Subgroups — #18075573

Order 4864.

404 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	#14562394	C₂	1	1	yes	maximal, characteristic, complemented
2	#14562391	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562370	C₂	1	2	yes	maximal
4	C₁₉⋊#1309	C₂	1	2	yes	maximal, complemented
5	C₁₉×#2931	C₂	1	1	yes	maximal, characteristic
6	#11926568	C₂²	1	2	yes	—
7	#11926568	C₂²	1	2	yes	—
8	#11926297	—	2	1	no	—
9	#11926297	—	2	1	no	—
10	#11926295	—	2	1	no	—
11	#11926295	—	2	1	no	—
12	#11926293	—	2	2	no	—
13	#11926288	—	2	2	no	—
14	C₂×C₁₉×#131	C₄	1	2	yes	—
15	C₁₉×#532	C₂²	1	1	yes	characteristic
16	C₁₉×#436	—	2	1	no	—
17	C₁₉×#436	—	2	1	no	—
18	C₈×D₄×C₁₉	—	2	1	no	—
19	C₈×D₄×C₁₉	—	2	1	no	—
20	C₁₉×#423	C₂²	1	1	yes	characteristic
21	C₁₉×#408	C₄	1	2	yes	—
22	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
23	C₂×#41×C₁₉	C₂×C₄	1	2	yes	—
24	C₁₉×#131	—	4	4	no	—
25	C₂×M₁₆×C₁₉	—	2	2	no	—
26	C₂×M₁₆×C₁₉	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₁₉	—	2	2	no	—
28	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
30	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
31	C₁₉×#127	D₄	1	1	yes	characteristic
32	C₁₉×#124	Q₈	1	2	yes	—
33	C₁₉×#121	—	2	1	no	—
34	C₁₉×#121	—	2	1	no	—
35	C₄×D₄×C₁₉	—	2	1	no	—
36	C₄×D₄×C₁₉	—	2	1	no	—
37	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
38	C₂×#64520	—	2	2	no	—
39	C₂×#64520	—	2	2	no	—
40	C₁₉×C₄⋊C₈	—	2	1	no	—
41	C₁₉×C₄⋊C₈	—	2	1	no	—
42	C₁₉×C₂²⋊C₈	—	2	2	no	—
43	C₁₉×C₂²⋊C₈	—	2	2	no	—
44	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
45	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
46	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
47	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
48	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
49	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
50	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
51	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
52	#41×C₁₉	—	4	2	no	—
53	#41×C₁₉	—	4	2	no	—
54	C₂×Q₈×C₁₉	C₄⋊C₄	1	2	yes	—
55	C₂×D₄×C₁₉	—	2	2	no	—
56	C₂×D₄×C₁₉	—	2	1	no	—
57	C₂×D₄×C₁₉	—	2	1	no	—
58	C₂×D₄×C₁₉	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₁₉	—	2	2	no	—
60	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
63	M₁₆×C₁₉	—	4	4	no	—
64	M₁₆×C₁₉	—	4	2	no	—
65	C₂×C₈×C₁₉	—	2	2	no	—
66	C₂×C₈×C₁₉	—	2	2	no	—
67	C₂×C₈×C₁₉	—	4	1	no	—
68	C₂×C₈×C₁₉	—	4	4	no	—
69	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₄⋊C₄×C₁₉	—	2	1	no	—
74	C₄⋊C₄×C₁₉	—	2	1	no	—
75	C₂²⋊C₄×C₁₉	—	2	2	no	—
76	C₂²⋊C₄×C₁₉	—	2	2	no	—
77	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
81	#64520	—	8	2	no	—
82	#64520	—	8	2	no	—
83	C₂×C₁₉⋊C₈	—	4	2	no	—
84	C₂×C₁₉⋊C₈	—	4	2	no	—
85	#8640	—	19	1	no	maximal, Sylow
86	C₂³×C₁₉	—	2	2	no	—
87	C₂³×C₁₉	#117	1	1	yes	characteristic
88	Q₈×C₁₉	—	4	2	no	—
89	D₄×C₁₉	—	4	2	no	—
