Subgroups — #14573744

Order 2432.

392 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927078	C₂	1	1	yes	maximal, characteristic, complemented
2	#11926885	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926868	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926851	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926844	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926766	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926759	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926706	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926689	C₂	1	1	yes	maximal, characteristic
10	#11926654	C₂	1	1	yes	maximal, characteristic, complemented
11	#11926649	C₂	1	1	yes	maximal, characteristic, complemented
12	#11926644	C₂	1	1	yes	maximal, characteristic, complemented
13	#11926642	C₂	1	1	yes	maximal, characteristic, complemented
14	#11926601	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×#41×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#41×C₁₉	C₂²	1	1	yes	characteristic
17	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₁₉	—	2	1	no	—
20	C₄×D₄×C₁₉	—	2	1	no	—
21	C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
22	C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
23	C₁₉×#117	—	2	1	no	—
24	C₁₉×#117	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₁₉	—	2	1	no	—
26	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
27	C₂×#64528	C₂²	1	1	yes	characteristic
28	C₂×#64527	C₂²	1	1	yes	characteristic
29	C₂×#64526	C₂²	1	1	yes	characteristic
30	C₂×#64525	C₂²	1	1	yes	characteristic
31	#10597376	C₂²	1	1	yes	characteristic
32	#10597375	—	2	1	no	—
33	C₄×C₂²⋊D₁₉	—	2	1	no	—
34	#10597370	C₂²	1	1	yes	characteristic
35	#10597322	—	2	1	no	—
36	#10597321	C₂²	1	1	yes	characteristic
37	C₄×D₇₆	C₂²	1	1	yes	characteristic
38	#10597318	—	2	1	no	—
39	#10597316	C₂²	1	1	yes	characteristic
40	C₄×Dic₃₈	C₂²	1	1	yes	characteristic
41	#10597292	C₂²	1	1	yes	characteristic
42	#10597292	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	#10597267	C₂²	1	1	yes	characteristic
48	#10597267	C₂²	1	1	yes	characteristic
49	#10597266	C₂²	1	1	yes	characteristic
50	#10597266	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	#10597260	C₂²	1	1	yes	characteristic
59	#41×C₁₉	—	2	2	no	—
60	#41×C₁₉	—	2	2	no	—
61	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₁₉	—	2	1	no	—
63	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	—	2	1	no	—
65	C₂²×C₄×C₁₉	—	2	1	no	—
66	C₂²×C₄×C₁₉	—	2	1	no	—
67	C₂²×C₄×C₁₉	D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
70	C₂×C₂²⋊D₁₉	—	2	1	no	—
71	C₂×D₇₆	C₂³	1	1	yes	characteristic
72	C₂×C₄×D₁₉	—	2	1	no	—
73	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₁₉	—	2	1	no	—
75	C₄⋊C₄×C₁₉	—	2	1	no	—
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	—	2	1	no	—
80	C₂²⋊C₄×C₁₉	—	2	1	no	—
81	C₄²×C₁₉	—	2	1	no	—
82	C₄²×C₁₉	—	2	1	no	—
83	C₄²×C₁₉	—	2	1	no	—
84	C₄²×C₁₉	D₄	1	1	yes	characteristic
85	C₄²×C₁₉	D₄	1	1	yes	characteristic
86	C₄²×C₁₉	C₂³	1	1	yes	characteristic
87	C₄²×C₁₉	C₂³	1	1	yes	characteristic
88	C₂²⋊Dic₁₉	—	2	1	no	—
89	#64528	—	2	2	no	—
90	#64527	—	2	2	no	—
91	#64526	—	2	2	no	—
92	#64525	—	2	2	no	—
93	(C₂×D₁₉)⋊C₄	—	2	1	no	—
94	(C₂×D₁₉)⋊C₄	—	2	1	no	—
95	(C₂×D₁₉)⋊C₄	—	2	1	no	—
96	(C₂×D₁₉)⋊C₄	—	2	1	no	—
97	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
98	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
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	C₄×Dic₁₉	—	2	1	no	—
104	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
