Subgroups — #14574714

Order 2432.

364 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927154	C₂	1	1	yes	maximal, characteristic, complemented
2	#11927118	C₂	1	1	yes	maximal, characteristic
3	#11926937	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926922	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926911	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926900	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926769	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926768	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926753	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926752	C₂	1	1	yes	maximal, characteristic, complemented
11	#11926708	C₂	1	1	yes	maximal, characteristic
12	#11926704	C₂	1	1	yes	maximal, characteristic
13	#11926700	C₂	1	1	yes	maximal, characteristic, complemented
14	#11926602	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₉×#451	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Q₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₂×#64541	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic
19	C₂×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic
20	C₁₉×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
21	#10597393	C₂²	1	1	yes	characteristic
22	#10597393	C₂²	1	1	yes	characteristic
23	#10597392	—	2	1	no	—
24	#10597392	—	2	1	no	—
25	Q₈×Dic₁₉	C₂²	1	1	yes	characteristic
26	#10597390	C₂²	1	1	yes	characteristic
27	#10597390	C₂²	1	1	yes	characteristic
28	#10597390	C₂²	1	1	yes	characteristic
29	C₂×#64528	C₂²	1	1	yes	characteristic
30	C₂×#64528	C₂²	1	1	yes	characteristic
31	C₂×#64528	C₂²	1	1	yes	characteristic
32	C₂×#64527	C₂²	1	1	yes	characteristic
33	#10597375	—	2	1	no	—
34	#10597375	—	2	1	no	—
35	#10597372	C₂²	1	1	yes	characteristic
36	#10597370	C₂²	1	1	yes	characteristic
37	C₂×Dic₇₆	C₂²	1	1	yes	characteristic
38	C₂×#64516	C₂²	1	1	yes	characteristic
39	#10597336	—	2	1	no	—
40	#10597336	C₂²	1	1	yes	characteristic
41	#10597335	—	2	1	no	—
42	#10597333	C₂²	1	1	yes	characteristic
43	#10597325	—	2	1	no	—
44	#10597325	—	2	1	no	—
45	C₁₉×#103	C₂²	1	1	yes	characteristic
46	C₁₉×#103	C₂²	1	1	yes	characteristic
47	C₁₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
48	#10597292	C₂²	1	1	yes	characteristic
49	#10597292	C₂²	1	1	yes	characteristic
50	C₂²⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
51	#10597278	C₂²	1	1	yes	characteristic
52	#10597273	C₂²	1	1	yes	characteristic
53	#10597272	C₂²	1	1	yes	characteristic
54	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
55	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
56	C₈×Dic₁₉	C₂²	1	1	yes	characteristic
57	#10597267	C₂²	1	1	yes	characteristic
58	#10597266	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₉	—	2	1	no	—
60	C₂×Q₈×C₁₉	—	4	1	no	—
61	C₂×Q₈×C₁₉	—	4	1	no	—
62	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
