	2	2	no	—
7	#11926508	C₄	1	4	yes	—
8	#11926508	C₄	1	4	yes	—
9	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
10	C₂×C₂²⋊(C₁₉⋊C₈)	C₂²	1	4	yes	—
11	C₂⁴×C₄×C₁₉	C₂²	1	1	yes	characteristic
12	C₂²×M₁₆×C₁₉	C₄	1	2	yes	—
13	C₁₉×#407	—	2	2	no	—
14	C₁₉×#407	C₄	1	2	yes	—
15	C₁₉×#407	C₂²	1	2	yes	—
16	C₂×C₁₉×C₂²⋊C₈	—	2	2	no	—
17	C₂⁵×C₁₉	Q₈	1	1	yes	characteristic
18	C₂³×C₄×C₁₉	—	4	4	no	—
19	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
22	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₉	D₄	1	2	yes	—
24	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₉	D₄	1	2	yes	—
27	C₂³×C₄×C₁₉	Q₈	1	1	yes	characteristic
28	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
29	C₂×M₁₆×C₁₉	—	2	2	no	—
30	C₂×M₁₆×C₁₉	—	4	4	no	—
31	C₂×M₁₆×C₁₉	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₁₉	—	2	2	no	—
33	C₂×#64520	—	2	4	no	—
34	C₂×#64520	—	2	4	no	—
35	C₂×#64520	—	4	8	no	—
36	C₂²×C₁₉⋊C₈	—	2	4	no	—
37	C₁₉×C₂²⋊C₈	—	2	4	no	—
38	C₁₉×C₂²⋊C₈	—	2	4	no	—
39	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
40	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
41	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
42	C₂²⋊(C₁₉⋊C₈)	C₂×C₄	1	8	yes	—
43	C₂⁴×C₁₉	—	4	2	no	—
44	C₂⁴×C₁₉	—	4	4	no	—
45	C₂⁴×C₁₉	C₄⋊C₄	1	2	yes	—
46	C₂⁴×C₁₉	C₄⋊C₄	1	2	yes	—
47	C₂⁴×C₁₉	C₄⋊C₄	1	1	yes	characteristic
48	C₂⁴×C₁₉	C₄⋊C₄	1	1	yes	characteristic
49	C₂⁴×C₁₉	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₉	—	2	4	no	—
51	C₂²×C₄×C₁₉	—	2	8	no	—
52	C₂²×C₄×C₁₉	—	4	4	no	—
53	C₂²×C₄×C₁₉	—	4	2	no	—
54	C₂²×C₄×C₁₉	—	4	2	no	—
55	C₂²×C₄×C₁₉	—	4	4	no	—
56	C₂²×C₄×C₁₉	—	4	2	no	—
57	C₂²×C₄×C₁₉	—	4	2	no	—
58	C₂²×C₄×C₁₉	—	4	4	no	—
59	C₂²×C₄×C₁₉	—	4	2	no	—
60	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
63	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
78	M₁₆×C₁₉	—	4	4	no	—
79	M₁₆×C₁₉	—	4	4	no	—
80	C₂×C₈×C₁₉	—	2	4	no	—
81	C₂×C₈×C₁₉	—	4	4	no	—
82	#64520	—	4	8	no	—
83	#64520	—	4	8	no	—
84	C₂×C₁₉⋊C₈	—	2	8	no	—
85	C₂×C₁₉⋊C₈	—	4	8	no	—
86	#8055	—	19	1	no	maximal, Sylow
87	C₂³×C₁₉	—	2	4	no	—
88	C₂³×C₁₉	—	2	2	no	—
89	C₂³×C₁₉	—	2	4	no	—
90	C₂³×C₁₉	—	2	4	no	—
91	C₂³×C₁₉	—	2	4	no	—
92	C₂³×C₁₉	—	4	2	no	—
93	C₂³×C₁₉	—	4	1	no	—
94	C₂³×C₁₉	—	4	1	no	—
95	C₂³×C₁₉	—	4	2	no	—
96	C₂³×C₁₉	—	4	2	no	—
97	C₂³×C₁₉	—	4	2	no	—
98	C₂³×C₁₉	—	4	2	no	—
99	C₂³×C₁₉	—	4	16	no	—
100	C₂³×C₁₉	#95	1	1	yes	characteristic
101	C₂³×C₁₉	#95	1	2	yes	—
102	C₂³×C₁₉	#95	1	1	yes	characteristic
103	C₂³×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
104	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₉	—	2	4	no	—
106	C₂×C₄×C₁₉	—	2	2	no	—
107	C₂×C₄×C₁₉	—	2	4	no	—
108	C₂×C₄×C₁₉	—	2	2	no	—
109	C₂×C₄×C₁₉	—	2	4	no	—
110	C₂×C₄×C₁₉	—	2	2	no	—
111	C₂×C₄×C₁₉	—	2	4	no	—
112	C₂×C₄×C₁₉	—	2	2	no	—
113	C₂×C₄×C₁₉	—	2	2	no	—
114	C₂×C₄×C₁₉	—	2	4	no	—
115	C₂×C₄×C₁₉	—	4	2	no	—
116	C₂×C₄×C₁₉	—	4	2	no	—
117	C₂×C₄×C₁₉	—	4	2	no	—
118	C₂×C₄×C₁₉	—	4	2	no	—
119	C₂×C₄×C₁₉	—	4	1	no	—
