Subgroups — #14575103

Order 2432.

346 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927166	C₂	1	1	yes	maximal, characteristic, complemented
2	#11927163	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926835	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926832	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926829	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926827	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926799	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926796	C₂	1	1	yes	maximal, characteristic
9	#11926776	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926770	C₂	1	1	yes	maximal, characteristic, complemented
11	#11926768	C₂	1	1	yes	maximal, characteristic, complemented
12	#11926761	C₂	1	1	yes	maximal, characteristic, complemented
13	#11926756	C₂	1	1	yes	maximal, characteristic, complemented
14	#11926569	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₉×#464	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic
17	C₂×SD₁₆×C₁₉	C₂²	1	1	yes	characteristic
18	C₁₉×#124	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
20	#10597393	C₂²	1	1	yes	characteristic
21	#10597392	C₂²	1	1	yes	characteristic
22	#10597392	C₂²	1	1	yes	characteristic
23	Q₈×Dic₁₉	C₂²	1	1	yes	characteristic
24	C₂×#64527	C₂²	1	1	yes	characteristic
25	#10597384	C₂²	1	1	yes	characteristic
26	D₄×Dic₁₉	C₂²	1	1	yes	characteristic
27	C₂×#64525	C₂²	1	1	yes	characteristic
28	C₂×#64515	C₂²	1	1	yes	characteristic
29	C₂×C₈×D₁₉	C₂²	1	1	yes	characteristic
30	#10597342	—	2	1	no	—
31	#10597341	C₂²	1	1	yes	characteristic
32	#10597340	—	2	1	no	—
33	#10597334	C₂²	1	1	yes	characteristic
34	#10597332	—	2	1	no	—
35	#10597331	—	2	1	no	—
36	#10597325	—	2	1	no	—
37	#10597324	—	2	1	no	—
38	#10597322	—	2	1	no	—
39	#10597321	—	2	1	no	—
40	C₄×D₇₆	C₂²	1	1	yes	characteristic
41	#10597318	C₂²	1	1	yes	characteristic
42	#10597315	C₂²	1	1	yes	characteristic
43	C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
44	C₁₉×#103	C₂²	1	1	yes	characteristic
45	C₁₉×#102	C₂²	1	1	yes	characteristic
46	#10597292	C₂²	1	1	yes	characteristic
47	#10597292	C₂²	1	1	yes	characteristic
48	#10597292	C₂²	1	1	yes	characteristic
49	#10597289	C₂²	1	1	yes	characteristic
50	#10597278	C₂²	1	1	yes	characteristic
51	#10597278	C₂²	1	1	yes	characteristic
52	#10597274	C₂²	1	1	yes	characteristic
53	#10597274	C₂²	1	1	yes	characteristic
54	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
55	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
56	#10597265	C₂²	1	1	yes	characteristic
57	#10597265	C₂²	1	1	yes	characteristic
58	C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
