Subgroups — #18088167

Order 4864.

650 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉⋊#338	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926301	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926300	C₂	1	2	yes	maximal
4	#14563331	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	C₁₉⋊#338	—	2	2	no	—
9	#11926301	—	2	2	no	—
10	#11926300	—	2	4	no	—
11	#11926298	C₄	1	8	yes	complemented
12	C₂²×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
14	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
15	C₁₉×#405	—	2	2	no	—
16	C₁₉×#405	—	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	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₁₉	—	2	2	no	—
21	C₂×M₁₆×C₁₉	—	2	2	no	—
22	C₂×M₁₆×C₁₉	D₄	1	2	yes	—
23	C₂×M₁₆×C₁₉	D₄	1	2	yes	—
24	C₂²×C₈×C₁₉	—	2	1	no	—
25	C₂²×C₈×C₁₉	—	2	1	no	—
26	C₂²×C₈×C₁₉	—	2	2	no	—
27	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
31	C₁₉×#117	—	2	1	no	—
32	C₁₉×#117	—	2	1	no	—
33	C₁₉×#117	D₄	1	2	yes	—
34	C₁₉×#117	Q₈	1	2	yes	—
35	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
36	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
37	C₂×C₄²×C₁₉	C₂×C₄	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	—	2	2	no	—
43	C₂×C₄⋊Dic₁₉	—	2	2	no	—
44	C₂×C₄×Dic₁₉	C₂×C₄	1	4	yes	—
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	—	2	2	no	—
51	C₂²×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
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	2	no	—
56	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	#41	1	2	yes	—
64	C₂²×C₄×C₁₉	#41	1	2	yes	—
65	C₂²×C₄×C₁₉	#41	1	2	yes	—
66	C₂²×Dic₁₉	—	2	4	no	—
67	M₁₆×C₁₉	—	2	4	no	—
68	M₁₆×C₁₉	—	2	4	no	—
69	C₂×C₈×C₁₉	—	2	1	no	—
70	C₂×C₈×C₁₉	—	2	1	no	—
71	C₂×C₈×C₁₉	—	2	1	no	—
72	C₂×C₈×C₁₉	—	2	1	no	—
73	C₂×C₈×C₁₉	—	2	2	no	—
74	C₂×C₈×C₁₉	—	2	2	no	—
75	C₂×C₈×C₁₉	—	2	4	no	—
76	C₂×C₈×C₁₉	—	2	4	no	—
77	C₂×C₈×C₁₉	C₄²	1	2	yes	—
78	C₂×C₈×C₁₉	C₄²	1	2	yes	—
79	C₂×C₈×C₁₉	C₂²⋊C₄	1	4	yes	—
80	C₄⋊C₄×C₁₉	—	2	2	no	—
81	C₄⋊C₄×C₁₉	—	2	2	no	—
82	C₂²⋊C₄×C₁₉	—	2	2	no	—
83	C₂²⋊C₄×C₁₉	—	2	2	no	—
84	C₄²×C₁₉	—	2	2	no	—
85	C₄²×C₁₉	C₄⋊C₄	1	4	yes	—
86	C₂²⋊Dic₁₉	—	4	2	no	—
87	C₂²⋊Dic₁₉	—	4	2	no	—
88	C₄⋊Dic₁₉	—	4	2	no	—
89	C₄⋊Dic₁₉	—	4	2	no	—
90	C₄×Dic₁₉	—	2	4	no	—
91	C₄×Dic₁₉	—	2	4	no	—
92	#64520	—	4	2	no	—
93	#64520	—	4	2	no	—
94	#64520	—	4	2	no	—
95	#64520	—	4	2	no	—
96	C₂×C₁₉⋊C₈	—	2	4	no	—
97	C₂×C₁₉⋊C₈	—	2	4	no	—
98	C₂×C₁₉⋊C₈	—	2	4	no	—
99	#10800	—	19	1	no	maximal, Sylow
100	C₂³×C₁₉	—	4	2	no	—
101	C₂³×C₁₉	#117	1	2	yes	—
102	C₂³×C₁₉	#117	1	1	yes	characteristic
103	C₂³×C₁₉	#121	1	1	yes	characteristic
104	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂³×C₁₉	#126	1	2	yes	—
106	C₂×C₄×C₁₉	—	2	4	no	—
107	C₂×C₄×C₁₉	—	4	4	no	—
108	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₉	#117	1	2	yes	—
112	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
113	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
114	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
