Subgroups — #18088190

Order 4864.

674 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×#11926315	C₂	1	2	yes	maximal
2	C₂×#11926304	C₂	1	2	yes	maximal
3	C₂×C₁₉⋊#338	C₂	1	1	yes	maximal, characteristic
4	C₂×#11926301	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×M₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	C₂²	1	2	yes	—
7	C₂×#10597372	C₂²	1	2	yes	—
8	#11926315	C₄	1	8	yes	complemented
9	#11926304	—	2	4	no	—
10	C₁₉⋊#338	—	2	2	no	—
11	#11926301	—	2	2	no	—
12	C₂²×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
15	C₄×M₁₆×C₁₉	—	2	2	no	—
16	C₄×M₁₆×C₁₉	—	2	2	no	—
17	C₂×C₁₉×#97	C₄	1	2	yes	complemented
18	C₂×C₄×C₈×C₁₉	C₄	1	2	yes	complemented
19	C₂³×C₄×C₁₉	—	2	1	no	—
20	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
21	C₂×M₁₆×C₁₉	—	2	1	no	—
22	C₂×M₁₆×C₁₉	—	2	1	no	—
23	C₂×M₁₆×C₁₉	—	2	2	no	—
24	C₂×M₁₆×C₁₉	—	2	2	no	—
25	C₂×M₁₆×C₁₉	—	2	2	no	—
26	C₂×M₁₆×C₁₉	C₂×C₄	1	4	yes	—
27	C₂×M₁₆×C₁₉	D₄	1	2	yes	—
28	C₂×M₁₆×C₁₉	D₄	1	2	yes	—
29	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₉	—	2	1	no	—
32	C₂×C₄²×C₁₉	—	2	1	no	—
33	C₂×C₄²×C₁₉	—	2	1	no	—
34	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₉	D₄	1	2	yes	—
37	C₂×C₄²×C₁₉	Q₈	1	2	yes	—
38	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
39	#10597372	—	2	2	no	—
40	#10597372	—	2	2	no	—
41	#10597372	—	2	2	no	—
42	#10597372	C₂×C₄	1	4	yes	—
43	C₂×C₄⋊Dic₁₉	—	2	2	no	—
44	C₂×C₄×Dic₁₉	—	2	2	no	—
45	C₂×#64520	—	2	2	no	—
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	C₂²×C₁₉⋊C₈	—	2	2	no	—
52	C₁₉×#97	—	2	4	no	—
53	C₄×C₈×C₁₉	—	2	4	no	—
54	C₂⁴×C₁₉	#41	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	—	2	1	no	—
56	C₂²×C₄×C₁₉	—	2	1	no	—
57	C₂²×C₄×C₁₉	—	2	2	no	—
58	C₂²×C₄×C₁₉	—	2	2	no	—
59	C₂²×C₄×C₁₉	—	4	4	no	—
60	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	#41	1	2	yes	—
68	C₂²×C₄×C₁₉	#41	1	2	yes	—
69	C₂²×C₄×C₁₉	#41	1	2	yes	—
70	C₂²×Dic₁₉	—	4	2	no	—
71	M₁₆×C₁₉	—	2	4	no	—
72	M₁₆×C₁₉	—	2	4	no	—
73	M₁₆×C₁₉	—	2	4	no	—
74	M₁₆×C₁₉	—	2	4	no	—
75	C₂×C₈×C₁₉	—	2	2	no	—
76	C₂×C₈×C₁₉	—	2	2	no	—
77	C₂×C₈×C₁₉	—	2	2	no	—
78	C₂×C₈×C₁₉	—	2	2	no	—
79	C₂×C₈×C₁₉	C₄²	1	4	yes	—
80	C₂×C₈×C₁₉	C₂²⋊C₄	1	4	yes	—
81	C₄²×C₁₉	—	2	1	no	—
82	C₄²×C₁₉	—	2	1	no	—
83	C₄²×C₁₉	—	2	2	no	—
84	C₄²×C₁₉	—	2	2	no	—
85	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
86	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
87	C₂²⋊Dic₁₉	—	2	4	no	—
88	C₂²⋊Dic₁₉	—	4	2	no	—
89	C₄⋊Dic₁₉	—	2	4	no	—
90	C₄⋊Dic₁₉	—	4	2	no	—
91	C₄×Dic₁₉	—	2	4	no	—
92	C₄×Dic₁₉	—	4	2	no	—
93	#64520	—	2	4	no	—
94	#64520	—	2	4	no	—
95	#64520	—	4	2	no	—
96	#64520	—	4	2	no	—
97	C₂×C₁₉⋊C₈	—	2	4	no	—
98	C₂×C₁₉⋊C₈	—	4	2	no	—
99	C₂×C₁₉⋊C₈	—	4	2	no	—
100	#10805	—	19	1	no	maximal, Sylow
101	C₂³×C₁₉	—	4	2	no	—
102	C₂³×C₁₉	#117	1	2	yes	—
103	C₂³×C₁₉	#117	1	1	yes	characteristic
104	C₂³×C₁₉	#121	1	1	yes	characteristic
105	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂³×C₁₉	#126	1	2	yes	—
107	C₂×C₄×C₁₉	—	2	4	no	—
108	C₂×C₄×C₁₉	—	2	2	no	—
109	C₂×C₄×C₁₉	—	2	2	no	—
110	C₂×C₄×C₁₉	—	4	4	no	—
111	C₂×C₄×C₁₉	—	4	4	no	—
112	C₂×C₄×C₁₉	—	4	2	no	—
113	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₉	#117	1	2	yes	—