60	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₉	—	2	1	no	—
63	C₂²×Dic₁₉	—	2	1	no	—
64	C₂×D₇₆	—	2	1	no	—
65	C₂×D₇₆	C₂³	1	1	yes	characteristic
66	C₂×C₄×D₁₉	—	2	1	no	—
67	C₂×C₄×D₁₉	C₂³	1	1	yes	characteristic
68	C₂×Dic₃₈	—	2	1	no	—
69	C₂×Dic₃₈	—	2	1	no	—
70	Q₁₆×C₁₉	—	4	1	no	—
71	SD₁₆×C₁₉	—	4	1	no	—
72	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₁₉	—	2	1	no	—
75	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
76	C₂²⋊C₄×C₁₉	—	2	1	no	—
77	C₂²⋊C₄×C₁₉	—	2	1	no	—
78	C₄²×C₁₉	—	2	1	no	—
79	C₄²×C₁₉	C₂³	1	1	yes	characteristic
80	C₂²⋊Dic₁₉	—	2	1	no	—
81	C₂²⋊Dic₁₉	—	2	1	no	—
82	#64527	—	4	1	no	—
83	#64525	—	4	1	no	—
84	(C₂×D₁₉)⋊C₄	—	2	1	no	—
85	(C₂×D₁₉)⋊C₄	—	2	1	no	—
86	(C₂×D₁₉)⋊C₄	—	2	1	no	—
87	(C₂×D₁₉)⋊C₄	—	2	1	no	—
88	(C₂×D₁₉)⋊C₄	—	2	1	no	—
89	(C₂×D₁₉)⋊C₄	—	2	1	no	—
90	(C₂×D₁₉)⋊C₄	D₄	1	1	yes	characteristic
91	(C₂×D₁₉)⋊C₄	D₄	1	1	yes	characteristic
92	C₄⋊Dic₁₉	—	2	1	no	—
93	C₄⋊Dic₁₉	—	2	1	no	—
94	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
95	C₄⋊Dic₁₉	C₂³	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	D₄	1	1	yes	characteristic
101	Dic₁₉⋊C₄	D₄	1	1	yes	characteristic
102	C₄×Dic₁₉	—	2	1	no	—
103	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
104	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
105	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
106	#64515	—	4	1	no	—
107	C₈×D₁₉	—	4	1	no	—
108	C₂³×C₁₉	—	2	1	no	—
109	C₂²×D₁₉	—	2	1	no	—
110	C₂²×D₁₉	C₂×D₄	1	1	yes	characteristic
111	Q₈×C₁₉	—	2	1	no	—
112	Q₈×C₁₉	—	2	1	no	—
113	Q₈×C₁₉	—	2	1	no	—
114	Q₈×C₁₉	—	2	1	no	—
115	D₄×C₁₉	—	2	1	no	—
116	D₄×C₁₉	—	2	1	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	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
123	C₂²⋊D₁₉	—	8	1	no	—
124	C₂×Dic₁₉	—	2	1	no	—
125	C₂×Dic₁₉	—	2	1	no	—
126	C₂×Dic₁₉	—	2	1	no	—
127	C₂×Dic₁₉	—	8	1	no	—
128	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
131	D₇₆	—	2	1	no	—
132	D₇₆	—	2	1	no	—
133	D₇₆	—	8	1	no	—
134	C₄×D₁₉	—	2	1	no	—
135	C₄×D₁₉	—	2	1	no	—
136	C₄×D₁₉	—	8	1	no	—
137	Dic₃₈	—	8	1	no	—
138	Dic₃₈	—	8	1	no	—
139	C₈×C₁₉	—	2	1	no	—
140	C₈×C₁₉	—	2	1	no	—
141	C₁₉⋊C₈	—	2	1	no	—
142	C₁₉⋊C₈	—	2	1	no	—
143	#3263	—	19	1	no	maximal, Sylow
144	C₂²×C₁₉	—	4	1	no	—
145	C₂²×C₁₉	—	4	1	no	—
146	C₂²×C₁₉	—	4	1	no	—
147	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
148	C₂×D₁₉	—	2	1	no	—
149	C₂×D₁₉	—	2	1	no	—
150	C₂×D₁₉	—	2	1	no	—
151	C₂×D₁₉	—	4	1	no	—
152	C₂×D₁₉	—	4	1	no	—
153	C₂×D₁₉	—	4	1	no	—
154	C₄×C₁₉	—	2	1	no	—
155	C₄×C₁₉	—	2	1	no	—
156	C₄×C₁₉	—	4	1	no	—
157	C₄×C₁₉	—	4	1	no	—
158	C₄×C₁₉	—	4	1	no	—
159	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
160	C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
