Subgroups — #18075718

Order 4864.

432 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563655	C₂	1	2	yes	maximal
2	C₄×#10597274	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562394	C₂	1	1	yes	maximal, characteristic, complemented
4	#14561537	C₂	1	2	yes	maximal, complemented
5	C₁₉×#2965	C₂	1	1	yes	maximal, characteristic
6	C₂×#10597274	—	2	1	no	—
7	#11926568	C₂²	1	2	yes	—
8	C₂×#10597294	—	2	2	no	—
9	C₄×C₄⋊Dic₁₉	C₂²	1	2	yes	—
10	#11926307	—	2	1	no	—
11	#11926303	—	2	2	no	—
12	#11926297	—	2	1	no	—
13	#11926297	—	2	1	no	—
14	C₄×#41×C₁₉	C₂²	1	1	yes	characteristic
15	C₁₉×#446	C₄	1	1	yes	characteristic, complemented
16	C₁₉×#446	C₄	1	1	yes	characteristic, complemented
17	C₁₉×#436	C₄	1	1	yes	characteristic, complemented
18	C₁₉×#436	C₄	1	1	yes	characteristic, complemented
19	C₁₉×#435	—	2	1	no	—
20	C₁₉×#435	—	2	1	no	—
21	C₁₉×#431	—	2	1	no	—
22	C₁₉×#431	C₂²	1	1	yes	characteristic
23	C₄×M₁₆×C₁₉	—	2	1	no	—
24	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
25	C₂×#41×C₁₉	D₄	1	1	yes	characteristic
26	C₂×M₁₆×C₁₉	—	2	1	no	—
27	C₂×M₁₆×C₁₉	—	2	1	no	—
28	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
30	C₄×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₄×Q₈×C₁₉	Q₈	1	1	yes	characteristic
32	C₄×D₄×C₁₉	—	2	1	no	—
33	C₄×D₄×C₁₉	—	2	1	no	—
34	C₄×D₄×C₁₉	C₂×C₄	1	1	yes	characteristic
35	C₄×D₄×C₁₉	Q₈	1	1	yes	characteristic
36	C₁₉×#117	—	2	1	no	—
37	C₁₉×#117	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₉	—	2	1	no	—
39	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
40	C₂×C₄⋊Dic₁₉	—	2	2	no	—
41	C₂×C₄×Dic₁₉	—	2	2	no	—
42	C₂×#64520	—	2	2	no	—
43	C₁₉×C₄⋊C₈	—	2	1	no	—
44	C₁₉×C₄⋊C₈	—	2	1	no	—
45	C₁₉×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
46	C₁₉×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
47	C₁₉×C₂²⋊C₈	—	2	1	no	—
48	C₁₉×C₂²⋊C₈	—	2	1	no	—
49	C₁₉×C₂²⋊C₈	—	2	1	no	—
50	C₁₉×#97	—	2	1	no	—
51	C₁₉×#97	—	2	1	no	—
52	C₄×C₈×C₁₉	—	2	1	no	—
53	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
56	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
57	#10597294	—	4	4	no	—
58	C₁₉⋊#95	—	2	2	no	—
59	#10597274	—	2	4	no	—
60	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
61	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
62	#41×C₁₉	—	2	2	no	—
63	#41×C₁₉	—	2	2	no	—
64	C₂×Q₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
65	C₂×D₄×C₁₉	—	2	1	no	—
66	C₂×D₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	—	2	1	no	—
68	C₂²×C₄×C₁₉	—	2	1	no	—
69	C₂²×C₄×C₁₉	—	2	1	no	—
70	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
73	C₂²×Dic₁₉	—	4	2	no	—
74	M₁₆×C₁₉	—	4	2	no	—
75	M₁₆×C₁₉	—	4	2	no	—
76	C₂×C₈×C₁₉	—	2	2	no	—
77	C₂×C₈×C₁₉	—	2	1	no	—
78	C₂×C₈×C₁₉	—	2	1	no	—
79	C₂×C₈×C₁₉	—	4	1	no	—
80	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₄⋊C₄×C₁₉	—	2	1	no	—
85	C₄⋊C₄×C₁₉	—	2	1	no	—
86	C₄⋊C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
87	C₄⋊C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
88	C₂²⋊C₄×C₁₉	—	2	1	no	—
89	C₂²⋊C₄×C₁₉	—	2	1	no	—
90	C₂²⋊C₄×C₁₉	—	2	1	no	—
91	C₄²×C₁₉	—	2	1	no	—
92	C₄²×C₁₉	—	2	1	no	—
93	C₄²×C₁₉	—	2	1	no	—
94	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
97	C₄²×C₁₉	#41	1	1	yes	characteristic
98	C₄⋊Dic₁₉	—	2	4	no	—
99	C₄⋊Dic₁₉	—	4	2	no	—
100	C₄×Dic₁₉	—	4	2	no	—
101	C₄×Dic₁₉	—	4	2	no	—
102	#64520	—	4	2	no	—
103	#64520	—	4	2	no	—
104	C₂×C₁₉⋊C₈	—	4	2	no	—
105	#8669	—	19	1	no	maximal, Sylow
106	C₂³×C₁₉	—	2	1	no	—
107	C₂³×C₁₉	#121	1	1	yes	characteristic
108	Q₈×C₁₉	—	2	2	no	—
109	D₄×C₁₉	—	2	2	no	—
