Subgroups — #18078562

Order 4864.

524 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	#14569401	C₂	1	1	yes	maximal, characteristic
2	#14562308	C₂	1	2	yes	maximal, complemented
3	#14561949	C₂	1	1	yes	maximal, characteristic
4	#14561495	C₂	1	2	yes	maximal, complemented
5	C₁₉×#2905	C₂	1	1	yes	maximal, characteristic
6	C₂×#10597315	C₂²	1	1	yes	characteristic
7	C₂×#10597370	—	2	1	no	—
8	#11926597	C₂²	1	2	yes	—
9	#11926474	—	2	1	no	—
10	#11926474	C₄	1	2	yes	—
11	#11926469	—	2	1	no	—
12	#11926469	C₄	1	2	yes	—
13	#11926450	—	2	1	no	—
14	#11926410	C₂²	1	1	yes	characteristic
15	#11926291	—	2	4	no	—
16	#11926270	—	2	1	no	—
17	#11926268	—	2	1	no	—
18	C₂²×M₁₆×C₁₉	—	2	1	no	—
19	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
20	C₁₉×#424	—	2	4	no	—
21	C₁₉×#423	—	2	1	no	—
22	C₁₉×#423	C₂²	1	2	yes	—
23	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
24	C₂²×Dic₃₈	—	2	1	no	—
25	C₂×M₁₆×C₁₉	—	2	2	no	—
26	C₂×M₁₆×C₁₉	—	2	2	no	—
27	C₂×M₁₆×C₁₉	—	4	2	no	—
28	C₂×M₁₆×C₁₉	—	4	2	no	—
29	C₂²×C₈×C₁₉	—	2	2	no	—
30	C₁₉×#117	—	2	2	no	—
31	C₁₉×#117	—	2	2	no	—
32	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₁₉	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
38	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
39	C₂×C₄⋊Dic₁₉	—	2	1	no	—
40	C₂×C₄⋊Dic₁₉	—	2	1	no	—
41	C₂×C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
42	#10597370	—	4	2	no	—
43	#10597370	—	4	2	no	—
44	C₂×Dic₁₉⋊C₄	—	2	2	no	—
45	#10597315	—	2	2	no	—
46	#10597315	—	2	2	no	—
47	C₁₉×C₄⋊C₈	—	2	4	no	—
48	C₁₉×C₄⋊C₈	—	2	2	no	—
49	C₁₉×C₄⋊C₈	D₄	1	4	yes	—
50	C₁₉⋊#95	—	2	2	no	—
51	C₁₉⋊#95	—	2	2	no	—
52	C₁₉⋊#95	—	2	1	no	—
53	C₁₉⋊#95	C₂×C₄	1	2	yes	—
54	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
55	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
56	C₄×C₁₉⋊C₈	—	2	2	no	—
57	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	—	2	1	no	—
59	C₂²×C₄×C₁₉	—	2	1	no	—
60	C₂²×C₄×C₁₉	—	2	1	no	—
61	C₂²×C₄×C₁₉	—	2	1	no	—
62	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	D₈	1	2	yes	—
66	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
67	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×Dic₁₉	—	2	2	no	—
71	C₂²×Dic₁₉	—	2	2	no	—
72	C₂×Dic₃₈	—	4	1	no	—
73	C₂×Dic₃₈	—	4	1	no	—
74	C₂×Dic₃₈	—	4	1	no	—
75	C₂×Dic₃₈	—	4	1	no	—
76	C₂×Dic₃₈	—	4	1	no	—
77	C₂×Dic₃₈	—	4	1	no	—
78	M₁₆×C₁₉	—	4	4	no	—
79	M₁₆×C₁₉	—	4	2	no	—
80	M₁₆×C₁₉	—	4	2	no	—
81	C₂×C₈×C₁₉	—	2	4	no	—
82	C₂×C₈×C₁₉	—	4	4	no	—
83	C₄⋊C₄×C₁₉	—	2	2	no	—
84	C₄⋊C₄×C₁₉	—	2	2	no	—
85	C₂²⋊C₄×C₁₉	—	2	2	no	—
86	C₂²⋊C₄×C₁₉	—	2	2	no	—
87	C₄²×C₁₉	—	2	1	no	—
88	C₄²×C₁₉	—	2	1	no	—
89	C₄²×C₁₉	C₂×D₄	1	2	yes	—
90	C₄²×C₁₉	#41	1	2	yes	—
91	C₂²⋊Dic₁₉	—	4	4	no	—
92	C₄⋊Dic₁₉	—	2	4	no	—
93	C₄⋊Dic₁₉	—	4	2	no	—
94	C₄⋊Dic₁₉	—	4	1	no	—
95	C₄⋊Dic₁₉	—	4	1	no	—
96	Dic₁₉⋊C₄	—	4	4	no	—
97	C₂×C₁₉⋊C₈	—	4	2	no	—
98	#9195	—	19	1	no	maximal, Sylow
99	C₂³×C₁₉	—	2	1	no	—
100	C₂³×C₁₉	—	2	1	no	—
101	C₂³×C₁₉	—	4	1	no	—
102	C₂³×C₁₉	—	4	1	no	—
103	C₂³×C₁₉	#102	1	2	yes	—
104	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₉	—	2	1	no	—
106	C₂×C₄×C₁₉	—	2	1	no	—
107	C₂×C₄×C₁₉	—	2	2	no	—