90	D₄×C₁₉	—	4	4	no	—
91	D₄×C₁₉	—	4	2	no	—
92	D₄×C₁₉	—	4	1	no	—
93	D₄×C₁₉	—	4	1	no	—
94	C₂×C₄×C₁₉	—	2	1	no	—
95	C₂×C₄×C₁₉	—	2	1	no	—
96	C₂×C₄×C₁₉	—	2	2	no	—
97	C₂×C₄×C₁₉	—	2	1	no	—
98	C₂×C₄×C₁₉	—	2	1	no	—
99	C₂×C₄×C₁₉	—	2	2	no	—
100	C₂×C₄×C₁₉	—	4	2	no	—
101	C₂×C₄×C₁₉	—	4	2	no	—
102	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
105	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
106	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
108	C₈×C₁₉	—	2	1	no	—
109	C₈×C₁₉	—	2	1	no	—
110	C₈×C₁₉	—	2	1	no	—
111	C₈×C₁₉	—	4	2	no	—
112	C₈×C₁₉	#100	1	2	yes	—
113	C₁₉⋊C₈	—	8	2	no	—
114	C₁₉⋊C₈	—	8	2	no	—
115	#2931	D₁₉	1	1	yes	characteristic
116	#1567	—	19	1	no	—
117	#1566	—	19	1	no	—
118	#1560	—	19	2	no	—
119	#1309	—	19	2	no	—
120	C₂²×C₁₉	—	2	1	no	—
121	C₂²×C₁₉	—	2	1	no	—
122	C₂²×C₁₉	—	2	1	no	—
123	C₂²×C₁₉	—	4	2	no	—
124	C₂²×C₁₉	—	4	2	no	—
125	C₂²×C₁₉	—	4	2	no	—
126	C₂²×C₁₉	#380	1	1	yes	characteristic
127	C₄×C₁₉	—	2	1	no	—
128	C₄×C₁₉	—	2	1	no	—
129	C₄×C₁₉	—	2	1	no	—
130	C₄×C₁₉	—	4	2	no	—
131	C₄×C₁₉	C₂×#100	1	1	yes	characteristic
132	C₄×C₁₉	#413	1	1	yes	characteristic
133	C₄×C₁₉	#471	1	2	yes	—
134	C₄×C₁₉	#472	1	1	yes	characteristic
135	C₄×C₁₉	#473	1	1	yes	characteristic
136	C₂×#131	Dic₁₉	1	2	yes	—
137	#532	C₂×D₁₉	1	1	yes	characteristic
138	#436	—	2	1	no	—
139	#436	—	2	1	no	—
140	C₈×D₄	—	2	1	no	—
141	C₈×D₄	—	2	1	no	—
142	#423	—	19	2	no	—
143	#423	—	19	2	no	—
144	#423	C₂×D₁₉	1	1	yes	characteristic
145	#408	Dic₁₉	1	2	yes	—
146	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
147	#335	—	38	1	no	—
148	#335	—	38	1	no	—
149	#334	—	38	1	no	—
150	#334	—	38	1	no	—
151	#333	—	38	2	no	—
152	#331	—	38	2	no	—
153	C₂×C₁₉	—	4	1	no	—
154	C₂×C₁₉	—	8	2	no	—
155	C₂×C₁₉	#1800	1	1	yes	characteristic
156	C₂×C₁₉	#1815	1	1	yes	characteristic
157	C₂×C₁₉	#1816	1	1	yes	characteristic
158	C₂×#41	C₂×Dic₁₉	1	2	yes	—
159	#131	—	4	4	no	—
160	C₂×M₁₆	—	2	2	no	—
161	C₂×M₁₆	—	38	2	no	—
162	C₂×M₁₆	—	38	2	no	—
163	C₂×M₁₆	C₂×Dic₁₉	1	2	yes	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
166	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
167	C₄⋊Q₈	D₇₆	1	1	yes	characteristic
168	#127	D₇₆	1	1	yes	characteristic
169	#124	Dic₃₈	1	2	yes	—
170	#121	—	2	1	no	—
171	#121	—	2	1	no	—
172	C₄×D₄	—	2	1	no	—
173	C₄×D₄	—	2	1	no	—
174	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
175	C₄⋊C₈	—	2	1	no	—
176	C₄⋊C₈	—	2	1	no	—
177	C₄⋊C₈	—	38	2	no	—
178	C₄⋊C₈	—	38	2	no	—
179	C₄⋊C₈	—	38	2	no	—
180	C₄⋊C₈	—	38	2	no	—
181	C₂²⋊C₈	—	2	2	no	—
182	C₂²⋊C₈	—	2	2	no	—
183	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
184	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
185	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
186	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
187	C₁₉	#8640	1	1	yes	characteristic, complemented, Sylow
188	#41	—	4	2	no	—
