Subgroups — #18090927

Order 4864.

602 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14572199	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926314	C₂	1	2	yes	maximal
3	#14563332	C₂	1	1	yes	maximal, characteristic
4	#14563332	C₂	1	2	yes	maximal
5	C₁₉×#1740	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	C₂²	1	1	yes	characteristic
7	C₂²×#64520	C₂²	1	2	yes	—
8	C₂³×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
9	#11926571	—	2	4	no	—
10	#11926568	—	2	2	no	—
11	#11926314	—	2	4	no	—
12	#11926299	—	2	4	no	—
13	#11926299	—	2	2	no	—
14	C₂×C₁₉×#117	C₂²	1	2	yes	—
15	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
16	C₁₉×#342	C₄	1	4	yes	—
17	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
18	C₁₉×#117	—	2	2	no	—
19	C₁₉×#117	—	2	2	no	—
20	C₁₉×#117	—	2	2	no	—
21	C₁₉×#117	—	2	2	no	—
22	C₁₉×#117	—	2	2	no	—
23	C₁₉×#117	—	2	2	no	—
24	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
25	C₂×C₄⋊C₄×C₁₉	Q₈	1	2	yes	—
26	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
27	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₁₉	—	2	2	no	—
29	C₂×C₄²×C₁₉	D₄	1	2	yes	—
30	C₂×#64520	—	2	2	no	—
31	C₂×#64520	—	2	2	no	—
32	C₂×#64520	—	2	2	no	—
33	C₂×#64520	—	2	1	no	—
34	C₂×#64520	—	2	2	no	—
35	C₂×#64520	—	2	2	no	—
36	C₂×#64520	—	2	2	no	—
37	C₂×#64520	—	2	2	no	—
38	C₂×#64520	—	2	2	no	—
39	C₂×#64520	D₄	1	2	yes	—
40	C₂²×C₁₉⋊C₈	—	2	1	no	—
41	C₂²×C₁₉⋊C₈	—	2	2	no	—
42	C₂²×C₁₉⋊C₈	—	2	1	no	—
43	C₂²×C₁₉⋊C₈	—	2	2	no	—
44	C₂²×C₁₉⋊C₈	—	2	1	no	—
45	C₂²×C₁₉⋊C₈	D₄	1	4	yes	—
46	C₂²×C₁₉⋊C₈	Q₈	1	2	yes	—
47	C₄⋊(C₁₉⋊C₈)	—	2	8	no	—
48	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	—	2	4	no	—
50	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
51	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	#41	1	2	yes	—
57	C₂²×C₄×C₁₉	#41	1	2	yes	—
58	C₂²×C₄×C₁₉	#41	1	2	yes	—
59	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
61	C₄⋊C₄×C₁₉	—	2	4	no	—
62	C₄⋊C₄×C₁₉	—	4	2	no	—
63	C₄⋊C₄×C₁₉	—	4	2	no	—
64	C₂²⋊C₄×C₁₉	—	2	4	no	—
65	C₂²⋊C₄×C₁₉	—	2	4	no	—
66	C₂²⋊C₄×C₁₉	—	2	4	no	—
67	C₄²×C₁₉	—	2	2	no	—
68	C₄²×C₁₉	—	2	2	no	—
69	C₄²×C₁₉	—	4	2	no	—
70	C₄²×C₁₉	—	4	2	no	—
71	#64520	—	2	4	no	—
72	#64520	—	4	2	no	—
73	#64520	—	4	2	no	—
74	#64520	—	4	2	no	—
75	#64520	—	4	2	no	—
76	#64520	—	4	2	no	—
77	C₂×C₁₉⋊C₈	—	2	4	no	—
78	C₂×C₁₉⋊C₈	—	2	2	no	—
79	C₂×C₁₉⋊C₈	—	2	4	no	—
80	C₂×C₁₉⋊C₈	—	2	4	no	—
81	C₂×C₁₉⋊C₈	—	2	2	no	—
82	C₂×C₁₉⋊C₈	—	2	4	no	—
83	C₂×C₁₉⋊C₈	—	4	1	no	—
84	C₂×C₁₉⋊C₈	—	4	1	no	—
85	C₂×C₁₉⋊C₈	—	4	1	no	—
86	C₂×C₁₉⋊C₈	—	4	2	no	—
87	C₂×C₁₉⋊C₈	—	4	1	no	—
88	#11288	—	19	1	no	maximal, Sylow
89	C₂³×C₁₉	—	4	2	no	—
90	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂³×C₁₉	#117	1	2	yes	—
92	C₂³×C₁₉	C₄×D₄	1	2	yes	—
93	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
94	C₂³×C₁₉	C₄×Q₈	1	1	yes	characteristic
95	C₂×C₄×C₁₉	—	2	4	no	—
96	C₂×C₄×C₁₉	—	4	4	no	—
97	C₂×C₄×C₁₉	—	4	8	no	—
98	C₂×C₄×C₁₉	—	4	4	no	—
99	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
100	C₂×C₄×C₁₉	#121	1	2	yes	—
101	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