115	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
119	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×C₄×C₁₉	#123	1	2	yes	—
122	C₂×C₄×C₁₉	#123	1	2	yes	—
123	C₂×C₄×C₁₉	#124	1	2	yes	—
124	C₂×C₄×C₁₉	#125	1	2	yes	—
125	C₂×C₄×C₁₉	#126	1	2	yes	—
126	C₂×Dic₁₉	—	4	8	no	—
127	C₂×Dic₁₉	—	4	4	no	—
128	C₈×C₁₉	—	2	2	no	—
129	C₈×C₁₉	—	2	2	no	—
130	C₈×C₁₉	—	2	4	no	—
131	C₁₉⋊C₈	—	4	4	no	—
132	C₂×#405	D₁₉	1	1	yes	characteristic
133	C₂×#338	—	19	1	no	—
134	C₂×H(3,ℤ/4)	—	19	1	no	—
135	C₂×H(3,ℤ/4)	—	19	2	no	—
136	#1729	—	19	2	no	—
137	C₂²×C₁₉	—	2	4	no	—
138	C₂²×C₁₉	—	2	2	no	—
139	C₂²×C₁₉	—	4	1	no	—
140	C₂²×C₁₉	—	4	1	no	—
141	C₂²×C₁₉	—	4	2	no	—
142	C₂²×C₁₉	#376	1	1	yes	characteristic
143	C₂²×C₁₉	#380	1	1	yes	characteristic
144	C₂²×C₁₉	#383	1	1	yes	characteristic
145	C₂²×C₁₉	#387	1	2	yes	—
146	C₂²×C₁₉	#388	1	2	yes	—
147	C₄×C₁₉	—	4	4	no	—
148	C₄×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
149	C₄×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
150	C₄×C₁₉	#381	1	1	yes	characteristic
151	C₄×C₁₉	#382	1	1	yes	characteristic
152	C₄×C₁₉	#387	1	2	yes	—
153	C₄×C₁₉	#388	1	2	yes	—
154	Dic₁₉	—	8	4	no	—
155	C₂²×M₁₆	—	19	2	no	—
156	C₂²×M₁₆	C₂×D₁₉	1	1	yes	characteristic
157	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
158	C₂×#117	—	19	2	no	—
159	C₂×#117	C₂×D₁₉	1	1	yes	characteristic
160	#405	—	2	2	no	—
161	#405	—	2	2	no	—
162	C₂×#97	Dic₁₉	1	2	yes	complemented
163	C₂×C₄×C₈	Dic₁₉	1	2	yes	complemented
164	#338	—	38	2	no	—
165	H(3,ℤ/4)	—	38	2	no	—
166	H(3,ℤ/4)	—	38	4	no	—
167	#336	—	19	8	no	—
168	C₂×C₁₉	—	2	2	no	—
169	C₂×C₁₉	—	2	1	no	—
170	C₂×C₁₉	—	2	1	no	—
171	C₂×C₁₉	—	2	2	no	—
172	C₂×C₁₉	#1434	1	1	yes	characteristic
173	C₂×C₁₉	#1747	1	2	yes	—
174	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	38	2	no	—
178	C₂×M₁₆	—	38	2	no	—
179	C₂×M₁₆	—	38	2	no	—
180	C₂×M₁₆	—	38	2	no	—
181	C₂×M₁₆	—	38	2	no	—
182	C₂×M₁₆	—	38	2	no	—
183	C₂×M₁₆	C₂²⋊D₁₉	1	2	yes	—
184	C₂×M₁₆	C₂²⋊D₁₉	1	2	yes	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	—	2	1	no	—
187	C₂²×C₈	—	2	1	no	—
188	C₂²×C₈	—	19	4	no	—
189	C₂²×C₈	C₄×D₁₉	1	4	yes	—
190	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
191	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
192	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
193	#117	—	2	1	no	—
194	#117	—	2	1	no	—
195	#117	—	38	2	no	—
196	#117	—	38	2	no	—
197	#117	—	38	2	no	—
198	#117	—	38	2	no	—
199	#117	Dic₃₈	1	2	yes	—
200	#117	D₇₆	1	2	yes	—
201	C₂×C₄⋊C₄	—	2	1	no	—
202	C₂×C₄⋊C₄	—	38	2	no	—
203	C₂×C₂²⋊C₄	—	2	1	no	—
204	C₂×C₂²⋊C₄	—	38	2	no	—
205	C₂×C₄²	—	19	4	no	—
206	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
207	#97	—	2	4	no	—
208	C₄×C₈	—	2	4	no	—
209	C₁₉	#10800	1	1	yes	characteristic, complemented, Sylow
210	C₂⁴	#64541	1	1	yes	characteristic
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	38	4	no	—
213	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
215	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
216	C₂²×C₄	#64541	1	1	yes	characteristic
217	C₂²×C₄	#64541	1	2	yes	—
