Subgroups — #18090817

Order 4864.

634 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10597274	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926312	C₂	1	2	yes	maximal
3	C₂×#11926309	C₂	1	1	yes	maximal, characteristic
4	#14563356	C₂	1	2	yes	maximal
5	C₁₉×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	C₂²	1	2	yes	—
7	C₂²×C₄⋊Dic₁₉	C₂²	1	2	yes	—
8	C₂×#10597274	—	2	4	no	—
9	C₂×#10597274	—	2	2	no	—
10	#11926312	—	2	4	no	—
11	#11926309	C₄	1	4	yes	complemented
12	#11926306	—	2	4	no	—
13	C₂²×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
15	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
16	C₁₉×#336	—	2	2	no	—
17	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
18	C₂³×Dic₁₉	—	2	2	no	—
19	C₂×M₁₆×C₁₉	—	2	1	no	—
20	C₂×M₁₆×C₁₉	—	2	2	no	—
21	C₂×M₁₆×C₁₉	—	2	2	no	—
22	C₂×M₁₆×C₁₉	Q₈	1	2	yes	—
23	C₂²×C₈×C₁₉	—	2	2	no	—
24	C₂²×C₈×C₁₉	—	2	2	no	—
25	C₂²×C₈×C₁₉	—	2	1	no	—
26	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₁₉	D₄	1	2	yes	—
28	C₂²×C₈×C₁₉	Q₈	1	2	yes	—
29	C₁₉×#117	—	2	2	no	—
30	C₁₉×#117	—	2	2	no	—
31	C₂×C₄⋊C₄×C₁₉	D₄	1	2	yes	—
32	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
33	C₂×C₄²×C₁₉	—	2	1	no	—
34	C₂×C₄⋊Dic₁₉	—	2	2	no	—
35	C₂×C₄⋊Dic₁₉	—	2	2	no	—
36	C₂×C₄⋊Dic₁₉	—	2	2	no	—
37	C₂×C₄⋊Dic₁₉	—	2	2	no	—
38	C₂×C₄⋊Dic₁₉	—	2	2	no	—
39	C₂×C₄⋊Dic₁₉	D₄	1	4	yes	—
40	C₂×#64520	—	2	2	no	—
41	C₂×#64520	—	2	2	no	—
42	C₂×#64520	—	2	2	no	—
43	C₂×#64520	—	2	2	no	—
44	C₂×#64520	—	2	2	no	—
45	C₂×#64520	C₂×C₄	1	4	yes	—
46	C₂²×C₁₉⋊C₈	—	2	2	no	—
47	#10597274	—	2	8	no	—
48	C₂⁴×C₁₉	#41	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	—	2	2	no	—
50	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
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	1	yes	characteristic
57	C₂²×C₄×C₁₉	#41	1	2	yes	—
58	C₂²×C₄×C₁₉	#41	1	2	yes	—
59	C₂²×C₄×C₁₉	#41	1	2	yes	—
60	C₂²×Dic₁₉	—	2	4	no	—
61	C₂²×Dic₁₉	—	4	2	no	—
62	C₂²×Dic₁₉	—	4	2	no	—
63	C₂²×Dic₁₉	—	8	4	no	—
64	M₁₆×C₁₉	—	2	4	no	—
65	M₁₆×C₁₉	—	4	2	no	—
66	C₂×C₈×C₁₉	—	2	4	no	—
67	C₂×C₈×C₁₉	—	2	2	no	—
68	C₂×C₈×C₁₉	—	2	4	no	—
69	C₂×C₈×C₁₉	—	2	2	no	—
70	C₂×C₈×C₁₉	—	2	2	no	—
71	C₂×C₈×C₁₉	—	4	1	no	—
72	C₂×C₈×C₁₉	—	4	1	no	—
73	C₂×C₈×C₁₉	C₄⋊C₄	1	4	yes	—
74	C₄⋊C₄×C₁₉	—	2	4	no	—
75	C₂²⋊C₄×C₁₉	—	4	2	no	—
76	C₄²×C₁₉	—	4	1	no	—
77	C₄²×C₁₉	—	4	1	no	—
78	C₄⋊Dic₁₉	—	2	4	no	—
79	C₄⋊Dic₁₉	—	2	4	no	—
80	C₄⋊Dic₁₉	—	4	2	no	—
81	C₄⋊Dic₁₉	—	4	2	no	—
82	#64520	—	2	4	no	—
83	#64520	—	2	4	no	—
84	#64520	—	4	2	no	—
85	#64520	—	4	2	no	—
86	C₂×C₁₉⋊C₈	—	2	4	no	—
87	C₂×C₁₉⋊C₈	—	4	2	no	—
88	C₂×C₁₉⋊C₈	—	4	2	no	—
89	#11271	—	19	1	no	maximal, Sylow
90	C₂³×C₁₉	—	2	4	no	—
91	C₂³×C₁₉	C₄×Q₈	1	1	yes	characteristic
92	C₂³×C₁₉	#121	1	1	yes	characteristic
93	C₂³×C₁₉	#123	1	2	yes	—
94	C₂³×C₁₉	#125	1	1	yes	characteristic
95	C₂³×C₁₉	#126	1	2	yes	—
96	C₂×C₄×C₁₉	—	4	2	no	—
97	C₂×C₄×C₁₉	—	4	4	no	—
98	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₉	#117	1	2	yes	—
100	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
102	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
103	C₂×C₄×C₁₉	#121	1	2	yes	—
