Subgroups — #18073670

Order 4864.

502 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926265	C₂	1	2	yes	maximal
4	C₁₉⋊#1273	C₂	1	2	yes	maximal
5	C₄×C₁₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₁₉⋊C₈)	C₄	1	2	yes	—
7	C₂×C₄×C₁₉⋊C₈	C₄	1	2	yes	—
8	C₂×C₄×C₁₉⋊C₈	C₄	1	2	yes	—
9	C₂×C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
12	#11926299	C₄	1	2	yes	—
13	#11926265	—	2	4	no	—
14	C₂×C₁₉×C₄⋊C₈	C₄	1	2	yes	—
15	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
16	C₄³×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#336	C₄	1	2	yes	—
18	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
19	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
26	C₂²×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
27	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
28	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
29	C₁₉×C₄⋊C₈	C₈	1	8	yes	complemented
30	C₄×C₈×C₁₉	—	2	2	no	—
31	C₄×C₈×C₁₉	—	2	2	no	—
32	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
33	C₄×C₁₉⋊C₈	—	2	4	no	—
34	C₄×C₁₉⋊C₈	—	2	2	no	—
35	C₄×C₁₉⋊C₈	—	2	1	no	—
36	C₄×C₁₉⋊C₈	—	2	1	no	—
37	C₄×C₁₉⋊C₈	—	2	2	no	—
38	C₄×C₁₉⋊C₈	—	2	4	no	—
39	C₄×C₁₉⋊C₈	D₄	1	2	yes	—
40	C₄×C₁₉⋊C₈	Q₈	1	2	yes	—
41	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
43	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
44	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
45	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₉	—	2	4	no	—
49	C₂×C₈×C₁₉	—	2	4	no	—
50	C₂×C₈×C₁₉	C₂×C₈	1	8	yes	—
51	C₄²×C₁₉	—	2	2	no	—
52	C₄²×C₁₉	—	2	2	no	—
53	C₄²×C₁₉	—	2	2	no	—
54	C₄²×C₁₉	C₂×C₈	1	4	yes	—
55	C₄²×C₁₉	M₁₆	1	4	yes	—
56	C₄²×C₁₉	M₁₆	1	4	yes	—
57	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₁₉	#41	1	1	yes	characteristic
60	C₄²×C₁₉	#41	1	1	yes	characteristic
61	C₂×C₁₉⋊C₈	—	2	2	no	—
62	C₂×C₁₉⋊C₈	—	2	2	no	—
63	C₂×C₁₉⋊C₈	—	2	2	no	—
64	C₂×C₁₉⋊C₈	—	2	2	no	—
65	C₂×C₁₉⋊C₈	—	2	4	no	—
66	C₂×C₁₉⋊C₈	—	2	4	no	—
67	C₂×C₁₉⋊C₈	—	2	4	no	—
68	C₂×C₁₉⋊C₈	C₄⋊C₄	1	4	yes	—
69	C₂×C₁₉⋊C₈	M₁₆	1	4	yes	—
70	#8261	—	19	1	no	maximal, Sylow
71	C₂³×C₁₉	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₁₉	—	2	1	no	—
73	C₂×C₄×C₁₉	—	2	1	no	—
74	C₂×C₄×C₁₉	—	2	1	no	—
75	C₂×C₄×C₁₉	—	2	1	no	—
76	C₂×C₄×C₁₉	—	2	1	no	—
77	C₂×C₄×C₁₉	—	2	1	no	—
78	C₂×C₄×C₁₉	C₄×C₈	1	2	yes	—
79	C₂×C₄×C₁₉	C₄×C₈	1	2	yes	—
80	C₂×C₄×C₁₉	#97	1	2	yes	—
81	C₂×C₄×C₁₉	#97	1	2	yes	—
82	C₂×C₄×C₁₉	#97	1	2	yes	—
83	C₂×C₄×C₁₉	#97	1	2	yes	—
84	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
86	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
88	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
89	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
90	C₂×C₄×C₁₉	C₂²×C₈	1	2	yes	—
91	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
92	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
