Subgroups — #14574830

Order 2432.

366 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927097	C₂	1	1	yes	maximal, characteristic, complemented
2	#11927089	C₂	1	1	yes	maximal, characteristic
3	#11926952	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926951	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926907	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926907	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926869	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926868	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926865	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926833	C₂	1	1	yes	maximal, characteristic, complemented
11	#11926832	C₂	1	1	yes	maximal, characteristic, complemented
12	#11926823	C₂	1	1	yes	maximal, characteristic, complemented
13	#11926822	C₂	1	1	yes	maximal, characteristic, complemented
14	#11926598	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₉×#482	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#64543	—	2	1	no	—
17	C₁₉×#121	C₂²	1	1	yes	characteristic
18	C₁₉×#117	C₂²	1	1	yes	characteristic
19	#10597386	C₂²	1	1	yes	characteristic
20	#10597385	—	2	1	no	—
21	D₄×Dic₁₉	C₂²	1	1	yes	characteristic
22	#10597376	C₂²	1	1	yes	characteristic
23	#10597372	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₁₉	C₂²	1	1	yes	characteristic
25	#10597339	C₂²	1	1	yes	characteristic
26	#10597336	C₂²	1	1	yes	characteristic
27	#10597336	C₂²	1	1	yes	characteristic
28	#10597334	C₂²	1	1	yes	characteristic
29	#10597334	C₂²	1	1	yes	characteristic
30	#10597334	C₂²	1	1	yes	characteristic
31	#10597331	—	2	1	no	—
32	(C₂²⋊D₁₉)⋊C₄	—	2	1	no	—
33	#10597324	—	2	1	no	—
34	#10597324	—	2	1	no	—
35	C₄×Dic₃₈	—	2	1	no	—
36	C₄×Dic₃₈	—	2	1	no	—
37	C₁₉×#107	C₂²	1	1	yes	characteristic
38	C₁₉×#106	C₂²	1	1	yes	characteristic
39	C₁₉×#102	C₂²	1	1	yes	characteristic
40	C₁₉×#102	C₂²	1	1	yes	characteristic
41	C₁₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
42	#10597289	C₂²	1	1	yes	characteristic
43	#10597289	C₂²	1	1	yes	characteristic
44	C₂²⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
45	#10597278	C₂²	1	1	yes	characteristic
46	#10597278	C₂²	1	1	yes	characteristic
47	#10597275	C₂²	1	1	yes	characteristic
48	#10597274	C₂²	1	1	yes	characteristic
49	#10597272	C₂²	1	1	yes	characteristic
50	#10597272	C₂²	1	1	yes	characteristic
51	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
52	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
53	#10597266	C₂²	1	1	yes	characteristic
54	#10597266	C₂²	1	1	yes	characteristic
55	#10597265	C₂²	1	1	yes	characteristic
56	#10597265	C₂²	1	1	yes	characteristic
57	#10597264	C₂²	1	1	yes	characteristic
58	#10597264	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₁₉	—	2	1	no	—
60	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₉	—	2	1	no	—
63	C₂×C₂²⋊D₁₉	—	2	1	no	—
64	C₂×C₂²⋊D₁₉	—	2	1	no	—
65	C₂²×Dic₁₉	—	2	1	no	—
66	C₂²×Dic₁₉	D₄	1	1	yes	characteristic
67	C₂²×Dic₁₉	D₄	1	1	yes	characteristic
68	#64543	—	4	4	no	—
69	C₂×D₇₆	C₂³	1	1	yes	characteristic
70	C₂×C₄×D₁₉	—	2	1	no	—
71	C₂×Dic₃₈	—	2	1	no	—
72	C₂×Dic₃₈	—	2	1	no	—
73	C₂×Dic₃₈	—	2	1	no	—
74	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
75	C₂×C₈×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	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
79	C₂²⋊C₄×C₁₉	—	2	1	no	—
80	C₂²⋊C₄×C₁₉	—	2	1	no	—
81	C₄²×C₁₉	—	2	1	no	—
82	C₂²⋊Dic₁₉	—	2	1	no	—
83	C₂²⋊Dic₁₉	—	2	1	no	—
84	(C₂×D₁₉)⋊C₄	—	2	1	no	—
85	(C₂×D₁₉)⋊C₄	—	2	1	no	—
86	C₄⋊Dic₁₉	—	2	1	no	—
87	C₄⋊Dic₁₉	—	2	1	no	—