218	C₂²×C₄	#64541	1	2	yes	—
219	C₂²×C₄	#64541	1	2	yes	—
220	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
223	M₁₆	—	2	4	no	—
224	M₁₆	—	2	4	no	—
225	M₁₆	—	76	2	no	—
226	M₁₆	—	76	2	no	—
227	M₁₆	—	76	2	no	—
228	M₁₆	—	76	2	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	38	4	no	—
238	C₂×C₈	—	38	4	no	—
239	C₂×C₈	—	38	4	no	—
240	C₂×C₈	C₄×Dic₁₉	1	2	yes	—
241	C₂×C₈	C₄×Dic₁₉	1	2	yes	—
242	C₂×C₈	C₂²⋊Dic₁₉	1	4	yes	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	76	2	no	—
246	C₄⋊C₄	—	76	2	no	—
247	C₂²⋊C₄	—	2	2	no	—
248	C₂²⋊C₄	—	2	2	no	—
249	C₂²⋊C₄	—	76	2	no	—
250	C₂²⋊C₄	—	76	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	38	4	no	—
253	C₄²	—	38	4	no	—
254	C₄²	C₄⋊Dic₁₉	1	4	yes	—
255	C₂³	—	4	2	no	—
256	C₂³	#10597318	1	2	yes	—
257	C₂³	#10597322	1	2	yes	—
258	C₂³	#10597370	1	1	yes	characteristic
259	C₂³	#10597372	1	1	yes	characteristic
260	C₂³	#10597376	1	1	yes	characteristic
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	76	8	no	—
264	C₂×C₄	—	76	4	no	—
265	C₂×C₄	C₄×Dic₃₈	1	2	yes	—
266	C₂×C₄	#10597316	1	2	yes	—
267	C₂×C₄	#10597318	1	2	yes	—
268	C₂×C₄	C₄×D₇₆	1	2	yes	—
269	C₂×C₄	#10597321	1	2	yes	—
270	C₂×C₄	#10597322	1	2	yes	—
271	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
272	C₂×C₄	#10597370	1	1	yes	characteristic
273	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
274	C₂×C₄	#10597372	1	1	yes	characteristic
275	C₂×C₄	#10597372	1	1	yes	characteristic
276	C₂×C₄	#10597372	1	1	yes	characteristic
277	C₂×C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
278	C₂×C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
279	C₂×C₄	#10597375	1	2	yes	—
280	C₂×C₄	#10597375	1	2	yes	—
281	C₂×C₄	#10597376	1	1	yes	characteristic
282	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
283	C₈	—	2	2	no	—
284	C₈	—	2	2	no	—
285	C₈	—	2	4	no	—
286	C₈	—	76	4	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	2	no	—
291	C₂²	—	4	1	no	—
292	C₂²	#11926390	1	1	yes	characteristic
293	C₂²	#11926404	1	1	yes	characteristic
294	C₂²	#11926412	1	1	yes	characteristic
295	C₂²	#11926428	1	2	yes	—
296	C₂²	#11926435	1	2	yes	—
297	C₄	—	4	4	no	—
298	C₄	—	152	4	no	—
299	C₄	C₄×C₂²⋊Dic₁₉	1	1	yes	characteristic
300	C₄	C₄×C₄⋊Dic₁₉	1	1	yes	characteristic
301	C₄	#11926407	1	1	yes	characteristic
302	C₄	#11926410	1	1	yes	characteristic
303	C₄	#11926428	1	2	yes	—
304	C₄	#11926435	1	2	yes	—
305	C₂	—	2	1	no	—
306	C₂	—	2	2	no	—
307	C₂	—	2	1	no	—
308	C₂	—	2	2	no	—
309	C₂	#14561862	1	1	yes	characteristic
310	C₂	#14563390	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
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	#10800
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
11. #11926298	C₄	semidirect
17. C₂×C₁₉×#97	C₄	semidirect
18. C₂×C₄×C₈×C₁₉	C₄	semidirect
162. C₂×#97	Dic₁₉	semidirect
163. C₂×C₄×C₈	Dic₁₉	semidirect
209. C₁₉	#10800	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₉⋊#338	2
C₂×#11926301	2
C₂×#11926300	2
#14563331	2
C₂×C₁₉×#405	2
#10800	19