120	C₂×C₄×C₁₉	—	4	2	no	—
121	C₂×C₄×C₁₉	—	4	1	no	—
122	C₂×C₄×C₁₉	#95	1	1	yes	characteristic
123	C₂×C₄×C₁₉	#95	1	2	yes	—
124	C₂×C₄×C₁₉	#95	1	1	yes	characteristic
125	C₂×C₄×C₁₉	#99	1	4	yes	—
126	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
127	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
129	C₈×C₁₉	—	4	4	no	—
130	C₁₉⋊C₈	—	4	8	no	—
131	C₂×#407	—	19	2	no	—
132	C₂×#407	D₁₉	1	1	yes	characteristic
133	#1298	—	19	4	no	—
134	C₂²×C₁₉	—	2	2	no	—
135	C₂²×C₁₉	—	2	2	no	—
136	C₂²×C₁₉	—	2	2	no	—
137	C₂²×C₁₉	—	2	2	no	—
138	C₂²×C₁₉	—	2	2	no	—
139	C₂²×C₁₉	—	2	2	no	—
140	C₂²×C₁₉	—	2	2	no	—
141	C₂²×C₁₉	—	2	2	no	—
142	C₂²×C₁₉	—	2	1	no	—
143	C₂²×C₁₉	—	2	1	no	—
144	C₂²×C₁₉	—	4	8	no	—
145	C₂²×C₁₉	—	4	4	no	—
146	C₂²×C₁₉	—	4	4	no	—
147	C₂²×C₁₉	—	4	8	no	—
148	C₂²×C₁₉	—	4	4	no	—
149	C₂²×C₁₉	#342	1	2	yes	—
150	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
151	C₂²×C₁₉	#430	1	4	yes	—
152	C₄×C₁₉	—	2	2	no	—
153	C₄×C₁₉	—	2	2	no	—
154	C₄×C₁₉	—	2	1	no	—
155	C₄×C₁₉	—	2	1	no	—
156	C₄×C₁₉	#342	1	2	yes	—
157	C₄×C₁₉	C₂×#99	1	2	yes	—
158	C₂⁴×C₄	C₂×D₁₉	1	1	yes	characteristic
159	C₂²×M₁₆	—	38	2	no	—
160	C₂²×M₁₆	Dic₁₉	1	2	yes	—
161	#407	—	2	2	no	—
162	#407	—	19	4	no	—
163	#407	—	19	4	no	—
164	#407	—	38	2	no	—
165	#407	—	38	2	no	—
166	#407	Dic₁₉	1	2	yes	—
167	#407	C₂×D₁₉	1	2	yes	—
168	C₂×C₂²⋊C₈	—	2	2	no	—
169	C₂×C₂²⋊C₈	—	19	4	no	—
170	C₂×C₂²⋊C₈	—	38	2	no	—
171	C₂×C₁₉	—	2	2	no	—
172	C₂×C₁₉	—	2	4	no	—
173	C₂×C₁₉	—	4	2	no	—
174	C₂×C₁₉	—	4	1	no	—
175	C₂×C₁₉	—	4	1	no	—
176	C₂×C₁₉	#1738	1	2	yes	—
177	C₂×C₁₉	C₂×#342	1	1	yes	characteristic
178	C₂⁵	Dic₃₈	1	1	yes	characteristic
179	C₂³×C₄	—	4	4	no	—
180	C₂³×C₄	Dic₃₈	1	1	yes	characteristic
181	C₂³×C₄	C₄×D₁₉	1	2	yes	—
182	C₂³×C₄	C₄×D₁₉	1	2	yes	—
183	C₂³×C₄	D₇₆	1	1	yes	characteristic
184	C₂³×C₄	D₇₆	1	1	yes	characteristic
185	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
186	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
187	C₂³×C₄	C₂²⋊D₁₉	1	2	yes	—
188	C₂³×C₄	C₂²⋊D₁₉	1	2	yes	—
189	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	4	4	no	—
192	C₂×M₁₆	—	38	4	no	—
193	C₂×M₁₆	—	38	4	no	—
194	C₂×M₁₆	—	76	8	no	—
195	C₂×M₁₆	C₂×Dic₁₉	1	4	yes	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	38	4	no	—
198	C₂²⋊C₈	—	2	4	no	—
199	C₂²⋊C₈	—	2	4	no	—
200	C₂²⋊C₈	—	19	8	no	—
201	C₂²⋊C₈	—	38	4	no	—
202	C₂²⋊C₈	—	38	4	no	—
203	C₂²⋊C₈	—	38	4	no	—
204	C₁₉	#8055	1	1	yes	characteristic, complemented, Sylow
205	C₂⁴	—	4	2	no	—
206	C₂⁴	—	4	4	no	—
207	C₂⁴	Dic₁₉⋊C₄	1	2	yes	—
208	C₂⁴	Dic₁₉⋊C₄	1	2	yes	—
209	C₂⁴	C₄⋊Dic₁₉	1	1	yes	characteristic
210	C₂⁴	C₄⋊Dic₁₉	1	1	yes	characteristic
211	C₂⁴	C₂×Dic₃₈	1	1	yes	characteristic
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	8	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	—	4	4	no	—
216	C₂²×C₄	—	4	2	no	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	—	4	4	no	—
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	—	4	4	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