117	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
118	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
119	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
124	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
126	C₂×C₄×C₁₉	#123	1	2	yes	—
127	C₂×C₄×C₁₉	#123	1	2	yes	—
128	C₂×C₄×C₁₉	#124	1	2	yes	—
129	C₂×C₄×C₁₉	#125	1	2	yes	—
130	C₂×C₄×C₁₉	#126	1	2	yes	—
131	C₂×Dic₁₉	—	4	4	no	—
132	C₂×Dic₁₉	—	8	4	no	—
133	C₈×C₁₉	—	2	4	no	—
134	C₈×C₁₉	—	2	4	no	—
135	C₁₉⋊C₈	—	4	4	no	—
136	C₂×C₄×M₁₆	D₁₉	1	1	yes	characteristic
137	C₂×#344	—	19	2	no	—
138	C₂×#339	—	19	2	no	—
139	C₂×#338	—	19	1	no	—
140	C₂×H(3,ℤ/4)	—	19	1	no	—
141	C₂²×C₁₉	—	2	4	no	—
142	C₂²×C₁₉	—	2	2	no	—
143	C₂²×C₁₉	—	4	1	no	—
144	C₂²×C₁₉	—	4	1	no	—
145	C₂²×C₁₉	—	4	2	no	—
146	C₂²×C₁₉	#376	1	1	yes	characteristic
147	C₂²×C₁₉	#380	1	1	yes	characteristic
148	C₂²×C₁₉	#383	1	1	yes	characteristic
149	C₂²×C₁₉	#387	1	2	yes	—
150	C₂²×C₁₉	#388	1	2	yes	—
151	C₄×C₁₉	—	2	4	no	—
152	C₄×C₁₉	—	4	2	no	—
153	C₄×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
154	C₄×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
155	C₄×C₁₉	#381	1	1	yes	characteristic
156	C₄×C₁₉	#382	1	1	yes	characteristic
157	C₄×C₁₉	#387	1	2	yes	—
158	C₄×C₁₉	#388	1	2	yes	—
159	Dic₁₉	—	8	4	no	—
160	C₂²×M₁₆	—	19	2	no	—
161	C₂²×M₁₆	C₂×D₁₉	1	1	yes	characteristic
162	C₂²×M₁₆	C₂×D₁₉	1	1	yes	characteristic
163	C₂×#117	—	19	2	no	—
164	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
165	C₄×M₁₆	—	2	2	no	—
166	C₄×M₁₆	—	2	2	no	—
167	C₂×#97	Dic₁₉	1	2	yes	complemented
168	C₂×C₄×C₈	Dic₁₉	1	2	yes	complemented
169	#344	—	19	8	no	—
170	#339	—	38	4	no	—
171	#338	—	38	2	no	—
172	H(3,ℤ/4)	—	38	2	no	—
173	C₂×C₁₉	—	2	2	no	—
174	C₂×C₁₉	—	2	1	no	—
175	C₂×C₁₉	—	2	1	no	—
176	C₂×C₁₉	—	2	2	no	—
177	C₂×C₁₉	#1434	1	1	yes	characteristic
178	C₂×C₁₉	#1841	1	2	yes	—
179	C₂³×C₄	—	2	1	no	—
180	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	2	1	no	—
185	C₂×M₁₆	—	2	1	no	—
186	C₂×M₁₆	—	19	4	no	—
187	C₂×M₁₆	—	38	2	no	—
188	C₂×M₁₆	—	38	2	no	—
189	C₂×M₁₆	—	38	2	no	—
190	C₂×M₁₆	—	38	2	no	—
191	C₂×M₁₆	—	38	2	no	—
192	C₂×M₁₆	C₄×D₁₉	1	4	yes	—
193	C₂×M₁₆	C₂²⋊D₁₉	1	2	yes	—
194	C₂×M₁₆	C₂²⋊D₁₉	1	2	yes	—
195	C₂²×C₈	—	38	2	no	—
196	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
197	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
198	#117	—	19	4	no	—
199	#117	—	38	2	no	—
200	#117	—	38	2	no	—
201	#117	—	38	2	no	—
202	C₂×C₄⋊C₄	—	38	2	no	—
203	C₂×C₂²⋊C₄	—	38	2	no	—
204	C₂×C₄²	—	2	1	no	—
205	C₂×C₄²	—	2	1	no	—
206	C₂×C₄²	—	2	1	no	—
207	C₂×C₄²	—	38	2	no	—
208	C₂×C₄²	Dic₃₈	1	2	yes	—
209	C₂×C₄²	D₇₆	1	2	yes	—
210	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
211	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
212	#97	—	2	4	no	—
213	C₄×C₈	—	2	4	no	—
214	C₁₉	#10805	1	1	yes	characteristic, complemented, Sylow
215	C₂⁴	#64541	1	1	yes	characteristic
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	4	4	no	—
221	C₂²×C₄	—	76	2	no	—
222	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
224	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
225	C₂²×C₄	#64541	1	1	yes	characteristic