161	Dic₁₉	—	2	1	no	—
162	Dic₁₉	—	2	1	no	—
163	Dic₁₉	—	2	1	no	—
164	Dic₁₉	—	4	1	no	—
165	Dic₁₉	—	4	1	no	—
166	Dic₁₉	—	4	1	no	—
167	#542	—	19	1	no	—
168	#542	—	19	1	no	—
169	#476	—	19	1	no	—
170	#476	—	19	1	no	—
171	#475	—	19	1	no	—
172	#475	—	19	1	no	—
173	#470	—	19	1	no	—
174	#469	—	19	1	no	—
175	#465	—	19	1	no	—
176	#465	—	19	1	no	—
177	#464	—	19	1	no	—
178	#464	D₁₉	1	1	yes	characteristic, complemented
179	#463	—	19	1	no	—
180	#463	—	19	1	no	—
181	#424	—	19	1	no	—
182	C₂×C₁₉	—	8	1	no	—
183	C₂×C₁₉	#537	1	1	yes	characteristic
184	C₂×C₁₉	#577	1	1	yes	characteristic
185	C₂×C₁₉	#577	1	1	yes	characteristic
186	D₁₉	—	4	1	no	—
187	D₁₉	—	8	1	no	—
188	C₂×Q₁₆	C₂×D₁₉	1	1	yes	characteristic
189	C₂×SD₁₆	—	19	1	no	—
190	C₂×SD₁₆	C₂×D₁₉	1	1	yes	characteristic
191	C₂×D₈	—	19	1	no	—
192	C₂×M₁₆	—	19	1	no	—
193	C₂²×C₈	—	19	1	no	—
194	C₄⋊Q₈	—	19	1	no	—
195	C₄⋊Q₈	—	19	1	no	—
196	#126	—	38	1	no	—
197	#126	—	38	1	no	—
198	#124	—	19	1	no	—
199	#124	—	38	1	no	—
200	#124	—	38	1	no	—
201	#124	C₂×D₁₉	1	1	yes	characteristic
202	#123	—	38	1	no	—
203	#123	—	38	1	no	—
204	C₂²⋊Q₈	—	19	1	no	—
205	C₂²⋊Q₈	—	19	1	no	—
206	C₂²⋊Q₈	—	38	1	no	—
207	C₂²⋊Q₈	—	38	1	no	—
208	#121	—	19	1	no	—
209	#121	—	19	1	no	—
210	C₄×Q₈	—	19	1	no	—
211	C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
212	C₄×D₄	—	19	1	no	—
213	C₄×D₄	—	19	1	no	—
214	#117	—	19	1	no	—
215	#106	—	19	1	no	—
216	#106	—	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	C₄⋊C₈	—	19	1	no	—
222	C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
223	#103	—	19	1	no	—
224	#103	—	19	1	no	—
225	#103	—	19	1	no	—
226	#103	C₂×D₁₉	1	1	yes	characteristic
227	#102	—	19	1	no	—
228	#102	—	19	1	no	—
229	#102	—	19	1	no	—
230	#102	C₂×D₁₉	1	1	yes	characteristic
231	C₁₉	#3263	1	1	yes	characteristic, complemented, Sylow
232	C₂×Q₈	—	38	1	no	—
233	C₂×Q₈	—	38	1	no	—
234	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
235	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
236	C₂×D₄	—	19	1	no	—
237	C₂×D₄	—	38	1	no	—
238	C₂×D₄	—	38	1	no	—
239	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
240	C₂²×C₄	—	19	1	no	—
241	C₂²×C₄	—	38	1	no	—
242	C₂²×C₄	—	38	1	no	—
243	Q₁₆	—	4	1	no	—
244	SD₁₆	—	4	1	no	—
245	SD₁₆	—	76	1	no	—
246	D₈	—	76	1	no	—
247	M₁₆	—	76	1	no	—
248	C₂×C₈	—	19	1	no	—
249	C₂×C₈	—	19	1	no	—
250	C₂×C₈	—	76	1	no	—
251	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
252	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	19	1	no	—
255	C₄⋊C₄	—	19	1	no	—
256	C₄⋊C₄	—	19	1	no	—
257	C₄⋊C₄	—	19	1	no	—
258	C₄⋊C₄	—	19	1	no	—
259	C₄⋊C₄	—	38	1	no	—
260	C₄⋊C₄	—	38	1	no	—
261	C₄⋊C₄	—	38	1	no	—
262	C₄⋊C₄	—	38	1	no	—
263	C₄⋊C₄	—	38	1	no	—