105	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
106	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
107	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
108	C₂³×C₁₉	—	2	1	no	—
109	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
110	C₂²×D₁₉	—	2	1	no	—
111	Q₈×C₁₉	—	2	1	no	—
112	Q₈×C₁₉	#41	1	2	yes	—
113	D₄×C₁₉	—	2	4	no	—
114	D₄×C₁₉	—	2	1	no	—
115	D₄×C₁₉	#41	1	2	yes	—
116	C₂×C₄×C₁₉	—	2	4	no	—
117	C₂×C₄×C₁₉	—	2	1	no	—
118	C₂×C₄×C₁₉	—	2	1	no	—
119	C₂×C₄×C₁₉	—	2	1	no	—
120	C₂×C₄×C₁₉	—	2	1	no	—
121	C₂×C₄×C₁₉	—	2	1	no	—
122	C₂×C₄×C₁₉	—	4	1	no	—
123	C₂×C₄×C₁₉	—	4	1	no	—
124	C₂×C₄×C₁₉	—	4	2	no	—
125	C₂×C₄×C₁₉	—	4	2	no	—
126	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
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	D₇₆	—	2	1	no	—
137	D₇₆	—	2	1	no	—
138	C₄×D₁₉	—	4	2	no	—
139	Dic₃₈	—	2	1	no	—
140	Dic₃₈	—	2	1	no	—
141	C₁₉⋊C₈	—	2	1	no	—
142	C₁₉⋊C₈	—	2	1	no	—
143	C₁₉⋊C₈	—	2	1	no	—
144	C₁₉⋊C₈	—	2	1	no	—
145	#3111	—	19	1	no	maximal, Sylow
146	C₂²×C₁₉	—	2	1	no	—
147	C₂²×C₁₉	—	2	2	no	—
148	C₂²×C₁₉	—	2	1	no	—
149	C₂²×C₁₉	—	2	1	no	—
150	C₂²×C₁₉	—	4	1	no	—
151	C₂²×C₁₉	—	4	1	no	—
152	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
153	C₂×D₁₉	—	4	1	no	—
154	C₂×D₁₉	—	4	1	no	—
155	C₂×D₁₉	—	4	1	no	—
156	C₄×C₁₉	—	2	1	no	—
157	C₄×C₁₉	—	2	2	no	—
158	C₄×C₁₉	—	2	1	no	—
159	C₄×C₁₉	—	2	1	no	—
160	C₄×C₁₉	—	2	1	no	—
161	C₄×C₁₉	—	2	1	no	—
162	C₄×C₁₉	—	4	1	no	—
163	C₄×C₁₉	—	4	1	no	—
164	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
165	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
166	Dic₁₉	—	4	1	no	—
167	Dic₁₉	—	4	1	no	—
168	Dic₁₉	—	4	1	no	—
169	#529	—	19	1	no	—
170	C₄×#41	D₁₉	1	1	yes	characteristic, complemented
171	#485	—	19	1	no	—
172	#482	—	19	1	no	—
173	#479	—	19	1	no	—
174	#478	—	19	1	no	—
175	#464	—	19	1	no	—
176	#463	—	19	1	no	—
177	#452	—	19	1	no	—
178	#449	—	19	1	no	—
179	#442	—	19	1	no	—
180	#441	—	19	1	no	—
181	#440	—	19	1	no	—
182	#440	—	19	1	no	—
183	#432	—	19	1	no	—
184	C₂×C₁₉	—	4	1	no	—
185	C₂×C₁₉	—	4	2	no	—
186	C₂×C₁₉	#525	1	1	yes	characteristic
187	C₂×C₁₉	#575	1	1	yes	characteristic
188	C₂×C₁₉	#575	1	1	yes	characteristic
189	D₁₉	—	8	1	no	—
190	C₂×#41	C₂×D₁₉	1	1	yes	characteristic
191	C₂×Q₁₆	—	19	1	no	—
192	C₂×SD₁₆	—	19	1	no	—
193	C₂×SD₁₆	—	19	1	no	—
194	C₂×D₈	—	19	1	no	—
195	#126	—	38	1	no	—
196	#125	—	19	1	no	—
197	#124	—	19	1	no	—
198	#123	—	38	1	no	—
199	C₂²⋊Q₈	—	19	1	no	—
200	#121	—	19	1	no	—
201	C₄×Q₈	—	19	1	no	—
202	C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
203	C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
204	C₄×D₄	—	2	1	no	—
205	C₄×D₄	—	2	1	no	—
206	C₄×D₄	—	19	1	no	—
207	C₄×D₄	—	38	1	no	—
208	C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
209	C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
210	#117	—	2	1	no	—
211	#117	—	38	1	no	—
212	#117	C₂×D₁₉	1	1	yes	characteristic
213	C₂×C₄²	—	2	1	no	—
214	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
215	#107	—	19	1	no	—
216	#107	—	19	1	no	—
217	#106	—	19	1	no	—
218	#106	—	19	1	no	—
219	C₄⋊C₈	—	19	1	no	—
220	C₄⋊C₈	—	19	1	no	—
221	#103	—	19	1	no	—
222	#103	—	19	1	no	—
223	#103	—	19	1	no	—
224	#103	—	19	1	no	—
225	#102	—	19	1	no	—
226	#102	—	19	1	no	—
227	#102	—	19	1	no	—
228	#102	—	19	1	no	—
229	C₂²⋊C₈	—	19	1	no	—
230	C₂²⋊C₈	—	19	1	no	—
231	#97	—	19	1	no	—
232	#97	—	19	1	no	—
233	C₁₉	#3111	1	1	yes	characteristic, complemented, Sylow
234	#41	—	2	2	no	—
235	#41	—	2	2	no	—
236	C₂×Q₈	—	19	1	no	—
237	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	19	1	no	—
