Subgroups — #18072122

Order 4864.

838 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	C₂×C₄×#64520	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₄⋊Dic₁₉	C₂	1	1	yes	maximal, characteristic
3	#14561474	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₉×#432	C₂	1	1	yes	maximal, characteristic
5	C₂²×#64520	—	2	1	no	—
6	C₂²×C₄⋊Dic₁₉	C₄	1	2	yes	—
7	C₂²×C₄×Dic₁₉	—	2	1	no	—
8	C₄×#64520	—	2	2	no	—
9	C₄×#64520	C₄	1	4	yes	complemented
10	C₂×#10597260	—	2	1	no	—
11	C₂×C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
12	C₄×C₄⋊Dic₁₉	—	2	1	no	—
13	C₄×C₄⋊Dic₁₉	—	2	1	no	—
14	C₄×C₄⋊Dic₁₉	C₄	1	2	yes	—
15	C₄×C₄⋊Dic₁₉	C₂²	1	2	yes	—
16	C₂×C₁₉⋊#95	—	2	1	no	—
17	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
18	C₁₉×#432	—	2	2	no	—
19	C₁₉×#432	C₄	1	4	yes	complemented
20	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₁₉×C₂²⋊C₈	—	2	1	no	—
22	C₂×C₁₉×#97	—	2	1	no	—
23	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
26	C₂³×Dic₁₉	—	2	2	no	—
27	C₂²×C₈×C₁₉	—	2	2	no	—
28	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₉	D₄	1	2	yes	—
34	C₂×C₄²×C₁₉	Q₈	1	2	yes	—
35	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
36	C₂×C₄⋊Dic₁₉	—	2	2	no	—
37	C₂×C₄⋊Dic₁₉	—	2	8	no	—
38	C₂×C₄⋊Dic₁₉	C₂×C₄	1	4	yes	—
39	C₂×C₄×Dic₁₉	—	2	2	no	—
40	C₂×C₄×Dic₁₉	—	2	2	no	—
41	C₂×C₄×Dic₁₉	—	4	4	no	—
42	C₂×#64520	—	2	8	no	—
43	C₂×#64520	—	2	4	no	—
44	C₂²×C₁₉⋊C₈	—	2	2	no	—
45	C₁₉×C₄⋊C₈	—	2	2	no	—
46	C₁₉×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
47	C₁₉×C₂²⋊C₈	—	2	4	no	—
48	C₁₉×#97	—	2	4	no	—
49	C₁₉⋊#95	—	2	2	no	—
50	C₁₉⋊#95	—	2	2	no	—
51	#10597260	—	2	4	no	—
52	C₄×C₁₉⋊C₈	—	2	2	no	—
53	C₄×C₁₉⋊C₈	C₂×C₄	1	4	yes	complemented
54	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	—	2	4	no	—
56	C₂²×C₄×C₁₉	—	2	4	no	—
57	C₂²×C₄×C₁₉	—	2	2	no	—
58	C₂²×C₄×C₁₉	—	2	2	no	—
59	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
60	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×Dic₁₉	—	2	8	no	—
72	C₂²×Dic₁₉	—	2	4	no	—
73	C₂²×Dic₁₉	—	4	4	no	—
74	C₂²×Dic₁₉	—	4	4	no	—
75	C₂×C₈×C₁₉	—	2	4	no	—
76	C₂×C₈×C₁₉	—	4	2	no	—
77	C₂×C₈×C₁₉	—	4	2	no	—
78	C₄²×C₁₉	C₄²	1	2	yes	—
79	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
80	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
81	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
82	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
83	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
84	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
88	C₄⋊Dic₁₉	—	2	8	no	—
89	C₄⋊Dic₁₉	—	2	4	no	—
90	C₄⋊Dic₁₉	M₁₆	1	8	yes	complemented
91	C₄×Dic₁₉	—	4	4	no	—
92	C₄×Dic₁₉	—	4	4	no	—
93	#64520	—	2	16	no	—
94	C₂×C₁₉⋊C₈	—	2	8	no	—
95	C₂×C₁₉⋊C₈	—	2	4	no	—
96	#7904	—	19	1	no	maximal, Sylow
97	C₂³×C₁₉	—	2	2	no	—
98	C₂³×C₁₉	—	2	1	no	—
99	C₂³×C₁₉	—	2	1	no	—
100	C₂³×C₁₉	#95	1	2	yes	—
101	C₂³×C₁₉	#104	1	4	yes	—
102	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₉	—	2	4	no	—
104	C₂×C₄×C₁₉	—	2	4	no	—
105	C₂×C₄×C₁₉	—	2	2	no	—
106	C₂×C₄×C₁₉	—	2	1	no	—
107	C₂×C₄×C₁₉	—	2	1	no	—
108	C₂×C₄×C₁₉	—	2	2	no	—
109	C₂×C₄×C₁₉	—	2	2	no	—
110	C₂×C₄×C₁₉	—	2	4	no	—
111	C₂×C₄×C₁₉	—	2	4	no	—
112	C₂×C₄×C₁₉	—	2	4	no	—
113	C₂×C₄×C₁₉	—	2	4	no	—
114	C₂×C₄×C₁₉	#95	1	2	yes	—
115	C₂×C₄×C₁₉	#95	1	2	yes	—
