Subgroups — #18087715

Order 4864.

998 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#64520	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926390	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926303	C₂	1	4	yes	maximal
4	C₄×C₁₉×#117	C₂	1	1	yes	maximal, characteristic
5	C₂²×#64520	C₂²	1	2	yes	—
6	C₂²×C₄×Dic₁₉	C₂²	1	2	yes	—
7	C₄×#64520	—	2	2	no	—
8	C₄×#64520	C₄	1	4	yes	complemented
9	C₂×#10597260	—	2	1	no	—
10	C₂×C₄×C₁₉⋊C₈	—	2	1	no	—
11	#11926390	—	2	2	no	—
12	#11926390	C₄	1	4	yes	complemented
13	C₂×C₁₉⋊#95	—	2	2	no	—
14	#11926303	C₄	1	16	yes	complemented
15	C₂×C₁₉×#117	C₂²	1	2	yes	—
16	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₁₉	C₄	1	4	yes	complemented
18	C₄×C₂²⋊C₄×C₁₉	—	2	2	no	—
19	C₁₉×#376	—	2	1	no	—
20	C₄³×C₁₉	—	2	1	no	—
21	C₂³×C₄×C₁₉	D₄	1	2	yes	—
22	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
23	C₂³×Dic₁₉	—	2	2	no	—
24	C₁₉×#117	—	2	4	no	—
25	C₁₉×#117	C₂×C₄	1	8	yes	—
26	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
27	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
28	C₂×C₄²×C₁₉	—	2	2	no	—
29	C₂×C₄²×C₁₉	—	2	4	no	—
30	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₁₉	D₄	1	2	yes	—
34	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₉	Q₈	1	2	yes	—
37	C₂×C₄×Dic₁₉	—	2	2	no	—
38	C₂×C₄×Dic₁₉	—	2	4	no	—
39	C₂×C₄×Dic₁₉	C₂×C₄	1	8	yes	—
40	C₂×C₄×Dic₁₉	C₂×C₄	1	4	yes	—
41	C₂×#64520	—	2	2	no	—
42	C₂×#64520	—	2	4	no	—
43	C₂×#64520	C₂×C₄	1	8	yes	—
44	C₂×#64520	C₂×C₄	1	4	yes	—
45	C₂²×C₁₉⋊C₈	—	2	2	no	—
46	C₁₉×#95	—	2	4	no	—
47	C₁₉⋊#95	—	2	8	no	—
48	#10597260	—	2	4	no	—
49	C₄×C₁₉⋊C₈	—	2	4	no	—
50	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₉	—	2	4	no	—
52	C₂²×C₄×C₁₉	—	4	4	no	—
53	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
54	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
57	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	#41	1	4	yes	—
65	C₂²×Dic₁₉	—	2	4	no	—
66	C₂²×Dic₁₉	—	2	8	no	—
67	C₂²×Dic₁₉	—	4	8	no	—
68	C₄⋊C₄×C₁₉	—	2	4	no	—
69	C₄⋊C₄×C₁₉	C₄²	1	8	yes	—
70	C₂²⋊C₄×C₁₉	—	2	8	no	—
71	C₄²×C₁₉	—	2	8	no	—
72	C₄²×C₁₉	—	4	8	no	—
73	C₄²×C₁₉	C₄²	1	4	yes	—
74	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
75	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
76	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
77	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
78	C₄²×C₁₉	C₄⋊C₄	1	4	yes	—
79	C₄×Dic₁₉	—	2	4	no	—
80	C₄×Dic₁₉	—	2	8	no	—
81	C₄×Dic₁₉	C₄²	1	8	yes	complemented
82	#64520	—	2	4	no	—
83	#64520	—	2	8	no	—
84	#64520	C₄²	1	8	yes	complemented
85	C₂×C₁₉⋊C₈	—	2	4	no	—
86	C₂×C₁₉⋊C₈	—	2	8	no	—
87	#10701	—	19	1	no	maximal, Sylow
88	C₂³×C₁₉	—	2	4	no	—
89	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂³×C₁₉	C₄×D₄	1	4	yes	—
93	C₂×C₄×C₁₉	—	2	4	no	—
94	C₂×C₄×C₁₉	—	2	4	no	—
95	C₂×C₄×C₁₉	—	2	8	no	—
96	C₂×C₄×C₁₉	—	4	4	no	—
97	C₂×C₄×C₁₉	—	4	4	no	—
98	C₂×C₄×C₁₉	—	4	8	no	—
99	C₂×C₄×C₁₉	#95	1	4	yes	—
100	C₂×C₄×C₁₉	#95	1	4	yes	—
101	C₂×C₄×C₁₉	C₂×C₄²	1	2	yes	—