102	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
103	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
104	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂×C₄×C₁₉	#123	1	2	yes	—
106	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
107	C₂×C₄×C₁₉	#123	1	2	yes	—
108	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
109	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
110	C₂×C₄×C₁₉	#124	1	2	yes	—
111	C₂×C₄×C₁₉	#125	1	2	yes	—
112	C₂×C₄×C₁₉	#125	1	2	yes	—
113	C₂×C₄×C₁₉	#125	1	1	yes	characteristic
114	C₂×C₄×C₁₉	#126	1	2	yes	—
115	C₂×C₄×C₁₉	#126	1	2	yes	—
116	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
117	C₁₉⋊C₈	—	4	2	no	—
118	C₁₉⋊C₈	—	4	4	no	—
119	C₁₉⋊C₈	—	4	2	no	—
120	#2905	—	19	1	no	—
121	C₂×#344	—	19	2	no	—
122	#1740	D₁₉	1	1	yes	characteristic
123	#1729	—	19	2	no	—
124	#1729	—	19	1	no	—
125	C₂²×C₁₉	—	2	2	no	—
126	C₂²×C₁₉	—	2	4	no	—
127	C₂²×C₁₉	—	4	2	no	—
128	C₂²×C₁₉	—	4	1	no	—
129	C₂²×C₁₉	—	4	1	no	—
130	C₂²×C₁₉	#382	1	1	yes	characteristic
131	C₂²×C₁₉	#385	1	1	yes	characteristic
132	C₂²×C₁₉	#387	1	2	yes	—
133	C₂²×C₁₉	#388	1	2	yes	—
134	C₂²×C₁₉	#389	1	1	yes	characteristic
135	C₄×C₁₉	—	4	4	no	—
136	C₄×C₁₉	—	8	4	no	—
137	C₄×C₁₉	#393	1	1	yes	characteristic
138	C₄×C₁₉	#394	1	1	yes	characteristic
139	C₄×C₁₉	#397	1	1	yes	characteristic
140	C₄×C₁₉	#398	1	2	yes	—
141	C₄×C₁₉	#400	1	1	yes	characteristic
142	C₄×C₁₉	#400	1	2	yes	—
143	C₂²×M₁₆	—	19	2	no	—
144	C₂²×M₁₆	—	19	1	no	—
145	C₂³×C₈	—	19	1	no	—
146	C₂×#117	C₂×D₁₉	1	2	yes	—
147	C₂×#117	C₂×D₁₉	1	1	yes	characteristic
148	#424	—	38	4	no	—
149	#423	—	38	2	no	—
150	#344	—	38	4	no	—
151	#342	Dic₁₉	1	4	yes	—
152	#336	—	38	4	no	—
153	#336	—	38	2	no	—
154	C₂×C₁₉	—	2	2	no	—
155	C₂×C₁₉	—	2	1	no	—
156	C₂×C₁₉	—	2	2	no	—
157	C₂×C₁₉	—	2	1	no	—
158	C₂×C₁₉	#1442	1	1	yes	characteristic
159	C₂×C₁₉	#1868	1	2	yes	—
160	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
161	C₂×M₁₆	—	19	2	no	—
162	C₂×M₁₆	—	38	2	no	—
163	C₂×M₁₆	—	38	2	no	—
164	C₂×M₁₆	—	38	2	no	—
165	C₂×M₁₆	—	38	2	no	—
166	C₂×M₁₆	—	38	2	no	—
167	C₂×M₁₆	—	38	2	no	—
168	C₂×M₁₆	—	38	1	no	—
169	C₂×M₁₆	—	38	2	no	—
170	C₂×M₁₆	—	38	2	no	—
171	C₂²×C₈	—	19	4	no	—
172	C₂²×C₈	—	19	2	no	—
173	C₂²×C₈	—	38	1	no	—
174	C₂²×C₈	—	38	1	no	—
175	C₂²×C₈	—	38	2	no	—
176	C₂²×C₈	—	38	2	no	—
177	C₂²×C₈	—	38	1	no	—
178	#117	—	2	2	no	—
179	#117	—	2	2	no	—
180	#117	—	2	2	no	—
181	#117	—	2	2	no	—
182	#117	—	2	2	no	—
183	#117	—	2	2	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	Dic₃₈	1	2	yes	—
186	C₂×C₂²⋊C₄	C₂×Dic₁₉	1	4	yes	—
187	C₂×C₂²⋊C₄	C₂×Dic₁₉	1	2	yes	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	D₇₆	1	2	yes	—
190	C₄⋊C₈	—	38	8	no	—
191	C₁₉	#11288	1	1	yes	characteristic, complemented, Sylow
192	C₂⁴	C₂²×Dic₁₉	1	1	yes	characteristic
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	C₄⋊Dic₁₉	1	4	yes	—
195	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
196	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
197	C₂²×C₄	#64541	1	2	yes	—
198	C₂²×C₄	#64541	1	2	yes	—
199	C₂²×C₄	D₄×D₁₉	1	2	yes	—
200	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
