Subgroups — #14576403

Order 2432.

344 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927279	C₂	1	1	yes	maximal, characteristic
2	#11926910	C₂	1	1	yes	maximal, characteristic
3	#11926906	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926901	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926848	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926841	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926831	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926766	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926766	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926744	C₂	1	1	yes	maximal, characteristic
11	#11926675	C₂	1	1	yes	maximal, characteristic, complemented
12	#11926649	C₂	1	1	yes	maximal, characteristic, complemented
13	#11926642	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#64527	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₉×#557	C₂	1	1	yes	maximal, characteristic, complemented
16	C₁₉×C₄⋊Q₈	—	2	1	no	—
17	C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₁₉×#125	—	2	1	no	—
19	C₁₉×#125	—	2	1	no	—
20	C₁₉×#125	C₂²	1	1	yes	characteristic
21	C₁₉×#125	C₂²	1	1	yes	characteristic
22	C₄×Q₈×C₁₉	—	2	1	no	—
23	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
24	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
25	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
26	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
27	#10597393	—	2	1	no	—
28	C₂×#64528	C₂²	1	1	yes	characteristic
29	C₂×#64527	C₂²	1	1	yes	characteristic
30	C₂×#64527	C₂²	1	1	yes	characteristic
31	#10597340	—	2	1	no	—
32	#10597340	—	2	1	no	—
33	#10597339	—	2	1	no	—
34	#10597321	C₂²	1	1	yes	characteristic
35	#10597321	C₂²	1	1	yes	characteristic
36	#10597320	C₂²	1	1	yes	characteristic
37	C₄×D₇₆	C₂²	1	1	yes	characteristic
38	C₄×D₇₆	C₂²	1	1	yes	characteristic
39	C₄×Dic₃₈	C₂²	1	1	yes	characteristic
40	#10597292	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	#10597266	C₂²	1	1	yes	characteristic
45	#10597266	C₂²	1	1	yes	characteristic
46	#10597266	C₂²	1	1	yes	characteristic
47	#10597266	C₂²	1	1	yes	characteristic
48	#10597266	C₂²	1	1	yes	characteristic
49	#10597266	C₂²	1	1	yes	characteristic
50	#10597265	C₂²	1	1	yes	characteristic
51	#10597265	C₂²	1	1	yes	characteristic
52	#10597264	C₂²	1	1	yes	characteristic
53	C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
54	C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
55	C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
56	#10597260	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	D₄	1	1	yes	characteristic
61	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
62	C₂×D₇₆	—	2	1	no	—
63	C₂×D₇₆	—	2	1	no	—
64	C₂×D₇₆	C₂³	1	1	yes	characteristic
65	C₂×D₇₆	C₂³	1	1	yes	characteristic
66	C₂×C₄×D₁₉	—	2	1	no	—
67	C₂×C₄×D₁₉	—	2	1	no	—
68	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
73	C₄⋊C₄×C₁₉	—	2	1	no	—
74	C₄⋊C₄×C₁₉	—	2	1	no	—
75	C₄⋊C₄×C₁₉	—	2	1	no	—
76	C₄⋊C₄×C₁₉	—	2	1	no	—
77	C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
78	C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
79	C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
80	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
83	C₄²×C₁₉	—	2	1	no	—
84	C₄²×C₁₉	—	2	1	no	—
85	C₄²×C₁₉	—	2	1	no	—
86	C₄²×C₁₉	C₂³	1	1	yes	characteristic
87	C₄²×C₁₉	C₂³	1	1	yes	characteristic
88	C₄²×C₁₉	C₂³	1	1	yes	characteristic
89	#64528	—	2	2	no	—
90	#64527	—	2	2	no	—
91	#64527	—	2	2	no	—
92	(C₂×D₁₉)⋊C₄	—	2	1	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	(C₂×D₁₉)⋊C₄	—	2	1	no	—