63	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
64	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	—	2	1	no	—
66	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₁₉	—	2	1	no	—
68	#64541	—	4	1	no	—
69	#64541	—	4	1	no	—
70	C₂×D₇₆	C₂³	1	1	yes	characteristic
71	C₂×C₄×D₁₉	—	2	1	no	—
72	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
73	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
74	Q₁₆×C₁₉	—	4	1	no	—
75	Q₁₆×C₁₉	—	4	1	no	—
76	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₁₉	—	2	1	no	—
79	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
80	C₂²⋊C₄×C₁₉	—	2	1	no	—
81	C₂²⋊Dic₁₉	—	2	1	no	—
82	C₂²⋊Dic₁₉	D₄	1	1	yes	characteristic
83	C₂²⋊Dic₁₉	D₄	1	1	yes	characteristic
84	#64528	—	4	1	no	—
85	#64528	—	4	1	no	—
86	#64528	—	4	1	no	—
87	#64527	—	4	1	no	—
88	(C₂×D₁₉)⋊C₄	—	2	1	no	—
89	(C₂×D₁₉)⋊C₄	—	2	1	no	—
90	(C₂×D₁₉)⋊C₄	—	2	1	no	—
91	(C₂×D₁₉)⋊C₄	—	2	1	no	—
92	C₄⋊Dic₁₉	—	2	1	no	—
93	C₄⋊Dic₁₉	—	2	1	no	—
94	C₄⋊Dic₁₉	D₄	1	1	yes	characteristic
95	C₄⋊Dic₁₉	D₄	1	1	yes	characteristic
96	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
97	Dic₁₉⋊C₄	—	2	1	no	—
98	Dic₁₉⋊C₄	—	2	1	no	—
99	Dic₁₉⋊C₄	—	2	1	no	—
100	Dic₁₉⋊C₄	—	2	1	no	—
101	Dic₁₉⋊C₄	—	2	1	no	—
102	Dic₁₉⋊C₄	—	2	1	no	—
103	Dic₁₉⋊C₄	—	2	1	no	—
104	Dic₁₉⋊C₄	—	2	1	no	—
105	Dic₁₉⋊C₄	—	2	1	no	—
106	C₄×Dic₁₉	—	2	1	no	—
107	C₄×Dic₁₉	—	2	1	no	—
108	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
109	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
110	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
111	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
112	Dic₇₆	—	4	1	no	—
113	#64516	—	4	1	no	—
114	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
115	C₂²×D₁₉	—	2	1	no	—
116	Q₈×C₁₉	—	2	1	no	—
117	Q₈×C₁₉	—	2	1	no	—
118	Q₈×C₁₉	—	2	1	no	—
119	Q₈×C₁₉	—	2	1	no	—
120	Q₈×C₁₉	—	2	1	no	—
121	Q₈×C₁₉	—	2	1	no	—
122	Q₈×C₁₉	—	4	2	no	—
123	C₂×C₄×C₁₉	—	2	1	no	—
124	C₂×C₄×C₁₉	—	2	1	no	—
125	C₂×C₄×C₁₉	—	2	1	no	—
126	C₂×C₄×C₁₉	—	2	1	no	—
127	C₂×C₄×C₁₉	—	2	1	no	—
128	C₂×C₄×C₁₉	—	2	1	no	—
129	C₂×C₄×C₁₉	—	4	2	no	—
130	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
132	C₂²⋊D₁₉	—	4	2	no	—
133	C₂×Dic₁₉	—	2	1	no	—
134	C₂×Dic₁₉	—	2	1	no	—
135	C₂×Dic₁₉	—	2	1	no	—
136	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
137	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
138	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
139	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
140	D₇₆	—	2	1	no	—
141	D₇₆	—	2	1	no	—
142	C₄×D₁₉	—	4	2	no	—
143	Dic₃₈	—	2	1	no	—
144	Dic₃₈	—	2	1	no	—
145	Dic₃₈	—	2	1	no	—
146	Dic₃₈	—	2	1	no	—
147	C₈×C₁₉	—	2	1	no	—
148	C₈×C₁₉	—	2	1	no	—
149	C₁₉⋊C₈	—	2	1	no	—
150	C₁₉⋊C₈	—	2	1	no	—
151	#3223	—	19	1	no	maximal, Sylow
152	C₂²×C₁₉	—	2	1	no	—
153	C₂²×C₁₉	—	2	1	no	—
154	C₂²×C₁₉	—	2	1	no	—
155	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
156	C₂×D₁₉	—	4	1	no	—
157	C₂×D₁₉	—	4	1	no	—