110	D₄×C₁₉	—	4	2	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	—	2	1	no	—
115	C₂×C₄×C₁₉	—	2	1	no	—
116	C₂×C₄×C₁₉	—	4	1	no	—
117	C₂×C₄×C₁₉	—	4	2	no	—
118	C₂×C₄×C₁₉	—	4	1	no	—
119	C₂×C₄×C₁₉	—	4	1	no	—
120	C₂×C₄×C₁₉	—	4	1	no	—
121	C₂×C₄×C₁₉	—	4	1	no	—
122	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
125	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
126	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
129	C₂×C₄×C₁₉	#137	1	2	yes	—
130	C₂×Dic₁₉	—	4	4	no	—
131	C₂×Dic₁₉	—	8	2	no	—
132	C₂×Dic₁₉	—	8	2	no	—
133	C₈×C₁₉	—	2	1	no	—
134	C₈×C₁₉	—	2	1	no	—
135	C₈×C₁₉	—	2	1	no	—
136	C₈×C₁₉	—	4	1	no	—
137	C₈×C₁₉	—	4	1	no	—
138	C₈×C₁₉	#104	1	2	yes	complemented
139	C₁₉⋊C₈	—	8	2	no	—
140	#2965	D₁₉	1	1	yes	characteristic
141	#1806	—	19	2	no	—
142	C₄×#106	—	19	1	no	—
143	#1567	—	19	1	no	—
144	#1303	—	19	2	no	—
145	C₂²×C₁₉	—	2	1	no	—
146	C₂²×C₁₉	—	2	1	no	—
147	C₂²×C₁₉	—	4	1	no	—
148	C₂²×C₁₉	—	4	1	no	—
149	C₂²×C₁₉	—	4	1	no	—
150	C₂²×C₁₉	#380	1	1	yes	characteristic
151	C₂²×C₁₉	#468	1	2	yes	—
152	C₄×C₁₉	—	2	1	no	—
153	C₄×C₁₉	—	2	1	no	—
154	C₄×C₁₉	—	2	1	no	—
155	C₄×C₁₉	—	4	1	no	—
156	C₄×C₁₉	—	4	1	no	—
157	C₄×C₁₉	—	4	1	no	—
158	C₄×C₁₉	C₂×#104	1	1	yes	characteristic
159	C₄×C₁₉	#421	1	1	yes	characteristic
160	C₄×C₁₉	#428	1	2	yes	—
161	C₄×C₁₉	#469	1	1	yes	characteristic
162	C₄×C₁₉	#470	1	1	yes	characteristic
163	Dic₁₉	—	8	4	no	—
164	C₄×#41	C₂×D₁₉	1	1	yes	characteristic
165	#446	Dic₁₉	1	1	yes	characteristic, complemented
166	#446	Dic₁₉	1	1	yes	characteristic, complemented
167	#436	Dic₁₉	1	1	yes	characteristic, complemented
168	#436	Dic₁₉	1	1	yes	characteristic, complemented
169	#435	—	2	1	no	—
170	#435	—	2	1	no	—
171	#431	—	2	1	no	—
172	#431	C₂×D₁₉	1	1	yes	characteristic
173	C₂×#106	—	38	1	no	—
174	#423	—	19	2	no	—
175	C₂×#104	—	38	2	no	—
176	C₄×M₁₆	—	2	1	no	—
177	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
178	C₄×C₄⋊C₄	—	19	2	no	—
179	#340	—	38	1	no	—
180	#339	—	38	2	no	—
181	#335	—	38	1	no	—
182	#335	—	38	1	no	—
183	C₂×C₁₉	—	2	2	no	—
184	C₂×C₁₉	—	8	1	no	—
185	C₂×C₁₉	#1806	1	1	yes	characteristic
186	C₂×C₁₉	#1809	1	1	yes	characteristic
187	C₂×C₁₉	#1814	1	1	yes	characteristic
188	C₂×#41	D₇₆	1	1	yes	characteristic
189	C₂×M₁₆	—	2	1	no	—
190	C₂×M₁₆	—	2	1	no	—
191	C₂×M₁₆	—	38	2	no	—
192	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
193	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
194	C₄×Q₈	Dic₃₈	1	1	yes	characteristic
195	C₄×Q₈	C₂×Dic₁₉	1	1	yes	characteristic
196	C₄×D₄	—	2	1	no	—
197	C₄×D₄	—	2	1	no	—
198	C₄×D₄	Dic₃₈	1	1	yes	characteristic
199	C₄×D₄	C₂×Dic₁₉	1	1	yes	characteristic
200	#117	—	2	1	no	—
201	#117	D₇₆	1	1	yes	characteristic
202	C₂×C₄⋊C₄	—	38	2	no	—
203	C₂×C₄²	—	2	1	no	—
204	C₂×C₄²	—	38	2	no	—
205	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
206	#106	—	38	4	no	—
207	C₄⋊C₈	—	2	1	no	—
208	C₄⋊C₈	—	2	1	no	—
209	C₄⋊C₈	—	38	2	no	—
210	C₄⋊C₈	—	38	2	no	—
211	C₄⋊C₈	C₂×Dic₁₉	1	1	yes	characteristic
212	C₄⋊C₈	C₂×Dic₁₉	1	1	yes	characteristic
213	#104	—	76	4	no	—
214	C₂²⋊C₈	—	2	1	no	—
215	C₂²⋊C₈	—	2	1	no	—
216	C₂²⋊C₈	—	2	1	no	—
217	#97	—	2	1	no	—
218	#97	—	2	1	no	—
219	C₄×C₈	—	2	1	no	—
220	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
221	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
222	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
223	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
224	#95	—	38	2	no	—
225	C₁₉	#8669	1	1	yes	characteristic, complemented, Sylow
226	#41	—	2	2	no	—
227	#41	—	2	2	no	—
