Subgroups — #14574162

Order 2432.

340 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	#11927174	C₂	1	1	yes	maximal, characteristic
2	#11927171	C₂	1	1	yes	maximal, characteristic
3	#11926953	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926952	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926951	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926945	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926943	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926941	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926940	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926585	C₂	1	1	yes	maximal, characteristic
11	#11926583	C₂	1	1	yes	maximal, characteristic
12	#11926581	C₂	1	1	yes	maximal, characteristic
13	#106×D₁₉	C₂	1	1	yes	maximal, characteristic
14	#11926501	C₂	1	1	yes	maximal, characteristic
15	C₁₉×#501	C₂	1	1	yes	maximal, characteristic, complemented
16	C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₁₉×#125	C₂²	1	1	yes	characteristic
18	#10597392	—	2	1	no	—
19	#10597390	—	2	1	no	—
20	C₂×#64515	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₁₉	C₂²	1	1	yes	characteristic
22	#10597343	—	2	1	no	—
23	#10597342	—	2	1	no	—
24	#10597342	—	2	1	no	—
25	#10597338	C₂²	1	1	yes	characteristic
26	#10597338	C₂²	1	1	yes	characteristic
27	C₄⋊C₄×D₁₉	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×D₁₉	C₂²	1	1	yes	characteristic
29	#10597336	C₂²	1	1	yes	characteristic
30	#10597336	C₂²	1	1	yes	characteristic
31	#10597335	—	2	1	no	—
32	#10597335	—	2	1	no	—
33	#10597334	—	2	1	no	—
34	#10597334	C₂²	1	1	yes	characteristic
35	#10597334	C₂²	1	1	yes	characteristic
36	#10597318	C₂²	1	1	yes	characteristic
37	#10597316	C₂²	1	1	yes	characteristic
38	#10597315	C₂²	1	1	yes	characteristic
39	C₁₉×#107	C₂²	1	1	yes	characteristic
40	C₁₉×#107	C₂²	1	1	yes	characteristic
41	C₁₉×#106	C₂²	1	1	yes	characteristic
42	C₁₉×#106	C₂²	1	1	yes	characteristic
43	C₁₉×#97	C₂²	1	1	yes	characteristic
44	#10597275	C₂²	1	1	yes	characteristic
45	#10597275	C₂²	1	1	yes	characteristic
46	#10597274	C₂²	1	1	yes	characteristic
47	#10597274	C₂²	1	1	yes	characteristic
48	#10597272	C₂²	1	1	yes	characteristic
49	C₈×Dic₁₉	C₂²	1	1	yes	characteristic
50	#10597265	C₂²	1	1	yes	characteristic
51	#10597265	C₂²	1	1	yes	characteristic
52	#10597265	C₂²	1	1	yes	characteristic
53	#10597265	C₂²	1	1	yes	characteristic
54	#10597265	C₂²	1	1	yes	characteristic
55	#10597265	C₂²	1	1	yes	characteristic
56	#10597264	C₂²	1	1	yes	characteristic
57	#10597264	C₂²	1	1	yes	characteristic
58	C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₉	—	2	1	no	—