88	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
89	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
90	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
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	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
101	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
102	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
103	C₂³×C₁₉	—	2	1	no	—
104	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
105	C₂²×D₁₉	—	2	1	no	—
106	D₄×C₁₉	—	2	1	no	—
107	D₄×C₁₉	—	2	1	no	—
108	D₄×C₁₉	—	4	2	no	—
109	C₂×C₄×C₁₉	—	2	1	no	—
110	C₂×C₄×C₁₉	—	2	1	no	—
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	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
116	C₂²⋊D₁₉	—	4	2	no	—
117	C₂²⋊D₁₉	—	4	2	no	—
118	C₂²⋊D₁₉	—	4	2	no	—
119	C₂×Dic₁₉	—	2	2	no	—
120	C₂×Dic₁₉	—	2	1	no	—
121	C₂×Dic₁₉	—	2	1	no	—
122	C₂×Dic₁₉	—	2	1	no	—
123	C₂×Dic₁₉	—	4	1	no	—
124	C₂×Dic₁₉	—	4	2	no	—
125	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
129	D₇₆	—	2	1	no	—
130	D₇₆	—	2	1	no	—
131	C₄×D₁₉	—	4	2	no	—
132	Dic₃₈	—	4	2	no	—
133	Dic₃₈	—	4	2	no	—
134	Dic₃₈	—	4	2	no	—
135	C₈×C₁₉	—	2	1	no	—
136	C₈×C₁₉	—	2	1	no	—
137	C₁₉⋊C₈	—	2	1	no	—
138	C₁₉⋊C₈	—	2	1	no	—
139	#3235	—	19	1	no	maximal, Sylow
140	C₂²×C₁₉	—	2	1	no	—
141	C₂²×C₁₉	—	2	1	no	—
142	C₂²×C₁₉	—	2	1	no	—
143	C₂²×C₁₉	—	4	1	no	—
144	C₂²×C₁₉	—	4	1	no	—
145	C₂²×C₁₉	—	4	1	no	—
146	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
147	C₂×D₁₉	—	4	1	no	—
148	C₂×D₁₉	—	4	1	no	—
149	C₂×D₁₉	—	4	1	no	—
150	C₄×C₁₉	—	2	1	no	—
151	C₄×C₁₉	—	4	1	no	—
152	C₄×C₁₉	—	4	1	no	—
153	C₄×C₁₉	—	4	1	no	—
154	C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
155	C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
156	Dic₁₉	—	2	1	no	—
157	Dic₁₉	—	2	1	no	—
158	Dic₁₉	—	2	1	no	—
159	Dic₁₉	—	4	1	no	—
160	Dic₁₉	—	4	1	no	—
161	Dic₁₉	—	4	1	no	—
162	Dic₁₉	AGL(1,ℤ/8)	1	2	yes	—
163	#533	—	19	1	no	—
164	#532	—	19	1	no	—
165	#501	—	19	1	no	—
166	#501	—	19	1	no	—
167	#490	—	19	1	no	—
168	#490	—	19	1	no	—
169	#482	—	19	1	no	—
170	#482	—	19	1	no	—
171	#482	—	19	1	no	—
172	#482	D₁₉	1	1	yes	characteristic, complemented
173	#476	—	19	1	no	—
174	#476	—	19	1	no	—
175	#474	—	19	1	no	—
176	#474	—	19	1	no	—
177	#432	—	19	1	no	—
178	C₂×C₁₉	—	4	1	no	—
179	C₂×C₁₉	—	8	1	no	—
180	C₂×C₁₉	#536	1	1	yes	characteristic
181	C₂×C₁₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
182	C₂×C₁₉	#575	1	1	yes	characteristic
183	D₁₉	—	8	1	no	—
184	C₂×#41	—	38	1	no	—
185	C₄⋊Q₈	—	19	1	no	—
186	C₄⋊Q₈	—	19	1	no	—
187	C₄⋊Q₈	—	19	1	no	—
188	#127	—	19	1	no	—
189	#125	—	19	1	no	—
190	#125	—	19	1	no	—
191	#124	—	38	1	no	—
192	C₂²⋊Q₈	—	38	1	no	—
193	C₂²⋊Q₈	—	38	1	no	—
194	#121	—	19	1	no	—
195	#121	—	38	1	no	—
196	#121	C₂×D₁₉	1	1	yes	characteristic
197	C₄×Q₈	—	38	1	no	—
198	C₄×Q₈	—	38	1	no	—
199	C₄×D₄	—	19	1	no	—
200	C₄×D₄	—	19	1	no	—
201	C₄×D₄	—	38	1	no	—
202	#117	—	19	1	no	—
203	#117	C₂×D₁₉	1	1	yes	characteristic
204	C₂×C₄²	—	19	1	no	—
205	#107	—	19	1	no	—
206	#107	—	19	1	no	—
207	#107	—	19	1	no	—
208	#107	C₂×D₁₉	1	1	yes	characteristic
209	#106	—	19	1	no	—
210	#106	—	19	1	no	—
211	#106	—	19	1	no	—
212	#106	C₂×D₁₉	1	1	yes	characteristic
213	C₄⋊C₈	—	19	1	no	—
214	C₄⋊C₈	—	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	#102	—	19	1	no	—
221	#102	C₂×D₁₉	1	1	yes	characteristic
222	#102	C₂×D₁₉	1	1	yes	characteristic
223	C₂²⋊C₈	—	19	1	no	—
224	C₂²⋊C₈	C₂×D₁₉	1	1	yes	characteristic
225	#97	—	19	1	no	—
226	#97	—	19	1	no	—
227	C₁₉	#3235	1	1	yes	characteristic, complemented, Sylow
