Subgroups — #18088342

Order 4864.

842 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×Dic₁₉	C₂	1	1	yes	maximal, characteristic
2	#14566038	C₂	1	2	yes	maximal
3	C₂×#11926301	C₂	1	1	yes	maximal, characteristic
4	C₂×#11926300	C₂	1	2	yes	maximal
5	C₂×C₁₉×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	C₂²	1	1	yes	characteristic
7	C₂×#10597372	C₂²	1	2	yes	—
8	C₂²×C₄×Dic₁₉	C₂²	1	1	yes	characteristic
9	M₁₆×Dic₁₉	—	2	2	no	—
10	M₁₆×Dic₁₉	—	2	2	no	—
11	C₂×#10597272	C₄	1	2	yes	complemented
12	C₂×C₈×Dic₁₉	C₄	1	2	yes	complemented
13	#11926413	—	2	2	no	—
14	#11926409	—	2	2	no	—
15	#11926402	—	2	4	no	—
16	C₄×Dic₁₉⋊C₄	C₄	1	8	yes	complemented
17	#11926301	—	2	2	no	—
18	#11926300	—	2	4	no	—
19	C₂²×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
20	C₂×C₁₉×#117	C₂²	1	2	yes	—
21	C₁₉×H(3,ℤ/4)	—	2	2	no	—
22	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
23	C₂³×Dic₁₉	D₄	1	2	yes	—
24	C₂×M₁₆×C₁₉	—	2	2	no	—
25	C₂×M₁₆×C₁₉	—	2	2	no	—
26	C₂×M₁₆×C₁₉	D₄	1	2	yes	—
27	C₂×M₁₆×C₁₉	Q₈	1	2	yes	—
28	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
29	C₁₉×#117	—	2	2	no	—
30	C₁₉×#117	—	2	2	no	—
31	C₁₉×#117	—	2	2	no	—
32	C₁₉×#117	—	2	2	no	—
33	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
34	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
35	C₂×C₄²×C₁₉	C₂×C₄	1	4	yes	—
36	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
37	#10597372	—	2	2	no	—
38	#10597372	—	2	2	no	—
39	#10597372	—	2	2	no	—
40	#10597372	—	2	2	no	—
41	C₂×C₄⋊Dic₁₉	—	2	2	no	—
42	C₂×Dic₁₉⋊C₄	—	2	8	no	—
43	C₂×C₄×Dic₁₉	C₂×C₄	1	4	yes	—
44	C₂×C₄×Dic₁₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×Dic₁₉	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×Dic₁₉	C₂×C₄	1	4	yes	—
47	C₂×C₄×Dic₁₉	D₄	1	2	yes	—
48	C₂×C₄×Dic₁₉	D₄	1	2	yes	—
49	C₂×C₄×Dic₁₉	Q₈	1	2	yes	—
50	C₂×#64520	—	2	2	no	—
51	C₂×#64520	—	2	2	no	—
52	C₂×#64520	D₄	1	2	yes	—
53	C₂×#64520	Q₈	1	2	yes	—
54	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
55	C₁₉⋊#95	—	2	8	no	—
56	#10597272	—	2	4	no	—
57	C₈×Dic₁₉	—	2	2	no	—
58	C₈×Dic₁₉	—	2	2	no	—
59	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	—	2	4	no	—
61	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	#41	1	2	yes	—
70	C₂²×C₄×C₁₉	#41	1	2	yes	—
71	C₂²×Dic₁₉	—	2	4	no	—
72	C₂²×Dic₁₉	—	4	4	no	—
73	C₂²×Dic₁₉	C₂²⋊C₄	1	4	yes	—
74	C₂²×Dic₁₉	C₂²⋊C₄	1	4	yes	—
75	C₂²×Dic₁₉	C₄⋊C₄	1	2	yes	—
76	C₂²×Dic₁₉	C₄⋊C₄	1	2	yes	—
77	M₁₆×C₁₉	—	2	4	no	—
78	M₁₆×C₁₉	—	2	4	no	—
79	C₂×C₈×C₁₉	—	2	2	no	—
80	C₂×C₈×C₁₉	—	2	2	no	—
81	C₂×C₈×C₁₉	C₄⋊C₄	1	4	yes	complemented
82	C₄⋊C₄×C₁₉	—	4	2	no	—
83	C₄⋊C₄×C₁₉	—	4	2	no	—
84	C₂²⋊C₄×C₁₉	—	4	2	no	—
85	C₂²⋊C₄×C₁₉	—	4	2	no	—
86	C₄²×C₁₉	—	2	4	no	—
87	C₄²×C₁₉	—	2	4	no	—
88	C₂²⋊Dic₁₉	—	4	2	no	—
89	C₂²⋊Dic₁₉	—	4	2	no	—
