Subgroups — #18081706

Order 4864.

422 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14564580	C₂	1	2	yes	maximal
2	C₄×#10597274	C₂	1	1	yes	maximal, characteristic
3	#14562134	C₂	1	2	yes	maximal
4	#14561924	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#3170	C₂	1	1	yes	maximal, characteristic
6	#11926597	C₂²	1	1	yes	characteristic
7	C₂×#10597274	—	2	1	no	—
8	C₄×#64520	C₂²	1	1	yes	characteristic
9	C₄×C₄⋊Dic₁₉	C₂²	1	2	yes	—
10	#11926307	—	2	1	no	—
11	#11926303	—	2	4	no	—
12	#11926281	—	2	4	no	—
13	#11926266	—	2	1	no	—
14	#11926265	—	2	1	no	—
15	C₁₉×#533	C₂²	1	2	yes	—
16	C₁₉×#500	C₄	1	1	yes	characteristic, complemented
17	C₁₉×#494	C₄	1	1	yes	characteristic, complemented
18	C₁₉×#487	C₄	1	2	yes	complemented
19	C₁₉×#467	—	2	1	no	—
20	C₁₉×#467	—	2	1	no	—
21	C₄×Q₁₆×C₁₉	—	2	1	no	—
22	C₄×Q₁₆×C₁₉	—	2	1	no	—
23	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
24	C₂×Q₁₆×C₁₉	—	2	1	no	—
25	C₂×Q₁₆×C₁₉	—	2	1	no	—
26	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
28	C₁₉×C₄⋊Q₈	C₂×C₄	1	2	yes	—
29	C₁₉×C₄⋊Q₈	C₂×C₄	1	2	yes	—
30	C₁₉×#125	—	2	2	no	—
31	C₁₉×C₂²⋊Q₈	—	2	2	no	—
32	C₁₉×C₂²⋊Q₈	—	2	2	no	—
33	C₄×Q₈×C₁₉	—	2	2	no	—
34	C₄×Q₈×C₁₉	—	2	2	no	—
35	C₁₉×#117	—	2	2	no	—
36	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
37	C₂×C₄⋊Dic₁₉	—	2	2	no	—
38	C₂×C₄×Dic₁₉	—	2	2	no	—
39	C₂×#64520	D₄	1	2	yes	—
40	C₁₉×#107	—	2	1	no	—
41	C₁₉×#107	—	2	1	no	—
42	C₁₉×#103	—	2	2	no	—
43	C₁₉×#103	—	2	2	no	—
44	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
47	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
48	C₁₉⋊#95	—	2	2	no	—
49	C₂²⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
50	#10597274	—	2	4	no	—
51	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
52	#10597260	—	2	1	no	—
53	C₄×C₁₉⋊C₈	—	2	1	no	—
54	C₄×C₁₉⋊C₈	—	2	1	no	—
55	C₂×Q₈×C₁₉	—	2	2	no	—
56	C₂×Q₈×C₁₉	—	2	2	no	—
57	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×Dic₁₉	—	4	2	no	—
61	Q₁₆×C₁₉	—	4	2	no	—
62	Q₁₆×C₁₉	—	4	2	no	—
63	C₂×C₈×C₁₉	—	2	2	no	—
64	C₂×C₈×C₁₉	—	4	1	no	—
65	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
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	C₄⋊C₄×C₁₉	—	2	2	no	—
70	C₄⋊C₄×C₁₉	—	2	2	no	—
71	C₄⋊C₄×C₁₉	—	2	2	no	—
72	C₄⋊C₄×C₁₉	—	2	2	no	—
73	C₄⋊C₄×C₁₉	—	4	2	no	—
74	C₄⋊C₄×C₁₉	—	4	2	no	—
75	C₂²⋊C₄×C₁₉	—	4	2	no	—
76	C₄²×C₁₉	—	4	2	no	—
77	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₁₉	#41	1	1	yes	characteristic
80	C₄²×C₁₉	#41	1	1	yes	characteristic
81	C₄⋊Dic₁₉	—	2	4	no	—
82	C₄⋊Dic₁₉	—	4	2	no	—
83	C₄×Dic₁₉	—	4	4	no	—
84	#64520	—	2	4	no	—
85	C₂×C₁₉⋊C₈	—	2	2	no	—
86	C₂×C₁₉⋊C₈	—	2	2	no	—
87	#9720	—	19	1	no	maximal, Sylow
88	C₂³×C₁₉	#127	1	1	yes	characteristic
89	Q₈×C₁₉	—	4	2	no	—
90	Q₈×C₁₉	—	4	2	no	—