108	C₂×C₄×C₁₉	—	2	2	no	—
109	C₂×C₄×C₁₉	—	2	2	no	—
110	C₂×C₄×C₁₉	—	2	2	no	—
111	C₂×C₄×C₁₉	—	2	1	no	—
112	C₂×C₄×C₁₉	—	2	1	no	—
113	C₂×C₄×C₁₉	—	4	1	no	—
114	C₂×C₄×C₁₉	—	4	1	no	—
115	C₂×C₄×C₁₉	—	4	1	no	—
116	C₂×C₄×C₁₉	—	4	1	no	—
117	C₂×C₄×C₁₉	#102	1	2	yes	—
118	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
120	C₂×C₄×C₁₉	#121	1	2	yes	—
121	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
126	C₂×Dic₁₉	—	4	2	no	—
127	C₂×Dic₁₉	—	4	2	no	—
128	C₂×Dic₁₉	—	4	2	no	—
129	C₂×Dic₁₉	—	4	2	no	—
130	C₂×Dic₁₉	—	4	2	no	—
131	C₂×Dic₁₉	—	4	2	no	—
132	Dic₃₈	—	8	1	no	—
133	Dic₃₈	—	8	1	no	—
134	Dic₃₈	—	8	1	no	—
135	Dic₃₈	—	8	1	no	—
136	C₈×C₁₉	—	4	4	no	—
137	C₁₉⋊C₈	—	8	2	no	—
138	#2905	D₁₉	1	1	yes	characteristic
139	#2621	—	19	1	no	—
140	#1546	—	19	2	no	—
141	#1458	—	19	1	no	—
142	#1286	—	19	2	no	—
143	C₂²×C₁₉	—	2	1	no	—
144	C₂²×C₁₉	—	2	1	no	—
145	C₂²×C₁₉	—	2	2	no	—
146	C₂²×C₁₉	—	4	2	no	—
147	C₂²×C₁₉	—	4	1	no	—
148	C₂²×C₁₉	—	4	1	no	—
149	C₂²×C₁₉	—	8	1	no	—
150	C₂²×C₁₉	#386	1	1	yes	characteristic
151	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
152	C₂²×C₁₉	#417	1	1	yes	characteristic
153	C₄×C₁₉	—	2	2	no	—
154	C₄×C₁₉	—	2	2	no	—
155	C₄×C₁₉	—	2	2	no	—
156	C₄×C₁₉	—	4	1	no	—
157	C₄×C₁₉	—	4	1	no	—
158	C₄×C₁₉	#447	1	1	yes	characteristic
159	C₄×C₁₉	#448	1	1	yes	characteristic
160	C₄×C₁₉	#480	1	1	yes	characteristic
161	C₄×C₁₉	#481	1	1	yes	characteristic
162	Dic₁₉	—	8	2	no	—
163	Dic₁₉	—	8	2	no	—
164	C₂²×M₁₆	—	2	1	no	—
165	C₂×C₄⋊Q₈	—	19	1	no	—
166	C₂×C₂²⋊Q₈	—	38	1	no	—
167	C₂×#117	C₂×D₁₉	1	1	yes	characteristic
168	#431	—	19	2	no	—
169	#424	—	2	4	no	—
170	#423	—	2	1	no	—
171	#423	C₂×D₁₉	1	2	yes	—
172	#398	—	19	2	no	—
173	#398	—	38	1	no	—
174	#397	—	19	2	no	—
175	#397	—	38	1	no	—
176	#391	—	38	1	no	—
177	#382	—	19	1	no	—
178	#332	—	38	4	no	—
179	#324	—	38	1	no	—
180	#323	—	38	1	no	—
181	C₂×C₁₉	—	2	1	no	—
182	C₂×C₁₉	—	2	1	no	—
183	C₂×C₁₉	—	4	2	no	—
184	C₂×C₁₉	#1874	1	1	yes	characteristic
185	C₂×C₁₉	#1909	1	1	yes	characteristic
186	C₂×C₁₉	#1915	1	1	yes	characteristic
187	C₂²×Q₈	—	38	1	no	—
188	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	4	2	no	—
192	C₂×M₁₆	—	4	2	no	—
193	C₂²×C₈	—	2	2	no	—
194	C₄⋊Q₈	—	38	2	no	—
195	C₄⋊Q₈	—	38	2	no	—
196	C₂²⋊Q₈	—	76	2	no	—
197	C₂²⋊Q₈	—	76	2	no	—
198	#117	—	2	2	no	—
199	#117	—	2	2	no	—
200	C₂×C₄⋊C₄	—	19	2	no	—
201	C₂×C₄⋊C₄	—	38	2	no	—
202	C₂×C₄⋊C₄	—	38	1	no	—
203	C₂×C₄⋊C₄	—	38	1	no	—
204	C₂×C₄⋊C₄	C₄×D₁₉	1	1	yes	characteristic
205	C₂×C₄⋊C₄	D₇₆	1	1	yes	characteristic
206	C₂×C₂²⋊C₄	—	38	2	no	—
207	C₂×C₂²⋊C₄	C₄×D₁₉	1	1	yes	characteristic
208	C₂×C₂²⋊C₄	D₇₆	1	1	yes	characteristic
209	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
210	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
211	C₄⋊C₈	—	2	4	no	—
212	C₄⋊C₈	—	2	2	no	—
213	C₄⋊C₈	—	38	2	no	—
214	C₄⋊C₈	C₂²⋊D₁₉	1	4	yes	—
215	C₂²⋊C₈	—	38	2	no	—
216	C₄×C₈	—	38	2	no	—
217	#95	—	19	2	no	—
218	#95	—	38	2	no	—
219	#95	—	38	2	no	—
220	#95	—	38	1	no	—
221	C₁₉	#9195	1	1	yes	characteristic, complemented, Sylow
222	C₂⁴	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
223	C₂×Q₈	—	76	1	no	—
224	C₂×Q₈	—	76	1	no	—
