Subgroups — #14574927

Order 2432.

320 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927172	C₂	1	1	yes	maximal, characteristic, complemented
2	#11926950	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926940	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926903	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926898	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926882	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926872	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926865	C₂	1	1	yes	maximal, characteristic
9	#11926860	C₂	1	1	yes	maximal, characteristic
10	#11926851	C₂	1	1	yes	maximal, characteristic, complemented
11	#11926844	C₂	1	1	yes	maximal, characteristic, complemented
12	#11926834	C₂	1	1	yes	maximal, characteristic, complemented
13	#11926829	C₂	1	1	yes	maximal, characteristic, complemented
14	#11926605	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₉×#542	C₂	1	1	yes	maximal, characteristic
16	C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₁₉×#126	—	2	1	no	—
18	C₁₉×#124	—	2	1	no	—
19	C₁₉×#124	C₂²	1	1	yes	characteristic
20	C₁₉×#123	—	2	1	no	—
21	C₁₉×C₂²⋊Q₈	—	2	1	no	—
22	C₁₉×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
23	C₁₉×#121	C₂²	1	1	yes	characteristic
24	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
26	C₁₉×#117	C₂²	1	1	yes	characteristic
27	#10597372	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
29	#10597336	—	2	1	no	—
30	#10597334	—	2	1	no	—
31	#10597324	—	2	1	no	—
32	#10597324	—	2	1	no	—
33	#10597316	C₂²	1	1	yes	characteristic
34	#10597316	C₂²	1	1	yes	characteristic
35	#10597315	C₂²	1	1	yes	characteristic
36	#10597315	C₂²	1	1	yes	characteristic
37	#10597292	C₂²	1	1	yes	characteristic
38	#10597292	C₂²	1	1	yes	characteristic
39	#10597292	C₂²	1	1	yes	characteristic
40	#10597292	C₂²	1	1	yes	characteristic
41	#10597289	C₂²	1	1	yes	characteristic
42	#10597289	C₂²	1	1	yes	characteristic
43	#10597289	C₂²	1	1	yes	characteristic
44	#10597289	C₂²	1	1	yes	characteristic
45	C₂²⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
46	C₂²⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
47	#10597265	C₂²	1	1	yes	characteristic
48	#10597265	C₂²	1	1	yes	characteristic
49	#10597265	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	#10597264	C₂²	1	1	yes	characteristic
54	#10597264	C₂²	1	1	yes	characteristic
55	C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
56	C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
57	#10597260	C₂²	1	1	yes	characteristic
58	C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₉	—	2	1	no	—
60	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₁₉	—	2	1	no	—
62	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	—	2	1	no	—
64	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
65	C₂²×Dic₁₉	—	2	1	no	—
66	C₂×Dic₃₈	—	2	1	no	—
67	C₂×Dic₃₈	—	2	1	no	—
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	D₄	1	1	yes	characteristic
72	C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
73	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
75	C₂²⋊C₄×C₁₉	—	2	1	no	—
76	C₂²⋊C₄×C₁₉	—	2	1	no	—
77	C₂²⋊C₄×C₁₉	—	2	1	no	—
78	C₂²⋊C₄×C₁₉	—	2	1	no	—
79	C₂²⋊C₄×C₁₉	—	2	1	no	—
80	C₂²⋊C₄×C₁₉	D₄	1	1	yes	characteristic
81	C₂²⋊C₄×C₁₉	D₄	1	1	yes	characteristic
82	C₄²×C₁₉	—	2	1	no	—
83	C₄²×C₁₉	C₂³	1	1	yes	characteristic
84	C₄²×C₁₉	C₂³	1	1	yes	characteristic
85	C₂²⋊Dic₁₉	—	2	1	no	—
86	C₄⋊Dic₁₉	—	2	1	no	—
87	C₄⋊Dic₁₉	—	2	1	no	—
