Subgroups — #18087580

Order 4864.

510 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14564441	C₂	1	1	yes	maximal, characteristic
2	#14564440	C₂	1	1	yes	maximal, characteristic
3	#14564440	C₂	1	1	yes	maximal, characteristic
4	#14563794	C₂	1	1	yes	maximal, characteristic, complemented
5	#14563762	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉⋊#1273	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1844	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10597320	C₂²	1	1	yes	characteristic
9	C₂×#10597264	—	2	2	no	—
10	C₂×#10597278	—	2	2	no	—
11	C₂×#10597266	—	2	2	no	—
12	C₂×#10597266	—	2	2	no	—
13	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
15	#11926414	C₂²	1	1	yes	characteristic
16	#11926306	—	2	2	no	—
17	C₂×C₁₉×#106	—	2	2	no	—
18	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
19	C₁₉×#384	C₂²	1	1	yes	characteristic
20	C₁₉×#384	C₂²	1	1	yes	characteristic
21	C₂²×D₇₆	—	2	1	no	—
22	C₂²×D₇₆	—	2	1	no	—
23	C₂²×D₇₆	—	2	1	no	—
24	C₂²×C₈×C₁₉	D₄	1	2	yes	—
25	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
26	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
27	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
28	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
30	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
31	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊Dic₁₉	—	2	1	no	—
33	C₂×C₄⋊Dic₁₉	—	2	1	no	—
34	C₂×C₄⋊Dic₁₉	—	2	1	no	—
35	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
36	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
37	#10597320	—	2	1	no	—
38	#10597320	—	2	1	no	—
39	#10597320	—	2	1	no	—
40	#10597320	Q₈	1	2	yes	—
41	C₁₉×#106	—	4	4	no	—
42	C₄×C₈×C₁₉	—	2	1	no	—
43	C₄×C₈×C₁₉	—	2	1	no	—
44	#10597278	—	4	4	no	—
45	#10597266	—	4	4	no	—
46	#10597266	—	4	4	no	—
47	#10597264	—	4	4	no	—
48	C₄×C₁₉⋊C₈	—	2	1	no	—
49	C₄×C₁₉⋊C₈	—	2	1	no	—
50	C₄×C₁₉⋊C₈	—	2	1	no	—
51	C₄×C₁₉⋊C₈	—	2	1	no	—
52	C₂³×D₁₉	—	4	1	no	—
53	C₂²×C₄×C₁₉	—	4	1	no	—
54	C₂²×C₄×C₁₉	—	4	1	no	—
55	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×Dic₁₉	—	4	1	no	—
59	C₂×D₇₆	—	2	2	no	—
60	C₂×D₇₆	—	2	2	no	—
61	C₂×D₇₆	—	2	2	no	—
62	C₂×D₇₆	—	4	2	no	—
63	C₂×D₇₆	—	4	2	no	—
64	C₂×D₇₆	—	4	2	no	—
65	C₂×D₇₆	—	4	2	no	—
66	C₂×D₇₆	—	4	2	no	—
67	C₂×D₇₆	—	4	2	no	—
68	C₂×D₇₆	—	4	1	no	—
69	C₂×D₇₆	—	4	1	no	—
70	C₂×D₇₆	—	4	1	no	—
71	C₂×C₈×C₁₉	—	2	2	no	—
72	C₂×C₈×C₁₉	—	2	2	no	—
73	C₂×C₈×C₁₉	D₈	1	4	yes	—
74	C₄⋊C₄×C₁₉	—	4	2	no	—
75	C₄⋊C₄×C₁₉	—	4	2	no	—
76	C₄⋊C₄×C₁₉	—	4	2	no	—
77	C₄⋊C₄×C₁₉	—	4	2	no	—
78	C₄⋊C₄×C₁₉	—	4	2	no	—
