Subgroups — #14574804

Order 2432.

334 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	#11927079	C₂	1	1	yes	maximal, characteristic
2	#125×D₁₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926943	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926901	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926899	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926881	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926876	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926864	C₂	1	1	yes	maximal, characteristic
9	#11926859	C₂	1	1	yes	maximal, characteristic
10	#11926854	C₂	1	1	yes	maximal, characteristic, complemented
11	#11926850	C₂	1	1	yes	maximal, characteristic, complemented
12	#11926847	C₂	1	1	yes	maximal, characteristic, complemented
13	#11926843	C₂	1	1	yes	maximal, characteristic, complemented
14	#11926595	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₉×#499	C₂	1	1	yes	maximal, characteristic, complemented
16	C₁₉×#125	C₂²	1	1	yes	characteristic
17	C₁₉×#125	C₂²	1	1	yes	characteristic
18	#10597344	—	2	1	no	—
19	#10597343	C₂²	1	1	yes	characteristic
20	#10597341	C₂²	1	1	yes	characteristic
21	#10597340	—	2	1	no	—
22	#10597339	—	2	1	no	—
23	#10597339	C₂²	1	1	yes	characteristic
24	#10597338	—	2	1	no	—
25	C₄⋊C₄×D₁₉	—	2	1	no	—
26	C₄⋊C₄×D₁₉	—	2	1	no	—
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	#10597333	C₂²	1	1	yes	characteristic
34	#10597321	C₂²	1	1	yes	characteristic
35	C₄²×D₁₉	C₂²	1	1	yes	characteristic
36	#10597316	C₂²	1	1	yes	characteristic
37	C₁₉×#107	C₂²	1	1	yes	characteristic
38	C₁₉×#107	C₂²	1	1	yes	characteristic
39	C₁₉×#106	C₂²	1	1	yes	characteristic
40	C₁₉×#106	C₂²	1	1	yes	characteristic
41	C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
42	#10597278	C₂²	1	1	yes	characteristic
43	#10597278	C₂²	1	1	yes	characteristic
44	(C₂×D₁₉)⋊C₈	C₂²	1	1	yes	characteristic
45	(C₂×D₁₉)⋊C₈	C₂²	1	1	yes	characteristic
46	#10597273	C₂²	1	1	yes	characteristic
47	#10597273	C₂²	1	1	yes	characteristic
48	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
49	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
50	#10597267	C₂²	1	1	yes	characteristic
51	#10597267	C₂²	1	1	yes	characteristic
52	#10597266	C₂²	1	1	yes	characteristic
53	#10597266	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×D₇₆	—	2	1	no	—
60	C₂×D₇₆	C₂³	1	1	yes	characteristic
61	C₂×C₄×D₁₉	—	2	1	no	—
62	C₂×C₄×D₁₉	—	2	1	no	—
63	C₂×C₄×D₁₉	—	2	1	no	—
64	C₂×C₄×D₁₉	D₄	1	1	yes	characteristic
65	C₂×C₄×D₁₉	D₄	1	1	yes	characteristic
66	C₂×C₄×D₁₉	C₂³	1	1	yes	characteristic
67	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₁₉	—	2	1	no	—
71	C₄⋊C₄×C₁₉	—	2	1	no	—
72	C₄⋊C₄×C₁₉	—	2	1	no	—
73	C₄⋊C₄×C₁₉	—	2	1	no	—
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	C₄²×C₁₉	C₂³	1	1	yes	characteristic
79	(C₂×D₁₉)⋊C₄	—	2	1	no	—
80	(C₂×D₁₉)⋊C₄	—	2	1	no	—
81	(C₂×D₁₉)⋊C₄	—	2	1	no	—
82	(C₂×D₁₉)⋊C₄	—	2	1	no	—
83	(C₂×D₁₉)⋊C₄	—	2	1	no	—
84	C₄⋊Dic₁₉	—	2	1	no	—
85	C₄⋊Dic₁₉	—	2	1	no	—
86	C₄⋊Dic₁₉	—	2	1	no	—
87	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
