Subgroups — #18069066

Order 4864.

414 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	C₄×C₁₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₁₉⋊C₈)	C₄	1	2	yes	—
5	C₂×C₄×C₁₉⋊C₈	—	2	1	no	—
6	C₂×C₄×C₁₉⋊C₈	—	2	1	no	—
7	C₂×C₄×C₁₉⋊C₈	—	2	1	no	—
8	C₂×C₄×C₁₉⋊C₈	C₄	1	2	yes	—
9	#11926299	—	2	1	no	—
10	C₂×C₁₉×C₄⋊C₈	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₁₉	—	2	1	no	—
12	C₄³×C₁₉	C₂²	1	1	yes	characteristic
13	C₁₉×#336	—	2	1	no	—
14	C₂²×C₈×C₁₉	—	2	2	no	—
15	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₉	Q₈	1	1	yes	characteristic
22	C₂²×C₁₉⋊C₈	—	2	2	no	—
23	C₂²×C₁₉⋊C₈	—	2	2	no	—
24	C₁₉×C₄⋊C₈	—	2	2	no	—
25	C₁₉×C₄⋊C₈	C₈	1	4	yes	complemented
26	C₄×C₈×C₁₉	—	4	1	no	—
27	C₄×C₈×C₁₉	—	4	1	no	—
28	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
29	C₄⋊(C₁₉⋊C₈)	C₈	1	4	yes	complemented
30	C₄×C₁₉⋊C₈	—	2	4	no	—
31	C₄×C₁₉⋊C₈	—	2	4	no	—
32	C₄×C₁₉⋊C₈	—	2	2	no	—
33	C₄×C₁₉⋊C₈	—	4	1	no	—
34	C₄×C₁₉⋊C₈	—	4	1	no	—
35	C₄×C₁₉⋊C₈	C₈	1	4	yes	complemented
36	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
37	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₉	—	2	4	no	—
44	C₂×C₈×C₁₉	—	4	2	no	—
45	C₂×C₈×C₁₉	—	4	2	no	—
46	C₄²×C₁₉	—	2	2	no	—
47	C₄²×C₁₉	—	2	2	no	—
48	C₄²×C₁₉	—	2	2	no	—
49	C₄²×C₁₉	C₂×C₈	1	2	yes	—
50	C₄²×C₁₉	C₂×C₈	1	2	yes	—
51	C₄²×C₁₉	C₂×C₈	1	2	yes	—
52	C₄²×C₁₉	M₁₆	1	2	yes	—
53	C₄²×C₁₉	M₁₆	1	2	yes	—
54	C₄²×C₁₉	M₁₆	1	2	yes	—
55	C₄²×C₁₉	D₈	1	1	yes	characteristic
56	C₄²×C₁₉	SD₁₆	1	1	yes	characteristic
57	C₄²×C₁₉	SD₁₆	1	1	yes	characteristic
58	C₄²×C₁₉	Q₁₆	1	1	yes	characteristic
59	C₂×C₁₉⋊C₈	—	2	4	no	—
60	C₂×C₁₉⋊C₈	—	2	4	no	—
61	C₂×C₁₉⋊C₈	—	2	4	no	—
62	C₂×C₁₉⋊C₈	—	2	4	no	—
63	C₂×C₁₉⋊C₈	—	4	2	no	—
64	C₂×C₁₉⋊C₈	—	4	2	no	—
65	#7015	—	19	1	no	maximal, Sylow
66	C₂³×C₁₉	#95	1	1	yes	characteristic
67	C₂×C₄×C₁₉	—	2	1	no	—
68	C₂×C₄×C₁₉	—	2	1	no	—
69	C₂×C₄×C₁₉	—	2	1	no	—
70	C₂×C₄×C₁₉	—	2	1	no	—
71	C₂×C₄×C₁₉	—	2	1	no	—
72	C₂×C₄×C₁₉	—	2	1	no	—
73	C₂×C₄×C₁₉	C₄×C₈	1	1	yes	characteristic
74	C₂×C₄×C₁₉	C₄×C₈	1	1	yes	characteristic
75	C₂×C₄×C₁₉	C₄×C₈	1	1	yes	characteristic
76	C₂×C₄×C₁₉	#97	1	1	yes	characteristic
77	C₂×C₄×C₁₉	#97	1	1	yes	characteristic
78	C₂×C₄×C₁₉	#97	1	1	yes	characteristic
79	C₂×C₄×C₁₉	C₂²⋊C₈	1	2	yes	—
80	C₂×C₄×C₁₉	C₂²⋊C₈	1	2	yes	—
81	C₂×C₄×C₁₉	C₂²⋊C₈	1	2	yes	—
82	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
83	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
84	C₂×C₄×C₁₉	#103	1	1	yes	characteristic
85	C₂×C₄×C₁₉	#103	1	1	yes	characteristic
86	C₂×C₄×C₁₉	#104	1	2	yes	—
87	C₂×C₄×C₁₉	#104	1	2	yes	—
88	C₂×C₄×C₁₉	C₄⋊C₈	1	2	yes	—
89	C₂×C₄×C₁₉	C₄⋊C₈	1	2	yes	—
90	C₂×C₄×C₁₉	C₄⋊C₈	1	2	yes	—
91	C₂×C₄×C₁₉	#106	1	1	yes	characteristic
92	C₂×C₄×C₁₉	#107	1	1	yes	characteristic
93	C₂×C₄×C₁₉	#108	1	2	yes	—
94	C₈×C₁₉	—	4	4	no	—
95	C₁₉⋊C₈	—	2	8	no	—
96	C₁₉⋊C₈	—	4	4	no	—
97	C₄×C₄⋊C₈	—	19	1	no	—
