Subgroups — #14574903

Order 2432.

308 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927156	C₂	1	1	yes	maximal, characteristic
2	#11926950	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926943	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926914	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926896	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926885	C₂	1	1	yes	maximal, characteristic
7	#11926885	C₂	1	1	yes	maximal, characteristic
8	#11926879	C₂	1	1	yes	maximal, characteristic
9	#11926874	C₂	1	1	yes	maximal, characteristic
10	#11926848	C₂	1	1	yes	maximal, characteristic, complemented
11	#11926848	C₂	1	1	yes	maximal, characteristic, complemented
12	#11926836	C₂	1	1	yes	maximal, characteristic, complemented
13	#11926826	C₂	1	1	yes	maximal, characteristic, complemented
14	#11926605	C₂	1	1	yes	maximal, characteristic
15	C₁₉×#544	C₂	1	1	yes	maximal, characteristic
16	C₁₉×C₄⋊Q₈	—	2	1	no	—
17	C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₁₉×#125	—	2	1	no	—
20	C₁₉×#125	C₂²	1	1	yes	characteristic
21	C₁₉×#125	C₂²	1	1	yes	characteristic
22	C₁₉×C₂²⋊Q₈	—	2	1	no	—
23	C₁₉×C₂²⋊Q₈	—	2	1	no	—
24	C₁₉×#117	C₂²	1	1	yes	characteristic
25	C₁₉×#117	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
27	#10597370	C₂²	1	1	yes	characteristic
28	#10597370	C₂²	1	1	yes	characteristic
29	#10597335	—	2	1	no	—
30	#10597335	—	2	1	no	—
31	#10597325	—	2	1	no	—
32	#10597325	—	2	1	no	—
33	#10597316	C₂²	1	1	yes	characteristic
34	#10597315	C₂²	1	1	yes	characteristic
35	C₄×Dic₃₈	C₂²	1	1	yes	characteristic
36	C₄×Dic₃₈	C₂²	1	1	yes	characteristic
37	C₂²⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
38	C₂²⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
39	#10597267	C₂²	1	1	yes	characteristic
40	#10597267	C₂²	1	1	yes	characteristic
41	#10597267	C₂²	1	1	yes	characteristic
42	#10597267	C₂²	1	1	yes	characteristic
43	#10597267	C₂²	1	1	yes	characteristic
44	#10597267	C₂²	1	1	yes	characteristic
45	#10597267	C₂²	1	1	yes	characteristic
46	#10597267	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	#10597264	C₂²	1	1	yes	characteristic
52	#10597264	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	—	2	1	no	—
61	C₂²×C₄×C₁₉	—	2	1	no	—
62	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
63	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
64	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₁₉	—	2	1	no	—
66	C₄⋊C₄×C₁₉	—	2	1	no	—
67	C₄⋊C₄×C₁₉	—	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	—	2	1	no	—
72	C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
73	C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
74	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
78	C₂²⋊C₄×C₁₉	—	2	1	no	—
79	C₂²⋊C₄×C₁₉	D₄	1	1	yes	characteristic
80	C₂²⋊C₄×C₁₉	D₄	1	1	yes	characteristic
81	C₄²×C₁₉	—	2	1	no	—
82	C₄²×C₁₉	C₂³	1	1	yes	characteristic
83	C₄²×C₁₉	C₂³	1	1	yes	characteristic
84	C₂²⋊Dic₁₉	—	2	1	no	—
85	C₂²⋊Dic₁₉	—	2	1	no	—
86	C₄⋊Dic₁₉	—	2	1	no	—
87	C₄⋊Dic₁₉	—	2	1	no	—
88	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
