Subgroups — #18076561

Order 4864.

420 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14572199	C₂	1	1	yes	maximal, characteristic
2	#14563619	C₂	1	1	yes	maximal, characteristic
3	#14562410	C₂	1	2	yes	maximal, complemented
4	#14561531	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1458	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	—	2	1	no	—
7	#11926571	—	2	4	no	—
8	#11926568	—	2	1	no	—
9	#11926568	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	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
13	#11926299	—	2	1	no	—
14	#11926297	—	2	4	no	—
15	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
16	C₁₉×#431	C₂²	1	2	yes	—
17	C₁₉×#324	C₄	1	2	yes	—
18	C₁₉×#323	C₄	1	2	yes	—
19	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
20	C₁₉×#117	—	2	2	no	—
21	C₁₉×#117	—	2	2	no	—
22	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₁₉	Q₈	1	1	yes	characteristic
24	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₁₉	Q₈	1	1	yes	characteristic
26	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
28	C₂×#64520	—	2	2	no	—
29	C₂×#64520	—	2	2	no	—
30	C₂×#64520	—	4	2	no	—
31	C₂×#64520	—	4	2	no	—
32	C₂²×C₁₉⋊C₈	—	2	2	no	—
33	C₂²×C₁₉⋊C₈	—	2	1	no	—
34	C₂²×C₁₉⋊C₈	—	2	1	no	—
35	C₁₉×C₄⋊C₈	—	2	2	no	—
36	C₁₉×C₂²⋊C₈	C₂×C₄	1	4	yes	—
37	C₄×C₈×C₁₉	—	2	2	no	—
38	C₂²⋊(C₁₉⋊C₈)	—	4	2	no	—
39	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
40	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
41	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
42	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
43	C₄⋊(C₁₉⋊C₈)	D₄	1	4	yes	—
44	C₂⁴×C₁₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₉	—	2	1	no	—
46	C₂²×C₄×C₁₉	—	2	1	no	—
47	C₂²×C₄×C₁₉	—	2	1	no	—
48	C₂²×C₄×C₁₉	—	2	1	no	—
49	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₉	D₈	1	2	yes	—
53	C₂²×C₄×C₁₉	Q₁₆	1	2	yes	—
54	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₁₉	—	2	4	no	—
58	C₄⋊C₄×C₁₉	—	2	2	no	—
59	C₄⋊C₄×C₁₉	—	2	2	no	—
60	C₂²⋊C₄×C₁₉	—	2	2	no	—
61	C₂²⋊C₄×C₁₉	—	2	2	no	—
62	C₄²×C₁₉	—	2	1	no	—
63	C₄²×C₁₉	—	2	1	no	—
64	C₄²×C₁₉	C₂×D₄	1	2	yes	—
65	C₄²×C₁₉	#41	1	2	yes	—
66	#64520	—	4	4	no	—
67	#64520	—	4	2	no	—
68	#64520	—	4	2	no	—
69	C₂×C₁₉⋊C₈	—	2	4	no	—
70	C₂×C₁₉⋊C₈	—	4	4	no	—
71	C₂×C₁₉⋊C₈	—	4	1	no	—
72	C₂×C₁₉⋊C₈	—	4	1	no	—
73	C₂×C₁₉⋊C₈	—	4	1	no	—
74	C₂×C₁₉⋊C₈	—	4	1	no	—
75	C₂×C₁₉⋊C₈	—	4	1	no	—
76	C₂×C₁₉⋊C₈	—	4	1	no	—
77	#8861	—	19	1	no	maximal, Sylow
78	C₂³×C₁₉	—	2	1	no	—
79	C₂³×C₁₉	—	2	1	no	—
80	C₂³×C₁₉	—	4	1	no	—
81	C₂³×C₁₉	—	4	1	no	—
82	C₂³×C₁₉	#107	1	2	yes	—
83	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₉	—	2	1	no	—
85	C₂×C₄×C₁₉	—	2	1	no	—
86	C₂×C₄×C₁₉	—	2	2	no	—
87	C₂×C₄×C₁₉	—	2	2	no	—
88	C₂×C₄×C₁₉	—	2	2	no	—
89	C₂×C₄×C₁₉	—	2	2	no	—
90	C₂×C₄×C₁₉	—	2	1	no	—