264	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	19	1	no	—
268	C₂²⋊C₄	—	19	1	no	—
269	C₂²⋊C₄	—	38	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	C₂²⋊C₄	—	38	1	no	—
276	C₂²⋊C₄	—	38	1	no	—
277	C₄²	—	2	1	no	—
278	C₄²	—	19	1	no	—
279	C₄²	—	38	1	no	—
280	C₄²	C₂²×D₁₉	1	1	yes	characteristic
281	C₂³	—	2	1	no	—
282	C₂³	—	19	1	no	—
283	C₂³	—	38	1	no	—
284	Q₈	—	2	1	no	—
285	Q₈	—	2	1	no	—
286	Q₈	—	2	1	no	—
287	Q₈	—	2	1	no	—
288	Q₈	—	152	1	no	—
289	Q₈	—	152	1	no	—
290	D₄	—	2	1	no	—
291	D₄	—	2	1	no	—
292	D₄	—	38	1	no	—
293	D₄	—	38	1	no	—
294	D₄	—	152	1	no	—
295	D₄	—	152	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	19	1	no	—
300	C₂×C₄	—	19	1	no	—
301	C₂×C₄	—	19	1	no	—
302	C₂×C₄	—	38	1	no	—
303	C₂×C₄	—	38	1	no	—
304	C₂×C₄	—	38	1	no	—
305	C₂×C₄	—	38	1	no	—
306	C₂×C₄	—	38	1	no	—
307	C₂×C₄	—	152	1	no	—
308	C₂×C₄	—	152	1	no	—
309	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
310	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
311	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	2	1	no	—
314	C₈	—	38	1	no	—
315	C₈	—	38	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	38	1	no	—
320	C₂²	—	38	1	no	—
321	C₂²	—	38	1	no	—
322	C₂²	—	76	1	no	—
323	C₂²	—	76	1	no	—
324	C₂²	—	76	1	no	—
325	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	38	1	no	—
332	C₄	—	38	1	no	—
333	C₄	—	38	1	no	—
334	C₄	—	76	1	no	—
335	C₄	—	76	1	no	—
336	C₄	—	76	1	no	—
337	C₄	#10597430	1	1	yes	characteristic
338	C₄	#10597436	1	1	yes	characteristic
339	C₂	—	8	1	no	—
340	C₂	—	76	1	no	—
341	C₂	—	152	1	no	—
342	C₂	#11927123	1	1	yes	characteristic
343	C₂	#11927351	1	1	yes	characteristic
344	C₂	#11927353	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×#464	C₂
socle	C₂²×C₁₉	C₂²×D₄
prime radical	#11926796	C₂
hypercenter	#464	D₁₉
hypocenter	C₁₉	#3263
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #464

Complements

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

N	H ≅ G/N	type
1. #11927166	C₂	semidirect
2. #11927163	C₂	semidirect
3. #11926835	C₂	semidirect
4. #11926832	C₂	semidirect
5. #11926829	C₂	semidirect
6. #11926827	C₂	semidirect
7. #11926799	C₂	semidirect
9. #11926776	C₂	semidirect
10. #11926770	C₂	semidirect
11. #11926768	C₂	semidirect
12. #11926761	C₂	semidirect
13. #11926756	C₂	semidirect
14. #11926569	C₂	semidirect
15. C₁₉×#464	C₂	semidirect
178. #464	D₁₉	semidirect
231. C₁₉	#3263	semidirect

Maximal subgroups

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

subgroup	index
#11927166	2
#11927163	2
#11926835	2
#11926832	2
#11926829	2
#11926827	2
#11926799	2
#11926796	2
#11926776	2
#11926770	2
#11926768	2
#11926761	2
#11926756	2
#11926569	2
C₁₉×#464	2
#3263	19