Subgroups — #18087708

Order 4864.

842 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926499	C₂	1	2	yes	maximal
2	C₂×C₁₉⋊#338	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926301	C₂	1	1	yes	maximal, characteristic
4	C₂×#11926300	C₂	1	2	yes	maximal
5	C₂×C₁₉×#405	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	C₂²	1	2	yes	—
7	C₂×#10597372	C₂²	1	2	yes	—
8	#11926499	—	2	4	no	—
9	#11926499	—	2	4	no	—
10	C₂×#10597272	C₄	1	4	yes	complemented
11	C₂×C₈×Dic₁₉	C₄	1	4	yes	complemented
12	C₁₉⋊#338	C₄	1	4	yes	complemented
13	#11926301	C₄	1	4	yes	complemented
14	#11926300	C₄	1	8	yes	complemented
15	C₂²×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
18	C₁₉×#405	—	2	4	no	—
19	C₂×C₁₉×#97	C₄	1	2	yes	complemented
20	C₂×C₄×C₈×C₁₉	C₄	1	2	yes	complemented
21	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₁₉	—	2	2	no	—
23	C₂×M₁₆×C₁₉	—	2	2	no	—
24	C₂×M₁₆×C₁₉	C₂×C₄	1	2	yes	—
25	C₂×M₁₆×C₁₉	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₉	D₄	1	2	yes	—
32	C₂²×C₈×C₁₉	D₄	1	4	yes	—
33	C₂²×C₈×C₁₉	Q₈	1	2	yes	—
34	C₁₉×#117	—	2	2	no	—
35	C₁₉×#117	C₂×C₄	1	4	yes	—
36	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
37	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
38	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
39	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
40	#10597372	—	2	2	no	—
41	#10597372	—	2	2	no	—
42	#10597372	C₂×C₄	1	4	yes	—
43	#10597372	C₂×C₄	1	4	yes	—
44	C₂×C₄⋊Dic₁₉	—	2	2	no	—
45	C₂×C₄×Dic₁₉	C₂×C₄	1	4	yes	—
46	C₂×#64520	—	2	2	no	—
47	C₂×#64520	—	2	2	no	—
48	C₂×#64520	—	2	2	no	—
49	C₂×#64520	—	2	2	no	—
50	C₂×#64520	C₂×C₄	1	4	yes	—
51	C₂×#64520	C₂×C₄	1	4	yes	—
52	C₂²×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
53	C₁₉×#97	—	2	4	no	—
54	C₄×C₈×C₁₉	—	2	4	no	—
55	#10597272	—	2	8	no	—
56	C₈×Dic₁₉	—	2	8	no	—
57	C₂⁴×C₁₉	#41	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	—	2	2	no	—
59	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	#41	1	2	yes	—
69	C₂²×Dic₁₉	—	2	4	no	—
70	M₁₆×C₁₉	—	2	4	no	—
71	M₁₆×C₁₉	—	2	4	no	—
72	C₂×C₈×C₁₉	—	2	2	no	—
73	C₂×C₈×C₁₉	—	2	2	no	—
74	C₂×C₈×C₁₉	C₄²	1	4	yes	—
75	C₂×C₈×C₁₉	C₄²	1	2	yes	—
76	C₂×C₈×C₁₉	C₄²	1	1	yes	characteristic
77	C₂×C₈×C₁₉	C₄²	1	1	yes	characteristic
78	C₂×C₈×C₁₉	C₂²⋊C₄	1	4	yes	—
79	C₂×C₈×C₁₉	C₂²⋊C₄	1	4	yes	—
80	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
81	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
82	C₂×C₈×C₁₉	C₄⋊C₄	1	4	yes	—
83	C₂×C₈×C₁₉	C₄⋊C₄	1	4	yes	—
84	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
85	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
86	C₄⋊C₄×C₁₉	—	2	4	no	—
87	C₂²⋊C₄×C₁₉	—	2	4	no	—