189	#41	—	4	2	no	—
190	C₂×Q₈	C₄⋊Dic₁₉	1	2	yes	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	—	2	1	no	—
194	C₂×D₄	C₄⋊Dic₁₉	1	2	yes	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	#64541	1	1	yes	characteristic
197	C₂²×C₄	#64541	1	1	yes	characteristic
198	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
199	M₁₆	—	4	4	no	—
200	M₁₆	—	4	2	no	—
201	M₁₆	—	152	2	no	—
202	M₁₆	—	152	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	4	1	no	—
206	C₂×C₈	—	4	4	no	—
207	C₂×C₈	—	76	2	no	—
208	C₂×C₈	—	76	2	no	—
209	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
210	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
211	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
212	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
213	C₄⋊C₄	—	2	1	no	—
214	C₄⋊C₄	—	2	1	no	—
215	C₂²⋊C₄	—	2	2	no	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
218	C₄²	C₂×D₇₆	1	1	yes	characteristic
219	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
220	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
221	C₂³	—	2	2	no	—
222	C₂³	#10597372	1	1	yes	characteristic
223	Q₈	—	4	2	no	—
224	D₄	—	4	2	no	—
225	D₄	—	4	4	no	—
226	D₄	—	4	2	no	—
227	D₄	—	4	1	no	—
228	D₄	—	4	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	#10597370	1	1	yes	characteristic
238	C₂×C₄	#10597370	1	1	yes	characteristic
239	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
240	C₂×C₄	#10597376	1	1	yes	characteristic
241	C₂×C₄	#10597376	1	1	yes	characteristic
242	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	4	2	no	—
247	C₈	—	152	2	no	—
248	C₈	—	152	2	no	—
249	C₈	#10597290	1	2	yes	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	4	2	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	#11926404	1	1	yes	characteristic
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	4	2	no	—
261	C₄	C₂×#10597290	1	1	yes	characteristic
262	C₄	#11926528	1	1	yes	characteristic
263	C₄	#11926807	1	2	yes	—
264	C₄	#11926814	1	1	yes	characteristic
265	C₄	#11926819	1	1	yes	characteristic
266	C₂	—	4	1	no	—
267	C₂	—	8	2	no	—
268	C₂	#14563627	1	1	yes	characteristic
269	C₂	#14563694	1	1	yes	characteristic
270	C₂	#14563699	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926404
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#2931	C₂
socle	C₂²×C₁₉	#380
prime radical	C₁₉×#532	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2931	D₁₉
hypocenter	C₁₉	#8640
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931

Complements

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

N	H ≅ G/N	type
1. #14562394	C₂	semidirect
2. #14562391	C₂	semidirect
4. C₁₉⋊#1309	C₂	semidirect
187. C₁₉	#8640	semidirect

Maximal subgroups

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

subgroup	index
#14562394	2
#14562391	2
#14562370	2
C₁₉⋊#1309	2
C₁₉×#2931	2
#8640	19