102	C₂×C₄×C₁₉	C₂×C₄²	1	2	yes	—
103	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₁₉	C₂×C₄²	1	2	yes	—
105	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₉	#117	1	2	yes	—
109	C₂×C₄×C₁₉	#117	1	2	yes	—
110	C₂×C₄×C₁₉	#117	1	2	yes	—
111	C₂×C₄×C₁₉	C₄×D₄	1	4	yes	—
112	C₂×C₄×C₁₉	C₄×D₄	1	4	yes	—
113	C₂×C₄×C₁₉	C₄×Q₈	1	4	yes	—
114	C₂×Dic₁₉	—	2	8	no	—
115	C₂×Dic₁₉	—	4	8	no	—
116	C₂×Dic₁₉	—	4	16	no	—
117	C₁₉⋊C₈	—	2	8	no	—
118	C₂×C₄×M₁₆	—	19	1	no	—
119	C₄×#117	D₁₉	1	1	yes	characteristic
120	C₂×#376	—	19	1	no	—
121	C₂×#339	—	19	4	no	—
122	C₂²×C₁₉	—	2	2	no	—
123	C₂²×C₁₉	—	2	2	no	—
124	C₂²×C₁₉	—	2	8	no	—
125	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
126	C₂²×C₁₉	C₄×C₂²⋊C₄	1	2	yes	—
127	C₂²×C₁₉	C₄×C₂²⋊C₄	1	2	yes	—
128	C₂²×C₁₉	C₄×C₄⋊C₄	1	2	yes	—
129	C₂²×C₁₉	#421	1	4	yes	—
130	C₄×C₁₉	—	2	4	no	—
131	C₄×C₁₉	—	2	4	no	—
132	C₄×C₁₉	—	4	8	no	—
133	C₄×C₁₉	C₄³	1	1	yes	characteristic
134	C₄×C₁₉	#376	1	1	yes	characteristic
135	C₄×C₁₉	C₄×C₂²⋊C₄	1	2	yes	—
136	C₄×C₁₉	C₄×C₄⋊C₄	1	2	yes	—
137	C₄×C₁₉	C₄×C₄⋊C₄	1	2	yes	—
138	Dic₁₉	—	4	8	no	—
139	C₂²×M₁₆	—	19	2	no	—
140	C₂×#117	C₂×D₁₉	1	2	yes	—
141	C₂²×C₄²	—	19	2	no	—
142	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
143	C₄×M₁₆	—	19	4	no	—
144	C₄×M₁₆	—	38	2	no	—
145	C₂×#97	—	38	1	no	—
146	C₂×C₄×C₈	—	38	1	no	—
147	C₄×C₄⋊C₄	Dic₁₉	1	4	yes	complemented
148	C₄×C₂²⋊C₄	—	2	2	no	—
149	#376	—	2	1	no	—
150	#376	—	19	4	no	—
151	#376	—	38	2	no	—
152	C₂×#95	—	38	2	no	—
153	C₄³	—	2	1	no	—
154	#339	—	19	16	no	—
155	C₂×C₁₉	—	2	2	no	—
156	C₂×C₁₉	—	2	2	no	—
157	C₂×C₁₉	C₄×#95	1	1	yes	characteristic
158	C₂×C₁₉	#1733	1	2	yes	—
159	C₂×C₁₉	#1759	1	4	yes	—
160	C₂³×C₄	—	38	2	no	—
161	C₂³×C₄	C₂²⋊D₁₉	1	2	yes	—
162	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
163	C₂×M₁₆	—	19	8	no	—
164	C₂×M₁₆	—	19	4	no	—
165	C₂×M₁₆	—	38	2	no	—
166	C₂×M₁₆	—	38	4	no	—
167	C₂²×C₈	—	38	2	no	—
168	#117	—	2	4	no	—
169	#117	C₄×D₁₉	1	8	yes	—
170	C₂×C₄⋊C₄	C₂×Dic₁₉	1	4	yes	—
171	C₂×C₂²⋊C₄	—	2	2	no	—
172	C₂×C₄²	—	2	2	no	—
173	C₂×C₄²	—	2	4	no	—
174	C₂×C₄²	—	19	8	no	—
175	C₂×C₄²	—	19	4	no	—
176	C₂×C₄²	—	38	2	no	—
177	C₂×C₄²	—	38	4	no	—
178	C₂×C₄²	Dic₃₈	1	2	yes	—
179	C₂×C₄²	C₄×D₁₉	1	2	yes	—
180	C₂×C₄²	C₄×D₁₉	1	2	yes	—
181	C₂×C₄²	D₇₆	1	2	yes	—
182	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
183	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
184	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
185	#97	—	38	4	no	—
186	C₄×C₈	—	38	4	no	—
187	#95	—	2	4	no	—
188	#95	—	38	8	no	—
189	C₁₉	#10701	1	1	yes	characteristic, complemented, Sylow
190	C₂⁴	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	4	4	no	—
193	C₂²×C₄	—	38	4	no	—
194	C₂²×C₄	—	38	8	no	—
195	C₂²×C₄	—	76	8	no	—
196	C₂²×C₄	Dic₁₉⋊C₄	1	4	yes	—
197	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	4	yes	—
198	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
199	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
200	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₄×D₁₉	1	4	yes	—