88	C₄²×C₁₉	—	2	2	no	—
89	C₄²×C₁₉	C₄²	1	4	yes	—
90	C₂²⋊Dic₁₉	—	2	4	no	—
91	C₂²⋊Dic₁₉	—	2	4	no	—
92	C₄⋊Dic₁₉	—	2	4	no	—
93	C₄⋊Dic₁₉	—	2	4	no	—
94	C₄×Dic₁₉	—	2	4	no	—
95	C₄×Dic₁₉	C₄²	1	8	yes	—
96	#64520	—	2	4	no	—
97	#64520	—	2	4	no	—
98	#64520	—	2	4	no	—
99	#64520	—	2	4	no	—
100	C₂×C₁₉⋊C₈	—	2	4	no	—
101	C₂×C₁₉⋊C₈	—	2	4	no	—
102	C₂×C₁₉⋊C₈	C₄²	1	8	yes	—
103	#10699	—	19	1	no	maximal, Sylow
104	C₂³×C₁₉	—	4	2	no	—
105	C₂³×C₁₉	#117	1	2	yes	—
106	C₂³×C₁₉	#117	1	1	yes	characteristic
107	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
108	C₂³×C₁₉	C₄×D₄	1	2	yes	—
109	C₂³×C₁₉	C₄×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₁₉	—	2	4	no	—
111	C₂×C₄×C₁₉	—	4	4	no	—
112	C₂×C₄×C₁₉	C₂×C₄²	1	2	yes	—
113	C₂×C₄×C₁₉	C₂×C₄²	1	2	yes	—
114	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
115	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
119	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₉	#117	1	2	yes	—
122	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
126	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
129	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
130	C₂×Dic₁₉	—	2	8	no	—
131	C₂×Dic₁₉	—	4	8	no	—
132	C₈×C₁₉	—	2	4	no	—
133	C₈×C₁₉	#95	1	4	yes	—
134	C₈×C₁₉	#95	1	2	yes	—
135	C₈×C₁₉	#95	1	2	yes	—
136	C₁₉⋊C₈	—	2	8	no	—
137	C₂×#405	—	19	2	no	—
138	C₂×#405	D₁₉	1	1	yes	characteristic
139	C₂×#338	—	19	1	no	—
140	C₂×H(3,ℤ/4)	—	19	1	no	—
141	C₂×H(3,ℤ/4)	—	19	2	no	—
142	C₂²×C₁₉	—	2	4	no	—
143	C₂²×C₁₉	—	2	2	no	—
144	C₂²×C₁₉	—	4	1	no	—
145	C₂²×C₁₉	—	4	2	no	—
146	C₂²×C₁₉	—	4	1	no	—
147	C₂²×C₁₉	#376	1	1	yes	characteristic
148	C₂²×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
149	C₂²×C₁₉	C₄×C₂²⋊C₄	1	2	yes	—
150	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
151	C₂²×C₁₉	C₄×C₄⋊C₄	1	2	yes	—
152	C₄×C₁₉	—	4	4	no	—
153	C₄×C₁₉	C₄³	1	1	yes	characteristic
154	C₄×C₁₉	C₂×#95	1	1	yes	characteristic
155	C₄×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
156	C₄×C₁₉	C₄×C₂²⋊C₄	1	2	yes	—
157	C₄×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
158	C₄×C₁₉	C₄×C₄⋊C₄	1	2	yes	—
159	Dic₁₉	—	4	8	no	—
160	C₂²×M₁₆	—	19	2	no	—
161	C₂²×M₁₆	C₂×D₁₉	1	1	yes	characteristic
162	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
163	C₂×#117	—	19	2	no	—
164	C₂×#117	C₂×D₁₉	1	1	yes	characteristic
165	#405	—	2	4	no	—
166	#405	—	38	4	no	—
167	#405	—	38	4	no	—
168	C₂×#97	—	19	4	no	—
169	C₂×#97	Dic₁₉	1	2	yes	complemented
170	C₂×C₄×C₈	—	19	4	no	—
171	C₂×C₄×C₈	Dic₁₉	1	2	yes	complemented
172	#338	—	19	4	no	—
173	H(3,ℤ/4)	—	19	4	no	—
174	H(3,ℤ/4)	—	19	8	no	—
175	C₂×C₁₉	—	2	2	no	—
176	C₂×C₁₉	—	2	1	no	—
177	C₂×C₁₉	—	2	1	no	—
178	C₂×C₁₉	—	2	2	no	—
