Subgroups — #18081897

Order 4864.

394 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14574149	C₂	1	1	yes	maximal, characteristic
2	#14562383	C₂	1	1	yes	maximal, characteristic
3	#14562133	C₂	1	2	yes	maximal
4	#14562037	C₂	1	2	yes	maximal
5	C₁₉×#1770	C₂	1	1	yes	maximal, characteristic
6	#11927101	C₂²	1	2	yes	—
7	#11926949	C₄	1	1	yes	characteristic, complemented
8	#11926949	C₄	1	1	yes	characteristic, complemented
9	#11926946	—	2	1	no	—
10	#11926946	—	2	1	no	—
11	#11926942	C₄	1	2	yes	complemented
12	#11926597	C₂²	1	2	yes	—
13	#11926582	—	2	1	no	—
14	#11926582	—	2	1	no	—
15	#11926297	—	2	1	no	—
16	#11926295	—	2	1	no	—
17	#11926280	—	2	4	no	—
18	#11926274	—	2	4	no	—
19	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₁₉×#97	—	2	1	no	—
21	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
23	C₁₉×#336	—	2	1	no	—
24	C₂²×C₈×C₁₉	—	2	1	no	—
25	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
28	C₂×C₄²×C₁₉	—	2	1	no	—
29	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
30	#10597372	—	2	2	no	—
31	#10597370	—	2	2	no	—
32	#10597370	—	2	2	no	—
33	#10597316	—	2	2	no	—
34	#10597315	C₂×C₄	1	2	yes	—
35	#10597315	C₂×C₄	1	2	yes	—
36	C₄×Dic₃₈	—	2	2	no	—
37	C₄×Dic₃₈	—	2	2	no	—
38	C₁₉×C₄⋊C₈	—	2	1	no	—
39	C₁₉×C₄⋊C₈	D₄	1	2	yes	—
40	C₁₉×#97	—	4	2	no	—
41	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
44	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
45	C₁₉×#95	D₄	1	2	yes	—
46	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
47	#10597275	—	2	1	no	—
48	#10597275	—	2	1	no	—
49	#10597275	—	2	1	no	—
50	#10597275	—	2	1	no	—
51	#10597274	—	2	2	no	—
52	#10597274	—	2	2	no	—
53	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
54	C₄×C₁₉⋊C₈	—	2	2	no	—
55	C₂²×C₄×C₁₉	—	2	2	no	—
56	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
59	C₂×Dic₃₈	—	2	2	no	—
60	C₂×Dic₃₈	—	2	2	no	—
61	C₂×C₈×C₁₉	—	2	1	no	—
62	C₂×C₈×C₁₉	—	2	1	no	—
63	C₂×C₈×C₁₉	—	2	2	no	—
64	C₂×C₈×C₁₉	—	4	1	no	—
65	C₂×C₈×C₁₉	—	4	1	no	—
66	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
71	C₂×C₈×C₁₉	Q₁₆	1	2	yes	—
72	C₄⋊C₄×C₁₉	—	4	2	no	—
73	C₄²×C₁₉	—	4	1	no	—
74	C₄²×C₁₉	—	4	1	no	—
75	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₁₉	#41	1	1	yes	characteristic
79	C₂²⋊Dic₁₉	—	4	2	no	—
80	C₄⋊Dic₁₉	—	2	2	no	—
81	C₄⋊Dic₁₉	—	2	2	no	—
82	C₄⋊Dic₁₉	—	2	2	no	—
83	C₄⋊Dic₁₉	—	2	2	no	—
84	Dic₁₉⋊C₄	—	4	2	no	—
85	Dic₁₉⋊C₄	—	4	2	no	—
86	C₄×Dic₁₉	—	4	2	no	—
87	C₂×C₁₉⋊C₈	—	4	2	no	—
88	#9752	—	19	1	no	maximal, Sylow
89	C₂³×C₁₉	#121	1	1	yes	characteristic
90	C₂×C₄×C₁₉	—	2	2	no	—
91	C₂×C₄×C₁₉	—	4	1	no	—
92	C₂×C₄×C₁₉	—	4	2	no	—
93	C₂×C₄×C₁₉	—	4	2	no	—
94	C₂×C₄×C₁₉	—	4	2	no	—
95	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
99	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
100	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
101	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
