Subgroups — #18087732

Order 4864.

926 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926500	C₂	1	2	yes	maximal
2	C₂²×C₈×Dic₁₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926309	C₂	1	1	yes	maximal, characteristic
4	C₂×#11926303	C₂	1	2	yes	maximal
5	C₁₉×#1735	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	C₂²	1	2	yes	—
7	C₂³×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄×Dic₁₉	C₂²	1	1	yes	characteristic
9	#11926500	—	2	4	no	—
10	#11926500	C₄	1	8	yes	complemented
11	C₂×#10597260	—	2	2	no	—
12	C₂×C₄×C₁₉⋊C₈	—	2	2	no	—
13	C₂×C₈×Dic₁₉	—	2	4	no	—
14	C₂×C₈×Dic₁₉	C₄	1	4	yes	complemented
15	#11926309	C₄	1	4	yes	complemented
16	#11926303	C₄	1	8	yes	complemented
17	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
18	C₂×C₁₉×#117	C₂²	1	2	yes	—
19	C₁₉×#340	C₄	1	4	yes	complemented
20	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
21	C₂³×Dic₁₉	—	2	1	no	—
22	C₂²×C₈×C₁₉	—	2	2	no	—
23	C₂²×C₈×C₁₉	—	2	2	no	—
24	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
27	C₁₉×#117	—	2	2	no	—
28	C₁₉×#117	—	2	2	no	—
29	C₁₉×#117	C₂×C₄	1	4	yes	—
30	C₁₉×#117	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
32	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
33	C₂×C₄²×C₁₉	—	2	2	no	—
34	C₂×C₄×Dic₁₉	—	2	2	no	—
35	C₂×C₄×Dic₁₉	—	2	1	no	—
36	C₂×C₄×Dic₁₉	C₂×C₄	1	4	yes	—
37	C₂×C₄×Dic₁₉	C₂×C₄	1	2	yes	—
38	C₂×#64520	—	2	2	no	—
39	C₂×#64520	—	2	2	no	—
40	C₂×#64520	—	2	2	no	—
41	C₂×#64520	C₂×C₄	1	4	yes	—
42	C₂×#64520	C₂×C₄	1	4	yes	—
43	C₂×#64520	C₂×C₄	1	4	yes	—
44	C₂²×C₁₉⋊C₈	—	2	2	no	—
45	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
46	C₂²×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
47	C₂²×C₁₉⋊C₈	D₄	1	4	yes	—
48	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
49	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
50	C₂²×C₁₉⋊C₈	Q₈	1	2	yes	—
51	C₈×Dic₁₉	—	2	8	no	—
52	#10597260	—	2	8	no	—
53	C₄×C₁₉⋊C₈	—	2	8	no	—
54	C₂⁴×C₁₉	#41	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	—	2	4	no	—
56	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	#41	1	2	yes	—
65	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
66	C₂²×Dic₁₉	—	2	8	no	—
67	C₂²×Dic₁₉	—	2	2	no	—
68	C₂²×Dic₁₉	—	2	4	no	—
69	C₂×C₈×C₁₉	—	2	2	no	—
70	C₂×C₈×C₁₉	—	2	2	no	—
71	C₂×C₈×C₁₉	—	2	4	no	—
72	C₂×C₈×C₁₉	—	2	4	no	—
73	C₂×C₈×C₁₉	C₄²	1	4	yes	—
74	C₄⋊C₄×C₁₉	—	2	4	no	—
75	C₄⋊C₄×C₁₉	C₄²	1	8	yes	—
76	C₂²⋊C₄×C₁₉	—	2	4	no	—
77	C₂²⋊C₄×C₁₉	—	2	4	no	—
78	C₄²×C₁₉	—	2	4	no	—
79	C₄²×C₁₉	—	2	4	no	—
80	C₄×Dic₁₉	—	2	2	no	—
81	C₄×Dic₁₉	—	2	4	no	—
82	C₄×Dic₁₉	C₄²	1	4	yes	—
83	#64520	—	2	4	no	—
84	#64520	—	2	4	no	—
85	#64520	—	2	4	no	—
86	#64520	C₄²	1	8	yes	—
87	C₂×C₁₉⋊C₈	—	2	4	no	—
88	C₂×C₁₉⋊C₈	—	2	4	no	—
89	C₂×C₁₉⋊C₈	—	2	4	no	—
90	C₂×C₁₉⋊C₈	C₄²	1	4	yes	—
91	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	4	yes	—
92	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	4	yes	—
93	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	4	yes	—
94	C₂×C₁₉⋊C₈	C₄⋊C₄	1	2	yes	—
95	C₂×C₁₉⋊C₈	C₄⋊C₄	1	2	yes	—
96	C₂×C₁₉⋊C₈	C₄⋊C₄	1	8	yes	—
97	#10706	—	19	1	no	maximal, Sylow
98	C₂³×C₁₉	—	2	4	no	—
99	C₂³×C₁₉	#117	1	2	yes	—
100	C₂³×C₁₉	#117	1	1	yes	characteristic
101	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
102	C₂³×C₁₉	C₄×D₄	1	2	yes	—
103	C₂³×C₁₉	C₄×Q₈	1	1	yes	characteristic
104	C₂×C₄×C₁₉	—	2	8	no	—
105	C₂×C₄×C₁₉	—	4	8	no	—