223	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
224	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
225	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
226	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
227	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
228	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
229	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
230	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
231	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
232	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
233	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
234	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
235	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
237	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
238	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
240	M₁₆	—	4	4	no	—
241	M₁₆	—	4	4	no	—
242	M₁₆	—	76	8	no	—
243	M₁₆	—	76	8	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	4	4	no	—
246	C₂×C₈	—	38	8	no	—
247	C₂×C₈	—	76	8	no	—
248	C₂³	—	2	4	no	—
249	C₂³	—	2	2	no	—
250	C₂³	—	2	4	no	—
251	C₂³	—	2	4	no	—
252	C₂³	—	2	4	no	—
253	C₂³	—	4	2	no	—
254	C₂³	—	4	1	no	—
255	C₂³	—	4	1	no	—
256	C₂³	—	4	2	no	—
257	C₂³	—	4	2	no	—
258	C₂³	—	4	2	no	—
259	C₂³	—	4	2	no	—
260	C₂³	—	4	16	no	—
261	C₂³	C₁₉⋊#95	1	1	yes	characteristic
262	C₂³	C₁₉⋊#95	1	2	yes	—
263	C₂³	C₁₉⋊#95	1	1	yes	characteristic
264	C₂³	C₂×Dic₁₉⋊C₄	1	2	yes	—
265	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	C₁₉⋊#95	1	1	yes	characteristic
284	C₂×C₄	C₁₉⋊#95	1	2	yes	—
285	C₂×C₄	C₁₉⋊#95	1	1	yes	characteristic
286	C₂×C₄	#10597291	1	4	yes	—
287	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
288	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	2	yes	—
289	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
290	C₈	—	4	4	no	—
291	C₈	—	76	8	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	4	8	no	—
303	C₂²	—	4	4	no	—
304	C₂²	—	4	4	no	—
305	C₂²	—	4	8	no	—
306	C₂²	—	4	4	no	—
307	C₂²	#11926311	1	2	yes	—
308	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
309	C₂²	#11926592	1	4	yes	—
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	#11926311	1	2	yes	—
315	C₄	C₂×#10597291	1	2	yes	—
316	C₂	—	2	2	no	—
317	C₂	—	2	4	no	—
318	C₂	—	4	2	no	—
319	C₂	—	4	1	no	—
320	C₂	—	4	1	no	—
321	C₂	#14563361	1	2	yes	—
322	C₂	C₂×#11926311	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₁₉×#407	C₂
socle	C₂²×C₁₉	C₂×#95
prime radical	C₂⁵×C₁₉	Q₈
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×#407	D₁₉
hypocenter	C₁₉	#8055
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ C₂×#407

Complements

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

N	H ≅ G/N	type
2. #14561522	C₂	semidirect
204. C₁₉	#8055	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926508	2
#14561522	2
C₂×C₁₉×#407	2
#8055	19