60	C₂×C₄×D₁₉	—	2	1	no	—
61	C₂×C₄×D₁₉	—	2	1	no	—
62	C₂×C₄×D₁₉	C₂³	1	1	yes	characteristic
63	C₂×Dic₃₈	—	2	1	no	—
64	C₂×Dic₃₈	—	2	1	no	—
65	C₂×Dic₃₈	—	2	1	no	—
66	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₁₉	—	2	1	no	—
69	C₄⋊C₄×C₁₉	—	2	1	no	—
70	C₄⋊C₄×C₁₉	—	2	1	no	—
71	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
75	C₄²×C₁₉	C₂³	1	1	yes	characteristic
76	(C₂×D₁₉)⋊C₄	—	2	1	no	—
77	(C₂×D₁₉)⋊C₄	—	2	1	no	—
78	(C₂×D₁₉)⋊C₄	D₄	1	1	yes	characteristic
79	(C₂×D₁₉)⋊C₄	D₄	1	1	yes	characteristic
80	C₄⋊Dic₁₉	—	2	1	no	—
81	C₄⋊Dic₁₉	—	2	1	no	—
82	C₄⋊Dic₁₉	—	2	1	no	—
83	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
84	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
85	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
86	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
87	Dic₁₉⋊C₄	—	2	1	no	—
88	Dic₁₉⋊C₄	—	2	1	no	—
89	Dic₁₉⋊C₄	—	2	1	no	—
90	Dic₁₉⋊C₄	—	2	1	no	—
91	Dic₁₉⋊C₄	—	2	1	no	—
92	Dic₁₉⋊C₄	—	2	1	no	—
93	Dic₁₉⋊C₄	—	2	1	no	—
94	Dic₁₉⋊C₄	—	2	1	no	—
95	Dic₁₉⋊C₄	D₄	1	1	yes	characteristic
96	Dic₁₉⋊C₄	D₄	1	1	yes	characteristic
97	C₄×Dic₁₉	—	2	1	no	—
98	C₄×Dic₁₉	—	2	1	no	—
99	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
100	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
101	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
102	#64515	Q₈	1	4	yes	complemented
103	C₈×D₁₉	Q₈	1	4	yes	complemented
104	C₂²×D₁₉	C₂×D₄	1	1	yes	characteristic
105	Q₈×C₁₉	—	8	1	no	—
106	C₂×C₄×C₁₉	—	2	1	no	—
107	C₂×C₄×C₁₉	—	2	1	no	—
108	C₂×C₄×C₁₉	—	2	1	no	—
109	C₂×C₄×C₁₉	—	2	1	no	—
110	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
113	C₂×Dic₁₉	—	2	1	no	—
114	C₂×Dic₁₉	—	2	1	no	—
115	C₂×Dic₁₉	—	2	1	no	—
116	C₂×Dic₁₉	—	2	1	no	—
117	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
118	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
119	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
120	C₄×D₁₉	—	8	1	no	—
121	C₄×D₁₉	—	8	1	no	—
122	C₄×D₁₉	C₂×Q₈	1	2	yes	—
123	C₄×D₁₉	C₂×Q₈	1	2	yes	—
124	Dic₃₈	—	8	1	no	—
125	Dic₃₈	—	8	1	no	—
126	Dic₃₈	—	8	1	no	—
127	C₈×C₁₉	C₂×Q₈	1	2	yes	—
128	C₈×C₁₉	C₂×Q₈	1	2	yes	—
129	C₁₉⋊C₈	C₂×Q₈	1	2	yes	—
130	C₁₉⋊C₈	C₂×Q₈	1	2	yes	—
131	#3163	—	19	1	no	maximal, Sylow
132	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
133	C₂×D₁₉	—	2	1	no	—
134	C₂×D₁₉	—	2	1	no	—
135	C₂×D₁₉	C₄⋊Q₈	1	2	yes	—
136	C₄×C₁₉	—	2	1	no	—
137	C₄×C₁₉	—	2	1	no	—
138	C₄×C₁₉	—	4	1	no	—
139	C₄×C₁₉	—	4	1	no	—
140	C₄×C₁₉	—	4	1	no	—
141	C₄×C₁₉	—	4	1	no	—
142	C₄×C₁₉	C₂²×Q₈	1	1	yes	characteristic
143	C₄×C₁₉	C₂²×Q₈	1	1	yes	characteristic
144	Dic₁₉	—	2	1	no	—
145	Dic₁₉	—	2	1	no	—