116	C₂×C₄×C₁₉	#95	1	2	yes	—
117	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
118	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₂×M₁₆	1	4	yes	—
124	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
125	C₂×C₄×C₁₉	#137	1	2	yes	—
126	C₂×Dic₁₉	—	2	8	no	—
127	C₂×Dic₁₉	—	4	16	no	—
128	C₂×Dic₁₉	—	4	4	no	—
129	C₂×Dic₁₉	—	4	4	no	—
130	C₈×C₁₉	—	4	4	no	—
131	C₁₉⋊C₈	—	2	8	no	—
132	C₂×#432	D₁₉	1	1	yes	characteristic
133	C₂×C₄×M₁₆	—	19	1	no	—
134	C₂×C₄×C₄⋊C₄	—	19	1	no	—
135	#1279	—	19	4	no	—
136	C₂²×C₁₉	—	2	2	no	—
137	C₂²×C₁₉	—	2	2	no	—
138	C₂²×C₁₉	—	2	4	no	—
139	C₂²×C₁₉	—	2	4	no	—
140	C₂²×C₁₉	#339	1	4	yes	—
141	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
142	C₂²×C₁₉	#407	1	2	yes	—
143	C₂²×C₁₉	#418	1	1	yes	characteristic
144	C₂²×C₁₉	#419	1	1	yes	characteristic
145	C₂²×C₁₉	C₂×#104	1	2	yes	—
146	C₄×C₁₉	—	2	2	no	—
147	C₄×C₁₉	—	2	1	no	—
148	C₄×C₁₉	—	2	2	no	—
149	C₄×C₁₉	—	2	1	no	—
150	C₄×C₁₉	—	2	2	no	—
151	C₄×C₁₉	C₄×M₁₆	1	2	yes	—
152	C₄×C₁₉	#417	1	2	yes	—
153	C₄×C₁₉	#423	1	2	yes	—
154	C₄×C₁₉	#428	1	2	yes	—
155	Dic₁₉	—	4	8	no	—
156	C₂²×M₁₆	—	38	1	no	—
157	C₂²×C₄⋊C₄	—	19	2	no	—
158	C₂²×C₄²	—	38	1	no	—
159	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
160	#432	—	2	2	no	—
161	#432	Dic₁₉	1	4	yes	complemented
162	C₂×C₄⋊C₈	C₂×D₁₉	1	2	yes	—
163	C₂×C₂²⋊C₈	—	2	1	no	—
164	C₄×M₁₆	—	19	4	no	—
165	C₄×M₁₆	—	38	2	no	—
166	C₂×#97	—	2	1	no	—
167	C₂×#97	—	38	1	no	—
168	C₂×C₄×C₈	—	19	2	no	—
169	C₄×C₄⋊C₄	—	19	2	no	—
170	C₄×C₄⋊C₄	—	19	2	no	—
171	C₄×C₄⋊C₄	—	38	1	no	—
172	C₄×C₄⋊C₄	—	38	1	no	—
173	C₂×#95	—	38	1	no	—
174	C₂×C₁₉	—	2	4	no	—
175	C₂×C₁₉	#1490	1	2	yes	—
176	C₂×C₁₉	#1491	1	2	yes	—
177	C₂×C₁₉	#1728	1	1	yes	characteristic
178	C₂×C₁₉	C₂×#339	1	1	yes	characteristic
179	C₂×C₁₉	#1736	1	1	yes	characteristic
180	C₂³×C₄	—	38	2	no	—
181	C₂³×C₄	C₄×D₁₉	1	1	yes	characteristic
182	C₂³×C₄	D₇₆	1	1	yes	characteristic
183	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
184	C₂×M₁₆	—	38	8	no	—
185	C₂×M₁₆	—	38	4	no	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	38	2	no	—
188	C₂×C₄⋊C₄	—	19	4	no	—
189	C₂×C₄⋊C₄	—	38	8	no	—
190	C₂×C₄⋊C₄	—	38	2	no	—
191	C₂×C₄²	—	38	2	no	—
192	C₂×C₄²	—	38	2	no	—
193	C₂×C₄²	—	76	4	no	—
194	C₂×C₄²	Dic₃₈	1	2	yes	—
195	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
196	C₂×C₄²	C₄×D₁₉	1	2	yes	—
197	C₂×C₄²	D₇₆	1	1	yes	characteristic
198	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
199	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
200	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
201	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
202	C₄⋊C₈	—	2	2	no	—
203	C₄⋊C₈	C₂×Dic₁₉	1	4	yes	complemented
204	C₂²⋊C₈	—	2	4	no	—
205	#97	—	2	4	no	—
206	#97	—	38	4	no	—
207	C₄×C₈	—	19	4	no	—
208	C₄×C₈	—	38	2	no	—
209	#95	—	38	2	no	—
210	#95	—	38	2	no	—
211	C₁₉	#7904	1	1	yes	characteristic, complemented, Sylow
212	C₂⁴	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	38	8	no	—
218	C₂²×C₄	—	38	4	no	—
219	C₂²×C₄	—	76	4	no	—
220	C₂²×C₄	—	76	4	no	—
221	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
222	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
223	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
224	C₂²×C₄	C₄⋊Dic₁₉	1	2	yes	—