102	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
103	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
104	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
105	C₂×Dic₁₉	—	4	2	no	—
106	C₂×Dic₁₉	—	4	2	no	—
107	Dic₃₈	—	4	4	no	—
108	Dic₃₈	—	8	2	no	—
109	C₈×C₁₉	—	2	1	no	—
110	C₈×C₁₉	—	2	1	no	—
111	C₈×C₁₉	—	4	2	no	—
112	C₈×C₁₉	#103	1	2	yes	—
113	C₈×C₁₉	#103	1	2	yes	—
114	C₁₉⋊C₈	—	8	2	no	—
115	#3160	—	19	1	no	—
116	#1770	D₁₉	1	1	yes	characteristic
117	#1564	—	19	1	no	—
118	#1503	—	19	2	no	—
119	#1481	—	19	2	no	—
120	C₂²×C₁₉	—	2	1	no	—
121	C₂²×C₁₉	—	2	1	no	—
122	C₂²×C₁₉	#390	1	1	yes	characteristic
123	C₂²×C₁₉	#461	1	1	yes	characteristic
124	C₂²×C₁₉	#462	1	1	yes	characteristic
125	C₄×C₁₉	—	2	1	no	—
126	C₄×C₁₉	—	2	1	no	—
127	C₄×C₁₉	—	2	1	no	—
128	C₄×C₁₉	—	8	2	no	—
129	C₄×C₁₉	C₂×#103	1	1	yes	characteristic
130	C₄×C₁₉	#419	1	1	yes	characteristic
131	C₄×C₁₉	#444	1	2	yes	—
132	C₄×C₁₉	#461	1	1	yes	characteristic
133	C₄×C₁₉	#462	1	1	yes	characteristic
134	Dic₁₉	—	8	2	no	—
135	Dic₁₉	—	8	2	no	—
136	#533	—	19	2	no	—
137	#500	—	19	1	no	—
138	#500	—	19	1	no	—
139	#499	—	38	1	no	—
140	#499	—	38	1	no	—
141	#498	—	19	2	no	—
142	#431	—	19	2	no	—
143	#427	—	38	1	no	—
144	#427	—	38	1	no	—
145	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
146	C₂×#97	—	2	1	no	—
147	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
148	C₄×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
149	#336	—	2	1	no	—
150	#335	—	38	1	no	—
151	#334	—	38	1	no	—
152	#328	—	38	4	no	—
153	Q₈⋊C₈	—	38	4	no	—
154	C₂×C₁₉	—	2	1	no	—
155	C₂×C₁₉	—	2	1	no	—
156	C₂×C₁₉	#1975	1	1	yes	characteristic
157	C₂×C₁₉	#1976	1	1	yes	characteristic
158	C₂×C₁₉	#1981	1	1	yes	characteristic
159	C₂²×C₈	—	2	1	no	—
160	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
161	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
162	C₄⋊Q₈	—	19	2	no	—
163	C₄⋊Q₈	—	19	2	no	—
164	#125	—	38	2	no	—
165	C₂²⋊Q₈	—	38	2	no	—
166	C₂²⋊Q₈	—	38	2	no	—
167	C₄×Q₈	—	38	2	no	—
168	C₄×Q₈	—	38	2	no	—
169	#117	—	38	2	no	—
170	C₂×C₄⋊C₄	—	2	1	no	—
171	C₂×C₄²	—	2	1	no	—
172	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
173	#107	—	38	1	no	—
174	#107	—	38	1	no	—
175	#107	—	38	1	no	—
176	#107	—	38	1	no	—
177	#106	—	38	2	no	—
178	#106	—	38	2	no	—
179	C₄⋊C₈	—	2	1	no	—
180	C₄⋊C₈	—	38	2	no	—
181	C₄⋊C₈	D₇₆	1	2	yes	—
182	C₂²⋊C₈	—	38	2	no	—
183	#97	—	4	2	no	—
184	C₄×C₈	—	38	2	no	—
185	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
186	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
187	C₄×C₈	D₇₆	1	1	yes	characteristic
188	C₄×C₈	D₇₆	1	1	yes	characteristic
189	#95	D₇₆	1	2	yes	—
190	C₁₉	#9752	1	1	yes	characteristic, complemented, Sylow
191	C₂×Q₈	—	38	2	no	—
192	C₂×Q₈	—	38	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
195	C₂²×C₄	#64541	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	4	1	no	—