104	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
105	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
106	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
108	C₂×C₄×C₁₉	#123	1	2	yes	—
109	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
110	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
111	C₂×C₄×C₁₉	#124	1	2	yes	—
112	C₂×C₄×C₁₉	#125	1	2	yes	—
113	C₂×C₄×C₁₉	#125	1	2	yes	—
114	C₂×C₄×C₁₉	#126	1	2	yes	—
115	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
116	C₂×Dic₁₉	—	4	4	no	—
117	C₂×Dic₁₉	—	8	4	no	—
118	C₂×Dic₁₉	—	8	4	no	—
119	C₂×Dic₁₉	—	8	4	no	—
120	C₈×C₁₉	—	2	4	no	—
121	C₈×C₁₉	—	4	2	no	—
122	C₁₉⋊C₈	—	4	4	no	—
123	C₂²×#106	—	19	1	no	—
124	C₂×#343	—	19	2	no	—
125	C₂×#341	—	19	1	no	—
126	#1736	—	19	2	no	—
127	#1729	D₁₉	1	1	yes	characteristic
128	C₂²×C₁₉	—	2	4	no	—
129	C₂²×C₁₉	—	2	2	no	—
130	C₂²×C₁₉	—	2	2	no	—
131	C₂²×C₁₉	—	2	2	no	—
132	C₂²×C₁₉	—	2	2	no	—
133	C₂²×C₁₉	#387	1	2	yes	—
134	C₂²×C₁₉	#389	1	1	yes	characteristic
135	C₂²×C₁₉	#394	1	1	yes	characteristic
136	C₂²×C₁₉	#397	1	1	yes	characteristic
137	C₂²×C₁₉	#400	1	2	yes	—
138	C₂²×C₁₉	#471	1	4	yes	—
139	C₄×C₁₉	—	8	2	no	—
140	C₄×C₁₉	#382	1	1	yes	characteristic
141	C₄×C₁₉	#385	1	1	yes	characteristic
142	C₄×C₁₉	#388	1	2	yes	—
143	C₄×C₁₉	#393	1	1	yes	characteristic
144	C₄×C₁₉	#398	1	2	yes	—
145	C₄×C₁₉	#400	1	1	yes	characteristic
146	Dic₁₉	—	8	4	no	—
147	C₂²×M₁₆	—	19	2	no	—
148	C₂²×M₁₆	C₂×D₁₉	1	1	yes	characteristic
149	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
150	C₂×#117	C₂×D₁₉	1	1	yes	characteristic
151	C₂²×C₄⋊C₄	—	19	2	no	—
152	C₂×#106	—	38	4	no	—
153	C₂×#106	—	38	2	no	—
154	#343	—	38	4	no	—
155	#341	—	19	4	no	—
156	#340	—	38	4	no	—
157	#336	—	2	2	no	—
158	C₂×C₁₉	—	2	2	no	—
159	C₂×C₁₉	—	2	2	no	—
160	C₂×C₁₉	#1442	1	1	yes	characteristic
161	C₂×C₁₉	#1951	1	2	yes	—
162	C₂×C₁₉	#2046	1	2	yes	—
163	C₂×C₁₉	#2079	1	2	yes	—
164	C₂³×C₄	—	38	2	no	—
165	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
166	C₂×M₁₆	—	2	1	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	19	4	no	—
170	C₂×M₁₆	—	38	2	no	—
171	C₂×M₁₆	—	38	2	no	—
172	C₂×M₁₆	—	38	2	no	—
173	C₂×M₁₆	—	38	2	no	—
174	C₂×M₁₆	—	38	2	no	—
175	C₂×M₁₆	Dic₃₈	1	2	yes	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	2	1	no	—
179	C₂²×C₈	—	38	2	no	—
180	C₂²×C₈	Dic₃₈	1	2	yes	—
181	C₂²×C₈	C₄×D₁₉	1	2	yes	—
182	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
183	#117	—	2	2	no	—
184	#117	—	2	2	no	—
185	C₂×C₄⋊C₄	—	19	4	no	—
186	C₂×C₄⋊C₄	—	38	2	no	—
187	C₂×C₄⋊C₄	—	38	2	no	—
188	C₂×C₄⋊C₄	—	38	2	no	—
189	C₂×C₄⋊C₄	—	38	2	no	—
190	C₂×C₄⋊C₄	—	38	2	no	—
191	C₂×C₄⋊C₄	D₇₆	1	2	yes	—
192	C₂×C₂²⋊C₄	—	2	1	no	—
193	C₂×C₄²	—	2	1	no	—
194	#106	—	38	8	no	—
195	C₁₉	#11271	1	1	yes	characteristic, complemented, Sylow
196	C₂⁴	#64541	1	1	yes	characteristic
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	38	4	no	—
199	C₂²×C₄	—	76	2	no	—
200	C₂²×C₄	—	76	2	no	—
201	C₂²×C₄	—	152	4	no	—
202	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
203	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