228	#41	—	76	4	no	—
229	C₂×Q₈	—	38	1	no	—
230	C₂×Q₈	—	38	1	no	—
231	C₂×Q₈	—	38	1	no	—
232	C₂×D₄	—	2	1	no	—
233	C₂×D₄	—	19	1	no	—
234	C₂×D₄	—	38	1	no	—
235	C₂×D₄	—	38	1	no	—
236	C₂×D₄	—	38	1	no	—
237	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
238	C₂²×C₄	—	19	1	no	—
239	C₂²×C₄	—	19	1	no	—
240	C₂²×C₄	—	38	1	no	—
241	C₂²×C₄	—	38	1	no	—
242	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
243	C₂×C₈	—	19	1	no	—
244	C₂×C₈	—	19	1	no	—
245	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
246	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
247	C₄⋊C₄	—	19	1	no	—
248	C₄⋊C₄	—	19	1	no	—
249	C₄⋊C₄	—	19	1	no	—
250	C₄⋊C₄	—	38	1	no	—
251	C₄⋊C₄	—	38	1	no	—
252	C₄⋊C₄	—	38	1	no	—
253	C₄⋊C₄	—	38	1	no	—
254	C₄⋊C₄	—	38	1	no	—
255	C₄⋊C₄	—	38	1	no	—
256	C₄⋊C₄	—	38	1	no	—
257	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
258	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
259	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
260	C₂²⋊C₄	—	2	1	no	—
261	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	—	38	1	no	—
263	C₂²⋊C₄	—	38	1	no	—
264	C₂²⋊C₄	—	38	1	no	—
265	C₂²⋊C₄	—	38	1	no	—
266	C₄²	—	2	1	no	—
267	C₄²	—	19	1	no	—
268	C₄²	—	19	1	no	—
269	C₄²	—	19	1	no	—
270	C₄²	—	19	1	no	—
271	C₄²	—	38	1	no	—
272	C₂³	—	2	1	no	—
273	C₂³	—	38	1	no	—
274	C₂³	D₄×D₁₉	1	1	yes	characteristic
275	Q₈	—	76	2	no	—
276	Q₈	—	76	2	no	—
277	Q₈	—	76	2	no	—
278	D₄	—	2	1	no	—
279	D₄	—	2	1	no	—
280	D₄	—	4	2	no	—
281	D₄	—	38	1	no	—
282	D₄	—	38	1	no	—
283	D₄	—	76	2	no	—
284	D₄	—	76	2	no	—
285	D₄	—	76	2	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	19	1	no	—
292	C₂×C₄	—	19	1	no	—
293	C₂×C₄	—	19	1	no	—
294	C₂×C₄	—	19	1	no	—
295	C₂×C₄	—	38	2	no	—
296	C₂×C₄	—	38	1	no	—
297	C₂×C₄	—	38	1	no	—
298	C₂×C₄	—	38	1	no	—
299	C₂×C₄	—	76	2	no	—
300	C₂×C₄	—	76	2	no	—
301	C₂×C₄	—	76	1	no	—
302	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
303	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	38	1	no	—
307	C₈	—	38	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	76	1	no	—
315	C₂²	—	76	1	no	—
316	C₂²	—	76	1	no	—
317	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	19	2	no	—
323	C₄	—	38	1	no	—
324	C₄	—	38	1	no	—
325	C₄	—	38	1	no	—
326	C₄	—	76	1	no	—
327	C₄	—	76	1	no	—
328	C₄	—	76	1	no	—
329	C₄	#10597436	1	1	yes	characteristic
330	C₄	#10597436	1	1	yes	characteristic
331	C₂	—	4	1	no	—
332	C₂	—	8	1	no	—
333	C₂	—	152	1	no	—
334	C₂	#11927116	1	1	yes	characteristic
335	C₂	AGL(1,ℤ/8)×D₁₉	1	1	yes	characteristic
336	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×#482	C₂
socle	C₂²×C₁₉	C₂²×D₄
prime radical	#11927089	C₂
hypercenter	#482	D₁₉
hypocenter	C₁₉	#3235
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #482

Complements

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

N	H ≅ G/N	type
1. #11927097	C₂	semidirect
3. #11926952	C₂	semidirect
4. #11926951	C₂	semidirect
5. #11926907	C₂	semidirect
6. #11926907	C₂	semidirect
7. #11926869	C₂	semidirect
8. #11926868	C₂	semidirect
9. #11926865	C₂	semidirect
10. #11926833	C₂	semidirect
11. #11926832	C₂	semidirect
12. #11926823	C₂	semidirect
13. #11926822	C₂	semidirect
14. #11926598	C₂	semidirect
15. C₁₉×#482	C₂	semidirect
172. #482	D₁₉	semidirect
227. C₁₉	#3235	semidirect

Maximal subgroups

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

subgroup	index
#11927097	2
#11927089	2
#11926952	2
#11926951	2
#11926907	2
#11926869	2
#11926868	2
#11926865	2
#11926833	2
#11926832	2
#11926823	2
#11926822	2
#11926598	2
C₁₉×#482	2
#3235	19