90	C₄⋊Dic₁₉	—	4	2	no	—
91	C₄⋊Dic₁₉	—	4	2	no	—
92	Dic₁₉⋊C₄	—	4	16	no	—
93	C₄×Dic₁₉	—	2	4	no	—
94	C₄×Dic₁₉	—	2	4	no	—
95	C₄×Dic₁₉	C₄²	1	2	yes	—
96	C₄×Dic₁₉	C₄²	1	2	yes	—
97	C₄×Dic₁₉	C₂²⋊C₄	1	1	yes	characteristic
98	C₄×Dic₁₉	C₂²⋊C₄	1	1	yes	characteristic
99	C₄×Dic₁₉	C₂²⋊C₄	1	2	yes	—
100	C₄×Dic₁₉	C₄⋊C₄	1	4	yes	—
101	C₄×Dic₁₉	C₄⋊C₄	1	4	yes	—
102	#64520	—	2	4	no	—
103	#64520	—	2	4	no	—
104	C₂×C₁₉⋊C₈	—	2	2	no	—
105	C₂×C₁₉⋊C₈	—	2	2	no	—
106	C₂×C₁₉⋊C₈	C₄⋊C₄	1	4	yes	complemented
107	#10837	—	19	1	no	maximal, Sylow
108	C₂³×C₁₉	—	4	2	no	—
109	C₂³×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
110	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂³×C₁₉	#124	1	2	yes	—
112	C₂³×C₁₉	#127	1	1	yes	characteristic
113	C₂³×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
114	C₂×C₄×C₁₉	—	4	8	no	—
115	C₂×C₄×C₁₉	—	4	4	no	—
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×C₄⋊C₄	1	1	yes	characteristic
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	1	yes	characteristic
122	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
126	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
127	C₂×C₄×C₁₉	#121	1	2	yes	—
128	C₂×C₄×C₁₉	#121	1	2	yes	—
129	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
130	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
131	C₂×C₄×C₁₉	#125	1	2	yes	—
132	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
133	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
134	C₂×Dic₁₉	—	2	8	no	—
135	C₂×Dic₁₉	—	2	4	no	—
136	C₂×Dic₁₉	—	4	4	no	—
137	C₂×Dic₁₉	—	4	2	no	—
138	C₂×Dic₁₉	—	4	8	no	—
139	C₂×Dic₁₉	—	4	4	no	—
140	C₂×Dic₁₉	#95	1	2	yes	—
141	C₂×Dic₁₉	#95	1	4	yes	—
142	C₂×Dic₁₉	#95	1	2	yes	—
143	C₈×C₁₉	—	2	4	no	—
144	C₁₉⋊C₈	—	2	4	no	—
145	C₂×C₄×M₁₆	—	19	1	no	—
146	#2285	—	19	2	no	—
147	C₂×H(3,ℤ/4)	—	19	1	no	—
148	C₂×H(3,ℤ/4)	—	19	2	no	—
149	C₂×H(3,ℤ/4)	D₁₉	1	1	yes	characteristic
150	C₂²×C₁₉	—	2	4	no	—
151	C₂²×C₁₉	—	2	2	no	—
152	C₂²×C₁₉	—	4	2	no	—
153	C₂²×C₁₉	—	4	1	no	—
154	C₂²×C₁₉	—	4	1	no	—
155	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
156	C₂²×C₁₉	#382	1	1	yes	characteristic
157	C₂²×C₁₉	#384	1	1	yes	characteristic
158	C₂²×C₁₉	#390	1	2	yes	—
159	C₂²×C₁₉	#391	1	2	yes	—
160	C₄×C₁₉	—	8	4	no	—
161	C₄×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
162	C₄×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
163	C₄×C₁₉	#380	1	1	yes	characteristic
164	C₄×C₁₉	#380	1	1	yes	characteristic
165	C₄×C₁₉	#389	1	2	yes	—
166	C₄×C₁₉	#389	1	2	yes	—
167	Dic₁₉	—	2	4	no	—
168	Dic₁₉	—	2	2	no	—
169	Dic₁₉	—	8	4	no	—
170	Dic₁₉	H(3,ℤ/4)	1	4	yes	complemented
171	C₂²×M₁₆	—	19	1	no	—
172	C₂²×M₁₆	C₂×D₁₉	1	1	yes	characteristic
173	C₂×#117	—	19	2	no	—
174	C₂×#117	C₂×D₁₉	1	2	yes	—
175	C₂²×C₄²	—	19	1	no	—
176	C₄×M₁₆	—	38	2	no	—