240	C₂×D₄	—	38	1	no	—
241	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	38	1	no	—
246	C₂²×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
247	C₂²×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
248	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
249	Q₁₆	—	38	2	no	—
250	SD₁₆	—	38	2	no	—
251	SD₁₆	—	38	2	no	—
252	D₈	—	38	2	no	—
253	C₂×C₈	—	19	1	no	—
254	C₂×C₈	—	19	1	no	—
255	C₂×C₈	—	19	1	no	—
256	C₂×C₈	—	19	1	no	—
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	19	1	no	—
260	C₄⋊C₄	—	19	1	no	—
261	C₄⋊C₄	—	38	1	no	—
262	C₄⋊C₄	—	38	1	no	—
263	C₄⋊C₄	—	38	1	no	—
264	C₄⋊C₄	—	38	1	no	—
265	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
266	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	2	1	no	—
269	C₂²⋊C₄	—	2	1	no	—
270	C₂²⋊C₄	—	38	1	no	—
271	C₂²⋊C₄	—	38	1	no	—
272	C₂²⋊C₄	—	38	1	no	—
273	C₂²⋊C₄	—	38	1	no	—
274	C₂²⋊C₄	—	38	1	no	—
275	C₄²	—	2	1	no	—
276	C₄²	—	2	1	no	—
277	C₄²	—	2	1	no	—
278	C₄²	—	38	1	no	—
279	C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
280	C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
281	C₄²	C₂²×D₁₉	1	1	yes	characteristic
282	C₄²	C₂²×D₁₉	1	1	yes	characteristic
283	C₂³	—	2	1	no	—
284	C₂³	—	38	1	no	—
285	C₂³	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
286	Q₈	—	2	1	no	—
287	Q₈	—	38	1	no	—
288	Q₈	—	38	1	no	—
289	Q₈	#64541	1	2	yes	—
290	D₄	—	2	4	no	—
291	D₄	—	2	1	no	—
292	D₄	—	38	1	no	—
293	D₄	—	38	1	no	—
294	D₄	—	76	2	no	—
295	D₄	#64541	1	2	yes	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	38	1	no	—
307	C₂×C₄	—	38	1	no	—
308	C₂×C₄	—	38	1	no	—
309	C₂×C₄	—	76	2	no	—
310	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
311	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
312	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
313	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
314	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
315	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
316	C₈	—	38	1	no	—
317	C₈	—	38	1	no	—
318	C₈	—	38	1	no	—
319	C₈	—	38	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	76	1	no	—
327	C₂²	—	76	1	no	—
328	C₂²	—	76	1	no	—
329	C₂²	C₂²×C₂²⋊D₁₉	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	76	1	no	—
339	C₄	—	76	1	no	—
340	C₄	—	76	1	no	—
341	C₄	C₂×#64541	1	1	yes	characteristic
342	C₄	C₂×#64541	1	1	yes	characteristic
343	C₂	—	4	1	no	—
344	C₂	—	4	2	no	—
345	C₂	—	152	1	no	—
346	C₂	#11927055	1	1	yes	characteristic
347	C₂	#11927342	1	1	yes	characteristic
348	C₂	#11927342	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₉
commutator, cosocle	C₂×C₄×C₁₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉
Fitting	C₄×#41×C₁₉	C₂
socle	C₂²×C₁₉	C₂²×D₄
prime radical	#11926689	C₂
hypercenter	C₄×#41	D₁₉
hypocenter	C₁₉	#3111
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #11927078	C₂	semidirect
2. #11926885	C₂	semidirect
3. #11926868	C₂	semidirect
4. #11926851	C₂	semidirect
5. #11926844	C₂	semidirect
6. #11926766	C₂	semidirect
7. #11926759	C₂	semidirect
8. #11926706	C₂	semidirect
10. #11926654	C₂	semidirect
11. #11926649	C₂	semidirect
12. #11926644	C₂	semidirect
13. #11926642	C₂	semidirect
14. #11926601	C₂	semidirect
15. C₄×#41×C₁₉	C₂	semidirect
170. C₄×#41	D₁₉	semidirect
233. C₁₉	#3111	semidirect

Maximal subgroups

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

subgroup	index
#11927078	2
#11926885	2
#11926868	2
#11926851	2
#11926844	2
#11926766	2
#11926759	2
#11926706	2
#11926689	2
#11926654	2
#11926649	2
#11926644	2
#11926642	2
#11926601	2
C₄×#41×C₁₉	2
#3111	19