Subgroups — #18075733

Order 4864.

386 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#10597276	C₂	1	1	yes	maximal, characteristic, complemented
2	#14562391	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562074	C₂	1	2	yes	maximal
4	#14561526	C₂	1	2	yes	maximal, complemented
5	C₁₉×#2965	C₂	1	1	yes	maximal, characteristic
6	C₂×#10597276	—	2	1	no	—
7	#11926568	C₂²	1	2	yes	—
8	C₄×#64520	C₂²	1	2	yes	—
9	#11926309	—	2	1	no	—
10	#11926295	—	2	1	no	—
11	#11926295	—	2	1	no	—
12	Q₈⋊(C₁₉⋊C₈)	—	2	2	no	—
13	D₄⋊(C₁₉⋊C₈)	—	2	2	no	—
14	C₄×#41×C₁₉	C₂²	1	1	yes	characteristic
15	C₁₉×#446	C₄	1	1	yes	characteristic
16	C₁₉×#446	C₄	1	1	yes	characteristic
17	C₁₉×#436	C₄	1	1	yes	characteristic
18	C₁₉×#436	C₄	1	1	yes	characteristic
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	Q₈	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	D₄	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	D₄	1	1	yes	characteristic
36	C₁₉×#117	—	2	1	no	—
37	C₁₉×#117	Q₈	1	1	yes	characteristic
38	C₂×C₄²×C₁₉	—	2	1	no	—
39	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
40	C₂×#64520	—	2	2	no	—
41	C₂×#64520	—	2	2	no	—
42	C₁₉×C₄⋊C₈	—	2	1	no	—
43	C₁₉×C₄⋊C₈	—	2	1	no	—
44	C₁₉×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₁₉×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
46	C₁₉×C₂²⋊C₈	—	2	1	no	—
47	C₁₉×C₂²⋊C₈	—	2	1	no	—
48	C₁₉×C₂²⋊C₈	—	2	1	no	—
49	C₁₉×#97	—	2	1	no	—
50	C₁₉×#97	—	2	1	no	—
51	C₄×C₈×C₁₉	—	2	1	no	—
52	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
55	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
56	#10597276	—	2	4	no	—
57	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
58	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
59	#10597260	—	2	2	no	—
60	C₄×C₁₉⋊C₈	—	2	2	no	—
61	#41×C₁₉	—	4	2	no	—
62	C₂×Q₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
63	C₂×D₄×C₁₉	—	2	1	no	—
64	C₂×D₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	—	2	1	no	—
66	C₂²×C₄×C₁₉	—	2	1	no	—
67	C₂²×C₄×C₁₉	—	2	1	no	—
68	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
71	M₁₆×C₁₉	—	4	2	no	—
72	M₁₆×C₁₉	—	4	2	no	—
73	C₂×C₈×C₁₉	—	2	2	no	—
74	C₂×C₈×C₁₉	—	2	1	no	—
75	C₂×C₈×C₁₉	—	2	1	no	—
76	C₂×C₈×C₁₉	—	4	1	no	—
77	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
81	C₄⋊C₄×C₁₉	—	2	1	no	—
82	C₄⋊C₄×C₁₉	—	2	1	no	—
83	C₄⋊C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
84	C₄⋊C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
85	C₂²⋊C₄×C₁₉	—	2	1	no	—
86	C₂²⋊C₄×C₁₉	—	2	1	no	—
87	C₂²⋊C₄×C₁₉	—	2	1	no	—
88	C₄²×C₁₉	—	2	1	no	—
89	C₄²×C₁₉	—	2	1	no	—
90	C₄²×C₁₉	—	2	1	no	—
91	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₁₉	#41	1	1	yes	characteristic
95	#64520	—	2	4	no	—
96	#64520	—	4	2	no	—
97	#64520	—	8	2	no	—
98	C₂×C₁₉⋊C₈	—	4	2	no	—
99	C₂×C₁₉⋊C₈	—	4	2	no	—
100	#8672	—	19	1	no	maximal, Sylow
101	C₂³×C₁₉	—	2	1	no	—
102	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
103	Q₈×C₁₉	—	2	2	no	—