158	C₂×D₁₉	—	4	1	no	—
159	C₄×C₁₉	—	2	1	no	—
160	C₄×C₁₉	—	4	1	no	—
161	C₄×C₁₉	—	4	1	no	—
162	C₄×C₁₉	—	4	1	no	—
163	C₄×C₁₉	—	4	1	no	—
164	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
165	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
166	Dic₁₉	—	2	1	no	—
167	Dic₁₉	—	2	1	no	—
168	Dic₁₉	—	2	1	no	—
169	Dic₁₉	—	2	1	no	—
170	Dic₁₉	—	4	1	no	—
171	Dic₁₉	—	4	1	no	—
172	Dic₁₉	—	4	1	no	—
173	#541	—	19	1	no	—
174	#536	—	19	1	no	—
175	#497	—	19	1	no	—
176	#494	—	19	1	no	—
177	#491	—	19	1	no	—
178	#488	—	19	1	no	—
179	#464	—	19	1	no	—
180	#464	—	19	1	no	—
181	#462	—	19	1	no	—
182	#462	—	19	1	no	—
183	#452	—	19	1	no	—
184	#452	—	19	1	no	—
185	#451	—	19	1	no	—
186	#451	D₁₉	1	1	yes	characteristic, complemented
187	#433	—	19	1	no	—
188	C₂×C₁₉	—	4	1	no	—
189	C₂×C₁₉	#535	1	1	yes	characteristic
190	C₂×C₁₉	#578	1	1	yes	characteristic
191	C₂×C₁₉	#578	1	1	yes	characteristic
192	D₁₉	—	8	1	no	—
193	C₂×#41	—	19	1	no	—
194	C₂²×Q₈	C₂×D₁₉	1	1	yes	characteristic
195	C₂×Q₁₆	—	19	1	no	—
196	C₂×Q₁₆	—	19	1	no	—
197	C₂×Q₁₆	—	19	1	no	—
198	C₂×Q₁₆	—	19	1	no	—
199	C₂×Q₁₆	C₂×D₁₉	1	1	yes	characteristic
200	C₂×Q₁₆	C₂×D₁₉	1	1	yes	characteristic
201	C₂×SD₁₆	—	19	1	no	—
202	C₂×SD₁₆	—	19	1	no	—
203	C₄⋊Q₈	—	19	1	no	—
204	C₄⋊Q₈	—	19	1	no	—
205	C₄⋊Q₈	—	19	1	no	—
206	#126	—	38	1	no	—
207	#126	—	38	1	no	—
208	#125	—	19	1	no	—
209	#125	—	38	1	no	—
210	#125	—	38	1	no	—
211	#124	—	19	1	no	—
212	#124	—	19	1	no	—
213	#123	—	38	1	no	—
214	#123	—	38	1	no	—
215	C₂²⋊Q₈	—	19	1	no	—
216	C₂²⋊Q₈	—	38	1	no	—
217	C₂²⋊Q₈	—	38	1	no	—
218	C₂²⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
219	C₄×Q₈	—	19	1	no	—
220	C₄×Q₈	—	19	1	no	—
221	#117	—	19	1	no	—
222	C₄⋊C₈	—	19	1	no	—
223	C₄⋊C₈	—	19	1	no	—
224	#103	—	19	1	no	—
225	#103	—	19	1	no	—
226	#103	—	19	1	no	—
227	#103	—	19	1	no	—
228	#103	C₂×D₁₉	1	1	yes	characteristic
229	#103	C₂×D₁₉	1	1	yes	characteristic
230	#102	—	19	1	no	—
231	#102	—	19	1	no	—
232	C₂²⋊C₈	—	19	1	no	—
233	C₂²⋊C₈	C₂×D₁₉	1	1	yes	characteristic
234	#97	—	19	1	no	—
235	C₄×C₈	—	19	1	no	—
236	C₁₉	#3223	1	1	yes	characteristic, complemented, Sylow
237	#41	—	76	1	no	—
238	#41	—	76	1	no	—
239	C₂×Q₈	—	2	1	no	—
240	C₂×Q₈	—	4	1	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×Q₈	—	19	1	no	—
243	C₂×Q₈	—	19	1	no	—
244	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
245	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
246	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
247	C₂×D₄	—	19	1	no	—
248	C₂×D₄	—	38	1	no	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	38	1	no	—
251	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
252	Q₁₆	—	4	1	no	—
253	Q₁₆	—	4	1	no	—
254	Q₁₆	—	76	1	no	—
255	Q₁₆	—	76	1	no	—
256	Q₁₆	—	76	1	no	—
257	Q₁₆	—	76	1	no	—
258	SD₁₆	—	76	1	no	—
259	SD₁₆	—	76	1	no	—
260	C₂×C₈	—	19	1	no	—
261	C₂×C₈	—	19	1	no	—
262	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