225	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
226	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
227	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
228	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
229	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
230	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
233	M₁₆	—	38	16	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	38	8	no	—
238	C₂×C₈	—	38	4	no	—
239	C₄⋊C₄	—	19	8	no	—
240	C₄⋊C₄	—	38	8	no	—
241	C₄⋊C₄	—	38	4	no	—
242	C₄²	—	76	4	no	—
243	C₄²	—	76	4	no	—
244	C₄²	C₄×Dic₁₉	1	2	yes	—
245	C₄²	Dic₁₉⋊C₄	1	2	yes	—
246	C₄²	Dic₁₉⋊C₄	1	2	yes	—
247	C₄²	C₄⋊Dic₁₉	1	2	yes	—
248	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
249	C₄²	C₂²⋊Dic₁₉	1	2	yes	—
250	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
251	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
252	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
253	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
254	C₂³	—	2	1	no	—
255	C₂³	—	2	1	no	—
256	C₂³	—	2	2	no	—
257	C₂³	#10597282	1	4	yes	—
258	C₂³	C₁₉⋊#95	1	2	yes	—
259	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	38	8	no	—
272	C₂×C₄	—	76	16	no	—
273	C₂×C₄	—	76	4	no	—
274	C₂×C₄	—	76	4	no	—
275	C₂×C₄	C₁₉⋊#95	1	2	yes	—
276	C₂×C₄	C₁₉⋊#95	1	2	yes	—
277	C₂×C₄	C₁₉⋊#95	1	2	yes	—
278	C₂×C₄	M₁₆×D₁₉	1	4	yes	—
279	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
280	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
281	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
282	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
283	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
284	C₂×C₄	#10597395	1	2	yes	—
285	C₂×C₄	#10597397	1	2	yes	—
286	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
287	C₈	—	4	4	no	—
288	C₈	—	38	8	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	4	no	—
292	C₂²	—	2	4	no	—
293	C₂²	#11926305	1	4	yes	—
294	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
295	C₂²	#11926507	1	2	yes	—
296	C₂²	#11926548	1	1	yes	characteristic
297	C₂²	#11926552	1	1	yes	characteristic
298	C₂²	C₂×#10597282	1	2	yes	—
299	C₄	—	2	2	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	2	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	2	no	—
304	C₄	—	76	8	no	—
305	C₄	M₁₆×Dic₁₉	1	2	yes	—
306	C₄	#11926545	1	2	yes	—
307	C₄	#11926567	1	2	yes	—
308	C₄	#11926587	1	2	yes	—
309	C₂	—	2	4	no	—
310	C₂	#14562077	1	2	yes	—
311	C₂	#14562081	1	2	yes	—
312	C₂	#14563328	1	1	yes	characteristic
313	C₂	C₂×#11926305	1	1	yes	characteristic
314	C₂	#14563356	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#432	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#432	D₁₉
hypocenter	C₁₉	#7904
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#432

Complements

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

N	H ≅ G/N	type
3. #14561474	C₂	semidirect
9. C₄×#64520	C₄	semidirect
19. C₁₉×#432	C₄	semidirect
46. C₁₉×C₄⋊C₈	C₂×C₄	semidirect
53. C₄×C₁₉⋊C₈	C₂×C₄	semidirect
90. C₄⋊Dic₁₉	M₁₆	semidirect
161. #432	Dic₁₉	semidirect
203. C₄⋊C₈	C₂×Dic₁₉	semidirect
211. C₁₉	#7904	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#64520	2
C₂×C₄×C₄⋊Dic₁₉	2
#14561474	2
C₂×C₁₉×#432	2
#7904	19