88	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
96	C₄×Dic₁₉	—	2	1	no	—
97	C₄×Dic₁₉	D₄	1	1	yes	characteristic
98	C₄×Dic₁₉	D₄	1	1	yes	characteristic
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	C₂²×D₁₉	—	2	1	no	—
103	C₂²×D₁₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₉	—	2	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	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
111	C₂×Dic₁₉	—	2	1	no	—
112	C₂×Dic₁₉	—	2	1	no	—
113	C₂×Dic₁₉	—	2	1	no	—
114	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
115	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
116	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
117	D₇₆	—	2	1	no	—
118	D₇₆	—	2	1	no	—
119	D₇₆	—	4	2	no	—
120	C₄×D₁₉	—	2	2	no	—
121	C₄×D₁₉	—	2	1	no	—
122	C₄×D₁₉	—	2	1	no	—
123	C₄×D₁₉	—	4	1	no	—
124	C₄×D₁₉	—	4	2	no	—
125	C₄×D₁₉	—	4	2	no	—
126	C₄×D₁₉	—	4	2	no	—
127	Dic₃₈	—	2	1	no	—
128	Dic₃₈	—	2	1	no	—
129	C₈×C₁₉	—	2	1	no	—
130	C₈×C₁₉	—	2	1	no	—
131	C₁₉⋊C₈	—	2	1	no	—
132	C₁₉⋊C₈	—	2	1	no	—
133	#3232	—	19	1	no	maximal, Sylow
134	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
135	C₂×D₁₉	—	2	1	no	—
136	C₂×D₁₉	—	2	1	no	—
137	C₂×D₁₉	—	4	1	no	—
138	C₂×D₁₉	—	4	1	no	—
139	C₂×D₁₉	—	4	1	no	—
140	C₂×D₁₉	#135	1	2	yes	—
141	C₄×C₁₉	—	2	1	no	—
142	C₄×C₁₉	—	2	1	no	—
143	C₄×C₁₉	—	4	1	no	—
144	C₄×C₁₉	—	4	1	no	—
145	C₄×C₁₉	—	4	1	no	—
146	C₄×C₁₉	—	4	1	no	—
147	C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
148	C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
149	Dic₁₉	—	2	1	no	—
150	Dic₁₉	—	2	1	no	—
151	Dic₁₉	—	2	1	no	—
152	Dic₁₉	—	4	1	no	—
153	Dic₁₉	—	4	1	no	—
154	Dic₁₉	—	4	1	no	—
155	#529	—	19	1	no	—
156	C₂×#125	—	19	1	no	—
157	#499	—	19	1	no	—
158	#499	D₁₉	1	1	yes	characteristic, complemented
159	#488	—	19	1	no	—
160	#488	—	19	1	no	—
161	#484	—	19	1	no	—
162	#483	—	19	1	no	—
163	#481	—	19	1	no	—
164	#480	—	19	1	no	—
165	#479	—	19	1	no	—
166	#479	—	19	1	no	—
167	#478	—	19	1	no	—
168	#478	—	19	1	no	—
169	#431	—	19	1	no	—
170	C₂×C₁₉	#536	1	1	yes	characteristic
171	C₂×C₁₉	C₂×#135	1	1	yes	characteristic
172	C₂×C₁₉	#575	1	1	yes	characteristic
173	D₁₉	—	2	2	no	—
174	D₁₉	—	8	1	no	—
175	#126	—	38	1	no	—
176	#125	—	19	1	no	—
177	#125	—	19	1	no	—
178	#125	—	19	1	no	—
179	#125	—	38	1	no	—
180	#125	—	38	1	no	—
181	#125	C₂×D₁₉	1	1	yes	characteristic
182	#125	C₂×D₁₉	1	1	yes	characteristic
183	#124	—	19	1	no	—
184	#123	—	38	1	no	—
185	C₂²⋊Q₈	—	19	1	no	—
186	#121	—	19	1	no	—
187	C₄×Q₈	—	19	1	no	—
188	C₄×D₄	—	19	1	no	—
189	C₄×D₄	—	38	1	no	—
190	#117	—	38	1	no	—
191	C₂×C₄⋊C₄	—	19	1	no	—
192	C₂×C₄⋊C₄	—	19	1	no	—
193	C₂×C₄⋊C₄	—	38	1	no	—
194	C₂×C₄⋊C₄	—	38	1	no	—
195	C₂×C₄²	—	19	1	no	—
196	#107	—	19	1	no	—
197	#107	—	19	1	no	—
198	#107	C₂×D₁₉	1	1	yes	characteristic
199	#107	C₂×D₁₉	1	1	yes	characteristic
200	#106	—	19	1	no	—
201	#106	—	19	1	no	—
202	#106	C₂×D₁₉	1	1	yes	characteristic
203	#106	C₂×D₁₉	1	1	yes	characteristic
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	C₄×C₈	—	19	1	no	—
217	C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