98	C₄×C₄⋊C₈	D₁₉	1	1	yes	characteristic
99	C₄²×C₈	—	19	1	no	—
100	C₂²×C₁₉	#336	1	1	yes	characteristic
101	C₂²×C₁₉	#336	1	1	yes	characteristic
102	C₂²×C₁₉	#336	1	1	yes	characteristic
103	C₂²×C₁₉	#339	1	1	yes	characteristic
104	C₂²×C₁₉	#339	1	1	yes	characteristic
105	C₂²×C₁₉	#340	1	1	yes	characteristic
106	C₂²×C₁₉	#341	1	1	yes	characteristic
107	C₄×C₁₉	—	2	2	no	—
108	C₄×C₁₉	—	2	2	no	—
109	C₄×C₁₉	—	2	2	no	—
110	C₄×C₁₉	C₈²	1	1	yes	characteristic
111	C₄×C₁₉	#322	1	1	yes	characteristic
112	C₄×C₁₉	#322	1	1	yes	characteristic
113	C₄×C₁₉	#322	1	1	yes	characteristic
114	C₄×C₁₉	D₄⋊C₈	1	2	yes	—
115	C₄×C₁₉	D₄⋊C₈	1	2	yes	—
116	C₄×C₁₉	Q₈⋊C₈	1	2	yes	—
117	C₄×C₁₉	Q₈⋊C₈	1	2	yes	—
118	C₄×C₁₉	#334	1	2	yes	—
119	C₄×C₁₉	#335	1	2	yes	—
120	C₂×C₄⋊C₈	—	19	2	no	—
121	C₂×C₄⋊C₈	Dic₁₉	1	2	yes	—
122	C₂×C₄×C₈	—	2	1	no	—
123	C₂×C₄×C₈	—	19	2	no	—
124	C₂×C₄×C₈	—	38	1	no	—
125	C₂×C₄×C₈	—	38	1	no	—
126	C₂×C₄×C₈	—	38	1	no	—
127	C₄³	C₂×D₁₉	1	1	yes	characteristic
128	#336	—	2	1	no	—
129	#336	—	38	1	no	—
130	C₂×C₁₉	#1273	1	1	yes	characteristic
131	C₂×C₁₉	#1276	1	1	yes	characteristic
132	C₂×C₁₉	#1277	1	1	yes	characteristic
133	C₂×C₁₉	#1278	1	1	yes	characteristic
134	C₂×C₁₉	#1278	1	1	yes	characteristic
135	C₂×C₁₉	#1279	1	1	yes	characteristic
136	C₂×C₁₉	#1279	1	1	yes	characteristic
137	C₂²×C₈	—	2	2	no	—
138	C₂²×C₈	—	38	2	no	—
139	C₂²×C₈	—	38	2	no	—
140	C₂×C₄²	Dic₃₈	1	1	yes	characteristic
141	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
142	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
143	C₂×C₄²	D₇₆	1	1	yes	characteristic
144	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
145	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
146	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
147	C₄⋊C₈	—	2	2	no	—
148	C₄⋊C₈	—	19	4	no	—
149	C₄⋊C₈	—	38	2	no	—
150	C₄⋊C₈	C₁₉⋊C₈	1	4	yes	complemented
151	C₄×C₈	—	4	1	no	—
152	C₄×C₈	—	4	1	no	—
153	C₄×C₈	—	19	4	no	—
154	C₄×C₈	—	38	4	no	—
155	C₄×C₈	—	38	4	no	—
156	C₄×C₈	—	38	2	no	—
157	C₄×C₈	—	76	1	no	—
158	C₄×C₈	—	76	1	no	—
159	C₁₉	#7015	1	1	yes	characteristic, complemented, Sylow
160	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
161	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
162	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
163	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
164	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
165	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
166	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
167	C₂×C₈	—	2	4	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	—	4	2	no	—
170	C₂×C₈	—	38	4	no	—
171	C₂×C₈	—	38	4	no	—
172	C₂×C₈	—	38	4	no	—
173	C₂×C₈	—	38	4	no	—
174	C₂×C₈	—	76	2	no	—
175	C₂×C₈	—	76	2	no	—
176	C₄²	—	2	2	no	—
177	C₄²	—	2	2	no	—
178	C₄²	—	2	2	no	—
179	C₄²	C₈×D₁₉	1	2	yes	—