206	C₂²×C₄	#64541	1	1	yes	characteristic
207	C₂²×C₄	D₄×D₁₉	1	2	yes	—
208	C₂²×C₄	#64543	1	2	yes	—
209	C₂²×C₄	#64543	1	2	yes	—
210	C₂²×C₄	#64545	1	2	yes	—
211	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
212	M₁₆	—	2	4	no	—
213	M₁₆	—	4	2	no	—
214	M₁₆	—	38	4	no	—
215	M₁₆	—	38	4	no	—
216	M₁₆	—	76	2	no	—
217	M₁₆	—	76	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	38	4	no	—
226	C₂×C₈	—	76	2	no	—
227	C₂×C₈	—	76	2	no	—
228	C₂×C₈	Dic₁₉⋊C₄	1	4	yes	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	38	4	no	—
231	C₄⋊C₄	—	38	4	no	—
232	C₄⋊C₄	—	76	2	no	—
233	C₄⋊C₄	—	76	2	no	—
234	C₂²⋊C₄	—	4	2	no	—
235	C₄²	—	4	1	no	—
236	C₄²	—	4	1	no	—
237	C₂³	—	2	4	no	—
238	C₂³	C₄×Dic₃₈	1	1	yes	characteristic
239	C₂³	#10597316	1	1	yes	characteristic
240	C₂³	#10597325	1	2	yes	—
241	C₂³	#10597340	1	2	yes	—
242	C₂³	#10597376	1	1	yes	characteristic
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	76	4	no	—
246	C₂×C₄	—	152	4	no	—
247	C₂×C₄	—	152	4	no	—
248	C₂×C₄	—	152	4	no	—
249	C₂×C₄	#10597315	1	1	yes	characteristic
250	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
251	C₂×C₄	#10597324	1	2	yes	—
252	C₂×C₄	#10597324	1	2	yes	—
253	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
254	C₂×C₄	#10597331	1	2	yes	—
255	C₂×C₄	#10597332	1	2	yes	—
256	C₂×C₄	#10597336	1	2	yes	—
257	C₂×C₄	#10597336	1	2	yes	—
258	C₂×C₄	#10597338	1	2	yes	—
259	C₂×C₄	#10597341	1	2	yes	—
260	C₂×C₄	#10597344	1	2	yes	—
261	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
262	C₂×C₄	#10597370	1	1	yes	characteristic
263	C₂×C₄	#10597381	1	1	yes	characteristic
264	C₂×C₄	#10597382	1	1	yes	characteristic
265	C₂×C₄	#10597383	1	1	yes	characteristic
266	C₂×C₄	#10597393	1	1	yes	characteristic
267	C₈	—	2	4	no	—
268	C₈	—	4	2	no	—
269	C₈	—	76	4	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	4	no	—
275	C₂²	#11926431	1	2	yes	—
276	C₂²	#11926442	1	1	yes	characteristic
277	C₂²	#11926461	1	1	yes	characteristic
278	C₂²	#11926469	1	1	yes	characteristic
279	C₂²	#11926486	1	2	yes	—
280	C₂²	#11926807	1	4	yes	—
281	C₄	—	8	2	no	—
282	C₄	—	152	4	no	—
283	C₄	#11926409	1	1	yes	characteristic
284	C₄	#11926416	1	1	yes	characteristic
285	C₄	#11926436	1	2	yes	—
286	C₄	#11926456	1	1	yes	characteristic
287	C₄	#11926476	1	2	yes	—
288	C₄	#11926483	1	1	yes	characteristic
289	C₂	—	2	2	no	—
290	C₂	—	2	2	no	—
291	C₂	#14561891	1	1	yes	characteristic
292	C₂	#14564415	1	2	yes	—
293	C₂	#14564898	1	2	yes	—
294	C₂	#14565080	1	2	yes	—

	subgroup	G/H
center	C₂³	#10597376
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₁₉×#1729	C₂
socle	C₂³×C₁₉	#121
prime radical	C₂⁴×C₁₉	#41
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	#1729	D₁₉
hypocenter	C₁₉	#11271
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #1729

Complements

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

N	H ≅ G/N	type
11. #11926309	C₄	semidirect
195. C₁₉	#11271	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#10597274	2
C₂×#11926312	2
C₂×#11926309	2
#14563356	2
C₁₉×#1729	2
#11271	19