89	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
96	C₄×Dic₁₉	—	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	C₂×D₄	1	1	yes	characteristic
103	Q₈×C₁₉	—	8	1	no	—
104	Q₈×C₁₉	—	8	1	no	—
105	C₂×C₄×C₁₉	—	2	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	—	2	1	no	—
111	C₂×C₄×C₁₉	—	8	1	no	—
112	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
118	C₂×Dic₁₉	—	2	1	no	—
119	C₂×Dic₁₉	—	2	1	no	—
120	C₂×Dic₁₉	—	2	1	no	—
121	C₂×Dic₁₉	—	2	1	no	—
122	Dic₃₈	—	2	1	no	—
123	Dic₃₈	—	2	1	no	—
124	Dic₃₈	—	2	1	no	—
125	Dic₃₈	—	2	1	no	—
126	C₁₉⋊C₈	—	2	1	no	—
127	C₁₉⋊C₈	—	2	1	no	—
128	C₁₉⋊C₈	—	2	1	no	—
129	C₁₉⋊C₈	—	2	1	no	—
130	#3242	—	19	1	no	maximal, Sylow
131	C₂²×C₁₉	—	2	1	no	—
132	C₂²×C₁₉	—	2	1	no	—
133	C₂²×C₁₉	—	2	1	no	—
134	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
135	C₄×C₁₉	—	2	1	no	—
136	C₄×C₁₉	—	2	1	no	—
137	C₄×C₁₉	—	2	1	no	—
138	C₄×C₁₉	—	2	1	no	—
139	C₄×C₁₉	—	2	1	no	—
140	C₄×C₁₉	—	4	1	no	—
141	C₄×C₁₉	—	4	1	no	—
142	C₄×C₁₉	—	4	1	no	—
143	C₄×C₁₉	—	4	1	no	—
144	C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
145	C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
146	Dic₁₉	—	4	1	no	—
147	Dic₁₉	—	4	1	no	—
148	Dic₁₉	—	4	1	no	—
149	Dic₁₉	—	4	1	no	—
150	#544	D₁₉	1	1	yes	characteristic
151	#541	—	19	1	no	—
152	#501	—	19	1	no	—
153	#499	—	19	1	no	—
154	#491	—	19	1	no	—
155	#487	—	19	1	no	—
156	#485	—	19	1	no	—
157	#485	—	19	1	no	—
158	#484	—	19	1	no	—
159	#483	—	19	1	no	—
160	#479	—	19	1	no	—
161	#479	—	19	1	no	—
162	#477	—	19	1	no	—
163	#475	—	19	1	no	—
164	#433	—	19	1	no	—
165	C₂×C₁₉	—	4	1	no	—
166	C₂×C₁₉	#536	1	1	yes	characteristic
167	C₂×C₁₉	#577	1	1	yes	characteristic
168	C₂×C₁₉	#578	1	1	yes	characteristic
169	C₄⋊Q₈	—	2	1	no	—
170	C₄⋊Q₈	—	19	1	no	—
171	C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
172	C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
173	#126	—	38	1	no	—
174	#126	—	38	1	no	—
175	#125	—	2	1	no	—
176	#125	—	19	1	no	—
177	#125	—	38	1	no	—
178	#125	—	38	1	no	—
179	#125	C₂×D₁₉	1	1	yes	characteristic
180	#125	C₂×D₁₉	1	1	yes	characteristic
181	C₂²⋊Q₈	—	2	1	no	—
182	C₂²⋊Q₈	—	2	1	no	—
183	C₂²⋊Q₈	—	19	1	no	—
184	C₂²⋊Q₈	—	19	1	no	—
185	C₄×Q₈	—	19	1	no	—
186	C₄×Q₈	—	19	1	no	—
187	#117	C₂×D₁₉	1	1	yes	characteristic
188	#117	C₂×D₁₉	1	1	yes	characteristic
189	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
190	#107	—	19	1	no	—
191	#107	—	19	1	no	—
192	#107	—	19	1	no	—
193	#107	—	19	1	no	—
194	#106	—	19	1	no	—
195	#106	—	19	1	no	—
196	#106	—	19	1	no	—
197	#106	—	19	1	no	—
198	C₄⋊C₈	—	19	1	no	—
199	C₄⋊C₈	—	19	1	no	—
200	#103	—	19	1	no	—
201	#103	—	19	1	no	—
202	#103	—	19	1	no	—
203	#103	—	19	1	no	—
204	#103	—	19	1	no	—
205	#103	—	19	1	no	—
206	#103	—	19	1	no	—
207	#103	—	19	1	no	—
208	C₂²⋊C₈	—	19	1	no	—
209	C₂²⋊C₈	—	19	1	no	—
210	#97	—	19	1	no	—
211	C₄×C₈	—	19	1	no	—
212	C₁₉	#3242	1	1	yes	characteristic, complemented, Sylow
213	C₂×Q₈	—	2	1	no	—
214	C₂×Q₈	—	2	1	no	—
215	C₂×Q₈	—	19	1	no	—
216	C₂×Q₈	—	19	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
219	C₂×C₈	—	19	1	no	—
220	C₂×C₈	—	19	1	no	—