179	C₂×C₁₉	C₄×#95	1	1	yes	characteristic
180	C₂×C₁₉	#1747	1	2	yes	—
181	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	19	4	no	—
185	C₂×M₁₆	—	19	4	no	—
186	C₂×M₁₆	—	38	2	no	—
187	C₂×M₁₆	—	38	2	no	—
188	C₂×M₁₆	—	38	2	no	—
189	C₂×M₁₆	—	38	2	no	—
190	C₂×M₁₆	C₄×D₁₉	1	2	yes	—
191	C₂×M₁₆	C₄×D₁₉	1	2	yes	—
192	C₂²×C₈	—	19	4	no	—
193	C₂²×C₈	Dic₃₈	1	2	yes	—
194	C₂²×C₈	C₄×D₁₉	1	2	yes	—
195	C₂²×C₈	C₄×D₁₉	1	2	yes	—
196	C₂²×C₈	D₇₆	1	2	yes	—
197	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
198	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
199	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
200	C₂²×C₈	C₂²⋊D₁₉	1	4	yes	—
201	#117	—	2	2	no	—
202	#117	—	19	4	no	—
203	#117	—	19	4	no	—
204	#117	—	38	2	no	—
205	#117	—	38	2	no	—
206	#117	C₄×D₁₉	1	4	yes	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	—	38	2	no	—
209	C₂×C₂²⋊C₄	—	2	1	no	—
210	C₂×C₂²⋊C₄	—	38	2	no	—
211	C₂×C₄²	—	19	4	no	—
212	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
213	#97	—	2	4	no	—
214	#97	—	38	8	no	—
215	C₄×C₈	—	2	4	no	—
216	C₄×C₈	—	38	8	no	—
217	C₁₉	#10699	1	1	yes	characteristic, complemented, Sylow
218	C₂⁴	#64541	1	1	yes	characteristic
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	38	4	no	—
221	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
223	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
225	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
226	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
227	C₂²×C₄	#64541	1	1	yes	characteristic
228	C₂²×C₄	#64541	1	2	yes	—
229	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
231	M₁₆	—	2	4	no	—
232	M₁₆	—	2	4	no	—
233	M₁₆	—	38	4	no	—
234	M₁₆	—	38	4	no	—
235	M₁₆	—	38	4	no	—
236	M₁₆	—	38	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	19	8	no	—
240	C₂×C₈	—	38	4	no	—
241	C₂×C₈	—	38	4	no	—
242	C₂×C₈	C₄×Dic₁₉	1	4	yes	—
243	C₂×C₈	C₄×Dic₁₉	1	1	yes	characteristic
244	C₂×C₈	C₄×Dic₁₉	1	2	yes	—
245	C₂×C₈	C₄×Dic₁₉	1	1	yes	characteristic
246	C₂×C₈	Dic₁₉⋊C₄	1	4	yes	—
247	C₂×C₈	Dic₁₉⋊C₄	1	4	yes	—
248	C₂×C₈	C₄⋊Dic₁₉	1	2	yes	—
249	C₂×C₈	C₄⋊Dic₁₉	1	2	yes	—
250	C₂×C₈	(C₂×D₁₉)⋊C₄	1	4	yes	—
251	C₂×C₈	(C₂×D₁₉)⋊C₄	1	4	yes	—
252	C₂×C₈	C₂²⋊Dic₁₉	1	2	yes	—
253	C₂×C₈	C₂²⋊Dic₁₉	1	2	yes	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	38	4	no	—
256	C₄⋊C₄	—	38	4	no	—
257	C₂²⋊C₄	—	2	4	no	—
258	C₂²⋊C₄	—	38	4	no	—
259	C₂²⋊C₄	—	38	4	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	19	8	no	—
262	C₄²	—	38	4	no	—
263	C₄²	C₄×Dic₁₉	1	4	yes	—
264	C₂³	—	4	2	no	—
265	C₂³	C₄×Dic₃₈	1	1	yes	characteristic
266	C₂³	#10597318	1	2	yes	—
267	C₂³	C₄×D₇₆	1	1	yes	characteristic