177	C₄×M₁₆	—	38	2	no	—
178	C₂×#97	—	19	2	no	—
179	C₂×C₄×C₈	—	19	2	no	—
180	#384	—	38	2	no	—
181	#382	—	38	2	no	—
182	#380	—	38	4	no	—
183	C₄×C₄⋊C₄	—	19	8	no	—
184	H(3,ℤ/4)	—	2	2	no	—
185	H(3,ℤ/4)	—	38	4	no	—
186	H(3,ℤ/4)	—	38	2	no	—
187	C₂×C₁₉	—	2	2	no	—
188	C₂×C₁₉	—	2	1	no	—
189	C₂×C₁₉	—	2	1	no	—
190	C₂×C₁₉	—	2	2	no	—
191	C₂×C₁₉	C₄⋊#95	1	1	yes	characteristic
192	C₂×C₁₉	C₂×H(3,ℤ/4)	1	2	yes	—
193	C₂³×C₄	—	19	2	no	—
194	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
195	C₂×M₁₆	—	2	2	no	—
196	C₂×M₁₆	—	2	2	no	—
197	C₂×M₁₆	—	19	2	no	—
198	C₂×M₁₆	—	19	2	no	—
199	C₂×M₁₆	—	38	2	no	—
200	C₂×M₁₆	—	38	2	no	—
201	C₂×M₁₆	Dic₃₈	1	2	yes	—
202	C₂×M₁₆	C₂²⋊D₁₉	1	2	yes	—
203	C₂²×C₈	—	19	2	no	—
204	C₂²×C₈	C₄×D₁₉	1	2	yes	—
205	#117	—	2	2	no	—
206	#117	—	2	2	no	—
207	#117	—	2	2	no	—
208	#117	—	2	2	no	—
209	#117	—	38	2	no	—
210	#117	—	38	2	no	—
211	#117	—	38	2	no	—
212	#117	—	38	2	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	—	38	8	no	—
215	C₂×C₄⋊C₄	—	38	2	no	—
216	C₂×C₂²⋊C₄	—	2	2	no	—
217	C₂×C₂²⋊C₄	—	38	2	no	—
218	C₂×C₄²	—	19	4	no	—
219	C₂×C₄²	—	19	1	no	—
220	C₂×C₄²	—	19	1	no	—
221	C₂×C₄²	—	19	2	no	—
222	C₂×C₄²	—	19	2	no	—
223	C₂×C₄²	—	19	2	no	—
224	C₂×C₄²	—	19	4	no	—
225	C₂×C₄²	C₄×D₁₉	1	4	yes	—
226	#97	—	38	4	no	—
227	C₄×C₈	—	38	2	no	—
228	C₄×C₈	—	38	2	no	—
229	#95	—	38	8	no	—
230	C₁₉	#10837	1	1	yes	characteristic, complemented, Sylow
231	C₂⁴	D₄×D₁₉	1	1	yes	characteristic
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	19	4	no	—
234	C₂²×C₄	—	19	2	no	—
235	C₂²×C₄	—	19	4	no	—
236	C₂²×C₄	—	19	2	no	—
237	C₂²×C₄	—	38	4	no	—
238	C₂²×C₄	—	76	4	no	—
239	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
241	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
242	C₂²×C₄	#64541	1	2	yes	—
243	C₂²×C₄	D₄×D₁₉	1	2	yes	—
244	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
245	C₂²×C₄	#64543	1	2	yes	—
246	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
247	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
248	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
249	M₁₆	—	2	4	no	—
250	M₁₆	—	2	4	no	—
251	M₁₆	—	38	4	no	—
252	M₁₆	—	38	4	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	19	4	no	—
256	C₂×C₈	—	38	2	no	—
257	C₂×C₈	—	38	2	no	—
258	C₂×C₈	Dic₁₉⋊C₄	1	4	yes	complemented
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	—	76	16	no	—
262	C₄⋊C₄	—	76	2	no	—
263	C₄⋊C₄	—	76	2	no	—
264	C₂²⋊C₄	—	4	2	no	—
265	C₂²⋊C₄	—	4	2	no	—
266	C₂²⋊C₄	—	76	2	no	—
267	C₂²⋊C₄	—	76	2	no	—
268	C₄²	—	2	4	no	—
269	C₄²	—	2	4	no	—
270	C₄²	—	19	2	no	—
271	C₄²	—	19	2	no	—
272	C₄²	—	19	1	no	—
273	C₄²	—	19	1	no	—
274	C₄²	—	19	2	no	—
275	C₄²	—	19	4	no	—
276	C₄²	—	19	4	no	—
277	C₄²	—	38	4	no	—
278	C₄²	—	38	4	no	—
279	C₂³	—	4	2	no	—