98	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
99	Dic₁₉⋊C₄	—	2	1	no	—
100	C₄×Dic₁₉	—	2	1	no	—
101	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
102	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
103	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
104	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
105	C₂²×D₁₉	—	2	1	no	—
106	C₂²×D₁₉	—	2	1	no	—
107	Q₈×C₁₉	—	2	1	no	—
108	Q₈×C₁₉	—	4	1	no	—
109	Q₈×C₁₉	—	4	2	no	—
110	Q₈×C₁₉	#41	1	2	yes	—
111	C₂×C₄×C₁₉	—	2	1	no	—
112	C₂×C₄×C₁₉	—	2	1	no	—
113	C₂×C₄×C₁₉	—	2	1	no	—
114	C₂×C₄×C₁₉	—	2	1	no	—
115	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
122	C₂×Dic₁₉	—	2	1	no	—
123	C₂×Dic₁₉	—	2	1	no	—
124	D₇₆	—	2	1	no	—
125	D₇₆	—	2	1	no	—
126	D₇₆	—	2	1	no	—
127	D₇₆	—	2	1	no	—
128	D₇₆	—	4	2	no	—
129	D₇₆	—	8	1	no	—
130	C₄×D₁₉	—	8	1	no	—
131	C₄×D₁₉	—	8	1	no	—
132	Dic₃₈	—	2	1	no	—
133	Dic₃₈	—	2	1	no	—
134	C₁₉⋊C₈	—	2	1	no	—
135	C₁₉⋊C₈	—	2	1	no	—
136	C₁₉⋊C₈	—	2	1	no	—
137	C₁₉⋊C₈	—	2	1	no	—
138	#3371	—	19	1	no	maximal, Sylow
139	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
140	C₂×D₁₉	—	4	1	no	—
141	C₂×D₁₉	—	4	1	no	—
142	C₂×D₁₉	—	4	1	no	—
143	C₂×D₁₉	—	4	1	no	—
144	C₂×D₁₉	—	4	1	no	—
145	C₂×D₁₉	—	4	1	no	—
146	C₄×C₁₉	—	2	2	no	—
147	C₄×C₁₉	—	2	1	no	—
148	C₄×C₁₉	—	2	1	no	—
149	C₄×C₁₉	—	2	1	no	—
150	C₄×C₁₉	—	2	1	no	—
151	C₄×C₁₉	—	2	1	no	—
152	C₄×C₁₉	—	2	1	no	—
153	C₄×C₁₉	—	4	1	no	—
154	C₄×C₁₉	—	4	1	no	—
155	C₄×C₁₉	—	4	1	no	—
156	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
157	C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
158	Dic₁₉	—	4	1	no	—
159	Dic₁₉	—	4	1	no	—
160	#559	—	19	1	no	—
161	#557	D₁₉	1	1	yes	characteristic, complemented
162	#490	—	19	1	no	—
163	#489	—	19	1	no	—
164	#488	—	19	1	no	—
165	#479	—	19	1	no	—
166	#478	—	19	1	no	—
167	#476	—	19	1	no	—
168	#464	—	19	1	no	—
169	#464	—	19	1	no	—
170	#460	—	19	1	no	—
171	#446	—	19	1	no	—
172	#441	—	19	1	no	—
173	#440	—	19	1	no	—
174	C₄×SD₁₆	—	19	1	no	—
175	C₂×C₁₉	#548	1	1	yes	characteristic
176	C₂×C₁₉	#575	1	1	yes	characteristic
177	C₂×C₁₉	#577	1	1	yes	characteristic
178	D₁₉	—	8	1	no	—
179	D₁₉	—	8	1	no	—
180	C₂×Q₁₆	—	19	1	no	—
181	C₂×SD₁₆	—	19	1	no	—
182	C₂×SD₁₆	—	19	1	no	—
183	C₄⋊Q₈	—	2	1	no	—
184	C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
185	#127	—	19	1	no	—
186	#125	—	2	1	no	—
187	#125	—	2	1	no	—
188	#125	C₂×D₁₉	1	1	yes	characteristic
189	#125	C₂×D₁₉	1	1	yes	characteristic
190	#124	—	19	1	no	—
191	#124	—	19	1	no	—
192	#124	—	38	1	no	—
193	#123	—	38	1	no	—
194	#123	—	38	1	no	—
195	C₄×Q₈	—	2	1	no	—
196	C₄×Q₈	—	19	1	no	—
197	C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
198	C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
199	C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
200	C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
201	C₄×D₄	—	19	1	no	—
202	C₄×D₄	—	19	1	no	—
203	C₄×D₄	—	38	1	no	—
204	#107	—	19	1	no	—
205	#106	—	19	1	no	—
206	#106	—	19	1	no	—
207	C₄⋊C₈	—	19	1	no	—
208	C₄⋊C₈	—	19	1	no	—
209	C₄⋊C₈	—	19	1	no	—
210	#103	—	19	1	no	—
211	#103	—	19	1	no	—
212	#103	—	19	1	no	—
213	#103	—	19	1	no	—
214	#102	—	19	1	no	—
215	#102	—	19	1	no	—
216	#102	—	19	1	no	—
217	#102	—	19	1	no	—
218	#102	—	19	1	no	—
219	#102	—	19	1	no	—
220	#97	—	19	1	no	—
221	#97	—	19	1	no	—
222	C₄×C₈	—	19	1	no	—
223	C₁₉	#3371	1	1	yes	characteristic, complemented, Sylow
224	C₂×Q₈	—	2	1	no	—
225	C₂×Q₈	—	19	1	no	—
226	C₂×Q₈	C₂²⋊D₁₉	1	1	yes	characteristic
227	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