218	C₁₉	#3232	1	1	yes	characteristic, complemented, Sylow
219	C₂×Q₈	—	19	1	no	—
220	C₂×D₄	—	19	1	no	—
221	C₂×D₄	—	38	1	no	—
222	C₂²×C₄	—	19	1	no	—
223	C₂²×C₄	—	19	1	no	—
224	C₂²×C₄	—	19	1	no	—
225	C₂²×C₄	—	38	1	no	—
226	C₂²×C₄	—	38	1	no	—
227	C₂²×C₄	—	38	1	no	—
228	C₂×C₈	—	19	1	no	—
229	C₂×C₈	—	19	1	no	—
230	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
231	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	19	1	no	—
237	C₄⋊C₄	—	19	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	—	38	1	no	—
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	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
252	C₂²⋊C₄	—	38	1	no	—
253	C₂²⋊C₄	—	38	1	no	—
254	C₂²⋊C₄	—	38	1	no	—
255	C₂²⋊C₄	—	38	1	no	—
256	C₂²⋊C₄	—	38	1	no	—
257	C₄²	—	19	1	no	—
258	C₄²	—	19	1	no	—
259	C₄²	—	19	1	no	—
260	C₄²	—	38	1	no	—
261	C₄²	C₂²×D₁₉	1	1	yes	characteristic
262	C₂³	—	19	1	no	—
263	C₂³	—	38	1	no	—
264	Q₈	—	38	1	no	—
265	Q₈	—	38	1	no	—
266	D₄	—	38	1	no	—
267	D₄	—	38	1	no	—
268	D₄	—	76	2	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	19	1	no	—
274	C₂×C₄	—	19	1	no	—
275	C₂×C₄	—	19	1	no	—
276	C₂×C₄	—	38	2	no	—
277	C₂×C₄	—	38	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	—	76	1	no	—
283	C₂×C₄	—	76	2	no	—
284	C₂×C₄	—	76	2	no	—
285	C₂×C₄	—	76	2	no	—
286	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
287	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
288	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
289	C₈	—	2	1	no	—
290	C₈	—	2	1	no	—
291	C₈	—	38	1	no	—
292	C₈	—	38	1	no	—
293	C₂²	—	19	2	no	—
294	C₂²	—	38	1	no	—
295	C₂²	—	38	1	no	—
296	C₂²	—	76	1	no	—
297	C₂²	—	76	1	no	—
298	C₂²	—	76	1	no	—
299	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	38	1	no	—
307	C₄	—	38	1	no	—
308	C₄	—	38	1	no	—
309	C₄	—	76	1	no	—
310	C₄	—	76	1	no	—
311	C₄	—	76	1	no	—
312	C₄	#10597433	1	1	yes	characteristic
313	C₄	#10597433	1	1	yes	characteristic
314	C₂	—	38	2	no	—
315	C₂	—	152	1	no	—
316	C₂	#11927119	1	1	yes	characteristic
317	C₂	#135×D₁₉	1	1	yes	characteristic
318	C₂	#11927341	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×#499	C₂
socle	C₂²×C₁₉	C₂²×D₄
prime radical	#10597341	C₂²
hypercenter	#499	D₁₉
hypocenter	C₁₉	#3232
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #499

Complements

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

N	H ≅ G/N	type
2. #125×D₁₉	C₂	semidirect
3. #11926943	C₂	semidirect
4. #11926901	C₂	semidirect
5. #11926899	C₂	semidirect
6. #11926881	C₂	semidirect
7. #11926876	C₂	semidirect
10. #11926854	C₂	semidirect
11. #11926850	C₂	semidirect
12. #11926847	C₂	semidirect
13. #11926843	C₂	semidirect
14. #11926595	C₂	semidirect
15. C₁₉×#499	C₂	semidirect
158. #499	D₁₉	semidirect
218. C₁₉	#3232	semidirect

Maximal subgroups

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

subgroup	index
#11927079	2
#125×D₁₉	2
#11926943	2
#11926901	2
#11926899	2
#11926881	2
#11926876	2
#11926864	2
#11926859	2
#11926854	2
#11926850	2
#11926847	2
#11926843	2
#11926595	2
C₁₉×#499	2
#3232	19