201	C₂×C₈	—	4	1	no	—
202	C₂×C₈	—	76	2	no	—
203	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
204	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
205	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
206	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
207	C₂×C₈	#64526	1	2	yes	—
208	C₂×C₈	#64528	1	2	yes	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	38	2	no	—
211	C₄⋊C₄	—	38	2	no	—
212	C₄⋊C₄	—	38	2	no	—
213	C₄⋊C₄	—	38	2	no	—
214	C₄⋊C₄	—	76	2	no	—
215	C₄⋊C₄	—	76	2	no	—
216	C₂²⋊C₄	—	76	2	no	—
217	C₄²	—	4	1	no	—
218	C₄²	—	4	1	no	—
219	C₄²	—	76	2	no	—
220	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
221	C₄²	C₂×D₇₆	1	1	yes	characteristic
222	C₄²	C₂×D₇₆	1	1	yes	characteristic
223	C₄²	#64541	1	1	yes	characteristic
224	C₂³	#10597376	1	1	yes	characteristic
225	Q₈	—	76	4	no	—
226	Q₈	—	152	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	76	2	no	—
233	C₂×C₄	—	76	2	no	—
234	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
235	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
236	C₂×C₄	#10597320	1	1	yes	characteristic
237	C₂×C₄	#10597321	1	1	yes	characteristic
238	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
239	C₂×C₄	#10597376	1	1	yes	characteristic
240	C₂×C₄	C₂×#64526	1	1	yes	characteristic
241	C₂×C₄	C₂×#64528	1	1	yes	characteristic
242	C₂×C₄	#10597397	1	1	yes	characteristic
243	C₂×C₄	#10597397	1	1	yes	characteristic
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	4	2	no	—
247	C₈	—	152	2	no	—
248	C₈	#10597267	1	2	yes	—
249	C₈	#10597267	1	2	yes	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	#11926446	1	1	yes	characteristic
253	C₂²	#11926749	1	1	yes	characteristic
254	C₂²	#11926754	1	1	yes	characteristic
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	8	2	no	—
259	C₄	—	152	2	no	—
260	C₄	—	152	2	no	—
261	C₄	C₂×#10597267	1	1	yes	characteristic
262	C₄	#11926552	1	1	yes	characteristic
263	C₄	#11926667	1	2	yes	—
264	C₄	#11926749	1	1	yes	characteristic
265	C₄	#11926754	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	#14564528	1	1	yes	characteristic
269	C₂	#14564533	1	1	yes	characteristic
270	C₂	#14564558	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597376
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1770	C₂
socle	C₂²×C₁₉	#390
prime radical	C₂³×C₁₉	#121
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1770	D₁₉
hypocenter	C₁₉	#9752
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1770

Complements

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

N	H ≅ G/N	type
7. #11926949	C₄	semidirect
8. #11926949	C₄	semidirect
11. #11926942	C₄	semidirect
190. C₁₉	#9752	semidirect

Maximal subgroups

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

subgroup	index
#14574149	2
#14562383	2
#14562133	2
#14562037	2
C₁₉×#1770	2
#9752	19