79	C₄⋊C₄×C₁₉	—	4	2	no	—
80	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
81	C₄²×C₁₉	#41	1	1	yes	characteristic
82	C₄²×C₁₉	#41	1	1	yes	characteristic
83	C₄²×C₁₉	#41	1	1	yes	characteristic
84	C₄⋊Dic₁₉	—	4	2	no	—
85	C₄⋊Dic₁₉	—	4	2	no	—
86	C₄⋊Dic₁₉	—	4	2	no	—
87	C₂×C₁₉⋊C₈	—	2	2	no	—
88	C₂×C₁₉⋊C₈	—	2	2	no	—
89	C₂×C₁₉⋊C₈	—	2	2	no	—
90	C₂×C₁₉⋊C₈	—	2	2	no	—
91	C₂×C₁₉⋊C₈	—	2	2	no	—
92	C₂×C₁₉⋊C₈	SD₁₆	1	4	yes	—
93	#10672	—	19	1	no	maximal, Sylow
94	C₂³×C₁₉	#127	1	1	yes	characteristic
95	C₂²×D₁₉	—	4	2	no	—
96	C₂²×D₁₉	—	8	1	no	—
97	C₂²×D₁₉	—	8	1	no	—
98	C₂²×D₁₉	—	8	1	no	—
99	C₂²×D₁₉	—	8	1	no	—
100	C₂²×D₁₉	—	8	1	no	—
101	C₂²×D₁₉	—	8	1	no	—
102	C₂×C₄×C₁₉	—	8	1	no	—
103	C₂×C₄×C₁₉	—	8	1	no	—
104	C₂×C₄×C₁₉	—	8	1	no	—
105	C₂×C₄×C₁₉	—	8	1	no	—
106	C₂×C₄×C₁₉	—	8	1	no	—
107	C₂×C₄×C₁₉	—	8	1	no	—
108	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
111	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
112	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
113	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
114	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
115	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
116	C₂×C₄×C₁₉	#135	1	2	yes	—
117	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
118	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
119	C₂×C₄×C₁₉	#137	1	2	yes	—
120	C₂×Dic₁₉	—	8	1	no	—
121	C₂×Dic₁₉	—	8	1	no	—
122	C₂×Dic₁₉	—	8	1	no	—
123	D₇₆	—	4	4	no	—
124	D₇₆	—	4	4	no	—
125	D₇₆	—	4	4	no	—
126	D₇₆	—	8	2	no	—
127	D₇₆	—	8	2	no	—
128	D₇₆	—	8	2	no	—
129	C₈×C₁₉	—	2	4	no	—
130	C₁₉⋊C₈	—	2	4	no	—
131	C₁₉⋊C₈	—	4	2	no	—
132	#1957	—	19	1	no	—
133	#1957	—	19	1	no	—
134	#1957	—	19	1	no	—
135	#1845	—	19	1	no	—
136	#1844	D₁₉	1	1	yes	characteristic, complemented
137	#1834	—	19	1	no	—
138	#1273	—	19	1	no	—
139	C₂²×C₁₉	#399	1	1	yes	characteristic
140	C₂²×C₁₉	#492	1	1	yes	characteristic
141	C₂²×C₁₉	#493	1	1	yes	characteristic
142	C₂²×C₁₉	#495	1	1	yes	characteristic
143	C₂²×C₁₉	#496	1	1	yes	characteristic
144	C₂²×C₁₉	#496	1	1	yes	characteristic
145	C₂²×C₁₉	#497	1	1	yes	characteristic
146	C₂×D₁₉	—	8	2	no	—
147	C₂×D₁₉	—	8	2	no	—
148	C₂×D₁₉	—	8	2	no	—
149	C₂×D₁₉	—	16	1	no	—
150	C₂×D₁₉	—	16	1	no	—
151	C₂×D₁₉	—	16	1	no	—
152	C₂×D₁₉	—	16	1	no	—
153	C₄×C₁₉	—	16	1	no	—
154	C₄×C₁₉	—	16	1	no	—
155	C₄×C₁₉	#474	1	2	yes	—
156	C₄×C₁₉	#476	1	2	yes	—
157	C₄×C₁₉	#478	1	2	yes	—
158	C₄×C₁₉	#480	1	2	yes	—
159	C₄×C₁₉	#481	1	2	yes	—
160	C₄×C₁₉	#482	1	2	yes	—
161	Dic₁₉	—	16	1	no	—
162	C₂×#127	—	19	1	no	—
163	C₂×#107	—	38	2	no	—