106	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₄×C₁₉	C₂×C₄²	1	2	yes	—
108	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₄×C₁₉	C₂×C₄²	1	2	yes	—
111	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
112	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
116	C₂×C₄×C₁₉	#117	1	2	yes	—
117	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
124	C₂×Dic₁₉	—	2	16	no	—
125	C₂×Dic₁₉	—	2	8	no	—
126	C₂×Dic₁₉	—	2	4	no	—
127	C₈×C₁₉	—	2	4	no	—
128	C₁₉⋊C₈	—	2	8	no	—
129	C₁₉⋊C₈	#95	1	4	yes	—
130	C₁₉⋊C₈	#95	1	4	yes	—
131	C₂×#405	—	19	2	no	—
132	C₂²×C₄×C₈	—	19	1	no	—
133	C₂×#341	—	19	1	no	—
134	#1735	D₁₉	1	1	yes	characteristic
135	C₂×#339	—	19	2	no	—
136	C₂²×C₁₉	—	2	4	no	—
137	C₂²×C₁₉	—	2	4	no	—
138	C₂²×C₁₉	—	2	2	no	—
139	C₂²×C₁₉	—	2	2	no	—
140	C₂²×C₁₉	#376	1	1	yes	characteristic
141	C₂²×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
142	C₂²×C₁₉	C₄×C₂²⋊C₄	1	2	yes	—
143	C₂²×C₁₉	C₄×C₄⋊C₄	1	2	yes	—
144	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
145	C₂²×C₁₉	#443	1	4	yes	—
146	C₄×C₁₉	—	4	8	no	—
147	C₄×C₁₉	C₄³	1	1	yes	characteristic
148	C₄×C₁₉	C₂×#95	1	1	yes	characteristic
149	C₄×C₁₉	C₄×C₂²⋊C₄	1	2	yes	—
150	C₄×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
151	C₄×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
152	C₄×C₁₉	C₄×C₄⋊C₄	1	2	yes	—
153	Dic₁₉	—	2	8	no	—
154	C₂²×M₁₆	—	19	2	no	—
155	C₂³×C₈	—	19	1	no	—
156	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
157	C₂×#117	C₂×D₁₉	1	2	yes	—
158	C₂²×C₄²	—	19	1	no	—
159	#405	—	19	8	no	—
160	#405	—	38	4	no	—
161	C₂×#97	—	38	2	no	—
162	C₂×C₄×C₈	—	19	4	no	—
163	C₂×C₄×C₈	—	38	2	no	—
164	C₂×C₄×C₈	—	38	4	no	—
165	#341	—	19	4	no	—
166	#340	Dic₁₉	1	4	yes	complemented
167	#339	—	19	8	no	—
168	C₂×C₁₉	—	2	1	no	—
169	C₂×C₁₉	—	2	1	no	—
170	C₂×C₁₉	—	2	2	no	—
171	C₂×C₁₉	C₄×#95	1	1	yes	characteristic
172	C₂×C₁₉	#1764	1	4	yes	—
173	C₂×C₁₉	#1772	1	2	yes	—
174	C₂³×C₄	—	38	1	no	—
175	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
176	C₂×M₁₆	—	19	4	no	—
177	C₂×M₁₆	—	19	4	no	—
178	C₂×M₁₆	—	19	4	no	—
179	C₂×M₁₆	—	38	2	no	—
180	C₂×M₁₆	—	38	2	no	—
181	C₂×M₁₆	—	38	2	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	19	2	no	—
185	C₂²×C₈	—	19	4	no	—
186	C₂²×C₈	—	19	4	no	—
187	C₂²×C₈	—	19	1	no	—
188	C₂²×C₈	—	19	1	no	—
189	C₂²×C₈	—	19	2	no	—
190	C₂²×C₈	—	38	2	no	—
191	C₂²×C₈	C₄×D₁₉	1	2	yes	—
192	C₂²×C₈	C₄×D₁₉	1	2	yes	—
193	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
194	#117	—	2	2	no	—
195	#117	—	2	2	no	—
196	#117	C₄×D₁₉	1	4	yes	—
197	#117	C₄×D₁₉	1	4	yes	—
198	C₂×C₄⋊C₄	C₂×Dic₁₉	1	4	yes	—
199	C₂×C₂²⋊C₄	—	2	2	no	—
200	C₂×C₄²	—	2	2	no	—
201	C₂×C₄²	—	19	4	no	—
202	C₂×C₄²	—	19	2	no	—
203	C₂×C₄²	—	38	2	no	—
204	C₂×C₄²	—	38	1	no	—
205	#97	—	38	8	no	—
206	C₄×C₈	—	38	8	no	—
207	C₄×C₈	—	38	8	no	—
208	C₁₉	#10706	1	1	yes	characteristic, complemented, Sylow
209	C₂⁴	#64543	1	1	yes	characteristic
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	38	8	no	—
212	C₂²×C₄	—	38	4	no	—
213	C₂²×C₄	—	38	2	no	—
214	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
216	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
218	C₂²×C₄	D₄×D₁₉	1	2	yes	—
219	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
220	C₂²×C₄	#64543	1	2	yes	—
221	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
222	C₂²×C₄	#64545	1	1	yes	characteristic