221	C₂×C₈	—	19	1	no	—
222	C₂×C₈	—	19	1	no	—
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	—	2	1	no	—
225	C₄⋊C₄	—	2	1	no	—
226	C₄⋊C₄	—	2	1	no	—
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	—	38	1	no	—
233	C₄⋊C₄	—	38	1	no	—
234	C₄⋊C₄	—	38	1	no	—
235	C₄⋊C₄	—	38	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	C₂²⋊D₁₉	1	1	yes	characteristic
241	C₄⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
242	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
243	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
244	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
245	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
246	C₂²⋊C₄	—	2	1	no	—
247	C₂²⋊C₄	—	38	1	no	—
248	C₂²⋊C₄	—	38	1	no	—
249	C₂²⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
250	C₂²⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
251	C₄²	—	2	1	no	—
252	C₄²	—	38	1	no	—
253	C₄²	—	38	1	no	—
254	C₄²	C₂²×D₁₉	1	1	yes	characteristic
255	C₄²	C₂²×D₁₉	1	1	yes	characteristic
256	C₂³	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
257	Q₈	—	8	1	no	—
258	Q₈	—	8	1	no	—
259	Q₈	—	38	1	no	—
260	Q₈	—	38	1	no	—
261	Q₈	—	38	1	no	—
262	Q₈	—	38	1	no	—
263	C₂×C₄	—	2	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	—	2	1	no	—
269	C₂×C₄	—	8	1	no	—
270	C₂×C₄	—	38	1	no	—
271	C₂×C₄	—	38	1	no	—
272	C₂×C₄	—	38	1	no	—
273	C₂×C₄	—	38	1	no	—
274	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
275	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
276	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
277	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
278	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
279	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
280	C₈	—	38	1	no	—
281	C₈	—	38	1	no	—
282	C₈	—	38	1	no	—
283	C₈	—	38	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	C₂²×C₂²⋊D₁₉	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	4	1	no	—
294	C₄	—	4	1	no	—
295	C₄	—	4	1	no	—
296	C₄	—	4	1	no	—
297	C₄	—	76	1	no	—
298	C₄	—	76	1	no	—
299	C₄	—	76	1	no	—
300	C₄	—	76	1	no	—
301	C₄	#10597433	1	1	yes	characteristic
302	C₄	#10597433	1	1	yes	characteristic
303	C₂	—	4	1	no	—
304	C₂	#11927118	1	1	yes	characteristic
305	C₂	#11927355	1	1	yes	characteristic
306	C₂	#11927359	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×#544	C₂
socle	C₂²×C₁₉	C₂²×D₄
prime radical	C₂³×C₁₉	C₂×D₄
hypercenter	#544	D₁₉
hypocenter	C₁₉	#3242
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #544

Complements

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

N	H ≅ G/N	type
2. #11926950	C₂	semidirect
3. #11926943	C₂	semidirect
4. #11926914	C₂	semidirect
5. #11926896	C₂	semidirect
10. #11926848	C₂	semidirect
11. #11926848	C₂	semidirect
12. #11926836	C₂	semidirect
13. #11926826	C₂	semidirect
212. C₁₉	#3242	semidirect

Maximal subgroups

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

subgroup	index
#11927156	2
#11926950	2
#11926943	2
#11926914	2
#11926896	2
#11926885	2
#11926879	2
#11926874	2
#11926848	2
#11926836	2
#11926826	2
#11926605	2
C₁₉×#544	2
#3242	19