104	D₄×C₁₉	—	2	2	no	—
105	D₄×C₁₉	—	4	2	no	—
106	C₂×C₄×C₁₉	—	2	1	no	—
107	C₂×C₄×C₁₉	—	2	1	no	—
108	C₂×C₄×C₁₉	—	2	2	no	—
109	C₂×C₄×C₁₉	—	2	2	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	—	4	2	no	—
114	C₂×C₄×C₁₉	—	4	1	no	—
115	C₂×C₄×C₁₉	—	4	1	no	—
116	C₂×C₄×C₁₉	—	4	2	no	—
117	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
120	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
121	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
123	C₈×C₁₉	—	2	1	no	—
124	C₈×C₁₉	—	2	1	no	—
125	C₈×C₁₉	—	2	1	no	—
126	C₈×C₁₉	—	4	1	no	—
127	C₈×C₁₉	—	4	1	no	—
128	C₈×C₁₉	#104	1	2	yes	—
129	C₁₉⋊C₈	—	4	4	no	—
130	C₁₉⋊C₈	—	8	2	no	—
131	#2965	D₁₉	1	1	yes	characteristic
132	C₄×#108	—	19	1	no	—
133	#1566	—	19	1	no	—
134	#1489	—	19	2	no	—
135	#1299	—	19	2	no	—
136	C₂²×C₁₉	—	2	1	no	—
137	C₂²×C₁₉	—	2	1	no	—
138	C₂²×C₁₉	—	2	1	no	—
139	C₂²×C₁₉	—	4	1	no	—
140	C₂²×C₁₉	—	4	1	no	—
141	C₂²×C₁₉	—	4	1	no	—
142	C₂²×C₁₉	#380	1	1	yes	characteristic
143	C₄×C₁₉	—	2	1	no	—
144	C₄×C₁₉	—	2	1	no	—
145	C₄×C₁₉	—	2	1	no	—
146	C₄×C₁₉	—	4	1	no	—
147	C₄×C₁₉	—	4	1	no	—
148	C₄×C₁₉	—	4	1	no	—
149	C₄×C₁₉	C₂×#104	1	1	yes	characteristic
150	C₄×C₁₉	#421	1	1	yes	characteristic
151	C₄×C₁₉	#430	1	2	yes	—
152	C₄×C₁₉	#471	1	2	yes	—
153	C₄×#41	C₂×D₁₉	1	1	yes	characteristic
154	#446	Dic₁₉	1	1	yes	characteristic
155	#446	Dic₁₉	1	1	yes	characteristic
156	#436	Dic₁₉	1	1	yes	characteristic
157	#436	Dic₁₉	1	1	yes	characteristic
158	#435	—	2	1	no	—
159	#435	—	2	1	no	—
160	#431	—	2	1	no	—
161	#431	C₂×D₁₉	1	1	yes	characteristic
162	C₂×#108	—	38	1	no	—
163	#423	—	19	2	no	—
164	C₄×M₁₆	—	2	1	no	—
165	C₄×M₁₆	—	19	2	no	—
166	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
167	#341	—	38	1	no	—
168	#334	—	38	1	no	—
169	#334	—	38	1	no	—
170	Q₈⋊C₈	—	38	2	no	—
171	D₄⋊C₈	—	38	2	no	—
172	C₂×C₁₉	—	4	1	no	—
173	C₂×C₁₉	—	8	1	no	—
174	C₂×C₁₉	#1806	1	1	yes	characteristic
175	C₂×C₁₉	#1811	1	1	yes	characteristic
176	C₂×C₁₉	#1815	1	1	yes	characteristic
177	C₂×#41	Dic₃₈	1	1	yes	characteristic
178	C₂×M₁₆	—	2	1	no	—
179	C₂×M₁₆	—	2	1	no	—
180	C₂×M₁₆	—	38	2	no	—
181	C₂×M₁₆	—	38	2	no	—
182	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
183	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
184	C₄×Q₈	D₇₆	1	1	yes	characteristic
185	C₄×Q₈	C₂×Dic₁₉	1	1	yes	characteristic
186	C₄×D₄	—	2	1	no	—
187	C₄×D₄	—	2	1	no	—
188	C₄×D₄	D₇₆	1	1	yes	characteristic
189	C₄×D₄	C₂×Dic₁₉	1	1	yes	characteristic
190	#117	—	2	1	no	—
191	#117	Dic₃₈	1	1	yes	characteristic
192	C₂×C₄²	—	2	1	no	—
193	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
194	#108	—	38	4	no	—
195	C₄⋊C₈	—	2	1	no	—
196	C₄⋊C₈	—	2	1	no	—
197	C₄⋊C₈	—	38	2	no	—
198	C₄⋊C₈	—	38	2	no	—
199	C₄⋊C₈	C₂×Dic₁₉	1	1	yes	characteristic
200	C₄⋊C₈	C₂×Dic₁₉	1	1	yes	characteristic
201	C₂²⋊C₈	—	2	1	no	—
202	C₂²⋊C₈	—	2	1	no	—
203	C₂²⋊C₈	—	2	1	no	—
204	#97	—	2	1	no	—
205	#97	—	2	1	no	—
206	#97	—	38	2	no	—
207	C₄×C₈	—	2	1	no	—
208	C₄×C₈	—	38	2	no	—