280	C₂³	C₂²⋊C₄×D₁₉	1	2	yes	—
281	C₂³	#10597331	1	2	yes	—
282	C₂³	#10597334	1	1	yes	characteristic
283	C₂³	C₄⋊C₄×D₁₉	1	1	yes	characteristic
284	C₂³	#10597386	1	1	yes	characteristic
285	C₂×C₄	—	4	8	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	19	2	no	—
288	C₂×C₄	—	19	2	no	—
289	C₂×C₄	—	19	4	no	—
290	C₂×C₄	—	38	8	no	—
291	C₂×C₄	—	38	4	no	—
292	C₂×C₄	—	76	4	no	—
293	C₂×C₄	—	76	2	no	—
294	C₂×C₄	—	76	8	no	—
295	C₂×C₄	—	76	4	no	—
296	C₂×C₄	C₄×Dic₃₈	1	2	yes	—
297	C₂×C₄	C₄²×D₁₉	1	1	yes	characteristic
298	C₂×C₄	#10597318	1	1	yes	characteristic
299	C₂×C₄	#10597323	1	1	yes	characteristic
300	C₂×C₄	#10597324	1	2	yes	—
301	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
302	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
303	C₂×C₄	#10597330	1	2	yes	—
304	C₂×C₄	#10597333	1	2	yes	—
305	C₂×C₄	#10597334	1	1	yes	characteristic
306	C₂×C₄	#10597335	1	2	yes	—
307	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
308	C₂×C₄	C₄⋊C₄×D₁₉	1	2	yes	—
309	C₂×C₄	#10597342	1	2	yes	—
310	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
311	C₂×C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
312	C₂×C₄	#10597385	1	2	yes	—
313	C₂×C₄	#10597390	1	1	yes	characteristic
314	C₈	—	2	4	no	—
315	C₈	—	38	4	no	—
316	C₂²	—	2	4	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	1	no	—
321	C₂²	#95×D₁₉	1	1	yes	characteristic
322	C₂²	#11926408	1	1	yes	characteristic
323	C₂²	#11926413	1	1	yes	characteristic
324	C₂²	#11926444	1	2	yes	—
325	C₂²	#11926448	1	2	yes	—
326	C₄	—	8	4	no	—
327	C₄	—	19	4	no	—
328	C₄	—	38	4	no	—
329	C₄	—	38	2	no	—
330	C₄	—	152	4	no	—
331	C₄	#11926395	1	1	yes	characteristic
332	C₄	C₄×Dic₁₉⋊C₄	1	1	yes	characteristic
333	C₄	#11926402	1	1	yes	characteristic
334	C₄	#11926403	1	1	yes	characteristic
335	C₄	#11926439	1	2	yes	—
336	C₄	#11926441	1	2	yes	—
337	C₂	—	2	2	no	—
338	C₂	—	2	1	no	—
339	C₂	—	2	1	no	—
340	C₂	—	2	2	no	—
341	C₂	Dic₁₉⋊#95	1	1	yes	characteristic
342	C₂	D₁₉×H(3,ℤ/4)	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₄²×D₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₁₉×H(3,ℤ/4)	C₂
socle	C₂²×C₁₉	C₂×#95
prime radical	C₂⁴×C₁₉	C₂×D₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×H(3,ℤ/4)	D₁₉
hypocenter	C₁₉	#10837
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×H(3,ℤ/4)

Complements

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

N	H ≅ G/N	type
11. C₂×#10597272	C₄	semidirect
12. C₂×C₈×Dic₁₉	C₄	semidirect
16. C₄×Dic₁₉⋊C₄	C₄	semidirect
81. C₂×C₈×C₁₉	C₄⋊C₄	semidirect
106. C₂×C₁₉⋊C₈	C₄⋊C₄	semidirect
170. Dic₁₉	H(3,ℤ/4)	semidirect
230. C₁₉	#10837	semidirect
258. C₂×C₈	Dic₁₉⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×Dic₁₉	2
#14566038	2
C₂×#11926301	2
C₂×#11926300	2
C₂×C₁₉×H(3,ℤ/4)	2
#10837	19