146	Dic₁₉	—	4	1	no	—
147	Dic₁₉	—	4	1	no	—
148	Dic₁₉	—	4	1	no	—
149	Dic₁₉	—	4	1	no	—
150	Dic₁₉	C₄⋊Q₈	1	2	yes	—
151	#544	—	19	1	no	—
152	#544	—	19	1	no	—
153	#501	—	19	1	no	—
154	#501	—	19	1	no	—
155	#501	—	19	1	no	—
156	#501	D₁₉	1	1	yes	characteristic, complemented
157	#499	—	19	1	no	—
158	#499	—	19	1	no	—
159	#498	—	19	1	no	—
160	#498	—	19	1	no	—
161	#428	—	19	1	no	—
162	#428	—	19	1	no	—
163	#427	—	19	1	no	—
164	C₂×#106	—	19	1	no	—
165	#405	—	19	1	no	—
166	C₂×C₁₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
167	C₂×C₁₉	#577	1	1	yes	characteristic
168	C₂×C₁₉	#577	1	1	yes	characteristic
169	D₁₉	—	2	2	no	—
170	C₂×M₁₆	—	19	1	no	—
171	C₂²×C₈	—	19	1	no	—
172	C₄⋊Q₈	—	19	1	no	—
173	C₄⋊Q₈	—	19	1	no	—
174	C₄⋊Q₈	—	19	1	no	—
175	C₄⋊Q₈	—	38	1	no	—
176	C₄⋊Q₈	—	38	1	no	—
177	C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
178	#125	—	19	1	no	—
179	#125	—	19	1	no	—
180	#125	—	19	1	no	—
181	#125	—	38	1	no	—
182	#125	—	38	1	no	—
183	#125	C₂×D₁₉	1	1	yes	characteristic
184	C₂²⋊Q₈	—	38	1	no	—
185	C₂²⋊Q₈	—	38	1	no	—
186	C₂²⋊Q₈	—	38	1	no	—
187	C₂²⋊Q₈	—	38	1	no	—
188	#117	—	19	1	no	—
189	#117	—	19	1	no	—
190	#117	—	19	1	no	—
191	C₂×C₄⋊C₄	—	19	1	no	—
192	C₂×C₄⋊C₄	—	19	1	no	—
193	#107	—	19	1	no	—
194	#107	—	19	1	no	—
195	#107	—	19	1	no	—
196	#107	—	19	1	no	—
197	#107	C₂×D₁₉	1	1	yes	characteristic
198	#107	C₂×D₁₉	1	1	yes	characteristic
199	#106	—	19	1	no	—
200	#106	—	19	1	no	—
201	#106	—	19	1	no	—
202	#106	—	19	1	no	—
203	#106	—	19	1	no	—
204	#106	—	19	1	no	—
205	#106	—	19	1	no	—
206	#106	—	19	1	no	—
207	#106	C₂×D₁₉	1	1	yes	characteristic
208	#106	C₂×D₁₉	1	1	yes	characteristic
209	#97	—	19	1	no	—
210	#97	C₂×D₁₉	1	1	yes	characteristic
211	C₄×C₈	—	19	1	no	—
212	C₄×C₈	—	19	1	no	—
213	C₁₉	#3163	1	1	yes	characteristic, complemented, Sylow
214	C₂×Q₈	—	2	1	no	—
215	C₂×Q₈	—	38	1	no	—
216	C₂×Q₈	—	38	1	no	—
217	C₂×Q₈	—	38	1	no	—
218	C₂²×C₄	—	19	1	no	—
219	C₂²×C₄	—	38	1	no	—
220	C₂²×C₄	—	38	1	no	—
221	M₁₆	—	19	4	no	—
222	C₂×C₈	—	19	1	no	—
223	C₂×C₈	—	19	1	no	—
224	C₂×C₈	—	19	4	no	—
225	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
226	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
227	C₄⋊C₄	—	2	1	no	—
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	19	1	no	—
231	C₄⋊C₄	—	19	1	no	—
232	C₄⋊C₄	—	19	1	no	—
233	C₄⋊C₄	—	19	1	no	—
234	C₄⋊C₄	—	19	1	no	—
235	C₄⋊C₄	—	19	1	no	—
236	C₄⋊C₄	—	38	1	no	—
237	C₄⋊C₄	—	38	1	no	—
238	C₄⋊C₄	—	38	1	no	—
239	C₄⋊C₄	—	38	1	no	—
240	C₄⋊C₄	—	38	1	no	—
241	C₄⋊C₄	—	38	1	no	—
242	C₄⋊C₄	—	38	1	no	—
243	C₄⋊C₄	—	38	1	no	—
244	C₄⋊C₄	—	38	1	no	—
245	C₄⋊C₄	—	38	1	no	—
246	C₄⋊C₄	—	38	1	no	—