91	Q₈×C₁₉	—	8	2	no	—
92	C₂×C₄×C₁₉	—	2	1	no	—
93	C₂×C₄×C₁₉	—	2	1	no	—
94	C₂×C₄×C₁₉	—	4	1	no	—
95	C₂×C₄×C₁₉	—	4	2	no	—
96	C₂×C₄×C₁₉	—	4	2	no	—
97	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
101	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
102	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
103	C₂×C₄×C₁₉	#137	1	4	yes	—
104	C₂×Dic₁₉	—	4	4	no	—
105	C₂×Dic₁₉	—	8	2	no	—
106	C₂×Dic₁₉	—	8	2	no	—
107	C₈×C₁₉	—	2	1	no	—
108	C₈×C₁₉	—	2	1	no	—
109	C₈×C₁₉	—	2	1	no	—
110	C₈×C₁₉	#104	1	2	yes	—
111	C₁₉⋊C₈	—	4	2	no	—
112	C₁₉⋊C₈	—	4	2	no	—
113	#3170	D₁₉	1	1	yes	characteristic
114	#1986	—	19	2	no	—
115	C₄×#106	—	19	1	no	—
116	#1503	—	19	2	no	—
117	#1451	—	19	1	no	—
118	C₂²×C₁₉	—	2	1	no	—
119	C₂²×C₁₉	—	2	1	no	—
120	C₂²×C₁₉	#390	1	1	yes	characteristic
121	C₂²×C₁₉	#497	1	2	yes	—
122	C₄×C₁₉	—	2	1	no	—
123	C₄×C₁₉	—	2	1	no	—
124	C₄×C₁₉	—	2	1	no	—
125	C₄×C₁₉	—	8	2	no	—
126	C₄×C₁₉	—	8	2	no	—
127	C₄×C₁₉	C₂×#104	1	1	yes	characteristic
128	C₄×C₁₉	#421	1	1	yes	characteristic
129	C₄×C₁₉	#441	1	2	yes	—
130	C₄×C₁₉	#470	1	2	yes	—
131	Dic₁₉	—	8	4	no	—
132	#533	C₂×D₁₉	1	2	yes	—
133	#500	Dic₁₉	1	1	yes	characteristic, complemented
134	#494	Dic₁₉	1	1	yes	characteristic, complemented
135	#487	Dic₁₉	1	2	yes	complemented
136	#467	—	2	1	no	—
137	#467	—	2	1	no	—
138	C₄×Q₁₆	—	2	1	no	—
139	C₄×Q₁₆	—	2	1	no	—
140	#431	—	19	1	no	—
141	C₂×#106	—	38	1	no	—
142	C₄×M₁₆	—	19	1	no	—
143	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
144	C₄×C₄⋊C₄	—	19	2	no	—
145	#340	—	38	1	no	—
146	#339	—	38	4	no	—
147	#328	—	38	4	no	—
148	#322	—	38	1	no	—
149	#322	—	38	1	no	—
150	C₂×C₁₉	—	2	2	no	—
151	C₂×C₁₉	#1971	1	1	yes	characteristic
152	C₂×C₁₉	#1976	1	1	yes	characteristic
153	C₂×C₁₉	#1980	1	1	yes	characteristic
154	C₂×Q₁₆	—	2	1	no	—
155	C₂×Q₁₆	—	2	1	no	—
156	C₂×M₁₆	—	19	2	no	—
157	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
158	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
159	C₄⋊Q₈	C₂×Dic₁₉	1	2	yes	—
160	C₄⋊Q₈	C₂×Dic₁₉	1	2	yes	—
161	#125	—	2	2	no	—
162	C₂²⋊Q₈	—	2	2	no	—
163	C₂²⋊Q₈	—	2	2	no	—
164	C₄×Q₈	—	2	2	no	—
165	C₄×Q₈	—	2	2	no	—
166	#117	—	2	2	no	—
167	C₂×C₄⋊C₄	—	38	2	no	—
168	C₂×C₄²	—	38	2	no	—
169	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
170	#107	—	2	1	no	—
171	#107	—	2	1	no	—
172	#106	—	38	4	no	—
173	C₄⋊C₈	—	38	1	no	—
174	#103	—	2	2	no	—
175	#103	—	2	2	no	—
176	C₂²⋊C₈	—	19	2	no	—
177	#97	—	38	1	no	—
178	C₄×C₈	—	38	1	no	—
179	C₄×C₈	—	38	1	no	—
180	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
181	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
182	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
183	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
184	#95	—	38	2	no	—