164	C₂×#106	—	2	2	no	—
165	C₂×#102	—	38	2	no	—
166	C₂×#102	—	38	2	no	—
167	C₂×#102	—	38	2	no	—
168	C₂×C₄×C₈	—	19	1	no	—
169	C₂×C₄×C₈	—	19	1	no	—
170	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
171	#384	—	19	1	no	—
172	#384	C₂×D₁₉	1	1	yes	characteristic
173	#384	C₂×D₁₉	1	1	yes	characteristic
174	#340	—	38	2	no	—
175	C₂×C₁₉	#1706	1	1	yes	characteristic
176	C₂×C₁₉	#1714	1	1	yes	characteristic
177	C₂×C₁₉	#1720	1	1	yes	characteristic
178	C₂×C₁₉	#1721	1	1	yes	characteristic
179	C₂×C₁₉	#2092	1	1	yes	characteristic
180	C₂×C₁₉	#2093	1	1	yes	characteristic
181	C₂×C₁₉	#2094	1	1	yes	characteristic
182	D₁₉	—	16	2	no	—
183	C₂²×D₄	—	38	1	no	—
184	C₂²×D₄	—	38	1	no	—
185	C₂²×D₄	—	38	1	no	—
186	C₂²×C₈	—	19	2	no	—
187	C₂²×C₈	—	19	2	no	—
188	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
189	#127	—	19	2	no	—
190	#127	—	38	1	no	—
191	#127	—	38	1	no	—
192	#127	—	38	1	no	—
193	C₂×C₄⋊C₄	—	2	1	no	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₄⋊C₄	—	2	1	no	—
199	C₂×C₄⋊C₄	—	38	1	no	—
200	C₂×C₄⋊C₄	—	38	1	no	—
201	C₂×C₄⋊C₄	—	38	1	no	—
202	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
203	#107	—	76	4	no	—
204	#106	—	4	4	no	—
205	#102	—	76	4	no	—
206	#102	—	76	4	no	—
207	#102	—	76	4	no	—
208	C₄×C₈	—	2	1	no	—
209	C₄×C₈	—	2	1	no	—
210	C₄×C₈	—	38	1	no	—
211	C₄×C₈	—	38	1	no	—
212	C₄×C₈	—	38	1	no	—
213	C₄×C₈	—	38	1	no	—
214	C₁₉	#10672	1	1	yes	characteristic, complemented, Sylow
215	C₂⁴	—	76	1	no	—
216	C₂×D₄	—	38	2	no	—
217	C₂×D₄	—	38	2	no	—
218	C₂×D₄	—	38	2	no	—
219	C₂×D₄	—	76	2	no	—
220	C₂×D₄	—	76	2	no	—
221	C₂×D₄	—	76	1	no	—
222	C₂×D₄	—	76	1	no	—
223	C₂×D₄	—	76	2	no	—
224	C₂×D₄	—	76	2	no	—
225	C₂×D₄	—	76	2	no	—
226	C₂×D₄	—	76	2	no	—
227	C₂×D₄	—	76	1	no	—
228	C₂²×C₄	—	4	1	no	—
229	C₂²×C₄	—	4	1	no	—
230	C₂²×C₄	—	76	1	no	—
231	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
232	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	19	4	no	—
237	C₂×C₈	—	38	2	no	—
238	C₂×C₈	—	38	2	no	—
239	C₂×C₈	—	38	2	no	—
240	C₂×C₈	—	38	2	no	—
241	C₂×C₈	—	38	2	no	—
242	C₂×C₈	#64525	1	4	yes	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	76	2	no	—
250	C₄⋊C₄	—	76	2	no	—
251	C₄⋊C₄	—	76	2	no	—
252	C₄²	#64541	1	1	yes	characteristic
253	C₄²	#64541	1	1	yes	characteristic
254	C₄²	Q₈×D₁₉	1	1	yes	characteristic
255	C₄²	#64545	1	1	yes	characteristic
256	C₂³	—	76	2	no	—
257	C₂³	—	152	1	no	—
258	C₂³	—	152	1	no	—
259	C₂³	—	152	1	no	—
260	C₂³	—	152	1	no	—
261	C₂³	—	152	1	no	—
262	C₂³	—	152	1	no	—
263	C₂³	#10597386	1	1	yes	characteristic