93	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
94	C₂×C₄×C₁₉	#131	1	2	yes	—
95	C₂×C₄×C₁₉	#131	1	2	yes	—
96	C₈×C₁₉	—	2	8	no	—
97	C₁₉⋊C₈	—	2	4	no	—
98	C₁₉⋊C₈	—	2	4	no	—
99	C₁₉⋊C₈	—	4	4	no	—
100	C₄×C₄⋊C₈	—	19	1	no	—
101	C₄×C₄⋊C₈	D₁₉	1	1	yes	characteristic
102	C₄²×C₈	—	19	1	no	—
103	C₂×#322	—	19	2	no	—
104	#1273	—	19	2	no	—
105	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
106	C₂²×C₁₉	C₂×C₄×C₈	1	1	yes	characteristic
107	C₂²×C₁₉	C₂×#97	1	1	yes	characteristic
108	C₂²×C₁₉	C₂×#97	1	1	yes	characteristic
109	C₂²×C₁₉	C₄×M₁₆	1	1	yes	characteristic
110	C₂²×C₁₉	#405	1	1	yes	characteristic
111	C₂²×C₁₉	#405	1	1	yes	characteristic
112	C₄×C₁₉	—	2	2	no	—
113	C₄×C₁₉	—	2	2	no	—
114	C₄×C₁₉	—	2	2	no	—
115	C₄×C₁₉	#322	1	4	yes	complemented
116	C₄×C₁₉	#423	1	2	yes	—
117	C₄×C₁₉	#432	1	2	yes	—
118	C₄×C₁₉	C₈×D₄	1	2	yes	—
119	C₄×C₁₉	#436	1	2	yes	—
120	C₄×C₁₉	C₈×Q₈	1	2	yes	—
121	C₄×C₁₉	#446	1	2	yes	—
122	C₂×C₄⋊C₈	—	19	2	no	—
123	C₂×C₄⋊C₈	Dic₁₉	1	2	yes	—
124	C₂×C₄×C₈	—	19	1	no	—
125	C₂×C₄×C₈	—	19	2	no	—
126	C₂×C₄×C₈	—	19	2	no	—
127	C₂×C₄×C₈	—	19	2	no	—
128	C₂×C₄×C₈	—	19	1	no	—
129	C₂×C₄×C₈	C₂×D₁₉	1	2	yes	—
130	C₄³	C₂×D₁₉	1	1	yes	characteristic
131	#336	—	19	2	no	—
132	#336	Dic₁₉	1	2	yes	—
133	#322	—	38	4	no	—
134	C₂×C₁₉	C₂×#322	1	1	yes	characteristic
135	C₂×C₁₉	C₈×M₁₆	1	1	yes	characteristic
136	C₂×C₁₉	#1451	1	1	yes	characteristic
137	C₂×C₁₉	#1452	1	1	yes	characteristic
138	C₂×C₁₉	#1767	1	1	yes	characteristic
139	C₂×C₁₉	C₈×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₁₉	#1770	1	1	yes	characteristic
141	C₂²×C₈	—	19	2	no	—
142	C₂²×C₈	—	19	2	no	—
143	C₂²×C₈	—	19	4	no	—
144	C₂²×C₈	C₂×Dic₁₉	1	4	yes	—
145	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
146	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
147	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
148	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
149	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
150	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
151	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
152	C₄⋊C₈	—	38	4	no	—
153	C₄⋊C₈	C₁₉⋊C₈	1	8	yes	complemented
154	C₄×C₈	—	2	2	no	—
155	C₄×C₈	—	2	2	no	—
156	C₄×C₈	—	19	2	no	—
157	C₄×C₈	—	19	2	no	—
158	C₄×C₈	—	38	4	no	—
159	C₄×C₈	—	38	4	no	—
160	C₄×C₈	—	38	2	no	—
161	C₄×C₈	—	38	1	no	—
162	C₄×C₈	—	38	1	no	—
163	C₄×C₈	—	38	2	no	—
164	C₁₉	#8261	1	1	yes	characteristic, complemented, Sylow
165	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
166	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
167	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
168	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
169	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
170	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