226	C₂²×C₄	#64541	1	2	yes	—
227	C₂²×C₄	#64541	1	2	yes	—
228	C₂²×C₄	#64541	1	2	yes	—
229	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
232	M₁₆	—	2	4	no	—
233	M₁₆	—	2	4	no	—
234	M₁₆	—	2	4	no	—
235	M₁₆	—	2	4	no	—
236	M₁₆	—	38	4	no	—
237	M₁₆	—	38	4	no	—
238	M₁₆	—	76	2	no	—
239	M₁₆	—	76	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	38	4	no	—
245	C₂×C₈	—	76	2	no	—
246	C₂×C₈	—	76	2	no	—
247	C₂×C₈	C₄×Dic₁₉	1	4	yes	—
248	C₂×C₈	C₂²⋊Dic₁₉	1	4	yes	—
249	C₄⋊C₄	—	38	4	no	—
250	C₄⋊C₄	—	76	2	no	—
251	C₂²⋊C₄	—	38	4	no	—
252	C₂²⋊C₄	—	76	2	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	38	4	no	—
258	C₄²	—	76	2	no	—
259	C₄²	C₄⋊Dic₁₉	1	2	yes	—
260	C₄²	C₄⋊Dic₁₉	1	2	yes	—
261	C₂³	—	4	2	no	—
262	C₂³	#10597318	1	2	yes	—
263	C₂³	#10597322	1	2	yes	—
264	C₂³	#10597370	1	1	yes	characteristic
265	C₂³	#10597372	1	1	yes	characteristic
266	C₂³	#10597376	1	1	yes	characteristic
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	76	4	no	—
274	C₂×C₄	—	152	4	no	—
275	C₂×C₄	C₄×Dic₃₈	1	2	yes	—
276	C₂×C₄	#10597316	1	2	yes	—
277	C₂×C₄	#10597318	1	2	yes	—
278	C₂×C₄	C₄×D₇₆	1	2	yes	—
279	C₂×C₄	#10597321	1	2	yes	—
280	C₂×C₄	#10597322	1	2	yes	—
281	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
282	C₂×C₄	#10597370	1	1	yes	characteristic
283	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
284	C₂×C₄	#10597372	1	1	yes	characteristic
285	C₂×C₄	#10597372	1	1	yes	characteristic
286	C₂×C₄	#10597372	1	1	yes	characteristic
287	C₂×C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
288	C₂×C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
289	C₂×C₄	#10597375	1	2	yes	—
290	C₂×C₄	#10597375	1	2	yes	—
291	C₂×C₄	#10597376	1	1	yes	characteristic
292	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
293	C₈	—	2	4	no	—
294	C₈	—	2	4	no	—
295	C₈	—	76	4	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	4	no	—
298	C₂²	—	4	1	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	4	2	no	—
301	C₂²	#11926390	1	1	yes	characteristic
302	C₂²	#11926404	1	1	yes	characteristic
303	C₂²	#11926412	1	1	yes	characteristic
304	C₂²	#11926428	1	2	yes	—
305	C₂²	#11926435	1	2	yes	—
306	C₄	—	2	4	no	—
307	C₄	—	4	2	no	—
308	C₄	—	152	4	no	—
309	C₄	C₄×C₂²⋊Dic₁₉	1	1	yes	characteristic
310	C₄	C₄×C₄⋊Dic₁₉	1	1	yes	characteristic
311	C₄	#11926407	1	1	yes	characteristic
312	C₄	#11926410	1	1	yes	characteristic
313	C₄	#11926428	1	2	yes	—
314	C₄	#11926435	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	#14561862	1	1	yes	characteristic
320	C₂	#14563784	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#10597372
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₄×M₁₆×C₁₉	C₂
socle	C₂²×C₁₉	#383
prime radical	C₂⁴×C₁₉	#41
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×C₄×M₁₆	D₁₉
hypocenter	C₁₉	#10805
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
8. #11926315	C₄	semidirect
17. C₂×C₁₉×#97	C₄	semidirect
18. C₂×C₄×C₈×C₁₉	C₄	semidirect
167. C₂×#97	Dic₁₉	semidirect
168. C₂×C₄×C₈	Dic₁₉	semidirect
214. C₁₉	#10805	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926315	2
C₂×#11926304	2
C₂×C₁₉⋊#338	2
C₂×#11926301	2
C₂×C₄×M₁₆×C₁₉	2
#10805	19