223	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
224	M₁₆	—	19	8	no	—
225	M₁₆	—	38	4	no	—
226	M₁₆	—	38	4	no	—
227	M₁₆	—	38	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	19	4	no	—
233	C₂×C₈	—	19	4	no	—
234	C₂×C₈	—	19	4	no	—
235	C₂×C₈	—	19	4	no	—
236	C₂×C₈	—	19	2	no	—
237	C₂×C₈	—	19	2	no	—
238	C₂×C₈	—	19	8	no	—
239	C₂×C₈	—	38	4	no	—
240	C₂×C₈	—	38	4	no	—
241	C₂×C₈	—	38	4	no	—
242	C₂×C₈	C₄×Dic₁₉	1	4	yes	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	C₄×Dic₁₉	1	8	yes	—
245	C₂²⋊C₄	—	2	4	no	—
246	C₂²⋊C₄	—	2	4	no	—
247	C₄²	—	2	4	no	—
248	C₄²	—	2	4	no	—
249	C₄²	—	19	4	no	—
250	C₄²	—	38	2	no	—
251	C₄²	—	38	4	no	—
252	C₂³	—	2	4	no	—
253	C₂³	#10597323	1	2	yes	—
254	C₂³	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
255	C₂³	#10597333	1	1	yes	characteristic
256	C₂³	#10597338	1	1	yes	characteristic
257	C₂³	D₄×Dic₁₉	1	1	yes	characteristic
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	4	8	no	—
260	C₂×C₄	—	38	16	no	—
261	C₂×C₄	—	38	8	no	—
262	C₂×C₄	—	38	4	no	—
263	C₂×C₄	C₄²×D₁₉	1	1	yes	characteristic
264	C₂×C₄	C₄²×D₁₉	1	2	yes	—
265	C₂×C₄	C₄²×D₁₉	1	1	yes	characteristic
266	C₂×C₄	#10597323	1	1	yes	characteristic
267	C₂×C₄	C₂²⋊C₄×D₁₉	1	2	yes	—
268	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
269	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
270	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
271	C₂×C₄	#10597333	1	2	yes	—
272	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
273	C₂×C₄	C₄⋊C₄×D₁₉	1	2	yes	—
274	C₂×C₄	#10597338	1	2	yes	—
275	C₂×C₄	#10597339	1	1	yes	characteristic
276	C₂×C₄	#10597339	1	2	yes	—
277	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
278	C₂×C₄	C₂×C₄×Dic₁₉	1	2	yes	—
279	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
280	C₂×C₄	Q₈×Dic₁₉	1	1	yes	characteristic
281	C₈	—	2	4	no	—
282	C₈	—	19	4	no	—
283	C₈	—	19	4	no	—
284	C₈	—	38	8	no	—
285	C₂²	—	2	4	no	—
286	C₂²	—	2	4	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	#11926389	1	1	yes	characteristic
290	C₂²	#11926391	1	1	yes	characteristic
291	C₂²	C₂²⋊C₄×Dic₁₉	1	2	yes	—
292	C₂²	#11926395	1	2	yes	—
293	C₂²	C₄⋊C₄×Dic₁₉	1	1	yes	characteristic
294	C₂²	#11926661	1	4	yes	—
295	C₄	—	4	8	no	—
296	C₄	—	38	8	no	—
297	C₄	C₄²×Dic₁₉	1	1	yes	characteristic
298	C₄	#95×D₁₉	1	1	yes	characteristic
299	C₄	#11926391	1	2	yes	—
300	C₄	C₂²⋊C₄×Dic₁₉	1	1	yes	characteristic
301	C₄	#11926395	1	1	yes	characteristic
302	C₄	C₄⋊C₄×Dic₁₉	1	2	yes	—
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	2	2	no	—
306	C₂	#95×Dic₁₉	1	1	yes	characteristic
307	C₂	#14563475	1	4	yes	—
308	C₂	#14563516	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₉
commutator	C₄×C₁₉	C₄³
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₁₉×#1735	C₂
socle	C₂³×C₁₉	C₄×D₄
prime radical	C₂⁴×C₁₉	#41
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	#1735	D₁₉
hypocenter	C₁₉	#10706
LCS	G ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #1735

Complements

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

N	H ≅ G/N	type
10. #11926500	C₄	semidirect
14. C₂×C₈×Dic₁₉	C₄	semidirect
15. #11926309	C₄	semidirect
16. #11926303	C₄	semidirect
19. C₁₉×#340	C₄	semidirect
166. #340	Dic₁₉	semidirect
208. C₁₉	#10706	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926500	2
C₂²×C₈×Dic₁₉	2
C₂×#11926309	2
C₂×#11926303	2
C₁₉×#1735	2
#10706	19