263	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	19	1	no	—
266	C₄⋊C₄	—	19	1	no	—
267	C₄⋊C₄	—	19	1	no	—
268	C₄⋊C₄	—	38	1	no	—
269	C₄⋊C₄	—	38	1	no	—
270	C₄⋊C₄	—	38	1	no	—
271	C₄⋊C₄	—	38	1	no	—
272	C₄⋊C₄	—	38	1	no	—
273	C₄⋊C₄	—	38	1	no	—
274	C₄⋊C₄	—	38	1	no	—
275	C₄⋊C₄	—	38	1	no	—
276	C₄⋊C₄	—	38	1	no	—
277	C₄⋊C₄	—	38	1	no	—
278	C₄⋊C₄	—	38	1	no	—
279	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
280	C₂²⋊C₄	—	2	1	no	—
281	C₂²⋊C₄	—	19	1	no	—
282	C₂²⋊C₄	—	19	1	no	—
283	C₂²⋊C₄	—	38	1	no	—
284	C₂²⋊C₄	—	38	1	no	—
285	C₂²⋊C₄	—	38	1	no	—
286	C₂²⋊C₄	—	38	1	no	—
287	C₂²⋊C₄	—	38	1	no	—
288	C₄²	—	19	1	no	—
289	C₄²	—	19	1	no	—
290	C₄²	—	38	1	no	—
291	C₄²	—	38	1	no	—
292	C₂³	—	38	1	no	—
293	C₂³	D₄×D₁₉	1	1	yes	characteristic
294	Q₈	—	2	1	no	—
295	Q₈	—	2	1	no	—
296	Q₈	—	2	1	no	—
297	Q₈	—	2	1	no	—
298	Q₈	—	2	1	no	—
299	Q₈	—	2	1	no	—
300	Q₈	—	4	2	no	—
301	Q₈	—	38	1	no	—
302	Q₈	—	38	1	no	—
303	Q₈	—	38	1	no	—
304	Q₈	—	38	1	no	—
305	D₄	—	38	1	no	—
306	D₄	—	38	1	no	—
307	D₄	—	76	2	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	19	1	no	—
316	C₂×C₄	—	19	1	no	—
317	C₂×C₄	—	19	1	no	—
318	C₂×C₄	—	19	1	no	—
319	C₂×C₄	—	38	1	no	—
320	C₂×C₄	—	38	1	no	—
321	C₂×C₄	—	38	1	no	—
322	C₂×C₄	—	76	2	no	—
323	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
324	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
325	C₈	—	2	1	no	—
326	C₈	—	2	1	no	—
327	C₈	—	38	1	no	—
328	C₈	—	38	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	76	1	no	—
333	C₂²	—	76	1	no	—
334	C₂²	—	76	1	no	—
335	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	38	1	no	—
342	C₄	—	38	1	no	—
343	C₄	—	38	1	no	—
344	C₄	—	38	1	no	—
345	C₄	—	76	1	no	—
346	C₄	—	76	1	no	—
347	C₄	—	76	1	no	—
348	C₄	#10597430	1	1	yes	characteristic
349	C₄	#10597430	1	1	yes	characteristic
350	C₂	—	4	1	no	—
351	C₂	—	152	1	no	—
352	C₂	#11927109	1	1	yes	characteristic
353	C₂	#11927357	1	1	yes	characteristic
354	C₂	#11927360	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×#451	C₂
socle	C₂²×C₁₉	C₂²×D₄
prime radical	C₂×#64541	C₂²
hypercenter	#451	D₁₉
hypocenter	C₁₉	#3223
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #451

Complements

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

N	H ≅ G/N	type
1. #11927154	C₂	semidirect
3. #11926937	C₂	semidirect
4. #11926922	C₂	semidirect
5. #11926911	C₂	semidirect
6. #11926900	C₂	semidirect
7. #11926769	C₂	semidirect
8. #11926768	C₂	semidirect
9. #11926753	C₂	semidirect
10. #11926752	C₂	semidirect
13. #11926700	C₂	semidirect
14. #11926602	C₂	semidirect
15. C₁₉×#451	C₂	semidirect
186. #451	D₁₉	semidirect
236. C₁₉	#3223	semidirect

Maximal subgroups

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

subgroup	index
#11927154	2
#11927118	2
#11926937	2
#11926922	2
#11926911	2
#11926900	2
#11926769	2
#11926768	2
#11926753	2
#11926752	2
#11926708	2
#11926704	2
#11926700	2
#11926602	2
C₁₉×#451	2
#3223	19