180	C₄²	C₈×D₁₉	1	2	yes	—
181	C₄²	#64515	1	2	yes	—
182	C₄²	#64515	1	2	yes	—
183	C₄²	C₂×C₁₉⋊C₈	1	2	yes	—
184	C₄²	#64520	1	2	yes	—
185	C₄²	#64525	1	1	yes	characteristic
186	C₄²	#64526	1	1	yes	characteristic
187	C₄²	#64527	1	1	yes	characteristic
188	C₄²	#64528	1	1	yes	characteristic
189	C₂³	C₁₉⋊#95	1	1	yes	characteristic
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	C₄×C₁₉⋊C₈	1	1	yes	characteristic
197	C₂×C₄	#10597260	1	1	yes	characteristic
198	C₂×C₄	C₄⋊(C₁₉⋊C₈)	1	2	yes	—
199	C₂×C₄	#10597264	1	1	yes	characteristic
200	C₂×C₄	#10597265	1	1	yes	characteristic
201	C₂×C₄	#10597266	1	1	yes	characteristic
202	C₂×C₄	#10597267	1	1	yes	characteristic
203	C₂×C₄	C₈×Dic₁₉	1	1	yes	characteristic
204	C₂×C₄	C₈×Dic₁₉	1	1	yes	characteristic
205	C₂×C₄	Dic₁₉⋊C₈	1	2	yes	—
206	C₂×C₄	Dic₁₉⋊C₈	1	2	yes	—
207	C₂×C₄	#10597272	1	1	yes	characteristic
208	C₂×C₄	#10597272	1	1	yes	characteristic
209	C₂×C₄	(C₂×D₁₉)⋊C₈	1	2	yes	—
210	C₂×C₄	(C₂×D₁₉)⋊C₈	1	2	yes	—
211	C₂×C₄	#10597279	1	2	yes	—
212	C₂×C₄	#10597282	1	2	yes	—
213	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	2	yes	—
214	C₂×C₄	#10597289	1	1	yes	characteristic
215	C₂×C₄	#10597292	1	1	yes	characteristic
216	C₂×C₄	#10597294	1	2	yes	—
217	C₈	—	4	4	no	—
218	C₈	—	38	8	no	—
219	C₈	—	76	4	no	—
220	C₂²	#11926298	1	1	yes	characteristic
221	C₂²	#11926298	1	1	yes	characteristic
222	C₂²	#11926299	1	1	yes	characteristic
223	C₂²	#11926303	1	1	yes	characteristic
224	C₂²	#11926305	1	1	yes	characteristic
225	C₂²	#11926306	1	1	yes	characteristic
226	C₂²	#11926308	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	2	no	—
229	C₄	—	2	2	no	—
230	C₄	C₈×C₁₉⋊C₈	1	1	yes	characteristic
231	C₄	#11926265	1	1	yes	characteristic
232	C₄	#11926266	1	1	yes	characteristic
233	C₄	#11926266	1	1	yes	characteristic
234	C₄	#11926271	1	2	yes	—
235	C₄	D₄⋊(C₁₉⋊C₈)	1	2	yes	—
236	C₄	#11926274	1	2	yes	—
237	C₄	Q₈⋊(C₁₉⋊C₈)	1	2	yes	—
238	C₄	#11926294	1	2	yes	—
239	C₄	#11926296	1	2	yes	—
240	C₂	C₁₉⋊#1273	1	1	yes	characteristic
241	C₂	#14561467	1	1	yes	characteristic
242	C₂	#14561469	1	1	yes	characteristic
243	C₂	#14561471	1	1	yes	characteristic
244	C₂	#14561473	1	1	yes	characteristic
245	C₂	#14561474	1	1	yes	characteristic
246	C₂	#14561476	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂²⋊D₁₉
commutator	C₄×C₁₉	C₈²
Frattini	C₄³	C₂×D₁₉
Fitting	C₄×C₁₉×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₁₉	#95
cosocle	C₄³×C₁₉	C₂²
hypercenter	C₄×C₄⋊C₈	D₁₉
hypocenter	C₁₉	#7015
LCS	G ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄² ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
25. C₁₉×C₄⋊C₈	C₈	semidirect
29. C₄⋊(C₁₉⋊C₈)	C₈	semidirect
35. C₄×C₁₉⋊C₈	C₈	semidirect
150. C₄⋊C₈	C₁₉⋊C₈	semidirect
159. C₁₉	#7015	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
C₄×C₁₉×C₄⋊C₈	2
#7015	19