228	C₂×Q₈	C₄⋊Dic₁₉	1	1	yes	characteristic
229	C₂×D₄	—	2	1	no	—
230	C₂×D₄	C₄⋊Dic₁₉	1	1	yes	characteristic
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	—	76	2	no	—
235	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
236	C₂²×C₄	#64541	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
238	M₁₆	—	4	2	no	—
239	M₁₆	—	4	2	no	—
240	M₁₆	—	76	2	no	—
241	M₁₆	—	76	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	—	76	2	no	—
247	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
248	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
249	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
250	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	38	4	no	—
254	C₄⋊C₄	—	76	2	no	—
255	C₄⋊C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
256	C₄⋊C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
257	C₂²⋊C₄	—	2	1	no	—
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	2	1	no	—
260	C₄²	—	2	1	no	—
261	C₄²	—	2	1	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	76	2	no	—
264	C₄²	—	76	2	no	—
265	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
266	C₄²	#64541	1	1	yes	characteristic
267	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
268	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
269	C₂³	—	2	1	no	—
270	C₂³	#10597376	1	1	yes	characteristic
271	Q₈	—	2	2	no	—
272	D₄	—	2	2	no	—
273	D₄	—	4	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	2	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	4	no	—
286	C₂×C₄	—	152	2	no	—
287	C₂×C₄	—	152	2	no	—
288	C₂×C₄	#10597354	1	2	yes	—
289	C₂×C₄	#10597355	1	2	yes	—
290	C₂×C₄	#10597370	1	1	yes	characteristic
291	C₂×C₄	#10597370	1	1	yes	characteristic
292	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
293	C₂×C₄	#10597372	1	1	yes	characteristic
294	C₂×C₄	#10597376	1	1	yes	characteristic
295	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	—	2	1	no	—
299	C₈	—	4	1	no	—
300	C₈	—	4	1	no	—
301	C₈	—	152	2	no	—
302	C₈	#10597294	1	2	yes	complemented
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	#11926404	1	1	yes	characteristic
309	C₂²	#11926790	1	2	yes	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	1	no	—
316	C₄	—	152	4	no	—
317	C₄	C₂×#10597294	1	1	yes	characteristic
318	C₄	#11926561	1	1	yes	characteristic
319	C₄	#11926586	1	2	yes	—
320	C₄	#11926797	1	1	yes	characteristic
321	C₄	#11926802	1	1	yes	characteristic
322	C₂	—	2	2	no	—
323	C₂	—	8	1	no	—
324	C₂	#14563655	1	1	yes	characteristic
325	C₂	#14563668	1	1	yes	characteristic
326	C₂	#14563689	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597376
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#2965	C₂
socle	C₂²×C₁₉	#380
prime radical	C₂×#41×C₁₉	D₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2965	D₁₉
hypocenter	C₁₉	#8669
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2965

Complements

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

N	H ≅ G/N	type
2. C₄×#10597274	C₂	semidirect
3. #14562394	C₂	semidirect
4. #14561537	C₂	semidirect
15. C₁₉×#446	C₄	semidirect
16. C₁₉×#446	C₄	semidirect
17. C₁₉×#436	C₄	semidirect
18. C₁₉×#436	C₄	semidirect
138. C₈×C₁₉	#104	semidirect
165. #446	Dic₁₉	semidirect
166. #446	Dic₁₉	semidirect
167. #436	Dic₁₉	semidirect
168. #436	Dic₁₉	semidirect
225. C₁₉	#8669	semidirect
302. C₈	#10597294	semidirect

Maximal subgroups

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

subgroup	index
#14563655	2
C₄×#10597274	2
#14562394	2
#14561537	2
C₁₉×#2965	2
#8669	19