225	C₂×Q₈	—	76	1	no	—
226	C₂×Q₈	—	76	1	no	—
227	C₂×Q₈	—	76	1	no	—
228	C₂×Q₈	—	76	1	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	38	2	no	—
234	C₂²×C₄	—	38	2	no	—
235	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
236	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
237	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
238	C₂²×C₄	#64525	1	2	yes	—
239	C₂²×C₄	#64527	1	2	yes	—
240	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
243	M₁₆	—	4	4	no	—
244	M₁₆	—	4	2	no	—
245	M₁₆	—	4	2	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	4	4	no	—
248	C₂×C₈	—	76	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	38	4	no	—
252	C₄⋊C₄	—	76	2	no	—
253	C₄⋊C₄	—	76	1	no	—
254	C₄⋊C₄	—	76	1	no	—
255	C₄⋊C₄	—	76	4	no	—
256	C₂²⋊C₄	—	2	2	no	—
257	C₂²⋊C₄	—	2	2	no	—
258	C₂²⋊C₄	—	76	4	no	—
259	C₄²	—	2	1	no	—
260	C₄²	—	2	1	no	—
261	C₄²	#64541	1	2	yes	—
262	C₄²	C₂×C₂²⋊D₁₉	1	2	yes	—
263	C₂³	—	2	1	no	—
264	C₂³	—	2	1	no	—
265	C₂³	—	4	1	no	—
266	C₂³	—	4	1	no	—
267	C₂³	#10597266	1	2	yes	—
268	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
269	Q₈	—	152	1	no	—
270	Q₈	—	152	1	no	—
271	Q₈	—	152	1	no	—
272	Q₈	—	152	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	76	2	no	—
286	C₂×C₄	—	76	2	no	—
287	C₂×C₄	—	76	2	no	—
288	C₂×C₄	—	76	2	no	—
289	C₂×C₄	—	76	2	no	—
290	C₂×C₄	—	76	2	no	—
291	C₂×C₄	#10597266	1	2	yes	—
292	C₂×C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
293	C₂×C₄	#10597375	1	1	yes	characteristic
294	C₂×C₄	#10597376	1	2	yes	—
295	C₂×C₄	C₂×#64525	1	1	yes	characteristic
296	C₂×C₄	#10597378	1	1	yes	characteristic
297	C₂×C₄	C₂×#64527	1	1	yes	characteristic
298	C₂×C₄	#10597388	1	1	yes	characteristic
299	C₂×C₄	#10597399	1	1	yes	characteristic
300	C₈	—	4	4	no	—
301	C₈	—	152	2	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	4	2	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	8	1	no	—
309	C₂²	#11926423	1	1	yes	characteristic
310	C₂²	C₂×#10597266	1	1	yes	characteristic
311	C₂²	#11926543	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	2	no	—
315	C₄	—	4	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	152	2	no	—
318	C₄	—	152	2	no	—
319	C₄	#11926678	1	1	yes	characteristic
320	C₄	#11926683	1	1	yes	characteristic
321	C₄	#11926857	1	1	yes	characteristic
322	C₄	#11926862	1	1	yes	characteristic
323	C₂	—	2	1	no	—
324	C₂	—	2	1	no	—
325	C₂	—	4	2	no	—
326	C₂	#14563952	1	1	yes	characteristic
327	C₂	#14564157	1	1	yes	characteristic
328	C₂	#14564187	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926423
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#2905	C₂
socle	C₂²×C₁₉	#386
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2905	D₁₉
hypocenter	C₁₉	#9195
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2905

Complements

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

N	H ≅ G/N	type
2. #14562308	C₂	semidirect
4. #14561495	C₂	semidirect
221. C₁₉	#9195	semidirect

Maximal subgroups

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

subgroup	index
#14569401	2
#14562308	2
#14561949	2
#14561495	2
C₁₉×#2905	2
#9195	19