202	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
205	C₂²×C₄	#64541	1	4	yes	—
206	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
208	M₁₆	—	19	8	no	—
209	M₁₆	—	38	4	no	—
210	M₁₆	—	38	8	no	—
211	C₂×C₈	—	38	4	no	—
212	C₂×C₈	—	38	8	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	C₄×Dic₁₉	1	8	yes	—
215	C₂²⋊C₄	—	2	8	no	—
216	C₄²	—	2	8	no	—
217	C₄²	—	4	8	no	—
218	C₄²	—	19	8	no	—
219	C₄²	—	38	4	no	—
220	C₄²	—	38	8	no	—
221	C₄²	C₄×Dic₁₉	1	4	yes	—
222	C₄²	Dic₁₉⋊C₄	1	2	yes	—
223	C₄²	Dic₁₉⋊C₄	1	2	yes	—
224	C₄²	C₄⋊Dic₁₉	1	4	yes	—
225	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
226	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
227	C₂³	—	2	4	no	—
228	C₂³	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
229	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
230	C₂³	C₄×C₂²⋊D₁₉	1	4	yes	—
231	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	8	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	8	no	—
238	C₂×C₄	—	38	8	no	—
239	C₂×C₄	—	76	8	no	—
240	C₂×C₄	—	76	16	no	—
241	C₂×C₄	C₁₉⋊#95	1	4	yes	—
242	C₂×C₄	C₁₉⋊#95	1	4	yes	—
243	C₂×C₄	C₄×Dic₃₈	1	4	yes	—
244	C₂×C₄	C₄²×D₁₉	1	2	yes	—
245	C₂×C₄	C₄²×D₁₉	1	2	yes	—
246	C₂×C₄	#10597318	1	2	yes	—
247	C₂×C₄	#10597318	1	2	yes	—
248	C₂×C₄	C₄×D₇₆	1	4	yes	—
249	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
250	C₂×C₄	C₂×C₄×Dic₁₉	1	2	yes	—
251	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
252	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
253	C₂×C₄	#10597372	1	2	yes	—
254	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
255	C₂×C₄	C₄×C₂²⋊D₁₉	1	4	yes	—
256	C₈	—	38	8	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	8	no	—
260	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
261	C₂²	C₄×C₂²⋊Dic₁₉	1	2	yes	—
262	C₂²	C₄×(C₂×D₁₉)⋊C₄	1	2	yes	—
263	C₂²	C₄×Dic₁₉⋊C₄	1	2	yes	—
264	C₂²	#11926561	1	4	yes	—
265	C₄	—	2	4	no	—
266	C₄	—	2	4	no	—
267	C₄	—	4	8	no	—
268	C₄	—	76	8	no	—
269	C₄	C₄²×Dic₁₉	1	1	yes	characteristic
270	C₄	#11926390	1	1	yes	characteristic
271	C₄	C₄×(C₂×D₁₉)⋊C₄	1	2	yes	—
272	C₄	C₄×C₄⋊Dic₁₉	1	2	yes	—
273	C₄	C₄×Dic₁₉⋊C₄	1	2	yes	—
274	C₂	—	2	2	no	—
275	C₂	—	2	2	no	—
276	C₂	C₄×C₁₉⋊#95	1	1	yes	characteristic
277	C₂	#14563344	1	2	yes	—
278	C₂	#14563445	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₉
commutator	C₄×C₁₉	C₄³
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₄×C₁₉×#117	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂×D₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₄×#117	D₁₉
hypocenter	C₁₉	#10701
LCS	G ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
8. C₄×#64520	C₄	semidirect
12. #11926390	C₄	semidirect
14. #11926303	C₄	semidirect
17. C₄×C₄⋊C₄×C₁₉	C₄	semidirect
81. C₄×Dic₁₉	C₄²	semidirect
84. #64520	C₄²	semidirect
147. C₄×C₄⋊C₄	Dic₁₉	semidirect
189. C₁₉	#10701	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#64520	2
C₂×#11926390	2
C₂×#11926303	2
C₄×C₁₉×#117	2
#10701	19