91	C₂×C₄×C₁₉	—	2	1	no	—
92	C₂×C₄×C₁₉	—	4	1	no	—
93	C₂×C₄×C₁₉	—	4	1	no	—
94	C₂×C₄×C₁₉	—	4	1	no	—
95	C₂×C₄×C₁₉	—	4	1	no	—
96	C₂×C₄×C₁₉	#107	1	2	yes	—
97	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
98	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
99	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
100	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
101	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
103	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
104	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
105	C₈×C₁₉	—	4	4	no	—
106	C₁₉⋊C₈	—	4	4	no	—
107	C₁₉⋊C₈	—	8	1	no	—
108	C₁₉⋊C₈	—	8	1	no	—
109	#2905	—	19	1	no	—
110	#1798	—	19	1	no	—
111	#1571	—	19	2	no	—
112	#1458	D₁₉	1	1	yes	characteristic
113	#1301	—	19	2	no	—
114	C₂²×C₁₉	—	2	1	no	—
115	C₂²×C₁₉	—	2	1	no	—
116	C₂²×C₁₉	—	2	2	no	—
117	C₂²×C₁₉	—	4	2	no	—
118	C₂²×C₁₉	—	4	1	no	—
119	C₂²×C₁₉	—	4	1	no	—
120	C₂²×C₁₉	—	8	1	no	—
121	C₂²×C₁₉	#385	1	1	yes	characteristic
122	C₂²×C₁₉	C₂×#107	1	1	yes	characteristic
123	C₂²×C₁₉	#428	1	1	yes	characteristic
124	C₄×C₁₉	—	2	2	no	—
125	C₄×C₁₉	—	2	2	no	—
126	C₄×C₁₉	—	2	2	no	—
127	C₄×C₁₉	—	4	1	no	—
128	C₄×C₁₉	—	4	1	no	—
129	C₄×C₁₉	#447	1	1	yes	characteristic
130	C₄×C₁₉	#451	1	1	yes	characteristic
131	C₄×C₁₉	#480	1	1	yes	characteristic
132	C₄×C₁₉	#484	1	1	yes	characteristic
133	C₂²×M₁₆	—	38	1	no	—
134	C₂×#117	C₂×D₁₉	1	1	yes	characteristic
135	#431	C₂×D₁₉	1	2	yes	—
136	#424	—	38	4	no	—
137	#423	—	19	2	no	—
138	#423	—	38	1	no	—
139	C₂×C₄⋊C₈	—	19	1	no	—
140	C₂×C₄⋊C₈	—	19	1	no	—
141	C₂×C₂²⋊C₈	—	38	1	no	—
142	#336	—	38	1	no	—
143	#335	—	38	4	no	—
144	#324	Dic₁₉	1	2	yes	—
145	#323	Dic₁₉	1	2	yes	—
146	C₂×C₁₉	—	2	1	no	—
147	C₂×C₁₉	—	2	1	no	—
148	C₂×C₁₉	—	4	2	no	—
149	C₂×C₁₉	#1852	1	1	yes	characteristic
150	C₂×C₁₉	#1858	1	1	yes	characteristic
151	C₂×C₁₉	#1868	1	1	yes	characteristic
152	C₂³×C₄	D₇₆	1	1	yes	characteristic
153	C₂×M₁₆	—	38	2	no	—
154	C₂×M₁₆	—	38	2	no	—
155	C₂×M₁₆	—	76	2	no	—
156	C₂×M₁₆	—	76	2	no	—
157	C₂²×C₈	—	38	2	no	—
158	C₂²×C₈	—	38	1	no	—
159	C₂²×C₈	—	38	1	no	—
160	#117	—	2	2	no	—
161	#117	—	2	2	no	—
162	C₂×C₄⋊C₄	Dic₃₈	1	1	yes	characteristic
163	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
164	C₂×C₂²⋊C₄	Dic₃₈	1	1	yes	characteristic
165	C₂×C₂²⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
166	C₂×C₄²	D₇₆	1	1	yes	characteristic
167	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
168	C₄⋊C₈	—	2	2	no	—
169	C₄⋊C₈	—	19	4	no	—
170	C₄⋊C₈	—	38	4	no	—
171	C₄⋊C₈	—	38	2	no	—
172	C₄⋊C₈	—	38	2	no	—
173	C₄⋊C₈	—	38	2	no	—
174	C₂²⋊C₈	—	76	2	no	—
175	C₂²⋊C₈	C₂×Dic₁₉	1	4	yes	—
176	C₄×C₈	—	2	2	no	—
177	C₁₉	#8861	1	1	yes	characteristic, complemented, Sylow
178	C₂⁴	C₄⋊Dic₁₉	1	1	yes	characteristic