228	C₂×D₄	—	19	1	no	—
229	C₂×D₄	—	19	1	no	—
230	C₂×D₄	—	38	1	no	—
231	C₂×D₄	—	38	1	no	—
232	C₂²×C₄	—	38	1	no	—
233	C₂²×C₄	—	38	1	no	—
234	Q₁₆	—	38	2	no	—
235	SD₁₆	—	38	2	no	—
236	SD₁₆	—	38	2	no	—
237	C₂×C₈	—	19	1	no	—
238	C₂×C₈	—	19	1	no	—
239	C₂×C₈	—	19	1	no	—
240	C₂×C₈	—	19	1	no	—
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	2	1	no	—
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	19	1	no	—
250	C₄⋊C₄	—	38	1	no	—
251	C₄⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
252	C₄⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
253	C₄⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
254	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
255	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
257	C₂²⋊C₄	—	38	1	no	—
258	C₂²⋊C₄	—	38	1	no	—
259	C₂²⋊C₄	—	38	1	no	—
260	C₂²⋊C₄	—	38	1	no	—
261	C₂²⋊C₄	—	38	1	no	—
262	C₂²⋊C₄	—	38	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	2	1	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	38	1	no	—
267	C₄²	C₂²×D₁₉	1	1	yes	characteristic
268	C₄²	C₂²×D₁₉	1	1	yes	characteristic
269	C₄²	C₂²×D₁₉	1	1	yes	characteristic
270	C₂³	—	38	1	no	—
271	C₂³	—	38	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	4	1	no	—
274	Q₈	—	4	2	no	—
275	Q₈	—	38	1	no	—
276	Q₈	—	38	1	no	—
277	Q₈	#64545	1	2	yes	—
278	D₄	—	38	1	no	—
279	D₄	—	38	1	no	—
280	D₄	—	38	1	no	—
281	D₄	—	38	1	no	—
282	D₄	—	76	2	no	—
283	D₄	—	152	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	38	1	no	—
289	C₂×C₄	—	38	1	no	—
290	C₂×C₄	—	152	1	no	—
291	C₂×C₄	—	152	1	no	—
292	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
293	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
294	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
295	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
296	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
297	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
298	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
299	C₈	—	38	1	no	—
300	C₈	—	38	1	no	—
301	C₈	—	38	1	no	—
302	C₈	—	38	1	no	—
303	C₂²	—	76	1	no	—
304	C₂²	—	76	1	no	—
305	C₂²	—	76	1	no	—
306	C₂²	—	76	1	no	—
307	C₂²	—	76	1	no	—
308	C₂²	—	76	1	no	—
309	C₂²	C₂²×C₂²⋊D₁₉	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	2	no	—
317	C₄	—	4	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	76	1	no	—
321	C₄	—	76	1	no	—
322	C₄	C₂×#64545	1	1	yes	characteristic
323	C₄	#10597433	1	1	yes	characteristic
324	C₂	—	152	1	no	—
325	C₂	—	152	1	no	—
326	C₂	#11927205	1	1	yes	characteristic
327	C₂	#11927342	1	1	yes	characteristic
328	C₂	#11927352	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×#557	C₂
socle	C₂²×C₁₉	C₂²×D₄
prime radical	#10597320	C₂²
hypercenter	#557	D₁₉
hypocenter	C₁₉	#3371
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #557

Complements

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

N	H ≅ G/N	type
3. #11926906	C₂	semidirect
4. #11926901	C₂	semidirect
5. #11926848	C₂	semidirect
6. #11926841	C₂	semidirect
7. #11926831	C₂	semidirect
8. #11926766	C₂	semidirect
9. #11926766	C₂	semidirect
11. #11926675	C₂	semidirect
12. #11926649	C₂	semidirect
13. #11926642	C₂	semidirect
14. C₄×#64527	C₂	semidirect
15. C₁₉×#557	C₂	semidirect
161. #557	D₁₉	semidirect
223. C₁₉	#3371	semidirect

Maximal subgroups

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

subgroup	index
#11927279	2
#11926910	2
#11926906	2
#11926901	2
#11926848	2
#11926841	2
#11926831	2
#11926766	2
#11926744	2
#11926675	2
#11926649	2
#11926642	2
C₄×#64527	2
C₁₉×#557	2
#3371	19