185	C₁₉	#9720	1	1	yes	characteristic, complemented, Sylow
186	C₂×Q₈	—	2	2	no	—
187	C₂×Q₈	—	2	2	no	—
188	C₂²×C₄	—	76	2	no	—
189	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
190	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
192	Q₁₆	—	4	2	no	—
193	Q₁₆	—	4	2	no	—
194	M₁₆	—	38	4	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	38	2	no	—
198	C₂×C₈	—	38	2	no	—
199	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
200	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
201	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
202	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₄⋊C₄	—	38	4	no	—
210	C₄⋊C₄	—	76	2	no	—
211	C₂²⋊C₄	—	4	2	no	—
212	C₄²	—	4	2	no	—
213	C₄²	—	76	4	no	—
214	C₄²	#64543	1	1	yes	characteristic
215	C₄²	#64543	1	1	yes	characteristic
216	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
217	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
218	C₂³	#10597386	1	1	yes	characteristic
219	Q₈	—	4	2	no	—
220	Q₈	—	4	2	no	—
221	Q₈	—	8	2	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	76	4	no	—
228	C₂×C₄	—	152	2	no	—
229	C₂×C₄	—	152	2	no	—
230	C₂×C₄	#10597364	1	4	yes	—
231	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
232	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
233	C₂×C₄	#10597382	1	1	yes	characteristic
234	C₂×C₄	#10597384	1	1	yes	characteristic
235	C₂×C₄	#10597384	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
237	C₈	—	2	1	no	—
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	76	2	no	—
241	C₈	—	76	2	no	—
242	C₈	#10597294	1	2	yes	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	#11926447	1	1	yes	characteristic
246	C₂²	#11926937	1	2	yes	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	8	2	no	—
251	C₄	—	8	2	no	—
252	C₄	—	152	4	no	—
253	C₄	C₂×#10597294	1	1	yes	characteristic
254	C₄	#11926561	1	1	yes	characteristic
255	C₄	#11926651	1	2	yes	—
256	C₄	#11926799	1	2	yes	—
257	C₂	—	2	2	no	—
258	C₂	#14564507	1	1	yes	characteristic
259	C₂	#14564534	1	1	yes	characteristic
260	C₂	#14564555	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597384
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#3170	C₂
socle	C₂²×C₁₉	#390
prime radical	C₂³×C₁₉	#127
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#3170	D₁₉
hypocenter	C₁₉	#9720
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
16. C₁₉×#500	C₄	semidirect
17. C₁₉×#494	C₄	semidirect
18. C₁₉×#487	C₄	semidirect
133. #500	Dic₁₉	semidirect
134. #494	Dic₁₉	semidirect
135. #487	Dic₁₉	semidirect
185. C₁₉	#9720	semidirect

Maximal subgroups

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

subgroup	index
#14564580	2
C₄×#10597274	2
#14562134	2
#14561924	2
C₁₉×#3170	2
#9720	19