268	C₂³	#10597372	1	1	yes	characteristic
269	C₂³	C₄×C₂²⋊D₁₉	1	2	yes	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	38	8	no	—
273	C₂×C₄	—	76	8	no	—
274	C₂×C₄	C₄×Dic₃₈	1	2	yes	—
275	C₂×C₄	C₄×Dic₃₈	1	1	yes	characteristic
276	C₂×C₄	C₄²×D₁₉	1	2	yes	—
277	C₂×C₄	C₄²×D₁₉	1	2	yes	—
278	C₂×C₄	#10597318	1	2	yes	—
279	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
280	C₂×C₄	C₄×D₇₆	1	2	yes	—
281	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
282	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
283	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
284	C₂×C₄	C₂×Dic₁₉⋊C₄	1	2	yes	—
285	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
286	C₂×C₄	#10597372	1	1	yes	characteristic
287	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	2	yes	—
288	C₂×C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
289	C₂×C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
290	C₂×C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
291	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
292	C₈	—	2	4	no	—
293	C₈	—	38	8	no	—
294	C₈	C₁₉⋊#95	1	4	yes	—
295	C₈	C₁₉⋊#95	1	2	yes	—
296	C₈	C₁₉⋊#95	1	2	yes	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	4	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	4	1	no	—
301	C₂²	—	4	2	no	—
302	C₂²	#11926390	1	1	yes	characteristic
303	C₂²	C₄×C₂²⋊Dic₁₉	1	1	yes	characteristic
304	C₂²	C₄×(C₂×D₁₉)⋊C₄	1	2	yes	—
305	C₂²	C₄×C₄⋊Dic₁₉	1	1	yes	characteristic
306	C₂²	C₄×Dic₁₉⋊C₄	1	2	yes	—
307	C₄	—	4	4	no	—
308	C₄	—	76	8	no	—
309	C₄	C₄²×Dic₁₉	1	1	yes	characteristic
310	C₄	C₂×C₁₉⋊#95	1	1	yes	characteristic
311	C₄	C₄×C₂²⋊Dic₁₉	1	1	yes	characteristic
312	C₄	C₄×(C₂×D₁₉)⋊C₄	1	2	yes	—
313	C₄	C₄×C₄⋊Dic₁₉	1	1	yes	characteristic
314	C₄	C₄×Dic₁₉⋊C₄	1	2	yes	—
315	C₂	—	2	1	no	—
316	C₂	—	2	2	no	—
317	C₂	—	2	1	no	—
318	C₂	—	2	2	no	—
319	C₂	C₄×C₁₉⋊#95	1	1	yes	characteristic
320	C₂	#14563390	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₄×Dic₁₉
commutator	C₄×C₁₉	C₄³
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₁₉×#405	C₂
socle	C₂²×C₁₉	#376
prime radical	C₂⁴×C₁₉	#41
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×#405	D₁₉
hypocenter	C₁₉	#10699
LCS	G ▷ C₄×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
10. C₂×#10597272	C₄	semidirect
11. C₂×C₈×Dic₁₉	C₄	semidirect
12. C₁₉⋊#338	C₄	semidirect
13. #11926301	C₄	semidirect
14. #11926300	C₄	semidirect
19. C₂×C₁₉×#97	C₄	semidirect
20. C₂×C₄×C₈×C₁₉	C₄	semidirect
169. C₂×#97	Dic₁₉	semidirect
171. C₂×C₄×C₈	Dic₁₉	semidirect
217. C₁₉	#10699	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926499	2
C₂×C₁₉⋊#338	2
C₂×#11926301	2
C₂×#11926300	2
C₂×C₁₉×#405	2
#10699	19