264	D₄	—	76	4	no	—
265	D₄	—	76	4	no	—
266	D₄	—	76	4	no	—
267	D₄	—	152	2	no	—
268	D₄	—	152	2	no	—
269	D₄	—	152	2	no	—
270	C₂×C₄	—	8	1	no	—
271	C₂×C₄	—	8	1	no	—
272	C₂×C₄	—	8	1	no	—
273	C₂×C₄	—	8	1	no	—
274	C₂×C₄	—	8	1	no	—
275	C₂×C₄	—	8	1	no	—
276	C₂×C₄	—	152	1	no	—
277	C₂×C₄	—	152	1	no	—
278	C₂×C₄	—	152	1	no	—
279	C₂×C₄	#10597340	1	1	yes	characteristic
280	C₂×C₄	#10597340	1	1	yes	characteristic
281	C₂×C₄	#10597342	1	1	yes	characteristic
282	C₂×C₄	#10597342	1	1	yes	characteristic
283	C₂×C₄	SD₁₆×D₁₉	1	2	yes	—
284	C₂×C₄	#10597360	1	2	yes	—
285	C₂×C₄	#10597360	1	2	yes	—
286	C₂×C₄	#10597362	1	2	yes	—
287	C₂×C₄	#10597375	1	1	yes	characteristic
288	C₂×C₄	C₂×#64525	1	2	yes	—
289	C₂×C₄	#10597388	1	2	yes	—
290	C₂×C₄	#10597392	1	1	yes	characteristic
291	C₈	—	2	4	no	—
292	C₈	—	38	4	no	—
293	C₈	—	76	2	no	—
294	C₂²	—	304	1	no	—
295	C₂²	—	304	1	no	—
296	C₂²	—	304	1	no	—
297	C₂²	—	304	1	no	—
298	C₂²	—	152	2	no	—
299	C₂²	—	152	2	no	—
300	C₂²	—	152	2	no	—
301	C₂²	#11926479	1	1	yes	characteristic
302	C₂²	#11926915	1	1	yes	characteristic
303	C₂²	#11926920	1	1	yes	characteristic
304	C₂²	#11926925	1	1	yes	characteristic
305	C₂²	#11926930	1	1	yes	characteristic
306	C₂²	#11926930	1	1	yes	characteristic
307	C₂²	#11926936	1	1	yes	characteristic
308	C₄	—	16	1	no	—
309	C₄	—	16	1	no	—
310	C₄	—	304	1	no	—
311	C₄	#11926821	1	2	yes	—
312	C₄	#11926833	1	2	yes	—
313	C₄	#11926843	1	2	yes	—
314	C₄	#11926857	1	2	yes	—
315	C₄	#11926864	1	2	yes	—
316	C₄	#11926870	1	2	yes	—
317	C₂	—	304	2	no	—
318	C₂	#14563225	1	1	yes	characteristic
319	C₂	#14563260	1	1	yes	characteristic
320	C₂	#14563293	1	1	yes	characteristic
321	C₂	#14563299	1	1	yes	characteristic
322	C₂	#14565157	1	1	yes	characteristic
323	C₂	#14565164	1	1	yes	characteristic
324	C₂	#14565165	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597386
commutator, cosocle	C₂×C₄²×C₁₉	C₂³
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1844	C₂
socle	C₂³×C₁₉	#127
prime radical	C₂×#10597320	C₂²
hypercenter	#1844	D₁₉
hypocenter	C₁₉	#10672
LCS	G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

Complements

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

N	H ≅ G/N	type
4. #14563794	C₂	semidirect
5. #14563762	C₂	semidirect
6. C₁₉⋊#1273	C₂	semidirect
7. C₁₉×#1844	C₂	semidirect
136. #1844	D₁₉	semidirect
214. C₁₉	#10672	semidirect

Maximal subgroups

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

subgroup	index
#14564441	2
#14564440	2
#14563794	2
#14563762	2
C₁₉⋊#1273	2
C₁₉×#1844	2
#10672	19