88	C₄⋊Dic₁₉	—	2	1	no	—
89	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
90	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
91	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
92	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
93	Dic₁₉⋊C₄	—	2	1	no	—
94	Dic₁₉⋊C₄	—	2	1	no	—
95	Dic₁₉⋊C₄	—	2	1	no	—
96	Dic₁₉⋊C₄	—	2	1	no	—
97	C₄×Dic₁₉	—	2	1	no	—
98	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
99	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
100	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
101	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
102	C₂³×C₁₉	—	2	1	no	—
103	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
104	Q₈×C₁₉	—	2	1	no	—
105	Q₈×C₁₉	—	2	1	no	—
106	Q₈×C₁₉	—	8	1	no	—
107	D₄×C₁₉	—	2	1	no	—
108	D₄×C₁₉	—	2	1	no	—
109	D₄×C₁₉	—	8	1	no	—
110	C₂×C₄×C₁₉	—	2	1	no	—
111	C₂×C₄×C₁₉	—	2	1	no	—
112	C₂×C₄×C₁₉	—	2	1	no	—
113	C₂×C₄×C₁₉	—	2	1	no	—
114	C₂×C₄×C₁₉	—	2	1	no	—
115	C₂×C₄×C₁₉	—	8	1	no	—
116	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
122	C₂×Dic₁₉	—	2	1	no	—
123	C₂×Dic₁₉	—	2	1	no	—
124	C₂×Dic₁₉	—	2	1	no	—
125	C₂×Dic₁₉	—	2	1	no	—
126	C₂×Dic₁₉	—	8	1	no	—
127	Dic₃₈	—	8	1	no	—
128	Dic₃₈	—	8	1	no	—
129	C₁₉⋊C₈	—	2	1	no	—
130	C₁₉⋊C₈	—	2	1	no	—
131	C₁₉⋊C₈	—	2	1	no	—
132	C₁₉⋊C₈	—	2	1	no	—
133	#3244	—	19	1	no	maximal, Sylow
134	C₂²×C₁₉	—	2	1	no	—
135	C₂²×C₁₉	—	2	1	no	—
136	C₂²×C₁₉	—	2	1	no	—
137	C₂²×C₁₉	—	4	1	no	—
138	C₂²×C₁₉	—	4	1	no	—
139	C₂²×C₁₉	—	4	1	no	—
140	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
141	C₄×C₁₉	—	2	1	no	—
142	C₄×C₁₉	—	2	1	no	—
143	C₄×C₁₉	—	2	1	no	—
144	C₄×C₁₉	—	2	1	no	—
145	C₄×C₁₉	—	2	1	no	—
146	C₄×C₁₉	—	4	1	no	—
147	C₄×C₁₉	—	4	1	no	—
148	C₄×C₁₉	—	4	1	no	—
149	C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
150	C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
151	Dic₁₉	—	4	1	no	—
152	Dic₁₉	—	4	1	no	—
153	Dic₁₉	—	4	1	no	—
154	Dic₁₉	—	4	1	no	—
155	#544	—	19	1	no	—
156	#542	D₁₉	1	1	yes	characteristic
157	#501	—	19	1	no	—
158	#498	—	19	1	no	—
159	#489	—	19	1	no	—
160	#488	—	19	1	no	—
161	#485	—	19	1	no	—
162	#483	—	19	1	no	—
163	#482	—	19	1	no	—
164	#481	—	19	1	no	—
165	#479	—	19	1	no	—
166	#478	—	19	1	no	—
167	#476	—	19	1	no	—
168	#475	—	19	1	no	—
169	#433	—	19	1	no	—
170	C₂×C₁₉	—	4	1	no	—
171	C₂×C₁₉	—	8	1	no	—
172	C₂×C₁₉	#536	1	1	yes	characteristic
173	C₂×C₁₉	#577	1	1	yes	characteristic
174	C₂×C₁₉	#577	1	1	yes	characteristic
175	C₄⋊Q₈	—	19	1	no	—
176	C₄⋊Q₈	—	19	1	no	—
177	C₄⋊Q₈	—	38	1	no	—
178	C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
179	#126	—	2	1	no	—
180	#125	—	19	1	no	—
181	#125	—	19	1	no	—
182	#125	—	38	1	no	—
183	#124	—	2	1	no	—
184	#124	C₂×D₁₉	1	1	yes	characteristic
185	#123	—	2	1	no	—
186	C₂²⋊Q₈	—	2	1	no	—
187	C₂²⋊Q₈	—	38	1	no	—
188	C₂²⋊Q₈	—	38	1	no	—
189	C₂²⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
190	#121	C₂×D₁₉	1	1	yes	characteristic
191	C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
192	C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
193	#117	—	19	1	no	—
194	#117	C₂×D₁₉	1	1	yes	characteristic
195	C₂×C₄⋊C₄	—	19	1	no	—
196	#107	—	19	1	no	—
197	#107	—	19	1	no	—
198	#106	—	19	1	no	—
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	C₄⋊C₈	—	19	1	no	—
205	C₄⋊C₈	—	19	1	no	—
206	#103	—	19	1	no	—
207	#103	—	19	1	no	—
208	#103	—	19	1	no	—
209	#103	—	19	1	no	—
210	#102	—	19	1	no	—
211	#102	—	19	1	no	—
212	#102	—	19	1	no	—
213	#102	—	19	1	no	—
214	C₂²⋊C₈	—	19	1	no	—