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	D₁₅₂	1	2	yes	—
184	C₂²×C₄	Dic₇₆	1	2	yes	—
185	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
186	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
187	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
188	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
190	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
191	M₁₆	—	76	4	no	—
192	M₁₆	—	76	2	no	—
193	M₁₆	—	76	2	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	38	4	no	—
196	C₂×C₈	—	76	4	no	—
197	C₂×C₈	—	76	1	no	—
198	C₂×C₈	—	76	1	no	—
199	C₂×C₈	—	76	1	no	—
200	C₂×C₈	—	76	1	no	—
201	C₂×C₈	—	76	1	no	—
202	C₂×C₈	—	76	1	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	2	2	no	—
205	C₂²⋊C₄	—	2	2	no	—
206	C₂²⋊C₄	—	2	2	no	—
207	C₄²	—	2	1	no	—
208	C₄²	—	2	1	no	—
209	C₄²	D₄×D₁₉	1	2	yes	—
210	C₄²	#64543	1	2	yes	—
211	C₂³	—	2	1	no	—
212	C₂³	—	2	1	no	—
213	C₂³	—	4	1	no	—
214	C₂³	—	4	1	no	—
215	C₂³	#10597275	1	2	yes	—
216	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	#10597275	1	2	yes	—
230	C₂×C₄	#10597324	1	2	yes	—
231	C₂×C₄	#10597328	1	1	yes	characteristic
232	C₂×C₄	#10597332	1	1	yes	characteristic
233	C₂×C₄	C₂×D₁₅₂	1	1	yes	characteristic
234	C₂×C₄	C₂×Dic₇₆	1	1	yes	characteristic
235	C₂×C₄	#10597354	1	1	yes	characteristic
236	C₂×C₄	#10597355	1	1	yes	characteristic
237	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
238	C₈	—	4	4	no	—
239	C₈	—	76	4	no	—
240	C₈	—	152	1	no	—
241	C₈	—	152	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	4	2	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	1	no	—
248	C₂²	—	8	1	no	—
249	C₂²	#11926419	1	1	yes	characteristic
250	C₂²	C₂×#10597275	1	1	yes	characteristic
251	C₂²	#11926586	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	4	1	no	—
256	C₄	—	4	1	no	—
257	C₄	#11926680	1	1	yes	characteristic
258	C₄	#11926702	1	1	yes	characteristic
259	C₄	#11926858	1	1	yes	characteristic
260	C₄	#11926880	1	1	yes	characteristic
261	C₂	—	2	1	no	—
262	C₂	—	2	1	no	—
263	C₂	—	4	2	no	—
264	C₂	#14563825	1	1	yes	characteristic
265	C₂	#14563857	1	1	yes	characteristic
266	C₂	#14563923	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597328
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1458	C₂
socle	C₂²×C₁₉	#385
prime radical	C₂⁴×C₁₉	C₄⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1458	D₁₉
hypocenter	C₁₉	#8861
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1458

Complements

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

N	H ≅ G/N	type
3. #14562410	C₂	semidirect
4. #14561531	C₂	semidirect
177. C₁₉	#8861	semidirect

Maximal subgroups

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

subgroup	index
#14572199	2
#14563619	2
#14562410	2
#14561531	2
C₁₉×#1458	2
#8861	19