247	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
248	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
249	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
250	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
251	C₂²⋊C₄	—	19	1	no	—
252	C₂²⋊C₄	—	19	1	no	—
253	C₂²⋊C₄	—	38	1	no	—
254	C₂²⋊C₄	—	38	1	no	—
255	C₄²	—	19	1	no	—
256	C₄²	—	38	1	no	—
257	C₄²	—	38	1	no	—
258	C₄²	C₂²×D₁₉	1	1	yes	characteristic
259	C₂³	—	19	1	no	—
260	Q₈	—	8	1	no	—
261	Q₈	—	152	1	no	—
262	Q₈	—	152	1	no	—
263	Q₈	—	152	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	19	1	no	—
269	C₂×C₄	—	19	2	no	—
270	C₂×C₄	—	19	2	no	—
271	C₂×C₄	—	19	1	no	—
272	C₂×C₄	—	19	1	no	—
273	C₂×C₄	—	38	1	no	—
274	C₂×C₄	—	38	1	no	—
275	C₂×C₄	—	38	1	no	—
276	C₂×C₄	—	38	1	no	—
277	C₂×C₄	—	152	1	no	—
278	C₂×C₄	—	152	1	no	—
279	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
280	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
281	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
282	C₈	—	19	2	no	—
283	C₈	—	19	2	no	—
284	C₈	Q₈×D₁₉	1	2	yes	—
285	C₈	Q₈×D₁₉	1	2	yes	—
286	C₂²	—	19	2	no	—
287	C₂²	—	38	1	no	—
288	C₂²	—	38	1	no	—
289	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	4	1	no	—
293	C₄	—	4	1	no	—
294	C₄	—	4	1	no	—
295	C₄	—	4	1	no	—
296	C₄	—	19	2	no	—
297	C₄	—	38	1	no	—
298	C₄	—	38	1	no	—
299	C₄	—	76	1	no	—
300	C₄	—	76	1	no	—
301	C₄	—	76	1	no	—
302	C₄	—	76	1	no	—
303	C₄	C₂×Q₈×D₁₉	1	1	yes	characteristic
304	C₄	C₂×Q₈×D₁₉	1	1	yes	characteristic
305	C₂	—	38	2	no	—
306	C₂	C₄⋊Q₈×D₁₉	1	1	yes	characteristic
307	C₂	#11927353	1	1	yes	characteristic
308	C₂	#11927354	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₉
commutator, cosocle	C₂×C₄×C₁₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉
Fitting	C₁₉×#501	C₂
socle	C₂²×C₁₉	C₂²×D₄
prime radical	C₂²×D₁₉	C₂×D₄
hypercenter	#501	D₁₉
hypocenter	C₁₉	#3163
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #501

Complements

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

N	H ≅ G/N	type
3. #11926953	C₂	semidirect
4. #11926952	C₂	semidirect
5. #11926951	C₂	semidirect
6. #11926945	C₂	semidirect
7. #11926943	C₂	semidirect
8. #11926941	C₂	semidirect
9. #11926940	C₂	semidirect
15. C₁₉×#501	C₂	semidirect
102. #64515	Q₈	semidirect
103. C₈×D₁₉	Q₈	semidirect
156. #501	D₁₉	semidirect
213. C₁₉	#3163	semidirect

Maximal subgroups

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

subgroup	index
#11927174	2
#11927171	2
#11926953	2
#11926952	2
#11926951	2
#11926945	2
#11926943	2
#11926941	2
#11926940	2
#11926585	2
#11926583	2
#11926581	2
#106×D₁₉	2
#11926501	2
C₁₉×#501	2
#3163	19