215	C₂²⋊C₈	—	19	1	no	—
216	#97	—	19	1	no	—
217	C₄×C₈	—	19	1	no	—
218	C₁₉	#3244	1	1	yes	characteristic, complemented, Sylow
219	C₂×Q₈	—	2	1	no	—
220	C₂×Q₈	—	38	1	no	—
221	C₂×Q₈	—	38	1	no	—
222	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
223	C₂×D₄	—	2	1	no	—
224	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	38	1	no	—
227	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
228	C₂×C₈	—	19	1	no	—
229	C₂×C₈	—	19	1	no	—
230	C₂×C₈	—	19	1	no	—
231	C₂×C₈	—	19	1	no	—
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	19	1	no	—
236	C₄⋊C₄	—	19	1	no	—
237	C₄⋊C₄	—	19	1	no	—
238	C₄⋊C₄	—	19	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	C₂²⋊D₁₉	1	1	yes	characteristic
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	C₂²⋊C₄	—	2	1	no	—
251	C₂²⋊C₄	—	2	1	no	—
252	C₂²⋊C₄	—	2	1	no	—
253	C₂²⋊C₄	—	2	1	no	—
254	C₂²⋊C₄	—	2	1	no	—
255	C₂²⋊C₄	—	38	1	no	—
256	C₂²⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
257	C₂²⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
258	C₄²	—	2	1	no	—
259	C₄²	—	38	1	no	—
260	C₄²	C₂²×D₁₉	1	1	yes	characteristic
261	C₄²	C₂²×D₁₉	1	1	yes	characteristic
262	C₂³	—	2	1	no	—
263	C₂³	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
264	Q₈	—	2	1	no	—
265	Q₈	—	2	1	no	—
266	Q₈	—	8	1	no	—
267	Q₈	—	152	1	no	—
268	Q₈	—	152	1	no	—
269	D₄	—	2	1	no	—
270	D₄	—	2	1	no	—
271	D₄	—	8	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	8	1	no	—
278	C₂×C₄	—	38	1	no	—
279	C₂×C₄	—	38	1	no	—
280	C₂×C₄	—	38	1	no	—
281	C₂×C₄	—	38	1	no	—
282	C₂×C₄	—	152	1	no	—
283	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
284	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
285	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
286	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
287	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
288	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
289	C₈	—	38	1	no	—
290	C₈	—	38	1	no	—
291	C₈	—	38	1	no	—
292	C₈	—	38	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	—	4	1	no	—
298	C₂²	—	4	1	no	—
299	C₂²	C₂²×C₂²⋊D₁₉	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	76	1	no	—
309	C₄	—	76	1	no	—
310	C₄	—	76	1	no	—
311	C₄	—	76	1	no	—
312	C₄	#10597436	1	1	yes	characteristic
313	C₄	#10597436	1	1	yes	characteristic
314	C₂	—	4	1	no	—
315	C₂	—	8	1	no	—
316	C₂	#11927115	1	1	yes	characteristic
317	C₂	#11927352	1	1	yes	characteristic
318	C₂	#11927355	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₉
commutator, cosocle	C₂×C₄×C₁₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉
Fitting	C₁₉×#542	C₂
socle	C₂²×C₁₉	C₂²×D₄
prime radical	C₁₉×#121	C₂²
hypercenter	#542	D₁₉
hypocenter	C₁₉	#3244
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #542

Complements

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

N	H ≅ G/N	type
1. #11927172	C₂	semidirect
2. #11926950	C₂	semidirect
3. #11926940	C₂	semidirect
4. #11926903	C₂	semidirect
5. #11926898	C₂	semidirect
6. #11926882	C₂	semidirect
7. #11926872	C₂	semidirect
10. #11926851	C₂	semidirect
11. #11926844	C₂	semidirect
12. #11926834	C₂	semidirect
13. #11926829	C₂	semidirect
14. #11926605	C₂	semidirect
218. C₁₉	#3244	semidirect

Maximal subgroups

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

subgroup	index
#11927172	2
#11926950	2
#11926940	2
#11926903	2
#11926898	2
#11926882	2
#11926872	2
#11926865	2
#11926860	2
#11926851	2
#11926844	2
#11926834	2
#11926829	2
#11926605	2
C₁₉×#542	2
#3244	19