201	C₂²×C₄	#64543	1	2	yes	—
202	C₂²×C₄	#64543	1	1	yes	characteristic
203	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
204	C₂²×C₄	#64545	1	1	yes	characteristic
205	M₁₆	—	38	4	no	—
206	M₁₆	—	76	2	no	—
207	M₁₆	—	76	2	no	—
208	M₁₆	—	76	2	no	—
209	M₁₆	—	76	2	no	—
210	M₁₆	—	76	2	no	—
211	C₂×C₈	—	38	4	no	—
212	C₂×C₈	—	38	4	no	—
213	C₂×C₈	—	38	4	no	—
214	C₂×C₈	—	38	2	no	—
215	C₂×C₈	—	38	4	no	—
216	C₂×C₈	—	38	2	no	—
217	C₂×C₈	—	76	1	no	—
218	C₂×C₈	—	76	1	no	—
219	C₂×C₈	—	76	1	no	—
220	C₂×C₈	—	76	2	no	—
221	C₂×C₈	—	76	1	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	4	2	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₂²⋊C₄	—	2	4	no	—
226	C₂²⋊C₄	—	2	4	no	—
227	C₂²⋊C₄	—	2	4	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	4	2	no	—
231	C₄²	—	4	2	no	—
232	C₂³	—	4	2	no	—
233	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
234	C₂³	#10597372	1	2	yes	—
235	C₂³	D₄×Dic₁₉	1	2	yes	—
236	C₂³	D₄×Dic₁₉	1	1	yes	characteristic
237	C₂³	Q₈×Dic₁₉	1	1	yes	characteristic
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	8	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	#10597316	1	2	yes	—
243	C₂×C₄	#10597321	1	1	yes	characteristic
244	C₂×C₄	#10597321	1	1	yes	characteristic
245	C₂×C₄	#10597324	1	2	yes	—
246	C₂×C₄	#10597325	1	2	yes	—
247	C₂×C₄	#10597328	1	1	yes	characteristic
248	C₂×C₄	#10597329	1	2	yes	—
249	C₂×C₄	#10597330	1	2	yes	—
250	C₂×C₄	#10597331	1	2	yes	—
251	C₂×C₄	#10597332	1	1	yes	characteristic
252	C₂×C₄	#10597334	1	1	yes	characteristic
253	C₂×C₄	#10597335	1	2	yes	—
254	C₂×C₄	#10597336	1	1	yes	characteristic
255	C₂×C₄	#10597340	1	2	yes	—
256	C₂×C₄	#10597341	1	1	yes	characteristic
257	C₂×C₄	#10597342	1	2	yes	—
258	C₂×C₄	#10597343	1	1	yes	characteristic
259	C₂×C₄	#10597344	1	2	yes	—
260	C₈	—	76	2	no	—
261	C₈	—	76	4	no	—
262	C₈	—	76	2	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	2	no	—
268	C₂²	#11926410	1	1	yes	characteristic
269	C₂²	#11926419	1	1	yes	characteristic
270	C₂²	#11926430	1	2	yes	—
271	C₂²	#11926437	1	2	yes	—
272	C₂²	#11926443	1	1	yes	characteristic
273	C₄	—	4	4	no	—
274	C₄	—	8	4	no	—
275	C₄	#11926457	1	1	yes	characteristic
276	C₄	#11926462	1	1	yes	characteristic
277	C₄	#11926471	1	1	yes	characteristic
278	C₄	#11926478	1	2	yes	—
279	C₄	#11926484	1	1	yes	characteristic
280	C₄	#11926486	1	2	yes	—
281	C₂	—	2	2	no	—
282	C₂	—	2	1	no	—
283	C₂	—	2	2	no	—
284	C₂	—	2	1	no	—
285	C₂	#14561892	1	1	yes	characteristic
286	C₂	#14563923	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#10597328
commutator, prime radical	C₂⁴×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₁₉×#1740	C₂
socle	C₂²×C₁₉	#385
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	#1740	D₁₉
hypocenter	C₁₉	#11288
LCS	G ▷ C₂⁴×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1740

Complements

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

N	H ≅ G/N	type
191. C₁₉	#11288	semidirect

Maximal subgroups

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

subgroup	index
#14572199	2
C₂×#11926314	2
#14563332	2
C₁₉×#1740	2
#11288	19