209	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
210	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
211	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
212	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
213	C₁₉	#8672	1	1	yes	characteristic, complemented, Sylow
214	#41	—	4	2	no	—
215	C₂×Q₈	C₄⋊Dic₁₉	1	1	yes	characteristic
216	C₂×D₄	—	2	1	no	—
217	C₂×D₄	C₄⋊Dic₁₉	1	1	yes	characteristic
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
222	C₂²×C₄	#64541	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
224	M₁₆	—	4	2	no	—
225	M₁₆	—	4	2	no	—
226	M₁₆	—	38	4	no	—
227	M₁₆	—	76	2	no	—
228	M₁₆	—	152	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	76	2	no	—
234	C₂×C₈	—	76	2	no	—
235	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
236	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
237	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
238	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
242	C₄⋊C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
243	C₂²⋊C₄	—	2	1	no	—
244	C₂²⋊C₄	—	2	1	no	—
245	C₂²⋊C₄	—	2	1	no	—
246	C₄²	—	2	1	no	—
247	C₄²	—	2	1	no	—
248	C₄²	—	2	1	no	—
249	C₄²	C₂×D₇₆	1	1	yes	characteristic
250	C₄²	#64541	1	1	yes	characteristic
251	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
252	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
253	C₂³	—	2	1	no	—
254	C₂³	#10597370	1	1	yes	characteristic
255	Q₈	—	2	2	no	—
256	D₄	—	2	2	no	—
257	D₄	—	4	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	#10597370	1	1	yes	characteristic
270	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
271	C₂×C₄	#10597372	1	1	yes	characteristic
272	C₂×C₄	#10597376	1	1	yes	characteristic
273	C₂×C₄	#10597376	1	1	yes	characteristic
274	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	2	1	no	—
278	C₈	—	4	1	no	—
279	C₈	—	4	1	no	—
280	C₈	—	76	4	no	—
281	C₈	—	152	2	no	—
282	C₈	#10597294	1	2	yes	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	4	1	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	1	no	—
289	C₂²	#11926404	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	4	1	no	—
294	C₄	—	4	1	no	—
295	C₄	—	4	1	no	—
296	C₄	C₂×#10597294	1	1	yes	characteristic
297	C₄	#11926561	1	1	yes	characteristic
298	C₄	#11926593	1	2	yes	—
299	C₄	#11926807	1	2	yes	—
300	C₂	—	4	1	no	—
301	C₂	—	8	1	no	—
302	C₂	#14563655	1	1	yes	characteristic
303	C₂	#14563677	1	1	yes	characteristic
304	C₂	#14563694	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	Q₈
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2965	D₁₉
hypocenter	C₁₉	#8672
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
1. C₄×#10597276	C₂	semidirect
2. #14562391	C₂	semidirect
4. #14561526	C₂	semidirect
213. C₁₉	#8672	semidirect

Maximal subgroups

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

subgroup	index
C₄×#10597276	2
#14562391	2
#14562074	2
#14561526	2
C₁₉×#2965	2
#8672	19