171	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	19	4	no	—
175	C₂×C₈	—	19	4	no	—
176	C₂×C₈	—	38	2	no	—
177	C₂×C₈	—	38	2	no	—
178	C₂×C₈	—	38	2	no	—
179	C₂×C₈	—	38	2	no	—
180	C₂×C₈	—	38	4	no	—
181	C₂×C₈	—	38	4	no	—
182	C₂×C₈	—	38	4	no	—
183	C₂×C₈	C₂×C₁₉⋊C₈	1	8	yes	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	2	2	no	—
187	C₄²	#64515	1	4	yes	—
188	C₄²	#64515	1	4	yes	—
189	C₄²	C₂×C₁₉⋊C₈	1	4	yes	—
190	C₄²	D₄×D₁₉	1	1	yes	characteristic
191	C₄²	#64543	1	1	yes	characteristic
192	C₄²	Q₈×D₁₉	1	1	yes	characteristic
193	C₄²	#64545	1	1	yes	characteristic
194	C₂³	C₂×C₄×Dic₁₉	1	1	yes	characteristic
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	C₄×C₁₉⋊C₈	1	2	yes	—
202	C₂×C₄	C₄×C₁₉⋊C₈	1	2	yes	—
203	C₂×C₄	#10597272	1	2	yes	—
204	C₂×C₄	#10597272	1	2	yes	—
205	C₂×C₄	#10597272	1	2	yes	—
206	C₂×C₄	#10597272	1	2	yes	—
207	C₂×C₄	#10597333	1	1	yes	characteristic
208	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
209	C₂×C₄	#10597338	1	1	yes	characteristic
210	C₂×C₄	#10597339	1	1	yes	characteristic
211	C₂×C₄	C₂×#64515	1	2	yes	—
212	C₂×C₄	C₂×#64515	1	2	yes	—
213	C₂×C₄	M₁₆×D₁₉	1	2	yes	—
214	C₂×C₄	#10597353	1	2	yes	—
215	C₂×C₄	C₂²×C₁₉⋊C₈	1	2	yes	—
216	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
217	C₂×C₄	Q₈×Dic₁₉	1	1	yes	characteristic
218	C₂×C₄	#10597394	1	2	yes	—
219	C₈	—	2	8	no	—
220	C₈	—	38	4	no	—
221	C₈	—	38	4	no	—
222	C₈	—	76	4	no	—
223	C₂²	C₄⋊C₄×Dic₁₉	1	1	yes	characteristic
224	C₂²	C₂×C₄×C₁₉⋊C₈	1	1	yes	characteristic
225	C₂²	C₂×#10597272	1	1	yes	characteristic
226	C₂²	C₂×#10597272	1	1	yes	characteristic
227	C₂²	M₁₆×Dic₁₉	1	1	yes	characteristic
228	C₂²	#11926500	1	1	yes	characteristic
229	C₂²	#11926502	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	#11926265	1	4	yes	complemented
234	C₄	#11926566	1	2	yes	—
235	C₄	#11926600	1	2	yes	—
236	C₄	D₄×C₁₉⋊C₈	1	2	yes	—
237	C₄	#11926621	1	2	yes	—
238	C₄	Q₈×C₁₉⋊C₈	1	2	yes	—
239	C₄	#11926671	1	2	yes	—
240	C₂	C₂×#11926265	1	1	yes	characteristic
241	C₂	M₁₆×C₁₉⋊C₈	1	1	yes	characteristic
242	C₂	#14561925	1	1	yes	characteristic
243	C₂	#14561928	1	1	yes	characteristic
244	C₂	#14563490	1	1	yes	characteristic
245	C₂	C₄⋊C₄×C₁₉⋊C₈	1	1	yes	characteristic
246	C₂	#14563505	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₁₉
commutator	C₂²×C₁₉	C₂×C₄×C₈
Fitting	C₄×C₁₉×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₄²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₄×C₄⋊C₈	D₁₉
hypocenter	C₁₉	#8261
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄² ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
29. C₁₉×C₄⋊C₈	C₈	semidirect
115. C₄×C₁₉	#322	semidirect
153. C₄⋊C₈	C₁₉⋊C₈	semidirect
164. C₁₉	#8261	semidirect
233. C₄	#11926265	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊(C₁₉⋊C₈)	2
C₄²×C₁₉⋊C₈	2
C₂×#11926265	2
C₁₉⋊#1273	2
C₄×C₁₉×C₄⋊C₈	2
#8261	19